An integrated guaranteed- and stochastic-service approach to inventory optimization in supply chains

European Journal of Operational Research 231 (2013) 109–119

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European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor

Production, Manufacturing and Logistics

An integrated guaranteed- and stochastic-service approach to inventory optimization in supply chains Steffen T. Klosterhalfen a,⇑, Daniel Dittmar b, Stefan Minner c a

Business School, University of Mannheim, 68131 Mannheim, Germany Schumpeter School of Business and Economics, University of Wuppertal, 42097 Wuppertal, Germany c TUM School of Management, Technische Universität München, 80333 Munich, Germany b

a r t i c l e

i n f o

Article history: Received 26 May 2012 Accepted 20 May 2013 Available online 30 May 2013 Keywords: Inventory Multi-echelon Guaranteed service Stochastic service Partitioning

a b s t r a c t Multi-echelon inventory optimization literature distinguishes stochastic- (SS) and guaranteed-service (GS) approaches as mutually exclusive frameworks. While the GS approach considers flexibility measures at the stages to deal with stockouts, the SS approach only relies on safety stock. Within a supply chain, flexibility levels might differ between stages rendering them appropriate candidates for one approach or the other. The existing approaches, however, require the selection of a single framework for the entire supply chain instead of a stage-wise choice. We develop an integrated hybrid-service (HS) approach which endogenously determines the overall cost-optimal approach for each stage and computes the required inventory levels. We present a dynamic programming optimization algorithm for serial supply chains that partitions the entire system into subchains of different types. From a numerical study we find that, besides implicitly choosing the better of the two pure frameworks, whose cost differences can be considerable, the HS approach enables additional pipeline and on-hand stock cost savings. We further identify drivers for the preferability of the HS approach. Ó 2013 Elsevier B.V. All rights reserved.

1. Introduction The practical relevance of multi-echelon inventory models has grown over the last decade as documented by the increasing number of successful deployments of software tools for multi-echelon inventory optimization (see Viswanathan (2007), Aberdeen Group). Examples include large multinational companies such as HP, Intel, or Philips. The underlying models represent extensions of the seminal works by Simpson (1958) and Clark and Scarf (1960), which form the basis of two competing research streams known as the ‘‘guaranteed-service (GS)’’ and the ‘‘stochastic-service (SS)’’ approaches (see Graves and Willems (2003)). Although both approaches solve the same practical inventory optimization problem in their core, they make a different assumption concerning the flexibility of the stages of a supply chain and thus the role of safety stock. The SS approach regards safety stock as the only means to deal with supply and demand uncertainty. As such, it treats the supply chain as being inflexible. It optimally allocates the stocks to the different stages by taking into account that occasional upstream material shortages cause delivery delays. Thus, the service of the supplying stages is stochastic. ⇑ Corresponding author. Tel.: +49 179 100 77 44; fax: +49 621 181 2042. E-mail addresses: [email protected] (S.T. Klosterhalfen), [email protected] (D. Dittmar), [email protected] (S. Minner). 0377-2217/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ejor.2013.05.032

In the GS approach, orders of any size can be met by the supplying stage(s) after a committed deterministic service time. This operation is based on the assumption that, if period demand exceeds a normal variability level, which would cause a material shortage, not only safety stock is available to cope with it, but also so-called operating flexibility (i.e. some sort of emergency measure like accelerated production), which provides the missing items at a negligible additional cost (see, e.g., van Houtum et al. (1996)). We point out that the GS approach, as we understand it in this paper, does not rely on the frequently used bounded-demand assumption. It deals with the original external customer demand by explicitly modeling how the flexibility measure works. Moreover, it allows for varying flexibility levels at the stages of the supply chain (see Dittmar and Klosterhalfen (2012)). This represents a modification and extension to most of the existing GS contributions, which assume a single flexibility level for the entire supply chain and do not address the operation of the flexibility measure explicitly (see, e.g., Graves and Willems (2000)). In any case, the incorporation of the flexibility element into a guaranteed-service model comes at a cost, since a stage is required to comply with the committed service time 100%. Both the complete negligence of a stage’s flexibility in the SS approach and the strict service guarantee in the GS approach, which results in restricted stock allocation options, leave cost-saving potentials untouched. The consequences of this neglect are less far-reaching in supply chains with extreme flexibility characteristics,

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i.e. (i) very little flexibility at all stages, which is very close to the underlying assumption of the SS approach or (ii) a lot of flexibility at all stages, which then outweighs the allocation disadvantage of the GS approach. However, for supply chains where the levels of flexibility differ between the stages both pure approaches may not be appropriate. In such settings, we are looking for a way to exploit the positive features of both pure approaches. One way to do this is presented in Lawson and Porteus (2000) (and later extensions, see Section 2 for details). They include an expediting option at each stage into the SS model. Speeding up missing items in the process is one possible interpretation of the implicitly assumed operating flexibility in the GS approach. Their model requires the manager of each stage to specify two basestock levels, one for regular ordering and one for expediting. That means, (s)he needs to keep track of two inventory positions in order to decide when and how much to expedite and how much to order regularly. This complicates the operation of the stages compared to the pure approaches where only one base-stock level is required at each stage and the following logic holds:  At a GS stage, the manager knows that all demand, no matter how large it is, has to be satisfied using flexibility measures, if necessary.  At an SS stage, the manager knows that (s)he can operate the stage in regular mode only. The usage of flexibility measures is never required. Therefore, we propose a different way of integration, which maintains the simple operation modes of the pure approaches and only requires a single base-stock level at each stage. We represent and integrate the flexibility element through GS stages and allow the entire supply chain to consist of stages (or subnetworks) of both types. The integrated hybrid-service (HS) approach optimally and endogenously determines which strategy (GS or SS) performs best at each individual stage, instead of choosing a single approach for the entire supply chain. Even though the HS model might not fully exploit all cost-saving potentials compared to the modeling idea of Lawson and Porteus (2000), the approach is easy to understand and implement in practice and still delivers additional cost savings over the pure approaches in many settings. In the remaining settings, the HS approach chooses the better of the two pure approaches, which it contains as special cases. The main contributions of this paper can be summarized as follows: 1. This work represents a first step towards the integration of the two existing multi-echelon inventory optimization frameworks, the GS and SS approaches, into a new hybrid-service (HS) framework, which extracts the best of both worlds. In order to keep the explanation of the underlying idea and the general modeling aspects with respect to the framework integration simple, we restrict the exposition to serial supply chains. 2. We develop a two-step optimization procedure for the partitioning of the supply chain into GS and SS subnetworks. First, we define so-called HS subnetworks, by which every potentially optimal partitioning pattern can be represented. For the optimization of the base-stock levels in these subnetworks, we extend the standard SS optimization procedure of Clark and Scarf (1960) to incorporate incoming and outgoing service times and costless expediting at the final stage. Second, we formulate a dynamic programming algorithm to obtain the overall optimal constellation of HS subnetworks together with the optimal service times. We review the literature in Section 2 and summarize both pure approaches in Section 3. Section 4 presents the combination of the

pure frameworks into a hybrid one and develops the optimization algorithm. Section 5 reports the results of a numerical comparison of the pure and hybrid approaches. Section 6 concludes the paper and addresses future research. All proofs are relegated to the Appendix.

