cost-effective, i.e., such that it reduces the total production costs during a products' life-cycle. .... to be taken into account at planning and control in hybrid systems2. ... relate to extensions of standard MRP-systems (see Brayman, 1992 and .... inventory holding costs in remanufacturable inventory per product per time-unit ch.
Production planning and inventory control for remanufacturable durable products. Erwin van der Laan, Marc Salomon, Rommert Dekker Erasmus Universiteit Rotterdam P.O. Box 1738 NL-3000 DR Rotterdam
Abstract In this paper we focus on production planning and inventory control in systems where manufacturing and remanufacturing operations occur simultaneously. Typical for these hybrid systems is, that both the output of the manufacturing process and the output of the remanufacturing process can be used to fulfil customer demands. This process interaction together with other process interactions and various sources of uncertainty make production planning and inventory control in hybrid systems more complex than production planning and inventory control in traditional systems. The objectives of this paper are to demonstrate some of the effects of remanufacturing on production planning and inventory control, and to indicate various actions that can be taken by remanufacturing companies to improve their cost-efficiency. In particular, the issues of how much and where to keep (safety) stocks, how much and when to manufacture and remanufacture, and how to reduce the various sources of system uncertainty will be addressed. In this context we study a relatively simple hybrid system, related to a single component durable product. The behaviour of this system is numerically evaluated both under a PUSH-remanufacturing strategy (in which all returned products are remanufactured as early as possible) and under a PULL-remanufacturing strategy (in which the timing of remanufacturing operations depends on a combination of market demands for new products and market supply of returned products). Keywords: Production planning and inventory control, manufacturing, remanufacturing, statistical re-order point models, computational experiments.
1
Introduction
Our research in the area of production planning and inventory control with remanufacturing was initiated by a consulting project which we carried out in the Netherlands for a large U.S. manufacturer of copiers (see Thierry et al., 1995). The manufacturer had developed a prototype of a new generation of copiers, which differed from older generations in that some components stemming from used copiers could be re-used in new copiers. The process that transforms used components into components that satisfy exactly the same quality and other standards as new components is called remanufacturing. The main motivations for the manufacturer to develop copiers with remanufacturable components were the (anticipation on) environmental laws that (will) apply in many European and other countries. These laws make product manufacturers responsible for the collection and further handling of their products and packaging materials after customer usage. Furthermore, in the near future it is expected that environmental laws will even be tightened in many countries, forcing manufacturers to design products and production processes such that waste is limited 1
and/or a significant percentage of product components can be re-used. Another incentive to remanufacture products is the existence of new technologies, which enable manufacturers to design products and production processes such that remanufacturing of used components becomes cost-effective, i.e., such that it reduces the total production costs during a products’ life-cycle. Finally, a last but important motivation to apply remanufacturing is the potential for sales increases resulting from the ’environmental friendly’ image of remanufacturing companies. The above led to the managements decision to further develop the remanufacturable prototype to a commercial product-line of copiers. During the development phase the manufacturer got confronted with many technical and organizational problems, concerning e.g product design and logistics. An important logistic problem was for instance the development of a new distribution network (see Thierry et al., 1995), in which both the supply of new copiers to the market and the collection of used copiers from the market could be handled efficiently. Our consulting project was regarding another category of logistic problems, i.e., production planning and inventory control problems. The project finally resulted in an advice on how to re-organize the production/inventory system for the specific situation at the manufacturer. Motivated by this and other consulting projects (see Thierry et al., 1995 and Wein, 1995), the authors have set-up a research project to investigate the effects of durable product remanufacturing on production planning and inventory control from a more theoretical perspective. In particular, the following questions are considered: 1. Which processes and goods-flows are present in hybrid manufacturing/remanufacturing systems? Based on the observations that we made in practice (see Thierry et al., 1995) we have developed the process/goods-flow model outlined in Section 2. 2. Which process variables and process characteristics make production planning and inventory control in hybrid manufacturing/remanufacturing systems more complex than in traditional manufacturing systems? Again, based on practical observations we list in Section 2 a number of process variables and process characteristics that complicate planning and control in hybrid systems particularly. 3. Which policies are currently available to coordinate manufacturing and remanufacturing operations simultaneously? To answer this question, we review in Section 3 the relevant literature on planning and control policies, and we discuss the specific system assumptions under which these policies apply. 4. What are the effects of typical process variables and process characteristics on overall system costs? To investigate these effects, we introduce in Section 4 a relatively simple, single-product single-component system. For this system we analyse two statistical re-order point strategies, which mainly differ in the way in which the timing of remanufacturing operations is controlled. In Section 5 we empirically evaluate the system behaviour in general, and the sensitivity of the re-order point strategies for different assumptions regarding process variables and process characteristics. 5. Which actions can be taken by a remanufacturer to improve the cost-efficiency in hybrid manufacturing/remanufacturing systems? Based on the outcomes of the empirical study we suggest in Section 6 some actions that may be taken by a remanufacturer to improve the cost-efficiency. 2
Furthermore, Section 6 summarizes the contributions of this paper and provides some directions for future research.
2
Processes, goods flows, and problem complexity
To structure the discussion on production planning and control in hybrid manufacturing/ remanufacturing systems, we describe in Section 2.1 a process/goods-flow model. As indicated earlier, this model has been developed based on the observations that we made while carrying out case-studies in industry. Using the model components, we discuss in Section 2.2. why planning and control in hybrid manufacturing/remanufacturing systems tends to be more complex than planning and control in traditional manufacturing systems.
2.1
Process/goods-flow model
In the process/goods-flow model we distinguish the following processes and goods-flows: • The manufacturing process. The activities in this process are outside procurement orderings to obtain raw-materials and sub-assemblies, and production/assembly operations to obtain serviceables, i.e., end-products and/or components that are sold in the firsthand market. This process differs from the manufacturing process in systems without remanufacturing in that serviceables may be a assembled from a composite of new components and remanufactured components. Consequently, the material flows that are input of this process are (outside procured) raw materials and sub-assemblies, and remanufactured components. Output of this process are serviceables. • The disassembly and testing process. Returned products first enter the disassembly and testing process. During disassembly returned products are decomposed into modules and/or components1 , while during testing it is determined whether modules and/or components satisfy the quality requirements for remanufacturing. Based on the testing outcome, remanufacturables are separated from non-remanufacturables. Remanufacturables enter the remanufacturing process, whereas non-remanufacturables may be handled according to alternative options, such as recycling, cannibalisation, refurbishing, or disposal (see Thierry et al., 1995). Summarizing, input of this process are material flows of used products that are returned from the market. Output of this process are two different material flows, i.e. • The remanufacturing and testing process. Purpose of this process is to transform remanufacturables into serviceables. The process usually consists of repair, replacement, and cleaning operations. Furthermore, after remanufacturing the remanufactured components are usually tested. Components that satisfy the quality requirements of serviceables may be sold directly in the first-hand market, or may serve as input to the manufacturing process. The non-serviceable components, i.e., the components that do not satisfy these quality requirements, may be handled according to alternative options. Summarizing, input of this process are remanufacturables. Output of this process are serviceable and non-serviceable components. 1 The module or component level to which returned products are disassembled is usually pre-determined by technical constraints set by product design and/or by production process design.
