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introduction to operations research

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

When you deal with a transportation problem where the supply and demand quantities have integer values, explain why the steps of the transportation simplex method guarantee that all the basic variables (allocations) in the BF solutions obtained must have integer values. Begin with why this occurs
Consider the general linear programming formulation of the transportation problem (see Table 8.6). Verify the claim in Sec. 8.2 that the set of (m n) functional constraint equations(m supply constraints and n demand constraints) has one redundant equation; i.e., any one equation can be reproduced
Consider the Northern Airplane Co. production scheduling problem presented in Sec. 8.1 (see Table 8.7). Formulate this problem as a general linear programming problem by letting the decision variables be xj number of jet engines to be produced in month j (j 1, 2, 3, 4). Construct the initial
Follow the instructions of Prob. 8.2-15 for the transportation problem formulated in Prob. 8.1-7a.
Reconsider the transportation problem formulated in Prob.D,I (a) Use each of the three criteria presented in Sec. 8.2 to obtain an initial BF solution, and time how long you spend for each one. Compare both these times and the values of the objective function for these solutions.C (b) Obtain an
Reconsider Prob. 8.1-6, Starting with Russell’s approximation method, interactively apply the transportation simplex method to obtain an optimal solution for this problem.
Reconsider Prob. 8.1-4, Starting with the northwest corner rule, interactively apply the transportation simplex method to obtain an optimal solution for this problem.
Reconsider Prob. 8.1-3b. Starting with the northwest corner rule, interactively apply the transportation simplex method to obtain an optimal solution for this problem.
Reconsider Prob. 8.1-2.(a) Use the northwest corner rule to obtain an initial BF solution.(b) Starting with the initial BF solution from part (a), interactively apply the transportation simplex method to obtain an optimal solution.
Interactively apply the transportation simplex method to solve the Northern Airplane Co. production scheduling problem as it is formulated in Table 8.9.D,I
The Energetic Company needs to make plans for the energy systems for a new building.The energy needs in the building fall into three categories:(1) electricity, (2) heating water, and (3) heating space in the building. The daily requirements for these three categories (all measured in the same
The Cost-Less Corp. supplies its four retail outlets from its four plants. The shipping cost per shipment from each plant to each retail outlet is given below.
Consider the prototype example for the transportation problem (the P & T Co. problem) presented at the beginning of Sec.8.1. Verify that the solution given there actually is optimal by applying just the optimality test portion of the transportation simplex method to this solution.
Consider the transportation problem having the following parameter table:(a) Notice that this problem has three special characteristics:(1) number of sources number of destinations, (2) each supply 1, and (3) each demand 1. Transportation problems with these characteristics are of a special
Consider the transportation problem having the following parameter table:Use each of the following criteria to obtain an initial BF solution.Compare the values of the objective function for these solutions.(a) Northwest corner rule.(b) Vogel’s approximation method.(c) Russell’s approximation
Consider the transportation problem having the following parameter table:(a) Use Vogel’s approximation method manually (don’t use the interactive procedure in IOR Tutorial) to select the first basic variable for an initial BF solution.(b) Use Russell’s approximation method manually to select
The MJK Manufacturing Company must produce two products in sufficient quantity to meet contracted sales in each of the next three months. The two products share the same production facilities, and each unit of both products requires the same amount of production capacity. The available production
Redo Prob. 8.1-7 when any distribution center may receive any quantity between 10 and 30 forklift trucks per week in order to further reduce total shipping cost, provided only that the total shipped to all three distribution centers must still equal 60 trucks per week.
The Move-It Company has two plants producing forklift trucks that then are shipped to three distribution centers. The production costs are the same at the two plants, and the cost of shipping for each truck is shown for each combination of plant and distribution center:
The Onenote Co. produces a single product at three plants for four customers. The three plants will produce 60, 80, and 40 units, respectively, during the next time period. The firm has made a commitment to sell 40 units to customer 1, 60 units to customer 2, and at least 20 units to customer 3.
