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

Consider the following problem.Maximize Z = 2x1 €“ x2 + 3x3,Subject toand x1 ‰¥ 0, x2 ‰¥ 0, x3 ‰¥ 0.
Read the referenced article that fully describes the OR study summarized in the application vignette presented in Sec. 7.2. 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.Maximize Z = €“5x1 + 5x2 + 13x3,Subject toand xj ‰¥ 0 (j = 1, 2, 3). If we let 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. 7.2-2. Suppose that the right-hand sides of the functional constraints are changed to 20 + 2θ (for constraints 1) and 90 – θ (for constraint 2), where θ can be assigned any positive or negative values.
Consider the following problem.Maximize Z = 2x1 + 5x2,Subject toand 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 within ±50 percent.
Reconsider the model given in Prob. 7.3-1. 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
Reconsider the model of Prob. 7.1-1. 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 = x1 + 2x2,Subject toand 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. (a) Use the graphical method to solve this
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
Reconsider Prob. 7.3-4. 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
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 Solver to do parts (a) to (g) below. Management
Reconsider the Union Airways problem and its spreadsheet model that was dealt with in Prob. 7.3-6. 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
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 example illustrating the use of robust optimization that was presented in Sec. 7.4. Wyndor management now feels that the analysis described in this example was overly conservative for three reasons: (1) it is unlikely that the true value of a parameter will turn out to be quite near
Consider the following problem.Maximize Z = c1x1 + c2x2,Subject toand x1 ‰¥ 0, x2 ‰¥ 0. The estimates and ranges of uncertainty for the parameters are shown in the next table. (a) Use the graphical method to solve this model when using the estimates of the parameters. (b) Now use
Follow the instructions of Prob. 7.4-2 when considering the following problem and the information provided about its parameters in the table below.Minimize Z = c1x1 + c2x2,subject to the constraints shown at the top of the next column.and x1 ‰¥ 0, x2 ‰¥ 0, (a) Use the graphical
Consider the following problem.Maximize Z = 5x1 + c2x2 + c3x3,Subject toand x1 ‰¥ 0, x2 ‰¥ 0, x3 ‰¥ 0. The estimates and ranges of uncertainty for the uncertain parameters are shown in the next table. (a) Solve this model when using the estimates of the parameters. (b)
Consider the following problem.Minimize W = 5y1 + 4y2,Subject toand 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 x5 and x6 denote the slack
Reconsider the example illustrating the use of chance constraints that was presented in Sec. 7.5. The concern is that there is some uncertainty about how much production time will be available for Wyndorís two new products when their production begins in the three plants a little later. Table 7.11
Consider the following constraint whose right-hand side b is assumed to have a normal distribution with a mean of 100 and some standard deviation σ. 30x1 + 20x2 ≤ b A quick investigation of the possible spread of the random variable b has led to the estimate that σ = 10. However, a subsequent
Suppose that a linear programming problem has 20 functional constraints in inequality form such that their right-hand sides (the bi) are uncertain parameters, so chance constraints with some a are introduced in place of these constraints. After next substituting the deterministic equivalents of
Consider the following problem.Maximize z = 20x1 + 30x2 + 25x3,Subject toand x1 ‰¥ 0, x2 ‰¥ 0, x3 ‰¥ 0, where b1, b2, and b3 are uncertain parameters that have mutually independent normal distributions. The mean and standard deviation of these parameters are (90, 3),
Reconsider the example illustrating the use of stochastic programming with recourse that was presented in Sec. 7.6. Wyndor management now has obtained additional information about the rumor that a competitor is planning to produce and market a special new product that would compete directly with
The situation is the same as described in Prob. 7.6- 1 except that Wyndor management does not consider the additional information about the rumor to be reliable. Therefore, they havenít yet decided whether their best estimate of the probability that the rumor is true should be 0.5 or 0.75 or
