ArticlePDF Available

Abstract and Figures

We consider a single stock-point for a repairable item facing Markov modulated Poisson demand. Repair of failed parts may be expedited at an additional cost to receive a shorter lead time. Demand that cannot be filled immediately is backordered and penalized. The manager decides on the number of spare repairables to purchase and on the expediting policy. We characterize the optimal expediting policy using a Markov decision process formulation and provide closed-form necessary and sufficient conditions that determine whether the optimal policy is a type of threshold policy or a no-expediting policy. We derive further asymptotic results as demand fluctuates arbitrarily slowly. In this regime, the cost of this system can be written as a weighted average of costs for systems facing Poisson demand. These asymptotics are leveraged to show that approximating Markov modulated Poisson demand by stationary Poisson demand can lead to arbitrarily poor results. We propose two heuristics based on our analytical results, and numerical tests show good performance with an average optimality gap of 0.11% and 0.33% respectively. Naive heuristics that ignore demand fluctuations have an average optimality gaps of more than 11%. This shows that there is great value in leveraging knowledge about demand fluctuations in making repairable expediting and stocking decisions.
Content may be subject to copyright.
A preview of the PDF is not available
... Song and Zipkin (2009) reinterpret and extend the work of Moinzadeh and Schmidt (1991) by showing that the same inventory system with a dual-index policy and stochastic lead times is a special type of product form queueing network with one or more overflow bypasses. The dual-index policy in the setting of Schmidt (1991) andSong andZipkin (2009) is in fact optimal for the special case where the regular repair lead time has a shifted exponential distribution and the base stock level for the regular inventory position is fixed (Arts et al. 2016). The policy that we consider for the central warehouse is equivalent to the dual-index policy of Song and Zipkin (2009). ...
... The expediting policy in our model falls into this latter category as it essentially changes the repair lead time of a part based on the current state of the repair pipeline. In a recent contribution, Arts et al. (2016) study an expediting policy similar to the present model, albeit in a single-echelon single-item setting under fluctuating demand. They remark that this expediting policy does not suffer from the tractability issues that other dynamic priority rules suffer from, while still providing the lead time flexibility inherent to this category of heuristic priority rules. ...
... Such processes can capture demand fluctuations for repairable spare parts over time, which might occur in practice due to for instance periodic inspections or revisions of equipment. Similar to Arts et al. (2016), assuming that demand for each repairable type is a Markov modulated Poisson process would then be a promising approach. OR spectrum 29(4) 699. ...
Article
Full-text available
Problem definition: We consider dual sourcing in a distribution network for spare parts consisting of one central warehouse and multiple local warehouses. Each warehouse keeps multiple types of repairable parts to maintain several types of capital goods. The repair shop at the central warehouse has two repair options for each repairable part: a regular repair option and an expedited repair option. Irrespective of the repair option, each repairable part uses a certain resource for its repair. In the design of these inventory systems, companies need to decide on stocking levels and expedite thresholds such that total stock investments are minimized while satisfying asset availability and expediting constraints. Academic/practical relevance: Although most companies have the possibility to expedite the repair of parts in short supply, no contributions have been made that incorporate such dynamic expediting policies in repairable investment decisions. Anticipating expediting decisions that will be made later leads to substantial reductions in repairable investments. Methodology: We use queueing theory to determine the performance of the central warehouse and subsequently find the performance of all local warehouses using binomial disaggregation. For the optimization problem, we develop a greedy heuristic and a decomposition and column generation based algorithm. Results: Both solution approaches perform very well with average optimality gaps of 2.38 and 0.27%, respectively, across a large test bed of industrial size. The possibility to expedite the repair of failed parts is effective in reducing stock investments with average reductions of 7.94% and even reductions up to 19.61% relative to the state of the art. Managerial implications: Based on a case study at Netherlands Railways, we show how managers can significantly reduce the investment in repairable spare parts when dynamic repair policies are leveraged to prioritize repair of parts whose inventory is critically low.
... This policy is dubbed overflow bypass by Song and Zipkin (2009) and its stationary distribution is shown to have a product form by Van Dijk (1993) and Jackson (1963). This policy is shown to be optimal by Arts et al. (2016) for deterministic repair times. In our study, we utilize the overflow bypass policy in a larger repairable control system including Markovian inspection and condemnation processes in addition to the repair processes with general lead time distributions. ...
