A First-Order Hybrid Petri Net Model for Supply Chain Management

A First Order Hybrid Petri Net Model for Supply Chain Management Mariagrazia Dotoli, Member, IEEE, Maria Pia Fanti*, Senior, IEEE, Giorgio Iacobellis, Agostino Marcello Mangini, Dipartimento di Elettrotecnica ed Elettronica, Politecnico di Bari, Bari, Italy Abstract— A Supply Chain (SC) is a network of independent manufacturing and logistics companies that perform the critical functions in the order fulfillment process. This paper proposes an effective and modular model to describe material, financial and information flow of SCs at the operational level based on first order hybrid Petri Nets (PNs), i.e., PNs that make use of first order fluid approximation. The proposed formalism enables the SC designer to choose suitable production rates of facilities in order to optimize the chosen objective function. The optimal mode of operation is performed based on the state knowledge of the obtained linear discrete-time, timevarying state variable model in order to react to unpredictable events such as the blocking of a supply or an accident in a transportation facility. A case study is modeled in the proposed framework and is simulated under three different closed loop control strategies that allow us to optimize appropriate performance indices. Note to Practitioners— The motivation of the present work is to provide a generic, simple and accurate model able to integrate material, financial and information flow description for use by decision makers in SC management at the operational level. To this aim, the paper proposes a model based on the first order hybrid PN formalism that provides the SC designers with important key features. First, the model exhibits good computational efficiency and easy implementation in engineering software packages. Second, some SC design parameters, such as the supplier and manufacturer production rates, are real continuous variables and can be determined by polynomial complexity optimization of suitable performance indices. Third, based on the knowledge of the system state and the information about the occurrence of typical operational 1 unpredictable events (e.g. facility blockings and goods demands), the model enables the decision makers to update some operational design parameters in a closed loop strategy. Future research aims to investigate on the decisional structure of the model in order to specify a decision support system based on the presented modeling framework and devoted to the management of the SC at the operational level. Keywords: supply chains, management, modeling, Petri nets, performance evaluation, simulation. I. INTRODUCTION The emergence of Supply Chains (SCs) is an outcome of the recent advances in logistics and information technology. SCs may be defined as complex networks interconnecting different independent manufacturing and logistics companies that perform the critical functions in the order fulfillment process [22, 28]. The analysis, design and management of SCs are currently active areas of research [15, 26, 27, 28]. Following a systematic way to organize decisions in automated manufacturing systems, an analogous guideline has been proposed to classify SC management decisions into three hierarchical levels according to the time horizon of the decisions [6, 20]: strategic (long-term), tactical (medium-term), and operational (short-term and real-time). The classes of SC problems encountered in the strategic level planning involve location-allocation decisions, demand planning, strategic alliances, new product development, supplier selection and pricing. This level of the hierarchy considers time horizons of a few years and requires approximate and aggregate data models. Tactical level planning basically refers to layout and network design, production/distribution coordination, equipment and material handling selection. Finally, operational level planning is short-term planning, which involves coordination across the stages and optimization of operational policies to reduce costs while improving services to customers. Different models have to be defined at each level of the decision hierarchy to describe the multiple aspects of the SC, with respect to different time horizons. While the development of formal decision models for the SC design at the strategic and tactical levels has been addressed in the related literature [9, 10, 13, 14, 19, 25, 27], research efforts are lagging * (Corresponding author) M.P. Fanti is with the Department of Elettrotecnica ed Elettronica, Politecnico di Bari, Via Re David 200, 70125, Bari, Italy (phone: +39-080-5963643; fax: +39-080-5963410; e-mail: [email protected]). 2 behind in the subject of modeling and controlling the operational performance of the SC. The description and control of the material flow in the network require using flexible and generic models. In the related literature, SCs are usually described as multi-echelon inventory systems [16, 23]. However, most multi-echelon inventory systems models do not explicitly take into account transportation operations and capacity constraints in SCs but simply assume a constant lead time between any two adjacent stocking locations. These models lack flexibility and generality in describing real-life SCs [5]. Hence, the problem of investigating about mathematical models that can describe material, information and financial flows of SCs in an integrated way is an open and relevant subject [5]. A. Literature review In this context, an effective SC model should focus on evaluating operational performance indices describing resources (cost, utilization and inventory), output (throughput, lead time) and flexibility (lead-time, lead time variability) [4] by integrating information and financial flows. To this aim, at the operational level, SCs can be viewed as Discrete Event Dynamical Systems (DEDSs), whose dynamics depends on the interaction of discrete events, such as customer demands, departure of parts or products from entities, arrival of transporters at facilities, start of assembly operations at manufacturers, arrival of finished goods at customers etc. [28]. Accordingly, the operational behavior of a SC may be captured employing discrete event simulation [17] and formal DEDS models. Among the available DEDS models, Petri Nets (PNs) may be singled out as a graphical and mathematical technique to describe concurrency and synchronization of SCs. In the context of models based on PNs proposed in the related literature, Desrochers et al. [7] suggest complex-valued token PNs to describe a two product pull SC with kanban control and determine the performance measures. Moreover, Elmahi et al. [11] employ timed event graphs that are a subclass of PNs, to model, analyze and control an elementary SC called supply link: the model allows establishing the system state equations in the max-plus algebra formalism, defining an optimal controller of the SC and measuring its performance. In addition, Von Mevius and Pibernik [30] propose an extension of PNs based on the XML standard for inter-organizational data exchange to model and manage SCs. Furthermore, a network-based technique extending PNs and proposing the so-called Trans-Net formalism for SC network modeling is presented by Wu and O’Grady [32]. To take into account the stochastic values of process and transportation times, Generalized Stochastic Petri Nets 3 (GSPNs) may be applied for SC modeling. In this direction, Viswanadham and Raghavan [28] employ GSPNs to describe a particular example of SC and determine the decoupling point location, i.e., the facility from which all finished goods are assembled after customer order confirmation. More recently, Dotoli and Fanti [8] propose a GSPN model, describing a generic SC at an operational level in a modular and simple way, which is applied to a case study. However, the mentioned models share the limitation that products are modeled by means of discrete quantities (i.e., tokens). This assumption is not realistic in large systems with a huge amount of material flow: by such formalisms the state space of the SC model is excessively large, so that inconveniences in the simulation and performance optimization often arise, leading to large computational efforts. Since SCs are DEDSs whose number of reachable states is very large, PN formalisms using fluid approximations provide an aggregate formulation to deal with complex systems, thus reducing the dimension of the state space [1, 21]. In this context, Chen et al. [5] propose an extended GSPN formalism, named batch deterministic and stochastic PNs, for modeling and performance evaluation of SCs. As applications an inventory system and an industrial SC are modeled and their performance is evaluated analytically and by simulation. However, such a paper does not face the basic aspect of the SC management and optimization problems at the operational level. B. Proposed approach This paper proposes an efficient modular model for SC management and control at the operational level in order to represent material, financial and information flow in an integrated framework. In particular, the presented model describes the SC dynamics and formalizes the system control, proposing several controllers able to optimize in a closed loop strategy some suitable performance indices. While the SC optimization models presented in the related literature determine the decision parameters off-line (e.g. see [13, 14, 15, 22, 25, 26, 27]) in order to design and manage the SC, the task of the presented model is selecting some operative SC parameters in short time on the basis of the knowledge of the system state and of the occasional and uncontrollable events that affect the SC behavior at the operational level. The model is based on First Order Hybrid Petri Nets (FOHPNs) [2, 3] that are a hybrid PN formalism including continuous places holding fluid, discrete places containing a nonnegative integer number of tokens and transitions, which are either discrete or continuous. 