Dynamical Systems and Operations Research: a Basic Model

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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 traditionally 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 processes 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 production 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 throughput 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”). To make our model more realistic we then must assume that our particles are “aging” and brand new particles which substitute the “dead” ones may be in some way different (“modernized”) particles. To handle this situation we need to allow that particles’ velocities depend on time as well, i.e., vi(x) = vi(x, t), where a time coordinate t carries the information about the number of already replaced particles, etc. Therefore, our system becomes non-autonomous. However, there is a simplified version of the OWNS model which is called the Bucket Brigade (BB-)model [2, 3]. The BB-model describes e.g., the motion of pickers in a warehouse with idealized customers. An ideal customer orders all sku’s in a warehouse. In this case a warehouse model, which realistically must be stochastic, because each customer makes a random order, becomes deterministic. So, instead the assumption that the last particle (car) leaves our narrow street and gets substituted by a new one entering it, we assume that the last particle at the moment when it reaches the end point, xN (t) = L, instantly jumps back to the position of the previous particle, i.e., xN (t + 0) = xN−1(t). All other particles (besides the first one) do the same and instantly jump to the positions of their predecessors, xi(t + 0) = xi−1(t), i = 2, 3,...,N − 1,N. Finally the first particle at this moment instantly jumps to the origin, i.e., x1(t + 0) = 0. We will call such moments of time the moments of reset and a time interval between two consecutive resets a production cycle. 212 LEONID A. BUNIMOVICH

It can happen that at a moment of reset t, when xN (t) = L, the last particle is blocking several other particles. Suppose that xN−k(t) = xN−k+1(t) = ··· = xN−1(t) = xN (t) = L. Then according to our definition of the dynamics of OWNS- model we must make at such moment instantly (k + 1) consecutive resets. (Then k last production cycles become zero.) At first, the (N − k)th particle jumps to a position of its predecessor, i.e., xN−k(t + 0) = xN−k−1(t). Then all first (N −k−2) particles jump into positions of their predecessors i.e., xj(t+0) = xj−1(t), j = 2,...,N − k − 1, and then x1(t + 0) = 0. After that the (N − k + 1)th particle instantly jumps to xN−k−1(t), the (N −k)th particle jumps to xN−k−2(t) and so on until the 2nd particle jumps to the origin (where it blocks the 1st particle). Then we repeat this procedure (k − 1) times more. Now the dynamics of our model is completely defined. The phase space of our system is the N-dimensional simplex DL(N) = {X = (x1, x2, . . . , xN ) : 0 ≤ x1 ≤ x2 ≤ · · · ≤ xN ≤ L}. This dynamical system with continuous time has several natural global Poincarésections which correspond to the moments of resets. We choose the most convenient one {x1 = 0}. Thus we will follow the evolution of any orbit of our system just at the moments when all resets have been made, i.e., all particles assumed new positions by jumping into positions of their immediate predecessor and x1 becomes equal 0. Now our dynamical system is completely defined. We denote by f : D˜L(N) → D˜L(N) the corresponding Poincarémap and D˜L(N) = {X ∈ DL(N), x1 = 0}. We now assume that the following conditions on velocities of all particles hold

1) 0 < vi(x) < ∞ at any point x ∈ [0,L]. 2) vi(x) is a continuous function in [0,L]. The first assumption is fulfilled basically in all applications. Indeed, any worker or asset moves with a finite velocity. The second assumption is also rather general one, but it may be violated in some models, e.g., if the last (Nth) asset in a line is allowed to fail (to be broken) and then it must immediately be removed from the operation. In the last case a dynamics becomes not completely defined. It can be easily defined though for any policy of substitution of such failed asset (part). Lemma 1. If the conditions 1), 2) hold then the map f is continuous.

Proof. Let Y = f(X), X,Y ∈ D˜L(N). Suppose that during the production cycle which moves particles from the configuration X to the configuration Y no one particle has been blocked. Then f is clearly continuous at the point X ∈ D˜L(N). The same is true if no one particle has been blocked at the moment when the last particle reached the end of the segment [0,L], i.e., at the moment of reset. Therefore, only the event of blocking at the end of a production cycle should be specially considered. 0 0 0 0 Suppose that xi(t ) = xi+1(t ), where t is such that xN (t ) = L. We assume at 0 first that xN−1(t ) 6= L. 0 0 Denote V = max max vi(x). Let kX − Xk < /V , where kX − Xk = 1≤i≤N 0≤x≤L 0 0 max |xi − xi|. Then kf(X ) − f(X)k < because of the definition of a reset. 1≤i≤N 0 0 0 Suppose now that xN−k(t ) = xN−k+1(t ) = ··· = xN (t ) for some 1 ≤ k ≤ N −1. Again assume that kx0 − xk < (k + 1)/V . Then, according to the definition of a reset at a point of a multiple blocking at the end of our segment, we have kf(X0) − f(X)k < , kf (2)(X0) − f (2)(X)k < , . . . , kf (k)(X0) − f (k)(X)k < where f (s) is sth iterate of f. DYNAMICAL SYSTEMS AND OPERATIONS RESEARCH 213

