Programming:

Model Formulation
and Graphical
Solution
Chapter 2

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Prentice Hall 2-1
Chapter Topics

 Model Formulation
 A Maximization Model Example
 Graphical Solutions of Linear Programming
Models
 A Minimization Model Example
 Irregular Types of Linear Programming
Models
 Characteristics of Linear Programming
Problems
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Linear Programming: An
Overview
 Objectives of business decisions frequently
involve maximizing profit or minimizing
costs.
 Linear programming uses linear algebraic
relationships to represent a firm’s
decisions, given a business objective, and
resource constraints.
 Steps in application:
1. Identify problem as solvable by linear
programming.
2. Formulate a mathematical model of the
unstructured problem.
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Model Components

 Decision variables - mathematical symbols

representing levels of activity of a firm.
 Objective function - a linear mathematical
relationship describing an objective of the firm, in
terms of decision variables - this function is to be
maximized or minimized.
 Constraints – requirements or restrictions placed
on the firm by the operating environment, stated in
linear relationships of the decision variables.
 Parameters - numerical coefficients and constants
used in the objective function and constraints.

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Summary of Model Formulation
Steps

Step 1 : Clearly define the decision

variables

Step 2 : Construct the objective

function

Step 3 : Formulate the constraints

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Prentice Hall 2-5
LP Model Formulation
A Maximization Example (1 of 4)

 Product mix problem - Beaver Creek Pottery Company

 How many bowls and mugs should be produced to
maximize profits given labor and materials constraints?
 Product resource requirements and unit profit:

Resource Requirements

Labor Clay Profit

Product
(Hr./Unit) (Lb./Unit) ($/Unit)

Bowl 1 4 40
Mug 2 3 50

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LP Model Formulation
A Maximization Example (2 of 4)

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Prentice Hall 2-7
LP Model Formulation
A Maximization Example (3 of 4)

Resource 40 hrs of labor per day

Availability: 120 lbs of clay
Decision x1 = number of bowls to produce per
day
Variables: x2 = number of mugs to produce per
day
Objective Maximize Z = $40x1 + $50x2
Function: Where Z = profit per day
Resource 1x1 + 2x2  40 hours of labor
Constraints: 4x1 + 3x2  120 pounds of clay

Non-Negativity as Prentice Hall

x  0; x2  0
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1 2-8
LP Model Formulation
A Maximization Example (4 of 4)

Complete Linear Programming Model:

Maximize Z = $40x1 + $50x2

subject to: 1x1 + 2x2  40

4x1 + 3x2  120
x1, x2  0

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Feasible Solutions

A feasible solution does not violate any of

the constraints:

Example: x1 = 5 bowls
x2 = 10 mugs
Z = $40x1 + $50x2 = $700

Labor constraint check: 1(5) + 2(10) = 25

< 40 hours
Clay constraint check: 4(5) + 3(10) = 50
<© 120
Copyright pounds
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Infeasible Solutions

An infeasible solution violates at least

one of the constraints:

Example: x1 = 10 bowls
x2 = 20 mugs
Z = $40x1 + $50x2 = $1400

Labor constraint check: 1(10) + 2(20) = 50

> 40 hours

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Graphical Solution of LP
Models

 Graphical solution is limited to linear

programming models containing only two
decision variables (can be used with three
variables but only with great difficulty).

 Graphical methods provide visualization of how

a solution for a linear programming problem is
obtained.

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Coordinate Axes
Graphical Solution of Maximization
Model

X2 is mugs

Maximize Z = $40x1 +
$50x2
subject to: 1x1 + 2x2 
40
4x1 + 3x2 
120
x1, x 2  0

X1 is bowls
Figure 2.2 Coordinates for
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Prentice Hall Graphical Analysis 2-13
Labor Constraint
Graphical Solution of Maximization
Model

Maximize Z = $40x1 +
$50x2
subject to: 1x1 + 2x2 
40
4x1 + 3x2 
120
x1, x 2  0

Figure 2.3 Graph of Labor

Prentice Hall
Constraint
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2-14
Labor Constraint Area
Graphical Solution of Maximization
Model

Maximize Z = $40x1 +
$50x2
subject to: 1x1 + 2x2 
40
4x1 + 3x2 
120
x1, x 2  0

Figure 2.4 Labor Constraint

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Prentice Hall Area 2-15
Clay Constraint Area
Graphical Solution of Maximization
Model