4 FOHPNs have been selected for SC modeling since they present several key features. First, their use typically leads to a considerable increase in computational efficiency with respect to place/transition models, since simulation of fluid models can often be performed much more efficiently than discrete ones. Second, fluid approximations provide an aggregated formulation to deal with complex systems, thus reducing the dimension of the state space. Third, the design parameters in fluid models are continuous; hence, gradient information may be employed to speed up optimization. In other words, the model allows us to define optimization problems of polynomial complexity in order to select suitable operational parameters that optimize appropriate performance indices. The proposed model is built using a modular approach based on the idea of the bottom-up methodology [33]. In particular, manufacturers are described by continuous transitions, buffers are continuous places and products are represented by continuous flows (fluids) routing from manufacturers, buffers and transporters. Moreover, transporters are modeled by stochastic transitions with a triangular distribution for the transportation time. Furthermore, discrete places and transitions describe the financial and information flows that are able to affect the system behavior by enabling, inhibiting or changing the material flow. Discrete exponential transitions model the information about the customer demands and the stochastic occurrence of unpredictable events in the system, such as the blocking of a supply or an accident in a transportation facility. Modeling SCs by the FOHPN framework allows us to tackle two main issues: system management and optimal mode of operation. The system management is realized by using PN structures that synthesize the well-known Make-ToStock management policy [31] and a standard inventory control rule [5]. Moreover, the optimal mode of operation is realized thanks to the obtained linear discrete-time, time-varying state variable model and the definition of different control strategies. More precisely, the controllers employ the real time knowledge of the system state and the information about the occurrence of the SC unpredictable discrete events (e.g., the blocking of a supply or a transport operation, the start of a request from retailers, etc.) in order to drive the overall system to exhibit satisfactory values of suitable performance indices. In particular, the controller selects in a closed loop control strategy the Instantaneous Firing Speeds (IFSs) of the continuous transitions by defining and solving appropriate optimization problems of polynomial complexity. A comparison between the proposed model based on FOHPNs and a traditional model employing discrete PNs performed for a simple example SC validates the chosen formalism, 5 showing its ability in modeling the system and evaluating its performance indices by the simulation. Moreover, a representative case study, including the typical SC elements, shows the efficiency of the modeling technique. In particular, the considered SC system is simulated under three different closed loop control strategies and the resulting dynamics is discussed, enlightening the flexibility and potential of the proposed model. The paper is organized as follows. Section II describes the structure and the dynamics of a generic SC at the operational level and Section III reports a brief overview of the FOHPN modeling formalism. Section IV presents and discusses the modular SC model under a standard operational management strategy, introducing a simple example to validate the FOHPN model in comparison with a classical discrete PN model. Moreover, Section V defines three different control policies. Section VI presents the FOHPN model of a SC case study and discusses the corresponding behavior under different operational conditions. A conclusion section closes the paper. II. THE SYSTEM DESCRIPTION A. The SC structure The SC structure is typically described by a set of facilities with materials that flow from the sources of raw materials to subassembly producers and onwards to manufacturers and consumers of finished products. Moreover, feedback paths may be present if disassemblers or recyclers are included in the SC. The SC facilities are connected by transporters of materials, semi-finished goods and finished products. More precisely, the entities of a SC are the following: 1-Suppliers: a supplier is a facility that provides raw materials, components and semifinished products to manufacturers that make use of them. 2-Manufacturers and assemblers: manufacturers and assemblers are facilities that transform input raw materials/components into desired output products. 3-Distributors: distributors are intermediate nodes of material flows representing agents with exclusive or shared rights for the marketing of an item. 4-Retailers or customers: retailers or customers are buyers of goods who eventually sell them to the general public. 5-Disassemblers or recyclers: entities of the disassembling stage feed recovered material, components or energy back to suitable upstream SC facilities. 6 6-Logistics and transporters: storage systems and transporters play a critical role in distributed manufacturing. The attributes of logistics facilities are storage and handling capacities, transportation times, operation and inventory costs. Here, part of the logistics, such as storage buffers, is considered pertaining to manufacturers, suppliers and customers. Moreover, transporters connect the different stages of the production process. The SC dynamics is traced by the flow of products between facilities (i.e., entities of types 1-5) and transporters (i.e., entities of type 6). Because of the large amount of material flowing in the system, we model a SC as a hybrid system: the continuous dynamics models the flow of products in the SC, the manufacturing and the assembling of different products and its storage in appropriate buffers. Hence, resources with limited capacities are represented by continuous states describing the amount of fluid material that the resource stores. Moreover, we consider also discrete uncontrollable events occurring stochastically in the system, such as: a) the blocking of a raw material supply, e.g. modeling the occurrence of labor strikes, accidents or stops due to the shifts; b) the blocking of a transport operation due to the shifts or to unpredictable events such as jamming of transportation routes, accidents, strikes of transporters etc.; c) the start of a request from the retailers. B. SC management and inventory control rules The operational SC dynamics depends on the considered planning and management methodology, which specifies the business model and determines the paths for the information and material flow in the SC, and on the corresponding inventory control rules governing each SC facility [28]. According to the Wortmann classification [31], three SC managing policies are followed in practice: Make-To-Stock (MTS), Make-To-Order (MTO) and Assemble-To-Order (ATO). In particular, in order to deliver on time the produced goods to end-users, the MTS strategy governs the system initiating production before the actual occurrence of demands, so that end customers are satisfied from stocks of inventory of finished goods. On the other hand, in the MTO technique customer orders trigger the flow of materials and the requirements at each production stage of the SC. Furthermore, the ATO policy can be viewed as a hybrid of the former two strategies, basically applying MTS in the first stages of the SC and MTO in the last stages [29]. This paper focuses on SCs governed by the MTS policy, which is typical of 7 standardized products with high volumes. The remaining strategies are not considered in detail, although the presented SC model could be straightforwardly adapted to systems governed by such managing policies. Together with the operational planning and management policy, inventory systems play a very important role in SC management. Inventory management addresses two fundamental issues: