Performance Analysis of Order Fulfillment for Low Demand Items in E-tailing

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1 Performance Analyss of Order Fulfllment for Low Demand Items n E-talng Pallav Chhaochhra, Stephen C Graves Massachusetts Insttute of Technology Abstract We study nventory allocaton and order fulfllment polces among warehouses for low-demand SKUs at an onlne retaler. A large e-taler strategcally stocks nventory for SKUs wth low demand. The motvatons are to provde a wde range of selectons and faster customer fulfllment servce. We assume the e-taler has the technologcal capablty to manage and control the nventory globally: all warehouses act as one to serve the global demand smultaneously. The e-taler wll utlze ts entre nventory, regardless of locaton, to serve demand. Thus, gven the global demand and an order fulfllment polcy, there are tradeoffs nvolvng nventory holdng costs, transshpment costs, and backorderng costs n determnng the optmal system nventory level and allocaton of nventory to warehouses. For the case of Posson demand and constant lead tme, we develop methods to approxmate the key system performance metrcs lke transshpment, backorders and average system nventory. We then use these results to develop gudelnes for nventory stockng and order fulfllment polces for onlne retalers. Index Terms Inventory allocaton, order fulfllment, low demand SKU, onlne retalers. I. ITRODUCTIO A large e-taler strategcally stocks nventory for SKUs wth low demand for several reasons. Xu [] lsts the followng reasons. One motvaton s to provde a wde range of selectons, snce such SKUs actually consttute a sgnfcant porton of the total SKUs. The second ncentve, of course, s to provde faster customer fulfllment servce. The thrd motvaton s to gan a compettve advantage from other onlne retaler. Suppose that an e-taler only drop-shps the low-demand SKUs, ts drop-shpper who serves many onlne retalers, may choose to satsfy a compettor s demand. For many of these SKUs, the e-taler may only stock a handful of nventory unts across all warehouses. Inventory plannng for low-demand SKUs s challengng because the dscrete effect s much more pronounced whle Manuscrpt receved ovember 8, Ths work was supported by the Sngapore MIT Allance. Stephen C Graves s the Abraham J Segel Professor of Management Scence at the Massachusetts Insttute of Technology, Cambrdge, MA 0239 USA (e-mal: sgraves@ mt.edu). Pallav Chhaochhra s a doctoral student at the Operatons Research Center, Massachusetts Insttute of Technology, Cambrdge, MA 0239 USA (e-mal: pallavc@ mt.edu). the current nventory models often assume all varables are contnuous. Effcent nventory plannng and order fulfllment for low-demand SKUs s mportant n the retalng settng. Often over 90% of a retaler s catalog comprses slow movng SKUs wth demand n the range of unts per week. Therefore, the mpact of nventory plannng for low-demand SKUs s very sgnfcant Research by Xu [], focused on the effect of nventory allocaton on outbound transportaton costs. Her model envsons an e-taler wth several warehouses n the system, wth the technologcal capablty to manage and control the nventory globally: all warehouses act as one to serve the global demand smultaneously. Specfcally, the e-taler wll utlze ts entre nventory, regardless of locaton, to serve demand. Gven that the e-taler stocks a certan number of unts of nventory n the system, Xu studed how best to allocate nventory to warehouses by consderng outbound transportaton costs from the warehouses to customers. Ther approach produced exact solutons for the 2-unt 2-locaton case, but was not tractable for the general mult-unt mult-locaton case. We extend the work done n [] by developng methods to calculate metrcs lke transshpment, backorders and average system nventory for specal cases of demand dstrbuton over the locatons. We use these exact results to develop an approxmaton for these performance metrcs for the case of a general demand dstrbuton over the locatons. We then use these results to develop gudelnes for nventory stockng and order fulfllment polces for onlne retalers. II. MODEL We start wth the -Unt -Locaton problem for a sngle tem. Suppose the e-taler decdes to stock exactly one unt at each of warehouses n the system. We want to fnd methods to estmate key performance metrcs lke transshpments, backorders and average system nventory for an -Unt -Locaton problem. We start wth the followng assumptons.

