Optimising a general repair kit problem with a service constraint

Transcription

1 Optmsng a general repar kt problem wth a servce constrant Marco Bjvank 1, Ger Koole Department of Mathematcs, VU Unversty Amsterdam, De Boelelaan 1081a, 1081 HV Amsterdam, The Netherlands Irs F.A. Vs Department of Informaton Systems and Logstcs, VU Unversty Amsterdam, De Boelelaan 1105, 1081 HV Amsterdam, The Netherlands Feld servces are a partcular type of after-sales servce performed at the customer s locaton where techncans repar malfunctonng machnes. The nventory decsons about whch spare part types to take to the repar ste and n what quanttes s called the repar kt problem. Ths problem s characterzed by an order-based performance measure snce a customer s only satsfed when all requred spare parts are avalable to fx the machne. As a result, the servce level n the decson makng process s defned as a job fll rate. In ths paper we derve a closed-form expresson for the expected servce level and total costs for the repar kt problem n a general settng, where multple unts of each part type can be used n a mult-perod problem. Such an all-or-nothng strategy s a new characterstc to nvestgate, but commonly used n practce. Namely, tems are only taken from the nventory when all tems to perform the repar are avalable n the rght quantty. We develop a new algorthm to determne the contents of the repar kt both for a servce and cost model whle ncorporatng ths new expresson for the job fll rate. We show that the algorthm fnds solutons whch dffer on average 0.2% from optmal costs. We perform a case study to test the performance of the algorthm n practce. Our approach results n servce level mprovements of more than 30% aganst smlar holdng costs. Keywords: nventory; repar kt problem; mult-perod; mult-tem; closed-form expresson 1

2 1 Introducton When customers buy a machne or pece of equpment they expect to get more than just the physcal product. They also expect to get after-sales servce regardng any malfunctonng of the machne durng ts lfe cycle. Nowadays, offerng a good after-sales servce s becomng a more and more mportant dfferentaton strategy among compettors n a busness market. One partcular servce that s offered s a local repar servce at the customer s faclty, for example, for copers or coffee machnes. Ths means that a servce company s called when the machne breaks down. On a daly bass, the servce company assgns repar jobs to techncans based on ther sklls. As a result, the techncans receve a lst ndcatng the order of the call ponts that they have to vst and they travel around to repar the broken machnes. Snce t s not known n advance whch parts of the machne have to be replaced, the repar person takes along a selecton of the spare parts n the car. Ths set of parts s referred to as the repar kt. The techncan can only complete a repar f all requred spare parts are avalable n the rght quantty n the repar kt. When one or more unts of the requred parts are unavalable, the techncan cannot fx the machne and has to return when the car s restocked wth all the tems that are requred to repar the machne. Ths extra vst s called a return-to-ft (RTF) vst. An mportant logstcs decson problem for techncans s to determne whch spare parts to put n the repar kt and n whch quanttes to avod return-to-ft vsts. In lterature ths problem s referred to as the repar kt problem. Notce that for an RTF vst t s known n advance whch parts are requred. Therefore, these repars are not consdered n the repar kt problem. Accordng to Bjvank [1] and as wll appear from a case study n Secton 6, companes usually base the contents of ther repar kts upon experences and practcal lmtatons (e.g., the capacty of the car and the amount of money spent to purchase the contents of the repar kt). It would be more effcent to have a systematc procedure to determne the contents of a repar kt. Such a procedure could be based on costs but also on a servce level granted to customers. We dstngush between two types of costs, namely holdng costs and return-to-ft costs. A fxed amount of holdng costs s ncurred for each unt that s stored n the repar kt of the car. RTF costs are nvolved when a techncan has to return because at least one of the requred parts s not avalable n the repar kt. These RTF costs usually consst of the actual labour and drvng costs, as well as costs due to loss of goodwll. As a result, a trade-off has to be made between holdng costs and RTF costs. Two knds of models are 2

3 developed n the lterature to make ths trade-off, namely cost models and servce models. In a cost model the sum of the expected holdng and RTF costs s mnmsed, whereas n a servce model the holdng costs are mnmsed subject to a servce level constrant. Ths latter model s preferred n practce due to the dffculty to quantfy the extra cost for an RTF vst. Moreover, ths type of model explctly ncorporates a customer servce level crteron such that a mnmum qualty of servce s guaranteed. Only a few papers address the repar kt problem. These papers mpose several unrealstc assumptons (see Secton 2 for detals). The objectve of ths paper s to solve the repar kt problem n a more general settng, where multple unts of each tem can be requred for a repar, multple repars are performed before the repar kt s restocked and no tems are taken from the repar kt when a repar s not fnshed. We show how the problem can be solved for both a servce and a cost model. We extend the work of Teunter [6] by frstly ntroducng an exact formulaton for the servce level, nstead of an approxmaton. In ths exact formulaton we assume that tems are only taken from the repar kt when a job can be completed. Ths representaton s n correspondence wth practce (see Secton 6). To our knowledge, we are the frst authors to nvestgate ths characterstc nstead of havng tems left behnd at the customer as commtted nventory n case of an RTF. Secondly, we propose a new soluton algorthm that ncludes ths new servce level formulaton. In Secton 2 we provde an overvew of exstng approaches and ther assumptons that deal wth the repar kt problem. We ntroduce our model and notatons n Secton 3. In ths secton we also derve a closed-form expresson to calculate the servce level. A new approach to solve both the servce model and the cost model s requred to nclude ths exact expresson. The algorthm to solve the repar kt problem n a more general settng s presented n Secton 4. The performance of the algorthm s compared to prevous results n Secton 5. In Secton 6 the results of a case study are presented n order to show the performance of the algorthm n real lfe stuatons. Fnally, we end wth our conclusons n Secton 7. 2 Development to a General Model In ths secton we show the context of our research by gvng a short crtcal overvew of papers addressng the repar kt problem, ncludng ther assumptons. As mentoned n Secton 1, two dfferent types of models have been consdered n the lterature to determne 3

