Question: Modify the function in Example 8.1

> Use a two-way PrecisionTree sensitivity analysis to examine the changes in both of the two previous problems simultaneously. Let the probability of technological success vary from 0.6 to 0.9 in increments of 0.05, and let the fixed cost of development va

> The fixed costs are split $4 million for development and $2 million for marketing. Perform a sensitivity analysis where the sum of these two fixed costs remains at $6 million but the split changes. Specifically, let the fixed cost of development vary fro

> In the product mix model it makes sense to change the maximum sales constraint to a “minimum sales” constraint, simply by changing the direction of the inequality Then the input values in row 23 can be considered customer demands that must be met. Make t

> Acme’s probability of technological success, 0.8, is evidently large enough to make “continue development” the best decision. How low would this probability have to be to make the opposite decision best?

> Use PrecisionTree to solve problem 7 of the previous section. Data from Problem 7: Sometimes a “single-stage” decision can be broken down into a sequence of decisions, with no uncertainty resolved between these decisions. Similarly, uncertainty can some

> Several decision criteria besides EMV are suggested in the section. For each of the following criteria, rank all three decisions in Figure 9.1 from best to worst. a. Look only at the worst possible outcome for each decision. b. Look only at the best poss

> You are given the following information concerning how a change in sales force effort impacts sales: ■ A 50% cut in sales force effort reduces sales to 48% of its current value. ■ Sales force effort of 0 reduces sales to 15% of its current value. ■ A 50%

> In the lawn mower production problem, experiment with the penalty cost for unsatisfied pickups in week 1. If this cost is sufficiently small, does the company ever produce fewer than seven models in week 1 and allow some week 1 pickups to be unsatisfied?

> In the lawn mower production problem, the model changeover cost dominates in the optimal objective value. Is this because we assumed such a large unit penalty cost, 200, for each model changeover? Explore this question by changing this unit penalty cost

> In the truck-loading problem, we assumed that any product could be loaded into any compartment. Suppose the following are not allowed: product 1 in compartment 2, product 2 in compartment 1, and product 3 in compartment 4. Modify the model appropriately,

> Based on Schrage (1997). The file P08_05.xlsx lists the size of the four main markets for Excel, Word, and the bundle of Excel and Word. (We assume that Microsoft is willing to sell Excel or Word separately, and it is willing to sell a package with Excel

> Suppose you are the new supply manager at FedEx. You need to choose three hubs for the company. Each of 28 cities will send all of its outgoing packages to one of the hubs. The packages will then be sent from the hubs to their final destinations. The fil

> Determine the optimal pricing policy if Menthos are sold in only a one-pack or a six-pack.

> Modify PC Tech’s product mix model so that there is no maximum sales constraint. (This is easy to do in the Solver dialog box. Just highlight the constraint and click the Delete button.) Does this make the problem unbounded? Does it change the optimal so

> A company has nine jobs that must be assigned to three ordered workstations. The file P08_39.xlsx lists the times required for each job, which are independent of the workstations they are assigned to. It also lists precedence relationships between the jo

> Based on Meneses et al. (2004). A string is a list of characters such as “1differ%”. The length of the string is the number of characters in the string. The distance between two strings is the number of positions in which the two strings differ. For exam

> A common approach to clustering is called multidimensional scaling (MDS). To apply MDS, we rank each pair of objects we want to cluster from least similar (higher number) to most similar (lower number). For example, in the file, we compared the similarit

> An important problem in manufacturing is the assembly line balancing problem. When setting up a manufacturing line, activities must be assigned to workstations. The maximum time spent at a workstation is called the cycle time. Minimizing the cycle time t

> Assume that a consumer’s purchase decision on an electric razor is based on four attributes, each of which can be set at one of three levels (1, 2, or 3). Using conjoint analysis (a type of analysis used in marketing research), our analysts have divided

> Suppose you are the ad manager for Fox NFL football. Thirty bids for ads on today’s game between the Packers and the Colts have been submitted. Information on these ads is given in the file P08_34.xlsx. For example, ad 1 is 23 seconds in length and will

