BASIC CONCEPTS OF LINEAR PROGRAMMING PDF (FORMULATION AND GRAPHICAL SOLUTIONS) in operation research [Quantitative Techniques for Management]

 List of contents:- 

LINEAR PROGRAMMING-I (FORMULATION AND GRAPHICAL SOLUTIONS)

INTRODUCTION

MEANING

DEFINITION

Objective Function

CONSTRAINTS

ASSUMPTIONS OF LINEAR PROGRAMMING (L P)

BUSINESS APPLICATIONS OF LINEAR PROGRAMMING

LINEAR PROGRAMMING-I (FORMULATION AND GRAPHICAL SOLUTIONS)

 

BASIC CONCEPTS OF LINEAR PROGRAMMING

INTRODUCTION

Linear programming is a technique to determine the optimal allocation of limited resources to meet the given objectives. The resources may be in the form of men, materials and machines, etc. and the objective may be to maximize or minimize a given function. There are certain restrictions on the total amount of each resource available, and on the quantity of each product made. Out of all permissible allocations of resources, one has to find the one which optimizes (maximizes or minimizes) the total profit or cost. This technique was first used by American economist George B. Dantzig in 1947.

 

MEANING

Linear programming consists of two words, 'linear' and 'programming'. The term 'linear' implies that all the relations among the variable in the particular problem are linear (i.e., of degree one) and given straight lines when plotted graphically. The term 'programming' refers to the procedure of determining a mathematical program or plan of action. Thus, linear programming is a mathematical technique for the analysis of optimum decisions subject to constraints in the form of linear inequalities. In other words, linear programming is a mathematical method that applies to those problems which require the solution of maximization or minimization problems subject to linear inequalities in terms of certain variables.

 

DEFINITION

According to -R. Dorfman, P. Samuelson, and R. Solow "Linear programming is the analysis of problems in which a linear function of a number of variables is to be maximized (or minimized) when those variables are subject to a number of constraints in the form of linear inequalities."

 

According to -Alpha C. Chiang "Linear programming is the simpler variety of programming problems in which the objective Junction, as well as the constraint inequalities, are all linear.

 

According to -David W. Pearce "Linear programming is a technique for the formulation and analysis of constrained optimization problems in which the objective function is a linear function, and is to be maximized or minimized subject to a number of linear inequality constraints."

 

Objective Function

An objective function of a linear programming problem states the determinants of the quantity to be maximized or to be minimized. Profits or revenues are objective functions when they are to be maximized and the cost is an objective function when it is to be minimized. An objective function has parts: (i) the primal or original problem, and (ii) the dual problem. If the primal of the objective function is to maximize revenue, then its dual will be the minimization of costs and vice-versa.

 

CONSTRAINTS

There must be certain constraints or restrictions on the variable of the function of the problem. Constraints are the limitations or bounds imposed on the solution, which are expressed in the form of inequalities.

 

1.     Non-Negativity Restrictions

The decision variables must not assume negative values which represent the impossible situation. Non-negativity restrictions are those which assume that there cannot be negative values of the variables involved in the study of linear programming problems. Thus all variables must take on values equal to or greater than zero.

 

2.     Feasible Region

The region which is common to all the constraints of a linear programming problem is called the feasible region of the given problem. In other words, the graph of the system of linear equations, comprising of constraints of the problem is the feasible region of the given problem.

3.     Feasible Solution

A feasible solution to a linear programming problem is the set of values of the variables which satisfies the set of constraints and the non-negative restrictions of the problem.

 

4.     Optimum Solution 

A feasible solution that satisfies both the conditions of the problem and also optimizes the objective function of the problem is called an optimum solution. The optimal solution is the best of the feasible solutions.

ASSUMPTIONS OF L P

 

The following four assumptions are necessary for all linear programming problems: 

1. 1. Linearity: All relationships in LPP, i.e., in both objective function and constraints are represented by straight lines. assumed to be linear. They

 

2. Additivity: The value of the objective function for the given values of the decision variables and the total sum of resources used must be equal to the sum of the contributions (profits or costs) earned from each decision variable.

 

3. Divisibility: Divisibility simply means that the solution need not be in whole numbers (integers). Instead, they are divisible and may take fractional values.

 

4. Certainty: This assumption means that all parameters are known with certainty and do not change during the period being studied.

 

5. Non-negative Variable: In LP problems, we assume that all variables are non-negative. Negative values of physical quantities are an impossible situation.

 

BUSINESS APPLICATIONS OF LINEAR PROGRAMMING

 

Linear programming is a technique of decision-making mostly used in business, industry, and in various other fields. Some of the applications of linear programming are as follows:

 

(i) Diet Problems: To determine the minimum requirement of nutrients subject to the availability of foods and their prices.

 

(ii) Manufacturing Problems: To find the number of items of each type that should be manufactured so as to maximize the profit subject to production restrictions imposed by limitations on the use of machinery and labor.

 

(iii) Transportation Problems: To find the least costly way of transporting shipments from the warehouses to customers.

 

(iv) Blending Problems: To determine the optimum amount of several constituents to use in producing a set of products while determining the optimum quantity of each product to produce.

 

(v) Assembling Problems: To have the best combination of basic components to produce goods according to certain specifications.

 

(vi) Production Problems: To decide the production schedule to satisfy demand and minimize cost in face of fluctuating rates and storage expenses.

 

(vii) Job Assigning Problems: To assign jobs to workers for maximum effectiveness and optimum results subject to restrictions of wages and other costs.

 

(viii) Trim-Loss Problem: To determine the best way to obtain a variety of smaller rolls of paper from a standard width of roll that is kept in stock and, at the same time, minimize wastage.

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