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

Reconsider Prob. 12.3-6(a). Use the BIP branch-andbound algorithm presented in Sec. 12.6 to solve this BIP model interactively.
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.
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. (b) Use this algorithm to solve this problem.
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 following nonlinear BIP problem:MaximizeSubject to xj is binary, for j = 1, 2, 3, 4.
Consider the Lagrangian relaxation described near the end of Sec. 12.6.
Read the referenced article that fully describes the OR study summarized in the application vignette presented in Sec. 12.7. Briefly describe how integer programming was applied in this study. Then list the various financial and nonfinancial benefits that resulted from this study.
Consider the following IP problem: Maximize Z = –3x1 + 5x2, Subject to 5x1 – 7x2 ≥ 3 and xj ≤ 3 xj ≥ 0 xj is integer, for j = 1, 2. (a) Solve this problem graphically. (b) Use the MIP branch-and-bound algorithm presented in Sec. 12.7 to solve this problem by hand. For each subproblem,
The board of directors of General Wheels Co. is considering six large capital investments. Each investment can be made only once. These investments differ in the estimated long-run profit (net present value) that they will generate as well as in the amount of capital required, as shown by the
Follow the instructions of Prob. 12.7-2 for the following IP model:Minimize Z = 2x1 + 3x2,Subject toAnd x1 ‰¥ 0, x2 ‰¥ 0 x1, x2 are integer. (a) Solve this problem graphically. (b) Use the MIP branch-and-bound algorithm presented in Sec. 12.7 to solve this problem by hand. For
Reconsider the IP model of Prob. 12.5-2. (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. (b) Now use the interactive procedure for this algorithm in your IOR Tutorial to solve this
Consider the IP example discussed in Sec. 12.5 and illustrated in Fig. 12.3. Use the MIP branch-and-bound algorithm presented in Sec. 12.7 to solve this problem interactively.
Reconsider Prob. 12.3-5a. Use the MIP branch-and bound algorithm presented in Sec. 12.7 to solve this IP problem interactively.
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
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 toand xj ‰¥ 0, for j = 1, 2, 3, 4 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 toand 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:Minimize Z = 5x1 + x2 + x3 + 2x4 + 3x5,Subject toand xj ‰¥ 0, for j = 1, 2, 3, 4, 5 xj is integer, for j = 1, 2, 3.
For each of the following constraints of pure BIP problems, use the constraint to fix as many variables as possible: (a) 4x1 + x2 + 3x3 + 2x4 ≤ 2 (b) 4x1 – x2 + 3x3 + 2x4 ≤ 2 (c) 4x1 – x2 + 3x3 + 2x4 ≥ 7
For each of the following constraints of pure BIP problems, use the constraint to fix as many variables as possible: (a) 20x1 – 7x2 + 5x3 ≤ 10 (b) 10x1 – 7x2 + 5x3 ≥ 10 (c) 10x1 – 7x2 + 5x3 ≤ –1
Reconsider Prob. 9.3-4, where a swim team coach needs to assign swimmers to the different legs of a 200-yard medley relay team. Formulate a BIP model for this problem. Identify the groups of mutually exclusive alternatives in this formulation.
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.
For each of the following constraints of pure BIP problems, identify which ones are made redundant by the binary constraints. Explain why each one is, or is not, redundant. (a) 2x1 + x2 + 2x3 ≤ 5 (b) 3x1 – 4x2 + 5x3 ≤ 5 (c) x1 + x2 + x3 ≥ 2 (d) 3x1 – x2 – 2x3 ≥ –4
In Sec. 12.8, at the end of the subsection on tightening constraints, we indicated that the constraint 4x1 – 3x2 + x3 + 2x4 ≤ 5 can be tightened to 2x1 – 3x2 + x3 + 2x4 ≤ 3 and then to 2x1 – 2x2 + x3 + 2x4 ≤ 3. Apply the procedure for tightening constraints to confirm these results.
Apply the procedure for tightening constraints to the following constraint for a pure BIP problem: 3x1 – 2x2 + x3 ≤ 3.
Apply the procedure for tightening constraints to the following constraint for a pure BIP problem: x1 – x2 + 3x3 + 4x4 ≥ 1.
Apply the procedure for tightening constraints to each of the following constraints for a pure BIP problem: (a) x1 + 3x2 – 4x3 ≤ 2. (b) 3x1 – x2 + 4x3 ≥ 1.
