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

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 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
For each of the following functions, show whether it is convex, concave, or neither. (a) f (x) = 10x – x2 (b) f (x) = x4 + 6x2 + 12x (c) f (x) = 2x3 – 3x2 (d) f (x) = x4 + x2 (e) f (x) = x3 + x4
Reconsider the linearly constrained convex programming model given in Prob. 13.4-7. (a) Use the separable programming technique presented in Sec. 13.8 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
Suppose that the separable programming technique has been applied to a certain problem (the €œoriginal problem€) to convert it to the following equivalent linear programming problem: Maximize Z = 5x11 + 4x12 + 2x13 + 4x21 + x22,subject toand What was the mathematical model for
For each of the following cases, prove that the key property of separable programming given in Sec. 13.8 must hold. (a) The special case of separable programming where all the gi(x) are linear functions. (b) The general case of separable programming where all the functions are nonlinear functions
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
Consider the following nonlinear programming problem: Maximize Z = 5x1 + x2, subject to 2x12 + x2 ≤ 13 x12 + 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. 13.8 to formulate an
Consider the following convex programming problem: Maximize Z = 32x1 – x41 + 4x2 – x22, Subject to x21 + x22 ≤ 9 and x1 ≥ 0, x2 ≥ 0. (a) Apply the separable programming technique discussed at the end of Sec. 13.8, with x1 = 0, 1, 2, 3 and x2 = 0, 1, 2, 3 as the breakpoint of the piecewise
Reconsider the integer nonlinear programming model given in Prob. 11.3-9. (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
Reconsider the linearly constrained convex programming model given in Prob. 13.6-5. 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. 13.6-5, and then use a second iteration
Reconsider the linearly constrained convex programming model given in Prob. 13.6-12. 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. 13.6-12, and then use a second iteration
Reconsider the linearly constrained convex programming model given in Prob. 13.6-13. Starting from the initial trial solution (x1, x2, x3) = (0, 0, 0), apply two iterations of the Frank- Wolfe algorithm.
For each of the following functions, use the test given in Appendix 2 to determine whether it is convex, concave, or neither. (a) f (x) = x1x2 – x21 – x22 (b) f (x) = 3x1 + 2x21 + 4x2 + x22 – 2x1x2 (c) f (x) = x21 + 3x1x2 + 2x22 (d) f (x) = 20x1 + 10x2 (e) f (x) = x1x2
Consider the quadratic programming example presented in Sec. 13.7. Starting from the initial trial solution (x1, x2) = (5, 5), apply eight iterations of the Frank-Wolfe algorithm.
Reconsider the quadratic programming model given in Prob. 13.7-4.
Reconsider the linearly constrained convex programming model given in Prob. 13.4-7. Starting from the initial trial solution (x1, x2) = (0, 0), use the Frank-Wolfe algorithm (four iterations) to solve this model (approximately).
Consider the following linearly constrained convex programming problem: Maximize f(x) = 3x1 x2 + 40x1 + 30x2 – 4x21 – x41 – 3x22 – x42, Subject to 4x1 + 3x2 ≤ 12 x1 + 2x2 ≤ 4 and x1 ≥ 0, x2 ≥ 0.
Consider the following linearly constrained convex programming problem: Maximize f(x) = 3x1 + 4x2 – x31 – x32, subject to x1 +x2 ≤ 1 and x1 ≥ 0, x2 ≥ 0.
Consider the following linearly constrained convex programming problem: Maximize f(x) = 4x1 – x41 + 2x2 – x22, Subject to 4x1 + 2x2 ≤ 5 And x1 ≥ 0, x2 ≥ 0.
Reconsider the linearly constrained convex programming model given in Prob. 13.9-8. (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?
Reconsider the linearly constrained convex programming model given in Prob. 13.9-9. Follow the instructions of parts (a), (b), and (c) of Prob. 13.9-10 for this model, except use (x1, x2) = (1/2, 1/2) as the initial trial solution and use r = 1, 10-2, 10-4, 10-6.
