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Modelling of Rail Guided Vehicles serving an automated parts-to-picker system Modelling of Rail Guided Vehicles serving an automated parts-to-picker system Modelling of Rail Vehicles serving an parts-to-picker system Modelling of Rail Guided Guided Vehicles an automated automated parts-to-picker system Calzavara Martina*, serving Persona Alessandro*, Sgarbossa Fabio*

Calzavara Martina*, Persona Alessandro*, Sgarbossa Fabio* Calzavara Martina*, Persona Sgarbossa Fabio* Alessandro*, Calzavara Martina*, Persona Alessandro*, Sgarbossa * Department of Management and Engineering, University of Padua, Stradella SanFabio* Nicola, 3 36100 Vicenza, Italy * Department of Management and Engineering,[email protected], University of Padua, [email protected]) San Nicola, 3 36100 Vicenza, Italy (e-mail: [email protected], * Department of Management and Engineering,[email protected], University of Padua, Stradella San Nicola, 3 36100 Vicenza, Italy (e-mail: [email protected], * Department of Management and Engineering, University of Padua, [email protected]) San Nicola, 3 36100 Vicenza, Italy (e-mail: [email protected], [email protected], [email protected]) (e-mail: [email protected], [email protected], [email protected]) Abstract: An automated parts-to-picker picking system usually consists of an automated warehouse, with Abstract: An automated parts-to-picker picking system of an automated Automatic Storage and Retrieval Systems (AS/RS) thatusually retrieveconsists the Stock Keeping Unitswarehouse, (SKUs) ofwith the Abstract: An automated parts-to-picker picking system usually consists of an automated warehouse, Automatic Storage and Retrieval Systems (AS/RS) that retrieve the Stock Keeping Units (SKUs) ofwith the Abstract: An automated parts-to-picker picking system usually consists of an automated warehouse, with various needed products from their stocking locations, and of a picking area, with human operators or Automatic Storage and Retrieval Systems (AS/RS) that and retrieve the Stockarea, Keeping Units (SKUs) of the various needed products from their stocking locations, of a picking with human operators or Automatic Storage and Retrieval Systems (AS/RS) that retrieve the Stock Keeping Units (SKUs) of the robots that pick the needed items in order to create a mixed shipping unit. The automated warehouse and various needed products from their stocking locations, and shipping of a picking area, with human operatorsand or robots that pick the needed items in order to create a mixed unit. The automated warehouse various needed products from their stocking locations, and of a picking area, with human operators or the picking area the areneeded connected byinanorder automated transportation system, which moves the SKUs from and the robots that pick items to create a mixed shipping unit. The automated warehouse the picking area are connected by an automated transportation system, which moves the SKUs from the robots that pick the needed items in order to create a mixed shipping unit. The automated warehouse and warehouse the are picking stations versa. The transportation system can moves be, for the example, ring rail the pickingto area connected byand an vice automated transportation system, which SKUsaafrom the warehouse the are picking stations and versa. The transportation canSKU be, for example, ring rail the picking area connected an vice automated transportation system, which moves SKUs from the conveyor onto which various Railby Guided Vehicles (RGVs) are able tosystem carry one at the a time. The present warehouse to the picking stations and vice versa. The transportation system can be, for example, a ring rail conveyor ontowhich various Rail Guided Vehicles (RGVs) arethen, able to carry one at example, a time. The present warehouse the picking stations and vice versa. The transportation system canSKU be, for a for ring rail paper proposes a preliminary simulative analysis and, a mathematical formulation this conveyor on which Rail Guided Vehicles (RGVs) able to one SKU formulation at a time. Thefor present paper proposes a various preliminary simulative analysis and,are a carry mathematical this conveyor on which various Guided Vehicles (RGVs) arethen, able to carry at a to time. Thea present transportation system, usefulRail to properly estimate the number of RGVs thatone areSKU required fulfill certain paper proposes a preliminary simulative analysis and, then, a mathematical formulation for this transportation system, useful to properly estimate the number of RGVs that are required to fulfill a certain paper proposes a preliminary simulative and,throughput then, a mathematical formulation for this picking throughput. Inuseful fact, ittoisproperly shown estimate that analysis the picking does not increase linearly with the transportation system, the number of RGVs that are required to fulfill a certain picking throughput. Inuseful fact, ittodue isproperly shown that theissues. picking throughput increase to linearly the transportation system, estimate the number of RGVsdoes that not are required fulfill awith certain number of vehicles employed, to congestion picking throughput. In fact, it due is shown that theissues. picking throughput does not increase linearly with the number of vehicles employed, to congestion picking throughput. In fact, it is shown that the picking throughput does not increase linearly with the number vehicles employed, due to congestion issues. © 2018, of IFAC (International Federation of Automatic Control)vehicles, Hosting picking by Elsevier Ltd. All rights reserved. Keywords: warehouse picking, parts-to-picker, rail guided throughput number of vehicles employed, due to congestion issues. Keywords: warehouse picking, parts-to-picker, rail guided vehicles, picking throughput Keywords: warehouse picking, parts-to-picker, rail guided vehicles, picking throughput Keywords: warehouse picking, parts-to-picker, rail guided vehicles, picking throughput The present paper aims at studying a specific kind of material 1. INTRODUCTION The presentsystem paper aims at studying a specific kind of for material 1. INTRODUCTION handling which is usually employed the The present paper aims at studying aa specific kind of material handling system which is usually employed for 1. INTRODUCTION The present paper aims at studying specific kind of material transportation of SKUs fromisthe usually automatedemployed warehousefor and the the In a warehouse, order picking is the activity required to handling 1. INTRODUCTION system which the of SKUs fromis the usually automated warehouse and the In a warehouse, orderfrom picking is the activity handlingarea system which employed for picking and vice versa, which uses Rail Guided Vehicles retrieve various items their storage locationsrequired to createto a transportation transportation of SKUs from the automated warehouse and the In a warehouse, order picking is the activity required to area and vice uses Railwarehouse Guided Vehicles retrieve various items from their locations to create transportation of SKUs from the automated and the In a warehouse, order picking is the activity