Ann Oper Res (217) 259:217 239 DOI 1.17/s1479-17-2542-z ORIGINAL PAPER Quantfyng sustanable control of nventory systems wth non-lnear backorder costs Lna Johansson 1 Fredrk Olsson 1 Publshed onlne: 3 May 217 The Author(s) 217. Ths artcle s an open access publcaton Abstract Tradtonally, when optmzng base-stock levels n spare parts nventory systems, t s common to base the decsons ether on a lnear shortage cost or on a certan target fll rate. However, n many practcal settngs the shortage cost s a non-lnear functon of the customer watng tme. In partcular, there may exst contracts between the spare parts provder and the customer, where the provder s oblged to pay a fxed penalty fee f the spare part s not delvered wthn a certan tme wndow. We consder a two-echelon nventory system wth one central warehouse and multple local stes. Focusng on spare parts products, we assume contnuous revew base stock polces. We frst consder a fxed backorder cost whenever a customer s tme n backorder exceeds a prescrbed tme lmt, second a general non-lnear backorder cost as a functon of the customer watng tme, and thrd a tme wndow servce constrant. We show from a sustanablty perspectve how our model may be used for evaluatng the expected CO 2 emssons assocated wth not satsfyng the customer demands on tme. Fnally, we generalze some known nventory models by dervng exact closed form expressons of nventory level dstrbutons. Keywords Inventory Mult-echelon Non-lnear backorder cost Tme wndow servce constrant Sustanablty 1 Introducton and lterature revew In many ndustres spare parts and aftersales n general are bg busness (Cohen et al. 26). For example, the US automotve aftermarket was estmated to be worth $188.6 bllon n 27 (US Automotve Parts Industry Annual Assessment 29). Another example s the B Fredrk Olsson fredrk.olsson@ml.lth.se Lna Johansson lna.johansson@ml.lth.se 1 Department of Industral Management and Logstcs, Lund Unversty, P.O. Box 118, 221 Lund, Sweden
218 Ann Oper Res (217) 259:217 239 avaton ndustry, whch stocks spare parts for several bllon US-dollars (Harrngton 27). However, although t may be very costly to nvest n expensve spare parts, t s crucal to have spare parts avalable when needed. Obvously, delays and downtmes of bottleneck producton equpment may be very costly. A partcularly strkng example s n the ol rg ndustry, where the producton downtme on an ol rg may ncur a cost of $2, per day (Turban 1988). For a general overvew of models for spare parts nventory control see, e.g., Basten and van Houtum (214). We consder a two-echelon nventory system wth N locatons and one central warehouse. All locatons and the warehouse apply (S 1, S) polces (.e., contnuous revew base stock control), whch s reasonable for low demand tems such as spare parts. Ths paper extends the lterature on spare parts nventory control n several new drectons. Three dfferent backorder/servce level structures are nvestgated n ths paper. In more precse terms, we consder: (1) a model wth pecewse constant backorder costs, (2) a model wth general non-lnear backorder costs, and fnally (3) a model wth tme wndow servce constrants. The motvaton for consderng these three cases stems from collaboraton and dscussons wth ndustry, and the state of the current research fronter. In connecton wth case (1), one man focus s also to develop an approach that can be used n order to explctly evaluate the expected CO 2 emssons assocated wth not satsfyng the customer demands on tme. We provde an exact analyss for all cases consdered. Consderng case (1), all unsatsfed demands are backordered and the customers at each locaton are satsfed f ther demands are met drectly or after an acceptable watng tme ω. That s, f a customer receves the requested unt wthn ths acceptable watng tme, no penalty cost s ncurred. If a customer, on the other hand, has to wat longer than ths gven tme lmt, a consderable fxed backorder/penalty cost has to be pad. Ths backorder cost s ndependent of the addtonal watng tme exceedng ω. Ths partcular scenaro s qute common n many practcal settngs. One example s Tetra Pak Techncal Servces whch provdes customers n the dary ndustry wth spare parts for packagng machnes. In such a case, havng to wat for a crtcal spare part means that the producton process s halted. Whle a short downtme may be acceptable, t s more crucal f the customer at the producton ste has to wat for the spare part longer than a certan crtcal tme lmt. Then, f the crtcal tme lmt s volated, the whole batch of the dary product must be dscarded due to the pershable nature of the product. In stuatons lke ths one, t s qute common that there exsts a servce agreement (or contract) n whch t s stpulated that the servce provder should pay a fxed penalty f the spare part s not delvered wthn a certan tme wndow. In ths case, the fxed penalty cost s drectly assocated wth the cost of dscardng the batch of the dary product (and to some extent also the set up cost for a new batch). Other smlar examples can be found n the agrculture sector, where a whole (or parts of) harvest may be lost f the harvest-machnes are down due to mssng crtcal spare parts. Smlarly as n the case wth dary products, a lost harvest ncur a large, well defned, fxed cost. Another man contrbuton of ths paper s to take a frst step n quantfyng how decson rules for the logstcs system affect the expected CO 2 emssons from a producton waste perspectve. As noted n Marklund and Berlng (217), there are few models that nclude emssons assocated wth not satsfyng customer demand n a tmely fashon. Marklund and Berlng (217) argue that ths s partcularly accentuated n the dstrbuton of spare parts, and they explctly state: The parts are often qute small suggestng qute lmted emssons assocated wth transportaton, nventory holdng and warehousng. However, not delverng them promptly may have serous consequences on costs and emssons. In fact, most lterature concernng how to control supply chan systems n order to reduce (or at least quantfy) CO 2 emssons have prmarly focused on transportaton ssues, see e.g. McKnnon
