Using pipeline information in a multi-echelon spare parts inventory system

Using pipeline information in a multi-echelon spare parts

inventory system

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Howard, C., Reijnen, I. C., Marklund, J., & Tan, T. (2010). Using pipeline information in a multi-echelon spare parts inventory system. (BETA publicatie : working papers; Vol. 330). Technische Universiteit Eindhoven.

Document status and date: Published: 01/01/2010

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Using pipeline information in a multi-echelon

spare parts inventory system

Christian Howard, Ingrid Reijnen, Johan Marklund, Tarkan Tan Beta Working Paper series 330

BETA publicatie WP 330 (working paper)

ISBN 978-90-386-2372-6 ISSN

NUR 804

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Using pipeline information in a multi-echelon

spare parts inventory system

Christian Howard ● Ingrid Reijnen ● Johan Marklund ● Tarkan Tan

Department of Industrial Management and Logistics, Lund University

Department of Industrial Engineering and Innovation Sciences, Eindhoven University of Technology christian.howard@iml.lth.se ● i.c.reijnen@tue.nl ● johan.marklund@iml.lth.se ● t.tan@tue.nl

Motivated by collaboration with a global spare parts service provider, we consider a two-echelon inventory system with multiple local warehouses, a so-called support warehouse, and a central warehouse with ample capacity. In case of stock-outs, the local warehouses can receive emergency shipments from the support warehouse or the central warehouse at an extra cost. The focus is on using information on orders in the replenishment pipeline, i.e. pipeline information, to achieve cost efficient policies for requesting emergency shipments. We introduce a policy where the request for an emergency shipment is based on the time until an outstanding order will reach the stock point considered. The goal is to determine how long one should wait for stock in the replenishment pipeline, before requesting an emergency shipment, and the cost effects of using pipeline information in this manner. In our analysis we utilize results from queuing theory and provide a decomposition technique that reduces a complex multi-echelon problem to more manageable single-echelon problems. Our results indicate that there is a significant benefit in using pipeline information. Based on data provided by the case company, we illustrate that the relative cost increase of ignoring pipeline information can be as high as 106%.

Keywords: Emergency shipments, Inventory, Multi-echelon, Pipeline information,

Spare parts

1. Introduction

The cost of downtime resulting from failed equipment and waiting for missing spare parts is a major concern for many companies. The research presented in this paper is motivated by collaboration with Volvo Parts Corporation, a global spare parts service provider with headquarters in Sweden. Volvo Parts is the supplier of aftermarket services for the Volvo Group, supporting the business areas: Volvo Trucks, Mack, Renault Trucks, Volvo Busses, Volvo Heavy Machinery and Volvo Penta. It follows that the core operational area for Volvo Parts is stock keeping and distribution of spare parts. These spare parts are distributed through central warehouses, positioned around the world, each one

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responsible for serving several local markets. On each local market they have a number of local warehouses (dealers/retailers) that, in turn, serve the end customer. This includes both service and repairs of the customers’ vehicles, as well as direct “over the counter” sales of the spare parts. The local warehouses replenish their stock by placing orders at the central warehouse.

A vehicle that is idle in the workshop due to a parts failure directly results in lost revenue for Volvo Parts’ customer, and this leads to a demand for high availability of spare parts at the local warehouse responsible for repairing the vehicle. To achieve this high availability, Volvo Parts combines regular stock replenishments from the central warehouse with emergency shipments. That is, a stock-out situation at a local warehouse is often resolved by sending an extra shipment from a source that does have the spare part in stock. Therefore, on most local markets, they have an additional stock point, referred to as a support warehouse. The support warehouse is also replenished by the central warehouse and its purpose is to provide these emergency shipments to the local warehouses in cases of stock-outs. The shipments are much faster than regular replenishments, but they come at higher cost. As a last resort, the central warehouse can also provide an emergency shipment to the local warehouse, should the support warehouse be out of stock as well. This type of system structure is by no means unique for Volvo Parts. It is, for instance, also utilized by some of their competitors.

At Volvo Parts, the general policy is to ask for an emergency shipment whenever a stock-out occurs. However, as recognized by the company, this is not necessarily the best strategy in terms of cost and service efficiency. For some spare parts it might be better to backorder the demand at the given stock point, in anticipation of the next incoming regular order, instead of requesting an emergency shipment. In particular, if there is a regular replenishment order from the central warehouse in close proximity to the local warehouse, the customer might actually receive the part sooner if she waits for this incoming shipment. At the same time Volvo Parts avoids the extra cost associated with an emergency shipment. This highlights the need for a replenishment policy that is more flexible regarding the use of emergency shipments, and that utilizes information on when outstanding orders in the replenishment pipeline from the central warehouse will be arriving. We refer to this as using pipeline information.

In this paper, we focus on a single local market and introduce a tolerance time for backordering customers. This tolerance time can be set individually for each product and stock point in the system and is designed to incorporate the possibility of waiting for incoming replenishment orders. When demand occurs at a specific local warehouse, and that warehouse is out of stock, the demand is backordered if there is a regular replenishment order arriving within the set tolerance time. If there is no regular order close enough in the pipeline, an emergency shipment is requested. The request first goes to the support warehouse which will meet the demand and send an emergency shipment if there is stock on hand or stock arriving within its own tolerance time. If this is not the case, the local warehouse requests an emergency shipment from the central warehouse instead. We assume that the

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central warehouse always can deliver and therefore, for the purpose of this work, it can be viewed as an external supplier. Hence, the multi-echelon nature of the model that we consider is derived from the support warehouse (upper echelon) supplying the local warehouses (lower echelons) with emergency shipments.

Our model assumes Poisson distributed customer demand and base-stock, or (S-1,S), policies at all stock points, which is reasonable for the slow moving items in Volvo Part’s product assortment. Moreover, we consider a customer waiting cost (e.g. based on loss of goodwill and penalty costs stipulated in service contracts etc.) per unit and time unit. Given this waiting cost, along with holding costs per unit and time unit and emergency shipment costs per unit, we provide a method for determining base-stock levels and tolerance times such that the expected system costs are minimized. One of the main advantages with our model is that when all tolerance times are set to zero this corresponds to the current situation at Volvo Parts, where emergency shipments are used whenever a stock-out occurs. Conversely, if all tolerance times are equal to the replenishment lead time, all demand will wait for regular replenishment and emergency shipments are never used. Our model can therefore provide structural results on suitable system configurations for different products (i.e. for which products should emergency shipments be used?), which is a key interest point for Volvo Parts. Numerical results based on data provided by Volvo Parts show that the penalty, i.e. the relative cost increase, of ignoring pipeline information and always requesting an emergency shipment can be as high as 106%.

The remainder of the paper is organized as follows: Section 2 provides a review of related literature. Section 3 presents the considered model in detail and discusses the assumptions made. Section 4 analyzes a single local warehouse in isolation and provides an exact method for cost evaluation and optimization of this single-echelon system. Based on these results an accurate heuristic for setting base-stock levels and tolerance times for all local warehouses and the support warehouse is presented. Section 5 evaluates the performance of the heuristic and provides managerial insights into the value of using pipeline information, both in the single - and multi-echelon case. Section 6 concludes.

