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Question: 1. For the Burger Palace example, perform

1. For the Burger Palace example, perform a complete analysis of efficiency improvement alternatives for unit S2, including determination of a composite reference unit. 2. For the Burger Palace example, perform a complete analysis of efficiency improvement alternatives for unit S5, including determination of a composite reference unit. 3. For the Burger Palace example, what is the effect of removing an inefficient unit from the analysis (e.g., S2)? 4. For the Burger Palace example, what is the effect of removing an efficient unit from the analysis (e.g., S6)? Supplement Data Envelopment Analysis (DEA): How can corporate management evaluate the productivity of a fast-food outlet, a branch bank, a health clinic, or an elementary school? The difficulties in measuring productivity are threefold. First, what are the appropriate inputs to the system (e.g., labor hours, material dollars) and the measures of those inputs? Second, what are the appropriate outputs of the system (e.g., checks cashed, certificate of deposits) and the measures of those outputs? Third, what are the appropriate ways of measuring the relationship between these inputs and outputs? Measuring Service Productivity The measure of an organization’s productivity, if viewed from an engineering perspective, is similar to the measure of a system’s efficiency. It can be stated as a ratio of outputs to inputs (e.g., miles per gallon for an automobile). To evaluate the operational efficiency of a branch bank, for example, an accounting ratio such as cost per teller transaction might be used. A branch with a high ratio in comparison with those of other branches would be considered less efficient, but the higher ratio could result from a more complex mix of transactions. For example, a branch opening new accounts and selling CDs would require more time per transaction than another branch engaged only in simple transactions such as accepting deposits and cashing checks. The problem with using simple ratios is that the mix of outputs is not considered explicitly. This same criticism also can be made concerning the mix of inputs. For example, some branches might have automated teller machines in addition to live tellers, and this use of technology could affect the cost per teller transaction. Broad-based measures such as profitability or return on investment are highly relevant as overall performance measures, but they are not sufficient to evaluate the operating efficiency of a service unit. For instance, one could not conclude that a profitable branch bank is necessarily efficient in its use of personnel and other inputs. A higher than-average proportion of revenue-generating transactions could be the explanation rather than the cost-efficient use of resources. The DEA Model Fortunately, a technique has been developed with the ability to compare the efficiency of multiple service units that provide similar services by explicitly considering their use of multiple inputs (i.e., resources) to produce multiple outputs (i.e., services). The technique, which is referred to as data envelopment analysis (DEA), circumvents the need to develop standard costs for each service, because it can incorporate multiple inputs and multiple outputs into both the numerator and the denominator of the efficiency ratio without the need for conversion to a common dollar basis. Thus, the DEA measure of efficiency explicitly accounts for the mix of inputs and outputs and, consequently, is more comprehensive and reliable than a set of operating ratios or profit measures. DEA is a linear programming model that attempts to maximize a service unit’s efficiency, expressed as a ratio of outputs to inputs, by comparing a particular unit’s efficiency with the performance of a group of similar service units that are delivering the same service. In the process, some units achieve 100 percent efficiency and are referred to as the relatively efficient units, whereas other units with efficiency ratings of less than 100 percent are referred to as inefficient units. Corporate management thus can use DEA to compare a group of service units to identify relatively inefficient units, measure the magnitude of the inefficiencies, and by comparing the inefficient with the efficient ones, discover ways to reduce those inefficiencies. The DEA linear programming model is formulated according to Charnes, Cooper, and Rhodes, and is referred to as the CCR Model. 9 Definition of Variables Let Ek, with k = 1, 2, . . . K, be the efficiency ratio of unit k, where K is the total number of units being evaluated. Let uj , with j = 1, 2, . . . , M, be a coefficient for output j, where M is the total number of output types considered. The variable uj is a measure of the relative decrease in efficiency with each unit reduction of output value. Let vi , with i = 1, 2, . . . , N, be a coefficient for input i, where N is the total number of input types considered. The variable vi is a measure of the relative increase in efficiency with each unit reduction of input value. Let Ojk be the number of observed units of output j generated by service unit k during one time period. Let Iik be the number of actual units of input i used by service unit k during one time period. Objective Function The objective is to find the set of coefficient u’s associated with each output and of v’s associated with each input that will give the service unit being evaluated the highest possible efficiency.
1. For the Burger Palace example, perform a complete analysis of efficiency improvement alternatives for unit S2, including determination of a composite reference unit. 2. For the Burger Palace example, perform a complete analysis of efficiency improvement alternatives for unit S5, including determination of a composite reference unit. 3. For the Burger Palace example, what is the effect of removing an inefficient unit from the analysis (e.g., S2)? 4. For the Burger Palace example, what is the effect of removing an efficient unit from the analysis (e.g., S6)? Supplement Data Envelopment Analysis (DEA): How can corporate management evaluate the productivity of a fast-food outlet, a branch bank, a health clinic, or an elementary school? The difficulties in measuring productivity are threefold. First, what are the appropriate inputs to the system (e.g., labor hours, material dollars) and the measures of those inputs? Second, what are the appropriate outputs of the system (e.g., checks cashed, certificate of deposits) and the measures of those outputs? Third, what are the appropriate ways of measuring the relationship between these inputs and outputs? Measuring Service Productivity The measure of an organization’s productivity, if viewed from an engineering perspective, is similar to the measure of a system’s efficiency. It can be stated as a ratio of outputs to inputs (e.g., miles per gallon for an automobile). To evaluate the operational efficiency of a branch bank, for example, an accounting ratio such as cost per teller transaction might be used. A branch with a high ratio in comparison with those of other branches would be considered less efficient, but the higher ratio could result from a more complex mix of transactions. For example, a branch opening new accounts and selling CDs would require more time per transaction than another branch engaged only in simple transactions such as accepting deposits and cashing checks. The problem with using simple ratios is that the mix of outputs is not considered explicitly. This same criticism also can be made concerning the mix of inputs. For example, some branches might have automated teller machines in addition to live tellers, and this use of technology could affect the cost per teller transaction. Broad-based measures such as profitability or return on investment are highly relevant as overall performance measures, but they are not sufficient to evaluate the operating efficiency of a service unit. For instance, one could not conclude that a profitable branch bank is necessarily efficient in its use of personnel and other inputs. A higher than-average proportion of revenue-generating transactions could be the explanation rather than the cost-efficient use of resources. The DEA Model Fortunately, a technique has been developed with the ability to compare the efficiency of multiple service units that provide similar services by explicitly considering their use of multiple inputs (i.e., resources) to produce multiple outputs (i.e., services). The technique, which is referred to as data envelopment analysis (DEA), circumvents the need to develop standard costs for each service, because it can incorporate multiple inputs and multiple outputs into both the numerator and the denominator of the efficiency ratio without the need for conversion to a common dollar basis. Thus, the DEA measure of efficiency explicitly accounts for the mix of inputs and outputs and, consequently, is more comprehensive and reliable than a set of operating ratios or profit measures. DEA is a linear programming model that attempts to maximize a service unit’s efficiency, expressed as a ratio of outputs to inputs, by comparing a particular unit’s efficiency with the performance of a group of similar service units that are delivering the same service. In the process, some units achieve 100 percent efficiency and are referred to as the relatively efficient units, whereas other units with efficiency ratings of less than 100 percent are referred to as inefficient units. Corporate management thus can use DEA to compare a group of service units to identify relatively inefficient units, measure the magnitude of the inefficiencies, and by comparing the inefficient with the efficient ones, discover ways to reduce those inefficiencies. The DEA linear programming model is formulated according to Charnes, Cooper, and Rhodes, and is referred to as the CCR Model. 9 Definition of Variables Let Ek, with k = 1, 2, . . . K, be the efficiency ratio of unit k, where K is the total number of units being evaluated. Let uj , with j = 1, 2, . . . , M, be a coefficient for output j, where M is the total number of output types considered. The variable uj is a measure of the relative decrease in efficiency with each unit reduction of output value. Let vi , with i = 1, 2, . . . , N, be a coefficient for input i, where N is the total number of input types considered. The variable vi is a measure of the relative increase in efficiency with each unit reduction of input value. Let Ojk be the number of observed units of output j generated by service unit k during one time period. Let Iik be the number of actual units of input i used by service unit k during one time period. Objective Function The objective is to find the set of coefficient u’s associated with each output and of v’s associated with each input that will give the service unit being evaluated the highest possible efficiency. Where e is the index of the unit being evaluated. This function is subject to the constraint that when the same set of input and output coefficients (uj ’s and yi ’s) is applied to all other service units being compared, no service unit will exceed 100 percent efficiency or a ratio of 1.0. Constraints Where all coefficient values are positive and nonzero. To solve this fractional linear programming model using standard linear programming software requires a reformulation. Note that both the objective function and all constraints are ratios rather than linear functions. The objective function in equation (3) Is restated as a linear function by arbitrarily scaling the inputs for the unit under evaluation to a sum of 1.0. Subject to the constraint that For each service unit, the constraints in equation (2) are similarly reformulated: Where Sample Size A question of sample size often is raised concerning the number of service units that are required compared with the number of input and output variables selected in the analysis. The following relationship relating the number of service units K used in the analysis and the number of input N and output M types being considered is based on empirical findings and the experience of DEA practitioners: K ≥ 2(N + M) Example 7.3 Burger Palace An innovative drive-in-only burger chain has established six units in several different cities. Each unit is located in a strip shopping center parking lot. Only a standard meal consisting of a burger, fries, and a drink is available. Management has decided to use DEA to improve productivity by identifying which units are using their resources most efficiently and then sharing their experience and knowledge with the less efficient locations. Table 7.5 summarizes data for two inputs: labor-hours and material dollars consumed during a typical lunch hour period to generate an output of 100 meals sold. Normally, output will vary among the service units, but in this example, we have made the outputs equal to allow for a graphical presentation of the units’ productivity. As Figure 7.19 shows, service units S, S3, and S6 have been joined to form an efficient-production frontier of alternative methods of using labor hours and material resources to generate 100 meals. As can be seen, these efficient units have defined an envelope that contains all the inefficient units—thus the reason for calling the process “data envelopment analysis.” Table 7.5: Summary of Outputs and Inputs for Burger Palace  Figure 7.19: Productivity Frontier of Burger Palace For this simple example, we can identify efficient units by inspection and see the excess inputs being used by inefficient units (e.g., S2 would be as efficient as S3 if it used $50 less in materials). To gain an understanding of DEA, however, we will proceed to formulate the linear programming problems for each unit, then solve each of them to determine efficiency ratings and other information. We begin by illustrating the LP formulation for the first service unit, S1, using equations (5), 6), and (7). max E(S1) = u1 100 subject to u1100 − v12 − v2200 ≤ 0 u1100 − v14 − v2150 ≤ 0 u1100 − v14 − v2100 ≤ 0 u1100 − v16 − v2100 ≤ 0 u1100 − v18 − v280 ≤ 0 u1100 − v110 − v250 ≤ 0 v1 2 + v2200 = 1 u1, u2, v2, ≥ 0 Similar linear programming problems are formulated (or, better yet, the S1 linear programming problem is edited) and solved for the other service units by substituting the appropriate output function for the objective function and substituting the appropriate input function for the last constraint. Constraints 1 through 6, which restrict all units to no more than 100 percent efficiency, remain the same in all problems. This set of six linear programming problems was solved with Excel Solver 7.0 in fewer than five minutes by editing the data file between each run. Because the output is 100 meals for all units, only the last constraint must be edited by substituting the appropriate labor and material input values from Table 7.5 for the unit being evaluated. The data file for unit 1 of Burger Palace using a linear programming Excel add-in is shown in Figure 7.20. The linear programming results for each unit are shown in Table 7.6 and summarized in Table 7.7. In Table 7.6, we find that DEA has identified the same units shown as being efficient in Figure 7.19. Units S2, S4, and S5 all are inefficient in varying degrees. Also shown in Table 7.6 and associated with each inefficient unit is an efficiency reference set. Each inefficient unit will have a set of efficient units associated with it that defines its productivity. As Figure 7.19 shows for inefficient unit S4, the efficient units S3 and S6 have been joined with a line defining the efficiency frontier. A dashed line drawn from the origin to inefficient unit S4 cuts through this frontier and, thus, defines unit S4 as inefficient. In Table 7.7, the value in parentheses that is associated with each member of the efficiency reference set (i.e., .7778 for S3 and .2222 for S6) represents the relative weight assigned to that efficient unit in calculating the efficiency rating for S4. These relative weights are the shadow prices that are associated with the respective efficient-unit constraints in the linear programming solution. (Note in Table 7.6 that for unit 4, these weights appear as opportunity costs for S3 and S6.). TABLE 7.6: LP Solutions for DEA Study of Burger Palace FIGURE 7.20: Excel Data File for DEA Analysis of Burger Palace Unit 1 TABLE 7.7: Summary of DEA Results The values for v1 and v2 that are associated with the inputs of labor-hours and materials, respectively, measure the relative increase in efficiency with each unit reduction of input value. For unit S4, each unit decrease in labor-hours results in an efficiency increase of 0.0555. For unit S4 to become efficient, it must increase its efficiency rating by 0.111 points. This can be accomplished by reducing labor used by 2 hours (i.e., 2 hours × 0.0555 = 0.111). Note that with this reduction in labor-hours, unit S4 becomes identical to efficient unit S3. An alternative approach would be a reduction in materials used by $16.57 (i.e., 0.111/0.0067 = 16.57). Any linear combination of these two measures also would move unit S4 to the productivity frontier defined by the line segment joining efficient units S3 and S6. Table 7.8 contains the calculations for a hypothetical unit C, which is a composite reference unit defined by the weighted inputs of the reference set S3 and S6. As Figure 7.19 shows, this composite unit C is located at the intersection of the productivity frontier and the dashed line drawn from the origin to unit S4. Thus, compared with this reference unit C, inefficient unit S4 is using excess inputs in the amounts of 0.7 labor-hour and 11.1 material dollars. Table 7.8: Calculation of Excess Inputs Used by Unit S4 DEA offers many opportunities for an inefficient unit to become efficient regarding its reference set of efficient units. In practice, management would choose a particular approach on the basis of an evaluation of its cost, practicality, and feasibility; however, the motivation for change is clear (i.e., other units actually are able to achieve similar outputs with fewer resources). DEA and Strategic Planning When combined with profitability, DEA efficiency analysis can be useful in strategic planning for services that are delivered through multiple sites (e.g., hotel chains). Figure 7.21 presents a matrix of four possibilities that arise from combining efficiency and profitability. Considering the top-left quadrant of this matrix (i.e., underperforming potential stars) reveals that units operating at a high profit may be operating inefficiently and, thus, have unrealized potential. Comparing these with similar efficient units could suggest measures that would lead to even greater profit through more efficient operations. FIGURE 7.21: DEA Strategic Matrix Star performers can be found in the top-right quadrant (i.e., benchmark group). These efficient units also are highly profitable and, thus, serve as examples for others to emulate in both operations efficiency and marketing success in generating high revenues. The lower-right quadrant (i.e., candidates for divestiture) contains efficient but unprofitable units. These units are limited in profit potential, perhaps because of a poor location, and should be sold to generate capital for expansion in new territories. It is not clear which strategy to employ with the lower-left quadrant units (i.e., problem branches). If profit potential is limited, investments in efficient operations might lead to a future candidate for divestiture.
Where e is the index of the unit being evaluated. This function is subject to the constraint that when the same set of input and output coefficients (uj ’s and yi ’s) is applied to all other service units being compared, no service unit will exceed 100 percent efficiency or a ratio of 1.0. Constraints
1. For the Burger Palace example, perform a complete analysis of efficiency improvement alternatives for unit S2, including determination of a composite reference unit. 2. For the Burger Palace example, perform a complete analysis of efficiency improvement alternatives for unit S5, including determination of a composite reference unit. 3. For the Burger Palace example, what is the effect of removing an inefficient unit from the analysis (e.g., S2)? 4. For the Burger Palace example, what is the effect of removing an efficient unit from the analysis (e.g., S6)? Supplement Data Envelopment Analysis (DEA): How can corporate management evaluate the productivity of a fast-food outlet, a branch bank, a health clinic, or an elementary school? The difficulties in measuring productivity are threefold. First, what are the appropriate inputs to the system (e.g., labor hours, material dollars) and the measures of those inputs? Second, what are the appropriate outputs of the system (e.g., checks cashed, certificate of deposits) and the measures of those outputs? Third, what are the appropriate ways of measuring the relationship between these inputs and outputs? Measuring Service Productivity The measure of an organization’s productivity, if viewed from an engineering perspective, is similar to the measure of a system’s efficiency. It can be stated as a ratio of outputs to inputs (e.g., miles per gallon for an automobile). To evaluate the operational efficiency of a branch bank, for example, an accounting ratio such as cost per teller transaction might be used. A branch with a high ratio in comparison with those of other branches would be considered less efficient, but the higher ratio could result from a more complex mix of transactions. For example, a branch opening new accounts and selling CDs would require more time per transaction than another branch engaged only in simple transactions such as accepting deposits and cashing checks. The problem with using simple ratios is that the mix of outputs is not considered explicitly. This same criticism also can be made concerning the mix of inputs. For example, some branches might have automated teller machines in addition to live tellers, and this use of technology could affect the cost per teller transaction. Broad-based measures such as profitability or return on investment are highly relevant as overall performance measures, but they are not sufficient to evaluate the operating efficiency of a service unit. For instance, one could not conclude that a profitable branch bank is necessarily efficient in its use of personnel and other inputs. A higher than-average proportion of revenue-generating transactions could be the explanation rather than the cost-efficient use of resources. The DEA Model Fortunately, a technique has been developed with the ability to compare the efficiency of multiple service units that provide similar services by explicitly considering their use of multiple inputs (i.e., resources) to produce multiple outputs (i.e., services). The technique, which is referred to as data envelopment analysis (DEA), circumvents the need to develop standard costs for each service, because it can incorporate multiple inputs and multiple outputs into both the numerator and the denominator of the efficiency ratio without the need for conversion to a common dollar basis. Thus, the DEA measure of efficiency explicitly accounts for the mix of inputs and outputs and, consequently, is more comprehensive and reliable than a set of operating ratios or profit measures. DEA is a linear programming model that attempts to maximize a service unit’s efficiency, expressed as a ratio of outputs to inputs, by comparing a particular unit’s efficiency with the performance of a group of similar service units that are delivering the same service. In the process, some units achieve 100 percent efficiency and are referred to as the relatively efficient units, whereas other units with efficiency ratings of less than 100 percent are referred to as inefficient units. Corporate management thus can use DEA to compare a group of service units to identify relatively inefficient units, measure the magnitude of the inefficiencies, and by comparing the inefficient with the efficient ones, discover ways to reduce those inefficiencies. The DEA linear programming model is formulated according to Charnes, Cooper, and Rhodes, and is referred to as the CCR Model. 9 Definition of Variables Let Ek, with k = 1, 2, . . . K, be the efficiency ratio of unit k, where K is the total number of units being evaluated. Let uj , with j = 1, 2, . . . , M, be a coefficient for output j, where M is the total number of output types considered. The variable uj is a measure of the relative decrease in efficiency with each unit reduction of output value. Let vi , with i = 1, 2, . . . , N, be a coefficient for input i, where N is the total number of input types considered. The variable vi is a measure of the relative increase in efficiency with each unit reduction of input value. Let Ojk be the number of observed units of output j generated by service unit k during one time period. Let Iik be the number of actual units of input i used by service unit k during one time period. Objective Function The objective is to find the set of coefficient u’s associated with each output and of v’s associated with each input that will give the service unit being evaluated the highest possible efficiency. Where e is the index of the unit being evaluated. This function is subject to the constraint that when the same set of input and output coefficients (uj ’s and yi ’s) is applied to all other service units being compared, no service unit will exceed 100 percent efficiency or a ratio of 1.0. Constraints Where all coefficient values are positive and nonzero. To solve this fractional linear programming model using standard linear programming software requires a reformulation. Note that both the objective function and all constraints are ratios rather than linear functions. The objective function in equation (3) Is restated as a linear function by arbitrarily scaling the inputs for the unit under evaluation to a sum of 1.0. Subject to the constraint that For each service unit, the constraints in equation (2) are similarly reformulated: Where Sample Size A question of sample size often is raised concerning the number of service units that are required compared with the number of input and output variables selected in the analysis. The following relationship relating the number of service units K used in the analysis and the number of input N and output M types being considered is based on empirical findings and the experience of DEA practitioners: K ≥ 2(N + M) Example 7.3 Burger Palace An innovative drive-in-only burger chain has established six units in several different cities. Each unit is located in a strip shopping center parking lot. Only a standard meal consisting of a burger, fries, and a drink is available. Management has decided to use DEA to improve productivity by identifying which units are using their resources most efficiently and then sharing their experience and knowledge with the less efficient locations. Table 7.5 summarizes data for two inputs: labor-hours and material dollars consumed during a typical lunch hour period to generate an output of 100 meals sold. Normally, output will vary among the service units, but in this example, we have made the outputs equal to allow for a graphical presentation of the units’ productivity. As Figure 7.19 shows, service units S, S3, and S6 have been joined to form an efficient-production frontier of alternative methods of using labor hours and material resources to generate 100 meals. As can be seen, these efficient units have defined an envelope that contains all the inefficient units—thus the reason for calling the process “data envelopment analysis.” Table 7.5: Summary of Outputs and Inputs for Burger Palace  Figure 7.19: Productivity Frontier of Burger Palace For this simple example, we can identify efficient units by inspection and see the excess inputs being used by inefficient units (e.g., S2 would be as efficient as S3 if it used $50 less in materials). To gain an understanding of DEA, however, we will proceed to formulate the linear programming problems for each unit, then solve each of them to determine efficiency ratings and other information. We begin by illustrating the LP formulation for the first service unit, S1, using equations (5), 6), and (7). max E(S1) = u1 100 subject to u1100 − v12 − v2200 ≤ 0 u1100 − v14 − v2150 ≤ 0 u1100 − v14 − v2100 ≤ 0 u1100 − v16 − v2100 ≤ 0 u1100 − v18 − v280 ≤ 0 u1100 − v110 − v250 ≤ 0 v1 2 + v2200 = 1 u1, u2, v2, ≥ 0 Similar linear programming problems are formulated (or, better yet, the S1 linear programming problem is edited) and solved for the other service units by substituting the appropriate output function for the objective function and substituting the appropriate input function for the last constraint. Constraints 1 through 6, which restrict all units to no more than 100 percent efficiency, remain the same in all problems. This set of six linear programming problems was solved with Excel Solver 7.0 in fewer than five minutes by editing the data file between each run. Because the output is 100 meals for all units, only the last constraint must be edited by substituting the appropriate labor and material input values from Table 7.5 for the unit being evaluated. The data file for unit 1 of Burger Palace using a linear programming Excel add-in is shown in Figure 7.20. The linear programming results for each unit are shown in Table 7.6 and summarized in Table 7.7. In Table 7.6, we find that DEA has identified the same units shown as being efficient in Figure 7.19. Units S2, S4, and S5 all are inefficient in varying degrees. Also shown in Table 7.6 and associated with each inefficient unit is an efficiency reference set. Each inefficient unit will have a set of efficient units associated with it that defines its productivity. As Figure 7.19 shows for inefficient unit S4, the efficient units S3 and S6 have been joined with a line defining the efficiency frontier. A dashed line drawn from the origin to inefficient unit S4 cuts through this frontier and, thus, defines unit S4 as inefficient. In Table 7.7, the value in parentheses that is associated with each member of the efficiency reference set (i.e., .7778 for S3 and .2222 for S6) represents the relative weight assigned to that efficient unit in calculating the efficiency rating for S4. These relative weights are the shadow prices that are associated with the respective efficient-unit constraints in the linear programming solution. (Note in Table 7.6 that for unit 4, these weights appear as opportunity costs for S3 and S6.). TABLE 7.6: LP Solutions for DEA Study of Burger Palace FIGURE 7.20: Excel Data File for DEA Analysis of Burger Palace Unit 1 TABLE 7.7: Summary of DEA Results The values for v1 and v2 that are associated with the inputs of labor-hours and materials, respectively, measure the relative increase in efficiency with each unit reduction of input value. For unit S4, each unit decrease in labor-hours results in an efficiency increase of 0.0555. For unit S4 to become efficient, it must increase its efficiency rating by 0.111 points. This can be accomplished by reducing labor used by 2 hours (i.e., 2 hours × 0.0555 = 0.111). Note that with this reduction in labor-hours, unit S4 becomes identical to efficient unit S3. An alternative approach would be a reduction in materials used by $16.57 (i.e., 0.111/0.0067 = 16.57). Any linear combination of these two measures also would move unit S4 to the productivity frontier defined by the line segment joining efficient units S3 and S6. Table 7.8 contains the calculations for a hypothetical unit C, which is a composite reference unit defined by the weighted inputs of the reference set S3 and S6. As Figure 7.19 shows, this composite unit C is located at the intersection of the productivity frontier and the dashed line drawn from the origin to unit S4. Thus, compared with this reference unit C, inefficient unit S4 is using excess inputs in the amounts of 0.7 labor-hour and 11.1 material dollars. Table 7.8: Calculation of Excess Inputs Used by Unit S4 DEA offers many opportunities for an inefficient unit to become efficient regarding its reference set of efficient units. In practice, management would choose a particular approach on the basis of an evaluation of its cost, practicality, and feasibility; however, the motivation for change is clear (i.e., other units actually are able to achieve similar outputs with fewer resources). DEA and Strategic Planning When combined with profitability, DEA efficiency analysis can be useful in strategic planning for services that are delivered through multiple sites (e.g., hotel chains). Figure 7.21 presents a matrix of four possibilities that arise from combining efficiency and profitability. Considering the top-left quadrant of this matrix (i.e., underperforming potential stars) reveals that units operating at a high profit may be operating inefficiently and, thus, have unrealized potential. Comparing these with similar efficient units could suggest measures that would lead to even greater profit through more efficient operations. FIGURE 7.21: DEA Strategic Matrix Star performers can be found in the top-right quadrant (i.e., benchmark group). These efficient units also are highly profitable and, thus, serve as examples for others to emulate in both operations efficiency and marketing success in generating high revenues. The lower-right quadrant (i.e., candidates for divestiture) contains efficient but unprofitable units. These units are limited in profit potential, perhaps because of a poor location, and should be sold to generate capital for expansion in new territories. It is not clear which strategy to employ with the lower-left quadrant units (i.e., problem branches). If profit potential is limited, investments in efficient operations might lead to a future candidate for divestiture.
Where all coefficient values are positive and nonzero. To solve this fractional linear programming model using standard linear programming software requires a reformulation. Note that both the objective function and all constraints are ratios rather than linear functions. The objective function in equation (3) Is restated as a linear function by arbitrarily scaling the inputs for the unit under evaluation to a sum of 1.0.
1. For the Burger Palace example, perform a complete analysis of efficiency improvement alternatives for unit S2, including determination of a composite reference unit. 2. For the Burger Palace example, perform a complete analysis of efficiency improvement alternatives for unit S5, including determination of a composite reference unit. 3. For the Burger Palace example, what is the effect of removing an inefficient unit from the analysis (e.g., S2)? 4. For the Burger Palace example, what is the effect of removing an efficient unit from the analysis (e.g., S6)? Supplement Data Envelopment Analysis (DEA): How can corporate management evaluate the productivity of a fast-food outlet, a branch bank, a health clinic, or an elementary school? The difficulties in measuring productivity are threefold. First, what are the appropriate inputs to the system (e.g., labor hours, material dollars) and the measures of those inputs? Second, what are the appropriate outputs of the system (e.g., checks cashed, certificate of deposits) and the measures of those outputs? Third, what are the appropriate ways of measuring the relationship between these inputs and outputs? Measuring Service Productivity The measure of an organization’s productivity, if viewed from an engineering perspective, is similar to the measure of a system’s efficiency. It can be stated as a ratio of outputs to inputs (e.g., miles per gallon for an automobile). To evaluate the operational efficiency of a branch bank, for example, an accounting ratio such as cost per teller transaction might be used. A branch with a high ratio in comparison with those of other branches would be considered less efficient, but the higher ratio could result from a more complex mix of transactions. For example, a branch opening new accounts and selling CDs would require more time per transaction than another branch engaged only in simple transactions such as accepting deposits and cashing checks. The problem with using simple ratios is that the mix of outputs is not considered explicitly. This same criticism also can be made concerning the mix of inputs. For example, some branches might have automated teller machines in addition to live tellers, and this use of technology could affect the cost per teller transaction. Broad-based measures such as profitability or return on investment are highly relevant as overall performance measures, but they are not sufficient to evaluate the operating efficiency of a service unit. For instance, one could not conclude that a profitable branch bank is necessarily efficient in its use of personnel and other inputs. A higher than-average proportion of revenue-generating transactions could be the explanation rather than the cost-efficient use of resources. The DEA Model Fortunately, a technique has been developed with the ability to compare the efficiency of multiple service units that provide similar services by explicitly considering their use of multiple inputs (i.e., resources) to produce multiple outputs (i.e., services). The technique, which is referred to as data envelopment analysis (DEA), circumvents the need to develop standard costs for each service, because it can incorporate multiple inputs and multiple outputs into both the numerator and the denominator of the efficiency ratio without the need for conversion to a common dollar basis. Thus, the DEA measure of efficiency explicitly accounts for the mix of inputs and outputs and, consequently, is more comprehensive and reliable than a set of operating ratios or profit measures. DEA is a linear programming model that attempts to maximize a service unit’s efficiency, expressed as a ratio of outputs to inputs, by comparing a particular unit’s efficiency with the performance of a group of similar service units that are delivering the same service. In the process, some units achieve 100 percent efficiency and are referred to as the relatively efficient units, whereas other units with efficiency ratings of less than 100 percent are referred to as inefficient units. Corporate management thus can use DEA to compare a group of service units to identify relatively inefficient units, measure the magnitude of the inefficiencies, and by comparing the inefficient with the efficient ones, discover ways to reduce those inefficiencies. The DEA linear programming model is formulated according to Charnes, Cooper, and Rhodes, and is referred to as the CCR Model. 9 Definition of Variables Let Ek, with k = 1, 2, . . . K, be the efficiency ratio of unit k, where K is the total number of units being evaluated. Let uj , with j = 1, 2, . . . , M, be a coefficient for output j, where M is the total number of output types considered. The variable uj is a measure of the relative decrease in efficiency with each unit reduction of output value. Let vi , with i = 1, 2, . . . , N, be a coefficient for input i, where N is the total number of input types considered. The variable vi is a measure of the relative increase in efficiency with each unit reduction of input value. Let Ojk be the number of observed units of output j generated by service unit k during one time period. Let Iik be the number of actual units of input i used by service unit k during one time period. Objective Function The objective is to find the set of coefficient u’s associated with each output and of v’s associated with each input that will give the service unit being evaluated the highest possible efficiency. Where e is the index of the unit being evaluated. This function is subject to the constraint that when the same set of input and output coefficients (uj ’s and yi ’s) is applied to all other service units being compared, no service unit will exceed 100 percent efficiency or a ratio of 1.0. Constraints Where all coefficient values are positive and nonzero. To solve this fractional linear programming model using standard linear programming software requires a reformulation. Note that both the objective function and all constraints are ratios rather than linear functions. The objective function in equation (3) Is restated as a linear function by arbitrarily scaling the inputs for the unit under evaluation to a sum of 1.0. Subject to the constraint that For each service unit, the constraints in equation (2) are similarly reformulated: Where Sample Size A question of sample size often is raised concerning the number of service units that are required compared with the number of input and output variables selected in the analysis. The following relationship relating the number of service units K used in the analysis and the number of input N and output M types being considered is based on empirical findings and the experience of DEA practitioners: K ≥ 2(N + M) Example 7.3 Burger Palace An innovative drive-in-only burger chain has established six units in several different cities. Each unit is located in a strip shopping center parking lot. Only a standard meal consisting of a burger, fries, and a drink is available. Management has decided to use DEA to improve productivity by identifying which units are using their resources most efficiently and then sharing their experience and knowledge with the less efficient locations. Table 7.5 summarizes data for two inputs: labor-hours and material dollars consumed during a typical lunch hour period to generate an output of 100 meals sold. Normally, output will vary among the service units, but in this example, we have made the outputs equal to allow for a graphical presentation of the units’ productivity. As Figure 7.19 shows, service units S, S3, and S6 have been joined to form an efficient-production frontier of alternative methods of using labor hours and material resources to generate 100 meals. As can be seen, these efficient units have defined an envelope that contains all the inefficient units—thus the reason for calling the process “data envelopment analysis.” Table 7.5: Summary of Outputs and Inputs for Burger Palace  Figure 7.19: Productivity Frontier of Burger Palace For this simple example, we can identify efficient units by inspection and see the excess inputs being used by inefficient units (e.g., S2 would be as efficient as S3 if it used $50 less in materials). To gain an understanding of DEA, however, we will proceed to formulate the linear programming problems for each unit, then solve each of them to determine efficiency ratings and other information. We begin by illustrating the LP formulation for the first service unit, S1, using equations (5), 6), and (7). max E(S1) = u1 100 subject to u1100 − v12 − v2200 ≤ 0 u1100 − v14 − v2150 ≤ 0 u1100 − v14 − v2100 ≤ 0 u1100 − v16 − v2100 ≤ 0 u1100 − v18 − v280 ≤ 0 u1100 − v110 − v250 ≤ 0 v1 2 + v2200 = 1 u1, u2, v2, ≥ 0 Similar linear programming problems are formulated (or, better yet, the S1 linear programming problem is edited) and solved for the other service units by substituting the appropriate output function for the objective function and substituting the appropriate input function for the last constraint. Constraints 1 through 6, which restrict all units to no more than 100 percent efficiency, remain the same in all problems. This set of six linear programming problems was solved with Excel Solver 7.0 in fewer than five minutes by editing the data file between each run. Because the output is 100 meals for all units, only the last constraint must be edited by substituting the appropriate labor and material input values from Table 7.5 for the unit being evaluated. The data file for unit 1 of Burger Palace using a linear programming Excel add-in is shown in Figure 7.20. The linear programming results for each unit are shown in Table 7.6 and summarized in Table 7.7. In Table 7.6, we find that DEA has identified the same units shown as being efficient in Figure 7.19. Units S2, S4, and S5 all are inefficient in varying degrees. Also shown in Table 7.6 and associated with each inefficient unit is an efficiency reference set. Each inefficient unit will have a set of efficient units associated with it that defines its productivity. As Figure 7.19 shows for inefficient unit S4, the efficient units S3 and S6 have been joined with a line defining the efficiency frontier. A dashed line drawn from the origin to inefficient unit S4 cuts through this frontier and, thus, defines unit S4 as inefficient. In Table 7.7, the value in parentheses that is associated with each member of the efficiency reference set (i.e., .7778 for S3 and .2222 for S6) represents the relative weight assigned to that efficient unit in calculating the efficiency rating for S4. These relative weights are the shadow prices that are associated with the respective efficient-unit constraints in the linear programming solution. (Note in Table 7.6 that for unit 4, these weights appear as opportunity costs for S3 and S6.). TABLE 7.6: LP Solutions for DEA Study of Burger Palace FIGURE 7.20: Excel Data File for DEA Analysis of Burger Palace Unit 1 TABLE 7.7: Summary of DEA Results The values for v1 and v2 that are associated with the inputs of labor-hours and materials, respectively, measure the relative increase in efficiency with each unit reduction of input value. For unit S4, each unit decrease in labor-hours results in an efficiency increase of 0.0555. For unit S4 to become efficient, it must increase its efficiency rating by 0.111 points. This can be accomplished by reducing labor used by 2 hours (i.e., 2 hours × 0.0555 = 0.111). Note that with this reduction in labor-hours, unit S4 becomes identical to efficient unit S3. An alternative approach would be a reduction in materials used by $16.57 (i.e., 0.111/0.0067 = 16.57). Any linear combination of these two measures also would move unit S4 to the productivity frontier defined by the line segment joining efficient units S3 and S6. Table 7.8 contains the calculations for a hypothetical unit C, which is a composite reference unit defined by the weighted inputs of the reference set S3 and S6. As Figure 7.19 shows, this composite unit C is located at the intersection of the productivity frontier and the dashed line drawn from the origin to unit S4. Thus, compared with this reference unit C, inefficient unit S4 is using excess inputs in the amounts of 0.7 labor-hour and 11.1 material dollars. Table 7.8: Calculation of Excess Inputs Used by Unit S4 DEA offers many opportunities for an inefficient unit to become efficient regarding its reference set of efficient units. In practice, management would choose a particular approach on the basis of an evaluation of its cost, practicality, and feasibility; however, the motivation for change is clear (i.e., other units actually are able to achieve similar outputs with fewer resources). DEA and Strategic Planning When combined with profitability, DEA efficiency analysis can be useful in strategic planning for services that are delivered through multiple sites (e.g., hotel chains). Figure 7.21 presents a matrix of four possibilities that arise from combining efficiency and profitability. Considering the top-left quadrant of this matrix (i.e., underperforming potential stars) reveals that units operating at a high profit may be operating inefficiently and, thus, have unrealized potential. Comparing these with similar efficient units could suggest measures that would lead to even greater profit through more efficient operations. FIGURE 7.21: DEA Strategic Matrix Star performers can be found in the top-right quadrant (i.e., benchmark group). These efficient units also are highly profitable and, thus, serve as examples for others to emulate in both operations efficiency and marketing success in generating high revenues. The lower-right quadrant (i.e., candidates for divestiture) contains efficient but unprofitable units. These units are limited in profit potential, perhaps because of a poor location, and should be sold to generate capital for expansion in new territories. It is not clear which strategy to employ with the lower-left quadrant units (i.e., problem branches). If profit potential is limited, investments in efficient operations might lead to a future candidate for divestiture.
Subject to the constraint that
1. For the Burger Palace example, perform a complete analysis of efficiency improvement alternatives for unit S2, including determination of a composite reference unit. 2. For the Burger Palace example, perform a complete analysis of efficiency improvement alternatives for unit S5, including determination of a composite reference unit. 3. For the Burger Palace example, what is the effect of removing an inefficient unit from the analysis (e.g., S2)? 4. For the Burger Palace example, what is the effect of removing an efficient unit from the analysis (e.g., S6)? Supplement Data Envelopment Analysis (DEA): How can corporate management evaluate the productivity of a fast-food outlet, a branch bank, a health clinic, or an elementary school? The difficulties in measuring productivity are threefold. First, what are the appropriate inputs to the system (e.g., labor hours, material dollars) and the measures of those inputs? Second, what are the appropriate outputs of the system (e.g., checks cashed, certificate of deposits) and the measures of those outputs? Third, what are the appropriate ways of measuring the relationship between these inputs and outputs? Measuring Service Productivity The measure of an organization’s productivity, if viewed from an engineering perspective, is similar to the measure of a system’s efficiency. It can be stated as a ratio of outputs to inputs (e.g., miles per gallon for an automobile). To evaluate the operational efficiency of a branch bank, for example, an accounting ratio such as cost per teller transaction might be used. A branch with a high ratio in comparison with those of other branches would be considered less efficient, but the higher ratio could result from a more complex mix of transactions. For example, a branch opening new accounts and selling CDs would require more time per transaction than another branch engaged only in simple transactions such as accepting deposits and cashing checks. The problem with using simple ratios is that the mix of outputs is not considered explicitly. This same criticism also can be made concerning the mix of inputs. For example, some branches might have automated teller machines in addition to live tellers, and this use of technology could affect the cost per teller transaction. Broad-based measures such as profitability or return on investment are highly relevant as overall performance measures, but they are not sufficient to evaluate the operating efficiency of a service unit. For instance, one could not conclude that a profitable branch bank is necessarily efficient in its use of personnel and other inputs. A higher than-average proportion of revenue-generating transactions could be the explanation rather than the cost-efficient use of resources. The DEA Model Fortunately, a technique has been developed with the ability to compare the efficiency of multiple service units that provide similar services by explicitly considering their use of multiple inputs (i.e., resources) to produce multiple outputs (i.e., services). The technique, which is referred to as data envelopment analysis (DEA), circumvents the need to develop standard costs for each service, because it can incorporate multiple inputs and multiple outputs into both the numerator and the denominator of the efficiency ratio without the need for conversion to a common dollar basis. Thus, the DEA measure of efficiency explicitly accounts for the mix of inputs and outputs and, consequently, is more comprehensive and reliable than a set of operating ratios or profit measures. DEA is a linear programming model that attempts to maximize a service unit’s efficiency, expressed as a ratio of outputs to inputs, by comparing a particular unit’s efficiency with the performance of a group of similar service units that are delivering the same service. In the process, some units achieve 100 percent efficiency and are referred to as the relatively efficient units, whereas other units with efficiency ratings of less than 100 percent are referred to as inefficient units. Corporate management thus can use DEA to compare a group of service units to identify relatively inefficient units, measure the magnitude of the inefficiencies, and by comparing the inefficient with the efficient ones, discover ways to reduce those inefficiencies. The DEA linear programming model is formulated according to Charnes, Cooper, and Rhodes, and is referred to as the CCR Model. 9 Definition of Variables Let Ek, with k = 1, 2, . . . K, be the efficiency ratio of unit k, where K is the total number of units being evaluated. Let uj , with j = 1, 2, . . . , M, be a coefficient for output j, where M is the total number of output types considered. The variable uj is a measure of the relative decrease in efficiency with each unit reduction of output value. Let vi , with i = 1, 2, . . . , N, be a coefficient for input i, where N is the total number of input types considered. The variable vi is a measure of the relative increase in efficiency with each unit reduction of input value. Let Ojk be the number of observed units of output j generated by service unit k during one time period. Let Iik be the number of actual units of input i used by service unit k during one time period. Objective Function The objective is to find the set of coefficient u’s associated with each output and of v’s associated with each input that will give the service unit being evaluated the highest possible efficiency. Where e is the index of the unit being evaluated. This function is subject to the constraint that when the same set of input and output coefficients (uj ’s and yi ’s) is applied to all other service units being compared, no service unit will exceed 100 percent efficiency or a ratio of 1.0. Constraints Where all coefficient values are positive and nonzero. To solve this fractional linear programming model using standard linear programming software requires a reformulation. Note that both the objective function and all constraints are ratios rather than linear functions. The objective function in equation (3) Is restated as a linear function by arbitrarily scaling the inputs for the unit under evaluation to a sum of 1.0. Subject to the constraint that For each service unit, the constraints in equation (2) are similarly reformulated: Where Sample Size A question of sample size often is raised concerning the number of service units that are required compared with the number of input and output variables selected in the analysis. The following relationship relating the number of service units K used in the analysis and the number of input N and output M types being considered is based on empirical findings and the experience of DEA practitioners: K ≥ 2(N + M) Example 7.3 Burger Palace An innovative drive-in-only burger chain has established six units in several different cities. Each unit is located in a strip shopping center parking lot. Only a standard meal consisting of a burger, fries, and a drink is available. Management has decided to use DEA to improve productivity by identifying which units are using their resources most efficiently and then sharing their experience and knowledge with the less efficient locations. Table 7.5 summarizes data for two inputs: labor-hours and material dollars consumed during a typical lunch hour period to generate an output of 100 meals sold. Normally, output will vary among the service units, but in this example, we have made the outputs equal to allow for a graphical presentation of the units’ productivity. As Figure 7.19 shows, service units S, S3, and S6 have been joined to form an efficient-production frontier of alternative methods of using labor hours and material resources to generate 100 meals. As can be seen, these efficient units have defined an envelope that contains all the inefficient units—thus the reason for calling the process “data envelopment analysis.” Table 7.5: Summary of Outputs and Inputs for Burger Palace  Figure 7.19: Productivity Frontier of Burger Palace For this simple example, we can identify efficient units by inspection and see the excess inputs being used by inefficient units (e.g., S2 would be as efficient as S3 if it used $50 less in materials). To gain an understanding of DEA, however, we will proceed to formulate the linear programming problems for each unit, then solve each of them to determine efficiency ratings and other information. We begin by illustrating the LP formulation for the first service unit, S1, using equations (5), 6), and (7). max E(S1) = u1 100 subject to u1100 − v12 − v2200 ≤ 0 u1100 − v14 − v2150 ≤ 0 u1100 − v14 − v2100 ≤ 0 u1100 − v16 − v2100 ≤ 0 u1100 − v18 − v280 ≤ 0 u1100 − v110 − v250 ≤ 0 v1 2 + v2200 = 1 u1, u2, v2, ≥ 0 Similar linear programming problems are formulated (or, better yet, the S1 linear programming problem is edited) and solved for the other service units by substituting the appropriate output function for the objective function and substituting the appropriate input function for the last constraint. Constraints 1 through 6, which restrict all units to no more than 100 percent efficiency, remain the same in all problems. This set of six linear programming problems was solved with Excel Solver 7.0 in fewer than five minutes by editing the data file between each run. Because the output is 100 meals for all units, only the last constraint must be edited by substituting the appropriate labor and material input values from Table 7.5 for the unit being evaluated. The data file for unit 1 of Burger Palace using a linear programming Excel add-in is shown in Figure 7.20. The linear programming results for each unit are shown in Table 7.6 and summarized in Table 7.7. In Table 7.6, we find that DEA has identified the same units shown as being efficient in Figure 7.19. Units S2, S4, and S5 all are inefficient in varying degrees. Also shown in Table 7.6 and associated with each inefficient unit is an efficiency reference set. Each inefficient unit will have a set of efficient units associated with it that defines its productivity. As Figure 7.19 shows for inefficient unit S4, the efficient units S3 and S6 have been joined with a line defining the efficiency frontier. A dashed line drawn from the origin to inefficient unit S4 cuts through this frontier and, thus, defines unit S4 as inefficient. In Table 7.7, the value in parentheses that is associated with each member of the efficiency reference set (i.e., .7778 for S3 and .2222 for S6) represents the relative weight assigned to that efficient unit in calculating the efficiency rating for S4. These relative weights are the shadow prices that are associated with the respective efficient-unit constraints in the linear programming solution. (Note in Table 7.6 that for unit 4, these weights appear as opportunity costs for S3 and S6.). TABLE 7.6: LP Solutions for DEA Study of Burger Palace FIGURE 7.20: Excel Data File for DEA Analysis of Burger Palace Unit 1 TABLE 7.7: Summary of DEA Results The values for v1 and v2 that are associated with the inputs of labor-hours and materials, respectively, measure the relative increase in efficiency with each unit reduction of input value. For unit S4, each unit decrease in labor-hours results in an efficiency increase of 0.0555. For unit S4 to become efficient, it must increase its efficiency rating by 0.111 points. This can be accomplished by reducing labor used by 2 hours (i.e., 2 hours × 0.0555 = 0.111). Note that with this reduction in labor-hours, unit S4 becomes identical to efficient unit S3. An alternative approach would be a reduction in materials used by $16.57 (i.e., 0.111/0.0067 = 16.57). Any linear combination of these two measures also would move unit S4 to the productivity frontier defined by the line segment joining efficient units S3 and S6. Table 7.8 contains the calculations for a hypothetical unit C, which is a composite reference unit defined by the weighted inputs of the reference set S3 and S6. As Figure 7.19 shows, this composite unit C is located at the intersection of the productivity frontier and the dashed line drawn from the origin to unit S4. Thus, compared with this reference unit C, inefficient unit S4 is using excess inputs in the amounts of 0.7 labor-hour and 11.1 material dollars. Table 7.8: Calculation of Excess Inputs Used by Unit S4 DEA offers many opportunities for an inefficient unit to become efficient regarding its reference set of efficient units. In practice, management would choose a particular approach on the basis of an evaluation of its cost, practicality, and feasibility; however, the motivation for change is clear (i.e., other units actually are able to achieve similar outputs with fewer resources). DEA and Strategic Planning When combined with profitability, DEA efficiency analysis can be useful in strategic planning for services that are delivered through multiple sites (e.g., hotel chains). Figure 7.21 presents a matrix of four possibilities that arise from combining efficiency and profitability. Considering the top-left quadrant of this matrix (i.e., underperforming potential stars) reveals that units operating at a high profit may be operating inefficiently and, thus, have unrealized potential. Comparing these with similar efficient units could suggest measures that would lead to even greater profit through more efficient operations. FIGURE 7.21: DEA Strategic Matrix Star performers can be found in the top-right quadrant (i.e., benchmark group). These efficient units also are highly profitable and, thus, serve as examples for others to emulate in both operations efficiency and marketing success in generating high revenues. The lower-right quadrant (i.e., candidates for divestiture) contains efficient but unprofitable units. These units are limited in profit potential, perhaps because of a poor location, and should be sold to generate capital for expansion in new territories. It is not clear which strategy to employ with the lower-left quadrant units (i.e., problem branches). If profit potential is limited, investments in efficient operations might lead to a future candidate for divestiture.
For each service unit, the constraints in equation (2) are similarly reformulated:
