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introduction to operations research

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

Consider the following problem.Minimize Z 3x1 2x2 4x3, subject to 2x1 x2 3x3 60 3x1 3x2 5x3 120 and x1 0, x2 0, x3 0.I (a) Using the Big M method, work through the simplex method step by step to solve the problem.I (b) Using the two-phase method, work through the simplex
Consider the following problem.Minimize Z 2x1 x2 3x3, subject to 5x1 2x2 7x3 420 3x1 2x2 5x3 280 and x1 0, x2 0, x3 0.I (a) Using the two-phase method, work through phase 1 step by step.C (b) Use a software package based on the simplex method to formulate and solve the phase
Consider the following problem.Maximize Z 90x1 70x2, subject to 2x1 x2 2 x1 x2 2 and x1 0, x2 0.(a) Demonstrate graphically that this problem has no feasible solutions.C (b) Use a computer package based on the simplex method to determine that the problem has no feasible solutions.I (c)
Consider the following problem.Minimize Z 2x1 3x2 x3, subject to x1 4x2 2x3 8 3x1 2x2 2x3 6 and x1 0, x2 0, x3 0.(a) Reformulate this problem to fit our standard form for a linear programming model presented in Sec. 3.2.I (b) Using the Big M method, work through the simplex
Consider the following problem.Minimize Z 3x1 2x2, subject to2x1 x2 10 3x1 2x2 6 x1 x2 6 and x1 0, x2 0.(a) Solve this problem graphically.(b) Using the Big M method, construct the complete first simplex tableau for the simplex method and identify the corresponding initial
Consider the following problem.Maximize Z 4x1 2x2 3x3 5x4, subject to 2x1 3x2 4x3 2x4 300 8x1 x2 x3 5x4 300 and xj 0, for j 1, 2, 3, 4.(a) Using the Big M method, construct the complete first simplex tableau for the simplex method and identify the corresponding initial
Consider the following problem.Maximize Z x1 x2 x3 x4, subject to x1 x2 3 x3 x4 2 and xj 0, for j 1, 2, 3, 4.Work through the simplex method step by step to find all the optimal BF solutions.4.6-1.* Consider the following problem.Maximize Z 2x1 3x2, subject to x1 2x2 4 x1 x2
Consider a two-variable linear programming problem whose CPF solutions are (0, 0), (6, 0), (6, 3), (3, 3), and (0, 2). (See Prob.3.2-2 for a graph of the feasible region.)(a) Use the graph of the feasible region to identify all the constraints for the model.(b) For each pair of adjacent CPF
Consider the following problem.Maximize Z 5x1 x2 3x3 4x4, subject to x1 2x2 4x3 3x4 20 4x1 6x2 5x3 4x4 40 2x1 3x2 3x3 8x4 50 and x1 0, x2 0, x3 0, x4 0.Work through the simplex method step by step to demonstrate that Z is unbounded.
Follow the instructions of Prob. 4.5-2 when the constraints are the following:2x1 x2 20 x1 2x2 20 and x1 0, x2 0.D,I
Suppose that the following constraints have been provided for a linear programming model with decision variables x1 and x2.x1 3x2 30 3x1 x2 30 and x1 0, x2 0.(a) Demonstrate graphically that the feasible region is unbounded.(b) If the objective is to maximize Z x1 x2, does the model
Work through the simplex method step by step to solve the following problem.Maximize Z x1 x2 2x3, subject to x1 2x2 x3 20 2x1 4x2 2x3 60 2x1 3x2 x3 50 and x1 0, x2 0, x3 0.
