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introduction to operations research

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

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.(Hint: Convert this form to our standard form assumed in this chapter by using the techniques presented in Sec. 4.6 and then applying the KKT
Use the KKT conditions to derive an optimal solution for each of the following problems.(a) Maximize f(x) x1 2x2 x2 3, subject to x1 x2 1 and x1 0, x2 0.(b) Maximize f(x) 20x1 10x2, subject to x1 2 x2 2 1 x1 2x2 2 and x1 0, x2 0.
Consider the following nonlinear programming problem:Maximize f(x) x2 x11 , subject to x1 x2 2 and x1 0, x2 0.(a) Use the KKT conditions to demonstrate that (x1, x2) (4, 2)is not optimal.(b) Derive a solution that does satisfy the KKT conditions.(c) Show that this problem is not a convex
Determine whether (x1, x2) (1, 2) can be optimal by applying the KKT conditions.
Consider the following linearly constrained optimization problem:Maximize f(x) ln(x1 1) x2 2, subject to x1 2x2 3 and x1 0, x2 0, where ln denotes the natural logarithm,(a) Verify that this problem is a convex programming problem.(b) Use the KKT conditions to derive an optimal solution.(c)
Consider the following convex programming problem:Maximize f(x) 12x1 x1 2 50x2 x2 2, subject to x1 10, x2 15, and x1 0, x2 0.(a) Use the KKT conditions for this problem to derive an optimal solution.(b) Decompose this problem into two separate constrained optimization problems involving
Reconsider the model given in Prob. 12.3-3. What are the KKT conditions for this model? Use these conditions to determine whether (x1, x2) (0, 10) can be optimal.
Use the KKT conditions to check whether (x1, x2) (1/2, 1/2) is optimal.
Reconsider Prob.
Reconsider the one-variable convex programming model given in Prob. 12.4-5. Use the KKT conditions to derive an optimal solution for this model.
Starting from the initial trial solution (x1, x2) (0, 0), interactively apply the gradient search procedure with 1 to solve (approximately) the following problem, and then apply the automatic routine for this procedure (with 0.01).Maximize f(x) x1x2 3x2 x1 2 x2 2.
Consider the following unconstrained optimization problem:Maximize f(x) 3x1x2 3x2x3 x1 2 6x2 2 x3 2.(a) Describe how solving this problem can be reduced to solving a two-variable unconstrained optimization problem.D,I (b) Starting from the initial trial solution (x1, x2, x3) (1, 1, 1),
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 x1 2 x1 4 2x1x2 x2 2.To complete this iteration, approximately solve for t* by manually applying two iterations of the
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 0.01).Maximize f(x) 6x1 2x1x2 2x2 2x1 2 x2 2.Then solve
Starting from the initial trial solution (x1, x2) (0, 0), interactively apply the gradient search procedure with 0.3 to obtain an approximate solution for the following problem, and then apply the automatic routine for this procedure (with 0.01).Maximize f(x) 8x1 x1 2 12x2 2x2 2
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 0.01).Maximize f(x) 60x1x2 15x1 2 80x2 2.Then solve f(x)
Consider the following unconstrained optimization problem:Maximize f(x) 2x1x2 x2 x1 2 2x2 2.D,I (a) Starting from the initial trial solution (x1, x2) (1, 1), interactively apply the gradient search procedure with 0.25 to obtain an approximate solution.(b) Solve the system of linear
Consider the following linearly constrained convex programming problem:Maximize f(x) 32x1 50x2 10x2 2 x2 3 x1 4 x2 4, subject to 3x1 x2 11 2x1 5x2 16 and x1 0, x2 0.Ignore the constraints and solve the resulting two one-variable unconstrained optimization problems. Use calculus to
Consider the problem of maximizing a differentiable function f(x) of a single unconstrained variable x. Let x0 and x0, 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 convex programming problem:Minimize Z x4 x2 4x,
Consider the following problem:Maximize f(x) 10x3 60x 2x6 3x4 12x2.I (a) Apply the bisection method to (approximately) solve this problem. Use an error tolerance 0.07 and find appropriate initial bounds by inspection.(b) Apply Newton’s method, with 0.001 and x1 1, to this
Consider the following problem:Maximize f(x) 48x5 42x3 3.5x 16x6 61x4 16.5x2.I (a) Apply the bisection method to (approximately) solve this problem. Use an error tolerance 0.08 and initial bounds x 1, x 4.(b) Apply Newton’s method, with 0.001 and x1 1, to this problem.
