WOOD 492 MODELLING FOR DECISION SUPPORT Lecture 3 Basics of the Simplex Algorithm

Last Class Introduction to Linear Programming Solving LPs with the graphical method Sept 10, 2012Wood Saba Vahid2

Example: Custom Cabinets company Sept 10, 2012Wood Saba Vahid3 x 1 =48, x 2 =12 Z=$2,520 Feasible Region

Why use a specialized algorithm? Sept 10, 20124Wood Saba Vahid Exhaustive search takes too long –Too many feasible solutions We want to ask many “what if” questions –So we run the model over and over We want to perform sensitivity analysis –What constraints are binding? –How much do the constraints cost us? –Which products are the most profitable? We can use Simplex Algorithm to solve LPs

Terminology Feasible solution –Solution where all constraints are satisfied –Many are possible Optimal solution –Feasible solution with highest (or lowest) objective function value –Can be unique, but there are many cases where there are ties Boundary equation –Constraint with inequality replaced by an equality –These define the feasible region Corner-point solution –Where two or more constraints intersect Sept 10, 2012Wood Saba Vahid5

Sept 10, 2012Wood Saba Vahid6 Feasible Region

Important properties of LP 1.An optimal solution is always at a feasible corner-point solution 2.If a feasible corner-point solution has an objective value higher than all the adjacent feasible corner-point solutions, then it is optimal 3.There is a finite number of feasible corner-point solutions for an LP Sept 10, 2012Wood Saba Vahid7 These properties make it possible to use the simplex algorithm which is very efficient in practice

Sept 10, 2012Wood Saba Vahid8 Feasible Region (0,0) Z=$0 (48,0) Z=$1920 (48,12) Z=$2520 (22,25) Z=$2130 demo

Simplex Algorithm Has two steps: 1.Start-up: Find any feasible corner-point solution 2.Iterate: Move repeatedly to adjacent feasible corner- point solutions with the highest improvement in objective values, until no better values are achieved by moving to an adjacent feasible corner-point solution. The final corner-point solution is the optimal solution. (it is possible to have more than one optimal solution) Excel Solver uses the Simplex algorithm for solving LPs Sept 10, 2012Wood Saba Vahid9 Cabinet LP Example

Assumptions of LP For a system to be modelled with an LP, 4 assumptions must hold : –Proportionality: Contribution of each activity (decision variable) to the Obj. Fn. is proportional to its value (represented by its coefficient in the Obj. Fn.), e.g. Z=3x 1 +2x 2, when x 1 is increased, its contribution to the Obj. (3x 1 ) is always increased three-fold. –Additivity: Every function in an LP (Obj. Fn. Or the constraints) is the linear sum of individual contributions of the respective activities (decision variables) –Divisibility: Activities can be run at fractional level, i.e., decision variables can have any level (not just integer values). –Certainty: Parameter values (coefficients in the functions) are known with certainty Sept 10, 2012Wood Saba Vahid10

Next Lecture Assumptions of LP More examples of LP matrixes and Solver Overview of Lab 1 Problem Quiz on Friday, Sept 14 Sept 10, 2012Wood Saba Vahid11