wharehouse = AGV storage area workstation = unloading = loading

Transcription

1 CORE DISCUSSION PAPER 2001/37 DESIGN AND PERFORMANCE ANALYSIS OF A HEAVILY LOADED MATERIAL HANDLING SYSTEM Philippe CHEVALIER 1,Yves POCHET 2 and Laurence TALBOT 3 September 2001 Abstract We consider the problem of designing a 2-stations Automated Guided Vehicle System (AGVS). The AGV System consists of a pool of vehicles that transports products from one station to the other station through an unidirectional guidepath. We seek a model to estimate the minimal number of vehicles needed to guarantee some target service level expressed in terms of mean waiting time. To achieve this we also need to design dispatching rules in order to utilize the vehicles in the best way. Our solution procedure starts by computing the necessary ll rate in order to respect the maximum mean waiting time. We use a Reorder Point Inventory Policy and Markov Chain Theory to determine the dispatching rules and to estimate the minimum number of vehicles required to guarantee the ll rate. Our computer simulations indicate that the model oers a good degree of approximation. Keywords: Automated Guided Vehicle System, Number of vehicles, Queueing theory, Markov Chain theory, simulations 1 CORE and IAG, Universite catholique de Louvain, Belgium. chevalier@poms.ucl.ac.be 2 CORE and IAG, Universite catholique de Louvain, Belgium. ypochet@core.ucl.ac.be 3 IAG, Universite catholique de Louvain, Belgium. talbot@poms.ucl.ac.be This text presents research results of the Belgian Program on Interuniversity Poles of Attraction initiated by the Belgian State, Prime Minister's Oce, Science Policy Programming. The scientic responsibility is assumed by the authors.

2 1 Introduction An Automated Guided Vehicle (AGV) is a driverless vehicle which canac- complish material handling tasks (i.e. load, transport and unload). An Automated Guided Vehicle System (AGVS) consists of a number of vehicles operating in a facility, usually controlled by a computer. The computer takes the dispatching and the routing decisions. AGV technology is a key factor in reducing material handling operating costs and increasing the reliability of material handling systems. However, the purchasing and installation costs are signicant, hence the design is an important decision that should be made carefully. In this paper, we analyze the design of a 2-stations system depicted in Figure 1. This research originated from a real case, which we do not describe in full details because of condentiality agreements. Imagine, for instance a transport system between a workstation and a warehouse. Products are processed at the workstation according to independent processes. After operations at the workstation, the products are transported by vehicles (AGVs) towards the warehouse. The warehouse sends also products loaded on vehicles towards the workstation, when they are required at the workstation. The vehicles are unloaded when they arrive at their destination and the empty vehicles join a storage area. We seek a model to estimate the minimal number of vehicles needed to guarantee some target service level expressed in terms of mean waiting time. To achieve this we also need to design dispatching rules in order to utilize the vehicles in the best way. Our solution procedure starts by computing the necessary ll rate in order to respect the maximum mean waiting times. The dispatching rules and the minimum number of vehicles are then determined to guarantee the ll rate. workstation wharehouse = AGV storage area = unloading = loading Figure 1: Environnement The content of the paper is as follows. Section 2 presents the AGV environment and the motivations of the analysis. Section 3 presents a literature review for similar AGVS design problems. In Section 4, we present our solution procedure. In section 5, we evaluate the quality and the relevance of 1

3 our approach. We conclude in section 6. 2 AGV Environment and research motivations Consider a 2-stations system depicted in Figure 2. Products arrive ateach loading station to be transported to the other station. For each station i the arrival process is characterized by a mean interarrival time 1 and i a variance i 2. A product waits in queue until a vehicle is available to transport it towards the other station. When arriving at the other station, the product is unloaded and leaves the system (and thus the environment analyzed). The emptied vehicle joins the storage area and becomes available for a transportation request at the other station. We introduce the following assumptions: The interarrival times form a renewal process (interarrival times are IID random variables). The arrival processes of the two stations are independent. The queue of products at the two stations have innite capacity. Only one product can be transported by a vehicle at a time. The products are dispatched according to the First-Come-First-Served rule. The loading time and the unloading time are neglected (equal to zero). The internal travel times within the two stations are neglected (equal to zero). We denote by the minimum time separating the sending of two vehicles at a station (technology limit). The travel time T between the two stations is deterministic and is independent of the load being conveyed by the vehicle. The vehicle storage area at each station is unlimited. We will only study the system in its steady state. The products are supposed to be transported as soon as possible. The objective is to minimize the average time separating the entry of the product in the waiting queue at one station and the unloading of this product from a vehicle at the other station. Since the travel time T between the two stations is constant, it is equivalent to minimize the average product waiting time before loading. Hence, the level of performance is measured by the product mean waiting time at station i, denoted by Wq i. We introduce the following notation: x i (t) =Thenumber of vehicles in the storage area of station i at time t. 2