required toa picking (RGVs) moving on aversa, fixedwhich path. In this case, RGVs are mixed shipping unit required by storage one or more customers (De picking area and vice versa, which uses Rail Guided Vehicles retrieve various items from their storage locations to create a moving onalternative aversa, fixedwhich path. In this RGVs are mixed shipping unit required by storage one or more customers (Dea (RGVs) picking area and vice uses Rail case, Guided Vehicles retrieve various items fromoftheir locations to create considered to be an of roll conveyors. Starting from Koster et al., 2007). Most the existing warehouse picking (RGVs) moving onalternative a fixed of path. this case, RGVsfrom are mixed shipping unit required by one or more customers (De considered to of be an roll In conveyors. Starting Koster etare al., still 2007). Most and of the existing warehouse picking moving on possible a fixed factors path. In thiscan case, RGVs the are mixed shipping unit required bypicker-to-parts, one or more customers (De (RGVs) the analysis the that influence systems manual with picking considered to be an alternative of roll conveyors. Starting from Koster et al., 2007). Most of the existing warehouse picking the analysis of the possible factors that can influence the systems are still manual and picker-to-parts, with picking considered to be an alternative of roll conveyors. Starting from Koster et al., 2007). Most of the existing warehouse design and the performances offactors these parts-to-picker systems, operators walking (or travelling) through the aisles to retrieve the analysis the possibleof that can influence the systems are still and picker-to-parts, with picking design and theof these parts-to-picker systems, operators walking (or travelling) through the aisles to retrieve the analysis ofperformances the possible simulative factors thatanalysis, can influence systems are still manual manual and picker-to-parts, with picking it is proposed a preliminary in orderthe to the items reported on their picking lists (Napolitano, 2012; operators walking (or travelling) through the aisles to retrieve design and the performances of these parts-to-picker systems, it is proposed preliminary of simulative analysis, in useful order to to the items reported on their picking lists (Napolitano, 2012; design and the aperformances these parts-to-picker systems, operators walking (or travelling) through the aisles tocould retrieve derive a proper mathematical formulation that can be Battini et al., 2015). However, in some contexts there be it is proposed amathematical preliminary formulation simulative analysis, in useful order to to the items reported their picking (Napolitano, 2012; a proper that can be Battini al., 2015).ofon However, in somelists contexts there could be derive it is proposed a preliminary simulative analysis, intoorder the items reported on their picking lists (Napolitano, 2012; estimate the right number of RGVs that are required fulfiltoa also theet possibility creating automated solutions, at different derive a proper mathematical formulation that can beto useful Battini etpossibility al., 2015).of However, in some contexts there could be estimate the right number of RGVs that are required fulfil also the creating automated solutions, at different apicking proper mathematical formulation can be that, usefuldue toa Battini al., 2015). However, in some contexts there could be derive certain throughput. InRGVs particular, itthat is shown levels ofetpossibility automation, with potential benefits of easing picking estimatepicking the right number ofIn that are required to fulfil a also theof of creating automated solutions, at different certain throughput. particular, it is shown that, due levels automation, with potential benefits of easing picking estimate the right number of RGVs that are required fulfil a also the possibility of creating automated solutions, at different RGVs congestion, the picking throughput does notto increase activities and improve picking performances (Azadeh et al., to certain picking throughput. In particular, it is shown that, due levels of automation, with potential benefits of easing picking RGVs congestion, the of picking throughput notitthat, increase activities and improve picking performances (Azadeh et al., to certain picking In particular, it isdoes shown due levels of automation, with potential benefits ofparts-to-picker, easing picking linearly with thethroughput. number installed vehicles. Hence, derives 2017). Automated picking systems are mainly to RGVswith congestion, the of picking throughput does notitincrease activities and improve picking performances (Azadeh et al., linearly the number installed vehicles. Hence, derives 2017). Automated picking systems are mainly parts-to-picker, to RGVs congestion, the picking throughput does not increase activities and an improve pickingStorage performances (AzadehSystem et al., that, to ensure a proper system design, it is important to usually with Automatic and Retrieval linearly the number of system installeddesign, vehicles. Hence, it derives 2017). Automated picking systems are mainly parts-to-picker, that, to with ensure a proper itsize is important to usually with an Automatic Storage Retrieval System linearly with the number of installed vehicles. Hence, it derives 2017). Automated picking systems are and mainly parts-to-picker, determine the correct trade-off between the of the RGVs (AS/RS) that retrieves the Stock Keeping Units (SKUs) of the usually with an Automatic Storage andUnits Retrieval System that, to aa proper system design, is to thepicking correct trade-off between theit of the RGVs (AS/RS) that retrieves thefrom Stock Keeping (SKUs) of the determine that, to ensure ensure proper system design, itsize is important important to usually with an Automatic Storage and locations, Retrieval System fleet and the performance. various needed products their stocking in order determine the correct trade-off between the size of the RGVs (AS/RS) that retrieves the Stock Keeping Units (SKUs) of the fleet and the picking performance. various needed products from their stocking locations, in order determine the correct trade-off between the size of the RGVs (AS/RS) that retrieves the Stock Keeping Units (SKUs) of the to deliver themproducts to human pickers, standinglocations, in their in picking and the picking various needed from their stocking order fleet to deliver them to human pickers, standing in their in picking The remainder of the performance. paper is structured as follows. In the next and the picking performance. various needed products from their stocking locations, order fleet stations. In such a system, the AS/RS and the picking stations remainder of the paper is structured as follows. the next to deliverInthem tosystem, human the pickers, standing in their stations picking The stations. such a AS/RS and the picking section, a brief literature review concerning materialIn handling to deliver them to human pickers, standing in their picking The remainder of the paper is structured as follows. Inhandling the next are connected through a proper conveyor, inpicking which