UNIFIED MODELLING FRAMEWORK OF MULTI-STAGE PRODUCTION-INVENTORY CONTROL POLICIES WITH LOT SIZING AND ADVANCE DEMAND INFORMATION

Transcription

1 Chapter 1 UNIFIED MODELLING FRAMEWORK OF MULTI-STAGE PRODUCTION-INVENTORY CONTROL POLICIES WITH LOT SIZING AND ADVANCE DEMAND INFORMATION George Liberopoulos Department of Mechanical and Industrial Engineering University of Thessaly, Pedion Areos, GR-38334, Volos, Greece glib@mie.uth.gr Isodoros Tsikis Department of Mechanical and Industrial Engineering University of Thessaly, Pedion Areos, GR-38334, Volos, Greece itsikis@mie.uth.gr 1

2 2 1. Introduction Every operations manager should be familiar with the terms reorder point policy (RPP), material requirements planning (MRP) and just in time (JIT). These terms have been used to describe three widely practiced approaches for coordinating the flow of material in multi-stage production-inventory systems. The literature advocating one or the other approach is voluminous. Each approach has its merits and its drawbacks; however, which approach is overall better remains a point of controversy among practitioners and researchers. In a growing literature that brings to light this controversy it is often pointed out that which approach is better? may not be the correct question to ask, since most real systems include all three approaches anyway. The main difficulty in comparing RPP, MRP and JIT systems is that they have emerged at different points in time, within different scientific cultures, and under different modelling assumptions. Thus, RPPs were developed for make-to-stock pure inventory systems, and MRP was developed as a computerized stage coordination tool in a deterministic, discrete-time setting, with advance demand information (ADI) in the form of a finite, planning horizon. Finally, the kanban system, the single technique most closely associated with JIT practices, was developed as a manual production control mechanism in Toyota s automobile production lines. The purpose of this chapter is not to study the controversy of RPP vs. MRP vs. JIT. Some of the important issues related to this controversy are discussed in [11] and references therein. Instead, the goal of this chapter is to (1) provide a unified modelling framework to facilitate the precise description and comparison of the dynamic behavior of simple production-inventory control policies with ADI, which can be characterized as RPP, MRP or JIT, (2) develop hybrid policies by combining simpler policies, and (3) bring to light properties of these policies. By exposing the dynamics and properties of production control policies in a common modelling framework, we hope to provide a connection between RPPs, MRP, and JIT and show that all three approaches are related and can coexist. The proposed modelling framework extends the framework for multistage production-inventory control mechanisms with lot sizing, developed in [10] and [11], to include policies that deal with perfect ADI. Most of the related literature review can be found in [11] and [9]. The latter reference is another chapter of this volume, which presents analytical and numerical results on single-stage and two-stage productioninventory control policies with ADI but no lot sizing.

3 Multi-Stage Production-Inventory Control Policies 3 Initially, we focus our attention on two of the most common RPPs, installation stock and echelon stock (Q, r) policies, to which we refer as IS and ES policies, respectively, for notational simplicity. We show that IS policies with ADI are equivalent to continuous-time MRP systems with fixed lot ordering quantity. MRP systems are used quite routinely in practice. A flaw of MRP systems, and RPPs in general, when used for production control, is that they assume infinite processing capacity, whereas actual manufacturing systems have finite capacity. In a capacitated manufacturing environment, the use of WIP-cap mechanisms, such as kanban-type policies, is more appropriate. In this chapter, we restrict our attention to two kanban-type policies, installation kanban and echelon kanban policies, to which we refer as IK and EK policies, respectively, for notational simplicity. An IK policy is closely related to what most researchers and practitioners understand as an ordinary kanban system, although there does not seem to be an agreed upon definition of what a kanban system exactly is [10]. According to an IK policy, each stage (or installation) has associated with it a number of installation kanbans, so that when a part leaves a stage, it releases an installation kanban of this stage, which can then be used to authorize the release of a new part into the stage. An EK policy with unit lot size is equivalent to what Buzacott and Shanthikumar [5] refer to as an integral control system. According to an EK policy, each stage has associated with it a number of echelon kanbans, so that when a part leaves the entire system, it releases one echelon kanban for each stage, which can then be used to authorize the release of a new part into that stage. Liberopoulos and Dallery [11] point out that when the number of echelon kanbans of the first stage is smaller than or equal to the number of echelon kanbans of all other stages, the resulting EK policy is equivalent to a make-to-stock CONWIP policy with lot sizing. They also note that EK policies use global information and may therefore have an advantage in terms of performance over IK policies, which use only local information. Both IK and EK policies, when used alone, however, have the disadvantages that (1) they do not communicate customer demand information to all upstream stages as quickly as IS and ES policies with ADI, (2) they use a single parameter to characterize both the base stock level and the number of kanbans at each stage, and (3) they can not take advantage of ADI. An obvious way to address the flaw of IS and ES policies with ADI, when used for production control, and the disadvantages of IK and EK policies, is to combine an IS or an ES policy with ADI with an IK or an EK policy to form a more sophisticated hybrid policy. In this chapter we restrict our attention to hybrid policies that result as combinations

