Slack Variables

The concept of slack variables must be introduced to solve linear programming problems algebraically. In the case of less-than-or-equal-to constraints, slack variables are used to increase the left side to equal the right side limits of the constraint conditions. In the illustrative problem, one slack variable is added to each constraint to account for excess capacity. The firm is faced with capacity constraints on input factors A, B, and C, so the algebraic specification of the problem contains three slack variables: SA, indicating the units of A that are not used in any given solution; SB, representing unused units of B; and SC, which measures the unused units of C.

With slack variables, each constraint equation becomes an equality rather than an inequality. After adding the relevant slack variable, the constraint on input A, 4QX + 2QY < 32, is

SA = 32 - 4QX - 2Qy is the amount of input A not used to produce X or Y. Similar equality constraints can be specified for inputs B and C. The equality form of the constraint on input B is

The constraint equation for input C is

The introduction of slack variables not only simplifies algebraic analysis, but slack variables' solution values also provide useful information. In a production problem, for example, slack variables with zero values at the optimal solution indicate inputs that are limiting factors and cause bottlenecks. Slack variables with positive values at the optimal solution indicate excess capacity in the related input factor. Slack variables cannot take on negative values, because this would imply that the amount of resource use exceeds available supply. The information provided by slack variable solution values is important in long-range planning and is a key benefit derived from algebraic solution methods.

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