4 λ1,σ1 T y2(t) c1(t) = AGV storage area = unloading x1(t) x2(t) = loading c2(t) y1(t) λ2,σ2 Figure 2: Environnement y i (t) =Thenumber of vehicles (loaded or empty) traveling towards station i at time t. c i (t) =Thenumber of products waiting at station i at time t. N i (t) = The cumulated number of products that arrived from time 0 to time t at station i. X = The total number of vehicles in the system = x 1 (t)+y 1 (t)+x 2 (t)+ y 2 (t) for all t. i; = Station preceding station i. The purchasing and the installation costs of vehicles are often signicant. So we seek a model to nd the minimal number of vehicles X needed to guarantee some predened minimal service level expressed in terms of mean waiting time for transportation requests. To be able to determine this minimal number of vehicles, we rst need to determine how to best utilize the vehicles. Therefore we must determine dispatching rules. Dispatching rules are used to manage the stock ofvehicles at the two stations. In order to prevent a shortage of vehicles at a station to cover transportation requests, the dispatching rules determine whether to send empty vehicles from one station towards another. In this way, available vehicles are divided among stations to provide the best service level. 2.1 Research Motivations In this section, we explain how the number of vehicles and the dispatching rules have an inuence on the mean waiting time. First of all, the number of vehicles will never need to exceed 2T. 2T is the total transportation time for a round trip and is the minimal time interval between two departures of vehicles (technology limit). Therefore, a lower bound on the mean waiting time is observed when there are 2T vehicles in the system and when one vehicle leaves each station every units of time. This lower bound is the mean waiting time of a G=D=1 system with service 3

5 time, denoted by WD(). But, the objective is to minimize the number of vehicles. When the number of vehicles is X, it is possible to send one vehicle from each station every 2T units of time. Therefore, we observe that the mean X waiting time of a G=D=1 system with service time 2T, denoted by WD( 2T ) X X is a level of performance easily reached. WD( 2T ) is an upper bound on the X optimal mean waiting time for a given number X of vehicles. Moreover, we observe that the role and the impact of the dispatching rules and the number of vehicles depend on the interarrival time distribution. Figure 3 represents two possible interarrival time distributions with the same mean interarrival time ( 1 ). In the rst case, the interarrival time distribution is characterized by a lot of randomness. This is the case, for instance, when the interarrival time process follows an exponential distribution. In the second case, the distribution is regular. This is the case when the interarrival times are relatively constant and correlated (which is contrary to our IID assumption). In the two cases, a decrease of the interarrival time means that a peak of transportation demand occurs. Let us assume that " is the number of arrivals during a peak. So, when the interarrival time distribution is random (respectively regular), we expect that " is small (respectively large). Instantaneous interarrival time Instantaneous interarrival time 1/ λ 1/λ ε t ε t Figure 3: Interarrival time distribution When the distribution is random (small "), we can benet from the randomness because, before the beginning of a peak, it is possible to constitute a stock of vehicles in order to cover the demand peak. A minimal number of vehicles is required in order to have enough vehicles in stock at the stations to cover the peak. The dispatching rules have the purpose of constituting this stock, and the quality of the dispatching rules is measured in terms of mean waiting time in this case. If the distribution is regular (large "), the dispatching rules are not useful because it is not possible to constitute a stock of vehicles, before the 4

6 beginning of the peak and large enough to cover the peak. In this case, a solution is to send the vehicles regularly, every 2T units of times and the X upper bound on the minimal mean waiting time will be reached. So, only the number of vehicles can inuence the performance of the system in this case. Figure 4 represents, for a given number X of vehicles, the evolution of the mean waiting time as a function of the length " of the demand peak. The mean waiting time approaches WD() when the demand peak is short (random) and approaches WD( 2T ) when the peak is long (regular). Between X these two extreme cases, the mean waiting time depends on the quality of the dispatching rules. Curve (2) represents the performance of a better dispatching rule than that of Curve (1). Mean waiting time WD(2T/X) (1) (2) WD() ε Figure 4: Evolution of the mean waiting time as a function of the peak length In conclusion, if the interarrival time distribution is regular, only the number of vehicles inuences the performance of the system. If the interarrival time distribution is random, the dispatching rules and the number of vehicles are both important. In this paper, we suppose that the interarrival times are IID random variables (see assumptions), so the interarrival time distribution is random and the objective is to determine the number of vehicles and the dispatching rules in order to achieve a given mean waiting time performance. Since this model is supposed to operate in a highly stochastic environment, we anticipate that the dispatching rules will play an important role. Better dispatching rules will allow to buy less vehicles, in order to achieve a given performance. 5