2. Literature Following the nomenclature by Graves and Willems (2003) our model integrates the two exisiting multi-echelon inventory frameworks: the stochastic-service (SS) and guaranteed-service (GS) approaches. The SS approach based on the pioneering work by Clark and Scarf (1960) assumes that safety stock is the only available buffer against demand fluctuations. Stock insufficiencies cause stochastic delays in the system’s material flow. The GS framework, which dates back to Simpson (1958), assumes that further countermeasures, beyond safety stock, are available to satisfy demand and always ensure 100% delivery service. Starting with the analysis of serial systems, the models in both research streams have gradually been extended to more complex network structures including general acyclic ones and optimization procedures have been developed. In the SS domain, extensive overviews are provided in de Kok and Fransoo (2003) and van Houtum (2006). For details about the GS research stream and optimization procedures we refer to Magnanti et al. (2006) and Humair and Willems (2011) and references therein. Both research streams have been mainly treated in isolation over the years. Only few works exist that compare, contrast, or even try to integrate them. One such comparison is presented in Graves and Willems (2003). They apply both approaches to a convergent system and a spanning-tree network and find that (under their assumptions) the GS model outperforms the SS model. For two-echelon divergent systems, Klosterhalfen and Minner (2010) provide a GS–SS model comparison and show that the superiority of any of the two approaches heavily depends on the specific parameter setting and cannot be established in general. Moreover, they present a cost-based method to determine the maximum level of demand variability to be covered by safety stock. Whereas both the above-mentioned works merely compare the two approaches, in this paper we are interested in integrating the features of both frameworks. Since we model the GS operating flexibility feature in the form of expediting (similar to Klosterhalfen and Minner (2010)), our approach is related to this literature stream as well. In fact, given this flexibility interpretation, expediting models as such can be regarded as a combination of both frameworks. Lawson and Porteus (2000), Muharremoglu and Tsitsiklis (2003), Kim et al. (2007), and Berling and Martínez-de Albeníz (2012) focus on expediting problems in serial systems. Lawson and Porteus (2000) analyze a periodic-review system with deterministic lead-times. They show that a top-down base-stock policy with two levels is optimal. Muharremoglu and Tsitsiklis (2003) extend this work by allowing a more general cost structure for expediting, i.e. supermodular instead of additive. Kim et al. (2007) provide an extension to stochastic lead times of the regular orders. Berling and Martínez-de Albeníz (2012) study expediting in a continuous-time continuous-stage setting. Their model provides decision support on the speed at which a unit is to be moved through the chain. Our approach differs from the previous contributions mainly in the way we integrate the flexibility option into the SS world. We do this through GS stages. This enables us to still employ a simple inventory control policy with only a single base-stock level. In this respect, our model is similar to Huggins and Olsen (2003). They study a two-stage serial system with a delivery guarantee at the upstream stage, which is ensured through expediting at a certain

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linear plus fixed cost. They show that at both stages threshold policies are optimal. Whereas the actual expediting usage is the outcome of the cost minimization in their model and, as such, is not restricted in any respect a priori, we consider a bound on the average number of items to be expedited. This bound is derived from the available flexibility at a stage. Further, our n-stage model does not require each stage to provide guaranteed service, but leaves this decision to be optimized within the model.

3.1. Common assumptions

ext

l

2 ð0; 1Þ

ð1Þ

where boext 2 (0, l) denotes the maximum permitted average backorders at the end of a period towards the external customer as specified by the company. The modified fill rate is easier-to-evaluate than the fill rate (b-service level), which is quite common in practice and is defined as ext

bext ¼ 1 

bonew

l

2 ð0; 1Þ:

þDðT i Þ  Bi Þþ

ð3Þ

BOSS 0

+

where (x) = max{0, x} and  0, since we assume that the external supplier has ample stock. The expected on-hand stock at stage i is computed as

i ¼ 1; . . . ; n:

We consider a serial production/inventory system with n stages, which are numbered from i = 1, . . . , n starting with the most upstream stage. Let hi, ji denote the subnetwork from stage i to j including both stages. To differentiate between subnetworks of different type a superscript SS, GS, or HS is added when necessary. All stages operate with a periodic-review base-stock policy with a common review period equal to the model’s underlying base time unit. Each stage performs a processing function and represents a potential location for holding stock after the process has finished. This means that also semi-finished items can be held at the corresponding stages of the system. The processing time at stage i is Ti. It is assumed to be deterministic and a multiple of the review period. No capacity constraints exist at any of the stages or processes. Period customer demand occurs at stage n and is represented by a stationary random variable D with mean l and standard deviation r. Let D(Ti) denote the demand random variable over Ti periods. For each unit of stock on-hand or in the pipeline of stage i at the end of a period a linear local holding cost of hi is incurred. There are no fixed ordering/setup costs. For notational convenience we assume that an item at a downstream stage requires exactly one item of the upstream stage that is connected to it. The objective is to determine an optimal local base-stock level for each stage, Bi, such that the total stock-holding cost in the entire supply chain is minimized subject to a service-level constraint at the final stage. As service-level type we assume a modified fill rate (c-service level):

bo

are random variables indicating the on-hand stock and the backorders at the end of a period at stage i = 1, . . . , n, which both depend on the local base-stock levels of the upstream stages, B1 to Bi. þ ! þ þ BOSS i ð B h1;ii Þ ¼ ððððDðT 1 Þ  B1 Þ þ DðT 2 Þ  B2 Þ þ . . . þ DðT i1 Þ  Bi1 Þ

h i  h i h i ! ! SS ! E OHSS  T i  l þ E BOSS i ð B h1;ii Þ ¼ Bi  E BOi1 ð B h1;i1i Þ i ð B h1;ii Þ

3. Pure multi-echelon inventory optimization approaches

cext ¼ 1 

111

ð2Þ

Whereas boext refers to all backorders and thus may also include ext backorders from previous periods, bonew denotes only the maximum permitted average new backorders at the end of a period. Hence, cext provides a lower bound to bext (see, e.g., van Houtum et al. (1996) or Zhang and Zhang (2007)). 3.2. Stochastic-service approach In the SS approach it is assumed that the only buffer against supply and demand uncertainty is safety stock, i.e. the production system is inflexible and safety stock needs to account for all contingencies. If this stock quantity is not sufficient, delays in the material flow occur. Consequently, the service of a stage, i.e. the ability to readily provide the requested materials, is stochastic. ! Let B h1;ii denote the vector of local base-stock levels of stages 1 ! SS ! to i, which are the decision variables. OHSS i ð B h1;ii Þ and BOi ð B h1;ii Þ

ð4Þ

Due to potential shortages at the supplying stage (‘‘stochastic serh i vice’’) given by the expected backorders, E BOSS i1 , the inventory poh i sition at stage i can only be raised to Bi  E BOSS i1 . Subtracting the expected demand during the replenishment time, Ti  l, and adding the expected backorders of stage i itself results in the expected onhand stock. The optimization problem, PSS h1;ni , is given as

PSS h1;ni

n  h i  X ! ! min C SS hi  E OHSS h1;ni ð B h1;ni Þ ¼ i ð B h1;ii Þ þ T i  l

s:t: 1 

h i ! E BOSS n ð B h1;ni Þ

l

ð5Þ

i¼1

¼ cext :

