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introduction to operations research

Introduction To Operations Research 9th Edition Frederick S. Hillier - Solutions

For the game having the following payoff table, use the graphical procedure described in Sec. 14.4 to determine the value of the game and the optimal mixed strategy for each player according to the minimax criterion.
Consider the game having the following payoff table.Use the graphical procedure described in Sec. 14.4 to determine the value of the game and the optimal mixed strategy for each player according to the minimax criterion. Check your answer for player 2 by constructing his payoff table and applying
Use the graphical procedure described in Sec. 14.4 to determine the optimal mixed strategy for each player according to the minimax criterion. Also give the corresponding value of the game.
Consider the odds and evens game introduced in Sec. 14.1 and whose payoff table is shown in Table 14.1. Use the graphical procedure described in Sec. 14.4 from the viewpoint of player 1(the evens player) to determine the optimal mixed strategy for each player according to the minimax criterion.
Consider the following parlor game between two players. It begins when a referee flips a coin, notes whether it comes up heads or tails, and then shows this result to player 1 only. Player 1 may then (i) pass and thereby pay $5 to player 2 or (ii) bet. If player 1 passes, the game is terminated.
Consider the odds and evens game introduced in Sec. 14.1 and whose payoff table is shown in Table 14.1.(a) Show that this game does not have a saddle point.(b) Write an expression for the expected payoff for player 1 (the evens player) in terms of the probabilities of the two players using their
Briefly describe what you feel are the advantages and disadvantages of the minimax criterion.
Two politicians soon will be starting their campaigns against each other for a certain political office. Each must now select the main issue she will emphasize as the theme of her campaign. Each has three advantageous issues from which to choose, but the relative effectiveness of each one would
Two companies share the bulk of the market for a particular kind of product. Each is now planning its new marketing plans for the next year in an attempt to wrest some sales away from the other company. (The total sales for the product are relatively fixed, so one company can increase its sales
Find the saddle point for the game having the following payoff table.Use the minimax criterion to find the best strategy for each player.Does this game have a saddle point? Is it a stable game?
Consider the game having the following payoff table? Strategy 1 Player 1 2 1 522 3 -2 Player 2 2 3 125 -7 -2 -5 -2 252 4 257
For the game having the following payoff table, determine the optimal strategy for each player by successively eliminating dominated strategies. (Indicate the order in which you eliminated strategies.)
Reconsider Prob. 14.1-1.(a) Use the concept of dominated strategies to determine the best strategy for each side.(b) Without eliminating dominated strategies, use the minimax criterion to determine the best strategy for each side.
Consider the following parlor game to be played between two players. Each player begins with three chips: one red, one white, and one blue. Each chip can be used only once.To begin, each player selects one of her chips and places it on the table, concealed. Both players then uncover the chips and
Two manufacturers currently are competing for sales in two different but equally profitable product lines. In both cases the sales volume for manufacturer 2 is three times as large as that for manufacturer 1. Because of a recent technological breakthrough, both manufacturers will be making a major
The labor union and management of a particular company have been negotiating a new labor contract. However, negotiations have now come to an impasse, with management making a “final”offer of a wage increase of $1.10 per hour and the union making a “final” demand of a $1.60 per hour
Use your IOR Tutorial to apply the basic algorithm for all three metaheuristics presented in this chapter to the traveling salesman problem described in Prob. 13.2-7, (Use 1-2-3-4-5-6-7-8-9-10-1 as the initial trial solution for the tabu search and simulated annealing algorithms.) Which
(Use 1-2-3-4-5-6-7-8-1 as the initial trial solution for the tabu search and simulated annealing algorithms.) Which metaheuristic happened to provide the best solution on this particular problem?
Use your IOR Tutorial to apply the basic algorithm for all three metaheuristics presented in this chapter to the traveling salesman problem described in Prob.
