# Linear Programming

Linear programming, or so-called “solver” PC software, can be used to figure out the best answer to an assortment of questions expressed in terms of functional relationships. In a fundamental sense, linear programming is a straightforward development from the more basic “what if” approach to problem solving. In a traditional “what-if” approach, one simply enters data or a change in input values in a computer spreadsheet and uses spreadsheet formulas and macros to calculate resulting output values. A prime advantage of the “what if” approach is that it allows managers to consider the cost, revenue, and profit implications of changes in a wide variety of operating conditions.

An important limitation of the “what if” method is that it can become a tedious means of searching for the best answer to planning and operating decisions. Linear programming can be thought of as performing “what-if in reverse.” All you do is specify appropriate objectives and a series of constraint conditions, and the software will determine the appropriate input values. When production goals are specified in light of operating constraints, linear programming can be used to identify the cost-minimizing operating plan. Alternatively, using linear programming techniques, a manager might find the profit-maximizing activity level by specifying production relationships and the amount of available resources.

Linear programming has proven to be an adept tool for solving problems encountered in a number of business, engineering, financial, and scientific applications. In a practical sense, typically encountered constrained optimization problems seldom have a simple rule-of-thumb solution. This chapter illustrates how linear programming can be used to quickly and easily solve real-world decision problems.

BASIC ASSUMPTIONS

Linear programming is a useful method for analyzing and solving certain types of management decision problems. To know when linear programming techniques can be applied, it is necessary to understand basic underlying assumptions.

inequality constraints

Many production or resource constraints faced by managers are inequalities. Constraints often limit the resource employed to less than or equal to (≤) some fixed amount available. In other instances, constraints specify that the quantity or quality of output must be greater than or equal to (≥) some minimum requirement. Linear programming handles such constraint inequalities easily, making it a useful technique for finding the optimal solution to many management decision problems. Atypical linear programming problem might be to maximize output subject to the constraint that no more than 40 hours of skilled labor per week be used. This labor constraint is expressed as an inequality where skilled labor ≤ 40 hours per week. Such an operating constraint means that no more than 40 hours of skilled labor can be used, but some excess capacity is permissible, at least in the short run. If 36 hours of skilled labor were fruitfully employed during a given week, the 4 hours per week of unused labor is called excess capacity.

Linearity Assumption

As its name implies, linear programming can be applied only in situations in which the relevant objective function and constraint conditions are linear. Typical managerial decision problems that can be solved using the linear programming method involve revenue and cost functions and their composite, the profit function. Each must be linear; as output increases, revenues, costs, and profits must increase in a linear fashion. For revenues to be a linear function of output, product prices must be constant. For costs to be a linear function of output, both returns to scale and input prices must be constant.

Constant input prices, when combined with constant returns to scale, result in a linear total cost function. If both output prices and unit costs are constant, then profit contribution and profits also rise in a linear fashion with output. Product and input prices are relatively constant when a typical firm can buy unlimited quantities of input and sell an unlimited amount of output without changing prices. This occurs under conditions of pure competition. Therefore, linear programming methods are clearly applicable for firms in perfectly competitive industries with constant returns to scale. However, linear programming is also applicable in many other instances. Because linear programming is used for marginal analysis, it focuses on the effects of fairly modest output, price, and input changes. For moderate changes in current operating conditions, a constant-returns-to-scale assumption is often valid. Similarly, input and output prices are typically unaffected by modest changes from current levels. As a result, sales revenue, cost, and profit functions are often linear when only moderate changes in operations are contemplated and use of linear programming methods is valid.

To illustrate, suppose that an oil company must choose the optimal output mix for a refinery with a capacity of 150,000 barrels of oil per day. The oil company is justified in basing its analysis on the \$25-per-barrel prevailing market price for crude oil, regardless of how much is purchased or sold. This assumption might not be valid if the companies were to quickly expand refinery output by a factor of 10, but within the 150,000 barrels per day range of feasible output, prices will be approximately constant. Up to capacity limits, it is also reasonable to expect that a doubling of crude oil input would lead to a doubling of refined output, and that returns to scale are constant. In many instances, the underlying assumption of linearity is entirely valid. In other instances in which the objective function and constraint conditions can be usefully approximated by linear relations, the linear programming technique can also be fruitfully applied. Only when objective functions and constraint conditions are inherently nonlinear must more complicated mathematical programming techniques be applied. In most managerial applications, even when the assumption of linearity does not hold precisely, linear approximations seldom distort the analysis.

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