Use the bisection method with an error tolerance 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 4.8.(b) Minimize f(x) 6x 7x2 4x3 x4, with x 4, x 1.
Consider the following problem:Maximize f(x) x3 2x 2x2 0.25x4.I (a) Apply the bisection method to (approximately) solve this problem. Use an error tolerance 0.04 and initial bounds x 0, x 2.4.(b) Apply Newton’s method, with 0.001 and x1 1.2, to this problem.
Consider the expressions in matrix notation given in Sec.12.7 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. 12.3, by identifying
Consider the following linear fractional programming problem:Maximize f(x) ,
Consider the following geometric programming problem:Minimize f(x) 2x1(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 part (a) is indeed a convex programming problem.
Consider the following nonlinear programming problem:Minimize Z x1 4 2x1 2 2x1x2 4x2 2, subject to 2x1 x2 10 x1 2x2 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
Consider the following constrained optimization problem:Maximize f(x) 120x 15x2 10x3, subject to x 0.Use just the first and second derivatives of f(x) to derive an optimal solution.
Show that this problem is a nonconvex programming problem.
Reconsider Prob.
Consider the following nonlinear programming problem:Minimize Z x4 1 2x2 2, subject to x2 1 x2 2 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 is a
Consider the following nonlinear programming problem:Maximize f(x) x1 x2, subject to x2 1 x2 2 1 and x1 0, x2 0.(a) Verify that this is a convex programming problem.(b) Solve this problem graphically.
Consider the following function:f(x) 5x1 2x2 2 x2 3 3x3x4 4x2 4 2x4 5 x2 5 3x5x6 6x2 6 3x6x7 x2 7.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.
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 12.2-7.* For each of the following functions, use the test given in Appendix 2 to determine whether it is convex,
Consider the following function:f(x) 240x 300x2 10x3.(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 variation of the Wyndor Glass Co. problem represented in Fig. 12.6, where the original objective function (see Sec. 3.1) has been replaced by Z 126x1 9x1 2 182x2 13x2 2.Demonstrate that (x1, x2) (8 3, 5) with Z 857 is indeed optimal by showing that the ellipse 857 126x1
Consider the variation of the Wyndor Glass Co. example represented in Fig. 12.5, where the second and third functional constraints of the original problem (see Sec. 3.1) have been replaced by 9x1 2 5x2 2 216. Demonstrate that (x1, x2) (2, 6) with Z 36 is indeed optimal by showing that the
Reconsider Prob. 12.1-4, 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.
Reconsider Prob. 12.1-2, Verify that this problem is a convex programming problem.
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
For the P & T Co. problem described in Sec. 8.1, 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. 12.3 and 12.4, showing the marginal cost and total cost for shipments of
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
Read the referenced article that fully describes the OR study summarized in the application vignette presented in Sec. 12.1.Briefly describe how nonlinear programming was applied in this study. Then list the various financial and nonfinancial benefits that resulted from this study.
From the bottom part of the selected references given at the end of the chapter, select three of these award-winning applications of integer programming. For each one, read the article and then write a one-page summary of the application and the benefits(including nonfinancial benefits) it provided.
From the bottom part of the selected references given at the end of the chapter, select one of these award-winning applications of integer programming. Read this article and then write a two-page summary of the application and the benefits (including nonfinancial benefits) it provided.
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
Problem 10.3-2 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.
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. 10.3-3, Formulate a compact constraint programming model for this problem.
Consider the problem of assigning swimmers to the different legs of a medley relay team that is presented in Prob. 8.3-4.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 Job Shop Co. example introduced in Sec. 8.3.Table 8.25 shows its formulation as an assignment problem. Use global constraints to formulate a compact constraint programming model for this assignment problem.
Consider the following problem.Maximize Z 100x1 3x2 1 400x2 5x2 2 200x3 4x2 3 100x4 2x4 4, subject to x1 ∈ {25, 30}, x2 ∈ {20, 25, 30, 35, 40, 50}, x3 ∈ {20, 25, 30}, x4 ∈ {20, 25}, all these variables must have different values, x2 x3 60, x1 x3 50.
