Multi-period supplier selection under price uncertainty

13 Pages·2014·0.38 MB·English

JournaloftheOperationalResearchSociety(2014)65,1636–1648 ©2014OperationalResearchSocietyLtd.Allrightsreserved.0160-5682/14
www.palgrave-journals.com/jors/
Multi-period supplier selection under price
uncertainty
AlperŞen1*,HandeYaman1,KemalGüler2andEvrenKörpeoğlu2
1BilkentUniversity,Ankara,Turkey;and2Hewlett-PackardLaboratories,PaloAlto,USA
Weconsideraproblemfacedbyaprocurementmanagerwhoneedstopurchasealargevolumeofmultipleitems
overmultipleperiodsfrommultiplesuppliersthatprovidebasepricesanddiscounts.Discountsarecontingenton
meetingvariousconditionsontotalvolumeorspend,andsomearetiedtofuturerealizationsofrandomeventsthat
canbemutuallyverified.Weformulateascenario-basedmulti-stagestochasticoptimizationmodelthatallowsusto
considerrandomeventssuchasadropinpricebecauseofthemostfavouredcustomerclauses,apricechangein
thespotmarketoranewdiscountoffer.Weproposecertainty-equivalentheuristicsandevaluatetheregretofusing
them.Weuseourmodelforthreebiddingeventsofalargemanufacturingcompany.Theresultsshowthatcon-
sidering most favored customer clauses in supplier offers may create substantial savings that may surpass the
savingsfromregulardiscountoffers.
JournaloftheOperationalResearchSociety(2014)65(11),1636–1648.doi:10.1057/jors.2013.111
Publishedonline25September2013
Keywords:supplierselection;quantitydiscounts;priceuncertainty;mostfavouredcustomer;multi-stagestochastic
programming
1.Introduction potentialvalueoftheprocurementfunction,theyalsomadethe
decisions faced by procurement managers more complex and
Procurement is a core strategic functionthat affects the profit-
challenging.Aprocurementmanagernowneedstodistributea
ability of a firm, as the cost of goods and services acquired
large volume of many items required at multiple locations
throughprocurement typically constitutesa majority of opera-
across many global suppliers that approach the firm with
ting expenses. A fundamental problem in procurement is
varioustermsandoffers.
supplierselectionorsourcing,thatis,howtoallocatethefirm’s
A complicating feature of this problem is regarding how
business across suppliers, considering factors such as cost,
the suppliers present their price offers in a procurement
quality,responsivenessandrisk.
environment. Many suppliers often exhibit economies of
Theindustryhasseentwoshiftsinsupplierselectionduring
scale and scope in their production activities. Some others
the past two decades. First, with the advent of the Internet,
have growth and market share targets for a specific group of
procurement organizations moved away from manual bidding
items. Suppliers express these internal efficiencies and pres-
processesandnegotiationstoelectronicsourcing.Forexample,
sures by offering discounts to the buyer. These offers are
between 2001 and 2006, Hewlett-Packard increased its total
usually complex and are contingent on meeting various
spend through e-sourcing events to US$30 billion, a 100-fold
conditions on total available market, volume or spend for a
increase (Carbone, 2004; Moody, 2006). Electronic sourcing
single item or a set of items, and the discounts may be
not only allowed firms to expand their supply pool to more
applicabletothesameoradifferentsetofitems.
competition,butalsomadethesupplierselectionprocessfaster
Itisnotpossibletoincorporatethesecomplexdiscountoffers
and more structured, enabling a simultaneous evaluation and
into a simple reverse auction. Ignoring these (potential) offers
negotiation of supplier offers. A second shift is an organi-
and selecting the lowest bidding supplier for each item or lot
zational change with which corporations started to procure
would lead to inefficiencies. We have recently seen efforts to
centrally to leverage economies of scale of their global
developtoolsthatwouldenablesupplierstoexpresstheseoffers
business. For example, Hewlett-Packard combined even its
and buyers to evaluate them. The success of the software tool
indirect and services procurement globally and started to use
CombineNet is described in Sandholm (2007). Between 2001
category-based sourcing to leverage its total spend of $16.5
and2006,447biddingeventsareadministeredtotallingaspend
billion (Avery, 2008). While these two shifts increased the
of$35billion.ItisbelievedthatCombineNetdeliveredsavings
of$4.4billionintheseevents.Bichleretal(2011)proposeda
*Correspondence:AlperŞen,DepartmentofIndustrialEngineering,Bilkent
bidding language and an optimization model to express and
UniversityBilkent,Ankara,06800,Turkey.
E-mail:[email protected] evaluatemorecomplexoffers.
AlperŞenetal—Multi-periodsupplierselectionunderpriceuncertainty 1637
Both of these efforts and many other research in this area agreement) shall be at least as low as the prices that the
assume that the important parameters of the problem are contractor has under any other contract instrument under like
perfectly known in advance. However, procurement environ- terms and conditions. If at any time the prices under any other
mentsarerepletewithuncertainties.Aprimaryuncertaintyisin contract instrument become lower than the prices in this BPA,
the volume that needs to be procured. Since the demand for thisBPAwillbemodifiedtoincludethelowerprices’.Govern-
end-productsisoftenvolatile,itisalsoveryhardtopredictthe mentsusuallyenforcethecompliancetotheseclausesstrictly.In
amountsofgoodsandservicesthatneedtobeprocuredtomake arecentsettlement,OracleacceptedtopaytheUSgovernment
them. For example, global shipments of personal computers $199.5millionafterafilesuitclaimingthatthecompanywasnot
declined in the first quarter of 2011, by 3.2% according to an providing the government discounts that were as deep as some
estimatebyInternationalDataCorp.andby1.1%accordingto othercustomerswerereceiving(Montalbano,2011).
