Many e-tailers providing attended home delivery, especially e-grocers, offer narrow delivery time slots to

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1 Vol. 45, No. 3, August 2011, pp ssn essn do /trsc INFORMS Tme Slot Management n Attended Home Delvery Nels Agatz Department of Decson and Informaton Scence, Rotterdam School of Management, Erasmus Unversty, 3062 PA DR Rotterdam, The Netherlands, nagatz@rsm.nl Ann Campbell Department of Management Scences, Tpple College of Busness, Unversty of Iowa, Iowa Cty, Iowa 52242, ann-campbell@uowa.edu Mortz Fleschmann Logstcs and Supply Chan Management, Unversty of Mannhem Busness School, Mannhem, Germany, mortz.fleschmann@bwl.un-mannhem.de Martn Savelsbergh CISRO Mathematcs, Informatcs, and Statstcs, North Ryde, NSW 1670 Australa martn.savelsbergh@csro.au Many e-talers provdng attended home delvery, especally e-grocers, offer narrow delvery tme slots to ensure satsfactory customer servce. The choce of delvery tme slots has to balance marketng and operatonal consderatons, whch results n a complex plannng problem. We study the problem of selectng the set of tme slots to offer n each of the zp codes n a servce regon. The selecton needs to facltate cost-effectve delvery routes, but also needs to ensure an acceptable level of servce to the customer. We present a fully automated approach that s capable of producng hgh-qualty delvery tme slot offerngs n a short amount of tme. Computatonal experments reveal the value of ths approach and the mpact of the envronment on the underlyng trade-offs. Key words: e-grocery; home delvery; tme slots; vehcle routng; contnuous approxmaton; nteger programmng Hstory: Receved: Aprl 2008; revsons receved: October 2009, August Publshed onlne n Artcles n Advance December 20, Introducton The success of many e-commerce busnesses hnges upon ther ablty to offer effcent and effectve lastmle delvery. Ths task s especally challengng n attended home delvery, partcularly for Internet grocers. The dramatc falures of e-grocery poneers such as Streamlne and Webvan underlne the mportance of a cost-effectve delvery operaton. Despte ntal falures, the onlne grocery market has recovered and has grown spectacularly snce 2003, showng an annual growth rate of 42% (Bernng, Ernst, and Hooker 2005). To provde a hgh servce level and to avod delvery falures as much as possble, t s customary n attended home delvery servces for the company to offer the customer a choce of narrow delvery tme slots. The specfc tme slot offerng does not only mpact the perceved customer servce but also the expected delvery effcency. In ths paper, we consder the tactcal problem of selectng the tme slots to offer n each zp code of the delvery regon. Ths selecton needs to offer an acceptable level of servce n each zp code and needs to be able to yeld cost-effectve daly delvery routes. Once such a tme slot schedule s put n place, there s the operatonal problem of managng the avalablty of the offered tme slots. We wll focus only on the tactcal, not the operatonal problem. Our research effort s motvated by operatons at Albert.nl, the Internet channel of Albert Hejn, the largest grocery supermarket chan n the Netherlands and a subsdary of Royal Ahold. Albert.nl s now the only full-scale Internet grocer n the Netherlands. Started n 2001, ther delvery servce s now avalable to around 60% of the country s households. From dedcated warehouses, the company uses small vans to delver a broad assortment of grocery products to each customer s home. The tme slot offerng at Albert.nl s typcal for those n the e-grocery ndustry (see, for example, Peapod.com and Tesco.com). The customer can choose a two-hour delvery slot from a menu of tme slots avalable n hs or her specfc zp code. Each tme slot has a correspondng delvery fee rangng from E4.95 to E8.95, dependng on the day and tme of delvery. On a day-to-day bass, when delvery capacty becomes tght, certan tme slots n certan zp codes may be closed. Closng a tme slot at 435

2 Agatz et al.: Tme Slot Management n Attended Home Delvery 436 Transportaton Scence 45(3), pp , 2011 INFORMS Albert.nl means that t s no longer avalable for the customer to choose (the slot turns grey on the webste). Routes are planned for each mornng and afternoon shft. After the order cutoff tme for a shft has passed and thus all orders are known, the delvery routes for that shft are planned usng a commercal vehcle routng package. The routng package properly takes delvery wndows, dwell tmes, and vehcle capactes nto account. Albert.nl uses a dfferentated tme slot offerng lnked to the demand characterstcs of each zp code. Ths means that a zp code wth low demand potental receves only a few weekly tme slots n an attempt to concentrate that demand, where hgh-demand zp codes receve a larger number of slots. The slots offered, regardless of the total number, must also exhbt a balance over the week between mornng and afternoon tme slots to ensure an acceptable level of servce and choce to the customers. Determnng the specfc tme slots for each zp code nvolves a careful trade-off between marketng and operatonal consderatons. After decdng the number of mornng and afternoon slots that wll satsfy marketng needs, referred to hereafter as servce requrements, the specfc tme slots are selected for each zp code to satsfy operatonal needs such as a low-cost delvery schedule. Because delvery vehcles may vst several zp codes durng a sngle tme slot and a delvery route spans multple tme slots, routng consderatons should play a vtal role n the constructon of a tme slot schedule. Assgnng specfc tme slots to a zp code should not be done n solaton but should be consdered jontly for neghborng slots. Ths results n a complex plannng problem. The tme slot schedule employed at Albert.nl s created manually. Because of ts complexty, ths task takes several weeks to complete. Clearly, ths s undesrable n an envronment that contnually changes as a result of a 30% annual growth rate. In partcular, ths manual process greatly lmts ther ablty to assess and compare potental future scenaros. Therefore, Albert.nl s consderng optons for automatng the tme slot schedule generaton. In ths paper, we present an approach for makng the tme slot decson that apples not only to Albert.nl but also for many other companes offerng attended delvery servces. More specfcally, we address the followng tme slot management problem (TSMP). Gven servce requrements and average weekly demands for each zp code n the delvery regon, determne the set of tme slots to offer n each zp code so as to mnmze expected delvery costs whle meetng the servce requrements. We explore a contnuous approxmaton approach for the TSMP. Ths type of approach focuses on a realstc evaluaton of a gven tme slot schedule and then uses relatvely smple optmzaton technques to mprove the schedule. Our man contrbutons are the dentfcaton and explanaton of a relevant plannng concern n the e-grocery busness that gves rse to challengng optmzaton problems; the development of a fully automated approach capable of producng hgh-qualty tme slot schedules much faster than the current manual process; a computatonal study analyzng the vablty and merts of the approach; and a computatonal study nvestgatng the mpact of changes n the envronment, such as ncreases n demand and demand varablty, more relaxed or more strngent servce requrements, and changes n the structure of the delvery tme slot template. The remander of the paper s organzed as follows. In 2, we summarze the relevant lterature. In 3, we ntroduce notaton and outlne our modelng framework. In 4, we present our contnuous approxmaton approach. In 5, we present an alternatve nteger programmng based approach that wll be used as a source of comparson n our computatonal experments. In 6, we descrbe the desgn of our computatonal experments and present the results. Fnally, n 7, we summarze our man nsghts and dscuss drectons for future research. 2. Lterature The choce of tme slots s one of many crtcal decsons n the desgn of an attended home delvery system, as dscussed n Agatz et al. (2008a) and Agatz, Fleschmann, and van Nunen (2008b). Other mportant ssues nclude the choce between store-based and warehouse-based delvery, nventory allocaton, prcng, and return handlng. Home delvery and the desgn and management of tme slots has receved lttle attenton n the dstrbuton and routng lterature. Tme slots are typcally assumed to be exogenous nformaton, e.g., specfed by the customer or set by the sales and marketng department. The rse of e-commerce, whch offers numerous opportuntes for the use and actve management of tme slots, s changng ths. Much of the early research on last-mle delvery prmarly examnes the mpact of dfferent tme slot lengths. For example, Punakv and Saranen (2001) assess the mpact of the tme slot length on transportaton costs for attended and unattended delveres. Ther results ndcate substantal effcency gans when tme constrants are relaxed. Completely flexble, unattended delvery reduces costs by up to a thrd, relatve to attended delvery wth two-hour tme slots. Ln and Mahmassan (2002) summarze the delvery polces employed by many onlne grocers n