7 3 Literature review The purchasing and installation costs of AGVS are signicant, and hence the design is an important decision that should be made carefully. For a given network layout, AGVS design is primarily concerned with the determination of the number of vehicles needed. The required number of vehicles is aected by several factors: number of transportations, travel time, network layout,... Newton (1985) estimates the number of vehicles needed in an AGVS by means of a simulation experiment. Wysk, Egbelu, Zhou and Ghosh (1987) and Egbelu (1987) use simulation to test the quality of solutions from analytical deterministic models based on the total transportation distance. van der Meer and de Koster (1997, 1998) used a simulation model to compare decentral control (the vehicle drives in assigned loops from workstation to workstation) with central control (a central controller is used for dispatching of loads or vehicles). They also classied dierent dispatching rules for uni-load vehicles. It was concluded that central control outperforms decentralized control. van der Meer and de Koster (1999) look at the robustness of the classication when multi-load vehicles are used, and also see how the classication holds up when loads are released in batches at the receiving station. In those papers, the vehicle is sent toaworkstation when it is free. They use a "push" approach. We use a "pull" approach when a workstation needs a vehicle, it calls one. Moreover, we try to provide an analytical model and use simulation only to validate our model. According to Askin and Standbridge (1993), considerable theory exists aiding the determination of the number of vehicles needed to support the material handling requirements. Although the actual problem is stochastic, most of them use a simpler deterministic model to estimate requirements. These models estimate the number of vehicles needed as the ratio between the total vehicle utilization time and the eective timeavehicle is available. The total utilization time can be divided into ve components: loaded travel time, load time, unload time, unloaded travel time and waiting and blocked time. Rajotia, Shanker and Batras (1998) describe the analytical model usually employed for the loaded travel time estimation. The loaded travel time required can easily be determined from the information available on material ow intensities and travel time matrix among various pairs of pick up/delivery (P/D) stations. Likewise, specications on load and unload times can be multiplied by the number of loads to nd total load and unload times. Maxwell and Muckstadt (1982) did pioneering work in analytical modeling of operation features of an AGVS. They estimate for instance the empty vehicle travel time by computing the net ow at each P/D station as the dierence between the total number of unit loads delivered there and the total number of unit loads picked up from there. It represents the num- 6

8 ber of empty trips into or out of that station. A standard transportation problem was formulated which assigned empty trips between various stations minimizing the total empty vehicle travel time. Authors such as Srinivasan, Bozer and Cho (1994), Mahadevan and Narendran (1990), Lin (1990), Malmborg (1990) proposed similar approaches to estimate the loaded and unloaded travel times and the number of vehicles. Askin and Standbridge (1993) discussed several research studies for the purpose of empty travel time estimation and proposed a new model. The model begins with the objective of minimizing empty vehicle trips. the frequency of such trips from/to a station is constrained by the total number of loads delivered at/picked up from that station. Because of these additional constraints, the number of empty trips is greater than the net ow as considered by Maxwell and Muckstadt (1982). They used simulation to validate the approach and the results indicate that the model underestimates the minimum AGV requirements but provides results close to the simulation results. Mahadevan and Narendran (1991) take into consideration the limited local buer. Mahadevan and Narendran (1994) demonstrate the use of a two stage approach for determining the number of vehicles, the layout of tracks, the dispatching rules for the vehicles and the provision of control zones and buers. The required number of vehicles is estimated using the model presented in Mahadevan and Narendran (1990) in the rst stage. In the second stage, the eects of AGV failures and AGV dispatching rules on the system performance are observed through simulation studies. The last component of vehicle utilization is waiting and blocking time. Waiting time refers to the time an AGV spends while waiting empty atadelivery station for assignment of the next load transportation task. Blocking time is the time an AGV remains in a blocked state because of trac congestion while it is traveling either loaded or empty. According to Askin and Standbridge (1998), it is not possible to compute such waiting and blocking times in advance because they are dependent upon the AGV eet size, a parameter which has to be determined rst. They depend also upon the dispatching and routing strategies, guidepath layout, and vehicles clearance procedures at crossings. Ko (1987) presented empirical approximation of vehicle idleness (empty travel time and idle waiting time) and blocking time factors. The range of values suggested is 10 to 15% of total loaded vehicle travel time for each of the two. In all these approaches, the number of vehicles is determined using deterministic approaches based on the mean transportation load. The performance is usually observed by simulation. Our approach is to impose a priori a system performance target, and develop analytical models to compute the number of vehicles required to achieve it, taking into account thestochastic aspect of the demand. Simulation is only used to validate our approach. Closer to our methodology, Tanchoco, Egbelu and Taghaboni (1987) model the AGVS as a closed queuing network in order to determine the 7