One of the constraints of a certain pure BIP problem is x1 + 3x2 + 2x3 + 4x4 ≤ 5. Identify all the minimal covers for this constraint, and then give the corresponding cutting planes.
One of the constraints of a certain pure BIP problem is 3x1 + 4x2 + 2x3 + 5x4 ≤ 7. Identify all the minimal covers for this constraint, and then give the corresponding cutting planes.
Generate as many cutting planes as possible from the following constraint for a pure BIP problem: 3x1 + 5x2 + 4x3 + 8x4 ≤ 10.
Generate as many cutting planes as possible from the following constraint for a pure BIP problem. 5x1 + 3x2 + 7x3 + 4x4 + 6x5 ≤ 9.
Vincent Cardoza is the owner and manager of a machine shop that does custom order work. This Wednesday afternoon, he has received calls from two customers who would like to place rush orders. One is a trailer hitch company which would like some custom-made heavy-duty tow bars. The other is a
Consider the following BIP problem:MaximizeSubject to and all xj binary.
Consider the following problem: Maximize Z = 3x1 + 2x2 + 4x3 + x4, Subject to x1 ∈ {1, 3}, x2 ∈ {1, 2}, x3 ∈ {2, 3}, x4 ∈ {1, 2, 3, 4}, all these variables must have different values, x1 + x2 + x3 + x4 ≤ 10.
Consider the following problem:MaximizeSubject to x1 ˆˆ {3, 6, 12}, x2 ˆˆ {3, 6}, x3 ˆˆ {3, 6, 9, 12}, x4 ˆˆ {6, 12}, x5 ˆˆ {9, 12, 15, 18}, all these variables must have different values, x1 + x3 + x4 + 25.
Consider the following problem:MaximizeSubject to x1 ˆˆ {25, 30}, x2 ˆˆ {20, 25, 30, 35, 40, 50}, x3 ˆˆ {20, 25, 30}, x4 ˆˆ {20, 25}, all these variables must have different values, x2 + x3 + 60, x1 + x3 + 50.
Consider the Job Shop Co. example introduced in Sec. 9.3. Table 9.25 shows its formulation as an assignment problem. Use global constraints to formulate a compact constraint programming model for this assignment problem.
Consider the problem of assigning swimmers to the different legs of a medley relay team that is presented in Prob. 9.3-4. The answer in the back of the book shows the formulation of this problem as an assignment problem. Use global constraints to formulate a compact constraint programming model for
Consider the problem of determining the best plan for how many days to study for each of four final examinations that is presented in Prob. 11.3-3. Formulate a compact constraint programming model for this problem.
Problem 11.3-2 describes how the owner of a chain of three grocery stores needs to determine how many crates of fresh strawberries should be allocated to each of the stores. Formulate a compact constraint programming model for this problem.
One powerful feature of constraint programming is that variables can be used as subscripts for the terms in the objective function. For example, consider the following traveling salesman problem. The salesman needs to visit each of n cities (city 1, 2, . . . ,n) exactly once, starting in city 1
Bentley Hamilton throws the business section of The New York Times onto the conference room table and watches as his associates jolt upright in their overstuffed chairs.Mr. Hamilton wants to make a point.He throws the front page of The Wall Street Journal on top of The New York Times and watches as
Reconsider Prob. 9.2-21 involving a contractor (Susan Meyer) who needs to arrange for hauling gravel from two pits to three building sites. Susan now needs to hire the trucks (and their drivers) to do the hauling. Each truck can only be used to haul gravel from a single pit to a single site. In
Plans are being made for an exhibit of up-and-coming modern artists at the San Francisco Museum of Modern Art. A long list of possible artists, their available pieces, and the display prices for these pieces has been compiled. There also are various constraints regarding the mix of pieces that can
Poor inventory management at the local warehouse for Furniture City has led to overstocking of many items and frequent shortages of some others. To begin to rectify this situation, the 20 most popular kitchen sets in Furniture City’s kitchen department have just been identified. These kitchen
As introduced in Case 4.3 and revisited in Case 7.3, the Springfield School Board needs to assign the middle school students in the city’s six residential areas to the three remaining middle schools. The new complication in that the school board has just made the decision to prohibit the
The Research and Development Division of the Progressive Company has been developing four possible new product lines. Management must now make a decision as to which of these four products actually will be produced and at what levels. Therefore, an operations research study has been requested to
Suppose that a mathematical model fits linear programming except for the restriction that |x1 – x2 | = 0, or 3, or 6. Show how to reformulate this restriction to fit an MIP model.
