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operations research an introduction
Operations Research Applications And Algorithms 4th Edition Wayne L. Winston - Solutions
5 Oilco has oil fields in San Diego and Los Angeles. The San Diego field can produce 500,000 barrels per day, and the Los Angeles field can produce 400,000 barrels per day.Oil is sent from the fields to a refinery, in either Dallas or Houston (assume each refinery has unlimited capacity). To refine
3 Fordco produces cars in Detroit and Dallas. The Detroit plant can produce as many as 6,500 cars, and the Dallas plant can produce as many as 6,000 cars. Producing a car costs $2,000 in Detroit and $1,800 in Dallas. Cars must be shipped to three cities. City 1 must receive 5,000 cars, city 2 must
2 a Find the dual of the LP that was used to find the length of the critical path for Example 6 of Section 8.4.b Show that the answer in part (a) is an MCNFP.c Explain why the optimal objective function value for the LP found in part (a) is the longest path in the project network from node 1 to
1 Formulate the problem of finding the shortest path from node 1 to node 6 in Figure 2 as an MCNFP. (Hint: Think of finding the shortest path as the problem of minimizing the total cost of sending 1 unit of flow from node 1 to node 6.)
14 Write a LINGO program that can be used to crash the project network of Example 6 with the crashing costs given in Table 14.
12 Given the information in Table 26, (a) draw the appropriate project network, and (b) find the critical path.
11 Determine the probabilities that 1–2–4 and 1–3–4 are critical paths for Figure 37.
10 A project is complete when activities A–E are completed. The predecessors of each activity are shown in Table 25. Draw the appropriate project diagram. (Hint:Don’t violate rule 4.)
9 Explain why an activity’s free float can never exceed the activity’s total float.
7 When an accounting firm audits a corporation, the first phase of the audit involves obtaining “knowledge of the business.” This phase of the audit requires the activities in Table 23.a Draw the project network and determine the critical path for the network, the total float for each activity,
6 Horizon Cable is about to expand its cable TV offerings in Smalltown by adding MTV and other exciting stations.The activities in Table 22 must be completed before the service expansion is completed.a Draw the project network and determine the critical path for the network, the total float for
4 The promoter of a rock concert in Indianapolis must perform the tasks shown in Table 19 before the concert can be held (all durations are in days).a Draw the project network.b Determine the critical path.c If the advance promoter wants to have a 99% chance of completing all preparations by June
3 Consider the project network in Figure 40. For each activity, you are given the estimates ofa, b, and m in Table 18. Determine the critical path for this network, the total float for each activity, the free float for each activity, and the probability that the project is completed within 40
2 A company is planning to manufacture a product that consists of three parts (A, B, and C). The company anticipates that it will take 5 weeks to design the three parts and to determine the way in which these parts must be assembled to make the final product. Then the company estimates that it will
16 During the next four months, a construction firm must complete three projects. Project 1 must be completed within three months and requires 8 months of labor. Project 2 must be completed within four months and requires 10 months of labor. Project 3 must be completed at the end of two months and
14 Suppose as many as 300 cars per hour can travel between any two of the cities 1, 2, 3, and 4. Set up a maximum-flow problem that can be used to determine how many cars can be sent in the next two hours from city 1 to city 4. (Hint: Have portions of the network represent t 0, t 1, and t 2.)
13 Suppose the total flow into a node of a network is restricted to 10 units or less. How can we represent this restriction via an arc capacity constraint? (This still allows us to use the Ford–Fulkerson method to find the maximum flow.)
12 Consider a network flow problem with several sources and several sinks in which the goal is to maximize the total flow into the sinks. Show how such a problem can be converted into a maximum-flow problem having only a single source and a single sink.
11 Suppose a network contains a finite number of arcs and the capacity of each arc is an integer. Explain why the Ford–Fulkerson method will find the maximum flow in the finite number of steps. Also show that the maximum flow from source to sink will be an integer.
