ch11 the art of modelling with spreadsheet

8/20/2019 Ch11 The Art of Modelling with Spreadsheet

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MANAGEMENT

SCIENCEThe Art of Modeling with Spreadsheets

STEPHEN G. POWELL

ENNETH !. "AE!

Co#pati$le with Anal%ti& Sol'er Platfor#(O)!TH E*ITION

CHAPTE! ++ POWE!POINT

INTEGE! OPTIMI,ATION


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INTRODUCTION

• The optimal solution of a linear program maycontain fractional decision variables, and thisis appropriate—or at least tolerable—in mostapplications

• In some cases it may be necessary to ensurethat some or all of the decision variables ta!eon integer values

• "ccommodating the re#uirement that

variables must be integers is the sub$ect ofinteger progra##ing

Chapter ++ Cop%right - /+0 1ohn Wile% 2Sons3 In&.


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INT%&%R '"RI"()%* "ND T+% INT%&%R *O)'%R

• *olver allos us to directly designate decisionvariables as integer values

• In integer linear programs, *olver employs analgorithm that chec!s all possible assignments of

integer values to variables, although some of theassignments may not have to be e-aminede-plicitly

• This procedure may re#uire the solution of a largenumber of linear programs. *olver can do this#uic!ly and reliably ith the simple- algorithm,and ill eventually locate a global optimum

• In the case of integer nonlinear programs, certaindi/culties can arise, although *olver ill alaysattempt to 0nd a solution


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D%*I&N"TIN& '"RI"()%* "* INT%&%R*


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*%TTIN& T+% TO)%R"NC% 1"R"2%T%R


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*O)'%R TI13 INT%&%R O1TION*

• The most important integer option is the Tolerance parameter

• The default value of the parameter is 45,

and e may leave this value undisturbedhile e debug our model

• Once e are convinced that our model isrunning correctly, e can set the Tolerance

parameter to 65 so that an optimal solutionis guaranteed


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(IN"R7 '"RI"()%* "ND (IN"R7 C+OIC%2OD%)*

• " binary variable, hich ta!es on the values8ero or one, can be used to represent a9go:no;go< decision

• =e can thin! in terms of discrete pro$ects,here the decision to accept the pro$ect isrepresented by the value >, and the decisionto re$ect the pro$ect is represented by thevalue 0


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T+% C"1IT") (UD&%TIN& 1RO()%2

• Companies, committees, and even householdsoften 0nd themselves facing a problem ofallocating a capital budget

• "s the problem arises in many 0rms, there is a

speci0ed budget for the year, to be invested inmulti;year pro$ects

• There are typically several proposed pro$ectsunder consideration

• The challenge is to determine ho to ma-imi8ethe value of the pro$ects selected, sub$ect tothe limitation on e-penditures represented bythe capital budget


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T+% C"1IT") (UD&%TIN& 1RO()%2

• In the classic version of the &apital$9dgeting pro$le#, each pro$ect isdescribed by to values3 the e-penditurere#uired and the value of the pro$ect

• "s a pro$ect is typically a multi;year activity,its value is represented by the net presentvalue ?N1'@ of its cash Aos over the pro$ectlife

• The e-penditure, combined ith thee-penditures of other pro$ects selected,cannot be more than the budget available


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D%*I&N"TIN& '"RI"()%* "* (IN"R7INT%&%R*


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T+% *%T CO'%RIN& 1RO()%2

• The set &o'ering pro$le# is a variation ofthe covering model in hich the variables areall binary

• In addition, the parameters in the constraints

are all 8eroes and ones• In the classic version of the set covering

problem, each pro$ect is described by asubset of locations that it 9covers<

•

The problem is to cover all locations ith aminimal number of pro$ects


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(IN"R7 '"RI"()%* "ND )O&IC")R%)"TION*+I1*

• =e sometimes encounter additionalconditions aBecting the selection of pro$ectsin problems li!e capital budgeting

•

These include relationships among pro$ects,0-ed costs, and #uantity discounts


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R%)"TION*+I1* "2ON& 1RO%CT*

• 1ro$ects can be related in any number ofays, 0ve of hich are as follos3 – "t least m pro$ects must be selected

– "t most n pro$ects must be selected

– %-actly k pro$ects must be selected

– *ome pro$ects are mutually e-clusive

– *ome pro$ects have contingency relationships


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R%)"TION*+I13 "T )%"*T M 1RO%CT* 2U*T (%*%)%CT%D

• y E y 4 F >

• 1ro$ect , or 1ro$ect 4, or both, ill beselected, thus satisfying the re#uirement of at

least one selection


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R%)"TION*+I13 "T 2O*T N 1RO%CT* 2U*T (%*%)%CT%D

• y G E y 4 H >

• 1ro$ect G, or 1ro$ect 4, or neither, but not bothill be selected, thus satisfying the

re#uirement of at most one selection


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R%)"TION*+I13 %"CT)7 K 1RO%CT* 2U*T (%*%)%CT%D

• y G E y 4 J >

• %-actly one of either 1ro$ect G or 1ro$ect 4 illbe selected, thus satisfying the re#uirement

of e-actly one selection


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R%)"TION*+I13*O2% 1RO%CT* +"'% CONTIN&%NC7R%)"TION*+I1*