Read the referenced article that fully describes the OR study summarized in the application vignette presented in Sec. 13.1. Briefly describe how nonlinear programming was applied in this study. Then list the various financial and nonfinancial benefits that resulted from this study.
Consider the following nonlinear programming problem: Minimize Z = x41 + 2x22, Subject to x21 + x22 ≥ 2. (No nonnegativity constraints.) (a) Use geometric analysis to determine whether the feasible region is a convex set. (b) Now use algebra and calculus to determine whether the feasible region
Reconsider Prob. 13.1-2. Verify that this problem is a convex programming problem.
Reconsider Prob. 13.1-4. Show that the model formulated is a convex programming problem by using the test in Appendix 2 to show that the objective function being minimized is convex.
Consider the variation of the Wyndor Glass Co. example represented in Fig. 13.5, where the second and third functional constraints of the original problem (see Sec. 3.1) have been replaced by 9x12 + 5x22 ≤ 216. Demonstrate that (x1, x2) = (2, 6) with Z = 36 is indeed optimal by showing that the
Consider the variation of the Wyndor Glass Co. problem represented in Fig. 13.6, where the original objective function (see Sec. 3.1) has been replaced by Z = 126x1 – 9x12 + 182x2 – 13x22. Demonstrate that (x1, x2) = (8/3, 5) with Z = 857 is indeed optimal by showing that the ellipse 857 =
Consider the following constrained optimization problem: Maximize f(x) = –6x + 3x2 – 2x3, Subject to x ≥ 0.
Consider the following nonlinear programming problem:Minimize Z = x41 + 2x21 + 2x1 x2 + 4x22,Subject toand x1 ‰¥ 0, x2 ‰¥ 0. (a) Of the special types of nonlinear programming problems described in Sec. 13.3, to which type or types can this particular problem be fitted? Justify
Consider the following geometric programming problem: Minimize f(x) = 2x1–2x2–1 + x2–2, Subject to 4x1x2 + x21x22 ≤ 12 And x1 ≥ 0, x2 ≥ 0. (a) Transform this problem to an equivalent convex programming problem. (b) Use the test given in Appendix 2 to verify that the model formulated in
Consider the following linear fractional programming problem:MaximizeSubject to and x1 ‰¥ 0, x2 ‰¥ 0. (a) Transform this problem to an equivalent linear programming problem. $Consider the following linear fractional programming problem: Maximize Subject to and x1 ‰¥ 0,$ $Consider the following linear fractional programming problem: Maximize Subject to and x1 ‰¥ 0,$
Consider the expressions in matrix notation given in Sec. 13.7 for the general form of the KKT conditions for the quadratic programming problem. Show that the problem of finding a feasible solution for these conditions is a linear complementarity problem, as introduced in Sec. 13.3, by identifying
Consider the product mix problem described in Prob. 3.1-11. Suppose that this manufacturing firm actually encounters price elasticity in selling the three products, so that the profits would be different from those stated in Chap. 3. In particular, suppose that the unit costs for producing products
Consider the following problem: Maximize f(x) = x3 + 2x – 2x2 – 0.25x4. (a) Apply the bisection method to (approximately) solve this problem. Use an error tolerance ϵ = 0.04 and initial bounds x = 0, x-bar = 2.4. (b) Apply Newton’s method, with ϵ = 0.001 and x1 = 1.2, to this problem.
Use the bisection method with an error tolerance ϵ = 0.04 and with the following initial bounds to interactively solve (approximately) each of the following problems. (a) Maximize f(x) = 6x – x2, with x = 0, x-bar = 4.8. (b) Minimize f(x) = 6x + 7x2 + 4x3 + x4, with x = – 4, = 1,
Consider the following problem: Maximize f(x) = 48x5 + 42x3 + 3.5 x – 16x6 – 61x4 – 16.5x2. (a) Apply the bisection method to (approximately) solve this problem. Use an error tolerance ϵ = 0.08 and initial bounds x = –1, = 4. (b) Apply Newton’s method, with ϵ = 0.001 and x1 = 1, to
Consider the following problem: Maximize f(x) = x3 + 30x – x6 – 2x4 – 3x2. (a) Apply the bisection method to (approximately) solve this problem. Use an error tolerance ϵ = 0.07 and find appropriate initial bounds by inspection. (b) Apply Newton’s method, with ϵ = 0.001 and x1 = 1, to this
Consider the following convex programming problem: Minimize Z = x4 + x2 – 4x, Subject to x ≤ 2 and x ≥ 0.