3
Hybrid manufacturing/remanufacturing system may further consist of a number of decoupling points (buffers or inventories) to store returned products, remanufacturables, non-remanufacturables, serviceables and non-serviceables. The actual existence and location of each of these stocking points differs however from situation to situation, and depends for instance on the existence of particular goods-flows and on the policy that is followed to plan and control the goods-flows.
2.2
Complexity of planning and control in hybrid systems
An important question in the context of hybrid manufacturing/remanufacturing systems is the question ”which process variables and process characteristics actually make planning and control in hybrid systems more complex than planning and control in traditional manufacturing systems?”. The answer to this question is, that in traditional manufacturing systems solely the manufacturing process and related goods-flows need to be planned and controlled, whereas in hybrid systems also the disassembly and remanufacturing processes and related goods-flows need to be planned and controlled. Moreover, these processes and goods-flows are often interrelated, due to the following system interactions: • Interactions between the timing of demands and returns. These interactions occur due to two different type of correlations, i.e demand-return correlation, and the return-demand correlation. The first type of correlation occurs since customers demand a new product which may be returned after a certain (unknown) period of usage in case of a regular purchase contract, or after a (known) period of usage in case of a lease contract. The second type of correlation occurs due to product replacements: a customers demands for a new product and may instantaneously return a used one. • Process/goods-flow interactions. This interaction occurs since some goods-flows that are output of a particular process serve as input to other processes. In this way, the disassembly/testing process is linked with the remanufacturing process and the manufacturing process is linked with the remanufacturing process. Particularly in the latter case, when during manufacturing the goods-flows of manufactured/outside procured components must be integrated with the goods-flows of remanufactured components, complex timing problems may arise. • Capacity interactions between manufacturing and remanufacturing operations. Scarce resources, such as labour and/or machinery may be shared by multiple processes. For example, the same machinery may be used both for carrying out remanufacturing and manufacturing operations. In addition to these interactions there are also other system characteristics that make planning and control in hybrid systems a complex task. The presence of more processes and more goodsflows result in the following additional complexities: • More stocking points. To balance the inputs and outputs of the involved processes, to protect against various sources of uncertainty, and to reduce the required level of coordination in process control (i.e., to keep planning and control manageable) there are in hybrid systems usually more stocking-points (inventory buffers) than in traditional systems.
4
• More uncertainties. In addition to the uncertainties that are present in traditional manufacturing systems, such as the timing and sizing of demand occurrences, the yield of the manufacturing process, and manufacturing and procurement lead-times, there are in hybrid systems also uncertainties concerning e.g. the timing and sizing of product returns, the quality of returned products, the yield of the remanufacturing process, and the lead-times in the disassembly and remanufacturing processes. • More complex cost-structures. An increase in the number of processes, goods-flows and stocking points results in an increase in the number of different cost components that have to be taken into account at planning and control in hybrid systems2 . Concluding, more processes, more goods-flows, more stocking points, more complex interaction mechanisms, more complex cost structures, and more uncertainties require more sophisticated planning and control methods than in traditional manufacturing systems.
3
Literature review
In the production planning and inventory control literature many articles have appeared that consider the simultaneous occurrence of product returns and product demands. However, a large number of these publications deal with systems in which returns and demands solely occur due to failure of products at the customer site. Usually it is assumed in these systems that defective products can always be repaired at the base, after which they can be re-used as spares to replace other defective products. The key issue in these repair systems is, to determine the optimal number of spare-parts in the system, such that the average time to replace a defective product at the customer site satisfies a service level constraint and total expected system costs are minimized. Since in most repair systems reviewed by Nahmias (1981) and Cho and Parlar (1991) the population of customers is assumed to be constant over time, and since it is assumed that every defective product can be repaired at the base, the tota Due to the absence of a manufacturing process in repair systems, the interaction between manufacturing and remanufacturing need not be modelled. Since this interaction is typical for the hybrid systems considered in this paper, the literature on repair models will not be further addressed here. Contrary to the number of publications involving repair systems, the number of publications dealing with hybrid manufacturing/remanufacturing systems is rather limited, and all existing publications relate to single-product, single-component systems. For more complex systems, in which a product consist of multiple components and/or multiple end-products are involved, no formal control models have been published in the literature3 . To structure the review, we make a common distinction between control models in which the system status is reviewed at discrete time periods (periodic review models), and continuous review models, in which the system status is continuously reviewed. Furthermore, we distinguish between PUSH type of control strategies and PULL type of control strategies. Under a PUSH type 2
It should be noted that the assignment of suitable costs to the cost components that appear in hybrid systems is far more complicated than the assignment of costs in traditional manufacturing systems, since the costs and benefits related to multiple re-use of products and/or components have to be taken into account. 3 The few publications that deal with manufacturing/remanufacturing systems for multi-component products relate to extensions of standard MRP-systems (see Brayman, 1992 and Flapper, 1994) to administer the processes. However, operating policies for production planning and inventory control are not discussed in the context of MRP.
