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

Consider the following problem.Maximize Z = €“x1 + 4x2,Subject to(No lower bound constraint for x1). (a) Solve this problem graphically. (b) Reformulate this problem so that it has only two functional constraints and all variables have nonnegativity constraints. (c) Work through the
Consider the following problem.Maximize Z = €“x1 + 2x2 +x3,Subject to(No nonnegativity constraints).(a) Reformulate this problem so that all variables have nonnegativity constraints.(b) Work through the simplex method step by step to solve the problem.(c) Use a computer package based on the
Repeat Prob. 4.1-4 for the following problem.Maximize Z = x1 + 2x2,Subject toandx1 ‰¥ 0, x2 ‰¥ 0.Prob. 4.1-4Consider the linear programming model (given in the back of the book) that was formulated for Prob. 3.2-3.(a) Use graphical analysis to identify all the corner-point solutions for this
This chapter has described the simplex method as applied to linear programming problems where the objective function is to be maximized. Section 4.6 then described how to convert a minimization problem to an equivalent maximization problem for applying the simplex method. Another option with
Consider the following problem.Maximize Z = €“ 2x1 + x2 €“ 4x3 + 3x4,Subject toandx2 ‰¥ 0, x3 0, x4 ‰¥ 0(no nonnegativity constraint for x1).(a) Reformulate this problem to fit our standard form for a linear programming model presented in Sec. 3.2.(b) Using the Big M method,
Consider the following problem.Maximize Z = 4x1 + 5x2 + 3x3,Subject toandx1 ‰¥ 0, x2 ‰¥ 0, x3 ‰¥ 0.Work through the simplex method step by step to demonstrate that this problem does not possess any feasible solutions.
Refer to Fig. 4.10 and the resulting allowable range for the respective right-hand sides of the Wyndor Glass Co. problem given in Sec. 3.1. Use graphical analysis to demonstrate that each given allowable range is correct.Fig 4.10
Reconsider the model in Prob. 4.1-5. Interpret the right-hand side of the respective functional constraints as the amount available of the respective resources.(a) Use graphical analysis as in Fig. 4.8 to determine the shadow prices for the respective resources.(b) Use graphical analysis to perform
You are given the following linear programming problem.Maximize Z = 4x1 + 2x2, subject toand x1 ‰¥ 0, x2 ‰¥ 0. D,I (a) Solve this problem graphically. (b) Use graphical analysis to find the shadow prices for the resources. (c) Determine how many additional units of resource 1
Consider the following problem.Maximize z = x1 €“ 7x2 + 3x3,Subject toand x1 ‰¥ 0, x2 ‰¥ 0, x3 ‰¥ 0. (a) Work through the simplex method step by step to solve the problem. (b) Identify the shadow prices for the three resources and describe their
Consider the following problem.Maximize Z = 2x1 €“ 2x2 + 3x3,Subject toand x1 ‰¥ 0, x2 ‰¥ 0, x3 ‰¥ 0. (a) Work through the simplex method step by step to solve the problem. (b) Identify the shadow prices for the three resources and describe their
Consider the following problem.Maximize Z = 5x1 + 4x2 €“ x3 + 3x4,Subject toand x1 ‰¥ 0, x2 ‰¥ 0, x3 ‰¥ 0, x4 ‰¥ 0. (a) Work through the simplex method step by step to solve the problem. (b) Identify the shadow prices for the two resources and describe
Use the interior-point algorithm in your IOR Tutorial to solve the model in Prob. 4.1-4. Choose α = 0.5 from the Option menu, use (x1, x2) = (0.1, 0.4) as the initial trial solution, and run 15 iterations. Draw a graph of the feasible region, and then plot the trajectory of the trial solutions
Describe graphically what the simplex method does step by step to solve the following problem.Maximize Z = 2x1 + 3x2,Subject toand x1 ‰¥ 0, x2 ‰¥ 0.
Repeat Prob. 4.9-1 for the model in Prob. 4.1-5.Repeat Prob.Use the interior-point algorithm in your IOR Tutorial to solve the model in Prob. 4.1-4. Choose α = 0.5 from the Option menu, use (x1, x2) = (0.1, 0.4) as the initial trial solution, and run 15 iterations. Draw a graph of the feasible
From the tenth floor of her office building, Katherine Rally watches the swarms of New Yorkers fight their way through the streets infested with yellow cabs and the sidewalks littered with hot dog stands. On this sweltering July day, she pays particular attention to the fashions worn by the various
AmeriBank will soon begin offering Web banking to its customers. To guide its planning for the services to provide over the Internet, a survey will be conducted with four different age groups in three types of communities. AmeriBank is imposing a number of constraints on how extensively each age
After deciding to close one of its middle schools, the Springfield school board needs to reassign all of next year’s middle school students to the three remaining middle schools. Many of the students will be bused, so minimizing the total busing cost is one objective. Another is to minimize the
Describe graphically what the simplex method does step by step to solve the following problem.Minimize Z = 5x1 + 7x2,Subject toand x1 ‰¥ 0, x2 ‰¥ 0.