when a stock should replenish its inventory (order timing choice) and how much it should order from suppliers for each replenishment (order size choice) [5]. In this paper, the (R,Q) inventory control rule is applied, where fixed quantities of parts that are Q in number are ordered any time the stock level drops below the reorder point R. Any time a withdrawal is made, a control system tracks the remaining inventory level of the buffer of products to determine whether it is time to reorder: in practice, thanks to automation and information systems, these reviews are continuous. At each review, the inventory level is compared with the pre-set reorder point R. In case the inventory level is higher than R, then no change in the inventory occurs. On the contrary, if the inventory level is lower than R, then a fixed quantity Q of products or lots of the considered items is ordered upstream, i.e., Q products or lots are manufactured if the considered stock level refers to an output product, or else they are ordered from an upstream facility in the SC. III. FIRST ORDER HYBRID PETRI NETS In this section we briefly outline the basics of the FOHPN formalism [2]. A. The FOHPN structure and marking A FOHPN is a bipartite digraph described by the seven-tuple PN=(P, T, Pre, Post, ∆, F, RS). The set of places P=Pd∪Pc is partitioned into a set of discrete places Pd (represented by circles) and a set of continuous places Pc (represented by double circles). The set of transitions T=Td∪Tc is partitioned into a set of discrete transitions Td and a set of continuous transitions Tc (represented by double boxes). Moreover, the set of discrete transitions Td=TI∪TS∪TD is further partitioned into a set of immediate transitions TI (represented by bars), a set of stochastic transitions TS (represented by boxes and including exponentially distributed transitions as well as transitions with triangular distribution) and a set of deterministic timed transitions TD (represented by black boxes). We also denote 8 Tt=TS∪TD, indicating the set of timed transitions. Function ∆: Tt→ \ + specifies the timing associated to timed transition. In particular, we associate to each tj∈TS the average firing delay ∆(tj)= δj=1/λj, where λj is the average firing rate of the transition. In case the transition is exponential, δj represents the expected value of the associated distribution, while in case it is triangular δj represents the modal value of such a distribution and we assume that the minimum and maximum values of the range in which the firing delay varies equal respectively dδj=0.8δj and Dδj=1.2δj. In addition, each tj∈TD is associated the constant firing delay ∆(tj)=δj. Moreover, function F: Tc→ \ +× \ ∞+ specifies the firing speeds associated to continuous transitions (we denote \ ∞+= \ +∪{+∞}). For any continuous transition tj∈Tc we let F(tj)=(Vmj,VMj), with Vmj≤VMj, where Vmj represents the minimum firing speed and VMj the maximum firing speed of the generic continuous transition. Finally, function RS: Td→ \ + associates a probability value called random switch to conflicting discrete transitions. Matrices Pre and Post are the pre-incidence and the post-incidence matrices, respectively, of dimension |P|×|T|. Note that symbol |A| denotes the cardinality of set A. Such matrices ⎧ Pc × T → \ + . specify the net digraph arcs and are defined as follows: Pre, Post : ⎨ ⎩ Pd × T → ` We require that for all t∈Tc and for all p∈Pd it holds Pre(p,t)=Post(p,t) (well-formed nets). Given a FOHPN and a transition t∈T, the following place sets may be defined: •t={p∈P: Pre(p,t)>0} (pre-set of t); t•={p∈P: Post(p,t)>0} (post-set of t). Moreover, the corresponding restrictions to discrete or continuous places are respectively defined as (d) t=•t∩Pd or (c) t=•t∩Pc. Similar notations may be used for pre-sets and post-sets of places. The incidence matrix of the net is defined as C=Post-Pre. The restriction of C to PX and TX (with X,Y∈{c, d}) is denoted by CXY. To extend the modeling capabilities of the FOHPN, we assume that for some t∈TS and for some p∈Pc the values Pre(p,t) and Post(p,t) may be stochastic, with constant distribution. Function ∆Pre: Pc×TS→ \ +× \ + (∆Post: Pc×TS→ \ +× \ +) specifies the minimum and the maximum values of the constant distribution associated to the elements of matrix Pre (Post). If ∆Pre(p,t)=(0,0) (∆Post(p,t)=(0,0)) then the element Pre(p,t) (Post(p,t)) is deterministic. We assume that when the stochastic firing delay is extracted for transition t∈TS, simultaneously the weights Pre(p,t) (Post(p,t)) are extracted for each p∈(c)t such that ∆Pre(p,t)≠(0,0) (for each p∈t(c) such that ∆Post(p,t)≠(0,0)). 9 ⎧P →` is a function that assigns to each discrete place a non-negative A marking m : ⎨ d + ⎩ Pc → \ number of tokens, represented by black dots, and to each continuous place a fluid volume; mi denotes the marking of place pi. The value of a marking at time τ is denoted by m(τ). The restrictions of m to Pd and to Pc are denoted by md and mc, respectively. A FOHPN system <PN,m(τ0)> is a FOHPN with initial marking m(τ0). The following statements rule the firing of continuous and discrete transitions: 1- a discrete transition t∈Td is enabled at m if for all pi∈•t, mi>Pre(pi,t); 2- a continuous transition t∈Tc is enabled at m if for all pi∈(d)t, mi>Pre(pi,t). Moreover, we say that an enabled transition t∈Tc is strongly enabled at m if for all places pi∈(c)t, mi>0; we say that transition t∈Tc is weakly enabled at m if for some pi∈(c)t, mi=0. In addition, for any continuous transition tj∈Tc its IFS is indicated by vj and it holds: 1- if tj is not enabled then vi=0; 2- if tj is strongly enabled, then it may fire with any firing speed vj∈[Vmj,VMj]; 3-if tj is weakly enabled, then it may fire with any firing speed vj∈[Vmj,Vj], where Vj≤VMj depends on the amount of fluid entering the empty input continuous place of ti. We denote by v(τ)=[v1(τ) v2(τ)… v|Tc|(τ)]T the IFS vector at time τ. Hence, any admissible IFS vector v at m is a feasible solution of the following set of linear constraints: VMj − v j ≥ 0 ∀t j ∈ Tε (m ) v j − Vmj ≥ 0 ∀t j ∈ Tε (m ) vj = 0 ∀t j ∈ Tυ (m ) ∑ ∀p ∈ Pε (m ) , t j∈Tε ( m ) C ( p, t j )v j ≥ 0 (1) where Tε (m ) ⊂ Tc ( Tυ (m ) ⊂ Tc ) is the subset of continuous transitions that are enabled (not enabled) at m and Pε (m ) = { pi ∈ Pc | mi = 0} is the subset of empty continuous places. In particular, the first three constraints in (1) follow from the firing rules of continuous transitions, while the last constraint in (1) imposes that if a continuous place is empty then its fluid content does not become negative. The set of all feasible solutions of (1) is denoted as S(PN,m). 10 B. The FOHPN dynamics The dynamics of the hybrid net combines both time-driven and event-driven dynamics. We define macro-events the events that occur when [3]: i) a discrete transition fires or the enabling/disabling of a continuous transition takes place; ii) a continuous place becomes empty; iii) a continuous place, whose marking is increasing, reaches a flow level that enables a set of discrete transitions; iv) a continuous place, whose marking is decreasing, reaches a flow level that disables a set of discrete transitions. The equation that governs the time-driven evolution of the marking of a place pi∈Pc is: m i (τ) = ∑ C ( p , t )v (τ) . t j ∈Tc i j (2) j Now, if τk and τk+1 are the occurrence times of two subsequent macro-events, we assume that within the time interval [τk,τk+1[ (macro-period) the IFS vector v(τk) is constant. Then the continuous behavior of an FOHPN for τ∈[τk,τk+1[ is described by: m c (τ) = m c (τk ) + Ccc v (τk )(τ − τ k ) (3) m d (τ) = m d (τk ). The evolution of the net at the firing of a discrete transition tj∈Td at m(τk-) yields the following marking: − m c (τk ) = m c (τk ) + Ccd σ(τk ) (4) − m d (τk ) = m d (τk ) + Cdd σ(τk ), where σ (τk ) is the firing count vector associated to the firing of transition tj at time τk. Moreover, we associate to each timed transition tj∈Tt a timer νj and we call ν(τk) the vector of timers associated to timed transitions at time τk. Hence, the timer evolution within the macro-period [τk,τk+1[ for each transition tj∈Tt is as follows: if t j is not enabled ⎧ ν j (τk ) = 0 , for j=1,…,|Tt|. ν j (τk +1 ) = ⎨ if t j is enabled ⎩ν j ( τ k ) + ( τ − τ k ) (5) Whenever tj is disabled or it fires, its timer is reset to zero. Equations (3)-(4)-(5) describe the dynamics of the FOHPN model. The overall state of the system at time τk is given by the marking of all places and by the values of all timers and it is 11 ⎡ m c ( τk ) ⎤ ⎡τ − τ ⎤ ⎢ ⎥ indicated by x (τk ) = ⎢ m d (τk ) ⎥ . Moreover, the system input is vector u( τ k ) = ⎢ k +1 k ⎥ , ⎣ σ ( τ k +1 ) ⎦ ⎢ ν (τk ) ⎥ ⎣ ⎦ collecting the length of the current macro-period and the transition (if any) that will fire at the end of such macro-period. A FOHPN system (3)-(4)-(5) can be described in the macro-period [τk,τk+1[ by a linear discrete-time time-varying state variable model of the following form: x (τk +1 ) = A(τk ) x (τk ) + B(τ k )u(τk ) , (6) where A ( τ k ) and B(τk ) are matrices