4 4 of an IK policy with an IS or an ES policy with ADI and omit hybrid policies that result as combinations of an EK policy with an IS or an ES policy with ADI, due to space considerations and because (1) IK policies are more conventional than EK policies, and (2) hybrids of EK policies have similar structural properties with hybrids of IK policies. First, we note that there exist more hybrids of IK policies when there is ADI than when there no ADI. Then, we note that some hybrid policies are special cases of other hybrid policies. In particular, we point out that the combination of an IK policy with an IS or an ES policy with ADI can be achieved in a synchronized or an independent way, leading to synchronized and independent hybrid policies, respectively, where synchronized policies can be further divided into policies with delay before synchronization (DBS) or delay after synchronization (DAS). This implies that there are six combinations of an IK policy with an IS or an ES policy with ADI. It turns out, however, that only three of them are distinct, the other three being special cases of the distinct policies. The three distinct hybrid policies are: (1) synchronized DAS IK/IS policies with ADI, (2) synchronized DAS IK/ES policies with ADI, and (3) independent IK/ES policies with ADI. We refer to these three hybrid policies as policies A, B and C, respectively. We then note that policy A is equivalent to the PAC system proposed by Buzacott and Shanthikumar [5], which is one of the first hybrid policies to appear in the literature. Policy C, on the other hand, is an extension of the extended kanban control system (EKCS), proposed by Dallery and Liberopoulos [6], with lot sizing and ADI. Finally, we develop evolution equations using max, + notation to mathematically describe the dynamic behavior and derive properties of the three hybrid policies with ADI. As was mentioned above, one of the main intended usages of the proposed framework is to provide a unified modelling representation that allows the precise description of the dynamic behavior and properties of different production-inventory control policies with ADI. We find that many definitions of control policies that we have encountered in the literature are ambiguous in describing the exact operation of the policies they refer to and/or their relationship to other policies. The framework of a queuing network representation that we use in this chapter is a precise tool for describing the operation of different policies and the connections between them. This framework can also be very useful for developing analytical or simulation models of control policies for performance evaluation purposes. The queuing network representations, however, especially of hybrid policies, can be quite complicated to the unaccustomed reader. This complexity is unavoidable and stems from two sources. Firstly, the

5 Multi-Stage Production-Inventory Control Policies 5 representations are purposely quite detailed in order not to leave any doubts to the reader about the exact operation of the control policies they model. Thus, they employ (1) synchronization stations to model the matching or batching of parts, demands and production authorizations (kanbans), and (2) delays in the processing of parts and in the transfer of demands and/or kanbans. An alternative but not less detailed representation tool would be a Petri net. Secondly, hybrid policies are inherently more complicated that RPPs and kanban policies because they are combinations of the latter policies. Not all hybrid policies, however, appear to be equally complicated. The simplest hybrid policy appears to be policy C, which as was mentioned above is an extension of an EKCS with lot sizing and ADI. Finally, the practical implementation of production control policies does not have to involve the physical transfer of real kanban cards and demand slips. The control policies can be implemented via electronic information transfers that take place every time the state of the queuing system changes (e.g., whenever a demand for a production lot arrives to a stage or a production lot leaves a stage). The remaining of this chapter is organized as follows. In Section 2, we present modelling assumptions that are common in all the control policies discussed in subsequent sections. In Section 3, we present models of IS and ES policies with ADI, and in Section 4, we present models of IK and EK policies. In Section 5, we present models of hybrid policies which result as combinations of an IK policy with an IS or an ES policy with ADI. In Section 6, we present properties of these hybrid policies, and in Section 7, we derive evolution equations, using max, + algebra, to describe their dynamic behavior. Finally, we conclude in Section 8. A list of notations and abbreviations is given in an Appendix. 2. Modelling Assumptions We consider an N-stage serial production-inventory system. Every stage consists of a work-in-process (WIP) facility, where parts are processed, followed by a finished goods (FG) output store, where processed parts are stored. We assume that the system has access to perfect ADI over a finite time horizon. More specifically, we assume that customers arrive randomly in time and that each customer places an order for a non-fixed number of end items, i.e. stage-n FG, to be delivered to him/her exactly T time units after the time of his/her arrival. The order can be neither cancelled nor modified, and T is referred to as the demand lead time.

6 6 The arrival of every customer demand triggers the consumption of an end-item from FG inventory and the placement, activation, and release of a replenishment production order to the WIP facility of each stage in the system. The consumption of an end-item from FG inventory is triggered T time units after the arrival time of the demand. If no end-items are available at that time, the demand is backordered. The placement, activation, and release of replenishment production orders to the facilities of each stage depend on the control policy in place. To speed up the replenishment process, FG inventory at some or all the stages may have been built up to a certain target level ahead of time, i.e. before any demands have arrived to the system. We have used the terms placement, activation, and release to describe the three different phases in the life of a replenishment order. These phases are defined as follows. When an order is placed at a stage, the stage receives the order information. When an order is activated, parts corresponding to the order are requested to be released into the WIP facility of the stage for processing. When an order is released, parts corresponding to the order are actually released into the WIP facility of the stage for processing. The placement, activation, and release of a replenishment order are indicated in Figures 1.1 and 1.2 in the queuing network representations of the IS and ES policies with ADI in Section 3. We assume that there is an infinite supply of raw parts feeding the first stage. FG inventory levels at all stages are followed continuously, and replenishments of FG inventory may be ordered at any time. There is a setup cost associated with placing and processing an order; therefore, orders are placed and released for processing in batches or lots. Demands that are waiting for the arrival of other demands to complete a lot are referred to as single demands (SD). More specifically, we assume that a replenishment order at stage n is placed for the least integer number of lot sizes Q n. We make the common assumption that the order lot sizes satisfy Q n = j n Q n+1, n = 1, 2,..., N, and Q N+1 = 1, (1.1) for some positive integers j n. Assumption (1.1) is necessary if the rationing policy is to satisfy all or nothing of a production order, because then the FG inventory at every stage should always consist of an integer number of downstream lot sizes, except for the last stage, where the rationing policy allows the partial satisfaction of a customer order as long as stock is available. Besides simplifying material handling, the integer ratio constraint (1.1) also simplifies the analysis significantly. The cost increase due to constraint (1.1) is likely to be insignificant due to the insensitivity of inventory costs to the choice of order quantities.

