Y. Yu (Yugang)
http://repub.eur.nl/ppl/5989/
List of Publicationsenhttp://repub.eur.nl/eur_logo_new.png
http://repub.eur.nl/
RePub, Erasmus University RepositoryAn exact method for scheduling a yard crane
http://repub.eur.nl/pub/76426/
Sun, 01 Jun 2014 00:00:01 GMT<div>A.H. Gharehgozli</div><div>Y. Yu</div><div>M.B.M. de Koster</div><div>J.T. Udding</div>
This paper studies an operational problem arising at a container terminal, consisting of scheduling a yard crane to carry out a set of container storage and retrieval requests in a single container block. The objective is to minimize the total travel time of the crane to carry out all requests. The block has multiple input and output (I/O) points located at both the seaside and the landside. The crane must move retrieval containers from the block to the I/O points, and must move storage containers from the I/O points to the block. The problem is modeled as a continuous time integer programming model and the complexity is proven. We use intrinsic properties of the problem to propose a two-phase solution method to optimally solve the problem. In the first phase, we develop a merging algorithm which tries to patch subtours of an optimal solution of an assignment problem relaxation of the problem and obtain a complete crane tour without adding extra travel time to the optimal objective value of the relaxed problem. The algorithm requires common I/O points to patch subtours. This is efficient and often results in obtaining an optimal solution of the problem. If an optimal solution has not been obtained, the solution of the first phase is embedded in the second phase where a branch-and-bound algorithm is used to find an optimal solution. The numerical results show that the proposed method can quickly obtain an optimal solution of the problem. Compared to the random and Nearest Neighbor heuristics, the total travel time is on average reduced by more than 30% and 14%, respectively. We also validate the solution method at a terminal.A decision-tree stacking heuristic minimising the expected number of reshuffles at a container terminal
http://repub.eur.nl/pub/76740/
Sat, 03 May 2014 00:00:01 GMT<div>A.H. Gharehgozli</div><div>Y. Yu</div><div>M.B.M. de Koster</div><div>J.T. Udding</div>
Reshuffling containers, one of the daily operations at a container terminal, is time consuming and increases a ships berthing time. We propose a decision-tree heuristic to minimise the expected number of reshuffles when arriving containers should be stacked in a block of containers with an arbitrary number of piles. The heuristic algorithm uses the optimal solutions of a stochastic dynamic programming model. Since the total number of states of the dynamic programming model increases exponentially, the model can only solve small-scale problems in a reasonable time. To solve large-scale problems, the heuristic uses the results of the exact model for small-scale problems to generate generalised decision trees. These trees can be used to solve problems with a realistic number of piles. The numerical experiments show the effectiveness of the algorithm. For small-scale problems, the trees can quickly make optimal decisions. For large-scale problems, the decision-tree heuristic significantly outperforms stacking policies commonly used in practice. Using the decision trees, we can compare the performance of a shared-stacking policy, which allows containers of multiple ships to be stacked on top of each other, with a dedicated-stacking policy. Shared-stacking appears to outperform dedicated-stacking.On the suboptimality of full turnover-based storage
http://repub.eur.nl/pub/76594/
Fri, 15 Mar 2013 00:00:01 GMT<div>Y. Yu</div><div>M.B.M. de Koster</div>
Full turnover-based storage has been widely claimed to outperform the commonly used ABC class-based storage policy in terms of the resulting storage and retrieval travel time. In the first paper describing the full turnover-based policy, simultaneously implicitly assuming shared storage (that is to say storage space that can be used by other products once the original product's inventory is depleted) was modelled, since no specific space was reserved to store the maximum inventory of a product. However, full turnover-based storage is a dedicated storage policy where the storage space allocated to one product must be able to accommodate its maximum inventory level. Since then, many authors have cited the results of the full turnover-based storage policy while often overlooking its incompatible shared-storage assumption. This paper adapts classic travel time models to accommodate full turnover-based dedicated storage. We consider different warehousing system configurations such as square-in-time (SIT) and non-SIT racks, and speed acceleration and deceleration effects to calculate the storage/retrieval (S/R) machine's travel time. Surprisingly, but in line with practice, the results of the adapted travel time models show that random and class-based storage normally outperform full turnover-based storage.Storage policies and optimal shape of a storage system
http://repub.eur.nl/pub/40304/
Tue, 01 Jan 2013 00:00:01 GMT<div>N. Zaerpour</div><div>M.B.M. de Koster</div><div>Y. Yu</div>