Reconsider the P & T Co. problem presented in Sec. 8.1.You now learn that one or more of the shipping costs per truckload given in Table 8.2 may change slightly before shipments begin.Use the Excel Solver to generate the Sensitivity Report for this problem. Use this report to determine the
The Versatech Corporation has decided to produce three new products. Five branch plants now have excess product capacity. The unit manufacturing cost of the first product would be $41, $39,$42, $38, and $39 in Plants 1, 2, 3, 4, and 5, respectively. The unit manufacturing cost of the second product
Tom would like 3 pints of home brew today and an additional 4 pints of home brew tomorrow. Dick is willing to sell a maximum of 5 pints total at a price of $3.00 per pint today and$2.70 per pint tomorrow. Harry is willing to sell a maximum of 4 pints total at a price of $2.90 per pint today and
The Childfair Company has three plants producing child push chairs that are to be shipped to four distribution centers. Plants 1, 2, and 3 produce 12, 17, and 11 shipments per month, respectively. Each distribution center needs to receive 10 shipments per month. The distance from each plant to the
Read the referenced article that fully describes the OR study summarized in the application vignette in Sec. 8.1. Briefly describe how the model for the transportation problem was applied in this study. Then list the various financial and nonfinancial benefits that resulted from this study.
Starting from the initial trial solution (x1, x2) (2, 2), use your IOR Tutorial to apply 15 iterations of the interior-point algorithm presented in Sec. 7.4 to the Wyndor Glass Co. problem presented in Sec. 3.1. Also draw a figure like Fig. 7.8 to show the trajectory of the algorithm in the
Consider the following problem.Maximize Z 2x1 5x2 7x3, subject to x1 2x2 3x3 6 and x1 0, x2 0, x3 0.I (a) Graph the feasible region.(b) Find the gradient of the objective function, and then find the projected gradient onto the feasible region.(c) Starting from the initial trial
Consider the following problem.Maximize Z 3x1 x2, subject to 3x1 2x2 45 6x1 x2 45 and x1 0, x2 0.I (a) Solve the problem graphically.
Consider the following problem.Maximize Z x1 2x2, subject to x1 x2 8 and x1 0, x2 0.C (a) Near the end of Sec. 7.4, there is a discussion of what the interior-point algorithm does on this problem when starting from the initial feasible trial solution (x1, x2) (4, 4). Verify the
Consider the following problem.Maximize Z 3x1 x2, subject to x1 x2 4 and x1 0, x2 0.I (a) Solve this problem graphically. Also identify all CPF solutions.C (b) Starting from the initial trial solution (x1, x2) (1, 1), perform four iterations of the interior-point algorithm presented in
Reconsider the example used to illustrate the interiorpoint algorithm in Sec. 7.4. Suppose that (x1, x2) (1, 3) were used instead as the initial feasible trial solution. Perform two iterations manually, starting from this solution. Then use the automatic procedure in your IOR Tutorial to check
Simultaneously use the upper bound technique and the dual simplex method manually to solve the following problem.Minimize Z 3x1 4x2 2x3,
Use the upper bound technique manually to solve the following problem.Maximize Z 2x1 5x2 3x3 4x4 x5, subject to x1 3x2 2x3 3x4 x5 6 4x1 6x2 5x3 7x4 x5 15 and 0 xj 1, for j 1, 2, 3, 4, 5.
Use the upper bound technique manually to solve the following problem.Maximize Z 2x1 3x2 2x3 5x4, subject to 2x1 2x2 x3 2x4 5 x1 2x2 3x3 4x4 5 and 0 xj 1, for j 1, 2, 3, 4.
Use the upper bound technique manually to solve the following problem.Maximize Z x1 3x2 2x3, subject to x2 2x3 1 2x1 x2 2x3 8 x1 1 x2 3 x3 2 and x1 0, x2 0, x3 0.
