Linear programming problems are concerned with efficient use of limited resources to meet desired objectives. These problems are characterized by large number of solutions that satisfy the basic conditions of each problem. The selection of a particular solution as the best solution to a problem depends on some objective that is implied in the statement of the problem.

Linear programming methods solve optimization problems having linear objective function and constraints. That is why this methods are extensively in operations research or transportation problem. Linear programming methods can also solve problems having some particular types of non-linear objective function and linear constraints which is explained in chapter 10.

Associated with every linear programming problem there always exists another programming problem which is based upon the same data and having the same solution. The original problem is called the primal problem while the associated one is called its dual problem. From both the theoretical and practical points of view, the theory of duality is one of the most important and interesting concepts in l ......

Sensitivity analysis refers to the study of the changes in the optimal solution and optimal value F(X) due to changes in the input data coefficients. For a discussion of the sensitivity analysis, we shall confine ourselves to the following changes in the data.

A linear programming problem in which all or some of the decision variables are constrained to some non-negative integer values is called an integer programming problem. This type of problem is of particular importance in business and industry where, quit often, the fractional solutions are unrealistic because the units are not divisible.

There is a multitude of operations research situations that can be conveniently modeled and solved as networks (nodes connected by branches). Consider the situation “the determination of the time schedule (starting and completion dates) for the activities of a construction project”. The solution of this situation, and others like it, is accomplished through a variety of network optimization algori ......

One of the earliest and most fruitful applications of linear programming problem has been the formulation and solution of the transportation problem as a linear programming problem.

The mathematical technique of optimizing a sequence of interrelated decisions over a period of times is called dynamic programming. It uses the idea of recurrence to solve a complex problem, broken into a series of sub-problems. The word dynamic has been used because time is explicitly taken into consideration. The objective in dynamic programming is to select a decision policy so as to optimize t ......

The term game now includes not only pleasurable activities of this kind, but also much more earnest competitive situations of ware and peace. It is not coincidental that the classic work on the theory of games was first published during the second world war. Many competitive situations are still too complex for the theory in its present state of development to solve.

Classical optimization techniques use differential calculus to determine points of maxima and minima of continuous and differentiable function. These techniques have limited scope in practical applications, since practical problems involve objective functions that are not continuous.

The queuing system consists of one or more queues and one or more servers, and operators under a set of procedures. Let us consider the reservation counter of an airlines where customers from different part of the world arrive and wait at the reservation counter. Depending on the server status, the incoming customer either waits at the queue or gets the turn to be served.

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