• y K L y 4 F 6

• If 1ro$ect 4 is selected, then pro$ect K must beas ell


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)INMIN& CON*TR"INT* "ND I%D CO*T*

• =e commonly encounter situations in hichactivity costs are composed of 0-ed costs andvariable costs, ith only the variable costsbeing proportional to activity level

• =ith an integer programming model, e canalso integrate the 0-ed component of cost


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• =e separate the 0-ed and variablecomponents of cost

• In algebraic terms, e rite cost as3Cost J Fy E cx here F represents the 0-ed cost, and c represents the linear variable cost

• The variables x and y are decision variables,here x is a normal ?continuous@ variable,

and y is a binary variable


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• To achieve consistent lin!ing of the tovariables, e add the folloing genericlin;ing &onstraint to the model3 x H My

here the number M represents an upperbound on the variable x

• In other ords, M is at least as large as anyvalue e can feasibly choose for x


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)INMIN& CON*TR"INT3 X H MY

• =hen y J 6 ?and therefore no 0-ed cost is incurred@, theright;hand side becomes 8ero, and *olver interprets theconstraint as x HJ 6

– *ince e also re#uire x FJ 6, these to constraints togetherforce x to be 8ero

–

Thus, hen y J 6, it ill be consistent to avoid the 0-edcost

• On the other hand, hen y J >, the right;hand side illbe so large that *olver does not need to restrict x at all,permitting its value to be positive hile e incur the0-ed cost

–

Thus, hen y J >, it ill be consistent to incur the 0-ed cost• Of course, because e are optimi8ing, *olver ill never

produce a solution ith the combination of y J > and x J6, because it ould alays be preferable to set y J 6


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*O)'%R TI13)O&IC") UNCTION* "ND INT%&%R1RO&R"22IN&

• %-perienced %-cel programmers might betempted to use the logical functions ?I, "ND,OR, etc@ to e-press certain relationships

•

Unfortunately, the linear solver does notalays detect the nonlinearity caused by theuse of logical functions, so it is important toremember never to use an I function in a

model built for the linear solver



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T+R%*+O)D )%'%)* "ND U"NTIT7DI*COUNT*

• Threshold le'el re


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T+R%*+O)D )%'%)*

• *uppose e have a variable x that is sub$ectto a threshold re#uirement )et m denote theminimum feasible value of x if it is non8ero

Then e can capture this structure in an

integer programming model by including thefolloing pair of constraints3 x L my F 6 x L My H 6here, as before, M is a large number that is

greater than or e#ual to any value x couldfeasibly ta!e


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PT+% "CI)IT7 )OC"TION 2OD%)

• The transportation model ?discussed inChapter >6@ is typically used to 0nd optimalshipping schedules in supply chains andlogistics systems

• The applications of the model can be vieedas tactical problems, in the sense that thetime interval of interest is usually short, say aee! or a month

• Over that time period, the supply capacities

and locations are unli!ely to change at all,and the demands can be predicted ithreasonable precision


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PT+% "CI)IT7 )OC"TION 2OD%)

• Over a longer time frame, a strategic versionof the problem arises In this setting, thedecisions relate to the selection of supplylocations as ell as the shipment schedule

• These decisions are strategic in the sensethat, once determined, they inAuence thesystem for a relatively long time interval

• The basic model for choosing supply locations

is called the fa&ilit% lo&ation #odel


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T+% C"1"CIT"T%D 1RO()%2

• Conceptually, e can thin! of this problem ashaving to stages

• In the 0rst stage, decisions must be madeabout ho many arehouses to open and

here they should be• Then, once e !no here the arehouses

are, e can construct a transportation modelto optimi8e the actual shipments

• The costs at sta!e are also of to types3 0-edcosts associated ith !eeping a arehouseopen and variable transportation costsassociated ith shipments from the openarehouses


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T+% UNC"1"CIT"T%D 1RO()%2

• Once e see ho to solve the facility locationproblem ith capacities given, it is notdi/cult to adapt the model to theuncapacitated case

• Obviously, e could choose a virtual capacityfor each arehouse that is as large as totaldemand, so that capacity ould neverinterfere ith the optimi8ation


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T+% "**ORT2%NT 2OD%)

• The facility location model, ith or ithoutcapacity constraints, clearly has direct applicationto the design of supply chains and the choice oflocations from a discrete set of alternatives

•(ut the model can actually be used in other typesof problems because it captures the essentialtrade;oB beteen 0-ed costs and variable costs

• "n e-ample from the 0eld of 2ar!eting is theassort#ent pro$le#, hich as!s hich items ina product line should be carried, hen customersare illing to substitute


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*U22"R7

• (inary variables can also be instrumental incapturing complicated logic in linear form so thate can harness the linear solver to 0nd solutions

• (inary variables ma!e it possible toaccommodate problem information on3 – Contingency conditions beteen pro$ects – 2utual e-clusivity among pro$ects – )in!ing constraints for consistency – Threshold constraints for minimum activity levels

•

=ith the capability of formulating these !inds ofrelationships in optimi8ation problems, ourmodeling abilities e-pand ell beyond the basiccapabilities of the linear and nonlinear solvers


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COPYRIGHT © 2013 JOHN WILEY & SONS, INC.

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ch11 the art of modelling with spreadsheet

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