Reconsider the model given in Prob. 13.3-3. (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?
Consider the example for applying SUMT given in Sec. 13.9. (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.
Consider the following function:Show that f (x) is convex by expressing it as a sum of functions of one or two variables and then showing (see Appendix 2) that all these functions are convex.
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?
Consider the following convex programming problem: Maximize f (x) = x1x2 – x1 – x12 – x2 – x22, subject to x2 ≥ 0.
Reconsider the quadratic programming model given in Prob. 13.7-4. 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, 10-2, 10-4, 10-6.
Reconsider the first quadratic programming variation of the Wyndor Glass Co. problem presented in Sec. 13.2 (see Fig. 13.6). 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, 10-2, 10-4.
Reconsider the convex programming model with an equality constraint given in Prob. 13.6-11. (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?
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
Consider the following nonconvex programming problem:Maximize f(x) = 3x1 x2 €“ 2x21 €“ x32,Subject toand 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?
Consider the following nonconvex programming problem: Minimize f (x) = sin 3x1 + cos 3x2 + sin(x1 + x2), subject to x12 – 10x2 ≥ – 1 10x1 + x22 ≤ 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
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 GRG Nonlinear solving method with the Multistart option to solve this problem. (b) Use Evolutionary
Consider the following nonconvex programming problem: Maximize Profit = 100x6 – 1,359x5 + 6,836x4 – 15,670x3 + 15,870x2 – 5,095x, subject to 0 ≤ x ≤ 5. (a) Formulate this problem in a spreadsheet, and then use the GRG Nonlinear solving method with the Multistart option to solve this
Consider the following nonlinear programming problem: Maximize f(x) = x1 + x2, Subject to x21 + x22 ≤ 0. (a) Verify that this is a convex programming problem. (b) Solve this problem graphically.
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
Reconsider the Wyndor Glass Co. problem introduced in Sec. 3.1. (a) Solve this problem using Solver.
Consider the following problem: Maximize Z = 4x1 – x12 + 10x2 – x22, subject to x12 + 4x22 ≤ 16 and x1 ≥ 0, x2 ≥ 0. (a) Is this a convex programming problem? Answer yes or no, and then justify your answer. (b) Can the modified simplex method be used to solve this problem? Answer yes or
Ever since the day she took her first economics class in high school, Lydia wondered about the financial practices of her parents. They worked very hard to earn enough money to live a comfortable middle-class life, but they never made their money work for them. They simply deposited their
A financial analyst is holding some German bonds that offer increasing interest rates if they are kept until their full maturity in three more years. They also can be redeemed at any time to obtain the original principal plus the accrued interest. The German federal government has just introduced a
This case continues Case 3.4 involving an advertising campaign for Super Grain Corporation’s new breakfast cereal. The analysis requested for Case 3.4 leads to the application of linear programming. However, certain assumptions of linear programming are quite questionable in this situation. In
Consider the traveling salesman problem shown below, where city 1 is the home city(a) List all the possible tours, except exclude those that are simply the reverse of previously listed tours. Calculate the distance of each of these tours and thereby identify the optimal tour.
Reconsider the example of a constrained minimum spanning tree problem presented in Sec. 14.2 (see Fig. 14.7(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
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
Reconsider the traveling salesman problem shown in Prob. 14.1-1. Using 1-2-3-4-5-1 as the initial trial solution, you are to follow the instructions below for applying the basic simulated annealing algorithm presented in Sec. 14.3 to this problem. (a) Perform the first iteration by hand. Follow the
Consider the following nonconvex programming problem. Maximize f(x) = x3 – 60x2 + 900x + 100, subject to 0 ≤ x ≤ 31. (a) Use the first and second derivatives of f(x) to determine the critical points (along with the end points of the feasible region) where x is either a local maximum or a
Consider the example of a nonconvex programming problem presented in Sec. 13.10 and depicted in Fig. 13.18.
