WOOD 492 MODELLING FOR DECISION SUPPORT

Lecture 3

Basics of the Simplex Algorithm

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Last Class

• Introduction to Linear Programming

• Solving LPs with the graphical method

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Example: Custom Cabinets company

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x1 =48, x2 =12 Z=$2,520

Feasible Region

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Why use a specialized algorithm?

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• 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

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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

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Feasible Region

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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

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These properties make it possible to use the simplex algorithm which is very efficient in practice

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Feasible Region

(0,0) Z=$0

(48,0) Z=$1920

(48,12) Z=$2520

(22,25) Z=$2130

demo

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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

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Cabinet LP Example

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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=3x1+2x2 , when x1 is increased, its contribution to the Obj. (3x1) 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

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Next Lecture

• Assumptions of LP• More examples of LP matrixes and Solver• Overview of Lab 1 Problem• Quiz on Friday, Sept 14

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