LINEAR OPTIMIZATION

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LINEAR OPTIMIZATIONLINEAR OPTIMIZATION
BYBY
Er. ASHISH BANSODEEr. ASHISH BANSODE
M.E. (WATER RESOURCES ENGINEERING)M.E. (WATER RESOURCES ENGINEERING)

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Introduction To Linear ProgrammingIntroduction To Linear Programming
Today many of the resources needed as inputs toToday many of the resources needed as inputs to
operations are in limited supply.operations are in limited supply.
Operations managers must understand the impact ofOperations managers must understand the impact of
this situation on meeting their objectives.this situation on meeting their objectives.
Linear programming (LP) is one way that operationsLinear programming (LP) is one way that operations
managers can determine how best to allocate theirmanagers can determine how best to allocate their
scarce resourcesscarce resources..

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Linear Programming (LP) in OMLinear Programming (LP) in OM
There are five common types of decisions inThere are five common types of decisions in
which LP may play a rolewhich LP may play a role
Product mixProduct mix
Production planProduction plan
Ingredient mixIngredient mix
TransportationTransportation
AssignmentAssignment

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LP Problems in OM: Product MixLP Problems in OM: Product Mix
ObjectiveObjective
To select the mix of products or services that resultsTo select the mix of products or services that results
in maximum profits for the planning periodin maximum profits for the planning period
Decision VariablesDecision Variables
How much to produce and market of each product orHow much to produce and market of each product or
service for the planning periodservice for the planning period
ConstraintsConstraints
Maximum amount of each product or serviceMaximum amount of each product or service
demanded; Minimum amount of product or servicedemanded; Minimum amount of product or service
policy will allow; Maximum amount of resourcespolicy will allow; Maximum amount of resources
availableavailable

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LP Problems in OM: Production PlanLP Problems in OM: Production Plan
ObjectiveObjective
To select the mix of products or services that resultsTo select the mix of products or services that results
in maximum profits for the planning periodin maximum profits for the planning period
Decision VariablesDecision Variables
How much to produce on straight-time labor andHow much to produce on straight-time labor and
overtime labor during each month of the yearovertime labor during each month of the year
ConstraintsConstraints
Amount of products demanded in each month;Amount of products demanded in each month;
Maximum labor and machine capacity available inMaximum labor and machine capacity available in
each month; Maximum inventory space available ineach month; Maximum inventory space available in
each montheach month

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Recognizing LP ProblemsRecognizing LP Problems
Characteristics of LP Problems in OMCharacteristics of LP Problems in OM
A well-defined single objective must be stated.A well-defined single objective must be stated.
There must be alternative courses of action.There must be alternative courses of action.
The total achievement of the objective must beThe total achievement of the objective must be
constrained by scarce resources or other restraints.constrained by scarce resources or other restraints.
The objective and each of the constraints must beThe objective and each of the constraints must be
expressed as linear mathematical functions.expressed as linear mathematical functions.

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Steps in Formulating LP ProblemsSteps in Formulating LP Problems
1.1. Define the objective. (min or max)Define the objective. (min or max)
2.2. Define the decision variables. (positive, binary)Define the decision variables. (positive, binary)
3.3. Write the mathematical function for the objective.Write the mathematical function for the objective.
4.4. Write a 1- or 2-word description of each constraint.Write a 1- or 2-word description of each constraint.
5.5. Write the right-hand side (RHS) of each constraint.Write the right-hand side (RHS) of each constraint.
6.6. WriteWrite <<, =, or, =, or >> for each constraint.for each constraint.
7.7. Write the decision variables on LHS of each constraint.Write the decision variables on LHS of each constraint.
8.8. Write the coefficient for each decision variable in eachWrite the coefficient for each decision variable in each
constraint.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 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?
Example: LP FormulationExample: LP Formulation

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Aluminum AlloyAluminum Alloy Steel AlloySteel Alloy
DeluxeDeluxe 2 32 3
ProfessionalProfessional 44 22
Pounds of each alloy needed per
frame

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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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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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Write all the decision variables on the left-hand sideWrite all the decision variables on the left-hand side
of each constraintof 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 in Final FormLP in Final Form
Max Z = 10x1 + 15x2
Subject To
2x1 + 4x2 < 100 ( aluminum constraint)
3x1 + 2x2 < 80 ( steel constraint)
x1 , x2 >> 0 (non-negativity constraints)

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Montana Wood Products manufacturers two-
high quality products, tables and chairs. Its profit is
$15 per chair and $21 per table. Weekly production is
constrained by available labor and wood. Each chair
requires 4 labor hours and 8 board feet of wood while
each 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?

