Questions and Answers of Introduction To Operations Research

For networks (a) and (b), use the augmenting path algorithm described in Sec. 9.5 to find the flow pattern giving the maximum flow from the source to the sink, given that the arc capacity from node i
The diagram to the right depicts a system of aqueducts that originate at three rivers (nodes R1, R2, and R3) and terminate at a major city(node T), where the other nodes are junction points in the
One track of the Eura Railroad system runs from the major industrial city of Faireparc to the major port city of Portstown. This track is heavily used by both express passenger and freight trains.The
Consider the maximum flow problem shown next, where the source is node A, the sink is node F, and the arc capacities are the numbers shown next to these directed arcs.(a) Use the augmenting path
Reconsider the maximum flow problem shown in Prob.
Formulate this problem as a minimum cost flow problem, including adding the arc A F.Use F 20.
Reconsider Prob 9.3-1.Now formulate this problem as a minimum cost flow problem by showing the appropriate network representation.
The Makonsel Company is a fully integrated company that both produces goods and sells them at its retail outlets. After production, the goods are stored in the company’s two warehouses until needed
The Audiofile Company produces boomboxes. However, management has decided to subcontract out the production of the speakers needed for the boomboxes. Three vendors are available to supply the
Consider the minimum cost flow problem shown above, where the bi values (net flows generated) are given by the nodes, the cij values (costs per unit flow) are given by the arcs, and the uij values
Reconsider the minimum cost flow problem formulated in Prob. 9.6-1.(a) Obtain an initial BF solution by solving the feasible spanning tree with basic arcs A B, A C, A F, B D, and E F, where two of
Reconsider the minimum cost flow problem formulated in Prob. 9.6-2.(a) Obtain an initial BF solution by solving the feasible spanning tree that corresponds to using just the two rail lines plus
Reconsider the minimum cost flow problem formulated in Prob. 9.6-3.Starting with the initial BF solution that corresponds to replacing the tractor every year, use the network simplex method yourself
For the P & T Co. transportation problem given in Table 8.2, consider its network representation as a minimum cost flow problem presented in Fig. 8.2. Use the northwest corner rule to obtain an
Consider the Metro Water District transportation problem presented in Table 8.12.(a) Formulate the network representation of this problem as a minimum cost flow problem. (Hint: Arcs where flow is
Consider the transportation problem having the following parameter table:Formulate the network representation of this problem as a minimum cost flow problem. Use the northwest corner rule to obtain
Consider the minimum cost flow problem shown below, where the bi values are given by the nodes, the cij values are given by the arcs, and the finite uij values are given in parentheses by the arcs.
Reconsider Prob.
Christine has done more detailed planning for this project and so now has the following expanded activity list:Construct the new project network.Activity Activity Description Predecessors Duration A
Construct the project network for a project with the following activity list.Immediate Estimated Activity Predecessors Duration A — 1 months B A 2 months C B 4 months D B 3 months E B 2 months F C
You and several friends are about to prepare a lasagna dinner. The tasks to be performed, their immediate predecessors, and their estimated durations are as follows:Tasks that Task Task Description
Consider Christine Phillip’s project involving planning and coordinating next spring’s sales management training program for her company as described in Prob.
After constructing the project network, she now is ready for the following steps.(a) Find all the paths and path lengths through this project network. Which of these paths is a critical path?(b) Find
Refer to the activity list given in Prob. 10.2-2 as Christine Phillips does more detailed planning for next spring’s sales man agement training program for her company. After constructing the
Reconsider the Reliable Construction Co. project introduced in Sec. 10.1, including the complete project network obtained in Fig. 10.7 at the end of Sec. 10.3. Note that the estimated durations of
Follow the instructions for Prob. 10.3-6 except use the optimistic estimates in Table 10.4 instead.
Follow the instructions for Prob. 10.3-6 except use the crash times given in Table 10.7 (Sec. 10.5) instead.
Reconsider Prob. 10.4-2.For each of the 10 activities, here are the three estimates that led to the estimates of the mean and variance of the duration of the activity (rounded to the nearest integer)
In particular, enter the three estimates for each activity, and the template immediately will display the estimates of the means and variances of the activity durations. After indicating each path of
The Lockhead Aircraft Co. is ready to begin a project to develop a new fighter airplane for the U.S. Air Force. The company’s contract with the Department of Defense calls for project completion
The Lockhead Aircraft Co. is ready to begin a project to develop a new fighter airplane for the U.S. Air Force. The company’s contract with the Department of Defense calls for project completion
Reconsider the Tinker Construction Co. problem presented in Prob. 10.5-1.While in college, Sean Murphy took an OR course that devoted a month to linear programming, so Sean has decided to use linear
Reconsider the Electronic Toys Co. problem presented in Prob. 10.4-5.Sharon Lowe is concerned that there is a significant chance that the vitally important deadline of 57 days will not be met.
