C = monthly cost of production, $; and the subscript refers to the machine. Find the most
economical manner of allocating production among the three machines.
Calculation Procedure:
- Write the equations of Incremental costs
The objective is to minimize the total cost of production. Since costs vary nonlinearly,
this situation does not lend itself to linear programming.
Assume that TV units have been produced on a given machine. The incremental (or
marginal) cost at that point is the cost of producing the (TV+ /)th unit. Let I= incremental
cost, $. If TV is large, / ^ dCldN. By differentiating the foregoing expressions, this approx-
imation gives IA = 0.3976TV 40 -^42 , IB = 0.5292TV 50 -^47 , Ic = 0.4590TV<?^53 , where the subscript
refers to the machine. Also, if TV is large, cost of producing TVth unit — cost of producing
(TV+l)thunit. - Establish the condition at which the total cost of production
is minimum
Arbitrarily set TV 4 = 150, NB = 250, TVC = 300, which gives a total of 700 units. The incre-
mental costs are IA = $3,2614, IB = $7.0901, and Ic = $9.4338. Also, when TV 4 = 151, then
IA = $3.2705. The total cost of production can be reduced by shifting 1 unit from machine
B to machine A and 1 unit from machine C to machine A, with the reduction being ap-
proximately $7.0901 + $9.4338 - ($3.2614 + 3.2705) = $9.9920. Thus, the arbitrary set of
TV values given above does not yield the minimum total cost.
Clearly the total cost of production is minimum when all three incremental costs are
equal (or as equal as possible, since TV is restricted to integral values). - Find the most economical allocation of production
At minimum total cost, IA=IB = Ic or 0.3976TV 40 -^42 = 0.5292TV 50 -^47 = 0.4590TVC°^53 , Eq. a;
and NA+ NB + NC = 700, Eq. b. By a trial-and-error solution, TV 4 = 468, TV 5 = 132, and
TVC=100.
Alternatively, proceed as follows: From Eq. a, TV 5 = 0.5442Af 40 -^8936 and TVC =
0.7627TV 40 -^7925. Substituting in Eq. b gives TV 4 + 0.5442TV 40 -^8936 + 0.7627TV^0 -^7925 = 700. As-
sign trial values to TV 4 until this equation is satisfied. The solution is TV 4 = 468, and the re-
maining values follow. - Devise a semigraphical method of solution
In Fig. 9, plot the incremental-cost curves. Pass an arbitrary horizontal line L through
these curves to obtain a set of TV values at which T 4 = IB = Ic. Scale the TV values, and find
their sum. Now displace the horizontal line until the sum of the TV values is 700.
Related Calculations: Allocation problems of this type usually are solved by ap-
plying Lagrange multipliers. However, as the previous solution demonstrates, the use of
simple economic logic can circumvent the need for abstract mathematical concepts.
OPTIMAL PRODUCTMIX
WITH NONLINEAR PROFITS
A firm manufactures three articles, A, B, and C, and it can sell as many units as it can pro-
duce. The monthly profits, exclusive of fixed costs, are PA - 4.75TV 4 - 0.005OTV 42 , PB =
2.6OTV 4 - 0.0014TV 52 , and Pc = 2.25TVC - 0.001OTV^, where TV = number of units produced