Optimal placement of warehouse cross aisles in a picker-to-part warehouse with class-based storage

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1 Optial placeent of warehouse cross aisles in a picker-to-part warehouse with class-based storage Paul Berglund and Rajan Batta Departent of Industrial and Systes Engineering, 438 Bell Hall University at Buffalo (SUNY), Buffalo, NY 14260, USA Revised: May Abstract Given a picker-to-part warehouse having a siple rectilinear aisle arrangeent with northsouth storage aisles and east-west travel aisles (or cross aisles ), this paper investigates the optial placeent of the cross aisles as a consequence of the probability density function of the order pick locations, as deterined by the storage policy. That is, for a given storage policy, what placeent of the cross aisles will result in a inial expected path length for the picker? An analytical solution procedure is developed for the optial placeent of a single iddle cross aisle given for a given storage policy. A siplifying assuption is ade as regards picker routing, but arbitrary non-rando storage policies are considered. The solution procedure is generalized to a ethod for ultiple cross aisles. Soe exaple probles are solved and a siulation study is used to easure the ipact of our siplifying assuptions. Keywords: Warehousing, aisle design, order-picking, aterial handling 13 1 Introduction A significant coponent in the operating cost of picker-to-part warehouses is picking tie, which has been estiated to contribute up to 55% of total operating costs (see for exaple Topkins et al. [2003]). In order to iniize this cost, a nuber of approaches are possible. Efficient picker 17 routing algoriths reduce the distance traveled to pick a given pick list. Class-based storage 18 policies reduce travel distance by concentrating ost frequently picked ites close to the I/O 1

2 19 point. These and other factors such as order batching, warehouse shape and so on have been studied extensively, both in isolation and in cobination with other factors. One factor which has been studied less extensively is layout design. The picker uses aisles to travel through the warehouse: storage aisles in which parts are picked and cross aisles which are used to travel fro storage aisle to storage aisle. Once the picker has ade all of the picks in a given storage aisle, he ust continue to travel through the storage aisle until he reaches the cross aisle via which he will travel to the next storage aisle. This is wasted travel in the sense that it adds no value; the less such travel, the shorter the picker s expected path will be. If it can be reduced, then picker travel will be reduced without losing value. As efficiently placed cross aisles are added, the expected distance fro a pick point to the closest cross aisle will be reduced, and the picker s total expected travel distance will decrease. However, as the nuber of cross aisles increases, the storage density of the warehouse decreases. Eventually, returns diinish to the point where adding cross aisles decreases picking efficiency. It should be noted that in practice cross aisle configurations ay be changed without incurring prohibitive costs. Product is often stored on shelving that consists of a nuber of odular units bolted together. Such shelving ay be reconfigured fairly easily so as to add a cross aisle or change the position of one. The greatest cost will be the effort of unloading the shelves and then reloading the once the shelves have been re-assebled in their new configuration. Therefore the optiu cross aisle positions is potentially valuable inforation, due to the practical possibility of acting on it. The optial positioning of aisles is conceptually siple: a cross aisle will provide a greater benefit if it is close to those locations where the ost picks are ade; to add a cross aisle in a seldo-visited area of a warehouse would be to trade storage capacity for only a sall benefit in picking efficiency. If our objective is to locate cross aisles so as to shorten the expected distance fro a pick point to the nearest cross aisle, we will prefer to locate cross aisles in areas where picking concentrations are high. Therefore a proper analysis of optial aisle placeent should take into account pick densities (storage policies). This paper therefore presents a solution ethod for the proble of where cross aisles should best be positioned to optiize picker travel distance for a given storage policy. 2

