Question: The Cubs are trying to determine which

> State University is scheduling 24 sections of a large computer skills course in the Fall semester. There are eight time slots for these sections, four on Monday/ Wednesday (MW) and four on Tuesday/Thursday (TR). In each time slot, three sections are sche

> Based on Spencer et al. (1990). When you lease 800 phone numbers from AT&T for telemarketing, AT&T uses an optimization model to tell you where you should locate calling centers to minimize your operating costs over a 10-year horizon. To illustrate the m

> It is currently the beginning of year 1. Gotham City is trying to sell municipal bonds to support improvements in recreational facilities and highways. The face values of the bonds and the due dates at which principal comes due are listed in the file P06

> A county is going to build two hospitals. There are nine cities in which the hospitals can be built. The number of hospital visits per year made by people in each city and the x-y coordinates of each city are listed in the file P06_83.xlsx. The county’s

> Four trucks are available to deliver milk to five grocery stores. The capacity and daily operating cost of each truck are shown in the file P06_82xlsx. The demand of each grocery store can be supplied by only one truck, but a truck can deliver to more th

> Newsome Construction is considering erecting three office buildings. The time (in years) required to complete each of them and the number of workers required to be on the job at all times are shown in the file P06_81.xlsx. After a building is completed,

> Based on Muckstadt and Wilson (1968). PSI believes it will need the amounts of generating capacity (in millions of kwh) shown in the file P06_80.xlsx during the next five years. The company has a choice of building (and then operating) power plants with

> Suppose the demand (in thousands) for a toaster is given by 100p22, where p is the price in dollars charged for the toaster. a. If the variable cost of producing a toaster is $10, what price maximizes profit? b. The elasticity of demand is defined as the

> As the Quality Sweaters problem is now modeled, if all inputs remain fixed except for the number mailed, profit will increase indefinitely as the number mailed increases. This hardly seems realistic—the company could become infinitely rich. Discuss reali

> Expand and then solve the capital budgeting model so that 20 investments are now possible. You can make up the data on cash requirements, NPVs, and the budget, but use the following guidelines: ■ The cash requirements and NPVs for the various investments

> Arthur Ross, Inc., must complete many corporate tax returns during the period February 15 to April 15. This year, the company must begin and complete the five jobs shown in the file P06_79.xlsx during this eight-week period. Arthur Ross employs four full

> Based on Eaton et al. The city of Springfield has been divided into eight districts. The time (in minutes) it takes an ambulance to travel from one district to another is shown in the file. This file also lists the populations of the districts. Suppose S

> On Monday morning, you have $5000 in cash on hand. For the next seven days, the following cash requirements must be met: Monday, $5000; Tuesday, $6000; Wednesday, $9000; Thursday, $2000; Friday, $7000; Saturday, $2000; Sunday, $3000. At the beginning of

> Based on Efroymson and Ray (1966). Stonecutters is a new bakery chain that sells bread to customers throughout the state of Indiana. Stonecutters is considering building bakeries in three locations: Evansville, Indianapolis, and South Bend. Each bakery c

> Indiana University’s Business School has two rooms that seat 50 students, one room that seats 100 students, and one room that seats 150 students. Classes are held five hours a day. At present, the four types of requests for rooms are listed in the file P

> Based on Bean et al. (1988). Simon’s Mall has 9000 square feet of space to rent and wants to determine the types of stores that should occupy the mall. The minimum number and maximum number of each type of store (along with the square footage of each typ

> Cousin Bruzie of radio station WABC schedules radio commercials in 60-second blocks. This hour, the station has sold time for commercials of 15, 16, 20, 25, 30, 35, 40, and 50 seconds. Determine the minimum number of 60-second blocks of commercials that

> Suppose you own 15 bronze coins worth a total of $150, 12 silver coins worth a total of $160, and 8 gold coins worth a total of $170. Develop a linear integer model to find a combination of coins worth exactly $110.

