Improving order-picking operation through efficient storage location assignment: A new approach

Journal Pre-proofs Improving order-picking operation through efficient storage location assignment: A new approach Meng Wang, Ren-Qian Zhang, Kun Fan PII: DOI: Reference:

S0360-8352(19)30655-2 https://doi.org/10.1016/j.cie.2019.106186 CAIE 106186

To appear in:

Computers & Industrial Engineering

Received Date: Revised Date: Accepted Date:

13 April 2019 3 October 2019 15 November 2019

Please cite this article as: Wang, M., Zhang, R-Q., Fan, K., Improving order-picking operation through efficient storage location assignment: A new approach, Computers & Industrial Engineering (2019), doi: https://doi.org/ 10.1016/j.cie.2019.106186

This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

© 2019 Published by Elsevier Ltd.

Improving order-picking operation through effi cient storage location assignment: A new approach Meng Wanga , Ren-Qian Zhanga,∗, Kun Fanb a School b School

of Economics and Management, Beihang University, Beijing 100191, China of Economics and Management, Beijing Forestry University, Beijing 100083, China

∗ Corresponding

author Email address: [email protected] (Ren-Qian Zhang)

Improving order-picking operation through efficient storage location assignment: A new approach

Abstract Fast development of online retail industry requires customer orders to be fulfilled within tight windows, where order-picking, the most time-consuming and laborintensive activity in warehouses, plays an important role. One of the basic ways to improve order-picking operation is assigning storage locations to appropriate items. The storage location assignment problem is in general NP-hard and is mainly solved by heuristics which usually suffer from limited solution quality or high computational effort, especially for large scale problems. In literature, most studies make the storage assignment decisions according to item properties, such as turnover or correlation, which are statistically extracted from item orders. These storage methods follow a data → concept → assign decision mechanism and may ignore useful data characteristics that are not conceptualized. This paper presents a new approach to improve the order-picking operation, which directly uses item orders to make the decisions without any statistical treatments, i.e., following a data → assign mechanism. The concept of good move pair is introduced to quickly find a better assignment through directly exploiting data characteristics of item orders, and an iterative algorithm is developed to minimize the total travel distance. We evaluate the algorithm on real data and numerical instances, and compare its performance with extant methods in the literature. The results show that the proposed method significantly outperforms other methods in most cases. We also extend the algorithm to the case of high-level warehouses and examine its effectiveness. Keywords: Logistics; Order-picking; Storage location assignment; Data characteristics

Preprint submitted to Journal of LATEX Templates

October 3, 2019

1. Introduction The online retail industry has achieved sharp growth in recent years. According to the statistics revealed by National Bureau of Statistics (2019), in 2018 the Chinese online retail industry increased 23.9% and reached 9 trillion 5

CNY. One of the biggest challenges that online retailers face in the rapid development is how to efficiently operate their warehouses or distribution centers, so that customer orders can be fulfilled within tight time windows. Among all warehouse activities, such as receiving, storage and packing, order-picking that retrieves the required items from their storage locations to fulfill specific

10

customer orders, is the most time-consuming and labor-intensive operation. According to Richards (2014), order-picking accounts for approximately 35% of the total operational costs in warehouses. This paper aims at improving the orderpicking operation via assigning storage locations to appropriate items, which is a tactical operation in warehouse management and planning (Gils et al., 2018).

15

Storage location assignment problem (SLAP) determines how to assign storage locations to appropriate items so that the order-picking efficiency is maximized. The problem is widely concerned since item layout within warehouse is a basic configuration for many other warehouse activities, such as order batching (Boysen et al., 2019) or picker routing (Scholz et al., 2016). SLAP is a gener-

20

alization of classic assignment problem, but its objective is more complex. The problem is proved to be NP-hard even in some special cases (Frazelle & Sharp, 1989). In literature SLAP is mainly solved by heuristics most of which suffer from limited solution quality or long computational time, especially for large scale instances. In this paper, we present a new approach for SLAP, which

25

significantly improves the solution of extent heuristics and is very efficient for large scale problems. 1.1. Motivation Sharp development of information technologies in recent years has enabled online retail companies to accumulate and analyze large amount of customer

2

30

orders which implicitly reflect customer buying habits and can help improve warehouse management (Nguyen et al., 2018). Modern storage assignment decisions largely rely on historical orders that are mainly used to statistically extract item properties based on which storage locations are assigned. One commonly used method is the full-turnover based storage policy which first

35

determines the turnover of items and then assigns the most popular items to storage locations nearest to the depot (Hausman et al., 1976). Instead of only considering individual item property, some researchers find that assigning the correlated items close to each other would significantly save the order-picking efforts (Frazelle & Sharp, 1989). Bindi et al. (2009) and Glock & Grosse (2012)

40

show that the correlation-based storage policies outperform other storage policies in their picking circumstances. One may notice that the aforementioned storage methods follow a data→concept→assign decision mechanism. This might make them not the best choice, since some useful data characteristics are not (fully) conceptualized, and thus are hard to be captured by this mechanism. For

45

example, if the demands for items A and B are equal, but A (B) is frequently requested by large (small) orders, it might be better to assign A (B) to storage location farther from (nearer to) the depot. Such data characteristic helps to improve the storage assignment, but cannot be utilized by the aforementioned storage methods. As a result, the data is not fully exploited. Moreover, efficien-

50

t storage methods for large scale problems and high-level warehouses are still limited (Most correlation-based storage methods in literature are addressing SLAPs within 1,000 items; the SLAP for high-level warehouses is mainly solved by class-based storage policies). To solve these problems and further improve the order-picking operation, this paper presents a new approach that makes

55

assignment decision by directly exploiting data characteristics implied in historical orders, and thus follows a data→assign decision mechanism which has three advantages: (1) make fully use of historical orders; (2) save the computational efforts for concepts (e.g. turnover, correlation, etc.) abstracting; (3) make it possible to efficiently solve large scale problems.

3

60

1.2. Contributions In this paper, we consider the SLAP in the picker-to-parts warehouse which is widely adopted in practice (Marchet et al., 2015). In this picking circumstance, order pickers travel along aisles to retrieve the requested items from storage locations to fulfill customer orders. The travel route is determined by

65

the S-shape routing procedure (Dijkstra & Roodbergen, 2017). We assume that historical orders are predetermined and known. Our goal is to fulfill these orders with minimized travel distance. This paper makes the following contributions: (1) We derive the S-shape route distance and formulate SLAP as an integer programming. Due to the model complexity, we approximate the objective

70

function with a formula that is easy to determine. Based on the approximation, a new concept is introduced to indicate how the objective value changes if any two items’ storage locations are swapped. This step is done by directly exploiting the data characteristics and thus follows the data→assign decision mechanism. A two-phase algorithm is developed based on the in-

75

troduced concept, which in the first phase assigns items to appropriate aisles and in the second phase assigns items to appropriate rows. We prove that the algorithm’s computational effort is polynomial. (2) We conduct computational experiments on real data that is collected from an online retailer and compare the results with extant storage methods

80

in literature. We observe that (a) The proposed approach that follows the data→assign decision mechanism significantly outperforms traditional methods that follow the data→concept→assign mechanism in both solution quality and running time; (b) The performance is mainly attributed to the first phase, i.e., assigning items to appropriate aisles; (c) The proposed

85

approach is capable of effciently solving large scale problems where the number of items is up to 4,000. (3) We extend the proposed approach to high-level picker-to-parts warehouses in which multilevel storage racks are used. In literature, the SLAP in such warehouses is mainly solved by class-based storage policies (Chan & Chan,

4

90

¨ urk, 2012; Pan et al., 2014). We consider the vertical as2011; Ene & Ozt¨ signment decision, modify the objective function as well as the approximated formula, and develop a three-phase algorithm. We also prove that the new algorithm’s computational effort remains polynomial. The algorithm’s effectiveness is examined.

95

1.3. Paper structure Remainder of this paper is organized as follows. We review related literature in section 2 and formulate the SLAP in section 3. The solution method is developed in section 4. Computational experiments are conducted and the results are discussed in section 5. Section 6 extends the proposed method to the

100

high-level warehouses. Finally, Section 7 concludes the paper.