1. For the Burger Palace example, perform a complete analysis of efficiency improvement alternatives for unit S2, including determination of a composite reference unit. 2. For the Burger Palace example, perform a complete analysis of efficiency improvement alternatives for unit S5, including determination of a composite reference unit. 3. For the Burger Palace example, what is the effect of removing an inefficient unit from the analysis (e.g., S2)? 4. For the Burger Palace example, what is the effect of removing an efficient unit from the analysis (e.g., S6)? Supplement Data Envelopment Analysis (DEA): How can corporate management evaluate the productivity of a fast-food outlet, a branch bank, a health clinic, or an elementary school? The difficulties in measuring productivity are threefold. First, what are the appropriate inputs to the system (e.g., labor hours, material dollars) and the measures of those inputs? Second, what are the appropriate outputs of the system (e.g., checks cashed, certificate of deposits) and the measures of those outputs? Third, what are the appropriate ways of measuring the relationship between these inputs and outputs? Measuring Service Productivity The measure of an organization’s productivity, if viewed from an engineering perspective, is similar to the measure of a system’s efficiency. It can be stated as a ratio of outputs to inputs (e.g., miles per gallon for an automobile). To evaluate the operational efficiency of a branch bank, for example, an accounting ratio such as cost per teller transaction might be used. A branch with a high ratio in comparison with those of other branches would be considered less efficient, but the higher ratio could result from a more complex mix of transactions. For example, a branch opening new accounts and selling CDs would require more time per transaction than another branch engaged only in simple transactions such as accepting deposits and cashing checks. The problem with using simple ratios is that the mix of outputs is not considered explicitly. This same criticism also can be made concerning the mix of inputs. For example, some branches might have automated teller machines in addition to live tellers, and this use of technology could affect the cost per teller transaction. Broad-based measures such as profitability or return on investment are highly relevant as overall performance measures, but they are not sufficient to evaluate the operating efficiency of a service unit. For instance, one could not conclude that a profitable branch bank is necessarily efficient in its use of personnel and other inputs. A higher than-average proportion of revenue-generating transactions could be the explanation rather than the cost-efficient use of resources. The DEA Model Fortunately, a technique has been developed with the ability to compare the efficiency of multiple service units that provide similar services by explicitly considering their use of multiple inputs (i.e., resources) to produce multiple outputs (i.e., services). The technique, which is referred to as data envelopment analysis (DEA), circumvents the need to develop standard costs for each service, because it can incorporate multiple inputs and multiple outputs into both the numerator and the denominator of the efficiency ratio without the need for conversion to a common dollar basis. Thus, the DEA measure of efficiency explicitly accounts for the mix of inputs and outputs and, consequently, is more comprehensive and reliable than a set of operating ratios or profit measures. DEA is a linear programming model that attempts to maximize a service unit’s efficiency, expressed as a ratio of outputs to inputs, by comparing a particular unit’s efficiency with the performance of a group of similar service units that are delivering the same service. In the process, some units achieve 100 percent efficiency and are referred to as the relatively efficient units, whereas other units with efficiency ratings of less than 100 percent are referred to as inefficient units. Corporate management thus can use DEA to compare a group of service units to identify relatively inefficient units, measure the magnitude of the inefficiencies, and by comparing the inefficient with the efficient ones, discover ways to reduce those inefficiencies. The DEA linear programming model is formulated according to Charnes, Cooper, and Rhodes, and is referred to as the CCR Model. 9 Definition of Variables Let Ek, with k = 1, 2, . . . K, be the efficiency ratio of unit k, where K is the total number of units being evaluated. Let uj , with j = 1, 2, . . . , M, be a coefficient for output j, where M is the total number of output types considered. The variable uj is a measure of the relative decrease in efficiency with each unit reduction of output value. Let vi , with i = 1, 2, . . . , N, be a coefficient for input i, where N is the total number of input types considered. The variable vi is a measure of the relative increase in efficiency with each unit reduction of input value. Let Ojk be the number of observed units of output j generated by service unit k during one time period. Let Iik be the number of actual units of input i used by service unit k during one time period. Objective Function The objective is to find the set of coefficient u’s associated with each output and of v’s associated with each input that will give the service unit being evaluated the highest possible efficiency. Where e is the index of the unit being evaluated. This function is subject to the constraint that when the same set of input and output coefficients (uj ’s and yi ’s) is applied to all other service units being compared, no service unit will exceed 100 percent efficiency or a ratio of 1.0. Constraints Where all coefficient values are positive and nonzero. To solve this fractional linear programming model using standard linear programming software requires a reformulation. Note that both the objective function and all constraints are ratios rather than linear functions. The objective function in equation (3) Is restated as a linear function by arbitrarily scaling the inputs for the unit under evaluation to a sum of 1.0. Subject to the constraint that For each service unit, the constraints in equation (2) are similarly reformulated: Where Sample Size A question of sample size often is raised concerning the number of service units that are required compared with the number of input and output variables selected in the analysis. The following relationship relating the number of service units K used in the analysis and the number of input N and output M types being considered is based on empirical findings and the experience of DEA practitioners: K ≥ 2(N + M) Example 7.3 Burger Palace An innovative drive-in-only burger chain has established six units in several different cities. Each unit is located in a strip shopping center parking lot. Only a standard meal consisting of a burger, fries, and a drink is available. Management has decided to use DEA to improve productivity by identifying which units are using their resources most efficiently and then sharing their experience and knowledge with the less efficient locations. Table 7.5 summarizes data for two inputs: labor-hours and material dollars consumed during a typical lunch hour period to generate an output of 100 meals sold. Normally, output will vary among the service units, but in this example, we have made the outputs equal to allow for a graphical presentation of the units’ productivity. As Figure 7.19 shows, service units S, S3, and S6 have been joined to form an efficient-production frontier of alternative methods of using labor hours and material resources to generate 100 meals. As can be seen, these efficient units have defined an envelope that contains all the inefficient units—thus the reason for calling the process “data envelopment analysis.” Table 7.5: Summary of Outputs and Inputs for Burger Palace  Figure 7.19: Productivity Frontier of Burger Palace For this simple example, we can identify efficient units by inspection and see the excess inputs being used by inefficient units (e.g., S2 would be as efficient as S3 if it used $50 less in materials). To gain an understanding of DEA, however, we will proceed to formulate the linear programming problems for each unit, then solve each of them to determine efficiency ratings and other information. We begin by illustrating the LP formulation for the first service unit, S1, using equations (5), 6), and (7). max E(S1) = u1 100 subject to u1100 − v12 − v2200 ≤ 0 u1100 − v14 − v2150 ≤ 0 u1100 − v14 − v2100 ≤ 0 u1100 − v16 − v2100 ≤ 0 u1100 − v18 − v280 ≤ 0 u1100 − v110 − v250 ≤ 0 v1 2 + v2200 = 1 u1, u2, v2, ≥ 0 Similar linear programming problems are formulated (or, better yet, the S1 linear programming problem is edited) and solved for the other service units by substituting the appropriate output function for the objective function and substituting the appropriate input function for the last constraint. Constraints 1 through 6, which restrict all units to no more than 100 percent efficiency, remain the same in all problems. This set of six linear programming problems was solved with Excel Solver 7.0 in fewer than five minutes by editing the data file between each run. Because the output is 100 meals for all units, only the last constraint must be edited by substituting the appropriate labor and material input values from Table 7.5 for the unit being evaluated. The data file for unit 1 of Burger Palace using a linear programming Excel add-in is shown in Figure 7.20. The linear programming results for each unit are shown in Table 7.6 and summarized in Table 7.7. In Table 7.6, we find that DEA has identified the same units shown as being efficient in Figure 7.19. Units S2, S4, and S5 all are inefficient in varying degrees. Also shown in Table 7.6 and associated with each inefficient unit is an efficiency reference set. Each inefficient unit will have a set of efficient units associated with it that defines its productivity. As Figure 7.19 shows for inefficient unit S4, the efficient units S3 and S6 have been joined with a line defining the efficiency frontier. A dashed line drawn from the origin to inefficient unit S4 cuts through this frontier and, thus, defines unit S4 as inefficient. In Table 7.7, the value in parentheses that is associated with each member of the efficiency reference set (i.e., .7778 for S3 and .2222 for S6) represents the relative weight assigned to that efficient unit in calculating the efficiency rating for S4. These relative weights are the shadow prices that are associated with the respective efficient-unit constraints in the linear programming solution. (Note in Table 7.6 that for unit 4, these weights appear as opportunity costs for S3 and S6.). TABLE 7.6: LP Solutions for DEA Study of Burger Palace FIGURE 7.20: Excel Data File for DEA Analysis of Burger Palace Unit 1 TABLE 7.7: Summary of DEA Results The values for v1 and v2 that are associated with the inputs of labor-hours and materials, respectively, measure the relative increase in efficiency with each unit reduction of input value. For unit S4, each unit decrease in labor-hours results in an efficiency increase of 0.0555. For unit S4 to become efficient, it must increase its efficiency rating by 0.111 points. This can be accomplished by reducing labor used by 2 hours (i.e., 2 hours × 0.0555 = 0.111). Note that with this reduction in labor-hours, unit S4 becomes identical to efficient unit S3. An alternative approach would be a reduction in materials used by $16.57 (i.e., 0.111/0.0067 = 16.57). Any linear combination of these two measures also would move unit S4 to the productivity frontier defined by the line segment joining efficient units S3 and S6. Table 7.8 contains the calculations for a hypothetical unit C, which is a composite reference unit defined by the weighted inputs of the reference set S3 and S6. As Figure 7.19 shows, this composite unit C is located at the intersection of the productivity frontier and the dashed line drawn from the origin to unit S4. Thus, compared with this reference unit C, inefficient unit S4 is using excess inputs in the amounts of 0.7 labor-hour and 11.1 material dollars. Table 7.8: Calculation of Excess Inputs Used by Unit S4 DEA offers many opportunities for an inefficient unit to become efficient regarding its reference set of efficient units. In practice, management would choose a particular approach on the basis of an evaluation of its cost, practicality, and feasibility; however, the motivation for change is clear (i.e., other units actually are able to achieve similar outputs with fewer resources). DEA and Strategic Planning When combined with profitability, DEA efficiency analysis can be useful in strategic planning for services that are delivered through multiple sites (e.g., hotel chains). Figure 7.21 presents a matrix of four possibilities that arise from combining efficiency and profitability. Considering the top-left quadrant of this matrix (i.e., underperforming potential stars) reveals that units operating at a high profit may be operating inefficiently and, thus, have unrealized potential. Comparing these with similar efficient units could suggest measures that would lead to even greater profit through more efficient operations. FIGURE 7.21: DEA Strategic Matrix Star performers can be found in the top-right quadrant (i.e., benchmark group). These efficient units also are highly profitable and, thus, serve as examples for others to emulate in both operations efficiency and marketing success in generating high revenues. The lower-right quadrant (i.e., candidates for divestiture) contains efficient but unprofitable units. These units are limited in profit potential, perhaps because of a poor location, and should be sold to generate capital for expansion in new territories. It is not clear which strategy to employ with the lower-left quadrant units (i.e., problem branches). If profit potential is limited, investments in efficient operations might lead to a future candidate for divestiture.
Where
1. For the Burger Palace example, perform a complete analysis of efficiency improvement alternatives for unit S2, including determination of a composite reference unit. 2. For the Burger Palace example, perform a complete analysis of efficiency improvement alternatives for unit S5, including determination of a composite reference unit. 3. For the Burger Palace example, what is the effect of removing an inefficient unit from the analysis (e.g., S2)? 4. For the Burger Palace example, what is the effect of removing an efficient unit from the analysis (e.g., S6)? Supplement Data Envelopment Analysis (DEA): How can corporate management evaluate the productivity of a fast-food outlet, a branch bank, a health clinic, or an elementary school? The difficulties in measuring productivity are threefold. First, what are the appropriate inputs to the system (e.g., labor hours, material dollars) and the measures of those inputs? Second, what are the appropriate outputs of the system (e.g., checks cashed, certificate of deposits) and the measures of those outputs? Third, what are the appropriate ways of measuring the relationship between these inputs and outputs? Measuring Service Productivity The measure of an organization’s productivity, if viewed from an engineering perspective, is similar to the measure of a system’s efficiency. It can be stated as a ratio of outputs to inputs (e.g., miles per gallon for an automobile). To evaluate the operational efficiency of a branch bank, for example, an accounting ratio such as cost per teller transaction might be used. A branch with a high ratio in comparison with those of other branches would be considered less efficient, but the higher ratio could result from a more complex mix of transactions. For example, a branch opening new accounts and selling CDs would require more time per transaction than another branch engaged only in simple transactions such as accepting deposits and cashing checks. The problem with using simple ratios is that the mix of outputs is not considered explicitly. This same criticism also can be made concerning the mix of inputs. For example, some branches might have automated teller machines in addition to live tellers, and this use of technology could affect the cost per teller transaction. Broad-based measures such as profitability or return on investment are highly relevant as overall performance measures, but they are not sufficient to evaluate the operating efficiency of a service unit. For instance, one could not conclude that a profitable branch bank is necessarily efficient in its use of personnel and other inputs. A higher than-average proportion of revenue-generating transactions could be the explanation rather than the cost-efficient use of resources. The DEA Model Fortunately, a technique has been developed with the ability to compare the efficiency of multiple service units that provide similar services by explicitly considering their use of multiple inputs (i.e., resources) to produce multiple outputs (i.e., services). The technique, which is referred to as data envelopment analysis (DEA), circumvents the need to develop standard costs for each service, because it can incorporate multiple inputs and multiple outputs into both the numerator and the denominator of the efficiency ratio without the need for conversion to a common dollar basis. Thus, the DEA measure of efficiency explicitly accounts for the mix of inputs and outputs and, consequently, is more comprehensive and reliable than a set of operating ratios or profit measures. DEA is a linear programming model that attempts to maximize a service unit’s efficiency, expressed as a ratio of outputs to inputs, by comparing a particular unit’s efficiency with the performance of a group of similar service units that are delivering the same service. In the process, some units achieve 100 percent efficiency and are referred to as the relatively efficient units, whereas other units with efficiency ratings of less than 100 percent are referred to as inefficient units. Corporate management thus can use DEA to compare a group of service units to identify relatively inefficient units, measure the magnitude of the inefficiencies, and by comparing the inefficient with the efficient ones, discover ways to reduce those inefficiencies. The DEA linear programming model is formulated according to Charnes, Cooper, and Rhodes, and is referred to as the CCR Model. 9 Definition of Variables Let Ek, with k = 1, 2, . . . K, be the efficiency ratio of unit k, where K is the total number of units being evaluated. Let uj , with j = 1, 2, . . . , M, be a coefficient for output j, where M is the total number of output types considered. The variable uj is a measure of the relative decrease in efficiency with each unit reduction of output value. Let vi , with i = 1, 2, . . . , N, be a coefficient for input i, where N is the total number of input types considered. The variable vi is a measure of the relative increase in efficiency with each unit reduction of input value. Let Ojk be the number of observed units of output j generated by service unit k during one time period. Let Iik be the number of actual units of input i used by service unit k during one time period. Objective Function The objective is to find the set of coefficient u’s associated with each output and of v’s associated with each input that will give the service unit being evaluated the highest possible efficiency. Where e is the index of the unit being evaluated. This function is subject to the constraint that when the same set of input and output coefficients (uj ’s and yi ’s) is applied to all other service units being compared, no service unit will exceed 100 percent efficiency or a ratio of 1.0. Constraints Where all coefficient values are positive and nonzero. To solve this fractional linear programming model using standard linear programming software requires a reformulation. Note that both the objective function and all constraints are ratios rather than linear functions. The objective function in equation (3) Is restated as a linear function by arbitrarily scaling the inputs for the unit under evaluation to a sum of 1.0. Subject to the constraint that For each service unit, the constraints in equation (2) are similarly reformulated: Where Sample Size A question of sample size often is raised concerning the number of service units that are required compared with the number of input and output variables selected in the analysis. The following relationship relating the number of service units K used in the analysis and the number of input N and output M types being considered is based on empirical findings and the experience of DEA practitioners: K ≥ 2(N + M) Example 7.3 Burger Palace An innovative drive-in-only burger chain has established six units in several different cities. Each unit is located in a strip shopping center parking lot. Only a standard meal consisting of a burger, fries, and a drink is available. Management has decided to use DEA to improve productivity by identifying which units are using their resources most efficiently and then sharing their experience and knowledge with the less efficient locations. Table 7.5 summarizes data for two inputs: labor-hours and material dollars consumed during a typical lunch hour period to generate an output of 100 meals sold. Normally, output will vary among the service units, but in this example, we have made the outputs equal to allow for a graphical presentation of the units’ productivity. As Figure 7.19 shows, service units S, S3, and S6 have been joined to form an efficient-production frontier of alternative methods of using labor hours and material resources to generate 100 meals. As can be seen, these efficient units have defined an envelope that contains all the inefficient units—thus the reason for calling the process “data envelopment analysis.” Table 7.5: Summary of Outputs and Inputs for Burger Palace  Figure 7.19: Productivity Frontier of Burger Palace For this simple example, we can identify efficient units by inspection and see the excess inputs being used by inefficient units (e.g., S2 would be as efficient as S3 if it used $50 less in materials). To gain an understanding of DEA, however, we will proceed to formulate the linear programming problems for each unit, then solve each of them to determine efficiency ratings and other information. We begin by illustrating the LP formulation for the first service unit, S1, using equations (5), 6), and (7). max E(S1) = u1 100 subject to u1100 − v12 − v2200 ≤ 0 u1100 − v14 − v2150 ≤ 0 u1100 − v14 − v2100 ≤ 0 u1100 − v16 − v2100 ≤ 0 u1100 − v18 − v280 ≤ 0 u1100 − v110 − v250 ≤ 0 v1 2 + v2200 = 1 u1, u2, v2, ≥ 0 Similar linear programming problems are formulated (or, better yet, the S1 linear programming problem is edited) and solved for the other service units by substituting the appropriate output function for the objective function and substituting the appropriate input function for the last constraint. Constraints 1 through 6, which restrict all units to no more than 100 percent efficiency, remain the same in all problems. This set of six linear programming problems was solved with Excel Solver 7.0 in fewer than five minutes by editing the data file between each run. Because the output is 100 meals for all units, only the last constraint must be edited by substituting the appropriate labor and material input values from Table 7.5 for the unit being evaluated. The data file for unit 1 of Burger Palace using a linear programming Excel add-in is shown in Figure 7.20. The linear programming results for each unit are shown in Table 7.6 and summarized in Table 7.7. In Table 7.6, we find that DEA has identified the same units shown as being efficient in Figure 7.19. Units S2, S4, and S5 all are inefficient in varying degrees. Also shown in Table 7.6 and associated with each inefficient unit is an efficiency reference set. Each inefficient unit will have a set of efficient units associated with it that defines its productivity. As Figure 7.19 shows for inefficient unit S4, the efficient units S3 and S6 have been joined with a line defining the efficiency frontier. A dashed line drawn from the origin to inefficient unit S4 cuts through this frontier and, thus, defines unit S4 as inefficient. In Table 7.7, the value in parentheses that is associated with each member of the efficiency reference set (i.e., .7778 for S3 and .2222 for S6) represents the relative weight assigned to that efficient unit in calculating the efficiency rating for S4. These relative weights are the shadow prices that are associated with the respective efficient-unit constraints in the linear programming solution. (Note in Table 7.6 that for unit 4, these weights appear as opportunity costs for S3 and S6.). TABLE 7.6: LP Solutions for DEA Study of Burger Palace FIGURE 7.20: Excel Data File for DEA Analysis of Burger Palace Unit 1 TABLE 7.7: Summary of DEA Results The values for v1 and v2 that are associated with the inputs of labor-hours and materials, respectively, measure the relative increase in efficiency with each unit reduction of input value. For unit S4, each unit decrease in labor-hours results in an efficiency increase of 0.0555. For unit S4 to become efficient, it must increase its efficiency rating by 0.111 points. This can be accomplished by reducing labor used by 2 hours (i.e., 2 hours × 0.0555 = 0.111). Note that with this reduction in labor-hours, unit S4 becomes identical to efficient unit S3. An alternative approach would be a reduction in materials used by $16.57 (i.e., 0.111/0.0067 = 16.57). Any linear combination of these two measures also would move unit S4 to the productivity frontier defined by the line segment joining efficient units S3 and S6. Table 7.8 contains the calculations for a hypothetical unit C, which is a composite reference unit defined by the weighted inputs of the reference set S3 and S6. As Figure 7.19 shows, this composite unit C is located at the intersection of the productivity frontier and the dashed line drawn from the origin to unit S4. Thus, compared with this reference unit C, inefficient unit S4 is using excess inputs in the amounts of 0.7 labor-hour and 11.1 material dollars. Table 7.8: Calculation of Excess Inputs Used by Unit S4 DEA offers many opportunities for an inefficient unit to become efficient regarding its reference set of efficient units. In practice, management would choose a particular approach on the basis of an evaluation of its cost, practicality, and feasibility; however, the motivation for change is clear (i.e., other units actually are able to achieve similar outputs with fewer resources). DEA and Strategic Planning When combined with profitability, DEA efficiency analysis can be useful in strategic planning for services that are delivered through multiple sites (e.g., hotel chains). Figure 7.21 presents a matrix of four possibilities that arise from combining efficiency and profitability. Considering the top-left quadrant of this matrix (i.e., underperforming potential stars) reveals that units operating at a high profit may be operating inefficiently and, thus, have unrealized potential. Comparing these with similar efficient units could suggest measures that would lead to even greater profit through more efficient operations. FIGURE 7.21: DEA Strategic Matrix Star performers can be found in the top-right quadrant (i.e., benchmark group). These efficient units also are highly profitable and, thus, serve as examples for others to emulate in both operations efficiency and marketing success in generating high revenues. The lower-right quadrant (i.e., candidates for divestiture) contains efficient but unprofitable units. These units are limited in profit potential, perhaps because of a poor location, and should be sold to generate capital for expansion in new territories. It is not clear which strategy to employ with the lower-left quadrant units (i.e., problem branches). If profit potential is limited, investments in efficient operations might lead to a future candidate for divestiture.
Sample Size A question of sample size often is raised concerning the number of service units that are required compared with the number of input and output variables selected in the analysis. The following relationship relating the number of service units K used in the analysis and the number of input N and output M types being considered is based on empirical findings and the experience of DEA practitioners: K ≥ 2(N + M) Example 7.3 Burger Palace An innovative drive-in-only burger chain has established six units in several different cities. Each unit is located in a strip shopping center parking lot. Only a standard meal consisting of a burger, fries, and a drink is available. Management has decided to use DEA to improve productivity by identifying which units are using their resources most efficiently and then sharing their experience and knowledge with the less efficient locations. Table 7.5 summarizes data for two inputs: labor-hours and material dollars consumed during a typical lunch hour period to generate an output of 100 meals sold. Normally, output will vary among the service units, but in this example, we have made the outputs equal to allow for a graphical presentation of the units’ productivity. As Figure 7.19 shows, service units S, S3, and S6 have been joined to form an efficient-production frontier of alternative methods of using labor hours and material resources to generate 100 meals. As can be seen, these efficient units have defined an envelope that contains all the inefficient units—thus the reason for calling the process “data envelopment analysis.” Table 7.5: Summary of Outputs and Inputs for Burger Palace 
1. For the Burger Palace example, perform a complete analysis of efficiency improvement alternatives for unit S2, including determination of a composite reference unit. 2. For the Burger Palace example, perform a complete analysis of efficiency improvement alternatives for unit S5, including determination of a composite reference unit. 3. For the Burger Palace example, what is the effect of removing an inefficient unit from the analysis (e.g., S2)? 4. For the Burger Palace example, what is the effect of removing an efficient unit from the analysis (e.g., S6)? Supplement Data Envelopment Analysis (DEA): How can corporate management evaluate the productivity of a fast-food outlet, a branch bank, a health clinic, or an elementary school? The difficulties in measuring productivity are threefold. First, what are the appropriate inputs to the system (e.g., labor hours, material dollars) and the measures of those inputs? Second, what are the appropriate outputs of the system (e.g., checks cashed, certificate of deposits) and the measures of those outputs? Third, what are the appropriate ways of measuring the relationship between these inputs and outputs? Measuring Service Productivity The measure of an organization’s productivity, if viewed from an engineering perspective, is similar to the measure of a system’s efficiency. It can be stated as a ratio of outputs to inputs (e.g., miles per gallon for an automobile). To evaluate the operational efficiency of a branch bank, for example, an accounting ratio such as cost per teller transaction might be used. A branch with a high ratio in comparison with those of other branches would be considered less efficient, but the higher ratio could result from a more complex mix of transactions. For example, a branch opening new accounts and selling CDs would require more time per transaction than another branch engaged only in simple transactions such as accepting deposits and cashing checks. The problem with using simple ratios is that the mix of outputs is not considered explicitly. This same criticism also can be made concerning the mix of inputs. For example, some branches might have automated teller machines in addition to live tellers, and this use of technology could affect the cost per teller transaction. Broad-based measures such as profitability or return on investment are highly relevant as overall performance measures, but they are not sufficient to evaluate the operating efficiency of a service unit. For instance, one could not conclude that a profitable branch bank is necessarily efficient in its use of personnel and other inputs. A higher than-average proportion of revenue-generating transactions could be the explanation rather than the cost-efficient use of resources. The DEA Model Fortunately, a technique has been developed with the ability to compare the efficiency of multiple service units that provide similar services by explicitly considering their use of multiple inputs (i.e., resources) to produce multiple outputs (i.e., services). The technique, which is referred to as data envelopment analysis (DEA), circumvents the need to develop standard costs for each service, because it can incorporate multiple inputs and multiple outputs into both the numerator and the denominator of the efficiency ratio without the need for conversion to a common dollar basis. Thus, the DEA measure of efficiency explicitly accounts for the mix of inputs and outputs and, consequently, is more comprehensive and reliable than a set of operating ratios or profit measures. DEA is a linear programming model that attempts to maximize a service unit’s efficiency, expressed as a ratio of outputs to inputs, by comparing a particular unit’s efficiency with the performance of a group of similar service units that are delivering the same service. In the process, some units achieve 100 percent efficiency and are referred to as the relatively efficient units, whereas other units with efficiency ratings of less than 100 percent are referred to as inefficient units. Corporate management thus can use DEA to compare a group of service units to identify relatively inefficient units, measure the magnitude of the inefficiencies, and by comparing the inefficient with the efficient ones, discover ways to reduce those inefficiencies. The DEA linear programming model is formulated according to Charnes, Cooper, and Rhodes, and is referred to as the CCR Model. 9 Definition of Variables Let Ek, with k = 1, 2, . . . K, be the efficiency ratio of unit k, where K is the total number of units being evaluated. Let uj , with j = 1, 2, . . . , M, be a coefficient for output j, where M is the total number of output types considered. The variable uj is a measure of the relative decrease in efficiency with each unit reduction of output value. Let vi , with i = 1, 2, . . . , N, be a coefficient for input i, where N is the total number of input types considered. The variable vi is a measure of the relative increase in efficiency with each unit reduction of input value. Let Ojk be the number of observed units of output j generated by service unit k during one time period. Let Iik be the number of actual units of input i used by service unit k during one time period. Objective Function The objective is to find the set of coefficient u’s associated with each output and of v’s associated with each input that will give the service unit being evaluated the highest possible efficiency. Where e is the index of the unit being evaluated. This function is subject to the constraint that when the same set of input and output coefficients (uj ’s and yi ’s) is applied to all other service units being compared, no service unit will exceed 100 percent efficiency or a ratio of 1.0. Constraints Where all coefficient values are positive and nonzero. To solve this fractional linear programming model using standard linear programming software requires a reformulation. Note that both the objective function and all constraints are ratios rather than linear functions. The objective function in equation (3) Is restated as a linear function by arbitrarily scaling the inputs for the unit under evaluation to a sum of 1.0. Subject to the constraint that For each service unit, the constraints in equation (2) are similarly reformulated: Where Sample Size A question of sample size often is raised concerning the number of service units that are required compared with the number of input and output variables selected in the analysis. The following relationship relating the number of service units K used in the analysis and the number of input N and output M types being considered is based on empirical findings and the experience of DEA practitioners: K ≥ 2(N + M) Example 7.3 Burger Palace An innovative drive-in-only burger chain has established six units in several different cities. Each unit is located in a strip shopping center parking lot. Only a standard meal consisting of a burger, fries, and a drink is available. Management has decided to use DEA to improve productivity by identifying which units are using their resources most efficiently and then sharing their experience and knowledge with the less efficient locations. Table 7.5 summarizes data for two inputs: labor-hours and material dollars consumed during a typical lunch hour period to generate an output of 100 meals sold. Normally, output will vary among the service units, but in this example, we have made the outputs equal to allow for a graphical presentation of the units’ productivity. As Figure 7.19 shows, service units S, S3, and S6 have been joined to form an efficient-production frontier of alternative methods of using labor hours and material resources to generate 100 meals. As can be seen, these efficient units have defined an envelope that contains all the inefficient units—thus the reason for calling the process “data envelopment analysis.” Table 7.5: Summary of Outputs and Inputs for Burger Palace  Figure 7.19: Productivity Frontier of Burger Palace For this simple example, we can identify efficient units by inspection and see the excess inputs being used by inefficient units (e.g., S2 would be as efficient as S3 if it used $50 less in materials). To gain an understanding of DEA, however, we will proceed to formulate the linear programming problems for each unit, then solve each of them to determine efficiency ratings and other information. We begin by illustrating the LP formulation for the first service unit, S1, using equations (5), 6), and (7). max E(S1) = u1 100 subject to u1100 − v12 − v2200 ≤ 0 u1100 − v14 − v2150 ≤ 0 u1100 − v14 − v2100 ≤ 0 u1100 − v16 − v2100 ≤ 0 u1100 − v18 − v280 ≤ 0 u1100 − v110 − v250 ≤ 0 v1 2 + v2200 = 1 u1, u2, v2, ≥ 0 Similar linear programming problems are formulated (or, better yet, the S1 linear programming problem is edited) and solved for the other service units by substituting the appropriate output function for the objective function and substituting the appropriate input function for the last constraint. Constraints 1 through 6, which restrict all units to no more than 100 percent efficiency, remain the same in all problems. This set of six linear programming problems was solved with Excel Solver 7.0 in fewer than five minutes by editing the data file between each run. Because the output is 100 meals for all units, only the last constraint must be edited by substituting the appropriate labor and material input values from Table 7.5 for the unit being evaluated. The data file for unit 1 of Burger Palace using a linear programming Excel add-in is shown in Figure 7.20. The linear programming results for each unit are shown in Table 7.6 and summarized in Table 7.7. In Table 7.6, we find that DEA has identified the same units shown as being efficient in Figure 7.19. Units S2, S4, and S5 all are inefficient in varying degrees. Also shown in Table 7.6 and associated with each inefficient unit is an efficiency reference set. Each inefficient unit will have a set of efficient units associated with it that defines its productivity. As Figure 7.19 shows for inefficient unit S4, the efficient units S3 and S6 have been joined with a line defining the efficiency frontier. A dashed line drawn from the origin to inefficient unit S4 cuts through this frontier and, thus, defines unit S4 as inefficient. In Table 7.7, the value in parentheses that is associated with each member of the efficiency reference set (i.e., .7778 for S3 and .2222 for S6) represents the relative weight assigned to that efficient unit in calculating the efficiency rating for S4. These relative weights are the shadow prices that are associated with the respective efficient-unit constraints in the linear programming solution. (Note in Table 7.6 that for unit 4, these weights appear as opportunity costs for S3 and S6.). TABLE 7.6: LP Solutions for DEA Study of Burger Palace FIGURE 7.20: Excel Data File for DEA Analysis of Burger Palace Unit 1 TABLE 7.7: Summary of DEA Results The values for v1 and v2 that are associated with the inputs of labor-hours and materials, respectively, measure the relative increase in efficiency with each unit reduction of input value. For unit S4, each unit decrease in labor-hours results in an efficiency increase of 0.0555. For unit S4 to become efficient, it must increase its efficiency rating by 0.111 points. This can be accomplished by reducing labor used by 2 hours (i.e., 2 hours × 0.0555 = 0.111). Note that with this reduction in labor-hours, unit S4 becomes identical to efficient unit S3. An alternative approach would be a reduction in materials used by $16.57 (i.e., 0.111/0.0067 = 16.57). Any linear combination of these two measures also would move unit S4 to the productivity frontier defined by the line segment joining efficient units S3 and S6. Table 7.8 contains the calculations for a hypothetical unit C, which is a composite reference unit defined by the weighted inputs of the reference set S3 and S6. As Figure 7.19 shows, this composite unit C is located at the intersection of the productivity frontier and the dashed line drawn from the origin to unit S4. Thus, compared with this reference unit C, inefficient unit S4 is using excess inputs in the amounts of 0.7 labor-hour and 11.1 material dollars. Table 7.8: Calculation of Excess Inputs Used by Unit S4 DEA offers many opportunities for an inefficient unit to become efficient regarding its reference set of efficient units. In practice, management would choose a particular approach on the basis of an evaluation of its cost, practicality, and feasibility; however, the motivation for change is clear (i.e., other units actually are able to achieve similar outputs with fewer resources). DEA and Strategic Planning When combined with profitability, DEA efficiency analysis can be useful in strategic planning for services that are delivered through multiple sites (e.g., hotel chains). Figure 7.21 presents a matrix of four possibilities that arise from combining efficiency and profitability. Considering the top-left quadrant of this matrix (i.e., underperforming potential stars) reveals that units operating at a high profit may be operating inefficiently and, thus, have unrealized potential. Comparing these with similar efficient units could suggest measures that would lead to even greater profit through more efficient operations. FIGURE 7.21: DEA Strategic Matrix Star performers can be found in the top-right quadrant (i.e., benchmark group). These efficient units also are highly profitable and, thus, serve as examples for others to emulate in both operations efficiency and marketing success in generating high revenues. The lower-right quadrant (i.e., candidates for divestiture) contains efficient but unprofitable units. These units are limited in profit potential, perhaps because of a poor location, and should be sold to generate capital for expansion in new territories. It is not clear which strategy to employ with the lower-left quadrant units (i.e., problem branches). If profit potential is limited, investments in efficient operations might lead to a future candidate for divestiture.
Figure 7.19: Productivity Frontier of Burger Palace
1. For the Burger Palace example, perform a complete analysis of efficiency improvement alternatives for unit S2, including determination of a composite reference unit. 2. For the Burger Palace example, perform a complete analysis of efficiency improvement alternatives for unit S5, including determination of a composite reference unit. 3. For the Burger Palace example, what is the effect of removing an inefficient unit from the analysis (e.g., S2)? 4. For the Burger Palace example, what is the effect of removing an efficient unit from the analysis (e.g., S6)? Supplement Data Envelopment Analysis (DEA): How can corporate management evaluate the productivity of a fast-food outlet, a branch bank, a health clinic, or an elementary school? The difficulties in measuring productivity are threefold. First, what are the appropriate inputs to the system (e.g., labor hours, material dollars) and the measures of those inputs? Second, what are the appropriate outputs of the system (e.g., checks cashed, certificate of deposits) and the measures of those outputs? Third, what are the appropriate ways of measuring the relationship between these inputs and outputs? Measuring Service Productivity The measure of an organization’s productivity, if viewed from an engineering perspective, is similar to the measure of a system’s efficiency. It can be stated as a ratio of outputs to inputs (e.g., miles per gallon for an automobile). To evaluate the operational efficiency of a branch bank, for example, an accounting ratio such as cost per teller transaction might be used. A branch with a high ratio in comparison with those of other branches would be considered less efficient, but the higher ratio could result from a more complex mix of transactions. For example, a branch opening new accounts and selling CDs would require more time per transaction than another branch engaged only in simple transactions such as accepting deposits and cashing checks. The problem with using simple ratios is that the mix of outputs is not considered explicitly. This same criticism also can be made concerning the mix of inputs. For example, some branches might have automated teller machines in addition to live tellers, and this use of technology could affect the cost per teller transaction. Broad-based measures such as profitability or return on investment are highly relevant as overall performance measures, but they are not sufficient to evaluate the operating efficiency of a service unit. For instance, one could not conclude that a profitable branch bank is necessarily efficient in its use of personnel and other inputs. A higher than-average proportion of revenue-generating transactions could be the explanation rather than the cost-efficient use of resources. The DEA Model Fortunately, a technique has been developed with the ability to compare the efficiency of multiple service units that provide similar services by explicitly considering their use of multiple inputs (i.e., resources) to produce multiple outputs (i.e., services). The technique, which is referred to as data envelopment analysis (DEA), circumvents the need to develop standard costs for each service, because it can incorporate multiple inputs and multiple outputs into both the numerator and the denominator of the efficiency ratio without the need for conversion to a common dollar basis. Thus, the DEA measure of efficiency explicitly accounts for the mix of inputs and outputs and, consequently, is more comprehensive and reliable than a set of operating ratios or profit measures. DEA is a linear programming model that attempts to maximize a service unit’s efficiency, expressed as a ratio of outputs to inputs, by comparing a particular unit’s efficiency with the performance of a group of similar service units that are delivering the same service. In the process, some units achieve 100 percent efficiency and are referred to as the relatively efficient units, whereas other units with efficiency ratings of less than 100 percent are referred to as inefficient units. Corporate management thus can use DEA to compare a group of service units to identify relatively inefficient units, measure the magnitude of the inefficiencies, and by comparing the inefficient with the efficient ones, discover ways to reduce those inefficiencies. The DEA linear programming model is formulated according to Charnes, Cooper, and Rhodes, and is referred to as the CCR Model. 9 Definition of Variables Let Ek, with k = 1, 2, . . . K, be the efficiency ratio of unit k, where K is the total number of units being evaluated. Let uj , with j = 1, 2, . . . , M, be a coefficient for output j, where M is the total number of output types considered. The variable uj is a measure of the relative decrease in efficiency with each unit reduction of output value. Let vi , with i = 1, 2, . . . , N, be a coefficient for input i, where N is the total number of input types considered. The variable vi is a measure of the relative increase in efficiency with each unit reduction of input value. Let Ojk be the number of observed units of output j generated by service unit k during one time period. Let Iik be the number of actual units of input i used by service unit k during one time period. Objective Function The objective is to find the set of coefficient u’s associated with each output and of v’s associated with each input that will give the service unit being evaluated the highest possible efficiency. Where e is the index of the unit being evaluated. This function is subject to the constraint that when the same set of input and output coefficients (uj ’s and yi ’s) is applied to all other service units being compared, no service unit will exceed 100 percent efficiency or a ratio of 1.0. Constraints Where all coefficient values are positive and nonzero. To solve this fractional linear programming model using standard linear programming software requires a reformulation. Note that both the objective function and all constraints are ratios rather than linear functions. The objective function in equation (3) Is restated as a linear function by arbitrarily scaling the inputs for the unit under evaluation to a sum of 1.0. Subject to the constraint that For each service unit, the constraints in equation (2) are similarly reformulated: Where Sample Size A question of sample size often is raised concerning the number of service units that are required compared with the number of input and output variables selected in the analysis. The following relationship relating the number of service units K used in the analysis and the number of input N and output M types being considered is based on empirical findings and the experience of DEA practitioners: K ≥ 2(N + M) Example 7.3 Burger Palace An innovative drive-in-only burger chain has established six units in several different cities. Each unit is located in a strip shopping center parking lot. Only a standard meal consisting of a burger, fries, and a drink is available. Management has decided to use DEA to improve productivity by identifying which units are using their resources most efficiently and then sharing their experience and knowledge with the less efficient locations. Table 7.5 summarizes data for two inputs: labor-hours and material dollars consumed during a typical lunch hour period to generate an output of 100 meals sold. Normally, output will vary among the service units, but in this example, we have made the outputs equal to allow for a graphical presentation of the units’ productivity. As Figure 7.19 shows, service units S, S3, and S6 have been joined to form an efficient-production frontier of alternative methods of using labor hours and material resources to generate 100 meals. As can be seen, these efficient units have defined an envelope that contains all the inefficient units—thus the reason for calling the process “data envelopment analysis.” Table 7.5: Summary of Outputs and Inputs for Burger Palace  Figure 7.19: Productivity Frontier of Burger Palace For this simple example, we can identify efficient units by inspection and see the excess inputs being used by inefficient units (e.g., S2 would be as efficient as S3 if it used $50 less in materials). To gain an understanding of DEA, however, we will proceed to formulate the linear programming problems for each unit, then solve each of them to determine efficiency ratings and other information. We begin by illustrating the LP formulation for the first service unit, S1, using equations (5), 6), and (7). max E(S1) = u1 100 subject to u1100 − v12 − v2200 ≤ 0 u1100 − v14 − v2150 ≤ 0 u1100 − v14 − v2100 ≤ 0 u1100 − v16 − v2100 ≤ 0 u1100 − v18 − v280 ≤ 0 u1100 − v110 − v250 ≤ 0 v1 2 + v2200 = 1 u1, u2, v2, ≥ 0 Similar linear programming problems are formulated (or, better yet, the S1 linear programming problem is edited) and solved for the other service units by substituting the appropriate output function for the objective function and substituting the appropriate input function for the last constraint. Constraints 1 through 6, which restrict all units to no more than 100 percent efficiency, remain the same in all problems. This set of six linear programming problems was solved with Excel Solver 7.0 in fewer than five minutes by editing the data file between each run. Because the output is 100 meals for all units, only the last constraint must be edited by substituting the appropriate labor and material input values from Table 7.5 for the unit being evaluated. The data file for unit 1 of Burger Palace using a linear programming Excel add-in is shown in Figure 7.20. The linear programming results for each unit are shown in Table 7.6 and summarized in Table 7.7. In Table 7.6, we find that DEA has identified the same units shown as being efficient in Figure 7.19. Units S2, S4, and S5 all are inefficient in varying degrees. Also shown in Table 7.6 and associated with each inefficient unit is an efficiency reference set. Each inefficient unit will have a set of efficient units associated with it that defines its productivity. As Figure 7.19 shows for inefficient unit S4, the efficient units S3 and S6 have been joined with a line defining the efficiency frontier. A dashed line drawn from the origin to inefficient unit S4 cuts through this frontier and, thus, defines unit S4 as inefficient. In Table 7.7, the value in parentheses that is associated with each member of the efficiency reference set (i.e., .7778 for S3 and .2222 for S6) represents the relative weight assigned to that efficient unit in calculating the efficiency rating for S4. These relative weights are the shadow prices that are associated with the respective efficient-unit constraints in the linear programming solution. (Note in Table 7.6 that for unit 4, these weights appear as opportunity costs for S3 and S6.). TABLE 7.6: LP Solutions for DEA Study of Burger Palace FIGURE 7.20: Excel Data File for DEA Analysis of Burger Palace Unit 1 TABLE 7.7: Summary of DEA Results The values for v1 and v2 that are associated with the inputs of labor-hours and materials, respectively, measure the relative increase in efficiency with each unit reduction of input value. For unit S4, each unit decrease in labor-hours results in an efficiency increase of 0.0555. For unit S4 to become efficient, it must increase its efficiency rating by 0.111 points. This can be accomplished by reducing labor used by 2 hours (i.e., 2 hours × 0.0555 = 0.111). Note that with this reduction in labor-hours, unit S4 becomes identical to efficient unit S3. An alternative approach would be a reduction in materials used by $16.57 (i.e., 0.111/0.0067 = 16.57). Any linear combination of these two measures also would move unit S4 to the productivity frontier defined by the line segment joining efficient units S3 and S6. Table 7.8 contains the calculations for a hypothetical unit C, which is a composite reference unit defined by the weighted inputs of the reference set S3 and S6. As Figure 7.19 shows, this composite unit C is located at the intersection of the productivity frontier and the dashed line drawn from the origin to unit S4. Thus, compared with this reference unit C, inefficient unit S4 is using excess inputs in the amounts of 0.7 labor-hour and 11.1 material dollars. Table 7.8: Calculation of Excess Inputs Used by Unit S4 DEA offers many opportunities for an inefficient unit to become efficient regarding its reference set of efficient units. In practice, management would choose a particular approach on the basis of an evaluation of its cost, practicality, and feasibility; however, the motivation for change is clear (i.e., other units actually are able to achieve similar outputs with fewer resources). DEA and Strategic Planning When combined with profitability, DEA efficiency analysis can be useful in strategic planning for services that are delivered through multiple sites (e.g., hotel chains). Figure 7.21 presents a matrix of four possibilities that arise from combining efficiency and profitability. Considering the top-left quadrant of this matrix (i.e., underperforming potential stars) reveals that units operating at a high profit may be operating inefficiently and, thus, have unrealized potential. Comparing these with similar efficient units could suggest measures that would lead to even greater profit through more efficient operations. FIGURE 7.21: DEA Strategic Matrix Star performers can be found in the top-right quadrant (i.e., benchmark group). These efficient units also are highly profitable and, thus, serve as examples for others to emulate in both operations efficiency and marketing success in generating high revenues. The lower-right quadrant (i.e., candidates for divestiture) contains efficient but unprofitable units. These units are limited in profit potential, perhaps because of a poor location, and should be sold to generate capital for expansion in new territories. It is not clear which strategy to employ with the lower-left quadrant units (i.e., problem branches). If profit potential is limited, investments in efficient operations might lead to a future candidate for divestiture.
For this simple example, we can identify efficient units by inspection and see the excess inputs being used by inefficient units (e.g., S2 would be as efficient as S3 if it used $50 less in materials). To gain an understanding of DEA, however, we will proceed to formulate the linear programming problems for each unit, then solve each of them to determine efficiency ratings and other information. We begin by illustrating the LP formulation for the first service unit, S1, using equations (5), 6), and (7). max E(S1) = u1 100 subject to u1100 − v12 − v2200 ≤ 0 u1100 − v14 − v2150 ≤ 0 u1100 − v14 − v2100 ≤ 0 u1100 − v16 − v2100 ≤ 0 u1100 − v18 − v280 ≤ 0 u1100 − v110 − v250 ≤ 0 v1 2 + v2200 = 1 u1, u2, v2, ≥ 0 Similar linear programming problems are formulated (or, better yet, the S1 linear programming problem is edited) and solved for the other service units by substituting the appropriate output function for the objective function and substituting the appropriate input function for the last constraint. Constraints 1 through 6, which restrict all units to no more than 100 percent efficiency, remain the same in all problems. This set of six linear programming problems was solved with Excel Solver 7.0 in fewer than five minutes by editing the data file between each run. Because the output is 100 meals for all units, only the last constraint must be edited by substituting the appropriate labor and material input values from Table 7.5 for the unit being evaluated. The data file for unit 1 of Burger Palace using a linear programming Excel add-in is shown in Figure 7.20. The linear programming results for each unit are shown in Table 7.6 and summarized in Table 7.7. In Table 7.6, we find that DEA has identified the same units shown as being efficient in Figure 7.19. Units S2, S4, and S5 all are inefficient in varying degrees. Also shown in Table 7.6 and associated with each inefficient unit is an efficiency reference set. Each inefficient unit will have a set of efficient units associated with it that defines its productivity. As Figure 7.19 shows for inefficient unit S4, the efficient units S3 and S6 have been joined with a line defining the efficiency frontier. A dashed line drawn from the origin to inefficient unit S4 cuts through this frontier and, thus, defines unit S4 as inefficient. In Table 7.7, the value in parentheses that is associated with each member of the efficiency reference set (i.e., .7778 for S3 and .2222 for S6) represents the relative weight assigned to that efficient unit in calculating the efficiency rating for S4. These relative weights are the shadow prices that are associated with the respective efficient-unit constraints in the linear programming solution. (Note in Table 7.6 that for unit 4, these weights appear as opportunity costs for S3 and S6.). TABLE 7.6: LP Solutions for DEA Study of Burger Palace
1. For the Burger Palace example, perform a complete analysis of efficiency improvement alternatives for unit S2, including determination of a composite reference unit. 2. For the Burger Palace example, perform a complete analysis of efficiency improvement alternatives for unit S5, including determination of a composite reference unit. 3. For the Burger Palace example, what is the effect of removing an inefficient unit from the analysis (e.g., S2)? 4. For the Burger Palace example, what is the effect of removing an efficient unit from the analysis (e.g., S6)? Supplement Data Envelopment Analysis (DEA): How can corporate management evaluate the productivity of a fast-food outlet, a branch bank, a health clinic, or an elementary school? The difficulties in measuring productivity are threefold. First, what are the appropriate inputs to the system (e.g., labor hours, material dollars) and the measures of those inputs? Second, what are the appropriate outputs of the system (e.g., checks cashed, certificate of deposits) and the measures of those outputs? Third, what are the appropriate ways of measuring the relationship between these inputs and outputs? Measuring Service Productivity The measure of an organization’s productivity, if viewed from an engineering perspective, is similar to the measure of a system’s efficiency. It can be stated as a ratio of outputs to inputs (e.g., miles per gallon for an automobile). To evaluate the operational efficiency of a branch bank, for example, an accounting ratio such as cost per teller transaction might be used. A branch with a high ratio in comparison with those of other branches would be considered less efficient, but the higher ratio could result from a more complex mix of transactions. For example, a branch opening new accounts and selling CDs would require more time per transaction than another branch engaged only in simple transactions such as accepting deposits and cashing checks. The problem with using simple ratios is that the mix of outputs is not considered explicitly. This same criticism also can be made concerning the mix of inputs. For example, some branches might have automated teller machines in addition to live tellers, and this use of technology could affect the cost per teller transaction. Broad-based measures such as profitability or return on investment are highly relevant as overall performance measures, but they are not sufficient to evaluate the operating efficiency of a service unit. For instance, one could not conclude that a profitable branch bank is necessarily efficient in its use of personnel and other inputs. A higher than-average proportion of revenue-generating transactions could be the explanation rather than the cost-efficient use of resources. The DEA Model Fortunately, a technique has been developed with the ability to compare the efficiency of multiple service units that provide similar services by explicitly considering their use of multiple inputs (i.e., resources) to produce multiple outputs (i.e., services). The technique, which is referred to as data envelopment analysis (DEA), circumvents the need to develop standard costs for each service, because it can incorporate multiple inputs and multiple outputs into both the numerator and the denominator of the efficiency ratio without the need for conversion to a common dollar basis. Thus, the DEA measure of efficiency explicitly accounts for the mix of inputs and outputs and, consequently, is more comprehensive and reliable than a set of operating ratios or profit measures. DEA is a linear programming model that attempts to maximize a service unit’s efficiency, expressed as a ratio of outputs to inputs, by comparing a particular unit’s efficiency with the performance of a group of similar service units that are delivering the same service. In the process, some units achieve 100 percent efficiency and are referred to as the relatively efficient units, whereas other units with efficiency ratings of less than 100 percent are referred to as inefficient units. Corporate management thus can use DEA to compare a group of service units to identify relatively inefficient units, measure the magnitude of the inefficiencies, and by comparing the inefficient with the efficient ones, discover ways to reduce those inefficiencies. The DEA linear programming model is formulated according to Charnes, Cooper, and Rhodes, and is referred to as the CCR Model. 9 Definition of Variables Let Ek, with k = 1, 2, . . . K, be the efficiency ratio of unit k, where K is the total number of units being evaluated. Let uj , with j = 1, 2, . . . , M, be a coefficient for output j, where M is the total number of output types considered. The variable uj is a measure of the relative decrease in efficiency with each unit reduction of output value. Let vi , with i = 1, 2, . . . , N, be a coefficient for input i, where N is the total number of input types considered. The variable vi is a measure of the relative increase in efficiency with each unit reduction of input value. Let Ojk be the number of observed units of output j generated by service unit k during one time period. Let Iik be the number of actual units of input i used by service unit k during one time period. Objective Function The objective is to find the set of coefficient u’s associated with each output and of v’s associated with each input that will give the service unit being evaluated the highest possible efficiency. Where e is the index of the unit being evaluated. This function is subject to the constraint that when the same set of input and output coefficients (uj ’s and yi ’s) is applied to all other service units being compared, no service unit will exceed 100 percent efficiency or a ratio of 1.0. Constraints Where all coefficient values are positive and nonzero. To solve this fractional linear programming model using standard linear programming software requires a reformulation. Note that both the objective function and all constraints are ratios rather than linear functions. The objective function in equation (3) Is restated as a linear function by arbitrarily scaling the inputs for the unit under evaluation to a sum of 1.0. Subject to the constraint that For each service unit, the constraints in equation (2) are similarly reformulated: Where Sample Size A question of sample size often is raised concerning the number of service units that are required compared with the number of input and output variables selected in the analysis. The following relationship relating the number of service units K used in the analysis and the number of input N and output M types being considered is based on empirical findings and the experience of DEA practitioners: K ≥ 2(N + M) Example 7.3 Burger Palace An innovative drive-in-only burger chain has established six units in several different cities. Each unit is located in a strip shopping center parking lot. Only a standard meal consisting of a burger, fries, and a drink is available. Management has decided to use DEA to improve productivity by identifying which units are using their resources most efficiently and then sharing their experience and knowledge with the less efficient locations. Table 7.5 summarizes data for two inputs: labor-hours and material dollars consumed during a typical lunch hour period to generate an output of 100 meals sold. Normally, output will vary among the service units, but in this example, we have made the outputs equal to allow for a graphical presentation of the units’ productivity. As Figure 7.19 shows, service units S, S3, and S6 have been joined to form an efficient-production frontier of alternative methods of using labor hours and material resources to generate 100 meals. As can be seen, these efficient units have defined an envelope that contains all the inefficient units—thus the reason for calling the process “data envelopment analysis.” Table 7.5: Summary of Outputs and Inputs for Burger Palace  Figure 7.19: Productivity Frontier of Burger Palace For this simple example, we can identify efficient units by inspection and see the excess inputs being used by inefficient units (e.g., S2 would be as efficient as S3 if it used $50 less in materials). To gain an understanding of DEA, however, we will proceed to formulate the linear programming problems for each unit, then solve each of them to determine efficiency ratings and other information. We begin by illustrating the LP formulation for the first service unit, S1, using equations (5), 6), and (7). max E(S1) = u1 100 subject to u1100 − v12 − v2200 ≤ 0 u1100 − v14 − v2150 ≤ 0 u1100 − v14 − v2100 ≤ 0 u1100 − v16 − v2100 ≤ 0 u1100 − v18 − v280 ≤ 0 u1100 − v110 − v250 ≤ 0 v1 2 + v2200 = 1 u1, u2, v2, ≥ 0 Similar linear programming problems are formulated (or, better yet, the S1 linear programming problem is edited) and solved for the other service units by substituting the appropriate output function for the objective function and substituting the appropriate input function for the last constraint. Constraints 1 through 6, which restrict all units to no more than 100 percent efficiency, remain the same in all problems. This set of six linear programming problems was solved with Excel Solver 7.0 in fewer than five minutes by editing the data file between each run. Because the output is 100 meals for all units, only the last constraint must be edited by substituting the appropriate labor and material input values from Table 7.5 for the unit being evaluated. The data file for unit 1 of Burger Palace using a linear programming Excel add-in is shown in Figure 7.20. The linear programming results for each unit are shown in Table 7.6 and summarized in Table 7.7. In Table 7.6, we find that DEA has identified the same units shown as being efficient in Figure 7.19. Units S2, S4, and S5 all are inefficient in varying degrees. Also shown in Table 7.6 and associated with each inefficient unit is an efficiency reference set. Each inefficient unit will have a set of efficient units associated with it that defines its productivity. As Figure 7.19 shows for inefficient unit S4, the efficient units S3 and S6 have been joined with a line defining the efficiency frontier. A dashed line drawn from the origin to inefficient unit S4 cuts through this frontier and, thus, defines unit S4 as inefficient. In Table 7.7, the value in parentheses that is associated with each member of the efficiency reference set (i.e., .7778 for S3 and .2222 for S6) represents the relative weight assigned to that efficient unit in calculating the efficiency rating for S4. These relative weights are the shadow prices that are associated with the respective efficient-unit constraints in the linear programming solution. (Note in Table 7.6 that for unit 4, these weights appear as opportunity costs for S3 and S6.). TABLE 7.6: LP Solutions for DEA Study of Burger Palace FIGURE 7.20: Excel Data File for DEA Analysis of Burger Palace Unit 1 TABLE 7.7: Summary of DEA Results The values for v1 and v2 that are associated with the inputs of labor-hours and materials, respectively, measure the relative increase in efficiency with each unit reduction of input value. For unit S4, each unit decrease in labor-hours results in an efficiency increase of 0.0555. For unit S4 to become efficient, it must increase its efficiency rating by 0.111 points. This can be accomplished by reducing labor used by 2 hours (i.e., 2 hours × 0.0555 = 0.111). Note that with this reduction in labor-hours, unit S4 becomes identical to efficient unit S3. An alternative approach would be a reduction in materials used by $16.57 (i.e., 0.111/0.0067 = 16.57). Any linear combination of these two measures also would move unit S4 to the productivity frontier defined by the line segment joining efficient units S3 and S6. Table 7.8 contains the calculations for a hypothetical unit C, which is a composite reference unit defined by the weighted inputs of the reference set S3 and S6. As Figure 7.19 shows, this composite unit C is located at the intersection of the productivity frontier and the dashed line drawn from the origin to unit S4. Thus, compared with this reference unit C, inefficient unit S4 is using excess inputs in the amounts of 0.7 labor-hour and 11.1 material dollars. Table 7.8: Calculation of Excess Inputs Used by Unit S4 DEA offers many opportunities for an inefficient unit to become efficient regarding its reference set of efficient units. In practice, management would choose a particular approach on the basis of an evaluation of its cost, practicality, and feasibility; however, the motivation for change is clear (i.e., other units actually are able to achieve similar outputs with fewer resources). DEA and Strategic Planning When combined with profitability, DEA efficiency analysis can be useful in strategic planning for services that are delivered through multiple sites (e.g., hotel chains). Figure 7.21 presents a matrix of four possibilities that arise from combining efficiency and profitability. Considering the top-left quadrant of this matrix (i.e., underperforming potential stars) reveals that units operating at a high profit may be operating inefficiently and, thus, have unrealized potential. Comparing these with similar efficient units could suggest measures that would lead to even greater profit through more efficient operations. FIGURE 7.21: DEA Strategic Matrix Star performers can be found in the top-right quadrant (i.e., benchmark group). These efficient units also are highly profitable and, thus, serve as examples for others to emulate in both operations efficiency and marketing success in generating high revenues. The lower-right quadrant (i.e., candidates for divestiture) contains efficient but unprofitable units. These units are limited in profit potential, perhaps because of a poor location, and should be sold to generate capital for expansion in new territories. It is not clear which strategy to employ with the lower-left quadrant units (i.e., problem branches). If profit potential is limited, investments in efficient operations might lead to a future candidate for divestiture.