> A steel manufacturer needs to cool 17 pieces of steel. The weight and due date for each piece are listed in the file P08_33.xlsx. Processing and cooling a batch in the furnace takes five minutes regardless of the weight in the furnace. The furnace can ha

> You are the Democratic campaign manager for the state of Indiana. There are 15 fairly large cities in the state of Indiana. The numbers of Democrats and Republican voters in these cities (in thousands) are listed in the file P08_32.xlsx. The Democrats co

> Xerox is trying to determine how many maintenance centers are needed in the mid-Atlantic states. Xerox earns $500 profit (excluding the cost of running maintenance centers) on each copier sale. The sales of copiers in each major market (Boston, New York,

> Ten data sets must be assigned for storage to one of three disk drives. Each disk drive can store 150 GB (about 150,000 MB). The sizes of the data sets (in MB) are listed in the file P08_30.xlsx. When many people access a disk drive, there is a significa

> Other sensitivity analyses besides those discussed could be performed on the product mix model. Use SolverTable to perform each of the following. In each case keep track of the values in the decision variable cells and the objective cell, and discuss you

> Determine the best pricing policy if quantity discounts with a single-price breakpoint are used.

> Bus 99 serves towns 1 through 10. We assume that town k is |k – j| miles from town j. The numbers of people in the towns who want to take the bus each hour are listed in the file P08_29.xlsx. Bus 99 will make two stops and anyone who wants to take the bu

> During the next 12 months, the amounts of electric power needed (in thousands of kwh) are listed in the file. This power can be supplied using four generators. The generating capacity (in thousands of kwh), the operating cost, the startup cost, and the s

> A Wall Street firm is trying to package nine mortgages for sale. The sizes of the mortgages (in thousands of dollars) are listed in the file. To be sold, each package must consist of at least $1,000,000 in mortgages. What is the largest number of package

> Music radio WABC has commercials of the following lengths (in seconds): 15, 15, 20, 25, 30, 35, 40, 57. The commercials must be assigned to 60-second breaks. What is the fewest number of breaks that are needed to air all of the commercials?

> The file contains quarterly revenue for a sporting goods company over an 8-year period. It also contains quarterly “indicator” variables Q1, Q2, and Q3. Here Q1 is 1 for the first quarter of a fiscal year (July–September) and 0 otherwise. Q2 and Q3 are d

> Cook County needs to build two hospitals. There are nine cities where the hospitals can be built. The number of hospital visits made annually by the inhabitants of each city and the x and y coordinates of each city are listed in the file P08_24.xlsx. To

> The costs of producing product A, product B, or products A and B bundled together are $50, $90, and $140, respectively. The file P08_23.xlsx lists the sizes of the three market segments for these products and how much each of the segments is willing to p

> Eight students need to be assigned to four dorm rooms (two students to a room) at State University. Based on incompatibility measures, the “cost” incurred if two students room together is shown in the file. How should these students be assigned to rooms

> Nine jobs need to be completed within eight weeks. The number of weeks required to complete each job is given in the file P08_21.xlsx. For example, job 2 requires five weeks. Each job requires 40 hours of labor per week. Each week, 160 hours of regular t

> In a production scheduling problem like Pigskin’s, suppose the company must produce several products to meet customer demands. Would it suffice to solve a separate model for each product, as we did for Pigskin, or would one big model for all products be

> Fourteen jobs must be assigned to one of three identical machines. The goal is to minimize the total time needed to complete all 14 jobs. The machine capacities and times needed for the jobs are given in file P08_20.xlsx. For example, job 8 requires thre

> Modify the function in Example 8.1 so that it becomes f(x) = x sin(x) for 0 ≤ x ≤ 30. (Here, sin(x) is the sine function from trigonometry. You can evaluate it with Excel’s SIN function.) Plot a lot of points from 0 to 30 to see what the graph of this fu