2. Literature review

There are obvious connections between our work and the literature on lateral transshipment, dual supply, partial backordering, and multi-echelon systems. The lateral transshipment literature focuses on models where locations in the same echelon can share inventory by transferring items between the locations. Dual supply models analyze situations where a stock point can have more than one supplier to choose from, e.g., a regular supplier and an emergency supplier. The partial backordering literature is concerned with inventory policies that allow backordering under certain conditions, e.g., by setting

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a maximum number for the amount of customers that can be backordered. Multi-echelon models focus on systems with multiple levels, where an upper level supplies the lower levels.

For a recent overview of the lateral transshipment literature we refer to Paterson et al. (2009) and Wong et al. (2006). We particularly mention the papers by Kranenburg and van Houtum (2009) and Reijnen et al. (2009) because, similar to a support warehouse, they have designated suppliers of transshipments. Kranenburg and van Houtum consider a two level structure where lateral transshipments can only be supplied by the upper level. Reijnen et al. consider a structure where local warehouses can only receive a lateral transshipment from another local warehouse if the local warehouse can be reached within a predefined time limit. However, even though the authors of both papers recognize that the lateral transshipment time is non-negligible, they do not consider waiting for a replenishment order in the pipeline as an alternative to sending an emergency shipment.

A lateral transshipment model that incorporates the option of waiting for incoming orders is Yang and Decker (2010). They consider a structure similar to Reijnen et al. where it is assumed that customers are willing to wait at a local warehouse for a given amount of time. Here, a local warehouse will wait for incoming orders, instead of requesting a lateral transshipment, if the order will arrive within this given time limit. This is similar to our assumptions but there are important differences. Firstly, they regard the customer time limit as a given parameter and assume that the customer is satisfied if she receives the item within this time (which makes it similar to a service constraint). Although our tolerance times could be used in a similar way, we assume a customer waiting cost per time unit at each local warehouse and regard the time one should wait as a decision variable in our policy. Secondly, they place the restriction that all local warehouses have the same time limit, whereas we allow for different tolerance times at different locations. Lastly, in their work they assume that demand is backordered if no lateral transshipment can reach the local warehouse in time, while we consider an emergency shipment from the central warehouse as a last option. These differences mean that we use fairly unrelated methods of analysis.

Axsäter (2003a) suggests a heuristic decision rule for lateral transshipments that incorporates the remaining delivery times for outstanding orders. Although we also utilize this information, our use of tolerance times is quite different from Axsäter’s transshipment rule. Furthermore, we place an emphasis on determining replenishment policy parameters, whereas Axsäter uses simulation for cost evaluation and optimization of (R,Q) policies under the given transshipment rule.

Our work is also related to the literature on unidirectional lateral transshipment models (see e.g. Axsäter, 2003b and Olsson, 2009) because the emergency shipments exclusively occur in one direction (from the support warehouse to the local warehouses). The distinguishing feature with our work is that we consider the inventory in the pipeline before requesting an emergency (lateral) transshipment. Even though most papers recognize that there is some lateral transshipment lead time, they do not consider waiting for a replenishment order as an alternative to an emergency (lateral) transshipment.

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Another paper with an obvious relation to our current work is Axsäter et al. (2010), which also analyzes the described distribution system used by Volvo Parts. However, their work is focused on minimizing costs under fill rate service constraints and, therefore, they do not consider the customer waiting times explicitly. Moreover, they do not consider pipeline information, and their model assumes that the local warehouses always order from the support warehouse when a stock-out occurs.

An emergency shipment from the support warehouse can also be viewed as a delivery from a second supplier, implying a connection between our present work and that on dual supply models. In these models a single stock point has the option of using a second supplier that can deliver emergency shipments at an extra cost. The main difference, compared to our work, is that the emergency supplier is exogenous, while we include the support warehouse in our model. In the analysis of a single local warehouse in Section 4 we explore the differences and similarities to a dual supplier model analyzed by Moinzadeh and Schmidt (1991), and Song and Zipkin (2009). For a more extensive overview on dual supply models we refer to Minner (2003).

The third stream of literature that shares similarities with our work is that of partial backordering. In our model, a customer that waits for a regular replenishment to arrive is regarded as a backordered customer. Basing the decision to backorder on a tolerance time can therefore be viewed as a type of partial backordering. For partial backordering models concerning (S-1,S) policies and Poisson demand we refer to Das (1977) and Moinzadeh (1989) and references therein. What sets this literature apart from our current work is the focus on single-echelon models and the fact that partial backordering is the result of a given customer behavior, i.e. that customer are willing to wait for a certain amount of time before leaving the system. Therefore, the modeling techniques are different and these papers do not investigate the value of being able to choose when to backorder a demand. Models that do investigate the value of partial backordering are given in Chu et al. (2001) and Rabinowitz et al. (1995). They study a single-echelon system under Poisson demand where customers are backordered when a replenishment order is close enough in the pipeline. Although they illustrate that there is a large potential in allowing for partial backordering, their analysis assumes (R,Q) replenishment policies under the assumption that there can be at most one order outstanding. In many cases, when allowing for backorders, this assumption is likely to be violated and, hence, the analysis is approximate in these cases.

The focus on a two-level system with a single stock point in the upper echelon supplying multiple downstream facilities implies that there is a relation between our current work and the literature on continuous review distribution systems. Although there are some similarities, the main difference is that the support warehouse in our model handles emergency shipments, whereas the upper echelon in previous literature typically handles regular replenishment orders. As a result, the problem formulations and solution techniques differ. For a general overview of the literature on distribution systems we refer to Axsäter (2003c). A more recent overview of the literature on

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continuous review one warehouse multiple retailer systems is available in Axsäter and Marklund (2008).

3. Problem formulation

We consider a single item inventory model consisting of j local warehouses (index j {1,…,J}), a support warehouse (index j = 0), and a central warehouse with infinite capacity (Figure 1). All stock points apply continuous review base-stock, or (S-1,S), policies and the customer demand at local warehouse j follows a Poisson process with demand rate λj. These assumptions are reasonable for the

slow moving spare parts in Volvo Parts assortment. Demand is satisfied according to a First Come-First Serve (FCFS) rule.

Figure 1

For fulfillment of a demand we first consider the stock on hand and the orders in the replenishment pipeline (i.e. outstanding orders on route from the central warehouse) at local warehouse j, where the demand occurred. In case local warehouse j has available stock on hand, the customer leaves directly with an item. If the warehouse is out of stock and an unreserved replenishment order will arrive within Tj time units, then the demand is backordered and waits until the order arrives. By unreserved

we mean that there is no other customer demand backordered and waiting for the considered item. We refer to the decision variable Tj as the tolerance time at warehouse j. In case a customer demand is

satisfied from stock on hand or backordered at local warehouse j a new item is ordered from the central warehouse with constant lead time Lj, at the moment the demand occurs. If there is no stock on

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be satisfied by an emergency shipment. A demand waiting for an emergency shipment at a local warehouse is not viewed as a backorder at that stock point. This is because the responsibility for fulfillment of that demand is now shifted to the support warehouse and the central warehouse.

