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Introduction to Operations Research 10th edition Frederick S. Hillier, Gerald J. Lieberman - Solutions

Reconsider the assignment problem formulation of Option 2 for the Better Products Co. problem presented in Table 9.29. Suppose that the cost of having Plant 1 produce product 1 is reduced from 820 to 720. Solve this problem by manually applying the Hungarian algorithm. (You may use the
Manually apply the Hungarian algorithm (perhaps using the corresponding interactive procedure in your IOR Tutorial) to solve the assignment problem having the following cost table:
Manually apply the Hungarian algorithm (perhaps using the corresponding interactive procedure in your IOR Tutorial) to solve the assignment problem having the following cost table:
Manually apply the Hungarian algorithm (perhaps using the corresponding interactive procedure in your IOR Tutorial) to solve the assignment problem having the following cost table:
Alabama Atlantic is a lumber company that has three sources of wood and five markets to be supplied. The annual availability of wood at sources 1, 2, and 3 is 15, 20, and 15 million board feet, respectively. The amount that can be sold annually at markets 1, 2, 3, 4, and 5 is 11, 12, 9, 10, and 8
The supplement to this chapter on the book’s website presents a case study of how the Texago Corp. solved many transportation problems to help make its decision regarding where to locate its new oil refinery. Management now needs to address the question of whether the capacity of the new refinery
Reconsider the P & T Co. problem presented in Sec. 9.1. You now learn that one or more of the shipping costs per truckload given in Table 9.2 may change slightly before shipments begin. Use Solver to generate the Sensitivity Report for this problem. Use this report to determine the allowable range
This case focuses on a series of applications of the assignment problem for a pharmaceutical manufacturing company. The decision has been made to undertake five research and development projects to attempt to develop new drugs that will treat five specific types of medical ailments. Five senior
The Onenote Co. produces a single product at three plants for four customers. The three plants will produce 60, 80, and 40 units, respectively, during the next time period. The firm has made a commitment to sell 40 units to customer 1, 60 units to customer 2, and at least 20 units to customer 3.
The Move-It Company has two plants producing forklift trucks that then are shipped to three distribution centers. The production costs are the same at the two plants, and the cost of shipping for each truck is shown for each combination of plant and distribution center:A total of 60 forklift trucks
Redo Prob. 9.1-7 when any distribution center may receive any quantity between 10 and 30 forklift trucks per week in order to further reduce total shipping cost, provided only that the total shipped to all three distribution centers must still equal 60 trucks per week. In problem (a) Formulate this
The MJK Manufacturing Company must produce two products in sufficient quantity to meet contracted sales in each of the next three months. The two products share the same production facilities, and each unit of both products requires the same amount of production capacity. The available production
Consider the following directed network.(a) Find a directed path from node A to node F, and then identify three other undirected paths from node A to node F. (b) Find three directed cycles. Then identify an undirected cycle that includes every node. (c) Identify a set of arcs that forms a spanning
The Wirehouse Lumber Company will soon begin logging eight groves of trees in the same general area. Therefore, it must develop a system of dirt roads that makes each grove accessible from every other grove. The distance (in miles) between every pair of groves is as follows:Management now wishes to
The Premiere Bank soon will be hooking up computer terminals at each of its branch offices to the computer at its main office using special phone lines with telecommunications devices.The phone line from a branch office need not be connected directly to the main office. It can be connected
For the network shown below, use the augmenting path algorithm described in Sec. 10.5 to find the flow pattern giving the maximum flow from the source to the sink, given that the arc capacity from node i to node j is the number nearest node i along the arc between these nodes. Show your work.
Formulate the maximum flow problem as a linear programming problem.