Follow the instructions of Prob. 14.3-8 for the following nonconvex programming problem when starting with x = 25 as the initial trial solution. Maximize f(x) = x6 - 136x5 + 6800x4 - 155,000x3 + 1,570,000x2 - 5,000,000x, subject to 0 ≤ x ≤ 50. (a) Using x = 2.5 as the initial trial solution,
Follow the instructions of Prob. 14.3-8 for the following nonconvex programming problem when starting with (x1, x2) = (18, 25) as the initial trial solution. Maximize subject to x1 + 2x2 ≤ 110 3x1 + x2 ≤ 120 and 0 ≤ x1 ≤ 36, 0 ≤ x2 ≤ 50. (a) Using x = 2.5 as the initial trial
For each of the following pairs of parents, generate their two children when applying the basic genetic algorithm presented in Sec. 14.4 to an integer nonlinear programming problem involving only a single variable x, which is restricted to integer values over the interval 0 ≤ x ≤ 63. (Follow
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 following pairs of parents, generate their two children when applying the basic genetic algorithm presented in Sec. 14.4. (Follow the
Reconsider the nonconvex programming problem shown in Prob. 14.3-7. 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. 14.4 by hand. Follow the instructions given at the
Reconsider the example of a traveling salesman problem shown in Fig. 14.4. (a) When the sub-tour reversal algorithm was applied to this problem in Sec. 14.1, 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
Reconsider the traveling salesman problem shown in Prob. 14.1-1. (a) Perform the initialization step and the first iteration of the basic genetic algorithm presented in Sec. 14.4 by hand. Follow the instructions given at the beginning of the Problems section to obtain the needed random numbers.
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. 14.2-6. (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
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. 14.2-7. (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
Consider the traveling salesman problem shown below, where city 1 is the home city.
Reconsider the example of an unconstrained minimum spanning tree problem given in Sec. 10.4. 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 included.
Reconsider the traveling salesman problem shown in Prob. 14.1-1. Starting with 1-2-4-3-5-1 as the initial trial solution, apply the basic tabu search algorithm by hand to this problem.
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).
Consider the 10-city traveling salesman problem whose links have the associated distances shown in the following table.
Read the referenced article that fully describes the OR study summarized in the application vignette presented in Sec. 14.2. 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 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.This problem also has the following two
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
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.)
Consider the game having the following payoff table:Determine the optimal strategy for each player by successively eliminating dominated strategies. Give a list of the dominated strategies (and the corresponding dominating strategies) in the order in which you were able to eliminate them.
Consider the odds and evens game introduced in Sec. 15.1 and whose payoff table is shown in Table 15.1. (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
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. 15.1 and whose payoff table is shown in Table 15.1. Use the graphical procedure described in Sec. 15.4 from the viewpoint of player 1 (the evens player) to determine the optimal mixed strategy for each player according to the minimax criterion.
Reconsider Prob. 15.3-2. Use the graphical procedure described in Sec. 15.4 to determine the optimal mixed strategy for each player according to the minimax criterion. Also give the corresponding value of the game.
Consider the game having the following payoff table:Use the graphical procedure described in Sec. 15.4 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
For the game having the following payoff table, use the graphical procedure described in Sec. 15.4 to determine the value of the game and the optimal mixed strategy for each player according to the minimax criterion.
The A. J. Swim Team soon will have an important swim meet with the G. N. Swim Team. Each team has a star swimmer (John and Mark, respectively) who can swim very well in the 100- yard butterfly, backstroke, and breaststroke events. However, the rules prevent them from being used in more than two of
Consider the odds and evens game introduced in Sec. 15.1 and whose payoff table is shown in Table 15.1. (a) Use the approach described in Sec. 15.5 to formulate the problem of finding optimal mixed strategies according to the minimax criterion as two linear programming problems, one for player 1
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
Refer to the last paragraph of Sec. 15.5. Suppose that 3 were added to all the entries of Table 15.6 to ensure that the corresponding linear programming models for both players have feasible solutions with x3 ≥ 0 and y4 ≥ 0. Write out these two models. Based on the information given in Sec.
