The impact of transportation delays on repairshop capacity pooling and spare part inventories

In this paper, to serve different fleets of machines at different locations, we study whether repair shop pooling is more cost effective than having dedicated on-site repair shops for each fleet. When modeling the former alternative, we take transportation delays and related costs into account and represent it as a closed queueing network. This allows us to include on-site spare-part inventories that operate according to a continuous-review base-stock policy. We obtain the steady-state distribution of components at each location and the cost of the system with a pooled repair shop by applying the Mean-Value Analysis technique. Our numerical findings indicate that when transportation costs are reasonable, repair shop pooling is a better alternative.     

Average Cost Optimality for an Unreliable Two-Machine Flowshop with Limited Internal Buffer

We consider a production planning problem in a two-machine flowshop subject to breakdown and repair of machines and subject to nonnegativity and upper bound constraints on work-in-process. The objective is to choose machine production rates over time to minimize the long-run average inventory/backlog and production costs. For sufficiently large upper bound on the work-in-process, the problem is formulated as a stochastic dynamic program. We then establish a verification theorem and a partial characterization of the optimal control policy if it exists.     

Optimising a general repair kit problem with a service constraint

Field services are a particular type of after-sales service performed at the customer’s location where technicians repair malfunctioning machines. The inventory decisions about which spare part types to take to the repair site and in what quantities is called the repair kit problem. This problem is characterized by an order-based performance measure since a customer is only satisfied when all required spare parts are available to fix the machine. As a result, the service level in the decision making process is defined as a job fill rate. In this paper we derive a closed-form expression for the expected service level and total costs for the repair kit problem in a general setting, where multiple units of each part type can be used in a multi-period problem. Such an all-or-nothing strategy is a new characteristic to investigate, but commonly used in practice. Namely, items are only taken from the inventory when all items to perform the repair are available in the right quantity. We develop a new algorithm to determine the contents of the repair kit both for a service and cost model while incorporating this new expression for the job fill rate. We show that the algorithm finds solutions which differ on average 0.2% from optimal costs. We perform a case study to test the performance of the algorithm in practice. Our approach results in service level improvements of more than 30% against similar holding costs.     

A stochastic model for joint spare parts inventory and planned maintenance optimisation

Spare parts demands are usually generated by the need of maintenance either preventively or at failures. These demands are difficult to predict based on historical data of past spare parts usages, and therefore, the optimal inventory control policy may be also difficult to obtain. However, it is well known that maintenance costs are related to the availability of spare parts and the penalty cost of unavailable spare parts consists of usually the cost of, for example, extended downtime for waiting the spare parts and the emergency expedition cost for acquiring the spare parts. On the other hand, proper planned maintenance intervention can reduce the number of failures and associated costs but its performance also depends on the availability of spare parts. This paper presents the joint optimisation for both the inventory control of the spare parts and the Preventive Maintenance (PM) inspection interval. The decision variables are the order interval, PM interval and order quantity. Because of the random nature of plant failures, stochastic cost models for spare parts inventory and maintenance are derived and an enumeration algorithm with stochastic dynamic programming is employed for finding the joint optimal solutions over a finite time horizon. The delay-time concept developed for inspection modelling is used to construct the probabilities of the number of failures and the number of the defective items identified at a PM epoch, which has not been used in this type of problems before. The inventory model follows a periodic review policy but with the demand governed by the need for spare parts due to maintenance. We demonstrate the developed model using a numerical example.     

Optimal lateral transshipment policies for a two location inventory problem with multiple demand classes

We consider an inventory model for spare parts with two stockpoints, providing repairable parts for a critical component of advanced technical systems. As downtime costs for these systems are expensive, ready–for–use spare parts are kept in stock to be able to quickly respond to a breakdown of a system. We allow for lateral transshipments of parts between the stockpoints upon a demand arrival. Each stockpoint faces demands from multiple demand classes. We are interested in the optimal lateral transshipment policy. There are three ways in which a demand can by satisfied: from own stock, via a lateral transshipment, or via an emergency procedure. Using stochastic dynamic programming, we characterize and prove the structure of the optimal policy, that is, the policy for satisfying the demands which minimizes the average operating costs of the system. This optimal policy is a threshold type policy, with state-dependent thresholds at each stockpoint for every demand class. We show a partial ordering in these thresholds in the demand classes. In addition, we derive conditions under which the so-called hold back and complete pooling policies are optimal, two policies that are often assumed in the literature. Furthermore, we study several model extensions which fit in the same modeling framework.     

