1. The simplex method involves 6 steps to find the optimal solution for a linear programming problem. 2. The first step is to convert all equations (objective function and constraints) to standard form. This allows starting the problem at the origin while preserving mathematical rules. 3. For less than or equal to constraints, slack variables are added. For equal constraints, artificial variables are added. For greater than or equal constraints, surplus variables are subtracted and artificial variables are added. 4. The objective function is modified to include all slack, artificial, and surplus variables with coefficients of 0, except for artificial variables which use a very large coefficient M if maximizing or -M if minimizing.

1. Simplex Method
Part 1 of 4
Professor Dansereau

2. 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

3. 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

4. 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

5. 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 =)

6. 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.

7. 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

8. 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.

9. 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

10. 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

11. 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.

12. 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.

13. 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.

14. 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