Linear Programming

Linear programming (LP) is a mathematical technique used for optimally allocating the minimum or available resources to suitable activities on the basis of given criterion of optimality. The method used for solving such LP problems is called Simplex Method and this method can be applied to any programming problems that can be expressed in linear structure. The general form of any LP problem/model consists of the following three components:

Variables and their Relationships: The variables are usually represented by X1, X2, ... , Xn and these variables are also called as decision variables and the values of these decisions are always non-negative. That is, X1 ≥ 0, X2 ≥ 0, . . . . , Xn ≥ 0.

The variables are usually represented by X1, X2, ... , Xn and these variables are also called as decision variables and the values of these decisions are always non-negative. That is, X1 ≥ 0, X2 ≥ 0, . . . . , Xn ≥ 0. Objective Function: The objective function is a mathematical representation of the goal or objective of the problem. The most often used representation of the objective function (Z) is given as follows: where Z is a variable used to represent the measure of performance (such as profit, cost or time), Xi's are the decision variables, and Ci's are the parameters that yield the contribution to the respective Xi. It is important to be noted that the objective function would be either to maximize or minimize the value of the performance measure Z, that can be either a total profit or total cost or an equivalent measure. For instance, if our objective is to optimize the total profit, then our objective function would be to Maximize the total profit (i.e., Maximize Z). On the other hand, if our objective is to optimize the total cost, then our objective function would be to Minimize the total cost (i.e., Minimize Z).

The objective function is a mathematical representation of the goal or objective of the problem. The most often used representation of the objective function (Z) is given as follows: where Z is a variable used to represent the measure of performance (such as profit, cost or time), Xi's are the decision variables, and Ci's are the parameters that yield the contribution to the respective Xi. It is important to be noted that the objective function would be either to maximize or minimize the value of the performance measure Z, that can be either a total profit or total cost or an equivalent measure. For instance, if our objective is to optimize the total profit, then our objective function would be to Maximize the total profit (i.e., Maximize Z). On the other hand, if our objective is to optimize the total cost, then our objective function would be to Minimize the total cost (i.e., Minimize Z). Constraints: Constraints are the set of linear inequalities and/or linear equalities, used to express the restrictions / limitations of available resources, such as man power, machinery, etc. The LP models are mostly used in maximizing the total profit or minimizing the total cost based on optimum use of available and profitable (limited) resources. Presently, this model is used in various fields, such as R&D industries, Hospitals, Engineering fields, Companies that deals with selling FMCG products, etc. This model can be used only if the constraints and the objective functions can be expressed in linear form, which is a major disadvantage of this model. If the objective function and/or the constraints cannot be expressed in linear form, then the problem can be solved by the use of an alternative models such as Non-linear programming models.

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