Ann Oper Res (217) 259:217 239 219 (21) and Alkawaleet et al. (214). However, t s mportant to notce that supply chan polcy decsons also affect CO 2 emssons related to waste at the customer stes. For example, as noted above, a whole producton batch of a dary product may be wasted f the coordnaton of supply and demand for crtcal spares s not algned. Of course, there may exst cases where other types of backorder cost structures are more plausble than a model wth pecewse constant backorder costs. Therefore, n case (2) we generalze the structure of the backorder cost and focus on general non-lnear backorder costs as a functon of the watng tme. For many companes a qute long watng tme for a crtcal spare part may be severe, whle a relatvely short watng tme may not ncur very large costs. Hence, although our modelng technque can handle general non-lnear backorder costs, we focus on exponentally ncreasng backorder costs due to ts ntutve and appealng features from a practcal pont of vew. Another possble stuaton may be when the backorder cost s so called S-shaped. Ths means that the backorder cost as a functon of longer watng tmes s concave (nstead of convex as n the exponental case). We provde a model that can handle all such possble backorder cost structures (as mentoned, the exponental case should be vewed as a concrete example). In case (3) we consder a servce level nstead of a model wth backorder costs. Ths servce level s defned as the probablty of satsfyng a customer demand wthn a certan tme wndow. In all three cases we utlze nformaton about the tmng of outstandng orders for the central warehouse and the downstream locatons, respectvely. Early studes on contnuous revew mult-echelon nventory models nclude, e.g., Sherbrooke (1968), Graves (1985) andaxsäter (199). Sherbrooke (1968) consders a twoechelon nventory system wth multple local retalers and one central warehouse, all applyng (S 1, S) orderng polces where unsatsfed demands are backordered. Gven ths nventory system, he develops an approxmate method (the METRIC approxmaton), where the real stochastc leadtmes for the retalers are approxmated by ther mean values. Graves (1985) extends the results from Sherbrooke (1968) by dervng an exact soluton procedure. However, one of the man problems s that the soluton provded n Graves (1985) s not n closed form, whch means that the evaluaton of the nventory level probabltes wll be approxmate (due to necessary truncaton of nfnte seres). In order to ease the computatonal burden, Graves (1985) also presents correspondng approxmate dstrbutons. Gven the same nventory system as n Graves (1985), Axsäter (199) develops a dfferent exact soluton procedure. However, smlar to Graves (1985), Axsäter (199)does notether present a closed form soluton. Moreover, he does not derve the dstrbuton of the nventory levels for the local retalers. Instead, he derves a recursve soluton procedure n order to obtan the total average cost. In ths paper, we extend the analyss of Graves (1985) and Axsäter (199) by dervng closed form solutons of the probablty dstrbutons of the nventory levels and the customer watng tmes. For more nformaton concernng contnuous revew mult-echelon nventory systems see, e.g., Axsäter (1993). The lterature on mult-echelon nventory models wth tme wndow servce levels s relatvely lmted. Ettl et al. (2) consder a mult-echelon nventory system wth smlar servce requrements as n our model. They model ther nventory system as a M X /G/ queueng system, and consder both assembly and dstrbuton structures. Although Ettl et al. (2) n some aspects consder a more general model than we do, ther analyss requre assumptons lke leadtmes based on the exponental dstrbuton. Another related model s presented n Caggano et al. (27), where the problem n Ettl et al. (2) s extended to a mult-tem settng. A lmtaton of ths work s that the acceptable customer watng tmes at dfferent stes (ω n our case) are only allowed to be multples of the transportaton tmes between dfferent echelons. Another study on mult-echelon nventory models wth tme
22 Ann Oper Res (217) 259:217 239 wndow servce levels s Wong et al. (27), whch deals only wth average tme wndow constrants. One category of papers n the lterature that s qute related to our work s nventory models wth emergency supply. Two papers that fall nto ths category are Monzadeh and Schmdt (1991) and Monzadeh and Aggarwal (1997). The former studes a sngle locaton system, where the locaton apples an (S 1, S) polcy and has the opton to choose ether a normal order or an emergency order under a gven orderng polcy. Ths polcy takes all avalable nformaton regardng the nventory level and tmng of outstandng orders nto account. Monzadeh and Aggarwal (1997) extend the work done by Monzadeh and Schmdt (1991) by consderng a base-stock two-echelon nventory system wth one warehouse, multple retalers and an outsde suppler wth the possblty of emergency orders. Huang et al. (211) ntroduces a commtted servce tme, n whch t s acceptable for the customer to be served. After ths tme has passed, the retalers face a backorder cost (per unt and tme unt) and has the ablty to fll the demand wth an emergency order. In a more recent paper, Howard et al. (215) evaluate an approxmate two-echelon spare parts nventory model usng ppelne nformaton. In more detal, they consder an nventory system that ncludes a central warehouse actng as a suppler and a so called support warehouse. The purpose of the support warehouse s to provde emergency orders to the local warehouses, and as a last resort emergency transshpments can be sent drectly from the central warehouse. All of these papers mentoned above consder a standard unt backorder cost per tme unt. However, n our model we consder general non-lnear backorder costs, and n partcular pecewse constant backorder costs. Another related stream of lterature concerns lateral transshpments between warehouses, where a local warehouse wth no stock on hand can request an tem from another local warehouse f needed. Yang et al. (213) assumes, just as we do, that customers have a certan pre-specfed acceptable watng tme lmt. Wthn ths tme lmt, the local warehouse wll wat for an ncomng uncommtted tem. If the watng tme s too long, t wll request a lateral transshpment havng a shorter, but postve, leadtme. Olsson (215) studes a smlar model as Yang et al. (213), but uses a backorder cost per unt and tme unt nstead of a tme lmt. For a revew of papers studyng lateral transshpments see, e.g., Paterson et al. (211). The lterature dscussed so far s, n general, based on one-for-one orderng polces, whch s also the case we consder n ths paper. When nstead consderng batch-orderng polces n mult-echelon nventory settngs wth tme wndow servce levels, the model complexty wll ncrease consderably. In a sngle-echelon settng Axsäter (23b) consders an nventory system wth lateral transshpments where the locatons apply (R, Q) polces. In Axsäter (23b) an approxmate method s developed that uses nformaton about the resdual leadtmes of the tems n the system. Other relevant lterature concernng batch-orderng polces nclude, e.g., Axsäter (2) and Katehaks and Smt (212). In the followng secton we formulate our model n detal. Secton 3 presents the soluton procedure wth the dervaton of exact closed form expressons for varous performance characterstcs. In Sect. 4, cost structures are presented together wth cost evaluatons. In Sect. 5, we present cost optmzaton procedures for all three cases consdered, and n Sect. 6 numercal examples are presented and dscussed. In Sect. 6, we also gve an applcaton of how to use the theory developed n order to quantfy sustanablty measures, such as expected CO 2 emssons related to producton waste. Some concludng remarks are gven n Sect. 7.
