Modeling lateral transshipments in multiechelon repairable-item inventory systems with finite repair channels

doi:10.1016/S0305-0548(02)00078-3

Computers & Operations Research

Volume 30, Issue 9, August 2003, Pages 1401-1417

https://doi.org/10.1016/S0305-0548(02)00078-3 Get rights and content

Abstract

This article concerns the problem of determining the optimal spare inventory level for a multiechelon repairable-item inventory system, which has several bases and a central depot. When an item fails, the failed item is replaced immediately if a spare is available at the base. If there is no spare item available at the base, it is requested as a lateral transshipment to another base that has a spare in stock. If an item is not available from any of the bases, the item is backordered until a spare becomes available. We propose a model to describe the behavior of the system and develop an algorithm to find the spare inventory level at each base, which minimizes the total expected cost of the system. The algorithm is illustrated using examples of various sizes.

Scope and purpose

Repairable-items refer to components that are very expensive, critically important, and subject to infrequent failure. The navigational computer of a plane or subway cars are typical examples of the repairable-items. In multiechelon repairable-item inventory systems that cover extensive geographical regions, lateral transshipment between bases is often used for improved service levels. However, previous research of the lateral transshipments assumes infinite repair capacity and is less appropriate for most industrial systems, where it is resource constrained. In this paper, we propose a model and an algorithm to find the optimal spare level for the system with finite repair channels.

Introduction

Repairable-items are referred to as components, which are expensive, critically important, and subject to infrequent failures. When they fail, they should be repaired and reused after repair, since they can be fixed cost-effectively. They are common in the military and in a variety of commercial settings. Hydraulic pumps, navigational computers in aircraft, helicopter gearboxes, transportation equipment such as subway cars and buses, and high cost electronics are typical examples of repairable-items. Repairable-items have considerable economical implication, since the cost of the items is very high. They represent more than 20% of the total equipment assets, and amount to 9 billion pounds or 13 billion US dollars in the United Kingdom military, as of 2000 [1]. The US military had 46.4 billion dollars of repairable-item inventory in September of 2000 [2]. Furthermore, they have strategic importance, due to the indispensability of the items. For example, 10% of the military aircrafts are grounded, waiting for a failed repairable-item to be replaced at any point of time.

In multiechelon repairable-item inventory systems that cover extensive geographical regions, the lateral transshipment between bases is commonly practiced to provide improved service level for products. A lateral transshipment is made whenever a demand at a base causes a backorder and a spare on hand at some other base can be transshipped. The use of lateral transshipments has the effect of reducing the time that a demand is backordered. Consequently, the expected level and the cost of total backorders in the system can be reduced.

While the repairable inventory problem has its roots in military applications, it is extremely relevant today, for both the military and commercial sectors. For this reason, many researchers study numerous policies on the optimal stocking of spare inventory or on the variety of methods to estimate the operating characteristics of the system. Researches on a single or multiechelon system include the works of Sherbrooke [3], Gross et al. [4], Albright and Gupta [5], and Kim et al. [6], [7]. For a lateral transshipment, Das [8] develops a periodic review inventory model that consists of only two locations, and transshipment is allowed at pre-specified times during the review period. Hoadley and Heyman [9] consider a one-period multiechelon model that allows lateral transshipment between stocking points at the same echelon level. (For an extensive review of the articles on the various repairable-item systems, refer to [10]).

There are three articles that have immediate bearing on the objective of this research. Lee [11] develops a model to derive an approximation for the expected level of backorders and the quantity of emergency transshipments. His model has two critical limitations though. It allows transshipments between some of the identical bases only. The performance of the procedure seems to be unsatisfactory for the systems with a low to medium service level (fill-rate). Axsäter [12] suggests a method to estimate the same operating characteristics of a similar system. His approach puts more emphasis on modeling the demand at a base correctly and, in contrast to Lee's [11], can be applied to the case of nonidentical bases. He tests the method on a small system with three bases to show an improvement compared to Lee's method. Sherbrooke [13] does a simulation study to estimate expected backorders in a multiechelon system with lateral transshipments.