2. Literature review Our study is widely related to order-picking systems, storage assignments, as well as routing strategies. Thus, we review the related studies in these three aspects. 105

2.1. Order-picking systems There are many ways to retrieve items from storage locations, according to which various order-picking systems are designed, including: the picker-toparts, parts-to-picker, and pick-and-pass systems. The picker-to-parts system employs human pickers to walk along aisles to retrieve items, which requires

110

low investment and high manpower level (De Koster et al., 2007). To reduce the labor cost, the parts-to-picker systems use automated storage/retrieval system (AS/RS) or automated guided vehicles (AGV) to carry the required items to pickers. Boze & Aldarondo (2018) compare the performance of AS/RS and AGVs through a simulation model. The pick-and-pass system is developed

115

for high pick volumes (PV), which splits the storage area into several zones and equips each zone with one or more pickers, so that the picking operation can proceed simultaneously (Pan et al., 2015). Dallari et al. (2009) carry out 5

an survey on 68 distribution centers in Italy, and find that the picker-to-parts system is very popular when PV≤10,000 (order lines per day), whereas the 120

pick-and-pass system is widely adopted when PV≤100,000. A similar study is done by Marchet et al. (2015), which shows that about half of the investigated companies are (simultaneously) equipped with the picker-to-parts and parts-topicker systems. This paper considers the picker-to-parts system. 2.2. Routing strategies

125

Routing strategies usually plays an important role in resource assignment and planning problems (De et al., 2018). In warehouse with certain storage assignment, how to determine the picking route to fulfill an order is a sub-problem which is a special case of traveling salesman problem (TSP) and can be solved in polynomial time (Ratliff & Rosenthal, 1983). Roodbergen & De Koster (2001)

130

extend Ratliff & Rosenthal’s approach to a 2-block warehouse. Scholz et al. (2016) solves the routing problem through new mathematical formulations. However, in practice the picker routing problem is mainly solved by heuristics, since optimal routes may seem illogical to pickers who would deviate from the specified routes as a result (Glock et al., 2017). Commonly used rout-

135

ing heuristic include: S-shape, return, mid-point, largest gap, composite, etc. Petersen & Schmenner (1999) evaluate the performance of heuristic and optimal routes in a volume-based storage assignment, and indicate that the overall gap between the best heuristic solution for a given storage assignment and the optimal is no longer than 3%. Dijkstra & Roodbergen (2017) make a perfor-

140

mance comparison of four routing heuristics in their optimal class-based storage assignments. They show that the return routing is the best routing method for instances with a small expected order length, whereas the largest-gap routing is better when the expected order length is larger. Readers may refer to De et al. (2019a,b) for routing procedures involving environmental sustainabili-

145

ty. In this paper, we use the S-shape routing heuristic, since under such routing procedure each picker has only one traffic direction, and the aisle congestion (neglected in this paper) can be reduced or avoided when pick density is high 6

(De Koster et al., 2007). 2.3. Storage location assignment 150

Storage location assignment is a tactical configuration for warehouses, since the operational decisions, such as batching (Boysen et al., 2019) and routing decisions (Scholz et al., 2016), have to be done with the aid of item layout information. The simplest storage method is the random storage policy which randomly assigns incoming pallets to available storage locations. The random

155

storage policy makes highly use of space utilization since one storage location may be shared by different items, but leads to less efficiency in the order-picking operation. Unlike the random storage policy, dedicated storage policy assigns every item to a fixed storage location which is retained even though the item is out of stock. One advantage of this policy lies in the high order-picking efficien-

160

cy, since order pickers become familiar to item layout. Many item properties have been considered to support the assignment decisions for dedicated storage policies, including: popularity, volume, turnover, pick density, cube-per-order (COI), etc. (Petersen et al., 2005). Malmborg & Bhaskaran (1990) prove the optimality of the COI-based storage assignment in AS/RS with dual command

165

cycle. Petersen & Schmenner (1999) examine the volume-based storage assignment in a picker-to-parts warehouse, considering different routing strategies. The class-based storage policy combines the random and dedicated storage polices, which first divides items into several classes and assigns each class to a fixed storage area within which the random policy is adopted. Hausman et al.

170

(1976) compare the operating performance of the random, full turnover-based and class-based turnover assignments in the AS/RS with single command cycle. It shows that significant travel time reduction can be achieved based on the turnover-based rules. Muppani & Adil (2008) propose a simulated annealing method for the class formation problem that considers both order-picking cost

175

and storage-space cost. Yu et al. (2015) examine the class-based storage policy in the cases where each storage region contains only a finite number of items, and find that a small number of classes is optimal. Dijkstra & Roodbergen 7

(2017) present a dynamic programming approach for the optimal class-based storage assignments under four common routing heuristics. 180

In reality it is very frequent for some items to appear on the same order, and these items are thought to be correlated. Correlation-based (family grouping) storage strategies put correlated items close to each other so that the picking effort is reduced (Frazelle & Sharp, 1989). Bindi et al. (2009) develop and compare a set of correlation-based storage rules based on the application of similarity

185

coefficients and clustering techniques. Xiao & Zheng (2010) present a multistage heuristic for the SLAP with picking capacity constraints in a production warehouse, where parts belonging to one bill of material are highly correlated. Chuang et al. (2012) use a between-item association to measure item relationship and propose a two-stage clustering-assignment model. Based on association

190

rule mining, Chiang et al. (2014) propose a weighted support count to represent both the intensity and nature of relationships between items, and develop two heuristics for SLAP. The computational results show that the proposed methods achieve at most 13% reduction in travel distance. Pang & Chan (2017) present a data mining-based algorithm for the SLAP in a randomized warehouse by an-

195

alyzing the correlation between different items, which minimize the total travel distance for both put-away and order-picking operations. Zhang et al. (2019) apply data mining techniques to extract demand correlation pattern of items, based on which a new SLAP model is built. Two heuristics are proposed to solve the model.

200

Instead of the low-level warehouses, some studies focus on storage assignment in high-level warehouses where multilevel storage racks are used and the vertical assignment must be considered. Chan & Chan (2011) present a simulation study of SLAP in high-level warehouses and indicate that vertical ABC class-based storage policy can improve the picking performance in terms of total

205

¨ urk (2012) design the storage assignment and orderretrieval time. Ene & Ozt¨ picking system in the automotive industry, where the SLAP is solved by the ABC class-based storage policy, considering the batching and routing operations. Pan et al. (2014) propose a travel time estimation model for a high-level 8

picker-to-parts warehouse, in which the SLAP is solved by six common class210

based storage policies. 2.4. Research gaps Most studies on SLAP make the assignment decisions according to one or more item properties, such as turnover (Hausman et al., 1976) or correlation (Chiang et al., 2014), which are usually statistically extracted from customer

215

orders. These methods follow the data→concept→assign decision mechanism and may ignore some data characteristics that are useful but are not conceptualized. Thus, customer orders are not fully used. The extent to which the order data can improve the storage assignment needs to be further explored and storage method that directly uses the order data (without statistically treatments)

220

to make decisions is lacking. Moreover, the SLAP in high-level warehouse is mainly solved by the class-based storage policies (Chan & Chan, 2011), whereas the implication of other efficient storage assignments for the high-level warehouses needs to be evaluated. These research gaps motivate us to develop a new approach to resolve the SLAP which makes decisions through fully exploiting

225

item orders, and can be applied to high-level warehouses and large scale SLAPs.

3. Problem formulation Consider a picker-to-parts warehouse with multiple two-sided aisles and multiple storage locations per aisle side. The warehouse layout is shown in Fig.1. Suppose that there are N items to be assigned to L storage locations and our 230

objective is to find an solution (assignment) that minimizes the total travel distance, T D. We optimize T D because the travel behavior do not produce value, and according to Tompkins et al. (2010), almost half of the order pickers’ time is spent on travelling. Let X = {xi,l : i = 1, ...N, l = 1, ..., L} denote one assignment where xi,l = 1 if item i is assigned to location l; 0, otherwise.

235

Under assignment X, let ai , ri be the aisle, row index of the storage location to which item i is assigned. Note that in the strict sense, ai (X) is the more precise notation; however, the dependency on X is omitted for ease of notation. 9

[Figure 1 about here.] We assume that one storage location can hold only one type of item and 240

that one item type must be assigned to one storage location. In other words, no stock mixing or splitting is allowed. For the simplicity, we assume that the number of items is equal to the number of storage locations, i.e., N = L. This is without loss of generality, since dummy items that are required by no order can be created, when N < L. Suppose that historical orders O in certain period

245

are available. With respect to each order o ∈ O, the picking route is determined by the S-shape strategy. In real picking operation, several orders are usually batched as a pick-list, which is retrieved in one tour and then split into orders. For the sake of simplicity, this paper does not consider the batching operation and assumes that the picking capacity in one tour is unlimited. This is also

250

without loss of generality, since the pick-lists that are formed according to the picking capacity can be treated as ordinary orders. The following notations are used throughout the paper: N = {1, ..., N }: set of items, indexed by i; L = {1, ..., L}: set of storage locations, indexed by l;

255

O = {1, ..., O}: set of orders, indexed by o; A = {1, ..., A}: set of aisles, indexed by a; R = {1, ..., R}: set of rows, indexed by r; ai ∈ A: aisle number of storage location to which item i is assigned; ri ∈ R: row number of storage location to which item i is assigned;

260

v: distance between adjacent storage locations; h: distance between adjacent aisles. xi,l = 1: (decision variable) if item i is assigned to location l; 0, otherwise. The following parameters are used to calculate picking route for order o: Ao : set of visited aisles;

265

a ¯o : the farthest visited aisle; r¯o : the last visited location in aisle a ¯o ; 10

| · |: the cardinality of the given set. Under assignment X = {xi,l : i ∈ N , j ∈ L}, the travel distance D(X, o) for picking order o is calculated as follows. The set of visited aisles is Ao = {ai : i ∈ o}, where duplicated elements in Ao are removed. The farthest visited aisle is a ¯o = max {a : a ∈ Ao } and the row of the last picked item in aisle a ¯o is r¯o = max {ri : ai = a ¯o , i ∈ o}. Then D(X, o) is calculated as: |Ao | mod 2

D(X, o) = 2 (¯ ao − 1) h + |Ao | (R + 1) v + (2¯ ro − R − 1)

v.