The response time of a storage system is mainly influenced by its shape (configuration), the storage assignment and retrieval policies, and the location of the input/output (I/O) points. In this paper, we show that the optimal shape of a storage system, which minimises the response time for single command cycles, is independent of the used storage policy and the location of the I/O point. This means decisions on the shape of the storage system and storage policy can be decoupled. This simplifies system design since at the design stage little information on customer demand is available. The storage system can then be designed for optimality assuming a random storage policy. Such a system optimally accommodates other time-saving storage policies such as class-based or full-turnover-based storage without requiring configuration correction. Aggregated state dynamic programming for a multiobjective two-dimensional bin packing problem
http://repub.eur.nl/pub/37819/
Wed, 01 Aug 2012 00:00:01 GMT<div>Y. Liu</div><div>C. Chu</div><div>Y. Yu</div>
This paper studies a real-life multi-objective two-dimensional single-bin-size bin-packing problem arising in industry. A packing pattern is defined by one bin, a set of items packed into the bin and the packing positions of these items. A number of bins can be placed with the same packing pattern. The objective is not only to minimise the number of bins used, as in traditional bin-packing problems, but also to minimise the number of packing patterns. Based on our previous study of a heuristic stemming from dynamic programming by aggregating states to avoid the exponential increase in the number of states, we further develop this heuristic by decomposing a pattern with a number of bins at each step. Computational results show that this heuristic provides satisfactory results with a gap generally less than 20% with respect to the optimum. A vendor managed inventory supply chain with deteriorating raw materials and products
http://repub.eur.nl/pub/37686/
Sun, 01 Apr 2012 00:00:01 GMT<div>Y. Yu</div><div>Z. Wang</div><div>L. Liang</div>
Fast deteriorating raw materials such as raw milk, fruit and vegetables are commonly used to produce slowly deteriorating finished products such as milk powders, cheeses, and pastas. This paper studies a Vendor Managed Inventory (VMI) type supply chain where the manufacturing vendor decides how to manage the system-wide inventories of its fast deteriorating raw material and its slowly deteriorating product. The decision variables are a common replenishment cycle of the product and the replenishment frequency of the raw material. We assume the deteriorating rates are known constants and every retailers demand is deterministic. We develop an integrated model to calculate the total inventory and deterioration cost for such a system. We prove the convexity of the cost functions, and based on this a golden search algorithm is developed to find the optimal solution of the model. Our numerical results show that the deteriorating rate of the product may increase the total cost by more than 40% compared to the zero-deteriorating rate, while the deteriorating raw material has less impact on the total cost (commonly less than 5% in our numerical examples). This indicates that more attention should be paid to the product than the raw material. Further, an increase in the number of retailers can make the replenishment frequency of the raw material increase significantly but the common replenishment cycle of the product decreases a little. This indicates that adding a new retailer would not be felt strongly by the other retailers but would be felt by the supplier of the raw material. Sequencing heuristics for storing and retrieving unit loads in 3D compact automated warehousing systems
http://repub.eur.nl/pub/37684/
Wed, 01 Feb 2012 00:00:01 GMT<div>Y. Yu</div><div>M.B.M. de Koster</div>
Sequencing unit-load retrieval requests has been extensively reported on in the literature for conventional single-deep automated warehousing systems. A proper sequence can greatly reduce the makespan when carrying out a group of such requests. Although the sequencing problem is NP-hard, some very good heuristics exist. Surprisingly, the problem has not yet been investigated for compact (multi-deep) storage systems, which have greatly increased in popularity the last decade. This article studies how to sequence a group (or block) of storage and retrieval requests in a multi-deep automated storage system with the objective to minimize the makespan. Currently utilized sequencing heuristics for the multi-deep system are adapted in this article and in addition a new heuristic, Percentage Priority to Retrievals with Shortest Leg (PPR-SL), is proposed and evaluated. It is shown that the PPR-SL heuristic consistently outperforms all of the other heuristics. Generally, it can outperform the benchmark First-Come First-Served (FCFS) heuristic by between 20 and 70%. The nearest neighbor heuristic that performs very well in conventional single-deep storage systems appears to perform poorly in the multi-deep system, even worse than FCFS. In addition, based on FCFS and PPR-SL, robust rack dimensions that yield a short makespan, regardless of the number of storage and retrieval requests, are found. An integrated pricing and deteriorating model and a hybrid algorithm for a VMI (vendor-managed-inventory) supply chain
