ON OPTIMIZATION OF A HIGHLY RE-ENTRANT PRODUCTION SYSTEM

. We discuss the optimal control problem stated as the minimization in the L 2 -sense of the mismatch between the actual out-ﬂux and a demand forecast for a hyperbolic conservation law that models a highly re-entrant pro- duction system. The output of the factory is described as a function of the work in progress and the position of the switch dispatch point (SDP) where we separate the beginning of the factory employing a push policy from the end of the factory, which uses a quasi-pull policy. The main question we discuss in this paper is about the optimal choice of the input in-ﬂux, push and quasi-pull constituents, and the position of SDP.


1.
Introduction. The aim of this article is to analyze an optimal control problem (OCP) for a highly re-entrant production system which is described by a scalar nonlinear conservation law. Typically, in high-technological semi-conductor manufacturing, many machines are repeatedly used for similar processing operations. In such production lines, semi-conductor wafers return to the same set of machines many times. So, the product flow has a re-entrant character. Typically, the semiconductor systems are characterized by a very high volume (number of parts manufactured per unit time) and a very large number of consecutive production steps. This fact motivates to consider the scalar nonlinear conservation laws for the simulation of such processes. Partial differential equations, which are related with nonlinear conservation laws, are rather popular due to their superior analytic properties and availability of efficient numerical tools for simulation. For more detailed 416 CIRO D'APICE, PETER I. KOGUT AND ROSANNA MANZO discussions of these models we refer to [3,4,6,11,18,19,20,21,22,23,24,25,26,27,29,30,31].
From the optimization point of view, in manufacturing systems the natural control input is the in-flux (see, for instance, [15,17] where the inflow on the output has been studied). Specifically, re-entrant production creates the opportunity to set priority rules for the various stages of production competing for capacity at the same machines. This dispatch policy, as it was indicated in [5], typically allows for two models of operations -the so-called push and pull policies. A push policy, also known as first buffer first step, is typically assigned to the front of the factory. A pull policy gives priority to later or fixed production steps over the earlier production steps. The step where push policy switches to pull policy is called the push-pull point (PPP). Moving the PPP leads not only to a change in dispatch rules, but also it may have an effect on the total output. In view of this it makes s sense to consider the PPP as a control variable.
A modern introduction to the study of hyperbolic conservation laws and especially of the control systems governed by such laws can be found in [9]. Fundamental are questions of wellposedness, regularity properties of solutions, controllability, existence, uniqueness and regularity of optimal controls. Existence of solutions, regularity and wellposedness of nonlinear conservation laws have been widely studied under diverse sets of hypotheses, see e.g. [1,2,6,7,8,10,14,28,32,33] and the references therein. Concerning the manufacturing systems, an optimal control problem related to minimization of the error-signal that is the difference between a given demand forecast and the actual out-flux of manufacturing system, was studied in [17,34]. For the controllability, exponential stability, and feedback stabilization of highly re-entrant production systems and further results in this field, we refer to [12,14,15,16,17,31].
Here we consider an optimal control problem for a PDE model of a re-entrant system governed by nonlinear hyperbolic conservation law for the part density ρ(t, x) ∂ t ρ + ∂ x (ρV (ρ)) = 0 in Q = (0, T ) × (0, 1), where and H(x) stands for the Heaviside function whose value is zero for negative argument and one for positive argument. The characteristic feature of OCP, we deal with in this article, is the fact that this model depends explicitly both on the so-called switch dispatch point (SDP) which is located at a priori unknown position x * and velocity functions V 1 and V 2 which describe different types of policy in the regions [0, x * ] and [x * , 1].
