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introduction to operations research

Introduction To Operations Research 7th Edition Frederick S. Hillier, Gerald J. Lieberman - Solutions

Consider the following “fixed-charge” problem.Maximize Z 3x1 7x2 6f(x3), subject to x1 3x2 2x3 6 x1 x2 2x3 5 and x1 0, x2 0, x3 0, where f(x3) Use dynamic programming to solve this problem.
Imagine that you have $5,000 to invest and that you will have an opportunity to invest that amount in either of two invest ments (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
Consider the following BIP problem.Maximize Z 2x1 3x2 x3 4x4 3x5 2x6 2x7 x8 3x9, subject to 3x2 x4 x5 3 x1 x2 1 x2 x4 x5 x6 1 x2 2x6 3x7 x8 2x9 4x3 2x5 x6 2x7 2x8 x9 5 and all xj binary.Develop the tightest possible formulation of this problem by using the techniques of automatic
In Sec. 12.8, a pure BIP example with the constraint, 2x1 3x2 4, was used to illustrate the procedure for tightening constraints.Show that applying the procedure for generating cutting planes to this constraint yields the same new constraint, x1 x2 1.
Use the following set of constraints for the same pure BIP problem to fix as many variables as possible. Also identify the constraints which become redundant because of the fixed variables.3x3 x5 x7 1 x2 x4 x6 1 x1 2x5 2x6 2 x1 x2 x4 0
Reconsider the discrete nonlinear programming problem given in Prob. 12.4-5.(a) Use the following outline in designing the main features of a branch-and-bound algorithm for solving this problem (and similar problems) directly without reformulation.(i) Specify the tightest possible nonlinear
Use the MIP branch-and-bound algorithm presented in Sec. 12.7 to solve the following MIP problem interactively.Minimize Z 5x1 x2 x3 2x4 3x5, subject to x2 5x3 x4 2x5 2 5x1 x2 x4 x5 7 x1 x2 6x3 x4 4 and xj 0, for j 1, 2, 3, 4, 5 xj is integer, for j 1, 2, 3.
Use the MIP branch-and-bound algorithm presented in Sec. 12.7 to solve the following MIP problem interactively.Maximize Z 3x1 4x2 2x3 x4 2x5, subject to 2x1 x2 x3 x4 x5 3x1 3x2 x3 x4 2x5 2 2x1 x2 x3 x4 3x5 1 and xj 0, for j 1, 2, 3, 4, 5 xj is binary, for j 1, 2, 3.
Use the MIP branch-and-bound algorithm presented in Sec. 12.7 to solve the following MIP problem interactively.Maximize Z 5x1 4x2 4x3 2x4, subject to x1 3x2 2x3 x4 10 5x1 x2 3x3 2x4 15 x1 x2 x3 x4 6 and xj 0, for j 1, 2, 3, 4 xj is integer, for j 1, 2, 3.
A machine shop makes two products. Each unit of the first product requires 3 hours on machine 1 and 2 hours on machine 2.Each unit of the second product requires 2 hours on machine 1 and 3 hours on machine 2. Machine 1 is available only 8 hours per day and machine 2 only 7 hours per day. The profit
Follow the instructions of Prob. 12.7-3 for the IP model of Prob. 12.5-2.D,I
Reconsider the IP model of Prob. 12.5-1.(a) Use the MIP branch-and-bound algorithm presented in Sec.12.7 to solve this problem by hand. For each subproblem, solve its LP relaxation graphically.D,I (b) Now use the interactive routine for this algorithm in your OR Courseware to solve this problem.C
Follow the instructions of Prob. 12.7-1 for the following IP model.Minimize Z 2x1 3x2, subject to x1 x2 3 x1 3x2 6 and x1 0, x2 0 x1, x2 are integers.