3 Agatz et al.: Tme Slot Management n Attended Home Delvery Transportaton Scence 45(3), pp , 2011 INFORMS 437 the Unted States and use vehcle routng software to evaluate the performance of some of these polces on a few realstc nstances of the problem. Both unattended and attended polces are consdered along wth dfferent tme slot lengths. Another related stream of research focuses on the real-tme management of tme slots n attended home delvery. Ths work draws on revenue management concepts and apples them n a dstrbuton settng. In ths ven, both Bent and Van Hentenryck (2004) and Campbell and Savelsbergh (2005a) examne whch delveres to accept or reject. Ther approaches explot stochastc nformaton about future requests to decde on requests under consderaton. Bent and Van Hentenryck (2004) am to maxmze the number of accepted requests by controllng the tme slots offered, but they do not consder rejectng an expensve delvery to preserve resources for more, future delveres, n contrast wth Campbell and Savelsbergh (2005a). Campbell and Savelsbergh (2005b) study the use of ncentve schemes to reduce delvery costs by encouragng the selecton of tme slots that result n low-cost delvery routes. In contrast to the real-tme management of tme slots, our TSMP s tactcal n nature and allocates tme slots geographcally pror to actual order ntake. The resultng tme slot schedules serve as a startng pont for subsequent real-tme adjustments. Antcpatng the mpact of the offered tme slots on delvery routes s crucal when solvng the TSMP. Ths lnks the TSMP to the vast body of lterature on vehcle routng wth tme wndows. See Braysy and Gendreau (2005a, b) for an extensve revew. In partcular, the TSMP shares some aspects wth dynamc and stochastc vehcle routng. In dynamc vehcle routng, as dscussed n Jallet and Odon (1988), Bertsmas, Jallet, and Odon (1990), and Gendreau and Potvn (1998), not all nputs are known beforehand but become known concurrently wth the determnaton of the soluton. In stochastc routng, some components of the problem are stochastc, such as travel tmes (Kenyon and Morton 2003), demand (Dror, Laporte, and Trudeau 1989), or customers (Hvattum, Lokketangen, and Laporte 2006). Another problem wth features smlar to those of the TSMP s the perod vehcle routng problem wth servce choce (PVRP-SC), where vehcle routes need to be constructed over a perod of tme and servce frequency s a decson n the model (Francs, Smlowtz, and Tzur 2006; Francs and Smlowtz 2006). Gven the computatonal complexty of the vehcle routng problem alone, t s unrealstc to ncorporate a full routng model n the tactcal TSMP. Instead, routng costs resultng from tme slot selectons have to be approxmated. One way s to model routng costs usng the contnuous approxmaton method (Daganzo 1987a, b). Ths approach represents demand by contnuous functons and assesses system-wde costs by aggregatng over local cost estmates. For an overvew of contnuous approxmaton models n logstcs, see Daganzo (2005) and Langevn, Mbaraga, and Campbell (1996). Another opton s to explctly model routng decsons but on a more aggregate level than a full vehcle routng problem. Ths dea s the bass for the alternatve approach explored n Assumptons and Notaton The objectve of the TSMP s to mnmze the expected delvery costs for the customer orders ensung from the offered delvery tme slots. Ths requres assumptons on the demand response to a certan tme slot offerng. Throughout our analyss we make the followng two key assumptons, where demand s measured n number of customer orders: 1. The expected weekly demand for each zp code s known and ndependent of the set of offered tme slots. 2. The expected weekly demand for each zp code s dvded evenly over the set of offered tme slots. The frst assumpton supposes that customers exhbt a certan degree of flexblty, such that changng the tme slots offered does not result n lost sales. An emprcal analyss of hstorcal data from Albert.nl supports the valdty of ths assumpton for moderate changes n the tme slot offerng (Agatz 2009). The second assumpton mples that all tme slots are equally popular. Dfferentated delvery fees are ntroduced specfcally for ths reason. They counterbalance dfferences n tme slot popularty and smooth demand. Hstorcal data from Albert.nl show the effectveness of dfferentated delvery fees and ndcate that the equal tme slot popularty assumpton s reasonable. We further assume that all orders are of the same sze n terms of the number of totes. The sze of an order mpacts the number of orders that ft n a delvery vehcle. We conducted a few computatonal experments n whch we vared order szes and found that t had lttle or no mpact on the results. The marketng-mposed servce requrements lmt the set of feasble tme slot offerngs. Our models treat these servce requrements as exogenous. The servce requrements employed at Albert.nl specfy for each zp code the requred number of avalable delvery slots n each shft, e.g., one on Monday mornng and two on Tuesday afternoon. As a result of these specfc servce requrements and the above demand assumptons, the problem decomposes by shft. Thus, separate problems need to be solved for each shft. Our objectve functon consders fxed costs as well as varable costs, ncorporatng both dstance-related