Work through the simplex method step by step (in tabular form) to solve the following problem.Maximize Z 2x1 x2 x3, subject to 3x1 x2 x3 6 x1 x2 2x3 1 x1 x2 x3 2 and x1 0, x2 0, x3 0.D,I
Consider the following problem.Maximize Z 2x1 x2 x3, subject to x1 x2 3x3 4 2x1 x2 3x3 10 x1 x2 x3 7 and x1 0, x2 0, x3 0.D,I (a) Work through the simplex method step by step in algebraic form to solve this problem.D,I (b) Work through the simplex method step by step in tabular form
Consider the following problem.Maximize Z 3x1 5x2 6x3, subject to 2x1 x2 x3 4 x1 2x2 x3 4 x1 x2 2x3 4 x1 x2 x3 3 and x1 0, x2 0, x3 0.D,I (a) Work through the simplex method step by step in algebraic form.D,I (b) Work through the simplex method in tabular form.C (c) Use
Consider the following problem.Maximize Z 2x1 4x2 3x3, subject to 3x1 4x2 2x3 60 2x1 x2 2x3 40 x1 3x2 2x3 80 and x1 0, x2 0, x3 0.D,I (a) Work through the simplex method step by step in algebraic form.D,I (b) Work through the simplex method step by step in tabular form.C (c)
Repeat Prob. 4.4-4 for the following problem.Maximize Z 2x1 3x2, subject to x1 2x2 30 x1 x2 20 and x1 0, x2 0.
Consider the following problem.Maximize Z 2x1 x2, subject to x1 x2 40 4x1 x2 100 and x1 0, x2 0.(a) Solve this problem graphically in a freehand manner. Also identify all the CPF solutions.(b) Now repeat part (a) when using a ruler to draw the graph carefully.D (c) Use hand calculations
Repeat Prob. 4.3-2, using the tabular form of the simplex method.D,I,C
Repeat Prob. 4.3-1, using the tabular form of the simplex method.D,I,C
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) The simplex method’s rule for choosing the entering basic variable is used because it always leads to the best adjacent BF solution
Consider the following problem.Maximize Z 2x1 4x2 3x3, subject to x1 3x2 2x3 30 x1 x2 x3 24 3x1 5x2 3x3 60 and x1 0, x2 0, x3 0.You are given the information that x1 0, x2 0, and x3 0 in the optimal solution.(a) Describe how you can use this information to adapt the
Consider the following problem.Maximize Z 5x1 3x2 4x3, subject to 2x1 x2 x3 20 3x1 x2 2x3 30 and x1 0, x2 0, x3 0.You are given the information that the nonzero variables in the optimal solution are x2 and x3.(a) Describe how you can use this information to adapt the simplex
Work through the simplex method (in algebraic form)step by step to solve the following problem.Maximize Z x1 2x2 2x3, subject to 5x1 2x2 3x3 15 x1 4x2 2x3 12 2x1 4x2 x3 8 and x1 0, x2 0, x3 0.
Work through the simplex method (in algebraic form)step by step to solve the following problem.Maximize Z x1 2x2 4x3, subject to 3x1 x2 5x3 10 x1 4x2 x3 8 2x1 4x2 2x3 7 and x1 0, x2 0, x3 0.D,I
Work through the simplex method (in algebraic form)step by step to solve the following problem.Maximize Z 4x1 3x2 6x3, subject to 3x1 x2 3x3 30 2x1 2x2 3x3 40 and x1 0, x2 0, x3 0.D,I
Follow the instructions of Prob. 4.3-2 for the model in Prob.4.1-6.D,I
Follow the instructions of Prob. 4.2-1 for the model in Prob.4.1-6.D,I
Follow the instructions of Prob. 4.2-1 for parts (a), (b), and (c).(d) Repeat part (b) for the corner-point infeasible solutions and the corresponding basic infeasible solutions.(e) Repeat part (c) for the basic infeasible solutions.
Reconsider the model in Prob.
Label each of the following statements about linear programming problems as true or false, and then justify your answer.(a) For minimization problems, if the objective function evaluated at a CPF solution is no larger than its value at every adjacent CPF solution, then that solution is optimal.(b)
Describe graphically what the simplex method does step by step to solve the following problem.Minimize Z 5x1 7x2, subject to 2x1 3x2 42 3x1 4x2 60 x1 x2 18 and x1 0, x2 0.
Describe graphically what the simplex method does step by step to solve the following problem.Maximize Z 2x1 3x2, subject to 3x1 x2 1 4x1 2x2 20 4x1 x2 10 x1 2x2 5 and x1 0, x2 0.