Label each of the following statements about linear programming problems as true or false, and then justify your answer.(i). Use optimality test. In minimization problems, "better" means smaller. To see this, note that min Z = – max (–Z).(ii). CPF solutions are not the only possible optimal
The following statements give inaccurate paraphrases of the six solution concepts presented in Sec. 4.1. In each case, explain what is wrong with the statement. (a) The best CPF solution always is an optimal solution. (b) An iteration of the simplex method checks whether the current CPF solution is
Consider the following problem.Maximize z = 3x1 + 2x2.Subject toandx1 ‰¥ 0, x2 ‰¥ 0.(a) Solve this problem graphically. Identify the CPF solutions by circling them on the graph.(b) Identify all the sets of two defining equations for this problem. For each set, solve (if a solution exists) for
Label each of the following statements about linear programming problems as true or false, and then justify your answer.(a) If a feasible solution is optimal but not a CPF solution, then infinitely many optimal solutions exist.(b) If the value of the objective function is equal at two different
Consider the augmented form of linear programming problems that have feasible solutions and a bounded feasible region. Label each of the following statements as true or false, and then justify your answer by referring to specific statements (with page citations) in the chapter. (a) There must be at
Reconsider the model in Prob. 4.6-9. Now you are given the information that the basic variables in the optimal solution are x2 and x3. Use this information to identify a system of three constraint boundary equations whose simultaneous solution must be this optimal solution. Then solve this system
Reconsider Prob. 4.3-6. Now use the given information and the theory of the simplex method to identify a system of three constraint boundary equations (in x1, x2, x3) whose simultaneous solution must be the optimal solution, without applying the simplex method. Solve this system of equations
Consider the following problem.Maximize Z = 2x1 + 2x2 + 3x3,Subject toandx1 ‰¥ 0, x2 ‰¥ 0, x3 ‰¥ 0.Let x4 and x5 be the slack variables for the respective functional constraints. Starting with these two variables as the basic variables for the initial BF solution, you now are given the
Consider the following problem.Maximize Z = 3x1 + 4x2 + 2x3,Subject toandx1 ‰¥ 0, x2 ‰¥ 0, x3 ‰¥ 0.Let x4 and x5 be the slack variables for the respective functional constraints. Starting with these two variables as the basic variables for the initial BF solution, you now are given the
By inspecting Fig. 5.2, explain why Property 1b for CPF solutions holds for this problem if it has the following objective function. (a) Maximize Z = x3. (b) Maximize Z = - x1 + 2x3.
Consider the three-variable linear programming problem shown in Fig. 5.2.(a) Explain in geometric terms why the set of solutions satisfying any individual constraint is a convex set, as defined in Appendix 2.(b) Use the conclusion in part (a) to explain why the entire feasible region (the set of
Suppose that the three-variable linear programming problem given in Fig. 5.2 has the objective functionMaximize Z = 3x1 + 4x2 + 3x3.Without using the algebra of the simplex method, apply just its geometric reasoning (including choosing the edge giving the maximum rate of increase of Z) to determine
Consider the three-variable linear programming problem shown in Fig. 5.2.(a) Construct a table like Table 5.4, giving the indicating variable for each constraint boundary equation and original constraint.boundary equation and original constraint.(b) For the CPF solution (2, 4, 3) and its three
Repeat Prob. 5.1-1 for the model in Prob. 3.1-6.Repeat prob.Consider the following problem.Maximize z = 3x1 + 2x2.Subject toandx1 ‰¥ 0, x2 ‰¥ 0.(a) Solve this problem graphically. Identify the CPF solutions by circling them on the graph.(b) Identify all the sets of two defining equations for
The formula for the line passing through (2, 4, 3) and (4, 2, 4) in Fig. 5.2 can be written as(2, 4, 3) + α [(4, 2, 4) – (2, 4, 3)] = (2, 4, 3) + α (2, –2, 1), where 0 ≤ α ≤ 1 for just the line segment between these points. After augmenting with the slack variables x4, x5, x6, x7 for the
Consider a two-variable mathematical programming problem that has the feasible region shown on the graph, where the six dots correspond to CPF solutions. The problem has a linear objective function, and the two dashed lines are objective function lines passing through the optimal solution (4, 5)
Consider the following problem.Maximize Z = 8x1 + 4x2 + 6x3 + 3x4 + 9x5,Subject toAnd x1 ‰¥ 0, j = 1,.,5.You are given the facts that the basic variables in the optimal solution are x3, x1, and x5 and that(a) Use the given information to identify the optimal solution.(b) Use the given
Work through the matrix form of the simplex method step by step to solve the following problem.Maximize Z = 5x1 + 8x2 + 7x3 + 4x4 + 6x5,Subject toAnd xj ‰¥ 0, j = 1, 2, 3, 4, 5.
