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introduction to operations research
Introduction To Operations Research 7th Edition Frederick S. Hillier, Gerald J. Lieberman - Solutions
You are given the following payoff table (in units of dollars):You have the option of paying $100 to have research done to better predict which state of nature will occur. When the true state of nature is S1, the research will accurately predict S1 60 percent of the time (but will inaccurately
Reconsider the Goferbroke Co. prototype example, including its analysis in Sec. 15.3. With the help of a consulting geologist, some historical data have been obtained that provide more precise information on the likelihood of obtaining favorable seismic soundings on similar tracts of land.
Using Bayes’ decision rule, consider the decision analysis problem having the following payoff table (in units of thousands of dollars):.T (a) Which alternative should be chosen? What is the resulting expected payoff?(b) You are offered the opportunity to obtain information which will tell you
Betsy Pitzer makes decisions according to Bayes’ decision rule. For her current problem, Betsy has constructed the following payoff table (in units of dollars):T (a) Which alternative should Betsy choose?T (b) Find EVPI.(c) What is the most that Betsy should consider paying to obtain more
You are given the following payoff table (in units of thousands of dollars) for a decision analysis problem:T (a) According to Bayes’ decision rule, which alternative should be chosen?T (b) Find EVPI.(c) You are given the opportunity to spend $1,000 to obtain more information about which state of
Management of Silicon Dynamics now is considering doing full-fledged market research at a cost of $1 million to predict which of the two levels of demand is likely to occur. Previous experience indicates that such market research is correct two-thirds of the time.T (a) Find EVPI for this
An individual makes decisions according to Bayes’ decision rule. For her current problem, she has constructed the following payoff table, and she now wishes to maximize the expected payoff.The value of x currently is 50, but there is an opportunity to increase x by spending some money now.What is
The cost of a spare engine purchased now is $400,000, whereas the cost of a spare engine purchased at a later date is $900,000.Spares must always be supplied if they are demanded, and unused engines will be scrapped when the airplanes become obsolete. Holding costs and interest are to be neglected.
A new type of airplane is to be purchased by the Air Force, and the number of spare engines to be ordered must be determined. The Air Force must order these spare engines in batches of five, and it can choose among only 15, 20, or 25 spares. The supplier of these engines has two plants, and the Air
Dwight Moody is the manager of a large farm with 1,000 acres of arable land. For greater efficiency, Dwight always devotes the farm to growing one crop at a time. He now needs to make a decision on which one of four crops to grow during the upcoming growing season. For each of these crops, Dwight
You are given the following payoff table (in units of thousands of dollars) for a decision analysis problem:T (a) Which alternative should be chosen under the maximin payoff criterion?T (b) Which alternative should be chosen under the maximum likelihood criterion?T (c) Which alternative should be
Consider a decision analysis problem whose payoffs (in units of thousands of dollars) are given by the following payoff table:T (a) Which alternative should be chosen under the maximin payoff criterion?T (b) Which alternative should be chosen under the maximum likelihood criterion?T (c) Which
Warren Buffy decides that Bayes’decision rule is his most reliable decision criterion. He believes that 0.1 is just about right as the prior probability of an improving economy, but is quite uncertain about how to split the remaining probabilities between a stable economy and a worsening
Reconsider Prob.
Warren Buffy is an enormously wealthy investor who has built his fortune through his legendary investing acumen. He currently has been offered three major investments and he would like to choose one. The first one is a conservative investment that would perform very well in an improving economy and
Jean Clark is the manager of the Midtown Saveway Grocery Store. She now needs to replenish her supply of strawberries.Her regular supplier can provide as many cases as she wants. However, because these strawberries already are very ripe, she will need to sell them tomorrow and then discard any that
Silicon Dynamics has developed a new computer chip that will enable it to begin producing and marketing a personal computer if it so desires. Alternatively, it can sell the rights to the computer chip for $15 million. If the company chooses to build computers, the profitability of the venture
Which action (drill for oil or sell the land) should be chosen?
Should a seismic survey be conducted before an action is chosen?
An oil company deciding whether to drill for oil in a particular location. How likely is oil there? How much? How deep will they need to drill? Should geologists investigate the site further before drilling?
An agricultural firm selecting the mix of crops and livestock for the upcoming season.What will be the weather conditions? Where are prices headed? What will costs be?
A government contractor bidding on a new contract. What will be the actual costs of the project? Which other companies might be bidding? What are their likely bids?
