4 Alternative Formulations_1

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

Info iconThis preview shows pages 1–9. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 ....
View Full Document

Page1 / 48

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

This preview shows document pages 1 - 9. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online