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Maximize total weekly profit
x1 = number of chairs produced weekly
x2 = number of tables produced weekly
Max Z = 15x1 + 21x2

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Labor hours available
Board feet available
At least 40 tables
At least 4 chairs for every table
920
2400
40
4 to 1 ratio
< 920
< 2400
>> 4040
4 to 11

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Write all the decision variables on the left-hand side of eachWrite all the decision variables on the left-hand side of each
constraintconstraint
x1 x2 < 920
x1 x2 < 2400
x2 >> 4040
4 to 1 ratio  x1 / x2 ≥ 4/1
Write the coefficient for each decision in each constraint
+ 4x1 + 3x2 < 920
+ 8x1 + 12x2 < 2400
x2 >> 4040
x1≥4 x2

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Max Z = 15x1 + 21x2
Subject To
4x1 + 3x2 < 920 ( labor constraint)
8x1 + 12x2 < 2400 ( wood constraint)
x2 - 40 >> 0 (make at least 40 tables)0 (make at least 40 tables)
x1 - 4 x2 >> 0 (at least 4 chairs for every table)0 (at least 4 chairs for every table)

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The Sureset Concrete Company produces
concrete. Two ingredients in concrete are sand (costs
$6 per ton) and gravel (costs $8 per ton). Sand and
gravel together must make up exactly 75% of the
weight of the concrete. Also, no more than 40% of the
concrete 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?

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Minimize daily costs
x1 = tons of sand purchased
x2 = tons of gravel purchased
Min Z = 6x1 + 8x2

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75% must be sand and gravel
No more than 40% must be sand
At least 30% must be gravel
.75(2000)
.40(2000)
.30(2000)
= 1500
< 800800
>> 600600

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Write all the decision variables on the left-hand sideWrite all the decision variables on the left-hand side
of each constraintof each constraint
x1 x2 = 1500
x1< 800800
x2 >> 600600
Write the coefficient for each decision in each
constraint
+ x1 + x2 = 1500
+ x1 < 800
x2 >> 600600

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Min Z = 6x1 + 8x2
Subject To
x1 + x2 = 1500 ( mix constraint)
x1 < 800 ( mix constraint)
x2 >> 600 ( mix constraint )600 ( mix constraint )

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LP Problems in GeneralLP Problems in General
Units of each term in a constraint must be the same asUnits of each term in a constraint must be the same as
the RHSthe RHS
Units of each term in the objective function must beUnits of each term in the objective function must be
the same as Zthe same as Z
Units between constraints do not have to be the sameUnits between constraints do not have to be the same
LP problem can have a mixture of constraint typesLP problem can have a mixture of constraint types

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LP ProblemLP Problem
Galaxy Ind. produces two water guns, the Space RayGalaxy Ind. produces two water guns, the Space Ray
and the Zapper. Galaxy earns a profit of $3 for everyand the Zapper. Galaxy earns a profit of $3 for every
Space Ray and $2 for every Zapper. Space Rays andSpace Ray and $2 for every Zapper. Space Rays and
Zappers require 2 and 4 production minutes per unit,Zappers require 2 and 4 production minutes per unit,
respectively. Also, Space Rays and Zappers require .5respectively. Also, Space Rays and Zappers require .5
and .3 pounds of plastic, respectively. Givenand .3 pounds of plastic, respectively. Given
constraints of 40 production hours, 1200 pounds ofconstraints of 40 production hours, 1200 pounds of
plastic, Space Ray production can’t exceed Zapperplastic, Space Ray production can’t exceed Zapper
production by more than 450 units; formulate theproduction by more than 450 units; formulate the
problem such that Galaxy maximizes profit.problem such that Galaxy maximizes profit.

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R = # of Space Rays to produceR = # of Space Rays to produce
Z = # of Zappers to produceZ = # of Zappers to produce
Max Z = 3.00R + 2.00ZMax Z = 3.00R + 2.00Z
STST
2R + 4Z2R + 4Z ≤ 2400≤ 2400 can’t exceed available hours (40*60)can’t exceed available hours (40*60)
.5R + .3Z ≤ 1200.5R + .3Z ≤ 1200 can’t exceed available plasticcan’t exceed available plastic
R - S ≤ 450R - S ≤ 450 Space Rays can’t exceed Zappers by moreSpace Rays can’t exceed Zappers by more
than 450than 450
R, S ≥ 0R, S ≥ 0 non-negativity constraintnon-negativity constraint
LP ModelLP Model