7 Multi-Stage Production-Inventory Control Policies 7 In the presence of ADI, it may be cost effective to introduce a deliberate time delay between placing and activating an order, particularly if the demand lead time T is long. An order that has been placed but has not yet been activated is referred to as an outstanding demand (OD). An order that has been activated may not be immediately released due to the temporary lack of parts or production authorizations (kanbans), if such authorizations are necessary. An order that has been activated but has not yet been released is referred to as a backordered demand (BD). The deliberate delay between placing and activating a replenishment order depends on the so-called installation and echelon planned lead times associated with each stage. These lead times are design parameters, which are defined as follows. The installation planned lead time of stage n is denoted by l n and is a specified fixed control parameter that is related to the flow time of a typical part through the facility of the stage. It has the same meaning as the lead time in MRP systems. The echelon planned lead time of stage n is denoted by L n and is the sum of the installation planned lead times of the stage and all its downstream stages, i.e. N L n = l k, n = 1, 2,..., N. (1.2) k=n With the above definitions in mind, the time of activating a replenishment order at stage n is determined using an MRP time-phasing logic by offsetting the due date of the demand that triggered the order by the stage echelon planned lead time L n. This means that the order is activated without delay, if L n T, or with a delay equal to T L n with respect to the demand arrival time, if L n < T. In other words, the delay in activating an order, which id denoted by T n, is given by T n = max(0, T L n ), n = 1, 2,..., N. (1.3) 3. Installation Stock (IS) and Echelon Stock (ES) Policies with ADI Two of the most widely used RPPs are IS and ES policies. There are two common variants of IS and ES policies, depending on whether the reorder quantity is fixed or variable. In the first case they are usually referred to as (Q, r) policies, and in the second case they are usually referred to as (s, S) policies. In this section, we extend the definitions of IS and ES policies presented in [11] to include ADI. We restrict our attention to (Q, r) policies only, because (1) (Q, r) policies are perhaps more widely used than (s, S) policies, and (2) (s, S) policies have the similar structural properties to (Q, r) policies. With this in mind, henceforth,

8 8 when we refer to IS and ES policies, we shall mean IS and ES (Q, r) policies. When a multi-stage production-inventory system is controlled by an IS or an ES policy with ADI, every stage is controlled by a (Q, r)-rule based on its inventory position. This means that as soon as the inventory position of stage n falls at or below a reorder point r n, a replenishment order is placed for the least integer number of lot sizes Q n that raises the inventory position above r n. Once a replenishment order has been placed at stage n, it becomes an outstanding demand that will be activated after a time delay T n, given by expression (1.3). The difference between IS and ES policies with ADI lies in the definition of the inventory position. In an IS policy with ADI, the inventory position at stage n is defined as the installation stock at stage n, i.e. stock on hand (stage-n FG) plus outstanding orders (stage-n WIP + BD + OD) minus backorders (stage-(n + 1) BD + OD). In an ES policy with ADI, the inventory position at stage n is defined as the echelon stock at stage n, i.e. the sum of the installation stocks at stage n and all its downstream stages. In other words, the installation and echelon stock at stage n, which are denoted by i n and I n, respectively, are related as follows: N I n = i k, n = 1, 2,..., N, (1.4) k=n i n = I n I n+1, n = 1, 2,..., N. (1.5) With the above definitions in mind, the decision to place an order at each stage is based on local information in an IS policy, and on global information in an ES policy. The parameters Q n and r n are in general different for each stage. Queuing network model representations of a two-stage productioninventory system operating under an IS policy with ADI and an ES policy with ADI are shown in Figures 1.1 and 1.2, respectively. The symbolism used in Figure 1.1 and all other figures that follow in the rest of the paper is the same as that used in [11], except for the delay circles OD n, which are new elements related to ADI. The symbols in Figure 1.1 have the following meaning. The ovals represent WIP facilities, where manufacturing and/or transportation delays take place. The circles represent delays in the activation of orders. These delays are given by expression (1.3). The ovals and circles are named according to their content, and their initial value is indicated inside parentheses. For example, the delay in activating an order at stage 1 is represented by the circle OD 1. This means that when an order enters the delay circle OD 1, it stays there for exactly T 1 time