When requesting an emergency shipment, the local warehouse first contacts the support warehouse which applies the same policy: (i) satisfy the demand from stock on hand by sending a shipment to the local warehouse straight away, (ii) backorder the demand based on the pipeline inventory of reach within tolerance time T0 and send a shipment when the item arrives in stock, (iii)

deny the request for an emergency shipment. In case (iii), the local warehouse requests an emergency shipment from the central warehouse, which can always deliver. We assume that all emergency shipment lead times (including picking, packing, shipping and receiving) are constant, but they may vary between local warehouses. In the case of emergency shipments from the support warehouse, the time for waiting on items in the replenishment pipeline is not included in this lead time. Let s_j and

c

j be the lead times for emergency shipments to local warehouse j, from the support warehouse and

central warehouse, respectively. It follows that the maximum waiting time for a customer arriving at local warehouse j is given by max(Tj, T0 + sj,

c

j). Note that this holds for the case when Tj < Lj and

T0 < L0. In the special case of Tj = Lj we have complete backordering at local warehouse j and the

maximum waiting time is therefore Tj. Similarly, if T0 = L0 and Tj < Lj the maximum waiting time is

given by max(Tj, T0 + sj), because emergency shipments from the central warehouse will never be

used.

The relationship with the maximum waiting time means that the tolerance times can be set to fulfill a given customer service requirement. It is, for instance, common to have service contracts where a spare part is guaranteed to be delivered within a certain time limit. Another approach is to base the value of the tolerance times on the emergency shipment lead times. For example, with Tj = min( s_j, c_j) j {1,…,J} a local warehouse only backorders demand if this guarantees faster

delivery than an emergency shipment, and with T0 = c j− s j (assuming s j c j ) the support

warehouse only backorders demand if this saves time compared to an emergency shipment from the central warehouse. This approach is reasonable if the cost of waiting for a part is so high that choosing the quickest option is the only reasonable approach. However, this is not the case for all spare parts at Volvo Parts. In this work the focus is more general, where the system can choose tolerance times based on what is most cost efficient. This approach implies that we investigate the full potential in using tolerance times. Moreover, it means that we can provide Volvo Parts with a tool for finding the most efficient way of using emergency shipments.

We assume a customer waiting cost, bj, per unit and time unit at local warehouse j. This cost

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Furthermore, there is a fixed per unit cost, cj, associated with every emergency shipment from the

support warehouse to local warehouse j. Analogously, there is a fixed per unit cost, pj, for every

emergency shipment from the central warehouse, pj cj. Note that, because we consider constant

emergency shipment lead times, the customer waiting costs incurred from waiting on items in transport from the support warehouse (bj s_j), or the central warehouse (bj c_j), will be the same for all

customers at local warehouse j. These costs are therefore included directly in cj and pj, respectively.

However, waiting due to backordering, either at the support warehouse or a local warehouse, needs to be handled separately. We also consider inventory holding costs hj (j {0,…,J}) per unit and time unit

for stock on hand.

Let S = (S0, S1,…,SJ) be the vector of the support warehouse and local warehouse base-stock

levels, and let T = (T0, T1,…,TJ) be the vector of the tolerance times. We refer to the policy applied at

each stock point j as an (Sj,Tj) policy. For demand occurring at local warehouse j (j {0,…,J}) we

define:

αj = fraction of demand satisfied from stock on hand at local warehouse j

βj = fraction of demand backordered at local warehouse j

γj = fraction of demand satisfied from stock on hand at the support warehouse

δj = fraction of demand backordered at the support warehouse

θj = fraction of demand satisfied by the central warehouse

ψj = γj + δj + θj, i.e., fraction of demand satisfied through emergency shipments

EWj = expected waiting time for an item backordered at local warehouse j

EVj = expected waiting time at the support warehouse for an item requested at local

warehouse j and backordered at the support warehouse EILj

+

= expected inventory on hand at stock point j.

Note that all customer demand must eventually be satisfied, i.e. αj + βj + γj + δj + θj = 1. The objective

is to find the S and T (0 Tj Ljfor all j) that minimize the expected total system cost per time unit: J 1 j J 1 j j j j j j j j j J 1 j j j j J 1 j j j j j J 0 j j jEIL b EW c (c bEV) p h ) , ( C S T . (1)

In (1), the first term is the expected holding costs, the second term is the expected costs for backorders at the local warehouses, the third term is the expected cost for emergency shipments sent immediately from the support warehouse, the fourth term is the expected cost for emergency shipments sent, after a delay, from the support warehouse, and, finally, the last term is the expected cost for emergency shipments sent from the central warehouse.

The settings described above correspond well to the current setup of Volvo Parts’ distribution system for low demand spare parts. There is, however, an alternative interpretation of the options of demand fulfillment. From a modeling perspective, we can view the request for an emergency

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shipment from the support warehouse as demand being lost at the local warehouse, and instantly transferred to the support warehouse. The parameter cj can then be interpreted as the cost of

transferring the demand. In the same way, an emergency shipment from the central warehouse can be viewed as demand being lost for the support warehouse at an additional cost pj − cj. We can therefore

alternatively rearrange (1) as J 1 j J 1 j j j j j j j j j J 1 j j j j J 1 j j j j j J 0 j j jEIL b EW c b EV (p c ) h ) , ( CS T , (2)

where the third and fifth terms now are the costs for losing/transferring demand and the fourth term is the costs for backorders at the support warehouse.

4. Analysis

This section presents the analysis of the considered model. First, we focus on a local warehouse j in isolation, and show how the costs for a given (Sj,Tj) policy can be evaluated exactly. We then present

a method for optimizing these decision variables. Next, utilizing these results, we provide heuristic for cost evaluation and for finding near-optimal values of the decision variables in the multi-echelon system.

4.1 Single-echelon model

We focus on a single local warehouse with the aim to determine the expected costs per time unit for a given (S,T) policy. For notational convenience we suppress the index j in this section. Demand is satisfied either directly from stock on hand, after being backordered for at most T time units, or from an emergency shipment. In this context we make no distinction as to where the emergency shipment comes from, only that the entire cost for the emergency shipment can be quantified by the parameter p. The most straightforward interpretation of this is that the support warehouse is removed from the multi-echelon system, thus leaving the central warehouse as the only option for emergency shipments. We can therefore focus solely on the fraction of demand satisfied by emergency shipments, ψ, compared to γj, δj and θj in the multi-echelon model. Note that we can choose to view ψ as the fraction

of demand that is lost to the system, at a fixed per unit cost p.