The mean critical path gives an estimate that the project will finish in 51 days. However, Sharon knows from the earlier analysis that some of the pessimistic estimates are far larger than the means,
Good Homes Construction Company is about to begin the construction of a large new home. The company’s President, Michael Dean, is currently planning the schedule for this project. Michael has
Reconsider the Lockhead Aircraft Co. problem presented in Prob. 10.4-6 regarding a project to develop a new fighter airplane for the U.S. Air Force. Management is extremely concerned that current
The corresponding mean critical path provides an estimate that the project will finish in 100 weeks. However, management understands well that the high variability of activity durations means that
Reconsider Prob. 10.5-4 involving the Good Homes Construction Co. project to construct a large new home. Michael Dean now has generated the plan for how to crash this project (as given as an answer
Consider the following problem:Maximize Z 4x1 x1 2 10x2 x2 2, subject to x1 2 4x2 2 16 and x1 0, x2 0.(a) Is this a convex programming problem? Answer yes or no, and then justify your
Consider the following nonconvex programming problem.Minimize f(x) sin 3x1 cos 3x2 sin(x1 x2), subject to x1 2 10x2 1 10x1 x2 2 100 and x1 0, x2 0(a) If SUMT were applied to this problem,
Reconsider the convex programming model with an equality constraint given in Prob. 13.6-14.(a) If SUMT were to be applied to this model, what would be the unconstrained function P(x; r) to be
Consider the following nonconvex programming problem:Maximize f(x) 3x1x2 2x1 2 x2 2, subject to x1 2 2x2 2 4 2x1 x2 3 x1x2 2 x1 2x2 2 and x1 0, x2 0.(a) If SUMT were to be applied to this
Consider the following nonconvex programming problem:Maximize f(x) 1,000x 400x2 40x3 x4, subject to x2 x 500 and x 0.(a) Identify the feasible values for x. Obtain general expressions for the
Reconsider the first quadratic programming variation of the Wyndor Glass Co. problem presented in Sec. 13.2 (see Fig.13.6). Beginning with the initial trial solution (x1, x2) (2, 3), use the
Reconsider the quadratic programming model given in Prob. 13.7-4.Beginning with the initial trial solution (x1, x2)(1 2, 1 2), use the automatic routine in your OR Courseware to apply SUMT to this
Consider the following convex programming problem:Maximize f(x) x1x2 x1 x1 2 x2 x2 2, subject to x2 0.Beginning with the initial trial solution (x1, x2) (1, 1), use the automatic routine in
Use SUMT to solve the following convex programming problem:Minimize f(x)(x1 31)3 x2, subject to x1 1 and x2 0.(a) If SUMT were applied directly to this problem, what would be the unconstrained
Consider the example for applying SUMT given in Sec.13.10.(a) Show that (x1, x2) (1, 2) satisfies the KKT conditions.(b) Display the feasible region graphically, and then plot the locus of points
Reconsider the model given in Prob. 13.3-3.(a) If SUMT were to be applied directly to this problem, what would be the unconstrained function P(x; r) to be minimized at each iteration?(b) Setting r
Reconsider the linearly constrained convex programming model given in Prob. 13.9-11.Follow the instructions of parts (a),(b), and (c) of Prob. 13.10-1 for this model, except use (x1, x2)(1 2, 1 2) as
Reconsider the linearly constrained convex programming model given in Prob. 13.9-10.(a) If SUMT were to be applied to this problem, what would be the unconstrained function P(x; r) to be maximized at
Consider the following linearly constrained convex programming problem:Maximize f(x) 4x1 x1 4 2x2 x2 2, subject to 4x1 2x2 5 and x1 0, x2 0.(a) Starting from the initial trial solution (x1,
Consider the following linearly constrained convex programming problem:Maximize f(x) 3x1 4x2 x1 3 x2 2, subject to x1 x2 1 and x1 0, x2 0.(a) Starting from the initial trial solution (x1, x2)
Consider the following linearly constrained convex programming problem:Maximize f(x) 3x1x2 40x1 30x2 4x1 2 x1 4 3x2 2 x2 4, subject to 4x1 3x2 12 x1 2x2 4 and x1 0, x2 0.Starting from the
Reconsider the first quadratic programming variation of the Wyndor Glass Co. problem presented in Sec. 13.2 (see Fig.13.6). Starting from the initial trial solution (x1, x2) (0, 0), use three
Reconsider the quadratic programming model given in Prob. 13.7-4.D,I (a) Starting from the initial trial solution (x1, x2) (0, 0), use the Frank-Wolfe algorithm (six iterations) to solve the problem
Consider the quadratic programming example presented in Sec. 13.7. Starting from the initial trial solution (x1, x2) (5, 5), apply seven iterations of the Frank-Wolfe algorithm.