... Also, Feng et al. (2006) show that the dual index policy is only optimal under restrictive conditions when there are three distinct delivery modes. Scheller-Wolf et al. (2007), Arts et al. (2011), Veeraraghavan andScheller-Wolf (2008) and Sheopuri et al. (2010) consider heuristic policies for dual sourcing problems in a periodic review setting whereas Moinzadeh and Schmidt (1991), Song andZipkin (2009), andArts et al. (2016) consider continuous review inventory problems with two supply sources. The first two consider dual sourcing problems for a single item using a heuristic control policy. ...
... In many practical settings, repair processes consist of subprocesses that are usually too much convoluted for a representation with distinct stages. Yet, Erlang( , ) assumption provides a good modeling environment for repairable and queuing systems (Arts et al., 2016;Gross et al., 2008). ...
Article
Stockouts of repairable spares usually lead to significant downtime costs. Managers of Maintenance Repair Organizations (MROs) seek advance indicators of future stockouts which might allow them to take proactive actions that are beneficial for achieving target service levels with reasonable costs. Among such (proactive) actions, the most common, and the cheapest one is expediting existing repair processes. In this study, we develop an advance stockout risk estimation system for repairable spare parts. To the best of our knowledge, this is the first study to estimate the future stockout risk of a repairable part. The method considers different statistics, e.g. the number of ongoing repair processes, demand rate, repair time, etc. to estimate stockout risk of a repairable part for a given planning horizon. In our field tests with empirical data, the suggested method overperforms two heuristic approaches and achieves accuracy rates of 63% for 15 day-planning horizon and 83% for 45 days. We also suggest a repairable inventory control system including repair expediting, inspection and condemnation processes. To optimize the control parameters we suggest a simple algorithm considering two constraints: Target service level and maximum fraction of expedited demand. The algorithm is proved to be efficient for finding the optimum policy parameter in our tests with empirical data. Tests with empirical data suggest savings up to 8%. Both systems are implemented at an MRO as building blocks of a inventory control tower. The impact of the implementation is assessed with empirical simulations and verified from the financial indicators of the company.
... There are several papers on service differentiation (Kranenburg and van Houtum, 2008), customer differentiation and stock reservation at the central warehouse (Axsäter et al, 2007), and expediting (Arts et al. 2016). However, these papers focus on tactical planning (e.g., finding the optimal target or base stock levels). ...
... The work in Sethi and Cheng (1997) also addresses the optimality of re-ordering policies with an MMPP fluctuating demand, and considers extensions to the model of Song and Zipkin (1993) to account for cyclic demand (non-stationarity), ordering periods and service level constraints. Recent formulations of mathematical models that capture fluctuation in the demand process via an MMPP include Bhat and Krishnamurthy (2015), Nasr and Maddah (2015), Arts et al. (2016), Arts (2017), Chen et al. (2017), and Avci et al. (2019). ...
Article
Full-text available
Modeling the behavior of customer demand is a key challenge in inventory control, where an accurate characterization of the demand process often involves accounting for a wide range of statistical descriptors. This motivates the use of Markovian processes, due to their proven versatility in matching key components of point processes, to capture the behavior of customer demand. Accordingly, this work presents computational frameworks for continuous inventory models with a batch Markovian demand. A Markovian formulation of the system state-space is presented along with computational approaches to obtain key inventory performance measures. Compact matrix representations are considered for the steady-state solution of the system performance measures. The transient and non-stationary behavior of the inventory system is calculated by numerically integrating the corresponding set of Kolmogorov forward equations. A byproduct of this work is explicitly expressing the solution of the moments of the batch Markovian counting process by a compact matrix exponential equation. Numerical examples illustrate the computational efficiency of the mathematical frameworks when evaluating and comparing the performance of different re-ordering policies.
... In a spare parts setting, emergency sources may represent an expedited parts shipment from a central warehouse or a supplier (e.g., Howard et al., 2015), or an expedited repair (e.g., Arts et al., 2016). In both cases, spare parts demand is essentially treated as a lost sale at the local warehouse if it cannot be satisfied directly from inventory. ...
