Matrix-geometric solution for semi-open queuing network model of autonomous vehicle storage and retrieval system

https://doi.org/10.1016/j.cie.2013.12.002Get rights and content

Highlights

  • We could estimate the key performance measures of AVS/RS using the MGM.

  • Our extended procedure for aggregation of stations also works reasonably well.

  • This procedure can be used to develop a tool for performance calculations of AVS/RS.

Abstract

In this paper, we model the autonomous vehicle storage and retrieval system (AVS/RS) as a semi-open queuing network (SOQN) and apply a matrix-geometric method (MGM) for analyzing it. An AVS/RS is an automated material handling system for the high-rise pallet storage area of a warehouse and allows pallets to be stored and retrieved quickly and efficiently from their storage locations. It is an alternative to the traditional crane-based AS/RS (automated storage and retrieval system). A combination of lifts and autonomous vehicles store pallets into and retrieve them out of their respective rack storage locations. The crane based AS/RS typically utilizes aisle-captive, mast-mounted cranes that can access any storage location in an aisle via horizontal movement of the mast and vertical movement of the crane on the mast.

In an SOQN, it is assumed that an arriving job or customer is paired with another device and the two visit all the stations that must process the job in the appropriate sequence. After all operations are completed on the job, it exits the system, but the device returns back to a device pool and awaits the next customer. Sometimes a job may have to wait for a device to arrive at the pool or a device may have to wait for a job to arrive. Although closed queuing networks (CQNs) and open queuing networks (OQNs) model systems that require pairing of an incoming job with a device, unlike the SOQN, they ignore the time that a device waits for a job or the time that a job waits for a device.

In the context of an AVS/RS, the jobs correspond to storage/retrieval (S/R) transaction requests and the autonomous vehicles (AVs) correspond to the devices. Because an AV may sometimes have to wait for an S/R transaction or vice versa, we model the AVS/RS as an SOQN. We build the queuing network by deriving general travel times of pre-defined servers. We model the AVS/RS system as a single-class, multiple-server, SOQN. Then, we solve the network using the MGM and obtain its key performance measures. We apply the MGM technique for solving the SOQN model to a warehouse in France that uses AVS/RS.

Introduction

Warehouses are an important element of supply chains. Their main role is to buffer materials along the supply chain so that the demand nodes in the chain can cope with variability caused by external factors such as seasonality, pricing, promotions, economic conditions, transportation, manufacturing cycle time and other factors. Although most companies prefer to supply products to customers directly from their manufacturing plants, this may not always be possible. For instance, to be able to respond customer demand quickly and improve customer service, a company may need to operate a warehouse. Operating it efficiently then becomes important.

The basic operations in a warehouse are to receive goods from suppliers, store them, receive orders from customers, retrieve the required items in an order and ship them to customers (van den Berg & Zijm, 1999). Because constructing and maintaining storage space is expensive, warehouse managers prefer a high-rise storage area with a relatively small foot-print that maximize storage density by storing pallets in locations on racks that are accessible via narrow aisles. Multiple tiers of racks ranging from 50 to 100 feet are possible as shown in Fig. 1. Fast and efficient storage and retrieval of pallets from their rack locations is important for high throughput operations. With advances in technology, new automated material handling technologies providing greater responsiveness and additional flexibility in fulfilling orders have been developed. One of them is AVS/RS developed as an alternative system to the crane-based AS/RS (automated storage and retrieval system) traditionally used to process unit load transactions.

An AVS/RS (autonomous vehicle storage and retrieval system) uses a combination of lifts and autonomous vehicles to store pallets into and retrieve them out of their respective rack storage locations. An autonomous vehicle (AV) picks up a pallet to be stored from its staging area and if it must be deposited at a rack location that is at a higher tier than that of the staging area, it travels to an available lift, travels in the lift with the pallet to its respective location, travels along orthogonal aisles via two separate sets of motors and deposits the pallet in its designated storage location. The same steps are executed in reverse sequence for a retrieval transaction. If the storage or retrieval location is on the same tier where the AV is currently at, of course, there is no need for it to utilize the lift.

Fig. 1 illustrates an AVS/RS in a warehouse. Fig. 1a shows an AV tasked with storing a pallet executing a vertical movement via a lift. Here, because the transaction is a storage, the AV travels to the destination tier and then to the storage address to deposit the load. Fig. 1b shows the AV’s horizontal travel on the destination tier and orthogonal aisles.

Because of the difficulty and expense in altering the physical layout of the high-rise storage area and the configuration of the material handling system, it is important to design such systems in a way that it can efficiently handle current and future demand requirements. This requires evaluating numerous design configurations, evaluating their performance against alternate demand scenarios and selecting a few (one or two) that can be further evaluated via simulation for a detailed and thorough performance analysis. To be able to scores of alternate design scenarios, a fast, efficient and valid analytical model is required. The purpose of this paper is to develop such an analytical model for AVS/RS and solve it via the matric geometric method (MGM).

The paper is organized as follows. In the next section, we discuss the need for developing analytical models for ‘design concepting’, a term explained in that section. In Section 3, we review the literature pertaining to AVS/RS. In Section 4, we show the semi-open queuing network (SOQN) modeling of the AVS/RS. We list assumptions made in the model and the service time calculations of the network. After modeling the AVS/RS as an SOQN, we solve the network using the MGM for various design configurations of the system in Section 5. Conclusions are drawn in Section 6.

Section snippets

Design concepting

Malmborg (2002) first proposed the idea of analytical design concepting models for automated warehouse material handling systems. He developed some analytical models and showed their use in the context of a sample problem. Design concepting is that stage of the warehouse design process in which material-handling systems integrators and warehouse designers identify and evaluate alternate warehouse configurations and material handling systems, and based on this evaluation develop preliminary

Background and discussion

Malmborg (2003) proposed a state equation model for predicting the proportion of dual command (DC) cycles in AVS/RSs using interleaving. Malmborg extends his previous study from Malmborg (2002) by estimating proportion of dual command cycles to predict system utilization and throughput capacity. The proposed model also substitutes a prediction of the maximum practical queue size in place of the proportion of DC cycles to bootstrap the estimation of utilization and throughput. Despite some

SOQN Modeling of AVS/RS

A general SOQN modeling procedure based on several arrival and service rate scenarios via simple examples is shown by Jia and Heragu (2009). Application of SOQN on an AVS/RS has been studied by Roy et al. (2012) and Ekren et al. (2013).

An SOQN consists of jobs, a device and servers. Each arriving job is paired with a device and the two – job and device – visit the set of servers required to process the job in the specified sequence. In the SOQN modeling of an AVS/RS, the S/R transaction

Matrix-geometric method

The MGM was developed by Marcel Neuts in the 1980s (Neuts, 1980). It is a numerical approach to solve Markov processes having a special repetitive property called matrix-geometric property. For systems with large or possibly infinite number of states, exact solutions can only be obtained if one can utilize structural properties of equations. The repetition allows us to determine a recursive solution for the stationary state probabilities since it implies that if one knows the stationary

Conclusion

In this study, we present a comprehensive analytical MGM solution for SOQN model of an AVS/RS. First, the AVS/RS system is modeled as a SOQN by considering three stations (servers) in the network. Second, the mean service times and their SCV values of these pre-defined servers are derived. To be able to solve the SOQN via the MGM, we reduce the network to two stations network by aggregating the last two stations. We obtain load-dependent throughput rates for the aggregated stations and assume

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