4 the contents of a repar kt. The cost model has been ntroduced by Smth et al. [5], whle the servce model has been ntroduced by Graves [2]. The man dffculty n formulatng both types of models s to fnd an exact expresson for the probablty that a return-to-ft (RTF) vst wll occur. We defne ths as the probablty that a reparman s not able to fnsh the repar job due to an nsuffcent amount of requred tems n the repar kt. In the cost model, ths probablty s requred to determne the expected RTF costs. In the servce model, t expresses the servce perceved by the customers. Clearly, a hgher probablty for an RTF to occur results n lower customer servce and hgher expected RTF costs. The recprocal of ths probablty s referred to as the job fll rate. Thus, the job fll rate ndcates the fracton of jobs (or orders) performed wthout a stock out or RTF vst. The frst models that have been developed assume that the techncan returns to the warehouse after each job to restock the repar kt. Consequently, each job has the same probablty of beng completed durng ts frst vst. These problems are called sngle perod problems. Another assumpton made n these models s that at most one unt of each spare part can be used for the repar. Ths s referred to as sngle tem problems. When both assumptons hold (see, e.g., Smth et al. [5], Graves [2], Hausman [3]), at most one unt of each part type s added to the repar kt to obtan optmal contents. So far, all authors assume ndependence between the dfferent part type falures causng the breakdown. Smth et al. [5] solve the problem for the cost model wth an optmal margnal analyss procedure. Graves [2] transforms the servce model nto a knapsack problem. The cost model of Mamer and Smth [5] consders a sngle perod problem as well. However, they are the frst authors to allow more than one unt of each part type to be requred for a sngle job. They also relax the assumpton of ndependence between the falure probabltes by defnng a representatve collecton of job types where each job type corresponds to a set of demands for parts. They formulate the problem as a network problem and solve t wth a max flow/mn cut algorthm. Based on the network formulaton, ths technque s, however, only applcable to sngle perod problems. Heeremans and Gelders [4] are the frst to relax the assumpton of a sngle perod model. They ntroduce the noton of a tour nto ther mult-perod model. In a tour a sequence of jobs s performed before the repar kt s restocked. The number of jobs performed between two restock moments s called the tour sze. The authors assume t to be fxed, but they do not mpose an assumpton on the maxmum number of unts that can be used of any part type to complete the repar (.e. a mult-tem model). However, nstead of usng the 4

5 defnton of job fll rate as servce level, Heeremans and Gelders [4] use the fracton of tours performed wthout an RTF vst (.e. tour fll rate, Teunter [6]). Ths does not correspond to the servce perceved by customers. When tours can be of any fxed sze, Teunter [6] gves an exact expresson for the job fll rate under the assumpton that at most one unt of each part type s used n a repar. Secondly, the author provdes an approxmaton for the job fll rate n a mult-tem, mult-perod repar kt problem based on the tour fll rate defnton. Teunter [6] s the frst author to relax the ndependence assumpton of part type falures n a mult-perod problem. However, the assumpton s made that all requred tems that are avalable on stock are always used (or left behnd at the customer) despte the fact whether the repar can be fnshed. Ths s not n accordance wth current practces, n whch tems are only used when all requred tems are avalable to complete the repar. Otherwse, the tems that are avalable can be used n the subsequent repars of the tour. In comparson to prevous research, we derve a closed-form expresson for the job fll rate where tems are only taken from the repar kt when a job can be completed (see Secton 1). Ths requres dependency of the avalablty between the dfferent part types. We derve the expresson wthout makng any assumptons on the tour sze or on the maxmum number of unts that can be used durng a repar. We model both aspects as stochastc random varables wth a general dstrbuton functon. Ths more general problem settng s qute common n practce as notced by Bjvank [1]. The only assumpton we make n our defnton of a general problem settng s ndependence between the falures of the dfferent part types. Ths assumpton s of less relevance n a lot of practcal settngs as demonstrated, for example, by the case study n Secton 6. We dd not fnd any sgnfcant correlaton between the usage of part types wthn a repar. We also develop an mproved soluton algorthm n Secton 4 to determne the contents of the repar kt based on the cost and servce models whch ncorporate the new expresson for the job fll rate n ths more general settng. 3 Model Descrpton In ths secton, we use the same notatons as n Teunter [6]. Frst, we ntroduce defntons for the cost model and the servce model n Secton 3.1. Both models use the same expresson for the job fll rate. In Secton 3.2, we derve ths expresson for the repar kt problem n a general settng where tems are not kept asde when a job cannot be completed due to lack 5