(1)

Now, SLAP can be modeled as follows. min T D(X) = s.t.

∑ i∈N

∑ l∈L

∑ o∈O

D(X, o),

(2)

xi,l = 1,

∀l ∈ L,

(3)

xi,l = 1,

∀i ∈ N ,

(4)

xi,l ∈ {0, 1} ,

i ∈ N , l ∈ L.

(5)

Objective (2) minimizes the total travel distance. Constraints (3) and (4) ensure the assumptions that a storage location only holds one item type and 270

that an item type must be assigned to one storage location, respectively. The model shows similarities with the assignment problem (AP): both problems have the same structure of constraints. AP has a simple objective function, which makes the problem solvable in polynomial time. The objective function of SLAP, however, depends on the warehouse layout, orders to be picked, and the

275

item-to-location assignment itself, which makes the problem NP-hard even in the special case where there are at most two items in each order (Frazelle & Sharp, 1989).

11

4. Solution Methodology To solve the model (2-5), a natural idea is to keep swapping any two items 280

that results in the objective value decreasing, until no such item pair exists. To find the swapped items, a total of

N (N −1) 2

item pairs have to be tried for

just one swap, which leads to high computational efforts. Another difficulty is that we do not exactly know how many swapping operations are needed before termination. To fix these shortages, a data-based approach (DBA) is 285

developed, which in the first phase improves the initial solution by reassigning items across the aisles and subsequently reassigning items to appropriate rows within each aisle. The proposed DBA is able to quickly find better solutions at each iteration via exploiting the data characteristics of historical orders. Basic ideas and definitions are introduced in section 4.1. The two phases of DBA are

290

described in section 4.2 and 4.3. 4.1. Basic ideas and definitions Objective (2) is complex due to the existence of the third term that determines the travel distance in the last visited aisle when |Ao | is odd. We consider an approximation for the objective function, which can be easily determined by

295

Ao . 4.1.1. Move evaluation In this section, we discuss how T D changes if item i, which is originally located in aisle ai , is moved to aisle a′ . We denote the move as mi,a′ . For the sake of simplicity, an integer permutation αo = (αo1 , ..., αoA ) is introduced to rewrite order o, where element αoa , a ∈ A refers to the number of items (contained by o) that are located in aisle a. Evidently, aisle a will not be entered if αoa = 0, otherwise aisle a must be entered. Let δo and ϵo be the non-zero element quantity and the last non-zero element index of αo . According to the definitions, we have δo = |Ao | and ϵo = a ¯o . With δo and ϵo , the estimated travel distance eD(αo ) for picking order o can be calculated as follows, eD (αo ) = 2(ϵo − 1)h + δo (R + 1)v. 12

(6)

Let νi,a′ be the move value of mi,a′ , which is defined as the estimated T D changes as the move happens. νi,a′ can be determined as follows. The set Oi of involved orders that contains item i is first extracted from historical orders O: Oi = {o ∈ O : i ∈ o}. With respect to each involved order o ∈ Oi , if the move mi,a′ occurs, we can obtain a new permutation α ˜ o that is modified from αo by α ˜ o,ai = αo,ai − 1 and α ˜ oa′ = αoa′ + 1. With formula (6), the move value νi,a′ can be calculated as the summation of eD(αo ) changes for all involved orders, ∑ νi,a′ = [eD(˜ αo ) − eD(αo )]. (7) o∈Oi

Fig.2 gives an illustrative example, where item 8 is supposed to be moved 300

from aisle 5 to 3, i.e., m8,3 . T D varies due to the travel distance changes of the involved orders, O8 = {o1 , o2 , o3 }. For order o1 , eD increases since aisle 3 needs to be traversed for picking item 8 after the move (δ˜o1 increases). For order o2 and o3 , the eDs decrease since the farthest aisles that need to be reached are nearer to the depot after the move (˜ ϵo2 and ϵ˜o3 decrease). According to formula

305

(7), the move value ν8,3 is equal to -32, indicating that the move may be a good choice, resulting in a decrease of T D. Note that even though the real changes of T D may not be equal to the estimated value, the move evaluation (6-7) still provides an efficient way to search for better solutions and this is validated in section 4.1.2 and 5.3. [Figure 2 about here.]

310

4.1.2. Good move pair The move mi,a′ discussed in section 4.1 is evaluated in the situation where the involved item i is reassigned from the source aisle ai to the target aisle a′ , while items that are located in a′ remain unchanged. Therefore, the move mi,a′ can only be applicable by conducting a dual move that reassigns one item, say j, of the target aisle a′ (= aj ) to the source aisle ai , i.e., mj,ai . Then, the exact T D change ∆ by exchanging the storage locations of item i and j can be estimated as the summation of the involved move values (denoted by ∆e ), i.e., ∆ ≈ ∆e = νi,aj + νj,ai . 13

(8)

The following proposition provides the relationship between ∆ and ∆e . Proposition 1. If all items have the same probability to be required by one order and A ≥ R > 1, h ≥ 2v hold, then we have Pr[

∆ > 0] ≥ 0.92 ∆e

(9)

Proof. See Appendix A. Proposition 1 shows that if the estimated value satisfies: ∆e = νi,aj + νj,ai < 315

0, the current solution is likely to be improved by swapping the storage locations of item i and j, which is the inspiration of DBA. Any two moves that satisfy νi,aj + νj,ai < 0 are referred to as good move pair (GMP). The effectiveness of estimation (8) will be validated by simulation in section 5.3. 4.2. Phase 1: Assign items across aisles

320

Algorithm 1 describes the first phase of DBA, which improves the input solution by repeatedly finding and exchanging GMPs. Specifically, for any two aisles, say a1 and a2 , the move value set V1 (V2 ) of all possible moves from aisle a1 (a2 ) to a2 (a1 ) is first constructed (step 4). The available moves with the minimum value in V1 and V2 are subsequently selected and checked (steps 6-8).

325

For ease of notation, let mi1 ,a2 , mi2 ,a1 be the selected moves, where items i1 , i2 are located in aisle a1 , a2 , i.e., ai1 = a1 , ai2 = a2 . Once a GMP is found, we further evaluate how T D actually varies. The storage locations of item i1 and i2 will be exchanged, if and only if the new solution s′ has a better objective value, i.e. ∆ < 0 (steps 10-11); otherwise, go on to find other GMPs (steps 13,

330

16). The process repeats until no GMP can be found. It should be noted that it is necessary to evaluate how T D actually varies since a very small probability still exists (< 0.08 according to Proposition 1) with which T D would increase even though the estimated change ∆e < 0. Moreover, there might be an infinite loop in the algorithm when step 10 is abandoned. Proposition 2 describes the

335

computational effort of Algorithm 1.

14

Algorithm 1 Phase 1 of DBA Require: Initial solution, s0 Ensure: Improved solution, s 1:

Set s := s0 ;

2:

for each a1 = 1, ..., A − 1 do

3: 4:

for each a2 = a1 + 1, ..., A do Under s, construct: V1 = {νi,a′ : i ∈ N , ai = a1 , a′ = a2 }, V2 = {νi,a′ : i ∈ N , ai = a2 , a′ = a1 };

5:

Set F1 , F2 = ∅;

6:

Get νi1 ,a2 = min{νi,a′ ∈ V1 : i ∈ / F1 };

7:

Get νi2 ,a1 = min{νi,a′ ∈ V2 : i ∈ / F2 };

8:

if νi1 ,a2 + νi2 ,a1 ≤ 0 then

9: 10:

Get s′ by swapping i1 and i2 ; if T D(s′ ) ≤ T D(s) then Set s := s′ ;

11: 12:

else Set F2 := F2 ∪ {i2 } and go to step 7;

13: 14: 15: 16: 17: 18: 19:

end if else Set F1 := F1 ∪ {i1 } and go to step 6. end if end for end for

15

Proposition 2. The computational effort of Algorithm 1 is polynomial and bounded by O(N 2 |O|). Proof. See Appendix B. 4.3. Phase 2: Assign items within aisle 340

Items are assigned to appropriate aisles in Phase 1, while item rows are not considered. In this section, we assign items to appropriate rows. Let mai,r′ denote the move that reassigns item i located in row ri to the target row r′ within aisle a. Note that the superscript a refers to the aisle a within which the move occurs. To determine move value νi,r ′ , the involved

345

order set Oia is first extracted from Oi , where every involved order o ∈ Oia satisfies: (1) The moved item i must be included in order o and located in aisle a; (2) |Ao | must be odd; and (3) Aisle a must be the last (farthest) visited aisle, i.e., Oia = {o ∈ Oi : ai = ϵo = a, δo is odd}. Note that mai,r′ does not affect the route length for order o ∈ / Oia .