> Based on Fitzsimmons and Allen (1983). The State of Texas frequently audits companies doing business in Texas. Because these companies often have headquarters located outside the state, auditors must be sent to out-of-state locations. Each year, auditors

> You are thinking of starting Peaco, which will produce Peakbabies, a product that competeswith Ty’s Beanie Babies. In year 0 (right now), you will incur costs of $4 million to build a plant. In year 1, you expect to sell 80,000 Peakbabies for a unit pric

> Based on Zangwill (1992). Murray Manufacturing runs a day shift and a night shift. Regardless of the number of units produced, the only production cost during a shift is a setup cost. It costs $8000 to run the day shift and $6000 to run the night shift.

> Change the capital budgeting model so that 80% of each cost in row 5 is incurred right now, at the beginning of year 1, and the other 20% is incurred a year from now, at the beginning of year 2. You can assume that the NPVs don’t change. The available bu

> Based on Boykin (1985). Thompson Chemicals produces 359 million pounds of the chemical maleic anhydride annually. A total of four reactors are available to produce maleic anhydride. Each reactor can be run on one of three settings. The cost (in thousands

> Based on Westerberg, Bjorklund, and Hultman (1977). Newcor’s steel mill has received an order for 25 tons of steel. The steel must be 5% carbon and 6% molybdenum by weight. The steel is manufactured by combining three types of metal: steel ingots, scrap

> Based on Liggett (1973). A court decision has stated that the enrollment of each high school in Metropolis must be at least 20% black. The numbers of black students and white students in each of the city’s five school districts are listed. The distance (

> Spencer Electric ships 17,000 capacitors per month to its customers. The capacitors can be produced at three different plants. The production capacity, fixed monthly cost of operation, and variable cost of producing a capacitor at each plant are given in

> At Blair General Hospital, six types of surgical operations are performed. The types of operations each surgeon is qualified to perform (indicated by an X) are listed in the file P06_65.xlsx. Suppose that surgeons 1 and 2 dislike each other and cannot be

> State University must purchase 1200 computers from three vendors. Vendor 1 charges $250 per computer plus a total delivery charge of $5000. Vendor 2 charges $300 per computer plus a total delivery charge of $4000. Vendor 3 charges $350 per computer plus

> Based on Walker (1974). The Smalltown Fire Department currently has seven conventional ladder companies and seven alarm boxes. The two closest ladder companies to each alarm box are listed in the file. The town council wants to maximize the number of con

> Eight jobs need to be completed. Each job can be completed on any of six machines, and each machine can complete any number of jobs. If a machine is assigned to at least one job, the setup time listed in the file is required. (All times are in hours.) Th

> Suppose you are borrowing $25,000 and making monthly payments with 1% interest. Show that the monthly payments should equal $556.11. The key relationships are that for any month t (Ending month t balance) = (Ending month t - 1 balance) - ((Monthly paymen

> Bertallo Classic produces tomato sauce at five different plants. The tomato sauce is then shipped to one of three warehouses, where it is stored until it is shipped to one of the company’s four customers. The following inputs for the problem are given in

> Ford has four automobile plants. Each is capable of producing the Focus, Mustang, or Taurus, but it can produce only one of these cars. The fixed cost of operating each plant for a year and the variable cost of producing a car of each type at each plant

> As it currently stands, investment 7 in the capital budgeting model has the lowest ROI. Keeping this same ROI, can you change the cash requirement and NPV for investment 7 so that it is selected in the optimal solution? Does this lead to any general insi

> At a machine tool plant, five jobs must be completed each day. The time it takes to do each job depends on the machine used to do the job. If a machine is used at all, a setup time is required. The relevant times (in minutes) are given in the file P06_59

> During the next five periods, the demands listed in the file must be met on time. At the beginning of period 1, the inventory level is 0. During each period when production occurs, a setup cost of $10,000 and a per-unit production cost of $45 are incurre

> Eastinghouse sells air conditioners. The annual demand for air conditioners in each region of the country is as follows: East, 100,000; South, 150,000; Midwest, 110,000; and West, 90,000. Eastinghouse is considering building its air conditioners in four

> Toshovo Computer owns four production plants at which computer workstations are produced. The company can sell up to 40,000 computers per year at a price of $1500 per computer. For each plant, the production capacity, the production cost per computer, an

