Slides prepared
by John Loucks
ã 2002 South-Western/Thomson Learning TM
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Chapter 8
Strategic Allocation
of Resources
2
Overview
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Introduction
Recognizing Linear Programming (LP) Problems
Formulating LP Problems
Solving LP Problems
Real LP Problems
Interpreting Computer Solutions of LP Problems
Wrap-Up: What World-Class Companies Do
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Introduction
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Today many of the resources needed as inputs to
operations are in limited supply.
Operations managers must understand the impact of
this situation on meeting their objectives.
Linear programming (LP) is one way that operations
managers can determine how best to allocate their
scarce resources.
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Linear Programming (LP) in OM
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There are five common types of decisions in which
LP may play a role
Product mix
Ingredient mix
Transportation
Production plan
Assignment
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LP Problems in OM: Product Mix
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Objective
To select the mix of products or services that results
in maximum profits for the planning period
Decision Variables
How much to produce and market of each product or
service for the planning period
Constraints
Maximum amount of each product or service
demanded; Minimum amount of product or service
policy will allow; Maximum amount of resources
available
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LP Problems in OM: Ingredient Mix
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Objective
To select the mix of ingredients going into products
that results in minimum operating costs for the
planning period
Decision Variables
How much of each ingredient to use in the planning
period
Constraints
Amount of products demanded; Relationship
between ingredients and products; Maximum amount
of ingredients and production capacity available
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LP Problems in OM: Transportation
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Objective
To select the distribution plan from sources to
destinations that results in minimum shipping costs
for the planning period
Decision Variables
How much product to ship from each source to each
destination for the planning period
Constraints
Minimum or exact amount of products needed at each
destination; Maximum or exact amount of products
available at each source
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LP Problems in OM: Production Plan
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Objective
To select the mix of products or services that results
in maximum profits for the planning period
Decision Variables
How much to produce on straight-time labor and
overtime labor during each month of the year
Constraints
Amount of products demanded in each month;
Maximum labor and machine capacity available in
each month; Maximum inventory space available in
each month
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LP Problems in OM: Assignment
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Objective
To assign projects to teams so that the total cost for
all projects is minimized during the planning period
Decision Variables
To which team is each project assigned
Constraints
Each project must be assigned to a team; Each team
must be assigned a project
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Recognizing LP Problems
Characteristics of LP Problems in OM
 A well-defined single objective must be stated.
 There must be alternative courses of action.
 The total achievement of the objective must be
constrained by scarce resources or other restraints.
 The objective and each of the constraints must be
expressed as linear mathematical functions.
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Steps in Formulating LP Problems
1.
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8.
Define the objective.
Define the decision variables.
Write the mathematical function for the objective.
Write a 1- or 2-word description of each constraint.
Write the right-hand side (RHS) of each constraint.
Write <, =, or > for each constraint.
Write the decision variables on LHS of each constraint.
Write the coefficient for each decision variable in each
constraint.
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Example: LP Formulation
Cycle Trends is introducing two new lightweight
bicycle frames, the Deluxe and the Professional, to be
made from aluminum and steel alloys. The anticipated
unit profits are $10 for the Deluxe and $15 for the
Professional.
The number of pounds of each alloy needed per
frame is summarized on the next slide. A supplier
delivers 100 pounds of the aluminum alloy and 80
pounds of the steel alloy weekly. How many Deluxe
and Professional frames should Cycle Trends produce
each week?
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Example: LP Formulation
Deluxe
Professional
Aluminum Alloy
2
4
Steel Alloy
3
2
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Example: LP Formulation
Define the objective
Maximize total weekly profit
Define the decision variables
x1 = number of Deluxe frames produced weekly
x2 = number of Professional frames produced
weekly
Write the mathematical objective function
Max Z = 10x1 + 15x2
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Example: LP Formulation
Write a one- or two-word description of each constraint
Aluminum available
Steel available
Write the right-hand side of each constraint
100
80
Write <, =, > for each constraint
< 100
< 80
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Example: LP Formulation
Write all the decision variables on the left-hand side
of each constraint
x1 x2 < 100
x1 x2 < 80
Write the coefficient for each decision in each constraint
+ 2x1 + 4x2 < 100
+ 3x1 + 2x2 < 80
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LP Problems in General
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Units of each term in a constraint must be the same as
the RHS
Units of each term in the objective function must be
the same as Z
Units between constraints do not have to be the same
LP problem can have a mixture of constraint types
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Solving LP Problems
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Graphical Solution Approach - used mainly as a
teaching tool
Simplex Method - most common analytic tool
Transportation Method - one of the earliest methods
Assignment Method - occasionally used in OM
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Graphical Solution Approach
1. Formulate the objective and constraint functions.
2. Draw the graph with one variable on the horizontal
axis and one on the vertical axis.
3. Plot each of the constraints as if they were lines or
equalities.
4. Outline the feasible solution space.
. . . more
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Graphic Solution Approach
5. Circle the potential solution points -- intersections on
the inner (minimization) or outer (maximization)
perimeter of the feasible solution space.
6. Substitute each of the potential solution point values
of the two decision variables into the objective
function and solve for Z.