1. For the Burger Palace example, perform a complete analysis of efficiency improvement alternatives for unit S2, including determination of a composite reference unit. 2. For the Burger Palace example, perform a complete analysis of efficiency improvement alternatives for unit S5, including determination of a composite reference unit. 3. For the Burger Palace example, what is the effect of removing an inefficient unit from the analysis (e.g., S2)? 4. For the Burger Palace example, what is the effect of removing an efficient unit from the analysis (e.g., S6)? Supplement Data Envelopment Analysis (DEA): How can corporate management evaluate the productivity of a fast-food outlet, a branch bank, a health clinic, or an elementary school? The difficulties in measuring productivity are threefold. First, what are the appropriate inputs to the system (e.g., labor hours, material dollars) and the measures of those inputs? Second, what are the appropriate outputs of the system (e.g., checks cashed, certificate of deposits) and the measures of those outputs? Third, what are the appropriate ways of measuring the relationship between these inputs and outputs? Measuring Service Productivity The measure of an organization’s productivity, if viewed from an engineering perspective, is similar to the measure of a system’s efficiency. It can be stated as a ratio of outputs to inputs (e.g., miles per gallon for an automobile). To evaluate the operational efficiency of a branch bank, for example, an accounting ratio such as cost per teller transaction might be used. A branch with a high ratio in comparison with those of other branches would be considered less efficient, but the higher ratio could result from a more complex mix of transactions. For example, a branch opening new accounts and selling CDs would require more time per transaction than another branch engaged only in simple transactions such as accepting deposits and cashing checks. The problem with using simple ratios is that the mix of outputs is not considered explicitly. This same criticism also can be made concerning the mix of inputs. For example, some branches might have automated teller machines in addition to live tellers, and this use of technology could affect the cost per teller transaction. Broad-based measures such as profitability or return on investment are highly relevant as overall performance measures, but they are not sufficient to evaluate the operating efficiency of a service unit. For instance, one could not conclude that a profitable branch bank is necessarily efficient in its use of personnel and other inputs. A higher than-average proportion of revenue-generating transactions could be the explanation rather than the cost-efficient use of resources. The DEA Model Fortunately, a technique has been developed with the ability to compare the efficiency of multiple service units that provide similar services by explicitly considering their use of multiple inputs (i.e., resources) to produce multiple outputs (i.e., services). The technique, which is referred to as data envelopment analysis (DEA), circumvents the need to develop standard costs for each service, because it can incorporate multiple inputs and multiple outputs into both the numerator and the denominator of the efficiency ratio without the need for conversion to a common dollar basis. Thus, the DEA measure of efficiency explicitly accounts for the mix of inputs and outputs and, consequently, is more comprehensive and reliable than a set of operating ratios or profit measures. DEA is a linear programming model that attempts to maximize a service unit’s efficiency, expressed as a ratio of outputs to inputs, by comparing a particular unit’s efficiency with the performance of a group of similar service units that are delivering the same service. In the process, some units achieve 100 percent efficiency and are referred to as the relatively efficient units, whereas other units with efficiency ratings of less than 100 percent are referred to as inefficient units. Corporate management thus can use DEA to compare a group of service units to identify relatively inefficient units, measure the magnitude of the inefficiencies, and by comparing the inefficient with the efficient ones, discover ways to reduce those inefficiencies. The DEA linear programming model is formulated according to Charnes, Cooper, and Rhodes, and is referred to as the CCR Model. 9 Definition of Variables Let Ek, with k = 1, 2, . . . K, be the efficiency ratio of unit k, where K is the total number of units being evaluated. Let uj , with j = 1, 2, . . . , M, be a coefficient for output j, where M is the total number of output types considered. The variable uj is a measure of the relative decrease in efficiency with each unit reduction of output value. Let vi , with i = 1, 2, . . . , N, be a coefficient for input i, where N is the total number of input types considered. The variable vi is a measure of the relative increase in efficiency with each unit reduction of input value. Let Ojk be the number of observed units of output j generated by service unit k during one time period. Let Iik be the number of actual units of input i used by service unit k during one time period. Objective Function The objective is to find the set of coefficient u’s associated with each output and of v’s associated with each input that will give the service unit being evaluated the highest possible efficiency. Where e is the index of the unit being evaluated. This function is subject to the constraint that when the same set of input and output coefficients (uj ’s and yi ’s) is applied to all other service units being compared, no service unit will exceed 100 percent efficiency or a ratio of 1.0. Constraints Where all coefficient values are positive and nonzero. To solve this fractional linear programming model using standard linear programming software requires a reformulation. Note that both the objective function and all constraints are ratios rather than linear functions. The objective function in equation (3) Is restated as a linear function by arbitrarily scaling the inputs for the unit under evaluation to a sum of 1.0. Subject to the constraint that For each service unit, the constraints in equation (2) are similarly reformulated: Where Sample Size A question of sample size often is raised concerning the number of service units that are required compared with the number of input and output variables selected in the analysis. The following relationship relating the number of service units K used in the analysis and the number of input N and output M types being considered is based on empirical findings and the experience of DEA practitioners: K ≥ 2(N + M) Example 7.3 Burger Palace An innovative drive-in-only burger chain has established six units in several different cities. Each unit is located in a strip shopping center parking lot. Only a standard meal consisting of a burger, fries, and a drink is available. Management has decided to use DEA to improve productivity by identifying which units are using their resources most efficiently and then sharing their experience and knowledge with the less efficient locations. Table 7.5 summarizes data for two inputs: labor-hours and material dollars consumed during a typical lunch hour period to generate an output of 100 meals sold. Normally, output will vary among the service units, but in this example, we have made the outputs equal to allow for a graphical presentation of the units’ productivity. As Figure 7.19 shows, service units S, S3, and S6 have been joined to form an efficient-production frontier of alternative methods of using labor hours and material resources to generate 100 meals. As can be seen, these efficient units have defined an envelope that contains all the inefficient units—thus the reason for calling the process “data envelopment analysis.” Table 7.5: Summary of Outputs and Inputs for Burger Palace  Figure 7.19: Productivity Frontier of Burger Palace For this simple example, we can identify efficient units by inspection and see the excess inputs being used by inefficient units (e.g., S2 would be as efficient as S3 if it used $50 less in materials). To gain an understanding of DEA, however, we will proceed to formulate the linear programming problems for each unit, then solve each of them to determine efficiency ratings and other information. We begin by illustrating the LP formulation for the first service unit, S1, using equations (5), 6), and (7). max E(S1) = u1 100 subject to u1100 − v12 − v2200 ≤ 0 u1100 − v14 − v2150 ≤ 0 u1100 − v14 − v2100 ≤ 0 u1100 − v16 − v2100 ≤ 0 u1100 − v18 − v280 ≤ 0 u1100 − v110 − v250 ≤ 0 v1 2 + v2200 = 1 u1, u2, v2, ≥ 0 Similar linear programming problems are formulated (or, better yet, the S1 linear programming problem is edited) and solved for the other service units by substituting the appropriate output function for the objective function and substituting the appropriate input function for the last constraint. Constraints 1 through 6, which restrict all units to no more than 100 percent efficiency, remain the same in all problems. This set of six linear programming problems was solved with Excel Solver 7.0 in fewer than five minutes by editing the data file between each run. Because the output is 100 meals for all units, only the last constraint must be edited by substituting the appropriate labor and material input values from Table 7.5 for the unit being evaluated. The data file for unit 1 of Burger Palace using a linear programming Excel add-in is shown in Figure 7.20. The linear programming results for each unit are shown in Table 7.6 and summarized in Table 7.7. In Table 7.6, we find that DEA has identified the same units shown as being efficient in Figure 7.19. Units S2, S4, and S5 all are inefficient in varying degrees. Also shown in Table 7.6 and associated with each inefficient unit is an efficiency reference set. Each inefficient unit will have a set of efficient units associated with it that defines its productivity. As Figure 7.19 shows for inefficient unit S4, the efficient units S3 and S6 have been joined with a line defining the efficiency frontier. A dashed line drawn from the origin to inefficient unit S4 cuts through this frontier and, thus, defines unit S4 as inefficient. In Table 7.7, the value in parentheses that is associated with each member of the efficiency reference set (i.e., .7778 for S3 and .2222 for S6) represents the relative weight assigned to that efficient unit in calculating the efficiency rating for S4. These relative weights are the shadow prices that are associated with the respective efficient-unit constraints in the linear programming solution. (Note in Table 7.6 that for unit 4, these weights appear as opportunity costs for S3 and S6.). TABLE 7.6: LP Solutions for DEA Study of Burger Palace FIGURE 7.20: Excel Data File for DEA Analysis of Burger Palace Unit 1 TABLE 7.7: Summary of DEA Results The values for v1 and v2 that are associated with the inputs of labor-hours and materials, respectively, measure the relative increase in efficiency with each unit reduction of input value. For unit S4, each unit decrease in labor-hours results in an efficiency increase of 0.0555. For unit S4 to become efficient, it must increase its efficiency rating by 0.111 points. This can be accomplished by reducing labor used by 2 hours (i.e., 2 hours × 0.0555 = 0.111). Note that with this reduction in labor-hours, unit S4 becomes identical to efficient unit S3. An alternative approach would be a reduction in materials used by $16.57 (i.e., 0.111/0.0067 = 16.57). Any linear combination of these two measures also would move unit S4 to the productivity frontier defined by the line segment joining efficient units S3 and S6. Table 7.8 contains the calculations for a hypothetical unit C, which is a composite reference unit defined by the weighted inputs of the reference set S3 and S6. As Figure 7.19 shows, this composite unit C is located at the intersection of the productivity frontier and the dashed line drawn from the origin to unit S4. Thus, compared with this reference unit C, inefficient unit S4 is using excess inputs in the amounts of 0.7 labor-hour and 11.1 material dollars. Table 7.8: Calculation of Excess Inputs Used by Unit S4 DEA offers many opportunities for an inefficient unit to become efficient regarding its reference set of efficient units. In practice, management would choose a particular approach on the basis of an evaluation of its cost, practicality, and feasibility; however, the motivation for change is clear (i.e., other units actually are able to achieve similar outputs with fewer resources). DEA and Strategic Planning When combined with profitability, DEA efficiency analysis can be useful in strategic planning for services that are delivered through multiple sites (e.g., hotel chains). Figure 7.21 presents a matrix of four possibilities that arise from combining efficiency and profitability. Considering the top-left quadrant of this matrix (i.e., underperforming potential stars) reveals that units operating at a high profit may be operating inefficiently and, thus, have unrealized potential. Comparing these with similar efficient units could suggest measures that would lead to even greater profit through more efficient operations. FIGURE 7.21: DEA Strategic Matrix Star performers can be found in the top-right quadrant (i.e., benchmark group). These efficient units also are highly profitable and, thus, serve as examples for others to emulate in both operations efficiency and marketing success in generating high revenues. The lower-right quadrant (i.e., candidates for divestiture) contains efficient but unprofitable units. These units are limited in profit potential, perhaps because of a poor location, and should be sold to generate capital for expansion in new territories. It is not clear which strategy to employ with the lower-left quadrant units (i.e., problem branches). If profit potential is limited, investments in efficient operations might lead to a future candidate for divestiture.

