Linear Programming

9 Linear Programming

How is a linear programming problem formulated?

Factors exist that restrict the immediate attainment of almost any objective. For ex- ample, assume that the objective of the board of directors at Washington Hospital is to aid more sick people during the coming year. Factors restricting the attain- ment of that objective include number of beds in the hospital, size of the hospi- tal staff, hours per week the staff is allowed to work, and number of charity pa- tients the hospital can accept. Each factor reflects a limited or scarce resource and Washington Hospital must find a means of achieving its objective by efficiently and effectively allocating its limited resources.

Managers are always concerned with allocating scarce resources among com- peting uses. If a company has only one scarce resource, managers will schedule

Chapter 12 Relevant Costing

scarce resource. Most situations, however, involve several limiting factors that com- pete with one another during the process of striving to attain business objectives. Solving problems having several limiting factors requires the use of mathematical

mathematical

programming, which refers to a variety of techniques used to allocate limited re-

programming

sources among activities to achieve a specific goal or purpose. This appendix pro- vides an introduction to linear programming, which is one form of mathematical programming. 14

Basics of Linear Programming

Linear programming (LP) is a method used to find the optimal allocation of

linear programming

scarce resources in a situation involving one objective and multiple limiting factors. 15 The objective and restrictions on achieving that objective must be expressible as

linear equations. 16 The equation that specifies the objective is called the objective

objective function

function; typically, the objective is to maximize or to minimize some measure of performance. For example, a company’s objective could be to maximize contribution margin or to minimize product cost.

A constraint is any type of restriction that hampers management’s pursuit of

constraint

the objective. Resource constraints involve limited availability of labor time, machine time, raw material, space, or production capacity. Demand or marketing constraints restrict the quantity of product that can be sold during a time period. Constraints can also be in the form of technical product requirements. For example, manage- ment may be constrained in the production requirements for frozen meals by caloric or vitamin content.

A final constraint in all LP problems is a nonnegativity constraint. This con-

nonnegativity constraint

straint specifies that negative values for physical quantities are not allowed. Con- straints, like the objective function, are specified in mathematical equations and represent the limits imposed on optimizing the objective function.

Almost every allocation problem has multiple feasible solutions that do not

feasible solution

violate any of the problem constraints. Different solutions generally give different values for the objective function, although in some cases, a problem may have several solutions that provide the same value for the objective function. Solutions can be generated that contain fractional values. If solutions for variables must be restricted to whole numbers, integer programming techniques must be used to add

integer programming

additional constraints to the problem. The optimal solution to a maximization or

optimal solution

minimization goal is the one that provides the best answer to the allocation problem. Some LP problems may have more than one optimal solution.