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Introduction to Operations Research 10th edition Frederick S. Hillier, Gerald J. Lieberman - Solutions
Read Selected Reference A3 that describes an OR study done for Swift & Company. (a) Summarize the background that led to undertaking this study. (b) Describe the purpose of each of the three general types of models formulated during this study. (c) How many specific models does the company now use
Read Selected Reference A8 that describes an OR study done for the Rijkswaterstaat of the Netherlands. (Focus especially on pp. 3–20 and 30–32.) (a) Summarize the background that led to undertaking this study. (b) Summarize the purpose of each of the five mathematical models described on pp.
Read Selected Reference 5.(a) Identify the author’s example of a model in the natural sciences and of a model in OR.(b) Describe the author’s viewpoint about how basic precepts of using models to do research in the natural sciences can also be used to guide research on operations (OR).
Read the referenced article that fully describes the OR study summarized in the application vignette presented in Sec. 3.1. Briefly describe how linear programming was applied in this study. Then list the various financial and nonfinancial benefits that resulted from this study.
Weenies and Buns is a food processing plant which manufactures hot dogs and hot dog buns. They grind their own flour for the hot dog buns at a maximum rate of 200 pounds per week. Each hot dog bun requires 0.1 pound of flour. They currently have a contract with Pigland, Inc., which specifies that a
The Omega Manufacturing Company has discontinued the production of a certain unprofitable product line. This act created considerable excess production capacity. Management is considering devoting this excess capacity to one or more of three products; call them products 1, 2, and 3. The available
Consider the following problem, where the value of c1 has not yet been ascertained.Maximize Z = c1x1 + x2,Subject toandx1 ≥ ¥ 0, x2 ≥ ¥ 0.Use graphical analysis to determine the optimal solution(s) for (x1, x2) for the various possible values of c1 (–∞ < c1 < ∞).
Consider the following problem, where the value of k has not yet been ascertained.Maximize Z = x1 + 2x2,Subject toandx1 ≥ ¥ 0, x2 ≥ ¥ 0.The solution currently being used is x1 = 2, x2 = 3. Use graphical analysis to determine the values of k such that this solution actually is optimal.
Consider the following problem, where the values of c1 and c2 have not yet been ascertained.Maximize Z = c1x1 + c2x2,Subject toandx1 ≥ ¥ 0, x2 ≥ ¥ 0.Use graphical analysis to determine the optimal solution(s) for (x1, x2) for the various possible values of c1 and c2.
The following table summarizes the key facts about two products, A and B, and the resources, Q, R, and S, required to produce them.All the assumptions of linear programming hold.(a) Formulate a linear programming model for this problem.(b) Solve this model graphically.(c) Verify the exact value of
The shaded area in the following graph represents the feasible region of a linear programming problem whose objective function is to be maximized.Label each of the following statements as True or False, and then justify your answer based on the graphical method. In each case, give an example of an
This is your lucky day. You have just won a $20,000 prize. You are setting aside $8,000 for taxes and partying expenses, but you have decided to invest the other $12,000. Upon hearing this news, two different friends have offered you an opportunity to become a partner in two different
Use the graphical method to find all optimal solutions for the following model:Maximize Z = 500x1 + 300x2,Subject toandx1 ≥ 0, x2 ≥ 0
Use the graphical method to demonstrate that the following model has no feasible solutions.Maximize Z = 5x1 + 7x2,Subject toandx1 ≥ 0, x2 ≥ 0.
For each of the following constraints, draw a separate graph to show the nonnegative solutions that satisfy this constraint.(a) x1 + 3x2 ≤ 6(b) 4x1 + 3x2 ≤ 12(c) 4x1 + x2 ≤ 8(d) Now combine these constraints into a single graph to show the feasible region for the entire set of functional
Suppose that the following constraints have been provided for a linear programming model.andx1 ≥ ¥ 0, x2 ≥ ¥ 0.(a) Demonstrate that the feasible region is unbounded.(b) If the objective is to maximize Z = – x1 + x2, does the model have an optimal solution? If so, find it. If not, explain
Indicate why each of the four assumptions of linear programming (Sec. 3.3) appears to be reasonably satisfied for this problem. Is one assumption more doubtful than the others? If so, what should be done to take this into account?