Consider the Lagrangian relaxation described near the end of Sec. 12.6.(a) If x is a feasible solution for an MIP problem, show that x also must be a feasible solution for the corresponding Lagrangian relaxation.(b) If x* is an optimal solution for an MIP problem, with an objective function value
Five jobs need to be done on a certain machine. However, the setup time for each job depends upon which job immediately preceded it, as shown by the following table:The objective is to schedule the sequence of jobs that minimizes the sum of the resulting setup times.(a) Design a branch-and-bound
Consider the assignment problem with the following cost table:(a) Design a branch-and-bound algorithm for solving such assignment problems by specifying how the branching, bounding, and fathoming steps would be performed. (Hint: For the assignees not yet assigned for the current subproblem, form
Consider the following statements about any pure IP problem (in maximization form) and its LP relaxation. Label each of the statements as True or False, and then justify your answer.(a) The feasible region for the LP relaxation is a subset of the feasible region for the IP problem.(b) If an optimal
Reconsider Prob. 12.4-10(a). Use the BIP algorithm presented in Sec. 12.6 to solve this problem interactively
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 to3x1 6x2 7x3 9x4 9x5 10 x1 2x2 7x3 x4 3x5 0 and 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.Minimize Z 5x1 6x2 7x3 8x4 9x5, subject to 3x1 x2 x3 x4 2x5 2 x1 3x2 x3 2x4 x5 0x1 x2 3x3 x4 x5 1 and xj is binary, for j 1, 2, . . . , 5.
Label each of the following statements as True or False, and then justify your answer by referring to specific statements(with page citations) in the chapter.(a) Linear programming problems are generally much easier to solve than IP problems.(b) For IP problems, the number of integer variables is
Follow the instructions of Prob. 12.5-1 for the following BIP problem.Maximize Z 5x1 25x2, subject to3x1 30x2 27 3x1 x2 4 and x1, x2 are binary.
Follow the instructions of Prob. 12.5-1 for the following BIP problem.Maximize Z 2x1 5x2, subject to 10x1 30x2 30 95x1 30x2 75 and x1, x2 are binary.
Follow the instructions of Prob. 12.5-1 for the following IP problem.Maximize Z 220x1 80x2, subject to 5x1 2x2 16 2x1 x2 4x1 2x2 4 and x1 0, x2 0 x1, x2 are integers.
Consider the following IP problem.Maximize Z 5x1 x2, subject tox1 2x2 4 x1 x2 1 4x1 x2 12 and x1 0, x2 0 x1, x2 are integers.(a) Solve this problem graphically.(b) Solve the LP relaxation graphically. Round this solution to the nearest integer solution and check whether it is feasible.
A U.S. professor will be spending a short sabbatical leave at the University of Iceland. She wishes to bring all needed items with her on the airplane. After collecting the professional items that she must have, she finds that airline regulations on space and weight for checked luggage will
Consider the project network for a PERT-type system shown in Prob. 11.2-3.Formulate a BIP model for the problem of finding a critical path (i.e., a longest path) for this project network.
Consider the following discrete nonlinear programming problem.Maximize Z 2x1 x2 1 3x2 3x2 2,(a) Reformulate this problem as a pure binary integer linear programming problem.C (b) Use the computer to solve the model formulated in part (a), and thereby identify an optimal solution for (x1, x2)
Consider the following integer nonlinear programming problem.Maximize Z 4x2 1 x3 1 10x2 2 x4 2, subject to x1 x2 3 and x1 0, x2 0 x1 and x2 are integers.This problem can be reformulated in two different ways as an equivalent pure BIP problem (with a linear objective function) with six
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
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 marginal
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.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
Suppose that a mathematical model fits linear programming except for the restrictions that 1. At least one of the following two inequalities holds:x1 x2 x3 x4 4 3x1 x2 x3 x4 3.2. At least two of the following four inequalities holds:5x1 3x2 3x3 x4 10 2x1 5x2 x3 3x4 10x1 3x2 5x3 3x4
Select three of the actual applications of BIP by a company or governmental agency mentioned in Sec. 12.2. 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. 12.2. Read the article describing the application in the referenced issue of Interfaces.Write a two-page summary of the application and its benefits.