Maximize Z = $40x1 +
$50x2
subject to: 1x1 + 2x2 
40
4x1 + 3x2 
120
x1, x 2  0

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Figure 2.5 Clay Constraint
Prentice Hall Area 2-16
Both Constraints
Graphical Solution of Maximization
Model

Maximize Z = $40x1 +
$50x2
subject to: 1x1 + 2x2 
40
4x1 + 3x2 
120
x1, x 2  0

Figure 2.6 Graph of Both Model

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Prentice Hall Constraints 2-17
Extreme (Corner) Point Solutions
Graphical Solution of Maximization
Model

Maximize Z = $40x1 +
$50x2
subject to: 1x1 + 2x2 
40
4x1 + 3x2 
120
x1, x 2  0

Figure 2.12 Solutions at All

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Prentice Hall Corner Points 2-18
Optimal Solution
Graphical Solution of Maximization
Model

Maximize Z = $40x1 +
$50x2
subject to: 1x1 + 2x2 
40
4x1 + 3x2 
120
x1, x 2  0

Figure 2.10 Identification of Optimal

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Prentice Hall
Solution Point 2-19
Slack Variables

 Standard form requires that all constraints

be in the form of equations (equalities).
 A slack variable is added to a  constraint
(weak inequality) to convert it to an
equation (=).
 A slack variable typically represents an
unused resource.
 A slack variable contributes nothing to
the objective function value.

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Linear Programming Model:
Standard Form

Max Z = 40x1 + 50x2 + 0s1

+ 0s2
subject to:1x1 + 2x2 + s1 =
40
4x1 + 3x2 + s2 =
120
x1, x 2, s 1, s 2  0
Where:
x1 = number of bowls
x2 = number of mugs
s1, s2 are slack variables
Figure
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Inc. Publishing as Solution Points A, B, and C
Prentice Hall 2-21
LP Model Formulation – Minimization
(1 of 8)
 Two brands of fertilizer available - Super-gro, Crop-
quick.
 Field requires at least 16 pounds of nitrogen and
24 pounds of phosphate.
 Super-gro costs $6 per bag, Crop-quick $3 per bag.
 Problem: How much of each brand to purchase to
minimize total cost of fertilizer
Chemicalgiven following data
Contribution
?
Nitrogen Phosphate
Brand
(lb/ bag) (lb/ bag)
Super-gro 2 4
Crop-quick 4 3

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LP Model Formulation – Minimization
(2 of 8)

Figure 2.15
Fertilizing farmer’s
field

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Prentice Hall 2-23
LP Model Formulation –
Minimization (3 of 8)
Decision Variables:
x1 = bags of Super-gro
x2 = bags of Crop-quick

The Objective Function:

Minimize Z = $6x1 + 3x2
Where: $6x1 = cost of bags of Super-
Gro
$3x2 = cost of bags of Crop-Quick

Model Constraints:
2x1 + 4x2  16 lb (nitrogen constraint)
4x1 + 3x2  24 lb (phosphate constraint)
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Prentice Hall x1, x2  0 (non-negativity constraint) 2-24
Constraint Graph – Minimization
(4 of 8)

Minimize Z = $6x1 + $3x2

subject to: 2x1 + 4x2  16
4x1 + 3x2  24
x1, x 2  0

Figure 2.16 Graph of Both Model

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Prentice Hall Constraints 2-25
Feasible Region– Minimization
(5 of 8)

Minimize Z = $6x1 + $3x2

subject to: 2x1 + 4x2  16
4x1 + 3x2  24
x1, x 2  0

Figure 2.17 Feasible Solution

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Prentice Hall Area 2-26
Optimal Solution Point –
Minimization (6 of 8)

Minimize Z = $6x1 + $3x2

subject to: 2x1 + 4x2  16
4x1 + 3x2  24
x1, x 2  0

Figure 2.18 Optimum

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Prentice Hall Solution Point 2-27
Surplus Variables – Minimization (7
of 8)
 A surplus variable is subtracted from a 
constraint to convert it to an equation (=).
 A surplus variable represents an excess
above a constraint requirement level.
 A surplus variable contributes nothing to
the calculated value of the objective
function.
 Subtracting surplus variables in the farmer
problem constraints:
2x1 + 4x2 - s1 = 16
(nitrogen)
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Graphical Solutions – Minimization
(8 of 8)

Minimize Z = $6x1 + $3x2 + 0s1

+ 0s2
subject to: 2x1 + 4x2 – s1 = 16
4x1 + 3x2 – s2 = 24
x1, x2, s1, s2  0

Figure 2.19 Graph of Fertilizer

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Prentice Hall Example 2-29
Irregular Types of Linear
Programming Problems

For some linear programming models, the

general rules do not apply.