of appropriate dimension. Hence, the behavior of the system can be described within the macro-period [τk,τk+1[ by the following equations: ⎡ m c (τk +1 ) ⎤ ⎡ I 0 ⎢ d ⎥ ⎢ ⎢ m (τk +1 ) ⎥ = ⎢ 0 I ⎢ ⎥ ⎣ ν (τk +1 ) ⎦ ⎣⎢ 0 0 0 ⎤ ⎡ m c (τk ) ⎤ ⎡Ccc v(τk ) Cdc ⎤ ⎡τ − τ ⎤ ⎢ d ⎥ ⎢ ⎥ 0 ⎥ ⎢ m (τk ) ⎥ + ⎢ 0 Cddσ (τk ) ⎥⎥ ⎢ k +1 k ⎥ . ⎣ σ (τk +1 ) ⎦ D (τk ) ⎦⎥ ⎣⎢ v (τk ) ⎦⎥ ⎣⎢ f (τk ) 0 ⎦⎥ (7) The elements of matrix D(τk) and vector f(τk) are elements equal to 0 or 1 and depend on the macro-event occurring at the sampling instant τk [3]. v1 V1 V2(a/b) 0 τ1 τ0 τ3 τ2 τ m1(τ) m1 p3 v2 t1 p4 V2 b t4 τ 0 t3 0 b a τ m2(τ) p 2 m2 p 1 m1 m2 0 a t2 τ ∆1 (a) ∆2 ∆3 (b) Fig. 1. An example of FOHPN (a) and its evolution (b). C. An example of FOHPN In this section we describe an example of FOHPN in order to clarify its dynamics. Consider the net in Fig. 1a. Places p1 and p2 are continuous and places p3 and p4 are discrete. Transitions t1 and t2 are continuous with firing speeds v1∈[0,V1] and v2∈[0,V2], respectively. We assume V1·b>V2·a (here a and b are the arc weights in Fig. 1a). In addition, 12 the discrete transitions t3 and t4 are exponentially distributed timed transitions with average firing rates λ3 and λ4, respectively. The net dynamics, depicted in Fig. 1b, is described as follows. Since place p4 is marked, transition t1 is enabled. Moreover, the initial markings of the continuous places are m1(τ0)>0 and m2(τ0)>0 so that transitions t1 and t2 are both strongly enabled and may fire according to the set of constraints (1): ⎧V1 − v1 ≥ 0 ⎪ ⎨V2 − v 2 ≥ 0 ⎪ v , v ≥ 0. ⎩ 1 2 (8) We assume v1=V1 and v2=V2. By (3), the continuous marking of the net during this first ⎧ m (τ ) = m1 (τ 0 ) − (V2 ⋅ a − V1 ⋅ b)(τ − τ 0 ) macro-period ∆1 is m c (τ ) = ⎨ 1 for τ>τ0 until the ⎩ m2 (τ ) = m2 (τ 0 ) − (V1 ⋅ b − V2 ⋅ a )(τ − τ 0 ) subsequent macro-event. Moreover, by (5) the timer vector is ν (τ ) = [0 τ − τ 0 ]T for τ>τ0, since t3 is disabled and t4 is enabled. Figure 1b shows the corresponding marking evolution and the IFSs of the net continuous transitions. In particular, we remark that the marking m1 increases while m2 decreases since it holds V1·b>V2·a. At time τ1 a macro-event occurs because place p2 becomes empty. Consequently, t1 becomes weakly enabled and the set of constraints (1) has to be re-written as follows: ⎧V1 − v1 ≥ 0 ⎪V − v ≥ 0 ⎪ 2 2 ⎨ v , v ⎪ 1 2 ≥0 ⎪⎩ v 2 ⋅ a − v1 ⋅ b ≥ 0. (9) Since t2 remains strongly enabled, its firing speed is assumed v2=V2. On the other hand, we choose the firing speed of t1 as v1=V2·(a/b). Therefore, during the subsequent macro-period of ⎧m (τ ) = m1 (τ1 ) for duration τ2-τ1, by (3) the continuous marking is expressed by m c (τ ) = ⎨ 1 ⎩ m2 (τ ) = 0 τ>τ1 until the subsequent macro-event (see Fig. 1b). Moreover, by (5) it holds ν (τ ) = [0 τ − τ1 ]T for τ>τ1. Next, suppose that at time τ 2 transition t4 fires and the macro-event updates the discrete markings to m3(τ2)=1 and m4(τ2)=0. Hence, t1 is disabled, i.e., v1=0, while t2 remains strongly enabled and we assume v2=V2. Then, during the macro-period [τ2,τ3[ the marking is given, as 13 ⎧ m (τ ) = m1 (τ 2 ) − V2 ⋅ a (τ − τ 2 ) (see Fig. 1b). Moreover, by (5) it holds in (5), by m c (τ ) = ⎨ 1 ⎩ m2 (τ ) = m2 (τ 2 ) + V2 ⋅ a (τ − τ 2 ) ν (τ ) = [τ − τ 2 0]T for τ>τ2. IV. THE SC MODEL Based on the idea of the bottom-up approach [33], this section proposes a modular FOHPN model to describe a SC. Such a method can be summarized in two steps: decomposition and composition. Decomposition consists in partitioning a system into several subsystems. In SCs this sub-division can be performed based on the determination of distributed system entities (i.e., suppliers, manufacturers, distributors, customers and transporters). All these subsystems are modeled by FOHPN modules. On the other hand, composition involves the interconnections of these sub-models into a complete model, representing the whole SC. In particular, manufacturers are described by continuous transitions, buffers are continuous places and products are represented by continuous flows (fluids) routing from manufacturers, buffers and transporters. Moreover, transporters are described by discrete stochastic transitions with a triangular distribution and the customers demand is modeled by exponential transitions. In addition, discrete exponential transitions model the information about the customer demands and the stochastic occurrence of unpredictable events in the system, such as the blocking of a supply or an accident in a transportation facility. Hence, the state of the SC model at the beginning of each macro-period is a vector x(τk) that includes the following sub-vectors: 1) the sub-vector mc(τk), collecting the markings of the continuous places, i.e., the buffer places and the associated capacity places (absent for infinite capacity buffers); 2) the sub-vector md(τk), collecting the markings of the discrete places, i.e., the places modeling choices, constraints and the operative states of entities; 3) the timers vector ν(τk), collecting the values of the timers of discrete timed transitions, i.e., the transitions associated to customer demands or transporters and the transitions modeling the blockings of supplies or transports due to unpredictable or external events. The following FOHPN modules model the individual subsystems composing the SC. 14 tT1 tTn … pC C - R -Q B B n CB - RB -Q1 t1 Q1 Qn CB - R B CB p’B 0 pB Q’1 Q’m Q’1 t Q’m … D 1 tDm Fig. 2. The FOHPN modeling the input buffers. A. The inventory management model of the input buffers In this section we describe the model of the input buffers of manufacturers and distributors managed by the (R,Q) policy. On the other hand, the output buffers are not managed by the (R,Q) policy since they are devoted just to providing the requested material. The basic quantities of the (R,Q) inventory management strategy are: the fixed order quantity Q; the lead time, i.e., the time between placing an order and receiving the goods in stock; the demand D, i.e., the number of units to be supplied from stock in a given time period; the reorder level R, i.e., the new orders take place whenever the stock level falls to R. Figure 2 shows the FOHPN model for the input buffers managed by the (R,Q) policy [14]. The continuous place pB denotes the input buffer of finite capacity CB. The complementary place p’B models the available buffer space so that at each time instant it holds mB+m’B=CB. Here and in the following models the assumed initial marking corresponds to empty buffers. Moreover, in the sequel we denote by Pb⊆Pc the set of the continuous places modeling the buffers and by P’b⊆Pc the set of places modeling the available buffer spaces. We assume that the buffer can receive demands from different facilities and can require the goods from different transporters. Transitions tTi∈TS with i=1,…,n represent the different kinds of transport operations and the continuous transitions tDi∈Tc with i=1,…,m model the demand of particular products, so that the corresponding demand to be fulfilled is Di=vDiQ’i with i=1,…,m. Hence, when mB>0 a transition tDi with i∈{1,…,m} may fire at the firing speed vDi so that the marking of place pB decreases with a constant slope vDi·Q’i. In this module the information flow is represented by the immediate transition t1∈TI and place pC∈Pd: as soon as mB falls below the level RB (or, equivalently, the marking m’B goes over CB–RB), the immediate transition t1 is enabled in order to send the information to the input facilities that have to provide material. When t1 fires, the choice place pC∈Pd becomes 15 marked and selects an input facility among the transitions tDi with i∈{1,…,m} that are enabled. Hence, new materials/products are requested by enabling one of the transitions tTi according to the value of the random switches RS(tTi) with i=1,…,n. Such random switches model the choice performed by a decision maker that selects the transporter on the basis of the knowledge of its attributes, e.g., distance, type, reliability, etc. If a particular transition tTi with i∈{1,…,n} is selected and fires after the lead time of average ∆(tTi)=1/λi, Qi products are received in the buffer and CB–RB–Qi units are restored in the buffer capacity. t’k p’k t1 pk tk pB 0 Q p’B CB Q tT Q Q Fig. 3. The FOHPN modeling the suppliers. B. The model of the SC entities The supplier model. Suppliers are modeled as a continuous transition and two continuous places (see Fig. 3). The continuous place pB∈Pb represents the raw material output buffer of finite capacity CB and the complementary place p’B∈P’b represents the available corresponding buffer space. Moreover, the continuous transition t1 models the arrival of raw material into the system. The occurrence of an event blocking the providing of raw material is represented by an exponentially distributed transition and two discrete places. In particular, place pk ∈ Pd models the operative state of the supplier and p’k ∈ Pd is the non-operative state. The blocking and the restoration of the raw material supply correspond to the firing of exponential transitions tk and t’k, respectively. For the sake of clarity, Fig. 3 depicts the transition tT ∈ TS that, as discussed later, models the transport operation. Here and in the following models the initial marking assumes that the entity is operative. 