Pawtucket University is planning to buy new copier machines for its library. Three members of its Operations Research Department are analyzing what to buy. They are considering two different models: Model A, a high-speed copier, and Model B, a lower-speed but less expensive copier. Model A can
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
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
Consider the assignment problem having the following cost table.The optimal solution is A-3, B-1, C-2, with Z 10.C (a) Use the computer to verify this optimal solution.(b) Reformulate this problem as an equivalent transportation problem by constructing the appropriate parameter table.C (c) Obtain
Reconsider Prob. 8.1-7.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
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
Reconsider Prob. 8.2-23.Now suppose that trucks (and their drivers) need to be hired to do the hauling, where each truck can only be used to haul gravel from a single pit to a single site. Each truck can haul 5 tons, and the cost per truck is five times the hauling cost per ton given earlier. Only
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. 8.1-3.Suppose that the sales forecasts have been revised downward to 240, 400, and 320 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. 8.2-24.
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).The numbers given in the parameter table are only estimates that may be somewhat inaccurate, so management now wishes to do some what-if
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
When you deal with a transportation problem where the supply and demand quantities have integer values, explain why the steps of the transportation simplex method guarantee that all the basic variables (allocations) in the BF solutions obtained must have integer values. Begin with why this occurs
Consider the general linear programming formulation of the transportation problem (see Table 8.6). Verify the claim in Sec.8.2 that the set of (m n) functional constraint equations (m supply constraints and n demand constraints) has one redundant equation; i.e., any one equation can be reproduced
Consider the Northern Airplane Co. production scheduling problem presented in Sec. 8.1 (see Table 8.7). Formulate this problem as a general linear programming problem by letting the decision variables be xj number of jet engines to be produced in month j (j 1, 2, 3, 4). Construct the initial
Consider the transportation problem having the following parameter table:(a) Using your choice of a criterion from Sec. 8.2 for obtaining the initial BF solution, solve this problem manually by the transportation simplex method. (Keep track of your time.)(b) Reformulate this problem as a general
Follow the instructions of Prob. 8.2-17 for the transportation problem formulated in Prob. 8.1-8a.
Reconsider the transportation problem formulated in Prob.8.1-7a.D,I (a) Use each of the three criteria presented in Sec. 8.2 to obtain an initial BF solution, and time how long you spend for each one. Compare both these times and the values of the objective function for these solutions.C (b) Obtain
Reconsider Prob. 8.1-6.Starting with Russell’s approximation method, interactively apply the transportation simplex method to obtain an optimal solution for this problem.
Reconsider Prob. 8.1-4.Starting with the northwest corner rule, interactively apply the transportation simplex method to obtain an optimal solution for this problem.
Reconsider Prob. 8.1-3.Starting with the northwest corner rule, interactively apply the transportation simplex method to obtain an optimal solution for this problem.
Reconsider Prob. 8.1-2b. Starting with the northwest corner rule, interactively apply the transportation simplex method to obtain an optimal solution for this problem.
Reconsider Prob. 8.1-1.(a) Use the northwest corner rule to obtain an initial BF solution.(b) Starting with the initial BF solution from part (a), interactively apply the transportation simplex method to obtain an optimal solution.
Interactively apply the transportation simplex method to solve the Northern Airplane Co. production scheduling problem as it is formulated in Table 8.9.
The Energetic Company needs to make plans for the energy systems for a new building.The energy needs in the building fall into three categories:(1) electricity, (2) heating water, and (3) heating space in the building. The daily requirements for these three categories (all measured in the same
Consider the transportation problem having the following parameter table:Use each of the following criteria to obtain an initial BF solution.In each case, interactively apply the transportation simplex method, starting with this initial solution, to obtain an optimal solution.Compare the resulting
Consider the transportation problem having the following parameter table:After several iterations of the transportation simplex method, a BF solution is obtained that has the following basic variables: x13 20, x21 25, x24 5, x32 25, x34 5, x42 0, x43 0, x45 20. Continue the transportation simplex
Consider the transportation problem formulation of Option 1 for the Better Products Co. problem presented in Table 8.28. Verify that the optimal solution given in Sec. 8.3 actually is optimal by applying just the optimality test portion of the transportation simplex method to this solution.
