Analysis of Business OperationsLINEAR PROGRAMMING TO MAXIMIZE
INCOME FROM JOINT PRODUCTS
A firm manufactures two articles, A and B. The unit cost of production, exclusive of fixed
costs, is $10 for A and $7 for B. The unit selling price is $16 for A and $13.50 for B. The
estimated maximum monthly sales potential of A is 9000 units; of B, 7000 units. It is the
policy of the firm to produce only as many units as can readily be sold. If production is
restricted to one article, the factory can turn out 13,000 units of A or 8500 units of B per
month. The capital allotted to monthly production after payment of fixed costs is
$100,000. What monthly production of each article will yield the maximum profit?
Calculation Procedure:
- Express the production constraints imposed by sales
and capital
Let NA and NB denote the number of articles A and B, respectively, produced monthly.
Then potential sales: NA < 9000, Eq. a; NB < 7000, Eq. b. Available capital: 1OW 4 + 1NB
<$ 100,000, Eq. c. - Determine the production constraint imposed
by the plant capacity
The number of months required to produce NA units of A is NA/13,000. Likewise, to pro-
duce NB units of B would be #0/8500. Then, W 4 /13,000 + Afc/8500 < 1, or S.5NA + 137V 5
< 110,500, Eq. d. - Express the monthly profit in equation form
Before fixed costs are deducted, the profit P = (16 - IQ)NA + (13.5 - I)N 39 or P = 6NA +
6.5NB9 Eq. e. - Construct a monthly production chart
Considering the expressions a to d above to be equalities, plot the straight lines represent-
ing them (Fig. 8).
Since these expressions actually establish upper limits to the values ofNA and NB, the
point representing the joint production of articles A and B must lie either within the shad-
ed area, which is termed the feasible region, or on one of its boundary lines. - Plot an equal-profit line
Assign the arbitrary value of $30,000 to P, and plot the straight line corresponding to Eq.
e above. Every point on this line (Fig. 8) represents a set of values for NA and N 8 for
which the profit is $30,000. This line is therefore termed an equal-profit line.
Next, consider that P assumes successively greater values. As P does so, the equal-
profit line moves away from the origin while remaining parallel to its initial position. - Maximize the profit potential
To maximize the profit, locate the point in Fig. 8 at which the equal-profit line, in its out-
ward displacement, is on the verge of leaving the feasible region. This is point g, which
lies at the intersection of the lines representing the equalities c and d. - Determine the number of units for maximum profit
Establish the coordinates of the maximum-profit point Q either by reading them from the