stations all the section, a brief literature review concerning material stations. In such a system, the AS/RS and the The remainder of theand paper is structured as follows. In the next are connected through amoved, proper conveyor, which all the systems, conveyors some existing mathematical models stations. In such a system, the AS/RS and stocking theinpicking stations section, a brief literature review concerning material handling stock keeping units are from the area to the systems, conveyors and somedevices existing mathematical models3 are connected through amoved, properfrom conveyor, in which all the section, a brief literature review concerning material handling stock keeping units are the stocking area to the for the dimensioning of such is presented. Section are connected through athe proper conveyor, inthe which all of systems, conveyors and some existing mathematical models picking area and, once picker has taken number the dimensioning of devices presented. Section stock keeping unitsonce are moved, fromhas thetaken stocking area to the systems, conveyors and some existing mathematical models3 picking area and, thestocking picker the number of for presents the setting of thesuch system underis study and the general stock keeping units are moved, from the(Roodbergen stocking area toVis, the for the dimensioning of such devices isstudy presented. Section 3 items he needs, back to the area and presents the setting of the system under and the general picking area and, once the picker has taken the number of for the dimensioning of such devices is presented. Section 3 items heThe needs, backonce to the area (Roodbergen and Vis, that have been obtained through the 3D simulation with picking area and, thestocking picker has takenhas thetonumber of results presents the setting of the system under study and the general 2009). dimensioning of such a system take into results that have been obtained through the 3D simulation with items he needs, back to the stocking area (Roodbergen and Vis, ® presents the setting of the system under study and the general 2009). dimensioning of such system hasoftoSKUs take into Applied AutoMod software. Subsequently, Section 4 reports items heThe needs, backof toorders, the stocking (Roodbergen and Vis, results that have been ® obtained through the 3D simulation with account the amount hence,aaarea the number that Applied software. Subsequently, Section 4 reports AutoMod 2009). The dimensioning of such system hasoftoSKUs take into results that have been obtained through simulation account the amount of orders, hence, the number that ® the mathematical model introduced forthe the3D calculation ofwith the ® 2009). The dimensioning of such a system has to take into have to be moved inofa orders, certain hence, time range. This isof translated in Applied AutoMod Subsequently, Section 4 reports ® software. the mathematical model introduced for the calculation ofright the account the amount the number SKUs that Applied AutoMod software. Subsequently, Section 4 reports have to be moved in a certain time range. This is translated in cycle time, of the system throughput and, hence, for the account the amount of orders, hence, the number of SKUs that the mathematical model introduced for thehence, calculation ofright the ahave certain throughput the system has to This warrant, in terms of time, of the system throughput and, for the to be moved in that a certain time range. is translated in cycle the mathematical model introduced for the calculation of the ahave certain throughput that the system has to warrant, in terms of dimensioning rail guided vehiclesand, conveyor toretrieved be moved instored a certain time This isit translated in cycle time, of of theaa system throughput hence,system. for theThen, right SKUs and per hour.range. Therefore, emerges the dimensioning of rail guided vehicles conveyor system. Then, aSKUs certain throughput that the system has to warrant, in terms of cycle time, of the system throughput and, hence, for the right retrieved and stored per hour. Therefore, it emerges the in Section 5, the model is applied to the simulation dataset, to aneed certain throughput thatthe thefactors systemthat has can to warrant, in limit termsthe of of amodel rail guided vehicles conveyor system. Then, ofretrieved understanding foster or in Sectionthe 5, the isformulations applied to the simulation dataset, to SKUsof and stored per hour.that Therefore, it emerges the dimensioning dimensioning of a rail guided vehicles conveyor system. Then, need understanding the factors can foster or limit the compare analytical to the output of the SKUs retrieved storedperformances. per hour. Therefore, it emerges in Sectionthe 5, the model isformulations applied to thetosimulation dataset, to reaching of the and required In foster particular, it has compare analytical the output of the need of understanding the factors that can or limit the in Section 5,and the to model is applied to the simulation dataset, to reaching of the required In foster particular, it has derive some general insights about the need of been understanding theperformances. factors that can or limit the simulation compare the analytical formulations to the output of the already demonstrated in literature, but also faced by the simulation toFinally, deriveformulations some general about reachingbeen of the required performances. In also particular, it has compare theandanalytical to insights thethe output of the already demonstrated in literature, but faced by the observed trends. in the last section, conclusions reaching of the required performances. In particular, it has and toFinally, deriveinsome general insights about the authors in practical industrial applications, that handling observed the last section, theproposed. conclusions already been demonstrated in literature, but that also the faced by the simulation and to derive some general insights about the authors in practical industrial applications, handling and some trends. suggestions for further researches are already been demonstrated in literature, but also the faced by the simulation system (e.g. the conveyor) between the automated warehouse observed trends. Finally, in the last section, the conclusions and some suggestions for further researches are proposed. authors in practical industrial applications, that the handling observed trends. Finally, in the last section, the conclusions system the stations conveyor) thea bottleneck automated warehouse authors in practical industrial applications, that the(Lee handling and the (e.g. picking canbetween represent et al., and some suggestions for further researches are proposed. system (e.g. the stations conveyor) between thea bottleneck automated (Lee warehouse and the picking can represent et al., and some suggestions for further researches are proposed. system (e.g. the conveyor) between the automated warehouse 1996; Schmidt and Jackman, 2000). and picking stations can 1996; Jackman, 2000). aa bottleneck and the theSchmidt pickingand stations can represent represent bottleneck (Lee (Lee et et al., al., 1996; Schmidt and Jackman, 2000). 1996; Schmidt and Jackman, 2000).

