Learning Objectives
A-1: Identify the critical features of linear programming (LP) and explain why it is critical for business operations success.
There are four critical features of linear programming. The objective of an LP problem expressed mathematically is referred to as an objective function. Every linear programming problem will have an objective that needs to be either maximized or minimized, such as the maximization of profits or the minimization of costs. Additionally, the decision variables are alternative courses of action. These variables are under the control of the decision maker who can use LP to decide what would be the best way to allocate the resources among these alternative courses of action to achieve the stated objective. Linear programming problems also include constraints. Constraints arise because resources are scarce and decisions have to be made within the limitations imposed by this scarcity. The last requirement of a linear programming problem is that the objective function and the constraints have linear relationships. That is, the effect of changing a decision variable is proportional to its magnitude.
The decisions that operations and supply chain managers make often deal with the effective use of limited company resources to achieve certain objectives. The limited availability of resources, usually referred to as constraints, can be in the form of labor, materials, time, equipment, energy, money, and so forth. Linear programming is a mathematical modeling technique that managers can use in decisions that involve optimizing an objective (such as maximizing profits or minimizing cost) subject to constraints such as limited resource availability. Linear programming has been applied in a variety of industries to address a variety of problems.
A-2: Solve LP problems with both maximization and minimization objectives, using graphical methods, and Excel solver.
A variety of business problems with both profit maximization and cost minimization objectives can be formulated and solved using the linear programming technique. Simple LP problems can be solved using the graphical approach. This approach is useful because it is relatively straightforward to apply, and it is intuitively appealing. Nevertheless, for more complex LP problems that involve more than two decision variables, the two-dimensional graphical approach is not appropriate. Complex problems will require more sophisticated solution approaches. There are a variety of software tools that allow us to formulate and solve more complex linear programming problems that involve multiple variables and constraint functions. Excel Solver is a good application software tool for solving these problems.
A-3: Perform sensitivity analysis on solutions to LP problems.
A key assumption of all linear programming models is that the input parameters such as the objective function coefficients and the righthand values of the constraints are assumed to be constant. In real-life decisions, however, the values of these parameters are estimates. Managers seek to determine how sensitive a current optimal solution is to changes in the input parameter values. Therefore, in addition to the optimal solution obtained by solving an LP problem, sensitivity analysis lets managers know what would be the impact of changes to the input parameter values, such as the objective function coefficients or the right-hand values of the constraint equations, or both.
A-4: Apply LP to other problems, including product-mixture, blending, and personnel scheduling situations.
Linear programming can be applied to more complex situations than simple profit or cost maximization or minimization using two variables, such as problems involving product mixture, blending, and personnel scheduling. As these problems have several variables and constraints, solving them requires tools such as Excel Solver.