SolutMan07 - Chapter 7 Integer Linear Programming Learning...

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Chapter 7 Integer Linear Programming Learning Objectives 1. Be able to recognize the types of situations where integer linear programming problem formulations are desirable. 2. Know the difference between all-integer and mixed integer linear programming problems. 3. Be able to solve small integer linear programs with a graphical solution procedure. 4. Be able to formulate and solve fixed charge, capital budgeting, distribution system, and product design problems as integer linear programs. 5. See how zero-one integer linear variables can be used to handle special situations such as multiple choice, k out of n alternatives, and conditional constraints. 6. Be familiar with the computer solution of MILPs. 7. Understand the following terms: all-integer mutually exclusive constraint mixed integer k out of n alternatives constraint zero-one variables conditional constraint LP relaxation co-requisite constraint multiple choice constraint Solutions: 7 - 1
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Chapter 7 1. a. This is a mixed integer linear program. Its LP Relaxation is Max 30 x 1 + 25 x 2 s.t. 3 x 1 + 1.5 x 2 400 1.5 x 1 + 2 x 2 250 x 1 + x 2 150 x 1 , x 2 0 b. This is an all-integer linear program. Its LP Relaxation just requires dropping the words "and integer" from the last line. 2. a. 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 Optimal solution to LP relaxation (1.43,4.29) x 2 x 1 5 x 1 + 8 2 = 41.47 b. The optimal solution to the LP Relaxation is given by x 1 = 1.43, x 2 = 4.29 with an objective function value of 41.47. Rounding down gives the feasible integer solution x 1 = 1, x 2 = 4. Its value is 37. c. 7 - 2
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Integer Linear Programming The optimal solution is given by x 1 = 0, x 2 = 5. Its value is 40. This is not the same solution as that found by rounding down. It provides a 3 unit increase in the value of the objective function. 3. a. 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 x 2 x 1 optimal solution to LP relaxation (also optimal integer solution) (4,1) x 1 + 2 = 5 b. The optimal solution to the LP Relaxation is shown on the above graph to be x 1 = 4, x 2 = 1. Its 7 - 3
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Chapter 7 value is 5. c. The optimal integer solution is the same as the optimal solution to the LP Relaxation. This is always the case whenever all the variables take on integer values in the optimal solution to the LP Relaxation. 4. a. 1 3 5 10 x 1 + 3 2 = 36.7 ti l so lu n to re la a n (3 .6 ,0 (2 .4 ,3 ) (0 ,5 .7 ) x x 1 The value of the optimal solution to the LP Relaxation is 36.7 and it is given by x 1 = 3.67, x 2 = 0.0. Since we have all less-than-or-equal-to constraints with positive coefficients, the solution obtained by "rounding down" the values of the variables in the optimal solution to the LP Relaxation is feasible. The solution obtained by rounding down is x 1 = 3, x 2 = 0 with value 30. Thus a lower bound on the value of the optimal solution is given by this feasible integer solution with value 30. An upper bound is given by the value of the LP Relaxation, 36.7. (Actually an upper bound of 36 could be established since no integer solution could have a value between 36 and 37.) 7 - 4
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Integer Linear Programming b.
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SolutMan07 - Chapter 7 Integer Linear Programming Learning...

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