Over the past 70 years, LP has been applied extensively to military, industrial, financial, marketing, accounting, and agricultural problems. Even though these applications are diverse, all LP problems have several properties and assumptions in common.

All problems seek to maximize or minimize some quantity, usually profit or cost. We refer to this property as the objective function of an LP problem. The major objective of a typical manufacturer is to maximize dollar profits. In the case of a trucking or railroad distribution system, the objective might be to minimize shipping costs. In any event, this objective must be stated clearly and defined mathematically. It does not matter, by the way, whether profits and costs are measured in cents, dollars, or millions of dollars.

The second property that LP problems have in common is the presence of restrictions, or constraints, that limit the degree to which we can pursue our objective. For example, the decision on how many units of each product in a firm’s product line to manufacture is restricted by available personnel and machinery. Selection of an advertising policy or a financial portfolio is limited by the amount of money available to be spent or invested. We want therefore to maximize or minimize a quantity (the objective function) subject to limited resources (the constraints).

There must be alternative courses of action to choose from. For example, if a company produces three different products, management may use LP to decide how to allocate among them its limited production resources (of personnel, machinery, and so on). Should it devote all manufacturing capacity to make only the first product, should it produce equal amounts of each product, or should it allocate the resources in some other ratio? If there were no alternatives to select from, we would not need LP.

The objective and constraints in LP problems must be expressed in terms of linear equations or inequalities. Linear mathematical relationships just mean that all terms used in the objective function and constraints are of the first degree (i.e., not squared, or to the third or higher power, or appearing more than once). Hence, the equation $2A+5B=10$ is an acceptable linear function, while the equation $2A^2+5B^3+3AB=10$ is not linear because the variable A is squared, the variable B is cubed, and the two variables appear again as a product of each other.

The term linear implies both proportionality and additivity. Proportionality means that if production of 1 unit of a product uses 3 hours, production of 10 units would use 30 hours. Additivity means that the total of all activities equals the sum of the individual activities. If the production of one product generated $3 profit and the production of another product generated $8 profit, the total profit would be the sum of these two, which would be $11.

We assume that conditions of certainty exist: that is, numbers in the objective and constraints are known with certainty and do not change during the period being studied.

We make the divisibility assumption that solutions need not be in whole numbers (integers). Instead, they are divisible and may take any fractional value. In production problems, we often define variables as the number of units produced per week or per month, and a fractional value (e.g., 0.3 chair) would simply mean that there is work in process. Something that was started in one week can be finished in the next. However, in other types of problems, fractional values do not make sense. If a fraction of a product cannot be purchased (for example, one-third of a submarine), an integer programming problem exists. Integer programming is discussed in more detail in Chapter 10.

Finally, we assume that all answers or variables are nonnegative. Negative values of physical quantities are impossible; you simply cannot produce a negative number of chairs, shirts, smart phones, or computers. However, there are some variables that can have negative values, such as profit, where a negative value indicates a loss. A simple mathematical operation can transform such a variable into two nonnegative variables, and that process can be found in books on linear programming. However, when working with linear programming in this book, we will work with only nonnegative variables. Table 7.1 summarizes these properties and assumptions.