4 Agatz et al.: Tme Slot Management n Attended Home Delvery 438 Transportaton Scence 45(3), pp , 2011 INFORMS Table 1 Notaton : Set of zp codes (ndexed by ) : Set of tme slots (ndexed by s) : Set of vehcles e : Expected demand for zp code for each tme slot Q: Vehcle capacty (n terms of the number of orders that can be accommodated) f: Vehcle fxed cost c n : Cost per mle for vehcle and drver usage when drvng wthn a zp code and wthn a tme slot c z : Cost per mle for vehcle and drver usage when drvng between dfferent zp codes wthn a tme slot c t : Cost per mle for vehcle and drver usage when drvng between zp codes n consecutve tme slots c 0 : Cost per mle for vehcle and drver usage when drvng to or from the depot d j : Dstance between centers of zp codes and j : Dwell tme per order and tme-related costs. We do not take nto account delvery fees and solely focus on mnmzng expected delvery costs. In Table 1, we ntroduce notaton that s common to the proposed contnuous approxmaton approach as well as the alternatve nteger programmng based approach that wll be used for comparson. Note that e s the weekly demand for zp code dvded by the number of tme slots that need to be offered n zp code. 4. Contnuous Approxmaton Approach At the heart of the contnuous approxmaton approach s a model for estmatng the delvery cost of a gven tme slot schedule for a set of zp codes. Based on ths delvery cost estmate, a local search s performed to teratvely mprove the tme slot schedule. The orgnal contnuous approxmaton approach proposed by Daganzo (1987a) does not consder tme slots and dstngushes two components of a vehcle tour, namely the stem dstance between the depot and the delvery area and the dstance between consecutve stops wthn the delvery area. To account for tme slots, we expand these deas and dstngush four components of vehcle tours (see Fgure 1): d 0 : the stem dstance to or from the depot to a stop n a tme slot; d n : the dstance between stops wthn a zp code wthn a tme slot; d z : the dstance between stops n dfferent zp codes wthn a tme slot; and d t : the dstance between stops n consecutve tme slots. The correspondng travel tmes are denoted by h 0, h n, h z, and h t. In lne wth the contnuous approxmaton methodology, we estmate the dstance values for these Fgure 1 d 0 d n zp 1 zp 2 d z Routng Components d t zp 3 Tme slot 1 e Tme slot 2 components of a vehcle tour based on local data. We estmate the dstance values from the perspectve of each zp-code tme slot combnaton s n the tme slot schedule. These local estmates are then used to compute a local estmate of the dstance per order (excludng the stem dstance to and from the depot) as follows: DPO s = 1 n s [ ( ( n t n t s 1 n z s d t n t s + s 1 nz s d z s n z s s + n n s 1 dn s ))] where n s denotes the estmated number of orders per route, n t s denotes the estmated number of tme slots covered by a route, n z s denotes the estmated number of zp codes vsted n tme slot on a route, and n n s denotes the estmated number of orders delvered n a zp code n a tme slot on a route. These n-values are agan estmated based on local data from the perspectve of a gven zp-code tmeslot s. By multplyng the dstance per order wth the cost per klometer, we obtan a local estmate for the dstance-related costs (excludng stem dstances). Ths, n turn, yelds a local estmate of the cost per order (excludng stem costs and fxed vehcle costs): CPO s = 1 [ ( ( n t n t s 1 n z s d t n s n t s ct + n z s 1 s d z s n z s cz s + n n s 1 dn s cn ))] We obtan an estmate of the expected delvery cost assocated wth a gven tme slot schedule by multplyng the expected number of orders e for a zp-code tme slot combnaton s wth the cost per order

5 Agatz et al.: Tme Slot Management n Attended Home Delvery Transportaton Scence 45(3), pp , 2011 INFORMS 439 CPO s, aggregatng over all zp codes and tme slots, and addng stem costs and fxed vehcle costs: expected delvery cost e CPO s +d 0 c 0 +fv s account for ths stuaton, we modfy s weghted densty: s = j mn 1 e j I js j a j to be a where d 0 s the estmated total stem dstance and v s the estmated number of vehcles requred. Note that we sum over all zp codes and all offered tme slots for that zp code. Also note that we omt labor costs for the dwell tmes because these costs are constant and ndependent of the offered schedule Estmatng Vehcle Tour Components Part I In ths subsecton, we dscuss how the vehcle tour components are estmated from the perspectve of a gven zp-code tme slot combnaton s. We start wth the vehcle tour components used n the calculaton of the cost per order Dstance Between Stops Wthn a Zp Code Wthn a Tme Slot ds n. Statstcal analyss of recent routng data at Albert.nl shows that the drvng dstance between two customers wthn the same zp code s smlar across zp codes; we use d n to represent ths dstance. The same s true for the travel tme between two customers wthn the same zp code, so we use h n to represent ths travel tme Dstance Between Stops n Dfferent Zp Codes Wthn a Tme Slot ds z. Daganzo (2005) approxmates the dstance between two consecutve stops of a route through a regon wth a slowly varyng demand densty by k/, where k s a dmensonless constant that s ndependent of the regon shape. We apply ths approach on a zp code level and consder the densty of open zp codes n tme slot s n the neghborhood of. Let be a collecton of zp codes n the neghborhood of zp code (ncludng zp code tself). In our calculatons, we take to be the set of zp codes wthn a gven maxmum dstance from the center of zp code. Let s be the densty of open zp codes n durng tme slot s. If a j denotes the surface area of zp code j and I js denotes whether zp code j s open durng tme slot s (I js = 1) or whether t s closed durng tme slot s (I js = 0), then j s = I js j a j The estmate of ds z s then k/ s. We mpose an upper bound d z on ths estmate to handle very low denstes s. In our calculatons, we set dz equal to twce the average dameter of a zp code. Thus, ds z = mn k/ s d z. The above calculaton of s does not take nto account that there may be open zp codes wth an expected demand per slot e smaller than one. To Ths can be nterpreted as treatng values of e smaller than one as probabltes of a demand occurrence Dstance Between Stops n Consecutve Tme Slots ds t. We take a smlar approach when computng ds t, but usng zp code denstes n precedng and succeedng tme slots. We set d pre = d n f tme slot s 1 s open for zp code,.e., for consecutve open tme slots we smply use the average dstance between stops wthn a zp code. Otherwse, we set d pre = mn d z k/ s 1, where s 1 s the densty of open zp codes n n tme slot s 1. Analogously, d suc = d n f tme slot s + 1 s open for zp code and d suc = mn d z k/ s+1 otherwse, where s+1 s the densty of open zp codes n n tme slot s + 1. Fnally, we set ds t = dpre (d suc ) f s s the frst (last) tme slot of the shft and ds t = dpre + d suc /2 otherwse Number of Stops n a Zp Code n a Tme Slot on a Route n n s. The number of stops n a zp code n a tme slot s lmted by the demand e and by the length of the tme slot l. Let n be the maxmum number of stops that can be made durng a tme slot of length l and recall that denotes the dwell tme. Then l h t s = n + n 1 h n s and thus n = l h t s + hn / + h n, where we subtract h t s from l to account for travel tme between consecutve tme slots. The number of stops n a zp code durng a tme slot n n s s therefore mn e n Number of Zp Codes Vsted n a Tme Slot on a Route n z s. Analogously, the number of zp codes vsted n a tme slot s lmted by the number of neghborng open zp codes n open and by the length of the tme slot l. Let z denote the maxmum number of zp codes that can be vsted durng a tme slot of length l. Then l h t s = z n n s + nn s 1 hn s + z 1 h z s, and thus z = l h t s + hz s / nn s + nn s 1 hn + h z s. We then set n z s = mn nopen z Number of Tme Slots Covered on a Route n t s. The number of tme slots covered on a delvery tour s lmted by the number of tme slots n the shft and by the number of orders that can be accommodated by the delvery vehcle Q. Therefore, we set n t s = mn Q/ nn s nz s Number of Stops on a Route n s. Usng the prevously estmated parameters, the number of stops on a delvery tour s n t s nz s nn s.