Follow the instructions of Prob. 13.4-8 for the traveling salesman problem described in Prob. 13.2-7,
Follow the instructions of Prob. 13.4-8 for the traveling salesman problem described in Prob. 13.2-6,
Reconsider the traveling salesman problem shown in Prob. 13.1-1.(a) Perform the initialization step and the first iteration of the basic genetic algorithm presented in Sec. 13.4 by hand. Follow the instructions given at the beginning of the Problems section to obtain the needed random numbers. Show
Table 13.9 shows the application of the basic genetic algorithm described in Sec. 13.4 to the example of a traveling salesman problem depicted in Fig. 13.4 through the initialization step and first iteration of the algorithm.(a) Use your IOR Tutorial to apply this same algorithm to this same
Follow the instructions of Prob. 13.4-4 for the nonconvex programming problem shown in Prob. 13.3-10 when both of the variables x1 and x2 are restricted to be integer.A
Follow the instructions of Prob. 13.4-4 for the nonconvex programming problem shown in Prob. 13.3-9 when the variable x is restricted to be an integer.A
Reconsider the nonconvex programming problem shown in Prob. 13.3-7. Suppose now that the variable x is restricted to be an integer.(a) Perform the initialization step and the first iteration of the basic genetic algorithm presented in Sec. 13.4 by hand. Follow the instructions given at the
Table 13.7 shows the application of the basic genetic algorithm described in Sec. 13.4 to an integer nonlinear programming example through the initialization step and the first iteration.(a) Use your IOR Tutorial to apply this same algorithm to this same example, starting from another randomly
For each of the following pairs of parents, generate their two children when applying the basic genetic algorithm presented ?13.4-2.* Consider an 8-city traveling salesman problem (cities 1, 2, . . . , 8) where city 1 is the home city and links exist between all pairs of cities. For each of the
Follow the instructions of Prob. 13.3-8 for the following nonconvex programming problem when starting with (x1, x2) (18, 25) as the initial trial solution.Maximize f(x1, x2) x5 1 81x4 1 2330x3 1 28,750x2 1150,000x1 0.5x5 2 65x4 22950x3 2 53,500x2 2 305,000x2, subject to x1 2x2 110 3x1 x2 120
Follow the instructions of Prob. 13.3-8 for the following nonconvex programming problem when starting with x 25 as the initial trial solution.Maximize f(x) x6 140x5 7000x4 160,000x3 1,600,000x2 5,000,000x, subject to 0 x 50.A
Consider the example of a nonconvex programming problem presented in Sec. 12.10 and depicted in Fig. 12.18.(a) Using x 2.5 as the initial trial solution, perform the first iteration of the basic simulated annealing algorithm presented in Sec. 13.3 by hand. Follow the instructions given at the
Consider the following nonconvex programming problem.Maximize f(x) x3 60x2 900x 100, subject to 0 x 31.
Because of its use of random numbers, a simulated annealing algorithm will provide slightly different results each time it is run. Table 13.6 shows one application of the basic simulated annealing algorithm in IOR Tutorial to the nonlinear programming example introduced in Sec. 13.1. Starting with
Follow the instructions of Prob. 13.3-3 for the traveling salesman problem described in Prob. 13.2-7, using 1-9-8-10-2-4-3-6-7-5-1 as the initial trial solution>
Follow the instructions of Prob. 13.3-3 for the traveling salesman problem described in Prob. 13.2-6, using 1-2-3-4-5-6-7-8-1 as the initial trial solution.
Using 1-4-2-3-5-1 as the initial trial solution, you are to follow the instructions below for applying the basic simulated annealing algorithm presented in Sec. 13.3 to this problem.(a) Perform the first iteration by hand. Follow the instructions given at the beginning of the Problems section to
Reconsider the traveling salesman problem shown in Prob.
Because of its use of random numbers, a simulated annealing algorithm will provide slightly different results each time it is run. Table 13.5 shows one application of the basic simulated annealing algorithm in IOR Tutorial to the example of a traveling salesman problem depicted in Fig. 13.4.
While applying a simulated annealing algorithm to a certain problem, you have come to an iteration where the current value of T is T 2 and the value of the objective function for the current trial solution is 30. This trial solution has four immediate neighbors and their objective function values
Consider the 10-city traveling salesman problem whose links have the associated distances shown in the following table.
Consider the 8-city traveling salesman problem whose links have the associated distances shown in the following table(where a dash indicates the absence of a link).
Reconsider the traveling salesman problem shown in Prob. 13.1-1. Starting with 1-5-3-2-4-1 as the initial trial solution, apply the basic tabu search algorithm by hand to this problem.
Reconsider the example of an unconstrained minimum spanning tree problem given in Sec. 9.4. Suppose that the following constraints are added to the problem.Constraint 1: Either link AD or link ET must be included.Constraint 2: At most one of the three links—AO, BC, and DE—can be
Reconsider the example of a constrained minimum spanning tree problem presented in Sec. 13.2 (see Fig. 13.7(a) for the data before introducing the constraints). Starting with a different initial trial solution, namely, the one with links AB, AD, BE, and CD, apply the basic tabu search algorithm
Consider the minimum spanning tree problem depicted below, where the dashed lines represent the potential links that could be inserted into the network and the number next to each dashed line represents the cost associated with inserting that particular link?