Ann Oper Res (217) 259:217 239 221 2 Model formulaton We consder a two-echelon contnuous revew nventory model wth one central warehouse and N local stes. All transportaton tmes are postve and constant. Customer demands follow ndependent Posson processes and occur only at the local stes. Snce we focus on spare parts products, we assume that replenshments are made accordng to base stock orderng polces. All unsatsfed demands are backordered. Furthermore, we assume that backorders at the stes and at the central warehouse are flled accordng to the FCFS (frst come - frst served) rule. In smlar models lke ths one, t s commonly assumed that base stock levels are based on a backorder cost per unt and tme unt or a prescrbed servce level. In those cases a target servce level s used, the servce level defnton s very often the so called fll rate (fracton of demand that can be satsfed drectly from stock on hand). However, n ths paper, we consder one step functon and one general non-lnear backorder cost structure, where the cost s a functon of the customer s tme n backorder. In addton, we also consder a tme wndow based servce level. Let us ntroduce some notatons for parameters and decson varables: λ customer arrval rate at local ste, L transportaton tme from the central warehoue to local ste, L transportaton tme from the suppler to the central warehouse, h holdng cost per unt and tme unt at local ste, h holdng cost per unt and tme unt at the warehouse, N number of local stes, S base-stock level at local ste, S base-stock level at the warehouse, We proceed by defnng the two cost structures and the tme wndow based servce level n more detal. In the subsequent analyss, let Y denote the tme a customer demand s backordered at ste. Note that Y L + L s stochastc and depends on the basestock levels at ste and the central warehouse. 3 Performance characterstcs Defne X as the lmtng (.e., the statonary case) age of the oldest unt at the warehouse not assgned to any watng customer, where the age s assumed to start when the unt s ordered from the outsde suppler. Smlarly, let X be the lmtng age of the oldest unt at ste ( = 1,...,N) not assgned to any watng customer, where the age of an tem s assumed to start when the unt s ordered from the warehouse. Note that, f the warehouse has zero stock on hand when a ste orders a unt, the ordered unt wll arrve at the ste after L + Z unt of tme, where Z s the stochastc delay at the warehouse. Obvously, we must have Z = L X,for X L. Hence, gven a stochastc delay of Z unts of tme at the warehouse, the oldest unt at ste s outstandng f < X < L + Z, and n stock f X L + Z. Let us proceed by notng that X Erlang(λ, S ) and X Erlang(λ, S ),where λ = λ 1 + +λ N. Ths s the case snce (S 1, S ) polces are used and that the demand process s pure Posson at the stes and at the warehouse, whch means that the well known
222 Ann Oper Res (217) 259:217 239 PASTA-property (Posson Arrvals See Tme Averages) holds. It should be noted that our technque of trackng the ages of the unts n the system yelds a rcher model than the unt trackng methodology frst presented n Axsäter (199). Ths s the case snce the unt trackng methodology n Axsäter (199) consders only what happens at tmes of customer arrvals. In our case, we have full nformaton about the ages of the (oldest) unts n the system at all tmes. Ths means that, unlke Axsäter (199), t s possble to generalze our model to ncorporate decson rules based on events whch are not customer arrvals. In any case, we have the densty functons, f X (t), and the dstrbuton functons, F X (t), f X (t) = λ S e λ t t S 1 (S 1)!, t (1) S 1 F X (t) = 1 e λ t (λ t) n, t (2) n! n= for =, 1,...,N. Usng(1) andz = L X, t s easy to obtan the densty of the stochastc delay, Z,as f Z (t) = f X (L t) = λ S e λ (L t) (L t) S 1, for t L. (3) (S 1)! The probablty mass n the pont Z = s found by usng (2), P{Z = } =P{X > L }=1 F X (L ) = S 1 n= e λ L (λ L ) n. (4) n! Let us contnue by dervng the probablty functon for the nventory level at the local ste, IL.Now,gventhedelayZ = z, the condtonal statonary probablty functon for IL follows as P{IL = k Z = z} = (λ (L + z)) S k e λ (L +z), (5) (S k)! Usng (3) (5), we may remove the condton on Z as follows P{IL = k} = (λ L ) S k L (S k)! e λ L P{Z = }+ P{IL = k Z = z} f Z (z)dz. (6) In order to be able to evaluate nventory level probabltes exactly, we provde the followng closed form expresson of the nventory level probablty functon: Proposton 1 For k S,S >, n:= S k, m := S 1, and μ := λ λ, the closed form expresson of P{IL = k} s gven by P{IL = k} = (λ L ) S k (S k)! e λ L P{Z = } n m ( )( ) +A ( 1) m k 2 n m L k 1 L k 2 (k 1, k 2 ), (7) k 1 = k 2 = k 1 k 2
Ann Oper Res (217) 259:217 239 223 where (k 1, k 2 ) = e μl (m + n k 1 k 2 )! A = m+n k 1 k 2 j= ( 1)m+n k 1 k 2 (m + n k 1 k 2 )! μ m+n k, 1 k 2 +1 λs k (S k)! λ S (S 1)! e λ L λ L. [ ( 1) m+n k 1 k 2 j μ m+n k 1 k 2 j+1 L j ] j! Proof See Appendx. Notce that, Proposton 1 s only defned for S >. However, for S = the problem degenerates to a sngle-echelon nventory system,.e., Z L. From (7) the average stock on hand at ste s obtaned as E { IL + } S = kp{il = k}, (8) k=1 and the average stock on hand at the central warehouse becomes E { IL + } S = k (λ L ) S k e λ L. (9) (S k)! k=1 Let us contnue by dervng an expresson for the probablty that an arrvng customer at ste has to wat longer than ω unts of tme. In ths case t s reasonable, from a practcal pont of vew, to assume that ω L. By condtonng on Z = z, we obtan the followng condtonal probablty P{Y >ω Z = z} =P{L + z X >ω }=P{X < L + z ω }=F X (L + z ω ) S 1 = 1 e λ (L +z ω ) (λ (L + z ω )) n. (1) n! n= Hence, by usng (3), (4), (1) and the law of total probablty we obtan L P{Y >ω }=P{X < L ω }P{Z = }+ P{X < L + z ω } f Z (z)dz. (11) Interestngly enough, smlar to Proposton 1, t s possble to derve a closed form expresson of the probablty n (11). Proposton 2 The probablty that an arrvng customer at ste has to wat longer than ω unts of tme s obtaned, n closed form, as P{Y >ω }=P{X < L ω }P{Z = }+I, (12)