350

Suppose that the move mai,r′ is applied; we estimate its move value as follows. For order o ∈ Oia , if o has only one item in the last visited aisle a, i.e. αoa = 1, the travel distance changes for picking o can be estimated as eDC(a, αo ) = 2(r′ − ri )v. If o has more than one item in aisle a, i.e., αoa > 1, we need to know how these items are distributed in this aisle. Another integer permutation

355

a a γoa = (γo1 , ..., γoR ) is introduced to describe the item distribution in aisle a, a where the element γor ∈ {0, 1, 2} refers to the number of items of o that are

located in aisle a and row r. After the move mai,r′ , a new permutation γ˜oa is a a a a modified from γoa by γ˜or ˜or = γor − 1. Let ηoa , η˜oa denote the ′ = γor ′ + 1 and γ i i

indexes of last non-zero element of γoa , γ˜oa . According to the definitions, 2vηoa , 360

2v η˜oa refer to the travel distance within aisle a before, after the move. If the a items of the target row r′ are also included in o, i.e., γor ′ = 2, obviously the

travel distance in this aisle will not change after the move, i.e., eDC(a, αo ) = 0. a If none of the items of the target row r′ is included in o, i.e., γor ′ = 0, then

we have eDC(a, αo ) = 2(η˜oa − ηoa )v. If only one item of the target row r′ is 365

a included in o, i.e. γor ′ = 1, we do not know whether this item will be swapped

16

with the moved one; therefore, the travel distance change can be estimated as eDC(a, αo ) = (˜ ηoa − ηoa )v. In summary, eDC(a, αo ) can be calculated by a formula (10) and the move value νi,r ′ can be calculated by formula (11). Now

we are able to use formula (8) to estimate the T D changes when any two items 370

are swapped within the same aisle.    2(r′ − r)v      2(η˜a − η a )v o o eDC(a, αo ) = a a   (˜ ηo − ηo )v      0 a νi,r ′ =

∑ o∈Oia

if αoa = 1; a if αoa > 1 and γor ′ = 0; a if αoa > 1 and γor ′ = 1;

(10)

a if αoa > 1 and γor ′ = 2.

eDC(a, αo ).

(11)

Algorithm 2 describes the second phase of DBA, which improves the input solution through reassigning items to appropriate rows within each aisle. Specifically, for each aisle a ∈ A, the set V of all possible move values in aisle a is constructed (step 3) and the move, say mai1 ,r2 , with the minimum negative move 375

value is selected as the first move (step 4). Among the moves from row r2 to row r1 (= ri1 ), the move mai2 ,r1 with the minimum move value is chosen as the second move (step 14). The summation of the first and second move values are subsequently checked (step 15). If ∆e > 0, the current move pair is not GMP and we go on to find another first move (step 23); otherwise, we further check

380

whether ∆ < 0 (step 17). We exchange the storage locations of item i1 and i2 if and only if ∆ < 0 (step 18). The process repeats for all aisles until no GMP can be found. Proposition 3 describes the computational effort of Algorithm 2. Proposition 3. The computational effort of Algorithm 2 is polynomial and bounded by O(2R2 N |O|).

385

Proof. See Appendix C. An initial solution is required to start DBA. Here the picking frequencybased storage strategy (PFS) that assigns items with higher frequency to storage locations closer to the depot, is employed as the initial solution. The initial 17

Algorithm 2 Phase 2 of DBA Require: Initial solution, s0 Ensure: Improved solution, s 1:

Set s := s0 ;

2:

for each a = 1, ..., A do

3:

a ′ Under s, construct V = {νi,r ′ : i ∈ N , ai = a, r ∈ R} and set F1 := ∅;

4:

a Get νia1 ,r2 = min{νi,r / F1 }; ′ ∈ V : i ∈

5:

if νia1 ,r2 ≥ 0 then

6:

if a = A then Stop and output s;

7: 8:

else Go back to step 2 and examine the next aisle;

9: 10: 11: 12:

end if else Set F2 := ∅;

13:

end if

14:

a ′ Get νia2 ,r1 = min{νi,r / F2 }; ′ ∈ V : ri = r2 , r = r1 , i ∈

15:

if ∆e = νia1 ,r2 + νia2 ,r1 < 0 then

16:

Get s′ by swapping i1 and i2 ;

17:

if T D(s′ ) < T D(s) then Set s := s′ and go back to step 3.

18: 19:

else Set F2 := F2 ∪ {i2 } and go back to step 14;

20: 21: 22: 23: 24: 25:

end if else Set F1 := F1 ∪ {i1 } and go back to step 4; end if end for

18

solution is then improved by running Algorithms 1 and 2 successively. The 390

improved solution is finally output. According to Propositions 1 and 2, DBA terminates within polynomial time. No parameter needs to be set.

5. Computational experiments We evaluate DBA on a real dataset and compare the computational results with that of extant methods for SLAP. All of the implemented algorithms are 395

coded in Java and run on a Windows 10 platform with an Intel i7-3770 CPU and 4.0 G ROM. 5.1. Test problems The real data is collected from an online retailer that supplies household products through the Internet. The dataset contains 24,870 orders ranging

400

from January 1 to December 31, 2013. We divide the dataset into four classes (quarters, Q1 -Q4 ) according to their dates. The order quantities of Q1 -Q4 are 5,069, 6,198, 6,216 and 7,387; the average item quantities per order are 22.13, 21.47, 20.47 and 21.35, respectively. There are a total of 4,039 products, and for the sake of simplicity, we only consider the top popular 4,000 products. No information about warehouse layouts or storage assignment exists. We test the proposed approach in a picker-to-parts warehouse with 40 two-sided aisles and 50 storage locations on each aisle side. Suppose that the distances of aisle-to-aisle and location-to-location are 4m and 1m, i.e. h = 4m, v = 1m. In the real picking process, several orders would usually be batched into a picklist, which will be retrieved in one tour and finally split into orders (re-bin). However, this paper concentrates on storage location assignment and assumes that orders are picked one by one. We adopt the picking frequency based storage strategy (PFS) as the benchmark storage method, where items with higher picking frequency will be assigned to storage locations nearer to the depot. The travel distance improvement (Impr) of a certain storage method over PFS is

19

Table 1: Simulation results

κ Dataset

(−∞, 0]

(0, 0.5]

(0.5, 1.5]

(1.5, +∞)

S1

0.0124

0.0172

0.9416

0.0288

S2

0.0024

0.0017

0.9893

0.0066

calculated and reported, Impr = 405

T D (PFS) − T D (·) × 100%, T D (PFS)

(12)

where · represents the methods to be examined. 5.2. Estimation validation This subsection presents simulation experiments to validate the objective approximation (8). The simulations are conducted with the aforementioned dataset and warehouse configuration. Under the PFS, two sets of item pairs are

410

sampled and swapped. Set S1 contains 10,000 item pairs, each of which comes from different aisles, i.e., S1 = {{i, j} : ai ̸= aj ; i, j ∈ N } and formulas (6-8) are used to obtain the estimated value ∆e . Set S2 contains 10,000 item pairs, both of which comes from the same aisle, i.e., S2 = {{i, j} : ai = aj ; i, j ∈ N } and formulas (8,10,11) are used for determining ∆e . At the same time, the real

415

T D changes ∆ is also calculated with respect to each swapped item pair and compared with ∆e . Table 1 shows the results of κ=∆/∆e . Note that samples satisfying ∆e = 0 are excluded. The simulation results show that in most cases 0.5 < κ ≤ 1.5, which confirms that the approximation (8), the basis of DBA, is of high precision, and that a GMP (∆e < 0) is likely to result in a better

420

assignment (∆ < 0). 5.3. Computational results We examine DBA on the datasets described in section 5.1. Since no good lower bound can be used to evaluate the solution, we implement three typical 20

Table 2: Computational results T D (m)

Impr (%)

CPU (s)

Dataset

PFS1

CAH

ASBH2

MIH

DBA

CAH

ASBH

MIH

DBA

CAH

ASBH

MIH

DBA

Q1

2,854,294

2,591,828

2,556,494

2,561,426

2,270,330

9.20

10.43

10.26

20.46

91.30

1,142.28

0.16

11.29

Q2

3,468,216

3,147,472

3,102,206

3,127,116

2,806,282

9.25

10.55

9.84

19.09

91.12

1,139.13

0.09

11.71

Q3

3,398,630

3,102,574

3,020,056

3,078,894

2,741,412

8.71

11.14

9.41

19.34

90.54

1,026.43

0.09

12.43

Q4

4,162,358

3,765,690

3,720,466

3,775,440

3,360,714

9.53

10.62

9.30

19.26

96.83

1,037.53

0.14

13.49

Avg.

3,470,875

3,151,891

3,099,806

3,135,719

2,794,685

9.17

10.69

9.70

19.54

92.45

1,086.34

0.12

12.23

1

The benchmark storage strategy.