A schematic representation of the local warehouse, for a given (S,T) policy, is provided in Figure 2. In Figure 2 we see that the replenishment pipeline can be separated into two parts. The first part is of length (L−T), and an order in this part of the pipeline cannot be reserved by an arriving customer. The second part is of length T, and an (unreserved) order in this part of the pipeline can be reserved for an incoming customer demand. We define the inventory position of the whole system, IP1, as the sum of the inventory level (stock on hand minus backorders), the number of items on order

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the pipeline (denoted by N2). Note that at any point in time IP1 = S holds. Similarly, we define IP2 as

the inventory level plus the number of items in the second part of the pipeline.

IL L-T T L N1 N2 IP2 IP1 Figure 2

This implies that IP2 = IP1−N1 and that 0 IP2 IP1. Furthermore, it means that a demand can be

satisfied at the local warehouse if IP2 > 0 and that demand is lost for the local warehouse (satisfied by

an emergency shipment) if IP2 = 0, or equivalently, N1 = S.

Following the notation in Section 2, our objective is to find the S and T that minimize the expected costs per time unit

p EW b hEIL ) T , S ( C .

In subsequent sections we refer to this single-echelon model as the Time Based Backordering (TBB) model.

4.1.1 Cost evaluation for a given (S,T) policy

We begin with the trivial case when S = 0. Because there will never be any unreserved items in the replenishment pipeline and all demand will be lost unless T = L we have

L T ; L b L T 0 ; p ) T , 0 ( C .

The result for S = 0 and T = L (i.e. complete backordering) follows from Little’s law. For the case with S 1, the TBB model can be represented as a queuing network, depicted in Figure 3.

Poisson(λ) N1<S N1 No Yes • / D / ∞ N2 • / D / ∞ Figure 3

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To the left in Figure 3 we see that customers arrive to the system according to a Poisson process, and to the right there are two queuing stations. The first station represents the first part of the replenishment pipeline, with deterministic service time L – T (and an infinite number of servers). The second station (also with an infinite number of servers) represents the next part of the pipeline with deterministic service time T. An arriving customer that encounters S customers in the first server (excluding herself), i.e. N1 = S, is lost to the system. Conversely, a customer that arrives when N1 < S

is either satisfied from stock on hand or backordered, and will therefore generate a replenishment order that goes directly into the pipeline (or equivalently into the first queuing station). Note that there are backorders in the system when N1 + N2 > S.

Analysis of this queuing network is difficult because there is no known product-form solution (and it is unlikely that one exists) to the steady state distribution of the occupancy of the system. However, we will show that this network can be analyzed by utilizing results from a similar queuing network, stemming from slightly different assumptions regarding the fulfillment of customer demand. Therefore, for the moment, assume that a customer facing a stock-out situation is now always backordered (never lost) at the local warehouse and that this always triggers a replenishment order. Furthermore, assume that the local warehouse has the option of choosing between two different suppliers, the first one with lead time L, and the second one with lead time T, T L. The local warehouse always places its orders at the first supplier, unless a stock-out occurs and an order from the second supplier can reach the local warehouse before an (unreserved) order from the first supplier. In this case an order is placed at the second supplier. This situation corresponds to a special case of the so-called “dual index policy” (see e.g. Song and Zipkin, 2009). In its general form, the dual index policy allows for the ordering from each supplier to be based on the two inventory positions IP1 and

IP2, respectively, but for our purposes the second supplier is only used when IP2 = 0 and an order

arrives. We will refer to this model as the Dual Supply model (DS) model, and it can be described by the queuing network depicted in Figure 4. In the DS model the number of orders in each part of the pipeline is denoted M1 and M2, respectively.

Poisson(λ) M1<S M1 No Yes • / D / ∞ M2 • / D / ∞ Figure 4

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Comparing Figure 3 and Figure 4, we see that the difference is that in the DS model customers that are blocked from entering the first station (which still has service time L − T) are not lost for the system, but expedited directly to the second station (with service time T). This strategy is sometimes referred to as “jump over blocking” in the queuing literature and it maintains the product form solution (see Lam, 1977), making it possible to derive the steady state distribution of the system. We will now show how this distribution can be used to obtain the performance measures for the TBB model, for given S and T.

As shown in Lam (1977), and Song and Zipkin (2009), the joint steady state distribution of M1

and M2 is given by ) T , m ( )) T L ( , k ( )) T L ( , m ( ) m , m ( S 2 0 k 1 2 1 DS ,

where (k, ) e k k! is the Poisson probability mass function. For tractability regarding the case with L = T (complete backordering), and the case with T = 0 (pure lost sales), we define 00 = 1. Recall that we wish to determine the expected inventory on hand (EIL+), the fraction of demand lost for the local warehouse (ψ), and the average waiting time for a backordered demand (EW), in the TBB model. For the DS model we define

αDS = fraction of demand satisfied from stock on hand

βDS = fraction of demand backordered in anticipation of an item (incoming from the

first supplier) already in the pipeline.

ψDS = fraction of demand backordered in anticipation of an incoming item from the

second supplier.

EIL+DS = expected inventory on hand

EWDS = expected waiting time for a backordered item.

We begin with the expected inventory on hand and introduce the following lemma:

Lemma 1

Given identical values for S, T, λ and L, the distribution of the on-hand inventories are identical in the TBB model and the DS model.

Proof

Consider both systems in a state with k units of stock on hand, k 1. Given this state, in both systems, an arriving customer receives an item from stock on hand, rendering k-1 units in stock. At the same time a regular replenishment order is placed, which will arrive in L time units. Hence, when k 1 both systems are equivalent. In the case when k = 0, and there is unreserved stock in the second part of the pipeline, i.e. N1 = M1 < S, the demand is backordered in both systems and k = 0 still holds. Once

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case as well. In the case when k = 0 and there are no unreserved items in the second part of the pipeline, i.e. N1 = M1 = S, in the TBB model, the demand is lost. The system will therefore remain in

this state until the unreserved order closest to the local warehouse reaches the second part of the pipeline, resulting in a transition to the state with k = 0 and N1 = S − 1. Given the same situation in the

DS model, the customer is backordered and an order for an item is placed with the second supplier, with lead time T. Nevertheless, because M1 = S implies that the unreserved item closest to the local

warehouse is farther than T time units away, the item from the second supplier will arrive at the local warehouse before any unreserved items, and leave the system immediately with the waiting customer. Therefore, k = 0 and M1 = S despite the order from the second supplier, and the transition to the state

with k = 0 and M1 = S − 1 will occur at the exact same time as in the TBB model, i.e. when the closest

unreserved item reaches the second part of the pipeline. Once this occurs, the systems will be identical again. ■

Lemma 1 implies that determining EIL+ is straightforward using the steady state distribution of the dual supply model:

Corollary 1 S 0 i i S 0 j DS DS (S i j) (i,j) EIL EIL .

Next we consider the fraction of demand that is lost to the system, ψ. We can determine this fraction directly, since it is equivalent to the probability that a customer is blocked at the first station in the queuing network in Figure 3. Studying this station in isolation it is clear that it is identical to the Erlang loss system and, hence, ψ is the Erlang loss probability, given by

S 0 k k S ! k )) T L ( ( ! S )) T L ( ( .

The relationship between ψ and ψDS is provided in Corollary 2.

