optimal solution would have been to produce 570 compacts and 1715 midsize cars

# Optimal solution would have been to produce 570

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optimal solution would have been to produce 570 compacts and 1,715 midsize cars. 1 2 3 1 2 3 6,000 0, 2,000, 0 0, 1, 0 Z x x x y y y 69 Suppose we want to ensure that > 0 implies . Then we include the following constraint in the formulation: Here, M is a large positive number, chosen large enough so that f < M and – g < M hold for all values of that satisfy the other constraints in the problem. If f > 0, y must be zero, g >= 0. This is called an if-then constraint. x x x n f ,..., , 2 1 0 ,..., , 2 1 x x x n g My g x x x n ,..., , 2 1 y M f x x x n 1 ,..., , 2 1 1 0 or y If-then constraint 70 1.3 The Branch-and-Bound Method for Solving Pure Integer Programming Problems In practice, most IPs are solved by some versions of the branch-and-bound procedure. Branch-and-bound methods implicitly (cleverly) enumerate all possible solutions to an IP. By solving a single subproblem, many possible solutions may be eliminated from consideration. Subproblems are generated by branching on an appropriately chosen fractional-valued variable. 71 An important observation If you solve the LP relaxation of a pure IP and obtain a solution in which all variables are integers , then the optimal solution to the LP relaxation is also the optimal solution to the IP. For example, the following IP: 1 2 1 2 1 2 max 3 2 . .2 6 , 0,int Z x x s t x x x x eger 72 Cont’d Then the LP relaxation is 1 2 1 2 1 2 max 3 2 . .2 6 , 0 Z x x s t x x x x The optimal solution to the LP relaxation is x 1 =0, x 2 =6, z=12. Because this solution gives integer values to all variables, the preceding observation implies that this solution is also the op- timal solution to the IP. 73 Cont’d Observe that the feasible region for the IP is a subset of the points in the LP relaxation’s feasible region. Thus, the optimal z-value for the IP cannot be larger than the optimal z-value for the LP relaxation. This means that the optimal z-value for the IP must be ≤12. But the point x 1 =0,x 2 =6 is also feasible for the IP. This implies that (x 1 =0,x 2 =6) must be optimal for the IP. 74 An example The Telfa Corporation manufactures tables and chairs. A table requires 1 hour of labor and 9 square board feet of wood, and a chair requires 1 hour of labor and 5 square board feet of wood. Currently, 6 hours of labor and 45 square board feet of wood are available. Each table contributes \$8 to profit, and each chair contributes \$5 to profit. Formulate and solve an IP to maximize Telfa’s profit. 75 Cont’d Let x 1 :=number of tables manufactured x 2 :=number of chairs manufactured. The IP model is 1 2 1 2 1 2 1 2 max 8 5 . . 6 9 5 45 , 0;int e Z x x s t x x x x x x ger 76 Cont’d The branch-and-bound method begins by solving the LP relaxation of the IP. If all the decision variables assume integer values in the optimal solution to the LP relaxation, then the optimal solution to the LP relaxation will also be the optimal solution to the IP.  • • • 