2 A-The system demand process s Posson wth rate λ. A-2 The demand process s splt nto ndependent processes, to. Wth probabltyα, a demand arrval s from regon ; wth probabltyα 2, a demand arrval s from regon 2; and so on. The α are non-negatve and sum to. A-3 The replenshment lead-tme for each warehouse s the same constant L. A-4 The nventory polcy s one-for-one replenshment at each warehouse: a replenshment s trggered at each demand epoch. A-5 Demand s backlogged when there s no on-hand nventory n the system. In the context of onlne retalng, the e-taler can utlze any warehouse or fulfllment center to serve the customer demand. Specfcally, a demand s always served by an onhand nventory unt n the system f there s any; f there are no on-hand nventory unts n the system, the demand s served by and trggers replenshment at the warehouse that has the next arrvng replenshment. We then have the followng assumptons on how the system operates for all stockng scenaros. A-6 When a customer arrves and ts closest warehouse has on-hand nventory, then ts closest warehouse serves the demand and trggers a replenshment. A-7 When a customer arrves and ts closest warehouse does not have nventory on-hand, the system wll assgn the demand to another warehouse f there s on-hand nventory elsewhere n the system. A warehouse wth onhand unt s chosen accordng to an order fulfllment polcy, P, to serve demand; ths assgnment trggers replenshment for the chosen warehouse. A-8 If a customer arrves and the system has no on-hand unts, then the polcy s to assgn the demand to the warehouse wth the next arrvng unassgned replenshment. Ths assgnment trggers another replenshment for the chosen warehouse. ote that assumpton A-8 s possble because we assume determnstc supply lead-tmes, so we know exactly when all future replenshments arrve. Also, assumpton A-7 and A-8 are analogous to an emergency transshpment. We defne the followng probablstc events: F, : Event that an order from regon s flled mmedately from on-hand stock at warehouse B, : Event that an order from regon s backordered and flled subsequently from a replenshment to warehouse A, : Event that an order from regon s flled from warehouse. Hence, A, = F, B, The system performance metrcs such as the fll rate and average nventory can be calculated for general cases f D L s known. System fll rate = - Pr[D L ] Average system nventory = Pr[ SI = ] = However, the amount of transshpment depends on the demand dstrbuton across the regons and the order fulfllment polcy n the system. Sectons III and IV estmate the transshpment for specal cases of balanced demand and extreme demand dstrbuton for a specfc order fulfllment polcy. We specfy a servce falure metrc and a method to estmate transshpment for general demand dstrbuton n Secton V. In Secton VI, we compare the performance of varous order fulfllment polces. We dscuss future research drectons and conclusons n Secton VII. III. BALACED DEMAD CASE We frst analyze the case n whch each of the - warehouses faces a demand rate of λ/ from ts local regon. We consder the order fulfllment polcy as stated n A-6 and A-7, wth the feature that f a customer arrves and ts closest warehouse does not have nventory on-hand, but one or more of the other warehouses do have nventory on hand, then a warehouse wth on-hand unt s randomly chosen (wth equal probablty) to fll the order. We call ths polcy P. Result : The probablty that an order from regon s flled from warehouse mmedately from stock under the above condton s gven by: otaton D L : Random varable for the system demand over the lead tme. E[D L ]= λl SI : Random varable for the on-hand system nventory SI = ( - D L ) +

3 ,, k = F = F SI = k Pr SI = k e where Pr SI = k = D = k = Pr k k Pr F, SI k = = = k for = k F, SI = k = for λl ( λl) k! k The probablty that an order from regon s backordered and flled from a replenshment to warehouse under the above condton s gven by:,, The results follow from applcaton of the Total Probablty Theorem, and propertes of the Posson process. The key nsght used here s that snce demand arrval s Posson wth equal rates for each regon, for a gven level of system nventory, each nventory state s equally lkely. For example, f =3 and SI=2, then the nventory states (,,0), (,0,) and (0,,) are equally lkely to occur. Hence, condtoned on SI=2, we can argue that the probablty that a demand from