ð6Þ

The objective function minimizes the costs of the expected on-hand and pipeline stock, Ti  l, in the entire system h1, ni, denoted as C SS h1;ni , subject to the fulfillment of the target service level, cext. The SS approach optimally solves the trade-off between possible shortages and costs for holding inventory at the different stages. Numerical procedures for the base-stock level optimization are discussed, e.g., in van Houtum and Zijm (1997). 3.3. Guaranteed-service approach 3.3.1. Assumptions The GS approach assumes that further countermeasures besides safety stock exist to cope with demand variability. Thus, safety stock is only used to cover demand variability up to a certain level (see, e.g., Graves (1988)). If demand exceeds this level, the company reverts to ‘‘operating flexibility’’ measures to make the requested items available in time. Due to this combination of safety stock and operating flexibility there are no stochastic delays in the material flow. The service time of a stage is the amount of time that elapses between the placement of an order by a downstream stage and the fulfillment of this order by the upstream stage. Let t denote the period index. Any order (irrespective of its size) placed by stage i (downstream stage) at the beginning of period t is delivered by stage i  1 (upstream stage) after the promised service time, STi1. Thus, this order is ready for processing at stage i at the beginning of period t + STi1, which takes another Ti periods. Consequently, the entire replenishment time of stage i is STi1 + Ti and the processed order is in stock at the beginning of period t + STi1 + Ti. By assumption, STi P 0, i = 1, . . . , n. In contrast to the SS approach, the GS approach requires an additional input parameter, namely the specification of the maximum demand level for safety stock coverage. There are basically two ways to operationalize this. The approach taken in most of the GS contributions is to bound the end-item demand and assume that only this bounded demand is propagated through the system and needs to be buffered through safety stock. The demand in excess of the bound is assumed to be handled outside the normal supply chain. It is argued that such a demand bound can be established by management. It expresses how often management is

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willing to resort to tactics other than safety stock to cover demand variability (see Graves and Willems (2000)). By restricting the demand inflow, this approach basically assumes a single flexibility level for the entire supply chain. Alternatively, varying safety stock coverage levels can be specified at the stages of the supply chain. That means, the unbounded original customer demand is propagated through the system and, depending on the individual flexibility of a stage, which is translated into the stage’s base-stock level, a certain portion of the demand is buffered through safety stock and the remaining portion is handled by the available flexibility measure (see Dittmar and Klosterhalfen (2012)). In practice, we encounter that stages actually do exhibit different levels of flexibility. We can exploit these potentials best by defining stage-specific safety stock coverage levels. Therefore, we pursue the latter approach. This ‘‘unbounded demand’’ approach requires the specification of two additional aspects: (i) the flexibility level of a stage and (ii) the operation of the flexibility measure. Concerning (i), we take up the idea of van Houtum et al. (1996) and assume that a certain amount of flexibility can be used without causing any additional costs (at least in the short run). It is this costless additional capacity that we exploit in the GS approach and according to which we set the safety stock coverage level. This level can be expressed as an internal service-level-like requirement. Similar to the maximum permitted average external backorders boext, which define cext above, we assume that management can specify at each upstream stage an upper limit for the average quantity, which can be made available in addition to the regular operation at a negligible cost weighted by the expediting timespan. This flex quantity, boi , results in a flexility level flex

cflex ¼1 i

boi

l

i ¼ 1; . . . ; n  1:

i ¼ 1; . . . ; n  1

ð8Þ

with D(x)  0 for x 6 0. Thus, the base-stock level must be at least as large as BUE determined as i

BUE i

n n o o ¼ min Bi jPr UEGS i ðST i1 ; ST i ; Bi Þ ¼ 0 P 1   Bi 2R

i ¼ 1; . . . ; n  1:

Assumption 1 (No expediting across upstream stages). The pipeline inventory of a stage is always sufficient to satisfy the expediting requirement at this stage. With regard to the speeding up of items, we make the following two additional assumptions: Assumption 2 (FIFO). Missing items are picked from the outstanding orders starting with the oldest one. Assumption 3 (Order splitting). Only the missing quantity is picked from the outstanding orders and the remaining part of the outstanding orders arrives according to the regular schedule. 3.3.2. Optimization model and procedure The expediting action reduces the pipeline inventory. The expected reduction at stage i corresponds to the expected potential backorders at the end of a period that would occur, if no expediting action was performed, because all these items require speeding up from pipeline inventory in order to comply with the guaranteedservice commitment. Due to Assumption 1, the latter quantity is computed as (for a given base-stock level and given incoming and outgoing service times)

h i þ E BOGS i ðST i1 ; ST i ; Bi Þ ¼E½ðDðST i1 þ T i  ST i Þ  Bi Þ  i ¼ 1; . . . ; n:

ð7Þ

A low level indicates a lot of slack capacity, whereas a high level reflects a less flexible process. At the final stage n we do not rely on any flexibility for the following reason. In contrast to the upstream stages, where we require guaranteed service, we assume that only a service of cext needs to be achieved towards the external customer as in the SS approach. Consequently, if we wanted to exploit flexibility at the final stage, we would have to specify two control parameters: a base-stock level for normal ordering and a second parameter that tells us when and how much to make available through the flexibility measure (see Dittmar and Klosterhalfen (2012) for modeling details). Since we want to keep the inventory control policy as simple as possible for the practitioner, we sacrifice the flexibility benefit at the final stage in favor of having only a single base-stock level. With respect to (ii), we follow the idea of Lawson and Porteus (2000) and model flexibility in the form of expediting. Whenever there is insufficient stock at a stage, the missing items are speeded up in the process. In theory, such an expediting action could affect the stock at upstream stages. In practice, expediting is often restricted to a stage’s own pipeline inventory due to various reasons. In order to reflect this restriction in our model, we limit the probability of upstream expediting by a predefined constraint  and choose it such that this event is basically negligible. The minimum base-stock level of a stage is set such that this constraint holds. The upstream expediting random variable at stage i is given as (see Dittmar and Klosterhalfen (2012)) þ UEGS i ðST i1 ; ST i ; Bi Þ ¼ ðDðST i1  ST i Þ  Bi Þ

Given an appropriate choice of , which we will elaborate on in the context of the numerical study in Section 5, we assume that the following is true:

ð9Þ

ð10Þ

Consequently, the expected pipeline inventory at stage i is derived as h i ( h i i ¼ 1; . . . ; n  1 T i  l  E BOGS i ðST i1 ; ST i ; Bi Þ ¼ E PIGS ðST ; ST ; B Þ : i1 i i i i¼n Ti  l ð11Þ At the final stage n we do not rely on expediting for the above-explained reasons. The expected on-hand stock at stage i = 1, . . . , n is computed as

h i E OHGS i ðST i1 ; ST i ; Bi Þ ¼ Bi  ðST i1 þ T i  ST i Þ  l h i þ E BOGS i ðST i1 ; ST i ; Bi Þ :

ð12Þ

The optimal base-stock level at a stage needs to ensure that both the available flexibility constraint and the upstream expediting probability constraint are met. Given BUE from (9), the average i quantity that can be expedited from pipeline inventory without violating the latter constraint is

h  i UE boi ¼ E BOGS ST i1 ; ST i ; BUE : i i

ð13Þ

Based on the available flexibility, the maximum average expediting flex quantity weighted by the expediting timespan is boi . At final stage n we do not use expediting, but we can allow for backorders of average size boext. Therefore, we derive the following target level constraint at stage i = 1, . . . , n

ctarget i

8 n o. < 1  min boflex ; boUE l i
ð14Þ

and postulate that the optimal base-stock level for given service times, BGS i ðST i1 ; ST i Þ, satisfies

S.T. Klosterhalfen et al. / European Journal of Operational Research 231 (2013) 109–119