6 of requred tems and no assumptons are made on ether the customer demand or the tour sze. The notatons are lsted n Table Model Defnton and Notaton As explaned n Secton 1, the repar kt problem deals wth the selecton of unts of dfferent part types that are put n the car of a techncan. A repar kt s denoted by S, whle n represents the number of unts of part type n repar kt S after t s restocked. The number of dfferent part types that are consdered to put n the repar kt s denoted by N, so S = [n 1,..., n N ]. As explaned n the ntroducton, the objectve functon n a cost model s to mnmse the expected total costs consstng of holdng and RTF costs. Calculatng the total holdng costs s trval, snce each part type has ts own fxed amount of holdng cost H per tour (.e., per replenshment cycle). The total holdng costs of repar kt S s denoted by C H (S) = n H. As stated n Secton 2, the expected total RTF costs s related to the expected job fll rate. We defne the tour sze as a stochastc random varable denoted by M wth a probablty dstrbuton functon P (M = m) and average E[M]. The maxmum number of jobs that can be performed n a tour equals M. For a gven repar kt S and tour sze m, the expected job fll rate s gven by γ job (S, m). The expected job fll rate of a repar kt S equals γ job (S) = M m=1 / P (M = m)mγ job (S, m) E[M]. (1) The expected number of RTF vsts equals the total expected number of jobs n a tour mnus the expected number of completed jobs n a tour. Hence, the expected RTF costs equal C RT F (S) = P RT F where P RT F M m=1 P (M = m)m ( 1 γ job (S, m) ) = P RT F E[M] ( 1 γ job (S) ), (2) denotes the penalty cost for a return-to-ft vst. In (2), the penalty cost s multpled wth the expected number of RTF vsts n a tour wth sze m and the probablty for ths to occur. The cost model s formulated as mnmse subject to n 0, C RT F (S) + C H (S) (3) 6

7 Input parameters N H P RT F β P (M = m) number of dfferent part types holdng cost per unt of part type per tour penalty cost per return-to-ft vst mnmum job fll rate n the servce model probablty that the number of jobs n a tour s m (wth expectaton E[M] and maxmum M) p job (j) probablty that j unts of part type are requred for a job (wth maxmum L max ) Stochastc varables M N m T r number of jobs n a tour (tour sze) number of unts of part type avalable n the repar kt to perform the m-th job number of unts of part type avalable n the repar kt to perform r jobs that are completed U r V m number of unts of part type used to complete r jobs number of completed jobs out of m jobs Other notatons S = [n 1,..., n N ] γ(m) γ job (S, m) γ job (S) C RT F (S) C H (S) q k repar kt wth n unts of part type probablty of fnshng the m-th job job fll rate for repar kt S and tour sze m job fll rate for repar kt S expected RTF costs for repar kt S expected holdng costs for repar kt S k-th quantty of part type to consder n the repar kt job (q k, q k+1 ) ncrease of the job fll rate when n ncreases from q k to q k+1 S repar kt used n the mprovement procedure S S repar kt after the mnmsaton procedure best found repar kt Table 1: The notatons used n ths paper. 7

8 and the servce model as mnmse subject to C H (S) γ job (S) β n 0. (4) A general demand process s consdered, where p job (j) represents the probablty of requrng j unts of part type to perform a job. At most L max one job. unts of part type are requred n For ease of understandng, we frst develop a closed-form expresson for γ job (S, m) n a mult-perod problem settng where L max the second part of that secton. = 1 n Secton 3.2. We relax ths assumpton at 3.2 Job Fll Rate n General Models Teunter [6] gves a closed-form expresson for the expected job fll rate wth a fxed tour sze M (.e., P (M = M) = 1) and at most a sngle unt s used for each tem (.e., L max = 1). Consequently, p job (0) + p job (1) = 1. For each of the M jobs the author calculates the expected probablty to successfully repar the machne. The average job fll rate s then found by addng these probabltes and dvdng the sum by M. The expected probablty to have enough unts avalable n the m-th job for part type depends upon the usage of that part type n the prevous m 1 jobs. At least one unt should be avalable for each of the requred part types after m 1 jobs to complete the m-th job. The probablty to use l unts of a partcular part type n m 1 repars equals the probablty to replace that part type n l out of the m 1 jobs. Ths latter s true, because at most one unt s used n one job. Ths probablty follows a bnomal dstrbuton functon. The assumpton of a fxed tour sze s relaxed by condtonng on the tour sze q (1 q M). Consequently, γ job (S, q) = 1 q where p = p job q N m=1 =1 (1 p ) + p mn{n 1,m 1} l=0 m 1 p l (1 p ) m 1 l l, (5) (1) and 1 p = p job (0). Ths equaton can be substtuted nto (1) to fnd the average job fll rate. Notce that ths expresson assumes that a requred part type s always removed from the repar kt, even f the job cannot be completed due to lack of other 8