Corollary 2

α = αDS, β = βDS and ψ = ψDS.

Proof

Lemma 1 implies that α = αDS. Furthermore, from Figure 4 it is clear that the first station in the DS

queuing model also is identical to the Erlang loss system and, hence, ψ = ψDS. Because all fractions

must add up to one it then follows directly that β = βDS. ■

Corollary 2 tells us that the fractions of customers waiting in anticipation of pipeline stock are equal in the two models, and the fraction of customers lost in the TBB model is equal to the fraction of customers using the second supplier in the DS model. We will utilize this to determine the last

14

performance measure, EW. Let EIL_DS denote the expected number of backorders in the DS model. The value of EIL_DS is obtained from the steady state distribution as

S 0 i jS i DS DS (i j S) (i,j) EIL .

We have the following lemma:

Lemma 2

T EIL

EW DS

Proof

We know that the performance measures of two systems are equal, except when it comes to the customer waiting times. In the TBB model we can only have waiting in anticipation of reserved pipeline stock, while in the DS model some customers will also be waiting for orders from the second supplier. However, because we know that the waiting time for an order from the second supplier is always T time units it holds that

T EW

EWDS . (3)

According to Little’s law

) ( EIL

EW DS

DS . (4)

Therefore, by combining (3) and (4), we obtain

T EIL T EW ) ( EW DS DS . ■ (5)

The result stated in Lemma 2 is very intuitive. If we multiply both sides with the factor λβ we see that the expected number of backorders in the TBB model is equal to the expected number of backorders in the DS model, minus the expected number of backorders due to waiting for orders from the second supplier.

4.1.2 Optimization of S and T

By counterexamples it can be shown that C(S,T) is neither convex in S, nor convex in T. Furthermore, examples show that C(S,T) is not even unimodal in S, implying that it is difficult to construct a simple optimization procedure. We therefore propose a procedure based on complete enumeration, where T is discretized. This means that one can come as close as desired to the optimal value of T by choosing a small enough step size in the search.

15

Let Δ represent the step size for T. Starting at T = 0 we increase T by Δ until we reach T = L. For each value of T, we start with S = 0 and increase this variable with one unit at a time, recording the resulting total cost in each step. To find an upper bound for S (given T), we utilize that: (i) EIL+ is increasing in S, (ii) EILDS is non-negative, and (iii) from Karush (1957) we know that the probability

ψ is decreasing in S. Therefore, by using (5) and rewriting the cost function as

) bT p ( bEIL hEIL p ) T EIL ( b hEIL p EW b hEIL ) T , S ( C DS DS

we conclude that, if p bT we can stop increasing S when hEIL+ is larger than the lowest total cost found so far for the given T. Correspondingly, if p < bT, we stop when hEIL+ + ψλ(p – bT) is larger than the lowest total cost found.

4.2 Multi-echelon model

Exact cost evaluation of the multi-echelon system is more complicated than in the single-echelon case. This is because the demand process at the support warehouse is the sum of J “overflow” demand processes, resulting from stock-outs at the local warehouses. These processes are difficult to characterize and we will therefore approximate them by independent Poisson processes. This commonly used approximation has been proven to work well in many situations (see e.g. Axsäter 1990, Kranenburg and van Houtum 2009, Reijnen et al. 2009, and Tiemessen et al. 2009) and, under this assumption, the cost evaluation of the multi-echelon model is straightforward using the results for the single-echelon model.

We wish to determine C(S,T), as specified in (2), for a given set of S and T. Recall that this means viewing the emergency shipments from the support warehouse as demand being lost at the local warehouses at a fixed per unit cost cj, and then possibly lost at the support warehouse at a fixed

per unit cost pj − cj. By applying the single-echelon model to each local warehouse j we can determine

the expected holding cost (hjEIL +

j), customer waiting costs (bjβjλjEWj), and the cost for transferring

demand to the support warehouse (cjψj) exactly. Next we determine the average demand rate at the

support warehouse as J

1 j j j

0 . The distribution of demand at the support warehouse is

unknown and therefore we use the approximation that the “overflow” demand from each local warehouse j follows a Poisson process. This implies that the demand at the support warehouse is given by a Poisson process with rate λ0. Combined with the assumption of First Come-First Serve at

the support warehouse, it also means that all demand that is transferred to the support warehouse will have equal average backorder waiting times and equal probabilities of being lost to the central warehouse. Therefore, we can determine the remaining system costs: the holding cost (h0EIL0

+

16

waiting cost (∑bjδjλjEVj) and transfer cost (∑(pj − cj)θjλj)), by applying the single-echelon model to

the support warehouse with the demand rate λ0, customer waiting cost b0 and lost sales cost p0, where

J 1 j j j J 1 j j j j 0 b b and _J 1 j j j J 1 j j j j j 0 ) c p ( p

are the average costs for a given demand at the support warehouse. Note that the expressions for b0

and p0 hold under Poisson demand and the First Come-First Serve assumption.

Turning to the optimization of S and T, we observe that deriving an exact procedure is difficult because the emergency shipment costs at different local warehouses are coupled to each other, through the support warehouse. We therefore propose a heuristic method which is based on decomposing this complex multi-echelon problem into more easily manageable single-echelon problems. The local warehouses are decoupled for a given fraction of demand lost for the support warehouse, ψ0. That is, if the fraction of emergency shipments that are sent from the support

warehouse and the central warehouse, respectively, are given, then the Sj and Tj at a specific local

warehouse j no longer affect the other local warehouses. This is because each local warehouse j will, on average, have a fixed cost for requesting emergency shipments (losing demand), ~ , which is p_j determined as ~pj (1 0)cj 0pj, j {1,…,J}. Hence, for a given ψ0, the variables Sj and Tj can be

optimized separately for each local warehouse j using the single-echelon method. Conversely, if the tolerance time T0 and the demand rate λ0 at the support warehouse are given, it is easy to find the S0

that results in a certain ψ0 (or close to ψ0, considering the issues of integrality). Consequently, our

heuristic enumerates over ψ0 and T0, and in each step the local warehouses are optimized separately

and, based on the resulting “overflow” demand streams, the support warehouse base-stock level is adjusted to match the given ψ0. The heuristic is given in algorithmic form in Appendix A.

The heuristic implies that all service levels at the support warehouse are evaluated and that local warehouse costs are minimized accordingly. Although the method is decentralized, implying that we do not consider asymmetrical solutions where one local warehouse might change its policy to benefit another local warehouse, our numerical tests indicate that it performs very well (see Section 5.2). Even though the enumeration over ψ0 and T0 does require some computational effort, the

solution times for most problems can be regarded as manageable. For instance, for the problems from Volvo Parts that are presented in Section 5.3, solution times varied between one and 15 minutes per problem, when using a laptop computer with a 2.4GHz processor.

