Multi-item spare parts systems with lateral transshipments and waiting time constraints

European Journal of Operational Research 171 (2006) 1071–1093 www.elsevier.com/locate/ejor

Multi-item spare parts systems with lateral transshipments and waiting time constraints H. Wong b

a,*

, G.J. van Houtum b, D. Cattrysse a, D. Van Oudheusden

a

a Katholieke Universite`it Leuven, Centre for Industrial Management, Celestijnenlaan 300A, 3001 Heverlee, Belgium Technische Universiteit Eindhoven, Faculty of Technology Management, P.O. Box 513, 5600 MB Eindhoven, The Netherlands

Available online 12 March 2005

Abstract This paper deals with the analysis of a multi-item, continuous review model of two-location inventory systems for repairable spare parts, used for expensive technical systems with high target availability levels. Lateral and emergency shipments occur in response to stockouts. A continuous review basestock policy is assumed for the inventory control of the spare parts. The objective is to minimize the total costs for inventory holding, lateral transshipments and emergency shipments subject to a target level for the average waiting time per demanded part at each of the two locations. A solution procedure based on Lagrangian relaxation is developed to obtain both a lower bound and an upper bound on the optimal total cost. The upper bound follows from a heuristic solution. An extensive numerical experiment shows an average gap of only 0.31% between the lower and upper bounds. The experiment also gives insights into the relative improvement achieved by applying lateral transshipments and or the system approach. We also apply the proposed model to actual data from an air carrier company.
2005 Elsevier B.V. All rights reserved. Keywords: Inventory; Emergency transshipments; Spare parts; Lagrangian relaxation

1. Introduction Equipment-intensive industries such as airlines, nuclear power plants, and various process and manufacturing plants using complex machines are often confronted with the difficult task of maintaining a high system availability, while at the same time there is a pressure to limit the spare parts inventories. A random failure of just one component can cause the system to go down. As the downtime can be very costly, spare

*

Corresponding author. Tel.: +32 16 322497; fax: +32 16 322986. E-mail address: [email protected] (H. Wong).

0377-2217/$ - see front matter
2005 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2005.01.018

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part inventories are required to keep the average stock-out time as low as possible. However, as most parts are quite expensive, maintaining an excessive number of spare parts should also be avoided. Lateral transshipments (also referred to as inventory pooling) represent an effective strategy to improve a companyÕs system availability while reducing the total system costs. Lateral transshipments are used to satisfy a demand at a location that is out of stock from another location with a surplus of on-hand inventory. Since costs for lateral transshipments are generally much lower than downtime costs, lateral transshipments can reduce total system costs. This research was originally motivated by an air carrier company, located in Brussels, who was interested in pooling its spare parts inventories with another company (see Timmers, 1999). Although a significant amount of research has been done studying various aspects of lateral transshipments in inventory systems, most of it deals with single-item problems in which only one item at a time is considered. Such problems are typical when we use an item approach. Under an item approach, inventory levels for each individual item are set independently. An alternative approach, denoted as the system approach by Sherbrooke (2004), considers all items in the system when making inventory-level decisions, and may lead to large reductions in inventory costs in comparison to an item approach (see also Thonemann et al., 2002; Rustenburg et al., 2003). The main purpose of this paper is to advance the existing literature on multi-item inventory systems with lateral transshipments. Archibald et al. (1997) is the only previous study dealing with such problems. They consider a two-location, multi-item, multi-period, periodic review inventory system with a limited storage space for all items together, i.e. the only connection between the problems for different items is due to the limited storage space. In contrast to their work, we analyze a two-location, multi-item, continuous-review system for repairable items with one-for-one stock replenishments and our optimization problem is to determine stocking policies for all items that minimize the total system cost subject to a target level for the average waiting time for an arbitrary request for a ready-for-use part at each of the two locations. In our model, the decisions with respect to different items are coupled because of the multi-item service measure that is used. Analyzing lateral transshipments in single-item inventory systems using a continuous review policy with one-for-one stock replenishments is done by Lee (1987), Axsa¨ter (1990), Sherbrooke (1992), Yanagi and Sasaki (1992), Alfredsson and Verrijdt (1999), Grahovac and Chakravarty (2001), Kukreja et al. (2001), and Wong et al. (2005a). One-for-one stock replenishments are common when we deal with (repairable) slow-moving and expensive items. Needham and Evers (1998), Evers (2001), and recently Xu et al. (2003) and Axsa¨ter (2003) consider continuous review (R, Q) policies. Other studies assume periodic review policies and they usually assume no order setup cost, so that an order-up-to or base-stock policy is appropriate. Examples are Gross (1963), Krishnan and Rao (1965), Das (1975), Hoadley and Heyman (1977), Cohen et al. (1986), Karmarkar (1987), Tagaras (1989, 1999), Robinson (1990), Tagaras and Cohen (1992), Archibald et al. (1997), Rudi et al. (2001), and Herer et al. (2002). Different from those examples, Herer and Rashit (1999) introduce the existence of non-negligible fixed and joint replenishment costs. In this paper we allow emergency supplies from an infinite source when demand at a location cannot be met by either the stock at the local warehouse or the stock at the other location. This type of emergency supply mode is similar to the one applied in Alfredsson and Verrijdt (1999). The only difference is that they consider a two-echelon system while we consider a single-echelon system. In their model, the emergency supply is sent from the central warehouse if it has stock on hand. If the central warehouse is out of stock, the emergency supply is sent from the plant having infinite capacity. Another work considering the emergency supply is by Archibald et al. (1997). They allow an emergency order even when the other location has inventory. For systems in which the down time is very costly, e.g. airline companies, the assumption of an emergency supply mode is more realistic than assuming backlogging. Most previous studies do not give exact analysis for the optimization problems with the notable exception of the models of Gross (1963), Krishnan and Rao (1965), Das (1975), Hoadley and Heyman (1977), Robinson (1990), Archibald et al. (1997), and Herer and Rashit (1999). In this paper we derive tight lower

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and upper bounds on the optimal costs for the multi-item problem. The study in this paper can be used as a building block for the analysis of more complex systems. Table 1 presents a brief review of the previous studies and shows the position of this paper in comparison to them. This paper is organized as follows. In Section 2, we present the problem formulation. We introduce the basic assumptions and the notation of the model, and we present the mathematical formulation of our problem. Section 3 describes our solution method that is based on Lagrangian relaxation. We describe how to find the best lower and upper bounds on the optimal objective function. The best lower bound is obtained by optimization of the Lagrange parameters, which is done by the subgradient optimization method. The best upper bound is determined by applying a local search procedure with the best feasible solution obtained from the subgradient optimization method as starting point. In Section 4 we describe the parameter settings and the results of our computational experiment. The main purpose of the experiment is to evaluate the tightness of the bounds and the economic benefits of applying lateral transshipments and the system approach. Section 5 presents a model application. We apply our model to actual data from the air carrier company that motivated this work. Finally, we summarize the results in Section 6 and conclude with directions for further research.