224 Ann Oper Res (217) 259:217 239 where S 1 I = 1 P{Z = } (n) n= (L ω ) k 1 L k 2 (n, k 1, k 2 ), n S 1 ( )( ) ( 1) S 1 k 2 n S 1 k 1 k 1 = k 2 = (n, k 1, k 2 ) = e (λ λ )L (n + S 1 k 1 k 2 )! n+s 1 k 1 k 2 j= ( 1)n+S 1 k 1 k 2 (n + S 1 k 1 k 2 )! (λ λ ) n+s k 1 k 2, (n) = λn n! λ S (S 1)! e λ (L ω) λ L. [ k 2 ( 1) n+s 1 k 1 k 2 j (λ λ ) n+s L j ] k 1 k 2 j j! Proof See Appendx. 4 Cost structures 4.1 Pecewse constant backorder costs Assume that a customer arrvng at ste ncurs a backorder cost that depends on the tme the demand s backordered. In more detal, denote B (Y ) as the backorder cost, as a functon of Y, ncurred by a customer arrvng at ste. Assume that a customer arrvng at ste has an acceptable watng tme of ω unts of tme for a demanded tem. As a start, n vew of the dscusson about servce contracts n the ntroducton, let us consder a partcular smple and mportant case concernng the backorder cost structure: { f Y ω B (Y ) =, (13) b f Y >ω. Hence, f an arrvng customer has to wat longer than ω for a spare part, then the servce provdng company s oblged to pay a fxed penalty cost b. Otherwse, no backorder cost s ncurred. Notce that, n ths settng, the assumpton of the FCFS-rule s not optmal but reasonable. For example, say that there are two watng customers at locaton and the customer frst n lne already has been watng for more than ω unts of tme, whle the second haswatedlessthanω. Then, when an tem then arrves at locaton from the warehouse, t would be more cost effcent to assgn the ncomng tem to the second customer (nstead of applyng the FCFS-rule). Also, n practce, usng a dfferent rule than FCFS could mean that a customer that already has exceeded the watng tme lmt would not be prortzed and mght end up wth a very long watng tme (whch n practce would mean that the company soon would be out of busness). Smlarly, the FCFS-rule s, of course, also an ssue for the cases presented n Sects. 4.2 and 4.3. In a more general case, assume that there are K dfferent tme lmts, ω ( j), j = 1, 2,...,K. We consder the followng backorder cost (or penalty cost) structure: f Y ω (1) B (Y ) = b ( j) f ω ( j), b (K ) f Y >ω (K ). < Y ω ( j+1), j = 1,...,K 1, (14)
Ann Oper Res (217) 259:217 239 225 That s, f an arrvng customer has to wat longer than a prescrbed tme lmt ω ( j), the servce provdng company has to pay a fx penalty cost b ( j). Here, we assume that ω (1) <ω (2) < <ω (K ) and b (1) < b (2) < < b (K ),.e. that the cost s ncreasng wth the customer watng tme. Ths cost structure can be extended to also nclude a standard lnear backorder cost per unt of tme. However, n ths paper we focus on the non-lnear cost expressons. 4.2 Exponental backorder costs Here we consder a more general type of non-lnear backorder cost structure that depends on the tme an arrvng customer has to spend n backorder. As mentoned n the ntroducton, although our soluton procedure can handle general non-lnear backorder costs, we focus on exponentally ncreasng backorder costs as a functon of the customer watng tme. That s, for Y >, let us defne: B (Y ) = c a Y, (15) where a > 1andc >, = 1, 2,...,N, are constants. In ths case, the backorder cost grows exponentally wth the customer s watng tme. Ths means that the backorder cost rapdly gets large when the customer s watng tme gets longer. 4.3 Tme wndow servce constrant Here we consder a case where we replace the backorder cost by a tme wndow servce constrant. In more detal, assume that there s an agreement that requres that the servce level should be at least l wthn ω unts of tme. In other words, the tme wndow servce level at locaton s defned as β = P{Y ω } l, {1,...,N}. Notce that, n vew of (14), ths tme wndow servce level can be extended to a more general servce level defnton where dfferent target servce levels may be defned for dfferent ntervals of watng tmes. For example, mmedate servce could be 95%, whle servce wthn 4 h could be 97%, etc. 4.4 Evaluaton of costs For the case where there s a backorder cost, let EBdenote the total system expected backorder cost, per unt of tme. Then, the expected total cost, EC, s obtaned as EC = h E { IL + } N + h E { IL + } + EB, (16) =1 where E { IL + } { and E IL + } are defned n (8)and(9), respectvely. In the followng analyss we state explctly how to evaluate EB for pecewse constant and exponental backorder costs. The cost mnmzaton problem for the case wth a tme wndow servce constrant s defned and analyzed n the optmzaton secton, see Sect. 5.2. Pecewse constant backorder costs For the specal case of a backorder cost wth just one acceptable tme lmt, ω,weobtaneb as N EB = λ b P{Y >ω }. (17) =1
226 Ann Oper Res (217) 259:217 239 For the more general cost structure n (14) t s reasonable to assume that ω (1) < ω (2) < <ω (K ) L. Hence, usng (11), the probablty for a customers tme n backorder to be n the nterval between two consecutve tme lmts can be wrtten as { } { } { } P ω ( j) < Y ω ( j+1) = P Y <ω ( j+1) P Y <ω ( j). (18) Gven these probabltes, the expected backorder cost per unt of tme follows as N K 1 { } { } EB = λ b ( j) P ω ( j) < Y ω ( j+1) + b (K ) P Y >ω (K ). (19) =1 j=1 Exponental backorder costs Gven the cost structure n (15), we derve the expected backorder cost EB. The tme n backorder, Y, depends both on the age of the oldest tem, X,andthe delay at the warehouse, Z. For a gven delay Z = z and age of the oldest tem X = t, we have Y = L + z t. Hence, the condtonal backorder cost for locaton becomes B (Y X = t, Z = z) = B (L + z t) = c a L +z t. (2) Usng (1) and(2), the condtonal expected backorder cost, per unt of tme, for locaton follows as L +z EB (Z = z) = λ B (L + z t) f X (t)dt. (21) Further, usng (3), (4) and(21), we get the (uncondtonal) expected backorder cost per unt of tme for locaton as EB = EB (Z = ) P{Z = }+ L whch n turn gves us the total expected backorder cost, EB = N EB. =1 EB (Z = z) f Z (z)dz, (22) Remark 1 Observe that when the backorder cost has a lnear structure, we have the specal case B (Y X = t, Z = z) = B (L + z t) = b (L + z t), whch s the backorder cost structure studed n Graves (1985)andAxsäter (199). Ths drectly mples that L +z EB (Z = z) = λ b (L + z t) f X (t)dt. 5 Optmzaton 5.1 Pecewse constant and general non-lnear backorder costs For a standard two-echelon nventory system wth lnear holdng and backorder costs, the optmzaton procedure s relatvely smple, see e.g. Axsäter (199). For example, wth lnear holdng and backorder costs t s easy to show that the total cost functon s convex n S, = 1,...,N, foragvens. Ths appealng property s, however, lost when consderng non-lnear costs, as n our case. Therefore, n our optmzaton procedure we wll use other characterstcs of the system when optmzng the base-stock levels. One such characterstc s descrbed n the followng remark, whch s easy to prove (we omt the detals):