9 requred parts. In the remander of ths secton, we correct for ths and relax the sngle tem assumpton. The bnomal dstrbuton of (5) cannot be used anymore when more than one unt of a partcular part type can be used n a sngle job. Therefore, a probablty dstrbuton functon has to be formulated to express the probablty that l unts of part type are avalable at the begnnng of the m-th job. Ths expresson should take the possblty nto account that not enough unts of a partcular part type were avalable n the repar kt to complete a job before the m-th job, but enough unts of the same part type are avalable to perform the m-th job. Take for nstance a stuaton n whch 3 unts of part type A are requred n the frst job, but only 2 unts are ntally avalable n the repar kt. Ths wll result n a return vst for ths frst job. Durng the second job only 2 unts of ths part type are requred. Snce no tems are taken from the repar kt at the frst job, the second job can be completed. Consequently, a stochastc varable N m s defned as the number of unts for part type that are avalable n the repar kt to perform the m-th job. An expresson for the probablty dstrbuton functon of ths random varable should be derved. Ths can be done by condtonng on the number of completed jobs V m out of m jobs and the number of unts used durng these jobs. Let ths latter varable be represented by U r for part type when r jobs are completed. When k unts are used n r jobs that are completed, then n k unts are left to perform the m-th job. When m = 1, 1, f l = n P (N 1 = l V 0 and r = 0 = r) = 0, otherwse. and when m > 1, P (N m = l V m 1 = r) = P (U r = n l T r = n ), f l n, r < m, where the probablty dstrbuton functon of U r depends on the number of tems remanng n the repar kt to perform the r completed jobs (denoted by T r ). For example, f L max = 3 and n = 2, then a job can only be completed f at most two unts of tem are demand (or used). Consequently, U r s only defned for 0, 1, and 2. When more than 2 unts are demanded, the job cannot be completed and s therefore not ncluded n U r. Therefore, we only consder the condtonal probabltes P (U r = u T r = j) for u j and j n. Gven the fact that a job can only be completed when all requred tems are avalable, we know for sure that the number of tems demanded s also used n jobs that are completed and not more 9

10 unts are demanded than avalable (otherwse the job cannot be completed). Consequently, p job (u), f r = 1, u j, P (U r = u T r = j) = mn{j,l max } P k=0 mn{l max,u} l=0 p job (k) p job (l) mn{j,l P max } k=0 p job (k) P (U r 1 = u l T r 1 = j l), f r > 1, u j, 1, f r = 0, u = 0, 0, otherwse. We dvde by k pjob (k) to normalze the dstrbuton functon such that u P (U r = u T r = j) = 1. Next, the probablty dstrbuton functon for the number of completed jobs out of m jobs has to be specfed, whch s denoted by V m. Frst, let us defne γ(m) as the probablty of completng the m-th job and γ(m V m 1 = r) as the probablty of completng the m-th job when r (< m) jobs have already been completed. The latter probablty depends on the number of unts requested for each part type and the avalablty of these unts, N L [ max n ] γ(m V m 1 = r) = p job (j) P (N m = l V m 1 = r), and γ(m) = or not, r<m =1 j=0 l=j γ(m V m 1 = r)p (V m 1 = r). Snce the m-th job can ether be completed 1, f m = 0, r = 0, 1 γ(1 0), f m = 1, r = 0, γ(1 0), f m = 1, r = 1, P (V m = r) = P (V m 1 = r)[1 γ(m r)], f m > 1, r = 0, P (V m 1 = r)[1 γ(m r)] + P (V m 1 = r 1)γ(m r 1), f m > 1, 0 < r m, 0, f r > m. To calculate the job fll rate for a gven repar kt S and tour sze q, the probabltes to fnsh each of the q jobs are added and dvded by q, smlar to (5), γ job (S, q) = 1 q q γ(m). m=1 (6) The mathematcal formulaton of the servce model and the cost model s fnshed when (6) s substtuted n (1) and put n (3) and (4), respectvely. 10

11 4 Algorthm In ths secton we develop an algorthm for the servce model. The algorthm for the cost model s presented n Appendx A, due to the fact that t conssts of the same steps as the algorthm for the servce model wth only a few mnor adjustments. Another reason to focus on the servce model s that not much research has been performed on ths model n the general settng dscussed n Secton 2. From (4), t can be notced that the servce model looks lke a knapsack problem. Therefore, a greedy margnal analyss procedure s used n the lterature to solve the servce model (see, e.g., Graves [2], Teunter [6]). Such a procedure starts wth an empty repar kt and adds one unt of a partcular part type n each teraton untl the predefned servce level s satsfed. Determnng whch part type to add s based upon a rato whch measures the relatve ncrease of the servce level (.e., the job fll rate) n relaton to the ncrease of the holdng costs. In prevous papers only one unt was added n each teraton. However, when we consder multple unts of the same part type to be used n one job, t s unlkely that addng just one unt s most benefcal n all subsequent teratons. Namely, there are a lot of practcal examples n whch t s requred to replace more than one unt of a part type to fx a job, whle replacng only one unt s less lkely. In such cases we would lke to add more than one unt at a tme to the repar kt. As a result, a new algorthm has to be developed to ncorporate these possbltes. The frst step of the algorthm conssts of the determnaton of the order of the number of unts to add to the repar kt for each part type. In the second step, a greedy procedure smlar to Teunter [6] s used to select the parts that are added to the repar kt based on the ncrease of the servce level. The thrd, and fnal, step of the algorthm conssts of mprovng the soluton of step 2 wth an mprovement and mnmsaton procedure. Each step wll be dscussed n more detal below. The result of ths algorthm s a near-optmal contents of the repar kt (see Secton 5 for numercal results). For the frst step, we ntroduce q k as the k-th quantty of part type to consder n the repar kt, where qk+1 > q k for all k. The values of q k are set such that the relatve ncrease of the servce level (or job fll rate) s decreasng for subsequent values of qk. Ths s translated n the property formulated n (7). job (q k, q k+1 ) q k+1 q k > job (qk+1, q k+2 ) qk+2, (7) q k+1 11