9 Multi-Stage Production-Inventory Control Policies 9 Figure 1.1. Queuing network model representation of a two-stage productioninventory system operating under an IS policy with ADI. Figure 1.2. Queuing network model representation of a two-stage productioninventory system operating under an ES policy with ADI. units before it is activated, i.e. before it departs from OD 1 and enters into BD 1, where T 1 is given by (1.3). Initially, OD 1 is empty, as is indicated by the (0) next to the symbol OD 1. The queues followed by vertical bars represent synchronization stations. The queues are named according to their content, and their initial value is indicated inside parentheses. For example, the queue representing the FG output store of stage 1 is named FG 1 and its initial value is i 0 n, i.e. i 0 n denotes the initial installation stock of stage 1. Every synchronization station has a marking on its side. This marking indicates the lot size needed in order to activate the synchronization station, i.e. the minimum number of customers that must be present in each queue to activate the synchronization station. For example, queues FG 1 and BD 2 are linked in a synchronization station marked with Q 2. This means

10 10 than as soon as there are at least Q 2 parts in FG 1 and Q 2 demands in BD 2, then exactly Q 2 parts depart from FG 1 and are released into WIP 2. At the same time, exactly Q 2 backordered demands depart from BD 2 and are discarded since they are satisfied. Another example is the synchronization station consisting of a single queue, SD 1, which is marked with Q 1. This marking means that as soon as there are at least Q 1 demands in SD 1, then exactly Q 1 demands depart from SD 1 and enter into the delay circle OD 1. The initial condition of SD 1 is r1 i + Q 1 i 0 1, where rn i denotes the installation stock reorder point of stage n. Notice that in both policies, if the demand lead time T is equal to zero, i.e. if there is no ADI, then all the delays T n are also equal to zero by (1.3), and therefore all the delay circles are redundant and can be eliminated. Even if the demand lead time T is not equal to zero, however, i.e. if there is ADI, the models in Figures 1.1 and 1.2 have exactly the same structure as the corresponding models without ADI, once we view OD n and BD n as a single queue. This is an important observation, because it implies that all of the properties of IS and ES policies without ADI developed in [1], [2] and [3] and reinterpreted in [11] carry over to the case where there is ADI. These properties are summarized below. An IS policy with ADI is always nested in the sense that when an order is placed at stage n, then orders must have been simultaneously placed at all downstream stages as well. The behavior of an IS policy with ADI does not depend on the initial installation stock positions i 0 n, but only on the echelon planned lead times L n, the installation stock reorder points rn, i and the reorder quantities Q n. An IS policy with ADI can always be replaced by an equivalent ES policy with ADI. Unlike IS policies with ADI, ES policies with ADI generally depend on the initial echelon stock positions In, e as well as on the echelon planned lead times L n, the echelon stock reorder points rn, e and the reorder quantities Q n. Also, unlike IS policies with ADI, ES policies with ADI are not always nested. An ES policy with ADI is nested if the initial installation stock inventory positions satisfy a certain condition. In this case, the resulting nested ES policy with ADI can be replaced by an equivalent IS policy with ADI. Even though the behavior of an IS policy with ADI does not depend on the initial installation stock positions i 0 n, the initial installation stock positions do play a role. Namely, they determine the number of demands for stage-n FG inventory that must arrive before a replenishment order of size Q n is placed at stage n. Without loss of generality, we may assume that i 0 n = r i n + k n Q n+1, n = 1, 2,..., N, (1.6)

11 Multi-Stage Production-Inventory Control Policies 11 where k n is an integer such that 1 k n j n. Assumption (1.6) guarantees that the inventory of stage n is at the reorder point exactly when placing an order. There are two extreme values for k n : one where k n = j n and another where k n = 1. If k n = j n, the initial installation stock positions are equal to their maximum levels, i.e. i 0 n = r i n +j n Q n+1 = r i n +Q n. In this case, exactly Q n demands, or j n lots of demands of size Q n+1, must arrive before a replenishment order of size Q n is placed at stage n. On the other hand, if k n = 1, the initial installation stock positions are equal to their minimum levels, i.e. i 0 n = r i n + Q n+1. In this case, exactly one lot of demands of size Q n+1 must arrive before a replenishment order of size Q n is placed at stage n. In the first case, replenishments cover past demands, whereas in the second case, they cover future demands. The second case corresponds exactly to the way an MRP system with fixed order quantity as its lot sizing rule operates in a continuous review setting. This is stated as follows. If the flow time of every replenishment order through the WIP facility of stage n is constant and equal to l n, and k n = 1 so that i 0 n = r i n+q n+1, for n = 1, 2,..., N, the resulting IS policy with ADI behaves exactly like an MPR system with fixed order quantity as its lot sizing rule. Given that an MRP system with fixed order quantity as its lot sizing rule is equivalent to an IS policy with ADI and that an IS policy with ADI is a special case of an ES policy with ADI, it follows that an MRP system with fixed order quantity is a special case of an ES policy with ADI; therefore, an ES policy with ADI may be viewed as a broader definition of an MRP system with fixed order quantity as its lot sizing rule. With this in mind, we presume that it is this broader definition of an MRP system that Asxäter and Rosling [2] have in mind, when they claim that any IS policy and any ES policy (without ADI) can be duplicated by an MRP system. Finally, in case of unit lot sizes, i.e. if Q n = 1, n = 1, 2,..., N + 1, an IS policy with ADI is identical to an ES policy with ADI, and they are both equivalent to what Karaesmen et al. refer to as a base stock policy with a release time parameter [8]. 4. Installation Kanban (IK) and Echelon Kanban (EK) Policies The original kanban system was developed as a manual information system for implementing JIT at Toyota s automobile production lines. The last two or three decades have seen a surge in the literature on