Corollary 1. A OWNS-model has at least one ﬁxed point.

The corollary immediately follows from Lemma 1 because the phase space D˜L(N) is a closed set. The following statement is just a simple observation. However, it is crucial for the understanding of the dynamics of OWNS-model and for a search for its optimal regimes with a maximal throughput. Lemma . ∗ ∗ ∗ ∗ 2 (Absence of Blocking Lemma) If X = (x2, x3, . . . , xN ) is a ﬁxed point of the BB-model then at this point there is no blocking.

∗ ∗ Proof. Suppose that on contrary xi = xi+1 for some i = 1, 2,...,N − 1. Let j be the minimal number and k the maximal number such that ∗ ∗ ∗ ∗ ∗ ∗ xj = xj+1 = ··· = xi = xi+1 = ··· = xk−1 = xk. (2) We know that X∗ is a ﬁxed point, i.e., f(X∗) = X∗. Let t be a moment of a reset, i.e., xN (t) = L. Then there are two possibilities:

1) xi(t) = xi+1(t), i.e., the ith particle was blocked at the moment of reset by the (i + 1)th particle. 2) xi(t) 6= xi+1(t). ∗ ∗ If 1) holds then, according to the rule of particles’ reset, xi+1 = xi if and only if i > j, i.e., at a moment of reset ith particle was also blocked. It means, again according to the rule of particles’ reset, that ∗ ∗ xi(t) = xi = xi+1 = xi+1(t). (3) The relation (3) implies that the ith and the (i + 1)th particles are idle and don’t move during the time between resets at points X∗ to f(X∗), i.e., during. But this contradicts to our assumption that v`(x) > 0 for all ` = 1, 2,...,N, and for all ∗ x ∈ [0,L] unless xN = L. However, the last relation as well cannot be true. Consider now the condition 2). It can hold only if either i) xi(t) = 0 or ii) xi(t) = xi−1(t). The condition i) means that the ith particle was idle throughout the time between the resets at the points X∗ to f(X∗) and we come again to the same contradiction as above. Finally, the case ii) can be considered just exactly at the same way as i). Observe that the Absence of Blocking Lemma just says that it can be no blocking at any fixed point of the dynamical system with a discrete time which corresponds to the BB-model. However, during an interval of a real (continuous) time, which corresponds to a fixed point of this system with discrete time, some particles in principle could be blocked. It is easy to see that it can happen if and only if at least for one pair of neighboring particles vi(x) > vi+1(x) in some nonempty subset A ⊂ [0,L] and vi(x) < vi+1(x) in another nonempty subset B ⊂ [0,L]. However the following proposition shows that the orbit of the BB-model with continuous time, which corresponds to the (unique) fixed point of the BB-model with discrete time, does not contain any blocking, i.e., it has no blocking times. Proposition 1. (Absence of Blocking at Any Time) Let X∗ be the fixed point of the BB-model with discrete time. Then the corresponding to X∗ orbit of the BB-model with continuous time does not have any moments of blocking.

∗ ∗ ∗ Proof. Denote by X (t) = (x2(t), . . . , xN (t)) the orbit of the BB-model with continuous time which corresponds to the ﬁxed point X∗. Suppose that there 0 ∗ 0 ∗ 0 exists such moment of time t that xi (t ) = xi+1(t ) for some i = 1, 2,...,N − 1. 214 LEONID A. BUNIMOVICH