Since the SDP divides the production line [0, 1] onto two parts [0, x * ] and [x * , 1] with different type of policies, in general, we cannot represent the right hand side of (2) in the form V (ρ) = λ which is the main constituent of models considered in [15,16,17,34]. Hence, the wellposedness, uniqueness and regularity properties of solutions of hyperbolic conservation law (1) with a nonlocal speed term (2) requires a separate analysis. In what follows we will show that in many aspects such analysis can be provided in the spirit of recent work [17]. Moreover, as follows from (2), we consider the push policy for the region [0, x * ], whereas the dependency on the rest production line [x * , 1] can be interpreted as a certain version of a pull policy -the so-called quasi-pull policy. However, the right choice of functions V 1 and V 2 is definitely open question (see, for instance, [5,17,34]). This fact motivates us to consider the functions V 1 and V 2 as controls too. As a result, we deal with an OCP for the nonlinear conservation law with a nonlocal character of the velocity and with three different control actions -the in-flux, the SDP, and the functions V 1 and V 2 . The paper is organized as follows. In Section 2 we give the precise statement of the OCP for a highly re-entrant production system. The aim of Section 3 is to give some preliminaries and auxiliary results that we make use for our further analysis. In Section 4 we prove the existence of a unique weak solution to the Cauchy problem associated with the re-entrant system under given control functions when the initial and boundary conditions we consider in L 1 (0, T ) and L 1 (0, 1) sense, respectively. We also study the main functional properties of the weak solutions and derive a priori estimates for them. Section 5 is addressed to the solvability of the original OCP. As for the optimality conditions for the given class of OCP, these aspects will be considered in the forthcoming paper.
2. Statement of the problem. Let α 2 > α 1 > 0 and α 3 > 0 be given constants. Let A ad be the following subset of C 1 ([0, ∞)) Following the concept of the continuous flow model, describing the flow of products through a factory like a fluid flow, we denote ρ(t, x) the product density at the stage x ∈ [0, 1] and time t ∈ [0, T ]. Here, x = 0 refers to the point of raw material and x = 1 to the finished product.
where H(x) stands for the Heaviside function and As follows from this definition F (t, x) can be associated with the flux (production rate) at the time t ∈ [0, T ] and stage x ∈ [0, 1] in the factory, whereas x * ∈ [0, 1] is the SDP. We consider the following statement of OCP for manufacturing system: subject to the constraints where , ρ 0 ∈ L 2 (0, 1), and y d ∈ L 2 (0, T ) are functions, and y(t) is the out-flux corresponding to the in-flux u ∈ L 2 + (0, T ), functions V 1 , V 2 , and initial data ρ 0 . We also suppose ρ 0 ∈ L 2 (0, 1) and y d ∈ L 2 (0, T ) are nonnegative almost everywhere, and the constant a in definition of the cost functional is such that max {a 1 , a 2 } ≤ a < +∞. As for the second and third terms in the cost functional (4), they are related to the regularity (and, hence, to the solvability) of the original problem.
3. Preliminaries and auxiliary results. It is easy to see that, for each admissible control (u, V 1 , V 2 , x * ), the Cauchy problem (5)-(7) can be represented in the form of a coupled system where the compatibility condition (16) means that the output flux at x = x * of the push region must be considered as the in-flux for the quasi-pull region.

Remark 1.
It is easy to note that the following representation for the solutions to the Cauchy problem (5)-(7) holds, where x = x * is the discontinuity point for the work in progress (wip) profile. However, the continuity assumption of the flux (16) guarantees the smooth solutions later on.