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The White Horse Apple Products Company purchases applesThe White Horse Apple Products Company purchases apples
from local growers and makes applesauce and apple juice. Itfrom local growers and makes applesauce and apple juice. It
costs $0.60 to produce a jar of applesauce and $0.85 tocosts $0.60 to produce a jar of applesauce and $0.85 to
produce a bottle of apple juice. The company has a policy thatproduce a bottle of apple juice. The company has a policy that
at least 30% but not more than 60% of its output must beat least 30% but not more than 60% of its output must be
applesauce.applesauce.
The company wants to meet but not exceed demand forThe company wants to meet but not exceed demand for
each product. The marketing manager estimates that theeach product. The marketing manager estimates that the
maximum demand for applesauce is 5,000 jars, plus anmaximum demand for applesauce is 5,000 jars, plus an
additional 3 jars for each $1 spent on advertising. Maximumadditional 3 jars for each $1 spent on advertising. Maximum
demand for apple juice is estimated to be 4,000 bottles, plus andemand for apple juice is estimated to be 4,000 bottles, plus an
additional 5 bottles for every $1 spent to promote apple juice.additional 5 bottles for every $1 spent to promote apple juice.
The company has $16,000 to spend on producing andThe company has $16,000 to spend on producing and
advertising applesauce and apple juice. Applesauce sells foradvertising applesauce and apple juice. Applesauce sells for
$1.45 per jar; apple juice sells for $1.75 per bottle. The$1.45 per jar; apple juice sells for $1.75 per bottle. The
company wants to know how many units of each to producecompany wants to know how many units of each to produce
and how much advertising to spend on each in order toand how much advertising to spend on each in order to
maximize profit.maximize profit.

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S = # jars apple Sauce to makeS = # jars apple Sauce to make
J = # bottles apple Juice to makeJ = # bottles apple Juice to make
SA = $ for apple Sauce AdvertisingSA = $ for apple Sauce Advertising
JA = $ for apple Juice AdvertisingJA = $ for apple Juice Advertising
Max Z = 1.45S + 1.75J - .6S - .85J – SA – JAMax Z = 1.45S + 1.75J - .6S - .85J – SA – JA
STST
SS ≥ .3(S + J)≥ .3(S + J) at least 30% apple sauceat least 30% apple sauce
S ≤ .6(S + J)S ≤ .6(S + J) no more than 60% apple sauceno more than 60% apple sauce
S ≤ 5000 + 3SAS ≤ 5000 + 3SA don’t exceed demand for apple saucedon’t exceed demand for apple sauce
J ≤ 4000 + 5JAJ ≤ 4000 + 5JA don’t exceed demand for apple juicedon’t exceed demand for apple juice
.6S + .85J + SA + JA ≤ 16000.6S + .85J + SA + JA ≤ 16000 budgetbudget
LP ModelLP Model

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A ship has two cargo holds, one fore and one aft. The fore cargo hold hasA ship has two cargo holds, one fore and one aft. The fore cargo hold has
a weight capacity of 70,000 pounds and a volume capacity of 30,000 cubica weight capacity of 70,000 pounds and a volume capacity of 30,000 cubic
feet. The aft hold has a weight capacity of 90,000 pounds and a volumefeet. The aft hold has a weight capacity of 90,000 pounds and a volume
capacity of 40,000 cubic feet. The shipowner has contracted to carry loadscapacity of 40,000 cubic feet. The shipowner has contracted to carry loads
of packaged beef and grain. The total weight of the available beef isof packaged beef and grain. The total weight of the available beef is
85,000 pounds; the total weight of the available grain is 100,000 pounds.85,000 pounds; the total weight of the available grain is 100,000 pounds.
The volume per mass of the beef is 0.2 cubic foot per pound, and theThe volume per mass of the beef is 0.2 cubic foot per pound, and the
volume per mass of the grain is 0.4 cubic foot per pound. The profit forvolume per mass of the grain is 0.4 cubic foot per pound. The profit for
shipping beef is $0.35 per pound, and the profit for shipping grain is $0.12shipping beef is $0.35 per pound, and the profit for shipping grain is $0.12
per pound. The shipowner is free to accept all or part of the availableper pound. The shipowner is free to accept all or part of the available
cargo; he wants to know how much meat and grain to accept in order tocargo; he wants to know how much meat and grain to accept in order to
maximize profit.maximize profit.