∗ ∗ 0 ∗ But it means that xi+1 < xi (t ) which contradicts to the assumption that X is the fixed point. According to Corollary 1 any OWNS-model satisfying conditions 1) and 2) has fixed points. For a BB-model one fixed point can be constructed directly. In fact, there is always such point in the phase space D˜L(N), where the initial positions of particles are such that for the time required for ith particle to get to the initial position of the (i + 1)th particle does not depend upon i, i = 1, 2,...,N − 1. In the production line interpretation of the BB-model it means that at this point the entire work is equally balanced between the productive elements. Lemma . ∗ ∗ ∗ 3 (Existence of a Balance Point) The point X = (x2, . . . , xN ), where ∗ 2 ∗ ∗ 2 ∗ ∗ 2 ∗ 2 (x2) (x3 − x2) (xN − xN−1) (L − xN ) ∗ = ∗ = ··· = ∗ = (4) R x2 R x3 R xN R L v1(x)dx ∗ v2(x)dx ∗ vN−1(x)dx ∗ vN (x)dx 0 x2 xN−1 xN is a fixed point of a BB-model. Proof of this lemma immediately follows from (1) and the conditions 1), 2). The next statement shows that any BB-model has only one fixed point, which is given by (4). Theorem 1. (Uniqueness of a Balance Point) A fixed point (4) is the only fixed point of a BB-model. Proof. The Absence of Blocking Lemma says that there is no blocking in any fixed point of any OWNS-model. Therefore it is also true for any BB-model because any BB-model is also a OWNS-model. One fixed point of a BB-model according to Lemma 3 is given by a balance point (4). ∗∗ ∗∗ ∗∗ ∗∗ Suppose that there is another fixed point X = (x2 , x3 , . . . , xN ) of a BB- model. Then it follows from the Absence of Blocking Lemma that ∗∗ 2 ∗∗ ∗∗ 2 ∗∗ ∗∗ 2 ∗∗ 2 (x2 ) (x3 − x2 ) (xN − xN−1) (L − xN ) ∗∗ = ∗∗ = ··· = ∗∗ = . (5) R x2 R x3 R xN R L v1(x)dx ∗∗ v2(x)dx ∗∗ vN−1(x)dx ∗∗ vN (x)dx 0 x2 xN−1 xN ∗∗ ∗ ∗∗ Suppose now that xN > xN . Then the time t defined by (5) is strictly less than ∗ ∗ ∗∗ the time t defined by (4). However, x1 = x1 = 0. Therefore, there exists at least ∗∗ ∗∗ ∗ ∗ one such i, 1 ≤ i ≤ N that xi+1 − xi > xi+1 − xi . But this inequality contradicts to (5). Hence, the Balance Point (4) is the unique fixed point of any BB-model. The Absence of Blocking Lemma and Theorem 1 suggest that one may look for a maximal throughput for the BB-model in its unique fixed point. However, such stationary states in the real systems are required to be stable and sufficiently robust. We will address these problems in the next section for some more narrow than OWNS but, as we believe, a basic model for many logistics processes. Here by a basic model we mean such one which, on one hand allows a rather complete investigation and, on another hand, captures some basic features of the processes in question.

3. The Basic Model and its dynamics. In this section we consider the simplest version of the BB-model. Namely we assume that velocities of all particles are constant, i.e., vi(x) = vi for any x ∈ [0,L] and i = 1, 2,...,N. (This model has been studied numerically in [3] for N = 2 and 3.) Such model we will call the Basic Model. It is easy to see that for a BB-model (and therefore for the Basic Model as well) the problem of maximizing of a throughput is equivalent to an optimal sequencing of particles in [0,L]. DYNAMICAL SYSTEMS AND OPERATIONS RESEARCH 215

For the Basic Model its unique fixed point (4) can be written in a somewhat simpler form Pk−1 v x∗ = L i=1 i (6) k PN i=1 vi ∗ ∗ ∗ ∗ k = 2, 3,...,N. It is easy to see that X = (x2, x3, . . . , xN ) is a fixed point and, moreover, the Basic Model has the maximal throughput at this point. N P Clearly the maximal throughput of the Basic Model equals vi. In fact, this i=1 system has the maximal throughput when no one particle becomes blocked by its successor, i.e., it never slows down. Exactly this situation occurs at the point X∗. However, in applications only stable fixed points are of interest. Moreover the basin of their attraction is important as well. A size of such basin shows how “robust” is this regime of motion. The main feature of the Basic Model is that if any particle gets blocked, then it remains blocked until the next reset. It is easy to see that the Basic Model can have infinitely many periodic points. Consider, for instance, the case of two particles (N = 2) with equal velocities (v1 = v2). Then the dynamics is trivial. The phase space D˜2(L) is just the segment [0,L]. It is easy to see that all orbits of this system have period two besides the one which corresponds to the initial position of the second particle right in the middle of the segment, i.e., x2 = L/2. Observe, that it is the point (6) for N = 2. Dynamics of the Basic Model is linear in the absence of blocking. Indeed, each particle in this case moves with a constant velocity. Thus, the Basic Model always has the unique fixed point given by the relation (4). Certainly, this point can be stable, unstable, or a neutral one. Let us consider the case N = 3 [3]. Then the fixed point (4) is stable in the region {v1/v3 < 1, 0 ≤ v2 < v1 + v3}. At the boundary of this region the fixed point becomes neutral. (These results were obtained in [3] numerically, but the corresponding computations are easy to do analytically as well.) One may wonder, whether the Basic Model has another regime with the maximal throughput besides the fixed point (4). Certainly, the first candidate to consider is the set of periodic points of this model. The proof of the following statement is essentially similar to the proof of the Absence of Blocking Lemma. Therefore we omit it.

