#5A

The Simplex method, PART 1

Example:

Beaver Creek Pottery Company

X1 = # of bowls to produce each day

X2 = # of mugs to produce each day

Zmax = \$40X1 + \$50X2 + 0S1 + 0S2

The company needs to know how many bowls and mugs to produce each day in order to maximize profit.

Subject to the following constraints:

X1 + 2X2 + 1S1 + 0S2 = 40 labor, hr

4X1 + 3X2 + 0S1 + 1S2 = 120 clay, lb

X1, X2, S1, S2 >= 0 nonnegativity

Initial Simplex Tableau for this Model

 Cj Basic Variables Quantity (RHS) Zj Cj - Zj

Step 1

(2nd row) Record the model variables:

• decision variables
• slack variables

Step 2

Determine the basic feasible solution. A basic feasible solution:

• satisfies the constraints
• contains as many variables with nonnegative values as there are model constraints (recall n-m)

Why is the following the basic feasible solution?

X1 = 0 and X2 = 0 and S1 = 40 and S2 = 120

Record the non-zero variables (basic variables) in this solution (2nd column) and the quantity of those variables (in the 3rd column).

Step 3

Record the Cj values (the objective function coefficients, representing the per unit contribution to profit or cost):

• (1st row) for all the decision variables
• (1st column) for the basic variables (those in the basic feasible solution)

Step 4

(4th-7th columns) Record the coefficients for the decision variables and the slack variables in the constraint equations.

Step 5

(5th row) Compute the Zj values (the gross profit). To do this, multiply each Cj value in the 1st column by each column value under Quantity, the decision variables, and the slack variables; then sum each set of values.

Example:

Zj for Quantity

0 * 40 = 0

0 * 120 = 0

0 + 0 = 0

Example:

Zj for X1

0 * 1 = 0

0 * 4 = 0

0 + 0 = 0

Step 6

(6th row) Compute the Cj – Zj values (net increase in profit associated with one additional unit of each variable). To do this, subtract the Zj row values from the Cj (top row) values.

Example:

Cj – Zj for X1

40 – 0 = 40

Step 7

Examine the profit represented by this solution. The profit (the Z value) is found in the Zj row under the Quantity column. Recall that the solution is to produce no bowls and no mugs. Why is this solution not optimal?

Therefore, we want to move to a solution point that will give a better solution. We want to produce some bowls or some mugs; therefore one of these nonbasic variables (variables not in the present basic feasible solution) will enter the solution and become basic.

And one of the variables must leave the solution. Why?

Step 8

Which variable shall enter the solution?

Choose the highest positive value in Cj – Zj row. It is 50, in this case, and belongs to X2.

X2 becomes the pivot column.

Step 9

Which variable shall leave the solution?

This analysis is performed by dividing the Quantity values of the basic solution variables by the pivot column variables. The leaving variable is the one with the smallest positive quotient.

for S1

40 / 2 = 20

for S2

120 / 3 = 40

S1 becomes the pivot row.

The value 2 at the intersection of the pivot row and the pivot column is called the pivot number.

We are now ready to the second simplex tableau and a better solution.