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introduction to operations research
Introduction To Operations Research 9th Edition Frederick S. Hillier - Solutions
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
Reconsider the Wyndor Glass Co. problem presented in Sec. 3.1. Management now has decided that only one of the two new products should be produced, and the choice is to be made on the basis of maximizing profit. Introduce auxiliary binary variables to formulate an MIP model for this new version of
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 marginal net revenue per airplane produced. Instead, the profits are given by the
Reconsider the Fly-Right Airplane Co. problem introduced in Prob.
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 cost is
Consider the two-variable IP example discussed in Sec. 11.5 and illustrated in Fig. 11.3.(a) Use a binary representation of the variables to reformulate this model as a BIP problem.C (b) Use the computer to solve this BIP problem. Then use this optimal solution to identify an optimal solution for
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
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
Suppose that a mathematical model fits linear programming except for the restrictions that 1. At least one of the following two inequalities holds:3x1 x2 x3 x4 12 x1 x2 x3 x4 15.2. At least two of the following three inequalities holds:2x1 5x2 x3 x4 30x1 3x2 5x3 x4 40 3x1 x2 3x3 x4
Suppose that a mathematical model fits linear programming except for the restriction that x1 x2 0, or 3, or 6. Show how to reformulate this restriction to fit an MIP model.
The Research and Development Division of the Progressive Company has been developing four possible new product lines.Management must now make a decision as to which of these four products actually will be produced and at what levels. Therefore, an operations research study has been requested to
Select three of the actual applications of BIP by a company or governmental agency mentioned in Sec. 11.2. Read the articles describing the applications in the referenced issues of Interfaces. For each one, write a one-page summary of the application and its benefits.
Select one of the actual applications of BIP by a company or governmental agency mentioned in Sec. 11.2. Read the article describing the application in the referenced issue of Interfaces.Write a two-page summary of the application and its benefits.
Read the referenced article that fully describes the OR study summarized in the application vignette presented in Sec. 11.2.Briefly describe how integer programming was applied in this study.Then list the various financial and nonfinancial benefits that resulted from this study.
Reconsider Prob. 8.2-21 involving a contractor (Susan Meyer) who needs to arrange for hauling gravel from two pits to three building sites.Susan now needs to hire the trucks (and their drivers) to do the hauling. Each truck can only be used to haul gravel from a single pit to a single site. In
Vincent Cardoza is the owner and manager of a machine shop that does custom order work. This Wednesday afternoon, he has received calls from two customers who would like to place rush orders. One is a trailer hitch company which would like some custom-made heavy-duty tow bars. The other is a
Reconsider Prob. 8.3-4, where a swim team coach needs to assign swimmers to the different legs of a 200-yard medley relay team. Formulate a BIP model for this problem. Identify the groups of mutually exclusive alternatives in this formulation.
The board of directors of General Wheels Co. is considering six large capital investments. Each investment can be made only once. These investments differ in the estimated long-run profit(net present value) that they will generate as well as in the amount of capital required, as shown by the
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
Reconsider the California Manufacturing Co. example presented in Sec. 11.1, 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
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,
Reconsider Example 7. 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
Suppose that the situation for the Hit-and-Miss Manufacturing Co. problem (Example 6) 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
Imagine that you have $10,000 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
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
Consider the following “fixed-charge” problem.Maximize Z 3x1 7x2 6f(x3),
Consider the following linear programming problem.Maximize Z 15x1 10x2, subject to x1 2x2 6 3x1 x2 8 and x1 0, x2 0.Use dynamic programming to solve this problem.
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.
Consider the following nonlinear programming problem.Maximize Z x3 1 4x2 2 16x3, subject to x1x2x3 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 programming to solve the
Consider the following nonlinear programming problem.Minimize Z x4 1 2x2 2subject to x2 1 x2 2 2.(There are no nonnegativity constraints.) Use dynamic programming to solve this problem.
Consider the following nonlinear programming problem.Maximize Z 2x2 1 2x2 4x3 x2 3subject to 2x1 x2 x3 4
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 36x1 9x2 1 6x3 136x2 3x3 2, subject to x1 x2 3 and x1 0, x2 0.Use dynamic programming to solve this problem.
Consider the following integer nonlinear programming problem.Maximize Z 32x1 2x2 1 30x2 20x3, subject to 3x1 7x2 5x3 20 and x1, x2, x3 are nonnegative integers.Use dynamic programming to solve this problem.
Consider the following integer nonlinear programming problem.Maximize Z 3x2 1 x3 1 5x2 2 x3 2, subject to x1 2x2 4 and x1 0, x2 0 x1, x2 are integers.Use dynamic programming to solve this problem.
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
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
Use dynamic programming to solve the Northern Airplane Co. production scheduling problem presented in Sec. 8.1 (see Table 8.7). Assume that production quantities must be integer multiples of 5.
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
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 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 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
Read the referenced article that fully describes the OR study summarized in the application vignette presented in Sec. 10.3.Briefly describe how dynamic programming was applied in this study. Then list the various financial and nonfinancial benefits that resulted from this study.
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.
Consider the following project network (as described in Sec. 9.8), 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
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
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.
From the bottom part of the selected references given at the end of the chapter, select three of these award-winning applications of network optimization models. For each one, read the article and then write a one-page summary of the application and the benefits (including nonfinancial benefits) it
From the bottom part of the selected references given at the end of the chapter, select one of these award-winning applications of network optimization models. Read this article and then write a two-page summary of the application and the benefits (including nonfinancial benefits) it provided.
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 (labeled
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, . . . ,
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 according
Reconsider the Tinker Construction Co. problem presented in Prob. 9.8-1, 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.