3 48 2 Literature Review Substantial work has been done on picking efficiency in picker-to-part warehouses, considering different cobinations of factors such as pick list size, routing policies, order batching, storage policies and so on. A good survey of the work which has been done in this area ay be found in de Koster et al. [2007]. A nuber of siulation studies have considered the efficiency of different cobinations of factors. Petersen [1999] studies the cobined effects of routing policies, pick list sizes and storage policies. Petersen and Aase [2004] does a siilar analysis for an extensive set of cobinations of order batching policies, storage policies and routing policies. Petersen [1997] considers the effects of pick list size, warehouse shape, routing policy and I/O point location, while assuing a rando storage policy. Petersen and Schenner [1999] studies different patterns of class-based storage in cobination with different routing policies, pick list sizes and I/O point locations. Soe analytical studies also exist. Le-Duc and de Koster [2005], assuing a warehouse with a single central cross aisle, a class-based storage policy and a return routing policy, coputes the effect on efficiency of warehouse shape, pick list size and storage policy. Caron et al. [1998], assuing the sae layout as Le-Duc and de Koster [2005] and class-based storage, calculates the efficiency of traversal and return routing policies. Caron et al. [2000], using the sae layout, finds the optial nuber of storage aisles as a consequence of pick list size and the shape of the ABC curve of a class-based storage policy, assuing a traversal routing strategy. Chew and Tang [1999] analyzes the effect of pick list size given class-based storage, assuing a traversal routing policy. Jarvis and McDowell [1991] calculates that for a full traversal routing policy the optial storage policy is a within aisle storage policy with the fastest-oving ites stored in aisles closest to the I/O point. Roodbergen and Vis [2006] finds the optial shape of a single-block warehouse with a rando storage policy, assuing either an S-shaped or largest gap routing heuristic. Less has been written specifically on the ipact of aisle layouts on picker travel distances. In fact, de Koster et al. [2007] notes that literature on layout design for low-level anual orderpicking systes is not abundant. Vaughan and Petersen [1999] uses a siulation study to calculate the optial nuber of evenly-spaced cross aisles in a warehouse, assuing an aisle-by-aisle routing policy and a unifor (rando) storage policy. Roodbergen and de Koster [2001a] extends this study by siulating a variety of routing policies in the sae setting. Thalayan [2008] uses 3

4 78 siulation to copare the effects on travel tie of a nuber of different factors, including the 79 nuber of cross aisles, storage policy and routing policy. Roodbergen and de Koster [2001b] investigates the benefit of a iddle cross aisle in cobination with a rando storage policy and varying pick list sizes. Roodbergen et al. [2008] develops a odel for calculating optial shape and nuber of evenly-spaced cross aisles for a warehouse with a rando storage policy, assuing an S-shaped routing heuristic. There are even fewer studies which consider layouts where the travel aisles are not evenly spaced. Gue and Meller [2009] uses an analytical approach to derive an unconventional but efficient aisle configuration for a unit-load warehouse assuing a rando storage policy. Pohl et al. [2009] coputes the efficiency of three different aisle configurations for dual-coand operation and rando storage. They also prove that, for this case, the optial position of a single east-west cross aisle in a warehouse with north-south storage aisles ust be between the center of the warehouse and the top cross aisle. This paper focuses on two factors: storage policy (represented here in the for of the distribution function of pick locations) and facility layout (ore specifically the question of cross aisle 93 position), and their effect on picker travel distances. An assuption is ade that pickers will be routed by a siple heuristic that will not always generate the shortest possible route. Given this routing policy and an arbitrary storage policy, an optial cross aisle position is calculated. Possible congestion effects are not considered. The reainder of this paper is organized as follows. First the warehouse odel is described. Then a procedure for coputing the expected picker path length as a function of the position of a single interior cross aisle is presented. This solution is extended to an arbitrary nuber of cross aisles. Soe exaple applications of the procedure are given. Siulations are perfored to estiate the effect of the siple routing heuristic on picker path lengths and on the resulting optial cross aisle positions Model Consider a picker-to-part warehouse with M vertical (or north-south ) storage aisles. Each stor- age aisle has B discrete pick locations of unifor size (e.g. one pallet width) on each side of the aisle, nubered 1, 2,..., B with the 1 st location being the southernost and the B th location 4

5 being the northernost. We will assue that storage aisles are narrow enough that the lateral oveent required to pick ites on both sides of the aisle ay be neglected. Therefore there ay be ore than one ite which for our purposes share the sae effective pick point (across the aisle fro each other, or, if ultiple level storage is used, above or below one another). Thus a pick list ay have ultiple ites at what for our purposes is the sae effective pick point. There are three lateral (or east-west ) cross aisles, one at y = 0 (i.e. south of all pick locations), one at y = B ( north of all pick locations) and one at y = h, where h is to be deterined (h being the nuber of pick locations south of the iddle cross aisle, where 0 < h < B). There is a single I/O point at y = 0, at soe x coordinate. The aount of north-south travel will be the sae regardless of where the I/O point is located, therefore we ay disregard the location of the I/O point when coputing the optial cross aisle position. The picker will have to be routed through the warehouse according to soe sort of heuristic, or routing policy. Many different routing policies are described in the literature, but only a few of these are appropriate for warehouses with ultiple cross aisles. Three such policies are aisle by aisle routing, described in Vaughan and Petersen [1999], the S-shaped heuristic and the largest gap heuristic (as adapted for ultiple cross aisles), both described in Roodbergen and de Koster [2001a]. We will begin by assuing an aisle by aisle routing odel as used in Vaughan and Petersen [1999]: the picker begins at the leftost storage aisle fro which ites ust be picked and picks all ites in that aisle, then proceeds to the nearest aisle to the right that has any ites to be picked, picks all ites in that aisle, and so on until all ites have been picked. (Pickers are able to turn around in storage aisles and to traverse the in either direction, but always ove west-to-east in travel aisles, except before aking the first pick or after aking the last.) The shortest path using this routing ay be calculated by dynaic prograing, however the necessary coputations are still coplex, and an analytical solution will be correspondingly difficult to obtain. In order to siplify this coputation sufficiently and allow us to develop an analytical solution, we will ake the additional siplifying naïve routing assuption that after aking the final pick in a given storage aisle, the picker then departs via the closest cross aisle to his current location, without considering the picking locations to be visited in subsequent aisles. If equidistant fro two cross aisles, the picker will choose the cross aisle closest to the I/O point. Note that this will 5