9 minimum number of vehicles required. The eectiveness of this modeling approach is compared to a simulation based method. Since their analysis ignores the time a vehicle spends traveling empty, it generally underestimates the actual number of vehicles required. Mantel and Landeweerd (1995) developed a hierarchical queuing network approach to determine the number of AGVs. Johnson and Brandeau (1993) propose analytical models for the design of an AGVS. Taking into account the stochastic nature of the demand process, they apply results from queuing theory to estimate system congestion. They investigate an AGVS that delivers material from a central storage depot to shop oor workcenters. The objective is to determine which workcenters to include in the AGV network and the number of vehicles required to service those workcenters, in order to maximize the benet of the network, subject to a constraint that the average waiting time at each station does not exceed a predened limit. The benet of each station is dened as the net present value of direct labor savings of delivering material via the AGVS minus the cost of constructing a pickup and dropo station at that workcenter. Their contribution is to solve the problem using an analytical model. The pool of vehicles is modeled as an M=G=c queuing system and the design model is formulated as a linear binary program with a set of nonlinear constraints to model average waiting time at each station. The constraints are expressed by an approximating queuing formula. The approximation attempts to adjust the M=M=c formula to account for service time variations. Thonemann and Brandeau (1997) introduce also an analytical model for the design of a multiple-vehicle AGVS with multiple-load capacity operating under a "gowhen-lled" dispatching rule. The AGVs deliver containers of material from a central depot to workcenters throughout the factory oor. The demand of the workcenters and the time until delivery are stochastic. They develop a nonlinear binary programming model to determine the optimal partition of workcenters into zones, the optimal number of AGVs to purchase, and the optimal subset of workcenters to service by the AGVS, subject to constraints on maximum allowable mean waiting time for material delivery. They develop an analytical expression for the mean waiting time and present an ecient branch-and-bound algorithm that solves the AGVS design model optimally. Johnson and Brandeau (1993) and Thonemann and Brandeau (1997) are closer to our methodology because they propose an analytical model taking into account the stochastic nature of the demand process and they apply results from queuing theory. But in our paper, we analyze and suggest dispatching rules in order to "best" utilize the vehicles to achieve a given mean waiting time with a smaller number of vehicles. Almeida and Kellert (2000) study job shop like exible manufacturing system (FMSs) with a discrete material handling device and machine transfer blocking. They propose an analytical queuing network model to evaluate 8

10 the quantitative steady-state performance of such FMSs. The FMS complex devices are structured in order to prevent deadlocks from occurring. So, in this paper, they suppose a light load. In our paper, the system is heavily loaded. Bozer and Kim (1996) determine optimal or near optimal transfer batch sizes in manufacturing systems and develop an analytical relationship, issued from queuing theory, between the material handling capacity and the expected work in process in a manufacturing system. The models developed by Almeida and Kellert (2000) and Bozer and Kim (1996) are not applicable to our problem because the aim is not the same. But the methodology is identical. They present an analytical model based on queuing theory and the results are validated against discrete event simulations. AGVS design is also concerned with dispatching rules. According to Taghaboni and Tanchoco (1988), dispatching involves the selection rule, or methodology, that is used for selecting a vehicle for pick up of delivery assignments. The problem addressed here is not exactly the same. More precisely, the dispatching rules are used here to organize or distribute the stock of empty vehicles in order to best cover the transportation demands. Due to the simple structure of the transportation system, the dispatching rules are not used to assign vehicles to specic transportation requests. 4 AGVS design model 4.1 Framework The level of performance is measured by the mean waiting times at the 2 stations, denoted by Wq 1 and Wq 2. As the purchasing and the installation costs of AGVs are signicant, we try to minimize the number of vehicles needed to guarantee a given target performance level. The waiting time (i.e. performance) depends on the probabilityofhaving a vehicle available at that station. Indeed, if a vehicle is available at the station, the time between the departures of successive vehicles is only limited by the technology limit (). If there is no vehicle at the station, the product has to wait for the arrival of a vehicle. So we introduce the concept of ll rate, the probability tohave avehicle available at the station to satisfy a request, at the moment the request occurs. The dispatching rules can be easily designed in order to guarantee a given ll rate by using a classical Reorder Point Inventory Policy (RPIP) from inventory management theory. As our primary objective is to guarantee directly a target mean waiting time, we needtoinvestigate the relationship between the ll rate and the mean waiting time. So, in the rst step of our model, we provide an expression for the mean waiting time in terms of the ll rate (subsection (4.2)). Then we design the dispatching rules using the RPIP to guarantee a given ll rate (or 9