http://repub.eur.nl/pub/30739/
Sat, 01 Oct 2011 00:00:01 GMT<div>Y. Yu</div><div>G.Q. Huang</div><div>Z.H. Liu</div><div>X. Zhang</div>
This paper studies a vendor-managed-inventory (VMI) supply chain where a manufacturer, as a vendor, procures a type of nondeteriorating raw material to produce a deteriorating product, and distribute it to multiple retailers. The price of the product offered by one retailer is also influenced by the prices offered by other retailers because consumers can choose the product from any of the retailers. This paper is one of the first papers that propose an integrated model to study the influence of pricing and deterioration on the profit of such a VMI system. A hybrid approach combining genetic algorithms and an analytical method is developed for efficiently determining the optimal price of the product of each retailer, the inventory policies of the product and the raw material. Our results of a detailed numerical study show that parameters related to the market and deterioration have significant influences on the profit of the VMI system. However, different from common intuition, we find that an increase in the substitution elasticity of the product among different retailers can bring an increase in the retail prices of the product, while the increase of the market scale can reduce the retail prices. Sequencing Heuristics for Storing and Retrieving Unit Loads in 3D Compact Automated Warehousing Systems
http://repub.eur.nl/pub/22722/
Thu, 17 Feb 2011 00:00:01 GMT<div>Y. Yu</div><div>M.B.M. de Koster</div>
Sequencing unit load retrieval requests has been studied extensively in literature for conventional single-deep automated warehousing systems. A proper sequence can greatly reduce the makespan when carrying out a group of such requests. Although the sequencing problem is NP-hard some very good heuristics exist. Surprisingly the problem has not yet been investigated for compact (multi-deep) storage systems, which have greatly increased in popularity the last decade. This paper studies how to sequence a group (or block) of storage and retrieval requests in a multi-deep automated storage system with the objective to minimize the makespan. We adapt well-known sequencing heuristics for the multi-deep system, and propose and evaluate a new heuristic: percentage priority to retrievals with shortest leg (PPR-SL). Our results show the PPR-SL heuristic consistently outperforms all the other heuristics. Generally, it can outperform the benchmark first-come first-served (FCFS) heuristic by 20-70%. The nearest neighbor (NN) heuristic that performs very well in conventional single-deep storage systems, appears to perform poorly in the multi-deep system; even worse than FCFS. In addition, based on FCFS and PPR-SL, we find robust rack dimensions yielding a short makespan, regardless of the number of storage and retrieval requests.Nash game model for optimizing market strategies, configuration of platform products in a Vendor Managed Inventory (VMI) supply chain for a product family
http://repub.eur.nl/pub/19234/
Sat, 16 Oct 2010 00:00:01 GMT<div>Y. Yu</div><div>G.Q. Huang</div>
This paper discusses how a manufacturer and its retailers interact with each other to optimize their product marketing strategies, platform product configuration and inventory policies in a VMI (Vendor Managed Inventory) supply chain. The manufacturer procures raw materials from multiple suppliers to produce a family of products sold to multiple retailers. Multiple types of products are substitutable each other to end customers. The manufacturer makes its decision on raw materials’ procurement, platform product configuration, product replenishment policies to retailers with VMI, price discount rate, and advertising investment to maximize its profit. Retailers in turn consider the optimal local advertising investments and retail prices to maximize their profits. This problem is modeled as a dual simultaneous non-cooperative game (as a dual Nash game) model with two sub-games. One is between the retailers serving in competing retail markets and the other is between the manufacturer and the retailers. This paper combines analytical, iterative and GA (genetic algorithm) methods to develop a game solution algorithm to find the Nash equilibrium. A numerical example is conducted to test the proposed model and algorithm, and gain managerial implications.Large scale stochastic inventory routing problems with split delivery and service level constraints
http://repub.eur.nl/pub/20321/
Thu, 01 Jul 2010 00:00:01 GMT<div>Y. Yu</div><div>C. Chu</div><div>H.X. Chen</div><div>F. Chu</div>