Ann Oper Res (217) 259:217 239 227 Remark 2 If the leadtme for locaton s constant, the holdng cost at locaton, H (S ),s ncreasng n S. Furthermore, H (S ) when S. Our optmzaton procedure s based on the property descrbed n Remark 2. Due to the non-convex behavor of the total cost functon, we derve upper and lower bounds for the optmal values of S and S, = 1,...,N, where we denote the optmal value of S for a gven S as S (S ). Smlarly, the overall optmal values of S and S are denoted as S and S, respectvely. Then, obvously, we have S S (S ) S,whereS and S are the lower and upper bounds, respectvely. In complete analogy, we also derve an upper bound, S, and a lower bound, S, for the optmal value of S. Hence, S s found by choosng S {S,...,S } such that EC(S ) = C (S ) + N ( C S, S (S ) ) (23) s mnmzed (where C ( ) s the average holdng costs per tme unt at the warehouse, and C (, ) s the average holdng and backorder costs at the local ste ). The computaton tme regardng the optmzaton procedure for fndng optmal base-stock levels was, on average, qute moderate (n general, less than 1 mn). =1 Lower and upper bounds for S Notce that, when S =, the leadtme for locaton becomes the shortest possble,.e., L. Hence, n order to obtan a lower bound for S, the general dea here s to mnmze C (S, S ) wth respect to S gven that S =.Now,snceC (S, S ),gvens =,s not convex n S we wll use Remark 2 for obtanng S := S ( ). Recall that when S =, the leadtme s constant (= L ) and therefore the characterstcs of H (S ) n Remark 2 may be used. The method for fndng S ( ) can be found below n Algorthm 1. The ntuton behnd Algorthm 1 s that we contnue ncreasng S untl the holdng cost becomes greater than the mnmum cost so far. Then, from Remark 1, we know that we can stop the search. In order to obtan an upper bound for S, we use exactly the same method as when dervng the correspondng lower bound. Notce that, the longest possble leadtme for locaton, L + L, s obtaned for S =. Hence, the dfference compared to the dervaton of the lower bound s that we, n Algorthm 1, sets = when dervng the upper bound for S. Furthermore, nstead of startng the optmzaton algorthm wth S =, we can here use the lower bound, S, as a startng pont for S. Optmzaton of S In vew of Algorthm 1, the problem of fndng the optmal S s relatvely straght-forward. As mentoned, the total cost functon s, n general, not convex n S. Therefore, we derve lower and ( upper bounds for ) the optmal S. A lower bound for S s found by optmzng EC S, S1, S 2,...,S N wth respect to S.Thats,fallS, = 1,...,N, arefxed and chosen as large as possble, a lower bound for S s obtaned. The optmzaton procedure s presented n Algorthm 2 below, and s qute smlar to Algorthm 1, wth only a few ( modfcatons. In complete ) analogy, an upper bound for S s found by optmzng EC S, S1, S 2,...,S N wth respect to S.
228 Ann Oper Res (217) 259:217 239 Algorthm 1 Computaton of S 1: procedure 2: S = ; S = ; C mn = ; 3: whle H (S )<C mn do 4: f C (S, S )<C mn then 5: C mn = C (S, S ); 6: S mn = S ; 7: end f 8: S = S + 1; 9: end whle 1: S = S mn ; 11: end procedure := S ( ) Algorthm 2 Computaton of S 1: procedure 2: S = ; S = S ; C mn = ; 3: whle H (S )<C mn do 4: f EC(S, S 1, S 2,...,S N )<C mn then 5: C mn = EC(S, S 1, S 2,...,S N ); 6: S mn = S ; 7: end f 8: S = S + 1; 9: end whle 1: S = S mn; 11: end procedure 5.2 Tme wndow servce constrant The optmzaton problem, when consderng a tme wndow servce constrant nstead of backorder costs, becomes N mn EC(S, S 1,...,S N ) = h E { IL + } S,S {1,...,N} (24) = subject to β = P{Y ω }=1 P{Y >ω } l, {1,...,N}, (25) where l s the target tme wndow servce level. The optmzaton procedure s qute smlar as n Sect. 5.1. Frst, notce that from (1) and(11) t follows that the tme wndow servce level β s strctly monotonc n S, {1,...,N},.e., P{Y ω S } > P{Y ω S 1}. Hence, for a gven S, the mnmum value of S whch satsfes the tme wndow constrant can easly be found. In order to fnd the optmal value of S, agan a very smlar procedure as n Sect. 5.1 can be developed. In short, gven S = arg mn S C (, S ) such that β l, a lower ( ) bound of the optmal S becomes snce mnmzng EC S, S1, S 2,...,S N wth respect to S, such that (25) s satsfed, gves the lower bound S =. Ths s the case snce ( ) EC S, S1, S 2,...,S N s obvously ncreasng n S. The upper bound S can be found ( ) n a smlar manner by mnmzng EC S, S1, S 2,...,S N wth respect to S, such that (25) s satsfed. That s, we start wth S = and ncrease S by one unt at a tme untl (25) s satsfed.
Ann Oper Res (217) 259:217 239 229 6 Applcatons and numercal experments In ths secton we frst evaluate the three cases consdered n Sect. 4 foranumberoftest problems. In all test problems we consder an nventory system wth two downstream stes,.e., N = 2. For the sake of smplcty let us also assume that the local stes are dentcal (although ths s not necessary from a modelng pont of vew). Secondly, we consder how our model may be used n order to quantfy envronmental effects assocated wth not satsfyng customer demands n a tmely fashon. We provde here a small numercal study concernng sustanablty aspects n connecton wth CO 2 emssons. 