The term ‛programming′ means planning and it refers to a particular plan of action amongst several alternatives for maximizing profit or minimizing cost etc. Programming problems deal with determining optimal allocation of limited resources to meet the given objectives, such as cost, maximum profit, highest margin or least time, when resources have alternative uses.

The term ‛linear’ means that all inequations or equations used and the function to be maximized or minimized are linear. That is why linear programming deals with that class of problems for which all relations among the variables involved are linear.

Formally, linear programming deals with the optimization (maximization or minimization) of a linear function of a number of variables subject to a ¹equations in variables involved.

The general form of a linear programming problem is

Optimize (Maximize or Minimize) Z = c1x1 + c2x2 + ……..+ cnxn Subject to

a11 x1 + a12x2 + ….. + a1n xn (≤ , = , ≥) b1

a21 x1+ a22x2+ ….. + a2nxn (≤ , = , ≥ ) b2

. . . .

am1 x1+ am2 x2 + … + amn xn {≤ , = , ≥ { bmC

x1, x2….., xn ≥ 0

OBJECTIVE FUNCTION:

IF C1, C2.., Cn are constants and X1,X2,…..,Xn are variables, then the linear function Z = C1X1 + C2X2 + … + CnXn which is to be maximized or minimized is called the objective function. The objective function describes the primary purpose of the formulation of a linear programming problem and it is always non- negative. In business applications, the profit function which is to be maximized or the cost function which is to be minimized is called objective function.

CONSTRAINTS:

The inequations or equations in the variable of a LPP which describe the conditions under which the...

...Q.1. What is a linearprogramming problem ? Discuss the steps and role of linearprogramming is solving management problems. Discuss and describe the role of liner programming in managerial decision-making bringing out limitations, if any.
Ans : LinearProgramming is a mathematical technique useful for allocation of scarce or limited resources to several competing activities on the basis of given criterion of optimality.
The usefulness of linearprogramming as a tool for optimal decision-making on resource allocation, is based on its applicability to many diversified decision problems. The effective use and application requires, as on its applicability to many diversified decision problems. The effective use and application requires, as a first step, the mathematical formulation of an LP model, when the problem is presented in words. Steps of linearprogramming model formulation are summarized as follows :
STEP 1 : Identify the Decision Variables
a) Express each constraint in words. For this you should first see whether the constraint is of the form >/ (at least as large as), of the form \< (no larger than) or of the form = (exactly equal to)
b) You should then verbally express the objective function
c) Steps (a) and (b) should then allow you to verbally identify the decision variables...

...RESEARCH PAPER ON
LINEARPROGRAMMING
Vikas Vasam
ID: 100-11-5919
Faculty: Prof. Dr Goran Trajkovski
CMP 561: Algorithm Analysis
VIRGINIA INTERNATIONAL UNIVERSITY
Introduction:
One of the section of mathematical programming is linearprogramming.
Methods and linearprogramming models are widely used in the optimization of processes in all sectors of the economy: the development of the production program of the company, its distribution on the performers, when placing orders between the performers and the time intervals, to determine the best range of products, in problems of perspective, current and operational planning and management, traffic planning, defining a plan of trade and distribution, in the problems of development and distribution of productive forces, bases and depots of material handling systems, resources, etc. especially widely used methods and linearprogramming model for solving problems are savings (choice of resource-saving technologies, preparation of mixes, nesting materials), production, transportation and other tasks.
Beginning of linearprogramming was initiated in 1939 by the Soviet mathematician and economist Kantorovich in his paper "Mathematical methods of organizing and planning production." The appearance of...

... LINEARPROGRAMMING
DATE;
5 JUNE, 14
UNIVERSITY OF CENTRAL PUNJAB
INTRODUCTION TO LINEARPROGRAMMINGLinearprogramming (LP; also called linear optimization) is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear relationships. Linearprogramming is a special case of mathematical programming.
It is a mathematical technique used in computer modeling to find the best possible solution in allocating limited resources (energy, materials, machines, money etc) to achieve maximum profit or minimum cost.
LinearProgramming is a method of expressing and optimizing a business problem with a mathematical model. It is one of the most powerful and widespread business optimization tools.
Linearprogramming can be used in very large variety of business problems. They include:
transportation distribution problems
production scheduling in oil & gas, manufacturing, chemical, etc industries
financial and tax planning
human resource planning
facility planning
fleet scheduling.
LINEARPROGRAMMING; an optimization technique capable of solving an amazingly...