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2. LITERATURE REVIEW A material-transport equipment is a material handling system which has the primary function of transporting material within a factory. Examples of material-transport systems are: conveyors, industrial vehicles, monorails, hoists, cranes (Tompkins et al., 2010). A typical classification of such systems considers their degree of automation (walking, riding, automated), the flow pattern (continuous, intermittent or synchronous, asynchronous), the flow path (fixed, variable) the location (underground, in-floor, floor level, overhead) and the throughput capacity (Le-Anh and De Koster, 2006; Roodbergen and Vis, 2009). In case there are many items that have to be moved frequently between specific points, always in the same direction and over a fixed path, usually the conveyor (e.g. roller, belt, chain) represents the best solution. An alternative to a traditional continuous conveyor is a Rail Guided Vehicles system, in which various vehicles move on a closed-loop rail path to retrieve and deliver the required items, which is often used for connecting an automated warehouse to a picking area (Lee et al., 1996; Dotoli and Fanti, 2002). The existing literature concerning RGVs system is not as wide as the one proposed for conveyor systems (Roy et al., 2016). For these latter systems, there are several contributions focused on understanding and modelling their actual operation, considering the proper design to warrant a certain throughput and to avoid possible congestions and queues of the material (Andriansyah, 2011; Claeys et al., 2015). However, it is important to underline that an RGVs system significantly differs from a conveyor system. In a RGVs system, in fact, the throughput mainly depends on the number of vehicles that are installed, which becomes a fundamental design driver. The so-called fleet sizing problem has been addressed several times for Automatic or Laser Guided Vehicles, considering that they work on a limited area, riding always on the same path (Arifin and Egbelu, 2000; Choobineh et al., 2012; Ferrara et al., 2014). However, these models are not easily applicable to a RGVs system serving a picking area, since for RGVs the operating context is usually different, the distances travelled are shorter and they are not allowed to overtake each other (Roodbergen and Vis, 2009). Moreover, due to the interaction of several operation aspects, the modelling of such a system presents a complexity that has often been addressed through simulative analyses. In this direction, one of the researches concerning the design and the operation of an RGVs system has been developed by Lee et al. (1996), through the proposal of a simulation model. In this case, the aim of the study is exactly the determination of the optimal number of RGVs able to warrant the maximum throughput of the system. It is demonstrated that the increase of the number of RGVs leads to an increase of the throughput but only for a certain number of vehicles. Besides this number, in fact, adding one more vehicle could cause congestions and additional costs. However, these outcomes are only based on the simulation results and they are not generalized by an analytical model. Dotoli and Fanti (2002; 2005) analyse a system similar to the one faced in the present paper, though focussing on the interaction between the RGVs conveyor system and the AS/RS of the automated warehouse, which is not the scope of the present paper. There, they propose a