11 waiting time Wq, through subsection (4.2)) (subsection (4.3)). Finally, we determine the number of vehicles in the system needed for the dispatching rules to work correctly (subsection (4.4)). 4.2 Finding the appropriate ll rate We suppose that the ll rate is given. We try to provide an expression for the mean waiting time Wq in terms of the ll rate. We consider each station independently in order to model the stations as a queuing process. This independence hypothesis will be satised if there are enough vehicles in the system. The products enter a waiting queue according to some interarrival times distribution with a mean interarrival time 1 and a variance 2. The service time is the time between successive departures of vehicles from the station. To estimate the mean waiting time we use the following approximation proposed by Marchal (1978) for a G/G/1 system, Wq = (2 + Var(S)) Var(S) 2(1 ; ) 1+ 2 Var(S) (4.1) where S is the service time distribution, Var(S) is the variance of S, = E(S) ande(s) is the expected service time. So, in order to compute Wq, we need to estimate the expected service time E(S) and the variance Var(S). We provide some approximations of E(S) and Var(S) Approximation of E(S) and Var(S) We assume here that the dispatching rule used will ensure a ll rate. Thus when a request (product) arrives, there is a probability of having at least one vehicle available and a probability 1; that the product should wait for the arrival of a vehicle. (1) With a probability, there is at least one vehicle available when a product arrives. In this case, the service time is distributed according to a constant distribution of parameter (technological limit). So, with probability, the expected service time E(S) is. (2) With a probability 1 ;, there is no vehicle available at the station when a product arrives. The product has to wait until a vehicle arrives. So, the service time distribution will correspond to the vehicle interarrival distribution at that station. The vehicles are either loaded or empty. We consider the worst case the second station does not send any product. In that case, using the RPIP, it sends only empty vehicles when the rst station requires one. The rst station requires a vehicle each time a product arrives at that station (see subsection (4.3)), consequently the time between arrivals of empty vehicles at 10

12 the rst station will have the same distribution as the time between arrivals of a product at that station. Therefore, we take the following approximation: with probability 1;, the expected service time distribution is 1. So, together for the two cases, the expected service time is E(S) =+(1; ) 1 : (4.2) The variance of the service time distribution is Var(S) =E(S 2 ) ; E(S) 2 : Following the same reasoning, we separate two cases: with probability the service time is, with probability 1;, the service time has the same distribution as S. Consequently, one obtains E(S 2 )= 2 +(1;)( ) And thus, Var(S) =(1 ; )( ; 1 )2 +(1; ) 2 : (4.3) Knowing E(S) and Var(S) from (4.2) and (4.3), we determine Wq() using (4.1), for any given ll rate. For a target mean waiting time Wq, we determine the appropriate ll rate as = minf : Wq() Wq g The expression (4.1) of Wq has been developed in terms of the ll rate. Wq depends on,, and 2, assuming that the ll rate can be guaranteed. Now, we want to determine dispatching rules able to achieve the ll rate. 4.3 Dispatching rules Knowing a target ll rate, the dispatching rules are used to guarantee the ll rate at each station using the classical Reorder Point Inventory Policy from inventory management theory. Namely, a station \orders"vehicles to the other station when its inventory positions (number of available vehicles) is too low to guarantee the ll rate. Knowing that any request for a vehicle at station i has to wait on average Wq i. When station 1 asks for an empty vehicle, it has to wait on average T + Wq 2 units of time (called order delivery or service time) for the vehicle to arrive. Therefore, to ensure a ll rate at station 1, we must make sure, with probability, that, at any time, there are enough vehicles traveling to station 1, or waiting at station 1, to cover the transportation demand of station 1 during T + Wq 2 units of time. 11

13 Recall that N i (t) is the cumulated number of products that arrived from time 0 to time t at station i. If the current time is t,(n 1 (t+t +Wq 2 );N 1 (t)) is the number of products that will arrive at station 1 during the next T + Wq 2 units of time. We want tomake sure that the number of vehicles traveling to station 1 is high enough so that the probability ofthe number of arrivals being greater does not exceed 1 ;. We dene S 1 as the smallest number which satises the inequality P (N 1 (t + T + Wq 2 ) ; N 1 (t)) S 1 1 ; : We want tohave at least S 1 vehicles available at station 1 at any time. We have the same expression for station 2. In general, for station i, S i is the smallest number which satises the following inequality: P (N i (t + T + Wq i; ) ; N i (t)) S i 1 ; : (4.4) In other words, the probability of not having enough vehicles (i.e. more demands than S i ) is at most 1 ;. The dispatching rules try to anticipate the possible demand by calling empty vehicles if and when necessary. So, we introduce the concept of net stock s i (t), to model the available vehicles at station i at time t. s i (t) is dened as the sum of vehicles at station i, plus the vehicles en route to station i, and minus the products waiting at station i (corresponding to vehicles not available to transport additional requests). Formally, s i (t) =x i (t)+y i (t) ; c i (t). The net stock s i (t) has to be higher than S i, at any time t, in order to satisfy the ll rate. This suggests the following dispatching rule: When the net stock of vehicles at station i, s i (t), is less than S i, the other station sends an empty vehicle to station i. 4.4 Number of AGVs Our dispatching rules suppose there are enough vehicles so that a request for an empty vehicle waits on average Wq i at station i. We will now tryto determine the minimum number of vehicles required to ensure this First observation It is obvious that the number of vehicles X must be higher or equal to S1+S2. Indeed X = x 1 (t)+y 1 (t)+x 2 (t)+y 2 (t) and the net stock s i (t) = x i (t)+y i (t) ; c i (t) must be at least S i,fori =1 2. But we need more than S1+S2 vehicles. Indeed, if the system has just S1+S2 vehicles, there is a risk to see the vehicles running without interruption only to satisfy the dispatching rules (empty transportation requests). If 12

