EM 540 Operations Research

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• EM 540 Operations Research/
• DecS 581 Operations Management

Chapter 7 Linear Programming Graphical and
Computer
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Farmer Dan

• Should I plant Corn, Beans or Hay?
• Corn is the most profitable per acre but, I need
  to buy a new harvester to do it.
• Beans sell for a good price, but they use a lot
  of scarce water and are labor intensive. Plus
  beans enrich the soil so, next years fertilizer
  costs will be reduced by 1/3rd.
• Hay is the easiest to grow and doesnt use much
  water. But, it has low yield and is hard to
  remove before next years crop?
• Water is limited. Money is limited. Labor is
  limited.

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High Profits
Plant Corn
Good Farmer
Lowest Water
Plant Hay
Farmer Dans Dilemma(s)
Farmer Dans Solution Know the Impact of
Trade-offs
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Chapter Outline

• 7.1 Introduction
• 7.2 Requirements of a Linear Programming Problem
• 7.3 Formulating LP Problems
• 7.4 Graphical Solution to an LP Problem
• 7.5 Solving Flair Furnitures LP Problem using QM
  for Windows and Excel

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Chapter Outline - continued

• 7.6 Solving Minimization Problems
• 7.7 Summary of the Graphical Solution Methods
• 7.8 Four Special Cases
• 7.9 Sensitivity Analysis in LP

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Learning Objectives

• Students will be able to
• Understand the basic assumptions and properties
  of linear programming (LP).
• Formulate small to moderate-sized LP problems.
• Graphically solve any LP problem with two
  variables by both the corner point and isoline
  methods.

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Learning Objectives - continued

• Understand special issues in LP - infeasibility,
  unboundedness, redundancy, and alternative
  optima.
• Understand the role of sensitivity analysis.
• Use Excel spreadsheets to solve LP problems,

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Examples of Successful LP Applications

• 1. development of a production schedule that will
  satisfy future demands for a firms production
  and at the same time minimize total production
  and inventory costs
• 2. selection of the product mix in a factory to
  make best use of machine-hours and labor-hours
  available while maximizing the firms products
• 3. determination of grades of petroleum products
  to yield the maximum profit
• 4. selection of different blends of raw materials
  to feed mills to produce finished feed
  combinations at minimum cost
• 5. determination of a distribution system that
  will minimize total shipping cost from several
  warehouses to various market locations

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Getting up to speed on equations and formula
Variable Value X 1 Y 4
Graph of limitedhours available
X Number of Chairs Y Number of Tables
X Y 5
Y -X 5 (Standard form ymxb)
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Take Another Equation
5 0
2X Y 6 or Y -2X 6
Graph of limitedVolume available
Y
0 5
X
X Number of Chairs Y Number of Tables
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Putting both Graphs on the Same Page
5 0
X Y 5
-X 0Y -1 X 1 Y 4
Hours available
Y
X
0 5
X Number of Chairs Y Number of Tables
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What is the Meaning?
We cant exceed the work time available. We cant
exceed the Space available. The optimum is 1
Chair and 4 Tables.
5 0
Volume available
Hours available
Y
X
0 5
X Number of Chairs Y Number of Tables
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Equations in Management
Y -2X 100
100 0
1
2X Y 100 Hours
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The 100 Hours is the amount of Paint Booth time
available (2 1/2 shifts, 5 days a week)
Y
0 100
X
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Equations in Management
In-Feasible Region (overload)
In Feasible Region (negative Chairs)
XHours to paint a chair YHours to paint a table
100 0
Feasible Region
2X Y 100 Hours
Y
Question?How many Chairs and How many Tables?
In Feasible Region (negative Tables)
0 100
X
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Inequality
2X Y lt 100 Hours X, Y gt0
100 0
XHours to paint a chair YHours to paint a table
Feasible Region
Y
If we can operate anywhere in the feasible
region, Where SHOULD we OPERATE our Paint Booth?
0 100
X
We need an objective! What are we trying to do?
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Maximize Profit!
2X Y lt 100 Hours
100 0
Feasible Region
If we make 7 for every Table and 5 for every
Chair?
Hum? 100 Tables _at_7 versus 50 Chairs _at_5 Hum?
Any question?
Y
Tables
0 100
X
Could we operate better producing both Tables AND
Chairs?
Chairs
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Minimize Costs?
2X Y lt 100 Hours
100 0
Feasible Region
If it costs 4 to paint a Table and 2 to paint a
Chair? What is the cheapest point in the feasible
region?
Y
Tables
0 100
X
Hum? 0 Tables _at_4 and 0 Chairs _at_2 Good idea?
Chairs
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Requirements of a Linear Programming Problem

• 1. All problems seek to maximize or minimize some
  quantity (the objective function).
• 2. The presence of restrictions or constraints,
  limits the degree to which we can pursue our
  objective.
• 3. There must be alternative courses of action to
  choose from.
• 4. The objective and constraints in linear
  programming problems must be expressed in terms
  of linear equations or inequalities.

