A multi-class closed queueing maintenance network model with a parts inventory system

Abstract

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.

Introduction

During the past decades, a substantial number of mathematical models have been developed related to various aspects of managing inventory systems. One of these models, a spares provisioning problem, was designed to determine the stock level of repairable-items such as expensive vehicles, electronic devices, and airplane engines. These repairable items are typically expensive and their individual failure rates usually relatively low. Their proper management, however, is imperative since the investment cost in spares is considerable.

Since Sherbrooke [1] developed a multi-echelon technique for repairable item control (METRIC), the spares provisioning problem for repairable items has been of interests. Muckstadt [2], Graves [3] and Schultz [4] are only a few among the researchers that have investigated the spares provisioning problem for repairable items (see [5 and 6] for surveys).

Most of the previous studies are based on the assumption that repair capacity is infinite. Lau et al. [7] considered a multi-echelon repairable item inventory system with infinite repair capacity and passivation. De Smidt-Destombes et al. [8] presented a spare parts model with cold-standby redundancy on system level under the assumption that a failure is immediately restored. Kranenburg and Van Houtum [9] studied a multi-item, single-stage spare parts inventory model with multiple customer classes. Hill et al. [10] studied a single-item, two-echelon inventory with lost sales and a batch ordering policy. Al-Rifaia and Rossettib [11] and Topan et al. [12] considered two-echelon inventory systems in which the central warehouse operates under a batch ordering policy and the local warehouses implement basestock policy. Due to substantial computing time required for exact analysis, most research has focused on developing approximation methods using queueing theory or Markov models.

For finite repair capacity models, Gross et al. [13] and Madu [14] proposed mathematical models using a closed queueing network which consists of one base and a two-stage repair depot. They proposed a heuristic method to decide the number of repairable items and the capacity of repair depot. Albright and Gupta [15] analyzed a multi-echelon, multi-indentured model that consists of a finite number of identical repairmen. Sleptchenko et al. [16] analyzed the trade-off between inventory levels and repair capacity in a multi-echelon, multi-indentured model. Díaz and Fu [17], Spanjers et al. [18], Jung et al. [19] and Kim et al. [20] developed an approximation method for a multi-item, multi-echelon model with finite repair capacity. Rappold and Van Roo [21] presented an approach to model and solve the joint problem of facility location, inventory allocation, and capacity investment when demand is stochastic. De Smidt-Destombes et al. [22], [23] considered k-out-of-N systems under block replacement sharing limited spares and repair capacity.

In the above-mentioned studies, it is assumed that repair of the failed items does not require parts. Therefore, when an item fails, it is directly sent to a repair depot. While this is a realistic assumption in some cases, in many real cases, parts are needed to repair failed items. Abboud and Daigle [24] introduced a spares provisioning problem incorporating a parts inventory system. They assumed that the system consists of a base, a parts inventory system and a repair depot. When an item in the base fails, the failed item is supplied with a part at the parts inventory system and sent to the repair depot. They assumed that (S−1,S) inventory policy is used at the parts inventory system. For this system, they developed a method using Little's result for the case that the value of S is zero or infinite. In this study, the model of Abboud and Daigle is extended by relaxing the following assumptions: First, while there is only one base in the model of Abboud and Daigle, our model assumes that there are multiple bases and the mean time before failure for items owned by one base differs from that for items owned by another base. Second, in our model we use a continuous review (S,Q) inventory policy. In the (S,Q) inventory policy, each time the accumulated demand for parts reaches level Q, a batch order of size Q is placed so that sum of the on-hand inventory and the on-order amount becomes a target number, S. The (S−1,S) policy is a special case of the (S,Q) inventory policy with Q=1. The advantage of the (S,Q) inventory policy over the (S−1,S) inventory policy is that the ordering cost can be significantly reduced by allowing batch orders. However, due to the characteristic of batch orders, the analysis becomes much more complicated.

The purpose of this paper is to develop a new approximate algorithm for the analysis of the proposed system based on a product-form approximation method [25]. In Section 2, the proposed system is defined and is modelled as a multi-class closed queueing network. The product-form approximation method is adapted to allow for the efficient analysis of the multi-class closed queueing network. 3 Maintenance network analysis algorithm, 4 Analysis of sub-network explain the algorithm in detail. Performance measures are calculated in Section 5. In Section 6, the results obtained from the algorithm are compared with those obtained by simulation and the cost effectiveness of the (S,Q) inventory policy is evaluated with numerical results. Finally, Section 7 concludes the paper.