However, the previous two mathematical models [11], [12] are far simpler than ours in the following two points. First and most importantly, they assume infinite repair capacity. Secondly, failures are not categorized into base and depot repairable and, as a result, base repair centers are excluded. Therefore, the models cannot represent the most important features of the real repairable-item systems.

In this paper, we present a model and an algorithm for a system with two levels of capacitated repairs and one level of stocking. The algorithm is to find an optimal spare inventory level at each base so as to minimize the whole system's operating cost per unit time. In the current state of knowledge, our algorithm is the first realistic optimization method for the problem.

This article is organized as follows. In the next section, the model and the probability distributions of the system are described. In Section 3, the algorithm for the model is detailed. The following section presents the results of computational experiments. We conclude with managerial implications and comments on the extension.

Section snippets

Model description

We consider a system with I bases, a central depot, and a single type of repairable-item. Each base owns its spare inventory and a base repair center. The central depot stocks no spares and only repairs failed items from bases. An infinite number of items is operating at each base. This infinite number of population assumption introduces an error by overestimating the arrival rate of the failed items. However, the low failure rates commonly observed in the items partially justify the

The algorithm

In this section, the algorithm we are proposing is presented.

Step 1:	Verify that the following steady-state conditions are satisfied.
$ρ_{d} =(1−α) ∑ i=1 I Λ_{i} /c_{d} μ_{d} <1 and ρ=α ∑ i=1 I Λ_{i} ∑ i=1 I c_{i} μ_{i} <1.$	If the conditions are met, go to Step 2. Otherwise, stop since the system cannot reach steady-state.
Step 2:	Let β_i=δ_i=0, S=(s₁,s₂,…,s_I)=0 and calculate P(b_i), P(D),P(k_i),P(l_i),P(z_i), respectively.
Step 3:	For each base with h_i/e_i⩽1, set s_i to the value satisfying
$∑ k=1 ∞ p(z_{i} =s_{i} +k)<h_{i} /e_{i} < ∑ k=0 ∞ p(z_{i} =s_{i} +k).$ Step 4:	For each base

Computational experiments

It is not possible to directly compare our algorithm with the previous ones on the lateral transshipments because, as mentioned earlier, we consider a more complicated system. Thus, we illustrate and test the accuracy and the speed of the algorithm using several newly constructed examples. The proposed algorithm is written in C and run on a Pentium III ( $933 MHz$ dual CPU) IBM compatible PC system. The input data is from the actual data values for the repairable-items used on aircraft in the US

Concluding remarks

Lateral transshipments are often used in multiechelon inventory systems to improve service performance. In this paper, we have presented a model for the multiechelon repairable inventory system with emergency lateral transshipments. An algorithm is also given for the determination of the local optimal spare levels in such a system. To the best of our knowledge, our approach is the first prescriptive optimization attempt for the lateral transshipment case with finite repair capacity.

Acknowledgements

This research was supported by the Brain Korea 21 project in 2001. The authors thank two anonymous referees for their helpful comments.

Jong S. Kim is a professor in the department of Industrial Engineering at Hanyang University in South Korea. He received his Doctor of Engineering and M.S. degrees from the University of California at Berkeley. His areas of research include mathematical programming, supply chain, and logistics.

References (15)

J.S Kim et al.
Optimal algorithm to determine the spare inventory level for a repairable-item inventory
Computers and Operations Research
(1996)
V.D.R Guide et al.
Repairable inventory theorymodels and applications
European Journal of Operational Research
(1997)
Spending review 2000, HM Treasury, UK, July...
Supply system inventory report, Department of Defense, USA, September 30,...
C.C Sherbrooke
METRICA multi-echelon technique for recoverable item control
Operations Research
(1968)
D Gross et al.
A network decomposition approach for approximating the steady-state behavior of Markovian multi-echelon repairable item inventory systems
Management Science
(1987)
S.C Albright et al.
Steady-state approximation of a multiechelon multi-Indentured repairable-item inventory system with a single repair facility
Naval Research Logistics
(1993)

There are more references available in the full text version of this article.