2

Best results in 10 runs.

methods of SLAP for performance comparison: clustering-assigning heuristic 425

(CAH, Xiao & Zheng, 2010), association seed based heuristic (ASBH, Chiang et al., 2014) and minimum increment heuristic (MIH, Zhang et al., 2019). We select the three methods due to their high performances and employment of the Sshape routing strategy. According to computational results of Xiao & Zheng (2010), the CAH parameters, T L and T U are set as 0.1 and 0.6. No parameter

430

needs to be set for MIH and ASBH. We apply CAH, ASBH, MIH and DBA to the datasets, and computational results are reported in Table 2. Results in Table 2 show that DBA with an average improvement of 19.54% significantly outperforms CAH, ASBH and MIH with averages of 9.17%, 10.69% and 9.70%, respectively. The dominant performance of DBA may be attributed

435

to its ability of exploiting data characteristics that provide insights for improving the order-picking operation; however, some characteristics are not conceptualized and thus are hard to be captured through traditional methods. For CAH and ASBH, however, only the relationship between item pairs is considered. Computational times of the examined methods are also reported. We observe

440

that MIH is the fastest method with an average running time of 0.12s, while ASBH is the slowest, with an average of 1,086.34s. The longer computational time of CAH and ASBH may be due to the costs of operations for determining similarities between any two items. The running time of DBA is less than 14s for all datasets, with an average of 12.23s which is much faster than that of CAH

445

and ASBH. The observation is consistent with Proposition 1 and 2. In summary, DBA that follows the data→assign decision mechanism, obtains the best per21

Table 3: Computational results on different warehouse shapes T D (m)

Impr (%)

CPU (s)

A

R

PFS1

CAH

ASBH2

MIH

DBA

CAH

ASBH

MIH

DBA

CAH

ASBH

MIH

DBA

10

200

9,130,578

7,568,860

8,229,266

7,209,182

7,033,288

17.10

9.87

21.04

22.97

153.96

1154.17

1.42

100.54

20

100

11,485,102

9,087,864

9,340,646

8,821,830

8,457,248

20.87

18.67

23.19

26.36

161.05

1127.86

1.23

65.42

25

80

12,005,888

10,025,678

10,193,724

9,784,612

9,135,860

16.49

15.09

18.50

23.91

158.02

1125.08

1.20

52.35

40

50

14,021,128

12,711,526

12,622,594

12,419,316

11,384,486

9.34

9.97

11.42

18.80

159.20

1120.30

1.19

38.69

50

40

15,527,024

14,351,248

14,083,662

14,150,646

12,713,490

7.57

9.30

8.86

18.12

157.81

1119.79

1.19

39.32

80

25

20,024,856

18,707,298

18,027,280

18,481,564

16,590,826

6.58

9.98

7.71

17.15

157.92

1109.79

1.19

42.70

100

20

22,734,856

21,276,486

20,585,616

21,123,838

19,011,180

6.41

9.45

7.09

16.38

163.75

1146.02

1.25

49.73

14,989,919

13,389,851

13,297,541

13,141,570

12,046,625

12.05

11.76

13.97

20.53

158.82

1129.00

1.24

55.54

Avg 1

The benchmark storage strategy.

2

Best results in 10 runs.

formance within short running time, and is much more powerful than the traditional PFS or correlation-based methods that follow the data→concept→assign mechanism. We believe that the advantages in both improvement and running 450

time make DBA very practical for implementing in industry. 5.4. Results on different warehouse shapes To test the DBA performance under different warehouse shapes, we change the number of aisles in the warehouse and apply DBA to solving the variant SLAPs. To make sure the number of storage locations is equal to 4,000, seven

455

warehouse shapes (A × R) are tested here, i.e., 10 × 200, 20 × 100, 25 × 80, 40 × 50, 50 × 40, 80 × 25 and 100 × 20. The entire dataset is used for each shape. The computational results are shown in Table 3. The results in Table 3 show that given N (= 2AR), improvements of the four methods deteriorate with the increase of A (except the case of A = 10, R = 200).

460

This phenomenon indicates that it is more eager for warehouses with fewer aisles to be optimized. We observe that T D increases when A grows, which means that fewer aisles usually result in less picking effort. We also observe that DBA significantly outperforms CAH and ASBH in both improvement and computational time for all warehouse shapes. Compared with MIH, DBA consumes

465

longer computational time and outputs much better performances for all shapes. Note that the running time of DBA is longer in the cases of large R, which is consistent with the result of Proposition 2, such that the computational effort 22

Table 4: Effectiveness of DBA T D (m) Instance

PFS

Q1

1

Impr (%) P1

P2

DBA

2,854,294

2,284,866

2,844,680

Q2

3,468,216

2,822,220

Q3

3,398,630

2,756,464

Q4

4,162,358

Avg

3,470,875

2

CPU (s)

P1

P2

DBA

P1

P2

DBA

2,270,330

19.95

0.34

20.46

4.04

4.63

11.32

3,458,298

2,806,282

18.63

0.29

19.09

4.57

5.56

11.97

3,390,584

2,741,412

18.89

0.24

19.34

4.30

4.48

11.32

3,378,042

4,150,388

3,360,714

18.84

0.29

19.26

5.48

5.69

12.38

2,810,398

3,460,988

2,794,685

19.08

0.29

19.54

4.60

5.09

11.75

1

The benchmark storage strategy.

2

DBA = P1 + P2.

of Phase 2 is proportional to R2 . In summary, DBA retains its efficiency for different warehouse shapes. 470

5.5. Effectiveness of DBA This section independently examines components of DBA, i.e., Phase 1 (P1) and Phase 2 (P2). The datasets used here are the same as those of section 5.3. Computational results are displayed in Table 4. Results in Table 4 show that P1 significantly outperforms P2 in improvement

475

and computational time for all datasets, which can be explained by formula (2), indicating that T D is proportional to the number of visited aisles and the last visited aisle, while the item row, however, only has an effect when the visited aisle number is odd. Therefore, it makes DBA more efficient if we run P1 before P2. We observe that the running time of P1 is slightly shorter than that of P2,

480

which can be explained by their computational efforts (see Proposition 1 and 2 for details). 5.6. Numerical experiments and results In addition to results from real data, DBA is also examined on numerical instances that are randomly generated, and a statistical analysis is presented.

485

The warehouse examined in this subsection has A = 20 aisles and R = 20 locations per aisle side, resulting in a total of L(N ) = 2AR = 800 storage locations. h and v are set to 4 and 1. 23

We generate instances following the well-known ABC phenomenon where items are divides into A items representing high-demand items, B the medium490

demand items, and C the low-demand items. It is typical that a small percentage of items contribute a large percentage of total demand. Here we assume that A/B/C contain 20/30/50(%) of items, and examine four scenarios of their contribution in percentage of total demand, i.e., 60/20.25/19.75, 70/15.76/14.24, 80/10.83/9.17, and 90/5.56/4.44(%). The four scenarios are respectively deter-

495

mined by ABC curves (the cumulative percentage demand of the top I × 100% popular items is described as I s with I, s ∈ (0, 1]) with shape factors s of 0.3174, 0.2216, 0.1386 and 0.0655 (Hausman et al., 1976). With respect to each scenario, we generate 100 instances yielding to a total of 400 instances. For each instances, a number of N · U [30, 50] orders are generated, where U [·, ·] rep-

500

resents discrete uniform distribution. For each order, its length is determined from U [1, 10] and then it is filled with A/B/C items with probabilities of corresponding demand contributions. We apply the four methods to the generated instances and compare their average Impr (Eq.(12)) for each scenario. [Figure 3 about here.]

505

The statistical results are displayed in Fig. 3 which indicates that DBA significantly outperforms other methods in all scenarios. We observe that the four methods’ improvements (≤ 7%) are much less than that from real data (≥ 11%), which may due to the fact that the numerical instances are generated only considering item demand, resulting in better performance of the bench-

510

mark method FPS and therefore lower improvement of the four methods. In particular, no item relationship is considered in the instance generation, which makes the performance of correlation-based methods such as ASBH and CAH, not ideal. We also observe that for each method, the improvement increases as the demand contribution of A items grows. This may because that some char-

515

acteristics such as item correlation, would be automatically generated when a small percentage of items account for a majority of demand. Average computational times for DBA, MIH, CAH and ASBH are respectively, 2.77s, 0.23s, 5.6s 24

and 9.33s. DBA obtains the best solution within short running time.

6. Extension to high-level warehouses 520

In previous sections, we develop DBA to address the SLAP in the low-level warehouse. In this section, we extend DBA to high-level warehouses where multilevel storage racks are used. The high-level warehouses is commonly adopted when items’ volume (weight) are small (light), and are close to each other. Suppose that the studied warehouse has A parallel two-sided aisles and R multilevel storage racks on each aisle side. Each storage rack has H levels. Therefore, there are L = 2ARH storage locations in total. Again, without loss of generality, we assume that N = L. Let k = 1, ..., H be the level index and ki the level index of the storage location to which item i is assigned. In a highlevel warehouse, to retrieve item i, a picker has to travel to the location (ai , ri ) and reach to the ki -th levels. The process repeats until all items in the order are retrieved. With the S-shape routing strategy and certain assignment X, we determine the travel distance for picking order o ∈ O as D (X, o) = 2 (¯ ao − 1) h + |Ao | (R + 1) v+ ∑ |A | mod 2 (2¯ ro − R − 1) o v + i∈o ki z,

(13)

where z is the vertical distance between any two adjacent levels. 525

6.1. DBA for high-level warehouses Most studies that develop efficient storage strategies mainly focus on the lowlevel warehouses, while the SLAP in high-level warehouses are mainly solved ¨ urk, with the class-based storage strategies (Chan & Chan, 2011; Ene & Ozt¨ 2012; Pan et al., 2014). Here we modify DBA and extend it to the high-level

530

warehouses. 6.1.1. Phase 1 Phase 1 is the most important component of DBA and can only be applied here by considering the vertical distance changes in the move value calculation.