14 station 1 sends a vehicle, its net stock becomes less than S1, hence station 1 ask for a vehicle. Station 2 sends this vehicle and its net stock becomes less than S2, and so on. This does not allow to satisfy the ll rate requirement. So, Let us dene n, as the number of vehicles above the minimum of S 1 + S 2. That is n = X ; (S 1 + S 2 ). We try to determine the minimum value of n needed for the dispatching rules to achieve jointly (i.e. at the two stations) a ll rate Number of vehicles en route To achieve a ll rate, we need - with probability - at least one vehicle available at each station. So, we want, with probability, more vehicles in the system than the vehicles en route. If we know the maximum number of vehicles en route, we can deduce the number of vehicles required. In this section, we try to approximate the number of vehicles en route. The numberof vehicles en route is the number of vehicles sent by the two stations. A station sends loaded vehicles at rate i ( i is given) and empty vehicles at the rate i (according to the dispatching rules). We assume that the time between successive vehicles on a link (between two stations) has the same type of distribution as the interarrival time process of demand, with the distribution being scaled to take the empty vehicles into account. That is, f l i(x) =f a i (x + ) where fi l () is the probability density function of times between vehicles on a link. fi a () is the probability density function of times between arrivals at station i. Let N 0 i (t) be the cumulated number of vehicles that arrived at station i according to the distribution fi l () from time 0 to time t. We want, with probability, more vehicles in the system than the number of vehicles en route. In other words, we want with a probability 1;, more vehicles en route than the numberofvehicles in the system. The number of vehicles en route at time t between station i; and station i is dened by Ni 0(t + T + Wq i; ) ; Ni 0(t). So we determine S0 i for each link to be the smallest number satisfying P (Ni(t 0 + T + Wq i; ) ; Ni(t)) 0 S 0 i 1 ; : (4.5) If we adopt the same reasoning for the two stations, we can thus say that, with probability, there is maximum S1 0 + S0 2 vehicles en route. In 13

15 conclusion, knowing the rates of empty vehicles ( 1 and 2 ), the minimum number of vehicles necessary to cover the total transportation demand is thus S S 0 2. In the following section, we develop an expression to estimate 1 and Estimation of i We try to estimate i, the rate at which station i sends empty vehicles. Station i sends empty vehicles to the other station when the other station requires an empty vehicle (we suppose that a station can send a vehicle as soon as the other station requires one). A station requires an empty vehicle when its net stock goes below the limit S i (when s i (t) S i ). We dene a stochastic process E(t) =s 1 (t);s 1. E(t) is the number of vehicles abovethe dispatching rule limit at station 1. E(t) will increase by one unit each time a product is sent from station 2 to station 1. E(t) will decrease by one unit each time a transportation request arrives at station 1. When E(t) = 0and station 1 receives a request then the net stock would go below the minimum threshold of the dispatching rule and consequently anemptyvehicle should be sent from station 2 to station 1. We have the same situation at the other direction when E(t) = n. Indeed, if E(t) = s 1 (t) ; S 1 = n, there is no vehicle above S 2 at station 2. Station 2 will require a vehicle if E(t) = n and if it receives a transportation request. So 2 = P (E(t) = 0) 1 and 1 = P (E(t) =n) 2. In order to approximate 1 and 2,we have to estimate the probability that E(t) = 0 and E(t) = n. To do that, we approximate the stochastic process E(t) by a Markov process M(t) where we assume the times between transitions are exponentially distributed with parameter 1 and 2 respectively. This corresponds to Poisson arrival processes for the products. Our simulation results (an example is given in section 5:1:3) show that this approximation gives satisfactory results. Figure 5 represents this Markov process. λ2 λ2 λ n λ1 λ1 λ1 Figure 5: Markov Process So, the expression of 1 and 2 are: 1 = P n 2 (4.6) 2 = P 0 1 (4.7) 14