12 12 kanban systems, but there seems to be no agreed upon definition of what a kanban system exactly is [10]. Motivated by the preceding discussion regarding IS and ES policies, Liberopoulos and Dallery [11] introduced the notions of installation kanbans and echelon kanbans, which led to the definitions of installation kanban (IK) and echelon kanban (EK) policies, respectively, in the case where there is no ADI. In both policies, the placement of a replenishment production order to the facility of each stage, triggered by the arrival of a customer demand, is initiated after the consumption of a part from FG inventory. In case there is ADI, since the consumption of a part from FG inventory is activated T time units after the arrival time of the demand, the demand lead time T is totally unexploited as far as the placement of the replenishment policy is concerned. This implies the following important fact. IK and EK policies can not take advantage of ADI. To put it differently, IK and EK policies with ADI behave exactly like IK and EK policies without ADI, respectively. Nevertheless, in the rest of this section, we will recall some of the basic facts about IK and EK policies without ADI, developed in [11], because we will use them later in Sections 5-7 in our discussion of hybrid policies with ADI. In a multi-stage production-inventory system controlled by an IK or an EK policy, every stage n has associated with it a finite number of authorization cards or kanbans. This number is equal to an integer multiple of the stage lot size Q n. A kanban may be either free or attached onto a part. A free stage-n kanban is used to signal a customer demand for one part at stage n. Kanbans, like parts, move in lots of size Q n. Specifically, when Q n free stage-n kanbans have accumulated at stage n, an order of equal size, i.e. Q n, is placed at stage n. If Q n parts are available in stage-(n 1) FG inventory, the free kanbans are attached onto the parts and the combined lot, i.e. the Q n parts plus their kanbans, is released into the WIP facility of stage n. The kanbans remain attached to the parts until the combined lot reaches a certain final FG output store. When a part exits that output store, because it is consumed by the next downstream stage or by a customer (if the final FG output store is the output store of the last stage), the kanban that was attached to it is detached and becomes free. This free kanban is used once again to signal a customer demand for one part at stage n so that when Q n free kanbans have accumulated, an order of equal size is placed at stage n. The difference between IK and EK policies lies in the definition of the final FG output store, i.e. the point after which kanbans are detached

13 Multi-Stage Production-Inventory Control Policies 13 from parts. In an IK policy, the final FG output store at stage n is the FG output store of stage n. In an EK policy, it is the FG output store of the last stage, i.e. stage N. This means that in an IK policy, a stage-n kanban follows a part through the WIP facility and the FG output store of stage n and is detached from the part after the part leaves the FG output store of stage n. In an EK policy, on the other hand, a stage-n kanban follows a part through the WIP facilities and FG output stores of stages n through N and is detached from the part after the part leaves the FG output store of stage N. This implies that in an IK policy, the decision to place an order at each stage is based on local information, whereas in an EK policy it is based on global information from all downstream stages. The kanbans used in IK and EK policies are referred to as installation kanbans and echelon kanbans, respectively. Note that in an IK policy, every part in the WIP facility or FG output buffer at stage n has attached onto it a stage-n installation kanban. In an EK policy, on the other hand, every part in the WIP facility or FG output buffer at stage n has attached onto it one echelon kanban from each of stages 1 through n. This means that in an EK policy, when an end item is consumed by a customer, N echelon kanbans are detached from the part and become free. The practical implementation of kanban policies does not have to involve the physical transfer of actual kanban cards. Kanban policies can be implemented via electronic information transfers that take place every time the system state changes (e.g., whenever a demand for a production lot arrives to a stage or a production lot leaves a stage). The queuing network model representations of a two-stage productioninventory system operating under an IK and an EK policy are shown in Figures 1.3 and 1.4, respectively. Next, we summarize some important facts about IK and EK policies, which were developed in [11]. Figure 1.3. Queuing network model representation of a two-stage productioninventory system operating under an IK policy.

14 14 Figure 1.4. Queuing network model representation of a two-stage productioninventory system operating under an EK policy. In an IK policy, the installation stock reorder point at stage n is defined as r i n = (K i n 1)Q n, where K i n is an integer such that K i n 1, and K i nq n is the number of installation kanbans at stage n. The behavior of an IK policy does not depend on the initial installation stock positions but only on the reorder quantities Q n and the integers K i n. In an IK policy, demand is communicated at a stage only when FG inventory is consumed by the next downstream stage or by a customer. A consequence of this is that an IK policy can not take advantage of ADI, as was mentioned earlier. Another consequence is that an IK policy is never nested in the sense that an IS policy is. A third consequence is that in an IK policy, the WIP + FG inventory at every stage is always bounded by the number of installation kanbans. In an EK policy, the echelon stock reorder point at stage n is defined as r e n = (K e n 1)Q n, where K e n is an integer such that K e n 1, and K e nq n is the number of echelon kanbans at stage n. Unlike IK policies, EK policies generally depend on the initial echelon stock positions as well as on the reorder quantities Q n and the integers K e n. An EK policy may never be nested in the sense that an ES policy may be nested. Nonetheless, under a certain condition, an EK policy may be partially nested in the sense that when an order is placed at stage n, then orders must have simultaneously been placed at all but the last downstream stages. Unlike a nested ES policy, which can always be replaced by an equivalent IS policy, a partially nested EK policy can never be replaced by an equivalent IK policy. In an EK policy, demand is communicated at a stage only when an end item from FG inventory is consumed by a customer. Finally, an EK policy with K e nq n K e 1 Q 1, for n = 2, 3,..., N, is equivalent to a make-to-stock CONWIP policy [12] with a WIP-cap of K e 1 Q 1.

