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practical management science

Practical Management Science, Revised 3rd Edition Wayne L Winston, S. Christian Albright - Solutions

During a week on a production line, each worker produces the number of units of glue shown in the file P06_51.xlsx. Each week, at least 120 units of glue 1, at least 150 units of glue 2, and at least 200 units of glue 3 must be produced. Determine how to minimize the total cost of meeting weekly
For a week of production, it costs $1000 to set up production line 1 and $2000 to set up production line
Glueco produces three types of glue on two different production lines. Each line can be used by up to seven workers at a time. Workers are paid $500 per week on production line 1 and $900 per week on production line
A company is considering opening warehouses in four cities: New York, Los Angeles, Chicago, and Atlanta.Each warehouse can ship 100 units per week. The weekly fixed cost of keeping each warehouse open is$400 for New York, $500 for Los Angeles, $300 for Chicago, and $150 for Atlanta. Region 1 of the
A total of 120 units of raw material are available. If any product 1 is produced, a setup cost of $10 is incurred; if any product 2 is produced, a setup cost of $20 is incurred.a. Determine how to maximize the manufacturer’s profit.b. Use SolverTable to analyze how a change in the setup cost for
A manufacturer can sell product 1 at a profit of $2 per unit and product 2 at a profit of $5 per unit. Three units of raw material are needed to manufacture 1 unit of product 1, and 6 units of raw material are needed to manufacture 1 unit of product
Because of excessive pollution on the Momiss River, the state of Momiss is going to build some pollution control stations. Three sites (1, 2, and 3) are under consideration. Momiss is interested in controlling the pollution levels of two pollutants (1 and 2). The state legislature requires that at
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 radiation that normal tissue can handle without
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 P06_45.xlsx. For example, if asset 1 is sold in year 2, the firm receives $20 million. To maintain a regular cash flow, Milkem must sell at
To graduate from Southeastern University with a major in operations research (OR), a student must complete at least two math courses, at least two OR courses, and at least two computer courses. Some courses can be used to fulfill more than one requirement:Calculus can fulfill the math
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 are given in the file P06_42.xlsx. If the space shuttle can carry up to 2600 pounds of items 1 through 3, which items should be taken on the space shuttle?
I am 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 I am considering moving on the truck are given in the file P06_41.xlsx. Which items should I bring to Indiana?
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: $1.20 per foot for a 64-inch coil and $1.00 per foot for a
Woodco sells 3-foot, 5-foot, and 9-foot pieces of lumber.Woodco’s customers demand 25 3-foot boards, 20 5-foot boards, and 15 9-foot boards. Woodco meets its demands by cutting up 17-foot boards. How can it satisfy its customers’ demands with the least amount of waste? Assume that all boards
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? For example, given the solution in Figure 6.28, this waste includes 2 inches from pattern 6, 12 inches from the extra 12-inch roll produced (in cell
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 service from Dallas and the other half from San
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 so that you can see the effect on the plants
In the Green Grass shipping model, we assumed that certain inputs (see the range B4:B7 in Figure 6.24) 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 shipping
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 locations are in Memphis, Houston, Cleveland,
How hard is it to expand a set-covering model to accommodate new cities? Answer this by modifying the model in Figure 6.18. (See the file LocatingHubs2.xlsx.)Add several cities that must be served: Memphis, Dallas, Tucson, Philadelphia, Cleveland, and Buffalo.You can look up the distances from
Set-covering models such as the original Western model in Figure 6.15 often have multiple optimal solutions.See how many alternative optimal solutions you can find. Of course, each must use three hubs because we know this is optimal. (Hint: Use various initial values in the changing cells and then
In the original Western set-covering model in Figure 6.15, 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 reasonable fixed costs, modify the
In the original Western set-covering model in Figure 6.15, 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 optimal solution change?
Modify the model appropriately and use Solver to find the optimal solution.
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, except that the small and large batch limits
In the last sheet of the file Fixed Cost Manufacturing.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 altogether, and eliminate the upper bounds in row 18 and the
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 we instead used a very large value such as 1,000,000 for this upper bound? Try it and see
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? (Note: You will have to uncheck the Assume Linear Model box.)
Modify the model as necessary and use Solver to find the new optimal solution.How do you know that the current optimal solution is not optimal for the modified model?