16 If the demands are the same at the two stations ( 1 = 2 ), all states are equiprobable, and P 0 = P 1 = P n = ::: = 1. If the demands are dierent n+1 at the two stations, then P n = n P 0 and P 0 = 1; where = 1; (n+1) Conclusion We nowhave two equations in order to nd the number of vehicles X. The rst one nds the rates i of empty vehicles from n (or X, using (4.6) and (4.7)) and the second one estimates the minimal value for X = S S 0 2 from i estimates (using expressions (4.5)). We solve this system in an iterative way. We propose the following simple procedure. (1) For a given waiting time Wq,we compute the corresponding ll rate using expression (4.1). (2) To achieve the ll rate, we compute S1 and S2, using expression (4.4). (3) We pose n and we compute 1 and 2, using expressions (4.6) and (4.7). (4) We compute S1 0 and S2 0 and X = S1 0 + S2 0, using expression (4.5). (5) If X ; (S 1 + S 2 ) >n, increase n and go back tothestep2. If X ; (S 1 + S 2 ) <n, decrease n and go back to the step 2. If X ; (S 1 + S 2 )=n, n is optimal, go to step 5. (6) X = S1+S2+n We did not study the theoretical convergence property of this procedure. But a very small number of iterations were needed in all test cases. 5 Evaluation of the quality and the relevance of the model In this section, we evaluate the quality and the relevance of our approach. First, we carry out simulations of the system using the Extend Software (see Extend v4.1.3, Imagine That, Inc., 1998, San Jose). Then, we show that a deterministic approach or a pure queuing theory approach can not achieve results of similar quality. 5.1 Evaluation of the quality of the model We carried out four series of simulations. The travel time is 100 units of time and the technology limit is one unit of time. The simulations vary in terms of interarrival time processes. In the rst simulation, the interarrival time processes at the two stations are exponential and asymmetric. In the second simulation, the interarrival time processes are exponential and symmetric. In the third simulation, the interarrival processes are non-exponential and 15

17 asymmetric. In the fourth simulation experiment, we evaluate the robustness of our solution. We compute, for dierent ll rates, the mean waiting time (Wq), the dispatching rules (S i ) and the number of vehicles (X) needed. In the simulation, we compare the observed ll rate ( obs ) to the given ll rate () and the observed mean waiting time (Wq obs ) to the computed mean waiting time (Wq). The four series of simulations relate to 10 runs of 10:000 units of time. The tables below present the computed and simulated quantities with their 95% condence intervals. (1) Asymmetric exponential demands The interarrival time processes at the two stations follow an exponential distribution, with a mean of 2 time units at station 1 and 4 time units at station 2. The simulation results are given in Table 1. objective design prediction observation S 1 S 2 X Wq 1 Wq 2 1 obs 2 obs Wq 1 obs Wq 2 obs 99% % % % % Table 1: Asymmetric exponential processes For instance, for a ll rate of 90%, station 2 sends a vehicle to station 1 if the net stock ofvehicles at station 1 is less than 60 vehicles. The simulation presents a ll rate of 94% and 95%, better than what we expected. The mean waiting time calculated (0:94 and 0:76) seems tobeagoodapproximation of the mean waiting time simulated (0:84 and 0:6). To achieve a ll rate of 100%, the number of vehicles would be 2T and the best level of performance WD() is reached by sending the vehicles at regular time intervals. In this case, 2T = 200, with a mean waiting time of 0:5. Our model proposes 134 vehicles for a ll rate of 99% and achieves a mean waiting time of 0:54 units of time. This result is interesting knowing that the purchasing and the installation costsofvehicles are often very large. (2) Symmetric exponential demands The interarrival time processes at the two stations follow an exponential distribution, with a mean of 2 time units. The corresponding results are presented in Table 2. The observed ll rates are larger than what we expected in our analysis. 16

18 objective design prediction observation S 1 S 2 X Wq 1 Wq 2 1 obs 2 obs Wq 1 obs Wq 2 obs 99% % % % % Table 2: Symmetric exponential processes The mean waiting time calculated seems to be a good approximation of the mean waiting time simulated. (3) Non-Poisson process The interarrival times of transportation requests follow a log-normal distribution with a mean time of 2 units of time for station 1 and a variance of 2, and a mean time of 4 units of time and a variance of 8 for station 2. Table 3 presents the results. objective design prediction observation S 1 S 2 X Wq 1 Wq 2 1 obs 2 obs Wq 1 obs Wq 2 obs 99% % % % % Table 3: Non-Poisson processes The observed ll rates are larger than what we expected for the large ll rates. According to Marchal (1978), the performance of the mean waiting time approximation deteriorates as the service times or interarrival times deviate further from exponentially. (4) Robustness 1 In real cases, the interarrival mean time varies with time around (t) the nominal rate 1. In our approach, we determine dispatching rules and the number of vehicles with a nominal rate, but if the mean varies with time, the level of performances should not be too degraded. We expect that the degradation will be proportional to the dierence between (t) and. We have simulated the worst case in which dispatching rules and the number of vehicles are based on exponential interarrival time processes with an interarrival mean time of 2 and 4 units of time, respectively at the two stations, but the real interarrival times have a mean of 4 and 2 units of time, respectively. In other words we expect the heavy ow to be from 1 to 2, but we observe the opposite. We see that the level of performance (Wq) is not too degraded. The simulation results are given in Table 4. The ll rate of station 2 is very small because there is constantly a 17