6 not always result in the picker choosing the optial route (see figure 1). See appendix A for a ore detailed discussion of the effects of the naïve routing assuption. The optial placeent of a single cross aisle will result in inial expected north-south travel (that is, travel in storage aisles), given the assuption that pickers will eploy our siplified routing strategy. Note that use of aisle by aisle routing ensures that for a given storage policy the expected aount of lateral or east-west travel (that is, travel in the cross aisles) will be the sae regardless of the value of h. Therefore we ay disregard this quantity when coputing the optial cross aisle position for a given storage policy. However it will be of interest when we copare the efficiency of one storage policy to that of another. 146 We will use the following ters (after Vaughan and Petersen [1999]): N is the pick list size M is the nuber of vertical (storage) aisles K is the nuber of pick locations to be visited in storage aisle X (t) is the t th pick location in aisle X is the largest ( northernost ) pick location in aisle X is the sallest ( southernost ) pick location in aisle Assue that pick locations are independent rando variables, and that each pick location will be distributed aong the different storage aisles according to soe arbitrary probability ass function g M (), and within each storage aisle = 1, 2,..., M according to soe set of M arbitrary probability ass functions (x). Thus a given pick will be at storage location x in storage aisle with probability g M () (x). Note that X and X are respectively the K th and 1 st order statistics for a discrete saple of K ites, and will thus have probability ass functions, given by Siotani [1956], of the for: (x) = (F X (x)) K (F X (x 1)) K 160 and (x) = (1 F X (x 1)) K (1 F X (x)) K 6

7 161 and their joint pf is,x (x, y) = [F X (x) F X (y 1)] K [F X (x) F X (y)] K [F X (x 1) F X (y 1)] K [F X (x 1) F X (y)] K As the forulas for these pfs include the value K they clearly depend on knowing the nuber of picks ade in storage aisle. There are therefore N M instances each of (x), (x) and,x (x, y). The algorith presented below is such that we will always know the appropriate values of K and will therefore know which pf to use at what tie. For the sake of siplicity of presentation this detail will be oitted, but we should ake clear that the pf being used ust in each case be the appropriate one given the nuber of picks in the storage aisle under consideration Algorith Define P as the north-south travel distance in aisle, and P as the total north-south travel distance. In order to find the value of h which iniizes E[P ], we ust find a way to copute E[P ] for a given value of h. We do this by conditioning over the ways the picks are distributed aong the storage aisles. The N picks ust be distributed aong the M storage aisles in soe way. That is, we have soe ordered set K = {K 1, K 2,..., K M } such that M =1 K = N. If we know all the different possible patterns of picks aong our aisles, and the probability of each pattern, then we can calculate E[P ] as follows: E[P ] = K K P r(k) E[P K] where K is the set of all possible patterns of picks aong our aisles. The enueration of the eleents of K and the calculation of their respective probabilities fro g M () is relatively straightforward To calculate E[P K], we consider the proble as a Markov reward process with three states 0, h and B, corresponding to the three cross aisles. The process will be considered to be in state 7

8 i {0, h, B} when the picker is in cross aisle i (using it to travel fro one storage aisle to the next). That is, when the picker enters storage aisle via cross aisle i, akes picks in and then departs via cross aisle j, we consider the Markov reward process to have transitioned fro state i to state j, with the reward being the expected north-south travel distance required to ake all of the picks in. We note that the initial state of the process will always be 0, and likewise once all picks have been ade the syste will end up in state 0. The expected total reward for a given pattern K will equal E[P K]. To copute the total expected reward we will need to calculate the relevant transition probability atrices, as well as the expected reward for each possible state transition, i.e. the expected path length in a particular storage aisle given the cross aisles via which the picker arrived and departed. Note that the path length will also depend on both the distribution of picks in, as deterined by (x), and K, the nuber of picks to be ade in, as deterined by the pattern K. We ake the following additional definitions: J is the 3-vector of probabilities that the picker will enter storage aisle via the three cross aisles. Note that J 1 = (1, 0, 0). R k is the 3-vector of expected rewards in aisle given that the picker will enter via each the three cross aisles and will ake k picks in the aisle. Note that R 0 = (0, 0, 0) T for all. T k is the 3 x 3 transition probability atrix for aisle given that k picks are ade in aisle. Note that T 0 is the 3 by 3 identity atrix for all. 202 The expected path length in storage aisle, R k, ust be coputed for each of the three cases: Case 0: storage aisle is entered at y = 0 Case h: storage aisle is entered at y = h Case B: storage aisle is entered at y = B The expected path lengths for the different cases are calculated as follows. Cases 0 and B are straightforward; the picker will proceed either up (if entering at y = 0) or down (if entering at y = B) the storage aisle until all picks have been ade and will then exit via the closest cross aisle to the final pick location, as shown in figure 2. 8