The next diagram depicts a system of aqueducts that originate at three rivers (nodes R1, R2, and R3) and terminate at a major city (node T), where the other nodes are junction points in the system.Using units of thousands of acre feet, the tables below the diagram show the maximum amount of water
The Texago Corporation has four oil fields, four refineries, and four distribution centers. A major strike involving the transportation industries now has sharply curtailed Texago€™s capacity to ship oil from the oil fields to the refineries and to ship petroleum products from the
One track of the Eura Railroad system runs from the major industrial city of Faireparc to the major port city of Portstown. This track is heavily used by both express passenger and freight trains. The passenger trains are carefully scheduled and have priority over the slow freight trains (this is a
Consider the maximum flow problem shown below, where the source is node A, the sink is node F, and the arc capacities are the numbers shown next to these directed arcs.(a) Use the augmenting path algorithm described in Sec. 10.5 to solve this problem.(b) Formulate and solve a spreadsheet model for
Read the referenced article that fully describes the OR study summarized in the application vignette presented in Sec. 10.6. Briefly describe how the model for the minimum cost flow problem was applied in this study. Then list the various financial and nonfinancial benefits that resulted from this
Reconsider the maximum flow problem shown in Prob. 10.5- 6. Formulate this problem as a minimum cost flow problem, including adding the arc A → F. Use F–bar = 20.
You need to take a trip by car to another town that you have never visited before. Therefore, you are studying a map to determine the shortest route to your destination. Depending on which route you choose, there are five other towns (call them A, B, C, D, E) that you might pass through on the way.
A company will be producing the same new product at two different factories, and then the product must be shipped to two warehouses. Factory 1 can send an unlimited amount by rail to warehouse 1 only, whereas factory 2 can send an unlimited amount by rail to warehouse 2 only. However, independent
Reconsider Prob. 10.3-3. Now formulate this problem as a minimum cost flow problem by showing the appropriate network representation.
The Makonsel Company is a fully integrated company that both produces goods and sells them at its retail outlets. After production, the goods are stored in the company€™s two warehouses until needed by the retail outlets. Trucks are used to transport the goods from the two plants to the
The Audiofile Company produces boomboxes. However, management has decided to subcontract out the production of the speakers needed for the boomboxes. Three vendors are available to supply the speakers. Their price for each shipment of 1,000 speakers is shown below.In addition, each vendor would
Consider the minimum cost flow problem shown below, where the bi values (net flows generated) are given by the nodes, the cij values (costs per unit flow) are given by the arcs, and the uij values (arc capacities) are given between nodes C and D. Do the following work manually.
Reconsider the minimum cost flow problem formulated in Prob. 10.6-2.
Reconsider the minimum cost flow problem formulated in Prob. 10.6-3.
Reconsider the minimum cost flow problem formulated in Prob. 10.6-4. Starting with the initial BF solution that corresponds to replacing the tractor every year, use the network simplex method yourself (you may use the interactive procedure in your IOR Tutorial) to solve this problem.
For the P & T Co. transportation problem given in Table 9.2, consider its network representation as a minimum cost flow problem presented in Fig. 9.2. Use the northwest corner rule to obtain an initial BF solution from Table 9.2. Then use the network simplex method yourself (you may use the
Consider the Metro Water District transportation problem presented in Table 9.12. (a) Formulate the network representation of this problem as a minimum cost flow problem.
At a small but growing airport, the local airline company is purchasing a new tractor for a tractor-trailer train to bring luggage to and from the airplanes. A new mechanized luggage system will be installed in 3 years, so the tractor will not be needed after that. However, because it will receive
Consider the minimum cost flow problem shown below, where the bi values are given by the nodes, the cij values are given by the arcs, and the finite uij values are given in parentheses by the arcs. Obtain an initial BF solution by solving the feasible spanning tree with basic arcs A → C, B →A,
The Tinker Construction Company is ready to begin a project that must be completed in 12 months. This project has four activities (A, B, C, D) with the project network shown next.The project manager, Sean Murphy, has concluded that he cannot meet the deadline by performing all these activities in
Reconsider the Tinker Construction Co. problem presented in Prob. 10.8-1. While in college, Sean Murphy took an OR course that devoted a month to linear programming, so Sean has decided to use linear programming to analyze this problem. (a) Consider the upper path through the project network.
Good Homes Construction Company is about to begin the construction of a large new home. The company€™s president, Michael Dean, is currently planning the schedule for this project. Michael has identified the five major activities (labeled A, B, . . . , E) that will need to be performed
The 21st Century Studios is about to begin the production of its most important (and most expensive) movie of the year. The movie€™s producer, Dusty Hoffmer, has decided to use PERT/CPM to help plan and control this key project. He has identified the eight major activities (labeled A, B,
The Lockhead Aircraft Co. is ready to begin a project to develop a new fighter airplane for the U.S. Air Force. The company€™s contract with the Department of Defense calls for project completion within 92 weeks, with penalties imposed for late delivery.The project involves 10 activities
Jake Nguyen runs a nervous hand through his once finely combed hair. He loosens his once perfectly knotted silk tie. And he rubs his sweaty hands across his once immaculately pressed trousers.Today has certainly not been a good day.Over the past few months, Jake had heard whispers circulating from
A rebel army is attempting to overthrow the elected government of the Russian Federation. The United States government has decided to assist its ally by quickly sending troops and supplies to the Federation. A plan now needs to be developed for shipping the troops and supplies most effectively.
