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 of linear 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 Excel’s 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 Company Company 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 Furniture’s 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 are two

unknown entities:

T - number of tables to be produced.

C - number of chairs 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 a single objective function.

In Flair’s problem, total profit may be expressed as: 

Using decision variables T and 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 from

selecting any specific subjective value for

decision variables.

• In Flair Furniture’s 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

• The non-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 for flair’s case:

 

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

Subject to constraints -

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

• Select any arbitrary value for Z.

– 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 involve minimizing objective such

as cost instead of maximizing profit function.

Examples:

– Restaurant may wish to develop work schedule to meet

staffing needs while minimizing total number of employees.

– Manufacturer may seek to distribute its products from

several factories to its many regional warehouses in such a

way as to minimize 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 Problem

Holiday Meal Turkey Ranch• Buy two brands of feed for good, low-cost diet for

turkeys. • Each feed may contain three nutritional ingredients

(protein, vitamin, and iron). • One pound of Brand A contains:

– 5 units of protein, – 4 units of vitamin, and – 0.5 units of iron.

• One pound of Brand 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 requirements for each

nutritional ingredient.

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Summary of Holiday Meal Turkey Ranch Data

Composition of Each Pound of Feed (Oz)

Minimum Monthly

Requirement Per Turkey (Oz)

Ingredient Brand A Brand B

Protein 5 10 90

Vitamin 4 3 48

Iron ½ 0 1 ½

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) ½A 1½ (iron constraint) A 0, B 0 (nonnegativity constraint)

Where: A denotes number of pounds of Brand A feed, and B denote number of pounds of Brand 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 touches feasible region at one point. With minimization, move down and left until it touches only one point in feasible region.

• Identify optimal solution as coordinates of point touched by highest possible isoprofit line or lowest 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 + 3Ysubject to:X + Y 202X + Y 30X 25X, Y 0

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2.7 Special Situations in Solving LP ProblemsInfeasibility: A condition that arises when an LP problem

has no solution that satisfies all of its constraints.

X + 2Y 6

2X + 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 runs parallel to a constraint in 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 optimal X and Y combination.

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2.8 Setting Up and Solving LP Problems Using Excel’s Solver

Procedure of Using Solver

1. Set up LP mathematic model• One objective function

• Decision variables

• Constraints

2. 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 (Click Tools, Solver)• Specifying Objective Function• Choosing Max or Min• Identifying Decision Variables• Adding Constraints• Options: Linear and Non-negative• Answer Report

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Example: Flair Furniture’s Problem

Using solver to solve Flair Furniture’s problem

Recall decision variables T ( Tables ) and

C ( Chairs ) in Flair Furniture problem:  

Maximize profit = $7T + $5C

Subject to constraints

4T + 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 Furniture’s 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 Furniture’s Problem

T C

Tables Chairs

Number of Units

Profit 7 5 Objective

Constraints

Carpentry Hours 4 3 <= 240

Painting Hours 2 1 <= 100

Chairs Limit 1 <= 60

LHS Sign RHS

1. Entering Data:

2. Entering Excel functions for Objective and Constraints

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Using Solver in Excel

In Solver (Click Tools, Solver)• LP model: Program 2.1 B, page 51

• Specifying Objective Function• Choosing Max or Min• 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 solution procedure such as simplex algorithm.

• Described how LP models can be set up on Excel and solved using Solver.