Graphical and Simplex Methods of Linear Programming
The graphical method is the more popular method to use because they are easy to use and understand. Working with only a few variables at a time they allow operations managers to compare projected demand to existing capacity. The graphical method is a trial and error approach that can be easily done by a manager or even a clerical staff. Since it is trial and error though, it does not necessarily generate the optimal plan. One downside of this method though is that it can only be used with two variables at the maximum. The graphical method is broken down into the following five steps: 1) Determine the demand in each period.

2) Determine the capacity for regular time, over time, and subcontracting each period. 3) Find labor costs, hiring and labor costs, and inventory holding costs. 4) Consider company policy that may apply to the workers or to stock levels 5) Develop alternative plans and examine their total costs. When a company has a LP problem with more than two variables it turns to the simplex method. This method can handle any number of variables as well as for certain give the optimal solution. In the simplex method we examine corner points in a methodical fashion until we arrive at the best solution which is either the highest profit or lowest cost. LP is used in a wide variety of companies in numerous applications. Airline companies use it to schedule their flights to maximize profit. Another use is for firms to figure out how much of a certain product to manufacture in order to maximize total profits. It also is used by hospitals in order to figure out the most economic diet for patients. It is also a useful tool to figure out labor scheduling for a specific time period. Other applications include product mix planning, distribution networks, truck routing, financial portfolios, and corporate restructuring. All LP problems have four properties in common. The first, LP problems seek to...

...1
Topics on "Operational Research" Mar. 2007, IST
LinearProgramming, an introduction
MIGUEL A. S. CASQUILHO IST, Universidade Técnica de Lisboa, Ave. Rovisco Pais, IST; 1049-001 Lisboa, Portugal
LinearProgramming is presented at an introductory level, mainly from the book by Hillier and Lieberman [2005], abridged and adapted to suit the objectives of the “Operational Research” course. It begins with segments of its third chapter.
Key words: linearprogramming; simplexmethod.
I. Fundamentals and scope
Based on a prototype example, LinearProgramming is presented, as well as the simplexmethod of resolution. This method was first presented by G. B. Dantzig in 1947 [MacTutor, 2007]. The text is based on the book by Hillier and Lieberman [2005], and begins with segments of the third chapter of the book.
II. Explanation of the simplexmethod 3 Introduction to LinearProgramming
(H&L 25)
The development of linearprogramming has been ranked among the most important scientific advances in the mid-20.th century, and we must agree with this assessment. Its impact since just 1950 has been extraordinary. Today it is a standard tool that has saved many thousands or millions of dollars for most...

...Managerial Decision Modeling w/ Spreadsheets, 3e (Balakrishnan/Render/Stair)
Chapter 2 LinearProgramming Models: Graphical and Computer Methods
2.1 Chapter Questions
1) Consider the following linearprogramming model:
Max X12 + X2 + 3X3
Subject to:
X1 + X2 ≤ 3
X1 + X2 ≤ 1
X1, X2 ≥ 0
This problem violates which of the following assumptions?
A) certainty
B) proportionality
C) divisibility
D) linearity
E) integrality
Answer: D
Page Ref: 22
Topic: Developing a LinearProgramming Model
Difficulty: Easy
2) Consider the following linearprogramming model:
Min 2X1 + 3X2
Subject to:
X1 + 2X2 ≤ 1
X2 ≤ 1
X1 ≥ 0, X2 ≤ 0
This problem violates which of the following assumptions?
A) additivity
B) divisibility
C) non-negativity
D) proportionality
E) linearity
Answer: C
Page Ref: 21
Topic: Developing a LinearProgramming Model
Difficulty: Easy
3) A redundant constraint is eliminated from a linearprogramming model. What effect will this have on the optimal solution?
A) feasible region will decrease in size
B) feasible region will increase in size
C) a decrease in objective function value
D) an increase in objective function value
E) no change
Answer: E
Page Ref: 36
Topic: Special Situations in Solving LinearProgramming Problems
Difficulty:...

...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 sum of the individual...

... 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 bylinear 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...

...
7
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.
LinearProgramming: The SimplexMethod
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.
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
M7-2
MODULE 7 • LINEARPROGRAMMING: THE SIMPLEXMETHOD
M7.1
Introduction
In Chapter 7 we looked at examples of...

...LinearProgramming History of linearprogramming goes back as far as 1940s. Main motivation for the need of linearprogramming goes back to the war time when they needed ways to solve many complex planning problems. The simplexmethod which is used to solve linearprogramming was developed by George B. Dantzig, in 1947. Dantzig, was one in who did a lot of work on linearprogramming, he was reconzied by several honours. Dantzig's discovery was through his personal contribution, during WWII when Dantzig was working in the pentagon with the miltary, one of his collegues challenged him, asking "speed up the planning process". Discovery of the simplexmethod was his solution.
Linearprogramming is a powerful tool to solve many problems that arise in many different areas of the outside world. Simplexmethod has been standard method of solving most Linearprogrammings since 1940s. Simplexmethod uses of maximizing and minimizing a linear function to find a feasible set, from then on determined as a miximizer or a minimizer. It can handle many hundreds of variables and in that way it's extremely powerful. These problems can actually be programmed in...

...The SimplexMethod: Learning Team A
Mike Smith, Todd Jones
Math212/Introduction to Finite Mathematics
February 1, 2011
The SimplexMethod: Learning Team A
Sam’s Hairbows and Accessories is a small company preparing for the next scheduled craft fair. The owners, Sam and Todd, both have full-time jobs in addition to owning the company so they are only able to spend a combined total of 80 hours labor to prepare for the fair in four weeks. Sam’s offers five main product lines: basic bows, elaborate bows, bug clips, flower clips, and headbands. Sam’s want to calculate the mix of products they should bring to the fair to maximize their potential profit.
Sam’s believes it is important to give their customers a variety of products. They want every product to make up at least 10% of the total items offered for sale but no more than 30%. Sam’s also knows from past festivals, that headbands are their biggest seller and want at least 15% of their product mix to be headbands. To fill their booth, they want to take at least 400 items. The cost, selling price, and labor requirements for each product are listed in Table 1.
Table 1
| Cost to Make | Selling Price | Labor Required (in minutes) |
Basic Bow | 0.27 | 2.50 | 7 |
Elaborate Bow | 1.07 | 4.00 | 15 |
Bug Clip | 0.22 | 2.50 | 10 |
Flower Clip | 0.94 | 3.00 | 5 |
Headband | 0.82 | 4.00 | 20 |
This problem, as outlined above, is an...

...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. Simplexmethod- is an algorithm for solving linearprogramming with any number of variables. Most real-world linearprogramming problems have more than two variables and thus are...

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