Reconsider Prob. 5.1-1. For the sequence of CPF solutions identified in part (e), construct the basis matrix B for each of the corresponding BF solutions. For each one, invert B manually, use this B–1 to calculate the current solution, and then perform the next iteration (or demonstrate that the
Work through the matrix form of the simplex method step by step to solve the model given in Prob. 4.1-5.
Work through the matrix form of the simplex method step by step to solve the model given in Prob. 4.7-6.
Consider the following problem.Maximize Z = x1 – x2 + 2x3,Subject toandx1 ≥ 0, x2 ≥ 0, x3 ≥ 0Let x4, x5, and x6 denote the slack variables for the respective constraints. After you apply the simplex method, a portion of the final simplex tableau is as follows:(a) Use the fundamental insight
Consider the following problem.Maximize Z = 4x1 + 3x2 + x3 + 2x4,Subject toandx1 ≥ 0, x2 ≥ 0, x3 ≥ 0, x4 ≥ 0.Let x5 and x6 denote the slack variables for the respective constraints. After you apply the simplex method, a portion of the final simplex tableau is as follows:(a) Use the
Consider the following problem.Maximize Z = 6x1 + x2 + 2x3,Subject toand x1 ‰¥ 0, x2 ‰¥ 0, x3 ‰¥ 0. Let x4, x5, and x6 denote the slack variables for the respective constraints. After you apply the simplex method, a portion of the final simplex tableau is as follows: Use
Consider the following problem.Maximize Z = 2x1 +3x2,Subject toandx1 ≥ 0, x2 ≥ 0.(a) Solve this problem graphically. Identify the CPF solutions by circling them on the graph.(b) Develop a table giving each of the CPF solutions and the corresponding defining equations, BF solution, and nonbasic
Consider the following problem.Maximize Z = 20x1 + 6x2 + 8x3,Subject toAnd x1 ‰¥ 0, x2 ‰¥ 0, x3 ‰¥ 0. Let x4, x5, x6, and x7 denote the slack variables for the first through fourth constraints, respectively. Suppose that after some number of iterations of the simplex
Consider the following problem.Maximize Z = c1x1 + c2x2 + c3x3,Subject toand x1 ‰¥ 0, x2 ‰¥ 0, x3 ‰¥ 0. Note that values have not been assigned to the coefficients in the objective function (c1, c2, c3), and that the only specification for the right-hand side of the
For iteration 2 of the example in Sec. 5.3, the following expression was shown:Derive this expression by combining the algebraic operations (in matrix form) for iterations 1 and 2 that affect row 0.
Most of the description of the fundamental insight presented in Sec. 5.3 assumes that the problem is in our standard form. Now consider each of the following other forms, where the additional adjustments in the initialization step are those presented in Sec. 4.6, including the use of artificial
Reconsider the model in Prob. 4.6-5. Use artificial variables and the Big M method to construct the complete first simplex tableau for the simplex method, and then identify the columns that will contain S* for applying the fundamental insight in the final tableau. Explain why these are the
Consider the following problem.Minimize Z = 2x1 + 3x2 + 2x3,Subject toAnd x1 ‰¥ 0, x2 ‰¥ 0, x3 ‰¥ 0. Let x4 and x6 be the surplus variables for the first and second constraints, respectively. Let x-bar5 and x-bar7 be the corresponding artificial variables. After you make
Consider the following problem.Maximize Z = 3x1 + 7x2 + 2x3,Subject toAnd x1 ≥ 0, x2 ≥ 0, x3 ≥ 0.You are given the fact that the basic variables in the optimal solution are x1 and x3.(a) Introduce slack variables, and then use the given information to find the optimal solution directly by
Consider the model given in Prob. 5.2-2. Let x6 and x7 be the slack variables for the first and second constraints, respectively. You are given the information that x2 is the entering basic variable and x7 is the leaving basic variable for the first iteration of the simplex method and then x4 is
Work through the revised simplex method step by step to solve the model given in Prob. 4.3-4.