Repeat Prob. 4.1-4 for the following problem.Maximize Z 3x1 2x2, subject to x1 3x2 4 x1 3x2 15 2x1 x2 10 and x1 0, x2 0.
Repeat Prob. 4.1-4 for the following problem.Maximize Z x1 2x2, subject to x1 3x2 8 x1 x2 4 and x1 0, x2 0.
Consider the linear programming model (given in the back of the book) that was formulated for Prob. 3.2-3.(a) Use graphical analysis to identify all the corner-point solutions for this model. Label each as either feasible or infeasible.(b) Calculate the value of the objective function for each of
Consider the following problem.Maximize Z 3x1 2x2, subject to 2x1 x2 6 x1 2x2 6 and x1 0, x2 0.(a) Use the graphical method to solve this problem. Circle all the corner points on the graph.(b) For each CPF solution, identify the pair of constraint boundary equations it satisfies.(c) For
Consider the following problem.Maximize Z x1 2x2, subject to x1 2 x2 2 x1 x2 3 and x1 0, x2 0.(a) Plot the feasible region and circle all the CPF solutions.(b) For each CPF solution, identify the pair of constraint boundary equations that it satisfies.(c) For each CPF solution, use this
One of the most important problems in the field of statistics is the linear regression problem. Roughly speaking, this problem involves fitting a straight line to statistical data represented by points—(x1, y1), (x2, y2), . . . , (xn, yn)—on a graph. If we denote the line by y a bx, the
Redo Prob. 7.5-6 with the following revised table:
Consider a preemptive goal programming problem with three priority levels, just one goal for each priority level, and just two activities to contribute toward these goals, as summarized in the following table:(a) Use the goal programming technique to formulate one complete linear programming model
Montega is a developing country which has 15,000,000 acres of publicly controlled agricultural land in active use. Its government currently is planning a way to divide this land among three basic crops (labeled 1, 2, and 3) next year. A certain percentage of each of these crops is exported to
Reconsider the original version of the Dewright Co. problem presented in Sec. 7.5 and summarized in Table 7.5. After further reflection about the solution obtained by the simplex method, management now is asking some what-if questions.(a) Management wonders what would happen if the penalty weights
The Research and Development Division of the Emax Corporation has developed three new products. A decision now needs to be made on which mix of these products should be produced.Management wants primary consideration given to three factors:total profit, stability in the workforce, and achieving an
Management of the Albert Franko Co. has established goals for the market share it wants each of the company’s two new products to capture in their respective markets. Specifically, management wants Product 1 to capture at least 15 percent of its market and Product 2 to capture at least 10 percent
Starting from the initial trial solution (x1, x2) (2, 2), use your OR Courseware to apply 15 iterations of the interior-point algorithm presented in Sec. 7.4 to the Wyndor Glass Co. problem presented in Sec. 3.1. Also draw a figure like Fig. 7.8 to show the trajectory of the algorithm in the
Consider the following problem.Maximize Z 2x1 5x2 7x3, subject to x1 2x2 3x3 6 and x1 0, x2 0, x3 0.(a) Graph the feasible region.(b) Find the gradient of the objective function, and then find the projected gradient onto the feasible region.(c) Starting from the initial trial solution
Consider the following problem.Maximize Z x1 x2, subject to x1 2x2 9 2x1 x2 9 and x1 0, x2 0.(a) Solve the problem graphically.(b) Find the gradient of the objective function in the original x1-x2 coordinate system. If you move from the origin in the direction of the gradient until you
Consider the following problem.Maximize Z x1 2x2, subject to x1 x2 8 and x1 0, x2 0.C (a) Near the end of Sec. 7.4, there is a discussion of what the interior-point algorithm does on this problem when starting from the initial feasible trial solution (x1, x2) (4, 4). Verify the results
Consider the following problem.Maximize Z 3x1 x2, subject to x1 x2 4 and x1 0, x2 0.(a) Solve this problem graphically. Also identify all CPF solutions.C (b) Starting from the initial trial solution (x1, x2) (1, 1), perform four iterations of the interior-point algorithm presented in Sec.