6 Agatz et al.: Tme Slot Management n Attended Home Delvery 440 Transportaton Scence 45(3), pp , 2011 INFORMS 4.2. Estmatng Vehcle Tour Components Part II In the prevous subsecton, we have shown how we compute local estmates for the vehcle tour components used n the computaton of the local estmate of the cost per order. What remans to be shown s how to estmate the number of vehcles requred (v) and the stem dstance (d 0 ) Number of Vehcles v. We start wth the estmate of the number of vehcles requred. The number of vehcles requred for servng the total demand of a shft depends on the vehcle capacty and on the number of orders that can be delvered n a tme slot,.e., on physcal capacty and on avalable tme. We address each of these constrants separately. We have local estmates for the number of orders per route (n s ). We take a demand-weghted average over these local estmates to obtan a global estmate of the number of orders per route. Dvdng the expected demand for a shft by ths global estmate of the number of order per route gves an estmate of the number of routes r. More precsely, let s denote the set of open zp codes n tme slot s. Then we have r = s s e s s e / = s s e 2 s s e n s s s e n s Smlarly, we estmate the number of vehcles requred to serve all orders n a gven tme slot v s, based on the expected demand per tme slot and the estmated number of stops per tme slot, wth v s = e 2 s e n z s s nn s The number of vehcles requred s the maxmum of the number of routes r and the number of vehcles requred to serve all orders n a tme slot v s. Ths s represented by v = max r max s v s Stem Dstance d 0. Next, we consder the (total) stem dstance d 0. We start by estmatng ds 0, the expected stem dstance traveled by a vehcle makng ts frst delvery n tme slot s. Ths s done by takng the average dstance between the depot and the open zp codes n tme slots s: d 0 s = j I js d 0j j I js Of course, the expected stem dstance traveled by a vehcle makng ts last delvery n tme slot s s the same. Next, we use the estmates of the number of vehcles requred n each tme slot (v s ) to estmate the number of vehcles that start ther trp n tme slot s and to estmate the number of vehcles that end ther trp n tme slot s. Let v 0 = 0 and v +1 = 0, then we estmate the (total) stem dstance as ( d 0 = max 0 vs+1 v s d 0 s+1 + max 0 v ) s v s+1 d 0 s s= Handlng Overlappng Tme Slots. A further complcaton arses f tme slots are overlappng, as s the case at Albert.nl. In our calculatons, we handle ths stuaton by creatng adjusted nonoverlappng slots. To ths end, we apporton the length of the overlappng part of two wndows to both of the ndvdual wndows, based on ther expected demand volumes. For example, consder two two-hour wndows that overlap for one hour. Furthermore, assume that the total expected demand from all open zp codes n the frst tme slot s twce as large as the total expected demand n the second tme slot. Then, we create a frst tme slot of sze 1.66 hours and a second slot of sze 1.33 hours Soluton Method The dscusson above shows how to estmate the expected costs of a gven tme slot allocaton. The next step s to optmze the tme slot allocaton based on ths evaluaton. Note that ths optmzaton problem s nonlnear and nonconvex. Therefore we use a smple greedy teratve mprovement heurstc. We can start from any feasble tme slot schedule. Observe that fndng a feasble schedule for the TSMP s not dffcult because any schedule that satsfes the servce requrements s feasble. In our experments, we begn wth the schedule currently n place at Albert.nl. Next, we determne for each zp code the tme slot allocaton for that zp code that results n the mnmum expected delvery cost (keepng the tme slot allocaton for all other zp codes fxed) by complete enumeraton. We then adjust the current tme slot schedule by mplementng the tme slot allocaton for the zp code that acheves the mnmum expected delvery cost. We repeat ths process as long as there s a reducton n expected delvery costs greater than some threshold or untl a maxmum number of teratons s reached. 5. Integer Programmng Alternatve The contnuous approxmaton model presented n the prevous secton does not model the operatonallevel routng decsons explctly. Instead, t ams to reflect them mplctly n the employed cost approxmatons. In ths secton, we complement ths approach wth a model that deals wth the embedded routng component of the TSMP n a more explct way and we wll compare the results obtaned by the contnuous approxmaton model wth the results of ths model n our computatonal experments. As wth the contnuous approxmaton model, the objectve s to construct a tme slot schedule that satsfes the servce requrements and mnmzes delvery costs. Intutvely, one could model the routng costs by ntroducng varables that represent, for each vehcle, the sequence of stops at varous zp codes wthn

7 Agatz et al.: Tme Slot Management n Attended Home Delvery Transportaton Scence 45(3), pp , 2011 INFORMS 441 each tme slot. However, because vehcles vst multple tme slots and may vst multple zp codes wthn a tme slot, the embedded routng component makes the nteger program very hard to solve. Even fndng feasble solutons n a reasonable amount of tme s qute dffcult wth modern, sophstcated, and powerful nteger programmng solvers. Therefore, we approxmate the routng costs usng a seed-based scheme (Fsher and Jakumar 1981). Even though the number of stops a vehcle makes durng a shft may be large, the number of tme slots n a shft s small; no more than four n the case of Albert.nl. Solvng routng problems wth at most four stops s much easer. Ths cost approxmaton s expressed graphcally n Fgure 2. The dashed crcles surround the customers served n a sngle tme slot wth the center dot representng the seed. Wth the use of seeds, we can approxmate the routng costs for a vehcle by the sum of two costs: (1) the cost of a route through the seeds assocated wth each tme slot plus (2) a smple estmate of the costs ncurred when vstng the customers wthn each sngle tme slot based on the dstance to the seed. Our nteger program has the followng varables to represent the dfferent decsons and assocated costs: z v s one f vehcle v s used and zero otherwse; y vs s one f zp code s the seed for vehcle v for tme slot s and zero otherwse; x vs s one f vehcle v vsts zp code n tme slot s and makes a delvery and zero otherwse; Xj vs s the dstance from zp code to zp code j f vehcle v serves customer n tme slot s and zp code j s the seed for tme slot s and zero otherwse; Yj vs s the dstance between the seed of tme slot s 1 and the seed of tme slot s for vehcle v f zp code s the seed of tme slot s 1 and zp code j s the seed of tme slot s and zero otherwse; u s s one f a partcular tme slot s s offered for a partcular zp code and zero otherwse; and q vs s the porton of the demand of zp code n tme slot s that s served by vehcle v. One zp code wll serve as the seed (or representatve) for the customers served n a partcular tme slot/vehcle combnaton, and ths decson s represented by the y varable. The set of zp codes served n a partcular tme slot/vehcle combnaton s captured by the x varables. The estmate of the cost to serve a set of zp codes n a tme slot s based on the sum of the dstances between the zp codes and the seed, whch are represented by the X varables. The cost of vstng the seeds assocated wth a vehcle,.e., the route through the seeds, s captured by the Y varables. Note that the desred tme slot schedule s mpled by the x varables. Slot s s offered n zp code f and only f x vs = 1 for some v. Gven the parameters defned n Table 1 and these varables, the objectve functon can be expressed as mn f z v + v V + Z j Z v V s Z j Z v V s s>1 c n X vs j + c t Y vs j j Z v V s c 0 X vs 0j The frst term represents the cost for usng vehcles/drvers. The second term represents the cost to travel between seeds. The last two sums represent the cost to serve zp codes from seeds and the depot from seeds, respectvely. Note that, as n the contnuous approxmaton model, we omt labor costs for the dwell tmes because they are constant. The contnuous approxmaton model presented n 4 evaluates the cost of a gven tme slot schedule. Here, though, the model needs not only to capture the costs but also needs to defne what makes a schedule feasble. To do so, we must nclude some new parameters: r : the number of slots that must be open for zp code n a shft,.e., the servce requrement for, t n : the travel tme between customers n the same zp code, and t t : a multpler to convert the dstance between zp codes nto a travel tme between zp codes. Next, we descrbe the constrants that defne feasble solutons. Frst, we examne the lnkage between the X and Y varables and the x and y varables. For the X varables, the lnkage s defned as follows: X vs j d j x vs + y vs j 1 Z j Z v V s Fgure 2 Routng wth Seeds The above s vald because t forces Xj vs to have nonzero value d j only f both x vs and yj vs equal one; otherwse, the rght-hand sde has a value less than or equal to zero. In the same way, we can defne the lnkage for the Y varables wth Y vs j d j y vs 1 +y vs j 1 Z j Z v V s s >1 Note that the above s only used for s > 1 because the cost to travel to the frst seed from the depot wll be handled by the X varables.

