Integrated scheduling of preventive maintenance and renewal projects for multi-unit systems with grouping and balancing

Accepted Manuscript Integrated scheduling of preventive maintenance and renewal projects for multiunit systems with grouping and balancing Farzad Pargar, Osmo Kauppila, Jaakko Kujala PII: DOI: Reference:

S0360-8352(17)30228-0 http://dx.doi.org/10.1016/j.cie.2017.05.024 CAIE 4755

To appear in:

Computers & Industrial Engineering

Received Date: Revised Date: Accepted Date:

6 December 2016 18 May 2017 21 May 2017

Please cite this article as: Pargar, F., Kauppila, O., Kujala, J., Integrated scheduling of preventive maintenance and renewal projects for multi-unit systems with grouping and balancing, Computers & Industrial Engineering (2017), doi: http://dx.doi.org/10.1016/j.cie.2017.05.024

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Integrated scheduling of preventive maintenance and renewal projects for multi-unit systems with grouping and balancing

Farzad Pargara,1, Osmo Kauppilaa, Jaakko Kujalaa a

1

Industrial Engineering and Management, Faculty of Technology, University of Oulu, Oulu, Finland

Corresponding author. Email: [email protected] 1

Integrated scheduling of preventive maintenance and renewal projects for multi-unit systems with grouping and balancing

ABSTRACT

This paper studies preventive maintenance and renewal scheduling for multi-unit systems. We develop an integrated optimization method to schedule preventive maintenance and renewal projects by grouping them and simultaneously finding the optimal balance between them. Grouping and balancing are resource utilization techniques to reduce the costs of maintenance without reducing the level of maintenance. We next model the problem as a pure integer linear programming formulation to minimize the costs of maintenance and renewal projects and their relevant preparatory works and downtime costs over the planning horizon. We use a numerical experiment with sensitivity analysis to illustrate the model, and the potential cost reductions that using such an integrated model may lead to. Applications of this model arise in the maintenance of railways, roadways, electricity distribution networks, and distributed pipeline assets. Due to the complexity of the model, two heuristic algorithms based on decomposition approach with problem-specific cutting planes are applied to tackle the problem. Experimental results show that our integrated optimization approach performs well in reducing the cost of preparatory works through proper scheduling. Finally, a case study for the maintenance of railway track shows that using our integrated approach reduces the maintenance and renewal costs by up to 14% as compared with the spreadsheet based approach used currently at the operation. Keywords: Maintenance; renewal; scheduling;

grouping; mathematical modeling; downtime

minimization 1. Introduction Maintenance is the work performed to keep a system in an appropriate condition and working order. It plays an important role in industries in which the loss resulting from a system failure is significant. Maintenance activities can be divided into preventive, corrective, and predictive maintenance (Moubray (1997); Murthy, Atrens, & Eccleston (2002)). Preventive maintenance (PM) involves a set of activities such as testing, measurement, and adjustment carried out to prevent unexpected system breakdowns. PM is often performed routinely based on time or cycles. Corrective maintenance (CM) is an unscheduled maintenance or repair performed after failures of the system have occurred and to restore the system to an operational state. CM could be either planned (run-to-failure) or unplanned. Predictive maintenance involves the use of modern measurement and signal processing methods to accurately predict and diagnose equipment condition during operation (Moubray (1997); Murthy et al. (2002)). An effective preventive maintenance program can reduce the probability of costly corrective replacement and repair, as well as avoid excessive maintenance, thus significantly cutting down servicing costs (Yang, Ma, & Zhao, 2017). The focus of the present paper is on scheduling preventive maintenance and preventive renewal activities. One of the main advantages of PM is that it can be planned ahead and performed at a convenient time. Often preparatory work such as shutting down a unit or transportation of the maintenance crew and machinery has to take place before maintenance can be performed (Chalabi, Dahane, Beldjilali, & Neki, 2016). PM activities are costly when frequently performed since they 2