16 CB3 – RB3 –Q3 CBn – RBn –Qn CB3 – RB3 CB2 – RB2 CB2 p’B2 Qn Q3 Q2 CB2 – RB2 –Q2 pB2 0 p’B3 CB3 0 Q3 Q2 CBn – RBn pB3 … CBn p’Bn 0 pBn Qn Q3 Qn Q2 tj p’B1 pB1 0 CB1 Q1 Q1 Fig. 4. The FOHPN modeling manufacturers and assemblers. The manufacturer and assembler model. Manufacturers and assemblers are modeled by the FOHPN shown by Fig. 4. More precisely, the continuous places pBi∈Pb and p’Bi∈P’b with i=2,…n describe the input buffers and the corresponding available capacities, respectively. Each buffer stores the input goods of a particular type. Analogously, the continuous places pB1 and p’B1 model the output buffer and its capacity, respectively. The production rate of the facility is modeled by the continuous transition tj with the assigned firing speed vj∈[Vmj,Vj]. Q p’k t’k Q pk tT1 Q Q ... tTn tk p1 pC1 t1 CB1-R1 CB1 p’B1 Q Q ... pCn CBn-Rn-Q tn CB1-R1-Q CBn-Rn pB1 0 ... p’Bn CBn pBn 0 Fig. 5. The FOHPN modeling the transporters. The transporter model. The transporters connecting the different facilities are modeled each by a set of timed transitions tTi for i=1,…,n with triangular distributions (see Fig. 5), according to [18]. Each transition describes the transport of items of a particular type from an upstream facility to a downstream one in an average time interval ∆(tTi)=δi. 17 In this module the information flow is represented by places p1∈Pd, pCi∈Pd with i=1,…,n and immediate transitions ti with i=1,…,n. More precisely, place p1∈Pd selects only one type of material by enabling only one transition ti with i∈{1,…,n} and disabling the remaining transitions. When the chosen transition ti* with i*∈{1,…,n} fires, the corresponding place pCi*∈Pd is marked and, by enabling the corresponding transition tTi*, sends the message about the replenishment request to the transporter. Moreover, the random stop and resume of the material transport are represented by two places pk,p’k∈Pd and two exponentially distributed transitions tk,t’k∈TS. The transporter capacity is Q and the places pBi∈Pb and p’Bi∈P’b with i=1,…n of Fig. 5 describe the n input buffers of the downstream facility (e.g., a manufacturer, a distributor, a retailer) and the corresponding available capacities, respectively. The shown initial marking assumes that no material has yet been selected for transportation. tT1 tTn … pC C - R -Q B B n CB - RB -Q1 t1 Q1 Qn CB - RB CB p’B 0 Q’1 pB Q’m Q’1 Q’m … tD1 tDm Fig. 6. The FOHPN modeling the distributors. The distributor model. The model of the distributors is represented by an input buffer managed by the customary (R,Q) inventory control rule. Hence, the model is similar to the FOHPN represented in Fig. 2, where each downstream continuous transition tDi with i=1,…,m is substituted by a stochastic timed transition representing a transport operation (see Fig. 6) and pc∈Pd as well as t1∈PI model the information flow about the request. 18 CB – R B CB – RB - Q CB 0 p’B Q pB Q1 Q1 t1 Q1 pF Q2 tL (1-µ)Q2 µQ2 pS pD Q3 tT Fig. 7. The FOHPN modeling the retailers under the MTS strategy. The retailer model. Considering in this paper the standard MTS strategy to manage system, the retailer is a customer that orders with a finite stochastic lead time a stochastic quantity of material. Hence, we model the retailer by the continuous place pF collecting all the obtained products and an input buffer modeled by places pB and p’B managed by the (R,Q) policy with a finite lead time and stochastic demand (see Fig. 7). Consequently, the model is similar to the FOHPN represented in Fig. 2 where all the downstream continuous transitions are substituted by one or more exponential transition (such as t1 in Fig. 7) modeling the time at which the retailer performs the request. In the retailer module the information about the timing and the quantity of the stochastically ordered material is described by each exponential transition  , with constant triggering the timing request (such as t1 in Fig. 7) and the stochastic weight Q 1 distribution specified by the couples ∆Pre(pB,t1)= ∆Post(p’B,t1)= ∆Post(pF,t1). The timed discrete transition with triangular distribution tL models the deterioration of the finished products used by the customer that are stored in the infinite capacity buffers pS and pD. In particular, pS collects the µQ2 products to be disassembled with µ∈[0,1], and pD the (1µ)Q2 goods to be discarded. In addition, transition tT represents the transport operation transferring products to the disassembler. 19 Q1 CB1 – RB1 –Q1 CB1 – RB1 p’B1 CB1 0 t1 Q2 Qn Q2 CB2 Q’2 0 Q’2 pB2 Qn Q3 Q3 p’B2 pB1 CB3 0 p’B3 Q’3 Q’3 … CBn pB p’Bn 0 pBn Q’n Q’n Fig. 8. The FOHPN modeling the disassemblers. The disassembler model. The disassembly facilities are modeled by the FOHPN shown by Fig. 8 that is the reverse of the manufacturer model reported in Fig. 4. More precisely, pB1 and p’B1 model the input buffer and its capacity, respectively, and the continuous places pBi and p’Bi with i=2,…n describe the output buffers and the corresponding available capacities, respectively. The continuous transition t1 models the disassembly rate of the facility. T1 t1 Transporter T1 p7 p15 p10 p6 t3 Q1·c4 c5 t2 t10 Q1 C1-R1 D Q1·c4+c5 C1-R1-Q1 p14 Distributor D p’1 t4 p1 p9 T2 Q2 Q2 p8 Q2·c2 t6 C2-R2-Q2 Transporter T2 p’2 c3 Q2 t5 C2-R2 t9 Q3 Q2·c2+c3 p2 Q3 Q3·c1 t7 Q3 R Retailer R p13 p11 p12 p3 (1-µ)Q4 p4 t8 µQ4 p5 (a) (b) Fig. 9. A set of SC entities (a) and the corresponding FOHPN model including the financial flow among such entities. C. The model of the financial flow The financial flow in the SC may be easily modelled by a set of discrete places, representing the completion of transportation operations and the availability of money after a financial 20 transfer, and by a set of exponentially distributed transitions that model payment operations [5]. As an example, let us consider the financial flow among several entities of a SC, namely a distributor (D), a retailer (R) and two transporters (T1,T2) connected as shown in Fig. 9a. The FOHPN model of the SC can be straightforwardly obtained merging the corresponding modules in Figs. 5 to 7 and the resulting FOHPN can straightforwardly be modified to insert the financial flows as in Fig. 9b using the discrete places p10, p11, p12, p13, p14, p15, the exponentially distributed transitions t9 and t10 and their associated arcs. In particular, the discrete markings m10 and m11 represent the number of transportation operations respectively executed by transporters T1 and T2, the markings m12, m13, m14 and m15 represent the money available in the various companies, while transitions t9 and t10 model payment operations from one company to another. When Q3 product units are withdrawn by the consumer (i.e., upon the firing of t7), then m12 is incremented of a value Q3·c1 that represents the money available in the retailer R, where c1 is the cost paid by the consumer for a single product. This money is used to pay the transporter T2 and the distributor D. Indeed, when t9 fires the markings m13 and m14 that represent the money available in T2 and D are respectively incremented of a value c3 and Q2·c2, while the m12 is decreased of Q2·c2+c3 units, where c3 is the cost of a single transport and c2 is the cost that the retailer has to pay to the distributor for a single product. In turn, the distributor D has to pay the transporter T1 and the upstream company. Naturally, the number of firings of transitions t9 and t10 depends by the discrete markings m10 and m11. D. Discussion on the proposed SC modeling formalism To validate the presented SC modeling formalism, in this section we perform a comparison between the proposed model based on FOHPNs and an analogous model based on stochastic PNs [28], which includes discrete places and discrete (immediate or timed stochastic) transitions. To this aim, we consider the simple SC system shown in Fig. 10 and constituted by two suppliers, a manufacturer, a retailer and three transporters: two semi-finished products, labeled A and B, are available by the suppliers S1 and S2 from contract manufacturer M, which produces the finished product C that is sold to retailer R (note the absence of disassembling processes). Composing the modules shown in Section IV-B, the system of Fig. 10 can be modeled by the FOHPN of Fig. 11a: the dashed rectangles depict the correspondence between each module and the entities of Fig. 10. Note that for the sake of straightforwardness the retailer sub-module in Fig. 11a is simplified with respect to the model 21 of Fig. 7, due to the absence of disassemblers in the SC layout of Fig. 10. The SC is further modeled, using the classical discrete PN formalism proposed in [28] for SC modeling, by the place/transition net of Fig. 11b where all places are discrete and the continuous transitions of Fig. 11a are substituted by stochastic timed discrete