Consider the prototype example for the transportation problem (the P & T Co. problem) presented at the beginning of Sec.8.1. Verify that the solution given there actually is optimal by applying just the optimality test portion of the transportation simplex method to this solution.
Consider the transportation problem having the following parameter table:(a) Notice that this problem has three special characteristics:(1) number of sources number of destinations, (2) each supply 1, and (3) each demand 1. Transportation problems with these characteristics are of a special type
Consider the transportation problem having the following parameter table:Use each of the following criteria to obtain an initial BF solution.Compare the values of the objective function for these solutions.(a) Northwest corner rule.(b) Vogel’s approximation method.(c) Russell’s approximation
Consider the transportation problem having the following parameter table:(a) Use Vogel’s approximation method manually (don’t use the interactive routine in your OR Courseware) to select the first basic variable for an initial BF solution.(b) Use Russell’s approximation method manually to
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
The Build-Em-Fast Company has agreed to supply its best customer with three widgets during each of the next 3 weeks, even though producing them will require some overtime work. The relevant production data are as follows:The cost per unit produced with overtime for each week is $100 more than for
Redo Prob. 8.1-7 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.
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:394 8 THE TRANSPORTATION AND
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.
Reconsider the P & T Co. problem presented in Sec. 8.1.You now learn that one or more of the shipping costs per truckload given in Table 8.2 may change slightly before shipments begin.Use the Excel Solver to generate the Sensitivity Report for this problem. Use this report to determine the
Suppose that England, France, and Spain produce all the wheat, barley, and oats in the world. The world demand for wheat requires 125 million acres of land devoted to wheat production.Similarly, 60 million acres of land are required for barley and 75 million acres of land for oats. The total amount
The Versatech Corporation has decided to produce three new products. Five branch plants now have excess product capacity. The unit manufacturing cost of the first product would be $31, $29,$32, $28, and $29 in Plants 1, 2, 3, 4, and 5, respectively. The unit manufacturing cost of the second product
Tom would like 3 pints of home brew today and an additional 4 pints of home brew tomorrow. Dick is willing to sell a maximum of 5 pints total at a price of $3.00 per pint today and$2.70 per pint tomorrow. Harry is willing to sell a maximum of 4 pints total at a price of $2.90 per pint today and
The Childfair Company has three plants producing child push chairs that are to be shipped to four distribution centers. Plants 1, 2, and 3 produce 12, 17, and 11 shipments per month, respectively. Each distribution center needs to receive 10 shipments per month. The distance from each plant to the
Repeat Prob. 4.9-1 for the model in Prob. 4.1-6.
Repeat Prob. 4.9-1 for the model in Prob. 4.1-5.
Choose 0.5 from the Option menu, use (x1, x2) (0.1, 0.4) as the initial trial solution, and run 15 iterations. Draw a graph of the feasible region, and then plot the trajectory of the trial solutions through this feasible region.
