Optimizing replenishment base on order structure in crane & shuttle based storage and retrieval system

: a new automatic warehouse sorting system, the crane & shuttle based storage and retrieval system (C&SBS/RS), is proposed in this paper. In C&SBS/RS, the crane-based storage and retrieval system (CBS/RS) in the pallet storage area provides the pallets picking, the shuttle-based storage and retrieval system (SBS/RS) in the tote storage area handles the cases and items picking. When the inventory in SBS/RS is lower than safety stock, SBS/RS initiates replenishment transaction. Besides, the order matrix is proposed to study order structure parameters, such as the order density, the order strength, the wave size and so on. Moreover, this paper analyzes the influence of order structure on the replenishment with four evaluation parameters, e.g., the workload of CBS/RS and SBS/RS, the number of used storage position in SBS/RS, the replenishment time. Numerical experiments are carried out to analysis the impact of the wave size and the proportion of high turnover SKU on those four evaluation parameters under multiple order structure, which is to help warehouse operation manager decide replenishment strategy parameters.


Introduction
The crane & shuttle based storage and Therefore, analyzing the effect of the order structure of this wave (e.g., the size of order wave, the density of order wave, the strength of order wave) on the replenishment efficiency becomes meaningful.  [1] provided an overview on the robotized and automated warehouse system, such as the crane-based storage and retrieval system, shuttle-based compact storage and retrieval system and robotic mobile fulfilment system, is a more comprehensive introduction about automated warehouse so far.
They focused on the research issues and modelling solutions related to the system analysis, design optimization and operations planning and control. Studies on AS/RS can be traced back to 1976 [2] , Hausman et al. studied the storage assignment and proposed that the class-based turnover assignment was the optimal storage assignment in AS/RS. Since then, research on AS/RS has gained momentum and hundreds of papers have been published.
In recent years, SBS/RS, as one type of autonomous vehicle-based storage and retrieval system (AVS/RS), has become very popular in practice and research. Malmborg [7][8][9] studied the storage and retrieval cycle time, system utilization, and throughput capacity for the tier-to-tier AVS/RS by optimization model. Fukunari and Malmborg [10] , Roy et al. [11][12] and Ekren et al. [13][14] , used queuing network to model AVS/RS and estimated resource utilization and throughput. Moreover, there is another type AVS/RS named tier-captive AVS/RS, where vehicles are dedicated to a single tier and cannot be transferred to another. Marchet et al. [15] 错误 ! 未找到引用源。 , Wang et al. [16] , Zou et al. [17] and Tappia et al. [18] proposed applicable queuing network to estimate the performance of the system and compare operation strategies.
However, previous studies hardly considered the impact of the order structure. In fact, different order structures are suitable for different automatic picking systems and operation strategies. SHEN [19] studies the order structure with the grid method to comparing sequential zoning and simultaneous zoning based on order cluster. WANG [20] studied the applicability analysis of AS/RS and Carousel system with different order structures.
The remainder of this paper is as follows.
we first describe the replenishment transaction

Main assumption
The assumption of C&SBS/RS in this study are listed below: • As most previous contribution (e.g. Ekren et al. [14] ; Lerher et al. [21] ), the system runs with random storage and retrieval policy.
Under the policy, the retrieval transaction and storage transaction can be assigned to any storage position with the same probability.
• We assume that the equipment in C&SBS/RS operates in dual-command cycles, i.e., a storage transaction and retrieval transaction in each cycle.
• The dwell point policy of cranes, RGVs, shuttles, and lifters follows the point-ofservice-completion (POSC); • The transporters manage the transaction queue in the first-come, first-served (FCFS) discipline.
• One storage position contains one pallet or one tote. Each pallet or tote can contain many cases depending on the SKU package size but only hold one SKU type.
• We assume that there is enough stock in CBS/RS to satisfy replenishment, i.e., there is no replenish to CBS/RS from outside 2.3 Main notation The main notation of this study are as follows:

Analyzing throughput of CBS/RS
Throughput of the CBS/RS represents the number of pallets that the system can retrieve per time unit. As in Marchet et al. [15] research, throughput is defined by the bottleneck of the system (i.e., the throughput of the system equals to the throughput of bottlenecks), which is related to the rack configuration and mechanical properties.
Throughput of CBS/RS is calculated as follows: Where ⅰ ( ) and ⅰ Where ( ) and ℎ ( ) represent the vertical and horizontal travel distance of cranes: The dual-command cycle of RGV is divided into 3 Replenishment strategy