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modelling based on Colored Time Petri Nets. Another interesting study which uses a simulative approach is by Lin et al. (2004). They investigate the behaviour of a double-loop of an interbay material handling system (for wafer fabrication) considering its throughput and transportation time. 3. SIMULATION SET UP In order to understand the behaviour of a RGV conveyor serving an automated parts-to-picker system, a 3D software simulation with Applied AutoMod ® has been set up. The simulation environment consisted in a rail ring conveyor positioned between an automated pallet warehouse with AS/RS and a picking area (Figure 1). The conveyor length was 𝐿𝐿=100 m, while the points of pallet- loading and unloading were situated symmetrically on both the longer sides of the ring, equally distributed. Two different configurations of the loading/unloading bays have been considered: one with three points both on the automated warehouse side and on the picking area, and one with five points on each side. The number of travelling RGVs have been varied in a range from 1 RGV to 20 RGVs: 1, 2, 4, 6, 10, 12, 14, 16, 18, 20. It has then been considered two values of RGV speed, 𝑣𝑣=1.0 m/s and 𝑣𝑣=1.5 m/s, and two values of stop time, needed for loading (or unloading) a pallet on (from) the RGV: 𝑡𝑡𝐿𝐿/𝑈𝑈 =5 seconds and 𝑡𝑡𝐿𝐿/𝑈𝑈 =10 seconds. The combination of all these parameters has led to 88 different simulation runs. Each simulation run referred to an operation timing of the system of 10 hours, in which the RGVs were continuously involved in retrieving and depositing pallets in the various points (i.e. there were always SKUs ready to be moved from and to the automated warehouse). In each loop, one RGV stopped one time at one of the points on the automated warehouse side and one time at one of the points on the picking area side. The points at which the RGVs had to stop were always different and chosen randomly. AUTOMATED WAREHOUSE