The management of a privately held company has made the decision to go public. Many interrelated steps need to be completed in the process of making the initial public offering of stock in the company. Management wishes to accelerate this process. Therefore, after you construct a project network to
Use the algorithm described in Sec. 10.3 to find the shortest path through each of the following networks, where the numbers represent actual distances between the corresponding nodes.(a)(b)
Formulate the shortest-path problem as a linear programming problem.
One of Speedy Airlines€™ flights is about to take off from Seattle for a nonstop flight to London. There is some flexibility in choosing the precise route to be taken, depending upon weather conditions. The following network depicts the possible routes under consideration, where SE and LN
The Quick Company has learned that a competitor is planning to come out with a new kind of product with a great sales potential. Quick has been working on a similar product that had been scheduled to come to market in 20 months. However, research is nearly complete and Quick€™s management
Read the referenced article that fully describes the OR study summarized in the application vignette presented in Sec. 10.3. Briefly describe how network optimization models were applied in this study. Then list the various financial and nonfinancial benefits that resulted from this study.
Reconsider the networks shown in Prob. 10.3-4. Use the algorithm described in Sec. 10.4 to find the minimum spanning tree for each of these networks.In problem(a)(b)
Consider the following network, where each number along a link represents the actual distance between the pair of nodes connected by that link. The objective is to find the shortest path from the origin to the destination.(a) What are the stages and states for the dynamic programming formulation of
Consider the following integer nonlinear programming problem. Maximize Z = 18x1 – x21 + 20x2 + 10x3, Subject to 2x1 + 4x2 + 3x3 ≤ 11 and x1, x2, x3 are nonnegative integers. Use dynamic programming to solve this problem.
Consider the following nonlinear programming problem.MaximizeSubject to x1 + x2 ‰¥ 3. and x1 ‰¥ 0, x2 ‰¥ 0. Use dynamic programming to solve this problem.
Re-solve the Local Job Shop employment scheduling problem (Example 4) when the total cost of changing the level of employment from one season to the next is changed to $100 times the square of the difference in employment levels.
Consider the following nonlinear programming problem. Maximize Z = 2x21 + 2x2 + 4x3 – x23 Subject to 2x1 + x2 + x3 ≤ 4 and x1 ≥ 0, x2 ≥ 0, x3 ≥ 0. Use dynamic programming to solve this problem.
Consider the following nonlinear programming problem. Minimize Z = x41 + 2x22 Subject to x21 + x22 ≥ 2. (There are no nonnegativity constraints.) Use dynamic programming to solve this problem.
Consider the following nonlinear programming problem. Maximize Z = x31 + 4x22 + 16x3, Subject to x1 x2 x3 =4 and x1 ≥ 1, x2 ≥ 1, x3 ≥ 1. (a) Solve by dynamic programming when, in addition to the given constraints, all three variables also are required to be integer. (b) Use dynamic
Consider the following nonlinear programming problem. Maximize Z = x1 (1 – x2) x3, Subject to x1 – x2 + x3 ≤ 1 and x1 ≥ 0, x2 ≥ 0, x3 ≥ 0. Use dynamic programming to solve this problem.
Read the referenced article that fully describes the OR study summarized in the application vignette presented in Sec. 11.3. Briefly describe how dynamic programming was applied in this study. Then list the various financial and nonfinancial benefits that resulted from this study.
