Optimization Solutions for Simplex Co. Midterm 2: Maximizing Profits, Exams of Engineering

Download Optimization Solutions for Simplex Co. Midterm 2: Maximizing Profits and more Exams Engineering in PDF only on Docsity! Version B IE 335: Operations Research - Optimization Fall 2008 Solutions to Midterm Exam 2 Problem 1. (10 points total = 5 parts  2 points) True or False. Circle the correct answer. No justification required. a. True False The Bellman-Ford algorithm can be used to find the longest path in an acyclic network with nonnegative arc lengths. b. True False In a graph with no negative dicycles, optimal paths must have optimal subpaths. c. True False Consider the following two linear programs: [A] max 2x1 C x2 s.t. x1 C x2  3 x2  1 x1; x2  0 [B] min 3v1 C v2 s.t. v1  2 v1 C v2  1 v1; v2  0 Suppose .x1 ; x  2 / is an optimal solution to [A], and .v  1 ; v  2 / is an optimal solution to [B]. Then 2x1 C x  2 D 3v  1 C v  2 : d. True False In a CPM project network for a given project, the length of the shortest path from the start node to the finish node gives the minimum time needed to complete the project. e. True False Interior point methods will always find an optimal basic feasible solution of a linear program, if an optimal solution exists. Solution. a. True. See Lecture 26 for an example. b. True. Prinicple of optimality for the shortest path problem (Lecture 24). c. True. [A] and [B] is a primal-dual pair. The statement follows by strong duality of linear program- ming (Lecture 18). d. False. The length of the longest path from the start node to the finish node gives the minimum time needed to complete the project (Lecture 26). e. False. Interior point methods may not converge to an optimal BFS if there are multiple optimal solutions (Lecture 22). 1 Version B Problem 2. (50 points total) Note that this problem has parts a through h. The Simplex Company produces 3 products: A, B, and C. The profit per unit sold for each of these 3 products is as follows (in dollars): Product A B C Price 3 4 5 Each product requires 2 resources: X and Y. The amounts of each resource required to produce each product are as follows: Product A Product B Product C Resource X 2 3 4 Resource Y 2 1 1 The company has 5 units of Resource X available, and 4 units of Resource Y available. Furthermore, the amount of product B that the company produces must be greater than or equal to the amount of product A it produces. The company would like to determine production levels of products A, B, and C so that its total profits are maximized. Assume that all the products that the Simplex Company produces are be sold. a. (10 points) Explain why the following linear program solves the Simplex Company’s problem: max 3xA C 4xB C 5xC s.t. 2xA C 3xB C 4xC  5 (1) 2xA C xB C xC  4 (2) xB xA  0 (3) xA; xB; xC  0 Assume that fractional units can be produced and sold. Be sure to explain what the decision variables represent, the objective function, the main constraints, and the variable-type constraints. b. (10 points) Write the dual of the linear program described in part a. Computer output for the linear program in part a is given below. The columns have identical meaning to the output we discussed in class. Variable sensitivity analysis: Optimal Bas Lower Upper Object Reduced Lower Upper Name Value Sts Bound Bound Coef Object Range Range -------------------------------------------------------------------------- Variable xA 1.000 BAS 0.000 +infin 3.000 0.000 2.667 +infin Variable xB 1.000 BAS 0.000 +infin 4.000 0.000 3.250 4.500 Variable xC 0.000 NBL 0.000 +infin 5.000 0.600 -infin 5.600 2 Version B Problem 3. (40 points total) Bellman Air has three daily flights from Indianapolis to Cleveland. From 10am until 6pm, the flights depart every 4 hours. The capacities of each flight, and the number of reservations for each flight at the start of today are as follows: Flight departure time 10am 2pm 6pm Total Capacity 120 140 110 370 Current reservations 130 160 110 400 If overbooking results in insufficient room for a passenger on a scheduled flight, Bellman Air can divert a passenger to a later flight. Bellman Air compensates any passenger delayed by more than 4 hours by paying $200 plus $30 for every hour of delay. So for example,  A passenger with reservations for the 10am flight who is diverted to the 2pm flight will not receive any compensation.  A passenger with reservations for the 10am flight who is diverted to the 6pm flight will receive 200C 8.30/ D 440 in compensation from Bellman Air. Bellman Air can always accommodate passengers delayed beyond the 6pm flight on the 11pm flight of another airline that always has a great deal of spare capacity (assume infinite capacity). a. (35 points) Show how to formulate the problem of determining the least costly passenger diversion strategy as a minimum cost flow problem by  drawing the corresponding network  indicating the net demands at each node  indicating the cost per unit of flow on each arc  indicating the capacity of each arc  describing what the flow on each arc represents Hint: define a network with 7 nodes. b. (5 points) Is there guaranteed to be an integer optimal solution to the minimum cost flow model you proposed in part a? Explain. Solution. a. Bellman Air’s problem reduces to assigning reservations to flights. We can do this using the following transportation-problem-type minimum cost flow model: 5 Version B 10am reservations 2pm reservations 6pm reservations 10am flight 2pm flight 6pm flight 11pm flight 130 160 110 120 140 110 30 0 0 440 D 200 C 8.30/ 590 D 200 C 13.30/ 0 0 470 D 200 C 9.30/ 0 350 D 200C 5.30/ The arc labels represent costs; all arcs have infinite capacity. The flow on arc (X reservations, Y flight) for each X 2 f10am, 2pm, 6pmg and Y 2 f10am, 2pm, 6pm, 11pmg represents the number of people with reservations on flight X put on flight Y. The demand on the 11pm flight node is 30: since there are 400 total reservations, and Bellman Air’s flights have a total capacity of 370, 30 reservations will need to be diverted to the 11pm flight. Note that this formulation implicitly assumes that Bellman Air’s flights are filled to capacity; since the cost of diverting a passenger to a later flight increases as the passenger’s delay increases, this assumption can be made without affecting the validity of the model. b. Yes, there is guaranteed to be an integer optimal solution to the minimum cost problem in part a, since all the data is integer. 6