RGVs CONVEYOR SYSTEM

PICKING AREA

Fig. 1. Rail Guided Vehicles conveyor system under study. For each simulation run, it has been counted the total number of loops done by the first RGV during the 10 hours. From this data it has been possible to derive the average time per loop 𝑡𝑡𝑠𝑠 : 𝑡𝑡𝑠𝑠 =

𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡

(1)

3600

(2)

𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛 𝑜𝑜𝑜𝑜 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙

as well as the single throughput per RGV 𝑄𝑄𝑠𝑠 , expressed in cycles per hour: 𝑄𝑄𝑠𝑠 =

1515

𝑡𝑡𝑠𝑠

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Then, defining 𝑁𝑁 as the number of RGVs working on the rail conveyor, the total throughput of the rail conveyor system is: 𝑄𝑄𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠 = 𝑁𝑁 ∙

3600 𝑡𝑡𝑠𝑠

(3)

As expected, the results of the simulation runs have shown that the total throughput does not increase linearly with the increasing of the number of RGVs. In fact, from a certain number of RGVs onwards, congestion issues emerge, slowing down the whole system. This is shown, for example, through the trends of the graphs in Figure 2 for the average cycle time 𝑡𝑡𝑠𝑠 and Figure 4 for the total throughput of the RGVs system 𝑄𝑄𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠 , varying the number of RGVs. 4. MODEL FOR RGV FLEET DIMENSIONING In this section the mathematical modelling for the evaluation of the performance of an RGVs conveyor system is showed.

First of all, considering the length of the conveyor 𝐿𝐿, the RGV maximum speed 𝑣𝑣 and acceleration 𝑎𝑎 and the pallet loading (unloading) time 𝑡𝑡𝐿𝐿/𝑈𝑈 , a theoretical cycle time is defined as 𝑡𝑡′ = + 2 ∙ + 2 ∙ 𝑡𝑡𝐿𝐿/𝑈𝑈 𝐿𝐿

𝑣𝑣

𝑣𝑣

𝑎𝑎

(4)

and, similarly to (2), the theoretical throughput of one RGV is 𝑄𝑄′ =

3600 𝑡𝑡′

(5)

Then, considering that a certain total throughput is required to the system 𝑄𝑄𝑟𝑟 , in terms of cycles that have to be performed per hour according to the pallets that have to be moved from one side to the other, the number of needed RGVs turns out to be: 𝑁𝑁′ = ⌈ 𝑟𝑟 ⌉ 𝑄𝑄

𝑄𝑄′

(6)

From the same data it can be calculated the average speed of a RGV: 𝑣𝑣̅ =

𝐿𝐿

𝑡𝑡′

(7)

and, then, the average interference space between two consequent RGVs: 𝑠𝑠 = 2 ∙ 𝑡𝑡𝐿𝐿/𝑈𝑈 ∙ 𝑣𝑣̅

(8)

𝑠𝑠𝑎𝑎 =

(9)

On the other side, the maximum available space per RGV depends on the number of RGVs that are employed and is calculated with: 𝐿𝐿

𝑁𝑁′

Moreover, the parameter introduced in (8) allows to calculate the threshold number of RGVs as: 𝑁𝑁𝑘𝑘 = ⌊ ⌋ 𝐿𝐿 𝑠𝑠

(10)

which represents the maximum number of RGVs that can work together on the rail conveyor without causing congestion issues.

It derives that the formulations of both the cycle time 𝑡𝑡 and of the total system throughput 𝑄𝑄𝑡𝑡𝑡𝑡𝑡𝑡 have to take into account this threshold number, as follows: 𝑡𝑡 = {

𝑡𝑡′ 𝑡𝑡′ + ∆𝑡𝑡

𝑖𝑖𝑖𝑖 𝑁𝑁′ ≤ 𝑁𝑁𝑘𝑘 𝑖𝑖𝑖𝑖 𝑁𝑁′ > 𝑁𝑁𝑘𝑘

(11)

where ∆𝑡𝑡 considers the positive slope of the straight line 𝑚𝑚, which can well approximate the trend of the cycle time shown in Figures 2 and 3: ∆𝑡𝑡 = 𝑚𝑚 ∙ (𝑁𝑁′ − (𝑁𝑁𝑘𝑘 + 1))

(12)

𝑄𝑄𝑡𝑡𝑡𝑡𝑡𝑡 = {

(13)

