In every field of daily life, we are always in search of a process which can optimize objective using the available resources. in an organized system, decision making is the main. the job of the managers of the system. Ge the system uses resources like manpower, material, money, machines, energy, time, etc. The job the managers is to carefully plan the strategy so that using the limited resources the objective of maximizing profit or minimizing the cost of production is fulfilled. The decision-making problem requires advance planning. A planning is the preparation of the plan of action for determining the number of units of different new products by subjecting them to the conditions between required and available different resources to fulfill the objective of optimum benefit. This plan of action is called programming.
The expressions for calculating required resources and total benefit for each unit of different products are linear. Therefore such problems are called linear programming problems, in short, written as linear programming problem. The ideal presentation of the real-life situation is known as the model. The conversion of numerical information of LPP in the form of mathematical expression is known as the mathematical model of linear programming problem.
In the formulation of mathematical model numbers for units, letters for variables and functions, operators (+, x, 4-) for operations and symbols, _>) for relations are used. The formation of linear programming problem involves the following five steps procedure.
Formulation of LPP
Defining the necessary variables, forming required conditions in them and stating formulated mathematical model of LPP is called formation of linear programming problem.
(1) Decision variable: The variables deciding the number of units of different new products are called decision variables.
Let n decision variables be X1, X2, ……….Xn
(ii) Non-negativity restrictions: The numbers representing units of different new products are always positive and are called non-negativity restrictions. Therefore non-negativity restrictions are X1, > 0, X2,>. 0, ….., Xn > 0.
(iii) Constraints: The condition between the linear expression of the number of resources required and the limited amount of resources available are called constraints.
Let ai1, ai2,…….ain be required number of units of resources b, i= 1, 2, …., m to produce each unit of n different new products.
Therefore the amount of resource of the kind b, required for producing x1 x2, ………., Xn units cot products is ai1 X1 + ai1 X2 + ……ain Xn.
Therefore the condition between the amount of required resource from the resource Bi is ai1 X1 + ai2 X2 +………..ain < or > Bi for i = 1, 2, ……m. These m linear inequalities in n decision variables are called constraints. The coefficients aij are called technological coefficients.
(iv) Objective function: A linear expression inn decision variables of total benefit, Which is to be optimized involving constraints is called the objective function. Let C1, C2, …….. Cn is the amounts of benefit on each unit of n different products.
Therefore total benefit on X1, X2, ……… Xn units of n different products is C1, X1, + C2X2 +………+ Cn Xn. it is represented by the function Z(X1, X2,………. Xn) or only by letter Z or C or P. Since the objective is to optimize the benefit. Therefore it is expressed as optimize Z = c1,x1, + c2,x2, +…….+ cnxn and called objective function. The coefficients ciare called contributory coefficients.
(v) Formulation of mathematical model: The structure of presenting linear programming problem in 11 decision variables and m constraints its expressed as follows.
Determine the values of the decision variables xi, x2, …xn , so as to Optimize (maximize or minimize) Z = c1, x1, + c2 x2 + + cn,xn
subject to the constraints
1) a11x1 + a12 X2 +…….+ a1n xn () b1,
2) a2 1X1 + a22 x2 +……..+a2n xn (< or >) b2,
Steps for the formation of LPP
Step I : Identify the different products and assign their number of units by decision variables x1, x2, • • • xn.
Step II : Introduce non-negativity restrictions x1 >0, x2 >0,……, xn >0.
Step III : Express the constraints in terms of decision variables as linear inequalities between required and available resources.
Step IV: Construct the objective function as the linear function of decision variables with an objective to maximize or minimize.
Step V : State the formulated mathematical model of LPP.
SOLVED PRACTICAL PROBLEMS
Example 1 : The diet for a sick person must contain at least (i) 4000 units of vitamins, (ii) 50 units of minerals and (iii) 1400 calories. Two foods F1 and F2 are available, their costs being Z 4/- and Z 8/- per unit respectively. One unit of F, contains 200 units of vitamins, 1 unit of minerals and 40 calories. One unit of F, contains 100 unit of vitamins, 2 units of minerals and 40 calories. It is required to know how many units of F1 and F2 should form the diet to meet the requirements at minimum cost. Formulate the problem as an linear programming problem.
Step I : x = units of food F, and y = units of food F, are used in the diet.
Step II : Since the number of units of foods F1 and F2 are used. They are non-negative numbers.
Step III : The information in tabular form is as follows
Step IV: The total cost says Z of diet is 4x + 8y, which is to be minimized.
Step V : Therefore the form of mathematical model of the given LPP is as follows Minimize Z = 4x + 8y
Subject to constraint 2x + y > 40
x + 2y > 50
x+y > 35 and the restrictions • x > 0, y_>_ 0.