Reconsider the linearly constrained convex programming model given in Prob. 13.6-16.Starting from the initial trial solution (x1, x2, x3) (0, 0, 0), apply two iterations of the FrankWolfe algorithm.
Reconsider the linearly constrained convex programming model given in Prob. 13.6-15.Starting from the initial trial solution (x1, x2) (0, 0), use one iteration of the Frank-Wolfe algorithm to obtain
Reconsider the linearly constrained convex programming model given in Prob. 13.6-6.Starting from the initial trial solution (x1, x2) (0, 0), use one iteration of the Frank-Wolfe algorithm to obtain
Reconsider the linearly constrained convex programming model given in Prob. 13.6-5.Starting from the initial trial solution (x1, x2) (1, 1), use one iteration of the Frank-Wolfe algorithm to obtain
Reconsider the integer nonlinear programming model given in Prob. 11.3-11.(a) Show that the objective function is not concave.(b) Formulate an equivalent pure binary integer linear programming model
Consider the following convex programming problem:Maximize Z 32x1 x1 4 4x2 x2 2, subject to x1 2 x2 2 9 and x1 0, x2 0.(a) Apply the separable programming technique discussed at the end of
Consider the following nonlinear programming problem(first considered in Prob. 11.3-23).Maximize Z 5x1 x2, subject to 2x1 2 x2 13 x1 2 x2 9 and x1 0, x2 0.(a) Show that this problem is a convex
The MFG Company produces a certain subassembly in each of two separate plants. These subassemblies are then brought to a third nearby plant where they are used in the production of a certain product.
For each of the following cases, prove that the key property of separable programming given in Sec. 13.8 must hold. (Hint:Assume that there exists an optimal solution that violates this property, and
Suppose that the separable programming technique has been applied to a certain problem (the “original problem”) to convert it to the following equivalent linear programming problem:Maximize Z
Reconsider Prob. 10.3-4 involving a project at Stanley Morgan Bank to install a new management information system.Ken Johnston already has obtained the earliest times, latest times, and slack for
Reconsider the linearly constrained convex programming model given in Prob. 13.4-7.(a) Use the separable programming technique presented in Sec.13.8 to formulate an approximate linear programming
The B. J. Jensen Company specializes in the production of power saws and power drills for home use. Sales are relatively stable throughout the year except for a jump upward during the Christmas
Reconsider the production scheduling problem of the Build-Em-Fast Company described in Prob. 8.1-9. The special restriction for such a situation is that overtime should not be used in any particular
The Dorwyn Company has two new products that will compete with the two new products for the Wyndor Glass Co. (described in Sec. 3.1). Using units of hundreds of dollars for the objective function,
The MFG Corporation is planning to produce and market three different products. Let x1, x2, and x3 denote the number of units of the three respective products to be produced. The preliminary
Reconsider the quadratic programming model given in Prob. 13.7-7.(a) Use the separable programming formulation presented in Sec.13.8 to formulate an approximate linear programming model for this
Jim Matthews, Vice President for Marketing of the J. R.Nickel Company, is planning advertising campaigns for two unrelated products. These two campaigns need to use some of the same resources.
