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A Multi-Item Approach to Repairable Stocking and Expediting in a Fluctuating Demand Environment

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Abstract

We consider a single inventory location where multiple types of repairable spare parts are kept for service and maintenance of several different fleets of assets. Demand for each part is a Markov modulated Poisson process (MMPP). Each fleet has a target for the maximum expected number of assets down for lack of a spare part. The inventory manager can meet this target by stocking repairables and by expediting the repair of parts. Expedited repairs have a shorter lead time. There are multiple repair shops (or departments) that handle the repair of parts and the load imposed on repair shops by expedited repairs is constrained. A dual-index policy makes stocking and expediting decisions that depend on demand fluctuations for each spare part type. We formulate the above problem as a non-linear non-convex integer programming problem and provide an algorithm based on column generation to compute feasible near optimal solutions and tight lower bounds. We show how to use the MMPP to model demand fluctuations in maintenance and other settings, including a moment fitting algorithm. We quantify the value of lead time flexibility and show that effective use of this flexibility can yield cost reductions of around 25%.
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... Instead of solving the original complex optimization problem, the problem is decomposed by spare part and the resulting single-item optimization problems are solved repeatedly. This method has recently received increasing attention in the multi-item spare parts inventory optimization literature , Wong et al. 2007, Topan et al. 2010, Alvarez et al. 2013, Alvarez et al. 2015, Arts 2017, Topan et al. 2017, and Drent and Arts 2020. Details on fundamentals and theoretical background of decomposition and column generation are provided by Dantzig and Wolfe (1960) and Desrosiers and Lübbecke (2005). ...
... Such an approach has recently been applied by Alvarez et al. (2013Alvarez et al. ( , 2015, Arts (2017), and Drent and Arts (2020). Alvarez et al. (2013Alvarez et al. ( , 2015 showed that it performs better than using the fractional solution of (RMP ) as a starting point for local search as done by Van Houtum (2007, 2008). ...
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Inventory optimization approaches typically optimize steady‐state performance, but do not consider the transition of an initial state to the optimized state. In this paper, we address this transition. Our research is motivated by a company that implemented an improved inventory policy for its spare parts division. The improved policy suggested new base stock levels for the majority of the parts. For parts with increased base stock levels, inventory increases were realized after the part lead times, but for low‐demand parts with decreased base stock levels, inventory reductions were slow. As a result, inventory cost increased over the first months after the new inventory policy had been introduced and exceeded the inventory budget substantially. To avoid such undesirable effects, base stock level changes must be phased in. We consider a multi‐item spare parts inventory system, initially operating under an item approach inventory policy that achieves identical fill rates for all parts. Our approach addresses the transition to a superior system approach inventory policy that maximizes the system fill rate. We model the inventory transition as a finite‐horizon optimization problem and apply column generation and a marginal analysis heuristic to determine transient base stock levels for all parts. Using data from the company that motivated our research, we illustrate how the transition can be controlled to quickly improve fill rates without exceeding the initial inventory budget.
... Performance Evaluation Optimization [16], [17], [21], [25], [27], [28], [30], [31], [33], [35] [8], [11], [20], [23], [29], [46], [51], [8] [15], [18], [19], [41], [45] [2], [14], [34], [36], [53] [3], [6], [50], [52] [5], [22], [37], [44] [26], [40], [49] [32], [42], [43] [4] ...
... Economy [2], [3], [4], [9], [15], [20], [28], [33], [47] International Journal Of Production Economics • [8], [14], [19], [23], [35] European Journal Of Operational Research [22], [32], [43] Annals Of Operations Research [36], [42], [45] Computers & Operations Research [21], [39] Asia-pacific Journal Of Operational Research • [18], [46] Or Spectrum [16], [41] Journal Of The Operational Research Society [30], [50] International Journal Of Production Research [31], [52] Omega [6], [17] Reliability Engineering & System Safety ...
... Decomposing this problem leads to relatively simple sub-problems per repairable type. This technique has been used extensively in recent contributions on spare part inventory optimization (e.g., Alvarez et al. 2013, Arts 2017, Kranenburg and Van Houtum 2007, Topan et al. 2017, Wong et al. 2007). Most contributions only consider an aggregated service level constraint that links the different repairable types. ...
... In this paper, repairable types are not only linked through such a service level constraint, but also through the maximally allowed mean fraction of expedited repairs over all repairable types that use the same repair resource. Arts (2017) considers a similar optimization model with linking constraints on both expedited repairs and service levels. ...
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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.
... Instead of solving the original complex optimization problem, the problem is decomposed by spare part and the resulting single-item optimization problems are solved repeatedly. This method has recently received increasing attention in the multi-item spare parts inventory optimization literature , Wong et al. 2007, Topan et al. 2010, Alvarez et al. 2013, Alvarez et al. 2015, Arts 2017, Topan et al. 2017, and Drent and Arts 2019. Details on fundamentals and theoretical background of decomposition and column generation are provided by Dantzig and Wolfe (1960) and Desrosiers and Lübbecke (2005). ...
... We achieve this by solving the final (RMP ) with the sequence setS i after column generation as an integer program, thus with enforced Integrality Constraints (12) and denote it by CG (column generation) approach. Such an approach has recently been applied by Alvarez et al. (2013Alvarez et al. ( , 2015, Arts (2017), and Drent and Arts (2019). Alvarez et al. (2013Alvarez et al. ( , 2015 showed that it performs better than using the fractional solution of (RMP ) as a starting point for local search as done by Van Houtum (2007, 2008). ...