Consider a problem with two decision variables, x1 and x2, which represent the levels of activities 1 and 2, respectively. For each variable, the permissible values are 0, 1, and 2, where the feasible combinations of these values for the two variables are determined from a variety of constraints.
Read the referenced article that fully describes the OR study summarized in the application vignette presented in Sec. 3.4. Briefly describe how linear programming was applied in this study. Then list the various financial and nonfinancial benefits that resulted from this study.
For each of the four assumptions of linear programming discussed in Sec. 3.3, write a one-paragraph analysis of how well you feel it applies to each of the following examples given in Sec. 3.4:(a) Design of radiation therapy (Mary).(b) Regional planning (Southern Confederation of Kibbutzim).(c)
For each of the four assumptions of linear programming discussed in Sec. 3.3, write a one-paragraph analysis of how well it applies to each of the following examples given in Sec. 3.4. (a) Reclaiming solid wastes (Save-It Co.). (b) Personnel scheduling (Union Airways). (c) Distributing goods
Use the graphical method to solve this problem:Maximize Z = 15x1 + 20x2,Subject toand x1 ‰¥ 0, x2 ‰¥ 0.
Use the graphical method to solve this problem:Minimize Z = 3x1 + 2x2,Subject tox1 + 2x2 ‰¤ 12andx1 ‰¥ 0, x2 ‰¥ 0.
Consider the following problem, where the value of c1 has not yet been ascertained.Maximize Z = c1x1 + 2x2,Subject toandx1 ‰¥ 0, x2 ‰¥ 0.Use graphical analysis to determine the optimal solution(s) for (x1, x2) for the various possible values of c1.
Consider the following model:Minimize Z = 40x1 + 50x2,Subject toandx1 ‰¥ 0, x2 ‰¥ 0.(a) Use the graphical method to solve this model.(b) How does the optimal solution change if the objective function is changed to Z = 40x1 + 70x2? (You may find it helpful to use the Graphical Analysis and
Consider the following objective function for a linear programming model:(a) Draw a graph that shows the corresponding objective function lines for Z = 6, Z = 12, and Z = 18.(b) Find the slope-intercept form of the equation for each of these three objective function lines. Compare the slope for
Ralph Edmund loves steaks and potatoes. Therefore, he has decided to go on a steady diet of only these two foods (plus some liquids and vitamin supplements) for all his meals. Ralph realizes that this isn't the healthiest diet, so he wants to make sure that he eats the right quantities of the two
Web Mercantile sells many household products through an online catalog. The company needs substantial warehouse space for storing its goods. Plans now are being made for leasing warehouse storage space over the next 5 months. Just how much space will be required in each of these months is known.
Larry Edison is the director of the Computer Center for Buckly College. He now needs to schedule the staffing of the center. It is open from 8 A.M. until midnight. Larry has monitored the usage of the center at various times of the day, and determined that the following number of computer
The Medequip Company produces precision medical diagnostic equipment at two factories. Three medical centers have placed orders for this month€™s production output. The table below shows what the cost would be for shipping each unit from each factory to each of these customers. Also shown are
Al Ferris has $60,000 that he wishes to invest now in order to use the accumulation for purchasing a retirement annuity in 5 years. After consulting with his financial adviser, he has been offered four types of fixed-income investments, which we will label as investments A, B, C, D.Investments A
The Metalco Company desires to blend a new alloy of 40 percent tin, 35 percent zinc, and 25 percent lead from several available alloys having the following properties:The objective is to determine the proportions of these alloys that should be blended to produce the new alloy at a minimum cost.(a)
A cargo plane has three compartments for storing cargo: front, center, and back. These compartments have capacity limits on both weight and space, as summarized below:Furthermore, the weight of the cargo in the respective compartments must be the same proportion of that compartment€™s weight
Oxbridge University maintains a powerful mainframe computer for research use by its faculty, Ph.D. students, and research associates. During all working hours, an operator must be available to operate and maintain the computer, as well as to perform some programming services. Beryl Ingram, the
Joyce and Marvin run a day care for preschoolers. They are trying to decide what to feed the children for lunches. They would like to keep their costs down, but also need to meet the nutritional requirements of the children. They have already decided to go with peanut butter and jelly sandwiches,
Read the referenced article that fully describes the OR study summarized in the application vignette presented in Sec. 3.5. Briefly describe how linear programming was applied in this study. Then list the various financial and nonfinancial benefits that resulted from this study.
