Interleaving models for the analysis of twin shuttle automated storage and retrieval systems

A system of state equations is formulated for describing the dynamic behaviour of the queue of storage and retrieval transactions in a storage aisle served by twin shuttle storage and retrieval (S/R) machines. Using estimates of order picking cycle times in conjunction with storage and retrieval transaction demand levels, an estimate of the probability distribution of storage queue states is derived for a given storage aisle. System performance measures under an opportunistic interleaving discipline including S/R machine utilization, transaction queue lengths, average cycle times and expected transaction waiting times are approximated using the state probability distribution. The modelling tools are demonstrated using sample problems where the performance of twin versus single shuttle systems is contrasted for alternative combinations of storage and retrieval machine travel speeds, storage rack configurations, and transactions demand scenarios.     

A discrete-time queueing network approach to performance evaluation of autonomous vehicle storage and retrieval systems

In this paper, we present a method for performance evaluation of autonomous vehicle storage and retrieval systems (AVS/RSs) with tier-captive single-aisle vehicles. A discrete-time open queueing network approach is applied. The data obtained from the evaluation of the lift and vehicle movements can be used directly as input for the general discrete service time distributions of the queueing network. Furthermore, the approach allows for the computation of the retrieval transaction time distribution as well as of the distribution of the number of transactions waiting to be stored. Consequently, not only expected values and variances but also quantiles of the performance measures can be obtained. Comparison to discrete-event simulation quantifies approximation errors resulting from the decomposition approach in the discrete-time domain. Moreover, the errors obtained by the discrete-time approach are compared to the errors obtained using a continuous-time open queueing network approach. Finally, it will be outlined how the model can be used for designing AVS/RSs according to given system requirements, such as storage capacity, throughput, height and length of the system as well as the 95% quantile of the retrieval transaction time.     

Heavy traffic analysis of a single vehicle loop in an automated storage and retrieval system

We consider an automated storage/retrieval system in which cargo moves between the storage/retrieval machines and the system entrance/exit stations through a single automated vehicle loop. Past studies indicated that the cargo waiting time in the loop is affected by the dispatching rules, which govern the sequence of the cargo to be handled. In this paper, we show that the loop configuration, which has received little research attention, also has a big impact on the cargo waiting time. When the first-come-first-served dispatching rule is used, we derive the relationship between the number of stations and the ratio of the average cargo retrieval time to the average cargo storage time. When the first-encountered-first-served dispatching rule is used, we show that even the arrangement of the input channel and the output channel of a station can have significant impact on the cargo waiting time. Furthermore, we derive a formula for the vehicle visit rate for each station under heavy traffic conditions. This formula helps to explain the phenomenon that the waiting times at different stations can be very different even when the loop is symmetrically designed and the cargo arrival rates to the stations are similar. In addition to analytical models, we use simulations to evaluate the performance of different loop configurations. Our research suggests that a substantial improvement can be achieved by making proper adjustments to the loop configuration.     

A decision model to determine the number of shuttles in a tier-to-tier SBS/RS

Shuttle-based storage and retrieval systems (SBS/RSs), which are designed to increase throughput capacity and flexibility, are a type of automated storage and retrieval system used for lightweight loads. SBS/RSs can increase throughput capacity by using multiple shuttles and elevators as storage and retrieval machines (SRMs). They can also facilitate improvements in flexibility since they are able to adjust the number of SRMs according to transaction demands. Thus, determining the number of shuttles is an important issue in tier-to-tier SBS/RSs. In this paper, a decision model to determine the number of shuttles is proposed. The model is based on the travel time model, and it considers parameters such as the physical configuration, velocity profile and the probability that the shuttle operates a dual command. Finally, the throughput capacity from the travel time model is compared with that from a simulation-based approach in order to verify the effects of the model. In addition, a critical discussion regarding the characteristics of the tier-to-tier system is provided.     