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5. Numerical experiments

The numerical experiments are divided into three separate studies. In Section 5.1 we focus on the single-echelon model and analyze the effects of using pipeline information in this context. The main motivation behind these tests is that it is easier to isolate and present the effects from specific parameters in a single-echelon setting. Moreover, because a local warehouse is an intricate part of any emergency (lateral) transshipment model, the insights gained about using pipeline information should be valuable for these types of models as well. We then focus on the multi-echelon model in Section 5.2 and validate the approximations used through simulation optimization based on complete enumeration. Finally, in Section 5.3, we provide results based on data provided by Volvo Parts.

In the following sections the optimal policy parameters in the single-echelon model are denoted by S* and T*, and the optimal base-stock level given that T = 0 is denoted by S'. Correspondingly, the best solutions found by our heuristic for the multi-echelon model are given by the vectors S* and T*, and by the vector S' when T = 0. We then define C(S*, T*) and C(S*, T*) as the total cost when using pipeline information, and C(S', 0) and C(S', 0) as the total cost when ignoring pipeline information. Furthermore, we define the penalty for ignoring pipeline information, ΔP, as

) ,T C(S ) ,T C(S ) 0 , S C( P _{* *} * *

in the single-echelon case. Because the cost evaluation is approximate in the multi-echelon case, we use simulation to obtain the total costs associated with each heuristic policy, rendering the corresponding penalty ) , ( C ) , ( C ) , ( C P _{* *} * * T S T S 0 S .

5.1 Single-echelon model

To analyze the penalty of ignoring pipeline information in the single-echelon case we consider a test bed featuring 3840 problem scenarios. Input data were chosen to cover a wide range of scenarios, and are given through all combinations of the values:

L {2, 4, 6, 8, 10, 12, 14, 16} h {0.01, 0.25, 0.5, 0.75, 1} b {1}

p {0.01, 0.1, 1, 2, 5, 8, 10, 12, 15, 20, 50, 100} λ {0.5, 1, 1.5, 2, 2.5, 3, 3.5, 4}.

We only consider a single value of b because different cost ratios are achieved by varying h and p. Furthermore, note that we have a greater number of problems where p is larger than b. Since p includes both the fixed costs associated with an emergency shipment, as well as the customer waiting

18

cost, this is likely to be the case in most practical situations (and it will always be the case if the emergency shipment lead time is longer than one time unit).

For all scenarios we calculated the penalty of ignoring pipeline information (ΔP), with the step size for optimizing T set to Δ = 0.01. The results show that this penalty can be quite high in many cases. The average ΔP over all 3840 scenarios was 65%, with a maximum of 414% and a minimum of zero. Figure 5 depicts the average ΔP as a function of p, L, λ and h.

Figure 5

From Figure 5 (Graph (a)) we see that, on average, ΔP increases with the cost for emergency shipments, as expected. In Graph (b) we see that the average ΔP decreases in L. Note, however, that the cost for ignoring pipeline information may increase in absolute terms as L increases. This is because it is, generally, more expensive to maintain an inventory system with long lead times. From Figure 5 (Graphs (c) and (d)) we can conclude that λ appears to have little effect on ΔP, while there is a positive correlation between ΔP and h. An increase in the holding cost means that it becomes more attractive to either backorder demand, or use emergency shipments. In the cases where pipeline stock is not considered, T is fixed to zero and we cannot achieve a combination of these two options. Therefore the penalty is likely to increase.

Because the penalty for ignoring pipeline information is increasing in p and h one can also conclude that the penalty must be decreasing in b. That is, if it is expensive to wait for orders in the replenishment pipeline it is not so costly to always request an emergency shipment. However, it is important to emphasize that this refers to a relative increase in the customer waiting cost, compared to the other costs. An absolute increase, will also lead to an increase in the emergency shipment cost, the size of which is determined by the underlying emergency shipment lead time. Apart from the interaction effects between the cost parameters, a 24 factorial design considering the highest and

0% 50% 100% 150% 0 25 50 75 100 Average ΔP p (a) 0% 50% 100% 150% 2 6 10 14 Average ΔP L (b) 0% 50% 100% 150% 0.5 1.5 2.5 3.5 Average ΔP λ (c) 0% 50% 100% 150% 0 0.25 0.5 0.75 1 Average ΔP h (d)

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lowest values of p, L, λ and h from our problem set did not reveal any other significant interaction effects.

Turning to the structure of solutions, the penalty ΔP is depicted against the optimal ratio T*/L in Figure 6.

Figure 6

In Figure 6 we see that the lowest ΔP-values are found when T*/L is close to zero, and the highest values of ΔP occur when T*/L is close to 1. It is, however, interesting to note that a high value for T*/L does not always result in a high value for ΔP. That is, just because the optimal tolerance time is to consider replenishment orders far away from the local warehouse does not necessarily imply that there is a high penalty for ignoring these orders completely. To explain this, Figure 7 depicts the fraction of demand that is backordered (β) against T*/L.

Figure 7

Perhaps somewhat counter-intuitive, Figure 7 shows that there is no obvious relationship between T*/L and β. However, these results are not so surprising if we recall that the values of β and ΔP are

0% 50% 100% 150% 200% 250% 300% 350% 400% 450% 0 0.2 0.4 0.6 0.8 1 ΔP T*/L 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 0.2 0.4 0.6 0.8 1 T*/L β

20

also influenced by S*. For example, consider a scenario where the holding cost is low, the customer waiting cost is high compared to the holding cost, and the emergency shipment cost is only slightly higher than the customer waiting cost. Because it is inexpensive to hold inventory, the optimal base-stock level will be high and we will not backorder (implying that β is low) or request emergency shipments very often. At the same time, since it is slightly less expensive to backorder, the optimal tolerance time will be high (implying that T*/L is high), but this will only have a marginal effect on the total cost. Hence, this implies that the solution can be to consider orders far away in the pipeline, while the fraction of demand backordered and the penalty for ignoring the pipeline stock are low, as our numerical tests show.

In Figure 8 we provide a few examples of how the cost function changes as we increase T, given that the base-stock level is fixed at S*.

Figure 8. Graph (e): L=8, h=0.5, b=1, λ=2. Graph (f): h=0.5, b=1, p=10, λ=2. Graph (g): L=8, h=0.5, b=1, p=2. Graph (h): L=2, b=1, p=15, λ=0.5

We observe in Figure 9 that the shape of the function may change dramatically depending on the input data. Moreover, the cost function appears to be fairly smooth around the optimum, at least as long as T* is not at the extreme points 0 or L. This implies that the choice of tolerance time is insensitive to small errors, as long as the article in question requires a mixture of emergency shipments and pipeline stock. In all cases studied, we have observed that the cost function appears to be unimodal in T. This could prove useful if one wishes to design heuristics for optimizing the tolerance time based on single-variable search methods, e.g., bisection search methods.

0 5 10 15 20 25 30 0 0.5 1 p = 1 p = 100 C(S*,T) T/L (e) 0 1 2 3 4 5 6 0 0.5 1 L = 2 L = 16 C(S*,T) T/L (f) 0 1 2 3 4 0 0.5 1 λ = 0.5 λ = 4 C(S*,T) T/L (g) 0 1 2 3 4 5 0 0.5 1 h = 0.25 h = 1 C(S*,T) T/L (h)

21

Clearly, the variable S will affect the proportion of demand that is satisfied with stock on-hand. The variable T, in turn, affects the ratio of demand fulfillment between the pipeline stock and the emergency shipments. However, as our numerical experiments indicate, it is still difficult to predict the properties of the optimal (S,T) policy for a given scenario. From a practical perspective, this highlights the need for methods, such as the one presented in this work, when designing control policies that use pipeline information.

