LP Formulation Ex

7/17/2019 LP Formulation Ex

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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 tooperations 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..

OT!" Linear Programming is presented inOT!" Linear Programming is presented in#upplement $ for %hapter 11. &e will focus on#upplement $ for %hapter 11. &e will focus on

formulation in this class.formulation in this class.


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Linear Programming (LP) in OMLinear Programming (LP) in OM

There are five common types of decisions in whichThere are five common types of decisions in whichLP may play a roleLP may play a role

Product miProduct mi

Production planProduction plan $ngredient mi$ngredient mi

TransportationTransportation

ssignmentssignment


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LP Problems in OM: Product MixLP Problems in OM: Product Mix

ObjectiveO

bjectiveTo select the mi of products or services that resultsTo select the mi of products or services that resultsin maimum profits for the planning periodin maimum profits for the planning period

+ecision ,ariables+ecision ,ariables

-ow much to produce and maret of each product or-ow much to produce and maret of each product orservice for the planning periodservice for the planning period

%onstraints%onstraints

/aimum amount of each product or service/aimum amount of each product or servicedemanded0 /inimum amount of product or servicedemanded0 /inimum amount of product or servicepolicy will allow0 /aimum amount of resourcespolicy will allow0 /aimum amount of resourcesavailableavailable


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LP Problems in OM: Production PlanLP Problems in OM: Production Plan

ObjectiveO

bjectiveTo select the mi of products or services that resultsTo select the mi of products or services that resultsin maimum profits for the planning periodin maimum profits for the planning period

+ecision ,ariables+ecision ,ariables

-ow much to produce on straight2time labor and-ow much to produce on straight2time labor andovertime labor during each month of the yearovertime labor during each month of the year

%onstraints%onstraints

mount of products demanded in each month0mount of products demanded in each month0/aimum labor and machine capacity available in/aimum labor and machine capacity available ineach month0 /aimum inventory space available ineach month0 /aimum inventory space available ineach montheach month


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Recognizing LP ProblemsRecognizing LP Problems

%haracteristics of LP Problems in O/%haracteristics of LP Problems in O/ well2defined single objective must be stated. well2defined 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 beconstrained 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

epressed as linear mathematical functions.epressed as linear mathematical functions.


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Steps in Formulating LP ProblemsSteps in Formulating LP Problems

!! +efine the objective. (min or ma)+efine the objective. (min or ma)"!"! +efine the decision variables. (positive5 binary)+efine the decision variables. (positive5 binary)

#!#! &rite the mathematical function for the objective.&rite the mathematical function for the objective.

$!$! &rite a 12 or '2word description of each constraint.&rite a 12 or '2word description of each constraint.%!%! &rite the right2hand side (6-#) of each constraint.&rite the right2hand side (6-#) of each constraint.

&!&! &rite&rite 775 85 or5 85 or 99for each constraint.for each constraint.

'!'! &rite the decision variables on L-# of each constraint.&rite the decision variables on L-# of each constraint.!! &rite the coefficient for each decision variable in each&rite the coefficient for each decision variable in each

constraint.constraint.


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%ycle Trends is introducing two new lightweightbicycle frames5 the +elue and the Professional5 to be

made from aluminum and steel alloys. The anticipated

unit profits are ;1< for the +elue and ;13 for the

Professional.The number of pounds of each alloy needed per

frame is summari=ed on the net slide. supplier

delivers 1 board feet of wood whileeach table reDuires * labor hours and 1' board feet of

wood. vailable wood is '


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xample: LP Formulationxample: LP Formulation

+efine the objective /aimi=e total weely profit

+efine the decision variables

18 number of chairs produced weely '8 number of tables produced weely

&rite the mathematical objective function

/a A 8 131B '1'


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&rite a one2 or two2word description of each constraint Labor hours available Eoard feet available t least < tables t least chairs for every table

&rite the right2hand side of each constraint @'< '


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&rite all the decision variables on the left2hand side of each&rite all the decision variables on the left2hand side of eachconstraintconstraint

1 ' 7 @'

The #ureset %oncrete %ompany producesconcrete. Two ingredients in concrete are sand (costs

;4 per ton) and gravel (costs ;> per ton). #and and

gravel together must mae up eactly :3H of the

weight of the concrete. lso5 no more than


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+efine the objective /inimi=e daily costs

+efine the decision variables

18 tons of sand purchased '8 tons of gravel purchased

&rite the mathematical objective function

/in A 8 41B >'


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'' #ubject To

1B ' 8 13


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LP Problems in *eneralLP Problems in *eneral

Inits of each term in a constraint must be the same asInits of each term in a constraint must be the same asthe 6-#the 6-#