Performance and cost analysis of a Geom/G/1 (G, SV) system

In this paper, a gate service Geom/G/1 queue system with single vacation is analyzed. Firstly, a gate service policy and single vacation are introduced in detail, the expected number of customers in the system at the beginning instant of a service period is obtained. Next, the Probability Generating Function (P.G.F.) of the stationary queue length is obtained by using regeneration cycle approach. Then the P.G.F. of the waiting time is derived under the condition of the independence between the arrival process and the waiting time. Moreover, the P.G.Fs. of the service period, the vacation period and the idle period are obtained, and the mean length of three periods are also obtained. Cost model is formulated to determine the optimal expected vacation length. Finally, based on numerical results, the relations of the performance measures and traffic intensity are discussed.     

Expected travel time and optimal boundary formulas for a two-class-based automated storage/retrieval system

Our paper studies a two-class-based rectangular-in-time automated storage/ retrieval system (AS/RS). We present explicit formulas for the optima! boundary of the two storage areas as well as for the expected single command cycle time for an optimally designed rack. In the basic model each crane handles a single aisle. These formulas provide the designer with a full picture of the quantitative effects of the various factors (i.e., access frequencies of the two storage areas, and dimensions of the rack) on the optimal boundary of the two storage areas and the achievable cycle lime in the warehouse. We also develop expected travel time formulas for the dual command AS/RS with two-class-based storage policies and obtain the optimal boundary with a one-dimensional search procedure. Similar developments (i.e., expected travel time formulas and optimal boundary search procedures) are discussed for AS/RS with a single command policy but with each crane handling multiple aisles.     

三维紧致化存储系统中出入口位置的优化研究

目的解决三维紧凑型存储系统的设计问题,寻求系统出入口的最佳位置。方法建立数学模型,并优化求解,研究自动化仓储系统出入口位置的设计问题。结果在双命令存储模式下,当三维紧致化仓储系统的系统出入口位置在货架底端1/2位置时,双命令行程期望时间最少。结论通过假设出入口的位置,列出共计24种双命令下存取货物的期望时间模型,并进行了求导分析,发现在双命令存储模式下系统出入口最优点即为货架底端中点处。     

Travel-time models and fill-grade factor analysis for double-deep multi-aisle AS/RSs

Double-deep multi-aisle automated storage/retrieval systems are increasingly applied for storing and retrieving unit loads, with advantages of increased space utilisation, reduced number of aisles and improved efficiency of storage rack (S/R) machines. In such systems, the retrieval process may consist of the rearrangement of blocking loads, based on the assumptions of uniformly distributed storage locations and random storage policy. We formulate analytical travel-time models of both single- and dual-command cycles under three rearrangement rules. We validate the analytical travel-time models by simulation and conduct numerical experiments to analyse the effect of the number of aisles an S/R machine serves, the fill-grade factor and the command cycles on the expected travel time of the S/R machine. The results show that the expected travel time of the S/R machine is increasing with the increase in the number of aisles an S/R machine serves and the increase in the fill-grade factor, and dual command cycle outperforms single-command cycle in terms of cycle time. To deal with the trade-off between the storage space cost and the operational cost of the S/R machine, we develop a decision model for finding an optimal fill-grade factor to minimise the total cost. We find the condition when an optimal fill-grade factor exists and show how to calculate it. Based on the decision model, we compare the performance of double-deep multi-aisle automated storage/retrieval system (AS/RSs) and single-deep single-aisle AS/RSs. The results show that double-deep multi-aisle AS/RSs outperform single-deep single-aisle AS/RSs in terms of total cost, although double-deep multi-aisle AS/RSs need more storage locations.     