1. For the Burger Palace example, perform a complete analysis of efficiency improvement alternatives for unit S2, including determination of a composite reference unit. 2. For the Burger Palace example, perform a complete analysis of efficiency improvement alternatives for unit S5, including determination of a composite reference unit. 3. For the Burger Palace example, what is the effect of removing an inefficient unit from the analysis (e.g., S2)? 4. For the Burger Palace example, what is the effect of removing an efficient unit from the analysis (e.g., S6)? Supplement Data Envelopment Analysis (DEA): How can corporate management evaluate the productivity of a fast-food outlet, a branch bank, a health clinic, or an elementary school? The difficulties in measuring productivity are threefold. First, what are the appropriate inputs to the system (e.g., labor hours, material dollars) and the measures of those inputs? Second, what are the appropriate outputs of the system (e.g., checks cashed, certificate of deposits) and the measures of those outputs? Third, what are the appropriate ways of measuring the relationship between these inputs and outputs? Measuring Service Productivity The measure of an organization’s productivity, if viewed from an engineering perspective, is similar to the measure of a system’s efficiency. It can be stated as a ratio of outputs to inputs (e.g., miles per gallon for an automobile). To evaluate the operational efficiency of a branch bank, for example, an accounting ratio such as cost per teller transaction might be used. A branch with a high ratio in comparison with those of other branches would be considered less efficient, but the higher ratio could result from a more complex mix of transactions. For example, a branch opening new accounts and selling CDs would require more time per transaction than another branch engaged only in simple transactions such as accepting deposits and cashing checks. The problem with using simple ratios is that the mix of outputs is not considered explicitly. This same criticism also can be made concerning the mix of inputs. For example, some branches might have automated teller machines in addition to live tellers, and this use of technology could affect the cost per teller transaction. Broad-based measures such as profitability or return on investment are highly relevant as overall performance measures, but they are not sufficient to evaluate the operating efficiency of a service unit. For instance, one could not conclude that a profitable branch bank is necessarily efficient in its use of personnel and other inputs. A higher than-average proportion of revenue-generating transactions could be the explanation rather than the cost-efficient use of resources. The DEA Model Fortunately, a technique has been developed with the ability to compare the efficiency of multiple service units that provide similar services by explicitly considering their use of multiple inputs (i.e., resources) to produce multiple outputs (i.e., services). The technique, which is referred to as data envelopment analysis (DEA), circumvents the need to develop standard costs for each service, because it can incorporate multiple inputs and multiple outputs into both the numerator and the denominator of the efficiency ratio without the need for conversion to a common dollar basis. Thus, the DEA measure of efficiency explicitly accounts for the mix of inputs and outputs and, consequently, is more comprehensive and reliable than a set of operating ratios or profit measures. DEA is a linear programming model that attempts to maximize a service unit’s efficiency, expressed as a ratio of outputs to inputs, by comparing a particular unit’s efficiency with the performance of a group of similar service units that are delivering the same service. In the process, some units achieve 100 percent efficiency and are referred to as the relatively efficient units, whereas other units with efficiency ratings of less than 100 percent are referred to as inefficient units. Corporate management thus can use DEA to compare a group of service units to identify relatively inefficient units, measure the magnitude of the inefficiencies, and by comparing the inefficient with the efficient ones, discover ways to reduce those inefficiencies. The DEA linear programming model is formulated according to Charnes, Cooper, and Rhodes, and is referred to as the CCR Model. 9 Definition of Variables Let Ek, with k = 1, 2, . . . K, be the efficiency ratio of unit k, where K is the total number of units being evaluated. Let uj , with j = 1, 2, . . . , M, be a coefficient for output j, where M is the total number of output types considered. The variable uj is a measure of the relative decrease in efficiency with each unit reduction of output value. Let vi , with i = 1, 2, . . . , N, be a coefficient for input i, where N is the total number of input types considered. The variable vi is a measure of the relative increase in efficiency with each unit reduction of input value. Let Ojk be the number of observed units of output j generated by service unit k during one time period. Let Iik be the number of actual units of input i used by service unit k during one time period. Objective Function The objective is to find the set of coefficient u’s associated with each output and of v’s associated with each input that will give the service unit being evaluated the highest possible efficiency. Where e is the index of the unit being evaluated. This function is subject to the constraint that when the same set of input and output coefficients (uj ’s and yi ’s) is applied to all other service units being compared, no service unit will exceed 100 percent efficiency or a ratio of 1.0. Constraints Where all coefficient values are positive and nonzero. To solve this fractional linear programming model using standard linear programming software requires a reformulation. Note that both the objective function and all constraints are ratios rather than linear functions. The objective function in equation (3) Is restated as a linear function by arbitrarily scaling the inputs for the unit under evaluation to a sum of 1.0. Subject to the constraint that For each service unit, the constraints in equation (2) are similarly reformulated: Where Sample Size A question of sample size often is raised concerning the number of service units that are required compared with the number of input and output variables selected in the analysis. The following relationship relating the number of service units K used in the analysis and the number of input N and output M types being considered is based on empirical findings and the experience of DEA practitioners: K ≥ 2(N + M) Example 7.3 Burger Palace An innovative drive-in-only burger chain has established six units in several different cities. Each unit is located in a strip shopping center parking lot. Only a standard meal consisting of a burger, fries, and a drink is available. Management has decided to use DEA to improve productivity by identifying which units are using their resources most efficiently and then sharing their experience and knowledge with the less efficient locations. Table 7.5 summarizes data for two inputs: labor-hours and material dollars consumed during a typical lunch hour period to generate an output of 100 meals sold. Normally, output will vary among the service units, but in this example, we have made the outputs equal to allow for a graphical presentation of the units’ productivity. As Figure 7.19 shows, service units S, S3, and S6 have been joined to form an efficient-production frontier of alternative methods of using labor hours and material resources to generate 100 meals. As can be seen, these efficient units have defined an envelope that contains all the inefficient units—thus the reason for calling the process “data envelopment analysis.” Table 7.5: Summary of Outputs and Inputs for Burger Palace  Figure 7.19: Productivity Frontier of Burger Palace For this simple example, we can identify efficient units by inspection and see the excess inputs being used by inefficient units (e.g., S2 would be as efficient as S3 if it used $50 less in materials). To gain an understanding of DEA, however, we will proceed to formulate the linear programming problems for each unit, then solve each of them to determine efficiency ratings and other information. We begin by illustrating the LP formulation for the first service unit, S1, using equations (5), 6), and (7). max E(S1) = u1 100 subject to u1100 − v12 − v2200 ≤ 0 u1100 − v14 − v2150 ≤ 0 u1100 − v14 − v2100 ≤ 0 u1100 − v16 − v2100 ≤ 0 u1100 − v18 − v280 ≤ 0 u1100 − v110 − v250 ≤ 0 v1 2 + v2200 = 1 u1, u2, v2, ≥ 0 Similar linear programming problems are formulated (or, better yet, the S1 linear programming problem is edited) and solved for the other service units by substituting the appropriate output function for the objective function and substituting the appropriate input function for the last constraint. Constraints 1 through 6, which restrict all units to no more than 100 percent efficiency, remain the same in all problems. This set of six linear programming problems was solved with Excel Solver 7.0 in fewer than five minutes by editing the data file between each run. Because the output is 100 meals for all units, only the last constraint must be edited by substituting the appropriate labor and material input values from Table 7.5 for the unit being evaluated. The data file for unit 1 of Burger Palace using a linear programming Excel add-in is shown in Figure 7.20. The linear programming results for each unit are shown in Table 7.6 and summarized in Table 7.7. In Table 7.6, we find that DEA has identified the same units shown as being efficient in Figure 7.19. Units S2, S4, and S5 all are inefficient in varying degrees. Also shown in Table 7.6 and associated with each inefficient unit is an efficiency reference set. Each inefficient unit will have a set of efficient units associated with it that defines its productivity. As Figure 7.19 shows for inefficient unit S4, the efficient units S3 and S6 have been joined with a line defining the efficiency frontier. A dashed line drawn from the origin to inefficient unit S4 cuts through this frontier and, thus, defines unit S4 as inefficient. In Table 7.7, the value in parentheses that is associated with each member of the efficiency reference set (i.e., .7778 for S3 and .2222 for S6) represents the relative weight assigned to that efficient unit in calculating the efficiency rating for S4. These relative weights are the shadow prices that are associated with the respective efficient-unit constraints in the linear programming solution. (Note in Table 7.6 that for unit 4, these weights appear as opportunity costs for S3 and S6.). TABLE 7.6: LP Solutions for DEA Study of Burger Palace FIGURE 7.20: Excel Data File for DEA Analysis of Burger Palace Unit 1 TABLE 7.7: Summary of DEA Results The values for v1 and v2 that are associated with the inputs of labor-hours and materials, respectively, measure the relative increase in efficiency with each unit reduction of input value. For unit S4, each unit decrease in labor-hours results in an efficiency increase of 0.0555. For unit S4 to become efficient, it must increase its efficiency rating by 0.111 points. This can be accomplished by reducing labor used by 2 hours (i.e., 2 hours × 0.0555 = 0.111). Note that with this reduction in labor-hours, unit S4 becomes identical to efficient unit S3. An alternative approach would be a reduction in materials used by $16.57 (i.e., 0.111/0.0067 = 16.57). Any linear combination of these two measures also would move unit S4 to the productivity frontier defined by the line segment joining efficient units S3 and S6. Table 7.8 contains the calculations for a hypothetical unit C, which is a composite reference unit defined by the weighted inputs of the reference set S3 and S6. As Figure 7.19 shows, this composite unit C is located at the intersection of the productivity frontier and the dashed line drawn from the origin to unit S4. Thus, compared with this reference unit C, inefficient unit S4 is using excess inputs in the amounts of 0.7 labor-hour and 11.1 material dollars. Table 7.8: Calculation of Excess Inputs Used by Unit S4 DEA offers many opportunities for an inefficient unit to become efficient regarding its reference set of efficient units. In practice, management would choose a particular approach on the basis of an evaluation of its cost, practicality, and feasibility; however, the motivation for change is clear (i.e., other units actually are able to achieve similar outputs with fewer resources). DEA and Strategic Planning When combined with profitability, DEA efficiency analysis can be useful in strategic planning for services that are delivered through multiple sites (e.g., hotel chains). Figure 7.21 presents a matrix of four possibilities that arise from combining efficiency and profitability. Considering the top-left quadrant of this matrix (i.e., underperforming potential stars) reveals that units operating at a high profit may be operating inefficiently and, thus, have unrealized potential. Comparing these with similar efficient units could suggest measures that would lead to even greater profit through more efficient operations. FIGURE 7.21: DEA Strategic Matrix Star performers can be found in the top-right quadrant (i.e., benchmark group). These efficient units also are highly profitable and, thus, serve as examples for others to emulate in both operations efficiency and marketing success in generating high revenues. The lower-right quadrant (i.e., candidates for divestiture) contains efficient but unprofitable units. These units are limited in profit potential, perhaps because of a poor location, and should be sold to generate capital for expansion in new territories. It is not clear which strategy to employ with the lower-left quadrant units (i.e., problem branches). If profit potential is limited, investments in efficient operations might lead to a future candidate for divestiture.