> McPherson Publishers is considering publishing five textbooks. The maximum number of copies of each textbook that can be sold, the variable cost of producing each textbook, the selling price of each textbook, and the fixed cost of a production run for ea

> Consider the Pigskin example. Find Pigskin’s optimal production policy if, in addition to the given production and holding costs, there is a fixed cost of $50,000 during any month in which there is positive production. Assume now that storage capacity is

> A product can be produced on four different machines. Each machine has a fixed setup cost, variable production cost per unit processed, and a production capacity, given in the file P06_53.xlsx. A total of 2000 units of the product must be produced. Deter

> You are entering the widget business. It costs $500,000, payable in year 1, to develop a prototype. This cost can be depreciated on a straight-line basis during years 1 to 5. Each widget sells for $40 and incurs a variable cost of $20. During year 1, the

> Avion Computer produces two types of computers: Pear computers and Apricot computers. The relevant data are given in the file P06_52.xlsx. The equipment cost is a fixed cost that is incurred if any of this type of computer is produced. A total of 4000 ch

> The TightBond company produces three types of glue on two different production lines. Each line can be used by up to 20 workers at a time. Workers are paid $500 per week on production line 1 and $900 per week on production line 2. For a week of productio

> A company is considering opening warehouses in four cities: New York, Los Angeles, Chicago, and Atlanta. Each warehouse can ship 15,000 units per week. The weekly fixed cost of keeping each warehouse open is $60,000 for New York, $50,000 for Los Angeles,

> In the capital budgeting model, investment 4 is tied for the largest ROI, but it is not selected in the optimal solution. How much NPV will be lost if Tatham is forced to select investment 4? Answer by solving a suitably modified model.

> A manufacturer can sell product 1 at a profit of $20 per unit and product 2 at a profit of $40 per unit. Three units of raw material are needed to manufacture one unit of product 1, and six units of raw material are needed to manufacture one unit of prod

> Because of excessive pollution on the Momiss River, the state of Momiss is going to build some pollution control stations. Three sites are under consideration. Momiss is interested in controlling the pollution levels of two pollutants. The state legislat

> Based on Sonderman and Abrahamson (1985). In treating a brain tumor with radiation, physicians want the maximum amount of radiation possible to bombard the tissue containing the tumors. The constraint is, however, that there is a maximum amount of radiat

> Based on Bean et al. (1987). Boris Milkem’s firm owns six assets. The expected selling price (in millions of dollars) for each asset is given in the file. For example, if asset 1 is sold in year 2, the firm receives $20 million. To maintain a regular cas

> To graduate from Southeastern University with a major in operations research (OR), a student must complete at least three math courses, at least three OR courses, and at least two computer courses. Some courses can be used to fulfill more than one requir

> The payback of a project is the number of years it takes before the project’s total cash flow is positive. Payback ignores the time value of money. It is interesting, however, to see how differing assumptions on project growth impact payback. Suppose, fo

> Coach Night is trying to choose the starting lineup for the basketball team. The team consists of seven players who have been rated on a scale of 1 (poor) to 3 (excellent) according to their ball handling, shooting, rebounding, and defensive abilities. T

> NASA must determine how many of three types of objects to bring on board the space shuttle. The weight and benefit of each of the items. If the space shuttle can carry up to 2000 pounds of items 1 through 3, how many of each item should be taken on the s

> You are moving from New Jersey to Indiana and have rented a truck that can haul up to 1100 cubic feet of furniture. The volume and value of each item you are considering moving on the truck. Which items should you bring to Indiana?

> You are given a group of possible investment projects for your company’s capital. For each project, you are given the NPV the project would add to the firm, as well as the cash outflow required by each project during each year. Given the information in t

> Solve the following modifications of the capital budgeting model. a. Suppose that at most two of projects 3, 5, and 6 can be selected. b. Suppose that if investment 5 is selected, then investment 1 must also be selected. c. Suppose that at least one of i

> Four projects are available for investment. The projects require the cash flows and yield the net present values (in millions) shown in the file. If $6 million is available now for investment, find the investment plan that maximizes NPV.

