# Multi-objective linear-programming problem

**DOI:**https://doi.org/10.1007/1-4020-0611-X_649

This problem has the usual set of linear-programming constraints (* Ax*v =

*v ≥;*

**b, x****0**) but requires the simultaneous optimization of more than one linear objective function, say

*p*of them. It can be written as “Maximize”

*v subject to*

**Cx***v =*

**Ax***v ≥*

**b, x****0,**where

*v is a*

**C***p*×

*n*matrix whose rows are the coefficients defined by the

*p*objectives. Here “Maximize” represents the fact that it is usually impossible to find a solution to

*v =*

**Ax***v ≥*

**b, x****0,**that simultaneously optimizes all the objectives. If there is such an (extreme) point, the problem is thus readily solved. Special multiobjective computational procedures are required to select a solution that is in effect a compromise solution between the extreme point solutions that optimize individual objective functions. The possible compromise solutions are taken from the set of efficient (nondominated) solutions. This problem is also called the vector optimization problem. Efficient solution; Multi-objective optimization; Multi-objective...