transitions. Supplier S1 Supplier S2 Stage 1 A B Transporter T1 Transporter T2 Logistics 1 B A Manufacturer M Stage 2 C Transporter T3 Logistics 2 C Retailer R Stage 3 Fig. 10. An example of SC. t2 t1 S1 p’1 C1 p1 p2 0 Q1 Q1 0 Q2 C2 Q2 Q1 C 1-R1 -Q1 C1 -R1 p’3 p3 p4 0 C3 C 2-R2 -Q2 0 C4 p’3 0 p5 p’5 C B4 Q3 0 CB 5 p5 Q3 Q3 M t6 C3-R3 -Q3 p’6 C 2-R2 p’4 Q3 T3 C6 C 2-R2 -Q2 t3 Q3 C3- R3 p'2 T2 Q2 p3 p4 CB 3 t3 C5 Q2 Q2 t4 t5 C1 -R1 p’4 p’5 CB 2 Q1 Q1 C1-R1-Q1 C2-R2 S2 p1 p2 T1 T2 Q2 CB 1 Q1 t4 t5 T1 p’1 p'2 t2 t1 S1 S2 t6 T3 C3-R3-Q3 R C3- R3 p6 p’6 CB 6 Q4 t7 R p6 t7 Q4 Q4 0 Q3 Q4 Q4 Q4 M p7 p7 (a) (b) Fig. 11. The model of the example SC by a FOHPN (a) and by a place/transition PN. 22 Table 1: Firing speed (average firing delay) of continuous (discrete) transitions in Figs. 11a-b Transition [Vmin, Vmax] [0, 4] (Fig. 10a) [0, 5] (Fig. 10a) [0, 7] (Fig. 10a) t1 t2 t3 t4 t5 t6 t7 Transition parameters Average firing delay [hours] 1/4 (Fig. 10b) 1/5 (Fig. 10b) 1/7 (Fig. 10b) 2 (Figs. 10a and 10b) 3 (Figs. 10a and 10b) 3 (Figs. 10a and 10b) 3 (Figs. 10a and 10b) Table 2: Capacities, reorder levels and fixed order quantities in Figs. 11a-b. Capacities C1, C2, C3, C4, C6 C5 [parts] 100 150 Reorder levels [parts] R1=18 R2=25 R3=10 Fixed order quantities [parts] Q1=50 Q2=45 Q3=60 Q4=5 Table 3: Transition throughput values obtained by the simulation of the PNs in Figs. 11a-b. Nets FOHPN in Fig. 10a PN in Fig. 10b TT1 1.69 1.67 Transition throughputs [parts per hour] TT2 TT3 TT4 TT5 TT6 1.69 1.69 0.03 0.04 0.03 1.67 1.67 0.03 0.04 0.03 TT7 0.34 0.33 Hence, the SC dynamics is analyzed in the two cases via numerical simulation using the data reported in Table 1 that shows the manufacturer production rates of the FOHPN in Fig. 11a, as well as the average firing delays of discrete stochastic transitions of the nets in Fig. 11a and Fig. 11b. In addition, Table 2 reports further data necessary to fully describe and simulate the system: the reorder levels, the fixed reorder quantities and the buffer capacities. Note that the initial markings of the continuous (discrete) places pi with i=1,…,7 and p’i with i=1,…,6 in Fig. 11a (Fig. 11b) are mi=0 for i=1,…,7 and m’i=Ci for i=1,…,6, where Ci indicates the i-th buffer capacity (see Table 2). In order to verify that the FOHPN behavior is similar to the dynamics of the timed discrete PN, the two models are simulated in the MATLAB environment [24]. We consider the transition throughput value TTi associated to each net transition ti with i=1,…,7, indicating the average number of transition firings in a time unit during the considered run time. Such a performance index is evaluated by a long simulation run of 30000 time units with a transient period of 100 time units, where we assume that one hour corresponds to a time unit, leading to estimates of the performance index with a 95% confidence interval. The obtained results are reported in Table 3, showing that the throughput values remain nearly unchanged in the evolution of the two models. 23 50 45 45 40 40 35 35 Marking of place p3 Marking of place p3 50 30 25 20 30 25 20 15 15 10 10 5 5 0 0 5 10 15 20 25 30 Time units 35 40 45 0 50 (a) 0 5 10 15 20 25 30 Time units 35 40 45 50 (b) Fig. 12. Evolution of marking m3 of the FOHPN in Fig. 11a (a) and of the PN in Fig. 11b (b). To further comment differences in the FOHPN and discrete PN evolution, let us observe the simulated behavior of marking m3 in the FOHPN model in Fig. 11a (see Fig.12a) and the analogous marking for the discrete PN model in Fig. 11b (see Fig.12b). Figures 12a-b clearly show that the changes in the evolution of m3 in the FOHPN are much more infrequent than the corresponding changes in the evolution of marking m3 in the discrete PN, i.e., the number of macro-events occurrences in the FOHPN model is much smaller than the number of discrete events occurrences in the discrete PN model. In other words, since in the FOHPN model events occur less frequently than in the discrete PN model, changes in the system state are rare in the former model, so that the system analysis is computationally more efficient under such a formalism. To comment the use of the two formalisms for SC modeling at the operational level, we remark the following points. With respect to discrete frameworks, fluid models have potential for the application of more analytical techniques for optimization and control, possibly at the price of losing some modeling or analysis capability, e.g. relaxing the model by fluidification [21]. Fortunately, in most practical cases, errors due to such a relaxation of discrete models happen to be not significant when relatively heavy traffic conditions are relaxed. Indeed, the results in Table 3 demonstrate that, given the PN model in Fig. 11b, the fluidification performed by the corresponding FOHPN model in Fig. 11a is reasonable, since throughput errors due to the model relaxation are not significant. In other words, the FOHPN evolution mimics the behavior of the discrete PN despite the fluidification relaxation characterizing the former net. Hence, the FOHPN formalism can be effectively employed to model the SC, just like the discrete PN formalism, while benefiting from its advantages. 24 Concerning the simulation cost we point out that the discrete event simulation has to update the PN marking at each event occurrence that is determined by the token displacement. On the contrary, the continuous markings have a continuous evolution, so that only at the macro events occurrence the discrete markings have to be updated (see Figs. 12a and 12b). Consequently, the computational cost of the simulation of the hybrid PN is reduced with respect to the discrete PN, due to the reduction of the number of discrete event occurrences. Carrying on the comparison of the presented model with the existing formalisms based on PNs and proposed for SC operational management, we remark three crucial benefits in using the FOHPN framework. First, the proposed formalism overcomes the difficulties arising from the use of discrete quantities representing parts flowing in the system typical of place/transition net models. Thanks to the fluid approximation, both the implementation and simulation of the system model are possible without an excessive computational effort. Indeed, the linear discrete-time time-varying dynamics of the FOHPN models let us infer efficient simulation algorithms (see for instance the algorithm proposed in [3]). Second, using such a FOHPN model enables the designer to give a systematic interpretation of complex systems such as SCs and to choose an appropriate SC dynamics (i.e., optimal IFSs) according to a given objective function (e.g. maximizing resource utilization or minimizing the work-inprocess). In the case of decision support systems for SC configuration and re-configuration, this characteristic of the presented model has a very high added value. Third, the proposed formalism is flexible and able to describe a generic SC and to apply different management rules. For instance, the inventory management policies can be applied by means of simple modifications of the buffer models [12] and different management strategies may be implemented by suitably governing in a push or pull way the SC facilities. V. THE SYSTEM CONTROL The linear time-varying system model (6) combines both time-driven and event driven system dynamics. The matrices A ( τ k ) and B(τk ) defined in (7) describe the system in the macro-period [τk,τk+1[ and depend on the macro-event occurring at the sampling instant τk. Moreover, the actual IFS vector v(τk) ∈S(PN, m ( τ k ) ) affects the input matrix B(τk ) value and the system inputs, because it influences the occurrence of the next macro-event. Consequently, the procedure devoted to select one v(τk) among all the admissible IFS vectors, which can be determined by an appropriate controlling function, is of crucial importance. 25 Hence, we propose some control strategies that select the vector v(τk) in each macro-period on the basis of the knowledge of the system state and in order to optimize a particular objective function. To this aim, we select a subset PY⊆Pb of the set of continuous places representing buffers and we define the system output by the vector y(τ)∈ \ q with τ∈[τk,τk+1[ equal to the marking of the continuous places in PY⊆Pb, with q=|PY|. Hence, it holds: y(τ)=E x(τ), (10) where matrix E simply provides the restriction of state x(τ) to the markings of the subset PY. The block diagram depicted in Fig. 13 shows the structure of the considered SC system under the proposed control. The block diagram shows that the state vector x(τk+1) is obtained by equation (6) and the state x(τk) is obtained by a block representing a delay element equal to ⎡τ − τ ⎤ the length zk= τk +1 − τk of the k-th macro-period. The input vector u(τk ) = ⎢ k +1 k ⎥ is ⎣ σ (τk +1 ) ⎦ determined by a decision system that computes the length of the current macro-period [τk,τk+1[ and the discrete transition (if any) that will fire at the end of the considered macro-period. Such a computation depends on the values of the IFS vector v(τk) representing the production rates of the facilities and, consequently, a particular operational mode of the system. To optimize the overall system behavior, the IFS vector v(τk) is selected by a controller that optimizes an objective function subject to the set of linear constraints (1). Hence, the inputs of the controller are the difference between the output vector y(τ)∈ \ q and a reference vector mr∈ \ q as well as the system state. In the following we define two controllers on the basis of different performance indices to be optimized and different values of the reference vector mr. mr + Control Action - v(τk) Decision Maker u(τk) B(τk) x(τk+1) + x(τk) zk + A(τk) Fig. 13. Block diagram of the SC system under control. 