25

Therefore, we modify formula (7) as follows. νi,a′ =

∑ o∈Oi

[eD(˜ αo ) − eD(αo )] + (H/2 − ki )|Oi |,

(14)

where Oi is the set of involved orders that contain the moved item i. In the high-level warehouse, one aisle has a total of 2RH items and to adjust this, the move values sets in step 4 of Algorithm 1 should be modified as 535

V1 = {νi,a′ : i ∈ N , ai = a1 , a′ = a2 , hi = 1, ..., H}, V2 = {νi,a′ : i ∈ N , ai = a2 , a′ = a1 , hi = 1, ..., H}, and the move value is determined by formula (14). The remaining steps of Algorithm 1 stays unchanged. In other words, the move values of items that are located on the high level (≥ 2) locations are calculated and considered. In this way, we need to calculate move values 4RH times

540

for each swapping operation, and in the worst case, at most 2RH swappings may occur for any two aisles. The computational efforts of Algorithm 1 become O(4RHf × 2RH ×

A(A−1) ) 2

= O(4R2 H 2 A2 f − 4R2 H 2 Af )) < O(N 2 f ) ≤

O(N 2 |O|), where f is the computational effort of calculating the move value only once, and O is the number of historical orders. Apparently, Algorithm 1 is 545

still polynomial for the high-level warehouses. 6.1.2. Phase 2 To apply Algorithm 2 in the high-level case, we consider the high-level warehouse as H independent low-level warehouses and run Algorithm 2 for each level h = 1, ..., H. In other words, within each aisle, we fix the level index and apply

550

the steps 3-24 of Algorithm 2 for H times to assign items to appropriate rows with their vertical locations unchanged. Since Algorithm 2 is run H times, we can determine the computational efforts as O(2R2 N H|O|), and evidently, it is still polynomial. 6.1.3. Phase 3

555

After assigning items to appropriate aisles and rows, the vertical assignment decisions need to be made. Phase 3 assigns items to appropriate levels according to their picking frequency, i.e., assigning items with higher picking frequency to

26

Table 5: Computational result in high-level warehouse T D (m)

Impr (%)

CPU (s)

Instance

PFS1

P1

P1+P2

P1+P3

DBA2

P1

P1+P2

P1+P3

DBA

P1

P1+P2

P1+P3

DBA

Q1

1,937,460

1,630,521

1,614,543

1,597,510

1,574,056

15.84

16.67

17.55

18.76

3.49

9.23

3.44

9.66

Q2

2,344,945

2,002,981

1,985,195

1,964,022

1,937,061

14.58

15.34

16.24

17.39

3.87

9.78

3.78

9.85

Q3

2,293,691

1,951,159

1,933,855

1,911,976

1,888,513

14.93

15.69

16.64

17.66

3.51

9.55

3.54

9.59

Q4

2,794,099

2,399,206

2,378,514

2,355,865

2,323,757

14.13

14.87

15.68

16.83

4.44

11.34

4.38

11.45

Avg

2,342,549

1,995,967

1,978,027

1,957,343

1,930,847

14.87

15.64

16.53

17.66

3.83

9.97

3.79

10.14

1

The benchmark storage strategy.

2

DBA = P1 + P2 +P3.

lower levels. The computational efforts of phase 3 is mainly the cost of the sorting process. For each (a, r) pair, there are H items, and the computational 560

efforts for sorting H integers can be bounded by O(H 2 ). Therefore, the computational effort of phase 3 is O(2ARH 2 ) = O(N H), which is also polynomial. 6.2. Computational experiments We test the modified DBA on the real data introduced in section 5.1. The warehouse configurations include: A = R = 20, H = 5, h = 4m, r = z = 1m.

565

PFS is still used as the benchmark storage strategy and the effectiveness of DBA is evaluated. The indices of the total travel distance (T D), improvement over PFS (Impr) and running time (CPU) are reported in Table 5. Results in Table 5 show that the modified DBA consistently obtains good performance in the high-level warehouses where the improvements range from

570

16.83% to 18.76% with an average of 17.66%. We observe that P1 contributes to the majority of total improvements. It is easy to determine the effectiveness of P2 if we compare the performance of P1+P2 v.s. P1, or P1+P3 v.s. DBA. We found that P2 provides an improvement of 0.74-1.21%. Similarly, we found that P3 contributes to approximately 1.66-2.02% of the total improvement achieved

575

by P1+P3 or DBA. The computational times are also reported. P1 only needs an average of 3.83s to complete. The running time of P2 can be observed by comparing P1 v.s. P1+P2, or P1+P3 v.s. DBA. We found that the running time of P2 is approximately 6.25s on average. The running time of P3 can be neglected since the computational times of P1 and P1+P3 (or P1+P2 and DBA)

27

580

are almost the same. The above observations are consistent with the computational effort analyses in the previous section. In summary, DBA can output competitive performance within a very short running time, and we believe that the approach can be applied to the high-level warehouses in industry.

7. Conclusions 585

This paper aims at improving the order picking operation through assigning items to appropriate storage locations. Instead of the traditional storage methods that follow a data→concept→assign decision mechanism, we propose a data-based approach (DBA) that makes the assignment decisions following a data→assign mechanism. Specifically, the total travel distance (T D) change by

590

swapping any two items is estimated, and the estimation precision is bounded. Then, the concept of good move pair (GMP) is introduced to find promising swapping options for better assignments. Based on GMP, a two-phase algorithm (DBA) is developed with polynomial computational efforts. We examine DBA against extant methods in literature on real data that is collected from an

595

online retailer and numerical instances that are randomly generated. The computational results indicate that DBA significantly outperforms other methods on both aspects of improvement and running time. Finally, we extend DBA to the high-level warehouses and the results show that DBA can still obtain the best performance within short running time. We believe that the advantages in

600

both improvement performance and running time make DBA very practical for companies to implement. Analyzing the computational results, we find some managerial insights as follows. (1) Historical orders provide many benefits for better storage assignments, and DBA that follows data→assign mechanism, indeed generates very competi-

605

tive solutions within short running times. The dominance may be due to its ability of exploiting data characteristics that are not fully conceptualized and are hard to be captured through traditional strategies. We believe that

28

developing data-based methods for other order-picking problems may be a potential choice; 610

(2) The proposed DBA is very useful and practical in developing efficient storage location assignment, since: (a) Its input data is easy to obtain for companies; (b) It has advantages in computational performance and running time, especially in solving large scale problems; (c) It can be applied to the high-level warehouses.

615

(3) Given the number of storage locations, warehouse shapes with fewer aisles and more rows usually result in less travel distance; (4) It is more eager for warehouses with fewer aisles to be improved through better storage assignments. DBA is developed to minimize the total order-picking distances in the single-

620

block and single-depot warehouses, and several limitations may be found as follows. First, the current DBA is not suitable for multi-block warehouses that adopt one or more cross-aisles in the middle. Second, this study assumes that there exists only one depot in the warehouse. In reality, however, some warehouses have a multi-depot setting, and the current version of DBA can not be

625

applied either. The aforementioned limitations provide us future research opportunities. To apply DBA to these aforementioned cases, the picking routes should be further analyzed and the move evaluation method should be revised, which is one of our future directions. Besides the order-picking route length, other item attributes

630

such as volume or weight, may also affects the order-picking efficiency and can be considered in storage decisions. Moreover, a decision supporting system can be developed for better warehouse management, where DBA works with other data-based techniques, such as demand or inventory prediction approaches, to capture more valuable insights from customer orders and then achieve smart

635

integrated decisions, such as ordering and replenishing (Alkahtani et al., 2019).

29

References Alkahtani, M., Choudhary, A., De, A., & Harding, J. A. (2019). A decision support system based on ontology and data mining to improve design using warranty data. Computers & Industrial Engineering, 128 , 1027–1039. 640

Bindi, F., Manzini, R., Pareschi, A., & Regattieri, A. (2009). Similarity-based storage allocation rules in an order picking system: an application to the food service industry. International Journal of Logistics Research and Applications, 12 , 233–247. Boysen, N., De Koster, R., & Weidinger, F. (2019). Warehousing in the e-

645

commerce era: A survey. European Journal of Operational Research, 277 , 396–411. Boze, Y. A., & Aldarondo, F. J. (2018). A simulation-based comparison of two goods-to-person order picking systems in an online retail setting. International Journal of Production Research, 56 , 3838–3858.

650

Chan, F. T. S., & Chan, H. K. (2011). Improving the productivity of order picking of a manual-pick and multi-level rack distribution warehouse through the implementation of class-based storage. Expert Systems with Applications, 38 , 2686–2700. Chiang, D. M.-H., Lin, C.-P., & Chen, M.-C. (2014). Data mining based storage

655

assignment heuristics for travel distance reduction. Expert Systems, 31 , 81– 90. Chuang, Y.-F., Lee, H.-T., & Lai, Y.-C. (2012). Item-associated cluster assignment model on storage allocation problems. Computers & Industrial Engineering, 63 , 1171–1177.

660

Dallari, F., Marchet, G., & Melacini, M. (2009). Design of order picking system. The International Journal of Advanced Manufacturing Technology, 42 , 1–12.