...1. Discuss why and how you would use a liner programming model for a project of your choice, either from your own work or as a hypothetical situation. Be sure that you stae your situation first, before you develpp the LP model
Linearprogramming is a modeling technique that is used to help managers make logical and informed decisions. All date and input factors are known with certainty. Linear program models are developed in three different steps:
Formulation
Solution
Interpretation
The formulation step deals with displaying the problem in a mathematical form. Once that is developed the solution stage solves the problem and finds the variable values. During the interpretation stage the sensitivity analysis gives managers the opportunity to answer hypothetical questions regarding the solutions that are generated.
There are four basic assumptions of linearprogramming and they are as follows:
Certainty
Proportionality
Additivity
Divisibility
Linearprogramming is the development of modeling and solution procedures which employ mathematical techniques to optimize the goals and objectives of the decision-maker. Programming problems determine the optimal allocation of scarce resources to meet certain objectives. LinearProgramming Problems are mathematical programming...

...LinearProgramming is a mathematical technique useful for allocation of scarce or limited resources to several competing activities on the basis of given criterion of optimality.The usefulness of linearprogramming as a tool for optimal decision-making on resource allocation, is based on its applicability to many diversified decision problems. The effective use and application requires, as on its applicability to many diversified decision problems. The effective use and application requires, as a first step, the mathematical formulation of an LP model, when the problem is presented in words. Steps of linearprogramming model formulation are summarized as follows :
STEP 1 : Identify the Decision Variables
a) Express
each constraint in words. For this you should first see whether the constraint is of the form >/ (at least as large as), of the form \< (no larger than) or of the form = (exactly equal to)
b) You should then verbally express the objective function
c) Steps (a) and (b) should then allow you to verbally identify the decision variables
If there are several decision alternatives available , then in order to identify the decision variables you need to ask yourself the question – what decisions must be made in order to optimize the objective function ?
Having accomplished step 1(a) through (c) decide the symbolic notation for the decision variables...

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7
MODULE
LinearProgramming:
The Simplex Method
LEARNING OBJECTIVES
After completing this chapter, students will be able to:
1. Convert LP constraints to equalities with slack,
surplus, and artificial variables.
2. Set up and solve LP problems with simplex tableaus.
3. Interpret the meaning of every number in a simplex
tableau.
4. Recognize special cases such as infeasibility,
unboundedness and degeneracy.
5. Use the simplex tables to conduct sensitivity
analysis.
6. Construct the dual problem from the primal problem.
CHAPTER OUTLINE
M7.1
M7.2
M7.3
M7.4
M7.5
M7.6
M7.7
Introduction
How to Set Up the Initial Simplex Solution
Simplex Solution Procedures
The Second Simplex Tableau
Developing the Third Tableau
Review of Procedures for Solving LP Maximization
Problems
Surplus and Artificial Variables
M7.8
M7.9
M7.10
M7.11
M7.12
M7.13
Solving Minimization Problems
Review of Procedures for Solving LP
Minimization Problems
Special Cases
Sensitivity Analysis with the Simplex Tableau
The Dual
Karmarkar’s Algorithm
Summary • Glossary • Key Equation • Solved Problems • Self-Test •
Discussion Questions and Problems • Bibliography
M7-1
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MODULE 7 • LINEARPROGRAMMING: THE SIMPLEX METHOD
Introduction
Recall that the...

...LinearProgramming Concept Paper
There are two types of linearprogramming:
1. LinearProgramming- involves no more than 2 variables, linearprogramming problems can be structured to minimize costs as well as maximize profits. Due to the increasing complexity of business organizations, the role of the management executive as a decision maker is becoming more and more difficult. Linearprogramming is a useful technique to solve such problems.
The necessary condition is that the data must be expressed in quantitative terms in the form of linear equations and inequalities. The general nature of the business problems in which linearprogramming can be effectively used are multifaceted. They include purchasing, transportation, job assignments, production scheduling and mixing. Linearprogramming provides a method of maximizing or minimizing a first degree function subject to certain environmental restrictions or constraints which are usually in the form of equations and inequalities.
2. Simplex method- is an algorithm for solving linearprogramming with any number of variables. Most real-world linearprogramming problems have more than two variables and thus are too complex for graphical solution. A...

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Chapter
8
LinearProgramming
CHAPTER OUTLINE
KEY TERMS
Linearprogramming
Production process
Feasible region
Optimal solution
Objective function
Inequality constraints
Nonnegativity constraints
Decision variables
Binding constraints
Slack variable
Simplex method
Primal problem
Dual problem
Shadow price
Duality theorem
Logistic management
8-1 Meaning, Assumptions, and Applications ofLinearProgramming •
The Meaning and Assumptions of LinearProgramming • Applications
of LinearProgramming
8-2 Some Basic LinearProgramming Concepts • Production Processes
and Isoquants in LinearProgramming • The Optimal Mix of Production
Processes
8-3 Procedure Used in Formulating and Solving LinearProgramming
Problems
8-4 LinearProgramming: Profit Maximization • Formulation of the
Profit Maximization LinearProgramming Problem • Graphic Solution of
the Profit Maximization Problem • Extreme Points and the Simplex
Method • Algebraic Solution of the Profit Maximization Problem •
Case Study 8-1: Maximizing Profits in Blending Aviation Gasoline and
Military Logistics by LinearProgramming • Case Study 8-2: Linear...