> The 30 teams in the NBA are each assigned to one of six divisions, where each division has five teams. Suppose the goal is to assign the teams to divisions so that the average distance among teams in the divisions is minimized. In other words, the goal i

> You are operating a Web site to match up sellers and buyers of a product. 35 sellers and 35 buyers have input their reservation prices, as listed in the file. For example, buyer 1 is willing to pay up to $8 for an item, and seller 1 is willing to accept

> Repeat Problem 15, but now assume there is a setup time for changing from any job to another job, and this setup time can depend on the jobs. For example, the setup time when changing from job 2 to job 4 can be different from the setup time when changing

> The traveling salesperson problem is notoriously difficult when the number of cities is even of moderate size. The file contains two sheets, one with a distance matrix for a 30-city problem and the other with a distance matrix for a 45-city problem. See

> An important problem in operations management is the job sequencing problem. Actually, there are many versions of this problem, but they all basically attempt to find the proper sequencing of jobs on a machine. Here is one version of the problem. There a

> Continuing the previous problem, determine the portfolio that minimizes the chance that you will lose money during any month, subject to a lower bound constraint on your expected monthly return. (The lower bound will depend on your data. It must not be a

> Visit biz.yahoo.com/r/. Under Research Tools, click on Historical Quotes, and then download the monthly returns on at least four stocks for the preceding 60 months. Use this data to determine the portfolio that maximizes the chance of beating the S&P 500

> Sales of a product over time often follow an S-shaped curve. Two functions that yield S-shaped curves are the Pearl (or logistic) curve Y = L / + ae-bt and the Gompertz curve Y = Le-be-kt Here, Y is annual sales, t is time (in years), L is the upper lim

> In a typical production scheduling model like Pigskin’s, if there are no production capacity constraints—the company can produce as much as it needs in any time period—but there are storage capacity constraints and demand must be met on time, is it possi

> The adoption level of a new product can often be modeled as an S-shaped curve called the Pearl (or logistic) curve. The equation of this curve is Y = L / + ae-bt where Y is the adoption level, L is an (unknown) upper limit on adoptions, t is time (in ye

> The file contains per capita data on annual advertising and annual unit sales in different regions of the country. Determine an S-shaped curve as described by Equation that can be used to determine how advertising influences sales.

> In the complementary-product pricing model, the elasticity of demand for suits is currently 22.5. Use SolverTable to see how the optimal price of suits and the optimal profit vary as the elasticity varies from 22.7 to 21.8 in increments of 0.1. Are the r

> Your family owns a large farm that can grow wheat, corn, cotton, alfalfa, barley, pears, and apples. Each product requires a certain amount of labor each month and a certain number of hours of machine time. You have just studied portfolio optimization an

> UE is going to invest $400 million to acquire companies in the auto and/or electronics industry. How would you apply portfolio optimization to determine which companies should be purchased?

> Consider the sports ratings model. If you were going to use the approach used there to forecast future sports contests, what problems might you encounter early in the season? How might you resolve these problems?

> Consider the sports ratings model. If you were going to give more recent games more weight, how might you determine whether the weight given to a game from k weeks ago should be, say, (0.95)k or (0.9)k?

> For the aggregate planning example, is it likely that the cost per worker of changing the size of the workforce during a month would be constant (as we assumed)? How could an NLP model account for a situation in which the cost per worker of changing the

> For the oil blending example, discuss where you think the assumptions of a linear model are most likely to break down. How might an NLP model look in this situation?

> In a typical product mix model, where a company must decide how much of each product to produce to maximize profit, there are sometimes customer demands for the products. We used upper-bound constraints for these: Don’t produce more than you can sell. Wo

> In some ordering problems, like the one for Sam’s Bookstore, whenever demand exceeds existing inventory, the excess demand is not lost but is filled by expedited orders—at a premium cost to the company. Change Sam’s model to reflect this behavior. Assume

> For the product mix examples, discuss where you think the assumptions of a linear model are most likely to break down. How might an NLP model look in this situation?

