Duration-of-Stay Storage Assignment under Uncertainty

1 INTRODUCTION . The rise of the modern era has been accompanied by ever-shortening product life cycles , straining the entire supply chain and demanding efficiency at every node . One integral part of any supply chain is warehousing ( storage ) ; warehouse operations often have major impacts downstream on the capability to deliver product on time . One of the largest cold storage companies in the world is looking to improve the efficiency of their warehouses by optimizing the scheduling of storage systems . According to Hausman et al . ( 1976 ) , the scheduling of labor in warehouses can be divided into three main components : • Pallet Assignment : The assignment of multiple items to the same pallet . • Storage Assignment : The assignment of pallets to a storage location . • Interleaving : The overarching rules for dealing with concurrent inbound and outbound requests . For this particular paper , we focus on the problem of storage assignment . Various papers such as Goetschalckx & Ratliff ( 1990 ) show labor efficiency to be a bottleneck . In a modern warehouse , the process of storage assignment usually involves forklift drivers moving inbound pallets from the staging area of the warehouse to the storage location , so a sub-optimal assignment system causes unnecessary long travel times to store the pallet . Unfortunately , the inefficiency is quadrupled when the return of the forklift and the retrieval of the pallet are considered . To increase the efficiency of the warehouse , we would thus like to minimize the total travel time needed to store a set of shipments from the staging area . Many different theoretical frameworks exist , and the details of such frameworks are contained in Appendix 9.1 . The ones of chief interest are turnover-based , class-based , and Duration-of-Stay ( DoS ) based strategies . Turnover-based strategies ( e.g . Hausman et al . ( 1976 ) , Yu & De Koster ( 2009 ) ) assign locations so that the travel distance is inversely proportional to the turnover of the product . Class-based strategies ( e.g . Hausman et al . ( 1976 ) , Schwarz et al . ( 1978 ) ) separate products into k classes , with each class assigned a dedicated area of storage . DoS-based strategies ( e.g . Goetschalckx & Ratliff ( 1990 ) , Chen et al . ( 2016 ) ) assign pallets to locations with travel distance proportional to the duration-of-stay . Simulation experiments in Goetschalckx & Ratliff ( 1990 ) and Kulturel et al . ( 1999 ) demonstrated that under complex stochastic environments , DoS-based strategies outperform other methodologies significantly . However , among all three categories , the most commonly used strategy is class-based , as pointed out by Yu et al . ( 2015 ) and Yu & De Koster ( 2013 ) . The authors and industry evidence suggest that this is due to the fact that class-based systems are relatively easy to implement , but also because DoS is not known in advance . To utilize a DoS system realistically would therefore require an accurate prediction model using the features available at shipment entry to the warehouse . However , even with the availability of modern high-powered predictive methods including Gradient Boosted Trees and Neural Networks , there has been no documented progress in employing DoS-based methods . This reflects the following significant challenges in a dynamic , real warehouse : • Multi-valuedness : It is common for 10+ identical pallets of the same product to arrive in a single shipment , and then leave the warehouse at different times depending on the consumption of the end consumer . This causes the ground truth for the DoS of a product entering at a given time to be ill-defined . • Data Sparsity : A large warehouse would have significant available historical DoS data , but such data is scattered across thousands of products/SKUs , and different operating conditions ( e.g . time of the year , day of the week , shipment size ) . Given strong variation of DoS in a warehouse , it is very unlikely that all environment-product combinations would exist in data for the historical average to be valid for future DoS . Furthermore , new SKUs are created relatively frequently , and the predictive algorithm needs to be robust against that as well . To solve such difficulties , we reformulate the DoS as a distribution and develop a new framework based on nonparametric estimation . Then we combine it with a parallel architecture of Residual Deep Convolutional Networks ( He et al . ( 2015 ) ) and Gated Recurrent Unit ( GRU ) networks ( Cho et al . ( 2014 ) ) to provide strong estimation results . As far as the authors know , this is the first documented attempt to predict DoS in warehousing systems . We further release the first public dataset of warehousing records to enable future research into this problem . This neural network is then integrated into the larger framework , which is being implemented in live warehouses . We illustrate how initial results from the ground show appreciable labor savings . Specifically , our contributions in this paper include : • We develop an novel end-to-end framework for optimizing warehouse storage assignment using the distribution of DoS . • We release a curated version of a large historical dataset of warehousing records that can be used to build and test models that predicts DoS . • We introduce a type of neural network architecture , ParallelNet , that achieves state-of-the-art performance in estimating the DoS distribution . • Most importantly , we present real-life results of implementing the framework with Parallel- Net in live warehouses , and show labor savings by up to 21 % . The structure of the paper is as followed . In Section 2 , we introduce the storage assignment problem formally , and how this leads to estimating DoS in a distributional way . In Section 3 , we develop the storage assignment framework . We would introduce the dataset in Section 4 and Section 5 contains the implementation with ParallelNet , and its results compared to strong baselines . Section 6 shows the computational results , while real-life evidence is provided in Section 7 . 