Linear programming Linear program optimization problem continuous variables

Linear programming • Linear program: optimization problem, continuous variables, single, linear objective function, all constraints linear equalities or inequalities • Applications – – Allocation models Operations planning models Limit load analysis in structues Dynamic linear programming: time-phased decision making 1

Matrix form • Basic solution (BS): solution of A X=b, n-m redundant variables zero (nonbasic variables), n constraints active. Remaining m variables non zero (basic variables) • Each BS corresponds to a vertex • BFS, non BFS 2

Possible solutions to a linear programming problem • • Unique solution Nonunique solution Unbounded solution No feasible solution 3

Simplex method Idea: Start from a vertex. Move to adjacent vertex so that F decreaces. Continue until no further improvement can be made. Facts • Optimum is a vertex • Vertex: BS • Moving from a vertex to adjacent vertex: swap a basic variable with a non basic variable 4

Simplex method • Variable with smallest negative cost coefficient will become basic • Select variable to leave set of basic variables so that a BFS is obtained • Design space convex 5

Tableau: canonical form x 1 x 2 … xm x m+1 x 1 1 0 0 0 x 2 0 1 0 0 0 … xn RHS a 1, m+1 a 1, n b 1 0 a 2, m+1 a 2, n b 2 0 1 am, m+1 a m, n bm 0 0 c m+1 cm F-F 0 … xm Basic variables Nonbasic variables 6

Tableau: swapping variables x 1 x 2 … xm x m+1 x 1 1 0 0 0 x 2 0 1 0 0 0 … x 2 leave xn RHS a 1, m+1 a 1, n b 1 0 a 2, m+1 a 2, n b 2 0 1 am, m+1 a m, n bm 0 0 c m+1 cm F-F 0 … xm Pivot element xm+1 enter 7

Example A, B, C: BS 8