Reconsider the example used to illustrate the interiorpoint algorithm in Sec. 7.4. Suppose that (x1, x2) (1, 3) were used instead as the initial feasible trial solution. Perform two iterations manually, starting from this solution. Then use the automatic routine in your OR Courseware to check
Simultaneously use the upper bound technique and the dual simplex method manually to solve the following problem.Minimize Z 3x1 4x2 2x3, subject to x1 x2 x3 15 x2 x3 10 and 0 x1 25, 0 x2 5, 0 x3 15.C
Use the upper bound technique manually to solve the following problem.Maximize Z 2x1 5x2 3x3 4x4 x5,subject to x1 3x2 2x3 3x4 x5 6 4x1 6x2 5x3 7x4 x5 15 and 0 xj 1, for j 1, 2, 3, 4, 5.
Use the upper bound technique manually to solve the following problem.Maximize Z 2x1 3x2 2x3 5x4, subject to 2x1 2x2 x3 2x4 5 x1 2x2 3x3 4x4 5 and 0 xj 1, for j 1, 2, 3, 4.
Use the upper bound technique manually to solve the following problem.Maximize Z x1 3x2 2x3, subject to x2 2x3 1 2x1 x2 2x3 8 x1 1 x2 3 x3 2 and x1 0, x2 0, x3 0.
Consider the following problem.Maximize Z 2x1 x2, subject to x1 x2 5 x1 10 x2 10 and x1 0, x2 0.(a) Solve this problem graphically.(b) Use the upper bound technique manually to solve this problem.(c) Trace graphically the path taken by the upper bound technique.
Use the upper bound technique manually to solve the Wyndor Glass Co. problem presented in Sec. 3.1.
Let Z* max nj1 cjxj, subject to nj1 aijxj bi, for i 1, 2, . . . , m, and xj 0, for j 1, 2, . . . , n(where the aij, bi, and cj are fixed constants), and let (y1*, y2*,..., y*m) be the corresponding optimal dual solution. Then let Z** max nj1 cjxj, subject to nj1 aijxj bi ki, for i
Consider the Z*() function shown in Fig. 7.2 for parametric linear programming with systematic changes in the bi parameters.(a) Explain why this function is piecewise linear.(b) Show that this function must be concave.
Consider the Z*() function shown in Fig. 7.1 for parametric linear programming with systematic changes in the cj parameters.(a) Explain why this function is piecewise linear.(b) Show that this function must be convex.
Consider Prob. 6.7-2.Use parametric linear programming to find an optimal solution as a function of over the following ranges of.(a) 0 20.(b) 20 0. (Hint: Substitute for, and then increase from zero.)
Use parametric linear programming to find an optimal solution for the following problem as a function of, for 0 30.Maximize Z() 5x1 6x2 4x3 7x4, subject to 3x1 2x2 x3 3x4 135 2 2x1 4x2 x3 2x4 78 x1 2x2 x3 2x4 30 and xj 0, for j 1, 2, 3, 4.Then identify the value of
Use the parametric linear programming procedure for making systematic changes in the bi parameters to find an optimal solution for the following problem as a function of, for 025.Maximize Z() 2x1 x2, subject to x1 10 2 x1 x2 25 x2 10 2 and x1 0, x2 0.Indicate graphically what this
Consider the following problem.Maximize Z() (10 )x1 (12 )x2 (7 2)x3, subject to x1 2x2 2x3 30 x1 x2 x3 20 and x1 0, x2 0, x3 0.I (a) Use parametric linear programming to find an optimal solution for this problem as a function of, for 0.(b) Construct the dual model for this
Use parametric linear programming to find the optimal solution for the following problem as a function of, for 0 20.Maximize Z() (20 4)x1 (30 3)x2 5x3, subject to 3x1 3x2 x3 30 8x1 6x2 4x3 75 6x1 x2 x3 45 and x1 0, x2 0, x3 0
Consider the following problem.Maximize Z 8x1 24x2, subject to x1 2x2 10 2x1 x2 10 and x1 0, x2 0.Suppose that Z represents profit and that it is possible to modify the objective function somewhat by an appropriate shifting of key personnel between the two activities. In particular, suppose
Use the dual simplex method manually to reoptimize for each of these two cases, starting from the revised final tableau.
