   Chapter 4.CRE, Problem 25CRE

Chapter
Section
Textbook Problem

MAXIMIZING PROFIT A company manufactures three products, A , B , and   C , on two machines, I and II. It has been determined that the company will realize a profit of $4 / unit of Product A ,$ 6 / unit of Product B , and $8 / unit of Product C . Manufacturing a unit of Product A requires 9 min on Machine I and 6 min on Machine II; manufacturing a unit of Product B requires 12 min on Machine I and 6 min on Machine II; manufacturing a unit of Product C requires 18 min on Machine I and 10 min on Machine II. There are 6 hr of machine time available on Machine I and 4 hr of machine time available on Machine II in each work shift. How many units of each product should be produced in each shift to maximize the company’s profit? What is the maximum profit? To determine To find: The number of units of each product to be produced in each shift to maximize the gains. Explanation Given: A company manufactures three products, A,B,andC, on two machines, I and II. It has been determined that the company will realize a profit of$4/unit of Product A, $6/unit of Product B, and$8/unit of Product C. Manufacturing a unit of Product A requires 9 min on Machine I and 6 min on Machine II; manufacturing a unit of Product B requires 12 min on Machine I and 6 min on Machine II; manufacturing a unit of Product C requires 18 min on Machine I and 10 min on Machine II. There is 6 hr of machine time available on Machine I and 4 hr of machine time available on Machine II in each work shift.

Approach:

The Simplex Method for Solving Non-standard Problems

1. If necessary, rewrite the problem as a maximization problem (recall that minimizing C is equivalent to maximizing C).

2. If necessary, rewrite all constraints (except x0,y0, .) using less than or equal to () inequalities.

3. Introduce slack variables and set up the initial simplex tableau.

4. Scan the upper part of the column of constants of the tableau for negative entries.

a. If there are no negative entries, complete the solution using the simplex method for problems in standard form.

b. If there are negative entries, proceed to Step 5.

5. a. Pick any negative entry in a row in which a negative entry in the column of constants occurs. The column containing this entry is the pivot column.

b. Compute the positive ratios of the numbers in the column of constants to the corresponding numbers in the pivot column (above the last row). The pivot row corresponds to the smallest ratio. The intersection of the pivot column and the pivot row determines the pivot element.

c. Pivot the tableau about the pivot element. Then return to Step 4.

Calculation:

Let x,y,andz are the number of units of products A,B,andC respectively that are produced in each shift to maximize the gains.

Maximize P=4x+6y+8z

subject to

9x+12y+18z3606x+6y+10z240

Introduce the slack variables uandv. Use the simplex method for nonstandard problems to obtain the following sequence of tableaus:

 Cj 4 6 8 0 0 Base Cb C x y z u v u 0 360 9 12 18 1 0 v 0 240 6 6 10 0 1 P 0 −4 −6 −8 0 0

The leaving variable is u an entering variable is z

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