Consider the following problem.Maximize Z 5x1 4x2 x3 3x4, subject to 3x1 2x2 3x3 x4 24 (resource 1)3x1 3x2 x3 3x4 36 (resource 2)and x1 0, x2 0, x3 0, x4 0.D,I (a) Work through the simplex method step by step to solve the problem.(b) Identify the shadow prices for the two
Consider the following problem.Maximize Z 2x1 4x2 x3, subject to 2x1 3x2 x3 30 (resource 1)2x1 x2 x3 10 (resource 2)4x1 2x2 2x3 40 (resource 3)and x1 0, x2 0, x3 0.D,I (a) Work through the simplex method step by step to solve the problem.(b) Identify the shadow prices for the three
Consider the following problem.Maximize Z 2x1 2x2 3x3, subject to x1 x2 x3 4 (resource 1)2x1 x2 x3 2 (resource 2)x1 x2 3x3 12 (resource 3)and x1 0, x2 0, x3 0.D,I (a) Work through the simplex method step by step to solve the problem.(b) Identify the shadow prices for the three
Consider the following problem.Maximize Z x1 7x2 3x3, subject to 2x1 x2 x3 4 (resource 1)4x1 3x2 x3 2 (resource 2)3x1 2x2 x3 3 (resource 3)and x1 0, x2 0, x3 0.D,I (a) Work through the simplex method step by step to solve the problem.(b) Identify the shadow prices for the three
You are given the following linear programming problem.Maximize Z 4x1 2x2, subject to 2x1 3x2 16 (resource 1)x1 3x2 17 (resource 2)x1 3x2 5 (resource 3)and x1 0, x2 0.(a) Solve this problem graphically.(b) Use graphical analysis to find the shadow prices for the resources.(c)
Repeat Prob. 4.7-2 for the model in Prob. 4.1-6.
Interpret the right-hand side of the respective functional constraints as the amount available of the respective resources.(a) Use graphical analysis as in Fig. 4.8 to determine the shadow prices for the respective resources.(b) Use graphical analysis to perform sensitivity analysis on this model.
Refer to Fig. 4.10 and the resulting allowable range to stay feasible for the respective right-hand sides of the Wyndor Glass Co. problem given in Sec. 3.1. Use graphical analysis to demonstrate that each given allowable range is correct.
Consider the following problem.Maximize Z 4x1 5x2 3x3, subject to x1 x2 2x3 20 15x1 6x2 5x3 50 x1 3x2 5x3 30 and x1 0, x2 0, x3 0.Work through the simplex method step by step to demonstrate that this problem does not possess any feasible solutions.
Consider the following problem.Maximize Z 2x1 x2 4x3 3x4, subject to x1 x2 3x3 2x4 4 x1 x2 x3 x4 1 2x1 x2 x3 x4 2 x1 2x2 x3 2x4 2 and x2 0, x3 0, x4 0(no nonnegativity constraint for x1).(a) Reformulate this problem to fit our standard form for a linear
This chapter has described the simplex method as applied to linear programming problems where the objective function is to be maximized. Section 4.6 then described how to convert a minimization problem to an equivalent maximization problem for applying the simplex method. Another option with
Consider the following problem.Maximize Z x1 2x2 x3, subject to 3x2 x3 120 x1 x2 4x3 80 3x1 x2 2x3 100(no nonnegativity constraints).(a) Reformulate this problem so that all variables have nonnegativity constraints.D,I (b) Work through the simplex method step by step to solve the
Consider the following problem.Maximize Z x1 4x2, subject to 3x1 x2 6 x1 2x2 4 x1 2x2 3(no lower bound constraint for x1).(a) Solve this problem graphically.(b) Reformulate this problem so that it has only two functional constraints and all variables have nonnegativity constraints.D,I
Consider the following problem.Maximize Z x1 4x2 2x3, subject to 4x1 x2 x3 5 x1 x2 2x3 10 and x2 0, x3 0(no nonnegativity constraint for x1).(a) Reformulate this problem so all variables have nonnegativity constraints.D,I (b) Work through the simplex method step by step to solve the
Follow the instructions of Prob. 4.6-10 for the following problem.Minimize Z 3x1 2x2 x3, subject to x1 x2 x3 7 3x1 x2 x3 10 and x1 0, x2 0, x3 0.
Follow the instructions of Prob. 4.6-10 for the following problem.Minimize Z 3x1 2x2 7x3, subject to x1 x2 x3 10 2x1 x2 x3 10 and x1 0, x2 0, x3 0.

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