increase the downtime of systems. It is desirable to reduce the costs of maintenance without reducing the level of maintenance (Budai, Huisman, & Dekker, 2006; Budai et al., 2006). One technique to reduce PM costs is to group the executions of maintenance and renewal projects (repair and replacement in the CM context, e.g. Dekker, Wildeman, & Van der Duyn Schouten (1997), Moghaddam & Usher (2011a)). Grouping allows savings on the preparatory work costs. However, it often implies that one deviates from previously planned execution times, possibly increasing costs due to performing a maintenance activity earlier than necessary (Budai, Dekker, & Nicolai, 2008). Another technique to reduce PM costs is to balance maintenance and renewal and determine the economic life of a system component. In general, a system component deteriorates with age, and the need for maintenance increases during the final periods of its technical life (Andrews, 2012). Accordingly, it can be economical to renew the component earlier than its physical life to prevent successive costly maintenance activities, or to be able to combine its renewal with other renewal projects. In this paper, finding a balance between maintenance and renewal is called “balancing”. This paper presents an optimization model that integrates grouping and balancing for planning preventive maintenance and renewal projects of multi-component assets which are spread out geographically. These assets are called multi-unit systems. A multi-unit system includes several multicomponent systems in sequence. In Figure 1 below, we show the structure of a typical multi-unit system, consisting of K units in which each unit may consist of I components. In a multi-unit system, each unit may have special maintenance requirements because of differentiation in factors such as the importance of the unit, its environment, and the component characteristics. The downtime cost of a multi-unit system is high, as the whole system will be out of service if a component in one unit is unavailable.

Figure 1. The structure of a typical multi-unit system with several components.

Our optimization model minimizes the costs of maintenance and renewal projects and their relevant preparatory works over a discrete number of time periods. In each period, it is assumed that one of the following three distinct actions can be performed for each component of the units (Moghaddam & Usher, 2011a): a) Do nothing: In this case, no action is to be performed on the component. This is often addressed to as leaving a component in a state of ‘bad-as-old’, where the component continues to age normally. b) Do maintenance: In this case, the component is maintained, which places it into a state somewhere between ‘good-as-new’ and ‘bad-as-old’. c) Do renewal: In this case, the component is removed from the unit and replaced by the new one, immediately placing it in a state of ‘good-as-new’, i.e. its age is effectively returned to time zero.

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The preventive maintenance and renewal scheduling problem (PMRSP) can be defined as designing the best sequence of actions (a), (b) or (c) for each component of the units over a planning horizon such that the total costs are minimized. The applications of our model are at a strategic level with time horizons of one to several years or at a tactical level covering a medium-long time horizon (weeks to a year). Although our optimization model is formulated for multi-unit systems, it can be easily adapted to any types of systems or even a single-unit system by assuming K=1. The maintenance of a railway, roadway or electricity network are the typical application areas of a multiunit system. Large benefits can be realized if preventive maintenance and renewal scheduling of these infrastructure assets can be improved (Lidén & Joborn, 2017). As an example of monetary volumes, the European countries are reported to allocate 15-25 billion EUR annually on maintenance and renewals for a railway system consisting of about 300 000 km of track giving an average of 70 000 EUR per km track and year (see EIM-EFRTC-CER Working Group (2012)). Therefore, a saving of millions of Euros may be obtained through small improvements in managing track maintenance and renewal (Acharya, Mishalani, Martland, & Eshelby, 1991). In Section 5.5, we perform a case study in the railway sector. According to the best of our knowledge, this study is the first work which integrates the grouping and balancing techniques for preventive maintenance and renewal scheduling of multi-unit systems. This paper presents a new hierarchical structure for setup costs which considers economic dependency between components of multi-unit systems. The developed setup configuration shows the potential benefits of grouping maintenance and renewal projects. In this paper, the economic life of the system components is determined by finding the optimal balance between maintenance and renewal while considering their effect on each other. These ideas are formulated in an integer optimization model which jointly schedule preventive maintenance and renewal projects. This paper includes two illustrative examples and a real-world case study to show the potential cost reductions that using our integrated model may result in. Example 1 (see Section 3.1) illustrates the benefits of grouping maintenance and renewal activities in multi-unit systems. Example 2 (see Section 4.3) visually demonstrates the impact of the model parameters such as the length of planning horizon and the system downtime cost on the structure of optimal preventive maintenance and renewal schedules. The rest of this paper is organized as follows. We review the related literature to the PMRSP in Section 2. In Section 3 the implications and importance of grouping and balancing are discussed and the model assumptions are presented. In Section 4, the mathematical model is investigated in detail. Numerical experiments with sensitivity analysis are carried out to gain more insight into the integrated model. Section 5 proposes heuristic algorithms to solve the problem and discusses the results of computational tests. The benefits of our integrated approach in cost savings are also reported. A real-world application of the proposed model and algorithms is presented. Finally, Section 6 concludes the research with summary and remarks. 2. Literature review Maintenance optimization models have been widely developed and used to optimize maintenance schedules for a variety of systems. Due to the complexity of analyzing systems with multiple units, most maintenance optimization studies consider a single-unit system with one or several components (Ko & Byon, 2016). The focus of the present research is on multi-unit systems with economically dependent components that provide the opportunity to group preventive maintenance and preventive renewal projects. The existing literature in this field is extensive, and it is beyond the scope of this article to discuss all the relevant research contributions. Instead, we refer the reader to related surveys for a 4