1

h  i E BOGS ST i1 ; ST i ; BGS i i ðST i1 ; ST i Þ

l

4.1. HS subnetwork definition and properties

¼ ctarget : i

ð15Þ

Through (15) there is a one-to-one relationship between the incoming and outgoing service times and the base-stock level of a stage for a given flexibility and/or service-level target. Consequently, we can write (10)–(12) as being only dependent on STi1 and STi. Hence, instead of searching for the optimal base-stock level at each stage, we can also try to find the optimal outgoing service time. Assuming that the external supplier delivers immediately (ST0 = 0) and that the external customer requires immediate demand satisfaction (STn = 0), the optimization problem is PGS h1;ni

n  h i h i X ! GS min C GS hi  E OHGS h1;ni ð ST h1;ni Þ ¼ i ðST i1 ; ST i Þ þ E PI i ðST i1 ; ST i Þ i¼1

s:t: 0 6 ST i 6 ST i1 þ T i ST 0 ; ST n ¼ 0:

113

ð16Þ ð17Þ ð18Þ

i ¼ 1; . . . ; n  1

! ST h1;ni denotes the vector of outgoing service times in the system from stage 1 to n. In contrast to the GS works with bounded demand, the objective function minimizes the cost of the average on-hand stock and the pipeline inventory in the entire supply chain, because both depend on the service times. Since the flexibility usage is free of any additional charge by assumption, no other costs need to be included. The constraints ensure that the outgoing service time of a stage does not exceed its replenishment time and is non-negative. 4. Hybrid multi-echelon inventory optimization approach In the hybrid-service (HS) approach, we jointly exploit the features of both pure approaches. For each stage, the HS approach chooses the optimal framework (SS or GS) with regard to the entire supply chain cost. This leads to a partitioning of the supply chain into SS and GS subnetworks. Definition 1 (HS system). An HS system is a supply chain consisting of subnetworks of type SS or GS or both.

In order to obtain the optimal HS system and optimal basestock levels, we proceed as follows: We define so-called HS subnetworks and show that every HS system can be partitioned into such HS subnetworks (Section 4.1). For the base-stock level optimization in each HS subnetwork, we formulate the optimization model (in Section 4.2) and adjust the standard SS optimization procedure to incorporate incoming and outgoing service times as well as expediting at the final stage (Section 4.3). Given the optimization procedure for the HS subnetworks, the overall optimal combination of HS subnetworks that results in the least total cost for the entire HS system is determined by a dynamic programming algorithm developed in Section 4.4. The online supplement contains an illustrative numerical example of the approach.

From the explanations of the pure approaches in the previous sections and the fact that operating flexibility is the only superior feature of a GS stage over an SS stage we can derive: Lemma 1. A GS stage can only exploit its operating flexibility and be superior to an SS stage, if it holds stock, i.e. if STi < STi1 + Ti, i = 1, . . . , n  1. Therefore, stockless GS stages can be replaced by SS stages without loosing any operating flexibility advantage. On the contrary, by doing so the stock allocation feature of the SS approach may be exploited. This directly leads to the definition of so-called HS subnetworks. Definition 2 (HS subnetwork). An HS subnetwork consists of a GS stage, say j, and the directly preceding SS subnetwork hi, j  1iSS, i = 1, . . . , j  1, and is denoted as hi, jiHS. If no SS subnetwork is preceding, the HS subnetwork is degenerated and only consists of the GS stage j, which is denoted as hj, jiHS. The pure systems are special cases of HS subnetworks. A pure SS system consists of a single HS subnetwork spanning from stage 1 to n (since no expediting takes place at the final GS stage n, see Section 3.3). A pure GS system consists of a series of degenerated HS subnetworks. In Fig. 1 an HS subnetwork hi, jiHS is depicted. From upstream, the GS predecessor, i  1, quotes a service time, STi1. Moreover, the comprised GS stage, j, quotes a service times, STj, to its successor. We derive the following condition on STj, which facilitates the later optimization: Lemma 2. Consider HS subnetwork hi, jiHS. For the outgoing service time, STj, it holds that

( ST j

6 ST i1 þ T j

i ¼ j; hi; jiHS

< Tj

i < j; hi; jiHS

:

ð19Þ

A (non-degenerated) HS subnetwork can exploit the features of both pure approaches. The comprised final GS stage can use the flexibility feature to ensure guaranteed-service towards a succeeding HS subnetwork. Thus, it enables a decoupling. At all stages within the HS subnetwork, the feature of the SS approach can be exploited. Since the comprised GS stage, say j, has to cope with potential material shortages at the preceding SS stage (similar to the stages in the pure SS approach), we can treat j as an SS stage when deriving the optimal base-stock levels in the HS subnetwork. However, we cannot simply apply the standard SS optimization model and procedure, because we need to account for expediting from pipeline inventory. Therefore, we extend the standard SS approach to incorporate an expediting option at the final stage of an HS subnetwork.

Fig. 1. HS subnetwork.

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4.2. HS subnetwork optimization model In order to account for expediting in the formulation of the HS subnetwork optimization model, we need to compute, at the final stage of the HS subnetwork, expressions for (i) the upstream expediting random variable in order to ensure that Assumption 1 holds and (ii) the expected pipeline inventory reduction, which is due to the expediting action. For the case of a degenerated HS subnetwork, we can simply employ the expressions derived for the pure GS approach in Section 3.3. Otherwise, we can apply a similar logic, the only difference being that the preceding stages are of type SS, not GS. The required expressions are specified in the following propositions. Proposition 1 (Upstream expediting random variable). Consider HS ! subnetwork hi, jiHS. For a given base-stock level vector, B hi;ji , and adjacent service times, STi1 and STj, the steady-state distribution of the upstream expediting random variable at stage j is

! UEHS j ðST i1 ; ST j ; B hi;ji Þ

8 < ðDðST i1  ST j Þ  Bj Þþ þ ¼  ! : NBOHS j1 ðST i1 ; ST j ; B hi;j1i Þ  Bj

i¼j i
h i ! E OHHS k ðST i1 ; ST j ; B hi;ki Þ h i 8 ! > Bk  E BOHS ðST i1 ; ST j ; B hi;k1i Þ > k1 > > > h i > > ! HS HS > > < T k  l þ E BOk ðST i1 ; ST j ; B hi;ki Þ k ¼ i; . . . ; j  1; hi; ji ¼ : h i ! > HS > > > Bj  E BOj1 ðST i1 ; ST j ; B hi;j1i Þ  ðT j  ST j Þ  l > > > > > h HS ! i : þE BOj ðST i1 ; ST j ; B hi;ji Þ k ¼ j; hi; jiHS ð28Þ Given these expressions, the optimal base-stock levels for HS subnetwork hi, jiHS (for given incoming and outgoing service times) result from solving the optimization problem:

PHS hi;ji ¼

! min C HS hi;ji ðST i1 ; ST j ; B hi;ji Þ j  h i h i X ! ! HS hk  E OHHS k ðST i1 ; ST j ; B hi;ki Þ þ E PI k ðST i1 ; ST j ; B hi;ki Þ k¼i

ð29Þ

ð20Þ where

! NBOHS m ðST i1 ; ST j ; B hi;mi Þ  þ ! þ ¼ NBOHS m1 ðST i1 ; ST j ; B hi;m1i Þ þ DðT m  ðST j  Tðm þ 1ÞÞ Þ  Bm m ¼ i; . . . ; j  1;

ð21Þ

! þ NBOHS i1 ðST i1 ; ST j ; B hi;j1i Þ ¼ DðST i1  ðST j  TðiÞÞ Þ; TðmÞ ¼

j1 X Tk

ð22Þ

m ¼ i; . . . ; j;