Consider the following problem.Maximize Z 2x1 3x2,
Let Z* max n j1 cjxj , subject ton j1 aijxj bi, for i 1, 2,..., m, and xj 0, for j 1, 2,..., n(where the aij, bi, and cj are fixed constants), and let (y1*, y2*,..., y*m) be the corresponding optimal dual solution. Then let Z** max n j1 cjxj , subject ton j1 aijxj bi ki, for i
Consider the Z*() function shown in Fig. 7.2 for parametric linear programming with systematic changes in the bi parameters.(a) Explain why this function is piecewise linear.(b) Show that this function must be concave.
Consider the Z*() function shown in Fig. 7.1 for parametric linear programming with systematic changes in the cj parameters.(a) Explain why this function is piecewise linear.(b) Show that this function must be convex.
Consider Prob. 6.7-3. Use parametric linear programming to find an optimal solution as a function of for 20 0.(Hint: Substitute for , and then increase from zero.)
Use parametric linear programming to find an optimal solution for the following problem as a function of , for 0 30.Maximize Z() 5x1 42x2 28x3 49x4,
Use the parametric linear programming procedure for making systematic changes in the bi parameters to find an optimal solution for the following problem as a function of , for 0 25.Maximize Z() 2x1 x2, subject to x1 10 2x1 x2 25 x2 10 2and x1 0, x2 0.Indicate graphically what
Consider the following problem.Maximize Z() (10 )x1 (12 )x2 (7 2)x3, subject to x1 2x2 2x3 30 x1 x2 x3 20 and x1 0, x2 0, x3 0.(a) Use parametric linear programming to find an optimal solution for this problem as a function of , for 0.(b) Construct the dual model
Use parametric linear programming to find the optimal solution for the following problem as a function of , for 0 20.Maximize Z() (20 4)x1 (30 3)x2 5x3, subject to 3x1 3x2 x3 10 8x1 6x2 4x3 25 6x1 x2 x3 15 and x1 0, x2 0, x3 0.
Consider the following problem.Maximize Z 8x1 24x2, subject to x1 2x2 10 2x1 x2 10 and x1 0, x2 0.Suppose that Z represents profit and that it is possible to modify the objective function somewhat by an appropriate shifting of key personnel between the two activities. In particular,
Consider part (a) of Prob. 6.7-2, Use the dual simplex method manually to reoptimize, starting from the revised final tableau.
Consider the example for case 1 of sensitivity analysis given in Sec. 6.7, where the initial simplex tableau of Table 4.8 is modified by changing b2 from 12 to 24, thereby changing the respective entries in the right-side column of the final simplex tableau to 54, 6, 12, and 2. Starting from this
Consider the following problem.Maximize Z 5x1 10x2,
Use the dual simplex method manually to solve the following problem.Minimize Z 7x1 2x2 5x3 4x4, subject to 2x1 4x2 7x3 x4 5 8x1 4x2 6x3 4x4 8 3x1 8x2 x3 4x4 4 and xj 0, for j 1, 2, 3, 4.
Use the dual simplex method manually to solve the following problem.Minimize Z 5x1 2x2 4x3, subject to 3x1 x2 2x3 4 6x1 3x2 5x3 10 and x1 0, x2 0, x3 0.
Consider the following problem.Maximize Z x1 2x2, subject to 2x1 x2 40 x2 15 2x1 x2 10 and x1 0, x2 0.I (a) Solve this problem graphically.(b) Use the dual simplex method manually to solve this problem.(c) Trace graphically the path taken by the dual simplex method.
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
Reconsider the Union Airways problem and its spreadsheet model that was dealt with in Prob. 6.8-6.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
Consider the Union Airways problem presented in Sec. 3.4, including the data given in Table 3.19. The Excel files for Chap. 3 include a spreadsheet that shows the formulation and optimal solution for this problem. You are to use this spreadsheet and the Excel Solver to do parts (a) to (g)
Reconsider Prob. 6.8-4, 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
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?