5
of control strategy the timing of the remanufacturing operations is completely return driven: as soon as a sufficient number of returned products is available in remanufacturable inventory, these products are batched, after which this batch is pushed into the remanufacturing process. Opposite to the PUSH type of control strategies, the timing of the remanufacturing operations under the PULL type of control strategies is return and demand driven. If, due to a demand occurrence, the serviceable inventory position becomes too low to satisfy expected future customer demands adequately, and if sufficient returned products are available in remanufacturable inventory, a batch from remanufacturable inventory is taken, which is pulled through the remanufacturing process to increase the serviceable inventory to an acceptable level. Both strategies coincide in that the timing of the manufacturing operations is solely based on the serviceable inventory position. Periodic review models The first model in this category has been proposed by Simpson (1978). It applies to a system with stochastic, dynamic, and correlated demands and returns, where the remanufacturing operation is controlled by a PULL-strategy, and remanufacturable products can be disposed of if necessary. The portion of demand that cannot be fulfilled by remanufacturing is fulfilled by outside procurement orderings. For each planning period within the finite planning horizon control parameters exist to indicate the amount of remanufacturable inventory that is remanufactured, the amount of remanufacturable inventory that is disposed of, and the amount of products that is outside procured. The cost function to be minimized consists of variable remanufacturing and outside procurement costs, inventory holding costs for remanufacturables and serviceables, backordering costs, and disposal costs. Clear limitations of this model are, that remanufacturing and outside procurement lead-times are assumed to be zero, and fixed manufacturing and outside procurement costs are not taken into account. Kelle and Silver (1989) formulate a model which differs from Simpsons model in that demands and returns are uncorrelated, all returned products are remanufactured (i.e., no disposal occurs), and remanufacturing is controlled by a PUSH-strategy. Furthermore, the cost structure includes fixed outside procurement costs, and service is modelled in terms of a service level constraint instead of backordering costs. The model has a single set of control parameters, i.e., the size of the outside procurement orderings in each planning period. As in Simpson’s model, lead-times are assumed to be negligible. Continuous review models The first continuous review model has been proposed by Heyman (1977). It applies to a market with uncertain but stationary and uncorrelated demands and returns. Every returned product is either disposed of immediately, or immediately remanufactured according to a PUSH-strategy. Remanufactured products enter serviceable inventory and outside procurement orders may be placed to fulfil that portion of customer demand which can not be fulfilled by remanufacturing. The policy that is applied to control this system is a single-parameter sd strategy, where sd is the serviceable inventory level at which returned products are disposed of instead of being remanufactured. The method that is suggested to determine the parameter sd that minimizes total average long-run costs uses results from queueing theory, and is exact, provided that demands and returns are Poisson processes. For more general demand and return processes Heyman presents approximation schemes to derive sd . As in the discret 6
Muckstadt and Isaac (1981) model a system which differs in many aspects from the system modelled by Heyman. The most important differences are, that Muckstadt and Isaac’s model applies to a situation with uncertain remanufacturing lead-times, finite remanufacturing capacities, and non-zero outside procurement lead-times. Furthermore, fixed outside procurement costs and backordering costs are included in the cost function. On the other hand, fixed remanufacturing costs are disregarded, and the option of product disposal does not exist. The system is controlled by a two parameter (sp , Qp ) PUSH-strategy, where sp is the inventory level at which an outside procurement ordering of size Qp is placed. The procedure that is suggested to determine the policy parameters which minimize total expected costs is approximative. Extensions of the Muckstadt and Isaac model to include disposal of returned products have been studied by Van der Laan et al. (1994). A deterministic model that includes the disposal option has been studied by Richter (1994). Finally, alternative models that could serve as a starting point to model and control hybrid manufacturing and remanufacturing systems stem from finance: the cash-balancing models. These models usually consider a local cash of a bank with incoming money flows stemming from customer deposits (returns), and outgoing money flows, stemming from customer withdrawals (demands). To satisfy the customer demands adequately, the possibility exists to increase the cash-level of the local cash by ordering money from the central bank (outside procurement). If the cash-level of the local cash becomes to high, it can be decreased by transferring money to the central bank (disposals). Objective in these models is, to determine the timing and sizing of the cash transactions, such that the sum of fixed and variable transaction costs, backordering costs, and interest costs related to the local cash are minimized. Constantinides and Richard (1978) have pointed out that under particular conditions the optimal control policy has the following structure: if the inventory level at the local cash becomes less than sp , a procurement order is placed at the central bank to raise the local cash level to Sp . If the local cash level exceeds sd , the local cash level is reduced to Sd by transferring money back to the central bank. The reason why we only briefly address the cash-balancing models here is, that they do not explicitly deal with some of the characteristics that are specific for hybrid manufacturing and remanufacturing systems. The most important limitation of the cash-balancing models is the absence of a remanufacturing process: every returned product (money) is instantaneously added to serviceable inventory (local cash), without being tested on quality and/or being remanufactured. For an extensive overview of cash-balancing models we further refer to Inderfurth (1982).
Literature on the effects of remanufacturing Contrary to this article, the articles that appeared until now in the literature focused mainly on algorithms to calculate an (approximately) optimal set of control parameters for a given operating strategy, rather than on an analysis of the effects of remanufacturing on system performance.
4
System assumptions and control strategies
To analyse the effects of remanufacturing on system performance we had to select system and control strategy combinations such that (i) mathematical analysis is possible, and (ii) all important effects that occur in practice could be represented. Unfortunately, conditions (i) and (ii) are impossible to satisfy simultaneously: constraint (i) limits the selection to relatively simple 7
systems, operating under relatively simple strategies, whereas (ii) forces to investigate complex real-life systems operating under optimal strategies. As a compromise we have selected to analyse a single-product hybrid system by means of which the influence of a number of typical process variables and process characteristics discussed in Section 2 could be evaluated. The specific assumptions regarding the system are outlined in Section 4.1. The notation that we use in the remainder of this paper and the cost-criterion that we define to measure the system performance are specified in Section 4.2. To control the system, we investigate two different continuous review operating strategies, one being a PUSH-strategy, the other one being a PULL-strategy. The PUSH and PULL-strategy are further outlined in Sections 4.3 and 4.4 respectively. The enumerative procedure that has been implemented to search for optimal strategy control parameters is discussed in Section 4.5. Finally, to further investigate the effects of specific assumptions regarding process variables and system characteristics, we analyse in Section 4.6 the operating strategies under more general assumptions.
4.1
System assumptions
To start out, we investigate a single-product, single-component system which is characterized as follows: • The remanufacturing process. All returned products that satisfy the quality requirements for remanufacturing enter the remanufacturing process, i.e, disposal of remanufacturables does not occur. The remanufacturing process has unlimited capacity, the remanufacturing lead-time is Lr , the fixed remanufacturing set-up costs are cfr , and the variable remanufacturing and work-in-process costs are cvr per product. After remanufacturing products enter serviceable inventory. • The manufacturing process. New products are manufactured. The manufacturing costs consist of a fixed component of cfm per set-up, and a variable component which includes work-in-process costs of cvm per product. Manufacturing capacity is unlimited, and the manufacturing lead-time is Lm . Manufactured products enter serviceable inventory. • Stocking points. There exist two stocking points in the system, one to keep remanufacturable inventory and one to keep serviceable inventory. The holding costs in remanufacturable (serviceable) inventory are chr (chs ) per product per time-unit. Stocking points have unlimited capacities. • Demands, returns, and backorders. To evaluate the influence of (uncertainties in) demands and returns on system performance we assume that unit sized demands and unit sized remanufacturable returns have exponentially distributed inter-occurrence times. The average time between two subsequent remanufacturable returns (demands) is λ1R ( λ1D ). Furthermore, the intensity of remanufacturable returns λR is smaller than the demand intensity λD , and demands and remanufacturable returns are uncorrelated. Demands that can not be fulfilled immediately are backordered. The above system is visualized in Figure 1. Insert Figure 1 about here
8
4.2
Notation and definition of system performance function
Table 1 below lists the notation that will be used in the remainder of this paper4 . Furthermore, for each of the time-dependent variables that appear in Table 1 (say V1 (t) and V2 (t, t + δ)) we define the long-run average (V 1 and V 2 ) as, t
Z V 1 = lim
t→∞
0
1 V1 (u)d(u), and V 2 (δ) = lim t→∞ t
Z 0
δ
1 V2 (t, t + u)d(u), and V 2 = lim V 2 (δ). δ→0 δ
cv.
=
Notation related to process (.) variable processing costs per product
cf.
=
fixed set-up costs per batch
L.