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.
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, C
Consider the Metro Water District transportation problem presented in Table 8.12,(a) Formulate the network representation of this problem as a minimum cost flow problem. (Hint: Arcs where flow is prohibited should be deleted.)D,I (b) Starting with the initial BF solution given in Table 8.19, use
For the P & T Co. transportation problem given in Table 8.2, consider its network representation as a minimum cost flow problem presented in Fig. 8.2. Use the northwest corner rule to obtain an initial BF solution from Table 8.2. Then use the network simplex method yourself (you may use the
Reconsider the minimum cost flow problem formulated Prob. 9.6-4, 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.
Reconsider the minimum cost flow problem formulated in Prob. 9.6-3,(a) Obtain an initial BF solution by solving the feasible spanning tree that corresponds to using just the two rail lines plus factory 1 shipping to warehouse 2 via the distribution center.D,I (b) Use the network simplex method
Reconsider the minimum cost flow problem formulated in Prob. 9.6-2.(a) Obtain an initial BF solution by solving the feasible spanning tree with basic arcs A B, A C, A F, B D, and E F, where two of the nonbasic arcs (E C and F D) are reverse arcs.D,I (b) Use the network simplex method
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.(a) Obtain an initial
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
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
Reconsider Prob. 9.3-3, Now formulate this problem as a minimum cost flow problem by showing the appropriate network representation.
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 the maximum flow problem shown in Prob. 9.5-6.Formulate this problem as a minimum cost flow problem, including adding the arc A F. Use F 20.
Read the referenced article that fully describes the OR study summarized in the application vignette presented in Sec. 9.6. 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
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. 9.5 to solve this problem.C (b) Formulate and solve a spreadsheet model for
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
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 refineries to
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.
Formulate the maximum flow problem as a linear programming problem.
For the network shown below, use the augmenting path algorithm described in Sec. 9.5 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.
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
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
Use the algorithm described in Sec. 9.4 to find the minimum spanning tree for each of these networks.
Reconsider the networks shown in Prob.
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 now
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 are
Formulate the shortest-path problem as a linear programming problem.
Use the algorithm described in Sec. 9.3 to find the shortest path through each of the following networks, where the numbers represent actual distances between the corresponding nodes.
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
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.
Read the referenced article that fully describes the OR study summarized in the application vignette presented in Sec. 9.3.Briefly describe how network optimization models were applied in this study. Then list the various financial and nonfinancial benefits that resulted from this study?
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
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:
Reconsider the assignment problem formulation of Option 2 for the Better Products Co. problem presented in Table 8.29. Suppose that the cost of having Plant 1 produce product 1
Reconsider Prob. 8.3-4, See its formulation as an assignment problem in the answers given in the back of the book. Manually apply the Hungarian algorithm to solve this problem. (You may use the corresponding interactive procedure in your IOR Tutorial.)
Reconsider the assignment problem presented in Prob. 8.3-2, Manually apply the Hungarian algorithm to solve this problem. (You may use the corresponding interactive procedure in your IOR Tutorial.)
Consider the linear programming model for the general assignment problem given in Sec. 8.3. Construct the table of constraint coefficients for this model. Compare this table with the one for the general transportation problem (Table 8.6). In what ways does the general assignment problem have more
Now assume that distribution centers 1, 2, and 3 must receive exactly 10, 20, and 30 units per week, respectively. For administrative convenience, management has decided that each distribution center will be supplied totally by a single plant, so that one plant will supply one distribution center
Starting with Vogel’s approximation method, interactively apply the transportation simplex method to solve the Job Shop Co. assignment problem as formulated in Table 8.26b. (As stated in Sec. 8.3, the resulting optimal solution has x14 1, x23 1, x31 1, x42 1, and all other xij 0.)
Consider the assignment problem formulation of Option 2 for the Better Products Co. problem presented in Table 8.29.(a) Reformulate this problem as an equivalent transportation problem with three sources and five destinations by constructing the appropriate parameter table.(b) Convert the optimal
The coach of an age group swim team needs to assign swimmers to a 200-yard medley relay team to send to the Junior Olympics. Since most of his best swimmers are very fast in more than one stroke, it is not clear which swimmer should be assigned to each of the four strokes. The five fastest swimmers
Reconsider Prob. 8.1-4, Suppose that the sales forecasts have been revised downward to 280, 400, and 350 units per day of products 1, 2, and 3, respectively, and that each plant now has the capacity to produce all that is required of any one product. Therefore, management has decided that each new
Four cargo ships will be used for shipping goods from one port to four other ports (labeled 1, 2, 3, 4). Any ship can be used for making any one of these four trips. However, because of differences in the ships and cargoes, the total cost of loading, transporting, and unloading the goods for the
Consider the assignment problem having the following cost table.(a) Draw the network representation of this assignment problem.(b) Formulate this problem as a transportation problem by constructing the appropriate parameter table.(c) Display this formulation on an Excel spreadsheet.C (d) Use the
Without generating the Sensitivity Report, adapt the sensitivity analysis procedure presented in Secs. 6.6 and 6.7 to conduct the sensitivity analysis specified in the four parts of Prob. 8.2-22,
Consider the transportation problem formulation and solution of the Metro Water District problem presented in Secs. 8.1 and 8.2 (see Tables 8.12 and 8.23).
A contractor, Susan Meyer, has to haul gravel to three building sites. She can purchase as much as 18 tons at a gravel pit in the north of the city and 14 tons at one in the south. She needs 10, 5, and 10 tons at sites 1, 2, and 3, respectively. The purchase price per ton at each gravel pit and the
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