30

De, A., Choudhary, A., & Tiwari, M. K. (2019a). Multiobjective approach for sustainable ship routing and scheduling with draft restrictions. IEEE Transactions on Engineering Management, 66 , 35–51. 665

De, A., Pratap, S., Kumar, A., & Tiwari, M. K. (2018). A hybrid dynamic berth allocation planning problem with fuel costs considerations for container terminal port using chemical reaction optimization approach. Annals of Operations Research, . De, A., Wang, J., & Tiwari, M. K. (2019b). Hybridizing basic variable neigh-

670

borhood search with particle swarm optimization for solving sustainable ship routing and bunker management problem. IEEE Transactions on Intelligent Transportation Systems, Accept. De Koster, R., Le-Duc, T., & Roodbergen, K. J. (2007). Design and control of warehouse order picking: A literature review. European Journal of Opera-

675

tional Research, 182 , 481–501. Dijkstra, A. S., & Roodbergen, K. J. (2017). Exact route-length formulas and a storage location assignment heuristic for picker-to-parts warehouses. Transportation Research Part E Logistics & Transportation Review , 102 , 38–59. ¨ urk, N. (2012). Storage location assignment and order picking Ene, S., & Ozt¨

680

optimization in the automotive industry. International Journal of Advanced Manufacturing technology, 60 , 787–797. Frazelle, E. A., & Sharp, G. P. (1989). Correlated assignment strategy can improve any order-picking operation. Industrial Engineering, 21 , 33–37. Gils, T. v., Ramaekers, K., Caris, A., & De Koster, R. B. (2018). Designing

685

efficient order picking systems by combining planning problems: State-of-theart classification and review. European Journal of Operational Research, 267 , 1–15.

31

Glock, C. H., & Grosse, E. H. (2012). Storage policies and order picking strategies in u-shaped order-picking systems with a movable base. International 690

Journal of Production Research, 50 , 4344–4357. Glock, C. H., Grosse, E. H., Elbert, R. M., & Franzke, T. (2017). Maverick picking: the impact of modifications in work schedules on manual order picking processes. International Journal of Production Research, 55 , 6344–6360. Hausman, W. H., Schwarz, L. B., & Graves, S. C. (1976). Optimal storage

695

assignment in automatic warehousing systems. Management Science, 22 , 629–638. Malmborg, C. J., & Bhaskaran, K. (1990). A revised proof of optimality for the cube-per-order index rule for stored item location. Applied Mathematical Modelling, 14 , 87–95.

700

Marchet, G., Melacini, M., & Perotti, S. (2015). Investigating order picking system adoption: a case-study-based approach. International Journal of Logistics: Research and Applications, 18 , 82–98. Muppani, V. R., & Adil, G. K. (2008). Efficient formation of storage classes for warehouse storage location assignment: A simulated annealing approach.

705

Omega, 36 , 609–618. National

Bureau

of

Statistics

(2019).

Sustainable

and

rapid

growth of new kinetic energy of economic development in china. http://www.stats.gov.cn/tjsj/sjjd/201907/t20190731 1683091.html ,

Ac-

cessed on Auguest 1st, 2019. 710

Nguyen, T., Zhou, L., Spiegler, V., Ieromonachou, P., & Lin, Y. (2018). Big data analytics in supply chain management: A state-of-the-art literature review. Computers & Operations Research, 98 , 254–264. Pan, J. C.-H., Shih, P.-H., Wu, M.-H., & Lin, J.-H. (2015). A storage assignment heuristic method based on genetic algorithm for a pick-and-pass warehousing

715

system. Computers & Industrial Engineering, 81 , 1–13. 32

Pan, J. C.-H., Wu, M.-H., & Chang, W.-L. (2014). A travel time estimation model for a high-level picker-to-part system with class-based storage policies. European Journal of Operational Research, 237 , 1054–1066. Pang, K.-W., & Chan, H.-L. (2017). Data mining-based algorithm for storage 720

location assignment in a randomised warehouse. International Journal of Production Research, 55 , 4035–4052. Petersen, C. G., & Schmenner, R. W. (1999). An evaluation of routing and volume - based storage policies in an order picking operation. Decision Sciences, 30 , 481–501.

725

Petersen, C. G., Siu, C., & Heiser, D. R. (2005). Improving order picking performance utilizing slotting and golden zone storage. International Journal of Operations & Production Management, 25 , 997–1012. Ratliff, H. D., & Rosenthal, A. S. (1983). Order-picking in a rectangular warehouse: A solvable case of the traveling salesman problem. Operations Re-

730

search, 31 , 507–521. Richards, G. (2014). Warehouse Management: A Complete Guide to Improving Efficiency and Minimizing Costs in the Modern Warehouse. London: Kogan Page. Roodbergen, K. J., & De Koster, R. (2001). Routing order pickers in a warehouse

735

with a middle aisle. European Journal of Operational Research, 133 , 32–43. Scholz, A., Henn, S., Stuhlmann, M., & W¨ascher, G. (2016). A new mathematical programming formulation for the single-picker routing problem. European Journal of Operational Research, 253 , 68–84. Tompkins, J. A., White, Y. A., Bozer, E. H., & Tanchoco, M. A. (2010). Facil-

740

ities Planning. 4th ed. Hoboken, NJ: Wiley. Xiao, J., & Zheng, L. (2010). A correlated storage location assignment problem in a single-block-multi-aisles warehouse considering bom information. International Journal of Production Research, 48 , 1321–1338. 33

Yu, Y., De Koster, R. B. M., & Guo, X. (2015). Class-based storage with a 745

finite number of items: Using more classes is not always better. Production and Operations Management, 24 , 1235–1247. Zhang, R.-Q., Wang, M., & Pan, X. (2019). New model of the storage location assignment problem considering demand correlation pattern. Computers & Industrial Engineering, 129 , 210–219.

34

750

Appendix A. Proof of Proposition 1 We prove the proposition with following assumptions: (1) All items have the same probability to be included in one order; (2) A ≥ R > 1, h ≥ 2v. The gap between ∆ and ∆e comes from the estimation formula (6), which

755

thinks that order pickers will return at the mid-point of the last visited aisle when the number of visited aisles |Ao | (or δo ) is odd. When δo is even, however, no error is generated since estimation formula (6) is equivalent to the exact formula (1). We mainly examine the cases where δo is odd. Supposing that the storage locations of item i and j are swapped, we are

760

going to calculate the summation of errors generated by the swapping and compare it with ∆e to make the proof possible. Note that we will put a tilde above variables to indicate their states after the swapping. For example, δo and δ˜o denote the number of visited aisles for retrieving order o before and after the swapping.

765

For orders with odd δo , let to = D(o, X) − eD(αo ) be the estimation error, which is a random variable and determined by the real turn-back point in the last visited aisle. According to Assumption 1, the point uniformly distributes in the last visited aisle and therefore the distribution of to can be determined. The expectation and variance are E[to ] = 0, V[to ] = 13 (R2 − 1)v 2 . Note that to is symmetrically distributed and therefore, −to has the same distribution. to

(1 − R)v

(3 − R)v

...

(R − 1)v

Pr

1 R

1 R

...

1 R

770

Since the estimation error is only generated in the cases where δo is odd, we split the involved order set of the swapped items, say Oi , into four classes Oik , k = 1, ..., 4 according to the parity of δo and δ˜o . The analytical results are displayed in Table A.1. Let Oik be the cardinality of Oik . Error summation in

35

Table A.1: Error analysis

Class

Order set

δo

δ˜o

Error from i per order

Error from j per order

1

Oi1 , Oj1 1 even

even

0

0

2

Oi2 , Oj2 2 even

odd

to

−to

3

Oi3 , Oj3 3

odd

even

−to

to

4

Oi4 , Oj4 4

odd

odd

to − t′o 5

t′o − to

1

Oj1 = {o : j ∈ o; δo is even; δ˜o is even} and so is the case for Oi1 .

2

Oj2 = {o : j ∈ o; δo is even; δ˜o is odd} and so is the case for Oi2 .

3

Oj3 = {o : j ∈ o; δo is odd; δ˜o is even} and so is the case for Oi3 .

4

Oj4 = {o : j ∈ o; δo is odd; δ˜o is odd} and so is the case for Oi4 .

5

t′o follows the same distribution of to .

the swapping can be calculated as follows. ∆= =

∑ ∑

∑ =

˜ o) − D(X, o)] [D(X,

o∈Oi1 ∪Oj1

[eD(˜ αo ) − eD(αo )]+

o∈Oi2 ∪Oj2

[eD(˜ αo ) − eD(αo ) + to ]+

o∈Oi3 ∪Oj3

[eD(˜ αo ) − eD(αo ) + to ]+

o∈Oi4 ∪Oj4

[eD(˜ αo ) − eD(αo ) + to + t′o ]

∑ ∑

o∈Oi ∪Oj

∑

o∈Oi ∪Oj

[eD(˜ αo ) − eD(αo )] + T

= ∆e + T where T is the summation of all to , t′o . When the involved order number is large enough, according to the Central Limit Theorem, T follows the normal ( ) distribution, i.e., T ∼ N 0, σT2 , where σT2 = (Oi2 + Oi3 + 2Oi4 + Oj2 + Oj3 + 2Oj4 )V[to ]. 775

To determine Pr[ ∆∆e ], we need to explore how ∆e distributes. According ∑ to formulas (6-8), we have ∆e = ϵo − ϵo )h + o∈Oi ∪Oj [qo ], where qo = 2(˜ 2(δ˜o − δo )(R + 1)v. Under Assumption 1, the last visited aisle of an order ϵo is uniformly distributed at 1, ..., A, where A is the number of aisles. Therefore, the distribution of ϵ˜o −ϵo is determined. Its expectation and variance are E[˜ ϵo −ϵo ] =

780

0, V[˜ ϵo − ϵo ] = 16 (A2 − 1).

