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Managerial Decision Modeling

with Spreadsheets

Chapter 2

Linear Programming Models:Graphical and Computer Methods

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Learning Objectives

Understand basic assumptions and properties oflinear programming (LP).

Use graphical solution procedures for LP

problems with only two variables to understand

how LP problems are solved.

Understand special situations such as

redundancy, infeasibility, unboundedness, and

alternate optimal solutions in LP problems.

Understand how to set up LP problems on a

spreadsheet and solve them using Excels solver.

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2.2 Development of a LP Model

LP applied extensively to problems areas - medical, transportation, operations,

financial, marketing, accounting,

human resources, and agriculture.

Development of all LP models can be examined

in three step process:

(1) formulation.

(2) solution.

(3) interpretation.

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2.3 Formulating a LP Problem

One of the most common LP application is

product mix problem.

Two or more products are usually produced using

limited resources - such as personnel, machines, raw

materials, and so on.

Profit firm seeks to maximize is based on profit

contribution per unit of each product.

Firm would like to determine - How many units of each product it should produce

Maximize overall profit given its limited resources.

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LP Example: Flair Furniture CompanyCompany Data and Constraints -

Flair Furniture Company produces tables and chairs.

Each table requires: 4 hours of carpentry and 2 hours of painting.

Each chair requires: 3 hours of carpentry and 1 hour of painting.

Available production capacity: 240 hours of carpentry time and 100

hours of painting time.

Due to existing inventory of chairs, Flair is to make no more than

60 new chairs.

Each table sold results in $7 profit, while each chair produced

yields $5 profit.

Flair Furnitures problem:

Determine best possible combination of tables and chairs to

manufacture in order to attain maximum profit.

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Decision Variables

Problem facing Flair is to determine how

many chairs and tables to produce to yield

maximum profit? In Flair Furniture problem, there aretwo

unknown entities:

T - number oftablesto be produced.

C - number ofchairs to be produced.

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Objective Function

Objective function states goal of problem. What major objective is to be solved?

Maximize profit!

An LP model must have asingle objective function.

In Flairs problem, total profit may be expressed as:

Using decision variables Tand C-

Maximize $7 T+ $5 C

($7 profit per table) x (number of tables produced) +

($5 profit per chair) x (number of chairs produced)

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Constraints

Denote conditions that prevent one fromselecting any specific subjective value for

decision variables.

In Flair Furnitures problem, there are

three restrictions on solution.

Restrictions 1 and 2 have to do with available

carpentry and painting times, respectively.

Restriction 3 is concerned with upper limit on

number of chairs.

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Constraints

There are 240 carpentry hours available.

4T + 3C < 240

There are 100 painting hours available.

2T + 1C

100

The marketing specified chairs limit constraint.

C 60

Thenon-negativity constraints.

T 0 (number of tables produced is 0)

C 0 (number of chairs produced is 0)

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2.4 Graphical Representation of Constraints

Complete LP model forflairs case:

Maximize profit = $7T+ $5C (objective function)

Subject toconstraints -4T+ 3C 240 (carpentry constraint)

2T+ 1C 100 (painting constraint)

C 60 (chairs limit constraint)

T 0 (non-negativity constraint on tables)

C 0 (non-negativity constraint on chairs)

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Isoprofit Line Solution Method

Write objective function: $7 T + $5 C = Z

Selectany arbitrary value forZ.

For example, one may choose a profit (Z ) of $210.

Z is written as: $7 T + $5 C = $210.

To plot this profit line:

Set T = 0 and solve objective function for C.

Let T= 0, then $7(0) + $5C= $210, or C= 42.

Set C = 0 and solve objective function for T.

Let C= 0, then $7T+ $5(0) = $210, or T= 30.

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Isoprofit Line Solution Method

Isoprofit lines ($350,$280, $210) are all

parallel.

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Optimal SolutionOptimal Solution:

Corner Point 4: T=30 (tables) and C=40 (chairs) with $410 profit

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Corner Point Solution Method

Point 1 (T= 0, C= 0)profit = $7(0) + $5(0) = $0

Point 2 (T= 0, C= 60)

profit = $7(0) + $5(60) = $300

Point 3 (T= 15, C= 60)

profit = $7(15) + $5(60) = $405

Point 4 (T = 30, C = 40)profit = $7(30) + $5(40) = $410

Point 5 (T= 50, C= 0)

profit = $7(50) + $5(0) = $350 .

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2.5 A Minimization LP Problem

Many LP problems involveminimizingobjective such

ascostinstead of maximizing profit function.

Examples:

Restaurant may wish to develop work schedule to meet

staffing needs whileminimizing total number ofemployees. Manufacturer may seek to distribute its products from

several factories to its many regional warehouses in such a

way as tominimize total shipping costs.

Hospital may want to provide its patients with a daily meal

plan that meets certain nutritional standards while

minimizing food purchase costs.

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Example of a Two Variable Minimization

LP ProblemHoliday Meal Turkey Ranch

Buy two brands of feed for good, low-cost diet forturkeys.

Each feed may contain three nutritional ingredients(protein, vitamin, and iron).

One pound ofBrand A contains:

5 units of protein,

4 units of vitamin, and

0.5 units of iron.

One pound ofBrand B contains:

10 units of protein,

3 units of vitamins, and

0 units of iron.

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Example of Two Variable Minimization

Linear Programming Problem

Holiday Meal Turkey Ranch

Brand A feed costs ranch $0.02 per pound, while

Brand B feed costs $0.03 per pound.

Ranch owner would like lowest-cost diet that meets

minimum monthly intake requirementsfor each

nutritionalingredient.