ð23Þ

k¼m

with D(x)  0, if x 6 0, and

Pb

k¼a T k

¼ 0, if a > b, by definition. ! For Assumption 1 to hold, the optimal base-stock levels, B hi;ji , in HS subnetwork hi, jiHS must satisfy

n  o !  Pr UEHS ST i1 ; ST j ; B hi;ji ¼ 0 P 1  : j

ð24Þ

Proposition 2 (Pipeline inventory reduction). Consider HS subnet! work hi, jiHS. For a given base-stock level vector, B hi;ji , and adjacent service times, STi1 and STj, the expected pipeline inventory reduction at stage j is equal to the expected potential backorders given as:

h ! i E BOHS j ðST i1 ; ST j ; B hi;ji Þ 8 þ > i¼j < E½ðDðST i1 þ T j  ST j Þ  Bj Þ   þ  ¼ ! HS > i
ð25Þ where the relevant backorder random variable in the HS subnetwork hi, jiHS is defined as: (

! BOHS k ðST i1 ; ST j ; B hi;ki Þ ¼

! BOSS k ¼ i; . . . ; j  1 k ð B hi;ki Þ

DðST i1 Þ

k¼i1

:

ð26Þ

Using Proposition 2 we obtain the expected pipeline inventory and on-hand stock at the end of a period at stage k = i, . . . , j of HS subnetwork hi, jiHS as:

h i ! E PIHS k ðST i1 ; ST j ; B hi;ki Þ h ( ! i T j  l  E BOHS k ¼ j < n; hi; jiHS j ðST i1 ; ST j ; B hi;ji Þ ¼ Tk  l otherwise

ð27Þ

s:t: 1 

h ! i E BOHS j ðST i1 ; ST j ; B hi;ji Þ

l

¼ ctarget : j

ð30Þ

4.3. HS subnetwork optimization procedure As explained in Section 3.3.2, if j = n, we set ctarget ¼ cext in PHS n hi;ji and optimize the base-stock levels according to the procedure described below. If j < n, ctarget is to be set equal to the maximum of j the available flexibility at stage j given by cflex and the amount of j flexibility that can be used without violating the upstream expediting probability constraint. Unlike in a pure GS system, in an HS subnetwork we cannot derive the latter quantity directly for a given set of adjacent service times, because it depends on all base-stock levels in the subnetwork as well (cf. (24)). Therefore, we use an iterative optimization procedure. We start with ctarget ¼ cflex and j j optimize the base-stock levels in the subnetwork according to the procedure described below. Thus, we ensure that the available flexibility constraint is met. If the optimized parameters also comply with the upstream expediting probability constraint (24), they represent the overall optimal solution. Otherwise, we increase ctarget gradually until constraint (24) holds. j For the optimization of the base-stock levels in PHS hi;ji , we exploit the established relationship between a c-service and/or flexibility level model and a penalty-cost model (see, e.g., van Houtum (2006) for details). That means, we work with the penalty-cost model and tune the penalty-cost parameter, e.g., via a bisection procedure, such that the desired c-target is reached. The derived base-stock levels will then also be optimal for the service-/flexibility-level model. Following the line of argument used to derive the decomposition result in the standard SS approach (see, e.g., van Houtum (2006)), we show that this property also holds for the optimization of an HS subnetwork hi, jiHS, i.e. an SS model with incoming and outgoing service times and (i) no expediting at the final stage j, if j = n, or (ii) expediting at the final stage, if j < n. As in the standard SS approach, we work with echelon base-stock levels instead of the local ! ones: Sk, k = i, . . . , j. S hi;ji denotes the vector of echelon base-stock levels of stages i to j. The local base-stock levels are derived from the echelon ones as Bk = Sk  Sk+1, k = i, . . . , j  1 and Bj = Sj. The echelon holding cost of stage k in HS subnetwork hi, jiHS is e e hk ¼ hk  hk1 ; k ¼ i þ 1; . . . ; j, and hk ¼ hk ; k ¼ i. We assume e hk > 0; k ¼ i; . . . ; j. p denotes the linear penalty cost per unit. Then,

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the average cost in the entire HS subnetwork hi, jiHS for j < n can be expressed as

BOk1 ¼

! C HS hi;ji ðST i1 ; ST j ; S hi;ji Þ  h i  h i e HS ¼ hi  Si  E BOHS i1  Siþ1  E BOi  h i  h i  e e  þ hi þ hiþ1  Siþ1  E BOHS  Siþ2  E BOHS i iþ1

 þ ðkÞ ðkÞ ¼ BOm1 þ DðT m Þ  ðSm  Smþ1 Þ k 6 m < j; BOm 8 þ < ðDðST j1 þ T j  ST j Þ  Sj Þ k¼i¼j ðkÞ þ : BOj ¼  ðkÞ : BO þ DðT j  ST j Þ  Sj otherwise j1

þ ... þ

j1  h i  h i X e HS hm  Sj1  E BOHS j2  Sj  E BOj1 m¼i

þ

j j  h i h i h i X h i X e e HS þ p  E BOHS  hm  Sj  E BOHS hm  E BOHS j1 þ ST j  l þ E BOj j j m¼i

m¼i

ð31Þ



h i   h i  e e HS þ ST j  l þ . . . ¼ hi  Si  E BOHS i1 þ ST j  l þ hiþ1  Siþ1  E BOi  h i   h i  h i e e HS HS þ hj1  Sj1  E BOHS j2 þ ST j  l þ hj  Sj  E BOj1 þ ST j  l þ p  E BOj

ð32Þ h

HS with BOHS k ; k ¼ i  1; . . . ; j  1 as defined in (26), and E BOj

i

as

specified in (25). At each stage k = i, . . . , j  1, the sum of the expected pipeline inventory and on-hand stock is given by the difference between the respective echelon base-stock levels minus the expected shortfall, i.e. the expected backorders of the preceding stage. Similarly, at the final stage j (fifth line in (31)), where the h ech-i elon base-stock level equals the local base-stock level, Sj  E BOHS j1 denotes the pipeline inventory plus the safety stock, if the outgoing service time was zero, STj = 0. Since we want to assign holding costs only to positive realizations of this expression and penalty costs to the ones, we add the expected backorders at that stage, h negative i E BOHS , in the holding cost expression plus a separate penalty-cost j expression. If the outgoing service time is positive, the stock level at j is not depleted by STj  l. Hence, we need to add this term in the holding cost expression at j, too. The final term in (31), represents the expected cost reduction due to expediting from pipeline inventory as derived in (25). After rearranging terms, (32) follows. For j = n, we can proceed in the same way. Here, the average cost for HS subnetwork hi, jiHS corresponds to (31) without the last term, because no expediting takes place at the final supply chain stage n. For the base-stock level optimization within HS subnetwork hi, jiHS we define subsystem k, k = i, . . . , j as the system consisting of stages k, . . . , j (including the pipeline inventory leading to stage k) with infinite supply at stage k  1, if k = i + 1, . . . , j, or backorders of BOHS echelon base-stock level policy for i1 ¼ DðST i1 Þ, if k = i. An ! subsystem k is denoted by S hk;ji , with Sm 2 R for all m = k, . . . , j. ! The corresponding average costs are denoted by Gk ð S hk;ji Þ. Note HS that, whereas C hi;ji denotes the average costs in the entire HS subnetwork from stage i to j, Gk only denotes the average costs in the subsystem from stage k to j with k = i, . . . , j. Subsystem k = i is identical to the full HS subnetwork hi, jiHS and thus Gi ¼ C HS hi;ji . Further, define BOðkÞ m as the backorders at the end of a period at stage m, k 6 m 6 j in subsystem k. Given the above explanations, we find: Lemma 3 (Average subsystem cost). Consider HS subnetwork hi, jiHS. Let i 6 k 6 j. For subsystem k with given incoming and outgoing service times, STi1 and STj, the average cost of an echelon base-stock ! level policy S hk;ji , with Sm 2 R for all m = k, . . . , j, is

