Operations Management - 5th EditionOperations Management - 5th Edition

Chapter 13 SupplementChapter 13 Supplement

Roberta Russell & Bernard W. Taylor, III

Linear ProgrammingLinear Programming

LP-LP-22

Lecture OutlineLecture Outline

What is LP? Where is LP used? LP Assumptions Model Formulation Examples

SolvingSolving

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A model consisting of linear relationshipsrepresenting a firm’s objective and resource

constraints

Linear Programming (LP)

LP is a mathematical modeling technique used to determine a level of operational activity in order to achieve an objective, subject to restrictions called

constraints

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Types of LPTypes of LP

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Types of LP (cont.)Types of LP (cont.)

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Types of LP (cont.)Types of LP (cont.)

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Common Elements to LPCommon Elements to LP

Decision variablesDecision variables Should completely describe the decisions to be Should completely describe the decisions to be

made by the decision maker (DM)made by the decision maker (DM)

Objective Function (OF)Objective Function (OF) DM wants to maximize or minimize some function of DM wants to maximize or minimize some function of

the decision variablesthe decision variables

ConstraintsConstraints Restrictions on resources such as time, money, Restrictions on resources such as time, money,

labor, etc.labor, etc.

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LP AssumptionsLP Assumptions

OF and constraints must be linearOF and constraints must be linear ProportionalityProportionality

Contribution of each decision variable is Contribution of each decision variable is proportional to the value of the decision proportional to the value of the decision variablevariable

AdditivityAdditivity Contribution of any variable is independent of Contribution of any variable is independent of

values of other decision variablesvalues of other decision variables

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LP Assumptions, cont’d.LP Assumptions, cont’d.

DivisibilityDivisibility Allow both integer and non-integer (real Allow both integer and non-integer (real

numbers)numbers)

CertaintyCertainty All coefficients are known with certaintyAll coefficients are known with certainty We are dealing with a deterministic worldWe are dealing with a deterministic world

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LP Model Formulation (NPS LP Model Formulation (NPS format)format)

IndicesIndices Domains and fundamental dimensions of the Domains and fundamental dimensions of the

modelmodel Examples: products, time period, region, …Examples: products, time period, region, …

DataData Input to the model – given in the problemInput to the model – given in the problem Indexed using indicesIndexed using indices Convention is UPPERCASEConvention is UPPERCASE

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LP Model FormulationLP Model Formulation

Decision variables Mathematical symbols representing levels of

activity of an operation The quantities to be determined, indexed

using indices Convention is lowercase

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LP Model Formulation, cont’d.LP Model Formulation, cont’d.

Objective function (OF) The quantity to be optimized 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

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LP Model Formulation, cont’d.LP Model Formulation, cont’d.

Constraint A linear relationship representing a

restriction on decision making Binding relationships Attach a word description to each set of

constraints Include bounds on variables

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LP Formulation: ExampleLP Formulation: Example

LaborLabor ClayClay RevenueRevenuePRODUCTPRODUCT (hr/unit)(hr/unit) (lb/unit)(lb/unit) ($/unit)($/unit)

BowlBowl 11 44 4040

MugMug 22 33 5050

There are 40 hours of labor and 120 pounds of clay There are 40 hours of labor and 120 pounds of clay available each dayavailable each day

Formulate this problem as a LP modelFormulate this problem as a LP model

RESOURCE REQUIREMENTSRESOURCE REQUIREMENTS

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LP Formulation: ExampleLP Formulation: Example

IndicesIndices pp = products { = products {b, mb, m}}

DataData REVENUEREVENUEpp = $ revenue per unit of = $ revenue per unit of pp made made LABORLABORpp = # of hours to produce a unit of = # of hours to produce a unit of pp CLAYCLAYpp = lbs of clay to produce a unit of = lbs of clay to produce a unit of pp TOTLABOR = total hours availableTOTLABOR = total hours available TOTCLAY = total lbs of clay availableTOTCLAY = total lbs of clay available

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LP Formulation: ExampleLP Formulation: Example

VariablesVariables numnumpp = units of = units of pp to produce to produce totrevtotrev = total revenue = total revenue

Objective FunctionObjective Function Max Max totrevtotrev = =

ConstraintsConstraints (labor constraint)(labor constraint)

(clay constraint)(clay constraint)

(non-negativity)(non-negativity)

pp

p numREVENUE *

p

pp TOTLABORnumLABOR *

p

pp TOTCLAYnumCLAY *

0pnum

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LP Formulation: ExampleLP Formulation: Example

Maximize Maximize totrevtotrev = 40 = 40 numnumbb + 50 + 50 numnummm

Subject toSubject to

numnumbb ++22numnummm40 40 (labor constraint)(labor constraint)

44numnumbb ++3num3nummm120 120 (clay constraint)(clay constraint)

numnumb b , , numnummm00

Solution is: Solution is:

numnumbb = 24 bowls = 24 bowls numnumm m = 8 mugs= 8 mugs

totrevtotrev = $1,360 = $1,360

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Bowls and Mugs SolvedBowls and Mugs Solved

Use OMTools > Linear ProgrammingUse OMTools > Linear Programming

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Another ExampleAnother Example

Joe’s Woodcarving, Inc. manufactures two Joe’s Woodcarving, Inc. manufactures two types of wooden toys: soldiers and trains. types of wooden toys: soldiers and trains.

Sale $Sale $ Raw CostRaw Cost Labor / OverheadLabor / Overhead Finishing LaborFinishing Labor Carpentry LaborCarpentry Labor

SoldierSoldier $27$27 $10$10 $14$14 2 hr2 hr 1 hr1 hr

TrainTrain $21$21 $9$9 $10$10 1 hr1 hr 1 hr1 hr

Unlimited supply of raw material, but only 100 Unlimited supply of raw material, but only 100 finishing hours and 80 carpentry hoursfinishing hours and 80 carpentry hours

Demand for trains unlimited, but at most 40 Demand for trains unlimited, but at most 40 soldiers can be sold each weeksoldiers can be sold each week

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Wooden Toys ExampleWooden Toys Example

IndicesIndices ??????

DataData ??????

VariablesVariables ??????

OFOF ??????

ConstraintsConstraints ??????

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Wooden Toys SolvedWooden Toys Solved