Travel time models for double-deep automated storage and retrieval systems

In this paper new analytical travel time models for the computation of cycle times for unit-load double-deep automated storage and retrieval systems (in continuation double-deep AS/RS) are presented. The proposed models consider the real operating characteristics of the storage and retrieval machine and the condition of rearranging blocking loads to the nearest free storage location during the retrieval process. With the assumption of the uniform distributed storage rack positions and the probability theory, the expressions of the single and modified dual command cycle have been determined. The proposed models enable the calculation of the mean cycle time for single and dual command cycles, from which the performance of the double-deep AS/RS can be evaluated. A simulation model of the selected double-deep AS/RS has been developed to compare the performances of the proposed analytical travel time models. The numerical analyses show that with regard to the examined type of double-deep AS/RS with a different fill-grade factor, the results of the proposed analytical travel time models correlate with the results of simulation models of double-deep AS/RS.     

自动存取系统及其仿真分析

自动存取系统是一种新的存取系统,其效率取决于库位地址的分配方式、存取方式以及堆垛机的运行方式等特性。自动存取系统相对于传统存取方式有优点也有缺点。其效率主要采用通过率来评价。对此,提出了用数学模型的方法来仿真分析,给出了模型的一些相应参数、重要假设和理论结果。最后,对一个实例运用Ａｗｅｓｉｍ软件仿真,给出仿真结果。     

Relative pre-positioning of storage/retrieval machines in automated storage/retrieval system to minimize expected system response time

This paper is focused on the control problem of determining the dwell point of an idle storage/retrieval (S/R) machine in an automated S/R system to improve the overall system service level as measured by the turnaround time of requests to the system. The emphasis of the paper is on the minimization of the expected response time over all retrieval and storage requests executed by the system. The model presented is developed under the operating condition that each machine serves several dedicated aisles of the system. The aisles are assumed to exist in the same region of the automated storage/retrieval system (AS/RS) and thus form a valid order S/R zone. A solution algorithm for determining the optimal dwell point location is developed. The paper assumes a dynamic environment where orders arrive over time and are to be serviced by the S/R machine. A dwell point or strategic point to pre-position the machine is to be determined whenever the machine becomes idle in anticipation of an incoming service request. A numerical example is provided to demonstrate how the technique can be used in an actual production setting to improve the overall order turnaround time.     

带有止步和中途退出的M/M/1/N多重休假排队系统的等待时间

本文研究了带有止步和中途退出的多重休假排队系统模型,该排队模型的稳态概率已在我们的另一篇文章中给出.本文的目的是在稳态概率的基础上推导系统的稳态等待时间分布.通过对进入系统的顾客可能中途退出的不同情况的概率分析,我们分别在服务员忙和休假的情况下得到了进入系统并接受服务的顾客的稳态条件等待时间分布,进而得到了进入系统并接受服务的顾客的稳态条件等待时间的概率密度.     

Travel-Time Models for Automated Storage/Retrieval Systems

Travel-time models are developed for automated storage/retrieval (AS/R) machines. The S/R machine travels simultaneously horizontally and vertically as it moves along a storage aisle. For randomized storage conditions expected travel times are determined for both single and dual command cycles. Alternative input/output (I/O) locations are considered. Additionally, various dwell-point strategies for the storage/retrieval machine are examined.     

Multi-shuttle automated storage/retrieval systems

Multi-shuttle automated storage/retrieval systems have been developed for use in factories and distribution centers because they are more efficient than single-shuttle systems (owing to less empty travel). This improved efficiency results in more agile support (flexible response, less waiting time, etc.) for the production system the storage/retrieval system serves. In this paper we develop analytical models to estimate the throughput in multi-shuttle systems. Throughput improvements greater than 100% are illustrated when triple-shuttle systems are compared with single-shuttle systems.     