The Royal Cola Company is considering developing a special new carbonated drink to add to its standard product line of drinks for a couple years or so (after which it probably would be replaced by another special drink). However, it is unclear whether the new drink would be profitable, so analysis
Consider the following problem.Minimize Z = 5x1 + c2x2,Subject toand x1 ‰¥ 0, x2 ‰¥ 0, where x1 represents the level of activity 1 and x2 represents the level of activity 2. The values of c2, a12, and a22 have not been determined yet. Only activity 1 needs to be undertaken soon
Refer to Sec. 3.4 (subsection entitled "Controlling Air Pollution") for the Nori & Leets Co. problem. After the OR team obtained an optimal solution, we mentioned that the team then conducted sensitivity analysis. We now continue this story by having you retrace the steps taken by the OR team,
The Ploughman family has owned and operated a 640-acre farm for several generations. The family now needs to make a decision about the mix of livestock and crops for the coming year. By assuming that normal weather conditions will prevail next year, a linear programming model can be formulated and
Consider the following problem.Maximize Z = 2x1 + 7x2 €“ 3x3,Subject toand 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:
This case is a continuation of Case 4.3, which involved the Springfield School Board assigning students from six residential areas to the city’s three remaining middle schools. After solving a linear programming model for the problem with any software package, that package’s sensitivity
After setting goals for how much the sales of three products should increase as a result of an upcoming advertising campaign, the management of the Profit & Gambit Co. now wants to explore the trade-off between advertising cost and increased sales. Your first task is to perform the associated
Reconsider the model of Prob. 7.2-4. Suppose that the right-hand sides of the functional constraints are changed to 30 + 3θ (for constraint 1) and 10 – θ (for constraint 1) where θ can be assigned any positive or negative values.
Consider the following problem.Maximize Z = 2x1 €“ x2 + x3,Subject toand 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:
Consider the Distribution Unlimited Co. problem presented in Sec. 3.4 and summarized in Fig. 3.13.
Consider the following problem.Maximize Z = c1x1 + c2x2,Subject toand 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
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. Use the formula b* = S*b-bar to find the allowable range for each bi. Then interpret each allowable range
Consider the following problem. Maximize Z = 8x1 + 24x2, Subject toAnd 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 the following problem.Maximize Z = x1 €“ x2,Subject toand x1 ‰¥ 0, x2 ‰¥ 0, (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.
Use the dual simplex method manually to solve the following problem.Minimize Z = 5x1 + 2x2 + 4x3,Subject toand x1 ‰¥ 0, x2 ‰¥ 0, x3 ‰¥ 0.
Use the dual simplex method manually to solve the following problem.Minimize Z = 7x1 + 2x2 + 5x3 +4x4,Subject toand xj ‰¥ 0, for j = 1, 2, 3, 4.
Consider the following problem.Maximize Z = 3x1 + 2x2,Subject toand x1 ‰¥ 0, x2 ‰¥ 0.
Consider the example for case 1 of sensitivity analysis given in Sec. 7.2, 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 part (a) of Prob. 7.2-2. Use the dual simplex method manually to reoptimize, starting from the revised final tableau.
Consider the following problem.Maximize Z = 2x1 + x2,Subject toand x1 ‰¥ 0, x2 ‰¥ 0. I (a) Solve this problem graphically. (b) Use the upper bound technique manually to solve this problem. (c) Trace graphically the path taken by the upper bound technique.
Use the upper bound technique manually to solve the following problem.Maximize Z = x1 + 3x2 €“ 2x3,Subject toand x1 ‰¥ 0, x2 ‰¥ 0, x3 ‰¥ 0.
Use the upper bound technique manually to solve the following problem.Maximize Z = 2x1 + 3x2 €“ 2x3 + 5x4,subject toand
Use the upper bound technique manually to solve the following problem.Maximize Z = 2x1 + 5x2 +3x3 + 4x4 + x5,Subject toand 0 ‰¤ xj ‰¤ 1, for j = 1, 2, 3, 4, 5
Use parametric linear programming to find the optimal solution for the following problem as a function of θ, for 0 ≤ θ ≤ 20. Maximize Z (θ) = (20 + 4θ)x1 + (30 - 3θ) x2 + 5x3, Subject to and x1 ≥ 0, x2 ≥ 0, x3 ≥ 0.