If large minivans are produced, the production level must be from 200 to
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 production level of
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 SolverTable that captures all changing cells and the target cell as outputs. You can choose the ranges of these inputs to make the results
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 put a lower bound of 300 on the production
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 its current value. (You can choose the range
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 Solver. Are the optimal integer values the
Make up an example, as described in Problem 7, with 20 possible investments. However, do it so the ratios of NPV to cash requirement are in a very tight range, from 3 to 3.2. Then use Solver to find the optimal solution when the Solver tolerance is set to its default value of 5%, and record the
The models in this chapter are often called combinatorial models because each solution is a combination of the various 0’s and 1’s, and only a finite number of such combinations exist. For the capital budgeting model in Figure 6.1, there are 7 investments, so there are 27 128 possible
Does your answer to part a change?Skill-Extending Problems
Suppose in the capital budgeting model in Figure 6.1 that each investment requires $2000 during year 2, and only $5000 is available for investment during year 2.a. Assuming that available money uninvested at the end of year 1 cannot be used during year 2, what combination of investments maximizes
Expand and then solve the capital budgeting model in Figure 6.1 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 can vary widely, but the ratio
As it currently stands, investment 7 in the capital budgeting model in Figure 6.1 has the lowest ratio of NPV to cash requirement, 2.5. Keeping this same ratio, 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
In the capital budgeting model in Figure 6.1, investment 4 has the largest ratio of NPV to cash requirement, 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.
Solve Problem 2 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 investments when at least one investment
Solve the previous problem using the input data in the file P06_03.xlsx.
In the capital budgeting model in Figure 6.1, we supplied the NPV for each investment. Suppose instead that you are given only the streams of cash inflows from each investment shown in the file P06_02.xlsx.This file also shows the cash requirements and the budget. You can assume that (1) all cash
Solve the following modifications of the capital budgeting model in Figure 6.1. (Solve each part independently of the others.)a. Suppose that at most two of projects 1 through 5 can be selected.b. Suppose that if investment 1 is selected, then investment 3 must also be selected.c. Suppose that at
In the crew-scheduling model in Example 5.7, there are only two cities. Suppose there are more than two cities. Is it possible to modify the network approach appropriately? Discuss how you would do it.
Powerhouse produces capacitors at three locations:Los Angeles, Chicago, and New York. Capacitors are shipped from these locations to public utilities in five regions of the country: northeast (NE), northwest(NW), midwest (MW), southeast (SE), and southwest(SW). The cost of producing and shipping a
I have put four valuable paintings up for sale. Four customers are bidding for the paintings. Customer 1 is willing to buy two paintings, but each other customer is willing to purchase at most one painting.The prices that each customer is willing to pay are given in the file P05_81.xlsx. Determine
Both customers 3 and 4 also want to buy as many of the remaining units as possible. The profit associated with shipping a unit from each plant to each customer is given in the file P05_80.xlsx. Determine how to maximize Kellwood’s profit.
Kellwood, a company that produces a single product, has three plants and four customers. The three plants will produce 3000, 5000, and 5000 units, respectively, during the next time period. Kellwood has made a commitment to sell 4000 units to customer 1, 3000 units to customer 2, and at least 3000
Based on Jacobs (1954). The Carter Caterer Company must have the following number of clean napkins available at the beginning of each of the next 4 days:day 1, 1500; day 2, 1200; day 3, 1800; day 4, 600.After being used, a napkin can be cleaned by one of two methods: fast service or slow service.
In the original RedBrand problem (Example 5.4), suppose that the company could add up to 100 tons of capacity, in increments of 10 tons, to any single plant.Use SolverTable to determine the yearly savings in cost from having extra capacity at the various plants.Assume that the capacity will cost
Based on Ravindran (1971). A library must build shelving to shelve 200 4-inch-high books, 100 8-inchhigh books, and 80 12-inch-high books. Each book is 0.5 inch thick. The library has several ways to store the books. For example, an 8-inch-high shelf can be built to store all books of height less
Eight students need to be assigned to four dorm rooms at Faber College. Based on “incompatibility measurements,”the cost incurred for any pair of students rooming together is shown in the file P05_76.xlsx. How should the students be assigned to the four rooms to minimize the total
At present, 500 long-distance calls must be routed from New York to Los Angeles (L.A.), and 400 calls must be routed from Philadelphia to L.A. On route to L.A. from Philadelphia or New York, calls are sent through Indianapolis or Cleveland, then through Dallas or Denver, and finally to L.A. The
Rework the previous problem under the assumption that Galveston has a refinery capacity of 150,000 barrels per day and Mobile has a refinery capacity of 180,000 barrels per day.