Work through the revised simplex method step by step to solve the model given in Prob. 4.7-5.
Consider the following problem.Maximize Z = 2x1 – x2 + x3,Subject toandx1 ≥ 0, x2 ≥ 0, x3 ≥ 0.After slack variables are introduced and then one complete iteration of the simplex method is performed, the following simplex tableau is obtained.(a) Identify the CPF solution obtained at
Work through the revised simplex method step by step to solve the model given in Prob. 3.1-6.
Consider the three-variable linear programming problem shown in Fig. 5.2. Discuss.
Consider the following problem.Minimize Z = 3x1 + 2x2,Subject toandx1 ≥ 0, x2 ≥ 0.(a) Identify the 10 sets of defining equations for this problem. For each one, solve (if a solution exists) for the corresponding corner-point solution, and classify it as a CPF solution or a corner-point
Reconsider the model in Prob. 3.1-5.
Each of the following statements is true under most circumstances, but not always. In each case, indicate when the statement will not be true and why.
Consider the original form (before augmenting) of a linear programming problem with n decision variables (each with a nonnegativity constraint) and m functional constraints. Label each of the following statements as true or false, and then justify your answer with specific references (including
Construct the dual problem for each of the following linear programming models fitting our standard form. (a) Model in Prob. 3.1-6 (b) Model in Prob. 4.7-5
Construct a pair of primal and dual problems, each with two decision variables and two functional constraints, such that the primal problem has no feasible solutions and the dual problem has an unbounded objective function.
Use the weak duality property to prove that if both the primal and the dual problem have feasible solutions, then both must have an optimal solution.
Consider the primal and dual problems in our standard form presented in matrix notation at the beginning of Sec. 6.1. Use only this definition of the dual problem for a primal problem in this form to prove each of the following results. (a) The weak duality property presented in Sec. 6.1.
Consider the primal and dual problems in our standard form presented in matrix notation at the beginning of Sec. 6.1. Let y* denote the optimal solution for this dual problem. Suppose that b is then replaced by . Let denote the optimal solution for the new primal problem. Prove that c≤ y*
For any linear programming problem in our standard form and its dual problem, label each of the following statements as true or false and then justify your answer. (a) The sum of the number of functional constraints and the number of variables (before augmenting) is the same for both the primal and
Consider the simplex tableaux for the Wyndor Glass Co. problem given in Table 4.8. For each tableau, give the economic interpretation of the following items: (a) Each of the coefficients of the slack variables (x3, x4, x5) in row 0 (b) Each of the coefficients of the decision variables (x1, x2) in
Consider the following problem.Maximize Z = 6x1 + 8x2,Subject toand x1 ‰¥ 0, x2 ‰¥ 0. (a) Construct the dual problem for this primal problem.
Consider the model with two functional constraints and two variables given in Prob. 4.1-5. Follow the instructions of Prob. 6.3-1 for this model. In problem (a) Construct the dual problem for this primal problem.
Consider the primal and dual problems for the Wyndor Glass Co. example given in Table 6.1. Using Tables 5.5, 5.6, 6.8, and 6.9, construct a new table showing the eight sets of nonbasic variables for the primal problem in column 1, the corresponding sets of associated variables for the dual problem
Suppose that a primal problem has a degenerate BF solution (one or more basic variables equal to zero) as its optimal solution. What does this degeneracy imply about the dual problem? Why? Is the converse also true?
Consider the linear programming model in Prob. 4.5-4. (a) Construct the primal-dual table and the dual problem for this model. (b) What does the fact that Z is unbounded for this model imply about its dual problem?
Consider the following problem. Maximize Z = 2x1 – 4x2, Subject to x1 – x2 ≤ 1 and x1 ≥ 0, x2 ≥ 0.
Consider the following problem.Maximize Z = 2x1 + 7x2 + 4x3Subject toand x1 ‰¥ 0, x2 ‰¥ 0, x3 ‰¥ 0. (a) Construct the dual problem for this primal problem. (b) Use the dual problem to demonstrate that the optimal value of Z for the primal problem cannot exceed 25.
Reconsider the model of Prob. 6.1-3b. (a) Construct its dual problem. (b) Solve this dual problem graphically.
Consider the model given in Prob. 5.3-10. (a) Construct the dual problem.
Consider the model given in Prob. 3.1-5. (a) Construct the dual problem for this model.
