Linear Programming in Operations Management

Linear programming is a mathematical, constrained optimization model that is used to maximize or minimize the linear functions of a large number of variables, subject to certain constraints. The technique specifies how to use limited resources to meet a particular objective such as maximization of profits or minimization of costs, when the resources have alternative uses. Since the output per unit of resource and the return per unit of output are known to the operations manager, he can determine the resource combination that optimizes the organization's objective. Table 4.1 outlines the types of linear programming problems and their typical features.

Linear programming is widely used in various industrial and military operations. For example, a firm can use linear programming techniques for selection of a specific investment from among several alternatives. Here, the objective function is usually the maximization of returns or the minimization of risks. The constraints in such situations will be the restrictions on the type of permissible investments, company policy, maximum permissible risk, state laws, etc. The development of high-speed computers and data-processing systems has made this technique increasingly easy to use.

Assumptions of Linear Programming

The assumptions made in linear programming models are as follows:

• Certainty

• Divisibility

• Additivity

• Proportionality

Additivity

The objective function and constraints include several decision variables. Here, we assume that the total value of the objective function and each constraint is equal to the sum of the individual contributions from each decision variable. It means that the model does not consider any synergistic or anti-synergistic effects among the decision variables while calculating the total value for the objective function.

Proportionality

In linear programming problems, we assume that the contribution of individual decision variables in the objective function is proportional to their numeric value. Suppose variable X_j represents the number of units of product j produced and C_j is the quantity of raw material utilized in producing a unit of the product, then producing 10 units of product j consumes 10 times the raw material quantity C_j. Hence, the raw material consumption per unit produced remains constant. This means that economies of scale do not play a role in linear programming problems.

Certainty

It is assumed that all the constants; C_j, A_ij and B_i (explained in the next section) have certain values. We assume that the solution is optimal for the problem only when the coefficients of variables have certain or definite values.     

Divisibility

In linear programming models, decision variables can be any non-negative and real numeric values within the range specified by the constraints. The problems that involve fractional values for the decision variables should also be solved in the same way in which the problems with decision variables as integers are solved. In order to avoid fractional values in the final solution, operations managers use integer programming, a technique similar to linear programming that allows only integer values in the result.

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