A Graphical Aid for the Initial Purchase of ‘Insurance Type’ Spares

Expensive renewable spares known as ‘insurance type’ spares are often a major concern in the design and setting up of industrial, commercial and military systems. These spares, though low in demand, are critical to the system's operation and their unavailability can lead to excessive downtime costs. Due to their nature, the (S-1, S) inventory control model provides an appropriate replenishment policy for this class of items, where S is the maximum number of spares in inventory. A (S-1, S) model with Exponential distribution of failure-free operating time at each of a finite number of machines and Exponential distribution of re-supply lead-time is developed. A graphical aid is presented which, for a given number of machines, indicates the range of the ratio {mean lead-time/mean failure-free operating time} for which a minimum S is required in order to satisfy a service level constraint on the service measure Pr[a spare is available at a machine stoppage due to part failure].     

The effect of lifetime buys on warranty repair operations

Lifetime buys are a common practice in the electronics and telecommunication industries. Under this practice, manufacturers procure their repair parts inventory in one order to support the spare part needs of a product for the duration of its warranty repair period. In this paper, we consider a repair operation in which defective items under warranty are returned to a manufacturer who either repairs these items using its spare parts inventory or replaces each defective unit with a new product. We show how fixed repair capability costs, variable repair costs, inventory holding costs, and replacement costs affect a firm's optimal repair and replacement decisions. The model is used to gain insights for products from a major mobile device manufacturer in the United States.     

On sourcing and stocking policies in a two-echelon,multiple location,repairable parts supply chain

This research develops policies to minimize spare part purchases and repair costs for maintaining a fleet of mission-critical systems that operate from multiple forward (base) locations within a two-echelon repairable supply chain with a central depot. We take a tactical planning perspective to support periodic decisions for spare part purchases and repair sourcing, where the repair capabilities of the various locations are overlapping. We consider three policy classes: a central policy, where all repairs are sourced to a central depot; a local policy, whereby failures are repaired at forward locations; and a mixed policy, where a fraction of the parts is repaired at the bases and the remainder is repaired at the depot. Parts are classified based on their repair cost and lead time. For each part class, we suggest a solution that is based on threshold policies or on the use of a heuristic solution algorithm that extends the industry standard of marginal analysis to determine spare parts positioning by including repair fraction sourcing. A validation study shows that the suggested heuristic performs well compared to an exhaustive search (an average 0.2% difference in cost). An extensive numerical study demonstrates that the algorithm achieves costs which are lower by about 7–12% on average, compared to common, rule-based sourcing policies.     

Optimal age-replacement model with age-dependent type of failure and random lead time based on a cumulative repair-cost limit policy

In this paper, we consider an age-replacement model with minimal repair based on a cumulative repair cost limit and random lead time for replacement delivery. A cumulative repair cost limit policy uses information about a system’s entire repair cost history to decide whether the system is repaired or replaced; a random lead time models delay in delivery of a replacement once it is ordered. A general cost model is developed for the average cost per unit time based on the stochastic behavior of the assumed system, reflecting the costs of both storing a spare and of system downtime. The optimal age for preventive replacement minimizing that cost rate is derived, its existence and uniqueness is shown, and structural properties are presented. Various special cases are included, and a numerical example is given for illustration. Because the framework and analysis are general, the proposed model extends several existing results.     

Preventive maintenance in a 1 out of n system: The uptime,downtime and costs

In repairable systems with redundancy, failed units can be replaced by spare units in order to reduce the system downtime. The failed units are sent to a repair shop or manufacturer for corrective maintenance and subsequently are returned for re-use. In this paper we consider a 1 out of n system with cold standby and we assume that repaired units are “as good as new”.When a unit has an increasing failure rate it can be advantageous to perform preventive maintenance in order to return it to its “as good as new” state, because preventive maintenance will take less time and tends to be cheaper. In the model we present we use age-replacement; a machine is taken out for preventive maintenance and replaced by a standby one if its age has reached a certain value, T_pm. In this paper we derive an approximation scheme to compute the expected uptime, the expected downtime and the expected costs per time unit of the system, given the total number of units and the age-replacement value, T_pm. Consequently the number of units and the value T_pm can be determined for maximum long-term economy.     