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BF = # lbs beef to load in fore cargo holdBF = # lbs beef to load in fore cargo hold
BA = # lbs beef to load in aft cargo holdBA = # lbs beef to load in aft cargo hold
GF = # lbs grain to load in fore cargo holdGF = # lbs grain to load in fore cargo hold
GA = # lbs grain to load in aft cargo holdGA = # lbs grain to load in aft cargo hold
Max Z = .35 BF + .35BA + .12GF + .12 GAMax Z = .35 BF + .35BA + .12GF + .12 GA
STST
BF + GFBF + GF ≤ 70000≤ 70000 fore weight capacity – lbsfore weight capacity – lbs
BA + GA ≤ 90000BA + GA ≤ 90000 aft weight capacity – lbsaft weight capacity – lbs
.2BF + .4GF ≤ 30000.2BF + .4GF ≤ 30000 for volume capacity – cubic feetfor volume capacity – cubic feet
.2BA + .4GA ≤ 40000.2BA + .4GA ≤ 40000 for volume capacity – cubic feetfor volume capacity – cubic feet
BF + BA ≤ 85000BF + BA ≤ 85000 max beef availablemax beef available
GF + GA ≤ 100000GF + GA ≤ 100000 max grain availablemax grain available
LP ModelLP Model

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In the summer, the City of Sunset Beach staffs lifeguardIn the summer, the City of Sunset Beach staffs lifeguard
stations seven days a week. Regulations require that citystations seven days a week. Regulations require that city
employees (including lifeguards) work five days a week andemployees (including lifeguards) work five days a week and
be given two consecutive days off. Insurance requirementsbe given two consecutive days off. Insurance requirements
mandate that Sunset Beach provide at least one lifeguard permandate that Sunset Beach provide at least one lifeguard per
8000 average daily attendance on any given day. The average8000 average daily attendance on any given day. The average
daily attendance figures by day are as follows: Sunday –daily attendance figures by day are as follows: Sunday –
58,000, Monday – 42,000, Tuesday – 35,000, Wednesday –58,000, Monday – 42,000, Tuesday – 35,000, Wednesday –
25,000, Thursday – 44,000, Friday – 51,000 and Saturday –25,000, Thursday – 44,000, Friday – 51,000 and Saturday –
68,000. Given a tight budget constraint, the city would like to68,000. Given a tight budget constraint, the city would like to
determine a schedule that will employ as few lifeguards asdetermine a schedule that will employ as few lifeguards as
possible.possible.

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X1 = number of lifeguards scheduled to begin on SundayX1 = number of lifeguards scheduled to begin on Sunday
X2 =X2 = ““ ““ ““ ““ “ Monday“ Monday
X3 =X3 = ““ ““ ““ ““ “ Tuesday“ Tuesday
X4 =X4 = ““ ““ ““ ““ “ Wednesday“ Wednesday
X5 =X5 = ““ ““ ““ ““ “ Thursday“ Thursday
X6 =X6 = ““ ““ ““ ““ “ Friday“ Friday
X7 =X7 = ““ ““ ““ ““ “ Saturday“ Saturday
LP ModelLP Model

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Min X1 + X2 + X3 + X4 + X5 + X6 + X7Min X1 + X2 + X3 + X4 + X5 + X6 + X7
STST
X1 + X4 + X5 + X6 +X7 ≥ 8 (Sunday)X1 + X4 + X5 + X6 +X7 ≥ 8 (Sunday)
X1 + X2 + X5 + X6 +X7 ≥ 6 (Monday)X1 + X2 + X5 + X6 +X7 ≥ 6 (Monday)
X1 + X2 + X3 + X6 +X7 ≥ 5 (Tuesday)X1 + X2 + X3 + X6 +X7 ≥ 5 (Tuesday)
X1 + X2 + X3 + X4 +X7 ≥ 4 (Wednesday)X1 + X2 + X3 + X4 +X7 ≥ 4 (Wednesday)
X1 + X2 + X3 + X4 +X5 ≥ 6 (Thursday)X1 + X2 + X3 + X4 +X5 ≥ 6 (Thursday)
X2 + X3 + X4 + X5 +X6 ≥ 7 (Friday)X2 + X3 + X4 + X5 +X6 ≥ 7 (Friday)
X3 + X4 + X5 + X6 +X7 ≥ 9 (Saturday)X3 + X4 + X5 + X6 +X7 ≥ 9 (Saturday)
All variables ≥ 0 and integerAll variables ≥ 0 and integer
LP ModelLP Model

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