Simultaneously use the upper bound technique and the dual simplex method manually to solve the following problem.Minimize Z = 3x1 + 4x2 + 2x3,Subject toAnd 0 ‰¤ x1 ‰¤ 25, 0 ‰¤ x2 ‰¤ 5, 0 ‰¤ x3 ‰¤ 15.
Reconsider the example used to illustrate the interiorpoint algorithm in Sec. 8.4. 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
Consider the following problem. Maximize Z = 3x1 + x2, Subject to x1 + x2 ≤ 4 and x1 ≥ 0, x2 ≥ 0.
Consider the following problem. Maximize Z = x1 + 2x2, Subject to x1 + x2 = 8 and x1 ≥ 0, x2 ≥ 0.
Consider the following problem.Maximize Z = x1 + x2,Subject toand x1 ‰¥ 0, x2 ‰¥ 0. (a) Solve the problem graphically.
Consider the following problem. Maximize Z = 2x1 +5x2 +7x3, Subject to x1 + 2x2 + 3x3 = 6 and x1 ≥ 0, x2 ≥ 0, x3 ≥ 0. (a) Graph the feasible region. (b) Find the gradient of the objective function, and then find the projected gradient onto the feasible region.
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. 8.4 to the Wyndor Glass Co. problem presented in Sec. 3.1. Also draw a figure like Fig. 8.8 to show the trajectory of the algorithm in the
Consider the following problem.Maximize Z(Î¸) = (10 €“ Î¸)x1 + (12 + Î¸) x2 + (7 + 2Î¸)x3,Subject toand x1 ‰¥ 0, x2 ‰¥ 0, x3 ‰¥ 0.
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 θ, for 0 ≤ θ ≤ 25. Maximize Z(θ) = 2x1 + x2, Subject to and x1 ≥ 0, x2 ≥ 0. Indicate graphically what this algebraic
Use parametric linear programming to find an optimal solution for the following problem as a function of θ, for 0 ≤ θ ≤ 30. Maximize Z (θ) = 5x1 + 6x2 + 4x3 + 7x4, Subject to and xj ≥ 0, for j = 1, 2, 3, 4. Then identify the value of θ that gives the largest optimal value of Z (θ).
Consider Prob. 7.2-3. Use parametric linear programming to find an optimal solution as a function of θ for -20θ ≤ θ ≤ 0.
Consider the Z*(θ) function shown in Fig. 8.1 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 the Z*(θ) function shown in Fig. 8.2 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.
Letsubject to and xj ‰¥ 0, for j = 1, 2,., n
Read the referenced article that fully describes the OR study summarized in the application vignette in Sec. 9.1. 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.
Interactively apply the transportation simplex method to solve the Northern Airplane Co. production scheduling problem as it is formulated in Table 9.9.
Reconsider Prob. 9.1-2.
Reconsider Prob. 9.1-3b. Starting with the northwest corner rule, interactively apply the transportation simplex method to obtain an optimal solution for this problem. In problem
Reconsider Prob. 9.1-4. Starting with the northwest corner rule, interactively apply the transportation simplex method to obtain an optimal solution for this problem. In problem Consider the transportation problem having the following parameter table:
Reconsider Prob. 9.1-6. Starting with Russell's approximation method, interactively apply the transportation simplex method to obtain an optimal solution for this problem. In problem After several iterations of the transportation simplex method, a BF solution is obtained that has the following
Reconsider the transportation problem formulated in Prob. 9.1-7a.
Follow the instructions of Prob. 9.2-15 for the transportation problem formulated in Prob. 9.1-7a.