8 The Hatfields, Montagues, McCoys, and Capulets are going on their annual family picnic. Four cars are available to transport the families to the picnic. The cars can carry the following number of people: car 1, four; car 2, three; car 3, three; and car 4, four. There are four people in each
7 Four workers are available to perform jobs 1–4.Unfortunately, three workers can do only certain jobs:worker 1, only job 1; worker 2, only jobs 1 and 2; worker 3, only job 2; worker 4, any job. Draw the network for the maximum-flow problem that can be used to determine whether all jobs can be
6 Seven types of packages are to be delivered by five trucks. There are three packages of each type, and the capacities of the five trucks are 6, 4, 5, 4, and 3 packages, respectively. Set up a maximum-flow problem that can be used to determine whether the packages can be loaded so that no truck
Five male and five female entertainers are at a dance. The goal of the matchmaker is to match each woman with a man in a way that maximizes the number of people who are matched with compatible mates. Table 10 describes the compatibility of the entertainers.Draw a network that makes it possible to
Fly-by-Night Airlines must determine how many connecting flights daily can be arranged between Juneau, Alaska, and Dallas, Texas. Connecting flights must stop in Seattle and then stop in Los Angeles or Denver. Because of limited landing space, Fly-by-Night is limited to making the number of daily
Sunco Oil wants to ship the maximum possible amount of oil (per hour) via pipeline from node so to node si in Figure 6. On its way from node so to node si, oil must pass through some or all of stations 1, 2, and 3. The various arcs represent pipelines of different diameters.The maximum number of
9 A company sells seven types of boxes, ranging in volume from 17 to 33 cubic feet. The demand and size of each box is given in Table 7. The variable cost (in dollars) of producing each box is equal to the box’s volume. A fixed cost of $1,000 is incurred to produce any of a particular box. If the
7 At the beginning of year 1, a new machine must be purchased. The cost of maintaining a machine i years old is given in Table 5.The cost of purchasing a machine at the beginning of each year is given in Table 6.There is no trade-in value when a machine is replaced.Your goal is to minimize the
6 It costs $40 to buy a telephone from the department store. Assume that I can keep a telephone for at most five years and that the estimated maintenance cost each year of operation is as follows: year 1, $20; year 2, $30; year 3,$40; year 4, $60; year 5, $70. I have just purchased a new telephone.
3 Formulate Problem 2 as a transshipment problem.
2 Find the shortest path from node 1 to node 5 in Figure 4.
1 Find the shortest path from node 1 to node 6 in Figure 3.
I have just purchased (at time 0) a new car for $12,000. The cost of maintaining a car during a year depends on its age at the beginning of the year, as given in Table 1. To avoid the high maintenance costs associated with an older car, I may trade in my car and purchase a new car. The price I
28 Three professors must be assigned to teach six sections of finance. Each professor must teach two sections of finance, and each has ranked the six time periods during which finance is taught, as shown in Table 88. A ranking of 10 means that the professor wants to teach that time, and a ranking
27 During the month of July, Pittsburgh resident B. Fly must make four round-trip flights between Pittsburgh and Chicago. The dates of the trips are as shown in Table 87.B. Fly must purchase four round-trip tickets. Without a discounted fare, a round-trip ticket between Pittsburgh and Chicago costs
24 A company can produce as many as 35 units/month.The demands of its primary customers must be met on time each month; if it wishes, the company may also sell units to secondary customers each month. A $1/unit holding cost is assessed against each month’s ending inventory. The relevant data are
23 A firm producing a single product has three plants and four customers. The three plants will produce 3,000, 5,000, and 5,000 units, respectively, during the next time period.The firm has made a commitment to sell 4,000 units to customer 1, 3,000 units to customer 2, and at least 3,000 units to
21 The Carter Caterer Company must have the following number of clean napkins available at the beginning of each of the next four days: day 1—15; day 2—12; day 3—18;day 4—6. After being used, a napkin can be cleaned by one of two methods: fast service or slow service. Fast service costs
20 During each of the next two months you can produce as many as 50 units/month of a product at a cost of $12/unit during month 1 and $15/unit during month 2. The customer is willing to buy as many as 60 units/month during each of the next two months. The customer will pay $20/unit during month 1,
18 During the next three quarters, Airco faces the following demands for air conditioner compressors: quarter 1—200;quarter 2—300; quarter 3—100. As many as 240 air compressors can be produced during each quarter. Production costs/compressor during each quarter are given in Table 78.The cost
16 For the Powerco problem, find the range of values of c23 for which the current basis remains optimal.
15 For the Powerco problem, find the range of values of c24 for which the current basis remains optimal.
14 Oilco has oil fields in San Diego and Los Angeles. The San Diego field can produce 500,000 barrels per day, and the Los Angeles field can produce 400,000 barrels per day.Oil is sent from the fields to a refinery, either in Dallas or in Houston (assume that each refinery has unlimited capacity).
12 Use the northwest corner method, the minimum-cost method, and Vogel’s method to find basic feasible solutions to the transportation problem in Table 73.
11 In Problem 10, suppose we increase si to 16 and d3 to 11. The problem is still balanced, and because 31 units(instead of 30 units) must be shipped, one would think that the total shipping costs would be increased. Show that the total shipping cost has actually decreased by $2, however.This is
10 Find the optimal solution to the balanced transportation problem in Table 72 (minimization).
8 Using the northwest corner method to find a bfs, find(via the transportation simplex) an optimal solution to the transportation (minimization) problem shown in Table 71.