5.2 Validation of the multi-echelon model

For the multi-echelon model we approximate the demand distribution at the support warehouse and we use a heuristic for determining S* and T* (or S' when T = 0). To provide evidence of the validity of these approximations, we compare our solutions to the ones obtained by complete enumeration using discrete-event simulation.

We consider 32 problems scenarios, evaluated both when using pipeline information and when ignoring pipeline information, which renders a total of 64 problems. We consider two groups of identical retailers in each problem. Furthermore, all input parameters are equal for all retailers except for the demand rates that vary between the two groups. More details about the testing are available in Appendix B.

Our analytical model found the exact same solutions as the simulation optimization in 63 out of the 64 problems. As shown in bold in Table B.1 (Appendix B), for problem 26 the analytical model found the solution S' = (2, 1, 4), when ignoring pipeline information. This is one unit less at the support warehouse than the solution obtained by the simulation optimization, S' = (3, 1 4), and this rendered a 0.9% increase in total costs.

Given the results presented in this section and Appendix B, our method appears to be accurate in determining base-stock levels and tolerance times in the multi-echelon system considered.

5.3 Results from the case company

For the multi-echelon model we use problem scenarios based on data from Volvo Parts Corporation, for the Spanish market. In total, 70 low demand spare parts were selected for which (S-1,S) policies is an appropriate choice. On the Spanish market, Volvo Parts has 63 local warehouses, but not all parts are sold at all locations. For the purpose of this work, we consider 12 local warehouses for each item and, hence, N = 12 for all problems. The demand rates at the local warehouses vary between 0.003 and 0.56 per day. The lead times for regular replenishments from the central warehouse are the same for the 12 local warehouses and are equal to 6 days. For the support warehouse, a regular replenishment from the central warehouse takes 3 days.

The holding costs are the same at all stock points and all the costs have been normalized so that holding costs are equal to one for all problems. The waiting costs are the same for all retailers, but

22

they vary depending on item. For the items considered, the customer waiting costs per unit and day are between 17 and 625 times larger than the holding cost. Emergency shipments from the support warehouse, which is situated close to Madrid, reach the local warehouses by truck within a day and, along with fixed transportation costs etc., the resulting waiting cost has been included in the emergency shipment cost. This cost varies depending on item and it is between 32 and 877 times the holding cost. The emergency shipments from the central warehouse also take one day, but are more expensive, between 76 and 8472 times the holding cost for the various items. The main reason for the higher costs of these shipments is that the central warehouse is situated in Gent, in Belgium and therefore they have to use air freight transport when providing emergency shipments from this location.

When facing a stock-out in Volvo Parts’ system one would typically be interested in knowing if the desired part will arrive within one day, or two days, and so forth. It is therefore reasonable to discretize the T-values to whole days. The problem scenarios and results from applying our model are presented in detail in Table C.1-C.3, in Appendix C. Note that the scenarios are sorted according to largest ΔP, and that the local warehouses are sorted according to largest λ within each scenario, in Table C.1-C.3. The average penalty of ignoring pipeline information over the 70 scenarios was 15.4%, with a maximum of 106% and a minimum of 2.4%. In Table C.1-C3 we see that our model identified 22 out of the 70 parts where the solution is complete backordering at the local warehouses (Tj

*

= 6 for j = 1,…,12). Using emergency shipments appears to be the wrong strategy for these spare parts and it follows that the ΔP-values are among the highest in these cases. Not surprisingly, the common attribute for these parts is that the costs of emergency shipments are relatively high compared to the waiting - and holding costs.

In concurrence with the results in Section 4.1, the values of ΔP are positively correlated with the ratio between the support warehouse emergency shipment cost and the customer waiting cost (c/b), as illustrated in Figure 9.

Figure 9

Although not shown here, we recorded a similar correlation with the corresponding ratio for the central warehouse emergency shipment cost (p/b). This is because c and p, generally, have a strong

0% 20% 40% 60% 80% 100% 120% 0 5 10 15 20 25 30 35 40 c/b ΔP

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positive correlation. That is, a part that is expensive to request from the support warehouse is also expensive to request from the central warehouse, and vice versa.

An interesting result, from Volvo Parts’ perspective, is that the solutions point to a rather infrequent use of emergency shipments from the central warehouse. From Table C1-C3 we deduce that these emergency shipments are only utilized in 20 of the scenarios considered. Furthermore, out of these 20 scenarios, there are only three cases where the central warehouse is the sole provider (i.e. S0

*

= 0 and T0 *

= 0 in scenarios 62, 68 and 70). This indicates that, in many cases, the high costs associated with emergency shipments from the central warehouse can be avoided by using pipeline information.

We now turn our attention to the problem scenarios where the values of ΔP are slightly lower; problems 42-50 in Table C.2 and problems 51-70 in Table C.3. We note that even though c and p are generally lower in these scenarios, there is not a single scenario where it is optimal for all stock points to always request an emergency shipment in a stock-out situation. In fact, many of these scenarios show solutions where the tolerance times at the local warehouses are equal to one day. This seems reasonable if we recall that one day is the lead time for an emergency shipment to reach the local warehouses. That is, even in cases where there is an obvious benefit in using emergency shipments frequently, one would typically want to backorder demand if the spare part in the replenishment pipeline reaches the local warehouse before an emergency shipment. This essentially means that we have identified the spare parts where the cost of waiting is so high that the quickest option is the most reasonable, as discussed in Section 3.

6. Summary and concluding remarks

In this paper we have studied the effects of using pipeline information in an inventory system with emergency shipments, consisting of a number of local warehouses, a support warehouse and a central warehouse with ample capacity. We have presented an exact method for determining costs and optimizing decision variables for a local warehouse in isolation, and provided an accurate heuristic for the multi-echelon system. Our results indicate that there is a significant benefit in using pipeline information. In a numerical study of 70 items from Volvo Part’s assortment of spare parts, the penalty (i.e. the relative cost increase) for ignoring pipeline information was 15.4% on average, with a maximum of 106% and a minimum of 2.4%. In addition to the Volvo examples our tests also suggest that the value of pipeline information is largely dependent on the cost structure. More specifically, the penalty for ignoring pipeline information increases with the cost for using emergency shipments and the cost for holding inventory.

From a managerial perspective we have found that the determination of suitable stock levels and the time one should wait for stock in the replenishment pipeline is a complex decision. Generally, it requires the use of advanced methods such as the one presented in this paper. For implementing our

24

proposed policy it is crucial that the pipeline information is available in the system. With modern technology, such as RFID and tracking and tracing systems, it is possible to have real-time information on incoming orders at all warehouses. Furthermore, at many companies a warehouse knows when an order will be arriving, even without these advanced information systems, because of strict routines and high delivery reliability. This is, i.e., the case at Volvo Parts.