Pooling of spare parts between multiple users: How to share the benefits?

Companies that maintain capital goods (e.g., airplanes or power plants) often face high costs, both for holding spare parts and due to downtime of their technical systems. These costs can be reduced by pooling common spare parts between multiple companies in the same region, but managers may be unsure about how to share the resulting costs or benefits in a fair way that avoids free riders. To tackle this problem, we study several players, each facing a Poisson demand process for an expensive, low-usage item. They share a stock point that is controlled by a continuous-review base stock policy with full backordering under an optimal base stock level. Costs consist of penalty costs for backorders and holding costs for on-hand stock. We propose to allocate the total costs proportional to players’ demand rates. Our key result is that this cost allocation rule satisfies many appealing properties: it makes all separate participants and subgroups of participants better off, it stimulates growth of the pool, it can be easily implemented in practice, and it induces players to reveal their private information truthfully. To obtain these game theoretical results, we exploit novel structural properties of the cost function in our (S − 1, S) inventory model.     

Optimal maintenance of two stochastically deteriorating machines with an intermediate buffer

We consider a manufacturing system in which an input generating installation transfers a raw material to a subsequent production unit. Both machines deteriorate stochastically with usage and may fail. For each machine the deteriorating process is described by some known transition probabilities between different degrees of deterioration. A buffer has been built between the two machines in order to cope with unexpected failures of the installation. A discrete-time Markov decision model is formulated for the optimal preventive maintenance of both machines. The maintenance times are geometrically distributed and the cost structure includes operating costs, storage costs, maintenance costs and costs due to the lost production. It is proved that for fixed buffer content and for fixed deterioration degree of one machine, the average-cost optimal policy initiates a preventive maintenance of the other machine if and only if its degree of deterioration exceeds some critical level. We study, by means of numerical results, the effect of the variation of some parameters on the optimal policy and on the minimum average cost. For the case in which the maintenance times follow continuous distributions, an approximate discrete-time Markov decision model is proposed.     

Optimal spare ordering policy for preventive replacement under cost effectiveness criterion

This paper presents a spare ordering policy for preventive replacement with age-dependent minimal repair and salvage value consideration. The spare unit for replacement is available only by order and the lead-time for delivering the spare due to regular or expedited ordering follows general distributions. To analyze the ordering policy, the failure process is modelled by a non-homogeneous Poisson process. By introducing the costs due to ordering, repairs, replacements and downtime, as well as the salvage value of an un-failed system, the expected cost effectiveness in the long run are derived as a criterion of optimality. It is shown, under certain conditions, there exists a finite and unique optimum ordering time which maximizes the expected cost effectiveness. Finally, numerical examples are given for illustration.     

Using repair priorities to reduce stock investment in spare part networks

In this paper, we examine the impact of repair priorities in spare part networks. Several heuristics for assigning priorities to items as well as optimising stock levels are developed, extending the well-known VARI-METRIC method. We model repair shops by multi-class, multi-server priority queues. A proper priority setting may lead to a significant reduction in the inventory investment required to attain a target system availability (usually 10–20%). The saving opportunities are particularly high if the utilisation of the repair shops is high and if the item types sharing the same repair shop have clearly different characteristics (price, repair time). For example, we find an investment reduction of 73% for a system with single server repair shops with an utilisation of 0.90 that handle five different item types.     

Optimal design of unreliable production–inventory systems with variable production rate

In this article, we study an economic manufacturing quantity (EMQ) problem for an unreliable production facility where the production rate is treated as a decision variable. As the stress condition of the machine changes with the production rate, the failure rate of the machine is assumed to be dependent on the production rate. The unit production cost is also taken as a function of the production rate, as the machine can be operated at different production rates resulting in different unit production costs. The basic EMQ model is formulated under general failure and general repair time distributions and the optimal production policy is derived for specific failure and repair time distributions viz., exponential failure and exponential repair time distributions. Considering randomness of the time to machine failure and corrective repair time, the model is extended to the case where certain safety stocks in inventory may be useful to improve service level to customers. Optimal production policies of the proposed models are derived numerically and the sensitivity of the optimal results with respect to those parameters which directly influence the machine failure and repair rates is also examined.     