=
processing lead-time
E. (t0 , t1 )
=
total number of products that enter process (.) in the time-interval (t0 , t1 ]
W. (t)
=
total number of products in work-in-process in (.) at time t
O. (t0 , t1 )
=
chr
=
total number of ordered batches from process (.) in time-interval (t0 , t1 ] Notation related to stocking points (inventories) inventory holding costs in remanufacturable inventory per product per time-unit
chs
=
inventory holding costs in serviceable inventory per product per time-unit
Is (t)
=
serviceable inventory position at time t. The serviceable inventory position is defined as the onhand serviceable inventory plus the number of products in manufacturing work-in-process plus the number of products in remanufacturing work-in-process minus the number of products in backorder at time t
IsOH (t)
=
number of products in on-hand serviceable inventory at time t
Isnet (t)
=
the net serviceable inventory at time t. The net serviceable inventory is defined as the number of products in on-hand serviceable inventory minus the number of products in backorder at time t
IrOH (t)
=
λD
=
number of products in remanufacturable on-hand inventory at time t Notation related to demands, returns, and backorders number of demanded products per time-unit (demand intensity)
D(t0 , t1 )
=
number of demanded products in the time-interval (t0 , t1 ]
2 cvD
=
squared coefficient of variation in the number of demanded products per time-unit (demand uncertainty)
λR
=
number of remanufacturable products that are returned per time-unit (remanufacturable return intensity)
2 cvR
=
squared coefficient of variation in the number of remanufacturable returned products per time-unit (remanufacturable return uncertainty)
ρRD
=
probability that a product return instantaneously induces a demand (return-demand correlation coefficient)
B(t)
=
number of products in backorder at time t
cb = backordering costs per product per time-unit Table 1. Definition of system notation.
The long-run average system costs per unit of time under control policy (.) are denoted by the function C(.). The function C(.) reads, OH
C(.) = chs I s
OH
+ chr I r
+ cvr E r + cfr Or + cvm E m + cfm Om + cb B
The cost components that appear in (1) are the following: 4
The notation regarding the policy control parameters is defined throughout the text.
9
(1)
OH
chs I s
:
long-run average inventory holding costs for serviceable inventory per unit of time,
chr I r
:
long-run average inventory holding costs for remanufacturable inventory per unit of time,
cvr E r
:
long-run average variable remanufacturing costs per unit of time,
cfr Or
:
long-run average fixed remanufacturing costs per unit of time,
cvm E m
:
long-run average variable manufacturing costs per unit of time.
cfm Om
:
long-run average fixed manufacturing costs per unit of time,
cb B
:
long-run average backordering costs per unit of time.
OH
4.3
Analysis of the (sm , Qm , Qr ) PUSH-strategy
The operating characteristics of the (sm , Qm , Qr ) PUSH-strategy are as follows: as soon as remanufacturable inventory contains Qr products, these products are batched and pushed into the remanufacturing process, reducing remanufacturable inventory to zero, and increasing the serviceable inventory position by Qr products. Manufacturing starts whenever the serviceable inventory position Is (t) drops below the level sm + 1. Manufacturing takes place in batches of Qm products. It should be noted that the (sm , Qm , Qr ) strategy generalizes the (sp , Qp ) strategy of Muckstadt and Isaac (1981) in that fixed remanufacturing costs can be taken into account by allowing to batch remanufacturing orders. On the other hand we assume that remanufacturing capacity is unlimited. Furthermore, our analysis to calculate the costs C(sm , Qm , Qr ) is exact rather than approximative. The calculation of the components of C(sm , Qm , Qr ) proceeds as follows: - The long-run average on-hand inventory of serviceables per unit of time is calculated using P OH net = inet } is the probability inet Pr{Isnet = inet the relation I s = inet s }, where Pr{Is s s >0 s that the net-inventory of serviceables equals inet in the long-run steady-state situation. To s calculate the probability distribution Pr{Isnet = inet }, we use the relationship that the nets inventory of serviceables at time t equals the inventory position at time t minus the number of products in outstanding manufacturing orderings at time t minus the remanufacturing work-in-process at time t, i.e., Isnet (t) = Is (t) − Wm (t) − Wr (t)
(2)
However, since the distributions of Wm (t) and Wr (t) are difficult to evaluate, we rewrite (2) as, (
Isnet (t)
=
Is (t − Lm ) + Er (t − Lm , t − Lr ) − D(t − Lm , t), Lr ≤ Lm Is (t − Lr ) + Em (t − Lr , t − Lm ) − D(t − Lr , t), Lr > Lm
(3)
To explain (3), we first consider the case Lr ≤ Lm . In this case, the net serviceable inventory at time t equals the net serviceable inventory at time t − Lm plus the remanufacturing work-in-process at time t−Lm (which arrive at or before t since Lr ≤ Lm ) plus the number of products in manufacturing work-in-process at t − Lm (which arrive in net serviceable 10
inventory at or before time t) plus the number of products that enter remanufacturing in the time-interval (t − Lm , t − Lr ] (which will have entered serviceable inventory at time t) minus the demands in the interval (t − Lm , t], i.e., Isnet (t) = Isnet (t − Lm ) + Wr (t − Lm ) + Wm (t − Lm ) + Er (t − Lm , t − Lr ) − D(t − Lm , t)
(4)
Substitution of the right-hand side of (2) at time t − Lm in (4) then yields (3) for the case Lr ≤ Lm . For the case Lr > Lm analogous arguments can be used to derive (3). Next, we further evaluate (3) to enable numerical analysis. First, we consider the case Lr ≤ Lm . For this case it can be verified that Er (t − Lm , t − Lr ) is correlated with Is (t − Lm ) since a low (high) number of products that enter the remanufacturing process in the interval [t − Lm , t − Lr ) relates to a relatively high (low) serviceable inventory at time t − Lm . Furthermore, the number of products that may enter the remanufacturing process in the interval [t − Lm , t − Lr ) (Er (t − Lm , t − Lr )) is correlated with the number of products that are available in remanufacturable on-hand inventory at the beginning of the interval (IrOH (t − Lm )). Taking into account these correlations, we obtain the limiting joint probability distribution Pr{Isnet = inet s } using the relation, Pr{Isnet = inet s } =
lim
r −1 X QX
t→∞
Pr{Is (t − Lm ) = is , IrOH (t − Lm ) = iOH , Er (t − Lm , t − Lr ) = er , D(t − Lm , t) = d} = r
Ω1 iOH =0 r
(5) lim
r −1 X QX
t→∞
Pr{Er (t − Lm , t − Lr ) = er |Is (t − Lm ) = is , IrOH (t − Lm ) = iOH }× r
Ω1 iOH =0 r
Pr{Is (t − Lm ) = is , IrOH (t − Lm ) = iOH } × Pr{D(t − Lm , t) = d} r
where Ω1 = {(is , er , d)|is +er −d = inet s }. The procedure to calculate Pr{Er (t−Lm , t−Lr ) = er |Is (t−Lm ) = is , IrOH (t−Lm ) = iOH r } is further outlined in Appendix A. The limiting joint OH probability distribution π1 (is , ir ) = limt→∞ Pr{Is (t − Lm ) = is , IrOH (t − Lm ) = iOH r } is calculated using a continuous time Markov-chain model M1 , with two-dimensional state variable X1 (t) = {Is (t), IrOH (t)|t > 0} and a two-dimensional state-space S1 = {sm + 1, . . . , ∞} × {0, . . . , Qr − 1}. The transition rate νs(1) ,s(2) related to a transition from state s(1) to state s(2) is defined as, ν(is ,iOH ),(is ,iOH +1) = λR r r
iOH < Qr − 1 r
ν(is ,iOH ),(is +Qr ,0) = λR r
iOH = Qr − 1 r
ν(is ,iOH ),(is −1,iOH ) = λD r r
is > sm + 1
ν(is ,iOH ),(sm +Qm ,iOH ) = λD is = sm + 1 r r Finally, the probability Pr{D(t − Lm , t) = d} = exp−λD Lm
11
(λD Lm )d . d!