12 where job (qk, q k+1 ) represents the ncrease of the job fll rate when the number of unts for part type ncreases from q k to q k+1. So, job (q k, q k+1) = γ job ([n 1,..., n = q k+1,..., n N ]) γ job ([n 1,..., n = q k,..., n N ]). The values of q k for a partcular part type can be found wth the followng pseudo-code: 1 n = 0, q 0 = 0, q 1 = 1, j = 2, k = 1 2 whle j L max M 3 whle job (q k 1,q k ) q k q k 1 4 k = k 1 5 end whle job (qk,j) j qk 6 qk+1 = j, k = k + 1, j = j end whle and k > 0 Ths completes the frst step of our algorthm. For the second step, we adjust the greedy procedure descrbed at the begnnng of ths secton such that the quanttes q k are consdered and multple unts of the same part type can be added to the repar kt n one teraton. The pseudo-code for ths step s found below. 1 n = 0 for all {1,..., N}, S = [n 1,..., n N ] (empty kt) 2 whle γ job (S) < β 3 = argmax { n <L max M} { } job (n,q1 ) (q1 n )H 4 n = q 1, S = [n 1,..., n,..., n N ] 5 k = 1 6 whle q1 < L max M and q 7 q k 8 end whle 9 q k 10 end whle = q k+1, k = k + 1 = q k+1 = Lmax M k+1 < Lmax M Lne 1 represents the ntalsaton. In lne 2 untl lne 10 tems are added to the repar kt untl the requred job fll rate s met. In lne 3 the part type s selected whch adds relatvely the most to the repar kt (.e., t has the hghest ncrease of the job fll rate wth 12

13 respect to the ncrease of the holdng costs). Lne 4 adds the unts q 1 of the selected part type to the repar kt S. In lne 5 untl lne 9 the orderng of q k such that q 1 represents the next quantty to consder for part type. s shfted one poston, Ths greedy procedure mmedately stops when the job fll rate s met. Even though the contents of the repar kt satsfes the servce level constrant after performng step 2 of the algorthm, the total holdng costs could be reduced when the last teraton s performed n a smarter way. Ths s the objectve of step 3 n our algorthm. The mprovement procedure starts wth removng the unts whch were added to the repar kt S n the last teraton, resultng n repar kt S. In order to satsfy the servce level constrant, tems have to be added to S wth the extra constrant C H (S ) < C H (S) to guarantee a better soluton. The same greedy procedure of step 2 can be used to nvestgate whether a soluton S exsts whch satsfes the job fll rate crteron wth lower holdng costs. In the prevous pseude-code, S has to be replaced by S and lne 3 of the pseudo-code should be replaced by 3 = argmax ( n <L max M, C H (S )+(q1 n )H <C H (S) ) { job (n,q 1 ) (q 1 n )H }. If such a soluton S exsts, t can be nvestgated for further mprovements by settng S to S and repeatng the mprovement procedure untl no new and better soluton s found. Besdes mprovng S we can also check whether unts of the current soluton S can be removed wthout replacng them wth other parts to reduce the holdng costs and stll satsfy the job fll rate crteron. Ths procedure s referred to as the mnmsaton of S. A backtrackng procedure s used to check whether the servce level s stll suffcent when one unt of the last added part type s removed and the one before, and so forth. The repar kt resultng from ths mnmsaton procedure s denoted by S. The overall best soluton S s found by performng the dfferent procedures n the order shown n Fgure 1. 5 Results In ths secton the performance of the algorthm descrbed n Secton 4 s tested by means of three test cases. Teunter [6] consders two knds of test cases: small nstances and large nstances. In the defnton of small nstances at most 8 dfferent part types are used and the maxmum tour sze s set to 4. For large nstances at most 100 dfferent part types are 13

