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 bounds can be zero or positive. If they are positive, then they indicate that some positive flow must occur on these arcs. Modify the model appropriately to handle these upper and lower bounds. You can make up the required bounds.
> 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 south
> Delko is considering hiring people for four types of jobs. The company would like to hire the number of people listed in the file P05_62.xlsx for each type of job. Delko can hire eight types of people. Each type is qualified to perform two or more types
> At the beginning of year 1, a new machine must be purchased. The cost of maintaining a machine, depending on its age, is given in the file P05_61.xlsx. The cost of purchasing a machine at the beginning of each year is given in this same file. There is no
> Suppose it costs $30,000 to purchase a new car. The annual operating cost and resale value of a used car are shown in the file. Assume that you presently have a new car. Determine a replacement policy that minimizes your net costs of owning and operating
> Assume that the number of units sold of a product is given by 100 2 0.5P 1 26√A, where P is the price (in dollars) charged for the product and A is the amount spent on advertising (in thousands of dollars). Each unit of the product costs $5 to produce. U
> Here is a problem to challenge your intuition. In the original Grand Prix example, reduce the capacity of plant 2 to 300. Then the total capacity is equal to the total demand. Run Solver on this model. You should find that the optimal solution uses all c
> It costs $300 to buy a lawn mower from a lawn supply store. Assume that you can keep a lawn mower for at most five years and that the estimated maintenance cost each year of operation is as follows: year 1, $90; year 2, $135; year 3, $175; year 4, $200;
> Each year, Data Corporal produces up to 10,000 computers in Boston and up to 6000 computers in Charlotte. There are customers in Los Angeles, New York, and Seattle who must receive 5300, 6700, and 3300 computers, respectively. Producing a computer costs
> Edsel Motors produces cars in Detroit and Dallas. The Detroit plant can produce up to 8500 cars, and the Dallas plant can produce up to 4000 cars. Producing a car costs $2000 in Detroit and $1800 in Dallas. Cars must be shipped to 12 cities. The costs of
> 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
> Sunco Oil produces oil at two wells. Well 1 can produce up to 150,000 barrels per day, and well 2 can produce up to 200,000 barrels per day. It is possible to ship oil directly from the wells to Sunco’s customers in Los Angeles and New York. Alternativel
> 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 blank means
> A company 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. The company are shipped by air to customers in Los
> 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 blank indicates that the contractor did not bid on the given job.) Contractor 2
> Based on Machol (1970). A swimming coach 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. The coach believes that ea
> The file gives the annual sales for Microsoft (in millions of dollars) for the years 1984–1993, where 1984 = year 1. a. Fit an exponential curve to these data. b. Assuming you are back in 1993, by what percentage do you estimate that Microsoft has grown
> 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 blank
> In the Grand Prix example with varying tax rates, the optimal solution uses all available plant capacity and more than satisfies customer demands. Will this always be the case? Experiment with the unit selling prices and/or tax rates to see whether the c
> Touche Young has eight auditors. Each can work up to 160 hours during the next month, during which time six projects must be completed. The hours required for each project and the amounts each auditor can be billed for each project are given in the file
> The Amorco Oil Company controls two oil fields. Field 1 can produce up to 22 million barrels of oil per day, and field 2 can produce up to 19 million barrels of oil per day. At field 1, it costs $42.50 to extract and refine a barrel of oil; at field 2 th
> 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 t
> The government is auctioning off oil leases at two sites. At each site, 150,000 acres of land are to be auctioned. Cliff Ewing, Blake Barnes, and Alexis Pickens are bidding for the oil. Government rules statethat no bidder can receive more than 45% of th
> A school system has 16 bus drivers that must cover 12 bus routes. Each driver can cover at most one route. The driver’s bids for the various routes are listed in the file P05_45.xlsx. Each bid indicates the amount the driver will charge the school system
> You 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, you are given the number of students, the mean family income, the percen
> A truck must travel from New York to Los Angeles. As shown, 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 tha
> 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_42.xlsx. The Supreme Court requires the schools in Busville to be racially balanced. Thus, each school must have
> Dataware is trying to determine whether to give a $10 rebate, cut the price $6, or have no price change on a software product. Currently, 40,000 units of the product are sold each week for $45 apiece. The variable cost of the product is $5. The most like
> One rather unrealistic assumption in the flight scheduling 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
> 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 eight planes. Let this percentage vary from 0% to 50% in incre
> In the Grand Prix example with varying tax rates, the optimal solution more than satisfies customer demands. Modify the model so that regions have not only lower limits on the amounts they require, but upper limits on the amounts they can sell. Assume th