! Gk ð S hk;ji Þ ¼

8 j  h i  h i X > > e ðkÞ ðkÞ > > hm  Sm E BOm1 þST j  l þpE BOj > <

0

otherwise

m¼k

m¼i

;

ð34Þ ð35Þ ð36Þ

The optimal echelon base-stock levels can be obtained from the minimization of the average costs in subsystems according to: Proposition 3 (Newsvendor-type equations). Let the one-period demand distribution F be continuous on (0, 1) with F(0) = 0. For k = i, i + 1, . . . , j in HS subnetwork hi, jiHS with given incoming and outgoing service times, STi1 and STj, under given optimal echelon base-stock levels Skþ1 ; . . . ; Sj for stages k þ 1; . . . ; j; Sk is such that

8 . P < p  jm¼k hem p n o > j : p þ m¼i hm p þ jm¼i hm j¼n

ð37Þ

ðkÞ

where BOj is given by the recursive formulas (34)–(36) with Sm replaced by Sm for all m. (Note that the formula for case j = n is exactly the one derived in Theorem 5 in van Houtum (2006), because HS subnetwork hi, niHS corresponds to a pure SS subnetwork from stage i to n.) 4.4. Dynamic program for HS system optimization We find the optimal partitioning of a supply chain into HS subnetworks and the respective optimal base-stock levels by dynamic programming with backward recursion starting at stage n. State space. The state variable, zk, represents the replenishment time of stage k, i.e. any uncovered processing times from preceding stages (including stage k).

zk 2 Z k ¼ fz 2 NjT k 6 z 6 M k g k ¼ 1; . . . ; n

ð38Þ

Pk

where M k ¼ m¼1 T m denotes the maximum replenishment time for stage k. Decision space. Consider stage k. We define a two-dimensional   decision variable uk ¼ u1k ; u2k . u1k indicates the final stage of the next downstream HS subnetwork, which might also be stage k itself in case of a degenerated one. u2k represents the outgoing service time of stage u1k .

8 2 f0; 1; . . . ; zk g if u1k ¼ k < n > > < n o 2 uk 2 0; 1; . . . ; T u1  1 if k < u1k < n : k > > : ¼0 otherwise

ð39Þ

Due to Lemma 2, we exclude u2k ¼ T u1 in (39) for u1k < n. Since we k assume immediate customer service, stage n chooses an outgoing service time of 0. The entire decision space of stage k for a given state, Uk(zk), is defined by all feasible combinations of the two decision variable elements. State transition equation. The state transition equation describes how the state of a succeeding stage of k depends on the state and the decision made by k. k

j
ð33Þ

with

DðST i1 Þ k ¼ i

zu1 þ1 ðuk Þ ¼ u2k þ T u1 þ1

! j j  h i  h i > X X > e e ðkÞ ðkÞ > > þST þ pþ j¼n h  S E BO  l h > m j m m E BOj m1 : m¼k

ðkÞ

k

ð40Þ

Value function. We calculate the minimum total cost of the current stage and the downstream part of the entire supply chain depending on the state we are in and the decision we make at the current stage k. Note that the entire cost of the HS subnetwork is assigned to stage k. The direct cost assigned to stage k, DCk, can be calculated as

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u1k  h  i X 2 ! DC k ðzk ; uk Þ ¼ hm  E OHHS m zk  T k ; uk ; B hk;u1 i

GS

RCS

h  i 2 ! : þE PIHS m zk  T k ; uk ; B hk;u1 i

ð41Þ

k

Given the incoming service time, zk  Tk, and the outgoing one, u2k , the optimal base-stock levels in the HS subnetwork ! hk; u1k iHS ; B hk;u1 i , result as the solution to PHS hk;u1 i . k k h  i h  i ! ! HS HS 2 2 E OHk zk  T k ; uk ; B hk;u1 i and E PIk zk  T k ; uk ; B hk;u1 i follow k

k

from (28) and (27), respectively. The value function is

v n ðzn Þ ¼ DC n ðzn ;ðn;0ÞÞ 8zn 2 Z n n  o v k ðzk Þ ¼ u min DC k ðzk ;uk Þ þ v u þ1 zu þ1 ðuk Þ 2U ðz Þ k

k

k

8zk 2 Z k ;k ¼ 1;...;n  1:

¼

1

RCSHS ¼

1

k

m¼k

1 k

ð42Þ

1 k

ð43Þ

5. Numerical study 5.1. Parameter settings We focus on serial supply chains of length n = 5 and choose the parameters such that a large range of supply chain characteristics is captured. External customer demand per period follows a Gamma distribution with a mean of 100 and coefficients of variation 0.2, 0.6, and 1.0 to reflect different levels of variability. Since operating flexibility might vary within a single supply chain, internal flexibility-level targets correspond to either 90%, 95%, or 99% and may differ between the stages. The external service-level target at the final stage is set equal to 90%, 95%, or 99%. Inspired by the numerical design in Graves and Willems (2005), three different holding-cost and processing-time patterns are analyzed: progres e   e

 sive hi ¼ T i ¼ i , linear hi ¼ T i ¼ nþ1 , and degressive 2 e (hi ¼ T i ¼ n  i þ 1) for i = 1, . . . , n. With regard to the choice of , Dittmar and Klosterhalfen (2012) study the -impact in serial GS supply chains with 4–7 stages. They find that, even though the service-level deviation at the final stage is slightly increasing in the number of stages, for  = 0.1% it is still basically negligible. Since we have at most 5 GS-stages in the supply chain, we set  = 0.1% in our computations, too. 5.2. Results In our analysis we use two performance measures. First, the optimality share (OS) indicates the fraction of instances where the optimal solution is of a certain type indicated by superscript m 2 {GS, SS, HS}. That means, the solution consists of stages of type GS or SS only, or stages of both types, GS and SS, which we denote by HS. The superscript is no indicator for the approach used for obtaining the optimal solution. Our new HS approach can produce solutions of all three types: a real hybrid-service solution (HS) or each of the pure solutions (GS, SS) as special cases of hybrid-service solutions. We use OSm pure ; m 2 fGS; SSg to indicate the optimality share, if only the pure solution types are available (computed with the existing pure approaches). OSm all ; m 2 fGS; SS; HSg indicates the optimality share, if an additional HS solution type is available SS (computed with our new HS approach). OSGS all and OSall illustrate in what percentage of the instances the HS approach produces the same result as the pure GS and SS approaches. OSHS all shows the share of instances where the optimal solution is of a real hybrid-service type. To quantify the cost savings of one solution type over another, i.e. GS vs. SS and vice versa, and HS vs. the better of the two (GS, SS), we introduce a second measure, the relative cost superiority (RCS) defined for each instance as

TC GS

!þ

TC SS

SS

; RCS ¼ TC HS

1

!þ

minfTC GS ; TC SS g

TC SS TC GS

!þ ;

