Matrix-geometric solution for semi-open queuing network model of autonomous vehicle storage and retrieval system
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
References (24)
- et al.
Approximate analysis of load dependent generally distributed queuing networks with low service time variability
European Journal of Operational Research
(2010) - et al.
Simulation based experimental design to identify factors affecting performance of AVS/RS
Computers and Industrial Engineering
(2010) - et al.
A network queuing approach for evaluation of performance measures in autonomous vehicle storage and retrieval systems
European Journal of Operational Research
(2009) - et al.
Design models for unit load storage and retrieval systems using autonomous vehicle technology and resource conserving storage and dwell point policies
Applied Mathematical Modelling
(2007) - Cai, X., Heragu, S. S., & Liu, Y. (2014). Modeling and evaluating the AVS/RS with tier-to-tier vehicles using semi-open...
- et al.
Modeling automated warehouses using semi-open queuing networks
- Ekren, B. Y., Heragu, S. S., Krishnamurthy, A., & Malmborg, C. J. (2009). Matrix-geometric solution for semi-open...
- Ekren, B. Y., & Heragu, S. S. (2011). Analytical and simulation models for performance evaluation of AVS/RS. Knovel...
- et al.
Simulation-based regression analysis for the rack configuration of an autonomous vehicle storage and retrieval system
International Journal of Production Research
(2010) - et al.
An approximate solution for semi-open queuing network model of autonomous vehicle storage and retrieval system
IEEE Transactions on Automation Science and Engineering
(2013)