2. Problem formulation In this section, we describe our model. Our model is developed for environments where many spare parts are kept in stock to obtain high availability of expensive technical systems, such as airplanes, nuclear power plants, and large computer systems. Assumptions are based on common practice in such environments. Our model is described in Section 2.1, its critical assumptions are discussed in Section 2.2, and the precise problem to be solved is formulated in Section 2.3. 2.1. Notation and assumptions We model the situation of two independent companies who keep spare parts on stock for their technical systems. Note that throughout this paper the words company and location are used interchangeably. The companies are indexed by j = 1, 2. We assume that both companies have a number of technical systems of the same type. These systems consist of components which are subject to failures. In total there are I different items (SKUs). These items are indexed by i = 1,2,. . .,I. Failures occur according to Poisson processes with constant rates. The total failure rate of components of item i at company j is given by mij (P0). If an item i does not occur in the configurations P of the technical systems at company j, then mij = 0. We assume I that mi1 + mi2 > 0 for all i. Further, M j ¼ i¼1 mij denotes the total failure rate at company j. We assume that Mj > 0 for j = 1,2. Company j has S ij ð2 N0 :¼ f0g [ NÞ spare parts of item i. We define Sj := (S1j,. . .,SIj). In total, both companies share Si spare parts of item i, where Si = Si1 + Si2. All parts are repairable and there is no condemnation. When a part of item i fails at company j, the failed part is replaced by a spare part. This means that the failed part is immediately removed and sent into repair. A ready-for-use part is put back into the system where the failed part belongs to, as soon as such a part is available. If company j has a ready-for-use part on stock then this can be done immediately. If not, then there is a waiting time for a ready-for-use part. In that case, if the other company has a ready-for-use part on stock, it sends a part by a lateral transshipment, and the average waiting time is equal to the average transshipment time ET tri ð> 0Þ. Notice that nothing is assumed for the distribution of the lateral transshipment times (i.e., any distribution with mean value ET tri is allowed). We assume that complete pooling is applied. This means a company offers its entire available inventory when the other company is experiencing a stockout. If also at the other company no ready-for-use part is available, an emergency supply mode is applied. This means that either the repair operation is expedited or the required part is ordered from an

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Table 1 A brief review of the literature on the multi-location inventory systems Paper

# Echelons

# Items

Periodic/cont.

Policies

Model analysis

Important remarks

Gross (1963)

Single

Single

Periodic

Order-up-to

Exact opt.

Krishnan and Rao (1965)

Single

Single

Periodic

Order-up-to

Exact opt.

Das (1975)

Single

Single

Periodic

Order-up-to

Exact opt.

Hoadley and Heyman (1977)

Two

Single

Periodic

Order-up-to

Exact opt.

Karmarkar (1987)

Single

Single

Periodic

General

Appr. opt.

Cohen et al. (1986)

Multi

Single

Periodic

Order-up-to

Appr. eval. Appr. opt.

• Allows the transshipments before the realization of demand • Derive optimal solution for two-location systems with identical cost parameters; allow the transshipment after demand is realized but before it must be satisfied • Extends GrossÕs model by permitting transshipments in the middle of each period • Allow balancing acts at the beginning of a period and emergency acts at the end of a period • Considers the multi-period problem; develops lower and upper bounds of the optimal costs • Allow transshipments at higher echelons

Lee (1987)

Two

Single

Continuous

(S
1, S)

Tagaras (1989)

Single

Single

Periodic

Order-up-to

Appr. eval. Appr. opt. Appr. opt.

Axsa¨ter (1990)

Two

Single

Continuous

(S
1, S)

Appr. eval.

Robinson (1990)

Single

Single

Periodic

General

Exact opt.

Tagaras and Cohen (1992)

Single

Single

Periodic

Order-up-to

Appr. eval. Appr. opt.

Yanagi and Sasaki (1992) Sherbrooke (1992)

Single

Single

Continuous

(S
1, S)

Appr. eval.

Two

Single

Continuous

(S
1, S)

Appr. eval.

• Assumes identical bases • Extends the model of Krishnan and Rao (1965) by using a more general cost structure • Allows non-identical bases, focus on the demand processes • Shows the optimality of a basestock ordering policy when the cost parameters are identical or there are only two locations; analyzes multi-period problems using dynamic programming model • Consider positive replenishment lead times and shows the superiority of complete pooling • Consider limited repair capacity • Estimates the expected backorders using a regression model

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Table 1 (continued) Paper

# Echelons

# Items

Periodic/cont.

Policies

Model analysis

Important remarks

Archibald et al. (1997)

Single

Multi

Periodic

Order-up-to

Exact eval.

• Allow lateral transshipments at any time during a period; model the problem as a Markov decision process • Considers the multiitem problem with limited storage space • Examine the interaction of relevant costs and transshipment policies; use a simulation-based optimization

Exact opt.

Needham and Evers (1998)

Two

Single

Continuous

(R, Q)

Appr. eval. Appr. opt.

Tagaras (1999)

Single

Single

Periodic

Order-up-to

Appr. eval.

Herer and Rashit (1999)

Single

Single

Periodic

General

Exact opt.

Alfredsson and Verrijdt (1999) Grahovac and Chakravarty (2001)

Two

Single

Continuous

(S
1, S)

Appr. eval.

Two

Single

Continuous

(S
1, S)

Appr. eval.

Kukreja et al. (2001)

Single

Single

Continuous

(S
1, S)

Appr. eval. Appr. opt.

Evers (2001)

Single

Single

Continuous

(R, Q)

Appr. opt.

Rudi et al. (2001)

Single

Single

Periodic

General

Exact opt.

Herer et al. (2002)

Single

Single

Periodic

Order-up-to

Appr. opt.

Xu et al. (2003)

Single

Single

Continuous

(R, Q)

Appr. eval.

• Studies several lateral transshipment policies using a simulation model • Consider fixed and joint replenishment costs; characterize the form of the optimal policy (single period) • Allow emergency ship ments from the depot and an outside supplier • Allow lateral transshipments before and after a location is out of stock • Relax the assumption of exponential repair time distribution • Develops heuristics for determining when transshipments should be made • Consider local decision making; focus on determining transshipment price • Prove the optimality of (s, S) policies; consider dynamic transshipments; use a simulation-based optimization • Introduce hold-back parameter which limit the level of outgoing transshipments (continued on next page)

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Table 1 (continued) Paper

# Echelons

# Items

Periodic/cont.

Policies

Model analysis

Important remarks

Axsa¨ter (2003)

Single

Single

Continuous

(R, Q)

Appr. eval.

Wong et al. (2005a)

Single

Single

Continuous

(S
1, S)

Appr. eval. Appr.opt.

This paper

Single

Multi

Continuous

(S
1, S)

Exact eval. Appr.opt.

• Allow lateral transshipments only in one direction • Consider pooling in a multi-hub system and delayed lateral transshipments • Considers the multi-item problem and employs waiting time constraints

outside supplier (e.g., an OEM or a third party supplier). A ready-for-use part becomes available after an tr average time ET em i ðP ET i Þ. Nothing is assumed for the distribution of the emergency supply lead times (i.e., any distribution with mean value ET em is allowed). Failed parts that are sent into repair are returned i as ready-for-use parts after exponential repair lead-times. The lead-times of different parts of the same item and of parts of different items are independent. The repair rate of a failed part of item i is given by li. We assume that in case a lateral transshipment (an emergency shipment) takes place from company j (the outside supplier) to the other company, the failed part will be returned to company j (the outside supplier), either immediately or upon completion of its repair. With this assumption, the number of parts in stock plus the number of parts in repair of item i at company j is always equal to Sij. At company j, there is a maximum level W max given for the average waiting time per request for a readyj for-use part. In this paper, we consider a service model rather than a cost model. In a service model, the objective is to minimize the total system costs subject to a set of service level constraints. In our case, the service level constraints are constraints on the maximum expected waiting time. In a cost model, however, the service constraints are replaced with penalty (downtime) costs. Although in general the cost models are analytically more tractable, they have a serious limitation in that the penalty costs are generally hard to specify. Thus service models are more acceptable from a practical point of view. For a systematic overview of possible relations between the two types of models for general inventory systems, see Van Houtum and Zijm (2000). Total system costs consist of holding costs, transshipment costs and emergency supply costs. Holding costs chi are counted for each spare part of item i. A cost ctri is counted for each lateral transshipment of part of item i. A cost cem is counted for each part coming from the emergency supply. The objective is i to find a policy (S1, S2) under which the total average costs are minimized subject to the waiting time constraints for the companies 1 and 2. 2.2. Discussion of critical assumptions There are several assumptions that are critical for our model and the analysis in the rest of the paper. Recall that our model is developed for environments with expensive technical systems such as airplanes. The main assumptions are as follows: (i) Failures occur according to Poisson processes with constant rates. Both companies are assumed to have a whole set of technical systems (e.g., airplanes) that operate at a rather constant rate (e.g., a rather constant number of flying hours per week). For expensive technical systems, failure rates are low in general and thus downtimes do not occur very often. Further, due to the use of lateral transshipments and emergency supplies, downtimes are never long. Therefore it is reasonable to assume constant failure rates per technical