14 greedy procedure: S mnmse(s): S best soluton S = S remove last added unts(s): S greedy procedure soluton S no soluton mnmse(s): S f C H (S ) < C H (S ), then S = S Fgure 1: The structure of the soluton procedure. consdered and the maxmum tour sze equals 12. As a thrd case, an addtonal settng s added whch s more representatve for realty. In ths thrd settng the number of dfferent part types ranges between 500 and 1,000. The test nstances are drawn from (dscrete) unform dstrbutons. In Appendx B we descrbe n more detal the specfc dstrbutons that are used for the dfferent parameters to randomly generate 1,000 examples for each test case. The results of all three cases are dscussed n ths secton. In our analyss we compare the repar kts obtaned by our algorthm wth the repar kts resultng from the algorthm of Teunter [6]. We have tested two aspects of our algorthm: (1) the mprovement and mnmsaton procedure (step 3 of our algorthm) and (2) the greedy procedure for the exact, closed-form expresson to calculate the job fll rate. To test the frst aspect, the contents of a repar kt s determned accordng to Teunter [6] and then our mprovement and mnmsaton procedure of Secton 4 modfes ths soluton. The relatve reducton of the total holdng costs for the dfferent nstances s shown n the frst row of Table 2. In the second row the relatve reducton of the holdng costs s presented when the entre algorthm of Secton 4 s used (ncludng the exact formula for the job fll rate) and compared to the outcome of Teunter s [6] algorthm. The thrd row presents the relatve devaton of the soluton found wth our algorthm compared to the optmal soluton, whch s found by enumeraton. Optmal solutons can only be found for the small nstances due to the complexty of the problem. Based on the results shown n Table 2, we conclude that the mprovement and mnmsaton procedure decreases the holdng costs on average by almost 5% for the small nstances. However, wth our closed-form expresson for the servce level we even fnd a decrease of 14

15 small large representatve average standard standard standard average average devaton devaton devaton approx. JFR + mprovements 4.68% 8.43% 0.45% 0.55% 1.05% 1.41% exact JFR + mprovements 5.83% 9.16% 1.30% 0.76% 14.18% 4.30% devaton from optmalty 0.25% 1.27% Table 2: Reducton of the total holdng costs over 1,000 nstances for each test settng, when the soluton procedure of Teunter [6] s complemented wth our mprovement and mnmsaton step and when t s compared to our procedure whch ncorporates the exact job fll rate (JFR). The fnal row shows the devaton of the results found wth our soluton procedure from optmalty. the holdng costs by 5.8% when the servce constrant s satsfed for the small nstances. Ths corresponds to an average devaton of 0.25% from the optmal soluton. For the large nstances the mprovements are less sgnfcant. The results for the representatve nstances show the most sgnfcant cost reductons. The reason that the representatve scenaro benefts the most from the exact job fll rate expresson s because of the dfferent prncples behnd the two algorthms. The algorthm of Teunter [6] adds unts to the repar kt based on the potental of each part type to ncrease the servce level, contrary to our algorthm whch adds unts that mmedately contrbute (relatvely) the most to the repar kt. In the representatve scenaro, the repar kt only contans at most one unt for most of the part types. The potental for each part type s, however, determned based on the contrbuton of addng more than one unt of that part type to the repar kt. Consequently, ths potental s not always realsed and other part types are selected n the next teratons of the algorthm. The average number of unts per part type n the repar kt s much larger for the small and large nstances. Therefore, the potental s a better representaton of the actual contrbuton of the part types n these two scenaros. Ths s also the reason why the mprovement and mnmsaton procedure of step 3 n our algorthm does not show bg mprovements n the representatve nstances. Table 3 shows a number of statstcs for the dfferent scenaros. The frst two rows show the sze of the repar kt and the average number of unts per part type n the repar kt. 15

16 small nstances large nstances representatve nstances average sze of repar kt average value of n frequences approx. JFR + mprovements s best 4.0% 0% 0.0% exact JFR + mprovements s best 19.4% 97.4% 93.6% same soluton 76.7% 2.6% 6.4% exact JFR + mprovements s optmal 89.3% - - p-values <1E-06 <1E-06 <1E-06 Table 3: Several statstcs about the solutons for the servce model n the dfferent test settngs. The sze of a repar kt s defned as the number of unts n the repar kt (.e., n ). The results for the representatve settng show repar kts wth the largest sze, but these repar kts also contan the most dfferent part types. Consequently, the repar kts of the representatve nstances contan on average 0.52 unts of each part type. Fgure 2 also shows ths relatonshp where mprovements are more sgnfcant when the average number of unts per part type (.e., n /N) s small. Table 3 also presents the frequences that the exact formula for the job fll rate results n a better soluton n comparson to the approxmaton procedure of Teunter [6]. The best soluton s found by the approxmate job fll rate procedure n 4.0% of the small nstances, whle the exact formulaton for the job fll rate fnds the best soluton n 19.4% of the nstances. In the remanng 76.7% of the small nstances, both methods result n the same soluton. Notce that we ncluded the mnmsaton and mprovement procedure n Teunter s [6] algorthm to obtan these results. Otherwse, the approxmaton procedure of Teunter would never have resulted n a better soluton. Table 3 also shows that our algorthm wth the exact job fll rate fnds the optmal soluton n 89.3% of the small nstances. For the large and more representatve cases the best found soluton s almost always found wth the exact job fll rate. Therefore, we conclude that the algorthm wth the exact expresson for the job fll rate, as formulated n Secton 3, sgnfcantly outperforms the algorthm wth the approxmaton of Teunter [6]. Ths can also be concluded when a Wlcoxon test s performed n whch the null 16