In Problem 55, assume that before being shipped to Los Angeles or New York, all oil produced at the wells must be refined at either Galveston or Mobile. To refine 1000 barrels of oil costs $5780 at Mobile and $6250 at Galveston. Assuming that both Mobile and Galveston have infinite refinery
Bloomington has two hospitals. Hospital 1 has four ambulances, and hospital 2 has two ambulances.Ambulance service is deemed adequate if there is only a 10% chance that no ambulance will be available when an ambulance call is received by a hospital. The average length of an ambulance service call
Based on Evans (1984). Currently, State University can store 200 files on hard disk, 100 files in computer memory, and 300 files on tape. Users want to store 300 word-processing files, 100 packaged-program files, and 100 data files. Each month a typical word processing file is accessed eight times;
Let the “cost” of merging an SCI record with a CPS record be ISCI ICPS FSSCI FSCPS where ISCI and ICPS are the incomes from the SCI and CPS records, and FSSCI and FSCPS are the family sizes.Determine the least expensive way to merge the SCI and CPS records.
Based on Glover and Klingman (1977). The government has many computer files that must be merged frequently. For example, consider the Survey of Current Income (SCI) and the Consumer Price Service(CPS) files, which keep track of family income and family size. The breakdown of records in each file is
Referring to the previous problem, suppose that Allied Freight can purchase and ship extra units to either warehouse for a total cost of $100 per unit and that all customer demand must be met. Determine how to minimize the sum of purchasing and shipping costs.
We are trying to help the MCSCC (Monroe County School Corporation) determine the appropriate high school district for each housing development in Bloomington. For each development, we are given the number of students, the mean family income, the percentage of minorities, and the distance to each
A truck must travel from New York to Los Angeles. As shown in Figure 5.46, several routes are available. The number associated with each arc is the number of gallons of fuel required by the truck to traverse the arc.Determine the route from New York to Los Angeles that uses the minimum amount of
The town of Busville has three school districts. The numbers of black students and white students in each district are shown in the file P05_62.xlsx. The Supreme Court requires the schools in Busville to be racially balanced. Thus, each school must have exactly 300 students, and each school must
It costs $200 to buy a lawn mower from a lawn supply store. Assume that I can keep a lawn mower for at most 5 years and that the estimated maintenance cost each year of operation is as follows: year 1, $50; year 2, $80; year 3, $140; year 4, $160; year 5, $180. I have just purchased a new lawn
Suppose it costs $10,000 to purchase a new car. The annual operating cost and resale value of a used car are shown in the file P05_59.xlsx. Assume that you presently have a new car. Determine a replacement policy that minimizes your net costs of owning and operating a car for the next six years.
Each year, Data Corporal produces up to 400 computers in Boston and up to 300 computers in Raleigh. Los Angeles customers must receive 400 computers, and 300 computers must be supplied to Austin customers.Producing a computer costs $350 in Boston and $400 in Raleigh. Computers are transported by
Fordco produces cars in Detroit and Dallas. The Detroit plant can produce up to 6500 cars, and the Dallas plant can produce up to 6000 cars. Producing a car costs $2000 in Detroit and $1800 in Dallas. Cars must be shipped to three cities. City 1 must receive 5000 cars, city 2 must receive 4000
It costs $10,000 to produce a car at each plant. The costs of shipping a car between various cities are listed in the file P05_56.xlsx. Assume that during a week, at most 50 cars can be shipped from a warehouse to any particular city. Determine how to meet Nash’s weekly demands at minimum cost.
Nash Auto has two plants, two warehouses, and three customers. The plants are in Detroit and Atlanta, the warehouses are in Denver and New York, and the customers are in Los Angeles, Chicago, and Philadelphia.Cars are produced at plants, then shipped to warehouses, and finally shipped to customers.
General Ford produces cars in Los Angeles and Detroit and has a warehouse in Atlanta. The company supplies cars to customers in Houston and Tampa.The costs of shipping a car between various points are listed in the file P05_54.xlsx, where a dash means that a shipment is not allowed. Los Angeles can
Widgetco manufactures widgets at two factories, one in Memphis and one in Denver. The Memphis factory can produce up to 150 widgets per day, and the Denver factory can produce up to 200 widgets per day. Widgets are shipped by air to customers in Los Angeles and Boston. The customers in each city
A company is taking bids on four construction jobs.Three contractors have placed bids on the jobs. Their bids (in thousands of dollars) are given in the file P05_52.xlsx. (A dash indicates that the contractor did not bid on the given job.) Contractor 1 can do only one job, but contractors 2 and 3
Based on Machol (1970). Doc Councilman is putting together a relay team for the 400-meter relay. Each swimmer must swim 100 meters of breaststroke, backstroke, butterfly, or freestyle, and each swimmer can swim only one race. Doc believes that each swimmer will attain the times given in the file
Five employees are available to perform four jobs.The time it takes each person to perform each job is given in the file P05_50.xlsx. Determine the assignment of employees to jobs that minimizes the total time required to perform the four jobs. (A dash indicates that a person cannot do that
Touche Young has three auditors. Each can work up to 160 hours during the next month, during which time three projects must be completed. Project 1 takes 130 hours, project 2 takes 140 hours, and project 3 takes 160 hours. The amount per hour that can be billed for assigning each auditor to each
The Amorco Oil Company controls two oil fields.Field 1 can produce up to 40 million barrels of oil per day, and field 2 can produce up to 50 million barrels of oil per day. At field 1, it costs $37.50 to extract and refine a barrel of oil; at field 2 the cost is $41.20.Amorco sells oil to two
At each site, 100,000 acres of land are to be auctioned. Cliff Ewing, Blake Barnes, and Alexis Pickens are bidding for the oil. Government rules state that no bidder can receive more than 40% of the land being auctioned. Cliff has bid $1000 per acre for site 1 land and $2000 per acre for site 2
The government is auctioning off oil leases at two sites: 1 and
The 7th National Bank has two check-processing sites.Site 1 can process 10,000 checks per day, and site 2 can process 6000 checks per day. The bank processes three types of checks: vendor, salary, and personal.The processing cost per check depends on the site, as listed in the file P05_46.xlsx.
