DISCRETE AND CONTINUOUS Website: http://math.smsu.edu/journal DYNAMICAL SYSTEMS–SERIES B Volume 1, Number 2, May 2001 pp. 209–218 DYNAMICAL SYSTEMS AND OPERATIONS RESEARCH: A BASIC MODEL Leonid Bunimovich School of Mathematics, Georgia Institute of Technology Atlanta,GA 30332 Abstract. Operations Research and Logistics are the areas, where tradition- ally only stochastic models were applied. However, recently this situation started to change, and dynamical systems are becoming to be recognized as the relevant models in manufacturing, managing supply chains, conditioned based maintenance, etc. We discuss the simplest basic model for these pro- cesses and prove some results on its global dynamics. The general approach to a management of such processes (Stabilization of a Target Regime or STR- method) is outlined and illustrated. 1. Introduction. A typical logistics system deals with a set of servers and a set of customers. Each customer should be serviced by some subset (or perhaps by all) servers. In a general situation some number of customers is waiting for a service at each server. These groups of customers form queues. The problem is to find such policy which minimizes a throughput in this system, i.e., an average time that a customer spends in a system (an average service time). By policies here one means a type how a service is organized. For instance, the most popular policy is first-in-first-out, which means that in each queue such customer has a priority which entered the system first. However, in many situations other policies are used. In concrete systems, servers and customers may have quite different meanings. For instance, in production lines customers usually are unfinished items, which are waiting to be processed at work stations (servers) which form this line. In other situation customers are the jobs which are waiting for servers to perform these jobs, e.g., a scheduled flight is waiting for an aircraft. The last system deals with a fleet of assets, which should be extensively used before a scheduled maintenance [12]. The same type of models are referred to as supply chains. In these models pro- duction of servers goes to customers (e.g., to the shops which sell these items). Such systems are difficult to manage. Therefore they include also buffers (warehouses) where the produced items are kept before shipping to the shops. These systems in practice form very large networks of such elements which makes their management to be very difficult. Moreover, even individual elements of such networks could be (and usually are) very complex. For instance, a typical warehouse contains tens of thousands stock keeping units (sku’s). Therefore, an optimal management of each element of a supply chain constitutes a separate complicated problem. All these systems are handled by operations research and logistics, where they traditionally are considered as stochastic systems. The large area called a queueing theory, of probability theory has been developed to address these problems. 1991 Mathematics Subject Classification. 34C35, 58F03, 58F05. Key words and phrases. Dynamical systems, operations research. 209 210 LEONID A. BUNIMOVICH However, recently the new approach to the optimal management of logistics systems has been developed. This approach is based on the ideas and methods of the theory of dynamical systems. This approach has been inspired by the experimental work [1, 2, 3] showed that deterministic models could be relevant for a modeling of logistics systems. One must mention here as well the pioneer paper [6], where it has been shown that different handling policies may lead to quite different types of behavior of a system. Moreover, the applications of the methods of nonlinear dynamics allowed to increase essentially a throughput in some production lines in the apparel industry and in warehouses [1, 2]. In this paper we prove some general results on a class of models which could be considered as the basic models of certain production lines, warehouses, condition based maintenance and supply chains. This basic model is relevant for any work sharing manufacturing, where each server can serve to any customer (or each worker has the skills to work at any work station). The basic model is the special case of the Bucket Brigade production lines [1, 2, 3, 5]. It is worthwhile to mention that the corresponding production lines were first modeled as stochastic systems [4, 11, 13]. But the approach based on their modeling as deterministic systems proved to be more efficient. Consider a production line where each item requires a single processing at each work station. (Much more complicated models arise in the re-entrant manufacturing [10], where each (or some) item must return to some work stations to be processed again. The first result on dynamics of the simplest models of this type were recently obtained in [8].) Suppose that the workers cannot pass each other and each worker carries a single item from station to station, waiting if necessary for the station to become available. In this model there are fewer workers than stations and workers are not allowed to work at the same station at the same time. When the last worker completes an item, he walks back to take over the time of his predecessor, who then walks back to take over the item of his predecessor and so on until the first worker walks back to the origin of the line to start a new item. This type of organizing workers on a flow line is called Bucket Brigade [1, 2, 3, 5, 9]. It is in use in apparel manufacturing and distribution warehousing. In this paper we prove some general results on the behavior of corresponding dynamical systems. The structure of the paper is the following. In Sect. 2 we consider a slightly more general than the Bucket Brigade model, which we call OWS (one way street) model and prove some results of its dynamics as well as on Bucket Brigades. Sect. 3 deals with a Basic Model and contains rather complete results on its global dynamics. The last Sect. 4 discusses the general algorithm of optimization of models of this type which is a generalization of the one proposed in [5]. 2. One way street (OWS) model and its dynamics. Consider a dynamical system generated by the motion of N particles in a straight segment [0,L], L > 0. Each particle moves with velocity vi(x) > 0, i = 1, 2,...,N, x ∈ [0,L]. Therefore it is the one way motion from the left to the right. Particles are allowed to pass each other. We will call this system a one way street model (OWS model). It describes the motion of assets in a fleet of assets along an axis of aging (from the appearance of a new asset till the time of its first maintenance), the motion of goods in a relatively simple supply chain, the motion of (unfinished) items along a production line (e.g., with parallel work stations) or the motion of a brigade of pickers, which works in several parallel aisles of a warehouse. The one way narrow street model (OWNS model) is a special case of OWS model where the particles are not allowed DYNAMICAL SYSTEMS AND OPERATIONS RESEARCH 211 to pass each other. Therefore, their positions xi(t), i = 1, 2,...,N, at any moment of time t satisfy the relations 0 ≤ x1(t) ≤ x2(t) ≤ · · · ≤ xn(t) ≤ L. If ith particle is faster than the (i+1)th particle then at some moment tˆpositions of these particles can become equal, i.e., xi(tˆ) = xi+1(tˆ). Such moments of time will be called blocking times. We assume that at any blocking time the ith particle (the faster one which is behind) instantly acquires the velocity of the (i + 1)th particle, i.e., vi(xi(tˆ)) becomes equal vi+1(xi(tˆ)) = vi+1(xi+1(tˆ)). After that these two particles move together with the velocity of (i+1)th particle 0 0 0 till the closest to xi(tˆ) point x > xi(tˆ), where vi+1(x ) > vi(x ). Such interval 0 [xi(tˆ), x ] we will call a blocking interval. In the same way we will refer to the time interval [t,ˆ t0], where Z x0 0 0 2. t = tˆ+ (x − xi(tˆ)) vi+1(x)dx (1) xi(tˆ) The main problem is to “organize” an evolution of this system so that its through- put be maximal, i.e., one needs to minimize the effect of blocking. Depending on the values of velocities vi(x), i = 1, 2,...,N, x ∈ [0,L] it may be possible that not just one but any (between one and N − 1) number of particles become blocked. This dynamics resembles a motion of cars in one-way narrow street, where a slower car is blocking all faster cars which move behind it. It is the reason why we name this system a OWNS-model. However, the dynamics of this model is not completely defined so far. To do so we need to say what happens when the last particle reaches the right end of a segment (street), i.e., xN (t) = L at some moment t. Here we have several possibilities. The one which must be chosen depends upon a real system (or a phenomenon) which we want to model. For instance, in condition based maintenance type of problems we may assume that [0,L] is a “life axis” of assets and an asset at the position L by a new asset placed at x = 0. This new asset could be a brand new one, or be the one that just has been serviced (repaired) and returned to work (to the “street”).