Simplex Method

Part 1 of 4Professor Dansereau

Example

We will use the following example to illustrate the Simplex Procedure

Decision Variables X1 and X2

Decision variables are products and typically organizations would have many products, to keep this example simple and to allow you to graph we will use only two.

Objective

Maximize profit

Example continued

Objective Function

Zmax = 250X1 + 400 X2

Constraints

We will use three but there could be many constraints

Budget → 2.5X1 + 3X2 <= 30

Work Hours → 8X1 + 4X2 <= 80

Material → 2X1 + 6X2 <= 48

Simplex Procedure

The simplex procedure has six steps

1.Place equations in standard form

2.Generate an initial feasible solution

3.Test for optimality

4.If not optimal, identify entering and leaving variables

5.Generate improved solution (move to next extreme point)

6.Go back to step 3 - repeat 3 to 6 till optimal solution is found

1.Place Equation in Standard Form

All equations (Objective Function and Constraints) need to be converted to Standard Form.

Simplex is a mathematical method to step around the extreme points till we find the optimal solution.

Conversion to Standard Form for all equations is necessary to facilitate starting at the origin while preserving the rules of mathematics.

Start the conversion with the constraints. How to convert depends on the logic of the equation (<=, >=, or =)

1.Standard Form - Less Than or Equal To

First let us consider <= equations.

Look back at the first constraint in the example. Budget → 2.5X1 + 3X2 <= 30. If we take the axis intercepts like we were going to graph out the constraint, they would be X1 = 12.

The extreme point at the origin (0,0), X1 = 0.

Mathematically we have a problem. If X1 = 12, then it can not equal 0. To get around this issue we add an imaginary variable called Slack. Then X1 + S1 = 12 when X1 = 12 and S1 = 0. When X1 + S1 = 0 at the origin, X1 = 0 and S1 = 12.

1.Standard Form - Less Than or Equal To

The same holds true for X2 variable.

Standard Form

To put a Less Than or Equal Equation Into Standard Form add Slack.

Constraint 1, Budget → 2.5X1 + 3X2 +S1 <= 30

1.Standard Form - Less Than or Equal To

Work Hours, 8X1 + 4X2 <= 80 → 8X1 + 4X2 + S2 <= 80

Material, 2X1 + 6X2 <= 48 → 2X1 + 6X2 + S3 <= 48

Note the subscript, S1. The 1 is because it is the first constraint. S1 represents the amount of slack in the budget constraint. S2 represents the amount of slack in Work Hours. S3 represents the amount of slack in Materials. Slack is assigned to the constraints.

1.Standard Form, Equal To

In a constraint for an “Equal To” equation, we again have trouble starting at zero. In this case we add an Artificial Variable, “A”. The subscript will follow the constraints.

For example say we had a fourth constraint where we must provide a good customer with four units per month of our second product, Customer → X2 = 4.

Standard Form

Customer, X2 = 4 Standard Form → Customer, X2 + A4 = 4

1.Standard Form, Greater Than or Equal To

For >= (Greater than or equal to) we have two variables. We add Artificial variable to allow us to start at zero and subtract a Surplus variable to adjust for greater than our intercept point. For example say we have a fifth constraint for management oversight, 2X1 + 4X2 >= 80.

Standard Form

Management, 2X1 + 4X2 >= 80 Std Form → 2X1 + 4X2 - S5 + A5 >= 80

1.Standard Form of Objective Function

After we have converted the constraints we roll all the slack, artificial, and surplus variables up into the Objective Function.

Budget, 2.5X1 + 3X2 <= 30 → Std Form 2.5X1 + 3X2 +S1 <= 30

Work Hours, 8X1 + 4X2 <= 80 → Std Form 8X1 + 4X2 + S2 <= 80

Material, 2X1 + 6X2 <= 48 → Std Form 2X1 + 6X2 + S3 <= 48

The objective function must be modified to include S1, S2, and S3.

1.Standard Form of Objective Function

Our O.F. is Zmax = 250X1 + 400X2

Standard form

Zmax = 250X1 + 400X2 + 0S1 + 0S2 + 0S3

●We make no profit from Slack.

●Therefore, Slack (and Surplus) variables are given the coefficient of zero.

1.Standard Form of Objective Function

If we had an artificial variable, we must include it in O.F.

We use the coefficient “M” for artificial. M stands for a very large number. If you do not like abstract variables, think of Million. One million is much larger than any of our other variables.

The sign of the coefficient depends of Objective. For a max problem substract, -MA4. For a min problem add, +MA4

Zmax = 250X1 + 400X2 + 0S1 + 0S2 + 0S3 - MA4

More to come we discuss mixed.

1.Standard Form - hack

●Pay attention to inequality○ Less than or equal to, <= Add Slack

○ Equal to, = Add Artificial

○ Greater than or equal to >= Subtract Surplus and Add Artificial

●Pay attention to Objective for Artificial○ Max Subtract M

○ Min Add M