36

ϵ˜o − ϵo

1−A

2−A

...

0

...

A−1

Pr

1 A2

2 A2

...

A A2

...

1 A2

The distribution of δ˜o − δo needs to be derived. Again, we split the involved orders into four classes as shown in Table A.1 and each class is analyzed as follows. Class 1: even to even. In this case, the number of visited aisles does not 785

change, since δo remains even after single swapping. We have δ˜o − δo = 0. Therefore, qo = 2h(˜ ϵo − ϵo ) and E[qo ] = 0, V[qo ] = 23 h2 (A2 − 1). Class 2: even to odd. Since δo turns from even to odd after one swapping, the number of visited aisles must increase or decrease by only one unit. According to Assumption 1, probabilities that δo increases or decreases should

790

be equal. Therefore, δ˜o − δo is uniformly distributed on -1 and 1. The random variable ϵ˜o − ϵo has the same distribution as in Class 1. We also observe that ϵ˜o − ϵo ≥ (≤)0 when δ˜o − δo = 1(−1). Therefore, the distribution of qo is determined as in Table A.2. Its expectation and variance are E (qo ) = 0, 2

V (qo ) = 23 h2 (A − 1) (2A − 1) + 2hv (A − 1) (R + 1) + (R + 1) v 2 Table A.2: Distribution of qo

795

δ˜o − δo

ϵ˜o − ϵo

qo

Pr

-1

1−A

2h(1 − A) − v(R + 1)

1 2A

-1

2−A

2h(2 − A) − v(R + 1)

1 2A

...

...

...

...

-1

0

−v(R + 1)

1 2A

1

0

v(R + 1)

1 2A

1

1

2h + v(R + 1)

1 2A

...

...

...

...

1

A−1

2h(A − 1) + v(R + 1)

1 2A

Class 3: odd to even. Same as Class 2. Class 4: odd to odd. Same as Class 1.

37

We derive the distribution of ∆e as: ∆e = =

∑ ∑

o∈Oi ∪Oj

[qo ]

o∈Oi1 ∪Oj1 ∪Oi4 ∪Oj4

[qo ] +

∑ o∈Oi2 ∪Oj2 ∪Oi3 ∪Oj3

[qo ]

=Q Again, according to the Central Limit Theorem, Q approximately follows the 2 2 normal distribution, i.e., Q ∼ N (0, σQ ), where σQ = (Oi1 + Oi4 + Oj1 + Oj4 )[ 23 h2 (A2 − 1)]+ 2

(Oi2 + Oi3 + Oj2 + Oj3 )[ 23 h2 (A − 1)(2A − 1) + 2hv(A − 1)(R + 1) + (R + 1) v 2 ]. With the distributions of T and Q, the estimation precision κ can be determined as, κ= 800

∆ ∆e

=1+

T Q.

Both T and Q follow the normal distribution with expectation of 0 and as a result, κ follows the Cauchy Distribution, i.e., κ ∼ C(1, λ), where λ = σT /σQ . Under Assumption 1, order numbers of the four classes are equal to each other and we have Oi1 = ... = Oi4 = 14 Oi , Oj1 = ... = Oj4 = 14 Oj , where Oi , Oj are numbers of involved orders of item i, j. We bound λ as follows, √ 1 2 2 3 (Oi +Oj )(R −1)v λ= 2 1 1 2 2 2 2 2 3 (Oi +Oj )h (A −1)+ 2 (Oi +Oj )( 3 h (A−1)(2A−1)+2hv(A−1)(R+1)+v (R+1) ) √ 1 2 2 3 (R −1)v = 2 . 1 2 1 2 2 2 2 3 h (A −1)+ 2 ( 3 h (A−1)(2A−1)+2hv(A−1)(R+1)+v (R+1) ) Under Assumption 2, A ≥ R and h ≥ 2v, thus we have: √ 1 2 2 3 (R −1)v λ≤ 4 2 2 −1)+ 1 8 v 2 (A2 −1)+4v 2 (A−1)(R+1)+v 2 (R+1)2 v (A ) 2(3 √3 1 2 −1)v 2 (R 3 ≤ . 4 2 v (R2 −1)+ 12 ( 83 v 2 (R2 −1)+4v 2 (R2 −1)+v 2 (R2 −1)) √3 √ 1 2 2 2 3 (R −1)v = = 31 2 2 31 (R −1)v 6

With the c.d.f. of C(1, λ), F(x) =

1 2

+ π1 arctan( x−1 λ ), we can describe the value

of κ as, Pr[κ > 0] = 1 − F(0) = 1 − [ 12 + ≥ 0.9210.

38

1 π

arctan( −1 λ )]

Appendix B. Proof of Proposition 2 Computation effort of Algorithm 1 is mainly spent on constructing the move value sets. Let f be the computational efforts of calculating move value for one 805

time and according to formula (7) we have f ≤ |Oi | ≤ |O|, where |O| is the total number of orders. For each swap operation, we have to calculate a total of 4R move values (2R for each set). In the worst case, the swap operation takes place for 2R times for any two aisles. Therefore, the computational effort of Algorithm 1 is O(4Rf × 2R ×

810

A(A−1) ) 2

= O(4R2 A2 f − 4R2 Af ) < O(N 2 f ) ≤ O(N 2 |O|),

where N = 2AR is the number of items. As we can see that Algorithm 1 is a polynomial algorithm and runs very fast even in large scale problems. Note that the efforts for searching GMPs and calculating T D(s′ ) are neglected, since they are minimal as compared with the effort for constructing move value sets.

Appendix C. Proof of Proposition 3 815

Similar to Algorithm 1, constructing the move value sets costs most of the computational effort of Algorithm 2. Let f be the computational effort of calculating one move value with formula (11) and then we have f ≤ |Oia | ≤ |O|. For each aisle a ∈ A, one item may be assigned to R possible rows and there are a total of 2R items in each aisle, resulting in the move value calculation for

820

2R2 times for one move value set. In the worst case, every item is assigned to a new row and the move value set is constructed for 2R times. The computational effort of Algorithm 2 is O(2R2 f × 2R × A) = O(2R2 N f ) ≤ O(2R2 N |O|), where N is the number of items and |O| the total number of orders. Again, the effort for searching GMPs and calculating T Ds are neglected, since they are minimal

825

as compared with the effort for constructing move value sets.

39

List of Figures 1 2 3

Warehouse layout and S-shape route . . . . . . . . . . . . . . . . An example for move evaluation . . . . . . . . . . . . . . . . . . Statistical results of numerical experiment . . . . . . . . . . . . .

40

41 42 43

( 2,8)

h

8 7

row

6

v

5 4 3 2 1 Depot

1

2

3

4

5

ailse

Figure 1: Warehouse layout and S-shape route

41

row

ľ ĸ

7 6 ķ 5 4 3 2 1

ľ

o1 = {ķĸĽľĿ}

ĺ Ĺ

A = 5, R = 7, h =4ˈv =1 Involved orders:

ļ Ŀ Ļ

Ľ

o2 = {ĸĹľ} o3 = {ĹĺĻļľ}

Depot

1

2

3 ailse

4

5

a o1 = ( 2,0,0,0,3) , d o1 =2,

o1

=5, 5, eD(a o1 ) = 48

a o = ( 2, 0,1, 0, 2 ) , d o = =3, 3,

o1

a o2 = ( 2,0,0,0,1) , d o2 =2,

o2

=5, 5, eD(a o2 ) = 48

= 2, a o = ( 2, 0,1, 0, 0 ) , d o =2,

o2

= a o = (1, 0, 4, 0, 0 ) , d o =2,

o3

a o3 = (1,0,3,0,1) , d o3 =3,

5, eD(a o3 ) = 56 o3 =5,

1

2

3

1

2

3

n 8,3 = å oÎ{o ,o ,o } éëeD (a o ) - eD (a o ) ùû = -32 1

2

3

Figure 2: An example for move evaluation

42

( ) =3, eD (a ) = 32 =3, =3, eD (a ) = 32 =3, =5 eD a o1 = 56 =5, o2

o3

ķĸĹĺĻļ

7 CAH

ASBH

MIH

DBA

6

Average Impr (%)

5

4

3

2

1

0 60/20.25/19.74

70/15.76/14.23

80/10.83/9.16

90/5.56/4.43

A/B/C items' contribution to total demand (%)

Figure 3: Statistical results of numerical experiment

43

> Order-picking efficiency is improved via efficient storage assignment > New concept is introduced for better assignment by exploiting order characteristics > A data-based approach (DBA) is developed for the solution > DBA outputs competitive results on real data with short running time > DBA is extended to the case of high-level warehouses

Acknowledgement The work is supported by the National Natural Science Foundation of China [No.71571006, 71502015].