Cost Analysis of the <Emphasis Type="Italic">M/M/R</Emphasis> Machine Repair Problem with Spares and Two Modes of Failure

In this paper we deal with the machine repair problem consisting of M operating machines with S spare machines, and R repairmen where machines have two failure modes under steady-state conditions. Spares are considered to be either cold-standby, warm-standby or hot-standby. The two failure modes have equal probability of repair. Failure time of the machines and repair time of the repairmen are assumed to follow a negative exponential distribution. A cost model is developed in order to determine the optimal values of the number of repairmen and the number of spares simultaneously, while maintaining a minimum specified level of system availability. Numerical results are presented in which several system characteristics are evaluated for three types of standby under optimal operating conditions.     

Optimization of component reliability in the design phase of capital goods

We introduce a quantitative model to support the decision on the reliability level of a critical component during its design. We consider an OEM who is responsible for the availability of its systems in the field through service contracts. Upon a failure of a critical part in a system during the exploitation phase, the failed part is replaced by a ready-for-use part from a spare parts inventory. In an out-of-stock situation, a costly emergency procedure is applied. The reliability levels and spare parts inventory levels of the critical components are the two main factors that determine the downtime and corresponding costs of the systems. These two levels are decision variables in our model. We formulate the portions of Life Cycle Costs (LCC) which are affected by a component’s reliability and its spare parts inventory level. These costs consist of design costs, production costs, and maintenance and downtime costs in the exploitation phase. We conduct exact analysis and provide an efficient optimization algorithm. We provide managerial insights through a numerical experiment which is based on real-life data.     

Joint spare parts inventory and reliability decisions under a service constraint

Reliability and inventory levels of spare parts are major factors that determine the service level for the maintenance of machines provided by original equipment manufacturers (OEMs). In general, decisions on reliability and stock levels are made separately in practice, and academic literature offers little guidance on how to jointly make these two decisions. In order to fill in the gap in the literature and provide guidance to OEMs, we jointly model reliability and inventory problems. We consider three different service measures: aggregate fill rate, average downtime per system per year and expected total number of long downs in a year. Our models minimize the sum of holding and emergency shipment costs subject to a limited reliability improvement budget and a target service level. We develop an algorithm that considers reliability and inventory decisions simultaneously, test our solution approach on real-life and randomly generated data sets and compare the results with an approach that considers reliability and inventory decisions sequentially. Numerical results show substantial benefits of integrating reliability and inventory decisions.     

Hierarchical decision making in production and repair/replacement planning with imperfect repairs under uncertainties

In this paper, we analyse an optimal production, repair and replacement problem for a manufacturing system subject to random machine breakdowns. The system produces parts, and upon machine breakdown, either an imperfect repair is undertaken or the machine is replaced with a new identical one. The decision variables of the system are the production rate and the repair/replacement policy. The objective of the control problem is to find decision variables that minimize total incurred costs over an infinite planning horizon. Firstly, a hierarchical decision making approach, based on a semi-Markov decision model (SMDM), is used to determine the optimal repair and replacement policy. Secondly, the production rate is determined, given the obtained repair and replacement policy. Optimality conditions are given and numerical methods are used to solve them and to determine the control policy. We show that the number of parts to hold in inventory in order to hedge against breakdowns must be readjusted to a higher level as the number of breakdowns increases or as the machine ages. We go from the traditional policy with only one high threshold level to a policy with several threshold levels, which depend on the number of breakdowns. Numerical examples and sensitivity analyses are presented to illustrate the usefulness of the proposed approach.     

Trade-off between inventory and repair capacity in spare part networks

The availability of repairable technical systems depends on the availability of (repairable) spare parts, to be influenced by (1) inventory levels and (2) repair capacity. In this paper, we present a procedure for simultaneous optimisation of these two factors. Our method is based on a modification of the well-known VARI-METRIC procedure for determining near-optimal spare part inventory levels and results for multi-class, multi-server queuing systems representing repair shops. The modification is required to avoid non-convexity problems in the optimisation procedure. To include part-time and overtime working, we allow for a non-integer repair capacity. To this end, we develop a simple approximation for queuing systems with a non-integer number of servers. Our computational experiments show that the near-optimal utilisation rate of the repair servers is usually high (0.80–0.98) and depends mainly on the relative price of the servers compared with inventory items. Further, the size of the repair shop (the minimal number of servers required for a stable system) plays its part. We also show that our optimisation procedure is robust for the choice of the step size for the server capacity.