ananother estimate by Gartner Inc.,while both trackingfirms Another contract clause that price uncertainty can affect is
previouslypredictedanincrease(Sherr,2011). what is called meet-the-competition-clause (MCC). An MCC
Anotherimportantuncertaintyisinthepricesofcomponents clause(sometimesalsoreferredtoasmeet-or-releaseclause)in
that need to be procured. For example, prices of many com- a procurement contract gives the seller an option to retain the
ponents that are used in personal computers fluctuate heavily customer’sbusinessbymatchinganylowerpriceofferthatmay
andtheseshiftsinpricesarealsoveryhardtopredict.Theprice becominginthefuture.Athirdcontracttypethatmayleadto
of the DRAM memory that Hewlett-Packard uses dropped by uncertaintyinpriceispriceindexing.Inthiscase,thecontract
over 90% in 2001, and then more than tripled in early 2002 price of a product is indexed to the price of a commodity or
(Nagalietal,2008).Supplieroffersusuallystateacommitment an official price index (such as consumer price index). For
to base prices and discounts that will be given if certain con- example, in the United Kingdom, 85% of natural gas is sold
ditionsaremet.However,priceuncertaintyinthemarketmay under long-term contracts in which prices are indexed to the
still have an impact on how a procurement manager would spot market (Neumann and von Hirschhausen, 2004). Uncer-
evaluate such offers through a number of ways. First, if the tainty in the commodity prices or price index obviously
procurementisthroughacontractedsupplier,thefutureprices createsuncertaintyonthepricesthatthesellerwouldchargeto
of this supplier to other customers may have an impact on an the buyer. Finally, if some of the products under considera-
existingcontractbasedoncertainclauses.Theseclausesusually tion are commodity-like products and can also be procured
refer to what is known as a most-favored-customer (MFC) from the spot market,anuncertainty in spot prices has a clear
status.AcustomerwhoobtainsaMFCstatusfromacompany and direct effect on how much the manager should procure
is guaranteed to receive the best price the company gives to fromthespotmarketorusingafixed-pricecontractnowandin
anyone. If the supplier lowers price to someone else, then the thefuture.
customer’s price will be lowered to match. In some cases, In this study, we develop a model that incorporates price
a customermayneedtopurchase a minimumvolume ofa set uncertaintytothesupplierselectionproblemwhenthesuppliers
of items over a specified time period to obtain this status. In offercomplexdiscountoffers.Theformalproblemweconsider
other cases, the customer (such asa government agency) may may be stated as follows. A buyer needs to purchase a large
demand MFC status for any contract without any condition. volumeofmultipleitemsoveraplanninghorizon(aquarteror
Lowest prices can be verified and contract compliance can be ayear)thatconsistsofmultipleperiods(months).Thedemand
ensuredthroughthird-partyaudits. for each item can be different in each period, but is known.
MFCclausesarecommonlyusedinprocurementcontractsin There are multiple suppliers that are qualified to offer all or
many industries. For example, a contract (Sample Business aportionoftheitemsinconsideration.Eachsupplierprovidesa
Contracts, 2012) between Cisco Systems Inc. and one of its basepriceforeachoffereditem.Inaddition,supplierspropose
suppliers,FrontierSoftwareDevelopmentInc.,stipulates‘Fron- various discount offers to the buyer that are contingent on
tier represents and warrants to Cisco that the product prices/ meetingvariousconditionsonasingleitemoragroupofitems,
license fees offered to Cisco under this agreement are no less overasingleperiodormultipleperiods.Anoffermayprovide
favorable than the product prices/license fees offered to any perunitdiscountsthatcanbeappliedtoasingleitemoragroup
other party purchasing or licensing similar quantities. In the of items, over a single period or multiple periods, and on all
event,Frontieroffersmorefavorableproductprices/licensefees units purchased (all-units discount) or units purchased above
toanyotherparty.FrontierwillpromptlynotifyCiscoofsuch athreshold(incrementaldiscounts).Alternatively,anoffermay
eventandoffersuchmorefavorableproductprices/licensefees providethebuyeralumpsum.Someoftheseoffersmaybetied
to Cisco. Commencing upon the date, such more favorable to realization of random events in the future that can be
productprices/licensefeeswereofferedtotheotherparty’. mutually verified. The buyer’s problem is to select suppliers
As an example of a contract that guarantees a govern- and determine the amount of each item to be procured from
ment agency to purchase at the lowest price, we note the each supplier in each period of the planning horizon to
following price reductions clause stated in a contract between minimizeitsexpectedprocurementandinventoryholdingcosts
Hewlett-Packard and US Department of Defense (DoD-ESI, whilesatisfyingitemdemandsineachperiod.Therecouldalso
2012): ‘The prices under this BPA (blanket purchase be capacity constraints that limit how much the buyer can
1638 JournaloftheOperationalResearchSocietyVol.65,No.11
procurefromasupplier.Inaddition,thebuyermayalsoenforce The rest of the paper is organized as follows. In Section 2,
certainsideconstraints(eg,enforceaminimumandmaximum we analyse a single-item, two-period problem with two
number of suppliers for each item) to properly manage other suppliers to gain insight into the trade-offs. In Section 3, we
procurementrisks.Whiletheprocurementdecisionsforthefirst reviewtheliteratureonsupplierselectionproblem.InSection4,
period are executed immediately, the decisions in the latter we present our model. In Section 5, we propose the two
periodswillbecontingentontherealizationofrandomevents certainty-equivalent heuristics. In Section 6, we discuss the
inthoseperiods(ie,recourses). implementation of our model at a major manufacturing com-
Weformulatethebuyer’sproblemasamulti-stagestochastic pany and analyse the effects of various model elements and
mixed-integer programme using a scenario tree. To the best parameters on the benefits of considering MFC terms. We
of our knowledge, this is the first model for the supplier concludeinSection7.
selection problem that simultaneously considers uncertainty
and discount offers of combinatorial nature. This is also one
ofthefirstmulti-periodmodelsandallowsthebuyertoconsider 2.Amotivatingexample
discountrulesdefinedovermultipleperiodsandcarryinventory
In order to explain the basic trade-offs in evaluating discount
from one period to another to be eligible for a favourable
offersunderuncertainty,andtoshowtheneedtouseaformal
discount. The formulation is also very general in two aspects.
stochasticmodeltosupportdecisionmakinginthiscontext,we
First, we can represent a variety of random events that have
providethefollowingstylizedexample.
direct or indirect effects on the discounts that the buyer gets
A company needs to procure an item over a two-period
from the suppliers. Second, we can represent many diffe- horizon.Thedemandsinthefirstandsecondperiodsareδ and
1
rentformsofsupplierofferswithverycomplexconditionsand δ , respectively. There are two suppliers that offer this item.