Consider the problem of maximizing a differentiable function f(x) of a single unconstrained variable x. Let x0 and 0, respectively, be a valid lower bound and upper bound on the same global maximum (if one exists). Prove the following general properties of the bisection method (as presented in
Consider the following linearly constrained convex programming problem:Maximize f(x) = 32x1 + 50x2 €“ 10x22 + x32 €“ x41 €“ x42,Subject toand x1 ‰¥ 0, x2 ‰¥ 0.
Consider the following unconstrained optimization problem: Maximize f(x) = 2x1 x2 + x2 – x21 – 2x22.
Starting from the initial trial solution (x1, x2) = (1, 1), interactively apply two iterations of the gradient search procedure to begin solving the following problem, and then apply the automatic routine for this procedure (with ϵ = 0.01). Maximize f(x) = 4x1 x2 - 2x21 - 3x22. Then solve ∆f(x)
Starting from the initial trial solution (x1, x2) = (0, 0), interactively apply the gradient search procedure with ϵ = 0.3 to obtain an approximate solution for the following problem, and then apply the automatic routine for this procedure (with ϵ = 0.01). Maximize f (x) = 8x1 - x12 - 12x2 - 2x22
For the P & T Co. problem described in Sec. 9.1, suppose that there is a 10 percent discount in the shipping cost for all truckloads beyond the first 40 for each combination of cannery and warehouse. Draw figures like Figs. 13.3 and 13.4, showing the marginal cost and total cost for shipments of
Starting from the initial trial solution (x1, x2) = (0, 0), interactively apply two iterations of the gradient search procedure to begin solving the following problem, and then apply the automatic routine for this procedure (with ϵ = 0.01). Maximize f(x) = 6x1 + 2x1 x2 - 2x2 - 2x21 - x22. Then
Starting from the initial trial solution (x1, x2) = (0, 0), apply one iteration of the gradient search procedure to the following problem by hand: Maximize f(x) = 4x1 + 2x2 + x21 - x41 - 2x1x2 - x22. To complete this iteration, approximately solve for t* by manually applying two iterations of the
Consider the following unconstrained optimization problem: Maximize f(x) = 3x1x2 + 3x2x3 – x21 – 6x22 – x23. (a) Describe how solving this problem can be reduced to solving a two-variable unconstrained optimization problem.
Starting from the initial trial solution (x1, x2) = (0, 0), interactively apply the gradient search procedure with ϵ = 1 to solve (approximately) the following problem, and then apply the automatic routine for this procedure (with ϵ = 0.01). Maximize f(x) = x1x2 + 3x2 - x21 - x22.
Reconsider the one-variable convex programming model given in Prob. 13.4-5. Use the KKT conditions to derive an optimal solution for this model.
Reconsider Prob. 13.2-9. Use the KKT conditions to check whether (x1, x2) = (1/√2, 1/ √2) is optimal.
Reconsider the model given in Prob. 13.3-3. What are the KKT conditions for this model? Use these conditions to determine whether (x1, x2) = (0, 10) can be optimal.
Consider the following convex programming problem: Maximize f(x) = 24x1 – x21 + 10x2 – x22, Subject to x1 ≤ 10, x2 ≤ 15, and x1 ≥ 0, x2 ≥ 0.
Consider the following linearly constrained optimization problem: Maximize f(x) = In (x1 + 1) – x22, Subject to x1 + 2x2 ≤ 3 and x1 ≥ 0, x2 ≥ 0. where In denotes the natural logarithm, (a) Verify that this problem is a convex programming problem.
Consider the nonlinear programming problem given in Prob. 11.3-11. Determine whether (x1, x2) = (1, 2) can be optimal by applying the KKT conditions.