17 mprovement 70% 60% 50% 40% 30% 20% 10% 0% -10% average number of unts per part type small nstances large nstances representatve nstances Fgure 2: The relatve mprovement plotted aganst the average number of unts n the repar kt per part type for the dfferent test settngs. hypothess specfes that Teunter s [6] algorthm performs better. Based on the p-values 2 shown n Table 3 we reject the null hypothess and conclude that our algorthm performs sgnfcantly better. The results for the cost model are dscussed n Appendx A by performng the same set of experments as descrbed here. In the next secton the performance of our algorthm s tested n a practcal settng. 6 Case study Besdes the test nstances of Secton 5, we performed a case study to get a better feelng for the performance of the algorthm n practce. In ths case study we solve the servce model and the cost model, but we also perform a senstvty analyss on the servce level. Ths s mportant snce the management of repar servce companes wants to know the mpact of a 2 The p-value of a test refers to the probablty of wrongly rejectng the null hypothess f t s n fact true. Small p-values suggest that the null hypothess s unlkely to be true. 17

18 partcular servce level crteron on the holdng costs and the sze of the repar kt. Ths case study s based on real data from Rcoh Europe. Rcoh s a leadng global manufacturer of offce automaton equpment. They offer products for busnesses and for personal use. Rcoh performs the after-sales servce to the customers as well. In ths case study, we looked at mult-functonal systems (combned coper/prnter/fax/etc.). Rcoh Europe, located n Amstelveen, s the regonal headquarter of Europe, Afrca, and the Mddle-East. Currently Rcoh Europe has subsdares and branches n fourteen countres and factores n France and the Unted Kngdom. Rcoh Netherlands s one of the subsdares. Rcoh Europe has about ffteen thousand dstnct types of servce parts for about three hundred dfferent mult-functonal systems. Rcoh Netherlands has more than 35 techncans drvng around wth a stock value of almost 6,000 Euros each. Rcoh charges RTF cost of 45 Euros f a repar cannot be performed n the frst vst. When we analyse the contents of the current repar kts used by the techncans we see rather low servce levels of 53%. Therefore, the expected total return-to-ft costs are qute hgh. An overvew of the current stuaton s shown n Table 4 as well as the results for applyng the cost model and the servce model and the assocated algorthms. The soluton of the cost model shows an ncrease of the holdng costs by 250%. Despte the fact that the total costs reduce sgnfcantly, ths soluton s undesrable for Rcoh because of hgh rsk of theft. However, a soluton wth less holdng costs and an mproved servce level can be found wth the servce model. Table 4 gves an overvew on the costs for dfferent values of the servce level. Based on these results t s possble to ncrease the servce level by 31% aganst current holdng costs. In Fgure 3 we consder the relatonshp between the total holdng costs and the servce level. It shows a rapd ncrease of the servce level when the sze of the repar kt s small. Ths concave relatonshp s what s to be expected based on the property expressed n (7). Fgure 3 also shows the dfferent costs when unts are added to the repar kt. It shows a clear trade-off between the holdng costs and RTF costs. Based on the results of ths case study we conclude that our closed-form expresson for the order fll rate and our algorthm work well n practce. It can help a company decde whch parts to put n the repar kt, but t can also help them to analyse ther current stock levels. 18

19 job fll rate holdng costs RTF costs total costs current contents 53.14% cost model 95.99% servce model 84% % % % % % % % % % % % % % % % Table 4: Results for the current contents of the repar kt used n ths case study, as well as the results for the cost model and the servce model. 7 Conclusons Customer-orented markets become more and more mportant and, therefore, after-sales servces as well. One partcular servce s a repar servce on locaton, n whch a customer s only satsfed when a repar s completed. Ths means that a techncan should have enough spare parts taken along to the customer. If one of the requred parts s mssng, the 19

20 fll rate (a) ncrease of servce level holdng cost costs/day (b) ncrease of costs * * ** * * * *** * *** * * * * ** * **** holdng cost RTF cost total cost * **** * *** * ************* sze of repar kt Fgure 3: The results for the case study: (a) the ncrease of the servce level when the holdng costs ncrease, (b) the dfferent costs when the number of tems n the repar kt ncreases. techncan has to return later and none of the requred parts that are avalable are taken out of the repar kt. Ths latter characterstc of the repar kt problem s not dealt wth n prevous lterature. In ths paper, we derved an exact, closed-form expresson for the servce level n a general settng where multple unts of each part type can be used n a job and multple jobs are performed before the car s restocked. We also developed two algorthms to solve the servce and cost model whch ncorporate ths exact, closed-form expresson for the job fll rate. The algorthm for the servce model conssts of three dfferent steps: (1) fndng an order for the quanttes of a partcular part type to be added to the repar kt, (2) gettng a reasonable good soluton for the servce model wth the use of a greedy, margnal analyss heurstc, and (3) mprovng the ntal soluton accordng to an mprovement and mnmsaton procedure. The algorthm for the cost model s almost smlar except for some mnor modfcatons. Based on test nstances we concluded that our algorthm performs sgnfcantly better compared to other exstng algorthms n the lterature. Especally when only a few unts are requred n the repar kt for each part type. It outperforms the other algorthms n almost all examples and t fnds solutons wthn a range of 0.2% from the optmal soluton. We have also tested the applcablty of the algorthm n practce. Based on a case study we have shown an ncrease of the servce level by more than 30% wthout an ncrease n the holdng costs. Ths shows that our algorthm and the closed-form expresson can be used to 20