2
discounts. For example, the model supports the separation of Supplier a charges μ per unit and offers a most favoured
a
items(periods)forwhichtheconditionsareimposedanditems
customer clause in the contract. Under this contract, if the
(periods) on which the discounts apply, pricing with multiple firm procures 100m percent of its demand from supplier a in
pricebreaksandincrementalorall-unitsdiscounts. Period1,itwillbenefitfromanypossiblereductioninprice(to
We also suggest two certainty-equivalent heuristics that
othercustomers)inPeriod2.Thepriceperunitwillbereduced
can be used for this problem. In both of these heuristics, by π with a probability γ and will remain constant with
a
adeterministicversionoftheproblemissolvedbysettingthe probability 1−γ. Supplier b charges μ per unit and offers a
b
prices in later periods to their expected values. The static volume discount contract. Under this contract, if the firm
heuristic solves the problem only once at the beginning of procures a total of ρ in two periods from supplier b, it will
the horizon and does not respond to the actual realizations receiveadiscountofπ perunit.Weassumethat
b
of events in the later periods. The dynamic heuristic, on the
other hand, resolves the problem at each period. We show δ +δ ⩾ρ>ð1-mÞδ +δ :
1 2 1 2
howtocomputetheperformanceoftheseheuristicsbyusing
That is, (i) the firm can always qualify for a volume discount
ourstochasticformulation.
from supplier b by buying enough; and (ii) the firm cannot
Our work is motivated by our experience with a major
qualifyforthevolumediscountofferfromsupplierbandbuy
manufacturingcompanythatconductsquarterlybiddingevents
enoughfromsupplieratobenefitfromapossiblepricedropat
to select suppliers for its global demand for various compo-
thesametime.Wealsoassumethat
nents.Inalloftheseprocurementevents,suppliers’bidsinclude
various volume discounts in addition to base prices for each μ <μ <μ -π +π ;
b a b b a
component.Someofthesediscountoffersareverycomplexin
nature and are contingent on meeting multiple conditions on that is, firm will choose supplier a over b if MFC clause in
multiple items. Furthermore, some of the suppliers offer con- supplieraisused,andsupplierboversuppliera,otherwise.If
tractswithMFCtermsthatguaranteethelowestpriceprovided thefirmbuysenoughfromsupplieratobenefitfromapotential
thatthemanufacturingcompanyprocuresenoughvolumefrom pricedrop,itsexpectedcostwillbe
them.Weusedourmulti-stagestochasticprogrammingmodel
toevaluateMFCstatusbenefitsandregulardiscountoffersfor Φa0 ¼μamδ1+μbð1-mÞδ1+γðμa-πaÞδ2+ð1-γÞμbδ2:
threebiddingeventsthattookplacein2010.Theresultsofthis
On the other hand, if the firm chooses to use supplier b and
preliminarystudyshowthatconsideringMFCtermsinaddition
benefitfromthevolumediscountoffer,itscostwillbe
to the regular discount offers may lead to substantial savings.
Insomeoftheevents,theincrementalsavingsbytakingMFC Φb ¼ðμ -π Þðδ +δ Þ:
b b 1 2
terms into account may be larger than the savings that can
Thefirmwilloptforsuppliera’sMFCclause(Φa<Φb)ifand
be obtained by only evaluating regular discount offers. The 0
onlyif
results also show that the price certainty-equivalent heuristics,
the static version in particular, fail to capture the benefits of γ>γ ¼πbðδ1+δ2Þ+ðμa-μbÞmδ1:
MFCtermsincontracts. 0 ðπa+μb-μaÞδ2
AlperŞenetal—Multi-periodsupplierselectionunderpriceuncertainty 1639
An alternative to using the stochastic formulation is to use frommultiplesuppliersthatoffertotalquantitydiscountsbased
a certainty-equivalent argument and assume that supplier ontotalpurchasequantities.Theauthorsprovethatthisproblem
a’s second period price will be the expected price μ −γπ . is NP-hard even for some specific discount structures. They
a a
In this case, we can write firm’s cost if it chooses to use the present a Mixed-Integer Linear Programming (MILP) model
MFCclauseas and a branch-and-bound procedure based on a minimum cost
Φa ¼μ mδ +μ ð1-mÞδ +ðμ -γπ Þδ : network flow problemreformulationof the continuousrelaxa-
1 a 1 b 1 a a 2
tion. They extendtheir results to variants of the problem with
Hence,thefirmwilloptforsuppliera’sMFCclause(Φa1<Φb) market share constraints, limited number of winning suppliers
ifandonlyif and multiple periods. Manerba and Mansini (2012) study the
π ðδ +δ Þ+ðμ -μ Þmδ +ðμ -μ Þδ sameproblemundercapacityconstraintsandpresentnewvalid
γ>γ ¼ b 1 2 a b 1 a b 2:
1 π δ inequalities. They propose a branch-and-cut algorithm and a
a 2
hybrid heuristic to provide an initial feasible solution to the
Denotingα=π (δ +δ )+(μ −μ )mδ ,β=(π +μ −μ )δ and
b 1 2 a b 1 a a b 2 exact approach. Qin etal(2012) study a distribution planning
λ=(μ −μ )δ , we have γ =α/β and γ =(α+λ)/(β+λ). It is
a b 2 0 1 problem where shipping companies offer total quantity dis-
theneasytoseethatγ isstrictlysmallerthanγ ifandonlyif
0 1 counts. Crama et al (2004) consider the supplier selection
γ <1,thatis,unlessitisneveroptimalforthefirmtoconsider
0 problem with alternative product recipes and Mansini et al
the supplier a, which offers the MFC clause. In this case for
(2012) incorporate transportation costs. A different setting
γ <γ ⩽γ , the firm’s optimal action is to procure mδ from
0 1 1 where suppliers offer their products in bundles is studied by
supplierainPeriod1.However,thisactionwillnotbetakenif
Murthy et al (2004). In this problem, decisions regarding the
a certainty-equivalent approach is used. In general, the deci-
purchase quantities for different items are related not only
sions taken using a certainty-equivalent approach would be through bundles but also through fixed costs of buying from
differentfromoptimaldecisions(usingstochasticformulations)
suppliers. ALagrangianrelaxation-basedheuristic isproposed
and therefore lead to expected costs that are higher than
tosolvethisproblem.SadrianandYoon(1994)andKatzetal
optimal.
(1994) present MILP formulations for the problem in the
presence of business volume discounts. Bichler et al (2011)
introduce a comprehensive bidding language that allows for
3.Literaturereview
elaborate discount structures. Total quantity and incremental
The impact of quantity discounts on replenishment and pro- quantitydiscounts,aswellaslumpsumdiscountsandmarkups,
curement decisions of a company is well studied in the with conditions on spend or purchase quantities can be
operations management literature (Munson and Rosenblatt, expressed with this bidding language. The authors present a
1998).Mostofthebasictextbooksinthisareaincludeasection MILPmodeltosolvethesupplierselectionproblemandreport
on extensions of the economic order quantity (EOQ) model the results of their experiments in solving the model under
that consider quantity discounts (eg, Silver et al, 1998, §5.5). differentscenarios.
