Supply chain focus dependent supplier selection problem

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G. Vanteddu et al. / Int. J. Production Economics 129 (2011) 204–216 207(Axsäter, 1993; Graves, 1985; Lee and Moinzadeh, 1987a, b). Forcapacitated models using a modified base stock policy, the readercan refer to Glasserman and Tayur (1995, 1996) and Kapuscinskiand Tayur (1999). For a thorough comparison of installation andechelon stock policies for multi-level inventory control, the readeris referred to Axsäter and Rosling (1993). They primarily considerserial and assembly systems and prove that for (Q, r) rules echelonstock policies are, in general, superior to installation stock policies.Advocating the necessity of models that include both costand responsiveness, which are the primary order winners in ourstrategic model, Moon and Choi (1998) suggest extending the leadtime reduction concept to other inventory models to justify theinvestment to reduce the lead times. Choi (1994) used an expeditingcost function to reduce the variance of supplier’s lead times. Asopposed to network design models that focus on the tradeoffbetween the fixed costs of locating facilities and variabletransportation costs between facilities and customers, Sourirajanet al. (2007) present a model for single product distribution networkdesign problem with lead times and service level requirements,which enables them to capture the tradeoff between lead times andinventory risk pooling benefits. The objective is to locate distributioncenters (DCs) in the network such that the sum of the location andinventory (pipeline and safety stocks) costs is minimized.One very interesting (Q, r) model with stochastic lead times thatcould serve as a building block in supply chain management isproposed by Bookbinder and Cakanyildirim (1999) as opposed toconstant lead time assumption in many other studies. Thedimensionless quantity CIR proposed in our model is similar in spiritto the ‘‘expediting factor’’ for the lead time proposed by them. Ryu andLee (2003) consider dual sourcing models with stochastic lead times inwhich lead times are reduced at a cost that can be viewed as aninvestment. They make use of the concept of ‘‘expediting factors’’proposed by Bookbinder and Cakanyildirim (1999) in their model. Theyanalyze (Q, r) models with and without lead time reduction andcompare the expected total cost per unit time for the two models.Even though, we did not consider the product mix flexibilityrelated issues in our model, the reader can refer to Upton (1997) forexploring the relationship between process range flexibility andstructure, infrastructure and managerial policy at the plant level.We assume information symmetry at all the stages in our model.The effect of information sharing for time series structure of thedemand on safety stocks is addressed in Gaur et al. (2005). Eventhough we did not address environmental issues, reader can referto Wolf and Seuring (2010) who analyze whether environmentalissues form a supplier selection criteria of companies when sourcingthird party logistics (3PL) services.3. Model developmentAt any stage in supply chain primarily there are two types ofinventory elements namely cycle inventory (in-process inventory)and safety inventory. The quantity of cycle inventory depends onthe average demand per period and the average cycle (processing)time and is given by the multiplied product of those two quantities.Among inventory cost elements, safety stock, which is maintainedto account for the internal and external variability in a supply chain,is vital in the sense that it directly affects customer satisfaction andsafety stock costs also constitute a significant portion of the cost ofgoods at a stage. We are considering both cycle stocks (in-processinventory) and safety stocks in our model. Responsiveness relatedcosts are the costs incurred on account of the volume relatedflexibility at any stage in a supply chain, the inclusion which intothe model is made possible by the inclusion of a new parametercalled coefficient of inverse responsiveness (CIR). The inclusion ofCIR also facilitates the explanation of the interrelationship betweeninventory costs and the responsiveness related costs.At any stage in a supply chain,Total cost¼Safety stock costs+Cycle stock costs+Responsivenessrelated costsThis section is devoted to developing total cost expression,made up of safety stock costs, cycle stock costs and responsivenessrelated costs, for a serial supply chain.3.1. Expression for safety stock costsThe content in Sections 3.1 and 3.2 is largely adopted fromVanteddu et al. (2007).We follow the building block model (Gravesand Willems, 2003) with installation base stock policies and acommon underlying review