Inits of each term in the objective function must beInits of each term in the objective function must be

the same as Athe same as A

Inits between constraints do not have to be the sameInits between constraints do not have to be the same

LP problem can have a miture of constraint typesLP problem can have a miture of constraint types


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LP ProblemLP Problem

Jalay $nd. produces two water guns5 the #pace 6ayJalay $nd. produces two water guns5 the #pace 6ayand the Aapper. Jalay earns a profit of ;* for everyand the Aapper. Jalay earns a profit of ;* for every

#pace 6ay and ;' for every Aapper. #pace 6ays and#pace 6ay and ;' for every Aapper. #pace 6ays and

Aappers reDuire ' and production minutes per unit5Aappers reDuire ' and production minutes per unit5

respectively. lso5 #pace 6ays and Aappers reDuire .3respectively. lso5 #pace 6ays and Aappers reDuire .3and .* pounds of plastic5 respectively. Jivenand .* pounds of plastic5 respectively. Jiven

constraints of < production hours5 1'


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6 8 of #pace 6ays to produce6 8 of #pace 6ays to produceA 8 of Aappers to produceA 8 of Aappers to produce

/a A 8 *.


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The &hite -orse pple Products %ompany purchases applesThe &hite -orse pple Products %ompany purchases applesfrom local growers and maes applesauce and apple juice. $tfrom local growers and maes applesauce and apple juice. $tcosts ;


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# 8 jars apple #auce to mae# 8 jars apple #auce to mae 8 bottles apple uice to mae 8 bottles apple uice to mae

# 8 ; for apple #auce dvertising# 8 ; for apple #auce dvertising

8 ; for apple uice dvertising 8 ; for apple uice dvertising

/a A 8 1.3# B 1.:3 2 .4# 2 .>3 # /a A 8 1.3# B 1.:3 2 .4# 2 .>3 #

#T#T

## G .*(# B )G .*(# B ) at least *


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ship has two cargo holds5 one fore and one aft. The fore cargo hold has a ship has two cargo holds5 one fore and one aft. The fore cargo hold has a

weight capacity of :


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EC 8 lbs beef to load in fore cargo holdEC 8 lbs beef to load in fore cargo hold

E 8 lbs beef to load in aft cargo holdE 8 lbs beef to load in aft cargo hold

JC 8 lbs grain to load in fore cargo holdJC 8 lbs grain to load in fore cargo hold

J 8 lbs grain to load in aft cargo holdJ 8 lbs grain to load in aft cargo hold

/a A 8 .*3 EC B .*3E B .1'JC B .1' J/a A 8 .*3 EC B .*3E B .1'JC B .1' J

#T#T

EC B JCEC B JC M :


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Q1 8 number of lifeguards scheduled to begin on #undayQ1 8 number of lifeguards scheduled to begin on #unday Q' 8Q' 8 RR RR RR RR R /ondayR /onday Q* 8Q* 8 RR RR RR RR R TuesdayR Tuesday Q 8Q 8 RR RR RR RR R &ednesdayR &ednesday Q3 8Q3 8 RR RR RR RR R ThursdayR Thursday

Q4 8Q4 8 RR RR RR RR R CridayR Criday Q: 8Q: 8 RR RR RR RR R #aturdayR #aturday

LP ModelLP Model


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/in Q1 B Q' B Q* B Q B Q3 B Q4 B Q:/in Q1 B Q' B Q* B Q B Q3 B Q4 B Q:#T#T

Q1 B Q B Q3 B Q4 BQ: G > (#unday)Q1 B Q B Q3 B Q4 BQ: G > (#unday)

Q1 B Q' B Q3 B Q4 BQ: G 4 (/onday)Q1 B Q' B Q3 B Q4 BQ: G 4 (/onday)

Q1 B Q' B Q* B Q4 BQ: G 3 (Tuesday)Q1 B Q' B Q* B Q4 BQ: G 3 (Tuesday)Q1 B Q' B Q* B Q BQ: G (&ednesday)Q1 B Q' B Q* B Q BQ: G (&ednesday)

Q1 B Q' B Q* B Q BQ3 G 4 (Thursday)Q1 B Q' B Q* B Q BQ3 G 4 (Thursday)

Q' B Q* B Q B Q3 BQ4 G : (Criday)Q' B Q* B Q B Q3 BQ4 G : (Criday)

Q* B Q B Q3 B Q4 BQ: G @ (#aturday)Q* B Q B Q3 B Q4 BQ: G @ (#aturday)ll variables G < and integerll variables G < and integer

LP ModelLP Model

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LP Formulation Ex