19 objective design prediction observation S 1 S 2 X Wq 1 Wq 2 1 obs 2 obs Wq 1 obs Wq 2 obs 99% % % % % Table 4: Robustness backlog of vehicles at this station. In fact a queue of products will appear at station 2 that will increase until station 2 calls in enough empty vehicles. The simulation shows that at some point this happens and that the number of products in queue (and thus the waiting time of products) does not grow indenitely. In conclusion, we observe that the suggested model oers a good degree of approximation for the ll rate and for the mean waiting time. Moreover the dispatching rule ensures that the system stays in a stable state even if the arrival process is not as expected. This observation gives us an idea for installing a self-adapting system. At regular intervals, we will observe the ll rate for the preceding period and if the ll rate observed exceeds the ll rate to be achieved, we decrease the dispatching rules limit. If not, we increase the dispatching rules limit. 5.2 Evaluation of the relevance of our model We conclude this evaluation section by showing that our results represent an improvement over a naive approach. Indeed, our model proposes to run the system with much less vehicles than the number of vehicles obtained through other simpler design approaches. For instance, in our real case, the rm decreased the number of vehicles by 30%. Moreover, our dispatching rules guarantee smooth operating conditions even with less vehicles. A deterministic approach, as the one proposed in Rajotia et al. (1998), Maxwell and Muckstadt (1982) and others, would place all the vehicles at equidistance on the loop and does not take into account the stochastic aspect of the arrival process. In this approach, the number of vehicles needed is 2T. For instance, if = 0:5 and T = 100 units of time, we should need 100 vehicles. The system with 100 vehicles will achieve a very bad level of performance. We need more than 100 vehicles. Similarly, the anticiped level of performance of a simple queuing approach, with Exponential interarrival time distribution, placing all vehicles at regular time intervalsisthemeanwaiting time of a M=D=1 model with a service time of 2T X. where = X 2T. Wq = 2( ; ) 18

20 In our example, to have a mean waiting time less than 1 unit of time, such a simple queuing approach needs 160 vehicles. In our approach, to achieve a mean waiting time Wq of 0:94 units of time, we only need 120 vehicles (see Table 1). So, we see that our model allows one to compute a smaller number of needed vehicles for the same performance, and proposes adequate dispatching rules to achieve the required performance. Of course, with respect to the existing literature, we have been able to improve the models used to estimate the number of vehicles because our transportation network is much simpler. It is the goal of further research to extend the models to more complex transportation settings. 6 Conclusions and practical implications In this paper we address a design problem of a 2-stations AGV system. The objective of this design phase is to achieve a given performance level. The level of performance is the mean waiting time at the stations. We use a combined approach based on three dierent operations research techniques: inventory management, queuing theory and stochastic processes. Our approach starts by determining a ll rate necessary to achieve the mean waiting time. To achieve a ll rate, each station has to maintain a sucient number of vehicles (at the station or traveling to the station) to cover the demand. To assure this, we determine dispatching rules, according to which a station asks empty vehicles to the other station. To assure that the dispatching rules work correctly, there must be enough vehicles in the system. We propose an iterative procedure to determine the minimum number of vehicles needed to allow the dispatching rules to work correctly. We simulate our model and we prove that our model oers better results than simpler deterministic and queuing approaches. Finally, our model lends itself well to a self-adapting system. Indeed, if the level of performance (Wq) is degraded, the dispatching rules and the number of vehicles can be modied. It is particularly interesting in the case where the mean interarrival process varies with time. Our design model is based on the assumption that there are two stations and that the product is always unloaded at the station following the loading station. One area of future research is to investigate more complex guidepath layouts. First, we want to extend our results to a closed loop of n stations. Secondly, we want to allow the products to be unloaded at any station. In this case, all the vehicles traveling to a station may not be considered to be available at the station when they arrive and we have to know the destination of the product. Finally, in some environments, the guidepath layout might not consist of closed loops. An interesting area for future research would allow the vehicles to use more general routes between stations. 19

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