9 Case 0: storage aisle is entered at y = 0. We ust travel up the storage aisle far enough to ake all picks (which aounts to traveling up to X ), whereupon we then leave by the closest exit point (0, h or B). Therefore the length of the optial path depends only on the value o. There are four possible sub-cases: If 0 < X h 2, the closest exit point to X is at y = 0. Then the shortest possible path length is (2 x 1) w b w a, where w a is the width of a cross aisle and w b is the width of a pick location If h 2 < X h, the closest exit point to X is at y = h, and no picks are ade at any locations h. Then the shortest possible path length is h w b w a If h < X Bh 2, the closest exit point to X is at y = h, and at least one pick is ade at soe location h. Then the shortest possible path length is (h 2 (x h) 1) w b 2 w a If Bh 2 < X B, the closest exit point to X is at y = B and the shortest possible path length is (B w b 2 w a ) 223 Fro this we can derive an expression for the expected path length in case 0: R k (1) = E[P K and Case 0] = (x) [(2 x 1) w b w a ] 1 x h 2 h 2 <x h h<x (Bh) 2 (Bh) 2 <x B (x) [h w b w a ] (x) [(h 2 (x h) 1) w b 2 w a ] (x) [B w b 2 w a ] Case h: storage aisle is entered at y = h. In this case, the values of both X and X are relevant, and to find the expected iniu path length we ust su over the doain of the joint pf o and X. Because we ay have soe picks above h and soe below, we ust calculate the path lengths of the two possible routes (either first aking all picks above h and then picks all below h, or else the reverse), and take the iniu of the two. 9

10 For all i, j such that 0 < i j B define P (i, j) as the iniu path length to pick all ites in aisle given that aisle was entered at y = h, X = i and X = j. For any given i, j P (i, j) is straightforwardly calculated as follows: Let P 1 (i, j) = (the distance fro the cross aisle at y = h to i) (the distance fro i to j) (the distance fro j to the closest cross aisle to j) 232 Let P 2 (i, j) = (the distance fro the cross aisle at y = h to j) (the distance fro j to i) (the distance fro i to the closest cross aisle to i) 233 Then P (i, j) = in{p 1 (i, j), P 2 (i, j)} and R k (2) = E[P K and Case 3] = B B i=1 j=i P(i, j),x (i, j) Note that if points i and j are either both above or both below y = h then the shorter of the two paths will always be the one that akes picks in the order of increasing distance fro h. This ay be considered a trivial case, for which the above forula will also copute the correct path length. Case B: storage aisle is entered at y = B. Then, analogously to case 0, the length of the optial path depends only on the value o, and is given by 10

11 R k (3) = E[P K and Case B] = (x) [B w b 2 w a ] 1 x h 2 h 2 <x h h<x (Bh) 2 (Bh) 2 <x B (x) [((B x) (h x) 1) w b 2 w a ] (x) [(B h) w b w a ] (x) [(2 (B x) 1) w b w a ] The transition probability atrices T k ay be coputed by following siilar reasoning. Define I 0 = {i the closest cross aisle to pick point i is at y = 0} I h = {i the closest cross aisle to pick point i is at y = h} I B = {i the closest cross aisle to pick point i is at y = B} 242 Then the first row of T k is coputed siilarly to case 0 above: T k (1, 1) = i I 0 (x) T k (1, 2) = i I h (x) T k (1, 3) = i I B (x) 243 and the third row of T k is coputed siilarly to case B above: 11