Next, we consider (3) for the case Lr > Lm . Here, Em (t − Lr , t − Lm ) depends on the serviceable inventory position Is (t − Lr ), on the number of products in on-hand remanufacturable inventory IrOH (t − Lr ), and on the demand D(t − Lm , t). Taking into account these correlations we write analogously to (5), Pr{Isnet = inet s } =
lim
r −1 X QX
t→∞ Ω2 iOH =0 r
(6)
π1 (is , iOH ) Pr{Em (t − Lr , t − Lm ) = em , D(t − Lr , t) = d| r Is (t − Lr ) = is , IrOH (t − Lr ) = iOH } r
where Ω2 = {(is , em , d)|is + em − d = inet s }. The calculations required to obtain the probability Pr{Em (t−Lr , t−Lm ) = em , D(t−Lr , t) = d|Is (t−Lr ) = is , IrOH (t−Lr ) = iOH r } are outlined in Appendix A. OH
- The long-run average on-hand remanufacturable inventory equals I r
=
Qr − 1 , 2
- The long-run average number of products that is remanufactured per unit of time equals E r = λR , - The long-run average number of remanufacturing set-ups per unit of time equals the longrun average number of products that is remanufactured (E r ), divided by the remanufacλR turing batch-size (Qr ), i.e., Or = Qr - The long-run average number of products that is manufactured per unit of time equals E m = λD − λR , - The long-run average number of manufacturing set-ups per unit of time equals the longrun average number of products manufactured (E m ), divided by the manufacturing that is λD − λR , batch-size (Qm ), i.e., Om = Qm - The long-run average number of backorders per unit of time follows from the relation P B = inet inet Pr{Isnet = inet s }. The right-hand-side of this relationship is calculated s 0 s from the analysis of (3) for the three parameter strategy, since other correlations are involved. First, we consider the case Lr ≤ Lm . In this case, Er (t − Lm , t − Lr ) is correlated with Is (t − Lm ) , with IrOH (t − Lm ), and with D(t − Lm , t). Taking into account these correlations Pr{Isnet = inet s } is calculated using the relation, Pr{Isnet = inet s } = lim
∞ X X
π2 (is , iOH ) Pr{Er (t − Lm , t − Lr ) = er , D(t − Lm , t) = d| r
t→∞
(7)
Ω1 iOH =0 r
Is (t − Lm ) = is , IrOH (t − Lm ) = iOH } r OH OH where π2 (is , iOH r ) = limt→∞ Pr{Is (t − Lm ) = is , Ir (t − Lm ) = ir } is obtained from a continuous time Markov-chain model M2 , with two dimensional state variable X2 (t) = {Is (t), IrOH (t)|t > 0} and a two-dimensional state-space S2 = {sm + 1, . . . , max(sm + Qm , Sr )} × {0, . . . , ∞}. The transition rate νs(1) ,s(2) related to a transition from state s(1) to state s(2) is defined as,
ν(is ,iOH ),(is ,iOH +1) = λR r r
is > sr or iOH < Sr − is − 1 r
ν(is ,iOH ),(Sr ,0) = λR r
= Sr − is − 1 sm + 1 ≤ is ≤ sr and iOH r
ν(is ,iOH ),(is −1,iOH ) = λD r r
is > sr + 1 or iOH < Sr − is + 1 r
OH ≥ S − i + 1 ν(is ,iOH r s ),(Sr ,iOH −(Sr −is +1)) = λD is = sr + 1 and ir r r
ν(is ,iOH ),(sm +Qm ,iOH ) = λD r r
< Sr − sm is = sm + 1 and iOH r
The calculations required to obtain the probability Pr{Er (t − Lm , t − Lr ) = er , D(t − Lm , t) = d|Is (t − Lm ) = is , IrOH (t − Lm ) = iOH r } of (7) are outlined in Appendix A. Next, we consider (3) for the case Lr > Lm . Here, the term Em (t−Lr , t−Lm ) is correlated with Is (t − Lr ), with IrOH (t − Lr ), and with D(t − Lr , t). Consequently, Pr{Isnet = inet s } can be calculated analogous to (6). - The long-run average remanufacturable inventory equals, OH Ir
max(sm +Qm ,Sr )
=
X
∞ X
is =sm +1
iOH =1 r
OH iOH r π2 (is , ir ).
- The long-run average number of batch set-ups in the remanufacturing process per unit of time can be obtained by using the PASTA property. The calculation of Or proceeds then 13
as follows: Or
=
λD Pr{demand induces remanufacturing set-up} + λR Pr{return induces remanufacturing set-up} =
=
λD
sr +1
X
∞ X
sr X
π2 (is , iOH ) + λR r
is =sm +1 iOH =Sr −is +1 r
π2 (is , Sr − is − 1)
is =sm +1
- The components E r , E m , Om , and B are calculated analogous to the three parameter strategy.
4.5
Optimization with respect to strategy control parameters
In the previous sections we outlined procedures to calculate the cost functions C(sm , Qm , Qr ) and C(sm , Qm , sr , Sr ) for arbitrary sets of control parameters. However, to analyze system behaviour, we are interested in the particular set of strategy parameters under which the cost ∗ functions are minimized. To find the minimum costs C P U SH = min(sm ,Qm ,Qr ) C(sm , Qm , Qr ) ∗ and C P U LL = min(sm ,Qm ,sr ,Sr ) C(sm , Qm , sr , Sr ) we have implemented an extensive enumerative search procedure. The results of this procedure indicate that for the problem instances considered so far (see Section 5) only a single local minimum exists. Although we did not succeed in constructing a formal proof for the existence of a single local minimum, the empirical observations suggest that for the optimization a simple linear search procedure would suffice.