𝑁𝑁′ ∙

𝑁𝑁′ ∙

3600

𝑡𝑡′ 3600

𝑡𝑡 ′+∆𝑡𝑡

𝑖𝑖𝑖𝑖 𝑁𝑁′ ≤ 𝑁𝑁𝑘𝑘

𝑖𝑖𝑖𝑖 𝑁𝑁′ > 𝑁𝑁𝑘𝑘

In the next Section, the model here presented is applied and compared to the results of the simulation runs, and some remarks are derived. 5. MODEL APPLICATION AND RESULTS COMPARISON The proposed model has been applied to the simulation dataset. The results shown in this section concern the cycle time of one RGV, expressed in seconds, and the system total throughput, measured in number of cycles (retrieval and storage of one SKU) performed per hour. In each one of the following figures, four graphs are shown, according to the as many combinations that come out from the two values of the RGV speed, 𝑣𝑣= 1.0 m/s and 1.5 m/s, and the two stop times, 𝑡𝑡𝐿𝐿/𝑈𝑈 =5 s and 10 s. Each graph displays the threshold number of RGVs 𝑁𝑁𝑘𝑘 , the theoretical values of the cycle time 𝑡𝑡′ or of the total throughput 𝑄𝑄′𝑡𝑡𝑡𝑡𝑡𝑡 , calculated considering that there are no congestions and interferences among the RGVs, the values obtained from the simulation runs (𝑡𝑡𝑠𝑠 or 𝑄𝑄𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠 ) and the ones calculated through the introduced formulation, 𝑡𝑡 or 𝑄𝑄𝑡𝑡𝑡𝑡𝑡𝑡 . For the calculation of 𝑡𝑡, the values of the various 𝑚𝑚 have been derived from the linear approximation of the curve related to the simulation results. In all cases, it can be noticed that the mathematical formulation represents a good approximation of the simulation results. Figures 2 and 3 show the trend of the cycle time (𝑡𝑡′, 𝑡𝑡𝑠𝑠 , 𝑡𝑡) varying the number of RGVs working simultaneously on the rail conveyor system. In Figure 2, the number of stop points for pallet loading/unloading is three, both on the automated warehouse and on the picking area side (3+3), while in Figure 3 the stop points are 5 on both sides (5+5). Looking at the graphs, the threshold number of RGVs 𝑁𝑁𝑘𝑘 varies in all of them but it is the same regardless of the number of stop points. In fact, as it can be seen also from (8) and (10), the threshold number of RGVs depends on 𝑡𝑡𝐿𝐿/𝑈𝑈 and 𝑣𝑣̅ but it does not depend on the number of stop points. Hence, the increasing of the cycle times, and then, the arising of congestion issues, start from the same number of RGVs. On the other side, the difference in the number of stop points influence the slope of the cycle time curves, that is higher for a lower number of points (3+3): if there are fewer points at which the RGVs can stop, it is more probable to have queues at them. As far as RGV

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speed maximum 𝑣𝑣 is concerned, it can be pointed out that, of course, a higher speed 𝑣𝑣 lowers 𝑁𝑁𝑘𝑘 and increases the average cycle times, keeping the same 𝑡𝑡𝐿𝐿/𝑈𝑈 . A similar effect, but with a higher impact, can be observed by increasing 𝑡𝑡𝐿𝐿/𝑈𝑈 .

Figures 4 and 5 report the trend of the total throughput of the system (𝑄𝑄′𝑡𝑡𝑡𝑡𝑡𝑡 , 𝑄𝑄𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠 , 𝑄𝑄𝑡𝑡𝑡𝑡𝑡𝑡 ), for 3+3 L/U points and 5+5 L/U points, respectively. Here, the graphs show how the performance of the system increases linearly until reaching 𝑁𝑁𝑘𝑘 ; after it, the curve has a lower slope. Also in this case it can be seen the greater influence of 𝑡𝑡𝐿𝐿/𝑈𝑈 with respect to 𝑣𝑣.