Another line of research focuses on sourcing, that is, how a Therearefewstudiesonthesupplierselectionproblemwith
company should select and allocate its spend to different multiple periods and dynamic demand. Tempelmeier (2002)
suppliers based on different factors such as cost, quality, lead presentsformulationsandaheuristicsolutionapproachforthe
timeandreliability(ChopraandMeindl,2013,§13). single-item problem with both total quantity discounts and
While the extensions of the EOQ model for the case of incrementaldiscounts.Moreover,vandeKlundertetal(2005)
quantitydiscountsandsingle-item,single-supplierproblemsare study the problem of selecting telecommunications carriers
usually tractable, the problem becomes difficult, even for the under total quantity discounts. The discounts are given based
case of a single item when there are multiple suppliers whose on the total call minutes over the planning horizon; however,
prices are functions of the quantity purchased. For example, lowerandupperboundsareimposedoncallminutesperperiod
consider a buyer who needs to purchase a predetermined routedviaeachcarrier.
amountof a singleitemfrom a setofsuppliers. Eachsupplier The multi-item problem with dynamic demand is studied by
offersacertainprice,butthepriceisvalidonlyifthequantity Stadtler(2007).Differentfromourstudy,discountrulesinvolve
purchased is in a specific interval, reflecting the cost and asingleitem,anddecisionsconcerningdifferentitemsdependon
capacity structure of the supplier. Chauhan et al (2005) show afixedcostofbuyingfromsuppliers.Xuetal(2000)studythe
thattheproblemisNP-hard. single supplier problem in the presence of business volume
For reviews on the supplier selection problem, we refer the discountsandsetupcosts.Rongetal(2012)studyaninteresting
reader to Benton and Park (1996), Munson and Rosenblatt multi-item problem where the discounts are due to ordering in
(1998) and Aissaoui et al (2007). Here we summarize some standardbatches(eg,fullpallets)andthebuyerneedstodetermine
examplesondifferentvariantsoftheproblem. theprocurementstrategyforeachitemtominimizeprocurement,
Moststudiesonsupplierselectionformultipleitemsconsider transportation,inventoryandmaterialhandlingcosts.
single-period problems. Goossens et al (2007) study the Most research onsupplier selection problems with discount
problem of deciding on purchase quantities for multiple items offers assume a deterministic setting. When the demand is
1640 JournaloftheOperationalResearchSocietyVol.65,No.11
stochastic, existence of a contract with volume discounts
entanglestheinventorydecisionsofabuyer.Forexample,see
Bassok and Anupindi (1997) who analyse a supply contract
ofasingleproductinwhichthecumulativeordersovermultiple
periodsshouldbelargerthanaprespecifiedquantityinorderto
qualifyforadiscount.Theauthorsextendtheirwork(Anupindi
and Bassok, 1998) to the case of multiple products with
businessvolumediscounts.
The present study extends the literature on multi-item
problems with multiple suppliers and dynamic demand by
consideringverygeneraldiscountrulesandpriceuncertainty.
Figure1 ScenariotreefortheexampleproblemofSection2.
4.Themulti-stagestochasticprogrammingmodel
volumeofproductsI ¼f1ginperiodsT ¼f1gpurchased
c1 c1
Aperfiiormds(Tbu.yFeor)renaecehdsitteompiro2cuIreaandsepteorifoidtetm2sTI,otvheerfiarmsethoasf fthroemsestuKprp1li¼erfake1xgc.eDeidsscoρuc1nt¼k1mpδr1o.vRiduelsear1diaslcloowunstdoifscπoku1n¼tsπina
to satisfy a demand denoted by δit (a deterministic quantity) per unit for products Ik1 ¼f1g bought in quantity above
without a backlog. The firm works with a set of suppliers N. θk1 ¼0 in periods Tk1 ¼f2g. Supplier b offers the discount
Foreachitemi,thereisasubsetofsuppliersNi (cid:2)N thatare ruler2.Thediscountruler2isavailableinnodes1and2andis
qualified. Supplier j in Ni charges a unit price μijt and has a contingentonconditionsinsetCr2 ¼fc2g.Conditionc2states
capacity κijt for item i in period t. The firm can also carry thatthevolumeofproductsIc2 ¼f1ginperiodsTc2 ¼f1;2g
inventory from period t to period t+1 incurring an inventory purchased from supplier b exceeds ρc2 ¼ρ. Rule r2 allows
holdingcostofηitforeachunitofitemi’sendinginventoryin discounts in the set Kr2 ¼fk2g. Discount k2 provides a
periodt. discount of πk2 ¼πb per unit for products Ik2 ¼f1g bought
AsetofdiscountrulesDandasetoflumpsumrebaterules inquantityaboveθk2 ¼0inperiodsTk2 ¼f1; 2g.
L are available. These rules involve a set of conditions C on Atthebeginningofhorizon,theordersareplacedforthefirst
orderquantities.Thequantityρ istheminimumorderquantity period. Recourse actions are taken at the beginning of each
c
that the firm needs to purchase from items in set I (cid:2)I over other period based on actual realizations. We define the
c
periodsTc(cid:2)T forconditionc2Ctobesatisfied.Letεbethe following decision variables. The quantity xijs is the order
setofconflictingpairsofrules. quantity for item i2I and supplier j2Ni at node s2V. Iis
WedefineR¼D∪L.Thebuyercanbenefitfromtheruler stands for the ending inventory for item i2I at node s2V.
offeredbysupplierjðrÞ2N ifitsatisfiestheconditionsC (cid:2)C The binary variable zs takes value 1 if rule r 2Rs applies at
r r
and if it does not benefit from any other rule in R (cid:3)R. If nodes2V andtakesvalue0otherwise.Finally,ys isthetotal
r k
r 2L, then a lump sum rebate of ω is offered. If r 2D, the amountofunitsofitemsinthesetIk thatarediscountedwith
r
supplier provides a set of discounts Kr. The discount k 2Kr discou(cid:2)nPtk 2KPr asapartofruler2(cid:3)D+s atnodes2V,thatis,
reduces the price by πk per unit for items in the set Ik (cid:2)I ysk ¼ i2Ik ^s2Ps:τð^sÞ2TkxijðrÞ^s-θk zsr.Nowwecanmodel
purchased over periods T (cid:2)T exceeding the quantity θ ourproblemasfollows:
k k
(I \I ¼;forallk ;k 2K ;k ≠k ). 0 1
k1 k2 1 2 r 1 2
X XX X XX X
corLreestpVonbdeintghetosethteofronootdaensdofVthbeesctheenasreiot otrfeneowdeitshinnoldaeye0r min γs@ μijτðsÞxijs+ ηiτðsÞIis- πkysk- ωrzsrA
t2T.Foragivennodes2V, ltetτ(s)beits layer,a(s)beits s2V i2I j2Ni i2I r2Dsk2Kr r2Ls
(1)
predecessorinthescenariotree,Ps bethesetofnodesonthe
X
pthaethrufrloesmatthethreootetrtmoinnoadlenosdaensd.γLsebteRitsspbreobthaebisleitty.oWf deisdceofiunnet s:t:Iis ¼IiaðsÞ+ xijs-δiτðsÞ 8i2I; s2V; (2)
rulesavailableatnodes.DefineDs ¼D\RsandLs ¼L\Rs. j2Ni
LEset¼EsEb\eðRthse´sRetsoÞf.pairsofconflictingrulesatnodes,thatis, xijs ⩽κijτðsÞ 8i2I; j2Ni; s2V; (3)
We show the construction of the scenario tree for the X X
example problem discussed in Section 1 in Figure 1. xijðrÞ^s⩾ρczsr 8s2V; r 2Rs; c2Cr; (4)
Here, N ¼fa; bg; T ¼f1; 2g; I ¼f1g, V ¼f0; 1; 2g; i2Ic^s2Ps:τð^sÞ2Tc
að1Þ¼að2Þ¼0,τ(0)=1,τ(1)=τ(2)=2.Supplieraoffersthe X X
discountruler .Thediscountruler isavailableonlyatnode2 ysk⩽ xijðrÞ^s-θkzsr
(if supplier a 1lowers its price to o1ther customers) contingent i2Ik^s2Ps:τð^sÞ2Tk (5)
on conditions in set Cr1 ¼fc1g. Condition c1 states that the 8s2V; r 2Ds; k 2Kr;
AlperŞenetal—Multi-periodsupplierselectionunderpriceuncertainty 1641
!