12 T k (3, 1) = i I 0 (x) T k (3, 2) = i I h (x) T k (3, 3) = i I B (x) As above, the only coplicated case is the iddle one, because we have to consider the shorter of two paths for the picker in aisle. Define 1 if P(i, ˆP 1 j) > P(i, 2 j) (i, j) = 0 otherwise 246 Then we can calculate the iddle row of T k by T(2, k 1) =,X (i, j) ˆP (i, j) i I 0 i<j B j I 0 0<i<j T(2, k 2) =,X (i, j) ˆP (i, j) i I h i<j B j I h 0<i<j T(2, k 3) =,X (i, j) ˆP (i, j),x (i, j) ˆP (j, i) i I B i<j B j I B 0<i<j,X (i, j) ˆP (j, i),x (i, i) i I 0,X (i, j) ˆP (j, i),x (i, i) i I h i I B,X (i, i) Once we have coputed R k and T k for all = 1, 2,..., M and k = 0, 1,..., N we are ready to calculate E[P K]. Note that there will have to be an additional calculation of R k ade for the case of the final (or easternost ) aisle with picks, because we will always exit that aisle via cross aisle 0 regardless of what picks are ade there, and therefore our expected path length will be different fro the usual case. The logic behind this calculation is siilar enough to the foregoing that we will oit the details. Note that J 1 = (1, 0, 0) and for > 1 we calculate J by J = J 1 T K Then the expected reward for aisle is given by 12

13 E[P K] = J R K and E[P K], the expected total north-south path length, is siply the su of the aisle-by-aisle expected path lengths: E[P K] = M E[P K] = =1 M =1 J R K Now as noted above we siply copute E[P ] by conditioning over all possible values of K The algorith will then consist of calculating R k and T k for all possible values of and k, and then using R k and T k to copute the expected path length for each pick pattern in K. Thus E[P ] ay be coputed as follows: { totalp ath is the total expected (north-south) path length } totalp ath 0 { is the storage aisle } for =1 to M do {k is the nuber of picks in the aisle} for k=0 to N do copute R k copute T k end for end for for all K K do {patternp ath is the total expected path length given pattern K} patternp ath 0 for =1 to M do if = 1 then else J (1, 0, 0) J J 1 T K

14 end if patternp ath patternp ath J R K end for totalp ath totalp ath (P r(k) patternp ath) end for {totalp ath will now be equal to E[P ]} The foregoing ay be straightforwardly extended to calculating the expected path lengths if we have two or ore floating cross aisles, at y = h 1, y = h 2, etc. If there are A cross aisles (including those at y = 0 and y = B) then the vectors J and R k ust be increased to size A and the atrices T k ust be increased to size A by A. The procedures for calculating J, R k and T k reain conceptually the sae, although the actual calculations are ore coplex as we ust consider a larger nuber of possible cross aisles via which we ight exit a given storage aisle. Calculations for the fixed aisles at y = 0 and y = B ay be coputed in a way quite siilar to cases 0 and B above, whereas calculations involving the ovable interior aisles at y = h 1, y = h 2, etc. are done as in case h. Once the J and R k values have been coputed, the procedure for calculating E[P ] will be identical to the case where A = Exaples 5.1 Coputing the optial position for a single cross aisle As we are able to copute the expected path length for each pattern, and we know the probability of each pattern, we are thus able to copute E[P ] for a given value of h. Once we know how to do this, the next step is to copute that value of h for which the picker s expected travel distance is iniized. We can do this by evaluating E[P ] for various values of h. If we wish to find the optial location of a single floating cross-aisle using the algorith outlined above, a single objective function evaluation reains sufficiently inexpensive that it is still feasible to use full enueration to find the solution. As an exaple, we do this for an across-aisle storage policy, where g M () is the unifor distribution and for each storage aisle (x) is the distribution function, so-called because 80% of picks are in the 20% of the aisle closest to the I/O point: 14

15 (x) = x x 50 0 otherwise Assue 50 pick locations per aisle (B = 50), 20 storage aisles (M = 20) and 5 picks per trip (N = 5), w a = 10 and w b = 5. For this case we obtain the results shown in Figure 3: the expected path length E[P ] as a function of h, the position of the interediate cross aisle. The inial value of E[P ] is , achieved when h = 8, and was found by full enueration in 155 illiseconds. We can also copare the expected north-south path length given optially positioned cross aisles with the expected path length for evenly-spaced cross aisles. Table 1 shows the percentage savings achieved by oving the iddle cross aisle to its optial position An exaple with dual-coand travel Dual-coand travel is the special case where the pick list size N is equal to 2. A theore proven by Pohl et al. [2009] states that the optial position of a single ovable cross aisle with dual coand-picking and a rando storage policy will be between the center of the warehouse and the top cross aisle (their Proposition 1). We calculate the optial cross-aisle position for rando storage and N = 2 for a nuber of warehouse sizes. The results, as shown in table 2, agree with the theore of Pohl et al. [2009] that the optial cross aisle position will be beyond the idpoint of the warehouse The coparison of three storage policies We now copute the optial cross aisle positions for three different volue-based storage policies, 323 and copare the resulting optial expected path lengths. The storage policies considered are diagonal storage, across-aisle storage and within-aisle storage, as shown in figure 4. These storage policies were evaluated in Petersen and Schenner [1999]. That study also considered perieter storage, but we will disregard perieter storage because it is clearly not suitable for the aisle-byaisle routing policy being used here. The other three storage policies were evaluated with storage classes A B and C, occupying 20, 30 and 50 percent of storage locations respectively. The skewness of the three classes was either high, ediu or low, defined as in table 3. A warehouse with 20 15