1. For the Burger Palace example, perform a complete analysis of efficiency improvement alternatives for unit S2, including determination of a composite reference unit. 2. For the Burger Palace example, perform a complete analysis of efficiency improvement alternatives for unit S5, including determination of a composite reference unit. 3. For the Burger Palace example, what is the effect of removing an inefficient unit from the analysis (e.g., S2)? 4. For the Burger Palace example, what is the effect of removing an efficient unit from the analysis (e.g., S6)? Supplement Data Envelopment Analysis (DEA): How can corporate management evaluate the productivity of a fast-food outlet, a branch bank, a health clinic, or an elementary school? The difficulties in measuring productivity are threefold. First, what are the appropriate inputs to the system (e.g., labor hours, material dollars) and the measures of those inputs? Second, what are the appropriate outputs of the system (e.g., checks cashed, certificate of deposits) and the measures of those outputs? Third, what are the appropriate ways of measuring the relationship between these inputs and outputs? Measuring Service Productivity The measure of an organization’s productivity, if viewed from an engineering perspective, is similar to the measure of a system’s efficiency. It can be stated as a ratio of outputs to inputs (e.g., miles per gallon for an automobile). To evaluate the operational efficiency of a branch bank, for example, an accounting ratio such as cost per teller transaction might be used. A branch with a high ratio in comparison with those of other branches would be considered less efficient, but the higher ratio could result from a more complex mix of transactions. For example, a branch opening new accounts and selling CDs would require more time per transaction than another branch engaged only in simple transactions such as accepting deposits and cashing checks. The problem with using simple ratios is that the mix of outputs is not considered explicitly. This same criticism also can be made concerning the mix of inputs. For example, some branches might have automated teller machines in addition to live tellers, and this use of technology could affect the cost per teller transaction. Broad-based measures such as profitability or return on investment are highly relevant as overall performance measures, but they are not sufficient to evaluate the operating efficiency of a service unit. For instance, one could not conclude that a profitable branch bank is necessarily efficient in its use of personnel and other inputs. A higher than-average proportion of revenue-generating transactions could be the explanation rather than the cost-efficient use of resources. The DEA Model Fortunately, a technique has been developed with the ability to compare the efficiency of multiple service units that provide similar services by explicitly considering their use of multiple inputs (i.e., resources) to produce multiple outputs (i.e., services). The technique, which is referred to as data envelopment analysis (DEA), circumvents the need to develop standard costs for each service, because it can incorporate multiple inputs and multiple outputs into both the numerator and the denominator of the efficiency ratio without the need for conversion to a common dollar basis. Thus, the DEA measure of efficiency explicitly accounts for the mix of inputs and outputs and, consequently, is more comprehensive and reliable than a set of operating ratios or profit measures. DEA is a linear programming model that attempts to maximize a service unit’s efficiency, expressed as a ratio of outputs to inputs, by comparing a particular unit’s efficiency with the performance of a group of similar service units that are delivering the same service. In the process, some units achieve 100 percent efficiency and are referred to as the relatively efficient units, whereas other units with efficiency ratings of less than 100 percent are referred to as inefficient units. Corporate management thus can use DEA to compare a group of service units to identify relatively inefficient units, measure the magnitude of the inefficiencies, and by comparing the inefficient with the efficient ones, discover ways to reduce those inefficiencies. The DEA linear programming model is formulated according to Charnes, Cooper, and Rhodes, and is referred to as the CCR Model. 9 Definition of Variables Let Ek, with k = 1, 2, . . . K, be the efficiency ratio of unit k, where K is the total number of units being evaluated. Let uj , with j = 1, 2, . . . , M, be a coefficient for output j, where M is the total number of output types considered. The variable uj is a measure of the relative decrease in efficiency with each unit reduction of output value. Let vi , with i = 1, 2, . . . , N, be a coefficient for input i, where N is the total number of input types considered. The variable vi is a measure of the relative increase in efficiency with each unit reduction of input value. Let Ojk be the number of observed units of output j generated by service unit k during one time period. Let Iik be the number of actual units of input i used by service unit k during one time period. Objective Function The objective is to find the set of coefficient u’s associated with each output and of v’s associated with each input that will give the service unit being evaluated the highest possible efficiency. Where e is the index of the unit being evaluated. This function is subject to the constraint that when the same set of input and output coefficients (uj ’s and yi ’s) is applied to all other service units being compared, no service unit will exceed 100 percent efficiency or a ratio of 1.0. Constraints Where all coefficient values are positive and nonzero. To solve this fractional linear programming model using standard linear programming software requires a reformulation. Note that both the objective function and all constraints are ratios rather than linear functions. The objective function in equation (3) Is restated as a linear function by arbitrarily scaling the inputs for the unit under evaluation to a sum of 1.0. Subject to the constraint that For each service unit, the constraints in equation (2) are similarly reformulated: Where Sample Size A question of sample size often is raised concerning the number of service units that are required compared with the number of input and output variables selected in the analysis. The following relationship relating the number of service units K used in the analysis and the number of input N and output M types being considered is based on empirical findings and the experience of DEA practitioners: K ≥ 2(N + M) Example 7.3 Burger Palace An innovative drive-in-only burger chain has established six units in several different cities. Each unit is located in a strip shopping center parking lot. Only a standard meal consisting of a burger, fries, and a drink is available. Management has decided to use DEA to improve productivity by identifying which units are using their resources most efficiently and then sharing their experience and knowledge with the less efficient locations. Table 7.5 summarizes data for two inputs: labor-hours and material dollars consumed during a typical lunch hour period to generate an output of 100 meals sold. Normally, output will vary among the service units, but in this example, we have made the outputs equal to allow for a graphical presentation of the units’ productivity. As Figure 7.19 shows, service units S, S3, and S6 have been joined to form an efficient-production frontier of alternative methods of using labor hours and material resources to generate 100 meals. As can be seen, these efficient units have defined an envelope that contains all the inefficient units—thus the reason for calling the process “data envelopment analysis.” Table 7.5: Summary of Outputs and Inputs for Burger Palace  Figure 7.19: Productivity Frontier of Burger Palace For this simple example, we can identify efficient units by inspection and see the excess inputs being used by inefficient units (e.g., S2 would be as efficient as S3 if it used $50 less in materials). To gain an understanding of DEA, however, we will proceed to formulate the linear programming problems for each unit, then solve each of them to determine efficiency ratings and other information. We begin by illustrating the LP formulation for the first service unit, S1, using equations (5), 6), and (7). max E(S1) = u1 100 subject to u1100 − v12 − v2200 ≤ 0 u1100 − v14 − v2150 ≤ 0 u1100 − v14 − v2100 ≤ 0 u1100 − v16 − v2100 ≤ 0 u1100 − v18 − v280 ≤ 0 u1100 − v110 − v250 ≤ 0 v1 2 + v2200 = 1 u1, u2, v2, ≥ 0 Similar linear programming problems are formulated (or, better yet, the S1 linear programming problem is edited) and solved for the other service units by substituting the appropriate output function for the objective function and substituting the appropriate input function for the last constraint. Constraints 1 through 6, which restrict all units to no more than 100 percent efficiency, remain the same in all problems. This set of six linear programming problems was solved with Excel Solver 7.0 in fewer than five minutes by editing the data file between each run. Because the output is 100 meals for all units, only the last constraint must be edited by substituting the appropriate labor and material input values from Table 7.5 for the unit being evaluated. The data file for unit 1 of Burger Palace using a linear programming Excel add-in is shown in Figure 7.20. The linear programming results for each unit are shown in Table 7.6 and summarized in Table 7.7. In Table 7.6, we find that DEA has identified the same units shown as being efficient in Figure 7.19. Units S2, S4, and S5 all are inefficient in varying degrees. Also shown in Table 7.6 and associated with each inefficient unit is an efficiency reference set. Each inefficient unit will have a set of efficient units associated with it that defines its productivity. As Figure 7.19 shows for inefficient unit S4, the efficient units S3 and S6 have been joined with a line defining the efficiency frontier. A dashed line drawn from the origin to inefficient unit S4 cuts through this frontier and, thus, defines unit S4 as inefficient. In Table 7.7, the value in parentheses that is associated with each member of the efficiency reference set (i.e., .7778 for S3 and .2222 for S6) represents the relative weight assigned to that efficient unit in calculating the efficiency rating for S4. These relative weights are the shadow prices that are associated with the respective efficient-unit constraints in the linear programming solution. (Note in Table 7.6 that for unit 4, these weights appear as opportunity costs for S3 and S6.). TABLE 7.6: LP Solutions for DEA Study of Burger Palace FIGURE 7.20: Excel Data File for DEA Analysis of Burger Palace Unit 1 TABLE 7.7: Summary of DEA Results The values for v1 and v2 that are associated with the inputs of labor-hours and materials, respectively, measure the relative increase in efficiency with each unit reduction of input value. For unit S4, each unit decrease in labor-hours results in an efficiency increase of 0.0555. For unit S4 to become efficient, it must increase its efficiency rating by 0.111 points. This can be accomplished by reducing labor used by 2 hours (i.e., 2 hours × 0.0555 = 0.111). Note that with this reduction in labor-hours, unit S4 becomes identical to efficient unit S3. An alternative approach would be a reduction in materials used by $16.57 (i.e., 0.111/0.0067 = 16.57). Any linear combination of these two measures also would move unit S4 to the productivity frontier defined by the line segment joining efficient units S3 and S6. Table 7.8 contains the calculations for a hypothetical unit C, which is a composite reference unit defined by the weighted inputs of the reference set S3 and S6. As Figure 7.19 shows, this composite unit C is located at the intersection of the productivity frontier and the dashed line drawn from the origin to unit S4. Thus, compared with this reference unit C, inefficient unit S4 is using excess inputs in the amounts of 0.7 labor-hour and 11.1 material dollars. Table 7.8: Calculation of Excess Inputs Used by Unit S4 DEA offers many opportunities for an inefficient unit to become efficient regarding its reference set of efficient units. In practice, management would choose a particular approach on the basis of an evaluation of its cost, practicality, and feasibility; however, the motivation for change is clear (i.e., other units actually are able to achieve similar outputs with fewer resources). DEA and Strategic Planning When combined with profitability, DEA efficiency analysis can be useful in strategic planning for services that are delivered through multiple sites (e.g., hotel chains). Figure 7.21 presents a matrix of four possibilities that arise from combining efficiency and profitability. Considering the top-left quadrant of this matrix (i.e., underperforming potential stars) reveals that units operating at a high profit may be operating inefficiently and, thus, have unrealized potential. Comparing these with similar efficient units could suggest measures that would lead to even greater profit through more efficient operations. FIGURE 7.21: DEA Strategic Matrix Star performers can be found in the top-right quadrant (i.e., benchmark group). These efficient units also are highly profitable and, thus, serve as examples for others to emulate in both operations efficiency and marketing success in generating high revenues. The lower-right quadrant (i.e., candidates for divestiture) contains efficient but unprofitable units. These units are limited in profit potential, perhaps because of a poor location, and should be sold to generate capital for expansion in new territories. It is not clear which strategy to employ with the lower-left quadrant units (i.e., problem branches). If profit potential is limited, investments in efficient operations might lead to a future candidate for divestiture.