;

where TCGS and TCSS denote the total cost of the optimal solutions to SS HS PGS is the total cost of the optimal h1;ni and Ph1;ni , respectively. TC solution of the HS system found by the dynamic programming algorithm of Section 4.4. The average RCS in Table 1 refers only to the portion of instances where the respective solution type is superior, i.e. the bold numbers in the OS columns. With regard to the comparison of the pure solution types and approaches only, we find that the GS outperforms the SS in 34.8% of the cases. In these instances the relative cost superiority is 2.0% on average and 12.9% at maximum. The SS is superior to the GS in almost two thirds of the cases with a relative cost superiority of 3.0% on average and 8.0% at maximum (see Table 1). The HS approach improves the best pure approach in 48.4% of the settings. In the remaining instances the HS optimization algorithm finds a pure solution to be optimal. Within the improved instances, the additional savings amount to 3.6% at most and 0.4% on average. Whenever a former GS optimum does not hold stock at each stage, the HS approach can make use of the SS feature to allocate stocks and improve the GS solution. Since holding stock at all stages is only optimal in a few settings, the OS of the GS drops to 2.6%. In contrast, SS optima still make up 49.0% of the optima after the HS type introduction. We also observe that within the 48.4% HS optima, all possible partitioning patterns, i.e. combinations of GS and SS subnetworks, can be found. Next, we analyze the effect of the different parameters on the OS and RCS in detail. We focus on the development of the HS figures knowing that the remaining share refers almost exclusively to the SS type, because pure GS optima hardly exist after the HS type introduction (cf. Table 1). In all upcoming figures, the dark-gray shaded bars indicate the OS of the HS and refer to the left-hand y-axis. The triangles illustrate the maximum RCS of the HS and refer to the right-hand y-axis. Final-stage service level. As the final-stage service-level target gets higher, the RCS as well as the OS of the HS increases (see Fig. 2). In turn, the SS share decreases. An optimal SS solution has to ensure that the service level is achieved solely by safety stock taking into account potential supply shortages. This results in high costs, if the final-stage service level is high. In contrast, the HS can exploit the GS flexibility feature to decouple and carry more stock upstream. Coefficient of variation of period demand. The coefficient of variation (CV) turns out to be a major driver for the OS (see Fig. 2). For low CV-levels (0.2), the HS can exploit the GS flexibility feature and thus pipeline stock reduction without having to increase the base-stock levels by too much in order to ensure the non-upstream expediting probability constraint. As the variability increases (0.6), the exploitation of this feature becomes more costly. It is advisable to rely more frequently on the SS stock allocation. For high CV-levels (1.0), using the GS flexibility feature is basically not beneficial any more. Here, the pure SS is dominant,

Table 1 Result summary. m

GS SS HS

RCSm

OSm ... Pure

All (%)

Avg (%)

Max (%)

34.8% 65.2% n.a.

2.6 49.0 48.4

2.0 3.0 0.4

12.9 8.0 3.6

S.T. Klosterhalfen et al. / European Journal of Operational Research 231 (2013) 109–119

117

Fig. 2. Optimality share (OS) and relative cost superiority (RCS) with respect to . . ..

Fig. 3. Optimality share (OS) and relative cost superiority (RCS) with respect to the flexibility-level targets.

leaving the HS with an OS of less than 20%. In terms of the RCS, we find that the HS can gain most in settings where the exploitation of both pure approach features is advantageous, i.e. for medium CVlevels (0.6). Holding cost. The holding-cost structure has basically no impact on the OS. The HS share is very similar to OSHS all in Table 1 for all instances (see Fig. 2). Within the optimal HS instances, however, we find that the holding-cost pattern affects the RCS. From degressive over linear to progressive patterns, the RCS increases with maximum values of 0.8%, 1.7%, and 3.6%, respectively. When value-adding is large, the joint exploitation of the features of both pure approaches in the HS approach is most relevant. In these settings, it is preferable to hold more stock upstream. The HS can do this most effectively by exploiting the flexibility feature in the upstream part of the supply chain to decouple while benefiting from the SS stock allocation feature in the downstream part. Processing time. The processing-time pattern does not have a noticeable impact on the OS, either (see Fig. 2). Within the HS optima, though, the maximum RCS decreases from degressive over linear to progressive patterns with 3.6%, 3.1%, and 1.1%, respec-

tively. The HS can use the GS flexibility feature to decouple only in case of upstream coverage. Holding stock at upstream stages becomes more advantageous with larger processing times in the upstream part of the supply chain compared to the downstream part due to the risk-pooling effect over time. Flexibility-level targets. We cluster all instances into disjoint sets defined by minimum–maximum pairs of the target values. For instance, cluster 0.9  0.99 contains all instances where the smallest flexibility target is 90% and the largest one is 99%. In contrast to the previous analyses, the number of instances in these clusters differs. Consequently, the finding of Table 1 that the OS of the GS is negligible after the HS type introduction does not apply here. Therefore, we visualize all three approaches in Fig. 3. In addition to the OS bars of the HS (dark-gray), we introduce light-gray shaded and black shaded bars for the GS and SS shares, respectively. The white bars in the back indicate the original optimality shares of the pure approaches before the HS type usage. The flexibility levels are a major driver for the RCS and the OS of the different solution types. Stages with low flexibility targets (i.e. high expediting capability) turn out to be prominent candidates

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for decoupling the HS system. That means, they become a final GS stage of an HS subnetwork, cf. Fig. 1. On the contrary, stages with low expediting capability often prefer to exploit the SS stock allocation feature as a non-final stage of an HS subnetwork. We observe that the HS can exploit both individual features best in settings where the minimum flexibility target is low and the maximum one is high. Here, the HS achieves the largest RCS and OS. On the contrary, in settings with all flexibility targets set to 99% the OS and thus RCS of the HS approach is 0%. In these settings, the GS flexibility feature is negligible and the SS allocation feature is dominant. 6. Conclusions and outlook We have presented an approach to combine two major streams in multi-echelon inventory optimization, the stochastic-(SS) and guaranteed-service (GS) approach. The integrated hybrid-service (HS) approach derives a cost-optimal partitioning of the supply chain into SS and GS subnetworks and calculates the optimal base-stock levels. By the stage-wise choice of an appropriate approach instead of only a single decision for the entire supply chain, the HS approach performs at least as well as the better of the two pure approaches and often realizes additional gains. In our numerical study for serial systems with 5 stages we find that the superiority of the pure GS (SS) over the SS (GS) solution in terms of total cost, i.e. pipeline and on-hand stock, can be quite large amounting to 12.9% (8.0%) at maximum. The HS approach not only mitigates the risk of choosing the ‘‘wrong’’ pure approach, but even improves the pure solutions in nearly 50% of all instances. The additional total cost savings account for 3.6% at most and 0.4% on average. The HS approach is most likely to improve the pure solutions in settings with high final-stage service levels, medium customer demand variability, and diverse flexibility-level targets (low minimum, large maximum) at the supply chain stages. The incorporation of capacity constraints and the HS model extension to other network structures are promising areas for future research. The expected increase in the computational complexity calls for the development of efficient (approximate) solution approaches for larger networks. Since the current paper does not consider any setup costs at the supply chain stages, this aspect, which most likely requires the use of a different replenishment policy, is also worth exploring in future research. Appendix A. Proofs

Proof Lemma 1. First part: The proof is by contradiction. No stock at a GS stage i means that the outgoing service time is equal to the replenishment time, STi = STi1 + Ti. The GS stage waits with the delivery of the order request to its successor until it itself has replenished this specific order. Such a GS stage never needs to use its operating flexibility to fulfill the orders (provided that Assumption 1 holds). As soon as the replenishment time of the GS stage i is larger than its outgoing service time, STi < STi1 + Ti, a just-in-time replenishment and delivery without any stock at i is no longer possible. Hence, operating flexibility may need to be used in some periods. Second part: Suppose a two-stage setting. The generalization to more stages is straightforward. If the first GS stage i does not hold any stock, i.e. STi = Ti, the shortage that the next downstream stage needs to take into account is D(Ti). This is exactly the same shortage resulting from an SS stage with a local þ base-stock level of 0, because BOSS i ¼ ðDðT i Þ  Bi Þ ¼ DðT i Þ. Thus, the material flow can be mimicked by an SS stage. If the first GS stage i holds stock, STi < Ti, it can make use of its operating flexibility, which results in a system behavior that the SS approach cannot mimic. h