> In the flight-scheduling model, use SolverTable to examine the effect of decreasing all net revenues by the same percentage, assuming that the company owns six planes. Let this percentage vary from 0% to 50% in increments of 10%. Discuss the changes that
> We illustrated how a machine replacement problem can be modeled as a shortest path problem. This is probably not the approach most people would think of when they first see a machine replacement problem. In fact, most people would probably never think in
> In the VanBuren machine replacement problem, suppose the company starts with a machine that is eight quarters old at the beginning of the first quarter. Modify the model appropriately, keeping in mind that this initial machine must be sold no more than f
> In the VanBuren machine replacement problem, the company’s current policy is to keep a machine at least four 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 polic
> 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 this policy is instead to keep a machine at least 5 quarters but no more than 10 quarters. Modify the
> 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, a
> In the VanBuren machine replacement problem, we assumed that the maintenance cost and salvage values are linear functions of age. Suppose instead that the maintenance cost increases by 50% each quarter and that the salvage value decreases by 10% each qua
> The file lists sales (in millions of dollars) of Dell Computer during the period 1987–1997 (where year 1 corresponds to 1987). a. Fit a power and an exponential trend curve to these data. Which fits the data better? b. Use your part a answer to predict 1
> Continuing the previous problem, suppose again that all arcs go in both directions, but suppose Maude’s objective is to find the shortest path from node 1 to node 7 (not node 10). Modify the spreadsheet model appropriately and solve.
> 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 10. I
> In Maude’s shortest path problem, suppose all arcs in the network are double-arrowed, that is, Maude can travel along each arc (with the same distance) in either direction. Modify the spreadsheet model appropriately. Is her optimal solution still the sam
> Suppose in the original Grand Prix example that the routes from plant 2 to region 1 and from plant 3 to region 3 are not allowed. (Perhaps there are no railroad lines for these routes.) How would you modify the original model to rule out these routes? Ho
> Continuing the previous problem, develop and optimize a sample model with your own choices of N, M, and L that barely stay within Solver’s limit. You can make up any input data. The important point here is the layout and formulas of the spreadsheet model
> 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 plant
> 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 uni
> 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
> Expand the RedBrand two-product spreadsheet model so that there are now three products competing for the arc capacity. You can make up the required input data.
> Repeat parts a–d of the problem 24 for a six-month European put option with exercise price $40. Again, assume a current stock price of $35, a risk-free rate of 5%, and an annual volatility of 40%. Data from Problem 24: A European call option on a stock
> In the original RedBrand problem, we assume a constant arc capacity, the same for all allowable arcs. Modify the model so that each arc has its own arc capacity. You can make up the required arc capacities.
> In the original RedBrand problem, the costs for shipping from plants or warehouses to customer 2 were purposely made high so that it would be optimal to ship to customer 1 and then let customer 1 ship to customer 2. Use SolverTable appropriately to do th
> In the original RedBrand problem, suppose the plants cannot ship to each other and the customers cannot ship to each other. Modify the model appropriately and reoptimize. How much does the total cost increase because of these disallowed routes?
> The optimal solution to the original Grand Prix problem indicates that with a unit shipping cost of $132, the route from plant 3 to region 2 is evidently too expensive—no autos are shipped along this route. Use SolverTable to see how much this unit shipp
> Modify the original RedBrand problem so that all flows must be from plants to warehouses and from warehouses to customers. Disallow all other arcs. How much does this restriction cost RedBrand, relative to the original optimal shipping cost?
> When we (the authors) originally developed the bus route assignment model, we included an arc capacity constraint: Flow # 1. After giving this further thought, we deleted this constraint as being redundant. Why could we do this? Specifically, why can’t o
> In the optimal solution to the bus route assignment problem, company 2 is assigned to bus routes 6 and 7. Suppose these two routes are far enough apart that it is infeasible for one company to service both of them. Change the model to accommodate this re
> In the optimal solution to the machine-to-job assignment problem, jobs 3 and 4 are assigned to different machines. Suppose there is an extra restriction that these jobs must be assigned to the same machine. Change the model to accommodate this restrictio
> In the optimal solution to the machine-to-job assignment problem, jobs 1 and 2 are both assigned to machine 4. Suppose there is an extra restriction that jobs 1 and 2 must be assigned to different machines. Change the model to accommodate this restrictio
> A European call option on a stock earns the owner an amount equal to the price at expiration minus the exercise price, if the price of the stock on which the call is written exceeds the exercise price. Otherwise, the call pays nothing. A European put opt
> We modeled the bus route assignment problem with the alternative form of the transportation model (as in Figure 5.8). Model it instead with the standard form. Discuss the pros and cons of these two approaches for this particular example.