For future research, we believe that significant advantages can be achieved by using pipeline information in more general systems where emergency shipments are utilized. Steps towards this would be to consider the inventory level at the central warehouse, batch ordering, structures with multiple support warehouses, and more general demand distributions.

Appendices

Appendix A – Heuristic for the multi-echelon model

Let Δj be the step sizes for the tolerance times Tj, j {0,…,J}, and let ₀ be a given value of ψ0, with

corresponding step size ε. Furthermore, denote S*

and T* as the best solution found so far by the heuristic, with resulting total cost C*. We then have the following algorithm:

Step 0. Initialize by setting C* = , T0 = 0.

Step 1. Set S0 = 0 and ₀= 1.

Step 2. Compute ~pj (1 0)cj 0pj, for all j {1,…,J}.

Step 3. For each j {1,…,J} optimize Sj and Tj and determine the

corresponding θj using the single-echelon optimization procedure, based on

p = ~ and Δ = Δpj j.

Step 4. Determine λ0 based on the θj-values obtained from the solutions in Step 3.

Step 5. Given λ0, find the highest S0 such that j(S0) 0 by increasing S0 one

unit at a time.

Step 6. Compute C(S,T) from the Sj and Tj (j {0,…,J}) obtained in Step 3 and

Step 5.

Step 7. If C(S,T) < C* then set C* = C(S,T), S* = S and T* = T. Step 8. Set _{0 0} and if ₀ 0 then return to Step 2. Step 9. Set T0 = T0 + Δ0. If T0 L0 then stop, otherwise return to Step 1.

Appendix B – Details on the model validation

The input data for the 32 scenarios are given by N = 12 or 24, (λ1, λ2) = (0.003, 0.02) or (0.01, 0.15), Lj = 4 or 8, bj = 100 or 400 (j {1,…,J)), where the 32 combinations of low and high values constitute

25

retailers N/2+1 to N are identical with demand rate λ2

. Remaining input parameters are the same for all problems: hj = 1 (j {0,…,J)), cj = 600 (j {1,…,J)) and the support warehouse lead time is equal

to half of the retailer lead time, that is, LS = 0.5Lj. For each problem we obtained S *

and T* (or S' when T = 0) from our analytical model with Δ = 2 and ε = 0.001. We then searched for the optimal solution according to the simulation model by performing a complete search over these variables. In lack of a computationally tractable upper bound for S, we used the ad-hoc approach of only considering S at most two units above the S* (or S') given by the analytical model. Each combination of variables was simulated for 5000 arrivals, excluding a system warm-up period of 500 arrivals. The 10 best candidate solutions were then simulated for an additional 1 800 000 arrivals, making it plausible that the final solution with the lowest total cost is the optimal solution.

The results from our analytical model are presented in Table B.1. Note that the solutions given by the simulation optimization only differed in problem 26.

26

Table B.1 Input data Using pipeline information Ignoring pipeline information # N L b c 100*λ T* S* S' 1 12 4 100 200 (0.3, 2) (2, 0, 2) (1, 0, 1) (1, 0, 1) 2 12 4 100 200 (1, 15) (0, 4, 4) (0, 1, 3) (1, 1, 3) 3 12 4 100 400 (0.3, 2) (0, 4, 4) (0, 1, 1) (1, 1, 1) 4 12 4 100 400 (1, 15) (0, 4, 4) (0, 1, 3) (0, 1, 4) 5 12 4 400 200 (0.3, 2) (0, 0, 0) (1, 0, 1) (1, 0, 1) 6 12 4 400 200 (1, 15) (0, 0, 0) (1, 1, 3) (1, 1, 3) 7 12 4 400 400 (0.3, 2) (0, 2, 0) (1, 1, 1) (1, 1, 1) 8 12 4 400 400 (1, 15) (0, 2, 2) (0, 1, 4) (0, 1, 4) 9 12 8 100 200 (0.3, 2) (4, 0, 2) (1, 0, 1) (2, 0, 1) 10 12 8 100 200 (1, 15) (4, 2, 2) (1, 1, 4) (2, 1, 4) 11 12 8 100 400 (0.3, 2) (2, 4, 4) (1, 1, 1) (0, 1, 2) 12 12 8 100 400 (1, 15) (0, 6, 6) (0, 1, 4) (1, 1, 5) 13 12 8 400 200 (0.3, 2) (2, 0, 0) (2, 0, 1) (2, 0, 1) 14 12 8 400 200 (1, 15) (2, 0, 0) (2, 1, 4) (2, 1, 4) 15 12 8 400 400 (0.3, 2) (0, 2, 2) (0, 1, 2) (0, 1, 2) 16 12 8 400 400 (1, 15) (0, 0, 0) (1, 1, 5) (1, 1, 5) 17 24 4 100 200 (0.3, 2) (2, 0, 2) (1, 0, 1) (2, 0, 1) 18 24 4 100 200 (1, 15) (0, 4, 4) (0, 1, 3) (2, 1, 3) 19 24 4 100 400 (0.3, 2) (0, 4, 4) (0, 1, 1) (1, 1, 1) 20 24 4 100 400 (1, 15) (0, 4, 4) (0, 1, 3) (1, 1, 4) 21 24 4 400 200 (0.3, 2) (2, 0, 0) (2, 0, 1) (2, 0, 1) 22 24 4 400 200 (1, 15) (2, 0, 0) (2, 1, 3) (2, 1, 3) 23 24 4 400 400 (0.3, 2) (0, 2, 0) (1, 1, 1) (1, 1, 1) 24 24 4 400 400 (1, 15) (0, 2, 2) (0, 1, 4) (1, 1, 4) 25 24 8 100 200 (0.3, 2) (4, 0, 2) (2, 0, 1) (2, 0, 1) 26 24 8 100 200 (1, 15) (4, 2, 2) (1, 1, 4) (2, 1, 4) 27 24 8 100 400 (0.3, 2) (2, 4, 4) (1, 1, 1) (0, 1, 2) 28 24 8 100 400 (1, 15) (2, 4, 4) (1, 1, 4) (1, 1, 5) 29 24 8 400 200 (0.3, 2) (0, 0, 0) (2, 0, 1) (2, 0, 1) 30 24 8 400 200 (1, 15) (2, 0, 0) (2, 1, 4) (2, 1, 4) 31 24 8 400 400 (0.3, 2) (0, 2, 2) (0, 1, 2) (0, 1, 2) 32 24 8 400 400 (1, 15) (0, 0, 0) (1, 1, 5) (1, 1, 5)

Note. L refers to the retailer lead time. The support warehouse lead time is equal to 0.5L, h = 1

and p = 600 in all problems.

Appendix C – Problem scenarios for the multi-echelon model

The multi-echelon model was applied with the step size for ψ0 set to ε = 0.001. In the following tables,

the values of ΔP were obtained by simulation. The standard deviations of the ΔP-values are always less than 1% of ΔP.