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Inventory optimization approaches typically optimize steady-state performance, but do not consider the transition of an initial state to the optimized state. In this paper, we address this transition. Our research is motivated by a company that implemented an improved inventory policy for its spare parts division. The improved policy suggested new base stock levels for the majority of the parts. For parts with increased base stock levels, inventory increases were realized after the part lead times, but for low-demand parts with decreased base stock levels, inventory reductions were slow. As a result, inventory cost increased over the first months after the new inventory policy was introduced and exceeded the inventory cost budget substantially. To avoid such undesirable effects, base stock level changes must be phased in. We consider a multi-item spare parts inventory system initially operating under an item approach inventory policy that achieves identical fill rates for all parts. Our approach addresses the transition to a superior system approach inventory policy that maximizes the system fill rate. We model the inventory transition as a finite-horizon optimization problem and apply column generation and a marginal analysis heuristic to determine transient base stock levels for all parts. Using data from the company that motivated our research, we illustrate how the transition can be controlled to quickly improve fill rates without exceeding the initial inventory budget.
... near here[/t] Considering the optimization of the performance indicators of the repair shops, it could be observed that Linear or Non-linear Programming (16 papers) are the algorithms most commonly used, accounting for 30% of all papers. The objective is to minimize either the expected waiting time in repair shops(Safaei et al. 2011;Liang et al. 2013;Aramon and Beck 2014;Dreyfuss et al. 2018;Al-Refaie et al. 2020;Driessen et al. 2020) or the spare parts inventory(Adan et al. 2009;Basten and van Houtum 2014;Jaber et al. 2014;van Jaarsveld et al. 2015;Taleizadeh et al. 2016;Arts 2017;Sleptchenko et al. 2019) or the maintenance costs(van Ommeren et al. 2006;Simeu-Abazi and Ahmad 2011;Shivasankaran et al. 2013;Turan et al. 2018;Sanchez et al. 2020). The advantage of both methods is that they are exact, even though the repair shop model needs to be simplified as much as possible to reduce calculation complexity. ...
Article
Maintenance activities are crucial for all manufacturing industries. To ensure availability and lifetime of production equipment, operations management and logistics support for maintenance need to evolve year after year. Besides, Centralised Maintenance Workshops are one of the most interesting approaches to reduce the cost and time required to repair faulty equipment. Generally known in the research community as ‘repair shops’, they aim to pool all the resources needed to repair defective equipment provided by different production sites. This paper aims to provide a comprehensive overview of repair shops and to present opportunities for future research with a focus on the circular economy context. The most relevant papers have been rigorously selected and analyzed, providing interesting reference materials on the subject. Repair shops are a set of workstations, operators, and spare parts inventories required to restore a group of failed production equipment. After detecting the origin of the failures, there are two options: either repair the equipment by restoring its defective components or replace the defective components with others in good working order. In the case of non-repairable components/equipment, circular strategies allow identification of components/equipment that could be restored and used to supply the spare parts warehouse.
... In the dual sourcing literature, studies on continuous review repair expediting are the closest to our repair expediting model as they consider the repair expediting using a threshold on the ''initial'' part of the repair process. Our study employs this control heuristic within a larger repairable inventory system which includes random inspection and condemnation processes and is optimized with respect to service level and expediting frequency constraints, which are also mentioned by Arts (2017). However, they focus on unconstrained cost optimization problem with no condemnation. ...
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.
... 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
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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.
... Optimization model types Note that a model with a stochastic setting does not necessarily lead to a stochastic optimization problem. For example, [35] considered a repairable spare parts inventory problem with a stochastic setting, i.e., the demand for each part is a Markov-modulated Poisson process, but the problem is formulated as a non-linear integer programming optimization problem. In the third column of Table 8, we classify the studies based on optimization model types. ...
Article
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Spare parts are held as inventory to support product maintenance in order to reduce downtime and extend the lifetime of products. Recently, spare parts inventory management has been attracting more attention due to the “right-to-repair” movement which requires that manufacturers provide sufficient spare parts throughout the life-cyle of their products to reduce waste so as to achieve sustainability. In this review, 148 papers regarding spare parts inventory management published from 2010 to 2020 are examined. The studies are classified based on two groups of perspectives. The first group includes the characteristics of spare parts, products, inventory systems, and supply chains, while the second group focuses on the characteristics of research methodologies and topics in the reviewed studies. The novelty of this literature review is three-fold. Firstly, we focus on analyzing the supply chain structure of different inventory networks for managing spare parts. Secondly, we classify the current literature based on analytics techniques, i.e., descriptive analytics, predictive analytics, and prescriptive analytics. Finally, the research gaps in this field are discussed from the perspective of reverse logistics, consumer durable goods, inventory network structure and policy, spare parts demand pattern modeling, and big data analytics.
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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.
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In this paper, we analyze a repair shop serving several fleets of machines that fail from time to time. To reduce downtime costs, a continuous-review spare machine inventory is kept for each fleet. A spare machine, if available on stock, is installed instantaneously in place of a broken machine. When a repaired machine is returned from the repair shop, it is placed in inventory for future use if the fleet has the required number of machines operating. Since the repair shop is shared by different fleets, choosing which type of broken machine to repair is crucial to minimize downtime and holding costs. The optimal policy of this problem is difficult to characterize, and, therefore, is only formulated as a Markov Decision Process to numerically compute the optimal cost and base-stock level for each spare machine inventory. As an alternative, we propose the dynamic Myopic(R) policy, which is easy to implement, yielding costs very close to the optimal. Most of the time it outperforms the static first-come-first-served, and preemptive-resume priority policies. Additionally, via our numerical study, we demonstrate that repair shop pooling is better than reserving a repair shop for each fleet.
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IntroductionThe Erlang Delay ModelLoss ModelsService-system DesignInsensitivityA Phase Method Queueing NetworksExercisesBibliographic NotesReferences
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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.