![Part (a) shows the state dependent threshold for n = 593 in the case where Lr has an Erlang(2) distribution (m = 2). Part (b) shows the state dependent thresholds for n = 1513 when Lr is Erlang(3) distributed (m = 3). Both cases are based on the problem instance with |Θ| = 1, λ1 = 1, E[Lr] = 4, e = 2, ce = 8, p = 10, and S = 8. In both cases, n coincides with the iteration in which average optimal policies are found within some precision.](https://www.researchgate.net/profile/Joachim-Arts/publication/303744142/figure/fig2/AS:667866860900360@1536243276523/Part-a-shows-the-state-dependent-threshold-for-n-593-in-the-case-where-Lr-has-an_Q320.jpg)
![Consider the problem instance with Poisson demand with rate λ = 3.43, e = 8, m = 1, E[Lr] = 15, p = 37.2 and ce = 116. This figure shows C(S) for this instance. Although the fact that C(S) is not unimodal is not immediately apparent from sub-figure (a), it is apparent from sub-figure (b). Any small deviation of any of the problem parameters will make C(S) convex.](https://www.researchgate.net/profile/Joachim-Arts/publication/303744142/figure/fig1/AS:370244330770432@1465284535198/Consider-the-problem-instance-with-Poisson-demand-with-rate-l-343-e-8-m-1-ELr_Q320.jpg)
![Optimal and heuristic policies. Part (a) shows the optimal state dependent threshold in conjunction with the best WDT policy for the case with λ = 1, E[Lr] = 4, e = 2, p = 10, ce = 8, S = 8 and m = 2. Part (b) shows the optimal state dependent threshold for in conjunction with the best heuristic policy for the same case except S = 12.](https://www.researchgate.net/profile/Joachim-Arts/publication/303744142/figure/fig3/AS:667866860904464@1536243276583/Optimal-and-heuristic-policies-Part-a-shows-the-optimal-state-dependent-threshold-in_Q320.jpg)
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Repairable Stocking and Expediting in a Fluctuating Demand Environment: Optimal Policy and Heuristics
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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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... They are used in after-sales services and in maintaining operating systems and capital goods within most companies, which renders their management an important lever to improve the service to customers and to reduce costs (Al Hanbali & van der Heijden 2013, Sleptchenko et al. 2018, Topan et al. 2020. The management of repairable spare parts inventory systems is challenging since it consists not only in controlling the inventories used to satisfy demand triggered by planned and unplanned maintenance activities but also in managing the repair shops that repair the spare parts often with a limited capacity and replenish the inventory points (Tiemessen & Van Houtum 2013, Arts et al. 2016, Drent & Arts 2021. The management of such systems require integrated design and control decisions. ...
... With regard to the literature that considers that the stock point can be supplied by a dual sourcing including a repair expediting, we refer to the research by Arts et al. (2016) and Drent & Arts (2021). This research builds on the early work on dual source inventory systems developed by Moinzadeh & Schmidt (1991), Song & Zipkin (2009) among others. ...
... A review on the inventory control of such multiple supply systems is provided by Svoboda et al. (2021). Arts et al. (2016) analyse a dual sourcing spare parts inventory system with an expedited sourcing. They make the assumption of a Markov modulated Poisson process and relax the assumptions that the demand process is a stationary Poisson process (often considered in the literature). ...
We study a repairable inventory system dedicated to a single component that is critical in operating a capital good. The system consists of a stock point containing spare components, and a dedicated repair shop responsible for repairing damaged components. Components are replaced using an age-replacement strategy, which sends components to the repair shop either preventively if it reaches the age-threshold, and correctively otherwise. Damaged components are replaced by new ones if there are spare components available, otherwise the capital good is inoperable. If there is free capacity in the repair shop, then the repair of the damaged component immediately starts, otherwise it is queued. The manager decides on the number of repairables in the system, the age-threshold, and the capacity of the repair shop. There is an inherent trade-off: A low (high) age-threshold reduces (increases) the probability of a corrective replacement but increases (decreases) the demand for repair capacity, and a high (low) number of repairables in the system leads to higher (lower) holding costs, but decreases (increases) the probability of downtime. We first show that the single capital good setting can be modelled as a closed queuing network with finite population, which we show is equivalent to a single queue with fixed capacity and state-dependent arrivals. For this queue, we derive closed-form expressions for the steady-state distribution. We subsequently use these results to approximate performance measures for the setting with multiple capital goods.
