Linear programming is a valuable technique for solving maximization or minimization problems in which inequality constraints are imposed on the decision maker. This chapter introduces graphic and analytic approaches for setting up, solving, and interpreting the solutions to such problems.
• Linear programming is a proven tool used to isolate the best solution, or optimal solution, to decision problems. The technique is ideally suited to solving decision problems that involve an objective function to be maximized or minimized, where the relevant objective function is subject to inequality constraints.
• Simple linear programming problems can be solved graphically using the relative distance method. The feasible space is the graphical region showing the linear programming problem solution space that is both technically and economically feasible.
• An equation that expresses the goal of a linear programming problem is called the objective function.
• The optimal solution to a linear programming problem occurs at the intersection of the objective function and a corner point of the feasible space. A corner point is a spot in the feasible space where the X-axis, Y-axis, or constraint conditions intersect.
• Slack variables indicate the amount by which constraint conditions are exceeded. 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 case of greater-than-or-equal-to constraints, slack variables are used to decrease the left side to equal the right side limits of the constraint conditions.
• The simplex solution method is an iterative method used to solve linear programming problems. In this procedure, computer programs find solution values for all variables at each corner point, then isolate that corner point with the optimal solution to the objective function.
• For every maximization problem in linear programming, there exists a symmetrical minimization problem; for every minimization problem, there exists a symmetrical maximization problem. These pairs of related maximization and minimization problems are known as the primal and dual linear programming problems.
• The primal solution is often described as a tool for short-run operating decisions, whereas the dual solution is often seen as a tool for long-range planning. Both provide management with valuable insight for the decision-making process.
• Shadow prices are implicit values or opportunity costs associated with linear programming problem decision variables. In the case of output, shadow prices indicate the marginal cost of a one-unit increase in output. In the case of the constraints, shadow prices indicate the marginal cost of a one-unit relaxation in the constraint condition.
During recent years, rapid advances in user-friendly computer software have allowed the widespread application of linear programming techniques to a broad range of complex managerial decision problems. With the background provided in this chapter, it is possible to apply this powerful technique to a wide array of problems in business, government, and the not-for-profit sector.
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