1. For the Burger Palace example, perform a complete analysis of efficiency improvement alternatives for unit S2, including determination of a composite reference unit. 2. For the Burger Palace example, perform a complete analysis of efficiency improvement alternatives for unit S5, including determination of a composite reference unit. 3. For the Burger Palace example, what is the effect of removing an inefficient unit from the analysis (e.g., S2)? 4. For the Burger Palace example, what is the effect of removing an efficient unit from the analysis (e.g., S6)? Supplement Data Envelopment Analysis (DEA): How can corporate management evaluate the productivity of a fast-food outlet, a branch bank, a health clinic, or an elementary school? The difficulties in measuring productivity are threefold. First, what are the appropriate inputs to the system (e.g., labor hours, material dollars) and the measures of those inputs? Second, what are the appropriate outputs of the system (e.g., checks cashed, certificate of deposits) and the measures of those outputs? Third, what are the appropriate ways of measuring the relationship between these inputs and outputs? Measuring Service Productivity The measure of an organization’s productivity, if viewed from an engineering perspective, is similar to the measure of a system’s efficiency. It can be stated as a ratio of outputs to inputs (e.g., miles per gallon for an automobile). To evaluate the operational efficiency of a branch bank, for example, an accounting ratio such as cost per teller transaction might be used. A branch with a high ratio in comparison with those of other branches would be considered less efficient, but the higher ratio could result from a more complex mix of transactions. For example, a branch opening new accounts and selling CDs would require more time per transaction than another branch engaged only in simple transactions such as accepting deposits and cashing checks. The problem with using simple ratios is that the mix of outputs is not considered explicitly. This same criticism also can be made concerning the mix of inputs. For example, some branches might have automated teller machines in addition to live tellers, and this use of technology could affect the cost per teller transaction. Broad-based measures such as profitability or return on investment are highly relevant as overall performance measures, but they are not sufficient to evaluate the operating efficiency of a service unit. For instance, one could not conclude that a profitable branch bank is necessarily efficient in its use of personnel and other inputs. A higher than-average proportion of revenue-generating transactions could be the explanation rather than the cost-efficient use of resources. The DEA Model Fortunately, a technique has been developed with the ability to compare the efficiency of multiple service units that provide similar services by explicitly considering their use of multiple inputs (i.e., resources) to produce multiple outputs (i.e., services). The technique, which is referred to as data envelopment analysis (DEA), circumvents the need to develop standard costs for each service, because it can incorporate multiple inputs and multiple outputs into both the numerator and the denominator of the efficiency ratio without the need for conversion to a common dollar basis. Thus, the DEA measure of efficiency explicitly accounts for the mix of inputs and outputs and, consequently, is more comprehensive and reliable than a set of operating ratios or profit measures. DEA is a linear programming model that attempts to maximize a service unit’s efficiency, expressed as a ratio of outputs to inputs, by comparing a particular unit’s efficiency with the performance of a group of similar service units that are delivering the same service. In the process, some units achieve 100 percent efficiency and are referred to as the relatively efficient units, whereas other units with efficiency ratings of less than 100 percent are referred to as inefficient units. Corporate management thus can use DEA to compare a group of service units to identify relatively inefficient units, measure the magnitude of the inefficiencies, and by comparing the inefficient with the efficient ones, discover ways to reduce those inefficiencies. The DEA linear programming model is formulated according to Charnes, Cooper, and Rhodes, and is referred to as the CCR Model. 9 Definition of Variables Let Ek, with k = 1, 2, . . . K, be the efficiency ratio of unit k, where K is the total number of units being evaluated. Let uj , with j = 1, 2, . . . , M, be a coefficient for output j, where M is the total number of output types considered. The variable uj is a measure of the relative decrease in efficiency with each unit reduction of output value. Let vi , with i = 1, 2, . . . , N, be a coefficient for input i, where N is the total number of input types considered. The variable vi is a measure of the relative increase in efficiency with each unit reduction of input value. Let Ojk be the number of observed units of output j generated by service unit k during one time period. Let Iik be the number of actual units of input i used by service unit k during one time period. Objective Function The objective is to find the set of coefficient u’s associated with each output and of v’s associated with each input that will give the service unit being evaluated the highest possible efficiency. Where e is the index of the unit being evaluated. This function is subject to the constraint that when the same set of input and output coefficients (uj ’s and yi ’s) is applied to all other service units being compared, no service unit will exceed 100 percent efficiency or a ratio of 1.0. Constraints Where all coefficient values are positive and nonzero. To solve this fractional linear programming model using standard linear programming software requires a reformulation. Note that both the objective function and all constraints are ratios rather than linear functions. The objective function in equation (3) Is restated as a linear function by arbitrarily scaling the inputs for the unit under evaluation to a sum of 1.0. Subject to the constraint that For each service unit, the constraints in equation (2) are similarly reformulated: Where Sample Size A question of sample size often is raised concerning the number of service units that are required compared with the number of input and output variables selected in the analysis. The following relationship relating the number of service units K used in the analysis and the number of input N and output M types being considered is based on empirical findings and the experience of DEA practitioners: K ≥ 2(N + M) Example 7.3 Burger Palace An innovative drive-in-only burger chain has established six units in several different cities. Each unit is located in a strip shopping center parking lot. Only a standard meal consisting of a burger, fries, and a drink is available. Management has decided to use DEA to improve productivity by identifying which units are using their resources most efficiently and then sharing their experience and knowledge with the less efficient locations. Table 7.5 summarizes data for two inputs: labor-hours and material dollars consumed during a typical lunch hour period to generate an output of 100 meals sold. Normally, output will vary among the service units, but in this example, we have made the outputs equal to allow for a graphical presentation of the units’ productivity. As Figure 7.19 shows, service units S, S3, and S6 have been joined to form an efficient-production frontier of alternative methods of using labor hours and material resources to generate 100 meals. As can be seen, these efficient units have defined an envelope that contains all the inefficient units—thus the reason for calling the process “data envelopment analysis.” Table 7.5: Summary of Outputs and Inputs for Burger Palace  Figure 7.19: Productivity Frontier of Burger Palace For this simple example, we can identify efficient units by inspection and see the excess inputs being used by inefficient units (e.g., S2 would be as efficient as S3 if it used $50 less in materials). To gain an understanding of DEA, however, we will proceed to formulate the linear programming problems for each unit, then solve each of them to determine efficiency ratings and other information. We begin by illustrating the LP formulation for the first service unit, S1, using equations (5), 6), and (7). max E(S1) = u1 100 subject to u1100 − v12 − v2200 ≤ 0 u1100 − v14 − v2150 ≤ 0 u1100 − v14 − v2100 ≤ 0 u1100 − v16 − v2100 ≤ 0 u1100 − v18 − v280 ≤ 0 u1100 − v110 − v250 ≤ 0 v1 2 + v2200 = 1 u1, u2, v2, ≥ 0 Similar linear programming problems are formulated (or, better yet, the S1 linear programming problem is edited) and solved for the other service units by substituting the appropriate output function for the objective function and substituting the appropriate input function for the last constraint. Constraints 1 through 6, which restrict all units to no more than 100 percent efficiency, remain the same in all problems. This set of six linear programming problems was solved with Excel Solver 7.0 in fewer than five minutes by editing the data file between each run. Because the output is 100 meals for all units, only the last constraint must be edited by substituting the appropriate labor and material input values from Table 7.5 for the unit being evaluated. The data file for unit 1 of Burger Palace using a linear programming Excel add-in is shown in Figure 7.20. The linear programming results for each unit are shown in Table 7.6 and summarized in Table 7.7. In Table 7.6, we find that DEA has identified the same units shown as being efficient in Figure 7.19. Units S2, S4, and S5 all are inefficient in varying degrees. Also shown in Table 7.6 and associated with each inefficient unit is an efficiency reference set. Each inefficient unit will have a set of efficient units associated with it that defines its productivity. As Figure 7.19 shows for inefficient unit S4, the efficient units S3 and S6 have been joined with a line defining the efficiency frontier. A dashed line drawn from the origin to inefficient unit S4 cuts through this frontier and, thus, defines unit S4 as inefficient. In Table 7.7, the value in parentheses that is associated with each member of the efficiency reference set (i.e., .7778 for S3 and .2222 for S6) represents the relative weight assigned to that efficient unit in calculating the efficiency rating for S4. These relative weights are the shadow prices that are associated with the respective efficient-unit constraints in the linear programming solution. (Note in Table 7.6 that for unit 4, these weights appear as opportunity costs for S3 and S6.). TABLE 7.6: LP Solutions for DEA Study of Burger Palace FIGURE 7.20: Excel Data File for DEA Analysis of Burger Palace Unit 1 TABLE 7.7: Summary of DEA Results The values for v1 and v2 that are associated with the inputs of labor-hours and materials, respectively, measure the relative increase in efficiency with each unit reduction of input value. For unit S4, each unit decrease in labor-hours results in an efficiency increase of 0.0555. For unit S4 to become efficient, it must increase its efficiency rating by 0.111 points. This can be accomplished by reducing labor used by 2 hours (i.e., 2 hours × 0.0555 = 0.111). Note that with this reduction in labor-hours, unit S4 becomes identical to efficient unit S3. An alternative approach would be a reduction in materials used by $16.57 (i.e., 0.111/0.0067 = 16.57). Any linear combination of these two measures also would move unit S4 to the productivity frontier defined by the line segment joining efficient units S3 and S6. Table 7.8 contains the calculations for a hypothetical unit C, which is a composite reference unit defined by the weighted inputs of the reference set S3 and S6. As Figure 7.19 shows, this composite unit C is located at the intersection of the productivity frontier and the dashed line drawn from the origin to unit S4. Thus, compared with this reference unit C, inefficient unit S4 is using excess inputs in the amounts of 0.7 labor-hour and 11.1 material dollars. Table 7.8: Calculation of Excess Inputs Used by Unit S4 DEA offers many opportunities for an inefficient unit to become efficient regarding its reference set of efficient units. In practice, management would choose a particular approach on the basis of an evaluation of its cost, practicality, and feasibility; however, the motivation for change is clear (i.e., other units actually are able to achieve similar outputs with fewer resources). DEA and Strategic Planning When combined with profitability, DEA efficiency analysis can be useful in strategic planning for services that are delivered through multiple sites (e.g., hotel chains). Figure 7.21 presents a matrix of four possibilities that arise from combining efficiency and profitability. Considering the top-left quadrant of this matrix (i.e., underperforming potential stars) reveals that units operating at a high profit may be operating inefficiently and, thus, have unrealized potential. Comparing these with similar efficient units could suggest measures that would lead to even greater profit through more efficient operations. FIGURE 7.21: DEA Strategic Matrix Star performers can be found in the top-right quadrant (i.e., benchmark group). These efficient units also are highly profitable and, thus, serve as examples for others to emulate in both operations efficiency and marketing success in generating high revenues. The lower-right quadrant (i.e., candidates for divestiture) contains efficient but unprofitable units. These units are limited in profit potential, perhaps because of a poor location, and should be sold to generate capital for expansion in new territories. It is not clear which strategy to employ with the lower-left quadrant units (i.e., problem branches). If profit potential is limited, investments in efficient operations might lead to a future candidate for divestiture.