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Summary of Holiday Meal Turkey

Ranch Data

Composition of EachPound of Feed (Oz)

MinimumMonthly

Requirement

Per Turkey (Oz)Ingredient Brand A Brand B

Protein 5 10 90

Vitamin 4 3 48

Iron 0 1

Cost per

pound

2 cents 3 cents

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Formulation of LP Problem:

Minimize cost (in cents) = 2A + 3BSubject to:

5A + 10B 90 (protein constraint)4A + 3B 48 (vitamin constraint)

A 1 (iron constraint)A 0, B 0 (nonnegativity constraint)

Where:A denotes number of pounds ofBrand A feed, andBdenote number of pounds ofBrand B feed.

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Corner Point Solution Method

Point 1 - coordinates (A = 3,B = 12)

cost of 2(3) + 3(12) = 42 cents. Point 2 - coordinates (A = 8.4, b = 4.8)

cost of 2(8.4) + 3(4.8) = 31.2 cents

Point 3 - coordinates (A = 18,B = 0)

cost of (2)(18) + (3)(0) = 36 cents.

Optimal minimal cost solution:

Corner Point 2, cost = 31.2 cents

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2.6 Summary of Graphical Solution Methods

1. Graph each constraint equation.

2. Identify feasible solution region, that is, area

that satisfies all constraints simultaneously.

3. Select one of two following graphical solution

techniques and proceed to solve problem.

1. Corner Point Method.

2. Isoprofit or Isocost Method.

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2.6 Summary of Graphical Solution

Methods (Continued)

Corner Point Method Determine coordinates of each of corner

points of feasible region by visual inspection

or solving equations. Compute profit or cost at each point by

substitution of values of coordinates into

objective function and solving for result. Identify an optimal solution as a corner point

with highest profit (maximization problem),

or lowest cost (minimization).

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2.6 Summary of Graphical Solution

Methods (continued)

Isoprofit or Isocost Method Select value for profit or cost, and draw isoprofit /isocost line to reveal its slope.

With a maximization problem, maintain same

slope and move line up and right until it touchesfeasible region at one point. With minimization,move down and left until it touches only onepoint in feasible region.

Identify optimal solution as coordinates of pointtouched by highest possible isoprofit line orlowest possible isocost line.

Read optimal coordinates and compute optimal

profit or cost.

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2.7 Special Situations in Solving LP Problems

Redundancy: A redundant constraint is constraint that

does not affect feasible region in any way.

Maximize

Profit

= 2X+ 3Y

subject to:

X+ Y 20

2X+ Y 30

X 25X, Y 0

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2.7 Special Situations in Solving LP Problems

Infeasibility: A condition that arises when an LP problem

has no solution that satisfies all of its constraints.

X+ 2Y

62X+ Y 8

X 7

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2.7 Special Situations in Solving LP Problems

Unboundedness: Sometimes an LP model will

not have a finite solution

Maximize profit

= $3X+ $5Y

subject to:

X 5

Y 10

X+ 2Y 10

X, Y 0

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Alternate Optimal Solutions

An LP problem may have more than one

optimal solution.

Graphically, when the isoprofit (or isocost)

line runsparallel to a constraintin problem

which lies in direction in which isoprofit (or

isocost) line is located.

In other words, when they have same slope.

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Example: Alternate Optimal Solutions

At profit level of $12, isoprofit line will rest directly on top of

first constraint line.

This means that any point along line between corner points 1 and

2 provides an optimalXand Ycombination.

2 8 S tti U d S l i LP P bl U i

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2.8 Setting Up and Solving LP Problems Using

Excels Solver

Procedure of Using Solver

1. Set up LP mathematic model

One objective function

Decision variables

Constraints2. Rewriting the model for Solver

RHS of constraints are numbers.

3. Entering information in Solver

The CD-ROM that accompanies this textbook contains

excel file for each example problem discussed here.

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Using Solver in Excel

1. Entering Data

Labels and Titles Parameters and Coefficients

Objective Function Use fixed cell address

Use =sumproduct(range1, range2) function

Constraints (Use =sumproduct() function)2. In Solver (ClickTools, Solver)

Specifying Objective Function

ChoosingMax orMin

Identifying Decision Variables

Adding Constraints Options: Linear and Non-negative

Answer Report

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Example: Flair Furnitures Problem

Using solver to solve Flair Furnitures problem

Recall decision variables T( Tables ) and

C( Chairs ) in Flair Furniture problem:

Maximize profit = $7T+ $5C

Subject to constraints4T+ 3C 240 (carpentry constraint)

2T+ 1C 100 (painting constraint)

C 60 (chairs limit constraint)T, C 0 (non-negativity)

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Example: Flair Furnitures Problem

Data File: 611_render_2-1.xls Entering Data: Program 2.1A, page 48

Coefficients

Excel Functions

In Solver:

LP model: Program 2.1 B, page 51.

LP options: Program 2.1 C, page 52.

Results options: Program 2.1 D, page 53.

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Example: Flair Furnitures Problem

T C

Tables Chairs

Number of Units

Profit 7 5 ObjectiveConstraints

Carpentry Hours 4 3

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Using Solver in Excel

In Solver (ClickTools, Solver)

LP model: Program 2.1 B, page 51 Specifying Objective Function

ChoosingMax orMin

Identifying Decision Variables Adding Constraints

LP options: Program 2.1 C, page 52 Options: Linear and Non-negative

Results options: Program 2.1 D, page 53 Answer Report

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Summary

Introduced a mathematical modeling technique

called linear programming (LP).

LP models used to find an optimal solution to

problems that have a series of constraints

binding objective value. Showed how models with only two decision

variables can be solved graphically.

To solve LP models with numerous decision

variables and constraints, one need a solutionprocedure such as simplex algorithm.

Described how LP models can be set up on

E l d l d i S l