Managerial Decision Modeling
with Spreadsheets
Chapter 2
Linear Programming Models:
Graphical and Computer Methods
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 Solution
Optimal 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)
Ingredient Brand A Brand B
Protein
5
10
Minimum
Monthly
Requirement
Per Turkey (Oz)
90
Vitamin
4
3
48
Iron
½
0
1½
2 cents
3 cents
Cost per
pound
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Formulation of LP Problem:
Minimize cost (in cents) = 2A + 3B
Subject 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
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profit or cost.
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  25
X, 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  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
•
•
•
2.
Use fixed cell address
Use “=sumproduct(range1, range2)” function
Constraints (Use “=sumproduct() function)
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
1. Entering Data:
T
C
Tables
Chairs
7
5
Objective
Carpentry Hours
4
3
<=
240
Painting Hours
2
1
<=
100
1
<=
60
Sign
RHS
Number of Units
Profit
Constraints
Chairs Limit
LHS
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.
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