2 SOLVING THE STORAGE ASSIGNMENT PROBLEM . The general storage assignment problem asks for an assignment function that outputs a storage location given a pallet and warehouse state so the total travel time for storage is minimized . We formalize such statement below and show how it naturally leads to the problem of estimating DoS . Let warehouseW have locations labeled { 1 , · · · , N } where the expected travel time from the loading dock to location i is ti . We assume pallets arrive in discrete time periods { 1 , 2 , · · · , T } where T is the lifetime of the warehouse and each of the pallets has an integer DoS in { 1 , · · · , P } . The warehouse uses an assignment functionA that assigns each arriving pallet to an available location in { 1 , · · · , N } . Define nAi ( t ) as the number of pallets ( 0 or 1 ) stored in location i in time period t under assignment function A . Then our optimization problem can be stated as : min A T∑ t=1 N∑ i=1 4tin A i ( t ) ( 1 ) The constant 4 is to allow for the 4 distances traveled during storage/retrieval . Given that future pallet arrivals could be arbitrary , we need additional assumptions to make the theoretical analysis tractable . In particular , we assume the warehouse is in steady state or perfect balance : Definition 1 . Let np ( t ) be the number of pallets arriving at time period t that has a DoS of p. A warehouse is in perfect balance iff for all p and t > p we have np ( t− p ) = np ( t ) . This means that in each time period t > p , the np ( t− p ) outgoing pallets with DoS of p are replaced exactly by incoming pallets with DoS of p , so the number of pallets with DoS of p in the warehouse remains constant at zp = ∑p i=1 np ( i ) for every period t > p. For a real-life large warehouse , this steady state assumption is usually a good approximation except when shock events happen . Under such assumption , ideally we only need zp positions to store all the pallets with DoS of p , or in total∑P p=1 zp locations . Goetschalckx & Ratliff ( 1990 ) showed that the DoS strategy below is optimal : Theorem 1 ( Goetschalckx & Ratliff ( 1990 ) ) . Assume the warehouse is in perfect balance , and P is large enough so that zp ≤ 1 for every p. Define W ( t ) = ∑ p < =t zp∑P p=1 zp as the cumulative distribution of pallet DoS . Let r = ∑P p=1 zp N be the average occupancy rate of the warehouse . Then the optimal solution to ( 1 ) , A∗ , would assign a pallet with DoS of p to the NrW ( p ) th location . In other words , the locations are arranged in increasing order of DoS . We can always make P large enough so that zp ≤ 1 by separating time periods to finer intervals . With Theorem 1 , we can implement A∗ in a real warehouse using three approximations : 1 . We approximate W ( t ) with the historical pallet cumulative DoS distribution Ŵ ( t ) . 2 . We approximate r with the historical average occupancy rate of the warehouse r̂ . 3 . We approximate the DoS p with a prediction p̂ . Both Ŵ ( t ) and r̂ are easily retrieved from the historical data , and N is known . We stress however that Ŵ ( t ) needs to be size-debiased . This is because pallets of DoS p would appear with a frequency ∝ pzp in a period of time , and thus the historical data needs to be corrected to match the definition above , which is∝ zp . This is done through standard theory of size-biased distributions ( Gove ( 2003 ) ) . Therefore , in the next subsection , we would focus on the task of generating a prediction p̂ of the DoS . 2.1 PREDICTING DURATION OF STAY . To utilize Theorem 1 above , we need to predict the DoS of a pallet at its arrival . However , in most real-life warehouses , a product Pi enters the warehouse with multiple zi > 1 identical pallets , which leave at different times ti1 , · · · tizi . Thus , assuming the product came in at time 0 , the DoS of incoming pallets of Pi could be any of the numbers ti1 , · · · tizi , which makes the quantity ill-defined . The uncertainty can further be very large - our collaborating company had an average variance of DoS within a shipment of 10 days when the median DoS was just over 12 days . Therefore , to account for such uncertainty , we would assume that for every shipment S which contains multiple pallets , the DoS of a random pallet follows a cumulative distribution FS ( t ) . Furthermore , we assume that such distribution can be identified using the characteristics XS of the shipment known at arrival . As in , there exists a function g such that : FS ( t ) = g ( XS ) ( t ) for all possible shipments S. This assumption is not as strong as it may seem - the most important variables that affect DoS usually are the time and the good that is being transported , both of which are known at arrival . As a simple example , if the product is ice cream and it is the summer , then we expect the DoS distribution to be right skewed , as ice cream is in high demand during the summer . Moreover , the experimental results in Section 6 are indicative that g exists and is highly estimable . If the above assumption holds true , then we can estimate g using machine learning . Assume we have an optimal estimate g̃ of g relative to a loss function l ( FS , g̃ ( XS ) ) denoting the loss between the actual and predicted distribution . Since by our assumption FS is identified by XS , we can not obtain any further information about DoS relative to this loss function . Thus , for each shipment with features XS , we take p̂ to be a random sample from the distribution F̃S = g̃ ( XS ) . In the next section , we would outline our storage assignment framework based on using such prediction p̂ .

Summary: The paper tackles the problem of assigning pallets to storage positions in a warehouse so as to minimize total travel time. Previous literature has shown that if durations of stay (DoS) is known in advance then the optimal storage position is inversely proportional to this. The paper proposes to predict a DoS distribution for each pallet using a 19 point CDF with a neural network. The authors show that the proposed network is better than a few other neural baselines and that it can predict the DoS reasonably well.