Many production decisions are more complex than the preceding example. Consider the problem of finding the optimal output mix for a multiproduct firm facing restrictions on productive facilities and other inputs. This problem, which is faced by a host of companies producing consumer and producer goods alike, is readily solved with linear programming techniques.
Consider a firm that produces products X and Y and uses inputs A, B, and C. To maximize total profit, the firm must determine optimal quantities of each product subject to constraints imposed on input availability. It is often useful to structure such a linear programming problem in terms of the maximization of profit contribution, or total revenue minus variable costs, rather than to explicitly maximize profits. Of course, fixed costs must be subtracted from profit contribution to determine net profits. However, because fixed costs are constant, maximizing profit contribution is tantamount to maximizing profit. The output mix that maximizes profit contribution also maximizes net profit.
An equation that expresses the goal of a linear programming problem is called the objective function. Assume that the firm wishes to maximize total profits from the two products, X and Y, during each period. If per-unit profit contribution (the excess of price over average variable costs) is $12 for product X and $9 for product Y, the objective function is
QX and QY represent the quantities of each product produced. The total profit contribution, π, earned by the firm equals the per-unit profit contribution of X times the units of X produced and sold, plus the profit contribution of Y times QY.
Constraint Equation Specification
Table specifies the available quantities of each input and their usage in the production of X and Y. This information is all that is needed to form the constraint equations.
The table shows that 32 units of input A are available in each period. Four units of A are required to produce each unit of X, whereas 2 units of A are necessary to produce 1 unit of Y. Because 4 units of A are required to produce a single unit of X, the total amount of A used to manufacture X can be written as 4QX. Similarly, 2 units of A are required to produce each unit of Y, so 2QY represents the total quantity of A used to produce product Y. Summing the quantities of A used to produce X and Y provides an expression for the total usage of A. Because this total cannot exceed the 32 units available, the constraint condition for input A is
The constraint for input B is determined in a similar manner. One unit of input B is necessary to produce each unit of either X or Y, so the total amount of B employed is 1QX + 1QY. The maximum quantity of B available in each period is 10 units; thus, the constraint requirement associated with input B is
Finally, the constraint relation for input C affects only the production of Y. Each unit of Y requires an input of 3 units of C, and 21 units of input C are available. Usage of C is given by the expression 3QY, and the relevant constraint equation is
Constraint equations play a major role in solving linear programming problems. One further concept must be introduced, however, before the linear programming problem is completely specified and ready for solution.
Non negativity Requirement
Because linear programming is merely a mathematical tool for solving constrained optimization problems, nothing in the technique itself ensures that an answer makes economic sense. In a production problem for a relatively unprofitable product, the mathematically optimal output level might be a negative quantity, clearly an impossible solution. In a distribution problem, an optimal solution might indicate negative shipments from one point to another, which again is impossible.
Inputs Available for Production of X and Y
To prevent economically meaningless results, a nonnegativity requirement must be introduced. This is merely a statement that all variables in the problem must be equal to or greater than zero. For the present production problem, the following expressions must be added:
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