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Linear Programming

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Introduction: Linear Programming deals with the optimization (max. or min.) of a function of variables, known as ‘objective function’, subject to a set of linear equations and/or inequations, known as ‘constraints’. The objective function may be profit, cost, production capacity or any other measure of effectiveness, which is to be obtained in the best possible or optimal manner. There are two words to be understood – ‘Linear’ and ‘Programming’. The word ‘linear’ refers to linear relationship among the variables in a model. Hence, a given change in one variable will always result in a proportional change in the other variable(s). For example, doubling the investment on a certain project will also double the rate of return. The word ‘programming’ refers to the mathematical modeling and solving a problem that involves the economic allocation of limited resources, by choosing a particular course of action or strategy among the various alternatives in order to achieve the desired result.

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Structure of Linear Programming Model: A linear programming problem may be defined as the problem of ‘maximizing or minimizing a linear function subject to linear constraints’. The constraints may be equalities or inequalities. Here is a simple example. Find numbers X 1 and X 2 that maximizes the sum (X 1 + X 2 ) subject to the constraints X 1 + 2X 2 ≤ 4 4X 1 + 2X 2 ≤ 12 −X 1 + X 2 ≤ 1 and, X 1 ≥ 0, X 2 ≥ 0 In this problem there are two unknowns, and five constraints. All the constraints are inequalities and they are all linear in the sense that each involves an inequality in some linear function of the variables. The last two constraints X 1 ≥ 0 and X 2 ≥ 0, are special. These are called ‘non-negativity constraints’ and are often found in linear programming problems. The other constraints are then called the ‘main constraints’. The function to be maximized (or minimized) is called the ‘objective function’ and is denoted by ‘Z’. Hence, the objective function is ‘maximize Z = X 1 + X 2 ’.

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General Form of LPP: The general linear programming model with ‘n’ decision variables and ‘m’ constraints can be stated in the following form: Find the values of X 1,X 2,……, X n so as to Maximize (or Minimize) Z = C 1 X 1 +C 2 X 2 +…….+C n X n subject to the constraints a 11 X 1 + a 12 X 2 +….+ a 1n X n (≤ or = or ≥) b 1 a 21 X1+ a 22 X 2 +….+ a 2n X n (≤ or = or ≥) b a m1 X 1 + a m2 X 2 +….+ a mn X n (≤ or = or ≥) b n and X 1,X 2,……, X n ≥ 0 where a ij ’s are the ‘technological coefficients’ or ‘input-output coefficients’; b i represents the ‘total availability of i th resource’ and C ij ’s are the ‘coefficients representing per unit contribution of decision variable X j ’. i = 1,2,3,…., m and j = 1,2,3,……, n.

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Steps in Formulation of LPP: Step1: Identify the decision variables and assign symbols x 1, x 2, x 3 …. to them. These decision variables are those quantities whose values we wish to determine. Step2: Identify the set of constraints and express them as linear equations/ inequations in terms of the decision variables. These constraints are the given conditions. Step3: Identify the objective function and express it as a linear function of decision variables. It might take the form of maximizing profit or production or minimizing cost. Step4: Add the non-negativity restrictions on the decision variables, as in the physical problems, negative values of decision variables have no valid interpretation.

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