Proof (Lemma 2). In order to find the upper bounds on the outgoing service time, STj, we first derive the maximum feasible outgoing service time of GS stage j in HS subnetwork hi, jiHS. This P maximum is ST j ¼ ST i1 þ jk¼i T k and follows for Bk = 0, k = i, . . . , j  1 (SS stages), because the resulting shortage  Pj1  BOSS can be translated into an incoming serj1 ¼ D ST i1 þ k¼i T k P vice time at GS stage j of ST j1 ¼ ST i1 þ j1 k¼i T k . Together with its own processing time Tj, the maximum outgoing service time is obtained. Due to Lemma 1, GS stage j must hold stock for at least one time unit, however. Consequently, we obtain an upper bound P of ST j < ST i1 þ jk¼i T k . However, in order to be able to quote an outgoing service time of STj > Tj at GS stage j, it needs to hold that BOSS j1 ¼ DðST j  T j Þ, i.e. stage j  1 must propagate a shortage to stage j that corresponds exactly to the demand over STj  Tj periods. Due to this predetermined backorder expression, we cannot benefit from the SS stock allocation feature within the HS subnetwork any more, because the basis of this feature is the shifting of stocks between the stages in a cost-optimal way via appropriate backorder expressions. This degree of freedom is no longer available. Consequently, we do not forgo any benefits by obtaining the identical outgoing service-time result through a series of degenerated HS subnetworks. Since this approach facilitates the optimization computations later one, we pursue it. In order to do so, we need to allow the GS stage of a degenerated HS subnetwork hj, jiHS to not hold any stock, i.e. STj = STj1 + Tj. Then, all cases in HS P subnetwork hi, jiHS, for which T j < ST j < ST i1 þ jk¼i T k , are not covered by this specific partitioning pattern, but by another one with an appropriate degenerated HS subnetwork constellation. Hence, for a non-degenerated HS subnetwork, hi, jiHS with i < j, it suffices to postulate STj < Tj. For a degenerated HS subnetwork, hj, jiHS, we find STj 6 STj1 + Tj. h Proof Proposition 1. Suppose the non-degenerated HS subnetwork hi, jiHS depicted in Fig. 1 with incoming service time, STi1, and outgoing service time, STj. Upstream expediting at stage j is needed, if the quantity that requires expediting is larger than the pipeline inventory. Without expediting, the planned pipeline inventory between stages j  1 and j at the end of period t is:

b HS ¼ PI j;t

T j 1

X

HS dtk þ BOHS j1;tT j  BOj1;t

ðA:1Þ

k¼0

where dt denotes the realized demand in period t. Using the cumulative processing time of stages m to j  1, T(m) (cf. (23)), the final backorder expression is

0 BOHS m;tTðmþ1Þ

¼

@BOHS m1;tTðmÞ

þ

TðmÞ1 X

1þ dtk  Bm A

m ¼ i; . . . ; j  1

k¼Tðmþ1Þ

ðA:2Þ BOHS i1;tTðiÞ ¼

ST i1X þTðiÞ1

dtk :

ðA:3Þ

k¼TðiÞ

(The other backorder expression in (A.1) results by simply shifting the time index by Tm+1.) Provided that Assumption 1 holds, the quantity to be expedited and thus the pipeline inventory reduction amounts to the backorders at stage j at the end of period t: T j 1

BOHS j;t

¼

BOHS j1;tT j

þ

X k¼0

dtk  Bj 

!þ

ST j 1

X

dtk

:

ðA:4Þ

k¼0

The last term is due to the outgoing service time, by which the demand satisfaction of these periods can be delayed. Consequently, the upstream expediting random variable is given as

S.T. Klosterhalfen et al. / European Journal of Operational Research 231 (2013) 109–119

 þ HS b HS ¼ UEHS j;t ¼ BOj;t  PI j;t

!þ

ST j 1

BOHS j1;t 

X

dtk  Bj

ðA:5Þ

:

k¼0

This is the equivalent to the GS expression (8), which results by PST j1 1 replacing BOHS dtk and computing the steady-state j1;t with k¼0 distribution. For m = j  1 in (A.2) it is apparent that the relevant demand periods for BOHS j1;t partly or fully overlap with the ones in Eq. (A.5) depending on the relation of STj and Tj1. If STj > Tj1, STj also influences the backorder calculation at the upstream stages of j  1. In order to take these overlaps into account, we define NBOHS j1;t as the net backorders at the end of period t:

119

where the BOðkÞ m are given by (34)–(36). (In (A.11), the last sum vanishes for k = j.) Starting at the final stage j, gj(Sj) has  a zero point, which gives Sj . Proceeding upstream, g j1 Sj1 ; Sj ¼ 0 gives Sj1 , and so on. Note that when we are at stage k, the last sum in (A.11) becomes equal to 0 for already determined optimal echelon order-up-to levels. Consequently, (37) follows. h

Appendix B. Supplementary material Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.ejor.2013.05.032.

ST j 1 HS NBOHS j1;t ¼ BOj1;t 

X

dtk :

ðA:6Þ

References

k¼0

Then, we can rewrite (A.5) as

UEHS j;t

 þ ¼ NBOHS j1;t  Bj

ðA:7Þ

0

TðmÞ1 X

@NBOHS with NBOHS m;tTðmþ1Þ ¼ m1;tTðmÞ þ

1þ dtk  Bm A

k¼maxfTðmþ1Þ;ST j g

m ¼ i; . . . ; j  1 NBOHS i1;tTðiÞ ¼

ðA:8Þ

ST i1X þTðiÞ1

dtk :

ðA:9Þ

k¼maxfTðiÞ;ST j g

P By definition, bk¼a dtk ¼ 0, if a > b. If STj P STi1 + T(i), no upstream expediting takes place, since all items are in the pipeline of stage j, before they need to be delivered to stage j + 1. (21) and (22) directly follow as the steady-state distributions of (A.8) and (A.9). From (A.7) and (8), (20) is easily derived, which distinguishes between a degenerated and a non-degenerated HS subnetwork. h Proof Proposition 2. From (A.4) we directly obtain the expression for the expected pipeline inventory reduction. In (25) we need to distinguish between a degenerated and a non-degenerated HS subnetwork because of a potential demand period overlap due to the condition on the outgoing service time in Lemma 2. In the non! degenerated case, BOHS j1 ðST i1 ; ST j ; B hi;j1i Þ and D(Tj  STj) refer to demands of different periods, since STj < Tj. This is not necessarily true for the degenerated case, where STj may exceed Tj. h Proof Proposition 3. We use partial derivatives defined as

! ! @ g k ð S hk;ji Þ : ¼ fGk ð S hk;ji Þg i 6 k 6 j; Sm 2 R for all @Sk m ¼ k; . . . ; j:

ðA:10Þ

Then, we can derive 8 j j n o X n o X > ! > e ðkÞ ðkÞ > > hm pPr BOj > 0  Pr BOm1 ¼ 0 g m ð S hm;ji Þ j < ! m¼k m¼kþ1 g k ð S hk;ji Þ ¼ ! j j j n o X n o > X e X e > ! > ðkÞ ðkÞ > hm Pr BOj > 0  Pr BOm1 ¼ 0 g m ð S hm;ji Þ j ¼ n > : hm  pþ m¼k

m¼i

m¼kþ1

ðA:11Þ

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