regon s served from warehouse s F, SI = 2 = 2 3, as ths wll happen for nventory states (,,0) and (,0,). Smlarly, condtoned on SI=2, a demand from regon s served by warehouse 2 or 3 only f the nventory state s (0,,), where each warehouse has an equal probablty; thus, we have F,2 SI = 2 = F,3 SI = 2 = 6. A customer order s back-ordered f and only f none of the warehouses n the system has nventory,.e, SI=0. Agan, snce demand s Posson wth equal rates for each regon, when the system nventory s 0, each warehouse s equally lkely to have the next arrvng unassgned replenshment unt. Thus: B, SI = 0 = Wth the above results, we can fnd the probablty a demand n regon s served by a transshpment from warehouse : Pr[F,]+Pr[B,] for. L B = B SI = 0 Pr SI = 0 [ = ] = [ ] where Pr SI 0 Pr DL and B, SI = 0 = IV. EXTREME DEMAD CASE We now suppose that all demand orgnates from one regon, e.g., α =, whle α = 0 for =2,.. We now analyze the case where one of the - warehouses faces a demand rate of λ from ts local regon, whle the other warehouses do not face any demand but stll carry nventory. The order fulfllment polcy for ths analyss s P, as descrbed n Secton III. Consder the demand arrval process wthα =. We defne a renewal as occurrng whenever an order s flled by warehouse ether mmedately from stock or as a backorder. We defne the nter-renewal nterval (M t ) as the number of demands that occur between renewal epochs. Then the countng process that looks at the number of orders served by warehouse, s a renewal process, and M t are IID RVs for renewals occurrng at t. Result 2: The probablty that an order from regon s flled from warehouse mmedately from stock under the above condton s gven by: F, = F, A, A, where F, A, = Pr[ DL < ] A =, + E[ M] = Pr[ L = ] + ( ) Pr[ L ] E M k D k D k = 0 The probablty that an order from regon s backordered and flled from a replenshment to warehouse under the above condton s gven by: B, = B, A, A, where B, A, = F, A, = Pr DL < A =, + E[ M] = Pr[ L = ] + ( ) Pr[ L ] E M k D k D k = 0 The results follow from applcaton of the Total Probablty Theorem and Renewal-Reward Theory [2]. Defne a reward functon, R(n) = f order n s served by warehouse. Then, by the Key Renewal Theorem,

4 where X E[ R( n)] Lm E[ R( t)] = n X s the average nter - renewal nterval, X = + E( M ) However, E[R(t)] s the expected rate of reward accumulaton, whch n ths model, s the probablty of an order beng assgned to warehouse to be fulflled ether mmedately from stock or from a replenshment unt when t arrves. Hence, the result follows. For the other warehouses, we can show that for : Pr[ DL < ] Pr F, F, =. ( Pr[ DL < ] ) Pr B, B, = As explanaton, we note that the fll rate from the nonlocal warehouses to serve demand n regon equals the system fll rate, net of the fll rate from warehouse ; by symmetry, we then dvde the total fll rate assocated wth the non-local warehouses equally across these - warehouses. Smlarly we fnd the probablty that a nonlocal warehouse serves a backordered demand from regon. Wth the above results, we can fnd the probablty a demand n regon s served by a transshpment from warehouse : Pr[F,]+Pr[B,] for. V. ITERPOLATIO METHOD For other cases of demand, exact solutons could not be found ether by smlar methods or usng the method n []. We expect that Pr[F,] to be monotoncally decreasng wth α for a gven system demand rate, λ (provded that demand n all other regons remans proportonal, and the order fulfllment polcy n place s P ). Thus, we propose to approxmate the Pr[F,] for other cases of demand usng some form of monotonc nterpolaton Usng the known results for α = and α =, we consdered a lnear nterpolaton and exponental nterpolaton approxmaton for Pr[F,] usng the values of Pr[F,] for the balanced and extreme demand dstrbuton cases. We compared ths approxmaton wth Monte-Carlo smulaton results for Pr[F,]. A sample case for a 4-unt 4-warehouse scenaro wth λ= and L=3 s shown below: TABLE I: COMPARISO OF SIMULATIO AD ITERPOLATIO RESULTS α - Regon orders flled from WH stock Smulaton Lnear approx Exp approx Regon orders flled from stock α Smulaton Lnear approx Exp approx Fg : Comparson of Smulaton and Interpolaton Results We observe that the exponental approxmaton performs slghtly better than the lnear approxmaton. In general, the approxmatons are both reasonably good, wthn 5% of the smulaton results. Furthermore, the