comprehensive review of previous work by Cho & Parlar (1991), Dekker et al. (1997), Wang (2002), Nicolai & Dekker (2008), and Van Horenbeek, Pintelon, & Muchiri (2010). Detailed description of maintenance policies such as failure-based maintenance and condition-based maintenance with different types of dependencies between the components are out of scope of this paper, as well as modeling component failure processes caused by internal-based deterioration and external environmental shocks. In the following, we aim to review the most recent studies, as well as others particularly related to our modeling and solution approach. 2.1. Group maintenance approaches Maintenance grouping can be divided into three categories: long-term (static), medium-term (dynamic) and short-term (opportunistic) (Chalabi et al., 2016). Static grouping assumes a stable situation with static rules for group maintenance throughout the planning horizon. It is suitable for strategic maintenance planning. In dynamic models, medium-term information such as new data on degradation and remaining useful life of components can be utilized to adapt maintenance planning. In dynamic grouping, planned PM activities can be combined with each other, and also with planned corrective maintenance activities. Opportunistic grouping accounts for short-term information (e.g. failures). Therefore, planned PM activities can be joint with each other, or even with unplanned corrective maintenance. The focus of this study is on the long-term grouping approach. The main reason for choosing this approach for the group maintenance is because of its practicality and the availability of the required data. Scheduling is the main approach used in static (stationary) grouping models. In this approach, it is typically assumed that the outcome and timing of the maintenance and renewal activities are known and deterministic. The idea of the scheduling approach is to plan maintenance or renewal projects over a finite horizon to minimize their processing costs and maximize the benefits of grouping them. Grouping of these projects can be done in particular ways; defining a set of activities that can be combined (e.g. Budai et al. (2006)), or following basic rules for grouping (e.g. Zhao, Chan, & Burrow (2009). Budai et al. (2006) considered a preventive maintenance scheduling problem for railway systems. They presented a mixed-integer linear programming model to group maintenance activities by minimizing the sum of the possession costs and the maintenance costs. The authors considered a maximum length for the planning of each routine work and the earliest and latest possible starting times for each maintenance project. They suggested some heuristic algorithms that determine the schedules of (cyclic) routine activities and (non-cyclic) unique projects for one track in a certain period. Compared to our problem, Budai et al. (2006) only considered the grouping of maintenance activities on a single track segment (one unit) in a network to reduce the disturbance of railway traffic. However, the railway track consists of several adjacent segments (units), and optimizing each segment independently from the adjacent segments could provide a non-optimum solution for the entire railway track. By using this approach and scheduling maintenance activities of each segment individually, the overall maintenance plan of the whole segments will be likely far from the optimal. Zhao et al. (2009) considered the problem of scheduling the renewal of track components within a fixed planning horizon for n segments of track. They developed a mixed integer program for maximizing the benefit of grouping renewal works of track components. They assumed that renewal activities could be combined if they had related components at the same segment or the same type of components at several adjacent segments. They assumed that the cost savings which resulted from grouping different renewal projects and the life cycle costs of individual components are known. Hence, the loss induced by the change of renewal time due to the grouping of the work can be calculated. They developed a genetic algorithm to solve the proposed problem. Caetano & Teixeira