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system, and thus also for the whole set of technical systems at a company. In addition, we assume that failures per item for the whole set of technical systems occur according to a Poisson process. This is quite common in the literature on spare parts inventory models (see e.g. Sherbrooke, 2004). Often this assumption is justified because lifetimes of parts are simply exponential. This assumption is also justified in case lifetimes are non-exponential but the set of technical systems is so large that the merged stream of failure processes of individual technical systems is close to Poisson. (ii) All parts are repairable and there is no condemnation. In fact, for our model it is only essential that the inventory positions be kept at constant levels. This is also obtained when: (i) items are consumable and onefor-one procurement is applied, or (ii) there is condemnation for repairable items and for each condemned part immediately a new part is procured. (iii) Lateral transshipments are faster and cheaper than emergency supplies. The two locations in our model should be at close distance of each other, otherwise it makes no sense to pool their spare parts inventories. We implicitly assume that emergency supplies are significantly longer and more expensive than lateral transshipments. Therefore, a lateral transshipment is always preferred above an emergency supply. (iv) Failed parts are returned to the company that fulfilled the demand for a ready-for-use part. For the sake of clarity, we incorporated this rule in our model description. What we really need is that inventory positions of all parts are kept at constant levels at both companies. Alternative rules under which this is obtained are also possible. E.g., in case of both a lateral transshipment and an emergency supply, it may also be that the one company leases a ready-for-use part supplied by another company until a readyfor-use part becomes available at this company itself. Then the leased ready-for-use part is sent back to the other company after a while and one pays a price per day that the part was leased. This leasing construction is common practice in the airline industry. (v) Repair lead-times are exponential. This assumption facilitates the analysis. It allows that the performance per item can be evaluated by a Markov process; see Section 2.3. The impact of this assumption is very small. For each item, the two locations together behave pretty much as an Erlang loss system (an M/G/c/c queue), for which it is known that the steady-state distribution is insensitive to the distribution of the service times. In fact, we would obtain an Erlang loss system under zero lateral transshipment times. Alfredsson and Verrijdt (1999) made a similar exponential assumption and justified the assumption by a sensitivity analysis. They assumed exponential shipment times for the regular supply channels of the local bases in their two-echelon model (this is equivalent to what we assume), and showed by simulation that almost no change in steady-state behavior is obtained when the exponential times are replaced by deterministic or lognormal times. (vi) Complete pooling is applied. This is part of the agreement between the two companies. In case companies do not want to share their last parts, one may introduce threshold parameters, and agree that a company does not supply a part by a lateral transshipment if the physical stock of the requested item is at or below the threshold level. This leads to an extended version of the model. A rule has to be added for how the values of the threshold parameters are chosen, or one may consider them as additional decision parameters. In principle, this extended model may be analyzed along the same lines as our current model. (vii) Inventory holding costs dominate. An implicit assumption is that inventory holding costs dominate. This means that inventory holding costs constitute at least 70%, say, of the yearly average total costs. Hence the focus is on the optimization of inventory levels rather than the use of lateral transshipments and emergency supplies. 2.3. Model formulation To formulate the problem, we define: bij = fraction of demand for item i at company j satisfied by its own stock; aij = fraction of demand for item i at company j satisfied by lateral transshipments; hij = fraction

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of demand for item i at company j satisfied by emergency supply; Wj = average waiting time per request for a ready-for-use part at company j. The system behavior with respect to an item i is independent of all other items and may be described by a two-dimensional Markov process. For each item i, we introduce the state xi = (xi1, xi2), where xij represents the physical stock of spare parts of item i at company j, and 0 6 xij 6 Sij. We define xi1
= (xi1
1, xi2), xi1+ = (xi1 + 1, xi2), xi2
= (xi1, xi2
1), xi2+ = (xi1, xi2 + 1). All possible transitions of the Markov process are as follows: Transition 1: a failure of a part of item i occurs at location j while xij > 0; the state transition is xi ! xij
; the transition rate is mij. Transition 2: a failure of a part of item i occurs at company j while xij = 0 and the other company j 0 has a positive stock of item i; the state transition is xi ! xij 0
; the transition rate is mij and this represents a lateral transshipment requested by company j. Transition 3: a failure of a part of item i occurs at company j while xi1 = xi2 = 0; an emergency supply is applied; the state transition is xi ! xi; the transition rate is mij. Transition 4: the repair of a part of item i belonging to company j is completed; the state transition is xi ! xij+; the transition rate is (Sij
xij)li. Fig. 1 shows the Markov process that is obtained when (Si1,Si2) = (2,1). A similar process is obtained for any other choice for (Si1,Si2). We define p as the steady-state probability vector and p(k,l) as the steady-state probability of being in state (k,l), 0 6 k 6 Si1, 0 6 l 6 Si2. Since the number of states in our problem is not large, a direct method based on Gaussian elimination can be applied to determine p. Obviously, bij + aij + hij = 1 for i = 1,. . .,I; j = 1,2. Since complete pooling is applied here, an emergency supply only takes place when all locations are out of stock. The fraction of demand for item i at company j satisfied by emergency supplies is thus equal to the probability of being in state (0, 0), i.e. hij = p(0,0). This shows that hij is the same for j = 1,2, i.e. hi1 = hi2 = hi for all i. (In the case where partial pooling or no pooling is applied, hij may be different for each location j). Notice that this fraction can also be obtained analytically by aggregation on the basis of total physical stock at the locations 1 and 2. With this aggregation, hi is also equal to the Erlang loss probability of an M/M/Si/Si queuing system. The fraction of demand for parts P i2 of item i at company P 1i1 satisfied by lateral transshipments from company 2 is given by ai1 ¼ Sl¼1 pð0;lÞ . Similarly, ai2 ¼ Sk¼1 pðk;0Þ . The fraction of demand that is satisfied by the local stock is obtained from bij = 1
aij
hi, j = 1,2.

m i2 2,1

µ,i

2,0

µi µi

m i1

m i1 +m i2

m i2 1,1

2µ i

1,0

µi

2µ i

m i1

m i1 +m i2

m i1 +m i2 0,1

µi

0,0

Fig. 1. Markov process for item i with (Si1,Si2) = (2,1).

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We now explain how to obtain the average waiting time. It is possible to aggregate the individual item service (waiting time) functions in several ways. Here, we employ a demand-weighted average service function as used in Cohen et al. (1992) and Thonemann et al. (2002). For a given stocking decision, the average waiting time per request for a ready-for-use part at company j can be expressed as: Wj ¼

I X

Prob fan arbitrary failing part at location j is of item ig

i¼1

 ðaverage waiting time for a ready-for-use part of item iÞ ¼ ¼

I X  mij
aij ET tri þ hi ET em : i Mj i¼1

I X  mij
bij 0 þ aij ET tri þ hi ET em i Mj i¼1

ð1Þ

With this notation, we can formulate our problem as follows: Problem P0: Minimize

2 X I X
j¼1

subject to

chi S ij þ ctri mij aij þ cem i mij hi



ð2Þ

i¼1

I X  mij
aij ET tri þ hi ET em ; 6 W max i j M j i¼1

j ¼ 1; 2:

ð3Þ

The constraints in (3) can be rewritten as I X


 mij aij ET tri þ hi ET em ; 6 M j W max i j

j ¼ 1; 2:

ð4Þ

i¼1

Problem P0 is an integer-programming problem with a non-linear objective function and non-linear constraints. Remark. Finding approximations for the fractions of demand satisfied by stock on hand, by lateral transshipments, and by emergency supply has been the focus of previous research; see Lee (1987), Axsa¨ter (1990), Sherbrooke (1992), Yanagi and Sasaki (1992), Alfredsson and Verrijdt (1999), Grahovac and Chakravarty (2001), Kukreja et al. (2001). Since we want to derive exact bounds for our model, we will work with exact expressions for these fractions instead of approximations.