> The Mayfree Appliance Company requires sheet metal for its appliances. The company can purchase long coils of sheet metal in two different widths: 65 inches and 40 inches. The company must purchase the coils by linear foot of length: $1.20per foot for a

> A company sells 3-foot, 5-foot, and 9-foot pieces of lumber. The company’s customers demand 25 3-foot boards, 20 5-foot boards, and 15 9-foot boards. The company meets its demands by cutting up 17-foot boards. How can it satisfy its customers’ demands wi

> In the cutting stock example, we minimized the total number of rolls cut. Do you get the same solution if you minimize the total inches of waste? Solve the problem with this objective.

> In the Green Grass shipping model, we currently assume that if a customer’s order is satisfied, it must be satisfied from a single plant. Suppose instead that it can be satisfied from more than one plant. For example, if the company decides to satisfy Da

> A software company is considering translating its program into French. Each unit of the program sells for $50 and incurs a variable cost of $10 to produce. Currently, the size of the market for the product is 300,000 units per year, and the English versi

> In the Green Grass shipping model, we assumed that the plants are already built, so that in each month, the only decision is whether to open particular plants (at a monthly fixed cost). Consider instead a general location-shipping model of this type wher

> In the United Copiers service center model, we assumed that a customer is serviced totally by a single service center. Suppose a customer can be serviced partly by multiple service centers. For example, the customer in Denver could get half of its servic

> In the Green Grass shipping model, use SolverTable to perform a sensitivity analysis on the fixed cost of opening a plant, letting it vary over some reasonable range that extends below and above the current value of $60,000. Keep track of enough outputs

> In the optimal solution to the Green Grass shipping model, the Miami customer’s order is not satisfied. Suppose that Green Grass decides, as a matter of policy, to satisfy each customer’s order (at the customer’s bid price). How much profit will the comp

> In the Green Grass shipping model, we assumed that certain inputs are the same for all plants or plant–customer combinations. Change this so that the unit production cost, the monthly fixed cost, and the monthly capacity can vary by plant, and the unit s

> Solve Problem 1 with the extra assumption that the investments can be grouped naturally as follows: 1–4, 5–8, 9–12, 13–16, and 17–20. a. Find the optimal investments when at most one investment from each group can be selected. b. Find the optimal investm

> In the United Copiers service center model, we used total distance traveled as the objective to minimize. Suppose in addition that there is an annual fixed cost of locating a service center in any city, where this fixed cost can vary across cities. There

> In the United Copiers service center model, we assumed that the potential locations of service centers are the same as existing customer locations. Change the model so that the customer locations are the ones given, but the only potential service center

> How hard is it to expand a set-covering model to accommodate new cities? Answer this by modifying the model. Add several cities that must be served: Memphis, Dallas, Tucson, Philadelphia, Cleveland, and Buffalo. You can look up the distances from these c

> Set-covering models such as the original Western model often have multiple optimal solutions. See how many alternative optimal solutions you can find. Of course, each must use three hubs because this is optimal.

> Consider a project with the following cash flows: year 1, 2$400; year 2, $200; year 3, $600; year 4, 2$900; year 5, $1000; year 6, $250; year 7, $230. Assume a discount rate of 15% per year. a. Find the project’s NPV if cash flows occur at the ends of th

> In the original Western set-covering model, we assumed that each city must be covered by at least one hub. Suppose that for added flexibility in flight routing, Western requires that each city must be covered by at least two hubs. How do the model and op

> In the original Western set-covering model, we used the number of hubs as the objective to minimize. Suppose instead that there is a fixed cost of locating a hub in any city, where these fixed costs can possibly vary across cities. Make up some reasonabl

> Suppose in the Dorian production model that no minimum production limits are placed on the individual vehicle types. However, minimum production limits are placed on all cars and on all minivans. Specifically, if Dorian produces any cars, regardless of s

> In the Dorian production model, suppose that the production quantity of compact cars must either be less than or equal to 100 (a small batch) or greater than or equal to 1000 (a large batch). The same statements hold for the other vehicle types as well,

> In the last sheet of the file Fixed Cost Manufacturing Finished.xlsx, we illustrated one way to model the Great Threads problem with IF functions that didn’t work. Try a slightly different approach here. Eliminate the binary variables in row 14 altogethe