8 Agatz et al.: Tme Slot Management n Attended Home Delvery 442 Transportaton Scence 45(3), pp , 2011 INFORMS The next set of constrants enforces that every tme slot has a seed, y vs = z v v V s Z Note that enforcng that every tme slot has a seed does not create unnecessary travel costs n the case where a vehcle serves no customers n a tme slot, because the seed can be chosen to be the same as the seed of the prevous (or subsequent) tme slot. We allow multple vehcles to vst the same zp code n the same tme slot. However, we should make sure that multple vsts to a partcular zp code n the same tme slot do not count toward the zp code s servce requrements. Therefore, we use the bnary varable u s n u s = r Z v V s x vs u s Z s Recall that we have assumed that the demand n a zp code s the same n every tme slot offered. If multple vehcles serve a zp code durng a partcular tme slot, then the zp code s demand e must be dstrbuted over the dfferent vehcles. Ths s defned by the followng constrants: q vs e u s Z s q vs v V mn e Q x vs v V Z s The next set of constrants enforces that the demand served by a sngle vehcle does not exceed the vehcle capacty Q: Qz v Z s q vs x vs v V To properly estmate the routng costs, especally the travel tme to and from the depot, we force the depot to be served n the frst and last tme slot of a partcular delvery perod,.e., tme slot 1 and tme slot ŝ: x v1 0 = zv v V x vŝ 0 = zv v V Recall from the contnuous approxmaton model that the maxmum number of delveres that a vehcle can make s constraned not only by the vehcle capacty but also by the length of each tme slot. Enforcng ths tme constrant s one of the most complex parts of the nteger programmng model. The stuaton s further complcated by the fact that the number and the length of the tme slots can vary by day and by shft. Fnally, there s a need to dstngush the frst tme slot, n between tme slots, and the last tme slot. Recall that travel from the depot and back to the depot can occur outsde of the tme wndows, so t does not need to be ncorporated wthn these tme wndow length constrants. As an example, let l 1 represent the wdth of the frst tme slot. Then the constrants to lmt the tme spent on delveres durng the frst tme slot s Z j Z t t X v1 j + Z Z j Z ( e + t n e 1 ) x v1 t t Y v2 j l 1 v V (1) The frst term captures the travel tme to the seed and the travel tme between the zp codes vsted durng the tme slot. The second term n constrant (1) captures the stop tme at a customer served n a zp code as well as the travel tme between customers served n a zp code. The fnal term n constrant (1) captures the travel tme from the seed of ths tme slot to the seed of the next tme slot. The 0.5 n the fnal term reflects the fact that half of ths travel tme s allocated to ths tme slot and the other half s allocated to the next tme slot. Constrants for the n between and fnal tme slots are structured smlarly. To handle overlappng tme wndows, we add a constrant that assures that the total length of the combned slots s not exceeded. As an example, consder the tme slots 8 a.m. 10 a.m. and 9 a.m. 11 a.m. (referred to below by 1 and 2, respectvely) and assume that the 8 a.m. 10 a.m. tme slot s the frst tme slot of the shft and that the 9 a.m. 11 a.m. tme slot s not the last tme slot of the shft. Let the total tme perod covered by the tme slots be denoted by l, whch s three hours n ths example. We add the followng constrant: Z j Z + Z t t X v1 j + X v2 j + Z j Z t t Y v1 j Y v2 j ( e + t t e 1 ) x v1 + ( e + t t e 1 ) x v2 l Z Last, we add symmetry breakng constrants. These constrants are not necessary for the correctness of the model, but help the nteger programmng solver fnd better solutons faster. These symmetry breakng constrants force the lower ndexed vehcles to be the ones wth the largest number of stops: x vs x v 1s v V v > 1 Z s Z s When only a subset of the vehcles s used, there are many equvalent solutons. By mposng z v z v 1 v V v > 1 the number of equvalent solutons s also sgnfcantly reduced.

9 Agatz et al.: Tme Slot Management n Attended Home Delvery Transportaton Scence 45(3), pp , 2011 INFORMS 443 Customer Depot Fgure 3 Geographcal Dstrbuton n Njmegen Regon 6. Computatonal Experments In ths secton, we report on the results of a computatonal study to evaluate the performance of the proposed models and to analyze the mpact of varous characterstcs of the envronment on ther performance. Recall that the TSMP s a tactcal plannng problem wth an embedded routng problem on the operatonal level. We evaluate alternatve tme slot schedules through smulaton. To ths end, we generate multple demand realzatons for each schedule and determne the correspondng detaled routng costs usng a commercal routng package. To evaluate the mpact of characterstcs of the envronment on the tme slot schedules, we vary ndvdual characterstcs and analyze the results. For our study, we use real-lfe data from Albert.nl. Specfcally, we focus on the Njmegen regon, a subset of the servce area of Albert.nl consstng of 30 three-dgt zp codes wth varyng demand denstes and coverng a total area of approxmately 1,000 km 2. Fgure 3 shows the geographc dstrbuton of demand n the regon. The area s served through a delvery hub n the cty of Njmegen. We consder a typcal mornng shft and a typcal afternoon shft, because they dffer n terms of total length, number of tme slots, and tme slot overlap (see Fgure 4). Furthermore, we use the current servce requrements at Albert.nl for the Monday mornng and the Tuesday afternoon. (We also performed smulatons for other weekdays and obtaned smlar results.) The employed servce requrements mply that n the mornng schedule 24 zp codes have to receve one tme slot, three zp codes have to receve three tme slots, and three zp codes do not receve any tme slot. In the afternoon schedule, 25 zp codes have to receve one tme slot and fve zp codes have to receve two tme slots. In our experments we evaluate the followng tme slot schedules: CA: The schedule produced by the contnuous approxmaton approach; IP: The schedule produced by the alternatve nteger programmng approach; ALBERT: The schedule that s currently n place at Albert.nl; ALL: The schedule n whch all tme slots are made avalable n all zp codes where Albert.nl currently offers at least one tme slot; NO-SLOT: The schedule n whch a sngle tme slot spannng the entre shft s made avalable n all zp codes where Albert.nl currently offers at least one tme slot. The schedules ALL and NO-SLOT are ncluded to provde bounds on the performance and to provde nsghts nto the cost-servce trade-off. Note that these two schedules do not use the servce requrements of Mornng shft 08:00 14:00 Afternoon shft 16:00 21:00 09:00 11:00 19:00 21:00 08:00 10:00 10:00 12:00 12:00 14:00 16:00 18:00 18:00 20:00 Fgure 4 Albert.nl Tme Slots