16 storage aisles was assued, with an I/O point in at botto center, and approxiately twice as wide as deep (not including the portion of warehouse depth due to cross aisles). The results were that across aisle storage was the ost efficient in all cases. (A subset of those results, for the ediu skewness level, are shown in table 4.) (The north-south distances were calculated using the algorith of section 4, and the east-west distances were calculated using a sipler forula given in appendix B.) For the sallest pick list sizes, diagonal storage was superior to within aisle storage, for larger pick list sizes within aisle storage was ore efficient than diagonal storage. It should be observed that this does not iply that across aisle storage is superior in all cases. The optial storage policy is dependent on the routing policy used; according to Jarvis and McDowell [1991] a within-aisle storage policy is preferable when traversal routing is used, and Le-Duc and de Koster [2005] finds that an across-aisle policy is superior when return routing is used. Another factor which should not be ignored is that, as seen in table 4, within-aisle storage results in less east-west travel but ore north-south travel, when copared to across-aisle storage. Thus the relative erits of the two storage policies will be sensitive to factors such as storage aisle widths. (Widening storage aisles will increase the path lengths for across-aisle storage ore than it will those for within-aisle storage.) The results for different nubers of cross aisles with across aisle storage and ediu skewness are shown in table 5. One result which is apparent here (and was observed for other storage policies and skewness levels as well) is that the the benefits for additional cross aisles decrease rapidly. The first cross-aisle brings a significant benefit, especially for larger pick list sizes. The second aisle added never gives as uch as a two per cent iproveent, and the third either yields a very sall iproveent or else ay even cause path lengths to increase. Also note that optial cross aisle positions are fairly insensitive to pick list size. For three cross aisles, the optial position for the iddle aisle is always at y = 10. However when an increase in pick list size does cause the optial aisle positions to change, they tend to do so in an abrupt fashion: for A = 4 and 2 N 5 the optial aisle positions are ( ), and for A = 4 and 5 < N 10 they are ( ). Table 6 shows the results for different skewness levels for across aisle storage with four cross aisles. As before, we see that optial cross aisle positions are insensitive to pick list size, and that this insensitivity is ore pronounced for higher skewness levels. It is also worth noticing that in this exaple the percentage savings resulting fro higher skewness levels increases as the pick list 16

17 360 size increases Conclusions and Further Work As noted by any researchers (e.g. Hausan et al. [1976]), volue-based storage policies decrease picker travel distances. Up to soe point of diinishing returns, the addition of interior cross-aisles reduce travel as well. Therefore it is useful to study the use of volue-based storage policies in a warehouse with interior cross aisles. As we have seen, in the absence of rando storage, the ost efficient cross aisle positions will not be equally-spaced. Furtherore, as the cost of adjusting cross aisle positions is not prohibitive, practitioners will be able to benefit fro knowing the axially efficient positions for cross aisles corresponding to storage policies in use, or storage policies under consideration. We have presented a ethod for calculating axially efficient cross aisle positions for a picker-to-part warehouse using arbitrary storage policies, subject to certain assuptions, ost notably an assuption regarding routing policy. This work could be developed into a tool that could be used to calculate optial cross aisle positions, and applied to a larger nuber of questions. For exaple, warehouse shape could be varied, as so could be the nuber of storage aisles. Other pick list sizes could also be considered. A ore difficult task would be to relax our routing assuptions. The siulation results (see appendix A) suggest that our predicted optial cross aisle positions will also be optial for aisleby-aisle routing in the absence of the naïve routing assuption. However this ay not be true for all possible routing policies. Other siple routing policies exist for which path lengths could be calculated analytically (e.g. traversal routing), but such policies do not as a rule take advantage of cross aisles. Optial routing is therefore the nut we need to crack. For optial routing however it sees unlikely that the expected path length for a given layout could be found except by siulation. The coputational effort required to find optial positions for several cross-aisles using siulation ight, however, prove prohibitively large. 384 References F. Caron, G. Marchet, and A. Perego. Routing policies and COI-based storage policies in picker- to-part systes. International Journal of Production Research, 36: ,