4.6
Relaxation of system assumptions
To further study the influences of process interactions and process uncertainties on system performance, we investigate the system defined in Section 4.1 under relaxation of the following assumptions: • Uncertainties in returns and demands. To evaluate the influences of uncertainties in the timing of demands and the timing and quality of returns in more detail, the assumption of exponentially distributed demand and return inter-occurrence times has been generalized to Coxian-2 distributed demand and return inter-occurrence times. The Coxian-2 distribution enables to fit a first and second moment of processes, rather than a first moment only. In this way, the behaviour of demand and return processes that appear in practice can be approximated more accurately. The process uncertainty for the returns is specified 2 (cv 2 ). The required modifications to in terms of the squared coefficient of variation cvR D calculate C(.) under Coxian-2 distributed demand and return inter-occurrence times are outlined in Appendix B. • Correlation between returns and demands. The correlation between returns and demands is modelled by the coefficient ρRD , which indicates the fraction of product returns that instantaneously induce a demand for a new product to replace the returned product. The introduction of correlations between returns and demands changes the calculation of C(sm , Qm , Qr ) and C(sm , Qm , sr , Sr ). For further details we refer to Appendix C. Although both the PUSH and the PULL-strategy can be evaluated in presence of leadtime uncertainty, the mathematical analysis to calculate the cost function (1) tends to
14
become very complex. Therefore, the issue of lead-time uncertainty will be addressed in a follow-up paper (see Van der Laan et al., 1995).
5
Empirical study
To evaluate the effects of remanufacturing on system costs we have set-up an empirical study which is organized as follows: first, a base-case scenario has been developed regarding process variables and process characteristics. Subsequently, data from the base-case scenario are varied to analyse their influence on system costs. The base-case scenario is defined below. Base-case scenario • Regarding the characteristics of testing, remanufacturing and manufacturing processes the following assumptions are made: fixed costs (cf. )
variable costs (cv. )
lead-times (L. )
remanufacturing
0
0
2
manufacturing
0
0
2
Process
In the base-case scenario we have chosen for a cost structure without variable costs, since variable costs do not influence the relative difference between the costs of the PUSH and ∗ ∗ PULL-strategy (C P U SH − C P U LL ). Furthermore, fixed costs have been set to zero. The influence of fixed costs will be addressed in subsequent scenario’s. • Demand and return processes are characterized as follows:
inter-occurrence distribution
Returns
Demands
exponential
exponential
intensity
λR
=
0.7
λD
=
1
uncertainty
2 cvR
=
1
2 cvD
=
1
correlation
ρRD = 0
Note that the base-case scenario corresponds to a situation which usually occurs before the end of the product life-cycle, since the return intensity (λR ) is lower than the demand intensity (λD ). Finally, the timing of returns and demands are uncorrelated (ρRD = 0), • Inventory holding costs are chr = 0.5 for remanufacturables and chs = 1 for serviceables, • Backordering costs are cb = 50. In the empirical study we have investigated the following effects, • the effect of the decision to remanufacture. Figures 2a–b show a comparison between the production and inventory related costs for the situation without product returns (λR = 0) and the situation with product returns and remanufacturing. In the situation without product returns the manufacturing system is controlled by a regular (s, S)-policy with 15
∗
corresponding costs C(s, S). The minimum costs min(s,S) C(s, S) are represented by C . The other curves in the Figure 2a (2b) correspond to the situation with remanufacturing, ∗ and show the relation between the return intensity λR and the total system costs C P U SH ∗ (C P U LL ) at different variable remanufacturing costs cvr for the situation with cvm = 10. Figures 2a–b indicate that once a company has decided to remanufacture, the total production and inventory related costs in the system with remanufacturing may dominate C ∗ , even when the return intensity λR is smaller than the demand intensity λD and all remanufacturing related cost components are smaller than all corresponding manufacturing related cost components (cvr < cvm , chr < chs ). This effect is caused by two sources of uncertainty that are not present in traditional manufacturing systems, i.e., uncertainty in the timing of product returns and uncertainty in the quality of returned products. These uncertainties induce a high variability in the output of the remanufacturing process, and require an increase in safety stocks to protect the system against high stock-out costs. As a result, total system costs increase, and may even dominate the total expected costs in systems without product returns and remanufacturing. • the life-cycle effect. The different stages of the product life-cycle can be represented by the ratio λR /λD . Typically, at the introductory stage the return intensity will be low compared to the demand intensity. In later stages λR will increase and may even become larger than λD at the end of the product life-cycle. Figures 2a–b show that ∗ when λR increases, the costs C (.) may first decrease and then start to increase. The cost decrease is due to the opportunity to fulfil part of the demand from remanufacturing, which reduces total variable manufacturing costs. Furthermore, since remanufacturable inventory holding costs are lower than serviceable inventory holding costs in the base-case scenario, the total inventory holding costs first decrease when the return intensity increases. However, a further increase of λR results in an increase in inventory holding costs due to the aforementioned variability in the output of the remanufacturing process. The return ∗ intensity at which C (.) reaches it This indicates that remanufacturing companies should at all stages of the product life-cycle consider whether each returned product that satisfies the quality requirements for remanufacturing should actually be remanufactured. If alternative options exist to handle part of the remanufacturables, they may under certain conditions be worthwhile to consider from an economical point of view. • the effect of return uncertainties. Figures 3a–c show the cost effect of two sources of return uncertainty, i.e., uncertainty in the timing of product returns and uncertainty in the quality of returned products. The two sources of uncertainty are aggregated and 2 corresponding to a Coxian-2 distribution function. represented by a single parameter cvR 2 . Since the magnitude of this As expected, total system costs increase with increasing cvR effect may become large, in particular when the return intensity in high, remanufacturing 2 as low as possible. To reduce the variability in companies should attempt to keep cvR the timing of product returns, it may be considered to have lease contracts with a fixed lease-period instead of regular purchase contracts. To reduce the variability in the quality of returned products the development of a more robust product design could be studied and maintenance contracts could be stimulated. An advantage of maintenance contracts is, that during the contract period the quality of products can be observed and controlled, resulting in a better estimate of the quality of a product at the time it is returned. • the effect of demand and return correlations. Figures 4a–c show the influence of cor16
relations between returns and demands (ρRD ) on total system costs at different return intensities. It is demonstrated that when ρRD increases, the total system costs decrease. Furthermore, the magnitude of this effect increases with an increasing return intensity. The cost reduction occurs since correlation between returns and demands reduces total system uncertainty. In this way, safety stocks can be reduced. The observed effect stresses the importance of collecting data to estimate ρRD . Both, disregarding and overestimating ρRD may lead to unnecessary high costs for remanufacturing companies. • the effect of manufacturing lead-times. Figures 5a–c show the influence of manufacturing lead-times on system costs. It can be seen that total system costs increase with increasing manufacturing lead-times. This effect occurs since lead-time increases result in an increase in the uncertainty of demands and returns during the lead-time. Consequently, higher safety stocks must be kept to protect the system against high stock-out costs, resulting in the observed cost effect. • the effect of control policies. The computational experiments confirm our initial expectation that if inventory holding costs for remanufacturables are low compared to the inventory holding costs for serviceables, the PULL-strategy with simultaneous control over serviceable and remanufacturable inventory is more cost-efficient than the PUSH-strategy with control over serviceable inventory only (see Figures 5a–c). Moreover, this effect becomes more dominant when the uncertainty in system variables such as demands and returns increases (see Figures 3–4). The explanation for the occurrence of this effect is, that with the PUSH-strategy all safety stock is kept in serviceable inventory, whereas under the PULLstrategy most of the safety stock is located in (less costly) remanufacturable inventory. This effect is further visualised in Figure 6. On the other hand, if inventory holding costs of remanufacturables are as high as inventory holding costs for serviceables, the PUSHstrategy is more efficient than the PULL-strategy, since the total system inventory can be slightly lower with the PUSH-strategy (see also Figure 6). • the effects of other system components. In the above discussion not all model components have been considered in-depth. The table below summarizes the cost effects of these components.