Consider the following equation of a line:20x1 + 40x2 = 400(a) Find the slope-intercept form of this equation.(b) Use this form to identify the slope and the intercept with the x2 axis for this line.(c) Use the information from part (b) to draw a graph of this line.
You are given the following data for a linear programming problem where the objective is to maximize the profit from allocating three resources to two nonnegative activities.Contribution per unit = profit per unit of the activity.(a) Formulate a linear programming model for this problem.(b) Use the
Ed Butler is the production manager for the Bilco Corporation, which produces three types of spare parts for automobiles. The manufacture of each part requires processing on each of two machines, with the following processing times (in hours):Each machine is available 40 hours per month. Each part
You are given the following data for a linear programming problem where the objective is to minimize the cost of conducting two nonnegative activities so as to achieve three benefits that do not fall below their minimum levels.(a) Formulate a linear programming model for this problem.(b) Use the
Fred Jonasson manages a family-owned farm. To supplement several food products grown on the farm, Fred also raises pigs for market. He now wishes to determine the quantities of the available types of feed (corn, tankage, and alfalfa) that should be given to each pig. Since pigs will eat any mix of
Maureen Laird is the chief financial officer for the Alva Electric Co., a major public utility in the midwest. The company has scheduled the construction of new hydroelectric plants 5, 10, and 20 years from now to meet the needs of the growing population in the region served by the company. To
The Philbrick Company has two plants on opposite sides of the United States. Each of these plants produces the same two products and then sells them to wholesalers within its half of the country. The orders from wholesalers have already been received for the next 2 months (February and March),
Reconsider Prob. 3.1-11.(a) Use MPL/Solvers to formulate and solve the model for this problem.(b) Use LINGO to formulate and solve this model.In problem
Reconsider Prob. 3.4-11.(a) Use MPL/Solvers to formulate and solve the model for this problem.(b) Use LINGO to formulate and solve this model.In problem
Reconsider Prob. 3.4-15.(a) Use MPL/Solvers to formulate and solve the model for this problem.(b) Use LINGO to formulate and solve this model.In problem
Reconsider Prob. 3.5-5.(a) Use MPL/Solvers to formulate and solve the model for this problem.(b) Use LINGO to formulate and solve this model.In problem
Use the graphical method to solve the problem:Maximize Z = 2x1 + x2,Subject tox2 ‰¤ 10and x1 ‰¥ 0, x2 ‰¥ 0,
Reconsider Prob. 3.5-6.(a) Use MPL/Solvers to formulate and solve the model for this problem.(b) Use LINGO to formulate and solve this model.In problem
A large paper manufacturing company, the Quality Paper Corporation, has 10 paper mills from which it needs to supply 1,000 customers. It uses three alternative types of machines and four types of raw materials to make five different types of paper. Therefore, the company needs to develop a detailed
Automobile Alliance, a large automobile manufacturing company, organizes the vehicles it manufactures into three families: a family of trucks, a family of small cars, and a family of midsized and luxury cars. One plant outside Detroit, MI, assembles two models from the family of midsized and luxury
This case focuses on a subject that is dear to the heart of many students. How should the manager of a college cafeteria choose the ingredients of a casserole dish to make it sufficiently tasty for the students while also minimizing costs? In this case, linear programming models with only two
California Children’s Hospital currently uses a confusing, decentralized appointment and registration process for its patients. Therefore, the decision has been made to centralize the process by establishing one call center devoted exclusively to appointments and registration. The hospital
The vice president for marketing of the Super Grain Corporation needs to develop a promotional campaign for the company’s new breakfast cereal. Three advertising media have been chosen for the campaign, but decisions now need to be made regarding how much of each medium should be used.