15 Multi-Stage Production-Inventory Control Policies 15 Liberopoulos and Dallery [11] mention that an important advantage of IK and EK policies over IS and ES policies is that the former policies impose an upper bound on the WIP + FG inventory, whereas the latter policies do not. One of the disadvantages of IK and EK policies, however, is that they do not communicate customer demand information to all upstream stages as quickly as IS and ES policies. This disadvantage has a direct impact on customer service since it implies longer customer response times, particularly if customer demand is highly variable. It also implies that the capacity of the system depends on the number of kanbans. Another disadvantage of IK and EK policies is that they can not exploit ADI, as was mentioned earlier. One way to overcome the disadvantages of kanban policies and increase customer service and system capacity is to uncouple (1) the actions of detaching a kanban and communicating demand information, and (2) the initial FG inventory and reorder point from the number of kanbans at every stage. This can be accomplished by combining an IK or an EK policy with an IS or an ES policy with ADI to form a more sophisticated hybrid policy. In the next section we will study such hybrid policies; however, we will limit our attention to hybrid policies that result as combinations of an IK policy with an IS or an ES policy with ADI and omit hybrid policies that result as combinations of an EK policy with an IS or an ES policy with ADI, due to space considerations and because (1) IK policies are more conventional than EK policies and (2) hybrids of EK policies have similar structural properties with hybrids of IK policies. The analysis of hybrids of IK policies with ADI can be extended with little effort to hybrids of EK policies with ADI. 5. Hybrid IK/IS and IK/ES Policies with ADI A hybrid IK/IS or IK/ES policy with ADI is a combination of an IK policy with an IS or an ES policy with ADI, respectively In a hybrid IK/IS or IK/ES policy with ADI, installation kanbans trace a closedloop trajectory within each stage and are detached from the FG output store of their stage as in an IK policy; however, when an installation kanban is detached from a part in FG inventory, it does not carry with it customer demand information to the previous stage, as is the case in an IK policy. Instead, demand is communicated according to the IS or ES with ADI in place. When there is no ADI, Liberopoulos and Dallery [11] differentiate between two types of IK/IS and IK/ES policies: synchronized and independent. The same differentiation holds when there is ADI; however, synchronized IK/IS and IK/ES policies with ADI are further divided

16 16 into policies with a delay before synchronization (DBS) or a delay after synchronization (DAS). The similarities and differences between independent, synchronized DAS and synchronized DBS IK/IS and IK/ES policies with ADI are presented next. In all three types of hybrid policies, i.e. independent, synchronized DAS and synchronized DBS IK/IS and IK/ES policies with ADI, the actions of detaching a kanban and communicating demand are uncoupled. Moreover, in all cases, the initial FG inventory and the reorder point are not determined by the number of kanbans, as is the case in IK policies. Finally, in all cases, customer demands are communicated according to the RPP (IS or ES) with ADI in place. The difference between the three types of hybrid policies has to do with the particular phase in the life of an order, i.e. the placement, activation or release phase (see the discussion in Section 2), that a stagen installation kanban authorizes once it is detached from a part in stage-n FG inventory. Thus, in a synchronized DAS IK/IS or IK/ES policy with ADI, when a stage-n installation kanban is detached from a part in stagen FG inventory, it is used to authorize the placement of a replenishment order for one part at stage n. In a synchronized DBS IK/IS or IK/ES policy with ADI, when a stage-n installation kanban is detached from a part in stage-n FG inventory, it is used to authorize the activation of a replenishment order for one part at stage n. Finally, in an independent IK/IS or IK/ES policy policy with ADI, when a stage-n installation kanban is detached from a part in stage-n FG inventory, it is used to authorize the release of a replenishment order for one part at stage n. In other words, in a synchronized IK/IS or IK/ES policy with ADI, the trajectory of installation kanbans is synchronized with either the placement (in the case of DAS) or activation (in the case of DBS) of orders, whereas in an independent IK/IS or IK/ES policy with ADI, the trajectory of installation kanbans is independent of the placement and activation of orders. In all three types of hybrid policies, the decision to authorize the placement, authorization or release of an order at each stage is based on local information, since it depends on the availability of installation kanbans. The decision to place an order at each stage, on the other hand, is based on local information, if the RPP in place is an IS policy with ADI, and on global information, if the RPP in place is an ES policy with ADI. With the above definitions in mind, there are six hybrid IK/IS and IK/ES policies with ADI: (1) synchronized DAS IK/IS policies with ADI, (2) synchronized DBS IK/IS policies with ADI, (3) independent IK/IS policies with ADI, (4) synchronized DAS IK/ES policies with ADI, (5) synchronized DBS IK/ES policies with ADI, and (6) independent IK/ES