Consider parts (a) and (b) of Prob.
Consider the example for case 1 of sensitivity analysis given in Sec. 6.7, where the initial simplex tableau of Table 4.8 is modified by changing b2 from 12 to 24, thereby changing the respective entries in the right-side column of the final simplex tableau to 54, 6, 12, and 2. Starting from this
Consider the following problem.Maximize Z 3x1 2x2, subject to 3x1 x2 12 x1 x2 6 5x1 3x2 27 and x1 0, x2 0.I (a) Solve by the original simplex method (in tabular form). Identify the complementary basic solution for the dual problem obtained at each iteration.(b) Solve the dual of this
Use the dual simplex method manually to solve the following problem.Minimize Z 7x1 2x2 5x3 4x4, subject to 2x1 4x2 7x3 x4 5 8x1 4x2 6x3 4x4 8 3x1 8x2 x3 4x4 4 and xj 0, for j 1, 2, 3, 4.
Use the dual simplex method manually to solve the following problem.Minimize Z 5x1 2x2 4x3, subject to 3x1 x2 2x3 4 6x1 3x2 5x3 10 and x1 0, x2 0, x3 0.
Consider the following problem.Maximize Z x1 x2, subject to x1 x2 8 x2 3x1 x2 2 and x1 0, x2 0.(a) Solve this problem graphically.(b) Use the dual simplex method manually to solve this problem.(c) Trace graphically the path taken by the dual simplex method.
Read the article footnoted in Sec. 2.1 that describes an OR study done for the San Francisco Police Department.(a) Summarize the background that led to undertaking this study.(b) Define part of the problem being addressed by identifying the six directives for the scheduling system to be
Read the article footnoted in Sec. 2.1 that describes an OR study done for the Health Department of New Haven, Connecticut.(a) Summarize the background that led to undertaking this study(b) Outline the system developed to track and test each needle and syringe in order to gather the needed data.(c)
Read the article footnoted in Sec. 2.2 that describes an OR study done for the Rijkswaterstaat of the Netherlands. (Focus especially on pp. 3–20 and 30–32.)(a) Summarize the background that led to undertaking this study.(b) Summarize the purpose of each of the five mathematical models described
Refer to pp. 18–20 of the article footnoted in Sec. 2.2 that describes an OR study done for the Rijkswaterstaat of the Netherlands. Describe an important lesson that was gained from model validation in this study.
Read the article footnoted in Sec. 2.5 that describes an OR study done for Texaco.(a) Summarize the background that led to undertaking this study.(b) Briefly describe the user interface with the decision support system OMEGA that was developed as a result of this study.(c) OMEGA is constantly being
Refer to the article footnoted in Sec. 2.5 that describes an OR study done for Yellow Freight System, Inc.(a) Referring to pp. 147–149 of this article, summarize the background that led to undertaking this study.(b) Referring to p. 150, briefly describe the computer system SYSNET that was
Refer to pp. 163–167 of the article footnoted in Sec. 2.5 that describes an OR study done for Yellow Freight System, Inc., and the resulting computer system SYSNET.(a) Briefly describe how the OR team gained the support of upper management for implementing SYSNET.(b) Briefly describe the
Read the article footnoted in Sec. 2.4 that describes an OR study done for IBM and the resulting computer system Optimizer.(a) Summarize the background that led to undertaking this study.(b) List the complicating factors that the OR team members faced when they started developing a model and a
Read Selected Reference 3. The author describes 13 detailed phases of any OR study that develops and applies a computer-based model, whereas this chapter describes six broader phases. For each of these broader phases, list the detailed phases that fall partially or primarily within the broader
Consider the following objective function for a linear programming model:Maximize Z 2x1 3x2(a) Draw a graph that shows the corresponding objective function lines for Z 6, Z 12, and Z 18.(b) Find the slope-intercept form of the equation for each of these three objective function lines. Compare the
Use the graphical method to solve the problem:Maximize Z 2x1 x2, subject to x2 10 2x1 5x2 60 x1 x2 18 3x1 x2 44 and x1 0, x2 0.