> In the exchange rate model, we found that the optimal unit revenue, when converted to dollars, is $85.71. Now change the problem so that the company is selling in Japan, not the United Kingdom. Assume that the exchange rate is 0.00965 ($/¥) and that the

> Your company is about to market a new golf club. You have convened a focus group of 100 golfers and asked them to compare your club to the clubs produced by your competitors. You have found, for example, that 30 customers in the focus group would purchas

> A triangle has a 5-inch side and a 12-inch side. To maximize the area of the triangle what should the third side be? Can you generalize this result?

> You can swim two miles per hour and run six miles per hour. You are walking north along South Beach and see someone drowning half a mile out in the ocean and one mile north of you. What combination of running and swimming is the quickest way to reach the

> City B is 10 miles downstream from city A. City A is 5 miles south of the river, and city B is 20 miles north of the river. The river is two miles wide. Where should a bridge be built across the river to make the travel distance between cities A and B as

> A cylindrical soda can has a volume of 20 cubic inches. What height and diameter minimize the surface area of the can? Can you generalize this result?

> You are given that the two non-hypotenuse sides of a right triangle add up to 10 inches. What is the maximum area of the triangle? Can you generalize this result?

> Find the minimum perimeter rectangle having area 64 square feet. Can you generalize this result?

> Kellpost Cereal Company sells four products: (1) Special L (a low-calorie, high-nutrition cereal); (2) Corn Bran (another low-calorie, high-nutrition cereal); (3) Admiral Smacks (a sugary cereal pitched at the children’s market); and (4) Honey Pops (anot

> In a typical product mix model, where a company must decide how much of each product to produce to maximize profit, discuss possible situations where there might not be any feasible solutions. Could these be realistic? If you had such a situation in your

> Based on Grossman and Hart (1983). A salesperson for Fuller Brush has three options: (1) quit, (2) put forth a low level of effort, or (3) put forth a high level of effort. Suppose for simplicity that each salesperson will sell $0, $5000, or $50,000 wort

> Each morning during rush hour, 10,000 people want to travel from New Jersey to New York City. If a person takes the commuter train, the trip lasts 40 minutes. If x thousand people per morning drive to New York, it takes 20 1 5x minutes to make the trip.

> Change the exchange rate model slightly so that the company is now a UK manufacturing company producing for a U.S. market. Assume that the unit cost is now £75, the demand function has the same parameters as before (although the price for this demand fun

> Most economies have a goal of maximizing the average consumption per period. Assume that during each year, an economy saves the same (to be determined) percentage S of its production. During a year in which the beginning capital level is K, a quantity K1

> Four items are for sale in the Dollar Value store. The product and sum of their prices is $7.11. What is the price of each item?

> The cost of producing x units of a product during a month is x1/2 dollars. Show that the minimum cost method of producing 40 units during the next two months is to produce all 40 units during a single month.

> Suppose that you are hiring a weather forecaster to predict the probability that next summer will be rainy or sunny. The following suggests a method that can be used to ensure that the forecaster is accurate. Suppose that the actual probability of next s

> An oil company must determine how many barrels of oil to extract during each of the next two years. If the company extracts x1 million barrels during year 1, each barrel can be sold for 80 - x1 dollars. If the company extracts x2 million barrels during y

> You have $50,000 to invest in three stocks. Let Ri be the random variable representing the annual return on $1 invested in stock i. For example, if Ri = 0.12, then $1 invested in stock i at the beginning of a year is worth $1.12 at the end of the year. T

> Consider three investments. You are given the following means, standard deviations, and correlations for the annual return on these three investments. The means are 0.12, 0.15, and 0.20. The standard deviations are 0.20, 0.30, and 0.40. The correlation b

> Suppose you have a linear optimization model where you are trying to decide which products to produce to maximize profit. What does the additive assumption imply about the profit objective? What does the proportionality assumption imply about the profit

> Monroe County is trying to determine where to place the county fire station. The locations of the county’s four major towns are as follows: (10, 20), (60, 20), (40, 30), and (80, 60) (see Figure 7.50). Town 1 averages 40 fires per year;