Consider the following problem.Maximize Z 5x1 x2 1 8x2 x2 2 10x3 x2 3 15x4 x2 4 20x5 x2 5, subject to x1 ∈ {3, 6, 12}, x2 ∈ {3, 6}, x3 ∈ {3, 6, 9, 12}, x4 ∈ {6, 12}, x5 ∈ {9, 12, 15, 18}, all these variables must have different values, x1 x3 x4 25.Use the techniques of
Consider the following problem.Maximize Z 10x1 30x2 40x3 30x4, subject to x1 ∈ {2, 3}, x2 ∈ {2, 4}, x3 ∈ {3, 4}, x4 ∈ {1, 2, 3, 4}, all these variables must have different values, x1 x2 x3 x4 10.Use the techniques of constraint programming (domain reduction, constraint propagation, a
Consider the following BIP problem.Maximize Z 2x1 3x2 x3 4x4 3x5 2x6 2x7 x8 3x9,
Generate as many cutting planes as possible from the following constraint for a pure BIP problem.5x1 3x2 7x3 4x4 6x5 9.
Generate as many cutting planes as possible from the following constraint for a pure BIP problem.3x1 5x2 4x3 8x4 10.
One of the constraints of a certain pure BIP problem is 25x1 15x2 20x3 10x4 35.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 x1 3x2 2x3 4x4 5.Identify all the minimal covers for this constraint, and then give the corresponding cutting planes.
In Sec. 11.8, a pure BIP example with the constraint, 2x1 3x2 4, was used to illustrate the procedure for tightening constraints. Show that applying the procedure for generating cutting planes to this constraint yields the same new constraint, x1 x2 1.
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.
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 the following constraint for a pure BIP problem.5x1 10x2 15x3 15.
In Sec. 11.8, 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.
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
Use the following set of constraints for the same pure BIP problem to fix as many variables as possible. Also identify the constraints which become redundant because of the fixed variables.3x3 x5 x7 1 x2 x4 x6 1 x1 2x5 2x6 2 x1 x2 x4 0
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
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
Use the MIP branch-and-bound algorithm presented in Sec. 11.7 to solve the following MIP problem interactively.Minimize Z 5x1 x2 x3 2x4 3x5, subject to x2 5x3 x4 2x5 2 5x1 x2 x4 x5 7 x1 x2 6x3 x4 4 and xj 0, for j 1, 2, 3, 4, 5 xj is integer, for j 1, 2, 3.
Use the MIP branch-and-bound algorithm presented in Sec. 11.7 to solve the following MIP problem interactively.Maximize Z 3x1 4x2 2x3 x4 2x5, subject to 2x1 x2 x3 x4 x5 3x1 3x2 x3 x4 2x5 2 2x1 x2 x3 x4 3x5 1 and xj 0, for j 1, 2, 3, 4, 5 xj is binary, for j 1, 2, 3.
Use the MIP branch-and-bound algorithm presented in Sec. 11.7 to solve the following MIP problem interactively.Maximize Z 20x1 10x2 25x3 20x4, subject to x1 x2 x3 2x4 12 3x1 x2 2x3 2x4 20 x1 2x2 5x3 3x4 30
A machine shop makes two products. Each unit of the first product requires 3 hours on machine 1 and 2 hours on machine 2.Each unit of the second product requires 2 hours on machine 1 and 3 hours on machine 2. Machine 1 is available only 8 hours per day and machine 2 only 7 hours per day. The profit
Reconsider Prob. 11.3-5a. Use the MIP branch-andbound algorithm presented in Sec. 11.7 to solve this IP problem interactively.
Consider the IP example discussed in Sec. 11.5 and illustrated in Fig. 11.3. Use the MIP branch-and-bound algorithm presented in Sec. 11.7 to solve this problem interactively.
Reconsider the IP model of Prob. 11.5-2.(a) Use the MIP branch-and-bound algorithm presented in Sec. 11.7 to solve this problem by hand. For each subproblem, solve its LP relaxation graphically.D,I (b) Now use the interactive procedure for this algorithm in your IOR Tutorial to solve this problem.C
Follow the instructions of Prob. 11.7-2 for the following IP model.Minimize Z 15x1 10x2, subject to 15x1 5x2 30 10x1 10x2 30 and x1 0, x2 0 x1, x2 are integers.
