optimal solution would have been to produce 570 compacts and 1,715 midsize cars.1231236,0000,2,000,00,1,0Zxxxyyy
69Suppose we want to ensure that > 0 implies . Then we include the following constraint in the formulation:Here, Mis a large positive number, chosen large enough so that f < Mand – 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.xxxnf,...,,210,...,,21xxxngMygxxxn,...,,21yMfxxxn1,...,,2110oryIf-then constraint
701.3 The Branch-and-Bound Method for Solving Pure Integer Programming ProblemsIn practice, most IPs are solved by some versions of the branch-and-boundprocedure. 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.
71An important observationIf 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:121212max32. .26,0,intZxxs txxxxeger
72Cont’dThen the LP relaxation is121212max32. .26,0Zxxs txxxxThe optimal solution to the LP relaxation is x1=0, x2=6, z=12.Because this solution gives integer values to all variables, thepreceding observation implies that this solution is also the op-timal solution to the IP.
73Cont’dObserve 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 x1=0,x2=6 is also feasible for the IP. This implies that (x1=0,x2=6) must be optimal for the IP.
74An exampleThe 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.
75Cont’dLet x1:=number of tables manufactured x2:=number of chairs manufactured. The IP model is12121212max85. .69545,0;int eZxxs t xxxxxxger
76Cont’dThe 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.