Cited by (34)

Two-echelon inventory location model with response time requirement and lateral transshipment
2022, Heliyon
Citation Excerpt :
considered a semi-conductor firm ASML under a reactive LT setting, and shows that incorporating Lateral Transshipments (LTs) resulted to annual savings of up to 50% savings of total service parts inventory cost. The use of LT has been considered in diverse multi-echelon environments in supply chain research. [23], [24], [25], [26], and [4] considered the use of LT in two-echelon inventory problems with continuous (S-1,S) replenishment policy. [6]
This study presents a new model for a two-echelon location-inventory system with response time constraints. This system controls inventory with a (S-1, S) policy and comprises of a finite collection of customers, a finite collection of service facilities and a single plant. This paper's main novelty is the incorporation of lateral transshipment into a two-echelon location-inventory system with response time requirement. By using a continuous-time Markov process approach, we determine expected on-hand inventory level in steady state, expected lateral transshipment level in steady state and expected backorder level in steady state. We utilize these steady state levels to formulate a mixed integer nonlinear programming model which incorporates lateral transshipment into an integrated location-inventory system with response time constraint. The model minimizes the total system cost and simultaneously determines: optimal location and number of service facilities, the optimal assignment of customers and base-stock level. We exploit the model's properties using Lagrange decomposition and we show that the model is convex. The model is tested on a real-world scenario using GAMS and our model returned lower costs following comparisons with a model without lateral transshipment. We also establish that lateral transshipment results to consistency of expected cost with varying response time requirement.
Inventory model for a multi-echelon system with unidirectional lateral transshipment
2016, Expert Systems with Applications
Inventory management constitutes a fundamental decision-making problem, especially in systems which must guarantee high availability levels. In complex multi-echelon networks, in case of shortage at a location, resupplying from a near location on the same echelon rather than from the original supplier at the upper echelon, would be a potential faster and then more profitable policy. A wide interest in literature shows the importance of this policy, i.e. the lateral transhipment (LTR), as a means to reduce the inventory costs. This paper deals with unidirectional LTR, which often represent a reasonable policy in scenarios where backorders have different effects on the system. Based on METRIC, this paper defines a system-approach model for determining the stock levels of repairable items in a complex network, by a genetic algorithm optimization process. The model considers non-zero maintenance time for each item and different skills of maintenance centres, in a multi-echelon single-indenture system, with unidirectional LTR allowed. The expert system developed in the paper assists the decision maker to define the inventory levels for the maintenance sites across the system, taking advantage of the LTR to reduce costs and enhance availability. A case study of a European airline shows the relevance of the developed expert system, also considering its reproducibility to face different industrial contexts.
Approximation analysis of multi-class closed queueing maintenance networks with a parts inventory system and two-phase Coxian time distributions
2014, Computers and Operations Research
Citation Excerpt :
Sleptchenko et al. [14] analyzed the trade-off between inventory levels and repair capacity in a multi-echelon multi-indentured model under the assumption of infinite items operating at each base. Díaz and Fu [15], Spanjers et al. [16], Jung et al. [17] and Kim et al. [13] developed an approximation method for a multi-item multi-echelon model with finite repair capacity and infinite items. Rappold and Van Roo [18] presented an approach to model and solve the joint problem of facility location, inventory allocation, and capacity investment when demands are stochastic, and an infinite number of items are operating at bases.
We consider a maintenance network where a set of bases is supported by a replacement parts inventory system and a centrally located repair depot. The ordering policy for the parts is the (S, Q) inventory policy. We extended the previous results to the network, where processing times at each node follow a two-phase Coxian distribution. The proposed network was modeled as a multi-class closed queueing network with a synchronization station. To make the analysis of the network computationally tractable, we developed a two-phase approximation method. In the first phase of the method, the proposed network was analyzed with the previous algorithm based on a product-form approximation. In the second phase, a sub-network was again analyzed with the procedure of a product-form approximation method such that the state space of the sub-network was reduced. In the analysis of a sub-network, a recursive method was also used to solve balance equations by exploiting the special structure of the Markov chain. The new algorithm provided a good estimation of the performance measures of interest. In addition to being accurate, the new algorithm is simple and converges rapidly.
Cost adjustment for single item pooling models using a dynamic failure rate: A calculation for the aircraft industry
2012, Transportation Research Part E: Logistics and Transportation Review
Citation Excerpt :
The calculation of the expected costs of lateral transshipments CTj (in $) is not a part of this paper, which focuses on a dynamic failure rate. In papers like (Jung et al. (2003), Lee et al. (2007) and Kranenburg and van Houtum (2009)), possible calculations of transshipment costs are presented. In the presented paper, a method to extend existing pooling models to a dynamically computed failure rate is shown.
This paper presents an analytical model for cost estimation in a single-item, multi-hub (S-1,S) inventory policy-pooling model for high-value spare parts in the aviation industry. The model extends existing, static pooling models by implementing a dynamic failure rate, using a maintenance free operating period (MFOP) as a measurement technique to increase availability of aircraft components. The gained results through a dynamic failure rate show significant effects for a reduction of total costs of ownership and achieving a better operational stock planning, which is demonstrated in a numerical application.
Enhanced lateral transshipments in a multi-location inventory system
2012, European Journal of Operational Research
In managing an inventory network, two approaches to the pooling of stock have been proposed. Reactive transshipments respond to shortages at a location by moving inventory from elsewhere within the network, while proactive stock redistribution seeks to minimize the chance of future stockouts. This paper is the first to propose an enhanced reactive approach in which individual transshipments are viewed as an opportunity for proactive stock redistribution. We adopt a quasi-myopic approach to the development of a strongly performing enhanced reactive transshipment policy. In comparison to a purely reactive approach to transshipment, service levels are improved while a reduction in safety stock levels is achieved. The aggregate costs incurred in managing the system are significantly reduced, especially so for large networks. Moreover, an optimal policy is determined for small networks and it is shown that the enhanced reactive policy substantially closes the gap to optimality.
A multi-class closed queueing maintenance network model with a parts inventory system
2011, Computers and Operations Research
The system we address is a maintenance network of repairable items where a set of bases is supported by a centrally located repair depot and a consumable replacement parts inventory system. If an item fails, a replacement part must be obtained at the parts inventory system before the failed item enters the repair depot. The ordering policy for the parts is the (S,Q) inventory policy. An approximation method for this system is developed to obtain performance measures such as steady-state probabilities of the number of items at each site and the expected backorders at the parts inventory system. The proposed system is modelled as a multi-class closed queueing network with a synchronization station and analyzed using a product-form approximation method. Particularly, the product-form approximation method is adapted so that the computational effort on estimating the parameters of the equivalent multi-class network is minimized. In analyzing a sub-network, a recursive method is used to solve balance equations by exploiting the special structure of the Markov chain. Numerical tests show that the approximation method provides fairly good estimation of the performance measures of interests.