Article
Full-text available
We investigate the benefits of on‐site printing at remote geographic locations, where access to spare parts is intermittent and supplies are replenished at fixed intervals. Organizations typically have no solution to spare parts shortages other than expensive expediting orders, or waiting for a part to arrive with the next replenishment. We investigate whether on‐site three‐dimensional (3D) printing of spare parts can bring relief. Our work extends dual‐sourcing literature with fixed order cycles by considering two emergency supply options: expediting regular parts and 3D printing lower quality parts. We model the replenishment and emergency supply decisions as a Markov decision process and find that the optimal inventory control policy consists of two thresholds that control when to expedite, when to print and when to wait for regular parts via the next replenishment. We apply our model to a case study of the Royal Netherlands Army (RNLA) for her United Nations peacekeeping mission in Mali. Our results show that on‐site 3D printing, much more so than expediting, leads to large operational cost savings through on‐site inventory reductions and increased asset availability, thus increasing the ability of the RNLA to operate in remote locations. These results extend to many other organizations that operate in remote locations, e.g., those in the mining and offshore industry.
... Aside from the various economic and financial areas previously mentioned, MMPPs are popular in the areas of network theory, telecommunications and data traffic modelling such as in Yoshihara et al. (2001), Salvador et al. (2003), Scott and Smyth (2003) and Casale et al. (2016). Finally, this approach is utilised in the literature on risk, inventory, reliability and queueing theory to address unrealistic Poisson process assumptions through modulation, for example, inÖzekici and Parlar (1999), Ching (1997), Landon et al. (2013) and Arts et al. (2016). ...
Preprint
Full-text available
The Markov-modulated Poisson process is utilised for count modelling in a variety of areas such as queueing, reliability, network and insurance claims analysis. In this paper, we extend the Markov-modulated Poisson process framework through the introduction of a flexible frequency perturbation measure. This contribution enables known information of observed event arrivals to be naturally incorporated in a tractable manner, while the hidden Markov chain captures the effect of unobservable drivers of the data. In addition to increases in accuracy and interpretability, this method supplements analysis of the latent factors. Further, this procedure naturally incorporates data features such as over-dispersion and autocorrelation. Additional insights can be generated to assist analysis, including a procedure for iterative model improvement. Implementation difficulties are also addressed with a focus on dealing with large data sets, where latent models are especially advantageous due the large number of observations facilitating identification of hidden factors. Namely, computational issues such as numerical underflow and high processing cost arise in this context and in this paper, we produce procedures to overcome these problems. This modelling framework is demonstrated using a large insurance data set to illustrate theoretical, practical and computational contributions and an empirical comparison to other count models highlight the advantages of the proposed approach.
... There are several papers on (i) dual sourcing and expediting (e.g., Arts, Basten, & van Houtum, 2016;Song & Zipkin, 2009;Veeraraghavan & Scheller-Wolf, 2008 ), (ii) lateral transshipments (e.g., Glazebrook, Paterson, Rauscher, & Archibald, 2015;Kranenburg & van Houtum, 2009;Paterson, Kiesmller, Teunter, & Glazebrook, 2011;Paterson, Teunter, & Glazebrook, 2012 ), and (iii) stock allocation (e.g., van der Heijden, Diks, & de Kok, 1997;Marklund & Rosling, 2012 ). The major difference from the vast majority of these three streams is three-fold: First, in contrast to most of these papers, which typically focus on tactical planning and investigating impact of expediting, lateral transshipments, and stock allocation on optimal stock levels, we make operational planning decisions using real-time information of the supply chain. ...
Article
In this paper, we investigate operational spare parts planning in a multi-item two-echelon distribution system, taking into account real-time supply information in the system. We consider a broad range of operational interventions, either reactive (to solve a shortage) or proactive (to avoid a shortage). These interventions particularly include lateral transshipments between warehouses (local warehouses), emergency shipments from the depot (central warehouse), and doing nothing and waiting for pipeline inventory. We propose an integrated approach to determine the optimal timing and size of each intervention type to minimize the total downtime and shipment costs associated with interventions. Data from a leading original equipment manufacturer of high-tech systems is used to test the performance of our approach. We find that our integrated approach reduces total downtime considerably with a very limited increase in total shipment costs. Proactive emergency shipments contribute most to downtime reduction. The benefit of our approach is higher for high demand parts. Allowing complete pooling between warehouses increases downtime savings and usage of proactive emergency shipments even further. Our approach is efficient enough to solve practical size problems. We also propose a heuristic based on a greedy algorithm, which is well known in the literature. We find that the gap between the heuristic and the optimal solution is relatively large.