6 The Gotham City police have just received three calls for police. Five cars are available. The distance (in city blocks)of each car from each call is given in Table 69. Gotham City wants to minimize the total distance cars must travel to respond to the three police calls. Use the Hungarian method
5 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; a typical
3 A company must meet the following demands for a product: January, 30 units; February, 30 units; March, 20 units. Demand may be backlogged at a cost of$5/unit/month. All demand must be met by the end of March.Thus, if 1 unit of January demand is met during March, a backlogging cost of 5(2) $10
1 Televco produces TV picture tubes at three plants. Plant 1 can produce 50 tubes per week; plant 2, 100 tubes per week; and plant 3, 50 tubes per week. Tubes are shipped to three customers. The profit earned per tube depends on the site where the tube was produced and on the customer who purchases
5 General Ford has two plants, two warehouses, and three customers. The locations of these are as follows:Plants: Detroit and Atlanta Warehouses: Denver and New York Customers: Los Angeles, Chicago, and Philadelphia Cars are produced at plants, then shipped to warehouses, and finally shipped to
4 Rework Problem 3 under the assumption that Galveston has a refinery capacity of 150,000 barrels per day and Mobile has one of 180,000 barrels per day. (Hint: Modify the method used to determine the supply and demand at each transshipment point to incorporate the refinery capacity restrictions,
2 Sunco Oil produces oil at two wells. Well 1 can produce as many as 150,000 barrels per day, and well 2 can produce as many as 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. Alternatively, Sunco could transport oil to
1 General Ford produces cars at L.A. and Detroit and has a warehouse in Atlanta; the company supplies cars to customers in Houston and Tampa. The cost of shipping a car between points is given in Table 60 (“—” means that a shipment is not allowed). L.A. can produce as many as 1,100 cars, and
10 Suppose cij is the smallest cost in row i and column j of an assignment problem. Must xij 1 in any optimal assignment?
9 Show that step 3 of the Hungarian method is equivalent to performing the following operations: (1) Add k to each cost that lies in a covered row. (2) Subtract k from each cost that lies in an uncovered column.
7 Any transportation problem can be formulated as an assignment problem. To illustrate the idea, determine an assignment problem that could be used to find the optimal solution to the transportation problem in Table 56. (Hint:You will need five supply and five demand points).
5 Greydog Bus Company operates buses between Boston and Washington, D.C. A bus trip between these two cities takes 6 hours. Federal law requires that a driver rest for four or more hours between trips. A driver’s workday consists of two trips: one from Boston to Washington and one from Washington
3 Tom Cruise, Freddy Prinze Jr., Harrison Ford, and Matt LeBlanc are marooned on a desert island with Jennifer Aniston, Courteney Cox, Gwyneth Paltrow, and Julia Roberts. The “compatibility measures” in Table 52 indicate how much happiness each couple would experience if they spent all their
time it takes each person to perform each job is given in Table 50. Determine the assignment of employees to jobs that minimizes the total time required to perform the four jobs.
Machineco has four machines and four jobs to be completed. Each machine must be assigned to complete one job. The time required to set up each machine for completing each job is shown in Table 43. Machineco wants to minimize the total setup time needed to complete the four jobs. Use linear
4 If s3 and d3 are both decreased by 2, what is the new optimal solution?
3 If s2 and d3 are both increased by 3, what is the new optimal solution?
2 Determine the range of values of c34 for which the current basis remains optimal.
1 Determine the range of values of c14 for which the current basis remains optimal.
Use the transportation simplex to solve Problems 1–8 in Section 7.1. Begin with the bfs found in Section 7.2
4 How should Vogel’s method be modified to solve a maximization problem?
3 Use Vogel’s method to find a bfs for Problems 5 and 6 of Section 7.1.
2 Use the minimum-cost method to find a bfs for Problems 4, 7, and 8 of Section 7.1. (Hint: For a maximization problem, call the minimum-cost method the maximumprofit method or the maximum-revenue method.)
1 Use the northwest corner method to find a bfs for Problems 1, 2, and 3 of Section 7.1.
12 Explain how each of the following would modify the formulation of the Sailco problem as a balanced transportation problem:a Suppose demand could be backlogged at a cost of$30/sailboat/month. (Hint: Now it is permissible to ship from, say, month 2 production to month 1 demand.)b If demand for a
10 Touche Young has three auditors. Each can work as many as 160 hours during the next month, during which time three projects must be completed. Project 1 will take 130 hours; project 2, 140 hours; and project 3, 160 hours.The amount per hour that can be billed for assigning each auditor to each
9 For the examples and problems of this section, discuss whether it is reasonable to assume that the proportionality assumption holds for the objective function.