View all citing articles on Scopus

Bong R. Jung is a doctoral student in Industrial Engineering at Hanyang University. He is also a major in the Republic of Korea Army. He received his M.S. degree in Industrial Engineering from the University of Missouri at Columbia.

Byung G. Sun is a major in the Republic of Korea Army. He received his M.S. degree in Industrial Management from the Korea Advanced Institute of Science & Technology and his Ph.D. degree from Korea University in South Korea.

Sun E. Ahn is an assistant professor in the department of Industrial Engineering at Hanyang University. He received his Doctoral degree from the University of California at Berkeley and received his M.S. degree in Industrial Engineering from Hanyang University. His areas of research include probability modeling, statistical data analysis, and reliability.

View full text

Modeling lateral transshipments in multiechelon repairable-item inventory systems with finite repair channels

Abstract

Scope and purpose

Introduction

Section snippets

Model description

The algorithm

Computational experiments

Concluding remarks

Acknowledgements

Computers and Operations Research

European Journal of Operational Research

METRICA multi-echelon technique for recoverable item control

Operations Research

A network decomposition approach for approximating the steady-state behavior of Markovian multi-echelon repairable item inventory systems

Management Science

Steady-state approximation of a multiechelon multi-Indentured repairable-item inventory system with a single repair facility

Naval Research Logistics