26 y(τk) E Controller 1 (C1): Flow maximization. We consider a controller that intends to maximize the sum of all the flow rates. Hence, given mr=0 and PY=Pb, it chooses the solution v* that maximizes the following performance index: maxv J1=maxv (1T·v), (11) s.t. v∈S(PN,m). The controller defined by the linear programming problem (11) chooses the values of the IFS vector to maximize the throughput of the system. Controller 2 (C2): Buffer inventory control. This controller has the objective of keeping the stocks in a set of buffers at a particular constant level. Hence, given the subset PY⊆Pb, the reference value mri with i=1,…,|PY| is the desired constant level of the buffer pi∈PY. The controller chooses the vector v* that minimizes the following performance index: 2 ⎡ ⎤ minv J2= min v ∑ ⎢ m ri − (mic (τk ) + ∑ C ( pi , t j )v j (τk )) ⎥ , pi ∈PY ⎢ t j ∈Tc ⎥⎦ ⎣ (12) s.t. v∈S(PN,m). The controller defined by the least square problem (12) selects the production rates to guarantee the chosen good level of the buffer inventories in order to protect the SC from uncertainties, such as variations of the nominal values of demand quantity and mix, transport delays, deliveries etc.. Indeed, the controller modifies the production rates with the task of setting to constant values the goods in the buffers. We remark that in each defined control strategy, since the set S(PN,m) corresponds to a particular system macro-state, the optimization scheme is myopic, in the sense that it generates a piece-wise optimal solution, i.e. a solution that is optimal only in a macro-period. Nevertheless, we remark that the objective of the proposed controllers is guaranteeing that the SC is able to react in real time to the occurrences of unpredictable events, such as the blocking of supply or transport operations and the start of retailer requests, by suitably changing basic SC parameters such as the IFSs. To this aim, the controller has to work in real time and has to base the choices on the system state knowledge. 27 Stage 1 Supplier S1 M,K Transporter T1 Supplier S2 Supplier S3 M,K C,H,M C,H Transporter Transporter T3 T2 Logistics 1 M,K C,H,M Manufacturer M1 Transporter T4 C,H M,K Stage 2 Manufacturer M2 PC PC Transporter T5 Transporter T6 Logistics 2 PC PC Distributor D1 Stage 3 H C PC PC Transporter T7 Transporter T8 Logistics 3 PC PC Retailer R1 Retailer R2 Stage 4 PC PC Transporter T10 Transporter T9 Logistics 4 PC PC De-manufacturer DM1 Stage 5 H C Transporter T11 Transporter T12 Logistics 5 Fig. 14. The structure of the case study SC. VI. A CASE STUDY We describe an example of SC whose target product is a desktop computer system. Figure 14 depicts the SC network, comprising three suppliers, two manufacturers, one distributor, two retailers and one disassembler. Moreover, twelve transporters connect the facilities. Each edge represents the flow of material and is labeled by the parts/products that are transported between the connected facilities: the Personal Computer or PC, the central processing unit or 28 C, the hard disk driver or H, the keyboard or K and the monitor or M. In particular, with reference to the SC layout of Fig. 14, products of type C, H, K, and M are semi-finished products obtained from suppliers S1, S2 and S3, while the PC is produced by manufacturer M1 (M2) with a bill of materials of C, H, M and K provided by suppliers S1 and S2 (S3). Moreover, retailers R1 and R2 acquire the finished product PC from distributor D1. In addition, the disassembly facility DM1 obtains the finished product PC from the retailers and supplies manufacturer M1 (M2) with the semi-finished product H (C). Note that the SC scheme in Fig. 14 includes two inter-twined productive chains with a remarkable advantage: if a transportation link is temporarily unavailable the productive cycle does not stop. A. The case study SC model We model the whole SC by properly merging the elementary modules described in Section IV. Figure 15 shows the FOHPN modeling the SC under the MTS policy and dashed rectangles depict the correspondence between each module and the entities of Fig. 14. The production is determined by the firing of the continuous transitions t1, …, t7 (modules S1, S2 and S3) that describe the input of the raw materials that can be interrupted by stochastic events only. Each input buffer is managed by the (R,Q) strategy and when the input buffer of manufacturer M1 (M2) requires a particular product, a request has to be sent to the corresponding transporter. Hence, places p60, p63, p66 and p67 (modules T1 and T2) and places p72, p74, p75 and p77 (modules T3 and T4) are introduced to select a particular transporter. For example, if place p60 (module T1) is marked then the transport modeled by t43 (module T1) is enabled. In addition, transitions t56 and t58 and place p63 (modules T1 and T2) are introduced since the buffer of M1 storing the semi-finished products monitors (denoted by p17 and p’17) can require material from S1 by T1 or from S2 by T2. Consequently, place p63 with transitions t56 and t58 model the choice. According to the SC scheme of Fig. 14, in the model of Fig 15 the supply of some semifinished products at the manufacturers (i.e., H at place p21 in M1 and C at place p27 in M2) may be obtained via two different paths, either by a supplier or by the disassembler. The corresponding replenishment transition (i.e., t60 of T2 or t82 of T11 for M1, t63 of T4 or t83 of T12 for M2) is selected assigning a higher priority to the less costly supply obtained by the disassembly facility (i.e., assigning a higher random switch to t82 and t83 than those assigned to the conflicting transitions t60 and t63). However, the corresponding transition is enabled via 29 the respective arc weights Q13 and Q14 only when the matching semi-finished product output buffer of the disassembly facility (i.e., place p96 or p97 of module DM1) contains sufficient material. Finally, note that for the sake of simplicity in the considered model of the SC case study the financial flows are neglected, since the performance indices to be evaluated are related to production in general and not to the economic behavior of the system. B. The simulation specification The SC dynamics is analyzed via numerical simulation using the data reported in Table 4 that shows the manufacturer production rates and the average firing delays of discrete stochastic transitions. In addition, Table 5 shows further data necessary to fully describe and simulate the system, namely the buffer capacities, the reorder levels and the fixed order quantities. The initial markings of continuous places pi∈Pb (p’i∈P’b) are mi=0, (m’i=Ci). Nevertheless, we remark that additional simulations (not reported for the sake of brevity) carried out for the FOHPN in Fig. 15 with different initial markings in such places show that, if the simulation run time is sufficiently long, the obtained results are nearly identical to those here reported. Moreover, equal probability values are assigned to random switches of conflicting transitions and the fraction of consumed goods to be recycled is set equal to µ=0.5 in both retailers (see modules R1 and R2 of Fig. 15). In order to analyze the system behavior, the following basic performance indices are selected [26]: i) the average system throughput T, i.e., the average number of products obtained in a time unit; ii) the average system inventory SI, i.e., the average amount of products stored in all the system buffers during the run time TP; iii) the average lead time LT=SI/T that is a measure of the time spent by the SC to convert the raw material in final products. Note that in the considered simulation experiments the SI performance index (and, consequently, the LT value) is calculated taking into account only the upstream buffers with respect to the retailers. 