> A company has five factories. The x- and y-coordinates of the location of each factory are given in the file. The company wants to locate a warehouse at a point that minimizes the sum of the squared distances of the plants from the warehouse. Where shoul

> Based on Kolesar and Blum (1973). Suppose that a company must service customers lying in an area of A square miles with n warehouses. Kolesar and Blum showed that when the warehouse(s) are located properly, the average distance between a warehouse and a

> In the exchange rate model, suppose the company continues to manufacture its product in the United States, but now it sells its product in the United States, the United Kingdom, and possibly other countries. The company can independently set its price in

> J&J has given you $12 million to spend on advertising Huggys diapers during the next 12 months. At the beginning of January, Huggys has a 30% market share. During any month, 10% of the people who purchase Huggys defect to brand X, and a fraction 0.2a1/2

> A company uses raw material to produce two products. For $175, a unit of raw material can be purchased and processed into 3 units of product 1 and 4 units of product 2. If x1 units of product 1 are produced, they can be sold at 300 - x1 dollars per unit.

> A beer company has divided Bloomington into two territories. If the company spends x1 dollars on promotion in territory 1, it can sell 60x1/21 cases of beer there; and if it spends x2 dollars on promotion in territory 2, it can sell 40x1/22 cases of beer

> Q&H Company advertises during soap operas and football games. Each soap opera ad costs $75,000, and each football game ad costs $130,000. If S soap opera ads are purchased, they will be seen by 4S1/2 million men and 22S1/2 million women. If F football ad

> A brewing company has $100,000 to spend on advertising in four markets. The sales revenue (in thousands of dollars) that can be created in each market by spending xi thousand dollars in market i is given in the file. a. To maximize its sales revenue, how

> A company manufactures two products. If it charges price pi for product i, it can sell qi units of product i, where q1 = 80 - 2p1 + p2 and q2 = 100 - 3p2 + p1. It costs $8 to produce a unit of product 1 and $15 to produce a unit of product 2. How many un

> Why is it generally necessary to add nonnegativity constraints to an optimization model? Wouldn’t Solver automatically choose nonnegative values for the decision variable cells?

> Two companies are producing widgets. It costs the first company q21 dollars to produce q1 widgets and the second company 0.75q22 dollars to produce q2 widgets. If a total of q widgets are produced, consumers will pay 300 - q dollars for each widget. If

> The cost per day of running a hospital is 300,000 + 0.75x2 dollars, where x is the number of patients served per day. What number of patients served per day minimizes the cost per patient per day of running the hospital if the hospital’s daily capacity i

> In the previous problem, what is the minimum-cost method of producing 100 machines? (Ignore the $10,000 budget constraint.)

> It costs a company $15 to purchase an hour of labor and $20 to purchase an hour of capital. If L hours of labor and K units of capital are available, then 0.075L2/3K1/3 machines can be produced. Suppose the company has $150,000 to purchase labor and capi

> Continuing Problem 3 in a slightly different direction, create a two-way SolverTable where the inputs are the elasticity and the production capacity, and the outputs are the optimal price and the optimal profit. (This actually creates two tables, one for

> If a monopolist produces q units, she can charge 400 - 4q dollars per unit. The variable cost is $60 per unit. a. How can the monopolist maximize her profit? b. If the monopolist must pay a sales tax of 5% of the selling price per unit, will she increase

> A company produces widgets at plants 1 and 2. It costs 135x1/2 dollars to produce x units at plant 1 and 260x1/3 dollars to produce x units at plant 2. Each plant can produce up to 1000 units. Each unit produced can be sold for $12. At most 1700 widgets

> Another way to derive a demand function is to break the market into segments and identify a low price, a medium price, and a high price. For each of these prices and market segments, we ask company experts to estimate product demand. Then we use Excel’s

> Suppose the current exchange rate is 115 yen per dollar. We currently have a demand for 50 units of our product when the unit price is 800 yen. The cost of producing and shipping the product to Japan is $6, and the current elasticity of demand is -2.5. F