3. Solution procedure Consider problem P0 with the constraints in (3) replaced by the constraints in (4). This problem is appropriate to be analyzed by Lagrange relaxation (see Porteus, 2002, Appendix B, for an excellent description of this technique applied to general constrained optimization problems). Both the objective function and the lefthand sides of the constraints in (4) are separable in terms per item. The constraints may be interpreted as budget constraints. At company j, j = 1,2, there is in total a budget M j W max available for delays. Under j Lagrange relaxation, an (artificial) price kj is attached to the use of delays at company j, j = 1,2, and corresponding cost terms are added to the objective function. These prices kj are the Lagrange parameters, and they can be tuned such that the usage of delays is close to the available budgets. In Section 3.1, we formulate the Lagrange relaxation. For each pair of given Lagrange parameters, a lower bound is obtained for the optimal value of problem P0. In Section 3.2, we show that, for given Lagrange parameters, the Lagrange relaxation decomposes into single-item problems, and we derive an exact

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solution procedure for these single-item problems. The best possible lower bound is obtained by optimization of the Lagrange parameters, for which a subgradient method is described in Section 3.3. Finally, in Section 3.4, we describe how a good heuristic solution is obtained for Problem P0. 3.1. The relaxed problem For a given vector k 2 R2 with kj P 0, j = 1,2, we formulate the following problem P1 that is obtained from problem P0 by relaxing the waiting time constraints: Problem P1: Minimize

2 X I X
j¼1

i¼1

chi S ij

þ

ctri mij aij

þ

cem i mij hi



þ

2 X

kj

j¼1

I X




mij aij ET tri

þ

hi ET em i



!
M jW

max j

:

ð5Þ

i¼1

Let Z P 0 denote the optimal cost of problem P0, let Z P 1 (k) denote the optimal cost of problem P1 for given k = (k1, k2), and let Z P 0 (S1, S2) denote the costs of problem P0 under a basestock policy (S1, S2). We define Wj(S1, S2) as the average waiting time for company j obtained under the stocking policy (S1, S2). Property 1 (i) Z P 0 P Z P 1 ðkÞ for all (k1, k2) P (0, 0). (ii) Z P 0 P Maxk Z P 1 ðkÞ. (iii) If for some k = (k1, k2) P (0, 0) the optimal solution for problem P1 is ðS 1 ; S 2 Þ and W j ðS 1 ; S 2 Þ 6 W max , j   j = 1,2, then ðS 1 ; S 2 Þ is feasible for problem P0, and Z P 0 ðS 1 ; S 2 Þ
Z P 0 6 k1 ðW max
W ðS ; S ÞÞ þ 1 1 2 1 k2 ðW max
W 2 ðS 1 ; S 2 ÞÞ. 2 (iv) If for some k = (k1, k2) P (0, 0) the optimal solution for problem P1 is ðS 1 ; S 2 Þ and for j = 1,2, W j ðS 1 ; S 2 Þ ¼ W max if kj > 0; W j ðS 1 ; S 2 Þ 6 W max if kj = 0, then ðS 1 ; S 2 Þ is the optimal stocking policy j j for problem P0. Proof (i) An optimal solution for problem P0 is also a feasible solution for problem P1 for any (k1, k2) P (0, 0). The cost of this solution for problem P1 is smaller than or equal to the cost of this solution for problem P0. The cost of an optimal solution of problem P1 in its turn is smaller than or equal to the cost of any feasible solution. (ii) This follows from (i). (iii) The first part follows from the problem definition in P0. The second part follows directly from (i). (iv) This follows immediately from (iii). This property also follows from EverettÕs result for general constrained optimization problems; see Everett (1963). h By Property 1(i), for any (k1, k2) we obtain a lower bound on Z P 0 . The maximum value of this lower bound over all considered values (k1, k2) is the best obtained lower bound (Property 1(ii)). Besides a lower bound, Property 1 provides us with an upper bound for the distance between the total costs of any feasible solution and the optimal solution (Property 1(iii)). In the next subsection, we describe how to solve problem P1 for given values of the Lagrange multipliers. 3.2. Determining an optimal solution of the relaxed problem We now describe a procedure for determining an optimal solution of the relaxed problem for given values of Lagrange multipliers. Notice that the optimality is required to ensure that the objective function of

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the relaxed problem forms a lower bound on the optimal objective function of the original problem. The objective function of the relaxed problem P1 may be rewritten as Minimize

I X 2 X
i¼1

2

  X em m
chi S ij þ ctri þ kj ET tri mij aij þ cem þ k ET h kj M j W max : j ij i i i j

j¼1

ð6Þ

j¼1

P2 Since the kj values are assumed to be given, the second factor
j¼1 kj M j W max is a constant and thus can be j ignored for optimization purposes. Thus, we are left with I independent single-item optimization problems. For each item i the optimization problem can be stated as follows:


 em mi1 hi þ chi S i2 þ ctri þ k2 ET tri mi2 ai2 Minimize chi S i1 þ ctri þ k1 ET tri mi1 ai1 þ cem i þ k1 ET i
 em mi2 hi : þ cem i þ k2 ET i

ð7Þ

Recall that ai1 and ai2 depend on the individual stock Si1 and Si2, while hi depends on the aggregate stock level Si. We can rewrite the objective function as follows: Minimize f ðS i Þ þ gðS i1 ; S i2 Þ;

ð8Þ

where


 em em em f ðS i Þ ¼ chi S i þ cem hi ; i mi1 þ k1 mi1 ET i þ ci mi2 þ k2 mi2 ET i


 tr

gðS i1 ; S i2 Þ ¼ ctri þ k1 ET i




 mi1 ai1 þ ctri þ k2 ET tri mi2 ai2 :

ð9Þ ð10Þ

It is known that hi is decreasing and convex as a function of Si (see Kranenburg and Van Houtum, 2004, Appendix A, for a proof, based on a result by Dowdy et al., 1984). Since the inventory holding cost increases linearly with Si, f(Si) is thus convex and either increasing or first decreasing and then increasing in Si. With regard to the function g(Si1,Si2), for each value of Si there exist (Si1,Si2) with Si1 + Si2 = Si such that g(Si1, Si2) is minimized. Let us define g ðS i Þ ¼ MinS i1 ;S i2 fgðS i1 ; S i2 ÞjS i1 þ S i2 ¼ S i g. Intuitively, g*(Si) will be first increasing and then decreasing in Si. When Si = 0, no lateral transshipments occur and g*(Si) = 0. Then, increasing Si will increase g*(Si). When Si is large enough, the need for a lateral transshipment diminishes and hence, increasing Si will then decrease g*(Si). Solving the problem in (8)–(10) is equivalent to minimizing f(Si) + g*(Si). The latter problem is solved as follows. We evaluate the costs f(Si) + g*(Si) for increasing values of Si. We start with Si = 0 and then increase Si incrementally by one. For each Si, we calculate the cost f(Si) + g*(Si) and keep track of the best solution obtained so far (g*(Si) is determined by evaluating all possible stock allocations (Si1,Si2) with Si1 + Si2 = Si). If we arrive at an Si for which f(Si) is larger than or equal to the minimum costs obtained so far, we may stop the procedure. This condition implies that for such an Si, f(Si) > f(Si
1). Since f(Si) is a convex lower bound function for f(Si) + g*(Si), we can conclude at this point that no better solutions can be obtained for larger values than the current Si. The best solution obtained so far is an optimal solution. The formal description of the optimization procedure is as follows. Optimization algorithm for the single-item problem. Step 1: Step 2: Step 3: Step 4: Step 5:

Initialization: Set Si = 0, Si1 = 0, Si2 = 0, min = f(0), S i1 ¼ S i1 ; S i2 ¼ S i2 . Set Si = Si + 1 and calculate f(Si). If f(Si) P min go to END. Otherwise set k = 0 and continue. Set Si1 = k and Si2 = Si
k; calculate f(Si) and g(Si1,Si2). If f(Si) + g(Si1,Si2) < min, set min = f(Si) + g(Si1,Si2), S i1 ¼ S i1 and S i2 ¼ S i2 . If k < Si, set k = k + 1 and go to Step 3. Otherwise go to Step 2. END

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3.3. Finding the tightest lower bound Once the Lagrangian problem P1 is solved for a given k, we know that the objective function ZP1(k) is a lower bound on the optimal cost of the original problem P0. The Lagrange multipliers giving the tightest lower bound are denoted by k ¼ ðk1 ; k2 Þ. The value of the best possible bound that can be obtained using Lagrangian relaxation is then given by ZP1(k*) where Z P 1 ðk Þ ¼ Max Z P 1 ðkÞ:

ð11Þ

kP0

The next step is to find the best Lagrange multipliers k1 and k2 . Since ZP1(k) is not differentiable in general, methods like steepest ascent, which depend on the gradient directions, are not applicable. The subgradient optimization method is usually used instead. It can be viewed as a generalization of the steepest ascent method in which the gradient direction is substituted by a subgradient-based direction (see e.g. Bazaraa et al., 1993). The subgradient optimization is an iterative procedure that has been effective in producing good multiplier values in a variety of Lagrangian-based optimization problems (see Fisher, 1985). We use this method also for solving our problem defined in (11). 3.3.1. Overview of the subgradient optimization method Let W kj , j = 1,2, be the expected waiting times that correspond to an optimal solution of the Lagrangian problem for the current vector of multipliers kk at iteration k of the subgradient method. Then the subgradient direction ck ¼ ðck1 ; ck2 Þ for kk is calculated as ckj ¼ W kj
W max j

for j ¼ 1; 2:

ð12Þ

At each iteration k, the vector of Lagrange multipliers for iteration (k + 1) is updated as   k k k ¼ max 0; k
t c kkþ1 j j j

for j ¼ 1; 2:

ð13Þ

In this formula, tk is a scalar stepsize and calculated from tk ¼ sk

b Z P 1 ðkk Þ
Z kck k22

;

ð14Þ

which is a stepsize updating procedure that has been used for widespread practical applications. Justificab is the objective function value of the tion for this formula is given in Held et al. (1974). In this formula, Z best known feasible solution of problem P0 (the best upper bound so far) and sk is a scalar chosen between 0 and 2. The value of sk is halved whenever there is no improvement in the value of the Lagrangian solution after a specified number of iterations. Usually k0 = 0 is the most natural choice as an initial solution. Here, we develop a simple bisectionbased procedure to obtain an initial solution that is expected to be close to the optimal solution k*. We describe the initialization procedure in the following part. 3.3.2. Initialization procedure From Property 1(iii), we know that the optimal vector k* will be found at iteration k where the subgrak max * dient direction (W k1
W max 1 ; W 2
W 2 ) is close to 0. Our estimation of the location of k is based on the following property. Property 2. Let ðS 1 ; S 2 Þk1 ;k2 denote the optimal solution of problem P1 at penalty values k1 and k2, and W j ðS 1 ; S 2 Þk1 ;k2 , j = 1,2, the corresponding achieved average waiting times.

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(i) For k01 > k1, W 1 ðS 1 ; S 2 Þk1 0;k2 6 W 1 ðS 1 ; S 2 Þk1 ;k2 . (ii) For k02 > k2, W 2 ðS 1 ; S 2 Þk1 ;k2 0 6 W 2 ðS 1 ; S 2 Þk1 ;k2 . Proof. See Theorem 2, Van Houtum and Zijm (2000).

h

For each value of k2 there exists a smallest value of k1, say k1 min, at which W max is satisfied. These points 1 form a function in k2. Similarly, we can have k2 min as a function of k1. From Property 2, it follows that of all the points (k1, k2) that give feasible solutions for problem P0, the point where the functions k1 min(k2) and k2 min(k1) cross each other (if they do cross) is the best estimate for k*. If both functions do not cross, we would expect that the location of k* is close to the point (k1, k2) where both functions come close to each other. Let us define (^ k1 min ; ^ k2 min ) to represent such point. We performed a computational experiment to learn about the behavior of the functions k1 min(k2), k2 min(k1), and the locations of (^ k1 min ; ^ k2 min ). Interesting results were obtained from this experiment and several possible situations are depicted in Fig. 2(a)–(f). For each situation, we plotted both functions k1 min(k2) and k2 min(k1) and also (^ k1 min ; ^ k2 min ). In each figure, (^k1 min ; ^k2 min ) is represented by a small circle. As expected, we found that in general, k1 min is decreasing in k2, and k2 min is decreasing in k1. The locations of (^ k1 min ; ^ k2 min ) can be classified as follows: (a) ^ k1 min  ^ k2 min (see Fig. 2(a) and (b)): this occurs when the two companies are identical so that the function k1 min(k2) and k2 min(k1) are symmetrical to each other. If the aggregate stocks are even numbers, both companies will have precisely the same number of stocks and both functions k1 min(k2) and k2 min(k1) lie on the same line as shown by Fig. 2(a).

λ2

λ2

λ2 λ 1min

λ 1min

λ 1min

(a)

λ 2min

λ 2min

λ 2min

λ1

λ1

(b)

λ2

λ2 λ 2m in

λ2 λ 1min

λ 1min

λ 1m in

(d)

λ 2min

λ 2min λ

1

λ1

(c)

(e)

λ1

Fig. 2. Some possibilities of k1 min(k2), k2 min(k1) and (^k1 min ; ^k2 min ).

(f)

λ1

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^1 min < ^ (b) k k2 min (see Fig. 2(c) and (d)): this occurs in the two following cases: (i) mi1 > mi2 for all i and max W max ¼ W max > W max 1 2 ; (ii) mi1 P mi2 for all i and W 1 2 . In those two cases, if ‘ is as an arbitrary value of Lagrange multiplier, we will find that k1 min(k2 = ‘) < k2 min(k1 = ‘). In the extreme case, we may find the situation where the first constraint is inactive and ^k1 min ¼ 0. (c) ^ k1 min > ^ k2 min (see Fig. 2(e) and (f)): this occurs in the two following cases: (i) mi1 < mi2 and max W max ¼ W max < W max 1 2 ; (ii) mi1 6 mi2 and W 1 2 . By symmetry, the same explanation as under (b) applies here too. Based on these characteristics, we now describe our initialization procedure. The main idea here is to obtain an initial solution which is expected to be close to the optimal solution of problem (13). In general, we do so by examining three points of (k1, k2). For the first point, k1 is set to zero, and we determine smallest value k2 for which both constraints are satisfied. For the second point, we have to find the smallest value k, where k1 = k2 = k, such that both constraints are satisfied. And for the third point, similar to the first point, we set k2 equal to zero and we determine smallest value k1 for which both constraints are satisfied. The point with the largest objective function value ZP1(k) is then selected as the initial solution for the subgradient optimization procedure. Those three points can be easily determined by applying a standard bisection method. Since the corresponding optimal solution to the selected point is feasible in problem P0, the total b in the subgradient method. costs function ZP0 of this solution is used as the initial value for Z 3.4. Finding the best upper bound In this section we describe the procedure to obtain a heuristic solution for the original problem P0 which provides an upper bound on the optimal total cost. In principal, we apply a greedy heuristic which consists of a first main step in which a feasible solution is generated, and a second step in which the solution is (slightly) improved by local search. 3.4.1. Obtaining a good feasible solution The subgradient method for obtaining the best lower bound can also be used to generate a good feasible solution of this heuristic. During the execution of the subgradient method, for each solution (S1, S2) that is feasible for problem P0, we evaluate the cost ZP0(S1, S2) and we keep track of the best solution obtained so far. If the final solution of the subgradient method is feasible, we may proceed with the local search procedure. Otherwise, we first apply the following greedy procedure which may give a better feasible solution than obtained so far. This procedure is iterative. At each iteration, we add one unit of stock for the item and the location that gives the largest decrease in distance to the set of feasible solutions per extra unit of cost. Let eij ¼ ð0; . . . ; 0; 1; 0; . . . ; 0Þ denote a stocking policy with one unit of stock for item i and location j and zero stocks for all otherP items and locations. We define for each solution (S1, S2) the distance to the set of 2 max þ feasible solutions as Þ where (a)+ = max(0,a). For each combination of j¼1 ðW j ðS 1 ; S 2 Þ
W j DW i i 2 {1,. . .,I} and j 2 {1,2}, we calculate the ratio rij ¼ DZ ij where: j J J X X þ i max þ ðW j ðS 1 ; S 2 Þ
W j Þ
ðW j ððS 1 ; S 2 Þ þ eij Þ
W max Þ ; DW j ¼ j j¼1