10 Agatz et al.: Tme Slot Management n Attended Home Delvery 444 Transportaton Scence 45(3), pp , 2011 INFORMS Albert.nl. Instead, the schedule ALL offers the hghest level of customer servce, and the schedule NO-SLOT offers the lowest level of customer servce. The servce levels wll be reflected n the delvery costs, so the NO-SLOT schedule serves as a lower bound, albet a weak lower bound, and the ALL schedule serves as an upper bound. We supplement the comparson outlned above wth values derved from the smulaton results of a few randomly generated feasble tme slot schedules. More precsely, we randomly generate fve feasble tme slot schedules, perform a smulaton for each of them, and average the statstcs obtaned from the smulaton runs. In our results tables, the resultng values wll be labeled wth RAND-ASGN. For each problem nstance, we generate 20 random demand nstances for our smulatons as follows. We start wth a lst of 448 customer addresses n the Njmegen area. For each demand nstance, we randomly select a number of addresses from ths lst for each zp code, based on the expected number of orders per tme slot n that zp code. Specfcally, we decde randomly and ndependently for each address whether or not to select t. We set the probablty of selecton of an address equal to the rato between the expected number of orders n the zp code to whch that address belongs and the total number of addresses n that zp code. Therefore, the total number of orders vares across demand realzatons, wth an average equal to the number of orders that was used as nput to the optmzaton models. For 20 dfferent streams of random numbers, ths results n 20 dfferent random sets of addresses. We use the same sets for each schedule, n order to reduce varablty across schedules to obtan more comparable results. Usng ths approach, demand realzatons for dfferent tme slot schedules dffer only n terms of the tme slots assocated wth each order. For example, f tme slot schedule CA has slots 2 and 4 open and tme slot schedule IP has slots 3 and 4 open, then the same customer addresses have to be vsted n slots 2 (CA) and 3 (IP) and n slots 4 (CA) and 4 (IP). For the ALL schedules, we determne the tme slot assocated wth an order by means of a unform random draw. For the NO-SLOT schedule, all orders are assocated wth a sngle tme slot spannng the entre shft. The contnuous approxmaton model s mplemented n Excel wth the heurstc mplemented n VBA. The tme slot schedules were produced on an Intel Pentum M 1.6 GHz machne. The heurstc takes about two mnutes to converge. The nteger programmng model s mplemented n AMPL wth CPLEX 9.0 as the solver. The tme slot schedules were produced on an Intel Pentum D CPU 3.20 GHz ( 2) machne. A lmt of eght hours of CPU tme was mposed. The lmt was always reached wthout provng optmalty. We used Shortec verson , the routng tool from ORTEC ( to construct the delvery routes and determne assocated costs for each demand realzaton. Smulaton experments were run on an Intel Xeon 3.0 GHz machne and take approxmately 25 mnutes of computng tme for 20 demand realzatons of a tme slot schedule Model Comparson The frst set of experments s amed at assessng dfferences n performance, f any, of the dfferent tme slot schedules. We compute the followng statstcs to compare the dfferent tme slots schedules: the average cost, the average number of vehcles used, the average dstance, and the average total tme (where the averages are taken over the 20 demand realzatons). For convenence, we also dsplay the percentage cost savngs relatve to ALL and provde the wdths of the 95% confdence ntervals for these savngs. For confdentalty, the cost fgures are normalzed by settng the cost for ALL for the mornng shft to 100. The results of the smulaton experments can be found n Tables 2 and 3. We observe that the gap between the upper bound (ALL) and the lower bound (NO-SLOTS) n terms of average cost s substantal. The absence of tme slots n NO-SLOTS allows for more cost-effectve delvery routes but, of course, s nconvenent for the customers as they have to be home durng the entre mornng or afternoon. Ths llustrates the core trade-off between servce and delvery cost and s n lne wth the results of Punakv and Saranen (2001) and Ln and Mahmassan (2002). Tme slot schedules provde a means for managng Table 2 Schedule Comparson Mornng vehcles Tme Dstance Cost Savngs (%) CI (%) ALL 5.3 1, RAND-ASG 4.8 1, ALBERT 4.5 1, CA 4.3 1, IP 4.6 1, NO-SLOTS 3.9 1, Table 3 Schedule Comparson Afternoon vehcles Tme Dstance Cost Savngs (%) CI (%) ALL 5.4 1, RAND-ASG 5.1 1, ALBERT 4.6 1, CA 4.5 1, IP 4.6 1, NO-SLOTS 3.9 1,

11 Agatz et al.: Tme Slot Management n Attended Home Delvery Transportaton Scence 45(3), pp , 2011 INFORMS 445 Depot 4P.M. 6P.M. 6P.M. 8P.M. 7P.M. 9P.M. Fgure 5 Afternoon Soluton CA ths trade-off by concentratng demand whle stll provdng acceptable servce levels. Next, we observe that the mornng and afternoon schedules currently n place at Albert.nl (ALBERT) are mproved on average by both CA and IP but that the savngs are not statstcally sgnfcant. Albert s current schedule seems to perform qute well. We also see that randomly constructed tme slot schedules (RAND-ASGN) already yeld a notceable mprovement over smply offerng all tme slots (ALL) but do not acheve the qualty of Albert.nl s current tme slot schedule and that of the tme slot schedules produced by our optmzaton models. We also observe, not surprsngly, that the average number of vehcles requred n NO-SLOT s less than the average number of vehcles requred by the optmzed schedules, whch n turn s less than the average number of vehcles requred by ALL (and RAND-ASGN). Ths agan reflects the trade-off between effcency and servce. In addton to comparng performance dfferences, t s nterestng to compare the actual tme slot schedules. Fgures 5 and 6 show the afternoon tme slot schedules produced by the contnuous approxmaton model and the nteger programmng model, Depot 4P.M. 6P.M. 6P.M. 8P.M. 7P.M. 9P.M. Fgure 6 Afternoon Soluton IP

