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

Linear programming 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 linear programming and they are as follows: Certainty
Proportionality
Additivity
Divisibility

Linear programming 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. Linear Programming Problems are mathematical programming problems where all of the relationships amongst the variables are linear.

Components of a LP Formulation are as follows:
Decision Variables
Objective Function
Constraints
Non-negativity Conditions

Decision variables represent unknown quantities. The solutions for these terms are what we would like to optimize. Objective function states the goal of the decision-maker. There are two types of objectives: Maximization

Minimization

Constraints put limitations on the possible solutions of the problem. The availability of scarce resources may be expressed as equations or inequalities which rule out certain combinations of variable values...

...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...

...TOPIC – LINEARPROGRAMMINGLinearProgramming is a mathematical procedure for determining optimal allocation of scarce resources.
Requirements of LinearProgramming
• all problems seek to maximize or minimize some quantity
• The presence of restrictions or constraints
• There must be alternative courses of action
• The objective and constraints inlinearprogramming must be expressed in terms of linear equations or
inequalities
Objective Function it maps and translates the input domain (the feasible region) into output range, with
the two-end values called the maximum and minimum values
Restriction Constraints it limits the degree to which we can pursue our objective
Decision Variables represents choices available to the decision maker in terms of amount of either inputs or outputs
Parameters these are the fixed values in which the model is solved
Basic Assumption of LinearProgramming
1. Certainty- figures or number in the objective and constraints are known with certainty and do not vary
1. Proportionality - for example 1:2 is equivalent to 5:10
1. Additivity - the total of all the activities equals the...

...LINEARPROGRAMMING
INTRODUCTION:
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 linearprogramming deals with that class of problems for which all relations among the variables involved are linear.
Formally, linearprogramming 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 linearprogramming 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 +...

...
LinearProgramming
After completing this chapter you should be able to:
identify a product which maximises the contribution per unit of scarce resource when there is only one scarce resource, and determine the optimum solution.
formulate an LP model to solve for the optimal product mix which maximises profits, or for cost minimisation problems to minimise costs.
solve 2 variable problems graphically.
use a spreadsheet to solve LP problems with any number of variables.
interpret the sensitivity reports of spreadsheet solutions to LP problems to test objective function coefficient sensitivity, determine shadow prices and RHS ranging.
perform throughput accounting and solve problems using the concept of the Theory of Constraints.
Introduction
In this chapter we continue with our profit planning, or product mix, decisions. We extend CVP analysis in the last chapter by introducing the notion of scarce resources. Although CVP analysis does not provide answers regarding optimum product mixes (mixes which maximise profits or minimise costs) one advantage of CVP analysis is that it focuses attention on products with high contribution margins. Managers and salespeople can often direct their efforts to increasing output and sales of high contribution margin products and thereby maximise the contribution towards fixed costs and profits. Unfortunately, it is not always desirable to attempt to maximise the sales of high...

...Duality in LinearProgramming
4
In the preceding chapter on sensitivity analysis, we saw that the shadow-price interpretation of the optimal
simplex multipliers is a very useful concept. First, these shadow prices give us directly the marginal worth
of an additional unit of any of the resources. Second, when an activity is ‘‘priced out’’ using these shadow
prices, the opportunity cost of allocating resources to that activity relative to other activities is determined.
Duality in linearprogramming is essentially a unifying theory that develops the relationships between a
given linear program and another related linear program stated in terms of variables with this shadow-price
interpretation. The importance of duality is twofold. First, fully understanding the shadow-price interpretation
of the optimal simplex multipliers can prove very useful in understanding the implications of a particular
linear-programming model. Second, it is often possible to solve the related linear program with the shadow
prices as the variables in place of, or in conjunction with, the original linear program, thereby taking advantage
of some computational efﬁciencies. The importance of duality for computational procedures will become
more apparent in later chapters on network-ﬂow problems and large-scale systems.
4.1
A PREVIEW OF DUALITY...

...DQ 17
A common form of the product-mix linearprogramming seeks to find the quantities of items in the product mix that maximizes profit in the presence of limited resources.
-True
Linearprogramming helps operations managers make decisions necessary to allocate resources.
-True
In linearprogramming, the unit profit or unit contribution associated with one decision variable can be affected by the quantity made of that variable or of any other variable in the problem.
-False
What combination of x and y will yield the optimum for this problem?
Minimize $3x + $15y, subject to
(1) 2x + 4y 12 and
(2) 5x + 2y 10.
-x = 0, y = 0
In linearprogramming, a statement such as "maximize contribution" becomes a(n)
-objective function.
Choosing the best alternative in the face of random states of nature is referred to as
-decision thoery
Linearprogramming is part of larger body of knowledge referred to as optimization
-True
One requirement of a linearprogramming problem is that the objective function must be expressed as a linear equation.
-True
Which of the following is not one of the steps in setting up a LP formulation>
-calculate the objective function
Which of the following represents valid constraints in...

...Spreadsheet Modeling and Excel Solver A mathematical model implemented in a spreadsheet is called a spreadsheet model. Major spreadsheet packages come with a built-in optimization tool called Solver. Now we demonstrate how to use Excel spreadsheet modeling and Solver to find the optimal solution of optimization problems. If the model has two variables, the graphical method can be used to solve the model. Very few real world problems involve only two variables. For problems with more than two variables, we need to use complex techniques and tedious calculations to find the optimal solution. The spreadsheet and solver approach makes solving optimization problems a fairly simple task and it is more useful for students who do not have strong mathematics background. The first step is to organize the spreadsheet to represent the model. We use separate cells to represent decision variables, create a formula in a cell to represent the objective function and create a formula in a cell for each constraint left hand side. Once the model is implemented in a spreadsheet, next step is to use the Solver to find the solution. In the Solver, we need to identify the locations (cells) of objective function, decision variables, nature of the objective function (maximize/minimize) and constraints. Example One (Linear model): Investment Problem Our first example illustrates how to allocate money to different bonds to...