1. 1 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

2. 2 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

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

4. 4 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

5. 5 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

6. 6 Tableau: canonical form x1x1 x2x2 …xmxm x m+1 …xnxn RHS x1x1 1000a 1,m+1 a 1,n b1b1 x2x2 0100a 2,m+1 a 2,n b2b2 … xmxm 0001a m,m+1 a m,n bmbm 0000c m+1 cmcm F-F 0 Basic variablesNonbasic variables

7. 7 Tableau: swapping variables x1x1 x2x2 …xmxm x m+1 …xnxn RHS x1x1 1000a 1,m+1 a 1,n b1b1 x2x2 0100a 2,m+1 a 2,n b2b2 … xmxm 0001a m,m+1 a m,n bmbm 0000c m+1 cmcm F-F 0 x m+1 enter x 2 leave Pivot element

8. 8 Example A, B, C: BS