Read the referenced article that fully describes the OR study summarized in the application vignette presented in Sec. 13.2.Briefly describe how tabu search was applied in this study. Then list the various financial and nonfinancial benefits that resulted from this study.
Consider the traveling salesman problem shown below, where city 1 is the home city
Reconsider the example of a traveling salesman problem shown in Fig. 13.4.(a) When the sub-tour reversal algorithm was applied to this problem in Sec. 13.1, the first iteration resulted in a tie for which of two sub-tour reversals (reversing 3-4 or 4-5) provided the largest decrease in the distance
Consider the traveling salesman problem shown below, where city 1 is the home city?
Consider the following problem:Maximize Z 4x1 x1 2 10x2 x2 2, subject to x1 2 4x2 2 16
Reconsider the Wyndor Glass Co. problem introduced in Sec. 3.1.C (a) Solve this problem using the standard Excel Solver.C (b) Starting with an initial solution of producing 0 batches of doors and 0 batches of windows, solve this problem using Evolutionary Solver.(c) Comment on the performance of
Because of population growth, the state of Washington has been given an additional seat in the House of Representatives, making a total of 10. The state legislature, which is currently controlled by the Republicans, needs to develop a plan for redistricting the state. There are 18 major cities in
Consider the following nonconvex programming problem:Maximize Profit 100x6 1,359x5 6,836x4 15,670x3 15,870x2 5,095x,
Consider the following nonconvex programming problem:Maximize Profit x5 13x4 59x3 107x2 61x, subject to 0 x 5.(a) Formulate this problem in a spreadsheet, and then use the Solver Table to solve this problem with the following starting points: x 0, 1, 2, 3, 4, and 5. Include the value of
Consider the following nonconvex programming problem.Minimize f(x) sin 3x1 cos 3x2 sin(x1 x2), subject to x1 2 10x2 1 10x1 x2 2 100 and x1 0, x2 0.(a) If SUMT were applied to this problem, what would be the unconstrained function P(x; r) to be minimized at each iteration?(b) Describe
Consider the following nonconvex programming problem:Maximize f(x) 3x1x2 2x1 2 x2 2, subject to x1 2 2x2 2 4 2x1 x2 3 x1x2 2 x1 2x2 2 and x1 0, x2 0.(a) If SUMT were to be applied to this problem, what would be the unconstrained function P(x; r) to be maximized at each iteration?D,C
Consider the following nonconvex programming problem:Maximize f(x) 1,000x 400x2 40x3 x4, subject to x2 x 500 and x 0.(a) Identify the feasible values for x. Obtain general expressions for the first three derivatives of f(x). Use this information to help you draw a rough sketch of f(x)
Reconsider the convex programming model with an equality constraint given in Prob. 12.6-11.(a) If SUMT were to be applied to this model, what would be the unconstrained function P(x; r) to be minimized at each iteration?D,C (b) Starting from the initial trial solution (x1, x2, x3) (3 2, 3 2,
Reconsider the first quadratic programming variation of the Wyndor Glass Co. problem presented in Sec. 12.2 (see Fig. 12.6). Beginning with the initial trial solution (x1, x2) (2, 3), use the automatic procedure in your IOR Tutorial to apply SUMT to this problem with r 102, 1, 102, 104.
Reconsider the quadratic programming model given in Prob. 12.7-4. Beginning with the initial trial solution (x1, x2) (1 2, 1 2), use the automatic procedure in your IOR Tutorial to apply SUMT to this model with r 1, 102, 104, 106.
Consider the following convex programming problem:Maximize f(x) x1x2 x1 x1 2 x2 x2 2, subject to x2 0.
Consider the following convex programming problem:Maximize f(x) 2x1 (x2 3)2, subject to x1 3 and x2 3.(a) If SUMT were applied to this problem, what would be the unconstrained function P(x; r) to be maximized at each iteration?(b) Derive the maximizing solution of P(x; r) analytically, and
Consider the example for applying SUMT given in Sec. 12.9.(a) Show that (x1, x2) (1, 2) satisfies the KKT conditions.(b) Display the feasible region graphically, and then plot the locus of points x1x2 2 to demonstrate that (x1, x2) (1, 2) with f(1, 2) 2 is, in fact, a global maximum.