18 F. Caron, G. Marchet, and A. Parego. Optial layout in low-level picker-to-part systes. Inter- national Journal of Production Research, 38(1): , E.P. Chew and L.C. Tang. Travel tie analysis for general ite location assignent in a rectangular warehouse. European Journal of Operational Research, 112: , R. de Koster, T. Le-Duc, and K. Roodbergen. Design and control of warehouse order picking: a literature review. European Journal of Operational Research, 182: , K.R. Gue and R. D. Meller. Aisle configurations for unit-load warehouses. IIE Transactions, 41 (3): , Warren H. Hausan, Leroy B. Schwarz, and Stephen C. Graves. Optial storage assignent in autoatic warehousing systes. Manageent Science, 22(6): , J.M. Jarvis and E.D. McDowell. Optial product layout in an order-picking warehouse. IIE Transactions, 23(1):93 102, T. Le-Duc and R. de Koster. Travel distance estiation and storage zone optiization in a 2- block class-based storage strategy warehouse. International Journal of Production Research, 43: , C.G. Petersen. An evaluation of order picking routeing policies. International Journal of Operations & Production Manageent, 17: , C.G. Petersen. The ipact of routing and storage policies on warehouse efficiency. International Journal of Operations & Production Manageent, 19: , C.G Petersen and G. Aase. A coparison of picking, storage and routing policies in anual order picking. International Journal of Production Econoics, 92:11 19, C.G. Petersen and R.W Schenner. An evaluation of routing and volue-based storage policies in an order picking warehouse. Decision Sciences, 30: , L.M. Pohl, R.D. Meller, and K.R. Gue. An analysis of dual-coand operations in coon 411 warehouse designs. Transportation Research Part E, 45(3): ,

19 K. Roodbergen and R. de Koster. Routing ethods for warehouses with ultiple cross-aisles. International Journal of Production Research, 39: , 2001a K. Roodbergen and R. de Koster. Routing order pickers in a warehouse with a iddle aisle. European Journal of Operational Research, 133:32 43, 2001b. 416 K. Roodbergen and I. Vis. A odel for warehouse layout. IIE Transactions, 38: , K. Roodbergen, G. Sharp, and I. Vis. Designing the layout structure of anual order picking areas in warehouses. IIE Transactions, 40: , M. Siotani. Order statistics for discrete case with a nuerical application to the binoial distri- bution. Annals of the Institute of Statistical Matheatics, 8:95 104, P. Thalayan. Coparative study of ite storage policies, vehicle routing strategies and warehouse layouts under congestion. Master s thesis, State University of New York at Buffalo, J.A. Topkins, J.A. White, Y.A. Bozer, and J.M.A. Tanchoco. Facilities Planning, third ed. John Wiley & Sons, Inc., T.S. Vaughan and C.G. Petersen. The effect of warehouse cross-aisles on order picking efficiency. International Journal of Production Research, 37(4): , A Siulation 428 Siulation was used to validate the results of the analytical study in two different ways. An estiate was ade of the increase in average path length due to the naïve routing assuption. The path length resulting fro the naïve routing assuption was copared to that resulting fro the aisle by aisle routing policy used by Vaughan and Petersen [1999], where storage aisles are visited in a strict left-to-right sequence, but the shortest possible path subject to that restriction was found (using a dynaic prograing algorith). For the saple proble with the distribution defined in section 5.1, cross aisles at 0, 6, 29, 44 and 50, fifty pick locations per storage aisle (B = 50), 20 storage aisles (M = 20) and a pick list size of five (N = 5), the results were that the average penalty of the naïve routing assuption was 8.64%, based on a saple of 100,000 picking trips. For the three-aisle case, the discrepancy for the optial configuration (cross 19

20 aisles at 0, 8 and 50) was 13.49%. It akes intuitive sense that the discrepancy should be larger in this case: when the cross aisles are fewer and therefore farther apart, the penalty for using the wrong cross aisle will be larger. But does this larger discrepancy between the expected path lengths when using the different routing assuptions cause us to find the wrong optial solution? Siulation was used to estiate the expected path length using the routing policy of Vaughan and Petersen [1999] for the different candidate solutions to the saple proble of the distribution, with N = 5 and M = 20, B = 50 and three cross aisles, varying the position of the interior cross aisle between 1 and 49. Figure 5 plots these siulated path lengths against the values calculated analytically using the naïve routing assuption. Although the routing assuption of Vaughan and Petersen [1999] resulted in significantly shorter path lengths, the two curves are quite siilar in shape and have the sae iniu point (at h = 8). 450 B East-West Path Length The expected east-west path length (total distance traveled in cross aisles) for a given storage policy and pick list size ay be calculated as follows. Given the aisle-by-aisle routing policy in use, the expected east-west path length depends only on the easternost and westernost storage aisle being visited (the actual pick points visited does not atter). Also, if we know the easternost and westernost storage aisle being visited we ay calculate the (exact) east-west path length. If we define P ew is the east-west path length. W i is the event that the westernost storage aisle visited is aisle i E j is the event that the easternost storage aisle visited is aisle j. p ij is the east-west path length given the events W i and E j. w s is the distance between storage aisles, easured idpoint-to-idpoint. I is the location of the I/O point, expressed as a nuber between 1 and M. That is, I = 10.5 eans that the I/O point is between storage aisles 10 and Then the expectation of P ew ay be calculated by conditioning over the values of easternost 20