Use the graphical method to solve the problem:Maximize Z = 10x1 + 20x2,Subject toand x1 ‰¥ 0, x2 ‰¥ 0,
The Whitt Window Company, a company with only three employees, makes two different kinds of hand-crafted windows: a wood-framed and an aluminum-framed window. The company earns $300 profit for each wood-framed window and $150 profit for each aluminum-framed window. Doug makes the wood frames and
The WorldLight Company produces two light fixtures (products 1 and 2) that require both metal frame parts and electrical components. Management wants to determine how many units of each product to produce so as to maximize profit. For each unit of product 1, 1 unit of frame parts and 2 units of
The Primo Insurance Company is introducing two new product lines: special risk insurance and mortgages. The expected profit is $5 per unit on special risk insurance and $2 per unit on mortgages.Management wishes to establish sales quotas for the new product lines to maximize total expected profit.
Consider the following problem.Maximize z = x1 + 2x2,Subject toandx1 ‰¥ 0, x2 ‰¥ 0.(a) Plot the feasible region and circle all the CPF solutions.(b) For each CPF solution, identify the pair of constraint boundary equations that it satisfies.(c) For each CPF solution, use this pair of
Reconsider the model in Prob. 4.1-4.(a) Introduce slack variables in order to write the functional constraints in augmented form.(b) For each CPF solution, identify the corresponding BF solution by calculating the values of the slack variables. For each BF solution, use the values of the variables
Reconsider the model in Prob. 4.1-5. Follow the instructions of Prob. 4.2-1 for parts (a), (b), and (c).(d) Repeat part (b) for the corner-point infeasible solutions and the corresponding basic infeasible solutions.(e) Repeat part (c) for the basic infeasible solutions.In problem(a) Introduce slack
Read the referenced article that fully describes the OR study summarized in the application vignette presented in Sec. 4.3. Briefly describe the application of the simplex method in this study. Then list the various financial and nonfinancial benefits that resulted from this study.
Work through the simplex method (in algebraic form) step by step to solve the model in Prob. 4.1-4.
Reconsider the model in Prob. 4.1-5.(a) Work through the simplex method (in algebraic form) by hand to solve this model.(b) Repeat part (a) with the corresponding interactive routine in your IOR Tutorial.(c) Verify the optimal solution you obtained by using a software package based on the simplex
Work through the simplex method (in algebraic form) step by step to solve the following problem.Maximize Z = 4x1 + 3x2 + 6x3,Subject toand x1 ‰¥ 0, x2 ‰¥ 0.
Work through the simplex method (in algebraic form) step by step to solve the following problem.Maximize Z = x1 + 2x2 + 4x3,Subject toand x1 ‰¥ 0, x2 ‰¥ 0.
Consider the following problem.Maximize Z = 5x1 + 3x2 + 4x3,Subject toandx1 ‰¥ 0, x2 ‰¥ 0, x3 ‰¥ 0.You are given the information that the nonzero variables in the optimal solution are x2 and x3.(a) Describe how you can use this information to adapt the simplex method to solve this problem
Consider the following problem.Maximize Z = 2x1 + 4x2 + 3x3,subject toandx1 ‰¥ 0, x2 ‰¥ 0, x3 ‰¥ 0.You are given the information that x1 > 0, x2 = 0, and x3 = 0 in the optimal solution.(a) Describe how you can use this information to adapt the simplex method to solve this problem in the
Label each of the following statements as true or false, and then justify your answer by referring to specific statements in the chapter.(a) The simplex method’s rule for choosing the entering basic variable is used because it always leads to the best adjacent BF solution (largest Z).(b) The
Consider the following problem.Maximize Z = 3x1 + 2x2,Subject toandx1 ‰¥ 0, x2 ‰¥ 0.D,I (a) Use the graphical method to solve this problem. Circle all the corner points on the graph.(b) For each CPF solution, identify the pair of constraint boundary equations it satisfies.(c) For each CPF
Repeat Prob. 4.3-2, using the tabular form of the simplex method. Repeat problem Work through the simplex method (in algebraic form) step by step to solve the model in Prob. 4.1-4.
