MILP. LP: max cx ' MILP: some integer. ILP: x integer BLP: x 0,1. x 1. x 2 2 2, c ,

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1 MILP LP: max cx ' s.t. Ax b x 0 MILP: some nteger x max 6x 8x s.t. x x x 7 x, x 0 c A 6 8, 0 b 7 ILP: x nteger BLP: x 0, x 4 x, cx * * x 06

2 Branch and Bound x 4 0 max 6x 8x s.t. xx x 7 x, x 0 x, x nteger x c 6 8 A, 0 b 7 UB, LB LP 6 UB, LB 0 x x 4 x x Incumbent UB, LB 6 UB, LB 6 x x UB 0, LB 6 Incumbent x x UB 0, LB 8 7 Incumbent UB 0, LB 0 gap 0 0 STOP Fathomed, nfeasble Fathomed, nfeasble 07

3 MILP Solvers 08

4 MILP Solvers x x Gomory cut Cplex (IBM, frst solver) Gurob (dev Robert Bxby) Xpress (used by LLamasoft) SAS/OR (part of SAS sys) Symphony (open source) Presolve: elmnate varables x x, x, x 0 and nteger x x 0 Cuttng planes: keeps all nteger solutons and cuts off LP solutons (Gomory cut) Heurstcs: fnd good ntal ncumbent soluton Parallel: use separate cores to solve nodes n B&B tree Speedup from : 0,000 computer speed 580,000 algorthm mprovements 09

5 MILP Formulaton of UFL mn j j N N jm s.t. x, jm where N my x, N ky 0 x, N, j M j jm j j y 0,, N cx k fxed cost of NF at ste N,..., n c varable cost from to serve EF j M,..., m j, f NF establshed at ste y 0, otherwse x fracton of EF j demand served from NF at ste. j y x, N, j M j 0

6 Capactated Faclty Locaton (CFL) mn ky j j N N jm s.t. x, jm N Ky fx, N 0 x, N, j M j j j jm j cx y 0,, N where k fxed cost of NF at ste N,..., n c varable cost from to serve EF j M,..., m j K capacty of NF at ste N,..., n f demand EF jm,..., m j, f NF establshed at ste y 0, otherwse xj fracton of EF j demand served from NF at ste. CFL does not have smple and effectve heurstcs, unlke UFL Other types of constrants: Fx NF at ste j: set LB and UB of x j to Convert UFL to p Medan: set all k to 0 and add constrant sum{y } = p

7 Matlog s Mlp Executng mp = Mlp creates a Mlp object that can be used to defne a MILP model that s then passed to a Solver Smlar syntax to math notaton for MILP AMPL and OPL algebrac modelng languages provde smlar capabltes, but Mlp ntegrated nto MATLAB

8 Ex: Illustrate Mlp syntax

9 Ex: UFL mn j j N N jm s.t. x, jm N my x, N ky 0 x, N, jm j jm j j y 0,, N cx 4

10 4 M M * (Weghted) Set Coverng,..., m, objects to be covered M M, N,..., n, subsets of M c cost of usng M n cover I arg mn c : M M, mn cost coverng of M I I I M 4 M M M I M *,...,6 N,...,5,,, 4, 5,, 5,,6, M 6 M M M M 4 5 c, for all N * I arg mn c : M M c, 4 I I I 5

11 mn N (Weghted) Set Coverng s.t. a x, j M N M j x 0,, N * cx,..., m, objects to be covered M M, N,..., n, subsets of M c cost of usng M n cover I arg mn c : M M, mn cost coverng of M I I I where a x j, f M s n cover 0, otherwse, f j M 0, otherwse. 6

12 Set Packng Maxmze the number of mutually dsjont sets Dual of Set Coverng problem Not all objects requred n a packng Lmted logstcs engneerng applcaton (c.f. bn packng) M M M 5 max N s.t. a x, j M N j x 0,, N x

13 mn M y s.t. Vy v x, M where y x j j j jm M x, jm j y 0,, M x 0,, M, jm j M v j * Bn Packng,..., m, objects to be packed volume of object j V volume of each bn B max v V B arg mn B : v V, B M, mn packng of M B j jb BB, f bn B s used n packng 0, otherwse, f object j packed nto bn B 0, otherwse. 8 j