period for all stages. A typical basestock policy works as follows.When the inventory position (i.e., onhand plus on order minus back orders) at stage i falls below somespecified base stock level B i , the stage places a replenishment orderthereby keeping the inventory position constant. Simchi-Levi andZhao (2005) attribute the popularity of base stock policy to the factthat it is simple, easily implementable, and because this policy hasbeen proven to be optimal or close to optimal in many cases. Forexample, in serial supply chains with zero setup costs and withoutcapacity constraints, because the installation base stock policy isequivalent to an echelon base stock policy under certain initialconditions (Axsäter and Rosling, 1993), it is indeed optimal in thesecases (Clark and Scarf, 1960).In serial systems, even modified basestock policy with capacity constraints is still close to optimal (Speckand vander Wal, 1991; Van Houtum et al., 1996)In an installation policy, each facility only needs the inputs fromthe immediate upstream (U/S) and downstream (D/S) facilities andmakes ordering decisions based on its local order and inventorystatus (Simchi-Levi and Zhao, 2005) as opposed to an echelon basestockpolicy, which is a centralized control scheme that allows for acentral decision maker to coordinate and control the actions at allstages in the SC (Graves and Willems, 2003).Even though our model assumes all the stages to be manufacturingstages, without loss of generality, a stage could be modeledas a distribution center (DC) as well. A pure transportation functioncan also be modeled with the building block concept, wherein thetransport time is the lead time with pipeline inventories.Orders are placed at discrete time intervals and each stage isconsidered as a building block (Graves, 1988) that generates astochastic lead time. A building block is typically a processor plus astock keeping facility (Simchi-Levi and Zhao, 2005). Depending onthe scope and granularity of the analysis being performed, the stagecould represent anything from a single step in manufacturing ordistribution process to a collection of such steps to an entireassembly and test operation (Graves and Willems, 2003). Demandis assumed to be stationary and uncorrelated across different timeperiods with no capacity constraints.Our model is designed as a decentralized supply chain (Gravesand Willems, 2003; Lee and Billington, 1993) to mimic the realitymore closely with each stage following a local base-stock policy.Buttressing the same view, Lee and Billington (1993) state thatorganizational barriers and restricted information flows betweenstages may result in complete centralized control of material flowin a supply chain to be not feasible or desirable.The primary distinction between centralized and decentralizedsupply chain is put in the following succinct form by Lee andBillington (1993) ‘‘Centralized control means that decisions on howmuch and when to produce are made centrally, based on materialand demand status of the entire system. Decentralized control, onthe other hand, refers to cases where each individual unit in thesupply chain makes decisions based on local information’’.
208G. Vanteddu et al. / Int. J. Production Economics 129 (2011) 204–216Assuming each building block operates independently using asimple installation policy, one can first characterize various buildingblocks such as serial, assembly, distribution, etc. and thenidentify the links among these building blocks (Simchi-Levi andZhao, 2005).We have chosen series system for the simplicity ofanalysis and primarily to develop certain insights that areinsensitive to the specific supply chain topology. Also, othernetworks such as assembly system can be reduced to anequivalent series system (Rosling, 1989). Most of the featuresare similar to the features of a serial system presented in Gallegoand Zipkin (1999) with some modifications.There are several stages or stocking points arranged in series.Thefirst stage receives supplies from an external source.Demand occurs only at the last stage. Demands that cannot befilled are immediately backlogged. There is one product, or moreprecisely, one per stage. To move units to a stage from itspredecessor, the goods must pass through a supply systemrepresenting production or warehousing activities. There is aninventory holding cost at each stage and our model does notconsider back order penalty cost, which could be easily included.The horizon is finite, all data are stationary, information iscentralized but control is decentralized.As in Gallego and Zipkin (1999), the numbering of the stagesfollows the flow of goods; stage one is the first and at the last stagedemand occurs. The external source, which supplies stage