1. For the Burger Palace example, perform a complete analysis of efficiency improvement alternatives for unit S2, including determination of a composite reference unit. 2. For the Burger Palace example, perform a complete analysis of efficiency improvement alternatives for unit S5, including determination of a composite reference unit. 3. For the Burger Palace example, what is the effect of removing an inefficient unit from the analysis (e.g., S2)? 4. For the Burger Palace example, what is the effect of removing an efficient unit from the analysis (e.g., S6)? Supplement Data Envelopment Analysis (DEA): How can corporate management evaluate the productivity of a fast-food outlet, a branch bank, a health clinic, or an elementary school? The difficulties in measuring productivity are threefold. First, what are the appropriate inputs to the system (e.g., labor hours, material dollars) and the measures of those inputs? Second, what are the appropriate outputs of the system (e.g., checks cashed, certificate of deposits) and the measures of those outputs? Third, what are the appropriate ways of measuring the relationship between these inputs and outputs? Measuring Service Productivity The measure of an organization’s productivity, if viewed from an engineering perspective, is similar to the measure of a system’s efficiency. It can be stated as a ratio of outputs to inputs (e.g., miles per gallon for an automobile). To evaluate the operational efficiency of a branch bank, for example, an accounting ratio such as cost per teller transaction might be used. A branch with a high ratio in comparison with those of other branches would be considered less efficient, but the higher ratio could result from a more complex mix of transactions. For example, a branch opening new accounts and selling CDs would require more time per transaction than another branch engaged only in simple transactions such as accepting deposits and cashing checks. The problem with using simple ratios is that the mix of outputs is not considered explicitly. This same criticism also can be made concerning the mix of inputs. For example, some branches might have automated teller machines in addition to live tellers, and this use of technology could affect the cost per teller transaction. Broad-based measures such as profitability or return on investment are highly relevant as overall performance measures, but they are not sufficient to evaluate the operating efficiency of a service unit. For instance, one could not conclude that a profitable branch bank is necessarily efficient in its use of personnel and other inputs. A higher than-average proportion of revenue-generating transactions could be the explanation rather than the cost-efficient use of resources. The DEA Model Fortunately, a technique has been developed with the ability to compare the efficiency of multiple service units that provide similar services by explicitly considering their use of multiple inputs (i.e., resources) to produce multiple outputs (i.e., services). The technique, which is referred to as data envelopment analysis (DEA), circumvents the need to develop standard costs for each service, because it can incorporate multiple inputs and multiple outputs into both the numerator and the denominator of the efficiency ratio without the need for conversion to a common dollar basis. Thus, the DEA measure of efficiency explicitly accounts for the mix of inputs and outputs and, consequently, is more comprehensive and reliable than a set of operating ratios or profit measures. DEA is a linear programming model that attempts to maximize a service unit’s efficiency, expressed as a ratio of outputs to inputs, by comparing a particular unit’s efficiency with the performance of a group of similar service units that are delivering the same service. In the process, some units achieve 100 percent efficiency and are referred to as the relatively efficient units, whereas other units with efficiency ratings of less than 100 percent are referred to as inefficient units. Corporate management thus can use DEA to compare a group of service units to identify relatively inefficient units, measure the magnitude of the inefficiencies, and by comparing the inefficient with the efficient ones, discover ways to reduce those inefficiencies. The DEA linear programming model is formulated according to Charnes, Cooper, and Rhodes, and is referred to as the CCR Model. 9 Definition of Variables Let Ek, with k = 1, 2, . . . K, be the efficiency ratio of unit k, where K is the total number of units being evaluated. Let uj , with j = 1, 2, . . . , M, be a coefficient for output j, where M is the total number of output types considered. The variable uj is a measure of the relative decrease in efficiency with each unit reduction of output value. Let vi , with i = 1, 2, . . . , N, be a coefficient for input i, where N is the total number of input types considered. The variable vi is a measure of the relative increase in efficiency with each unit reduction of input value. Let Ojk be the number of observed units of output j generated by service unit k during one time period. Let Iik be the number of actual units of input i used by service unit k during one time period. Objective Function The objective is to find the set of coefficient u’s associated with each output and of v’s associated with each input that will give the service unit being evaluated the highest possible efficiency. Where e is the index of the unit being evaluated. This function is subject to the constraint that when the same set of input and output coefficients (uj ’s and yi ’s) is applied to all other service units being compared, no service unit will exceed 100 percent efficiency or a ratio of 1.0. Constraints Where all coefficient values are positive and nonzero. To solve this fractional linear programming model using standard linear programming software requires a reformulation. Note that both the objective function and all constraints are ratios rather than linear functions. The objective function in equation (3) Is restated as a linear function by arbitrarily scaling the inputs for the unit under evaluation to a sum of 1.0. Subject to the constraint that For each service unit, the constraints in equation (2) are similarly reformulated: Where Sample Size A question of sample size often is raised concerning the number of service units that are required compared with the number of input and output variables selected in the analysis. The following relationship relating the number of service units K used in the analysis and the number of input N and output M types being considered is based on empirical findings and the experience of DEA practitioners: K ≥ 2(N + M) Example 7.3 Burger Palace An innovative drive-in-only burger chain has established six units in several different cities. Each unit is located in a strip shopping center parking lot. Only a standard meal consisting of a burger, fries, and a drink is available. Management has decided to use DEA to improve productivity by identifying which units are using their resources most efficiently and then sharing their experience and knowledge with the less efficient locations. Table 7.5 summarizes data for two inputs: labor-hours and material dollars consumed during a typical lunch hour period to generate an output of 100 meals sold. Normally, output will vary among the service units, but in this example, we have made the outputs equal to allow for a graphical presentation of the units’ productivity. As Figure 7.19 shows, service units S, S3, and S6 have been joined to form an efficient-production frontier of alternative methods of using labor hours and material resources to generate 100 meals. As can be seen, these efficient units have defined an envelope that contains all the inefficient units—thus the reason for calling the process “data envelopment analysis.” Table 7.5: Summary of Outputs and Inputs for Burger Palace  Figure 7.19: Productivity Frontier of Burger Palace For this simple example, we can identify efficient units by inspection and see the excess inputs being used by inefficient units (e.g., S2 would be as efficient as S3 if it used $50 less in materials). To gain an understanding of DEA, however, we will proceed to formulate the linear programming problems for each unit, then solve each of them to determine efficiency ratings and other information. We begin by illustrating the LP formulation for the first service unit, S1, using equations (5), 6), and (7). max E(S1) = u1 100 subject to u1100 − v12 − v2200 ≤ 0 u1100 − v14 − v2150 ≤ 0 u1100 − v14 − v2100 ≤ 0 u1100 − v16 − v2100 ≤ 0 u1100 − v18 − v280 ≤ 0 u1100 − v110 − v250 ≤ 0 v1 2 + v2200 = 1 u1, u2, v2, ≥ 0 Similar linear programming problems are formulated (or, better yet, the S1 linear programming problem is edited) and solved for the other service units by substituting the appropriate output function for the objective function and substituting the appropriate input function for the last constraint. Constraints 1 through 6, which restrict all units to no more than 100 percent efficiency, remain the same in all problems. This set of six linear programming problems was solved with Excel Solver 7.0 in fewer than five minutes by editing the data file between each run. Because the output is 100 meals for all units, only the last constraint must be edited by substituting the appropriate labor and material input values from Table 7.5 for the unit being evaluated. The data file for unit 1 of Burger Palace using a linear programming Excel add-in is shown in Figure 7.20. The linear programming results for each unit are shown in Table 7.6 and summarized in Table 7.7. In Table 7.6, we find that DEA has identified the same units shown as being efficient in Figure 7.19. Units S2, S4, and S5 all are inefficient in varying degrees. Also shown in Table 7.6 and associated with each inefficient unit is an efficiency reference set. Each inefficient unit will have a set of efficient units associated with it that defines its productivity. As Figure 7.19 shows for inefficient unit S4, the efficient units S3 and S6 have been joined with a line defining the efficiency frontier. A dashed line drawn from the origin to inefficient unit S4 cuts through this frontier and, thus, defines unit S4 as inefficient. In Table 7.7, the value in parentheses that is associated with each member of the efficiency reference set (i.e., .7778 for S3 and .2222 for S6) represents the relative weight assigned to that efficient unit in calculating the efficiency rating for S4. These relative weights are the shadow prices that are associated with the respective efficient-unit constraints in the linear programming solution. (Note in Table 7.6 that for unit 4, these weights appear as opportunity costs for S3 and S6.). TABLE 7.6: LP Solutions for DEA Study of Burger Palace FIGURE 7.20: Excel Data File for DEA Analysis of Burger Palace Unit 1 TABLE 7.7: Summary of DEA Results The values for v1 and v2 that are associated with the inputs of labor-hours and materials, respectively, measure the relative increase in efficiency with each unit reduction of input value. For unit S4, each unit decrease in labor-hours results in an efficiency increase of 0.0555. For unit S4 to become efficient, it must increase its efficiency rating by 0.111 points. This can be accomplished by reducing labor used by 2 hours (i.e., 2 hours × 0.0555 = 0.111). Note that with this reduction in labor-hours, unit S4 becomes identical to efficient unit S3. An alternative approach would be a reduction in materials used by $16.57 (i.e., 0.111/0.0067 = 16.57). Any linear combination of these two measures also would move unit S4 to the productivity frontier defined by the line segment joining efficient units S3 and S6. Table 7.8 contains the calculations for a hypothetical unit C, which is a composite reference unit defined by the weighted inputs of the reference set S3 and S6. As Figure 7.19 shows, this composite unit C is located at the intersection of the productivity frontier and the dashed line drawn from the origin to unit S4. Thus, compared with this reference unit C, inefficient unit S4 is using excess inputs in the amounts of 0.7 labor-hour and 11.1 material dollars. Table 7.8: Calculation of Excess Inputs Used by Unit S4 DEA offers many opportunities for an inefficient unit to become efficient regarding its reference set of efficient units. In practice, management would choose a particular approach on the basis of an evaluation of its cost, practicality, and feasibility; however, the motivation for change is clear (i.e., other units actually are able to achieve similar outputs with fewer resources). DEA and Strategic Planning When combined with profitability, DEA efficiency analysis can be useful in strategic planning for services that are delivered through multiple sites (e.g., hotel chains). Figure 7.21 presents a matrix of four possibilities that arise from combining efficiency and profitability. Considering the top-left quadrant of this matrix (i.e., underperforming potential stars) reveals that units operating at a high profit may be operating inefficiently and, thus, have unrealized potential. Comparing these with similar efficient units could suggest measures that would lead to even greater profit through more efficient operations. FIGURE 7.21: DEA Strategic Matrix Star performers can be found in the top-right quadrant (i.e., benchmark group). These efficient units also are highly profitable and, thus, serve as examples for others to emulate in both operations efficiency and marketing success in generating high revenues. The lower-right quadrant (i.e., candidates for divestiture) contains efficient but unprofitable units. These units are limited in profit potential, perhaps because of a poor location, and should be sold to generate capital for expansion in new territories. It is not clear which strategy to employ with the lower-left quadrant units (i.e., problem branches). If profit potential is limited, investments in efficient operations might lead to a future candidate for divestiture.
FIGURE 7.20: Excel Data File for DEA Analysis of Burger Palace Unit 1
1. For the Burger Palace example, perform a complete analysis of efficiency improvement alternatives for unit S2, including determination of a composite reference unit. 2. For the Burger Palace example, perform a complete analysis of efficiency improvement alternatives for unit S5, including determination of a composite reference unit. 3. For the Burger Palace example, what is the effect of removing an inefficient unit from the analysis (e.g., S2)? 4. For the Burger Palace example, what is the effect of removing an efficient unit from the analysis (e.g., S6)? Supplement Data Envelopment Analysis (DEA): How can corporate management evaluate the productivity of a fast-food outlet, a branch bank, a health clinic, or an elementary school? The difficulties in measuring productivity are threefold. First, what are the appropriate inputs to the system (e.g., labor hours, material dollars) and the measures of those inputs? Second, what are the appropriate outputs of the system (e.g., checks cashed, certificate of deposits) and the measures of those outputs? Third, what are the appropriate ways of measuring the relationship between these inputs and outputs? Measuring Service Productivity The measure of an organization’s productivity, if viewed from an engineering perspective, is similar to the measure of a system’s efficiency. It can be stated as a ratio of outputs to inputs (e.g., miles per gallon for an automobile). To evaluate the operational efficiency of a branch bank, for example, an accounting ratio such as cost per teller transaction might be used. A branch with a high ratio in comparison with those of other branches would be considered less efficient, but the higher ratio could result from a more complex mix of transactions. For example, a branch opening new accounts and selling CDs would require more time per transaction than another branch engaged only in simple transactions such as accepting deposits and cashing checks. The problem with using simple ratios is that the mix of outputs is not considered explicitly. This same criticism also can be made concerning the mix of inputs. For example, some branches might have automated teller machines in addition to live tellers, and this use of technology could affect the cost per teller transaction. Broad-based measures such as profitability or return on investment are highly relevant as overall performance measures, but they are not sufficient to evaluate the operating efficiency of a service unit. For instance, one could not conclude that a profitable branch bank is necessarily efficient in its use of personnel and other inputs. A higher than-average proportion of revenue-generating transactions could be the explanation rather than the cost-efficient use of resources. The DEA Model Fortunately, a technique has been developed with the ability to compare the efficiency of multiple service units that provide similar services by explicitly considering their use of multiple inputs (i.e., resources) to produce multiple outputs (i.e., services). The technique, which is referred to as data envelopment analysis (DEA), circumvents the need to develop standard costs for each service, because it can incorporate multiple inputs and multiple outputs into both the numerator and the denominator of the efficiency ratio without the need for conversion to a common dollar basis. Thus, the DEA measure of efficiency explicitly accounts for the mix of inputs and outputs and, consequently, is more comprehensive and reliable than a set of operating ratios or profit measures. DEA is a linear programming model that attempts to maximize a service unit’s efficiency, expressed as a ratio of outputs to inputs, by comparing a particular unit’s efficiency with the performance of a group of similar service units that are delivering the same service. In the process, some units achieve 100 percent efficiency and are referred to as the relatively efficient units, whereas other units with efficiency ratings of less than 100 percent are referred to as inefficient units. Corporate management thus can use DEA to compare a group of service units to identify relatively inefficient units, measure the magnitude of the inefficiencies, and by comparing the inefficient with the efficient ones, discover ways to reduce those inefficiencies. The DEA linear programming model is formulated according to Charnes, Cooper, and Rhodes, and is referred to as the CCR Model. 9 Definition of Variables Let Ek, with k = 1, 2, . . . K, be the efficiency ratio of unit k, where K is the total number of units being evaluated. Let uj , with j = 1, 2, . . . , M, be a coefficient for output j, where M is the total number of output types considered. The variable uj is a measure of the relative decrease in efficiency with each unit reduction of output value. Let vi , with i = 1, 2, . . . , N, be a coefficient for input i, where N is the total number of input types considered. The variable vi is a measure of the relative increase in efficiency with each unit reduction of input value. Let Ojk be the number of observed units of output j generated by service unit k during one time period. Let Iik be the number of actual units of input i used by service unit k during one time period. Objective Function The objective is to find the set of coefficient u’s associated with each output and of v’s associated with each input that will give the service unit being evaluated the highest possible efficiency. Where e is the index of the unit being evaluated. This function is subject to the constraint that when the same set of input and output coefficients (uj ’s and yi ’s) is applied to all other service units being compared, no service unit will exceed 100 percent efficiency or a ratio of 1.0. Constraints Where all coefficient values are positive and nonzero. To solve this fractional linear programming model using standard linear programming software requires a reformulation. Note that both the objective function and all constraints are ratios rather than linear functions. The objective function in equation (3) Is restated as a linear function by arbitrarily scaling the inputs for the unit under evaluation to a sum of 1.0. Subject to the constraint that For each service unit, the constraints in equation (2) are similarly reformulated: Where Sample Size A question of sample size often is raised concerning the number of service units that are required compared with the number of input and output variables selected in the analysis. The following relationship relating the number of service units K used in the analysis and the number of input N and output M types being considered is based on empirical findings and the experience of DEA practitioners: K ≥ 2(N + M) Example 7.3 Burger Palace An innovative drive-in-only burger chain has established six units in several different cities. Each unit is located in a strip shopping center parking lot. Only a standard meal consisting of a burger, fries, and a drink is available. Management has decided to use DEA to improve productivity by identifying which units are using their resources most efficiently and then sharing their experience and knowledge with the less efficient locations. Table 7.5 summarizes data for two inputs: labor-hours and material dollars consumed during a typical lunch hour period to generate an output of 100 meals sold. Normally, output will vary among the service units, but in this example, we have made the outputs equal to allow for a graphical presentation of the units’ productivity. As Figure 7.19 shows, service units S, S3, and S6 have been joined to form an efficient-production frontier of alternative methods of using labor hours and material resources to generate 100 meals. As can be seen, these efficient units have defined an envelope that contains all the inefficient units—thus the reason for calling the process “data envelopment analysis.” Table 7.5: Summary of Outputs and Inputs for Burger Palace  Figure 7.19: Productivity Frontier of Burger Palace For this simple example, we can identify efficient units by inspection and see the excess inputs being used by inefficient units (e.g., S2 would be as efficient as S3 if it used $50 less in materials). To gain an understanding of DEA, however, we will proceed to formulate the linear programming problems for each unit, then solve each of them to determine efficiency ratings and other information. We begin by illustrating the LP formulation for the first service unit, S1, using equations (5), 6), and (7). max E(S1) = u1 100 subject to u1100 − v12 − v2200 ≤ 0 u1100 − v14 − v2150 ≤ 0 u1100 − v14 − v2100 ≤ 0 u1100 − v16 − v2100 ≤ 0 u1100 − v18 − v280 ≤ 0 u1100 − v110 − v250 ≤ 0 v1 2 + v2200 = 1 u1, u2, v2, ≥ 0 Similar linear programming problems are formulated (or, better yet, the S1 linear programming problem is edited) and solved for the other service units by substituting the appropriate output function for the objective function and substituting the appropriate input function for the last constraint. Constraints 1 through 6, which restrict all units to no more than 100 percent efficiency, remain the same in all problems. This set of six linear programming problems was solved with Excel Solver 7.0 in fewer than five minutes by editing the data file between each run. Because the output is 100 meals for all units, only the last constraint must be edited by substituting the appropriate labor and material input values from Table 7.5 for the unit being evaluated. The data file for unit 1 of Burger Palace using a linear programming Excel add-in is shown in Figure 7.20. The linear programming results for each unit are shown in Table 7.6 and summarized in Table 7.7. In Table 7.6, we find that DEA has identified the same units shown as being efficient in Figure 7.19. Units S2, S4, and S5 all are inefficient in varying degrees. Also shown in Table 7.6 and associated with each inefficient unit is an efficiency reference set. Each inefficient unit will have a set of efficient units associated with it that defines its productivity. As Figure 7.19 shows for inefficient unit S4, the efficient units S3 and S6 have been joined with a line defining the efficiency frontier. A dashed line drawn from the origin to inefficient unit S4 cuts through this frontier and, thus, defines unit S4 as inefficient. In Table 7.7, the value in parentheses that is associated with each member of the efficiency reference set (i.e., .7778 for S3 and .2222 for S6) represents the relative weight assigned to that efficient unit in calculating the efficiency rating for S4. These relative weights are the shadow prices that are associated with the respective efficient-unit constraints in the linear programming solution. (Note in Table 7.6 that for unit 4, these weights appear as opportunity costs for S3 and S6.). TABLE 7.6: LP Solutions for DEA Study of Burger Palace FIGURE 7.20: Excel Data File for DEA Analysis of Burger Palace Unit 1 TABLE 7.7: Summary of DEA Results The values for v1 and v2 that are associated with the inputs of labor-hours and materials, respectively, measure the relative increase in efficiency with each unit reduction of input value. For unit S4, each unit decrease in labor-hours results in an efficiency increase of 0.0555. For unit S4 to become efficient, it must increase its efficiency rating by 0.111 points. This can be accomplished by reducing labor used by 2 hours (i.e., 2 hours × 0.0555 = 0.111). Note that with this reduction in labor-hours, unit S4 becomes identical to efficient unit S3. An alternative approach would be a reduction in materials used by $16.57 (i.e., 0.111/0.0067 = 16.57). Any linear combination of these two measures also would move unit S4 to the productivity frontier defined by the line segment joining efficient units S3 and S6. Table 7.8 contains the calculations for a hypothetical unit C, which is a composite reference unit defined by the weighted inputs of the reference set S3 and S6. As Figure 7.19 shows, this composite unit C is located at the intersection of the productivity frontier and the dashed line drawn from the origin to unit S4. Thus, compared with this reference unit C, inefficient unit S4 is using excess inputs in the amounts of 0.7 labor-hour and 11.1 material dollars. Table 7.8: Calculation of Excess Inputs Used by Unit S4 DEA offers many opportunities for an inefficient unit to become efficient regarding its reference set of efficient units. In practice, management would choose a particular approach on the basis of an evaluation of its cost, practicality, and feasibility; however, the motivation for change is clear (i.e., other units actually are able to achieve similar outputs with fewer resources). DEA and Strategic Planning When combined with profitability, DEA efficiency analysis can be useful in strategic planning for services that are delivered through multiple sites (e.g., hotel chains). Figure 7.21 presents a matrix of four possibilities that arise from combining efficiency and profitability. Considering the top-left quadrant of this matrix (i.e., underperforming potential stars) reveals that units operating at a high profit may be operating inefficiently and, thus, have unrealized potential. Comparing these with similar efficient units could suggest measures that would lead to even greater profit through more efficient operations. FIGURE 7.21: DEA Strategic Matrix Star performers can be found in the top-right quadrant (i.e., benchmark group). These efficient units also are highly profitable and, thus, serve as examples for others to emulate in both operations efficiency and marketing success in generating high revenues. The lower-right quadrant (i.e., candidates for divestiture) contains efficient but unprofitable units. These units are limited in profit potential, perhaps because of a poor location, and should be sold to generate capital for expansion in new territories. It is not clear which strategy to employ with the lower-left quadrant units (i.e., problem branches). If profit potential is limited, investments in efficient operations might lead to a future candidate for divestiture.
TABLE 7.7: Summary of DEA Results
1. For the Burger Palace example, perform a complete analysis of efficiency improvement alternatives for unit S2, including determination of a composite reference unit. 2. For the Burger Palace example, perform a complete analysis of efficiency improvement alternatives for unit S5, including determination of a composite reference unit. 3. For the Burger Palace example, what is the effect of removing an inefficient unit from the analysis (e.g., S2)? 4. For the Burger Palace example, what is the effect of removing an efficient unit from the analysis (e.g., S6)? Supplement Data Envelopment Analysis (DEA): How can corporate management evaluate the productivity of a fast-food outlet, a branch bank, a health clinic, or an elementary school? The difficulties in measuring productivity are threefold. First, what are the appropriate inputs to the system (e.g., labor hours, material dollars) and the measures of those inputs? Second, what are the appropriate outputs of the system (e.g., checks cashed, certificate of deposits) and the measures of those outputs? Third, what are the appropriate ways of measuring the relationship between these inputs and outputs? Measuring Service Productivity The measure of an organization’s productivity, if viewed from an engineering perspective, is similar to the measure of a system’s efficiency. It can be stated as a ratio of outputs to inputs (e.g., miles per gallon for an automobile). To evaluate the operational efficiency of a branch bank, for example, an accounting ratio such as cost per teller transaction might be used. A branch with a high ratio in comparison with those of other branches would be considered less efficient, but the higher ratio could result from a more complex mix of transactions. For example, a branch opening new accounts and selling CDs would require more time per transaction than another branch engaged only in simple transactions such as accepting deposits and cashing checks. The problem with using simple ratios is that the mix of outputs is not considered explicitly. This same criticism also can be made concerning the mix of inputs. For example, some branches might have automated teller machines in addition to live tellers, and this use of technology could affect the cost per teller transaction. Broad-based measures such as profitability or return on investment are highly relevant as overall performance measures, but they are not sufficient to evaluate the operating efficiency of a service unit. For instance, one could not conclude that a profitable branch bank is necessarily efficient in its use of personnel and other inputs. A higher than-average proportion of revenue-generating transactions could be the explanation rather than the cost-efficient use of resources. The DEA Model Fortunately, a technique has been developed with the ability to compare the efficiency of multiple service units that provide similar services by explicitly considering their use of multiple inputs (i.e., resources) to produce multiple outputs (i.e., services). The technique, which is referred to as data envelopment analysis (DEA), circumvents the need to develop standard costs for each service, because it can incorporate multiple inputs and multiple outputs into both the numerator and the denominator of the efficiency ratio without the need for conversion to a common dollar basis. Thus, the DEA measure of efficiency explicitly accounts for the mix of inputs and outputs and, consequently, is more comprehensive and reliable than a set of operating ratios or profit measures. DEA is a linear programming model that attempts to maximize a service unit’s efficiency, expressed as a ratio of outputs to inputs, by comparing a particular unit’s efficiency with the performance of a group of similar service units that are delivering the same service. In the process, some units achieve 100 percent efficiency and are referred to as the relatively efficient units, whereas other units with efficiency ratings of less than 100 percent are referred to as inefficient units. Corporate management thus can use DEA to compare a group of service units to identify relatively inefficient units, measure the magnitude of the inefficiencies, and by comparing the inefficient with the efficient ones, discover ways to reduce those inefficiencies. The DEA linear programming model is formulated according to Charnes, Cooper, and Rhodes, and is referred to as the CCR Model. 9 Definition of Variables Let Ek, with k = 1, 2, . . . K, be the efficiency ratio of unit k, where K is the total number of units being evaluated. Let uj , with j = 1, 2, . . . , M, be a coefficient for output j, where M is the total number of output types considered. The variable uj is a measure of the relative decrease in efficiency with each unit reduction of output value. Let vi , with i = 1, 2, . . . , N, be a coefficient for input i, where N is the total number of input types considered. The variable vi is a measure of the relative increase in efficiency with each unit reduction of input value. Let Ojk be the number of observed units of output j generated by service unit k during one time period. Let Iik be the number of actual units of input i used by service unit k during one time period. Objective Function The objective is to find the set of coefficient u’s associated with each output and of v’s associated with each input that will give the service unit being evaluated the highest possible efficiency. Where e is the index of the unit being evaluated. This function is subject to the constraint that when the same set of input and output coefficients (uj ’s and yi ’s) is applied to all other service units being compared, no service unit will exceed 100 percent efficiency or a ratio of 1.0. Constraints Where all coefficient values are positive and nonzero. To solve this fractional linear programming model using standard linear programming software requires a reformulation. Note that both the objective function and all constraints are ratios rather than linear functions. The objective function in equation (3) Is restated as a linear function by arbitrarily scaling the inputs for the unit under evaluation to a sum of 1.0. Subject to the constraint that For each service unit, the constraints in equation (2) are similarly reformulated: Where Sample Size A question of sample size often is raised concerning the number of service units that are required compared with the number of input and output variables selected in the analysis. The following relationship relating the number of service units K used in the analysis and the number of input N and output M types being considered is based on empirical findings and the experience of DEA practitioners: K ≥ 2(N + M) Example 7.3 Burger Palace An innovative drive-in-only burger chain has established six units in several different cities. Each unit is located in a strip shopping center parking lot. Only a standard meal consisting of a burger, fries, and a drink is available. Management has decided to use DEA to improve productivity by identifying which units are using their resources most efficiently and then sharing their experience and knowledge with the less efficient locations. Table 7.5 summarizes data for two inputs: labor-hours and material dollars consumed during a typical lunch hour period to generate an output of 100 meals sold. Normally, output will vary among the service units, but in this example, we have made the outputs equal to allow for a graphical presentation of the units’ productivity. As Figure 7.19 shows, service units S, S3, and S6 have been joined to form an efficient-production frontier of alternative methods of using labor hours and material resources to generate 100 meals. As can be seen, these efficient units have defined an envelope that contains all the inefficient units—thus the reason for calling the process “data envelopment analysis.” Table 7.5: Summary of Outputs and Inputs for Burger Palace  Figure 7.19: Productivity Frontier of Burger Palace For this simple example, we can identify efficient units by inspection and see the excess inputs being used by inefficient units (e.g., S2 would be as efficient as S3 if it used $50 less in materials). To gain an understanding of DEA, however, we will proceed to formulate the linear programming problems for each unit, then solve each of them to determine efficiency ratings and other information. We begin by illustrating the LP formulation for the first service unit, S1, using equations (5), 6), and (7). max E(S1) = u1 100 subject to u1100 − v12 − v2200 ≤ 0 u1100 − v14 − v2150 ≤ 0 u1100 − v14 − v2100 ≤ 0 u1100 − v16 − v2100 ≤ 0 u1100 − v18 − v280 ≤ 0 u1100 − v110 − v250 ≤ 0 v1 2 + v2200 = 1 u1, u2, v2, ≥ 0 Similar linear programming problems are formulated (or, better yet, the S1 linear programming problem is edited) and solved for the other service units by substituting the appropriate output function for the objective function and substituting the appropriate input function for the last constraint. Constraints 1 through 6, which restrict all units to no more than 100 percent efficiency, remain the same in all problems. This set of six linear programming problems was solved with Excel Solver 7.0 in fewer than five minutes by editing the data file between each run. Because the output is 100 meals for all units, only the last constraint must be edited by substituting the appropriate labor and material input values from Table 7.5 for the unit being evaluated. The data file for unit 1 of Burger Palace using a linear programming Excel add-in is shown in Figure 7.20. The linear programming results for each unit are shown in Table 7.6 and summarized in Table 7.7. In Table 7.6, we find that DEA has identified the same units shown as being efficient in Figure 7.19. Units S2, S4, and S5 all are inefficient in varying degrees. Also shown in Table 7.6 and associated with each inefficient unit is an efficiency reference set. Each inefficient unit will have a set of efficient units associated with it that defines its productivity. As Figure 7.19 shows for inefficient unit S4, the efficient units S3 and S6 have been joined with a line defining the efficiency frontier. A dashed line drawn from the origin to inefficient unit S4 cuts through this frontier and, thus, defines unit S4 as inefficient. In Table 7.7, the value in parentheses that is associated with each member of the efficiency reference set (i.e., .7778 for S3 and .2222 for S6) represents the relative weight assigned to that efficient unit in calculating the efficiency rating for S4. These relative weights are the shadow prices that are associated with the respective efficient-unit constraints in the linear programming solution. (Note in Table 7.6 that for unit 4, these weights appear as opportunity costs for S3 and S6.). TABLE 7.6: LP Solutions for DEA Study of Burger Palace FIGURE 7.20: Excel Data File for DEA Analysis of Burger Palace Unit 1 TABLE 7.7: Summary of DEA Results The values for v1 and v2 that are associated with the inputs of labor-hours and materials, respectively, measure the relative increase in efficiency with each unit reduction of input value. For unit S4, each unit decrease in labor-hours results in an efficiency increase of 0.0555. For unit S4 to become efficient, it must increase its efficiency rating by 0.111 points. This can be accomplished by reducing labor used by 2 hours (i.e., 2 hours × 0.0555 = 0.111). Note that with this reduction in labor-hours, unit S4 becomes identical to efficient unit S3. An alternative approach would be a reduction in materials used by $16.57 (i.e., 0.111/0.0067 = 16.57). Any linear combination of these two measures also would move unit S4 to the productivity frontier defined by the line segment joining efficient units S3 and S6. Table 7.8 contains the calculations for a hypothetical unit C, which is a composite reference unit defined by the weighted inputs of the reference set S3 and S6. As Figure 7.19 shows, this composite unit C is located at the intersection of the productivity frontier and the dashed line drawn from the origin to unit S4. Thus, compared with this reference unit C, inefficient unit S4 is using excess inputs in the amounts of 0.7 labor-hour and 11.1 material dollars. Table 7.8: Calculation of Excess Inputs Used by Unit S4 DEA offers many opportunities for an inefficient unit to become efficient regarding its reference set of efficient units. In practice, management would choose a particular approach on the basis of an evaluation of its cost, practicality, and feasibility; however, the motivation for change is clear (i.e., other units actually are able to achieve similar outputs with fewer resources). DEA and Strategic Planning When combined with profitability, DEA efficiency analysis can be useful in strategic planning for services that are delivered through multiple sites (e.g., hotel chains). Figure 7.21 presents a matrix of four possibilities that arise from combining efficiency and profitability. Considering the top-left quadrant of this matrix (i.e., underperforming potential stars) reveals that units operating at a high profit may be operating inefficiently and, thus, have unrealized potential. Comparing these with similar efficient units could suggest measures that would lead to even greater profit through more efficient operations. FIGURE 7.21: DEA Strategic Matrix Star performers can be found in the top-right quadrant (i.e., benchmark group). These efficient units also are highly profitable and, thus, serve as examples for others to emulate in both operations efficiency and marketing success in generating high revenues. The lower-right quadrant (i.e., candidates for divestiture) contains efficient but unprofitable units. These units are limited in profit potential, perhaps because of a poor location, and should be sold to generate capital for expansion in new territories. It is not clear which strategy to employ with the lower-left quadrant units (i.e., problem branches). If profit potential is limited, investments in efficient operations might lead to a future candidate for divestiture.
The values for v1 and v2 that are associated with the inputs of labor-hours and materials, respectively, measure the relative increase in efficiency with each unit reduction of input value. For unit S4, each unit decrease in labor-hours results in an efficiency increase of 0.0555. For unit S4 to become efficient, it must increase its efficiency rating by 0.111 points. This can be accomplished by reducing labor used by 2 hours (i.e., 2 hours × 0.0555 = 0.111). Note that with this reduction in labor-hours, unit S4 becomes identical to efficient unit S3. An alternative approach would be a reduction in materials used by $16.57 (i.e., 0.111/0.0067 = 16.57). Any linear combination of these two measures also would move unit S4 to the productivity frontier defined by the line segment joining efficient units S3 and S6. Table 7.8 contains the calculations for a hypothetical unit C, which is a composite reference unit defined by the weighted inputs of the reference set S3 and S6. As Figure 7.19 shows, this composite unit C is located at the intersection of the productivity frontier and the dashed line drawn from the origin to unit S4. Thus, compared with this reference unit C, inefficient unit S4 is using excess inputs in the amounts of 0.7 labor-hour and 11.1 material dollars. Table 7.8: Calculation of Excess Inputs Used by Unit S4
1. For the Burger Palace example, perform a complete analysis of efficiency improvement alternatives for unit S2, including determination of a composite reference unit. 2. For the Burger Palace example, perform a complete analysis of efficiency improvement alternatives for unit S5, including determination of a composite reference unit. 3. For the Burger Palace example, what is the effect of removing an inefficient unit from the analysis (e.g., S2)? 4. For the Burger Palace example, what is the effect of removing an efficient unit from the analysis (e.g., S6)? Supplement Data Envelopment Analysis (DEA): How can corporate management evaluate the productivity of a fast-food outlet, a branch bank, a health clinic, or an elementary school? The difficulties in measuring productivity are threefold. First, what are the appropriate inputs to the system (e.g., labor hours, material dollars) and the measures of those inputs? Second, what are the appropriate outputs of the system (e.g., checks cashed, certificate of deposits) and the measures of those outputs? Third, what are the appropriate ways of measuring the relationship between these inputs and outputs? Measuring Service Productivity The measure of an organization’s productivity, if viewed from an engineering perspective, is similar to the measure of a system’s efficiency. It can be stated as a ratio of outputs to inputs (e.g., miles per gallon for an automobile). To evaluate the operational efficiency of a branch bank, for example, an accounting ratio such as cost per teller transaction might be used. A branch with a high ratio in comparison with those of other branches would be considered less efficient, but the higher ratio could result from a more complex mix of transactions. For example, a branch opening new accounts and selling CDs would require more time per transaction than another branch engaged only in simple transactions such as accepting deposits and cashing checks. The problem with using simple ratios is that the mix of outputs is not considered explicitly. This same criticism also can be made concerning the mix of inputs. For example, some branches might have automated teller machines in addition to live tellers, and this use of technology could affect the cost per teller transaction. Broad-based measures such as profitability or return on investment are highly relevant as overall performance measures, but they are not sufficient to evaluate the operating efficiency of a service unit. For instance, one could not conclude that a profitable branch bank is necessarily efficient in its use of personnel and other inputs. A higher than-average proportion of revenue-generating transactions could be the explanation rather than the cost-efficient use of resources. The DEA Model Fortunately, a technique has been developed with the ability to compare the efficiency of multiple service units that provide similar services by explicitly considering their use of multiple inputs (i.e., resources) to produce multiple outputs (i.e., services). The technique, which is referred to as data envelopment analysis (DEA), circumvents the need to develop standard costs for each service, because it can incorporate multiple inputs and multiple outputs into both the numerator and the denominator of the efficiency ratio without the need for conversion to a common dollar basis. Thus, the DEA measure of efficiency explicitly accounts for the mix of inputs and outputs and, consequently, is more comprehensive and reliable than a set of operating ratios or profit measures. DEA is a linear programming model that attempts to maximize a service unit’s efficiency, expressed as a ratio of outputs to inputs, by comparing a particular unit’s efficiency with the performance of a group of similar service units that are delivering the same service. In the process, some units achieve 100 percent efficiency and are referred to as the relatively efficient units, whereas other units with efficiency ratings of less than 100 percent are referred to as inefficient units. Corporate management thus can use DEA to compare a group of service units to identify relatively inefficient units, measure the magnitude of the inefficiencies, and by comparing the inefficient with the efficient ones, discover ways to reduce those inefficiencies. The DEA linear programming model is formulated according to Charnes, Cooper, and Rhodes, and is referred to as the CCR Model. 9 Definition of Variables Let Ek, with k = 1, 2, . . . K, be the efficiency ratio of unit k, where K is the total number of units being evaluated. Let uj , with j = 1, 2, . . . , M, be a coefficient for output j, where M is the total number of output types considered. The variable uj is a measure of the relative decrease in efficiency with each unit reduction of output value. Let vi , with i = 1, 2, . . . , N, be a coefficient for input i, where N is the total number of input types considered. The variable vi is a measure of the relative increase in efficiency with each unit reduction of input value. Let Ojk be the number of observed units of output j generated by service unit k during one time period. Let Iik be the number of actual units of input i used by service unit k during one time period. Objective Function The objective is to find the set of coefficient u’s associated with each output and of v’s associated with each input that will give the service unit being evaluated the highest possible efficiency. Where e is the index of the unit being evaluated. This function is subject to the constraint that when the same set of input and output coefficients (uj ’s and yi ’s) is applied to all other service units being compared, no service unit will exceed 100 percent efficiency or a ratio of 1.0. Constraints Where all coefficient values are positive and nonzero. To solve this fractional linear programming model using standard linear programming software requires a reformulation. Note that both the objective function and all constraints are ratios rather than linear functions. The objective function in equation (3) Is restated as a linear function by arbitrarily scaling the inputs for the unit under evaluation to a sum of 1.0. Subject to the constraint that For each service unit, the constraints in equation (2) are similarly reformulated: Where Sample Size A question of sample size often is raised concerning the number of service units that are required compared with the number of input and output variables selected in the analysis. The following relationship relating the number of service units K used in the analysis and the number of input N and output M types being considered is based on empirical findings and the experience of DEA practitioners: K ≥ 2(N + M) Example 7.3 Burger Palace An innovative drive-in-only burger chain has established six units in several different cities. Each unit is located in a strip shopping center parking lot. Only a standard meal consisting of a burger, fries, and a drink is available. Management has decided to use DEA to improve productivity by identifying which units are using their resources most efficiently and then sharing their experience and knowledge with the less efficient locations. Table 7.5 summarizes data for two inputs: labor-hours and material dollars consumed during a typical lunch hour period to generate an output of 100 meals sold. Normally, output will vary among the service units, but in this example, we have made the outputs equal to allow for a graphical presentation of the units’ productivity. As Figure 7.19 shows, service units S, S3, and S6 have been joined to form an efficient-production frontier of alternative methods of using labor hours and material resources to generate 100 meals. As can be seen, these efficient units have defined an envelope that contains all the inefficient units—thus the reason for calling the process “data envelopment analysis.” Table 7.5: Summary of Outputs and Inputs for Burger Palace  Figure 7.19: Productivity Frontier of Burger Palace For this simple example, we can identify efficient units by inspection and see the excess inputs being used by inefficient units (e.g., S2 would be as efficient as S3 if it used $50 less in materials). To gain an understanding of DEA, however, we will proceed to formulate the linear programming problems for each unit, then solve each of them to determine efficiency ratings and other information. We begin by illustrating the LP formulation for the first service unit, S1, using equations (5), 6), and (7). max E(S1) = u1 100 subject to u1100 − v12 − v2200 ≤ 0 u1100 − v14 − v2150 ≤ 0 u1100 − v14 − v2100 ≤ 0 u1100 − v16 − v2100 ≤ 0 u1100 − v18 − v280 ≤ 0 u1100 − v110 − v250 ≤ 0 v1 2 + v2200 = 1 u1, u2, v2, ≥ 0 Similar linear programming problems are formulated (or, better yet, the S1 linear programming problem is edited) and solved for the other service units by substituting the appropriate output function for the objective function and substituting the appropriate input function for the last constraint. Constraints 1 through 6, which restrict all units to no more than 100 percent efficiency, remain the same in all problems. This set of six linear programming problems was solved with Excel Solver 7.0 in fewer than five minutes by editing the data file between each run. Because the output is 100 meals for all units, only the last constraint must be edited by substituting the appropriate labor and material input values from Table 7.5 for the unit being evaluated. The data file for unit 1 of Burger Palace using a linear programming Excel add-in is shown in Figure 7.20. The linear programming results for each unit are shown in Table 7.6 and summarized in Table 7.7. In Table 7.6, we find that DEA has identified the same units shown as being efficient in Figure 7.19. Units S2, S4, and S5 all are inefficient in varying degrees. Also shown in Table 7.6 and associated with each inefficient unit is an efficiency reference set. Each inefficient unit will have a set of efficient units associated with it that defines its productivity. As Figure 7.19 shows for inefficient unit S4, the efficient units S3 and S6 have been joined with a line defining the efficiency frontier. A dashed line drawn from the origin to inefficient unit S4 cuts through this frontier and, thus, defines unit S4 as inefficient. In Table 7.7, the value in parentheses that is associated with each member of the efficiency reference set (i.e., .7778 for S3 and .2222 for S6) represents the relative weight assigned to that efficient unit in calculating the efficiency rating for S4. These relative weights are the shadow prices that are associated with the respective efficient-unit constraints in the linear programming solution. (Note in Table 7.6 that for unit 4, these weights appear as opportunity costs for S3 and S6.). TABLE 7.6: LP Solutions for DEA Study of Burger Palace FIGURE 7.20: Excel Data File for DEA Analysis of Burger Palace Unit 1 TABLE 7.7: Summary of DEA Results The values for v1 and v2 that are associated with the inputs of labor-hours and materials, respectively, measure the relative increase in efficiency with each unit reduction of input value. For unit S4, each unit decrease in labor-hours results in an efficiency increase of 0.0555. For unit S4 to become efficient, it must increase its efficiency rating by 0.111 points. This can be accomplished by reducing labor used by 2 hours (i.e., 2 hours × 0.0555 = 0.111). Note that with this reduction in labor-hours, unit S4 becomes identical to efficient unit S3. An alternative approach would be a reduction in materials used by $16.57 (i.e., 0.111/0.0067 = 16.57). Any linear combination of these two measures also would move unit S4 to the productivity frontier defined by the line segment joining efficient units S3 and S6. Table 7.8 contains the calculations for a hypothetical unit C, which is a composite reference unit defined by the weighted inputs of the reference set S3 and S6. As Figure 7.19 shows, this composite unit C is located at the intersection of the productivity frontier and the dashed line drawn from the origin to unit S4. Thus, compared with this reference unit C, inefficient unit S4 is using excess inputs in the amounts of 0.7 labor-hour and 11.1 material dollars. Table 7.8: Calculation of Excess Inputs Used by Unit S4 DEA offers many opportunities for an inefficient unit to become efficient regarding its reference set of efficient units. In practice, management would choose a particular approach on the basis of an evaluation of its cost, practicality, and feasibility; however, the motivation for change is clear (i.e., other units actually are able to achieve similar outputs with fewer resources). DEA and Strategic Planning When combined with profitability, DEA efficiency analysis can be useful in strategic planning for services that are delivered through multiple sites (e.g., hotel chains). Figure 7.21 presents a matrix of four possibilities that arise from combining efficiency and profitability. Considering the top-left quadrant of this matrix (i.e., underperforming potential stars) reveals that units operating at a high profit may be operating inefficiently and, thus, have unrealized potential. Comparing these with similar efficient units could suggest measures that would lead to even greater profit through more efficient operations. FIGURE 7.21: DEA Strategic Matrix Star performers can be found in the top-right quadrant (i.e., benchmark group). These efficient units also are highly profitable and, thus, serve as examples for others to emulate in both operations efficiency and marketing success in generating high revenues. The lower-right quadrant (i.e., candidates for divestiture) contains efficient but unprofitable units. These units are limited in profit potential, perhaps because of a poor location, and should be sold to generate capital for expansion in new territories. It is not clear which strategy to employ with the lower-left quadrant units (i.e., problem branches). If profit potential is limited, investments in efficient operations might lead to a future candidate for divestiture.
DEA offers many opportunities for an inefficient unit to become efficient regarding its reference set of efficient units. In practice, management would choose a particular approach on the basis of an evaluation of its cost, practicality, and feasibility; however, the motivation for change is clear (i.e., other units actually are able to achieve similar outputs with fewer resources). DEA and Strategic Planning When combined with profitability, DEA efficiency analysis can be useful in strategic planning for services that are delivered through multiple sites (e.g., hotel chains). Figure 7.21 presents a matrix of four possibilities that arise from combining efficiency and profitability. Considering the top-left quadrant of this matrix (i.e., underperforming potential stars) reveals that units operating at a high profit may be operating inefficiently and, thus, have unrealized potential. Comparing these with similar efficient units could suggest measures that would lead to even greater profit through more efficient operations. FIGURE 7.21: DEA Strategic Matrix
1. For the Burger Palace example, perform a complete analysis of efficiency improvement alternatives for unit S2, including determination of a composite reference unit. 2. For the Burger Palace example, perform a complete analysis of efficiency improvement alternatives for unit S5, including determination of a composite reference unit. 3. For the Burger Palace example, what is the effect of removing an inefficient unit from the analysis (e.g., S2)? 4. For the Burger Palace example, what is the effect of removing an efficient unit from the analysis (e.g., S6)? Supplement Data Envelopment Analysis (DEA): How can corporate management evaluate the productivity of a fast-food outlet, a branch bank, a health clinic, or an elementary school? The difficulties in measuring productivity are threefold. First, what are the appropriate inputs to the system (e.g., labor hours, material dollars) and the measures of those inputs? Second, what are the appropriate outputs of the system (e.g., checks cashed, certificate of deposits) and the measures of those outputs? Third, what are the appropriate ways of measuring the relationship between these inputs and outputs? Measuring Service Productivity The measure of an organization’s productivity, if viewed from an engineering perspective, is similar to the measure of a system’s efficiency. It can be stated as a ratio of outputs to inputs (e.g., miles per gallon for an automobile). To evaluate the operational efficiency of a branch bank, for example, an accounting ratio such as cost per teller transaction might be used. A branch with a high ratio in comparison with those of other branches would be considered less efficient, but the higher ratio could result from a more complex mix of transactions. For example, a branch opening new accounts and selling CDs would require more time per transaction than another branch engaged only in simple transactions such as accepting deposits and cashing checks. The problem with using simple ratios is that the mix of outputs is not considered explicitly. This same criticism also can be made concerning the mix of inputs. For example, some branches might have automated teller machines in addition to live tellers, and this use of technology could affect the cost per teller transaction. Broad-based measures such as profitability or return on investment are highly relevant as overall performance measures, but they are not sufficient to evaluate the operating efficiency of a service unit. For instance, one could not conclude that a profitable branch bank is necessarily efficient in its use of personnel and other inputs. A higher than-average proportion of revenue-generating transactions could be the explanation rather than the cost-efficient use of resources. The DEA Model Fortunately, a technique has been developed with the ability to compare the efficiency of multiple service units that provide similar services by explicitly considering their use of multiple inputs (i.e., resources) to produce multiple outputs (i.e., services). The technique, which is referred to as data envelopment analysis (DEA), circumvents the need to develop standard costs for each service, because it can incorporate multiple inputs and multiple outputs into both the numerator and the denominator of the efficiency ratio without the need for conversion to a common dollar basis. Thus, the DEA measure of efficiency explicitly accounts for the mix of inputs and outputs and, consequently, is more comprehensive and reliable than a set of operating ratios or profit measures. DEA is a linear programming model that attempts to maximize a service unit’s efficiency, expressed as a ratio of outputs to inputs, by comparing a particular unit’s efficiency with the performance of a group of similar service units that are delivering the same service. In the process, some units achieve 100 percent efficiency and are referred to as the relatively efficient units, whereas other units with efficiency ratings of less than 100 percent are referred to as inefficient units. Corporate management thus can use DEA to compare a group of service units to identify relatively inefficient units, measure the magnitude of the inefficiencies, and by comparing the inefficient with the efficient ones, discover ways to reduce those inefficiencies. The DEA linear programming model is formulated according to Charnes, Cooper, and Rhodes, and is referred to as the CCR Model. 9 Definition of Variables Let Ek, with k = 1, 2, . . . K, be the efficiency ratio of unit k, where K is the total number of units being evaluated. Let uj , with j = 1, 2, . . . , M, be a coefficient for output j, where M is the total number of output types considered. The variable uj is a measure of the relative decrease in efficiency with each unit reduction of output value. Let vi , with i = 1, 2, . . . , N, be a coefficient for input i, where N is the total number of input types considered. The variable vi is a measure of the relative increase in efficiency with each unit reduction of input value. Let Ojk be the number of observed units of output j generated by service unit k during one time period. Let Iik be the number of actual units of input i used by service unit k during one time period. Objective Function The objective is to find the set of coefficient u’s associated with each output and of v’s associated with each input that will give the service unit being evaluated the highest possible efficiency. Where e is the index of the unit being evaluated. This function is subject to the constraint that when the same set of input and output coefficients (uj ’s and yi ’s) is applied to all other service units being compared, no service unit will exceed 100 percent efficiency or a ratio of 1.0. Constraints Where all coefficient values are positive and nonzero. To solve this fractional linear programming model using standard linear programming software requires a reformulation. Note that both the objective function and all constraints are ratios rather than linear functions. The objective function in equation (3) Is restated as a linear function by arbitrarily scaling the inputs for the unit under evaluation to a sum of 1.0. Subject to the constraint that For each service unit, the constraints in equation (2) are similarly reformulated: Where Sample Size A question of sample size often is raised concerning the number of service units that are required compared with the number of input and output variables selected in the analysis. The following relationship relating the number of service units K used in the analysis and the number of input N and output M types being considered is based on empirical findings and the experience of DEA practitioners: K ≥ 2(N + M) Example 7.3 Burger Palace An innovative drive-in-only burger chain has established six units in several different cities. Each unit is located in a strip shopping center parking lot. Only a standard meal consisting of a burger, fries, and a drink is available. Management has decided to use DEA to improve productivity by identifying which units are using their resources most efficiently and then sharing their experience and knowledge with the less efficient locations. Table 7.5 summarizes data for two inputs: labor-hours and material dollars consumed during a typical lunch hour period to generate an output of 100 meals sold. Normally, output will vary among the service units, but in this example, we have made the outputs equal to allow for a graphical presentation of the units’ productivity. As Figure 7.19 shows, service units S, S3, and S6 have been joined to form an efficient-production frontier of alternative methods of using labor hours and material resources to generate 100 meals. As can be seen, these efficient units have defined an envelope that contains all the inefficient units—thus the reason for calling the process “data envelopment analysis.” Table 7.5: Summary of Outputs and Inputs for Burger Palace  Figure 7.19: Productivity Frontier of Burger Palace For this simple example, we can identify efficient units by inspection and see the excess inputs being used by inefficient units (e.g., S2 would be as efficient as S3 if it used $50 less in materials). To gain an understanding of DEA, however, we will proceed to formulate the linear programming problems for each unit, then solve each of them to determine efficiency ratings and other information. We begin by illustrating the LP formulation for the first service unit, S1, using equations (5), 6), and (7). max E(S1) = u1 100 subject to u1100 − v12 − v2200 ≤ 0 u1100 − v14 − v2150 ≤ 0 u1100 − v14 − v2100 ≤ 0 u1100 − v16 − v2100 ≤ 0 u1100 − v18 − v280 ≤ 0 u1100 − v110 − v250 ≤ 0 v1 2 + v2200 = 1 u1, u2, v2, ≥ 0 Similar linear programming problems are formulated (or, better yet, the S1 linear programming problem is edited) and solved for the other service units by substituting the appropriate output function for the objective function and substituting the appropriate input function for the last constraint. Constraints 1 through 6, which restrict all units to no more than 100 percent efficiency, remain the same in all problems. This set of six linear programming problems was solved with Excel Solver 7.0 in fewer than five minutes by editing the data file between each run. Because the output is 100 meals for all units, only the last constraint must be edited by substituting the appropriate labor and material input values from Table 7.5 for the unit being evaluated. The data file for unit 1 of Burger Palace using a linear programming Excel add-in is shown in Figure 7.20. The linear programming results for each unit are shown in Table 7.6 and summarized in Table 7.7. In Table 7.6, we find that DEA has identified the same units shown as being efficient in Figure 7.19. Units S2, S4, and S5 all are inefficient in varying degrees. Also shown in Table 7.6 and associated with each inefficient unit is an efficiency reference set. Each inefficient unit will have a set of efficient units associated with it that defines its productivity. As Figure 7.19 shows for inefficient unit S4, the efficient units S3 and S6 have been joined with a line defining the efficiency frontier. A dashed line drawn from the origin to inefficient unit S4 cuts through this frontier and, thus, defines unit S4 as inefficient. In Table 7.7, the value in parentheses that is associated with each member of the efficiency reference set (i.e., .7778 for S3 and .2222 for S6) represents the relative weight assigned to that efficient unit in calculating the efficiency rating for S4. These relative weights are the shadow prices that are associated with the respective efficient-unit constraints in the linear programming solution. (Note in Table 7.6 that for unit 4, these weights appear as opportunity costs for S3 and S6.). TABLE 7.6: LP Solutions for DEA Study of Burger Palace FIGURE 7.20: Excel Data File for DEA Analysis of Burger Palace Unit 1 TABLE 7.7: Summary of DEA Results The values for v1 and v2 that are associated with the inputs of labor-hours and materials, respectively, measure the relative increase in efficiency with each unit reduction of input value. For unit S4, each unit decrease in labor-hours results in an efficiency increase of 0.0555. For unit S4 to become efficient, it must increase its efficiency rating by 0.111 points. This can be accomplished by reducing labor used by 2 hours (i.e., 2 hours × 0.0555 = 0.111). Note that with this reduction in labor-hours, unit S4 becomes identical to efficient unit S3. An alternative approach would be a reduction in materials used by $16.57 (i.e., 0.111/0.0067 = 16.57). Any linear combination of these two measures also would move unit S4 to the productivity frontier defined by the line segment joining efficient units S3 and S6. Table 7.8 contains the calculations for a hypothetical unit C, which is a composite reference unit defined by the weighted inputs of the reference set S3 and S6. As Figure 7.19 shows, this composite unit C is located at the intersection of the productivity frontier and the dashed line drawn from the origin to unit S4. Thus, compared with this reference unit C, inefficient unit S4 is using excess inputs in the amounts of 0.7 labor-hour and 11.1 material dollars. Table 7.8: Calculation of Excess Inputs Used by Unit S4 DEA offers many opportunities for an inefficient unit to become efficient regarding its reference set of efficient units. In practice, management would choose a particular approach on the basis of an evaluation of its cost, practicality, and feasibility; however, the motivation for change is clear (i.e., other units actually are able to achieve similar outputs with fewer resources). DEA and Strategic Planning When combined with profitability, DEA efficiency analysis can be useful in strategic planning for services that are delivered through multiple sites (e.g., hotel chains). Figure 7.21 presents a matrix of four possibilities that arise from combining efficiency and profitability. Considering the top-left quadrant of this matrix (i.e., underperforming potential stars) reveals that units operating at a high profit may be operating inefficiently and, thus, have unrealized potential. Comparing these with similar efficient units could suggest measures that would lead to even greater profit through more efficient operations. FIGURE 7.21: DEA Strategic Matrix Star performers can be found in the top-right quadrant (i.e., benchmark group). These efficient units also are highly profitable and, thus, serve as examples for others to emulate in both operations efficiency and marketing success in generating high revenues. The lower-right quadrant (i.e., candidates for divestiture) contains efficient but unprofitable units. These units are limited in profit potential, perhaps because of a poor location, and should be sold to generate capital for expansion in new territories. It is not clear which strategy to employ with the lower-left quadrant units (i.e., problem branches). If profit potential is limited, investments in efficient operations might lead to a future candidate for divestiture.
Star performers can be found in the top-right quadrant (i.e., benchmark group). These efficient units also are highly profitable and, thus, serve as examples for others to emulate in both operations efficiency and marketing success in generating high revenues. The lower-right quadrant (i.e., candidates for divestiture) contains efficient but unprofitable units. These units are limited in profit potential, perhaps because of a poor location, and should be sold to generate capital for expansion in new territories. It is not clear which strategy to employ with the lower-left quadrant units (i.e., problem branches). If profit potential is limited, investments in efficient operations might lead to a future candidate for divestiture.