Simulation-based regression analysis for the rack configuration of an autonomous vehicle storage and retrieval system

In this paper, a simulation-based regression analysis for the rack configuration of an autonomous vehicle storage and retrieval system (AVS/RS) is presented. The aim of this study is to develop mathematical functions for the rack configuration of an AVS/RS that reflects the relationship between the outputs (responses) and the input variables (factors) of the system under various scenarios. In the regression model, we consider five outputs: the average cycle time of storage and retrieval transactions, the average waiting time for vehicle transactions, the average waiting time of vehicles (transactions) for the lift, the average utilisation of vehicles and the average utilisation of the lifts. The input variables are the number of tiers, aisles and bays that determine the size of the warehouse. Thirty regression models are developed for six warehouse scenarios. The simulation model of the system is developed using ARENA 12.0 commercial software and the statistical analyses are completed using MINITAB statistical software. Two different approaches are used to fit the regression functions–stepwise regression and the best subsets. After obtaining the regression functions, we optimise them using the LINGO software. We apply the approach to a company that uses AVS/RS in France.     

一般限量服务M/G/1型单重休假排队系统的随机分解

本文研究了M／G／1型一般限量服务单重休假排队系统,根据稳态下M／G／1型非空竭服务休假排队系统的队长的随机分解的一般结构,提出简便算法一结构分析法,并且在先到先服务规则下(FCFS)求得稳态下队长的随机分解的概率母函数(GFP)和服务时间的随机分解的Laplace变换(LST),还充分地讨论了系统平稳的条件及其概率含义。     

Optimal storage rack design for a 3-dimensional compact AS/RS

In this paper, we consider a newly designed compact three-dimensional automated storage and retrieval system (AS/RS). The system consists of an automated crane taking care of the pallets' movements in the horizontal and vertical direction. A gravity or powered conveying mechanism takes care of the pallets’ depth movement in the rack. Our research objective is to analyze the system performance and optimally dimension the system. For single-command cycles, the crane's expected retrieval travel time is the same for gravity and powered conveyors; we give a closed-form expression. From the expected travel time, we calculate the optimal ratio between three dimensions that minimizes the travel time for a random storage strategy. In addition, we derive an approximate travel time expression for dual command cycles for the system with powered and gravity conveyors, respectively, and use it to optimize the system dimensions. Finally, we illustrate the findings of the study by a practical example.     

Optimal routing in an automated storage/retrieval system with dedicated storage

We address the sequencing of requests in an automated storage/retrieval system with dedicated storage. We consider the block sequencing approach, where a set of storage and retrieval requests is given beforehand and no new requests come in during operation. The objective for this static problem is to find a route of minimal total travel time in which all storage and retrieval requests may be performed. The problem of sequencing a list of retrievals is equivalent to the Traveling Salesman Problem (TSP), and thus NP-hard in general. We show that the special case of sequencing under the dedicated storage policy can be solved in polynomial time. The results apply to systems with arbitrary positions of the input and output stations. Tlus generalizes the models in the literature, where only combined input/output stations are considered. Furthermore we identify a single command area in the rack. At the end we evaluate the model against heuristic procedures.     

Analytical Results for Miniload Throughput and the Distribution of Dual Command Travel Time

We determine the throughput, the maximum rate at which a miniload automated storage and retrieval system can process requests, as a'function of the dimension of the storage racks, the speed of the s/r machine and the speed of die picker. As a byproduct, we determine the distribution function of me travel time for a storage/retrieval machine that stores one container and retrieves another container (“dual command cycle”) before returning to the I/O point. Our emphasis is on obtaining closed-form, computable expressions.     

On the analysis of single server finite queue with state dependent arrival and service processes: M(n)/G(n)/1/K

This paper discusses the analysis of a single server finite queue with Poisson arrival and arbitrary service time distribution, wherein the arrival rates are state dependent which are all distinct or all equal, service times are conditioned on the system length at the moment of service initiation. The analytic analysis of the queue is carried out and the final results have been presented in the form of recursive equations which can be easily programmed on any PC to obtain the distributions of number of customers in the system at arbitrary, departure and pre-arrival epochs. It is shown that the method works for all service time distributions including the non-phase type and also for low and high values of the model parameters. Some performance measures, and relations among the state probabilities at arbitrary, departure and pre-arrival epochs are also discussed. Furthermore, it is shown that results for a number of queueing models can be obtained from this model as special cases. To demonstrate the effectiveness of our method some numerical examples have been presented.