one hasample stock and it responds immediately to orders.We have assumed that the service level targets required at eachof the players are exogenous, i.e. dictated by the immediate D/Splayer or the final customer. Following Graves and Willems (2003)treatment of stochastic service model in supply chains, LetFðk 1 Þ,...,Fðk n Þ be the service levels for corresponding safetyfactors k 1 ,...,k n where F(k i ) represents the cumulativedistribution function for a standard normal variable. k j is themultiplying factor for the variability term for stage j, whichalong with the average in-process inventory would give theexpected base stock to achieve a given service level. Let theprocessing time at stage j be a random variable t j (measured intime periods) with mean L j and variance s j 2 .The stochastic service model (Graves and Willems, 2003;Simchi-Levi and Zhao, 2005; Lee and Billington, 1993; Ettl et al.,2000) assumes delivery or service time between stages to varybased on the material availability at that stage and each stage in thesupply chain maintains a base stock sufficient to meet its servicelevel target (Graves and Willems, 2003). If D i is the random delayat the preceding stage i, then the replenishment cycle time atstage j equalsg j ¼ t j þD iWe have adopted the procedure for calculating this delay due tothe stock out at the preceding stage as presented in Graves andWillems (2003) and Ettl et al.(2000) with a modification that takesinto account the fact that there is only one player at the precedingstage. We are assuming that the expected value of this delay issimply equivalent to the probability of stock out at the precedingstage p i times its average processing time.Therefore, expected replenishment cycle time at stage j is given byE½g j Š¼L j þp i L ið2Þwherep i ¼ 1 Fðk i Þ ð3ÞAssuming that the demand is Nðm,s 2 Þ, where m is the meandemand during one time period and s 2 is the demand varianceduring one time period, to satisfy average demand m, given theð1Þaverage replenishment cycle time from (2), ‘average cycle stock’ isgiven bymðL j þð1 Fðk i ÞÞL i Þ ð4ÞAssuming the independence of processing times at a stage andbetween the stages we realize thats 2 D i¼ s 2 i ð1 Fðk iÞÞ ð5Þs 2 g j¼ s 2 j þs2 i ð1 Fðk iÞÞ ð6ÞWhen the demands are uncorrelated between time periods,then the mean and variance of the demand during replenishmentperiod for stage j denoted by the continuous random variable W J areobtained as follows by slightly modifying the equations to accountfor the portion of the lead time variability transferred from thepreceding stage (see for example, Eppen and Martin, 1988; Feller,1960).m Wj¼ mE½g j Šs 2 W j¼ s 2 E½g j Šþs 2 g jm 2Now, we introduce another parameter ‘c j ’ termed as the‘coefficient of inverse responsiveness (CIR)’ defined as the ratioof the average demand to the rate of production (throughput) p j .c j ¼ m=p jð8ÞAssuming that there is enough capacity at all the stages tosatisfy a given demand, c j r1 at all the times. When c j ¼1, expectedreplenishment cycle time is given by (2). We also assume that thereis some upper bound above which rate of production cannot beincreased further. CIR at a stage is similar to the ‘‘expediting factor’’proposed by Bookbinder and Cakanyildirim (1999). They define the‘‘expediting factor’’ t as the constant of proportionality betweenrandom variables T ~ (the expedited lead time) and T (ordinary leadtime). For expedited orders (to1) shorter than average lead timecan be obtained at a cost. Similarly, longer mean lead time results ina rebate for the customer when (t41). By considering a model withthree decision variables (Q, r, t), they show that the expected costper unit time is jointly convex in the decision variables and obtainthe global best solution.Keeping the average cycle stock constant (given in (4)), fromLittle’s law, at expedited rates of production (c j o1), replenishmentcycle time mean and variability for player j, when operating at a CIRlevel c j will be equivalent toE½g j Š¼ðL j þð1 Fðk i ÞÞL i Þc j ð9Þs 2 g j¼ðs 2 j þs2 i ð1 Fðk iÞÞÞc j ð10ÞAs c j decreases, average replenishment cycle time for player jdecreases, which means player j is becoming more responsive.Because of this inverse relationship between c j and responsiveness atstage j, c j is termed as ‘coefficient of inverse responsiveness’ at stage j.In light of Eqs. (9) and (10), we assume that demand duringreplenishment period W j for stage j, is normally distributed as follows:m Wj¼ mE½g j Šqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis Wj ¼ s 2 E½g j Šþðs 2 g jÞm 2ð7Þð11ÞThe above equation considers both the demand variability andthe replenishment cycle time variability. Replenishment cycle time
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