Consider the following problem.Maximize Z = x1 + x2,Subject toand x2 ‰¥ 0 (x1 unconstrained in sign). (a) Use the SOB method to construct the dual problem. (b) Use Table 6.12 to convert the primal problem to our standard form given at the beginning of Sec. 6.1, and construct the
Consider the primal and dual problems in our standard form presented in matrix notation at the beginning of Sec. 6.1. Use only this definition of the dual problem for a primal problem in this form to prove each of the following results. (a) If the functional constraints for the primal problem Ax
Construct the dual problem for the linear programming problem given in Prob. 4.6-3.
Consider the following problem.Minimize Z = x1 + 2x2,Subject toAnd x1 ‰¥ 0, x2 ‰¥ 0. (a) Construct the dual problem.
Consider the two versions of the dual problem for the radiation therapy example that are given in Tables 6.15 and 6.16. Review in Sec. 6.4 the general discussion of why these two versions are completely equivalent. Then fill in the details to verify this equivalency by proceeding step by step to
For each of the following linear programming models, give your recommendation on which is the more efficient way (probably) to obtain an optimal solution: by applying the simplex method directly to this primal problem or by applying the simplex method directly to the dual problem instead.
For each of the following linear programming models, use the SOB method to construct its dual problem. (a) Model in Prob. 4.6-7 (b) Model in Prob. 4.6-16
Consider the model with equality constraints given in Prob. 4.6-2. (a) Construct its dual problem. (b) Demonstrate that the answer in part (a) is correct (i.e., equality constraints yield dual variables without nonnegativity constraints) by first converting the primal problem to our standard form
Consider the model without nonnegativity constraints given in Prob. 4.6-14. (a) Construct its dual problem. (b) Demonstrate that the answer in part (a) is correct (i.e., variables without nonnegativity constraints yield equality constraints in the dual problem) by first converting the primal
Consider the dual problem for the Wyndor Glass Co. example given in Table 6.1. Demonstrate that its dual problem is the primal problem given in Table 6.1 by going through the conversion steps given in Table 6.13.
Consider the following problem.Minimize Z = €“x1 €“ 3x2,Subject toand x1 ‰¥ 0, x2 ‰¥ 0. (a) Demonstrate graphically that this problem has an unbounded objective function. (b) Construct the dual problem.
Consider the following problem.Maximize Z = €“ x1 €“ 2x2 €“ x3,Subject toand x1 ‰¥ 0, x2 ‰¥ 0, x3 ‰¥ 0. (a) Construct the dual problem.
Consider the following problem.Maximize Z = 2x1 + 6x2 + 9x3,Subject toand x1 ‰¥ 0, x2 ‰¥ 0, x3 ‰¥ 0. (a) Construct the dual problem for this primal problem.
Follow the instructions of Prob. 6.1-5 for the following problem.Maximize Z = x1 €“ 3x2 + 2x3,Subject toand x1 ‰¥ 0, x2 ‰¥ 0, x3 ‰¥ 0. (a) Construct the dual problem for this primal problem.
Consider the following problem.Maximize Z = x1 + 2x2,Subject toand x1 ‰¥ 0, x2 ‰¥ 0.
Construct and graph a primal problem with two decision variables and two functional constraints that has feasible solutions and an unbounded objective function. Then construct the dual problem and demonstrate graphically that it has no feasible solutions.
Construct a pair of primal and dual problems, each with two decision variables and two functional constraints, such that both problems have no feasible solutions. Demonstrate this property graphically.
Consider the following problem.Maximize Z = 3x1 + x2 +4x3,Subject toand x1 ‰¥ 0, x2 ‰¥ 0, x3 ‰¥ 0.
Consider Variation 5 of the Wyndor Glass Co. model (see Fig. 7.5 and Table 7.8), where the changes in the parameter values given in Table 7.5 are c-bar2 = 3, a-bar22 = 3, and a-bar32 = 4. Verify both algebraically and graphically that the allowable range for c1 is c1 ≥ 9/4.
For the problem given in Table 7.5, find the allowable range for c2. Show your work algebraically, using the tableau given in Table 7.5. Then justify your answer from a geometric viewpoint, referring to Fig. 7.2.
For the original Wyndor Glass Co. problem, use the last tableau in Table 4.8 to do the following. (a) Find the allowable range for each bi. (b) Find the allowable range for c1 and c2. (c) Use a software package based on the simplex method to find these allowable ranges.
For Variation 6 of the Wyndor Glass Co. model presented in Sec. 7.2, use the last tableau in Table 7.9 to do the following. (a) Find the allowable range for each bi. (b) Find the allowable range for c1 and c2. (c) Use a software package based on the simplex method to find these allowable ranges.

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