6.1 Numercal evaluaton of dfferent backorder cost structures For the frst cost structure wth a pecewse constant backorder cost functon, we let K = 1,.e., there s a sngle acceptable customer watng tme lmt. The customer arrval ntensty s ether λ =.1orλ =.5. The holdng costs are, for smplcty, assumed to be the same at all locatons. In more detal, we consder the settngs h = h =.5 andh = h = 1. Moreover, we let the backorder cost take values n b {1, 1, 5, 1}. Theman purpose s to study the effect of the backorder cost n the form of a step functon. Notce that, n ths numercal study we consder several dfferent ratos of b and h. A low rato corresponds to a stuaton wth relatvely expensve spare parts. When ths rato s relatvely hgh, the backorder costs are consderably larger than the holdng costs. Ths may be the case f the spare parts are relatvely nexpensve and/or t s very costly f the spare part s not avalable when needed. In all problems the transportaton tme to the central warehouse s L = 1. In Table 1 we have L = 1, and n Table 2 we set L = 5. The customer s acceptable watng tme s ether 1, 3 or 5% of the transportaton tme to the local stes. As we can see n Tables 1 and 2, the optmal base-stock levels at the local stes n many cases tend to ncrease when ω gets lower. On the other hand, the optmal base-stock level at the central warehouse tends to, for most cases, decrease wth lower ω. Hence, when the acceptable tme lmt, ω, s rather low, nventory should be pushed to the downstream locatons. The ntutve explanaton s that when ω s low, t s relatvely lkely that an arrvng customer s watng tme wll exceed ω f there s no stock on hand at the downstream stes. Therefore, the base-stock levels at the local stes should be relatvely hgh n these cases. On the other hand, f ω s relatvely long t s not that crucal to keep stock close to the customers. As expected, the total expected cost ncreases wth hgher backorder costs and lower acceptable watng tmes. The results from Tables 1 and 2 also reflect the dffculty of fndng structural optmzaton propertes of the system. For example, n Table 1, t s mportant to notce that the optmal S s n general not monotonc n ω. Such characterstcs also make t hard for practtoners to develop heurstc optmzaton procedures whch often rely on convexty results. Hence, snce the optmzaton results mght be counterntutve for practtoners, t s mportant to use the exact optmzaton procedure developed n Sect. 5. However, t s nterestng to note that the optmal total amount of tems n the system, S + N =1 S, s non-ncreasng n ω for all cases consdered n Tables 1 and 2. Ths result s n lne wth ntuton snce when ω ncreases, customers acceptable watng tmes ncrease, and we may lower the total amount of stock n the system and stll provde suffcently hgh customer servce. However, due to the non-monotonc property dscussed above, t s very dffcult to argue exactly how the total stock s allocated among the local stes and the central warehouse. Apart from the test problems where b = 1 n Tables 1 and 2, the probablty of exceedng the acceptable tme lmt s qute small. In these problems the backorder cost, b,slarge
23 Ann Oper Res (217) 259:217 239 Table 1 Optmal base-stock levels, expected total cost and probablty of exceedng the acceptable tme lmt ω b h = h =.5 h = h = 1 λ =.1 λ =.5 λ =.1 λ =.5 S, S EC P{Y >ω } S, S EC P{Y >ω } S, S EC P{Y >ω } S, S EC P{Y >ω }.1 L 1 1, 1 1.51.4513 1, 3 3.92.1311, 1 1.99.6638 8, 3 6.16.2618.3 L 1 1, 1 1.49.442 1, 3 3.78.1173, 1 1.98.657 1, 2 5.97.2476.5 L 1 1, 1 1.47.4289 1, 3 3.66.15, 1 1.97.651 1, 2 5.7.2212.1 L 1 3, 2 2.96.277 14, 4 6.24.73 2, 2 5.21.688 12, 4 11.21.219.3 L 1 3, 2 2.91.251 13, 4 6.5.14 2, 2 5.13.648 12, 4 1.88.186.5 L 1 3, 2 2.86.228 15, 3 5.87.86 2, 2 5.6.611 14, 3 1.47.144.1 L 5 3, 3 3.88.48 15, 5 7.57.11 4, 2 6.94.114 14, 5 14.12.22.3 L 5 5, 2 3.83.43 16, 4 7.34.17 4, 2 6.75.95 15, 4 13.56.31.5 L 5 4, 2 3.7.79 16, 4 7.9.12 4, 2 6.6.79 15, 4 13.16.23.1 L 1 4, 3 4.19.14 16, 5 8.9.6 3, 3 7.77.48 15, 5 15.14.11.3 L 1 4, 3 4.15.13 15, 5 7.88.9 5, 2 7.67.43 16, 4 14.75.17.5 L 1 5, 2 4.1.31 17, 4 7.61.6 4, 2 7.39.79 16, 4 14.18.12 L = 1 and L = 1
Ann Oper Res (217) 259:217 239 231 Table 2 Optmal base-stock levels, expected total cost and probablty of exceedng the acceptable tme lmt ω b h = h =.5 h = h = 1 λ =.1 λ =.5 λ =.1 λ =.5 S, S EC P{Y >ω } S, S EC P{Y >ω } S, S EC P{Y >ω } S, S EC P{Y >ω }.1 L 1, 2 1.63.4253 8, 6 4.38.1659, 1 1.98.7654 7, 5 6.64.3513.3 L 1, 2 1.56.398 8, 6 3.85.1131, 1 1.93.748 7, 5 5.83.271.5 L 1, 2 1.49.3554 9, 5 3.3.996, 1 1.87.7134 8, 4 5.6.2577.1 L 1 2, 3 3.44.461 11, 8 7.28.126 2, 3 5.95.461 1, 8 13..196.3 L 1 2, 3 3.21.349 12, 7 6.5.98 1, 3 5.62.769 1, 7 11.53.244.5 L 1 2, 3 3.3.258 12, 6 5.71.116 3, 2 5.22.577 11, 6 1.2.188.1 L 5 3, 4 4.44.44 13, 9 8.95.19 2, 4 8.9.18 12, 9 16.53.31.3 L 5 2, 4 4.27.77 13, 8 8.7.21 3, 3 7.61.16 12, 8 14.81.36.5 L 5 4, 3 3.97.46 14, 7 7.19.14 3, 3 7.4.13 12, 7 13.13.42.1 L 1 3, 4 4.89.44 12, 1 9.59.11 3, 4 8.89.44 13, 9 17.9.19.3 L 1 3, 4 4.56.28 13, 9 8.7.7 2, 4 8.55.77 13, 8 16.14.21.5 L 1 3, 4 4.35.17 13, 8 7.77.8 4, 3 7.93.46 14, 7 14.38.14 L = 1 and L = 5
232 Ann Oper Res (217) 259:217 239 Table 3 Optmal base-stock levels and expected total cost L = 1 and L = 2 a h = h =.5 h = h = 1 λ =.1 λ =.5 λ =.1 λ =.5 S, S EC S, S EC S, S EC S, S EC 1.1 1,.5 6, 2 1.48 1,.57 6, 1 1.75 1.5 2, 1 1.48 9, 3 2.64 3, 2.4 9, 2 3.93 2 4, 1 2.15 11, 3 3.51 3, 1 3.91 1, 3 5.88 4 7, 1 3.64 15, 3 5.38 6, 1 6.75 14, 3 9.85 compared to the holdng cost h. Moreover, as expected, we see that the probablty of exceedng the acceptable tme lmt s sgnfcantly hgher when the backorder cost tends to be relatvely low. When the tems are relatvely expensve to keep n stock we note that the optmal base-stock levels are low, especally when the customer arrval rate s low. For our second cost structure wth an exponentally ncreasng backorder cost functon, we evaluate our model for a set of test problems where we let c = 1anda n (15) be one of four values, a {1.1, 1.5, 2, 4}. As before, the customer arrval rate s ether λ =.1 or λ =.5. The holdng costs are the same at all locatons and ether h = h =.5 or h = h = 1. The transportaton tmes are L = 1 and L = 2. We consder dfferent ratos of a and h where a lower rato corresponds to relatvely expensve spare parts and vce versa. As seen n Table 3, the optmal base-stock levels at the central warehouse ncrease rather rapdly wth ncreasng values of a. It s also nterestng to notce that the base-stock levels at the local stes are kept at relatvely low levels, although a ncreases sgnfcantly. The ntuton behnd ths behavor s that when a s relatvely large, long customer watng tmes wll be very costly, whle short watng tmes are not so expensve. Notce that, n order to suppress long customer watng tmes t may be wse to allocate stock to the central warehouse