... Different from the literature reviewed above that focus on demand quantity information solely, a number of IC models assume richer information about future demands: both the quantity and arrival time of a future demand. They often model future demands using continuous stochastic arrival processes, such as a compound Poisson process (Zhao 2009), Markov modulated Poisson process (Arts et al. 2016), and time-dependent phase-type process (Nasr and Elshar 2018). For example, Arts et al. (2016) investigate a repairable stocking system, in which the demand of a repairable item follows a Markov modulated Poisson process and failed parts can be expedited with extra cost to shorten waiting time. ...
... They often model future demands using continuous stochastic arrival processes, such as a compound Poisson process (Zhao 2009), Markov modulated Poisson process (Arts et al. 2016), and time-dependent phase-type process (Nasr and Elshar 2018). For example, Arts et al. (2016) investigate a repairable stocking system, in which the demand of a repairable item follows a Markov modulated Poisson process and failed parts can be expedited with extra cost to shorten waiting time. ...
Disaster response is critical to save lives and reduce damages in the aftermath of a disaster. Fundamental to disaster response operations is the management of disaster relief resources. To this end, a local agency (e.g., a local emergency resource distribution center) collects demands from local communities affected by a disaster, dispatches available resources to meet the demands, and requests more resources from a central emergency management agency (e.g., Federal Emergency Management Agency in the U.S.). Prior resource management research for disaster response overlooks the problem of deciding optimal quantities of resources requested by a local agency. In response to this research gap, we define a new resource management problem that proactively decides optimal quantities of requested resources by considering both currently unfulfilled demands and future demands. To solve the problem, we take salient characteristics of the problem into consideration and develop a novel deep learning method for future demand prediction. We then formulate the problem as a stochastic optimization model, analyze key properties of the model, and propose an effective solution method to the problem based on the analyzed properties. We demonstrate the superior performance of our method over prevalent existing methods using both real world and simulated data. We also show its superiority over prevalent existing methods in a multi-stakeholder and multi-objective setting through simulations.
... Dual-sourcing systems operating in discrete time as well as their variants (e.g., continuous review models, multi-echelon systems, etc.) have been widely studied. For a broad overview of this rich field, we refer the interested reader to the surveys of Minner (2003) and Svoboda et al. (2021), as well as to the recent studies of, e.g., Gong et al. (2014), Boute and Van Mieghem (2015), Arts et al. (2016), Sapra (2017), Song et al. (2017), Drent and Arts (2021), and the references therein. Here we confine ourselves to contributions most relevant to our research, a large part of which revolves around heuristic policies. ...
We consider a single-echelon inventory system under periodic review with two suppliers facing stochastic demand, where excess demand is backlogged. The expedited supplier has a shorter lead time than the regular supplier but charges a higher unit price. We introduce the Projected Expedited Inventory Position (PEIP) policy, and we show that the relative difference between the long run average cost per period of this policy and the optimal policy converges to zero when both the shortage cost and the cost premium for expedited units become large, with their ratio held constant. A corollary of this result is that several existing heuristics are also asymptotically optimal in this non-trivial regime. We show through an extensive numerical investigation that the PEIP policy outperforms the current best performing heuristic policies in literature.
We study the robust production and maintenance control for a production system subject to degradation. A periodic maintenance scheme is considered, and the system production rate can be dynamically adjusted before maintenance, serving as a proactive way of degradation management. Optimal control of the degradation rate aims to strike a balance between the risk of failure and the production profit. We first consider the scenario in which the degradation rate increases linearly with the production rate. Different from the existing literature that posits a parametric stochastic degradation process, we suppose that the degradation increment during a period lies in an uncertainty set, and our objective is to minimize the maintenance cost in the worst case. The resulting model is a robust mixed‐integer linear program. We derive its robust counterpart and establish structural properties of the optimal production plan. These properties are then used for real‐time condition‐based control of the production rate through reoptimization. The model is further generalized to the nonlinear production‐degradation relation. Based on a real production‐degradation dataset from an extruder system, we conduct comprehensive numerical experiments to illustrate the application of the model. Numerical results show that our model significantly outperforms existing methods in terms of the mean and variance of cost rate when degradation model misspecification is presented.
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In the realm of disaster response operations, effective resource management is crucial. This research introduces an innovative approach that proactively determines the optimal quantities of resources that should be requested by local agencies. This determination is based on both current and anticipated demands, thereby ensuring a more efficient and effective response to disasters. The approach first utilizes a method that combines deep learning and temporal point process for predicting irregularly spaced future demands, and then, it formulates the resource allocation problem faced with randomly arrived demands as a stochastic optimization model. The superiority of this approach over existing resource allocation methods is demonstrated using both real-world data and simulated scenarios. The findings highlight the need for a shift from reactive to proactive strategies. Moreover, the research emphasizes the potential of advanced techniques, such as deep learning and stochastic optimization, in disaster management. These techniques can provide valuable tools for policy makers and practitioners in the field, enabling them to make more informed and effective decisions. Policies that encourage the adoption of such optimized resource allocation strategies could lead to more effective disaster response operations.