Consider the following problem.Maximize Z x1 2x2, subject to x1 3x2 8 (resource 1)x1 x2 4 (resource 2)and x1 0, x2 0, where Z measures the profit in dollars from the two activities and the right-hand sides are the number of units available of the respective resources.
Reconsider the model given in Prob. 6.8-1, While doing sensitivity analysis, you learn that the estimates of the right-hand sides of the two functional constraints are accurate only to within 50 percent. In other words, the ranges of likely values for these parameters are 5 to 15 for the first
Consider the following problem.Maximize Z 2x1 5x2, subject to x1 2x2 10 (resource 1)x1 3x2 12 (resource 2)and x1 0, x2 0, where Z measures the profit in dollars from the two activities.While doing sensitivity analysis, you learn that the estimates of the unit profits are accurate only to
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 first
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
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
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(a) Suppose that a new
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.Maximize Z() 2x1 4x2 5x3,subject to x1 3x2 2x3 5 x1 2x2 3x3 6 2 and 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.
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
Reconsider the model of Prob.
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
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 for each bi.(b) Find the allowable range for c1 and c2.C (c) Use a software package based on the simplex method to find these allowable ranges.
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 (c) Use a software package based on the simplex method to find these allowable ranges.
For the problem given in Table 6.21, find the allowable range 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 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 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 for each bi. Then interpret each allowable range graphically.I
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.
Consider the Distribution Unlimited Co. problem presented in Sec. 3.4 and summarized in Fig. 3.13.Although Fig. 3.13 gives estimated unit costs for shipping through the various shipping lanes, there actually is some uncertainty about what these unit costs will turn out to be. Therefore, before
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 x4
Reconsider the model of Prob. 6.7-4, 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
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:
Reconsider the model of Prob. 6.7-2, 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)
Consider the following problem.Maximize Z 5x1 5x2 13x3, subject tox1 x2 3x3 20 12x1 4x2 10x3 90 and xj 0 (j 1, 2, 3).
Read the referenced article that fully describes the OR study summarized in the application vignette presented in Sec. 6.7.Briefly describe how sensitivity analysis was applied in this study.Then list the various financial and nonfinancial benefits that resulted from the study.
Consider the following problem.Minimize W 5y1 4y2 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 x5 and x6 denote the slack variables for this dual problem, the resulting final
Reconsider the model of Prob. 6.6-1, 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
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(a) Identify the optimal solution from this set of equations.(b) Construct the dual problem.
Reconsider part (d) of Prob. 6.7-6. Use duality theory directly to determine whether the original optimal solution is still optimal.
Consider the model 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 (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 (e) of Prob. 6.7-2(b) The change in part (g) of Prob. 6.7-2
Consider the model of Prob.
Consider the following problem.Minimize Z 5x1 15x2, subject to2x1 4x2 83x1 3x2 24 and x1 0, x2 0.I (a) Demonstrate graphically that this problem has an unbounded objective function.(b) Construct the dual problem.I (c) Demonstrate graphically that the dual problem has no feasible
Consider the dual problem for the Wyndor Glass Co. example given in Table 6.1. Demonstrate that its dual problem is the?
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 problem
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
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 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
Consider the following problem.Minimize Z 5x1 10x2, subject to4x1 2x2 4 5x1 10x2 10 and x1 0, x2 0.(a) Construct the dual problem.I (b) Use graphical analysis of the dual problem to determine whether the primal problem has feasible solutions and, if so, whether its objective function is
Construct the dual problem for the linear programming problem given in Prob. 4.6-3.
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 b
Consider the following problem.Maximize Z 5x1 4x2, subject to 2x1 3x2 10 x1 2x2 20 and 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
Suppose that you also want information about the dual problem when you apply the matrix form of the simplex method(see Sec. 5.2) to the primal problem in our standard form.(a) How would you identify the optimal solution for the dual problem?(b) After obtaining the BF solution at each iteration, how

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