17 Multi-Stage Production-Inventory Control Policies 17 policies with ADI. Similarly to [11], however, it can be shown that only three of them are distinct, the other three being special cases of the distinct policies (see [13] for details). The three distinct hybrid policies are: A Synchronized DAS IK/IS policies with ADI, B Synchronized DAS IK/ES policies with ADI, C Independent IK/ES policies with ADI. In what follows, we will restrict our attention to these three policies only, to which we will henceforth refer as policies A, B and C, respectively, for notational simplicity. Queuing network model representations of a two-stage production-inventory system operating under the three distinct hybrid polices A, B and C are shown in Figures 1.5, 1.6 and 1.7, respectively. Notice that the model in Figure 1.5 is a combination of the models in Figures 1.1 and 1.3. Similarly, the models in Figures 1.6 and 1.7 are combinations of the models in Figures 1.2 and 1.3. Figure 1.5. Queuing network model representation of a two-stage productioninventory system operating under a synchronized DAS IK/IS policy (policy A). A new element in Figures , with respect to all previous figures, is the set of queues FK n, which contain free stage-n kanbans. In all three hybrid policies, the total number of installation kanbans at stage n is KnQ i n, where Kn i is an integer such that Kn i 1, as was the case in IK policies. Initially, a number of these kanbans is attached onto an equal number of parts in the FG output buffer of stage n, defining the initial installation stock FG inventory position, i 0 n, and consequently the initial echelon stock FG inventory position, In, 0 at stage n, for all n. The

18 18 Figure 1.6. Queuing network model representation of a two-stage productioninventory system operating under a synchronized DAS IK/ES policy (policy B). Figure 1.7. Queuing network model representation of a two-stage productioninventory system operating under an independent IK/ES policy (policy C). remaining installation kanbans, i.e. K i nq n i 0 n kanbans, are stored in queue FK n as free installation kanbans, which are available to authorize the placement or release of an equal number of orders at stage n. The queuing network representations of the three hybrid policies A, B and C are more complicated than the simple RPPs and kanban policies discussed in Sections 3 and 4, but their complexity varies. Clearly, policy

19 Multi-Stage Production-Inventory Control Policies 19 C is less complicated than policies A and B. This is because in policy C, the transfer of demands and kanbans is totally uncoupled, whereas in policies A and B, it is indirectly coupled. Specifically, in policies A and B the transfer of demands is coupled with the placement of orders and the placement of orders is coupled with the return of free installation kanbans. This implies that in policies A and B, the communication of demands from a stage n to the previous upstream stage n 1 can be blocked due to the lack of free stage-n kanbans in queue FK n. Notice that in all three policies, if the demand lead time T is equal to zero, i.e. if there is no ADI, then all the delays T n are also equal to zero by (1.3), which means that all the delay circles are redundant and can be eliminated. In this case, it is not difficult to see that the behavior of policy B is identical to that of policy C. Even if the demand lead time T is not equal to zero, however, i.e. if there is ADI, the models of the hybrid policies with ADI have the exactly the same structure as the models of the corresponding hybrid policies without ADI. Specifically, the model of policy A has exactly the same structure as the model of the synchronized IK/IS policy without ADI, and the models of policies B and C have exactly the same structure as the model of the independent IK/ES policy without ADI, once we view OD n and BD n as a single queue. This is an important observation, because it implies that all of the properties of synchronized IK/IS policies without ADI and independent IK/ES policies without ADI developed in [11] carry over to the case where there is ADI. These properties are summarized in the following section. 6. Properties of Hybrid Policies A, B and C In hybrid policy A, the installation stock reorder point at stage n is defined as r i n = (R i n 1)Q n, where R i n is an integer such that 1 R i n K i n. We assume that the initial installation stock FG inventory positions satisfy (R i n 1)Q n < i 0 n R i nq n, for all n. Without loss of generality, we further assume that i 0 n = (R i n 1)Q n + k n Q n+1, where k n is an integer such that 1 k n j n. This assumption is equivalent to (1.6) and guarantees that the inventory of stage n is at the reorder point exactly when ordering. It also guarantees that i 0 n Q n+1, which is necessary in order for the system not to come to a deadlock. Under some fairly non-restrictive assumptions on the customer demand arrival process, we may further assume that the initial installation stock positions are equal to their maximum level, i.e. i 0 n = R i nq n. In this case, all the initial SD positions will be zero. This implies that policy A does not depend on the initial installations stock positions but only on parameters L n, Q n,

20 20 Kn i and Rn. i Policy A can not be nested in the sense that an IS policy with ADI is, except when Kn i =, for all n, as we will see below. Policy A includes an IK policy and an IS policy with ADI as special cases. Specifically, policy A with Kn i = Rn, i for all n, is equivalent to an IK policy, i.e. a policy which does not exploit ADI, as was mentioned in Section 4. Policy A with Kn i =, for all n, is equivalent to an IS policy with ADI (and hence to an MRP system with fixed order quantity), with installation stock reorder points equal to rn i = (Rn i 1)Q n, and is therefore nested. Any other policy A with Kn i such that Rn i < Kn i <, for all n, is not nested, imposes an upper bound on the WIP + FG inventory, just as an IK does, and exploits ADI for better replenishment control, just as an IS does; however, as was mentioned earlier, it does not take full advantage of ADI, since the communication of demands from a stage n to the previous upstream stage n 1 may be blocked due to the lack of free stage-n kanbans in queue FK n. Policy A is not new. Buzacott and Shanthikumar [5] introduced a system for coordinating multi-stage production-inventory systems, which they called production authorization card (PAC) system. The PAC system depends on four parameters per stage: the initial installation stock position, the number of installation kanbans, the order lot size, and the time delay when placing an order. Buzacott and Shanthikumar [5] demonstrate how through the appropriate choice of parameters the PAC system can be specialized into a wide variety of classical coordination approaches, such as kanban, base stock, etc. The PAC system is an extended version of one of the first hybrid policies to appear in the literature, called generalized kanban control system (GKCS), which was developed independently by Buzacott [4] and Zipkin [14]. More specifically, the PAC system is a GKCS without lot sizing and ADI. Buzacott [4] divided the GKCS system into two cases. In the first case, the number of installation kanbans at each stage is greater than or equal to the initial installation stock position. In the second case, the number of installation kanbans at each stage is smaller than the initial installation stock position. He referred to the first system as a backorderd kanban system and to the second case as a reserve stock kanban system. Liberopoulos and Dallery [10] argued that the backordered kanban system is indeed a new stage coordination policy, whereas the reserve stock kanban system is a classical IK policy, i.e. a policy that limits the WIP + FG inventory at every stage, with an additional constraint on WIP inventory alone. For this reason, they identified the GKCS, and by extension the PAC system, with the backordered kanban system only. We will follow the same approach here so that henceforth when we refer to the PAC system we will strictly mean the backordered kanban

