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Unformatted text preview: IEE 598  4. (I.1) Alternative Formulations Muhong Zhang DEPARTMENT OF INDUSTRIAL ENGINEERING Feb. 04, 2010 Alternative Formulations Example: (Uncapacitated) facility location problem There are a set N = { 1 , ..., n } of potential facility locations and a set of clients I = { 1 , ..., m } . To build a facility at location j costs f j . Each client i ∈ I has certain demand b i . The unit cost of shipping product from facility j to client i is c ij . Goal: Choose a subset of the locations at which to place facilities and determine how to ship product from facilities to clients to minimize to total cost. Zhang IEE 376 Introduction to OR 2 / 17 Alternative Formulations Example: (Uncapacitated) facility location problem There are a set N = { 1 , ..., n } of potential facility locations and a set of clients I = { 1 , ..., m } . To build a facility at location j costs f j . Each client i ∈ I has certain demand b i . The unit cost of shipping product from facility j to client i is c ij . Goal: Choose a subset of the locations at which to place facilities and determine how to ship product from facilities to clients to minimize to total cost. Variables: x j : indicator of opening facility j , j ∈ N ; y ij : units of demand sent from facility j to client i , i ∈ I , j ∈ N . Objective: min ∑ i ∈ I ∑ j ∈ N c ij y ij + ∑ j ∈ N f j x j s.t. ∑ j ∈ N y ij = b i , i ∈ I ∑ i ∈ I y ij ≤ Mx j , j ∈ N ( M = ∑ i ∈ I d i ) y ij ∈ R + , x i ∈ B , i ∈ I , j ∈ N Zhang IEE 376 Introduction to OR 2 / 17 Alternative Formulations Use variables z ij : fraction of demand of i satisfied from facility j . Constraints: Zhang IEE 376 Introduction to OR 3 / 17 Alternative Formulations Use variables z ij : fraction of demand of i satisfied from facility j . Constraints: ∑ j ∈ N z ij = 1 , i ∈ I ( UFL 2 ) ∑ i ∈ I z ij ≤  I  x j , j ∈ N Alternatively: ∑ j ∈ N z ij = 1 , i ∈ I ( UFL 3 ) z ij ≤ x j , j ∈ N , i ∈ I Zhang IEE 376 Introduction to OR 3 / 17 Strong Formulations For two formulations A and B of a problem ( P ) . Let P A and P B denote the feasible region of the LP relaxations of A and B . Zhang IEE 376 Introduction to OR 4 / 17 Strong Formulations For two formulations A and B of a problem ( P ) . Let P A and P B denote the feasible region of the LP relaxations of A and B . Formulation A is said to be at least as strong as B if P A ⊆ P B . Zhang IEE 376 Introduction to OR 4 / 17 Strong Formulations For two formulations A and B of a problem ( P ) . Let P A and P B denote the feasible region of the LP relaxations of A and B . Formulation A is said to be at least as strong as B if P A ⊆ P B . If P A ⊂ P B , A is stronger than B ....
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This note was uploaded on 05/01/2011 for the course IEE 598 taught by Professor Hillary during the Spring '10 term at USC.
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