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Basic Assumptions of Linear Programming

• Certainty (Numbers are known exactly)
• Proportionality (Relationships are linear)
• Additivity (You can combine like elements)
• Divisibility (You can have fractions)
• Non-negativity (No negative Resources)

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Flair Furniture Company Data - Table 7.1
Hours Required to Produce One Unit
Available Hours This Week
X1 Tables
X2 Chairs
Department
Carpentry Painting/Varnishing
4 2
3 1
240 100
Profit/unit Constraints
7
5
Maximize
Objective
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Flair Furniture Company Constraints
120 100 80 60 40 20 0
Painting/Varnishing
Number of Chairs
Carpentry
20 40 60 80 100
Number of Tables
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Flair Furniture Company Feasible Region
120 100 80 60 40 20 0
Painting/Varnishing
Number of Chairs
Carpentry
Feasible Region
20 40 60 80 100
Number of Tables
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Flair Furniture Company Isoprofit Lines
120 100 80 60 40 20 0
Painting/Varnishing
Number of Chairs
Carpentry
20 40 60 80 100
Zero Profit7X1 5X20
Number of Tables
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Possible Points
Isoprofit Lines
120 100 80 60 40 20 0
Painting/Varnishing
7X1 5X2 350
Solution (X1 50, X2 0)
Number of Chairs
Carpentry
20 40 60 80 100
Number of Tables
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Possible Points
Isoprofit Lines
120 100 80 60 40 20 0
Painting/Varnishing
7X1 5X2 400
Solution (X1 0, X2 80)
Number of Chairs
Carpentry
20 40 60 80 100
Number of Tables
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Flair Furniture Company Optimal Solution
Isoprofit Lines
120 100 80 60 40 20 0
Painting/Varnishing
7X1 5X2 410
Solution (X1 30, X2 40)
Number of Chairs
Carpentry
20 40 60 80 100
Number of Tables
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Flair Furniture Company Optimal Solution
120 100 80 60 40 20 0
Corner Points
Painting/Varnishing
1. 0 2. 400 3. 410 4. 350
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Number of Chairs
Carpentry
3
1
4
20 40 60 80 100
Number of Tables
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Flair Furniture - QM for Windows
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Flair Furniture - Excel
Carpenter Maximum
Painter Maximum
Non Negativity on Tables
Non Negativity on Chairs
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Holiday Meal Turkey Ranch
3
2



Minimize
X
X
2
1
90
10
5
³


to
Subject
X
X

2
1
48
3
4
³

(B)




X
X
2
1
1
1

1
³
(C)





X

1
2
2
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Holiday Meal Turkey Problem
Corner Points
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Holiday Meal Turkey Problem
Isoprofit Lines
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Special Cases in LP

• Infeasibility
• Unbounded Solutions
• Redundancy
• Degeneracy
• More Than One Optimal Solution

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A Problem with No Feasible Solution
X2
8 6 4 2 0
Region Satisfying 3rd Constraint
X1
2 4 6 8
Region Satisfying First 2 Constraints
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A Solution Region That is Unbounded to the Right
X2
Problem trying to Maximize!
15 10 5 0
X1 gt 5
Feasible Region
X1 2X2 gt 10
X1
5 10 15
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A Problem with a Redundant Constraint
X2
30 25 20 15 10 5 0
Redundant Constraint
2X1 X2 lt 30
X1 lt 25
X1 X2 lt 20
Feasible Region
X1
5 10 15 20 25 30
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An Example of Alternate Optimal Solutions
8 7 6 5 4 3 2 1 0
Maximize Profit
A
X2
B
1 2 3 4 5 6 7 8
X1
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Revisiting the LP Assumptions

• Number are Exact (7.3 hours/wk? hardly ever!)
• Numbers never change (who are you kidding!)
• Numbers Add (Mechanics can do Electrical?)
• Objects can be Divided (Deliver 2.37 planes)
• Numbers non-Negative (Never negative Profit?)

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Fuzzy Areas
X2
8 6 4 2 0
X1
2 4 6 8
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Sensitivity Analysis

• Changes in the Objective Function Coefficient
• Changes in Resources (RHS)
• Changes in Technological Coefficients

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Changes in the Technological Coefficients for
High Note Sound Co.
(a) Original Problem
X2
60 40 20 0
3X1 1X2 lt 60
Optimal Solution
Stereo Receivers
a
2X1 4X2 lt 80
b
c
X1
20 40
CD Players
CD Players
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Changes in the Technological Coefficients for
High Note Sound Co.
(a) Original Problem
X2
60 40 20 0
3X1 1X2 lt 60
Optimal Solution
Stereo Receivers
a
2X1 4X2 lt 80
b
c
X1
20 40
CD Players
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Bottom Line

• Linear Programming is an excellent tool for
  optimization.
• Graphical is limited to two dimensions.
• Visual Solutions build concepts of constraints
  and opportunities
• Sensitivity analysis is necessary
• The world has 3,000,000 dimensions.
• Luckily, LP can do much for us.
• Dr Holt (Mid Term Exam Handout Next Week)