XX InadditiontoMFCtermsthatareexplainedbyanexample
ys ⩽ κ -θ zs 8s2V; r2Ds; k2K ; (6)
k ijðrÞt k r r inFigure1,variousotheruncertaintiescanbemodelled.Forthe
i2Ikt2Tk case of index pricing, various scenarios can be created for
thevalueoftheindexinthelaterperiods.Foreachscenario,we
zs+zs ⩽1 8s2V; fr; r0g2Es; (7)
r r0 define a discount rule that provides a discount in the amount
of price difference without a condition. One can also model
xijs ⩾0 8i2I; j2Ni; s2V; (8) spot price uncertainty by defining a dummy supplier for spot
purchases and creating a scenario for each possible price
Iis ⩾0 8i2I; s2V; (9) changeinthespotmarket.Wecanthendefineadiscountrule
for every terminal node whose path from the root node has
zs 2f0; 1g 8s2V; r2Rs; (10) a price change. These rules will also have no conditions and
r
will provide a discount in the amount of price change for
ys ⩾0 8s2V; r 2Ds; k 2K : (11) allunitspurchasedinperiodsafterthepricechangetookplace.
k r
Finally, any potential regular discount offer in the future
Theobjectivefunction(1)isequaltotheexpectedtotalcost. (eg, if the buyer thinks that there is a chance that one of the
Constraints(2)areinventorybalanceequations.Thecapacities supplierswillofferanewdiscountinthemiddleoftheplann-
ofsuppliersarerespectedduetoconstraints(3).Constraints(4) ing horizon) can be easily incorporated in the model pro-
impose the minimum order quantity conditions for the rules. vided that the conditions and discounts can be properly
Conditions on minimum spend can be modelled similarly. estimated.
Constraints(5),(6)and(11)PcompuPtetheamountofdiscounted
units (we assume that i2Ik ^s2Ps:τð^sÞ2TkxijðrÞ^s ⩾θk is
implied by the conditions for discount rule r and discount 5.Heuristics
k 2K ).If,inscenarios∈S,discountruler 2Dsapplies,then
r
A usual approach in practice to solve problems under uncer-
constraints (6) is redundant and the maximum of 0 and the
tainty is to use certainty-equivalent heuristics. These heuri-
right hand side of constraint (5) is equal to the amount of
stics solve a deterministic version of the problem in which
discountedunits.Ifthisruledoesnotapply,thenconstraint(5)
isredundantandconstraints(6)and(11)forceys to0.Finally, therandomelementsarereplacedbytheiraveragevalues.The
kr
constraints(7)ensurethatconflictingrulesdonotapplyatthe sameapproachcanalsobeusedforourproblem.Forexample,
for the case of a possible price drop in a given future period
sametime.
The formulation can be strengthened by replacing conflict that the buyer will also benefit owing to MFC clauses in the
contract, one can set the discount amount to be equal to the
constraints(7)withinequalitiescorrespondingtocliquesinthe
conflictgraphGs ¼ðRs; EsÞ.Insomecases,thesameinforma- difference between the current price and the expected price in
thatperiod.Notethattobeeligibleforthisdiscount,thebuyer
tion can be used to strengthen constraints (4). We sketch this
still has to abide by the rules of obtaining MFC status. For
withaverysimpleexample.Supposethatweconsiderasingle
the case of spot price uncertainty, one can use the expected
periodproblemwhereSupplier1offerstotalquantitydiscounts
spot price in each period as the price for that period in the
for Item 1. The unit price reduces by a factor for every 1000
deterministic formulation. For the case of index pricing, one
items purchased and the capacity is 4000. To handle this
discount, we define four discount rules r∈{1, …, 4}, each can index the product prices to the expected price of the
commodityorindex.
with aPsingle condition. We replace constraiPnt (4) with
x ⩾ 4 1000rz0andusethecliqueinequality 4 z0⩽1. We consider two versions of the certainty-equivalent heur-
11T0heMIrL¼P1formulartiongivenin(1–11)canbeuserd¼to1 mrodel istic. In the first version, certainty-equivalent deterministic
problem is solved only once at the beginning of the planning
variousformsofregulardiscountoffersandotherdiscountsthat
horizon and the decisions are never changed. In the second
arecontingentontherealizationofrandomevents.Wediscuss
version,thedeterministicproblemisre-solvedatthebeginning
someofthesehere.First,traditionalquantitydiscountsschemes
ofeachperiodafterrandomeventsforthatperiodareobserved.
canbeeasilymodelled.Consider,forexample,anincremental
Wenextexplainhowonecancomputethesolutionandobtain
discountrulerthatrequiresconditioncandappliesadiscountk.
theexpectedcostforeachheuristic.