21 Multi-Stage Production-Inventory Control Policies 21 system with lot sizing and ADI. With this in mind, a PAC system is equivalent to policy A, where the queues termed store, requisition tags, process tags and order tags in [5] are equivalent to queues FG n, BD n, FK n and SD n in Figure 1.5. A similar analysis can be carried out on hybrid policies B and C. We assume that in both policies B and C, the initial echelon stock FG inventory position at stage n satisfies (Rn e 1)Q n < In 0 RnQ e n, for all n, where Rn e is an integer such that Q n RnQ e n Rn+1 e Q n+1 KnQ i n or j n RnQj e n Rn+1 e Ke nj n, by (1.1), for n = 1, 2,..., N 1, and 1 RN e Ki N. Unlike policy A, policies B and C generally depend on the initial echelon stock positions, In, 0 as well as on the parameters L n, Q n, Kn e and Rn. e Under certain conditions, both policies may be nested in the sense that an ES policy with ADI may be nested. A nested synchronized policy B or C, however, can not be replaced by an equivalent policy A, because as was already mentioned above, policy A can never be nested (except when Kn i =, for all n). Both policies B and C include IK policies and ES policies with ADI as special cases. Specifically, a policy B or C with KnQ i n = RnQ i n = RnQ e n Rn+1 e Q n+1 or Knj i n = RnQj e n Rn+1 e, by (1.1), n = 1, 2,..., N 1, and Kn i = RN e, is equivalent to an IK policy. A policy B or C with Kn i =, for all n, is equivalent to an ES policy with ADI with echelon stock reorder points equal to rn e = (Rn e 1)Q n. Any other policy B or C with RnQj e n Rn+1 e < Ke nj n < imposes an upper bound on the WIP + FG inventory, just as an IK policy does, and exploits ADI for better replenishment control, just as an ES policy with ADI does. The notion of an independent IK/EK policy with ADI is not new. The idea of combining a local-information kanban system and a globalinformation RPP was introduced by Dallery and Liberopoulos [6]. They defined a control system that combines a base stock policy and a kanban policy in the case of unit customer demand, unit lot sizes, and no ADI, and called it extended kanban control system (EKCS). An independent IK/ES policy with ADI is an extension of an EKCS with lot sizing and ADI. 7. Evolution Equations of Hybrid Policies A, B and C Based on our discussions above, there are two limiting cases where policies A, B and C are equivalent to each other. In the first limiting case, all three policies are equivalent to an IK policy. In the second limiting case, policy A is equivalent to an IS policy with ADI, and policies B and C are equivalent to an ES policy with ADI, where, as was men-

22 22 tioned in Section 3, an ES policy with ADI can always be replaced by an IS policy with ADI. In any other case, policies A, B and C are never equivalent to each other. This means that if we take an IS policy with ADI and an equivalent nested ES policy with ADI, superimpose on each policy the same IK policy and synchronize the trajectory of installation kanbans with the placement of orders (in policies A and B) or the release of orders (in policy C), the resulting policies A, B and C will not be equivalent to each other. Although this remains to be seen, it would not be surprising if in many cases, policy C turned out to outperform the other two policies, because (1) policy C, like policy B, uses global information, whereas policy A uses local information, and (2) policy C totally uncouples the transfer of demands and kanbans, eliminating the possibility of blocking of demands due to the lack of free kanbans, whereas policies A and B indirectly couple the transfer of demands with the trajectory of kanbans. Moreover, as we observed earlier, policy C has the added advantage that it is less complicated than policies A and B. The dynamics of the three hybrid policies can be described in exact mathematical terms by recursive evolution equations that utilize the operators + and max only. These equations relate the timing of a particular event in a policy to the timings of events that must precede it. To elaborate, let the timings of different events be defined as follows: D (n 1,n),i The time that the ith part is released from stage n 1 to stage n, D (n 1,n),i The time that the ith order is placed from stage n to stage n 1, D n,i The time that the ith part completes processing in WIP n and is stored in FG n, D d,i The time that the ith demand arrives to the system. Now, suppose that WIP n consists of a single machine, and let σ n,i be the processing time of the ith part at the machine in WIP n. Following the methodology in [6], we can develop recursive evolution equations relating the timings of different events in the system (for details, see [13]). Specifically, the time that the ith part completes processing in WIP n and is stored in FG n is given by D n,i = σ n,i + max ( D (n 1,n),i, D n,i 1 ), (1.7)