DZ ij

P0

¼ Z ððS 1 ; S 2 Þ þ

j¼1

eij Þ

P0


Z ðS 1 ; S 2 Þ:

One unit of stock is then added for the combination of item and location with the highest ratio. This procedure is repeated until a feasible solution is obtained. (Remark: although we have positive DZ ij in most cases, it is also possible to have negative DZ ij . In that case, the combination of item and location with negative DZ ij should receive a higher priority. If there are several combinations, we can select the combination giving the maximum decrease in distance).

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3.4.2. Local search Under the solution obtained by the above-described procedure, the average waiting time for one or both locations may be somewhat lower than the target levels. Hence the solution may be improved by a local search. At each iteration step feasible neighboring solutions are considered and the cheapest one among them is selected as next solution. A solution is defined to be a neighbor if it is obtained by adding or removing one part, or by replacing one part at one location by the same or another part at the same or another location. 4. Computational experiment In this section we present and discuss our numerical findings. Our main focus of inquiry will span: the tightness of the bounds; the extent that lateral transshipments and emergency supplies are applied; the economic benefits gained by applying a system approach and a pooling policy. Table 2 shows all parameter values for the experiment. All parameter values were selected such that they are realistic for real-life situations, at least for the airline industry. In our experiment the ratios of demand and repair rates for each item were generated randomly from a uniform distribution (notice that only these ratios matter for our problem, and not the precise values for mi1, mi2 and li). Two uniform distributions were used representing two different variability levels of this ratio across items. From the first distribution, we could have the situation where the maximum value is five times the minimum value while from the second distribution, the ratio of 5 is increased to 30. The same value has been taken for the repair rates of all items, that is, li = 0.03 unit/day. Similarly, the values of the inventory holding costs were generated in the same way. We focus on systems where the inventory holding cost is dominant in comparison to the lateral transshipment or emergency supply cost. In this experiment the average inventory holding cost is set at 40 times as much as the lateral transshipment cost. Further, the ratio of the emergency supply cost to the lateral transshipment cost is set at two levels (5 and 10). With regard to the average maximum waiting time, two levels (0.1 and 0.25 days) were evaluated and equal values were used by the two locations. The lateral transshipment lead-time and the emergency supply lead-time depend of course on the distance between the two companies and the location of the source of emergency supplies. We used two levels for the lateral transshipment lead-time (0.1 and 0.25 days) while we fix the emergency supply lead-time at two days. It might also be of interest to see the effect of the size difference of the companies (in their demands). Two levels (1 and 3) were used for the ratio of demand between company 1 and company 2. With all combinations of the parameters described in Table 2, we have in total 3 · 2 · 1 · 2 · 2 · 1 · 2 · 2 · 2 = 192 settings. We generated 20 instances per setting, which means 3840 instances were evaluated in this experiment. For each instance, we computed and recorded the following performance measures: Table 2 Parameter values for the computational experiment Name of the parameter Number of items (I) Inventory holding costs (chi ) Transshipment costs (ctr i ) Emergency supply costs (cem i ) Lateral transshipment lead time (ET tr i ) Emergency supply lead time (ET em i ) Maximum waiting time (W max ¼ W max 2 )  1 Demand rates mi1 þ mi2 Repair rates li   Demand rates at company 1 mi1 Demand rates at company 2 mi2

Unit

Number of values

Values

$/unit/year

3 2

$ $ days days days

1 2 2 1 2

20, 50, 100 U[5000,15000], U[1000,19000] 250 1250, 2500 0.1, 0.25 2 0.25, 0.1

2

U[0.5,2.5], U[0.1,3.0]

2

1, 3

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• The relative gap between the upper and lower bounds (%GAP): %GAP ¼

upper bound
lower bound  100: lower bound

• The average fraction of demand satisfied by lateral transshipments for both locations (
a1 and a
2 ) and the average fraction of demand satisfied by emergency supplies (
h). Each of these measures is calculated PI mij as a
demand-weighted average of the individual fraction values of all items. Thus we have a ¼ j i¼1 M j aij for PI i2 j = 1,2 and
h ¼ i¼1 mMi11 þm h. þM 2 i • The relative cost savings obtained when a pooling policy is applied instead of a Ôno-poolingÕ policy (%SAVE1). These savings are obtained by comparing the cost of the Ôno-pooling with item-approachÕ policy to the cost of the Ôpooling with item-approachÕ policy: %SAVE1 ¼

costðno-pooling with item-approachÞ
costðpooling with item-approachÞ  100: costðno-pooling with item-approachÞ

The cost of the Ôno-pooling with item-approachÕ policy is calculated by treating each location and each item independently (and with W max as target average waiting time per item). Notice that emergency supplies are j still applied under the no-pooling policy. The cost of the Ôpooling with item approachÕ is obtained by treating each item independently and for each item we solve an optimization problem of a two-location pooling system. • The relative cost savings gained when a system approach is applied instead of an item approach (%SAVE2). These savings are obtained by comparing the cost of the Ôno-pooling with item-approachÕ policy to the cost of the Ôno-pooling with system-approachÕ policy: %SAVE2 ¼

costðno-pooling with item-approachÞ
costðno-pooling with system-approachÞ  100: costðno-pooling with item-approachÞ

To calculate the cost of the Ôno-pooling with system-approachÕ, we treated each location independently. For each location, a similar technique using Lagrangian relaxation is used to solve the multi-item optimization problem. • The relative cost savings gained when applying the Ôpooling with system-approachÕ policy instead of the Ôno-pooling with item-approachÕ policy (%SAVE3): %SAVE3 ¼

costðno-pooling with item-approachÞ
costðpooling with system-approachÞ  100: costðno-pooling with item
approachÞ

The results of our experiment are presented in Table 3(a)–(h). Each table records the average values for: the total cost of the heuristic solution, the fraction of demand satisfied by lateral transshipments for both locations (
a1 and
a2 ), the fraction of demand satisfied by emergency supplies (
h), %GAP, %SAVE1, %SAVE2 and %SAVE3. To get some insights into the robustness of the proposed solution methodology, for the 20 instances per setting we have also computed the sample standard deviation of the %GAP and the maximum value of the %GAP. In the tables, we have listed the averages of these standard deviations and the maximums over all maximum %GAPÕs per setting. In Table 3(a)–(g) we show how the performance measures are influenced by the system parameters. With an exception for the average standard deviation of the %GAP, all average values reported in Table 3(a)–(f) are calculated based on 1920 instances. The average standard deviation, however, is calculated based on 96 sample standard deviations. Different from Table 3(a)–(f), Table 3(g) reports the performance measures for three levels of parameter I (number of items) instead of two levels. So in this table, the average standard deviation of the %GAP is calculated