Theconditionanddiscountareappliedonthesameitemsetand
sameperiodset,thatis,I ¼I andT ¼T .Thresholdsare
c k c k
thenalsosettobethesame,thatis,ρ =θ .Theruleisdefined
c k 5.1.Staticcertainty-equivalent(SCE)heuristic
in all terminal nodes. Multiple price breaks can be modelled
usingmultiplerulesthataredisjoint.Anall-unitsdiscountrule In this heuristic, the problem is solved only once at the
canbemodelledsimilarly,exceptthatnowwesetθ =0.One beginning of the horizon (at node 0) and the solution is
k
can also separate the periods (items) for which the conditions followed, regardless of the actual realizations of the random
areimposedandtheperiods(items)forwhichthediscountsare eventsthroughouttheplanninghorizon.Inordertocomputethe
appliedon. solutionforthecertainty-equivalentheuristic,onecansolvethe
1642 JournaloftheOperationalResearchSocietyVol.65,No.11
followingmathematicalprogramme. 5.2.Dynamiccertainty-equivalent(DCE)heuristic
XXX XX XX X
min μ x + η I - π0y - ω0z In this heuristic, the problem is re-solved at the beginning of
ijt ijt it it k k r r
eachperiodtoaccountforrealizationsofallrandomeventsand
i2I j2Nit2T i2I t2T r2Dk2Kr r2L
(12) prior decisions until that node. At node 0, we solve the same
modelgivenin(12–22).Wethenneedtore-solvetheproblem
X
s:t:Iit ¼Ii;t-1+ xijt-δit 8i2I; t2T; (13) ateverynode(goinglayerbylayer)aswell.Atnodes,weneed
tosolvethefollowingmathematicalprogramme.
j2Ni
XXX XX
xijt ⩽κijt 8i2I; j2Ni; t 2T; (14) min μijtxijt+ ηitIit
XX
xijðrÞt ⩾ρczr 8r 2R; c2Cr; (15) i2IXj2NXi t2T Xi2I t2T
i2XIct2XTc - πskyk- ωsrzr ð23Þ
y ⩽ x -θ z 8r 2D; k 2K ; (16) r2Dk2Kr r2L
k ijðrÞt k r r
i2Ikt2Tk s:t:ð13-22Þ;
!
XX
yk⩽ κijðrÞt-θk zr 8r 2D; k 2Kr; (17) xijτð^sÞ ¼x^sij^s 8^s2Ps; i2I;j2Ni: (24)
i2Ikt2Tk
z +z ⩽1 8fr; r0g2E; (18) The last set of constraints enforces that the order quantity
r r0
decisionstakenbeforenodes(nodesinthepathfromnode0to
xijt⩾0 8i2I; j2Ni; t2T; (19) node s, Ps) are followed. Note also that we use πsk for the
expecteddiscountgivenbydiscountkandωs fortheexpected
Iit⩾0 8i2I; t2T; (20) lump sum amount given by rule r. These rerflect the fact that
z 2f0; 1g 8r2R; (21) expectations are taken at node s (conditioning on the fact that
r
wearealreadyatnodes).Ifadiscountoraruledependsonlyon
yk⩾0 8r 2D; k 2Kr: (22) random events that materialize in time periods before and at
nodes(t⩽τ(s)),thismeansthatweknowtheseparameterswith
Notethattheformulationaboveisthesameasthemodelin
(1–11),exceptthatthescenariosareremoved.Inthisformula- certaintyatnodes.
tainodn,Ixijitsisththeeeondrdinergqiunavnetnittyorfyorfoitreimtemifrioimnpsuerpipoldietr.jTihnepberinioadryt inpLeertioxsidjttboebtthaeinoepdtibmyasloolrvdienrgqmuaondteitly(2fo3r,i1t3em–2i2f,r2o4m)astunpopdlieersj.
it
variablez takesvalue1ifruler2Rappliesandtakesvalue0 InordertocomputetheexpectedcostoftheDCEheuristic,we
r
otherwise.Thevariabley isthetotalamountofitemsintheset needtosolvethefollowingprogramme.
k 0 1
I that are discounted with discount k 2K as a part of rule
k r X XX X XX X
r 2D.Theonlyparametersthataredifferentfromthemodelin min γs@ μijτðsÞxijs+ ηiτðsÞIis- πkysk- ωrzsrA
(1–11) are π0k, which stands for the expected value of the s2V i2I j2Ni i2I r2Dsk2Kr r2Ls
discountgiven bythediscountkand ω0, whichstands forthe
expectedlumpsumdiscountgivenbythreruler.Bothofthese s:t:ð2-11Þ;
expectationsaretakenatnode0(unconditionalexpectation). x ¼xs 8s2V; i2I; j2N :
ijs ijτðsÞ i
Letx0 betheoptimalorderquantityforitemifromsupplierj
ijt
inperiodtobtainedbysolving(12–22)(superscript0isusedto The last constraint enforces that order quantity decisions at
denotethatthis is the certainty-equivalentsolutionobtained at eachnodearetakenaccordingtothesolutionofthecertainty-
node0).Notethattheoptimalvalueofthemodelin(12–22)is equivalentmodelatthatnode.
notthetrueexpectedvalueobtainedbyfollowingtheordering
decisions obtained by solving (12–22). To calculate the
expected cost of the certainty-equivalent heuristic, we need to 6.Application
reinstate the scenarios and solve the following mathematical
We apply our model in a preliminary study at a major
programme.
manufacturing company. The company needs to procure var-
0 1
ious components that are used in the assembly of a large
X XX X XX X
min γs@ μijτðsÞxijs+ ηiτðsÞIis- πkysk- ωrzsrA number of end-products in various manufacturing sites across
s2V i2I j2Ni i2I r2Dsk2Kr r2Ls the globe. Sourcing decisions are usually taken quarterly by
holdingbiddingevents.Wetestourmodelandtheimplications
s:t:ð2-11Þ; of its use in three bidding events that took place in 2010.
These events are held for 40–45 items (components) and
xijs ¼x0ijτðsÞ 8s2V; i2I; j2Ni: involve3–5 suppliers. For most of the items, there are two or
AlperŞenetal—Multi-periodsupplierselectionunderpriceuncertainty 1643
more suppliers that are competing for the buyer’s business. discount offers are ignored). First,noticethatregular discount
Each supplier provides a base price for each offered item. In offers lead to savings in the range of 0.93–3.23% for the
addition,suppliersalsoofferdiscountsthatreflecttheirecono- company. In absolute terms, these savings are substantial for
mies ofscale in costsand market share targets. Each discount thecompany.
offer requires a minimum volume or a minimum spend on an Optimizing sourcing decisions by also considering MFC
item or a group of items and provides a discount on a set of clausesleadstoimportantadditionalsavingsforbiddingevents
items(whichcanbedifferentfromthesetthattheconditionsare 1 and 3. Even under a scenario when the price drops are not
imposed on). The discounts are either incremental or all-units verylikely,MFCclausesleadtoadditionalsavingsupto0.48%
discounts. ofthetotalspend.Whenthepricedropprobabilityismedium,
Inadditiontotheseusualdiscountoffers,atleastonesupplier the additional savings for bidding events 1 and 3 are in the
ineachbiddingeventprovidesmostfavouredcustomerbenefits rangeof0.55–1.20%.Whenthepricedropsareconsideredvery
initsoffers.IfthesupplierofferingtheMFCclausereducesthe likely, the incremental savings are in the range 1.05–1.92%.