A financial firm investing in securities. Which are the market sectors and individual securities with the best prospects? Where is the economy headed? How about interest rates? How should these factors affect the investment decisions?
A manufacturer introducing a new product into the marketplace. What will be the reaction of potential customers? How much should be produced? Should the product be test marketed in a small region before deciding upon full distribution? How much advertising is needed to launch the product
Consider the following problem.Maximize Z 3x1 7x2 2x3, subject to2x1 2x2 x3 103x1 x2 x3 20 and 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
Consider the following problem.Maximize Z 2x1 4x2 3x3, subject to x1 3x2 2x3 20 x1 5x2 2x3 10 and x1 0, x2 0, x3 0.Let x4 be the artificial variable for the first constraint. Let x5 andx6 be the surplus variable and artificial variable, respectively, for the second
Consider the following problem.Minimize Z 2x1 3x2 2x3, subject to x1 4x2 2x3 8 3x1 2x2 2x3 6 and x1 0, x2 0, x3 0.Let x4 and x6 be the surplus variables for the first and second constraints, respectively. Let x5 and x7 be the corresponding artificial variables. After you
Use artificial variables and the Big M method to construct the complete first sim-plex tableau for the simplex method, and then identify the columns that will contains S* for applying the fundamental insight in the final tableau. Explain why these are the appropriate columns.
For iteration 2 of the example in Sec. 5.3, the following expression was shown:Final row 0 [3, 5 0, 0, 0 0] [0, 3 2, 1] .Derive this expression by combining the algebraic operations (in matrix form) for iterations 1 and 2 that affect row 0.
Consider the following problem.Maximize Z c1x1 c2x2 c3x3, subject to x1 2x2 x3 b 2x1 x2 3x3 2b and 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
You are using the simplex method to solve the following linear programming problem.Maximize Z 6x1 5x2 x3 4x4, subject to 3x1 2x2 3x3 x4 120 3x1 3x2 x3 3x4 180 and x1 0, x2 0, x3 0, x4 0.You have obtained the following final simplex tableau where x5 and x6 are the slack
Consider the following problem.Maximize Z 20x1 6x2 8x3, subject to 8x1 2x2 3x3 200 4x1 3x2 3x3 100 2x1 3x2 x3 50 2x1 3x2 x3 20 and 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
Consider the following problem.Maximize Z x1 x2 2x3, subject to x1 x2 3x3 15 2x1 x2 x3 2x1 x2 x3 4 and x1 0, x2 0, x3 0.Let x4, x5, and x6 denote the slack variables for the respective constraints. After the simplex method is applied, a portion of the final simplex tableau is
Consider the following problem.Maximize Z 6x1 x2 2x3, subject to2x1 2x2 1 2x3 24x1 2x2 3 2x3 32x1 2x2 1 2x3 1 and 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
Consider the following problem.Maximize Z 4x1 3x2 x3 2x4, subject to 4x1 2x2 x3 x4 5 3x1 x2 2x3 x4 4 and x1 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
Work through the revised simplex method step by step to solve each of the following models:(a) Model given in Prob. 3.1-5.(b) Model given in Prob. 4.7-8.D 5.3-1.* Consider the following problem.Maximize Z x1 x2 2x3, subject to 2x1 2x2 3x3 5 x1 x2 x3 3 x1 x2 x3 2 and x1 0, x2 0,
Work through the revised simplex method step by step to solve the model given in Prob. 4.7-6.I
Work through the revised simplex method step by step to solve the model given in Prob. 4.1-5.I
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 B1 to calculate the current solution, and then perform the next iteration (or demonstrate that the current solution is
Reconsider Prob.
