Formulation of Linear Programming Problems

The formulation of a linear programming problem is the most vital and difficult aspect of solving a real problem. Even though there is no fixed pattern in formulating a linear programming problem, operations managers can follow the procedure explained below:

Identify the Decision Variables

The decision-maker should first identify the variables that are under his control and can be changed in order to optimize the objective function. These variables are called decision variables and they should be defined completely and precisely.

Define the Objective Function

The objective of the problem and the criteria for evaluating alternative solutions should be well defined. The objective is generally written as a linear function of the decision variables, each multiplied by an appropriate coefficient.               

Identify and Express Relevant Constraints

Once the decision variables and the objective function are defined, the operations manager should identify the constraints that affect the objective function.

The above process of formulation is generally iterative. Table 4.3 illustrates the steps involved in the formulation of a linear programming model for the problem given in the Table 4.2.

Thus, the general form of a linear programming problem can be stated as below:

Maximize Z = C₁ x₁ + C₂ x₂+...+ C_n x_n

Subject to the constraints;

A₁₁ x₁+A₁₂ x₂+ ... +A_1n x_n ≤ b₁

A₂₁ x₂+A₂₂ x₂+...+A_2n x_n ≤ b₂

A_m1x₁+A_m2x₂ +...A_mn x_n ≤ b_m

x₁, x₂, ...x_n ≥ 0.

Where, x₁, x₂, x_3, ...x_n are a set of variables whose numerical values are to be determined, and

C_ij, A_ij, and b_i are the numeric coefficients that are specified in the problem.

It can be noted that Z is a linear function of variables x_i, i.e. when the value of a variable x_i increases by unity, the value of Z increases by C_i.

The linear programming model can also be used to minimize the objective function. In this case, the constraints are written with a sign '≥'. The constraints can also be written as linear equalities. Thus, the resulting set of decision variables (values for the n variables, x₁, x₂, x₃...x_n) optimizes (either maximize or minimize) the objective function, subject to m constraints and the non-negativity conditions of x_j variables.              

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