price in the middle of the quarter to other customers, it would Under some scenarios, incremental savings through MFC
extendthepricereductiontothebuyerinconsiderationaswellif clausesaremorethansavingsthatarepossiblewithonlyregular
thebuyerhasalreadyprocuredaminimumfractionofitsdemand discount offers. Once again, these additional percentage gains
(for this item or a group of items) from the supplier until that correspond to substantial monetary savings for the company.
time. In order to model these possible discounts, we split the Inevent2,thebasepricesandregulardiscountoffersgivenby
quarter into two periods and created scenarios to represent theMFCsuppliersareeitheralreadyverycompetitive(leading
possiblepricereductionsinitemsforwhichMFCclausesapply. the buyer to select them even without MFC clauses) or very
All MILPs in this section were implemented and solved uncompetitive (leading the buyer to select other suppliers
using Gurobi (2012) version 5.0 ona notebookcomputer that despitethepossibleMFCbenefits).
has an Intel Dual-Core i7 6.20GHz processor, 4 GB memory
andaWindows7operatingsystem.Forallproblemsinourcase
6.1.Detailedanalysis
study,thesolutiontimeswerelessthan2seconds.
Table1summarizesourtestsforthesethreebiddingevents. In order to test our model and the performance of heuristics
The buyergetsprice and discount offers from the suppliersin proposed in Section 5 in various other settings, we study
two rounds. The suppliers submit initial bids in round 1 and biddingevent1aandvariationsofitinmoredetail.Thisevent
revise them after getting some feedback from the buyer’s was held for 44 items. There were four qualified suppliers,
procurementorganization.Werepresenteachroundseparately named in the following as A, B, C and D. Table 2 shows the
inTable1(iaandibstandforthefirstandsecondroundoffers number of items offered by different groups of suppliers. For
for bidding event i, respectively). Our baseline for each event example, four items are offered by all suppliers, while three
disregardsalldiscountsoffers(regularandMFCclauses).This items are offered by suppliers A, B and C but not D. For 29
corresponds to selecting the supplier that offers the minimum items,2ormoresupplierscompeteforthebuyer’sspend.There
basepriceforeachitem. maybeconsiderabledifferencesbetweenthepricesofferedby
In our first test for each bidding event, we run a version of competingsuppliers.Inthisevent,themaximumbidcanbeas
our model where we ignore all MFC clauses. In the last three muchas34.71%higherthantheminimumbid.Onaverage,the
tests, we run our model considering the MFC clauses in maximumbidsare8.29%higherthantheminimumbids.Note
thecontracts.Foreachevent,weconsiderthreesub-scenarios: thatthereare15itemsthatareofferedbyasinglesupplier(13
thepricesfortheitemscoveredinMFCclausesmaydropwith by supplier A and 2 by supplier B), but they still cannot be
alowprobability,amediumprobabilityandahighprobability. removedfromthemodelandsolvedindependentlysincesome
TheentriesinTable1showthereductionintotalprocurement ofthediscountoffersandMFCconditionsinvolvetheseitems
costs (as a percentage of total procurement costs when all togetherwithotheritemsofferedbymultiplefirms.
Table1 TheimpactofdiscountoffersandMFCclausesontotalspend
Test Events
1a 1b 2a 2b 3a 3b
IgnoreMFCclauses(regulardiscountoffersonly) 0.93 1.26 2.29 2.31 3.22 3.23
ConsiderMFCclauseswhentheprobability
ofapricedropis
Low 0.97 1.67 2.29 2.31 3.70 3.71
Medium 1.48 2.29 2.29 2.31 4.42 4.42
High 1.98 2.91 2.30 2.32 5.14 5.13
1644 JournaloftheOperationalResearchSocietyVol.65,No.11
Table2 Suppliersandtheirofferingsinbiddingevent1
Suppliers A,B,C,D A,B,C A,B A,C A,D A C
Numberofitems 4 3 11 4 7 13 2
Figure2 ScenariotreeforExperiment1.
Apart from base prices, suppliers propose various discount
offers.Inparticular,supplierAsubmitsfour,supplierBsubmits Table3 ResultsforExperiment1
three and supplier C submits one discount offers. The offers
Discountamount5% Discountamount10%
involvemultipleconditionsonhowmuchthebuyershouldbuy
fromasetofitemstoqualifyforthe discount.Thenumberof
γ J* Δ Δ J* Δ Δ
itemsintheconditionsetsisbetween6and21.Thediscounts SCE DCE SCE DCE
usually apply to the same set of items. However, some offers 0.1 0.00 0.00 0.00 0.00 0.00 0.00
mayapplyadiscountonitemsthatarenotintheconditionset. 0.2 0.00 0.00 0.00 3.81 100.00 100.00
0.3 0.00 0.00 0.00 21.99 84.12 0.00
Each offer provides a price reduction between 3% and 7% of
0.4 0.00 0.00 0.00 40.17 46.79 0.00
thebaseprice.
0.5 3.36 100.00 100.00 58.35 32.72 0.00
In addition to these eight discount offers, supplier A is 0.6 10.54 66.84 0.00 76.53 23.35 0.00
offeringanMFCtermforfouritemsinitscontract.According 0.7 17.72 29.82 0.00 94.71 14.15 0.00
tothisterm,ifthebuyerprocuresa certainpercentageoftotal 0.8 24.90 14.15 0.00 112.89 7.91 0.00
0.9 32.08 5.49 0.00 131.06 3.41 0.00
demand fromsupplierA,supplierAwillensurethatthebuyer
Avg 17.72 43.26 20.00 67.44 39.06 12.50
will get the lowest price throughout the quarter. That is, if
supplierAdropsthepriceforsomeoralloftheitemsduringthe
quarter,thebuyerwillalsobenefitfromthesepricedrops.We
consider various scenarios for the drop in prices for these pricesremainthesamewithprobability1−γ.Thecorrespond-
fouritems. ingscenariotreeisdepictedinFigure2.