Lemma 4. Let vi 6= vj if i 6= j, i = 1, 2,...,N. Then there is a blocking at any periodic point of the Basic Model with a period p > 1. The main question now about the dynamics of the Basic Model is how big could be a basin of a stable fixed point. The following theorem shows that a (locally) stable fixed point of the Basic Model attracts all points in the phase space. Theorem 2. Let the fixed point (4) of the BB-model is (locally) stable. Then it attracts almost every point in the phase space. ∗ Proof. Let X be a stable fixed point of a map f : D˜L(N) → D˜L(N), which defines a BB-model. Then there exists a number λ, 0 < λ < 1, and an open neighborhood U of the point X∗ such that f(U) ⊂ U and kf(X) − X∗k < λkX − X∗k for any X ∈ U. It is easy to see that any event of blocking corresponds to the boundary ∂D˜L(N) of the phase space. Observe that D˜L(N) is a simplex and therefore ∂D˜L(N) consists of a finite number of convex closed sets (faces of D˜L(N)). 216 LEONID A. BUNIMOVICH

∗ k A point X ∈ D˜L(N) may not be attracted to X iff its orbit {f X}, k > 0, hits ∂D˜L(N) infinitely many times. Therefore such orbit must hit some face C of D˜L(N) infinitely many times. Suppose that there exists an infinite sequence of ∞ T nj positive integers n1 < n2 < ··· nj < ··· , such that C˜ = (F C) 6= ∅. A set C˜ is i=1 convex because F is continuous. Let FC : C → C is the map induced by F . It is easy to see that FC is continuous. Consider the restriction of FC to C˜. Then it must have a fixed point Y ∈ C˜. Then Y must be a periodic point and therefore a sequence {ni} should be periodic as well. Hence Y is a periodic point of F what contradicts to the existence of a stable fixed point X∗. The following statement is the immediate corollary of Theorem 2. Theorem 3. A sequencing of particles in the Basic Model provides the maximal N P throughput vi iff the corresponding fixed point is stable. i=1 Theorem 3 states that there could be several different sequencing of particles which provide the maximal throughput in the Basic Model. One of such sequencing is from slowest to fastest. It is obvious that for such sequencing it can be no blocking in the Basic Model. Remark. It is easy to see that Theorem 3 does not hold for the BB-model. In this case not all sequencings of workers with the stable fixed point provide the same throughput. There are at least two practical consequences of the Theorem 3. One has to do with the motion of pickers in warehouses, where there are no constraints on movements of pickers (besides the one that an order of pickers must be preserved). In this case the sequencing of pickers from slowest to fastest is always preferable one, if the sku’s and orders are uniformly distributed. The application of this sequencing resulted in an essential increase of pick rates [2]. Another practical consequence of Theorem 3 deals with the condition based maintenance (CBM). It is well known that probability that any given asset (or part) will fail increases with time. Therefore the usage of the older assets (parts) should be more intensive. This will increase a rate of usage (“productivity”) of assets in a fleet during their lifetime (from the beginning of their performance till the time of a maintenance or replacement).