Article
We study performance improvement in multi-echelon, closed loop spare part supply chains using operational interventions based on real-time status information. Our objective is to minimize the total cost relevant costs, consisting of intervention costs and the backorder costs. In this paper, we focus on proactive interventions, aiming to avoid stockouts. We assume that all reactive interventions are fixed. Proactive interventions that we study include lateral transshipments, emergency shipments, stock reservations, expediting part repairs, and early new buys of parts. These interventions are invoked by using alert generation, when the supply chain status deviates from the plan. We propose heuristic rules to generate alerts. We also develop heuristic rules for the choice of operational interventions. We model and test our heuristics in a simulation test bed, based on data of a global IT company by using the case data in Germany. Numerical experiments reveal the following key insights: (i) downstream interventions – proactive lateral and emergency shipments – have most impact in reducing costs, (ii) communicating losses in the supply chain (no returns, failed repairs) for early new buys has positive impact on fill rates at negligible costs, and (iii) expedite repair and stock reservations using the proposed rules is not profitable.
Article
The Markov-modulated Poisson process is utilised for count modelling in a variety of areas such as queueing, reliability, network and insurance claims analysis. In this paper, we extend the Markov-modulated Poisson process framework through the introduction of a flexible frequency perturbation measure. This contribution enables known information of observed event arrivals to be naturally incorporated in a tractable manner, while the hidden Markov chain captures the effect of unobservable drivers of the data. In addition to increases in accuracy and interpretability, this method supplements analysis of the latent factors. Further, this procedure naturally incorporates data features such as over-dispersion and autocorrelation. Additional insights can be generated to assist analysis, including a procedure for iterative model improvement. Implementation difficulties are also addressed with a focus on dealing with large data sets, where latent models are especially advantageous due the large number of observations facilitating identification of hidden factors. Namely, computational issues such as numerical underflow and high processing cost arise in this context and in this paper, we produce procedures to overcome these problems. This modelling framework is demonstrated using a large insurance data set to illustrate theoretical, practical and computational contributions and an empirical comparison to other count models highlight the advantages of the proposed approach.
Article
This paper aims to provide a comprehensive review on Markovian arrival processes (MAPs), which constitute a rich class of point processes used extensively in stochastic modelling. Our starting point is the versatile process introduced by Neuts (1979) which, under some simplified notation, was coined as the batch Markovian arrival process (BMAP). On the one hand, a general point process can be approximated by appropriate MAPs and, on the other hand, the MAPs provide a versatile, yet tractable option for modelling a bursty flow by preserving the Markovian formalism. While a number of well-known arrival processes are subsumed under a BMAP as special cases, the literature also shows generalizations to model arrival streams with marks, nonhomogeneous settings or even spatial arrivals. We survey on the main aspects of the BMAP, discuss on some of its variants and generalizations, and give a few new results in the context of a recent state-dependent extension.
Article
The problem of determining the optimal ordering policies under stochastic demand is examined when two supply options, air and surface, are available, with different costs and different delivery times. Assuming mild conditions on the holding-penalty cost functions, linear ordering costs and backlogging, some sufficient conditions are obtained to show when it is optimal to order nothing by air and others to show when it is optimal to order nothing by surface. Explicit formulas are derived for the optimal orders in the case when air delivery time is kappa periods and surface delivery time is kappa plus 1 periods, respectively, under more general conditions than before.
Article
We develop an approximate model of an inventory control system in which there exist two options for resupply, with one having a shorter lead-time. We assume that demand and the fixed ordering costs are small relative to the holding cost so that a one-for-one ordering policy is appropriate. We consider a policy for placing emergency orders that uses information about the age of outstanding orders. We derive the steady-state behavior of this policy and present some computational results.
Article
Stocks of spare parts, located at appropriate locations, can prevent long downtimes of technical systems that are used in the primary processes of their users. Since such downtimes are typically very expensive, generally system-oriented service measures are used in spare parts inventory control. Examples of such measures are system availability and the expected number of backorders over all spare parts. This is one of the key characteristics that distinguishes such inventory control from other fields of inventory control. In this paper, we survey models for spare parts inventory control under system-oriented service constraints. We link those models to two archetypical types of spare parts networks: networks of users who maintain their own systems, for instance in the military world, and networks of original equipment manufacturers who service the installed base of products that they have sold. We describe the characteristics of these networks and refer back to them throughout the survey. Our aim is to bring structure into the large body of related literature and to refer to the most important papers. We discuss both the single location and multi-echelon models. We further focus on the use of lateral and emergency shipments, and we refer to other extensions and the coupling of spare parts inventory control models to related problems, such as repair shop capacity planning. We conclude with a short discussion of application of these models in practice.