In the crew-scheduling model, there are exactly as many flights departing from Chicago as departing from New York. Suppose more flights are departing from one city than from the other. How would you model this? Illustrate by assuming that there is an extra flight from Chicago that leaves at 11 A.M.
Modify the model so that there must be at least 1 hour of downtime between consecutive flights.
One rather unrealistic assumption in the flightscheduling model is that a given plane can fly two consecutive flights with no downtime. For example, it could fly flight 5903 that gets into Washington, D.C.at time 14 and then fly flight 7555 that leaves Washington, D.C. at time
In the flight-scheduling model, use SolverTable to examine the effect of increasing both the fixed cost per plane and the overnight cost by the same percentage, assuming that the company owns 8 planes. Let this percentage vary from 0% to 50% in increments of 10%. Discuss the changes that occur in
In the crew-scheduling problem, suppose that two extra flights are added to the current list. The first leaves Chicago at 11 A.M. and arrives in New York at 2 P.M. The second leaves New York at 6 P.M. and arrives in Chicago at 8 P.M. (Remember that all quoted times are EST.) Modify the model to
The required downtime in the crew-scheduling problem is currently assumed to be 1 hour. Suppose we instead require it to be 2 hours. How does the model need to be modified? What is the new optimal solution?
In the crew-scheduling problem, suppose (as in the sensitivity analysis we discussed) that the first Chicago flight, C1, is delayed by 2 hours—that is, its departure and arrival times move up to 8 A.M. and 12 P.M., respectively. How does the model need to be modified? What is the new optimal
In the VanBuren machine replacement problem, the company’s current policy is to keep a machine at least 4 quarters but no more than 12 quarters.Suppose instead that the company imposes no upper limit on how long it will keep a machine; its only policy requirement is that a machine must be kept at
Does the new optimal solution go through node 11?
Assume that Maude’s objective is still to get from node 1 to node
How difficult is it to add nodes and arcs to an existing shortest path model? Answer by adding a new node, node 11, to Maude’s network. Assume that node 11 is at the top of the network, geographically, with double-arrowed arcs joining it to nodes 2, 5, and 7 with distances 45, 22, and
Is it the same as before? Should you have known the answer to this question before making any changes to the original model? Explain.
In Maude’s shortest path problem, suppose all arcs in the current network from higher-numbered nodes to lower-numbered nodes, such as from node 6 to node 5, are disallowed. Modify the spreadsheet model and find the shortest path from node 1 to node
Consider a modification of the original RedBrand problem where there are N plants, M warehouses, and L customers. Assume that the only allowable arcs are from plants to warehouses and from warehouses to customers. If all such arcs are allowable—all plants can ship to all warehouses and all
In the RedBrand problem with shrinkage, change the assumptions. Now instead of assuming that there is some shrinkage at the warehouses, assume that there is shrinkage in delivery along each route. Specifically, assume that a certain percentage of the units sent along each arc perish in
How difficult is it to expand the original RedBrand model? Answer this by adding a new plant, two new warehouses, and three new customers, and modify the spreadsheet model appropriately. You can make up the required input data.
In the RedBrand two-product problem, we assumed that the unit shipping costs are the same for both products. Modify the spreadsheet model so that each product has its own unit shipping costs. You can assume that the original unit shipping costs apply to product 1, and you can make up new unit
Continuing the previous problem, make the problem even more general by allowing upper bounds (arc capacities)and lower bounds for the flows on the allowable arcs. Some of the upper bounds can be very large numbers, effectively indicating that there is no arc capacity for these arcs, and the lower

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