Linear Programming

Linear Programming (LP) is a mathematical technique developed during World War II for the purpose of planning schedules adequately or distributing men optimally. The method that was developed by George Dantzig (1947) is called Simplex method of optimization, which provides a standard procedure for solving any programming problem that can be expressed in a linear form.

The general form of this LP model is usually represented with three components, namely
1. Decision variables and their relationships,
2. Objective Function, and
3. Constraints.

Here, the values of the decisions are always non-negative and the objective function is to either maximize or minimize the performance measure (say Z). The constraints are the set of linear equalities or inequalities used to incorporate the conditions or restrictions of available resources given in the problem.

Suppose, we have n decision variables with m constraints, where our objective is to optimize (maximize or minimize) the performance variable Z. Here, the mathematical formulation of the general linear programming problem can be given as follows:


 
where X_i's are the decision variable, C_i's are the coefficient of X_i , and b_j's are the total availability of jth resource; a_ji's denotes the amount of resources utilized per unit of decision variable X_i. LP models are used in various fields for attaining the optimum use of successful resources to maximize the profit or to minimize the cost. It is mainly used to improve the quality of decisions.

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