In the context of fast‐fashion business, we study the joint control of production, inventory, and unilateral transshipment in dual supply chains in which one supply chain is a responsive supply chain for fashion products and the other is an efficient supply chain for basic products in a fluctuating demand environment (FDE). Our research focuses on: (1) formulating an appropriate model for this complex problem; (2) characterizing the structured optimal policy, which can be used to develop effective heuristic policies; and (3) generalizing our studies for more applications. To address these issues, we formulate a discrete‐time finite‐horizon stochastic dynamic program model in which FDE is represented by a world state evolving in accordance with the discrete‐time Markov chain. The supply chain for the fashion product is modeled by a two‐stage tandem production system with a make‐to‐stock (MTS) stage for the fabric and a subsequent MTS stage for finished fashions. In parallel, the supply chain for the basic products is also modeled by a two‐stage tandem production system with a MTS stage for the fabric and a subsequent make‐to‐order (MTO) stage for finished basics. Further, the two supply chains are coupled via the unilateral transshipment of fabric with random transshipment time. By value iteration, we characterize the optimal policy for the single‐product problem with a set of monotone switching surfaces. The results are extended to the multi‐product multi‐supplier case. Further, based on the structure of the optimal policy, we develop a heuristic policy for the multi‐dimensional dynamic program model. From a practice perspective, we demonstrate the value of transshipment and controlled transshipment, analytically and numerically, and therefore justify the establishment of two separate local manufacturing bases in dual supply chains. Additionally, to achieve both quick response and efficiency, we show that the total cost can be reduced significantly if sufficient fabric is prepared just in time (JIT) for in‐season production at the beginning of a season.
Operations Management in a Repair Facility
Capital goods are machines or products that are used in the production of goods or service deliveries. Maintenance service providers keep spare parts on stock so that, whenever a critical component of a capital good breaks down, the broken part is replaced with a spare part to prevent long and costly downtime. In “Joint Inventory and Scheduling Control in a Repair Facility,” Özkan and van Houtum study inventory and repair scheduling decisions of a maintenance service provider for repairable capital goods. In case of a stock-out, the service provider should decide whether to back-order the demand or execute an emergency repair, which is an urgent but expensive repair operation for a broken part. The authors derive a simple and intuitive decision rule stating if the emergency repairs are necessary to achieve a close-to-optimal system performance. Moreover, they propose a simple, intuitive, and easy-to-implement heuristic control policy that performs well in numerical experiments.
We consider a single-product, two-source inventory system with Poisson demand and backlogging. Inventory can be replenished through a normal supply source, which consists of a two-stage tandem queue with exponential production time at each stage. We can also place an emergency order by skipping the first stage, for a fee. There is no fixed order cost. There are linear order, holding, and back-order costs. Through a new approach, we obtain optimal ordering policies for the discounted or long-run average cost and also characterize near-optimal heuristic policies. The approach consists of four steps. The first step is to establish an equivalent system, in the sense that it has the same optimal policy as the original system. The second step is to construct a tandem queueing system, where costs are charged in accord with the equivalent system’s cost structure. The third step derives an optimal control of the service rate at each server so as to minimize the tandem queue’s system-wide cost. The fourth and final step is to translate the queue’s optimal policy to an optimal policy for the equivalent system and hence the original system.
We study an inventory system with multiple supply sources and expediting options. The replenishment lead times from each supply source are stochastic, representing congestion and disruption. We construct a family of smart ordering and expediting policies that utilize real-time supply information. Such dynamic policies are generally difficult to evaluate, because the corresponding supply system is a tandem queue with state-dependent arrivals and routing, whose queue-length steady-state distribution is usually not in product form. Our main result is to identify two appealing special cases of the general policy, Policy-M and Policy-E, which possess simple product-form solutions and lead to closed-form performance measures. Policy-M retains full sourcing flexibility, but ignores upstream congestion in making expediting decisions. Policy-E only orders from the normal, farthest source, but makes expediting decisions based on both upstream and downstream information. A numerical study shows that the best Policy-M leads to a lower average cost than the best Policy-E in almost all cases. Also, implementing the best Policy-M parameters, the general policy only performs slightly better than Policy-M. These observations reveal the value of combining sourcing flexibility with some, but limited, dynamic expediting. Our findings are equally applicable to the equivalent tandem queue. They thus may aid dynamic routing and expediting decisions for online retailers and logistics providers, among others.
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.
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.
In this paper, 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.
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.