Consider the following problem.Maximize Z 8x1 4x2 6x3 3x4 9x5, subject to x1 2x2 3x3 3x4 x5 180 (resource 1)4x1 3x2 2x3 x4 x5 270 (resource 2)x1 3x2 2x3 x4 3x5 180 (resource 3)and xj 0, j 1, . . . , 5.You are given the facts that the basic variables in the optimal
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) [(4, 2, 4) (2, 4, 3)] (2, 4, 3) (2, 2, 1), where 0 1 for just the line segment between these points.After augmenting with the slack variables x4, x5, x6, x7 for the respective functional
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.(b) For the CPF solution (2, 4, 3) and its three adjacent CPF solutions (4, 2, 4), (0, 4, 2),
Consider the following problem.Maximize Z 3x1 4x2 2x3, subject to x1 x2 x3 20 x1 2x2 x3 30 and x1 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
Consider the following problem.Maximize Z 2x1 2x2 3x3, subject to 2x1 x2 2x3 4 x1 x2 x3 3 and x1 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
Consider the linear programming problem given in Table 6.1 as the dual problem for the Wyndor Glass Co. example.Coefficient of:Basic Right Iteration Variable Eq. Z x1 x2 x3 x4 x5 x6 Side Z (0) 1 0 1 3 0 2 0 20 x4 (1) 0 0 4 5 1 3 0 30 1 x1 (2) 0 1 1 2 0 1 0 10 x6 (3) 0 0 2 3 0 1 1 10
Consider the following problem.Maximize Z 2x1 x2 x3, subject to 3x1 x2 x3 60 x1 x2 2x3 10 x1 x2 x3 20 and x1 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.222 5
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(a) Solve this problem graphically. Identify
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.(a) The best CPF solution is an optimal solution.(b) An optimal solution is a CPF solution.(c) A CPF solution is the only optimal solution if none of
Reconsider the model in Prob. 3.1-4.(a) Identify the 15 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 infeasible solution.(b) For each corner-point solution,
Reconsider the model in Problem 4.6-3.(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 infeasible solution.(b) For each corner-point solution,
Consider the following problem.Minimize Z x1 2x2, subject tox1 x2 152x1 x2 902x1 x2 30 and x1 0, x2 0.(a) Solve this problem graphically.(b) Develop a table giving each of the CPF solutions and the corresponding defining equations, BF solution, and nonbasic variables.
Consider the linear programming problem given in Table 6.1 as the dual problem for the Wyndor Glass Co. example.(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
Consider the three-variable linear programming problem shown in Fig. 5.2.(a) Construct a table like Table 5.1, giving the set of defining equations for each CPF solution.(b) What are the defining equations for the corner-point infeasible solution (6, 0, 5)?(c) Identify one of the systems of three
Consider the following problem.Maximize Z 2x1 x2 x3, subject to 3x1 x2 x3 60 x1 x2 2x3 10 x1 x2 x3 20 and x1 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)
Consider the following problem.Maximize Z 2x1 3x2, subject to3x1 x2 14x1 2x2 204x1 x2 10x1 2x2 5 and x1 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
Repeat Prob. 5.1-1 for the model in Prob. 3.1-5.
Consider the following problem.Maximize Z 3x1 2x2, subject to 2x1 x2 6 x1 2x2 6 and x1 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
Consider the following problem.Maximize Z 2x1 x2 3x3, subject to x1 x2 x3 3 x1 2x2 x3 1 x1 2x2 x3 2 and x1 0, x2 0, x3 0.Suppose that the Big M method (see Sec. 4.6) is used to obtain the initial (artificial) BF solution. Let x4 be the artificial slack variable for the
Consider the Wyndor Glass Co. problem described in Sec.3.1. Suppose that, in addition to considering the introduction of two new products, management now is considering changing the production rate of a certain old product that is still profitable. Refer to Table 3.1. The number of production hours
Consider the following problem.Maximize Z 3x1 5x2 2x3, subject to2x1 2x2 x3 53x1 x2 x3 10 and x1 0, x2 0, x3 0.Let x4 and x5 be the slack variables for the respective functional constraints. After we apply the simplex method, the final simplex tableau is Parametric linear
Consider the following problem.Maximize Z 9x1 8x2 5x3, subject to 2x1 3x2 x3 4 5x1 4x2 3x3 11 and x1 0, x2 0, x3 0.Let x4 and x5 denote the slack variables for the respective functional constraints. After we apply the simplex method, the final simplex tableau is D,I (a) Suppose
Consider the following problem.Maximize Z 10x1 4x2, subject to 3x1 x2 30 2x1 x2 25 and x1 0, x2 0.Let x3 and x4 denote the slack variables for the respective functional constraints. After we apply the simplex method, the final simplex tableau is Now suppose that both of the following