Reconsider the model given in Prob. 12.3-3.(a) If SUMT were to be applied directly to this problem, what would be the unconstrained function P(x; r) to be minimized at each iteration?(b) Setting r 100 and using (x1, x2) (5, 5) as the initial trial solution, manually apply one iteration of the
Reconsider the linearly constrained convex programming model given in Prob. 12.9-9. Follow the instructions of parts (a), (b), and (c) of Prob. 12.9-10 for this model, except use (x1, x2) (1 2, 1 2)as the initial trial solution and use r 1, 102, 104, 106.
Reconsider the linearly constrained convex programming model given in Prob. 12.9-8,(a) If SUMT were to be applied to this problem, what would be the unconstrained function P(x; r) to be maximized at each iteration?
Consider the following linearly constrained convex programming problem:Maximize f(x) 4x1 x1 4 2x2 x2 2, subject to 4x1 2x2 5 and x1 0, x2 0.(a) Starting from the initial trial solution (x1, x2) (1 2, 1 2), apply four iterations of the Frank-Wolfe algorithm.(b) Show graphically how
Consider the following linearly constrained convex programming problem:Maximize f(x) 3x1 4x2 x1 3 x2 2, subject to x1 x2 1 and x1 0, x2 0.(a) Starting from the initial trial solution (x1, x2) (1 4, 1 4), apply three iterations of the Frank-Wolfe algorithm.(b) Use the KKT conditions
Consider the following linearly constrained convex programming problem:Maximize f(x) 3x1x2 40x1 30x2 4x1 2 x1 4 3x2 2 x2 4, subject to 4x1 3x2 12 x1 2x2 4 and x1 0, x2 0.Starting from the initial trial solution (x1, x2) (0, 0), apply two iterations of the Frank-Wolfe algorithm.
Reconsider the linearly constrained convex programming model given in Prob. 12.4-7. Starting from the initial trial solution (x1, x2) (0, 0), use the Frank-Wolfe algorithm(four iterations) to solve this model (approximately).
Reconsider the quadratic programming model given in Prob. 12.7-4.(a) Starting from the initial trial solution (x1, x2) (0, 0), use the Frank-Wolfe algorithm (six iterations) to solve the problem (approximately).(b) Show graphically how the sequence of trial solutions obtained in part (a) can be
Consider the quadratic programming example presented in Sec. 12.7. Starting from the initial trial solution (x1, x2) (5, 5), apply eight iterations of the Frank-Wolfe algorithm.
Reconsider the linearly constrained convex programming model given in Prob. 12.6-13. Starting from the initial trial solution (x1, x2, x3) (0, 0, 0), apply two iterations of the FrankWolfe algorithm.
Reconsider the linearly constrained convex programming model given in Prob. 12.6-12. Starting from the initial trial solution (x1, x2) (0, 0), use one iteration of the Frank-Wolfe algorithm to obtain exactly the same solution you found in part (c)of Prob. 12.6-12, and then use a second iteration
Reconsider the linearly constrained convex programming model given in Prob. 12.6-5. Starting from the initial trial solution (x1, x2) (0, 0), use one iteration of the Frank-Wolfe algorithm to obtain exactly the same solution you found in part (b)of Prob. 12.6-5, and then use a second iteration to
Reconsider the integer nonlinear programming model given in Prob. 10.3-9.(a) Show that the objective function is not concave.(b) Formulate an equivalent pure binary integer linear programming model for this problem as follows. Apply the separable programming technique with the feasible integers as
Consider the following convex programming problem:Maximize Z 32x1 x1 4 4x2 x2 2, subject to x1 2 x2 2 9 and x1 0, x2 0.(a) Apply the separable programming technique discussed at the end of Sec. 12.8, with x1 0, 1, 2, 3 and x2 0, 1, 2, 3 as the breakpoint of the piecewise linear
Consider the following nonlinear programming problem:Maximize Z 5x1 x2, subject to 2x1 2 x2 13 x1 2 x2 9 and x1 0, x2 0.(a) Show that this problem is a convex programming problem.(b) Use the separable programming technique discussed at the end of Sec. 12.8 to formulate an approximate linear
The MFG Company produces a certain subassembly in each of two separate plants. These subassemblies are then brought to a third nearby plant where they are used in the production of a certain product. The peak season of demand for this product is approaching, so to maintain the production rate
For each of the following cases, prove that the key property of separable programming given in Sec. 12.8 must hold. (Hint:Assume that there exists an optimal solution that violates this property, and then contradict this assumption by showing that there exists a better feasible solution.)(a) The