The owner of a chain of three grocery stores has purchased five crates of fresh strawberries. The estimated probability distribution of potential sales of the strawberries before spoilage differs among the three stores. Therefore, the owner wants to know how to allocate five crates to the three
A college student has 7 days remaining before final examinations begin in her four courses, and she wants to allocate this study time as effectively as possible. She needs at least 1 day on each course, and she likes to concentrate on just one course each day, so she wants to allocate 1, 2, 3, or 4
The sales manager for a publisher of college textbooks has six traveling salespeople to assign to three different regions of the country. She has decided that each region should be assigned at least one salesperson and that each individual salesperson should be restricted to one of the regions, but
A political campaign is entering its final stage, and polls indicate a very close election. One of the candidates has enough funds left to purchase TV time for a total of five prime-time commercials on TV stations located in four different areas. Based on polling information, an estimate has been
A backgammon player will be playing three consecutive matches with friends tonight. For each match, he will have the opportunity to place an even bet that he will win; the amount bet can be any quantity of his choice between zero and the amount of money he still has left after the bets on the
Imagine that you have $5,000 to invest and that you will have an opportunity to invest that amount in either of two investments (A or B) at the beginning of each of the next 3 years. Both investments have uncertain returns. For investment A you will either lose your money entirely or (with higher
Suppose that the situation for the Hit-and-Miss Manufacturing Co. problem (Example 5) has changed somewhat. After a more careful analysis, you now estimate that each item produced will be acceptable with probability 2/3, rather than 1/2, so that the probability of producing zero acceptable items in
Reconsider Example 6. Suppose that the bet is changed as follows: “Starting with two chips, she will not have at least five chips after five plays of the game.” By referring to the previous computational results, make additional calculations to determine the new optimal policy for the
The Profit & Gambit Co. has a major product that has been losing money recently because of declining sales. In fact, during the current quarter of the year, sales will be 4 million units below the break-even point. Because the marginal revenue for each unit sold exceeds the marginal cost by $5,
Consider the following project network (as described in Sec. 10.8), where the number over each node is the time required for the corresponding activity. Consider the problem of finding the longest path (the largest total time) through this network from start to finish, since the longest path is the
Consider the following statements about solving dynamic programming problems. Label each statement as true or false, and then justify your answer by referring to specific statements in the chapter.
A county chairwoman of a certain political party is making plans for an upcoming presidential election. She has received the services of six volunteer workers for precinct work, and she wants to assign them to four precincts in such a way as to maximize their effectiveness. She feels that it would
Use dynamic programming to solve the Northern Airplane Co. production scheduling problem presented in Sec. 9.1 (see Table 9.7). Assume that production quantities must be integer multiples of 5.
A company will soon be introducing a new product into a very competitive market and is currently planning its marketing strategy. The decision has been made to introduce the product in three phases. Phase 1 will feature making a special introductory offer of the product to the public at a greatly
Consider an electronic system consisting of four components, each of which must work for the system to function. The reliability of the system can be improved by installing several parallel units in one or more of the components. The following table gives the probability that the respective
Consider the following integer nonlinear programming problem.Maximize Z = 3x21 €“ x31 + 5x22 €“ x32,Subject tox1 + 2x2 ‰¤ 4andSue dynamic programming to solve this problem.
Reconsider the California Manufacturing Co. example presented in Sec. 12.1. The mayor of San Diego now has contacted the company’s president to try to persuade him to build a factory and perhaps a warehouse in that city. With the tax incentives being offered the company, the president’s staff
Suppose that a mathematical model fits linear programming except for the restrictions that1. At least one of the following two inequalities holds:2. At least two of the following three inequalities holds: Show how to reformulate these restrictions to fit an MIP model.
The Toys-R-4-U Company has developed two new toys for possible inclusion in its product line for the upcoming Christmas season. Setting up the production facilities to begin production would cost $50,000 for toy 1 and $80,000 for toy 2. Once these costs are covered, the toys would generate a unit
Northeastern Airlines is considering the purchase of new long-, medium-, and short-range jet passenger airplanes. The purchase price would be $67 million for each long-range plane, $50 million for each medium-range plane, and $35 million for each short-range plane. The board of directors has
Consider the two-variable IP example discussed in Sec. 12.5 and illustrated in Fig. 12.3. (a) Use a binary representation of the variables to reformulate this model as a BIP problem.