A stochastic inventory routing problem (SIRP) is typically the combination of stochastic inventory control problems and NP-hard vehicle routing problems, which determines delivery volumes to the customers that the depot serves in each period, and vehicle routes to deliver the volumes. This paper aims to solve a large scale multi-period SIRP with split delivery (SIRPSD) where a customer’s delivery in each period can be split and satisfied by multiple vehicle routes if necessary. This paper considers SIRPSD under the multi-criteria of the total inventory and transportation costs, and the service levels of customers. The total inventory and transportation cost is considered as the objective of the problem to minimize, while the service levels of the warehouses and the customers are satisfied by some imposed constraints and can be adjusted according to practical requests. In order to tackle the SIRPSD with notorious computational complexity, we first propose an approximate model, which significantly reduces the number of decision variables compared to its corresponding exact model. We then develop a hybrid approach that combines the linearization of nonlinear constraints, the decomposition of the model into sub-models with Lagrangian relaxation, and a partial linearization approach for a sub model. A near optimal solution of the model found by the approach is used to construct a near optimal solution of the SIRPSD. Randomly generated instances of the problem with up to 200 customers and 5 periods and about 400 thousands decision variables where half of them are integer are examined by numerical experiments. Our approach can obtain high quality near optimal solutions within a reasonable amount of computation time on an ordinary PC.Performance evaluation of dynamic scheduling approaches in vehicle-based internal transport systems
http://repub.eur.nl/pub/20308/
Mon, 01 Feb 2010 00:00:01 GMT<div>T. Le-Anh</div><div>M.B.M. de Koster</div><div>Y. Yu</div>
is paper studies the performance of static and dynamic scheduling approaches in vehicle-based internal transport (VBIT) systems and is one of the first to systematically investigate under which circumstances, which scheduling method helps in improving performance. In practice, usually myopic dispatching heuristics are used, often using look-ahead information. We argue more advanced scheduling methods can help, depending on circumstances. We introduce three basic scheduling approaches (insertion, combined and column generation) for the static problem. We then extend these to a dynamic, real-time setting with rolling horizons. We propose two further real-time scheduling approaches: dynamic assignment with and without look-ahead. The performances of the above five scheduling approaches are compared with two of the best performing look-ahead dispatching rules known from the literature. The performance of the various approaches depends on the facility layout and work distribution. However, column generation, the combined heuristic, and the assignment approach with look-ahead consistently outperform dispatching rules. Column generation can require substantial calculation time but delivers very good performance if sufficient look-ahead information is available. For large scale systems, the combined heuristic and the dynamic assignment approach with look ahead are recommended and have acceptable calculation times.Linearization and Decomposition Methods for Large Scale Stochastic Inventory Routing Problem with Service Level Constraints
http://repub.eur.nl/pub/18041/
Sat, 23 Jan 2010 00:00:01 GMT<div>Y. Yu</div><div>C. Chu</div><div>H.X. Chen</div><div>F. Chu</div>
A stochastic inventory routing problem (SIRP) is typically the combination of stochastic inventory control problems and NP-hard vehicle routing problems, for a depot to determine delivery volumes to its customers in each period, and vehicle routes to distribute the delivery volumes. This paper aims to solve a large scale multi-period SIRP with split delivery (SIRPSD) where a customer’s delivery in each period can be split and satisfied by multiple vehicles if necessary. The objective of the problem is to minimize the total inventory and transportation cost while some constraints are given to satisfy other criteria, such as the service level to limit the stockout probability at each customer and the service level to limit the overfilling probability of the warehouse of each customer. In order to tackle the SIRPSD with notorious computational complexity, we propose for it an approximate model, which significantly reduces the number of decision variables compared to its corresponding exact model. We develop a hybrid approach that combines the linearization of nonlinear constraints, the decomposition of the model into sub-models with Lagrangian relaxation, and a partial linearization approach for a sub model. A near optimal solution of the model can be found by the approach, and then be used to construct a near optimal solution of the SIRPSD. Numerical examples show that, for an instance of the problem with 200 customers and 5 periods that contains about 400 thousands decision variables where half of them are integer, our approach can obtain high quality near optimal solutions with a reasonable computational time on an ordinary PC.On the Suboptimality of Full Turnover-Based Storage
http://repub.eur.nl/pub/16898/
Wed, 07 Oct 2009 00:00:01 GMT<div>Y. Yu</div><div>M.B.M. de Koster</div>