Consider the following parametric linear programming problem, where the parameter must be nonnegative:Maximize Z() (5 2)x1 (2 )x2 (3 )x3, subject to 4x1 x2 2x3 5 53x1 x2 2x3 10 10and x1 0, x2 0, x3 0.Let x4 be the surplus variable for the first functional
Consider the following parametric linear programming problem.Maximize Z() 2x1 4x2 5x3, subject to x1 3x2 2x3 5 x1 2x2 3x3 6 2and x1 0, x2 0, x3 0, where can be assigned any positive or negative values. Let x4 and x5 be the slack variables for the respective functional
Consider the following parametric linear programming problem.Maximize Z() (10 4)x1 (4 )x2 (7 )x3, subject to 3x1 x2 2x3 7 (resource 1), 2x1 x2 3x3 5 (resource 2), and x1 0, x2 0, x3 0, where can be assigned any positive or negative values. Let x4 and x5 be the slack
Suppose that you now have the option of making trade-offs in the profitability of the first two activities, whereby the objective function coefficient of x1 can be increased by any amount by simultaneously decreasing the objective function coefficient of x2 by the same amount. Thus, the alternative
Consider Variation 5 of the Wyndor Glass Co. model presented in Sec. 6.7, where c2 3, a22 3, a32 4, and where the other parameters are given in Table 6.21. Starting from the resulting final tableau given at the bottom of Table 6.24, construct a table like Table 6.26 to perform parametric linear
Consider the following problem.Maximize Z 3x1 4x2 8x3, subject to 2x1 3x2 5x3 9 x1 2x2 3x3 5 and x1 0, x2 0, x3 0.Let x4 and x5 denote the slack variables for the respective functional constraints. After we apply the simplex method, the final simplex tableau is While doing
Now use a software package based on the simplex method to generate sensitivity analysis information preparatory to doing parts (a) and (c) below.C (a) Suppose that the estimates for c1 and c2 are correct but the estimates for both b1 and b2 are incorrect. Consider the following four cases where the
Consider the following problem.Maximize Z 2x1 5x2, subject to x1 2x2 10 x1 3x2 12 and x1 0, x2 0.Let x3 and x4 denote the slack variables for the respective functional constraints. After we apply the simplex method, the final simplex tableau is While doing postoptimality analysis, you
Consider the Union Airways problem presented in Sec.3.4, including the data given in Table 3.19.Management now is considering increasing the level of service provided to customers by increasing one or more of the numbers in the rightmost column of Table 3.19 for the minimum number of agents needed
David, LaDeana, and Lydia are the sole partners and workers in a company which produces fine clocks. David and LaDeana each are available to work a maximum of 40 hours per week at the company, while Lydia is available to work a maximum of 20 hours per week.The company makes two different types of
Ken and Larry, Inc., supplies its ice cream parlors with three flavors of ice cream: chocolate, vanilla, and banana. Because of extremely hot weather and a high demand for its products, the company has run short of its supply of ingredients: milk, sugar, and cream. Hence, they will not be able to
For Variation 6 of the Wyndor Glass Co. model presented in Sec. 6.7, use the last tableau in Table 6.25 to do the following.(a) Find the allowable range to stay feasible for each bi.(b) Find the allowable range to stay optimal for c1 and c2.C (c) Use a software package based on the simplex method
For the original Wyndor Glass Co. problem, use the last tableau in Table 4.8 to do the following.(a) Find the allowable range to stay feasible for each bi.(b) Find the allowable range to stay optimal for c1 and c2.C (c) Use a software package based on the simplex method to find these allowable
For the problem given in Table 6.21, find the allowable range to stay optimal for c2. Show your work algebraically, using the tableau given in Table 6.21. Then justify your answer from a geometric viewpoint, referring to Fig. 6.3.
Consider the following problem.Maximize Z 3x1 x2 2x3, subject to x1 x2 2x3 20 2x1 x2 x3 10 and x1 0, x2 0, x3 0.Let x4 and x5 denote the slack variables for the respective functional constraints. After we apply the simplex method, the final simplex tableau is(a) Perform
Consider Variation 5 of the Wyndor Glass Co. model (see Fig. 6.6 and Table 6.24), where the changes in the parameter values given in Table 6.21 are c2 3, a22 3, and a32 4. Verify both algebraically and graphically that the allowable range to stay optimal for c1 is c1 9 4.