A stockbroker, Richard Smith, has just received a call from his most important client, Ann Hardy. Ann has $50,000 to invest and wants to use it to purchase two stocks. Stock 1 is a solid blue-chip security with a respectable growth potential and little risk involved. Stock 2 is much more
Consider the following nonlinear programming problem:MaximizeSubject to x1 €“ x2 ‰¤ 2 and x1 ‰¥ 0, x2 ‰¥ 0. (a) Use the KKT conditions to demonstrate that (x1, x2) = (4, 2) is not optimal.
Use the KKT conditions to derive an optimal solution for each of the following problems. (a) Maximize f(x) = x1 + 2x2 - x32, subject to x1 + x2 ≤ 1 and x1 ≥ 0, x2 ≥ 0. (b) Maximize f(x) 20x1 + 10x2, Subject to and x1 ≥ 0, x2 ≥ 0.
What are the KKT conditions for nonlinear programming problems of the following form? Minimize f(x) Subject to gi(x) ≥ bi, for i = 1, 2, . . . ,m and x ≥ 0,
Consider the following nonlinear programming problem: Minimize Z = 2x1 + x22, subject to x1 + x2 = 10 and x1 ≥ 0, x2 ≥ 0. (a) Of the special types of nonlinear programming problems described in Sec. 13.3, to which type or types can this particular problem be fitted? Justify your answer. (b)
Consider the following linearly constrained programming problem: Minimize f(x) = x31 + 4x22 + 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
Consider the following linearly constrained convex programming problem: Minimize Z = x21 – 6x1 + x32 – 3x2, Subject to x1 + x2 ≤ 1 and x1 ≥ 0, x2 ≥ 0. (a) Obtain the KKT conditions for this problem.
Consider the following linearly constrained convex programming problem: Maximize f(x) = 8x1 – x21 + 2x2 + x3, Subject to x1 + 3x2 + 2x3 ≤ 12 and x1 ≥ 0, x2 ≥ 0, x3 ≥ 0.
Use the KKT conditions to determine whether (x1, x2, x3) = (1, 1, 1) can be optimal for the following problem: Minimize Z = 2x1 + x32 + x23, Subject to x21 + 2x22 + x23 ≥ 4 and x1 ≥ 0, x2 ≥ 0, x3 ≥ 0.
Reconsider the model given in Prob. 13.2-10. What are the KKT conditions for this problem? Use these conditions to determine whether (x1, x2) = (1, 1) can be optimal.
Reconsider the linearly constrained convex programming model given in Prob. 13.4-7. Use the KKT conditions to determine whether (x1, x2) = (2, 2) can be optimal.
Consider the following function: f (x) = 48x – 60x2 + x3. (a) Use the first and second derivatives to find the local maxima and local minima of f (x). (b) Use the first and second derivatives to show that f (x) has neither a global maximum nor a global minimum because it is unbounded in both
Consider the quadratic programming example presented in Sec. 13.7. (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, xTQx 0 for all
Consider the following quadratic programming problem: Maximize f (x) = 8x1 – x12 + 4x2 – x22, subject to x1 + x2 ≤ 2 and x1 ≥ 0, x2 ≥ 0.
Consider the following quadratic programming problem: Maximize f(x) = 20x1 – 20x12 + 50x2 – 50x22 + 18x1x2, subject to x1 + x2 ≤ 6 x1 + 4x2 ≤ 18 and x1 ≥ 0, x2 ≥ 0. Suppose that this problem is to be solved by the modified simplex method. (a) Formulate the linear programming problem
Consider the following quadratic programming problem: Maximize f (x) = 2x1 + 3x2 – x12 – x22, subject to x1 + x2 ≤ 2 and x1 ≥ 0, x2 ≥ 0.
Reconsider the first quadratic programming variation of the Wyndor Glass Co. problem presented in Sec. 13.2 (see Fig. 13.6). Analyze this problem by following the instructions of parts (a), (b), and (c) of Prob. 13.7-4.
Reconsider Prob. 13.1-4 and its quadratic programming model. (a) Display this model [including the values of R(x) and V(x)] on an Excel spreadsheet. (b) Use Solver (or ASPE) and its GRG Nonlinear solving method to solve this model for four cases: minimum acceptable expected return = 13, 14, 15,
The management of the Albert Hanson Company is trying to determine the best product mix for two new products. Because these products would share the same production facilities, the total number of units produced of the two products combined cannot exceed two per hour. Because of uncertainty about
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

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