In the past thirty years the full turnover-based storage policy as described by Hausman et al. (1976, Management Science 22(6)) has been widely claimed to outperform the commonly used ABC class-based storage policy, in terms of the resulting average storage and retrieval machine travel time. In practice however, ABC storage is the dominant policy. Hausman et al. (1976) model the turnover-based policy under the unrealistic assumption of shared storage, i.e. the storage space allocated to one product can only accommodate its average inventory level; no specific space is reserved to store the maximum inventory of a product. It appears that many authors citing Hausman et al.’s results overlook this assumption and use the resulting storage and retrieval machine travel times as if it were valid for full turnover-based storage. Full turnover-based storage is a dedicated storage policy where the storage space allocated to one product must accommodate its maximum inventory level. This paper adapts the travel time model of Hausman et al. to accommodate full turnover-based dedicated storage. Surprisingly, the result of the adapted travel time model is opposite to that of Hausman et al. (1976) but, in line with practice, it supports that ABC (2- or 3-) class-based storage normally outperforms full turnover-based storage.Stackelberg game-theoretic model for optimizing advertising, pricing and inventory policies in vendor managed inventory (VMI) production supply chains
http://repub.eur.nl/pub/76565/
Sat, 01 Aug 2009 00:00:01 GMT<div>Y. Yu</div><div>G.Q. Huang</div><div>L. Liang</div>
This paper discusses how a manufacturer and its retailers interact with each other in order to optimize their individual net profits by adjusting product marketing (advertising and pricing) and inventory policies in an information-asymmetric VMI (vendor managed inventory) supply chain. The manufacturer produces and supplies a single product at the same wholesale price to multiple retailers who then sell the product in dispersed and independent markets at retail prices. The demand rate in each market is an increasing and concave function of the advertising investments of both local retailers and the manufacturer, but a decreasing and convex function of the retail prices. The manufacturer determines its wholesale price, its advertising investment, replenishment cycles for the raw materials and finished product, and backorder quantity to maximize its profit. Retailers in turn consider the replenishment policies and the manufacturer's promotion policies and determine the optimal retail prices and advertisement investments to maximize their profits. This problem is modeled as a Stackelberg game where the manufacturer is the leader and retailers are followers. An algorithm has been proposed to search the Stackelberg equilibrium. A numerical study has been conducted to demonstrate how the algorithm works and to understand the influences of decision variables and/or parameters. Several research questions are examined, including under what circumstances the retailers and manufacturer should increase their advertising expenditures and/or reduce the retail prices and what actions should be taken if the prices of raw materials or their holding costs increase.Nash Game Model for Optimizing Market Strategies, Configuration of Platform Products in a Vendor Managed Inventory (VMI) Supply Chain for a Product Family
http://repub.eur.nl/pub/15029/
Mon, 02 Mar 2009 00:00:01 GMT<div>Y. Yu</div><div>G.Q. Huang</div>
This paper discusses how a manufacturer and its retailers interact with each other to optimize their product marketing strategies, platform product configuration and inventory policies in a VMI (Vendor Managed Inventory) supply chain. The manufacturer procures raw materials from multiple suppliers to produce a family of products sold to multiple retailers. Multiple types of products are substitutable each other to end customers. The manufacturer makes its decision on raw materials’ procurement, platform product configuration, product replenishment policies to retailers with VMI, price discount rate, and advertising investment to maximize its profit. Retailers in turn consider the optimal local advertising and retail price to maximize their profits. This problem is modeled as a dual simultaneous non-cooperative game (as a Nash game) model with two sub-games. One is between the retailers serving in competing retail markets and the other is between the manufacturer and the retailers. This paper combines analytical, iterative and GA (genetic algorithm) methods to develop a game solution algorithm to find the Nash equilibrium. A numerical example is conducted to test the proposed model and algorithm, and gain managerial implications.A Stackelberg game and its improvement in a VMI system with a manufacturing vendor
http://repub.eur.nl/pub/13660/
Sun, 01 Feb 2009 00:00:01 GMT<div>Y. Yu</div><div>F. Chu</div><div>H.X. Chen</div>