Transcribed Image Text:

max E₂ = U₁0₁e + U₂0₂e + ville + V₂/2e+ +UMO Me + VN Ne U₁Ole+ U₂O₂e + vile +V₂lze + ... ... +UMOME + VN Ne ≤1.0 k = 1, 2, ..., K max E₂ = u₁O₁e + U₂O₂e + +UMO Me Ville + V₂/2e+ + VN/Ne = 1 4₁01k +4₂0₂k + + UMOMK - (V₁/1K + V₂/2k +··· + VNẴNK) ≤0 k = 1, 2, ..., K u, 20 4,20 j= 1, 2, ..., M i = 1, 2, ..., N Service Unit 1 2 3 4 5 6 Meals Sold 100 100 100 100 100 100 Labor-Hours 2 4 4 6 8 10 Material Dollars 200 150 100 100 80 50 Material dollars 200 150 100 50 2 S₁ (2, 200) 4 S₂ (4, 150) S3 (4, 100) с (5.3, 88.9) Labor hours S4 (6, 100) 6 S5 ● (8,80) 1 8 DO S₂ (10, 50) 10 No. 12345 Variables Names U1 V1 V2 S1 S2 Summarized results for unit 1 Opportunity costs Solutions +1.0000000 +.16666667 +.00333333 0 0 0 0 +1.0000000 No. 6 7 0 +16.666670 Maximized objective function = 100 8 9 10 Variables Names S3 S4 S5 S6 A7 Solutions 0 +33.333336 +60.000000 +83.333336 0 Iterations = 4 Page: 1 Opportunity costs 00 0 0 +100.00000 No. GAWN →ő Variables Names 55 25 U1 V1 V2 S1 S2 Summarized results for unit 2 Opportunity costs Solutions +.85714287 +.14285715 +.00285714 0 0 0 0 +0.28571430 No. 6 7 8 Variables 9 +14.285717 0 10 Maximized objective function = 85.71429 Names S3 S4 S5 S6 A7 Solutions Page: 1 Opportunity costs 0+.71428573 +28.571430 +51.428574 +71.428574 0 0 0 0 +85.714287 Iterations = 4 No. 1 2 3 4 5 Variables Names U1 V1 V2 S1 S2 Summarized results for unit 3 Opportunity costs Solutions +1.0000000 +.06250000 +.00750000 +62.500000 +37.500008 Maximized objective function = 100 00 0 0 0 No. 6 7 8 9 10 Variables Names S3 S4 S5 S6 A7 Solutions Page: 1 Opportunity costs 0 +1.0000000 +12.500000 +10.000001 0 0 +100.00000 Iterations = 3 No. 12345 Variables Names U1 V1 V2 S1 S2 Summarized results for unit 4 Opportunity costs Solutions +.88888890 +.05555556 0 0 No. 0 0 0 67809 Variables +.00666667 +55.555553 +33.333340 10 Maximized objective function = 88.88889 Names S3 S4 S5 S6 A7 Solutions 0+.77777779 +11.111112 +8.8888893 Page: 1 Opportunity costs Iterations = 3 0 0 0 +.22222224 0 +88.888885 No. 12345 Variables Names U1 V1 V2 S1 S2 Summarized results for unit 5 Opportunity costs Solutions +.90909088 +.05681818 +.00681818 +56.818180 +34.090916 0 0 No. 0 0 0 Maximized objective function = 90.90909 67809 O 10 Variables Names S3 S4 S5 S6 A7 Solutions Page: 1 Opportunity costs O +.45454547 +11.363637 +9.0909100 0 0 0+.54545450 0 +90.909088 Iterations = 4 No. 12345 Variables Names U1 V1 V2 S1 S2 Summarized results for unit 6 Solutions +1.0000000 +.06250000 +.00750000 +62.500000 +37.500008 Opportunity costs 0 0 0 0 0 No. 6 7 8 Variables 9 10 Maximized objective function = 100 Names S3 S4 S5 S6 A7 Solutions 0 +12.500000 +10.000001 Page: 1 Iterations = 4 Opportunity costs 0 0 0 0 +1.0000000 0 +100.00000 Microsoft Excel - DEA Burger Palace Ble Edit Yew Insert Format Tools Qata Window Help BV - to 24 PL 1 Variables 2 Value 12 13 14 3 Objective 4 Function E(S1) 10 Constraints 11 Unit 1 Unit 2 Unit 3 15 16 PPRANAANPASARI E4 - A 17 Inputs 18 Nonnegative 19 Nonnegative 20 Nonnegative 21 Solver Parameters 22 26 28 23 24 Equal To: 29 Unit 4 Unit 5 Unit 6 Pol B SetTarget Cl ==100 C2 C U1 0.01 100 U1 100 100 100 100 100 100 0 1 BE14 By Changing Cells: $C$2:$E$2 Subject to the Constraints: $G$11<$1$11 $G$12 <-$1$12 $G$13<$1$13 $G$14<$1$14 $G$15 $1$IS Tacar 0 0 D E VI V2 0.166667 0.003333 KOO -200 -150 -100 -100 NGONO-O -2 -4 -8 -10 2 0 0 Value of: -50 200 0 0 1 Adid Change Delete F |||||| Arial G Value 0 -0.16667 0 -0.33333 -06 -0.83333 1 0.01 Solve Close Options H >= 0.166667 >= 0.003333 > Best All OF << RHS 0 OOOOOO. 0 0 0 0 0 1 OOO 0 0 0 BIU J K Service Unit S₁ S₂ S3 S4 S5 S6 Efficiency Rating (E) 1.000 0.857 1.000 0.889 0.901 1.000 Efficiency Reference Set N.A. S, (0.2857) S3 (0.7143) N.A. S3 (0.7778) S6 (0.2222) S3 (0.4545) S6 (0.5454) N.A. Relative Labor- Hour Value (V₁) 0.1667 0.1428 0.0625 0.0555 0.0568 0.0625 Relative Material Value (V₂) 0.0033 0.0028 0.0075 0.0067 0.0068 0.0075 Outputs and Inputs Meals Labor-hours Material ($) Composite Reference Unit C Reference Set S3 S8 (0.7778) x 100 + (0.2222) x 100 = 100 (0.7778) x 4 + (0.2222) x 10 = (0.7778) x 100 + (0.2222) x 50 = 5.3 88.9 S4 100 6 100 Excess Inputs Used 0 0.7 11.1 High Profit Low Underperforming potential stars Problem branches Low Benchmark group Candidates for divestiture High Efficiency

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> 1. Describe the service organization culture at Amy’s Ice Cream. 2. What are the personality attributes of the employees who are sought by Amy’s Ice Cream? 3. Design a personnel selection procedure for Amy’s Ice Cream using abstract questioning, a situat

> 1.How does Amazon.com illustrate the sources of service sector growth? Comment on information technology, the Internet as an enabler, innovation, and changing demographics. 2.What generic approach(s) to service design does Amazon.com illustrate, and what

> 1. Prepare a service blueprint for Commuter Cleaning. 2. What generic approach to service system design is illustrated by Commuter Cleaning, and what competitive advantages does this design offer? 3. Using the data in Table 3.5, calculate a break-eve

> 1. Describe the growth strategy of Federal Express. How did this strategy differ from those of its competitors? 2. What risks were involved in the acquisition of Tiger International? 3. In addition to the question of merging FedEx and Flying Tigers pilot

> 1. Prepare a service blueprint for the 100 Yen Sushi House operation. 2. What features of the 100 Yen Sushi House service delivery system differentiate it from the competition, and what competitive advantages do they offer? 3. How has the 100 Yen Sushi

> 1. Prepare a run chart on each of the incident categories. Does she have reason to be concerned about burglaries? What variable might you plot against burglaries to create a scatter diagram to determine a possible explanation? 2. What is unusual about th

> Conduct a Google search on “project finance” and find employment opportunities in project finance. What is the role of finance in projects?

> Could firms in the “world-class service delivery” stage of competitiveness be described as “learning organizations’?

> Discuss the difference between time variance, cost variance, and schedule variance.

> Explain why the PERT estimate of expected project duration always is optimistic. Can we get any feel for the magnitude of this bias?

> Are Gantt charts still viable project management tools? Explain.

> Illustrate the four stages of team building from your own experience.

> Give an example that demonstrates the trade-off inherent in projects among cost, time, and performance.

> Identify dependent and independent demand for an airline and a hospital.

2.99

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