Reconsider Prob. 13.1-3 and its quadratic programming model.(a) Display this model [including the values of R(x) and V(x)] on an Excel spreadsheet.(b) Solve this model for four cases: minimum
Reconsider the first quadratic programming variation of the Wyndor Glass Co. problem presented in Sec. 13.2 (see Fig.13.6). Analyze this problem by following the instructions of parts(a), (b), and
Consider the following quadratic programming problem.Maximize f(x) 2x1 3x2 x1 2 x2 2, subject to x1 x2 2 and x1 0, x2 0.(a) Use the KKT conditions to derive an optimal solution directly.(b)
Consider the following quadratic programming problem:Maximize f(x) 20x1 20x1 2 50x2 5x2 2 18x1x2, subject to x1 x2 6 x1 4x2 18 and x1 0, x2 0.Suppose that this problem is to be solved by the
Consider the following quadratic programming problem:Maximize f(x) 8x1 x1 2 4x2 x2 2, subject to x1 x2 2 and x1 0, x2 0.(a) Use the KKT conditions to derive an optimal solution.(b) Now
Consider the quadratic programming example presented in Sec. 13.7.(a) Use the test given in Appendix 2 to show that the objective function is strictly concave.(b) Verify that the objective function
Use the KKT conditions to determine whether (x1, x2, x3) (1, 1, 1) can be optimal for the following problem:Minimize Z 2x1 x2 3 x3 2, subject to x1 2 2x2 2 x3 2 4 and x1 0, x2 0, x3 0.
Consider the following linearly constrained convex programming problem:Maximize f(x) 8x1 x1 2 2x2 x3, subject to x1 3x2 2x3 12 and x1 0, x2 0, x3 0.(a) Use the KKT conditions to demonstrate
Consider the following linearly constrained convex programming problem:Minimize Z x1 2 6x1 x2 3 3x2, subject to x1 x2 1 and x1 0, x2 0.(a) Obtain the KKT conditions for this problem.(b) Use
Consider the following linearly constrained programming problem:Minimize f(x) x1 3 4x2 2 16x3, subject to x1 x2 x3 5 and x1 1, x2 1, x3 1.(a) Convert this problem to an equivalent nonlinear
Consider the following nonlinear programming problem:Minimize Z 2x1 2 x2 2, subject to x1 x2 10 and x1 0, x2 0.(a) Of the special types of nonlinear programming problems described in Sec. 13.3,
What are the KKT conditions for nonlinear programming problems of the following form?Minimize f(x), subject to gi(x) bi, for i 1, 2, . . . , m and x 0.(Hint: Convert this form to our standard
Reconsider the nonlinear programming model given in Prob. 11.3-16.(a) Use the KKT conditions to determine whether (x1, x2, x3)(1, 1, 1) can be optimal.(b) If a specific solution satisfies the KKT
Use the KKT conditions to derive an optimal solution for each of the following problems.(a) Maximize f(x) x1 2x2 x2 3, subject to x1 x2 1 and x1 0, x2 0.(b) Maximize f(x) 20x1 10x2, subject to
Consider the following nonlinear programming problem:Maximize f(x)x2 x1 1, subject to x1 x2 2 and x1 0, x2 0.(a) Use the KKT conditions to demonstrate that (x1, x2) (4, 2)is not optimal.(b)
Consider the nonlinear programming problem given in Prob. 11.3-14.Determine whether (x1, x2) (1, 2) can be optimal by applying the KKT conditions.
Consider the following convex programming problem:Maximize f(x) 10x1 2x1 2 x1 3 8x2 x2 2, subject to x1 x2 2 and x1 0, x2 0.(a) Use the KKT conditions to demonstrate that (x1, x2) (1, 1)is
Consider the following linearly constrained optimization problem:Maximize f(x) ln(x1 1) x2 2, subject to x1 2x2 3 and x1 0, x2 0, where ln denotes the natural logarithm,(a) Verify that this
Consider the following linearly constrained optimization problem:Maximize f(x) ln(1 x1 x2), subject to x1 2x2 5 and x1 0, x2 0, where ln denotes the natural logarithm.(a) Verify that this problem
Consider the following convex programming problem:Maximize f(x) 24x1 x1 2 10x2 x2 2, subject to x1 8, x2 7, and x1 0, x2 0.(a) Use the KKT conditions for this problem to derive an optimal
Use the KKT conditions to derive an optimal solution for this model.
Reconsider the one-variable convex programming model given in Prob.
Consider the following unconstrained optimization problem:Maximize f(x) 3x1x2 3x2x3 x1 2 6x2 2 x3 2.(a) Describe how solving this problem can be reduced to solving a two-variable unconstrained
Starting from the initial trial solution (x1, x2) (0, 0), apply one iteration of the gradient search procedure to the following problem by hand:Maximize f(x) 4x1 2x2 x1 2 x1 4 2x1x2 x2 2.To
Starting from the initial trial solution (x1, x2) (0, 0), interactively apply two iterations of the gradient search procedure to begin solving the following problem, and then apply the automatic

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