Use the graphical method to solve the problem:Maximize Z 10x1 20x2, subject tox1 2x2 15 x1 x2 12 5x1 3x2 45 and x1 0, x2 0.
The Whitt Window Company is a company with only three employees which makes two different kinds of hand-crafted windows: a wood-framed and an aluminum-framed window. They earn$60 profit for each wood-framed window and $30 profit for each aluminum-framed window. Doug makes the wood frames, and can
The Apex Television Company has to decide on the number of 27- and 20-inch sets to be produced at one of its factories.Market research indicates that at most 40 of the 27-inch sets and 10 of the 20-inch sets can be sold per month. The maximum number of work-hours available is 500 per month. A
The Omega Manufacturing Company has discontinued the production of a certain unprofitable product line. This act created considerable excess production capacity. Management is considering devoting this excess capacity to one or more of three products; call them products 1, 2, and 3. The available
Consider the following problem, where the value of c1 has not yet been ascertained.Maximize Z c1x1 x2, subject to x1 x2 6 x1 2x2 10 and x1 0, x2 0.Use graphical analysis to determine the optimal solution(s) for(x1, x2) for the various possible values of c1( c1 ).
Consider the following problem, where the value of k has not yet been ascertained.Maximize Z x1 2x2, subject tox1 x2 2 x2 3 kx1 x2 2k 3, where k 0 and x1 0, x2 0.The solution currently being used is x1 2, x2 3. Use graphical analysis to determine the values of k such that this
Consider the following problem, where the values of c1 and c2 have not yet been ascertained.Maximize Z c1x1 c2x2, subject to 2x1 x2 11x1 2x2 2 and x1 0, x2 0.Use graphical analysis to determine the optimal solution(s) for(x1, x2) for the various possible values of c1 and c2. (Hint:
Use the graphical method to find all optimal solutions for the following model:Maximize Z 500x1 300x2, subject to 15x1 5x2 300 10x1 6x2 240 8x1 12x2 450 and x1 0, x2 0.
Use the graphical method to demonstrate that the following model has no feasible solutions.Maximize Z 5x1 7x2, subject to 2x1 x2 1x1 2x2 1 and x1 0, x2 0.
Suppose that the following constraints have been provided for a linear programming model.x1 3x2 303x1 x2 30 and x1 0, x2 0.(a) Demonstrate that the feasible region is unbounded.(b) If the objective is to maximize Z x1 x2, does the model have an optimal solution? If so, find it. If
Reconsider Prob.
Use the graphical method to solve this problem:Maximize Z 15x1 20x2, subject to x1 2x2 10 2x1 3x2 6 x1 x2 6 and x1 0, x2 0.
Use the graphical method to solve this problem:Minimize Z 3x1 2x2, subject to x1 2x2 12 2x1 3x2 12 2x1 x2 8 and x1 0, x2 0.
Consider the following problem, where the value of c1 has not yet been ascertained.Maximize Z c1x1 2x2, subject to 4x1 x2 12 x1 x2 2 and x1 0, x2 0.Use graphical analysis to determine the optimal solution(s) for(x1, x2) for the various possible values of c1.
Consider the following model:Minimize Z 40x1 50x2, subject to 2x1 3x2 30 x1 x2 12 2x1 x2 20 and x1 0, x2 0.(a) Use the graphical method to solve this model.(b) How does the optimal solution change if the objective function is changed to Z 40x1 70x2?(c) How does the optimal
Ralph Edmund loves steaks and potatoes. Therefore, he has decided to go on a steady diet of only these two foods (plus some liquids and vitamin supplements) for all his meals. Ralph realizes that this isn’t the healthiest diet, so he wants to make sure that he eats the right quantities of the two

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