30 S2 S1 t10 p43 p42 t11 p’1 Q1 T1 t24 t1 t2 M K t14 t13 p44 p46 p45 p3 p1 t12 p’3 t15 Q1 Q1 p’5 t3 t4 M C t43 t44 t45 t46 t26 p59 t28 Q3 t57 C15-R1 p15 p’15 Q1 C17-R2 K p67 Q3 p72 Q2 t60 Q4 Q4 C C25-R5-Q3 C23-R6 p’21 p75 t63 C25-R5 C27-R7 M H p21 p23 p’23 p77 p74 t62 t61 C21-R4 p19 K p25 p’25 t64 C21-R4 Q5 p27 p’27 C p’29 p29 H C27-R7-Q14 Q14 PC Q13 C21-R4-Q13 p’33 p33 Q5 p78 C29-R8 t9 PC p31 p76 Q3 C23-R6-Q3 Q2 M1 T5 t50 C29-R8-Q4 Q2 C19-R3 p79 t30 p71 t31 t8 t32 T4 Q4 t48 t49 p73 p66 p’31 Q4 Q4 T3 C19-R3-Q2 C17-R2-Q2 M p’19 p17 p’17 t22 p’13 t29 p65 t59 Q1 p55 p13 p70 t47 C21-R4-Q2 C17-R2-Q1 t55 H p11 p69 p62 t58 C t23 p54 C27-R7-Q4 p64 t56 p63 t7 Q3 Q3 p68 T2 C15-R1-Q1 t6 Q4 t27 p60 t21 p’9 Q3 t25 p61 p51 t18 Q2 p58 t42 p52 p9 Q2 p57 p56 H p53 p’11 Q2 Q2 p49 t20 t19 p50 t5 t16 p’7 p7 p5 Q2 Q2 Q1 S3 t17 p48 p47 Q6 Q6 t51 p80 M2 C27-R7 t34 p81 T6 t52 t35 t33 p82 D1 C35-R9-Q6 t65 Q5 C35-R9 C35-R9-Q5 PC p35 p’35 t36 p84 t53 t37 C37-R10-Q7 C37-R10 p’37 Q9 T8 t54 Q7 Q8 PC PC t39 C39-R11-Q8 C39-R11 p39 p37 Q9 t38 p86 p85 Q8 Q7 p83 T7 Q6 Q8 Q7 p’39 Q10 Q10 t40 R1 t41 Q9 Q15 p87 Q16 (1-µ)Q15 p113 p112 t66 µQ15 t67 (1-µ)Q16 p88 t68 T9 p90 p91 µQ16 p89 Q11 Q12 t70 p93 t72 p92 t71 T10 t73 t69 Q11 p94 C95-R12-Q12 t74 C95-R12 DM1 C95-R12-Q11 PC p’95 Q13 t76 p’96 t75 p99 H p96 Q13 p’97 p102 Q14 p97 C Q14 Q13 Q14 p98 t82 Q12 p95 Q13 T11 R2 Q10 p41 t78 p100 Q14 t80 p101 t83 t79 t81 t77 Fig. 15. The FOHPN modeling the case study SC. 31 p103 T12 Table 4: Firing speed (average firing delay) of continuous (discrete) transitions in Fig. 15. Continuous transitions t1 t5 t7 t2 t3 t4 t6 t8 t9 t75 [Vmin, Vmax] Exponential [0, 4] [0, 5] [0, 6] [0, 7] [0, 6] [0, 7] t22 t40 t72 t84 t16 t26 t34 t10 t14 t18 t76 t24 t28 t32 t36 t38 t80 t20 t30 t41 t85 t12 t68 t13 t69 t21 t31 t11 t15 t19 t25 t29 t33 t81 t37 t39 t77 t17 t27 t35 t23 t73 Discrete transitions Average firing Triangular delay [hours] 2 t53 3 t42 t43 t70 4 t47 t48 t78 4 t54 t71 t79 4 t44 t45 t52 5 t46 t49 t50 t51 6 18 t66 19 t67 20 20 20 21 22 Average firing delay [hours] 1 2 2 2 3 3 3 60 60 Table 5: Capacities, reorder levels and fixed order quantities in Fig. 15. Capacities C1, C5, C11, C15 C23, C25 C31 C37, C39 C3, C9, C13 C7, C27 C17, C19, C29 C33 C21 C35 C95, C96, C97 [parts] 100 100 150 70 100 100 100 150 100 120 80 Reorder levels [parts] Fixed order quantities [parts] R1=18 Q1=50 R2=25 Q2=45 R3=25 Q3=55 R4=25 Q4=40 R5=15 Q5=15 R6=15 Q6=15 R7=20 Q7=30 R8=20 Q8=25 R9=30 Q9=2 R10=10 Q10=5 R11=10 Q11=35 Q12=45 R12=10 Q13=40 Q14=40 Q15=25 Q16=35 The FOHPN model of the case study is implemented and simulated in the MATLAB environment [24]. Indeed, the modularity of the model suggests using an efficient software such as MATLAB, that allows to model systems with a large number of places and transitions. Moreover, such a matrix-based software appears particularly appropriate for simulating the dynamics of FOHPNs based on the matrix formulation of the marking update. Furthermore, the MATLAB software is able to integrate modeling and simulation of hybrid systems with the execution of control and optimization algorithms. As regards additional details on the model implementation in MATLAB, at the beginning of each macro-period (e.g., at the time instant τk), the program defines and solves one of the optimization problems 32 (11) and (12) on the basis of the knowledge of the system state x(τk). Having obtained the values of the production rates v(τk), the program determines the state and input matrices A ( τ k ) and B(τk ) respectively, as well as the occurrence time and type of the next macroperiod. Hence, equation (7) provides the new state x(τk+1) and the procedure is iterated. The simulation study is performed considering controllers C1 and C2 in Section V. In particular, controller C2 is implemented considering two different subsets PY: C2-1 with PY={p31, p33} and C2-2 with PY={ p1, p3, p5, p7, p9, p11, p13, p31, p33}. In these two cases the setting points are the following: mr1=mr3=mr5=mr7=mr9=mr11=mr13=60 and mr31=mr33=80. Hence, controller C2-1 aims at limiting the inventory of the costly SC finished products, without controlling the stocks neither of raw material nor of semi-finished products. On the other hand, controller C2-2 has the objective of limiting all stocks in the productive chain. All the indices are evaluated by a simulation run of 600 time units with a transient period of 100 time units, so that the run time TP equals 500 hours if we associate one time unit to one hour. The estimates of the performance indices are deduced by 50 independent replications with a 95% confidence interval. Besides, we evaluate the percentage value of the confidence interval half width to assess the accuracy of the performance index estimation: the half width of the confidence interval, being about 3% in the worst case, confirms the sufficient accuracy of the performance indices estimation. C. The simulation results Figures 16, 17 and 18 report the SC performance indices, i.e., throughput, system inventory and lead time, respectively, obtained employing the three different controllers. In particular, Fig. 16 shows that the throughput value obtained under C1 is greater than the corresponding values obtained under C2-1 and C2-2, because C1 maximizes the flow rates. On the other hand, it is apparent in Fig. 16 that, even though controllers C2-1 and C2-2 are aimed at controlling the inventory of some buffers, they both provide a very good value of the throughput, too. In addition, Fig. 17 shows that, as expected, the highest storage level is obtained when the SC is governed by the C1 policy, while controllers C2-1 and C2-2 lead to better inventory values. Besides, Fig. 18 shows that the lead time obtained under C2-2 is lower than the corresponding values obtained with the other policies, due to the high throughput and the medium system inventory obtained under C2-2. Indeed, controller C2-1 aims at limiting stocks in two buffers only, while C2-2 has to constrain the inventory levels in 33 the whole SC. Moreover, C1 aims at maximizing production regardless of the stock levels. 2.00 1.75 1.80 1.72 1.74 C2-1 C2-2 Throughput [parts per hour] 1.60 1.40 1.20 1.00 0.80 0.60 0.40 0.20 0.00 C1 Controller Fig. 16. Average throughput under the three controllers. 1400.00 1278.20 1194.60 1200.00 System Inventory [parts] 1026.30 1000.00 800.00 600.00 400.00 200.00 0.00 C1 C2-1 C2-2 Controller Fig. 17. Average system inventory under the three controllers. 800.00 731.29 694.22 700.00 588.61 Lead Time [hours] 600.00 500.00 400.00 300.00 200.00 100.00 0.00 C1 C2-1 Controller Fig. 18. Average lead time under the three controllers. 34 C2-2 inventory level [parts] 150 100 50 C1 C2-1 C2-2 0 100 200 300 400 Time [hours] 500 600 Fig. 19. Evolution of marking m31 of the FOHPN in Fig. 15 under the three controllers. As an example, we report in Fig. 19 the evolution of the marking of the finished products buffer p31 of manufacturer M1. The figure shows that under controller C1 the buffer is almost always full and its marking equals the available capacity, which is expected since under C1 the material flow in the SC is maximized. In addition, controllers C2-1 and C2-2 both tend to keep the stock levels around the imposed set-point value (which is of 80 parts for the considered buffer modeled by place p31). However, under C2-2 the time necessary for the buffer to reach such a set-point level is longer than the time required with C2-1, since the former controller has to constrain not only the finished product buffer levels but also the stocks in the upstream supplier buffers. Summing up, the simulation results show that managing the SC by controller C1 guarantees the highest productivity (and highest inventory). On the other hand, managing the case study by controller C2 imposes almost constant stock values in selected buffers, even under stochastic variations of the SC operative conditions, such as faults, production blockages etc., while still leading to satisfactory performances in terms of throughput and lead time. Hence, the SC may be more efficiently controlled by C2 rather than by C1, since under the former controller on one hand the stock-out phenomenon can not occur, even under exceptional and unforeseen demands (as it may happen with any control policy that tends to minimize the work-in-process), while on the other hand buffers can not saturate (as with C1) so that storage costs are sustainable. 35 VII. CONCLUSIONS The paper focuses on the problem of modeling and controlling at the operational level Supply Chains (SCs) that are emerging networks of business entities, very complex to describe and manage. The SC system is described by a modular model based on the first order hybrid Petri Net (PN) formalism: a fluid approximation of material and products is proposed and discrete unpredictable events occurring stochastically (i.e., blocking of suppliers, manufacturers, transporters, etc.) are modeled by the discrete event dynamics. Information flows are easily described in the proposed framework and financial flows may be straightforwardly represented by a discrete PN sub-model. The formalism can effectively describe SCs by a linear discrete-time, time-varying state variable model and enables the designer to choose decision variables by closed loop control policies. Indeed, we propose different controllers that employ the knowledge of the system state in order to drive the overall system to exhibit a satisfactory performance in terms of throughput and product inventory levels. 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