6
∗
Model component
Effect on C (.)
2 Increase in demand uncertainty (cvD ) Increase in remanufacturing lead-times (Lr ) Increase in variable remanufacturing costs (cvr ) and variable manufacturing costs (cvm ) Increase in fixed remanufacturing set-up costs (cfr ) and fixed manufacturing set-up costs (cfm ) Increase in backordering costs (cb )
Increase Increase Increase
Effect on relative cost difference ∗ ∗ C P U SH − C P U LL See return uncertainty See manufacturing lead-times Not effected
Increase
Rather insensitive
Increase
Rather insensitive
Conclusions and directions for further research
The main objective of this paper was, to demonstrate some of the effects of remanufacturing on production planning and inventory control, and to indicate some of the actions that remanufacturing companies could take to improve their cost-efficiency. To do so, we have analysed a hybrid
17
system related to a single component durable product. Below we summarize the conclusions and recommendations that resulted from the analysis: - Remanufacturing companies should be aware of the fact that total production and inventory related costs may become higher than in traditional systems without remanufacturing. This effect is caused by various sources of uncertainty that do not occur in the traditional systems. Adequate protection against these uncertainties often requires an increase in total system inventories. - Remanufacturing companies should at all stages of the product life-cycle investigate whether all returned products that satisfy the (quality) requirements for remanufacturing should actually be remanufactured. If other options to handle returned products apply, they may under some conditions be more cost-effective. - Remanufacturing companies should investigate the possibility to have lease contracts with their customers instead of regular purchase contracts. Leasing reduces the uncertainty in the timing of product returns, and may result in lower total system costs. - Remanufacturing companies should derive reliable estimates on the correlation between the timing of product returns and the timing of product demands. Having reliable estimates reduces total system uncertainty and results in significantly lower total system costs. - Remanufacturing companies should attempt to control the quality of their products during the period of usage in the market, since a better quality control results in lower uncertainty regarding the ability to remanufacture a product at the time that it will be returned. The quality control could take place by having maintenance contracts with the customer. Furthermore, a more robust product design reduces the variability in the quality of returned products. - Remanufacturing companies should investigate which production, remanufacturing and inventory control policy to implement. The results of this study indicate that if a company values remanufacturable inventory as high as serviceable inventory, it may be cost-effective to implement a PUSH-strategy in which most of the stock is kept in serviceable inventory. On the other hand, if remanufacturable inventory is valued lower than serviceable inventory, a PULL-strategy in which part of the stock is kept in remanufacturable inventory may be more cost-effective. However, even in this case the benefits in costs must be traded-off against the disadvantage of needing more coordination between manufacturing and remanufacturing operations than under the PUSH-strategy. The latter makes the PUSH-strategy in some practical settings better suitable for implementation. Although the above conclusions and recommendations are based on the analysis of a relatively simple system, we found that they also apply to the more complex systems that we met in practice. However, in more complex systems a number of additional problems may occur that are not adequately modelled in the system considered in this paper. These problems include for instance the planning and control of multi-component products, the presence of capacity interactions between manufacturing and remanufacturing operations, and the choice between remanufacturing and alternative options to handle returned products (e.g. disposal). Together with technical issues, such as to identify the structure of optimal control policies and to improve the efficiency of search procedures for finding optimal control parameters, they constitute 18
a fruitful area for further research. Acknowledgements. The authors very much appreciate the helpful comments by their colleagues Peter Tielemans and Roelof Kuik from Erasmus University, and by Luk Van Wassenhove from INSEAD.
Appendix A Calculation of conditional probabilities appearing in (5), (6), and (7) In this appendix we show how the conditional probabilities, - Pr{Er (t − Lm , t − Lr ) = er |Is (t − Lm ) = is , IrOH (t − Lm ) = iOH r }, - Pr{Em (t − Lr , t − Lm ) = em , D(t − Lr , t) = d|Is (t − Lr ) = is , IrOH (t − Lr ) = iOH r , - Pr{Er (t − Lm , t − Lr ) = er , D(t − Lm , t) = d|Is (t − Lm ) = is , IrOH (t − Lm ) = iOH r }, that appear respectively in (5), (6), and (7) are calculated. To carry out the calculations we apply transient analysis to evaluate the system state of a Markov-Chain model at time t = τ , given the initial state of the system at time t = 0. Transient analysis is based on the technique of uniformization, which allows to transform a continuous-time Markov-Chain model into an equivalent discrete-time Markov-Chain model (see Tijms, 1986). To demonstrate the general idea behind the unformization technique, consider a continuous time Markov-Chain model {X(t)|t > 0} with discrete state space S, where X(t) = s indicates that the system is in state s ∈ S at time t. Furthermore, νs(0) ,s(1) is the rate by which transitions P occur from state s(0) to state s(1) , and νs(0) = s(1) ∈S νs(0) ,s(1) . To transform the continuous time Markov-Chain model {X(t)|t > 0} into an equivalent discrete-time Markov-Chain model (1) {Xn |n = 0, 1, 2, . . .}, we calculate the one-step discretized transition probabilities ps(0) ,s(1) as,
(1)
ps(0) ,s(1) =
ν (0) s ν ps(0) ,s(1) ,
s(0) 6= s(1) , (8)
1 − νs(0) p (0) (1) , s(0) = s(1) . s ,s ν
where ps(0) ,s(1) =
νs(0) ,s(1) νs(0)
, and the constant ν is chosen such that ν = maxs(0) ∈S {νs(0) }.
The conditional probability that the system will be in state s(1) at time t = τ , given that the system was in state s(0) at time t = 0 is denoted by ps(1) |s(0) (τ ). This probability is calculated as follows, ps(1) |s(0) (τ ) =
∞ X n=0
exp−ντ
(ντ )n (n) p n! s(0) ,s(1)
(9)
19
(n)
where the n-step discretized transition probability ps(0) ,s(1) is recursively calculated using the relation, (n)
ps(0) ,s(1) =
X
(n−1)
ps(0) ,s(2) ps(2) ,s(1) .