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. 11.7 to solve this problem by hand. For each subproblem, solve its LP
Read the referenced article that fully describes the OR study summarized in the application vignette presented in Sec. 11.7.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 Lagrangian relaxation described near the end of Sec. 11.6.(a) If x is a feasible solution for an MIP problem, show that x also must be a feasible solution for the corresponding Lagrangian relaxation.(b) If x* is an optimal solution for an MIP problem, with an objective function value
Consider the following nonlinear BIP problem.Maximize Z 80x1 60x2 40x3 20x4 (7x1 5x2 3x3 2x4)2
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:
Consider the assignment problem with the following cost table:(a) Design a branch-and-bound algorithm for solving such assignment problems by specifying how the branching, bounding, and fathoming steps would be performed. (Hint: For the assignees not yet assigned for the current subproblem, form
Consider the following statements about any pure IP problem (in maximization form) and its LP relaxation. Label each of the statements as True or False, and then justify your answer.(a) The feasible region for the LP relaxation is a subset of the feasible region for the IP problem.
Reconsider Prob. 11.4-8(a). Use the BIP algorithm presented in Sec. 11.6 to solve this problem interactively.
Reconsider Prob. 11.3-6(a). Use the BIP branch-andbound algorithm presented in Sec. 11.6 to solve this BIP model interactively.
Use the BIP branch-and-bound algorithm presented in Sec. 11.6 to solve the following problem interactively.Maximize Z 3x1 3x2 5x3 2x4 x5, subject to x1 2x2 7x3 3x4 x5 015x1 30x2 35x3 45x4 45x5 50 and xj is binary, for j 1, 2, . . . , 5.
Use the BIP branch-and-bound algorithm presented in Sec. 11.6 to solve the following problem interactively.Minimize Z 5x1 6x2 7x3 8x4 9x5, subject to 3x1 x2 x3 x4 2x5 2 x1 3x2 x3 2x4 x5 0x1 x2 3x3 x4 x5 1 and xj is binary, for j 1, 2, . . . , 5.
Use the BIP branch-and-bound algorithm presented in Sec. 11.6 to solve the following problem interactively.Maximize Z 2x1 x2 5x3 3x4 4x5, subject to 3x1 2x2 7x3 5x4 4x5 6 x1 x2 2x3 4x4 2x5 0 and xj is binary, for j 1, 2, . . . , 5.
Label each of the following statements as True or False, and then justify your answer by referring to specific statements in the chapter.(a) Linear programming problems are generally considerably easier to solve than IP problems.(b) For IP problems, the number of integer variables is generally more
Follow the instructions of Prob. 11.5-2 for the following BIP problem.Maximize Z 5x1 25x2, subject to3x1 30x2 27 3x1 x2 4 and x1, x2 are binary.
Follow the instructions of Prob. 11.5-2 for the following BIP problem.Maximize Z 10x1 25x2, subject to 19x1 6x2 15 5x1 15x2 15 and x1, x2 are binary.
Follow the instructions of Prob. 11.5-2 for the following IP problem.Maximize Z 220x1 80x2, subject to 5x1 2x2 16 2x1 x2 4x1 2x2 4 and x1 0, x2 0 x1, x2 are integers.
Consider the following IP problem.Maximize Z 5x1 x2, subject tox1 2x2 4 x1 x2 1 4x1 x2 12
Read the referenced article that fully describes the OR study summarized in the application vignette presented in Sec. 11.5.Briefly describe how integer programming was applied in this study. Then list the various financial and nonfinancial benefits that resulted from this study.
Suppose that a state sends R persons to the U.S. House of Representatives. There are D counties in the state (D R), and the state legislature wants to group these counties into R distinct electoral districts, each of which sends a delegate to Congress. The total population of the state is P, and
The management of Sunny Skies Unlimited now has decided that the decision on the locations of the fire stations should be based mainly on costs.The cost of locating a fire station in a tract is $300,000 for tract 1, $350,000 for tract 2, $600,000 for tract 3, $450,000 for?
An increasing number of Americans are moving to a warmer climate when they retire. To take advantage of this trend, Sunny Skies Unlimited is undertaking a major real estate development project. The project is to develop a completely new retirement community (to be called Pilgrim Haven) that will
Speedy Delivery provides two-day delivery service of large parcels across the United States. Each morning at each collection center, the parcels that have arrived overnight are loaded onto several trucks for delivery throughout the area. Since the competitive battlefield in this business is speed
Consider the following special type of shortest-path problem (see Sec. 9.3) where the nodes are in columns and the only paths considered always move forward one column at a time.
Consider the following integer nonlinear programming problem.Maximize Z 4x2 1 x3 1 10x2 2 x4 2, subject to x1 x2 3 and x1 0, x2 0 x1 and x2 are integers.This problem can be reformulated in two different ways as an equivalent pure BIP problem (with a linear objective function) with six

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