 Special types of problems include those

with:
 Multiple optimal solutions
 Infeasible solutions
 Unbounded solutions

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Multiple Optimal Solutions Beaver
Creek Pottery

The objective function is

parallel to a constraint
line.
Maximize Z=$40x1 + 30x2
subject to: 1x1 + 2x2  40
4x1 + 3x2 
120
x1, x 2  0
Where:
x1 = number of bowls
x2 = number of mugs
Figure 2.20 Example with Multiple
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Prentice Hall Optimal Solutions 2-31
An Infeasible Problem

Every possible solution

violates at least one
constraint:
Maximize Z = 5x1 + 3x2
subject to: 4x1 + 2x2  8
x1  4
x2  6
x1, x 2  0

Figure 2.21 Graph of an Infeasible

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Prentice Hall Problem 2-32
An Unbounded Problem

Value of the objective

function increases
indefinitely:
Maximize Z = 4x1 + 2x2
subject to: x1  4
x2  2
x1, x 2  0

Figure 2.22 Graph of an Unbounded

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Prentice Hall Problem 2-33
Characteristics of Linear
Programming Problems

 A decision amongst alternative courses of action

is required.
 The decision is represented in the model by
decision variables.
 The problem encompasses a goal, expressed as
an objective function, that the decision maker
wants to achieve.
 Restrictions (represented by constraints) exist
that limit the extent of achievement of the
objective.
 The objective and constraints must be definable
by linear mathematical functional relationships.
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Properties of Linear
Programming Models
 Proportionality - The rate of change (slope) of
the objective function and constraint equations is
constant.
 Additivity - Terms in the objective function and
constraint equations must be additive.
 Divisibility -Decision variables can take on any
fractional value and are therefore continuous as
opposed to integer in nature.
 Certainty - Values of all the model parameters
are assumed to be known with certainty (non-
probabilistic).
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Problem Statement
Example Problem No. 1 (1 of 3)

■ Hot dog mixture in 1000-pound batches.

■ Two ingredients, chicken ($3/lb) and beef
($5/lb).
■ Recipe requirements:
at least 500 pounds of
“chicken”
at least 200 pounds of
“beef”
■ Ratio of chicken to beef must be at least 2 to
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Solution
Example Problem No. 1 (2 of 3)
Step 1:
Identify decision variables.
x1 = lb of chicken in mixture
x2 = lb of beef in mixture
Step 2:
Formulate the objective function.
Minimize Z = $3x1 + $5x2
where Z = cost per 1,000-lb batch
$3x1 = cost of chicken
Prentice Hall $5x2 = cost of beef
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2-37
Solution
Example Problem No. 1 (3 of 3)

Step 3:
Establish Model Constraints
x1 + x2 = 1,000 lb
x1  500 lb of chicken
x2  200 lb of beef
x1/x2  2/1 or x1 - 2x2  0
x1, x 2  0
The Model: Minimize Z = $3x1 + 5x2
subject to: x1 + x2 = 1,000 lb
x1  500
x2  200
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as1 - 2x2  0
Prentice Hall 2-38
Example Problem No. 2 (1 of 3)
reading exercise on page 58
Solve the following
model graphically:
Maximize Z = 4x1 + 5x2
subject to: x1 + 2x2 
10
6x1 + 6x2 
36
x1  4
x1, x 2  0

Step 1: Plot the

constraints as equations Figure 2.23 Constraint
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Prentice Hall Equations 2-39
Example Problem No. 2 (2 of 3)

Maximize Z = 4x1 + 5x2

subject to: x1 + 2x2 
10
6x1 + 6x2 
36
x1  4
x1, x 2  0
Step 2: Determine the
feasible solution space

Figure 2.24 Feasible Solution Space and

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Prentice Hall Extreme Points 2-40
Example Problem No. 2 (3 of 3)

Maximize Z = 4x1 + 5x2

subject to: x1 + 2x2 
10
6x1 + 6x2 
36
x1  4
x1, x 2  0
Step 3 and 4:
Determine the solution
points and optimal
solution
Figure 2.25 Optimal Solution
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Prentice Hall Point 2-41