> Modify the bus route assignment model, assuming that company 1 decides to place bids on routes 7 and 8 (in addition to its current bids on other routes). The bids on these two routes are $5200 and $3300. Does the optimal solution change?
> In the machine-to-job assignment problem, the current capacities of the machines are 1, 2, 1, 2, and 1. If you could increase one of these capacities by 1, which would you increase? Why?
> Modify the machine-to-job assignment model under the assumption that only three of the four jobs must be completed. In other words, one of the four jobs does not have to be assigned to any machine. What is the new optimal solution?
> One possible solution method for the machine-to job assignment problem is the following heuristic procedure. Assign the machine to job 1 that completes job 1 quickest. Then assign the machine to job 2 that, among all machines that still have some capacit
> In the original Grand Prix example, the total capacity of the three plants is 1550, well above the total customer demand. Would it help to have 100 more units of capacity at plant 1? What is the most Grand Prix would be willing to pay for this extra capa
> Budget Auto produces inexpensive cars. Each car is sold for $12,500. The raw material in a car costs $8000. Labor time and robot time are needed to produce cars. A worker can do the needed labor on, at most, 100 cars per month; a robot can complete the n
> Broker Sonya Wong is currently trying to maximize her profit in the bond market. Four bonds are available for purchase and sale at the bid and ask prices shown in the file P04_97.xlsx. Sonya can buy up to 1000 units of each bond at the ask price or sell
> An automobile manufacturing company has a $500,000 advertising budget. To increase its automobile sales, the company is considering advertising in newspapers and on television. The more the company uses a particular medium, the less effective each additi
> Mondo Motorcycles is determining its production schedule for the next four quarters. Demands for motorcycles are forecasted to be 400 in quarter 1; 700 in quarter 2; 500 in quarter 3; and 200 in quarter 4. Mondo incurs four types of costs, as described h
> Georgia McBeal is trying to save for her retirement. She believes she can earn 10% on average each year on her retirement fund. Assume that at the beginning of each of the next 40 years, Georgia will allocate x dollars to her retirement fund. If at the b
> Bexter Labs produces three products: A, B, and C. Bexter can sell up to 2000 units of product A, up to 3000 units of product B, and up to 5000 units of product C. Each unit of product C uses two units of A and three units of B and incurs $7 in processing
> Molecular Products produces three chemicals: B, C, and D. The company begins by purchasing chemical A for a cost of $650 per 100 liters. For an additional cost of $320 and the use of three hours of skilled labor, 100 liters of A can be transformed into 4
> Capsule Drugs manufactures two drugs. The drugs are produced by blending two chemicals. By weight, drug 1 must contain at least 75% chemical 1, and drug 2 must contain at least 45% chemical 1. Drug 1 sells for $10 per ounce, and drug 2 sells for $7 per o
> Based on Magoulas and Marinos-Kouris (1988). An oil company produces two products: regular and premium gasoline. Each product contains 0.15 gram of lead per liter. The two products are produced from these six inputs: reformate, fluid catalytic cracker ga
> You can calculate the risk index of an investment by taking the absolute values of percentage changes in the value of the investment for each year and averaging them. Suppose you are trying to determine the percentages of your money to invest in several
> In the employee scheduling example, suppose that each full-time employee works eight hours per day. Thus, Monday’s requirement of 17 employees can be viewed as a requirement of 8(17) 5 136 hours. The company can meet its daily labor requirements by using
> You own a wheat warehouse with a capacity of 20,000 bushels. At the beginning of month 1, you have 6000 bushels of wheat. Each month, wheat can be bought and sold at the prices per bushel listed in the file. The sequence of events during each month is as
> The Internal Revenue Service (IRS) has determined that during each of the next 12 months it will need the numbers of supercomputers given in the file. To meet these requirements, the IRS rents supercomputers for a period of one, two, or three months. It
> During the next four quarters, Dorian Auto must meet (on time) the following demands for cars: 4000 in quarter 1; 2000 in quarter 2; 5000 in quarter 3; 1000 in quarter 4. At the beginning of quarter 1, there are 300 autos in stock. The company has the ca
> Based on Charnes and Cooper (1955). A small company is trying to determine employee salary based on the following attributes: effectiveness, responsibility, initiative, experience, education, self-expression, planning ability, intelligence, and the abili