Repeat Prob. 4.3-3, using the tabular form of the simplex method.
Consider the following problem.Maximize Z = 2x1 + x2,Subject toandx1 ‰¥ 0, x2 ‰¥ 0.(a) Solve this problem graphically in a freehand manner. Also identify all the CPF solutions.(b) Now use IOR Tutorial to solve the problem graphically.(c) Use hand calculations to solve this problem by the
Repeat Prob. 4.4-3 for the following problem.Maximize Z = 2x1 + 3x2,Subject toand x1 ‰¥ 0, x2 ‰¥ 0.
Consider the following problem.Maximize Z = 2x1 + 4x2 + 3x3,Subject toand x1 ‰¥ 0, x2 ‰¥ 0, x3 ‰¥ 0. (a) Work through the simplex method step by step in algebraic form. (b) Work through the simplex method step by step in tabular form. (c) Use a software package based on
Consider the following problem.Maximize Z = 3x1 + 5x2 + 6x3,Subject toand x1 ‰¥ 0, x2 ‰¥ 0, x3 ‰¥ 0, (a) Work through the simplex method step by step in algebraic form. (b) Work through the simplex method in tabular form. (c) Use a computer package based on the simplex
Work through the simplex method step by step (in tabular form) to solve the following problem.Maximize Z = 2x1 €“ x2 + x3,Subject toand x1 ‰¥ 0, x2 ‰¥ 0, x3 ‰¥ 0.
Work through the simplex method step by step to solve the following problem.Maximize Z = €“ x1 + x2 +2x3,Subject toand x1 ‰¥ 0, x2 ‰¥ 0, x3 ‰¥ 0.
Consider the following statements about linear programming and the simplex method. Label each statement as true or false, and then justify your answer.(a) In a particular iteration of the simplex method, if there is a tie for which variable should be the leaving basic variable, then the next BF
Suppose that the following constraints have been provided for a linear programming model with decision variables x1 and x2.andx1 ‰¥ 0, x2 ‰¥ 0.(a) Demonstrate graphically that the feasible region is unbounded.(b) If the objective is to maximize Z=–x1 + x2, does the model have an optimal
A certain linear programming model involving two activities has the feasible region shown below.The objective is to maximize the total profit from the two activities. The unit profit for activity 1 is $1,000 and the unit profit for activity 2 is $2,000.(a) Calculate the total profit for each CPF
Follow the instructions of Prob. 4.5-2 when the constraints are the following:andx1 ‰¥ 0, x2 ‰¥ 0,In problem(a) Demonstrate graphically that the feasible region is unbounded.(b) If the objective is to maximize Z = – 4 x1 – x2, does the model have an optimal solution? If so, find it. If
Consider the following problem.Maximize Z = 5x1 + x2 + 3x3 + 4x4,Subject toand x1 ‰¥ 0, x2 ‰¥ 0, x3 ‰¥ 0, x4 ‰¥ 0. Work through the simplex method step by step to demonstrate that Z is unbounded.
A basic property of any linear programming problem with a bounded feasible region is that every feasible solution can be expressed as a convex combination of the CPF solutions (perhaps in more than one way). Similarly, for the augmented form of the problem, every feasible solution can be expressed
Using the facts given in Prob. 4.5-5, show that the following statements must be true for any linear programming problem that has a bounded feasible region and multiple optimal solutions: (a) Every convex combination of the optimal BF solutions must be optimal. (b) No other feasible solution can be
Consider a two-variable linear programming problem whose CPF solutions are (0, 0), (6, 0), (6, 3), (3, 3), and (0, 2). (See Prob. 3.2-2 for a graph of the feasible region.) (a) Use the graph of the feasible region to identify all the constraints for the model. (b) For each pair of adjacent CPF
Consider the following problem.Maximize Z = x1 + x2 + x3 +x4,Subject toandxj ‰¥ 0, for j = 1, 2, 3, 4.Work through the simplex method step by step to find all the optimal BF solutions.