Sincethetotalprocurementspendsareusuallyverylarge, The results are given in Table 3 for different values of γ
wemeasuretheeffectofMFCclausesonprocurementcosts andπ.Averagesarereportedinthelastrowofthetablewhere
as a percentage of the savings obtained through regular the averages are taken over instances for which J*>0. First,
discount offers. Let S be the total procurement spend in the noticethatMFCtermsleadtoimportantsavingsinthisbidding
absence of any discount offers. Let D be the procurement eventand,asexpected, thebenefitsincreaseastheprobability
spend when only regular discount offers are utilized (those orthemagnitudeofpricedropsincrease.Theconditionofthe
offersthataregranted,regardlessofthepricedropstoother MFC clause is on the amount ordered in the first period.
customers). In this particular event, S−D is about 0.9321% Consequently, the SCE and DCE heuristics behave the same
of S. That is, regular discount offers lead to 0.9321% cost wayinthefirstperiod.Thedynamicheuristichasthepossibility
savings. Let D*betheoptimal expectedprocurementspend to correctitsdecisionbyrecourse inthe secondperiod.When
whenMFCclausesarealsoused.Thenwedenotetheeffect γ is small, neither the stochastic optimization model nor the
ofMFCtermsas heuristics choose to fulfill the condition of the MFC clause,
D-D(cid:4) leading to zero optimality gap for the heuristics. However,
J(cid:4) ¼100´ S-D : (25) when γ is large enough so that it is optimal to satisfy the
conditionoftheMFCclause,thenthegainobtainedusingthe
That is, J* measures the additional benefit of considering
stochastic optimization model is greater than that of the SCE
MFC clauses as a percentage of savings through regular
heuristic in all cases. The DCE heuristic obtains the same
discountoffers.
amountofreductionasourstochasticprogrammingmodelifit
Let D and D be the expected procurement spend if
SCE DCE optsforsatisfyingtheMFCconditioninthefirstperiod.Hence,
SCE and DCE heuristics are used, respectively. Then, we
thereisadifferencebetweenthecostreductionsobtainedwith
measure the regrets of these heuristics (given that MFC terms
thesetwomethodsonlywhenthestochasticmodelchoosesto
providesavings,ie,D*<D)asfollows
satisfy the MFC conditionand the dynamicheuristic doesnot
(when γ=0.5 for a 5% decrease and when γ=0.2 for a 10%
D -D(cid:4) D -D(cid:4)
Δ ¼100´ SCE Δ ¼100´ DCE : (26) decrease).Inthesecases,Δ =100%.Onefinalobservationis
SCE D-D(cid:4) DCE D-D(cid:4) DCE
thattheperformanceoftheSCEheuristicisalwaysworsethan
orequaltothatoftheDCEheuristic.
Wefirstconsidertwo-periodinstanceswithdifferentscenario Wecanalsouseourmodeltofullycharacterizetheregionsin
trees.Inoursimplestexperiment,twoscenariosareconsidered: whichthebuyershouldoptforthesupplier’sMFCclauseand
thepricesdropbyπunitsinPeriod2withprobabilityγorthe procure enough to satisfy the condition for MFC status. This
AlperŞenetal—Multi-periodsupplierselectionunderpriceuncertainty 1645
Figure4 ScenariotreeforExperiment2.
Table4 ResultsforExperiment2
γ γ J* Δ Δ γ γ J* Δ Δ
1 2 SCE DCE 1 2 SCE DCE
0.1 0.1 0.00 0.00 0.00 0.3 0.3 43.53 30.36 0.00
0.2 0.1 0.00 0.00 0.00 0.4 0.3 50.71 22.59 0.00
Figure 3 Indifference curve for selecting the MFC supplier— 0.3 0.1 7.18 100.00 100.00 0.5 0.3 57.89 16.75 0.00
Experiment1. 0.4 0.1 14.36 75.66 0.00 0.1 0.4 47.35 35.98 0.00
0.5 0.1 21.54 42.26 0.00 0.2 0.4 54.53 28.01 0.00
0.1 0.2 10.99 100.00 100.00 0.3 0.4 61.71 21.90 0.00
0.2 0.2 18.17 80.78 0.00 0.4 0.4 68.89 14.85 0.00
can be accomplished by solving the model repeatedly to 0.3 0.2 25.35 50.96 0.00 0.5 0.4 76.07 8.00 0.00
generate indifference curves. Figure 3 shows the indifference 0.4 0.2 32.53 34.30 0.00 0.1 0.5 65.53 26.45 0.00
curve for Experiment 1. For every value of γ, we find the 0.5 0.2 39.71 23.67 0.00 0.2 0.5 72.71 19.32 0.00
0.1 0.3 29.17 57.38 0.00 0.3 0.5 79.89 12.40 0.00
percentageprice dropat which buying enoughfrom the MFC
0.2 0.3 36.35 41.20 0.00 0.4 0.5 87.07 6.62 0.00
supplier to qualify for a discount would lead to expected cost Avg 45.51 38.61 9.09
thatisequaltowhatthebuyerwouldgetbyonlyusingregular
discount offers. This curve can be used to guide the buyer’s
decision,giventheestimatesfortheprobabilityandmagnitude
of a price drop. On the indifference curve (eg, when the
probabilityofpricedropis0.4andmagnitudeofpricedropis
5.54%), the buyer is indifferent between satisfying the MFC
conditionandnot.Iftheprobabilityand/orthemagnitudeofthe
price drop are higher, the buyer should satisfy the MFC
condition.
We next run another two-period experiment in which the
pricesmaydropindifferentamounts:5%or10%oftheoriginal
price.ThescenariotreeforthisexperimentisgiveninFigure4.
The results for various values of γ and γ are provided in
1 2
Table 4.Once again,the possibility of a MFC statusprovides
important benefits. The performances of the SCE and DCE
heuristics are similar to those seen for Experiment 1. SCE
heuristic may lead to significant suboptimality in a variety of Figure5 ScenariotreeforExperiment3.
settings especially when the price drop probability is small.
DCEheuristic,ontheotherhand,captureseitherallornoneof
theadditionalsavingspossiblewithMFCterms. Our final experiment models three periods. Reflecting the
Inthethirdexperiment,wegrouptheitemsintotwogroups regularpatterninpractice,thedemandineachoftheperiods1
and assume that each group’s price drops or stays the same and2isassumedtobe30%ofthetotaldemand,whiledemand
independentlyfromthe other group. The scenariotreeforthis inPeriod3isassumedtobe40%.Figure6showsthescenario
experimentisgiveninFigure5. treeforthisexperiment.InordertoobtaintheMFCstatusand
Theresultsforvariousvaluesofγ ,γ andγ areprovidedin benefit from a possible price drop in Period 2, a minimum
1 2 3
Table5.Theresultsarenotstructurallydifferentfromthefirst amountshouldbepurchasedinPeriod1.Inordertoobtainthe
two experiments, except that we now have more instances MFCstatusandbenefitfromapossiblepricedropinperiod3,
where the certainty-equivalent heuristic cannot capture any of the sum of purchases in periods 2 and 3 should be above
thebenefitsofMFCterms. anotherthreshold.