error seems systematc wth the approxmaton overestmatng Pr[F,] when α [,] and underestmatng Pr[F,] when α [ 0, ]. However, ths approxmaton method does not account for the effects due to the demand dstrbuton across the warehouses. For nstance, consder a 3-unt 3-locaton scenaro. If α = 0.33, α2 = 0.67, α3 = 0, then the F, α = 0.33, α2 = 0.67, α3 = 0 s clearly not equal to F, α = 0.33, α2 = 0.33, α3 = We expect that ths approxmaton performs bests when local demand faced at the other warehouses s equal. If demand at other warehouses s not equal, then under order fulfllment polcy P, the approxmaton overestmates the probablty of local order fulfllment from stock. We defne another performance metrc, Servce Falure, as the probablty that an order s not flled mmedately by ts local warehouse. Servce Falure for regon, SF = - Pr[F,] Servce Falure for system, SF α Pr[ = F, ] We can estmate the Servce Falure for the system qute accurately usng the above formula despte the errors n estmatng Pr[F,]. Ths s due to the cancellaton of the systematc errors n the approxmaton of Pr[F,] as some α [,] whle other α [ 0, ]. We approxmate the probablty of a backorder flled by ts local warehouse, Pr[B,], as beng almost equal for each warehouse n the system, then:

5 Pr[ B, ] Pr[ DL ] Thus, we can estmate the probablty of transshpment for each regon and for the system as: TS SF B = Pr, α TS = TS = We compared these estmates of Servce Falure and Transshpment for the system wth Monte-Carlo smulaton results obtaned over dfferent Fll Rates, system confguratons and demand dstrbuton spreads. We fnd a generally good ft wth errors below 5% for the most part. TABLE II: SERVICE FAILURE APPROXIMATIO ERROR PERCETAGE (RELATIVE TO SIMULATIO RESULTS) Statstcs taken over 00 smulaton runs of 500k orders Low Fll Rate 3U-3L 4U-4L 5U-5L 3U-3L 4U-4L 5U-5L 3U-3L 4U-4L 5U-5L 99% % % % % TABLE III: TRASSHIPMET APPROXIMATIO ERROR PERCETAGE (RELATIVE TO SIMULATIO RESULTS) Statstcs taken over 00 smulaton runs of 500k orders Low Medum Medum Hgh Hgh Fll Rate 3U-3L 4U-4L 5U-5L 3U-3L 4U-4L 5U-5L 3U-3L 4U-4L 5U-5L 99% % % % % warehouses do have nventory on hand, then a warehouse wth an on-hand unt s chosen by a certan rule to fll the order. In the frst type of polcy, say P 2, the warehouse s randomly chosen but wth hgher probabltes for warehouses facng lower local demand rates. In the second type of polcy, say P 3, the warehouse s chosen from a prorty lst that orders the warehouses accordng to ther local demand rates, wth lower local demand havng a hgher prorty. Results from Monte-Carlo smulaton ndcate that polcy P 3 has the best system-wde performance n terms of total transshpments but results n more varablty n the local performance measures. Overall, there was not a sgnfcant dfference n system performance between these polces. VII. COCLUSIO We have extended the work done n [] by developng methods to calculate performance metrcs lke transshpment, backorders and average system nventory for specal cases of the demand dstrbuton across the locatons. We have also developed approxmatons for these metrcs n the case of general demand dstrbuton across the locatons for -unts -locatons. Comparng these approxmatons wth Monte-Carlo smulaton results ndcate that these are good estmates. We are currently usng these performance metrcs to develop gudelnes for nventory stockng and order fulfllment polces for onlne retalers,.e, gven a certan system demand rate and lead tme, we ntend to develop gudelnes for the optmal nventory holdng confguraton across the warehouses and the optmal order fulfllment polcy to servce demand. REFERECES [] P. Xu, Order Fulfllment n Onlne Retalng: What goes where, Doctoral Dssertaton at MIT, pp , September [2] R. G. Gallager, "Dscrete Stochastc Processes," pp. 57-8, Kluwer Academc Publshers, 996 VI. ORDER FULFILLMET POLICIES We have only consdered the order fulfllment polcy, P, n the prevous cases due to ts tractablty. However, we fnd that other order fulfllment polces can perform better than P n terms of reducng the Transshpments n the system, wthout affectng the Average System Inventory and Backorders. We consder two other types of Order Fulfllment polces that have the same rules as stated n A-6 and A-7, but now wth a dfferent fulfllment polcy when a customer arrves and ts closest warehouse does not have nventory on-hand. When one or more of the other