Following [13], we adopt the following definition of a weak solution to the problem (12)- (16). 1], and V 1 , V 2 ∈ A ad be given. We say that a pair (ρ 1 , ρ 2 ) ∈ C 0 ([0, T ]; L 1 (0, x * ) × L 1 (x * , 1)) is a weak solution to the Cauchy problem (12)- (16) if for every τ ∈ [0, T ] and every test We make use of a few of auxiliary results. In particular, the next Lemmas give existence of characteristics to the original Cauchy problem and their regularity what is a crucial point for our further analysis. 1], and V 1 , V 2 ∈ A ad be given and such that V i (s) = V i (0) for all s < 0. Then there exists δ ∈ [0, T ] independent of x and y such that the Cauchy problem : and Let us show that there exists a constant κ ∈ (0, 1) such that for all ξ i , ζ i ∈ Ω δ and δ > 0 small enough. Since F maps into Ω δ provided δ < α −1 2 , it follows from (22) that F (ξ, ζ) : Ω δ × Ω δ → Ω δ × Ω δ is a contraction mapping. Then, by the Banach fixed point theorem, there exists a unique pair (ξ x , ζ z ) such that F (ξ x , ζ z ) = (ξ x , ζ z ), i.e. (ξ x , ζ z ) is the unique solution to the Cauchy problem (19). Moreover, as follows from definition of the set A ad and the fact that V 1 , V 2 ∈ A ad , the unique fixed pair (ξ x , ζ z ) for F is in Let ξ i , ζ i (i = 1, 2) be arbitrary elements of Ω δ . Then (20) implies the estimate We define ξ, ξ ∈ C 0 ([0, δ]) by Since ξ i are monotonically increasing functions, it follows that the inverse functions ξ −1 and ξ −1 are well defined. Then, changing the order of integrations in (23) and following in many aspects [17], we obtain where for the term ξ −1 (y) − ξ −1 (y) we have the following estimate for each y ∈ [0, ξ(t)] (this inference is based on (21) and the definition of ξ and ξ) Combining the above results, we finally get By analogy, it can be shown that As a result, the inequality (23) implies Since ρ 0 ∈ L 1 (0, 1), it follows that there exists δ ∈ (0, T ) small enough such that In view of estimate (24), this immediately leads us to inequality (22).
Our next intention is to study the properties of the mappings x → ξ x (t) and z → ζ z (t).
are uniformly Lipschitz.
Proof. Let x, y ∈ [0, x * ] be arbitrary points. Then, in view of definition of the class A ad , we can derive from the first equation of (19) the estimate As a result, by Gronwall-Bellman inequality, we see that , and x * ∈ [0, 1], we introduce the following couple of functions , the functions ρ 1,x and ρ 2,z , defined by (28)- (29) are such that Proof. We only prove the inclusion ρ 2 ∈ C([0, δ]; L 1 (x * , 1)), since the second one in (30) can be established by analogy. Let ε > 0 be an arbitrary value. Our aim it to show that there exists θ = θ(ε) > 0 such that, for arbitrary points s, t ∈ [0, δ], we have Since and ρ 0 ∈ L 1 (0, 1) and v ∈ L 1 (0, T ), it is easy to conclude from monotonicity property of ζ z ∈ C 1 ([0, δ]) and condition ζ z (0) = 0 that there exists a value θ 2 (ε) > 0 such that J 2 < ε/3. Now we show that the same conclusion can be obtained with respect to the term where the constant C(k) depends on k ∈ N but does not depend on t and s. Hence, in view of the strong convergence ρ k 0 → ρ 0 in L 1 (x * , 1) and monotonicity of It remains to estimate the first term in the right hand side of (31). Let {v k } k∈N ⊂ C 1 ([0, T ]) be a strongly convergent sequence to v in L 1 (0, T ). Then Since and it follows from definition of function ζ z (see the Cauchy problem (19)) that To estimate the right hand side in (32), we change the order of integration. As a result, we obtain Taking into account that we can conclude from (33) the following estimate It remains to estimate the last term in (32). Following in the similar manner, we change the order of integration. As a result, we obtain Since we deduce from (35) that Thus, combining the estimates (32), (34), and (36), we get and, hence, where the constant D(k) depends on k ∈ N but does not depend on t and s. As follows from (37), for k ∈ N large enough there exists a value θ 1 (ε) > 0 such that J 1 < ε/3. As a result, we arrive at the following conclusion: for a given ε > 0 and all t, s ∈ [0, δ] such that |s − t| < θ = min{θ 1 (ε), θ 2 (ε), θ 3 (ε)}, the estimate As a consequence of Lemma 3.3, we have the following important property.
Corollary 1. If, in additional to the assumptions of Lemma 3.4, ρ 0 ∈ L ∞ (0, 1), then the mappings Proof. It is easy to check that the following relations . As a result, for any x, y ∈ [0, x * ], we have The continuity of the mapping z → ρ 2,z (t, ·) L 1 (x * ,1) can be shown in a similar way.