Generally, these graphs clearly show that the increase of the number of RGVs not necessarily leads to an increase of the system throughput, due to congestion phenomenon. In fact, a positive linear correlation exists only until the reaching of the threshold number 𝑁𝑁𝑘𝑘 . Hence, if a throughput higher than 𝑄𝑄𝑡𝑡𝑡𝑡𝑡𝑡 (𝑁𝑁𝑘𝑘 ) is needed, it could be useful to investigate other ways to improve such throughput, besides increasing the number of RGVs. For example, this exploratory study shows that an action on the reduction of 𝑡𝑡𝐿𝐿/𝑈𝑈 gives higher benefits compared to having faster vehicles. Furthermore, given a certain configuration of the system in terms of length of the rail and number of stop points, these graphs allow to understand the best choices that can allow the reaching of the desired throughput 𝑄𝑄𝑟𝑟 . For example, looking at Figure 5 and considering 𝑄𝑄𝑟𝑟 =300 cycles/h, it can be seen that this is achievable with 10 RGVs with no congestion issues for 𝑣𝑣= 1.0 m/s and 𝑡𝑡𝐿𝐿/𝑈𝑈 =5 s, and with 8 RGVs for 𝑣𝑣= 1.5 m/s and 𝑡𝑡𝐿𝐿/𝑈𝑈 =5 s. On the other side, if 𝑡𝑡𝐿𝐿/𝑈𝑈 =10 s, 12 RGVs are needed if 𝑣𝑣= 1.0 m/s while 10 RGVs are necessary if 𝑣𝑣= 1.0 m/s. Therefore, in this particular case, the first and the fourth settings warrant the same performance of number of cycles per hour with the 𝑁𝑁𝑘𝑘

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same number of RGVs, even if in the latter case 𝑁𝑁′ > 𝑁𝑁𝑘𝑘 and there are congestion problems. 6. CONCLUSIONS AND FUTURE RESEARCHES In this paper, an explorative study concerning Rail Guided Vehicles serving an automated parts-to-picker system has been presented. Starting from a simulation setting developed in Applied AutoMod®, a mathematical formulation for the estimation of the system throughput and of the right number of RGVs to employ has been proposed. In particular, it has been shown that the system throughput 𝑄𝑄𝑡𝑡𝑡𝑡𝑡𝑡 does not increase linearly with the number of RGVs 𝑁𝑁, due to congestion phenomena. Therefore, it derives that the increase of the system throughput does not necessarily imply the increase of the number of RGVs: in some cases, other actions, like the improvement of the operation of the vehicles, should be investigated. Moreover, the graphs that have been derived have shown that the same throughput can be obtained with different system configurations, leading to the need of a tradeoffs analysis. Due to the narrowness of the simulation dataset, this study represents only a first step for a research that should be widened. Future researches in this sense should consider further simulation runs, by changing more parameters and by assigning them more values. For example, further congestion issues could emerge with the increase of the number of stop points, although in the presented cases the effect was the opposite (i.e. more stop points led to higher throughput). Therefore, it could be interesting to define a ‘stop points threshold’, similar to the one that has been proposed here for the number of RGVs.

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Fig. 5. Total throughput of the RGVs system (𝑄𝑄′𝑡𝑡𝑡𝑡𝑡𝑡 theoretical, 𝑄𝑄𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠 simulation, 𝑄𝑄𝑡𝑡𝑡𝑡𝑡𝑡 formulation) varying the number of RGVs, 5+5 L/U points. REFERENCES Andriansyah, R. (2011). Order-picking workstations for automated warehouses (Doctoral dissertation, PhD Thesis Eindhoven University of Technology). Arifin, R., & Egbelu, P. J. (2000). Determination of vehicle requirements in automated guided vehicle systems: a statistical approach. Production Planning & Control, 11(3), 258-270. Azadeh, K., de Koster, M. B. M. & Roy, D. (2017) Robotized Warehouse Systems: Developments and Research Opportunities. Available at: https://ssrn.com/abstract=2977779 Battini, D., Calzavara, M., Persona, A., & Sgarbossa, F. (2015). Order picking system design: the storage assignment and travel distance estimation (SA&TDE) joint method. International Journal of Production Research, 53(4), 1077-1093. Choobineh, F. F., Asef-Vaziri, A., & Huang, X. (2012). Fleet sizing of automated guided vehicles: a linear programming approach based on closed queuing networks. International Journal of Production Research, 50(12), 3222-3235. Claeys, D., Adan, I., & Boxma, O. (2015). Stochastic bounds for order flow times in warehouses with remotely located order-picking workstations. (Report Eurandom; Vol. 2015008). Eindhoven: Eurandom. De Koster, R., Le-Duc, T., & Roodbergen, K. J. (2007). Design and control of warehouse order picking: A literature review. European Journal of Operational Research, 182(2), 481-501. Dotoli, M., & Fanti, M. P. (2002). Modeling of an AS/RS serviced by rail-guided vehicles with colored Petri nets: a control perspective. In Systems, Man and Cybernetics, 2002 IEEE International Conference on (Vol. 3, pp. 6-pp). IEEE. Dotoli, M., & Fanti, M. P. (2005). A coloured Petri net model for automated storage and retrieval systems serviced by rail-

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