21 465 and westernost storage aisle visited, given pick list size N, as follows: M M E[P ew N] = p ij P r((w i E j ) N) i=1 j=i 466 Furtherore, p ij is easily calculated for each i and j: p ij = w s { I i i j I j } 467 And P r((w i E j ) N) ay be calculated as follows. If i = j, then P r((w i E j ) N) = (g M (i)) N 468 otherwise [ j P r((w i E j ) N) = g M () =i ] N N 1 k=1 N k l=1 ( j 1 (g M (i)) k (g M (j)) l =i1 ) N k l (N )( ) N k g M () k l 21

22 Figure 1: In the above exaple, the naïve routing ethod (shown on the left) results in a longer path length than the optial routing (shown on the right). 22

23 Figure 2: In the case where we enter a storage aisle at y = 0 or y = B, we ake all picks and then exit via the closest cross aisle to our last pick location. 23

24 Figure 3: Expected north-south path length E[P ] as a function of cross-aisle location (h) for the distribution, with 50 pick locations per aisle, 20 storage aisles and 5 picks per trip, w a =10 and w b =5. 24

25 north-south north-south Path length, Path length, percent N h=8 h=25 difference savings Table 1: Savings of optial cross-aisle position copared to centered cross-aisle for an across-aisle storage policy with two storage classes (the distribution), with B=50, M =25, and N =2 thru 10, based on 1,000,000 siulation runs per value of N. The optial aisle position of h=8 saved between 6 and 13.9 percent as copared to the centered cross aisle at h=25. N M P h N M P h Table 2: Optial cross-aisle position h and inial expected path length P for dual-coand operation for the unifor distribution, with B=50 and M between 2 and 30. The value of h is always greater than 25, which agrees with proposition 1 of Pohl et al. [2009] percent percent percent Skewness in class A in class B in class C Low Mediu High Table 3: Three different skewness levels used. 25

26 26 Figure 4: Diagonal, across aisle and within aisle storage with three storage classes. The I/O point is at botto center. After Petersen and Schenner [1999].

27 Figure 5: Siulation results for the distribution, with N = 5 and M = 20, 50 pick locations per aisle and 3 cross-aisles, with the position of the iddle aisle varying between 1 and 49. The higher of the two curves shows the analytical result for E[P ] given the naïve routing assuption, and lower curve shows the siulation result for the sae aisle configurations, given the routing assuption used by Vaughan and Petersen [1999]. Although the naïve routing assuption results in a significant penalty (longer trip lengths), the optial aisle position is the sae in both cases (h = 8). 27

28 Diagonal Storage Across Aisle Storage Within Aisle Storage Path Length Path Length Path Length north- east- north- east- north- east- A N south west total south west total south west total Table 4: Coparison of diagonal, across-aisle and within aisle storage policies with ediu skewness level. N is the pick list size and A is the nuber of cross aisles. Here M = 20, B = 50, w a = 8, w b = 2.5 and the center-to-center distance between adjacent storage aisles is

29 optial north- eastaisle south west total percent A N CPU positions distance distance distance savings 2 < < < < < < < < < < < < < < < < < < Table 5: Optial cross aisle positions for an across-aisle storage policy with ediu skewness level. N is the pick list size and A is the nuber of cross aisles. The savings is the percentage reduction in expected travel distance resulting in the addition of a cross aisle. Here M = 20, B = 50, w a = 8, w b = 2.5 and the center-to-center distance between adjacent storage aisles is CPU is the nuber of seconds required to calculate the optial aisle positions. 29

30 optial Path Length aisle north- east- percent Skew N positions south west total savings Low Mediu High Table 6: Coparison of the across-aisle storage policy with four cross aisles and different skewness levels. N is the pick list size. Here M = 20, B = 50, w a = 8, w b = 2.5 and the center-to-center distance between adjacent storage aisles is For ediu skewness, the percent savings is how uch was saved in coparison with low skewness (for the sae pick list size), and for high skewness, the percent savings is how uch was saved in coparison with ediu skewness (for the sae pick list size). 30