Order structure
We define the order matrix to descript the order structure according to SHEN [19] . The order matrix is combined by all orders in one wave, rows of order matrix represent the order and columns represent the SKU as shown in Where, ∈ {1,2, ⋯ , } represents the ith order ∈ {1,2, ⋯ , } represents the jth SKU represents the requested quantity of the jth SKU in the ith order The total request quantity of the jth SKU in this wave is: The request frequency matrix is defined by the following: Where, ∈ {1,2, ⋯ , } represents the ith order, ∈ {1,2, ⋯ , } represents the jth SKU, represents whether the jth SKU is requested in the ith order.
Therefore, the total request frequency of the jth SKU in this wave is: The following is several indicators to reflect the order structure: i. The size of wave represents the total number of orders and equals to ; ii. The density of order represents the number range of request SKU in one order and is calculated by: iii. The strength of order represents the request quantity rage of one SKU in each order and equals to Then the order matrix can be represented by the above three indicators:

Replenishment strategy
In Step3: SKU main-data matching. SKU main date includes the full quantity of a tote, the full quantity of a pallet and the SKU type.
The SKU type is divided according to the turnover rate for a while (e.g., one year, one month, one quarter). The SKU type matrix is defined by the following: Where, ∈ {1,2, ⋯ , } represents the jth SKU, represents whether the jth SKU belongs to the high turnover type. Hence the workload of replenishment (i.e., the number of replenishment task) can be calculated as follows: , and , represent the workload of replenishment in CBS/RS and SBS/RS respectively and can be calculated by using equation (33)  Hence it is obvious that the optimal object is the minimum replenishment time.      The effect of order strength is shown in Fig 4~Fig 6. The following observations on the replenishment transaction of C&SBS/RS can be made based on these results: (1). For replenishment workload of CBS/RS, as the number of high turnover SKU increases, the replenishment workload of CBS/RS decreases; as the wave size increases, the extent of reduction becomes larger; as the wave size decreases, the replenishment workload of CBS/RS decreases; as the number of high turnover SKU increases, the extent of reduction becomes smaller; as the order strength increases, the replenishment workload of CBS/RS increases, whose extent of reduction decreases due to the decrease of the wave size; the extent of reduction of the replenishment workload of CBS/RS decreases with result of increasing of the number of high turnover SKU; as the order density increases, the replenishment workload of CBS/RS increases; when the order strength is small, the extent of reduction of the replenishment workload of CBS/RS increases due to the decrease in wave size, the extent of reduction of the replenishment workload of CBS/RS increases with result of increasing of the number of high turnover SKU; when the order intensity is larger, the extent of reduction of the replenishment workload of CBS/RS decreases due to the decrease in wave size, the extent of reduction of the replenishment workload of CBS/RS increases with result of increasing of the number of high turnover SKU.
(2). For replenishment workload of SBS/RS, as the number of high turnover SKU increases, the replenishment workload of SBS/RS has increased slightly, with little change overall; as the wave size decreases, the replenishment workload of SBS/RS decreases; as the order strength increases, the replenishment workload of SBS/RS increases, whose extent of reduction increases due to the decrease of the wave size; as the order density increases, the replenishment workload of SBS/RS increases; when the order strength and density are small, the extent of reduction of the replenishment workload of SBS/RS increases due to the number of high turnover SKU increases; conversely, the extent of reduction of the replenishment workload of SBS/RS decreases with the result of increasing of the number of high turnover SKU in a bigger order strength and density scenario.
(3). For the number of used storage position, as the number of high turnover SKU increases, the number of used storage position increases, whose extent of reduction decreases due to the order size increases; as the wave size decreases, the number of used storage position decreases, whose extent of reduction is almost independent of the wave size; as the order strength increases, the number of used storage position increases, whose extent of reduction increases due to the decrease of the wave size and decreases with the result of increasing of the number of high turnover SKUs; as the order density increases, the number of used storage position increases, whose extent of increase decreases.
(4). For the replenishment time, the replenishment time changes slightly when the order density is smaller. Conversely, if the order density increases, the number of high turnover SKU has a significant impact on the replenishment time, while the size of the wave does not. The result shows that as the increase of order strength, the changing trend of replenishment time is more and more tended to the changing trend of replenishment workload of SBS/RS, because of the replenishment time of SBS/RS is more than it in CBS/RS.

Conclusions
We introduce a new automatic warehouse sorting system, i.e., C&SBS/RS, and mainly study the system's replenishment strategy. The throughput model is established by analyzing the working process and the system structure. Then the order matrix is established to study the influence of order structure on replenishment strategy. In the experiment study, we analyze the impact of order size and order density on relevant parameters (e.g., the wave size and the proportion of high turnover SKU) in replenishment strategy. The warehouse operation manager can reduce the wave size and increase the proportion of high turnover SKU to minimize the replenishment time, which can increase the picking throughput of the system.
In additionally, the order structure can be explored in the future. Some rules and indicators can be described by order density, order strength, order size and other parameters to guide the determination of relevant parameters of the replenishment strategy.    effect of γ=U [1,50]