8 The Ayatola 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 $3 to extract and refine a barrel of oil; at field 2, the cost is $2. Ayatola sells oil to two
6 A bank has two sites at which checks are processed. Site 1 can process 10,000 checks per day, and site 2 can process 6,000 checks per day. The bank processes three types of checks: vendor, salary, and personal. The processing cost per check depends on the site (see Table 11). Each day, 5,000
4 Steelco manufactures three types of steel at different plants. The time required to manufacture 1 ton of steel(regardless of type) and the costs at each plant are shown in Table 8. Each week, 100 tons of each type of steel (1, 2, and 3) must be produced. Each plant is open 40 hours per week.a
2 Referring to Problem 1, suppose that extra units could be purchased and shipped to either warehouse for a total cost of $100 per unit and that all customer demand must be met. Formulate a balanced transportation problem to minimize the sum of purchasing and shipping costs.
1 A company supplies goods to three customers, who each require 30 units. The company has two warehouses.Warehouse 1 has 40 units available, and warehouse 2 has 30 units available. The costs of shipping 1 unit from warehouse to customer are shown in Table 7. There is a penalty for each unmet
Sailco Corporation must determine how many sailboats should be produced during each of the next four quarters (one quarter is three months). Demand is as follows: first quarter, 40 sailboats; second quarter, 60 sailboats; third quarter, 75 sailboats; fourth quarter, 25 sailboats.Sailco must meet
Two reservoirs are available to supply the water needs of three cities. Each reservoir can supply up to 50 million gallons of water per day. Each city would like to receive 40 million gallons per day. For each million gallons per day of unmet demand, there is a penalty.At city 1, the penalty is
Powerco has three electric power plants that supply the needs of four cities.† Each power plant can supply the following numbers of kilowatt-hours (kwh) of electricity: plant 1—35 million; plant 2—50 million; plant 3—40 million (see Table 1). The peak power demands in these cities, which
1 Farmer Jones must determine how many acres of corn and wheat to plant this year. An acre of wheat yields 25 bushels of wheat and requires 10 hours of labor per week.An acre of corn yields 10 bushels of corn and requires 4 hours of labor per week. All wheat can be sold at $4 a bushel, and all corn
2 Answer these questions about Problem 1.a Is (x1 = 2, x2 = 3) in the feasible region?b Is (x1= 4, x2 = 3) in the feasible region?c Is (x1 = 2, x2 = -1) in the feasible region?d Is (x1 = 3, x2 = 2) in the feasible region?
3 Using the variables x1 = number of bushels of corn produced and x2 = number of bushels of wheat produced, reformulate Farmer Jones’s LP.
4 Truckco manufactures two types of trucks: 1 and 2.Each truck must go through the painting shop and assembly shop. If the painting shop were completely devoted to painting Type 1 trucks, then 800 per day could be painted;if the painting shop were completely devoted to painting Type 2 trucks, then
5 Why don’t we allow an LP to have< or > constraints?
1 Graphically solve Problem 1 of Section 3.1.
2 Graphically solve Problem 4 of Section 3.1.
3 Leary Chemical manufactures three chemicals: A, B, and C. These chemicals are produced via two production processes: 1 and 2. Running process 1 for an hour costs $4 and yields 3 units of A, 1 of B, and 1 of C. Running process 2 for an hour costs $1 and produces 1 unit of A and 1 of B.To meet
4 For each of the following, determine the direction in which the objective function increases a z = 4x1 - x2 b z=-x+2x2 czx13x2
5 Furnco manufactures desks and chairs. Each desk uses 4 units of wood, and each chair uses 3. A desk contributes $40 to profit, and a chair contributes $25. Marketing restrictions require that the number of chairs produced be at least twice the number of desks produced. If 20 units of wood are
6 Farmer Jane owns 45 acres of land. She is going to plant each with wheat or corn. Each acre planted with wheat yields $200 profit; each with corn yields $300 profit. The labor and fertilizer used for each acre are given in Table 1.One hundred workers and 120 tons of fertilizer are available.Use
An auto company manufactures cars and trucks. Each vehicle must be processed in the paint shop and body assembly shop. If the paint shop were only painting trucks, then 40 per day could be painted. If the paint shop were only painting cars, then 60 per day could be painted. If the body shop were
Suppose that auto dealers require that the auto company in Example 3 produce at least 30 trucks and 20 cars. Find the optimal solution to the new LP.
Graphically solve the following LP: max z= = 2x1 - x2 s.t. x1-x21 2x1 + x2 = 6 X1, X2 0
5 True or false: For an LP to be unbounded, the LP’s feasible region must be unbounded.
6 True or false: Every LP with an unbounded feasible region has an unbounded optimal solution.
7 If an LP’s feasible region is not unbounded, we say the LP’s feasible region is bounded. Suppose an LP has a bounded feasible region. Explain why you can find the optimal solution to the LP (without an isoprofit or isocost line) by simply checking the z-values at each of the feasible
Identify which of Case apply to the following LP: max z=x1 + x2 s.t. x1 + x2 4 x1-x25 X1, X20
Identify which of Case apply to the following LP: max z = 4x1 + x2 s.t. 8x1 + 2x2 16 5x + 2x2 12
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