Consider the following problem.Maximize Z = 2x1 + 3x2.Subject toandx1 ‰¥ 0, x2 ‰¥ 0.(a) Solve this problem graphically.(b) Using the Big M method, construct the complete first simplex tableau for the simplex method and identify the corresponding initial (artificial) BF solution. Also identify
Consider the following problem.Maximize Z = 4x1 + 2x2 + 3x3 + 5x4,Subject toandxj ‰¥ 0, for j = 1, 2, 3, 4.(a) Using the Big M method, construct the complete first simplex tableau for the simplex method and identify the corresponding initial (artificial) BF solution. Also identify the initial
Consider the following problem.Minimize Z = 2x1 +3x2 + x3,Subject toandx1 ‰¥ 0, x2 ‰¥ 0, x3 ‰¥ 0.(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, work through the simplex method step by step to solve the
For the Big M method, explain why the simplex method never would choose an artificial variable to be an entering basic variable once all the artificial variables are nonbasic.
Consider 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 model. Label each as either feasible or infeasible.(b) Calculate the value of the objective function for each of
Consider the following problem.Maximize Z = 90x1 + 70x2,Subject toandx1 ‰¥ 0, x2 ‰¥ 0.(a) Demonstrate graphically that this problem has no feasible solutions.(b) Use a computer package based on the simplex method to determine that the problem has no feasible solutions.(c) Using the Big M
Follow the instructions of Prob. 4.6-5 for the following problem.Minimize Z = 5,000x1 + 7,000x2Subject toandx1 ‰¥ 0, x2 ‰¥ 0.(a) Demonstrate graphically that this problem has no feasible solutions.(b) Use a computer package based on the simplex method to determine that the problem has no
Consider the following problem.Maximize Z = 2x1 +5x2 + 3x3,Subject toandx1 ‰¥ 0, x2 ‰¥ 0, x3 ‰¥ 0.(a) Using the Big M method, construct the complete first simplex tableau for the simplex method and identify the corresponding initial (artificial) BF solution. Also identify the initial
Consider the following problem.Minimize Z = 2x1 + x2 +3x3,Subject toand x1 ‰¥ 0, x2 ‰¥ 0, x3 ‰¥ 0. (a) Using the two-phase method, work through phase 1 step by step. (b) Use a software package based on the simplex method to formulate and solve the phase 1 problem. (c)
Consider the following problem.Minimize Z = 3x1 + 2x2 + 4x3,Subject toand x1 ‰¥ 0, x2 ‰¥ 0, x3 ‰¥ 0.(a) Using the Big M method, work through the simplex method step by step to solve the problem.(b) Using the two-phase method, work through the simplex method step by step to solve the
Follow the instructions of Prob. 4.6-9 for the following problem.Minimize Z = 3x1 + 2x2 + 7x3Subject toand x1 ‰¥ 0, x2 ‰¥ 0, x3 ‰¥ 0.(a) Using the Big M method, work through the simplex method step by step to solve the problem.(b) Using the two-phase method, work through the simplex method
Label each of the following statements as true or false, and then justify your answer.(a) When a linear programming model has an equality constraint, an artificial variable is introduced into this constraint in order to start the simplex method with an obvious initial basic solution that is
Consider the following problem.Maximize Z = x1 + 4x2 + 2x3,Subject toand x2 ‰¥ 0, x3 ‰¥ 0. (no nonnegativity constraint for x1). (a) Reformulate this problem so all variables have nonnegativity constraints. (b) Work through the simplex method step by step to solve the problem. (c)
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