(10)
s(2) ∈S
The conditional probabilities ps(1) |s(0) (τ ) are then used to calculate the probabilities that appear in (5), (6), and (7). As an example we show how the conditional probability Pr{Er (t − Lm , t − Lr ) = er |Is (t − Lm ) = is , IrOH (tn− Lm ) = iOH r } that appears o in (5) is calculated. The underlying Markov-Chain model OH X(t) = Is (t), Ir (t), Er (0, t)|t > 0 has a three dimensional state space S = {sm +1, . . . , ∞}× {0, . . . , Qr − 1} × {0, . . . , ∞}, where X(t) = (is , iOH r , er ) if at time t the inventory position is is , the number of products in on-hand remanufacturable inventory is iOH r , and the number of products entering the remanufacturing process in the interval (0, t) is er . The transition rates for this model are as follows, ν(is ,iOH ,er ),(is +1,iOH +1,er ) r r
< Qr − 1 = λR , iOH r
ν(is ,iOH ,er ),(is +Qr ,0,er +Qr ) r
= λR , iOH = Qr − 1 r
ν(is ,iOH ,er ),(is −1,iOH ,er ) r r
= λD , is > sm + 1
ν(is ,iOH ,er ),(sm +Qm ,iOH ,er ) = λD , is = sm + 1. r r Using the uniformization technique enables to calculate, lim Pr Er (t − Lm , t − Lr ) = er |Is (t − Lm ) = is , IrOH (t − Lm ) = iOH r
=
t→∞
∞ ∞ X X
p(k,`,er )|(is ,iOH ,0) (Lm − Lr ) r
k=sm +1 `=0
where the conditional probabilities p(k,`,er )|(is ,iOH ,0) (Lm − Lr ) is obtained according to (9) with r ν = λR + λD . The probabilities Pr{Em (t−Lr , t−Lm ) = em , D(t−Lr , t) = d|Is (t−Lr ) = is , IrOH (t−Lr ) = iOH r and Pr{Er (t − Lm , t − Lr ) = er , D(t − Lm , t) = d|Is (t − Lm ) = is , IrOH (t − Lm ) = iOH r } are obtained analogously.
Appendix B Modeling Coxian-2 distributed demand and return inter-occurrence times In this appendix we sketch how C(sm , Qm , Qr ) and C(sm , Qm , sr , Sr ) can be evaluated under Coxian-2 distributed demand and/or return inter-occurrence times. Before doing so, we first introduce the Coxian-2 distribution function more formally. A random variable X is Coxian-2 distributed if,
20
(
X=
X1 with probability p, X1 + X2 with probability 1 − p.
where X1 and X2 are independent exponentially distributed random variables with parameters γ1 and γ2 respectively. Furthermore, 0 ≤ p ≤ 1, and γ1 , γ2 > 0. It should be noted that the Coxian-2 distribution reduces to an exponential distribution if p = 1 and to an Erlang-2 distribution if p = 0. Under a Gamma normalization, an arbitrary distribution function with first moment E(X) and 2 can be approximated by a Coxian-2 distribution with, squared coefficient of variation cvX
v u
!
u cv 2 − 1 4 2 γ2 X 2 , γ2 = γ1 = 1+t − γ1 , p = (1 − γ2 EX) + . 2 EX EX γ1 cvX + 1
(11)
and with a third moment equal to a Gamma distribution with first moment E(X) and squared 2 , provided that cv 2 ≥ 1 (see Tijms 1986, pages 399-400). coefficient of variation cvX X 2 The Coxian-2 arrival process can be formulated as a Markov-Chain model {Y (t)|t > 0}, with state space S = {1, 2}. These states can be interpreted as being the states in a closed queueing network with two serial service stations and a single customer. The customer requires service from the first station only with probability p, and from both stations with probability 1 − p. The state Y (t) = 1 (Y (t) = 2) corresponds to the situation that the customer is being served by station one (two) at time t. The process is cyclical in that after service completion the customer enters the first service station again. The transition rates in this process are as follows, ν1,1 = pγ1 , ν1,2 = (1 − p)γ1 , ν2,1 = γ2 . The analysis of C(sm , Qm , Qr ) and C(sm , Qm , sr , Sr ) under Coxian-2 distributed demand and/or return inter-occurrence times solely requires a modification of the underlying Markov-Chain models M1 and M2 respectively. To demonstrate this modification, we adapt M1 to ac2 ≥ count for Coxian-2 distributed return inter-occurrence times, with E(X) = λ1R and cvR 0 0 1 2 . The adapted Markov-Chain model M1 has a three-dimensional state variable X1 (t) = 0 {(Is (t), IrOH (t), Y (t))|t > 0}, with state space S1 = {sm + 1, . . . , ∞} × {0, . . . , Qr − 1} × {1, 2}. 0 Note that in M1 every return of a remanufacturable product corresponds to a service comple0 tion in the Markov-Chain model {Y (t)|t > 0}. Furthermore, the transition rates in M1 are as follows, ν(is ,iOH ,1),(is ,iOH +1,1) = pγ1 , r r ν(is ,iOH ,1),(is ,iOH ,2) r r
iOH < Qr − 1 r
= (1 − p)γ1 ,
ν(is ,iOH ,2),(is ,iOH +1,1) = γ2 , r r
iOH < Qr − 1 r
ν(is ,iOH ,1),(is +Qr ,0,1) r
iOH = Qr − 1 r
= pγ1 ,
21
iOH = Qr − 1 r
ν(is ,iOH ,2),(is +Qr ,0,1) r
= γ2 ,
ν(is ,iOH ,y),(is −1,iOH ,y) r r
= λD , is > sm + 1
ν(is ,iOH ,y),(sm +Qm ,iOH ,y) = λD , is = sm + 1 r r where γ1 , γ2 , and p are calculated according to (11). The modifications required to model Coxian-2 distributed demands proceed analogously for the PUSH and for the PULL strategy.
Appendix C Correlation between the timing of return and demand occurrences Correlations between the timing of return and demand occurrences are modelled by modifying the Markov-Chain model M1 for the PUSH-strategy and the Markov-Chain model M2 for the PULL-strategy. As an example we extend the PUSH-strategy for the situation with correlated returns and 00 00 demands. The modified Markov-Chain model M1 has a two-dimensional state-variable X1 (t) = 00 {Is (t), IrOH (t)} and a state space S1 . The transition rates of M1 are as follows, ν(is ,iOH ),(is ,iOH +1) r r
= (1 − ρRD )λR , iOH < Qr − 1, r
ν(is ,iOH ),(is +Qr ,0) r
= (1 − ρRD )λR , iOH = Qr − 1, r
ν(is ,iOH ),(is −1,iOH +1) r r
= ρRD λR ,
is > sm + 1,
< Qr − 1 iOH r
ν(is ,iOH ),(is +Qr −1,0) r
= ρRD λR ,
is > sm + 1,
= Qr − 1 iOH r
ν(is ,iOH ),(sm +Qm ,iOH +1) = ρRD λR , r r
is = sm + 1,
iOH < Qr − 1 r
ν(is ,iOH ),(sm +Qr ,0) r
= ρRD λR ,
is = sm + 1,
iOH = Qr − 1 r
ν(is ,iOH ),(is −1,iOH ) r r
= λD − ρRD λR , is > sm + 1
ν(is ,iOH ),(sm +Qm ,iOH ) r r
= λD − ρRD λR , is = sm + 1
Modification of M2 for correlations between returns and demands proceeds analogous.
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