Consider the game having the following payoff table:(a) Use the approach described in Sec. 15.5 to formulate the problem of finding optimal mixed strategies according to the minimax criterion as a linear programming problem. (b) Use the simplex method to find these optimal mixed strategies.
Follow the instructions of Prob. 15.5-3 for the game having the following payoff table:(a) Use the approach described in Sec. 15.5 to formulate the problem of finding optimal mixed strategies according to the minimax criterion as a linear programming problem. (b) Use the simplex method to find
Follow the instructions of Prob. 15.5-3 for the game having the following payoff table:(a) Use the approach described in Sec. 15.5 to formulate the problem of finding optimal mixed strategies according to the minimax criterion as a linear programming problem. (b) Use the simplex method to find
Section 15.5 presents a general linear programming formulation for finding an optimal mixed strategy for player 1 and for player 2. Using Table 6.14, show that the linear programming problem given for player 2 is the dual of the problem given for player 1.
Consider the linear programming models for players 1 and 2 given near the end of Sec. 15.5 for variation 3 of the political campaign problem (see Table 15.6). Follow the instructions of Prob. 15.5-6 for these two models.
Consider variation 3 of the political campaign problem (see table 15.6). refer to the resulting linear programming model for player 1 given near the end of sec. 15.5. ignoring the objective function variable x3, plot the feasible region for x1 and x2 graphically (as described in sec. 3.1). (hint:
Consider the linear programming model for player 1 given near the end of Sec. 15.5 for variation 3 of the political campaign problem (see Table 15.6). Verify the optimal mixed strategies for both players given in Sec. 15.5 by applying an automatic routine for the simplex method to this model to
Consider the general m × n, two-person, zero-sum game. Let pij denote the payoff to player 1 if he plays his strategy i (i = 1, . . . , m) and player 2 plays her strategy j ( j = 1, . . . , n). Strategy 1 (say) for player 1 is said to be weakly dominated by strategy 2 (say) if p1j ≤ p2j for j =
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
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?
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?
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
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
Briefly describe what you feel are the advantages and disadvantages of the minimax criterion.
Reconsider Prob. 15.1-1. (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.
Read the referenced article that fully describes the OR study summarized in the application vignette presented in Sec. 16.2. Briefly describe how decision analysis was applied in this study. Then list the various financial and nonfinancial benefits that resulted from this study
An athletic league does drug testing of its athletes, 10 percent of whom use drugs. This test, however, is only 95 percent reliable. That is, a drug user will test positive with probability 0.95 and negative with probability 0.05, and a nonuser will test negative with probability 0.95 and positive
Management of the Telemore Company is considering developing and marketing a new product. It is estimated to be twice as likely that the product would prove to be successful as unsuccessful. It it were successful, the expected profit would be $1,500,000. If unsuccessful, the expected loss would be
The Hit-and-Miss Manufacturing Company produces items that have a probability p of being defective. These items are produced in lots of 150. Past experience indicates that p for an entire lot is either 0.05 or 0.25. Furthermore, in 80 percent of the lots produced, p equals 0.05 (so p equals 0.25 in
Consider two weighted coins. Coin 1 has a probability of 0.3 of turning up heads, and coin 2 has a probability of 0.6 of turning up heads. A coin is tossed once; the probability that coin 1 is tossed is 0.6, and the probability that coin 2 is tossed is 0.4. The decision maker uses Bayes€™
There are two biased coins with probabilities of landing heads of 0.8 and 0.4, respectively. One coin is chosen at random (each with probability 1/2) to be tossed twice. You are to receive $100 if you correctly predict how many heads will occur in two tosses. (a) Using Bayes’ decision rule, what
Read the referenced article that fully describes the OR study summarized in the application vignette presented in Sec. 16.3. Briefly describe how decision analysis was applied in this study. Then list the various financial and nonfinancial benefits that resulted from this study.
Reconsider Prob. 16.2-2. Management of Silicon Dynamics now is considering doing full-fledged market research at a cost of $1 million to predict which of the two levels of demand is likely to occur. Previous experience indicates that such market research is correct two-thirds of the time. Assume

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