Consider the transportation problem having the following parameter table:
Consider the Northern Airplane Co. production scheduling problem presented in Sec. 9.1 (see Table 9.7). 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
Consider the general linear programming formulation of the transportation problem (see Table 9.6). Verify the claim in Sec. 9.2 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
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
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
A contractor, Susan Meyer, has to haul gravel to three building sites. She can purchase as much as 18 tons at a gravel pit in the north of the city and 14 tons at one in the south. She needs 10, 5, and 10 tons at sites 1, 2, and 3, respectively. The purchase price per ton at each gravel pit and the
Consider the transportation problem formulation and solution of the Metro Water District problem presented in Secs. 9.1 and 9.2 (see Tables 9.12 and 9.23). The numbers given in the parameter table are only estimates that may be somewhat inaccurate, so management now wishes to do some what-if
Without generating the Sensitivity Report, adapt the sensitivity analysis procedure presented in Secs. 7.1 and 7.2 to conduct the sensitivity analysis specified in the four parts of Prob. 9.2-22.
Consider the transportation problem having the following parameter table:
Consider the transportation problem having the following parameter table:
Consider the transportation problem having the following parameter table:
Consider the transportation problem having the following parameter table:
Consider the prototype example for the transportation problem (the P & T Co. problem) presented at the beginning of Sec. 9.1. 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:
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
Consider the transportation problem having the following parameter table:
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.Plants 1, 2, 3, and 4 make 10, 20, 20, and 10 shipments per month, respectively. Retail outlets 1, 2, 3, and 4 need to receive 20, 10, 10,
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
Consider the assignment problem having the following cost table.(a) Draw the network representation of this assignment problem. (b) Formulate this problem as a transportation problem by constructing the appropriate parameter table. (c) Display this formulation on an Excel spreadsheet.
Four cargo ships will be used for shipping goods from one port to four other ports (labeled 1, 2, 3, 4). Any ship can be used for making any one of these four trips. However, because of differences in the ships and cargoes, the total cost of loading, transporting, and unloading the goods for the
Reconsider Prob. 9.1-4. Suppose that the sales forecasts have been revised downward to 240, 400, and 320 units per day of products 1, 2, and 3, respectively, and that each plant now has the capacity to produce all that is required of any one product. Therefore, management has decided that each new
The coach of an age group swim team needs to assign swimmers to a 200-yard medley relay team to send to the Junior Olympics. Since most of his best swimmers are very fast in more than one stroke, it is not clear which swimmer should be assigned to each of the four strokes. The five fastest swimmers
Consider the assignment problem formulation of Option 2 for the Better Products Co. problem presented in Table 9.29. (a) Reformulate this problem as an equivalent transportation problem with three sources and five destinations by constructing the appropriate parameter table.
Starting with Vogel's approximation method, interactively apply the transportation simplex method to solve the Job Shop Co. assignment problem as formulated in Table 9.26b. (As stated in Sec. 9.3, the resulting optimal solution has x14 = 1, x23 = 1, x31 = 1, x42 = 1, and all other xij = 0.)
Reconsider Prob. 9.1-7. Now assume that distribution centers 1, 2, and 3 must receive exactly 10, 20, and 30 units per week, respectively. For administrative convenience, management has decided that each distribution center will be supplied totally by a single plant, so that one plant will supply
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 $31, $29, $32, $28, and $29 in Plants 1, 2, 3, 4, and 5, respectively. The unit manufacturing cost of the second
Consider the assignment problem having the following cost table.
Consider the linear programming model for the general assignment problem given in Sec. 9.3. Construct the table of constraint coefficients for this model. Compare this table with the one for the general transportation problem (Table 9.6). In what ways does the general assignment problem have more
Reconsider the assignment problem presented in Prob. 9.3-2. Manually apply the Hungarian algorithm to solve this problem. (You may use the corresponding interactive procedure in your IOR Tutorial.)
Reconsider Prob. 9.3-4. See its formulation as an assignment problem in the answers given in the back of the book. Manually apply the Hungarian algorithm to solve this problem. (You may use the corresponding interactive procedure in your IOR Tutorial.)

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