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Table 3 Summary of the computational results (a) Performance measures with respect to

mi li

Total costs
a2 ;
h a1 ;
%GAP (average) %GAP (st dev; max) %SAVE1 %SAVE2 %SAVE3 (b) Performance measures with respect to

mi1 mi2

Total costs
a2 ;
h a1 ;
%GAP (average) %GAP (st dev; max) %SAVE1 %SAVE2 %SAVE3

U[0.5, 2.5]

U[0.1, 3.0]

3,450,837 0.11/0.13/0.08 0.33 0.22/1.65 20.39 11.92 26.84

3,462,899 0.11/0.13/0.08 0.29 0.28/1.51 20.01 12.10 27.07

1

3

3,453,442 0.12/0.12/0.08 0.31 0.25/1.14 20.60 11.88 27.33

3,460,294 0.10/0.14/0.08 0.32 0.25/1.65 19.80 12.14 26.58

U[5000, 15000]

U[1000, 19000]

3,505,595 0.11/0.13/0.08 0.32 0.28/1.46 20.18 10.68 25.87

3,408,141 0.11/0.13/0.08 0.31 0.22/1.65 20.22 13.34 28.04

(c) Performance measures with respect to chi Total costs
a2 ;
h a1 ;
%GAP (average) %GAP (st dev; max) %SAVE1 %SAVE2 %SAVE3 (d) Performance measures with respect to

cem i ctr i

Total costs
a1 ;
a2 ;
h %GAP (average) %GAP (st dev; max) %SAVE1 %SAVE2 %SAVE3

5

10

3,372,093 0.12/0.12/0.09 0.38 0.24/1.65 20.40 12.68 27.46

3,541,643 0.11/0.13/0.07 0.24 0.26/1.51 20.00 11.34 26.45

0.1

0.25

3,420,014 0.11/0.13/0.08 0.30 0.23/1.65 21.15 12.08 27.66

3,493,722 0.12/0.12/0.08 0.32 0.27/1.14 19.25 11.94 26.25

(e) Performance measures with respect to ET tr I Total costs
a2 ;
h a1 ;
%GAP (average) %GAP (st dev; max) %SAVE1 %SAVE2 %SAVE3 (f) Performance measures with respect to Wmax 0.1 Total costs
a1 ;
a2 ;
h

3,683,292 0.09/0.10/0.04

0.25 3,230,444 0.13/0.15/0.11 (continued on next page)

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Table 3 (continued)

%GAP (average) %GAP (st dev; max) %SAVE1 %SAVE2 %SAVE3

0.1

0.25

0.35 0.25/1.46 21.23 11.96 27.86

0.27 0.25/1.65 19.17 12.06 26.05

(g) Performance measures with respect to I Total costs
h a1 ;
a2 ;
%GAP (average) %GAP (st dev; max) %SAVE1 %SAVE2 %SAVE3 (h) Average results for all instances Total costs
h a1 ;
a2 ;
%GAP (average) %GAP (st dev; max) %SAVE1 %SAVE2 %SAVE3

20

50

100

1,225,521 0.11/0.13/0.08 0.36 0.29/1.46 20.21 11.81 26.93

3,024,790 0.11/0.13/0.08 0.30 0.27/1.65 20.15 11.93 26.87

6,120,293 0.11/0.13/0.08 0.28 0.20/1.14 20.24 12.29 27.06 3,456,868 0.11/0.13/0.08 0.31 0.25/1.65 20.20 12.01 26.95

based on 64 sample standard deviations while all other values are calculated based on 1280 instances. We also computed the overall average values, which are reported in Table 3(h). The main observations drawn from this experiment are summarized as follows: (i) The total cost values increase almost linearly with the number of items as shown in Table 3(g). The ratio of the emergency supply and lateral transshipment costs, the lateral transshipment lead time, and the maximum waiting time are factors that also influence the magnitude of the total cost while the effects of other parameters are not significant. To have a description about the contribution of each type of cost to the total cost, for each instance we calculated the percentage of inventory holding cost, lateral transshipment cost and emergency supply cost. The average percentages for the three types of costs respectively are 90.8%, 1.7%, and 8.1%. As expected, the total cost is dominated by the inventory holding cost. For settings in this experiment, we observed that the magnitudes of the aggregate stock levels (for two locations) for each item range from zero to eight units. A very small percentage of the lateral transshipment cost indicates the potential of lateral transshipments in improving the system performance without requiring any significant additional costs. (ii) The average fraction of demand satisfied by lateral transshipments are 0.11 and 0.13 for each of the two companies while the average fraction of demand satisfied by emergency supplies is 0.08. The size difference of the two companies influences the fraction of demand satisfied by lateral transshipments at both companies. Table 3(b) shows that both companies have equal fractions when their demands are identical. When their demands are relatively different, the company with a higher demand has a smaller fraction. This is reasonable since more stocks tend to be located at the company with a higher demand. As a result, the company with a lower demand will ask for more lateral transshipments. The target maximum waiting time is a significant parameter influencing the size of all fractions. As shown in Table 3(f), all fractions are lower when the target maximum waiting time is decreased. Decreasing the target maximum waiting time results in higher stock levels thereby reducing the fraction of demand satisfied by emergency supplies. As previously

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explained, by increasing the stock level (starting from zero), the fraction of demand satisfied by lateral transshipments will first increase but then turn to decrease when the system has reached a relatively high service level. The choice of the target maximum waiting time in our experiment has put the system to achieve high service levels as indicated by low values of
h. Therefore, Increasing stock levels also results in the decrease of both
a1 and
a2 . (iii) The proposed solution approach performs quite satisfactorily as indicated by very small gaps between the lower and upper bounds. The overall average of the %GAP is 0.31% with an average sample standard deviation of 0.25% and a maximum value of 1.65%. We also observed that the %GAP tends to decrease when the number of items is larger. Such behavior is common for many optimization problems with integer decision variables, e.g. Knapsack problems, where the heuristic can provide a reasonable approximation if the number of items is large enough. The %GAP is rather insensitive for the other parameters. (iv) This experiment shows that allowing lateral transshipments between two locations can significantly reduce the total cost. An average savings (%SAVE1) of 20.20% is obtained when we move from the Ôno pooling with item approachÕ policy to the Ôpooling with item approachÕ policy. We observed that these savings are sensitive to two parameters. The first parameter is the lateral transshipment lead-time. Intuitively, as the lateral transshipment lead-time increases, the savings resulting from pooling diminish as more stocks would be needed to satisfy the maximum waiting time constraints. The second parameter is the average maximum waiting time. The system with low maximum waiting time constraints (Wmax = 0.1) obtains higher savings than the system with higher waiting time constraints (Wmax = 0.25). This shows that pooling becomes more attractive when the cooperating companies set higher target service levels. (v) The relative savings of moving from an item approach to a system approach (%SAVE2) are also significant. The overall average of these savings is 12.01%. It is also shown that the variability of the inventory holding costs across items has a significant impact on the %SAVE2 as indicated in Table 3(c). The savings increase when the variability of the inventory holding costs is higher. This result is in line with the findings of Thonemann et al. (2002). From this experiment we observed that the savings gained by allowing lateral transshipments (%SAVE1) are higher than the savings gained by applying a system approach instead of an item approach (%SAVE2). However, it is notable to mention here that the magnitude the %SAVE2 could be much higher in different settings where a greater variability of the holding costs is observed. Thonemann et al. (2002), for example, indicated much higher benefits of a system approach (savings up to 25%) for settings with a high skewness of the inventory holding costs. (vi) This experiment also show the importance of applying both lateral transshipments and the system approach in making inventory decisions. The overall average of the third type of savings (%SAVE3) measured by comparing the total cost of the Ôitem approach with no poolingÕ policy with the Ôsystem approach with poolingÕ policy is 26.95%, which is higher than %SAVE1 and %SAVE2. It is also shown that these savings are sensitive to those parameters influencing the %SAVE1 (the lateral transshipment lead-time and the average maximum waiting time) and the %SAVE2 (the variability of the holding costs). The average computation times for solving the problems in Table 3 are 2.8, 7.6 and 14.9 minutes for I = 20, 50 and 100 respectively, using a PC with a 333-MHz Pentium II processor.