4. Optimal management of the OWNS-systems. In this section we discuss the general algorithm of optimization of the performance of OWNS-systems. It has been already mentioned that, in difference with the BB-model, in OWNS-models there are some additional constraints. Recall that the only constraint in the BB- model is that an order of particles is preserved. One may think on the additional to this one constraint as of some potential field distributed along the segment [0,L]. OWNS-models are much more complicated than the BB-models. For instance, they can have more than one stationary states (fixed points). Consider, for example, a TSS production line (where TSS stands for Toyota Sewing Products Management System). In this model the segment [0,L] gets par- M S titioned into M ≥ N nonoverlapping intervals Ii, i = 1, 2,...,M, Ii = [0,L]. i=1 The TSS constraint is that no two particles can be at the same moment of time at the one and the same interval Ii. This model is proved to be relevant to the DYNAMICAL SYSTEMS AND OPERATIONS RESEARCH 217 production lines in the apparel industry, where the intervals Ii are referred to as work stations [1]. Thus no two workers are allowed to work at the same work station at the same time. Instead if a worker (particle) finished to work at the ith work station Ii and the next work station Ii+1 at this moment is occupied by his successor (the particle which number is greater by one) then this worker is blocked at the left-most point of the interval Ii+1 until his successor finishes work at this (i + 1)th work station. The reset of workers (particles) at the time when the last one reaches the end of the interval [0,L] (production line) is the same as in the BB-model. It has been proven in [1] that a TSS-model has only one fixed point if the workers (particles) are sequenced in the increasing order of their velocities (from slowest to fastest). However, as the following example shows, this fixed point may not provide the highest production rate.

Example. Consider a TSS production line [0,L] with three work stations [0, L/8), [L/8, 5L/8), and [5L/8,L). For TSS-model the “lengths” of work stations are defined by a work content, or by a time that is required for work processing at the given work station by some “ideal” worker. Suppose that the velocities of workers are v1 = 1, v2 = 3, v3 = 4. Consider first the sequencing of workers at this line from slowest to fastest. ∗ ∗ The corresponding fixed point is {x2 = L/8, x3 = 13L/32}. The corresponding 16 12 production rate at this point equals v1 19 + v2 19 + v3. Let now exchange the order of workers and consider instead the sequence v1, v3, v2. ∗∗ ∗∗ Then a fixed point becomes {x2 = L/8, x3 = 5L/8}. (Observe that now the index 2(3) refers to a position of the worker with velocity v3(v2).) The production rate ∗∗ ∗∗ at the fixed point (x2 , x3 ) is the maximal possible one, i.e, v1 + v2 + v3. Another type of constraint appears in a hoist problem [7]. In this model instead of work stations there are tanks, and workers are substituted by printed circuit boards (PCB). A constraint is that there are upper and lower bounds on times which PCB can spend in each tank. In this system the problem is again to maximize a throughput. In [5] has been proposed the general algorithm of maximizing a throughput in systems appeared in a work-sharing manufacturing. In this type of manufacturing each productive element (worker, robot, asset, etc.) can “work” in any place of the corresponding “production line.” Therefore the problem of a maximization of a throughput can be formulated as a problem of optimal sequencing of productive elements in a production line. This algorithm consists of two steps: 1) Find all such sequencings of productive elements in a production line which have a stable stationary state (stable point). 2) Choose among the sequencings found at the step 1) the one which is the most consistent with a configuration of a production line. By a configuration of a production line we mean here its partition into work stations, into tanks with the corresponding time windows, etc. A measure of “con- sistency” depends upon a concrete model. For instance, for TSS production lines an optimal sequencing corresponds to a stable fixed point, where positions of workers are as close as possible (with respect to a work time) to the boundaries of work stations in this line (see the Example in this section). The above algorithm is the special case of the general approach of a stabilization of a target regime (STR-method). The STR-method is, in a sense, a generalization 218 LEONID A. BUNIMOVICH of the popular control of chaos approach (see e.g., the book [5]). In fact in the control of chaos a goal is to find a regular (periodic) orbit and to make such orbit stable by applying some procedure, which consists of a sequence of small perturbations of a controlled system. In the systems studied by the operations research or by logistics chaotic regimes are also usually unwanted. However, in such systems always there is a room for a global (rather than a local) control. Some types of such control are usually referred to as policies (first-in-first-out, last-in-first-out, etc.). The important feature of the logistics systems is that it is usually not known a priori which policy is an optimal one but it is known instead which regimes are unwanted and should be avoided. Thus the problem is to find such policy which makes stable a stationary regime located in the complement of the basin of attraction of the unwanted one. In the BB-model such unwanted regime is blocking. More generally, any con- gestion should be avoided. For instance, it is the event to avoid, when a large portion of assets must be sent for a maintenance (repair) simultaneously or almost simultaneously. Indeed, it will put the entire fleet of assets in a risk that its task will not be fulfilled. Thus the STR-method seems to be relevant to a rather general class of logistics models. Its implementation in concrete cases should be based on the analysis of a global dynamics of the corresponding model as has been illustrated in this paper by the analysis of dynamics of the BB-models.

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