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Linear Programming
Module 2
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Introduction To Linear Programming
Today many of the resources needed asinputs to operations are in limited supply.
Operations managers must understand theimpact of this situation on meeting their
objectives. Linear programming (LP) is one way that
operations managers can determine howbest to allocate their scarce resources.
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A model consisting of linear relationships
representing a firms objective and resourceconstraints
Linear Programming (LP)
LP is a mathematical modeling technique used todetermine a level of operational activity in order toachieve an objective, subject to restrictions called
constraints
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Linear Programming (LP) in OM
There are five common types of decisions inwhich LP may play a role
Product mix
Production plan
Ingredient mix
Transportation
Assignment
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LP Model Formulation Decision variables
Depends on the type of the LP problem
These variables are controllable
The variables can be the quantities of the resources to be
allocated or the number of units to be produced or sold The decision maker has to determine the value of these
variables
Objective function
a linear relationship reflecting the objective of an operation
most frequent objective of business firms is to maximize profit most frequent objective of individual operational units (such as
a production or packaging department) is to minimize cost
Constraint
a linear relationship representing a restriction on decision
making
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LP Problems on Product Mix
Objective
To select the mix of products or services thatresults 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 servicedemanded; Minimum amount of product orservice policy will allow; Maximum amount ofresources available
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LP Problems on Production Plan Objective
To select the mix of products or services thatresults in maximum profits for the planning period
Decision Variables
How much to produce on straight-time labor andovertime labor during each month of the year
ConstraintsAmount of products demanded in each month;Maximum labor and machine capacity available ineach month; Maximum inventory space available
in each month
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Recognizing LP ProblemsCharacteristics of LP Problems
A well-defined single objective must be
stated.There must be alternative courses of action.
The total achievement of the objective mustbe constrained by scarce resources or other
restraints.
The objective and each of the constraintsmust be expressed as linear mathematical
functions.
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Steps in Formulating LP Problems
1. Define the objective. (min or max)
2. Define the decision variables. (positive, binary)3. Write the mathematical function for the objective.
4. Write a 1- or 2-word description of each constraint.
5. Write the right-hand side (RHS) of each constraint.
6. Write for each constraint.
7. Write the decision variables on LHS of each constraint.
8. Write the coefficient for each decision variable in each
constraint.
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Cycle Trends is introducing two new lightweight
bicycle frames, the Deluxe and the Professional, to be
made from aluminum and steel alloys. The anticipatedunit 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 supplierdelivers 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?
Example: LP Formulation
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Aluminum Alloy Steel Alloy
Deluxe 2 3Professional 2 4
Pounds of each alloy needed per frame
Example: LP Formulation
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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 producedweekly
Write the mathematical objective function
Max Z = 10x1 + 15x2
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Example: LP FormulationWrite a one- or two-word description of each
constraint
Aluminum availableSteel 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 constraintx1 x2 < 100x1 x2 < 80
Write the coefficient for each decision in eachconstraint
+ 2x1 + 4x2 < 100
+ 3x1 + 2x2 < 80
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Example: LP Formulation
LP in Final Form
Max Z = 10x1 + 15x2Subject To
2x1 + 4x2 < 100 ( aluminum constraint)
3x1 + 2x2 < 80 ( steel constraint)x1 , x2 > 0 (non-negativity constraints)
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LP Formulation: Example
Maximize Z= 40 x1 + 50 x2
Subject to
x1 + 2x2 40 hr (labor constraint)
4x1 + 3x2 120 lb (clay constraint)
x1 , x2 0
Solution is x1 = 24 bowls x2 = 8 mugs
Revenue = 1,360
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Montana Wood Products manufacturers two-high quality products, tables and chairs. Its profit is
Rs15 per chair and Rs 21 per table. Weekly production
is constrained by available labor and wood. Each chair
requires 4 labor hours and 8 board feet of wood whileeach table requires 3 labor hours and 12 board feet of
wood. Available wood is 2400 board feet and available
labor is 920 hours. Management also requires at least
40 tables and at least 4 chairs be produced for every
table produced. To maximize profits, how many chairs
and tables should be produced?
Example: LP Formulation
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Example: LP Formulation
Define the objective Maximize total weekly profit
Define the decision variables
x1 = number of chairs produced weekly x2 = number of tables produced weekly
Write the mathematical objective function
Max Z = 15x1
+ 21x2
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Example: LP Formulation
Write a one- or two-word description of each constraint
Labor hours available
Board feet available
At least 40 tables
At least 4 chairs for every table
Write the right-hand side of each constraint
920 2400
40
4 to 1 ratio
Write for each constraint
< 920 < 2400
> 40
4 to 1
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Example: LP Formulation
Write all the decision variables on the left-hand side of eachconstraint
x1 x2 < 920
x1 x2 < 2400
x2 > 40
4 to 1 ratio x1 / x2 4/1
Write the coefficient for each decision in each constraint
+ 4x1 + 3x2 < 920 + 8x1 + 12x2 < 2400
x2 > 40
x1 4 x2
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The Sureset Concrete Company producesconcrete. Two ingredients in concrete are sand (costs
Rs 6 per ton) and gravel (costs Rs 8 per ton). Sand and
gravel together must make up exactly 75% of the
weight of the concrete. Also, no more than 40% of theconcrete can be sand and at least 30% of the concrete
be gravel. Each day 2000 tons of concrete are
produced. To minimize costs, how many tons of gravel
and sand should be purchased each day?