5. Model application As already mentioned, this research was originally motivated by an air carrier company located in Brussels who wanted to cooperate with another company to pool their spare parts inventories. We performed a pilot study to help management of the company to have an idea on the cost advantages of using a system approach and a pooling strategy. A potential partner considered at that time was a company located in Liege which is approximately at two hours driving distance from Brussels. However, since complete

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information of the partner company was not available, we assumed that both companies are identical. We selected a sample of 32 expensive parts of which the prices are at least €25,000. The average maximum waiting time, as desired by the company, was set at two hours (1/12 days). The yearly inventory holding costs are approximated at 20% of the unit prices. The unit transportation cost was set at €50 per hour. Emergency supplies are currently applied by the company in case of stock-outs. The emergency supply lead-time was set at the average level of one day with the cost of €500. By allowing lateral transshipments between the two companies, emergency supplies will only take place when both companies are out of stock.

Table 4 Data and solutions for the model application Part #

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

Data

Policy

Part title

mi (day
1)

li (day
1)

Price (€)

Current

Item, no pooling

Item, pooling

System, no pooling

System, pooling

Flap elec. control Flap hydr. control Fuel control Autopilot computer Engine display APU generator gearbox Cold air unit Multi process unit Actuator flap nacelle Pitch computer Radar Roll computer Rudder servo TMS computer Generator Flap actuator outboard Roll splr. servo Cabin pressure control Instr. switching unit Prop. control unit Distance bearing Servo altimeter Speed brake hydr. act. Navigation selector Yaw actuator Display processor unit Flap indicator switch Flap screw jack EFIS display TMS display Discharge valve Air speed indicator

0.0229 0.0029 0.0143 0.0343 0.0114 0.0029 0.0029 0.0400 0.0029 0.0171 0.0086 0.0057 0.0057 0.0400 0.0457 0.0457 0.0114 0.0086 0.0257 0.0171 0.0143 0.0400 0.0086 0.0171 0.0114 0.0886 0.0029 0.0171 0.0286 0.0200 0.0086 0.0143

0.0141 0.0204 0.0263 0.0303 0.0417 0.0084 0.0119 0.0500 0.0217 0.0294 0.0345 0.0185 0.0167 0.0147 0.0164 0.0357 0.0139 0.0159 0.0500 0.0125 0.0130 0.0278 0.0417 0.0370 0.0167 0.0476 0.0213 0.0500 0.0196 0.0167 0.0137 0.0192

143,450 113,625 94,575 89,900 74,875 55,750 50,292 47,775 45750 45,325 44,100 43,350 41,325 40,225 37,550 34,800 33,925 32,475 31,425 30,750 30,025 29,000 28,700 28,325 27,625 27,425 27,375 27,050 26,075 25,650 25,250 25,000

3 2 1 4 2 1 1 3 3 1 3 1 2 3 4 4 2 3 1 3 3 4 1 3 3 6 1 2 4 2 2 4

3 2 2 3 1 1 2 3 2 3 2 1 2 6 6 4 3 3 2 3 3 4 2 2 3 4 2 2 4 4 2 3

2, 1, 2, 2, 1, 1, 2, 2, 1, 2, 1, 1, 1, 5, 4, 3, 2, 2, 2, 3, 3, 3, 1, 1, 2, 4, 1, 1, 3, 3, 2, 2,

2 0 1 3 1 0 1 3 1 2 1 1 1 6 6 4 2 2 3 3 3 4 2 2 2 6 1 2 4 4 2 3

1, 0, 1, 2, 1, 0, 1, 3, 1, 1, 1, 1, 1, 4, 4, 3, 2, 1, 2, 2, 3, 4, 1, 2, 2, 5, 1, 2, 3, 3, 2, 2,

Total costs
a1 ;
a2
h %Savings

3 1 1 3 1 1 1 2 1 2 2 1 2 4 5 3 2 2 2 3 2 3 1 2 2 3 1 2 3 2 2 2

1,467,400 1,553,100 1,170,200 1,244,700 – – 0.12, 0.12 – 0.11 0.06 0.05 0.08 Item + no pooling ! item + pooling Item + no pooling ! system + no pooling Item + no pooling ! system + pooling Current ! system + pooling

1 0 1 2 1 0 0 2 0 2 1 1 1 5 5 3 2 2 2 3 2 3 1 1 2 5 1 1 4 3 2 3

965,000 0.11, 0.11 0.07 24.65 19.85 37.86 34.23

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Table 4 shows all data and solutions for this model application. We present the spare parts stocking levels under four policies: (i) no pooling with item-approach, (ii) pooling with item approach, (iii) no-pooling with system approach, and (iv) pooling with system approach. In addition to the spare parts stocking levels of those four policies, we also report the current stocking levels applied at the company (currently, an emergency supply is carried out when the company faces a stock-out; pooling is not applied). Notice that under the no pooling policy, the stocking policy of the two companies are exactly the same. For each of the policies, we present the average fractions of demand satisfied by lateral transshipments (in case pooling is applied) and by emergency supplies. Using the Ôno pooling with item approachÕ policy as the base policy, the relative savings of the other three policies are 24.65%, 19.85%, and 37.86% for %SAVE1, %SAVE2, and %SAVE3 respectively. The total cost of the current stocking policy is lower than the cost of the Ôno pooling with item approachÕ policy. We observe, however, that the current stocking policy leads to a higher average waiting time than allowed (the average waiting time is equal to 0.11 · 1 = 0.11 day, which is more than two hours). A significant savings (34.23%) are obtained if the company applied the proposed stocking policy (pooling with system approach).

6. Conclusions and directions for further research In this paper we considered a multi-item, continuous review model of a two-location inventory system for repairable spare parts in which lateral and emergency shipments can occur in response to stockouts. In the system that we analyzed the failed part at one location is replaced by a ready-for-use spare part that can come either from the local warehouse, or from the other location as a lateral transshipment, or from an emergency supply. Additionally, our formulation addresses the concern for service performance by stating constraints in terms of maximum average waiting time for a demanded ready-for-use spare part. We have developed a solution procedure based on Lagrangian relaxation that provides tight bounds for the optimal total costs. Our computational results give us some important insights into the tightness of the bounds and into the effect of the system parameters on the system performance that might be of interest from a managerial point of view. The main insights are: (1) The quality of our heuristic solution is quite good as indicated by a very small percentage of the gap between the costs of our heuristic solution and the lower bounds on the optimal costs. (2) Significant cost savings are obtained either by applying the pooling policy or by applying the system approach; the maximum savings are realized when both are applied together. (3) The relative cost savings of the pooling policy increase when the lateral transshipment lead-time is shorter and service constraints are tighter. (4) The relative cost savings of using the system approach increase when the variability of the inventory holding costs across items increases. Our work can be extended in several directions. One possible extension is to consider more than two, say N companies. The main difficulty is defining a policy for lateral transshipments when there may be more than one company that can be the source of the lateral transshipments. A very reasonable policy is to source the lateral transshipment from the closest neighbor company. In fact, under such pre-specified policy one can apply the same optimization procedure as in this paper. However, as the number of companies grows, the development of a more efficient heuristic is needed and becomes an interesting topic for further research. In the meantime we have developed a greedy heuristic for solving problems that involve several companies (see Wong et al., 2005b). It is shown that besides its simplicity, this heuristic is able to provide close-to-optimal solutions. Another topic that might be of interest is the analysis of the cost or profit sharing problem amongst the participating companies in a pooling group. In settings where all companies are independent (and do not belong to a single parent firm), all companies would also have self interests to maximize their benefits of the pooling. In this case, the competitive behavior of all companies should be considered. Game theoretic models are required to analyze such a problem.

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