Vendor managed inventory (VMI) is an inventory management strategy to let a vendor manage his retailers’ inventories, which makes the vendor have the opportunity to obtain some inventory and market-related information of his retailers. This paper discusses how the vendor can take advantage of this information for increasing his own profit by using a Stackelberg game in a VMI system. The vendor here is a manufacturer who procures raw materials to produce a finished product and supplies it at the same wholesale price to multiple retailers. The retailers then sell the product in independent markets at retail prices. Solution procedures are developed to find the Stackelberg game equilibrium that each enterprise is not willing to deviate from for maximizing his own profit. The equilibrium makes the manufacturer benefited, and the retailers’ profits maximized. The equilibrium can then be improved for further benefiting the manufacturer and his retailers if the retailers are willing to cooperate with the manufacturer by using a cooperative contract. Finally, a numerical example and the corresponding sensitivity analysis are given to illustrate that: (1) the manufacturer can benefit from his leadership, and monopolize the added profit of the VMI system in some cases; (2) The manufacturer can further improve his own profit, and then the retailers’ profits by the cooperative contract, as compared to the Stackelberg equilibrium; (3) market and raw material related parameters have significant influence on every enterprise’s net profit.Open Location Management in Automated Warehousing Systems
http://repub.eur.nl/pub/14615/
Fri, 30 Jan 2009 00:00:01 GMT<div>Y. Yu</div><div>M.B.M. de Koster</div>
A warehouse needs to have sufficient open locations to be able to store incoming shipments of various sizes. In combination with ongoing load retrievals open locations gradually spread over the storage area. Unfavorable positions of open locations negatively impact the average load retrieval times. This paper presents a new method to manage these open locations such that the average system travel time for processing a block of storage and retrieval jobs in an automated warehousing system is minimized. We introduce the effective storage area (ESA), a well-defined part of the locations closest to the depot; where only a part of the open locations –the effective open locations-, together with all the products, are stored. We determine the optimal number of effective open locations and the ESA boundary minimizing the average travel time. Using the ESA policy, the travel time of a pair of storage and retrieval jobs can be reduced by more than 10% on average. Its performance depends hardly on the number or the sequence of retrievals. In fact, in case of only one retrieval, applying the policy leads already to beneficial results. Application is also easy; the ESA size can be changed dynamically during storage and retrieval operations.Optimal zone boundaries for two-class-based compact three-dimensional automated storage and retrieval systems
http://repub.eur.nl/pub/15031/
Tue, 20 Jan 2009 00:00:01 GMT<div>Y. Yu</div><div>M.B.M. de Koster</div>
Compact, multi-deep three-dimensional (3D), Automated Storage and Retrieval Systems (AS/RS) are becoming more common, due to new technologies, lower investment costs, time efficiency and compact size. Decision-making research on these systems is still in its infancy. This paper studies a particular compact system with rotating conveyors for the depth movement and a Storage/Retrieval (S/R) machine for the horizontal and vertical movement of unit loads. The optimal storage zone boundaries are determined for this system with two product classes: high- and low-turnover, by minimizing the expected S/R machine travel time. We formulate a mixed-integer non-linear programming model to determine the zone boundaries. A decomposition algorithm and a one-dimensional search scheme are developed to solve the model. The algorithm is complex, but the results are appealing since most of them are in closed-form and easy to apply to optimally layout the 3D AS/RS rack. The results show that the S/R machine travel time is significantly influenced by the zone dimensions, zone sizes and ABC curve skewness (presenting turnover patterns of different products). The presented results are compared with those under random storage and it is shown that significant reductions of the machine travel time are obtainable by using class-based storage.Designing an optimal turnover-based storage rack for a 3D compact automated storage and retrieval system
http://repub.eur.nl/pub/20982/
Thu, 01 Jan 2009 00:00:01 GMT<div>Y. Yu</div><div>M.B.M. de Koster</div>
Compact, multi-deep (3D) automated storage and retrieval systems (AS/RS) are becoming increasingly popular for storing products. We study such a system where a storage and retrieval (S/R) machine takes care of movements in the horizontal and vertical directions of the rack, and an orthogonal conveying mechanism takes care of the depth movement. An important question is how to layout such systems under different storage policies to minimize the expected cycle time. We derive the expected single-command cycle time under the full-turnover-based storage policy and propose a model to determine the optimal rack dimensions by minimizing this cycle time. We simplify the model, and analytically determine optimal rack dimensions for any given rack capacity and ABC curve skewness. A significant cycle time reduction can be obtained compared with the random storage policy. We illustrate the findings of the study by applying them in a practical example