Example: LP Formulation
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Example: LP Formulation
Define the objective Minimize daily costs
Define the decision variables
x1 = tons of sand purchased x2 = tons of gravel purchased
Write the mathematical objective function
Min Z = 6x1
+ 8x2
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Example: LP Formulation
Write a one- or two-word description of each constraint 75% must be sand and gravel
No more than 40% must be sand
At least 30% must be gravel
Write the right-hand side of each constraint
.75(2000)
.40(2000)
.30(2000)Write for each constraint
= 1500
< 800
> 600
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Example: LP Formulation
Write all the decision variables on the left-hand sideof each constraint
x1 x2 = 1500
x1
< 800
x2 > 600
Write the coefficient for each decision in each constraint
+ x1 + x2 = 1500
+ x1 < 800
x2 > 600
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Example: LP Formulation
LP in Final Form Min Z = 6x1 + 8x2
Subject To
x1 + x2 = 1500 ( mix constraint) x1 < 800 ( mix constraint)
x2 > 600 ( mix constraint )
x1
, x2
> 0 (non-negativity constraints)
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LP Problem
Galaxy Ind. produces two water guns, the Space Rayand the Zapper. Galaxy earns a profit of Rs3 for every
Space Ray and Rs2 for every Zapper. Space Rays and
Zappers require 2 and 4 production minutes per unit,
respectively. Also, Space Rays and Zappers require .5and .3 pounds of plastic, respectively. Given
constraints of 40 production hours, 1200 pounds of
plastic, Space Ray production cant exceed Zapper
production by more than 450 units; formulate theproblem such that Galaxy maximizes profit.
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R = # of Space Rays to produceZ = # of Zappers to produce
Max Z = 3.00R + 2.00Z
ST
2R + 4Z 2400 cant exceed available hours (40*60)
.5R + .3Z 1200 cant exceed available plastic
R - S 450 Space Rays cant exceed Zappers by more
than 450
R, S 0 non-negativity constraint
LP Model
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Graphical Solution Method
1. Plot model constraint on a set of coordinatesin a plane
2. Identify the feasible solution space on thegraph where all constraints are satisfied
simultaneously
3. Plot objective function to find the point onboundary of this space that maximizes (or
minimizes) value of objective function
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Graphical Solution: Example
4 x1 + 3 x2 120 lb
x1 + 2 x2 40 hr
Area common toboth constraints
50
40
30
20
10
0 |10
|60
|50
|20
|30
|40 x1
x2
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Computing Optimal Values x1 + 2x2 = 404x1 + 3x2 = 120
4x1 + 8x2 = 160-4x1 - 3x2 = -120
5x2 = 40
x2 = 8
x1 + 2(8) = 40
x1 = 24
4 x1 + 3 x2 120 lb
x1 + 2 x2 40 hr
40
30
20
10
0 |10
|20
|30
|40
x1
x2
Z= 50(24) + 50(8) = 1,360
248
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Extreme Corner Points
x1
= 224 bowls
x2 = 8 mugs
Z= 1,360 x1 = 30 bowls
x2 = 0 mugs
Z= 1,200
x1 = 0 bowls
x2 = 20 mugs
Z= 1,000
A
B
C|
20
|
30
|
40
|
10 x1
x2
40
30
20
10
0
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4x1 + 3x2 120 lb
x1 + 2x2 40 hr
40
30
20
10
0
B
|10
|20
|30
|40
x1
x2
C
A
Z= 70x1 + 20x2Optimal point:x1 = 30 bowls
x2 =0 mugs
Z= $2,100
Objective Function
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Minimization Problem
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 given following data ?
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Minimize Z= $6x1 + $3x2
subject to 2x1 + 4x2 16 lb of nitrogen
4x1 + 3x2 24 lb of phosphate
x1, x2 0
CHEMICAL CONTRIBUTION
Brand Nitrogen (lb/bag) Phosphate (lb/bag)
Gro-plus 2 4Crop-fast 4 3
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Graphical method
Complete model formulation:
minimize Z = $6x1 + 3x2
subject to
2x1 + 4x2 16 lb of
nitrogen
4x1 + 3x2 24 lb of
phosphatex1, x2 0
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A Minimization Model Example
Feasible Solution Area
minimize Z = $6x1 + 3x2
subject to
2x1 + 4x2 16 lb of nitrogen
4x1 + 3x2 24 lb of phosphate
x1, x2 0
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A Minimization Model Example
Optimal Solution Point
minimize Z = $6x1 + 3x2subject to
2x1 + 4x2 16 lb of nitrogen
4x1 + 3x2 24 lb of phosphate
x1, x2
0
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Irregular Types of Linear Programming
Problems
For some linear programming models, the general
rules do not apply. Special types of problems include those with:
1. Multiple optimal solutions
2. Infeasible solutions
3. Unbounded solutions
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Multiple Optimal Solutions
Objective function is parallel to a
constraint line:
maximize Z=$40x1 + 30x2
subject to
1x1 + 2x2 40 hours of labor
4x2
+ 3x2
120 pounds of clay
x1, x2 0
where x1 = number of bowls
x2 = number of mugs
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An Infeasible Problem
Every possible solution violates
at least one constraint:
maximize Z = 5x1 + 3x2
subject to
4x1 + 2x2
8x1 4
x2 6
x1, x2 0
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An Unbounded Problem
Value of objective function
increases indefinitely:
maximize Z = 4x1 + 2x2
subject to
x1 4
x2 2
x1, x2 0