Suppose that the separable programming technique has been applied to a certain problem (the “original problem”) to convert it to the following equivalent linear programming problem:Maximize Z 5x11 4x12 2x13 4x21 x22, subject to 3x11 3x12 3x13 2x21 2x22 25 2x11 2x12 2x13
Reconsider the linearly constrained convex programming model given in Prob. 12.4-7, (a) Use the separable programming technique presented in Sec. 12.8 to formulate an approximate linear programming model for this problem. Use x1 0, 1, 2, 3 and x2 0, 1, 2, 3 as the breakpoints of the piecewise
The B. J. Jensen Company specializes in the production of power saws and power drills for home use. Sales are relatively stable throughout the year except for a jump upward during the Christmas season. Since the production work requires considerable work and experience, the company maintains a
The Dorwyn Company has two new products that will compete with the two new products for the Wyndor Glass Co.(described in Sec. 3.1). Using units of hundreds of dollars for the objective function, the linear programming model shown below has been formulated to determine the most profitable product
The MFG Corporation is planning to produce and market three different products. Let x1, x2, and x3 denote the number of units of the three respective products to be produced. The preliminary estimates of their potential profitability are as follows.For the first 15 units produced of Product 1, the
Reconsider Prob. 12.1-4 and its quadratic programming model.(a) Display this model [including the values of R(x) and V(x)] on an Excel spreadsheet.(b) Solve this model for four cases: minimum acceptable expected return 13, 14, 15, 16.
Reconsider the first quadratic programming variation of the Wyndor Glass Co. problem presented in Sec. 12.2 (see Fig. 12.6). Analyze this problem by following the instructions of parts (a), (b), and (c) of Prob. 12.7-4.
Consider the following quadratic programming problem.Maximize f(x) 2x1 3x2 x1 2 x2 2, subject to x1 x2 2 and x1 0, x2 0.(a) Use the KKT conditions to derive an optimal solution directly.(b) Now suppose that this problem is to be solved by the modified simplex method. Formulate the linear
Consider the following quadratic programming problem:Maximize f(x) 250x1 25x1 2 100x2 100x2 2 90x1x2, subject to 20x1 5x2 90 10x1 10x2 60 and x1 0, x2 0.Suppose that this problem is to be solved by the modified simplex method.(a) Formulate the linear programming problem that is to be
Consider the following quadratic programming problem:Maximize f(x) 8x1 x1 2 4x2 x2 2, subject to x1 x2 2 and x1 0, x2 0.(a) Use the KKT conditions to derive an optimal solution.(b) Now suppose that this problem is to be solved by the modified simplex method. Formulate the linear
Consider the quadratic programming example presented in Sec. 12.7.(a) Use the test given in Appendix 2 to show that the objective function is strictly concave.(b) Verify that the objective function is strictly concave by demonstrating that Q is a positive definite matrix; that is, xT Qx 0 for all x
Use the KKT conditions to determine whether (x1, x2) (2, 2) can be optimal.
Reconsider the linearly constrained convex programming model given in Prob.
What are the KKT conditions for this problem? Use these conditions to determine whether (x1, x2) (1, 1) can be optimal.
Reconsider the model given in Prob.
Use the KKT conditions to determine whether (x1, x2, x3) (1, 1, 1) can be optimal for the following problem:Minimize Z 2x1 x2 3 x3 2, subject to x1 2 2x2 2 x3 2 4 and x1 0, x2 0, x3 0.
Consider the following linearly constrained convex programming problem:Maximize f(x) 8x1 x1 2 2x2 x3,
Consider the following linearly constrained convex programming problem:Minimize Z x1 2 6x1 x2 3 3x2, subject to x1 x2 1 and x1 0, x2 0.(a) Obtain the KKT conditions for this problem.(b) Use the KKT conditions to check whether (x1, x2) (1 2, 1 2) is an optimal solution.(c) Use the
Consider the following linearly constrained programming problem:Minimize f(x) x1 3 4x2 2 16x3, subject to x1 x2 x3 5 and x1 1, x2 1, x3 1.(a) Convert this problem to an equivalent nonlinear programming problem that fits the form given at the beginning of the chapter (second paragraph),
Consider the following nonlinear programming problem:Minimize Z 2x1 2 x2 2, subject to x1 x2 10 and x1 0, x2 0.(a) Of the special types of nonlinear programming problems described in Sec. 12.3, to which type or types can this particular problem be fitted? Justify your answer. (Hint: First

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