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Chapter – 17
Linear Programming
Various types of problems may have to face in business and economic activities. These problems happen because of the limited resources. Here we are concerned with the objective of getting maximum profit with limited investment or maximum production with limited resources etc. Such type of problems are referred to the problems of optimization.
These types of problems are tackled by the mathematical method known as the linear programming. Thus linear programming is the mathematical method of getting the optimal solution of the desired objective under certain conditions. The linear programming problem is abbreviated by LP problem or LPP.
Linear Programming Class 12 Mathematics Solutions PDF
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Mathematical Model of Linear Programming Problem
The purpose of the LP problem is to maximize or minimize the objective function for which it is formed. So, mathematically, the LP problem is stated in the following way;
$latex \text{Optimize } z = a x_{1} + b x_{2} \text{, Satisfying the condition}$
$latex a_{1} x_{1} + b_{1} x_{2} \, (\leq \text{ or } \geq) \, c_{1}$
$latex a_{2} x_{1} + b_{2} x_{2} \, (\leq \text{ or } \geq) \, c_{2}$
$latex \text{and } x_{1}, x_{2} \geq 0$.
The function z = ax1 + bx2 which is to be maximized or minimized is known as the objective function. The conditions (i) which the objective function has to satisfy are known as the constraints. The variables x1 and x2 are known as the decision variables.
Standard form of a LP problem
The LP problem may be of maximizing or minimizing the problem according as the objective function is to maximize or minimize respectively. The standard form of both types of problems are given below. A standard maximizing LP problem is LP problem of maximizing its objective function.
Simplex Method
The LP problem consisting of two decision variables can be solved even by graphical method but when the number of decision variables increases, it will not be convinient to to solve by graphical method. Then for such a problem, there is another method of solving the LP problem known as the simplex method. The method can be used for two or more decision variables. This is more effective and most commonly used iteration method to get the optimal solution of the LP problem.
Initially this method gives the zero value of the objective function and in each of its iteration, the value of the objective function changes upto the value where no iteration is needed. For the use of simplex method, we need the following terms.
Duality
The objective of the LP problem may be to maximize or to minimize the objective function. So, the LP problem may be of maximizing the problem or minimizing the problem. But it happens that the maximizing LP problem of one system is equal to the minimizing LP problem on the other system based on the same data and the vice versa. In such a situation, one problem is said to be the dual problem of the other. But generally, the given LP problem is known as the primal problem and other as the dual problem of the former.
If the given LP problem is of the maximization type, it can be solved by the method explained earlier. But if the given LP problem is of the minimization type, then it can also be solved by the same method (explained earlier) only after changing the given minimizing L.P problem into maximizing LP problem i.e. after finding the dual of the given minimizing LP Problem.
Principle of duality
If either of the two LP problems primal or its dual has a finite optimal solution (i.e. max, or min. value) then the other also has a finite optimal solution (i.e. min. or max. value) and the objective functions of these LP problems have the same optimal solution. That is, maxim value of one objective function = minimum value of the other objective.
This is the fundamental principle of duality. The meaning of above principle is “If the objective function Z of the given minimizing LP problem has the finite value, then the objective function Z, the dual of the given LP problem i.e. the objective function Z* of the maximizing LP problem will also have the finite solution and min. Z = max. Z*.
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