7. Select the solution point that optimizes Z.
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Example: Graphical Solution Approach
x2
3x1 + 2x2 < 80 (Steel)
Potential Solution Point
x1 = 0, x2 = 25, Z = $375.00
40
35
Optimal Solution Point
x1 = 15, x2 = 17.5, Z = $412.50
30
25
2x1 + 4x2 < 100 (Aluminum)
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15
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Feasible
Solution
Space
MAX 10x1 + 15x2 (Profit)
Potential Solution Point
x1 = 26.67, x2 = 0, Z = $266.67
5
5
10
15
20
25
30
35
40
45
50
x1
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Simplex Method
Examples of standard LP computer programs that use
the simplex method are:
 IBM’s Optimization Solutions Library
 POM Software Library
 GAMS
 MPL for Windows
 Solver -- available within spreadsheet packages such
as MicrosoftExcel, Lotus 1-2-3, and Quattro Pro
 What’s Best! and Premium Solver -- add-ons to
common spreadsheet packages
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Example: Excel/Solver Solution
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Partial Spreadsheet Showing Problem Data
A
1
2 Material
3 Aluminum
4
Steel
5 Profit/Bike
B
C
Material Requirements
Deluxe
Profess.
2
4
3
2
10
15
D
Amount
Available
100
80
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Example: Excel/Solver Solution
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Partial Spreadsheet Showing Formulas
A
B
C
Decision Variables
Deluxe
Professional
D
6
7
8
Bikes Made
9
10
=B5*C10+C5*D10
Maximized Total Profit
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12 Constraints
Amount Used
Amount Avail.
13 Aluminum
=B3*B8+C3*C8
<=
100
14 Steel
=B4*B8+C4*C8
<=
80
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Example: Excel/Solver Solution
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Partial Spreadsheet Showing Solution
A
6
7
8
9
10
11
12
13
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Bikes Made
B
C
Decision Variables
Deluxe
Professional
15
17.500
Maximized Total Profit
Constraints
Aluminum
Steel
Amount Used
100
80
D
412.500
<=
<=
Amount Avail.
100
80
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Transportation Method
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This method can solve a special form of LP problem,
including the classical transportation problem, with
these typical characteristics:
m sources and n destinations
number of variables is m x n
number of constraints is m + n (constraints are for
source capacity and destination demand)
costs appear only in objective function (objective
is to minimize total cost of shipping)
coefficients of decision variables in the constraints
are either 0 or 1
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Example: Transportation LP
National Packaging has plants in Tulsa, Memphis,
and Detroit that ship to the firm’s warehouses in San
Diego, Norfolk, and Pensacola. The three
warehouses require at least 4,000, 2,500, and 2,500
pounds of cardboard per week, respectively. The
plants each have 3,000 pounds of cardboard per week
available for shipment.
The shipping cost per pound from each plant to
each warehouse is shown on the next slide.
Formulate and solve an LP to determine the shipping
arrangements (source, destination, and quantity) that
will minimize the total weekly shipping cost.
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Example: Transportation LP
Source
Tulsa
Memphis
Detroit
Destination
San Diego Norfolk Pensacola
$12
20
30
$6
11
26
$5
9
28
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Example: Transportation LP
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Define the objective
Minimize the total weekly shipping cost
Define the decision variables
There are m x n = 3 x 3 = 9 decision variables.
xij represents the number of pounds of cardboard
to be shipped from plant i to warehouse j.
Tulsa
Memphis
Detroit
San Diego Norfolk Pensacola
x11
x12
x13
x21
x22
x23
x31
x32
x33
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Example: Transportation LP
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Write the mathematical function for the objective
Min Z = 12x11 + 6x12 + 5x13 + 20x21 + 11x22
+ 9x23 + 30x31 + 26x32 + 28x33
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Example: Transportation LP
Write the constraints
(1) x11 + x12 + x13 < 3000
(2) x21 + x22 + x23 < 3000
(3) x31 + x32 + x33 < 3000
(4) x11 + x21 + x31 > 4000
(5) x12 + x22 + x32 > 2500
(6) x13 + x23 + x33 > 2500
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(Plant 1 capacity in pounds)
(Plant 2 capacity in pounds)
(Plant 3 capacity in pounds)
(Wareh. 1 demand in pounds)
(Wareh. 2 demand in pounds)
(Wareh. 3 demand in pounds)
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Assignment Method
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This method can solve a special form of LP problem,
including the classical assignment problem, with
these typical characteristics:
is a special case of a transportation problem
the right-hand sides of constraints are all 1
the signs of the constraints are = rather than < or >
the value of all decision variables is either 0 or 1
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Example: Assignment LP
A sprinkler system installation company has three
residential projects to complete and three work teams
available and capable of completing the projects. The
work teams are not equally efficient at completing a
particular project.
Shown on the next slide are the estimated laborhours required for each team to complete each
project. How should the work teams be assigned in
order to minimize total labor-hours?
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Example: Assignment LP
Work Team
Alice, Ted
Gary, Marv
Tina, Sam
Projects
A B C
28 30 18
35 32 20
25 25 14
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Real LP Problems
Real-world LP problems often involve:
 Hundreds or thousands of constraints
 Large quantities of data
 Many products and/or services
 Many time periods
 Numerous decision alternatives
 … and other complications
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Computer Solutions of LP Problems
Computer output typically contains:
 Original-problem formulation
 Basis variables (variables in the final solution) and
their values
 Slack-variable values -- slack represents the amount
of a scarce resource that is not used by the decision
variables
 Shadow prices -- which indicate the impact on Z if
the constraints’ right-hand sides are changed
… more
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Computer Solutions of LP Problems
Computer output typically contains:
 Ranges of right-hand side values for which the
shadow prices are valid
 For each nonbasic variable: amount Z is reduced (in a
MAX problem) or increased (in a MIN problem) for
one unit of x in the solution
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Wrap-Up: World-Class Practice
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Managers at all levels use LP to solve complex
problems and aid in decision making
Some companies establish formal operations
research, management science, or operations
analysis departments
Other companies employ consultants
LP is applied to long, medium, and short range
decision making
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End of Chapter 8
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