nstead of the local stes. That s, nstead of allocatng relatvely large amount of stock to each local ste, the optmal stock polcy wll suggest that stock should be kept upstream, snce long watng tmes can be suppressed just as well from stock at the central warehouse. It s nterestng to note that ths result runs absolutely counter to tradtonal mult-echelon models wth fll rate constrants, see e.g., Muckstadt and Thomas (198), Axsäter (23a), and Hausman and Erkp (1994). Optmal solutons of such nventory systems very often show that nventory should be prortzed downstream, whle the fll rate at the central warehouse may be only about 5%. For the case wth tme wndow servce level constrants β l, we evaluate our model for test problems wth three dfferent levels of the tme wndow servce level that must be acheved, that s l {.9,.95,.98}. As before, the customer arrval rate s ether λ =.1 or λ =.5. The holdng costs are the same at all locatons and ether h = h =.5 or h = h = 1. The transportaton tmes are L = 1 and L = 2 and the acceptable watng tme ω s ether, 1, 3 or 5% of the transportaton tme L. In Table 4, we see that there often s lttle or no change n the optmal base-stock levels dependng on the value of ω. Wth ncreasng l, the base-stock levels also ncrease, as expected. Notce that, a tme wndow servce level constrant where ω = sthesameas the tradtonal fll rate defned as the fracton of demand that can be satsfed mmedately from stock on hand. Comparng the test problems n Table 4 where ω s zero to the the ones where ω has a larger value, t can be seen that optmzng the system usng the tradtonal fll rate even though customers are wllng to wat for a certan amount of tme may lead to ncreased costs, especally for hgher holdng costs or hgher customer arrval rates. For
Ann Oper Res (217) 259:217 239 233 Table 4 Optmal base-stock levels, expected total cost and tme wndow servce level ω l h = h =.5 h = h = 1 λ =.1 λ =.5 λ =.1 λ =.5 S, S EC β S, S EC β S, S EC β S, S EC β.9 2, 2 1.83.958 11, 4 3.55.9217 2, 2 3.65.958 11, 4 7.11.9217.1 L.9 2, 2 1.83.918 11, 4 3.55.9314 2, 2 3.65.918 11, 4 7.11.9314.3 L.9 2, 2 1.83.923 1, 4 3.9.92 2, 2 3.65.923 1, 4 6.17.92.5 L.9 2, 2 1.83.9291 11, 3 2.63.953 2, 2 3.65.9291 11, 3 5.26.953.95 3, 2 2.31.9532 13, 4 4.52.9598 3, 2 4.62.9532 13, 4 9.4.9598.1 L.95 3, 2 2.31.9572 12, 4 4.3.9531 3, 2 4.62.9572 12, 4 8.6.9531.3 L.95 3, 2 2.31.9646 12, 4 4.3.9665 3, 2 4.62.9646 12, 4 8.6.9665.5 L.95 3, 2 2.31.979 11, 4 3.55.966 3, 2 4.62.979 11, 4 7.11.966.98 3, 3 3.3.9918 13, 5 5.51.9868 3, 3 6.6.9918 13, 5 11.1.9868.1 L.98 3, 3 3.3.9926 12, 5 5.1.9826 3, 3 6.6.9926 12, 5 1.2.9826.3 L.98 4, 2 2.8.9825 12, 5 5.1.9876 4, 2 5.61.9825 12, 5 1.2.9876.5 L.98 4, 2 2.8.9875 13, 4 4.52.9859 4, 2 5.61.9875 13, 4 9.4.9859 L = 1 and L = 2
234 Ann Oper Res (217) 259:217 239 example, among the test problems, the cost ncrease can be as hgh as 35% (see the problems where h = 1, λ =.5, and ω =.5L ). 6.2 Applcaton: sustanable nventory control Let us n ths secton demonstrate how our model may be used for quantfyng CO 2 emssons emanatng from the customer producton ste. To ths end, we focus on the practcal case concernng packagng of dary products, as mentoned n Sect. 1. To be more specfc, let us assume that the dary product s mlk. Qute a few prevous studes have nvestgated the carbon footprnt related to mlk producton. Thoma et al. (213) conclude that the range of CO 2 emssons s approxmately.75 1.5 kg, per kg mlk produced. Hence, we assume that, on average, there s a one to one correspondence between the amount of mlk produced and the amount of CO 2 emssons (n unts of weght). Recall from the dscusson n Sect. 1 that we assume a whole mlk-batch s wasted f the producton downtme exceeds the crtcal tme lmt ω. Hence, the expected CO 2 emssons of producton waste n kg per unt of tme becomes E{CO 2 }= N λ P{Y >ω }M, (26) =1 where M s the average batch sze (n kg) at the producton faclty correspondng to the local nventory ste. In practce, M ranges from approxmately 1 3, kg. Smlar expressons for E{CO 2 } for other products than mlk may, of course, be evaluated by usng exactly the same modelng technque, and the evaluaton of E{CO 2 } n (26) may be of nterest n many other related applcatons. For example, f a CO 2 tax s ntroduced by the government for the specfc product produced, a correspondng model should take the average CO 2 cost nto account. Another related problem s government mposed restrctons on CO 2 emssons for specfc products. In such a case, our modelng technque may be used n order to evaluate f these CO 2 restrctons are satsfed or not, when decdng nventory target levels n spare part logstcs systems. In order to set the CO 2 emssons emanatng from customer producton stes n relaton to CO 2 emssons from transportaton, let us consder a small set of numercal examples. For smplcty, we consder the same test bed as n Table 1. That s, n Table 1, wepresent optmal base-stock levels and probabltes P{Y >ω } for a system wth N = 2 locatons (for a specfc parameter settng). Assumng M = 15, as a benchmark, and gven the probabltes P{Y >ω } n Table 1 (for the case h = h = 1) we lst, n Table 5, the correspondng expected CO 2 emssons of producton waste. As mentoned, to set the CO 2 emssons of producton waste n relaton to transportaton, t s nterestng to notce that the carbon ntensty (expressed as gco 2 per tonne-km) for heavy trucks s approxmately 2 (McKnnon 21). Ths corresponds to 5 tonne-km per kg CO 2. To llustrate ths relaton we also lst, n Table 5, the equvalent number of tonne-km of transportaton by heavy trucks. In Table 5, we can conclude that E{CO 2 } may be relatvely large even for qute low probabltes of exceedng the acceptable watng tme. Of course, t s also clear that when P{Y >ω } s very low (whch corresponds to a very hgh fxed penalty cost), then E{CO 2 } s also very low. However, t s mportant to realze that n some stuatons the CO 2 emssons related to (too long) producton downtme may be relatvely hgh, whle the downtme cost may be moderate. For example, n order to avod dscardng a dary product, excessve energy n terms of coolng may be consdered (for other types of products, excessve heatng has to be ntated n order to avod obsolescence).
