OPTIMIZAÇÃO E DECISÃO 09/10

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1 OPTIMIZAÇÃO E DECISÃO 09/10 PL #7 Integer Programming Alexandra Moutinho (from Hillier & Lieberman Introduction to Operations Research, 8 th edition) Problem 1 Pawtucket University is planning to buy new copier machines for its library. Three members of its Operations Research Department are analyzing what to buy. They are considering two different models: Model A, a high speed copier, and Model B, a lower speed but less expensive copier. Model A can handle 20,000 copies a day, and costs $6,000. Model B can handle 10,000 copies a day, but costs only $4,000. They would like to have at least six copiers so that they can spread them throughout the library. They also would like to have at least one high speed copier. Finally, the copiers need to be able to handle a capacity of at least 75,000 copies per day. The objective is to determine the mix of these two copiers that will handle all these requirements at minimum cost. a) Formulate an IP model for this problem. b) Use a graphical approach to solve this model. c) Use the computer to solve the model. Resolution: a) Let A be the number of Model A copiers to buy. Let B be the number of Model B copiers to buy. The problem formulation is as follows: Mini mize C 6,000A4,000B subject to AB 6 A 1 20,000 A10,000B 75,000 and A 0, B 0 A, B are integers. b) In the following figure, the dots indicate feasible points. As we can see, A,B2,4 is the optimal solution with a minimum cost of $28,000. A 1 20,000A10,000B 75,000 AB 6 C 6,000A4,000B 28,000 1

2 c) We use Excel to solve this problem, as shown in the following figure. The Excel Solver finds the optimal solution,, 2,4 with a minimum cost of $28,000. Problem 2 Consider the following IP problem: Maximize , subject to and 0, 0,, are integers. a) Use the MIP branch and bound algorithm presented to solve this problem by hand. For each subproblem, solve its LP relaxation automatically (Excel for example). b) Check your answer by using an automatic procedure to solve the problem. c) Use the interactive procedure for this algorithm in your IOR Tutorial to solve the problem. Resolution: a) Summary of the MIP Branch and Bound Algorithm. Initialization: Set. Apply the branching step, bounding step, fathoming step, and optimality test described below to the whole problem. If not fathomed, classify this problem as the one remaining subproblem for performing the first full iteration below. Steps for each iteration: 1. Branching: Among the remaining (unfathomed) subproblems, select the one that was created most recently. (Break ties according to which has the larger bound.) Among the integerrestricted variables that have a noninteger value in the optimal solution for the LP relaxation of the subproblem, choose the first one in the natural ordering of the variables to be the branching variable. Let be this variable and its value in this solution. Branch from the node for the subproblem to create two new subproblems by adding the respective constraints and 1, where is the greatest integer. Optimização e Decisão 09/10 PL #7 Integer Programming Alexandra Moutinho 2

3 2. Bounding: For each new subproblem, obtain its bound by applying the simplex method to its LP relaxation and using the value of for the resulting optimal solution. 3. Fathoming: For each new subproblem, apply the three fathoming tests given below, and discard those subproblems that are fathomed by any of the tests. Test 1: Its bound, where is the value of for the current incumbent. Test 2: Its LP relaxation has no feasible solutions. Test 3: The optimal solution for its LP relaxation has integer values for the integer restricted variables. (If this solution is better than the incumbent, it becomes the new incumbent and test 1 is reapplied to all unfathomed subproblems with the new larger.) Optimality test: Stop when there are no remaining subproblems; the current incumbent is optimal. Otherwise, perform another iteration. Initialization: Relaxing the integer constraints, the optimal solution of the LP relaxation of the whole problem is, 2.667,1.333 with an objective function value of This LP relaxation of the whole problem possesses feasible solutions and its optimal solution has noninteger values for and, so the whole problem is not fathomed and we are ready to move on to the first full iteration. Iteration 1: The only remaining (unfathomed) subproblem at this point is the whole problem, so we use it for branching and bounding. In the above optimal solution for its LP relaxation, both integerrestricted variables ( and ) are noninteger, so we select the first one ( ) to be the branching variable. Since in this optimal solution, we will create two new subproblems below by adding the respective constraints, and 1, where is the greatest integer, so 2. Subproblem 1: The original problem plus the additional constraint, Subproblem 2: 2. The original problem plus the additional constraint, 3. For subproblem 1, the optimal solution for its LP relaxation is, 2,3with 680. Since the solution, 2,3 is integer valued, subproblem 1 is fathomed by fathoming test 3 and this solution becomes the first incumbent. Incumbent 2,3 with 680. Now consider subproblem 2. It can be seen that the new constraint 3results in having no feasible solutions. Therefore, subproblem 2 is fathomed by fathoming test 2. At this point, there are no remaining (unfathomed) subproblems, so the optimality test indicates that the current incumbent is optimal for the original whole problem, so no additional iterations are needed., 2,3 with 680. b) As shown in the following spreadsheet, the Excel Solver finds the optimal solution,, 2,3 with 680, which is identical to the solution found in part a). Optimização e Decisão 09/10 PL #7 Integer Programming Alexandra Moutinho 3

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