12 Agatz et al.: Tme Slot Management n Attended Home Delvery 446 Transportaton Scence 45(3), pp , 2011 INFORMS respectvely. For each zp code, the fgure shows the assgned tme slot(s). We use the sze of the crcle to reflect the number of expected orders for the zp code. We observe that the dstnct modelng paradgms of the CA and the IP approach are reflected n dfferent characterstcs of the resultng schedules. In Fgure 5, we see that the contnuous approxmaton model assgns the same tme slot to zp codes that are geographcally clustered together. Ths s especally the case for the zp codes that receve the late afternoon tme slot. In contrast, we see n Fgure 6 that the nteger programmng model does not necessarly clusters geographcally close zp codes, but forms vehcle routes. A possble explanaton for the observed behavor s the fact that the contnuous approxmaton approach ams to ncrease the demand densty wthn a tme slot for each zp code. Therefore, t tends to construct large zp code clusters for each tme slot. Because the nteger programmng model explctly consders the ndvdual vehcle routes, t s more concerned wth the zp code groups that can be vsted by ndvdual vehcles over the entre tme perod. From an applcaton perspectve, these dfferences are nterestng, but the CA approach yelds much more practcal run tmes. We conclude from the above experments that our CA model acheves our man objectve, namely, supportng the desgn of effectve tme slot schedules n real-lfe scenaros. By automatng the generaton of tme slot schedules the models sgnfcantly reduce the plannng effort, compared to the current practce of manual plannng as well as the alternatve IPbased approach. In what follows, we consder what-f analyses of many dfferent scenaros and dscuss the nsghts that they yeld Changes n the Envronment As the e-grocery channel s stll a relatvely new dstrbuton channel, understandng the mpact of characterstcs of the envronment on ts performance s of enormous value to companes expermentng wth t. Therefore, the second part of our computatonal study focuses on these ssues. All of the percentage dfferences reported are relatve to the costs found n Tables 2 and 3 for the same schedule type Demand. The e-grocery market s experencng huge annual growth rates. Therefore, t s nterestng to study the mpact of an ncrease n demand on the resultng tme slot schedules and ther performance. We consder a scenaro n whch the servce requrements reman unchanged, but every zp code n the exstng servce area s experencng a 30% ncrease n demand. The results can be found n Tables 4 and 5. (We omt tme slot schedule ALBERT from these and further experments as t was not desgned for these envronments.) Table 4 Demand Increase Mornng vehcles Tme Dstance Cost Cost (%) ALL 6.6 1, RAND-ASG 5.9 1, CA 5.5 1, IP 5.8 1, NO-SLOTS 5.0 1, Table 5 Demand Increase Afternoon vehcles Tme Dstance Cost Cost (%) ALL 6.7 1, RAND-ASG 6.2 1, CA 5.9 1, IP 6.0 1, NO-SLOTS 4.8 1, We observe a cost ncrease of a lttle over 20% for a demand ncrease of 30%, whch demonstrates, as expected, that there are economes of scale. Ths supports the mportance of growth n order for a company to become more proftable Vehcle Capacty. An mportant consderaton when settng up any delvery operaton s the vehcle fleet and mx. As Albert.nl wants to operate a homogeneous fleet of vehcles, we focus on the mpact of vehcle capacty. We consder two scenaros: one n whch larger vehcles are used,.e., the vehcle capacty s ncreased by 25%, and one n whch smaller vehcles are used,.e., the vehcle capacty s decreased by 25%. The results can be found n Tables 6 and 7. Not surprsngly, a capacty reducton results n a cost ncrease whereas a capacty expanson results n a cost decrease. What s nterestng, however, s that the cost penalty of a capacty reducton s much stronger than the cost beneft of addtonal capacty. Table 6 Vehcle Capacty Change Mornng vehcles Tme Dstance Cost Cost (%) 25% ALL 5.7 1, RAND-ASG 5.4 1, CA 5.3 1, IP 5.3 1, NO-SLOTS 5.2 1, % ALL 5.3 1, RAND-ASG 4.7 1, CA 4.1 1, IP 4.5 1, NO-SLOTS 3.5 1,

13 Agatz et al.: Tme Slot Management n Attended Home Delvery Transportaton Scence 45(3), pp , 2011 INFORMS 447 Table 7 Vehcle Capacty Change Afternoon Table 8 Servce Level Adjustments Mornng vehcles Tme Dstance Cost Cost (%) vehcles Tme Dstance Cost Cost (%) 25% ALL 5.5 1, RAND-ASG 5.3 1, CA 4.9 1, IP 5.1 1, NO-SLOTS 4.7 1, % ALL RAND-ASG 5.0 1, CA 4.5 1, IP 4.7 1, NO-SLOTS 3.8 1, Reduce servce ALL 5.3 1, RAND-ASG 4.9 1, CA 4.4 1, IP 4.5 1, NO-SLOTS 3.9 1, Increase servce ALL 5.3 1, RAND-ASG 5.0 1, CA 5.1 1, IP 4.9 1, NO-SLOTS 3.9 1, Ths suggests a decreasng margnal value of capacty. To understand ths, note that the physcal vehcle capacty s only one of the factors that lmts the number of orders that a vehcle can delver. In partcular, recall from our models that tme constrants, based on the length of a tme slot and of a shft, form another constranng factor. The above results suggest that these tme constrants become more relevant wth ncreasng vehcle capacty. The extent to whch vehcle capacty s bndng depends on the shft and on the servce level. The optmzaton-based tme slot schedules and the NO- SLOT tme slot schedule appear to be more senstve to capacty changes because they can better explot capacty, thus achevng a hgher utlzaton. Moreover, the mornng shft seems to be more senstve to a reducton n vehcle capacty (10% versus 7%), most lkely because t covers a perod of sx hours, as compared to fve hours n the afternoon, whereby tme constrants are less tght Servce Level. A key challenge for e-grocers s to fnd a proper trade-off between servce level and delvery costs. A hgher servce level typcally comes at the expense of hgher delvery costs. We clearly see ths when comparng the smulatons results for the NO-SLOTS schedules wth those for the schedules that use the servce requrements of Albert.nl and wth those of the ALL slots schedules. We now nvestgate ths ssue further by varyng the mposed servce requrements. We consder two scenaros. In the frst, we ncrease the requred servce level by offerng more tme slots, more specfcally by offerng one addtonal tme slot n each zp code except n those zp codes where all tme slots are already offered. In the second, we reduce the servce level by offerng fewer tme slots, more specfcally by offerng one fewer tme slot n each zp code except n those zp codes where only a sngle tme slot s currently offered. Note that n the latter scenaro, the changes are mnor as most zp codes requre only a sngle tme slot to be offered n the base case. Note also that these changes do not affect the ALL and NO-SLOT schedules because they do not use the servce requrements. We only recall ther performance as a pont of reference. The results can be found n Tables 8 and 9. As expected, the mpact on cost of ncreasng the servce level s much hgher than the mpact of reducng the servce level, as far fewer changes were made to the servce level requrements. What may be more surprsng s that for the mornng shft a reducton of the servce level actually leads to a cost ncrease! Ths seems counterntutve, because one would expect to gan operatonal effcency from a decrease n customer servce. However, we have to realze that we only reduced the number of tme slots offered n zp codes n whch we used to offer more than one tme slot. These zp codes tend to be urban areas wth a relatvely hgh demand densty. A reducton n the number of tme slots offered forces demand to be concentrated n fewer tme slots, whch results n a less Table 9 Servce Level Adjustments Afternoon vehcles Tme Dstance Cost Cost (%) Reduce servce ALL 5.4 1, RAND-ASG 5.1 1, CA 4.5 1, IP 4.5 1, NO-SLOTS 3.9 1, Increase servce ALL 5.4 1, RAND-ASG 5.1 1, CA 5.0 1, IP 5.1 1, NO-SLOTS 3.9 1,