By Lemma 3.3, the following limits In view of this, we make use of the following notations   Then relations (38)-(39) and Lemma 3.3 imply the following integral representation for the limit functions ρ 1 and ρ 2 4. Existence of weak solutions to the Cauchy Problem (5)- (7). We begin this section with the following result.
Following the similar scheme, it can be shown that  Proof. In order to show that the distribution (45) defined in Theorem 4.1 is the unique solution to this problem, we make use of some ideas from [17,Theorem 3.2]. Let us assume, by contraposition, that there exists another distribution such that ρ(t, x) = ρ(t, x). It is worth to emphasize that, in general, it is unknown whether this function can be represented in the form like (42)-(43), because in this case Lemma 3.2 immediately leads to the conclusion 1], and, therefore, ρ 1 (t, γ) = ρ 1 (t, x) and ρ 2 (t, γ) = ρ 2 (t, x) almost everywhere in the corresponding domains.
As a consequence of Theorem 4.1, we have the following hidden regularity property of the weak solutions. 1)) be a weak solution to the Cauchy problem (5)-(7) for some τ ∈ (0, T ]. Then for given ρ 0 ∈ L ∞ (0, 1), u ∈ L 1 (0, T ), V 1 , V 2 ∈ A ad , and x * ∈ [0, 1], we have Proof. Let x ∈ (0, x * ) be an arbitrary point. Then, by the first mean value theorem for integration, we get As follows from (28), we have the following representation

Remark 2. Taking into account the fact that
and by Lemma 3.2 and relations (42)-(43) it is easy to deduce from definition of the set A ad and representations (75)-(76) the following estimates We are now in a position to prove the main result of this section.
It remains to note that inclusion (82) 2 is a direct consequence of Corollary 2.

5.
Existence of optimal solutions. In this section we focus on solvability of OCP (4)- (10). To begin with we note that unknown control functions V 1 and V 2 in (4)-(9) are supposed to be defined on domains [0, a 1 ] and [0, a 2 ], respectively, with constants a 1 and a 2 given by (11). As follows now from Theorem 4.3, the reason to define the constant a i in the way (11) comes from a priori estimates (83)-(84) and the fact that u L 1 (0,T ) ≤ √ T u L 2 (0,T ) .
We denote by Ξ the set of all admissible solutions for the OCP (4)-(10).
(87) We may always suppose that functions V 0 i ∈ C 1 ([0, a i ]), (i = 1, 2), are extended to R + in such way that V 0 i ∈ A ad for i = 1, 2. Before proceeding further, we note that the set U ad := w ∈ L 2 (0, T ) | w L 2 (0,T ) ≤ α 4 , w(x) ≥ 0 a.e. on (0, T ) is sequentially closed with respect to the weak convergence in L 2 (0, T ). Hence, (87) and the admissibility condition u k ∈ U ad for all k ∈ N imply that u 0 ∈ U ad .
By (93) and Corollary 2, the sequence {v k } k∈N is uniformly bounded in L 1 (0, T ). Moreover, as immediately follows from (3), Corollary 2, and a priori estimate (79), this sequence is equi-integrable. Indeed, for any τ 1 , τ 2 ∈ [0, T ] (τ 1 < τ 2 ), we have = α 2 by Lipschitz continuity of the functions ξ −1 k,x * k ∈ C 1 ([0, x 0 ]) (see the proof of Corollary 2). Hence, by Dunford-Pettis Criterion, the sequence {v k } k∈N is weakly compact in L 1 (0, T ). It means that there exists a function v 0 ∈ L 1 (0, T ) such that, up to a subsequence, we have  Moreover, in view of definition of the class A ad , ξ k and ζ k are uniformly bounded from above and below: Then it follows from Arzelà-Ascoli Theorem that there exist functions ξ 0 (t, x) and ζ 0 (t, x) such that, up to subsequences,