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Lecture 10

# Lecture 10 - Solving Integer Programming Branch and Bound...

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1 Solving Integer Programming Branch and Bound Method Dynamic Programming

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2 IP and Its LP Relaxation Given an IP, we can obtain an LP by removing the integrality constraints Referred to as the LP relaxation Upper bound for a maximization IP problem – Z IP = optimal objective function value to the IP – Z LP = optimal objective function value to the LP relaxation We have Z IP ≤ Z LP Because the feasible region of LP relaxation contains the feasible region of IP – Z LP is an upper bound of the IP
3 LP Relaxation Maximize 4 x 1 +9 x 2 +6 x 3 Subject to 5 x 1 +8 x 2 +6 x 3 ≤12 x 1 , x 2 , x 3 are binary variables For the LP relaxation (0≤ x i ≤1), we have x 1 =0, x 2 =1, x 3 =2/3, and Z LP =13 Thus, the optimal solution to the IP 13 – The actual Z IP =10

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4 Branch An IP may be gradually decomposed into a series of smaller IP problems • Example: x 1 is a binary variable, then we may have two smaller problems, IP1 and IP2: – Let Z IP1 be the optimal solution of IP1 with x 1 =0 – Let Z IP2 be the optimal solution of IP2 with x 1 =1 – Z IP = max(Z IP1 , Z IP2 )
5 Bound Suppose we already have Z IP1 for IP. Consider IP2: Solving the LP Relaxation of IP2 to obtain Z LP2 – If Z LP2 ≤ Z IP1 , we do not need to solve IP2, since Z IP2 ≤ Z LP2 ≤ Z IP1 , so Z IP1 must be the optimal solution to the original IP This is called BOUND For each smaller IP, we can solve its LP relaxation and see if we can fathom it directly, meaning that we will no longer need to solve it

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6 The Branch and Bound Tree k -1 k k+ 2 k +1 x k -1 =1 x k -1 =0 x k =1 x k = 0 x k+ 1 =1 x k +1 =0 x k +2 =1 x k +2 =0 k+ 3 k+ 4 k+ 5 k+ 6 k+ 7 A partial tree
7 The Branch and Bound Tree k -1 k k+ 2 k +1 x k -1 =3 x k -1 = 2 x k =7 x k = 6 x k+ 1 =2 x k +1 = 1 x k +2 =3 x k +2 =2 k+ 3 k+ 4 k+ 5 k+ 6 k+ 7 A partial tree x k- 1 =3.6 in the current LP Solution

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8 General Ideas Suppose we have a feasible solution of the IP at hand, the objective function value is Z B Z B is a lower bound of the problem Let Z k be objective function value of the LP
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Lecture 10 - Solving Integer Programming Branch and Bound...

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