14 Agatz et al.: Tme Slot Management n Attended Home Delvery 448 Transportaton Scence 45(3), pp , 2011 INFORMS balanced workload and thereby n a lower capacty utlzaton. Ths example hghlghts two opposng effects. Capacty utlzaton benefts from a well-balanced workload and therefore from spreadng demand over multple tme slots. However, the dstance between customers decreases when clusterng demand n fewer slots. The nterplay between both of these effects results n complex trade-offs, whch are very hard to make wthout any analytcal support tools. In both the mornng and afternoon shfts, we do see ncreases n cost assocated wth mprovng the servce level. Ths s not surprsng but shows a valuable aspect of developng such models. They reveal the cost assocated wth such changes, so companes can make better decsons wth regard to decdng the level of servce to offer to ther customers Tme Slot Template. The set of tme slots that may be offered, known as the tme slot template, has overlappng tme slots n both shfts (see Fgure 4). A tme slot template wth overlappng tme slots may be attractve for both the delvery company and the customer. Overlappng slots provde more routng flexblty to the company than shorter nonoverlappng slots. At the same tme, they provde a hgher servce to the customer than a sngle overall slot. Note that from a modelng perspectve the overlap s challengng. We nvestgate the mpact of overlappng tme slots by analyzng alternatve tme slot templates. We consder two such tme slot templates. The frst uses tme slots wth smaller wdths that do not overlap,.e., 8:00 9:20, 9:20 10:40, 10:40 12:00, 12:00 14:00 for the mornng and 16:00 18:00, 18:00 19:30, 19:30 21:00 for the afternoon. The second uses dfferent overlappng two-hour tme slots,.e., 8:00 10:00, 9:30 11:30, 11:00 13:00, 12:00 14:00 for the mornng and 16:00 18:00, 17:30 19:30, 19:00 21:00 for the afternoon. The results for these dfferent tme slot templates can be found n Tables 10 and 11. Note that the dfferent templates do not affect the NO-SLOTS scenaro, whch s just reported as a benchmark. As expected, we see that the use of nonoverlappng tme slots wth smaller wdths translates n ncreased costs, especally n the mornng shft. The results for the alternatve overlap template are less conclusve. In general, the cost effects appear to be smaller than for removng the overlap. Note that the total amount of overlap, the number of hours whch fall nto more than one slot, s the same as n the orgnal templates, however dstrbuted dfferently. The overlap was a dffcult ssue n terms of modelng for both CA and IP. The contnuous approxmaton model heurstcally adjusts the ndvdual tme slot lengths based on the order volumes and vrtually removes the overlap. Conceptually, the IP model Table 10 Tme Slot Template Changes Mornng vehcles Tme Dstance Cost (%) No overlap ALL 5.7 1, RAND-ASG 5.2 1, CA 4.5 1, IP 4.6 1, NO-SLOTS 4.0 1, Alternatve overlap ALL 5.1 1, RAND-ASG 4.7 1, CA 4.4 1, IP 4.7 1, NO-SLOTS 4.0 1, Table 11 Tme Slot Template Changes Afternoon vehcles Tme Dstance Cost (%) No overlap ALL 6.0 1, RAND-ASG 5.3 1, CA 4.7 1, IP 4.8 1, NO-SLOTS 3.9 1, Alternatve overlap ALL 5.4 1, RAND-ASG 4.8 1, CA 4.8 1, IP 4.6 1, NO-SLOTS 3.9 1, deals wth the ssue n a more accurate way by takng nto account the exact tmng of a zp code delvery. However, ths comes at the cost of an ncreased problem sze and t ntroduces symmetry n the model. But, we see that the overlap does translate to lower cost solutons, so t s mportant to consder t. 7. Conclusons In ths paper, we ntroduce the tme slot management problem (TSMP), a novel tactcal plannng problem whch s relevant to many busnesses that offer attended home delvery, especally e-grocery. The problem entals the decson of whch tme slots to offer n each zp code so as to mnmze expected delvery costs. The use of tme slots gves rse to complex nteractons between marketng and operatonal consderatons and requres sophstcated decson support tools. Ths paper presents a fully automated approach that s capable of producng hgh-qualty solutons wthn a short amount of tme. The approach, whch s based on contnuous approxmaton, estmates the route costs resultng from the tme slot selectons per zp code based on local nformaton. Our numercal

15 Agatz et al.: Tme Slot Management n Attended Home Delvery Transportaton Scence 45(3), pp , 2011 INFORMS 449 experments on real-lfe data from Albert.nl show the followng: Although narrow delvery tme slots are convenent for the customer, they greatly reduce routng effcency. Our experments ndcate an ncrease n delvery cost of up to 25% from usng two-hour tme slots nstead of tme slots that span an entre mornng or afternoon. Ths llustrates the core tradeoff between servce and delvery costs. The regonal dfferentaton of the delvery tme slot offerng leads to sgnfcant cost savngs as compared to smply offerng the full range of tme slots n the whole delvery regon. In partcular, the results from our numercal experments suggest up to 10% delvery cost savngs. There are clear economes of scale n the delvery operaton. The results show potental cost benefts from ncreasng the number of stops wthn the current delvery area. Ths llustrates the mportance of growth n order for an e-taler to become more proftable. The tme slot constrants and the constrants on physcal vehcle capacty nteract. Tme slot management decsons become more relevant when the vehcle capacty allows a vehcle tour to span several tme slots. The man trade-off n the tme slot schedule desgn s between demand clusterng and demand smoothng. The advantage of clusterng demand s that t mnmzes the dstance between successve stops, and thus travel costs. The dsadvantage s the potental underutlzaton of the vehcle capacty, and a correspondng ncrease n vehcle costs. There reman several nterestng drectons for future research. We see the jont optmzaton of the servce requrements and the tme slot schedule as one of the most challengng. Ths wll not only requre a more complete understandng of customer behavor, but also more sophstcated soluton approaches. We would also be nterested n further nvestgatng the IP approach to see f we can fnd hgh-qualty solutons n a shorter amount of tme. In general, we see great potental for addtonal research on the real-tme management of demand and the use of ncentves n attended home delvery. References Agatz, N. A. H Demand management n e-fulfllment. Erm Ph.D. Seres Research n Management, Rotterdam, The Netherlands, Agatz, N. A. H., A. M. Campbell, M. Fleschmann, M. Savelsbergh. 2008a. Challenges and opportuntes n attended home delvery. B. Golden, S. Raghavan, E.Wasl, eds. The Vehcle Routng Problem: Latest Advances and New Challenges. Sprnger Scence & Busness Meda, New York, Agatz, N. A. H., M. Fleschmann, J. A. E. E. van Nunen. 2008b. E-fulfllment and mult-channel dstrbuton A revew. Eur. J. Oper. Res. 187(2) Bent, R., P. Van Hentenryck Scenaro-based plannng for partally dynamc vehcle routng wth stochastc customers. Oper. Res. 52(6) Bernng, C., S. Ernst, N. H. 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