The Fly-Right Airplane Company builds small jet airplanes to sell to corporations for the use of their executives. To meet the needs of these executives, the company€™s customers sometimes order a custom design of the airplanes being purchased. When this occurs, a substantial start-up
Reconsider the Fly-Right Airplane Co. problem introduced in Prob. 12.3-7. A more detailed analysis of the various cost and revenue factors now has revealed that the potential profit from producing airplanes for each customer cannot be expressed simply in terms of a start-up cost and a fixed
Reconsider Prob. 3.1-11, where the management of the Omega Manufacturing Company is considering devoting excess production capacity to one or more of three products. (See the Partial Answers to Selected Problems in the back of the book for additional information about this problem.) Management now
Consider the following integer nonlinear programming problem:Maximize Z = 4x21 €“ x31 + 10x22 €“ x42,Subject tox1 + x2 ‰¤ 3andx1 ‰¥ 0, x2 ‰¥ 0x1 and x2 are integers.This problem can be reformulated in two different ways as an equivalent pure BIP
Consider the following special type of shortest-path problem (see Sec. 10.3) where the nodes are in columns and the only paths considered always move forward one column at a time.The numbers along the links represent distances, and the objective is to find the shortest path from the origin to the
Speedy Delivery provides two-day delivery service of large parcels across the United States. Each morning at each collection center, the parcels that have arrived overnight are loaded onto several trucks for delivery throughout the area. Since the competitive battlefield in this business is speed
A young couple, Eve and Steven, want to divide their main household chores (marketing, cooking, dishwashing, and laundering) between them so that each has two tasks but the total time they spend on household duties is kept to a minimum. Their efficiencies on these tasks differ, where the time each
An increasing number of Americans are moving to a warmer climate when they retire. To take advantage of this trend, Sunny Skies Unlimited is undertaking a major real estate development project. The project is to develop a completely new retirement community (to be called Pilgrim Haven) that will
Reconsider Prob. 12.4-7. The management of Sunny Skies Unlimited now has decided that the decision on the locations of the fire stations should be based mainly on costs. The cost of locating a fire station in a tract is $200,000 for tract 1, $250,000 for tract 2, $400,000 for tract 3, $300,000 for
Suppose that a state sends R persons to the U.S. House of Representatives. There are D counties in the state (D > R), and the state legislature wants to group these counties into R distinct electoral districts, each of which sends a delegate to Congress. The total population of the state is P, and
Read the referenced article that fully describes the OR study summarized in the application vignette presented in Sec. 12.5. Briefly describe how integer programming was applied in this study. Then list the various financial and nonfinancial benefits that resulted from this study.
Consider the following IP problem:Maximize Z = 5x1 + x2,Subject toand x1 ‰¥ 0, x2 ‰¥ 0 x1, x2 are integers. (a) Solve this problem graphically.
Follow the instructions of Prob. 12.5-2 for the following IP problem:Maximize Z = 220x1 + 80x2,Subject toand x1 ‰¥ 0, x2 ‰¥ 0 x1, x2 are integers.
Follow the instructions of Prob. 12.5-2 for the following BIP problem:Maximize Z = 2x1 + 5x2,Subject toand x1, x2 are binary.
Follow the instructions of Prob. 12.5-2 for the following BIP problem:Maximize Z = €“5x1 + 25x2,Subject toand x1, x2 are binary.
Label each of the following statements as True or False, and then justify your answer by referring to specific statements in the chapter: (a) Linear programming problems are generally considerably easier to solve than IP problems. (b) For IP problems, the number of integer variables is generally
Use the BIP branch-and-bound algorithm presented in Sec. 12.6 to solve the following problem interactively:Maximize Z = 2x1 €“ x2 + 5x3 €“ 3x4 + 4x5,Subject toand xj is binary, for j = 1, 2, . . . , 5.
A real estate development firm, Peterson and Johnson, is considering five possible development projects. The following table shows the estimated long-run profit (net present value) that each project would generate, as well as the amount of investment required to undertake the project, in units of
Use the BIP branch-and-bound algorithm presented in Sec. 12.6 to solve the following problem interactively:Minimize Z = 5x1 + 6x2 + 7x3 + 8x4 + 9x5,Subject toand xj is binary, for j = 1, 2, . . . , 5.
Use the BIP branch-and-bound algorithm presented in Sec. 12.6 to solve the following problem interactively:Maximize Z = 5x1 + 5x2 + 8x3 €“ 2x4 €“ 4x5,Subject toand xj is binary, for j = 1, 2, . . . , 5.

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