21 fnd the near-optmal contents of repar kts n practce. References [1] M. Bjvank. Car stock management - the need to close the gap between theory and practce. In D. Ksperska-Moon, edtor, Tenth ELA Doctorate Workshop, [2] S.C. Graves. A multple-tem nventory model wth a job completon crteron. Management Scence, 28(11): , [3] W.H. Hausman. On optmal repar kts under a job completon crteron. Management Scence, 28(11): , [4] D. Heeremans and L.F. Gelders. Multple perod repar kt problem wth a job completon crteron: A case study. European Journal of Operatonal Research, 81(2): , [5] S.A. Smth, J.C. Chambers, and E. Shlfer. Optmal nventores based on job completon rate for repars requrng multple tems. Management Scence, 26(8): , [6] R.H. Teunter. The multple-job repar kt problem. European Journal of Operatonal Research, 175(2): , A Cost Model The cost model s defned by (3). We use the algorthm for the servce model (see Secton 4) to solve the cost model n a general settng. Step 1 of the algorthm s smlar for both models. However, snce the cost model does not have any restrctons upon the servce level, the algorthm for the cost model needs a dfferent stoppng crteron. It also needs to keep track of the soluton wth the lowest expected total costs. The cost model does not need any mprovement steps, snce the algorthm does not stop mmedately. Therefore, step 3 of the algorthm for the servce model s removed for the cost model. Only step 2 of the algorthm for the servce model has to be adapted for the cost model. When we denote the soluton wth the lowest total costs by S, the pseudo-code for step 2 of the algorthm for the cost model s gven below. 21

22 1 n = 0 for all {1,..., N}, S = [n 1,..., n N ] (empty kt), S = S 2 whle C H (S) < C T (S ) 3 =argmax {:n <L max} M job (n,q 1 ) (q 1 n )H 4 n = q 1, S = [n 1,..., n,..., n N ] 5 f C T (S) < C T (S ) then 6 S = S 7 end f 8 k = 1 9 whle q1 < L max M and q 10 q k 11 end whle 12 q k 13 end whle = q k+1, k = k + 1 = q k+1 = Lmax M k+1 < Lmax M The new stoppng crteron n lne 2 s to stop addng tems when the holdng costs are hgher than (or equal to) the expected total costs of the best found soluton so far, where C T (S ) = C H (S ) + C RT F (S ). The performance for the cost model s tested wth the same set of experments as descrbed for the servce model: small, large, and representatve nstances (see Secton 5 and Appendx B). Snce there s no mprovement and mnmsaton procedure n the algorthm, we only compared the outcome of our algorthm to the outcome of the algorthm developed by Teunter [6]. Table 5 shows the relatve savngs on the expected total costs for all three test cases. Ths table also shows that there s hardly any devaton from the optmal soluton. Table 5 also shows the frequences how many tmes our algorthm results n a better soluton compared to the algorthm of Teunter [6]. B Settngs Sample Instances The test nstances are drawn from (dscrete) unform dstrbutons. Table 6 shows the specfc dstrbutons that are used for the dfferent parameters n the dfferent test settngs. We remark that P (M = m) > 0 for M 2 m M, M 9 m M and M 1 m 22

23 small nstances large nstances representatve nstances mprovement average 0.41% 0.50% 2.37% standard devaton 1.04% 0.52% 0.85% devaton from optmalty average 0.00% - - standard devaton 0.00% - - frequences approx. JFR s best 0% 0% 0% exact JFR s best 29.7% 87.9% 100% same soluton 70.3% 12.1% 0% exact JFR s optmal 97.8% - - average sze of repar kt average value of n Table 5: The results for the cost model. small nstances large nstances representatve nstances N dscrete unform[1,8] dscrete unform[1,100] dscrete unform[500,1000] L max unform[1,4] unform[1,4] unform[1,3] p job (j) unform[0,0.2/l max ] unform[0,0.2/l max ] unform[0,0.0005/l max ] H unform[0,0.35] unform[0,0.35] unform[0,0.05] M dscrete unform[3,6] dscrete unform[10,12] dscrete unform[2,3] P (M = m) unform[0,1/3] unform[0,1/10] unform[0,1/2] β unform[85%,95%] unform[85%,95%] unform[85%,95%] P unform[0,10] unform[0,100] unform[40,80] Table 6: The dstrbutons for the parameters used to generate the nstances for the dfferent test cases. M, respectvely, for each test case. To ensure that m P (M = m) = 1 we put the remanng probablty mass on the mddle tour sze. Also notce that p job (0) = 1 j pjob (j). 23