> You are thinking of opening a small copy shop. It costs $5000 to rent a copier for a year, and it costs $0.03 per copy to operate the copier. Other fixed costs of running the store will amount to $400 per month. You plan to charge an average of $0.10 per
> AdminaStar processes Medicare claims. At the beginning of month 1 they have a backlog of 40,000 difficult claims and 60,000 easy claims. The predicted claim volume for months 1 through 8 is listed in the file P04_85.xlsx. At the beginning of month 1, Adm
> Based on Brams and Taylor (2000). Suppose that Eli Lilly and Pfizer are going to merge. Merger negotiations must settle the following issues: ■ What will the name of the merged corporation be?\ ■ Will corporate headquarters be in Indianapolis (Lilly want
> Suppose you are planning for retirement. At the beginning of this year and each of the next 39 years, you plan to contribute some money to your retirement fund. Each year, you plan to increase your retirement contribution by $500. When you retire in 40 y
> A graduated payment mortgage (GPM) enables the borrower to have lower payments earlier in the mortgage and increased payments later on. The assumption is the borrower’s income will increase over time so that it will be easier for the borrower to meet all
> You want to take out a $450,000 loan on a 20-year mortgage with end-of-month payments. The annual rate of interest is 3%. Twenty years from now, you will need to make a $50,000 ending balloon payment. Because you expect your income to increase, you want
> During the next two months an automobile manufacturer must meet (on time) the following demands for trucks and cars: month 1, 400 trucks and 800 cars; month 2, 300 trucks and 300 cars. During each month at most 1000 vehicles can be produced. Each truck u
> How much influence can the employee requirements for one, two, or three days have on the weekly schedule in the employee scheduling example? Explore this in the following questions: a. Let Monday’s requirements change from 17 to 25 in increments of 1. Us
> The HiTech company produces Blu-Ray disc players. Estimated demands for the next four quarters are 5000, 10,000, 8000, and 2000. At the beginning of quarter 1, HiTech has 60 workers. It costs $2000 to hire a worker and $4000 to fire a worker. Workers are
> An oil company produces gasoline from five inputs. The cost, density, viscosity, and sulfur content, and the number of barrels available of each input are listed in the file. Gasoline sells for $75 per barrel. Gasoline can have a density of at most 0.95
> An automobile manufacturer needs to plan its production for the next year. Demands for the next 12 months are forecasted to be 940, 790, 360, 720, 270, 130, 160, 300, 990, 290, 280, and 790. Other relevant information is as follows: ■ Workers are paid $5
> You are thinking of opening a Broadway play, I Love You, You’re Mediocre, Now Get Better! It will cost $5 million to develop the show. There are 8 shows per week, and you project the show will run for 100 weeks. It costs $1000 to open the theater each ni
> A rock company uses five types of rocks to fill six orders. The phosphate content, availability of each type of rock, and the production cost per pound for each rock are listed in the file, as well as the size of each order and the minimum and maximum ph
> Based on Gaballa and Pearce (1979). Northwest Airlines has determined that it needs the number of ticket agents during each hour of the day listed in the file. Workers work nine-hour shifts, one hour of which is for lunch. The lunch hour can be either th
> A bank needs exactly two employees working each hour from 9 a.m. to 5 p.m. Workers can work the shifts and are paid the wages listed in the file. For example, a worker working 9 a.m. to 2 p.m. is paid $42.00. Find an assignment of workers that provides e
> A company produces two products: A and B. Product A sells for $11 per unit and product B sells for $23 per unit. Producing a unit of product A requires two hours on assembly line 1 and one unit of raw material. Producing a unit of product B requires two
> Based on Thomas (1971). A toy company produces toys at two plants and sells them in three regions. The current demands at these regions are given in the file. Each plant can produce up to 4500 units. Each toy sells for $30, and the cost of producing and
> Carrington Oil produces gas 1 and gas 2 from two types of crude oil: crude 1 and crude 2. Gas 1 is allowed to contain up to 4% impurities, and gas 2 is allowed to contain up to 3% impurities. Gas 1 sells for $72 per barrel, whereas gas 2 sells for $84 pe
> A feed company produces two types of cattle feed, both consisting totally of wheat and alfalfa. Feed 1 must contain at least 75% wheat, and feed 2 must contain at least 65% alfalfa. Feed 1 sells for $2.50 per pound, and feed 2 sells for $2.10 per pound.
> Modify the employee scheduling model so that employees are paid $10 per hour on weekdays and $15 per hour on weekends. Change the objective so that you now minimize the weekly payroll. (You can assume that each employee works eight hours per day.) Is the