Consider Variation 5 of the Wyndor Glass Co. model (see Fig. 6.6 and Table 6.24), where the changes in the parameter values given in Table 6.21 are c2 3, a22 3, and a32 4. Use the formula b* S*b to find the allowable range to stay feasible for each bi. Then interpret each allowable range
Consider the following problem.Maximize Z c1x1 c2x2, subject to 2x1 x2 b1 x1 x2 b2 and x1 0, x2 0.Let x3 and x4 denote the slack variables for the respective functional constraints. When c1 3, c2 2, b1 30, and b2 10, the simplex method yields the following final simplex tableau.(a) Use
Consider the Union Airways problem presented in Sec.3.4, including the data given in Table 3.19.Management is about to begin negotiations on a new contract with the union that represents the company’s customer service agents. This might result in some small changes in the daily costs per agent
After further negotiations with each vendor, management of the G. A. Tanner Co. has learned that either of them would be willing to consider increasing their supply of their respective subassemblies over the previously stated maxima (3,000 subassemblies of type A per day and 1,000 of type B per
Reconsider Prob.
One of the products of the G. A. Tanner Company is a special kind of toy that provides an estimated unit profit of $3. Because of a large demand for this toy, management would like to increase its production rate from the current level of 1,000 per day.However, a limited supply of two subassemblies
Consider the following problem.Maximize Z 2x1 x2 x3, subject to 3x1 2x2 2x3 15x1 x2 x3 3 x1 x2 x3 4 and x1 0, x2 0, x3 0.If we let x4, x5, and x6 be the slack variables for the respective constraints, the simplex method yields the following final set of equations:(0) Z 2x3
Suppose that we now want to apply parametric linear programming analysis to this problem. Specifically, the right-hand sides of the functional constraints are changed to 30 3 (for constraint 1)and 10 (for constraint 2), where can be assigned any positive or negative values.Express the
Consider the following problem.Maximize Z 2x1 7x2 3x3, subject to x1 3x2 4x3 30 x1 4x2 x3 10 and x1 0, x2 0, x3 0.By letting x4 and x5 be the slack variables for the respective constraints, the simplex method yields the following final set of equations:(0) Z x2 x3 2x5 20,(1)
Consider the following problem.Maximize Z 2x1 x2 x3, subject to 3x1 x2 x3 60 x1 x2 2x3 10 x1 x2 x3 20 and x1 0, x2 0, x3 0.Let x4, x5, and x6 denote the slack variables for the respective constraints. After we apply the simplex method, the final simplex tableau is Now you are
Suppose that we now want to apply parametric linear programming analysis to this problem. Specifically, the right-hand sides of the functional constraints are changed to 20 2 (for constraint 1)and 90 (for constraint 2), where can be assigned any positive or negative values.Express the
Consider the following problem.Minimize W 5y1 4y2, subject to 4y1 3y2 4 2y1 y2 3 y1 2y2 1 y1 y2 2 and y1 0, y2 0.Because this primal problem has more functional constraints than variables, suppose that the simplex method has been applied directly to its dual problem. If we let
You are now to conduct sensitivity analysis by independently investigating each of the following six changes in the original model. For each change, use the sensitivity analysis procedure to revise the given final set of equations (in tableau form) and convert it to proper form from Gaussian
Consider the following problem.Maximize Z 3x1 x2 4x3, subject to 6x1 3x2 5x3 25 3x1 4x2 5x3 20 and x1 0, x2 0, x3 0.The corresponding final set of equations yielding the optimal solution is(0) Z 2x2 1 5x4 3 5x5 17(1) x1 1 3x2 1 3x4 1 3x5 5 3(2) x2 x3 1
Use duality theory directly to determine whether the original optimal solution is still optimal.
Reconsider part (d) of Prob.
Use duality theory directly to determine whether the current basic solution remains optimal after each of the following independent changes.(a) The change in part (b) of Prob. 6.7-4(b) The change in part (d) of Prob. 6.7-4
Use duality theory directly to determine whether the current basic solution remains optimal after each of the following independent changes.(a) The change in part (c) of Prob. 6.7-3(b) The change in part ( f ) of Prob. 6.7-3
Use duality theory directly to determine whether the current basic solution remains optimal after each of the following independent changes.(a) The change in part (e) of Prob. 6.7-1(b) The change in part (g) of Prob. 6.7-1
Consider the following problem.Minimize Z x1 3x2, subject tox1 2x2 2x1 x2 4 and x1 0, x2 0.(a) Demonstrate graphically that this problem has an unbounded objective function.(b) Construct the dual problem.(c) Demonstrate graphically that the dual problem has no feasible solutions.
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 (see
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