> In the Great Threads model, we found an upper bound on production of any clothing type by calculating the amount that could be produced if all of the resources were devoted to this clothing type. a. What if you instead used a very large value such as 1,0

> Solve the problem 1 using the input data in the file P06_02.xlsx. Data from Problem 1: In the capital budgeting model in Figure 6.5, we supplied the NPV for each investment. Suppose instead that you are given only the streams of cash inflows from each i

> If Solver could handle IF functions correctly, how would you use them in the Dorian production example to create an arguably more natural model—without binary variables? Run Solver on your modified model. Do you get the correct solution?

> The optimal solution to the Dorian production model appears to be sensitive to the model inputs. For each of the following inputs, create a one-way Solver Table that captures all decision variable cells and the objective cell as outputs. You can choose t

> As the Dorian production model is currently stated, each vehicle type has a minimum production level; if this type is produced at all, its production quantity must be at least this minimum. Suppose that for large minivans, there is also a maximum product

> Assume the demand for a company’s drug Wozac during the current year is 50,000, and assume demand will grow at 5% a year. If the company builds a plant that can produce x units of Wozac per year, it will cost $16x. Each unit of Wozac is sold for $3. Each

> In the Dorian production model, the optimal solution calls for the minimum number of compact cars and midsize minivans to be produced, but for more than the minimum number of large minivans to be produced. If the large minivans are evidently that profita

> In the optimal solution to the Great Threads model, no pants are produced. Suppose Great Threads has an order for 300 pairs of pants that must be produced. Modify the model appropriately and use Solver to find the new optimal solution. (Is it enough to p

> In the optimal solution to the Great Threads model, the labor hour and cloth constraints are both binding—the company is using all it has. a. Use SolverTable to see what happens to the optimal solution when the amount of available cloth increases from it

> In the Great Threads model, we didn’t constrain the production quantities in row 16 to be integers, arguing that any fractional values could be safely rounded to integers. See whether this is true. Constrain these quantities to be integers and then run S

> Referring to the previous problem, if it is optimal for the company to produce sweatshirts, use SolverTable to see how much larger the fixed cost of sweatshirt machinery would have to be before the company would not produce any sweatshirts. However, if t

> How difficult is it to expand the Great Threads model to accommodate another type of clothing? Answer by assuming that the company can also produce sweatshirts. The rental cost for sweatshirt equipment is $1100; the variable cost per unit and the selling

> Telco, a national telemarketing firm, usually picks a number of sites around the country from which it makes its calls. As a service, AD&D’s telecommunication marketing department wants to help Telco choose the number and location of its sites. How can I

> The models in this chapter are often called combinatorial models because each solution is a combination of the various 0–1 values, and only a finite number of such combinations exist. For the capital budgeting model in Figure 6.5, there are seven investm

> In the capital budgeting model in Figure 6.5, we supplied the NPV for each investment. Suppose instead that you are given only the streams of cash inflows from each investment. This file also shows the cash requirements and the budget. You can assume th

> Modify the original Grand Prix example as follows. Increase the demands at the regions by 200 each, so that total demand is well above total plant capacity. This means that some demands cannot be supplied. Suppose there is a unit penalty cost at each reg

> A bond is currently selling for $1040. It pays the amounts listed in the file at the ends of the next six years. The yield of the bond is the interest rate that would make the NPV of the bond’s payments equal to the bond’s price. Use Excel’s Goal Seek to

> Unlike the small logistics models presented here, real-world logistics problems can be huge. Imagine the global problem a company like FedEx faces each day. Describe as well as you can the types of decisions and constraints it has. How large (number of d

> What is the relationship between transportation models and more general logistics models? Explain how these two types of linear optimization models are similar and how they are different.

> “It is essential to constrain all shipments in a transportation problem to have integer values to ensure that the optimal LP solution consists entirely of integer-valued shipments.” Is this statement true or false? Why?

> You have been assigned to ensure that each high school in the Indianapolis area is racially balanced. Explain how you would use a network model to help attain this goal.