Ann Oper Res (217) 259:217 239 235 Table 5 Examples of E{CO 2 } and the correspondng number of tonne-km of transportaton by heavy trucks λ =.1 λ =.5 P{Y >ω } E{CO 2 } (kg) Transportaton (km) P{Y >ω } E{CO 2 } Transportaton (km).6638 1991 9955.2618 3927 19,635.657 1971 9855.2476 3714 18,57.651 195 975.2212 3318 16,59.688 26 13.219 329 1645.648 194 97.186 279 1395.611 183 915.144 216 18.114 34 17.22 33 165.95 28 14.31 47 235.79 24 12.23 36 18.48 14 7.11 17 85.43 13 65.17 26 13.79 24 12.12 18 9 7 Conclusons In ths paper, we have presented an exact analyss of a two-echelon spare part nventory model wth new types of backorder cost structures. By usng the statonary age dstrbuton of the unts n the system we extended the results from Graves (1985)andAxsäter (199)by dervng exact closed form expressons for the nventory level dstrbutons (Proposton 1), and also for the customer watng tme dstrbutons (Proposton 2). Furthermore, we analyzed a model wth a pecewse constant backorder cost, where a sgnfcant fxed cost s ncurred f the customer watng tme exceeds a pre-specfed threshold value. Ths backorder cost structure was then generalzed to a general non-lnear backorder cost as a functon of the customer watng tme. As an example, we analyzed a model wth an exponentally ncreasng backorder cost as a functon of the watng tme. Moreover, a correspondng model wth tme wndow servce levels was explored. As a fnal step, we nvestgated how polcy decsons affect, ndrectly, the expected CO 2 emssons related to producton waste. These knds of ndrect consequences of polcy decsons, n terms of expected CO 2 emssons, have been largely gnored n the lterature. Instead, most papers have focused on CO 2 emssons from a transportaton pont of vew. For all cases consdered n ths paper, we also developed optmzaton procedures for the base-stock levels. Usng these optmzaton procedures, we presented a numercal study n order to nvestgate how the optmal polcy (gven the base-stock polcy structure) behaves. In partcular, contrary to most results from mult-echelon nventory models, t s nterestng to notce that nventory should be pushed to the central warehouse n cases where the cost for long watng tmes are sgnfcantly larger than for short ones. Possble future extensons may be to nclude some knds of emergency supply n order to avod stops n producton and producton waste. Another lne of research would be to generalze to more complex demand structures, such as compound Posson demand. Such an extenson would be rather straght-forward by keepng track of all ndvdual tems n an ordered batch. Admttedly, the computatonal effort assocated wth obtanng exact nventory level dstrbutons would, however, be sgnfcant.
236 Ann Oper Res (217) 259:217 239 Acknowledgements The authors thank two anonymous referees and Professor Johan Marklund for the useful remarks and suggestons. Open Access Ths artcle s dstrbuted under the terms of the Creatve Commons Attrbuton 4. Internatonal Lcense (http://creatvecommons.org/lcenses/by/4./), whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded you gve approprate credt to the orgnal author(s) and the source, provde a lnk to the Creatve Commons lcense, and ndcate f changes were made. Appendx Proof of Proposton 1 Consder the ntegral n (6). We have: L P{IL = k Z = z} f Z (z)dz = λ S e λ (L z) (L z) S 1 dz = (S 1)! L L (λ (L + z)) S k λs k (S k)! (L + z) S k (L z) S 1 e (λ λ )z dz (S k)! e λ (L +z) λ S (S 1)! e λ L λ L Now, for notatonal purposes we set n := S k, m := S 1, and μ := λ λ. Then, the well known bnomal theorem (see any textbook n Calculus) gves, (L + z) n (L z) m = Hence, we have the ntegral where L k 1 = k 2 = n k 1 = k 2 = k 1 m ( ) n L k 1 z n k 1 k 1 k 2 ( m k 2 ) L k 2 ( z)m k 2. A (L + z) S k (L z) S 1 e μz dz L n m ( ) ( ) = A n L k 1 z n k 1 m L k 2 ( z)m k 2 e μz dz = A n k 1 = k 2 = m ( )( ( 1) m k 2 n m A = k 1 k 2 ) L L k 1 L k 2 z m+n k 1 k 2 e μz dz, λs k (S k)! λ S (S 1)! e λ L λ L. What remans s to calculate the ntegral = L z m+n k 1 k 2 e μz dz. For ths task, let us frst consder an arbtrary postve nteger M and a constant a. Then, by successve ntegraton by parts we obtan the followng ndefnte ntegral z M e az dz = z M a eaz M a z M 1 e az dz = =e az M! M j= ( 1) M j a M j+1 z j j!. (27)
Ann Oper Res (217) 259:217 239 237 Hence, by usng (27), we have the followng defnte ntegral = L z m+n k 1 k 2 e μz dz = e μl (m + n k 1 k 2 )! m+n k 1 k 2 j= [ ( 1) m+n k 1 k 2 j μ m+n k 1 k 2 j+1 L j ] j! ( 1)m+n k 1 k 2 (m + n k 1 k 2 )! μ m+n k. 1 k 2 +1 To conclude, P{IL = k} = (λ L ) S k n m ( )( (S k)! e λ L P{Z = }+A ( 1) m k 2 n m k 1 = k 2 = k 1 k 2 ) L k 1 L k 2. Proof of Proposton 2 Consder the ntegral n (11), and let us once agan defne μ := λ λ. Denotng ths ntegral as I,weget: I = = = L L L P{X < L + z ω } f Z (z)dz S 1 1 n= e λ (L +z ω ) (λ (L + z ω )) n λ S n! e λ (L z) (L z) S 1 dz (S 1)! L S 1 λ n f Z (z)dz n! λ S (S n= 1)! e λ (L ω) λ L (L + z ω ) n (L z) S 1 e μz dz. By defnng the functon (n) as, (n) = λn n! λ S (S 1)! e λ (L ω) λ L, we obtan a smlar soluton as n the proof of Proposton 1: S 1 I = 1 P{Z = } (n) n= L (L + z ω ) n (L z) S 1 e μz dz n= S 1 n S 1 ( )( ) = 1 P{Z = } (n) ( 1) S 1 k 2 n S 1 k 1 L k 1 = k 2 = k 2 (L ω ) k 1 L k 2 ) z n+s 1 k 1 k 2 e μz dz. (28) By usng (27), the ntegral n (28) becomes L z n+s 1 k 1 k 2 e μz dz = e μl (n + S 1 k 1 k 2 )! n+s 1 k 1 k 2 j= [ ( 1) n+s 1 k 1 k 2 j μ n+s k 1 k 2 j L j j! ] ( 1)n+S 1 k 1 k 2 (n + S 1 k 1 k 2 )! μ n+s k 1 k 2.