4 Alternative Formulations_1

4 Alternative Formulations_1 - IEE 598 4(I.1 Alternative...

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IEE 598 - 4. (I.1) Alternative Formulations Muhong Zhang DEPARTMENT OF INDUSTRIAL ENGINEERING Feb. 04, 2010
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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
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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
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Alternative Formulations Use variables z ij : fraction of demand of i satisfied from facility j . Constraints: Zhang IEE 376 Introduction to OR 3 / 17
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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
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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
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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
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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 . Zhang IEE 376 Introduction to OR 4 / 17
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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 .
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