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(2015) presented an optimization model that integrates ballast, rail, and sleeper degradation models in a mixed integer linear programming model. This model links the renewal decisions of these components with their condition and can be seen as an integrated version of the model presented by Zhao et al. (2009, 2009). Hou et al. (2016) developed a preventive replacement scheduling model for multicomponent systems with lifetime limited components. They formulated a mixed-integer non-linear mathematical model and solved it with a genetic algorithm to minimize total preventive replacement costs. In dynamic maintenance grouping models, the rolling horizon is the main approach used. It takes a static long-term tentative maintenance plan and updates it at regular time intervals by incorporating dynamic short-term information. Wildeman, Dekker, & Smit (1997) introduced the first rolling horizon approach to find the optimal maintenance planning in polynomial time. It has been shown that an optimal group must balance between the penalty costs due to the changes of tentative execution dates and the additional gain due to the changes of total planned shutdown costs. This research has drawn much attention and several works have been developed based on dynamic grouping approach. We can quote, for instance the work presented in Bouvard, Artus, Berenguer, & Cocquempot (2011), Van Horenbeek & Pintelon (2013a,b), Do Van, Barros, Bérenguer, Bouvard, & Brissaud (2013), and Do Van, Vu, Barros, & Bérenguer (2015). These articles optimally determine the grouped maintenance planning by minimizing similar penalty cost functions. The rolling horizon approach as a dynamic predictive maintenance policy can incorporate non-stationary situations with stochastic parameters such as the deterioration rate, threshold level of failure, and quality improvement after imperfect maintenance. As a grouping strategy, opportunistic maintenance seeks to take advantage of system downtime induced by failure or environmental conditions. This is done via conducting maintenance before the fixed PM time or condition threshold (Ab-Samat & Kamaruddin, 2014). Laggoune, Chateauneuf, & Aissani (2010) studied the opportunistic replacement policy for multi-component systems, where the expected total cost was minimized under a general lifetime distribution. Rokstad & Ugarelli (2015) implemented opportunistic maintenance in the renewal water infrastructure assets. They showed that more benefits are obtained by grouping the renewal of water pipes that are spatially close. Recent reviews on dynamic and opportunistic maintenance for multi-component systems can be found in Ab-Samat & Kamaruddin (2014) and Sheikhalishahi, and Azadeh, & Pintelon (2017). These previous works deal with economic dependence between system components (either explicitly or implicitly). As a consequence, the maintenance costs have a general economy-of-scale structure (Dekker et al., 1997). The term economies of scale is used to indicate that grouping maintenance activities is cheaper than performing maintenance on components separately (Maaroufi, Chelbi, & Rezg, 2013). This economic benefit mainly depends on the value of the setup cost. Hence, the grouping strategy has to be optimized with a more subtle compromise between setup cost savings, system downtime cost, and the structure of system (Vu, Do, Barros, & Bérenguer, 2014). In many industrial settings, complex setup structures exist and different setup activities have to be performed at various levels before maintenance can take place. However, there are few examples of modeling maintenance of multi-component systems with multiple setup activities and a hierarchical (tree-like) setup structure. These setup activities may be combined when several components are maintained concurrently. Kobbacy & Murthy (2008) specifically state about models that consider multiple setup activities over a finite horizon: “We have found one article in this category ... this is the first attempt to model a maintenance problem with a hierarchical setup structure.” In that article, van Dijkhuizen (2000) studied the problem of clustering preventive maintenance jobs in multiple setups multi-component production system. He models the multi-component production system as a tree with leaves representing components that require maintenance within specified frequencies. Each component is maintained preventively at an integer multiple of a certain basis interval, and corrective 6

maintenance is carried out in between whenever necessary. An integer programming formulation is then used to find the maintenance frequencies that minimize the average cost per unit of time. 2.2. Replacement models In general, all infrastructures and system components deteriorate with time and use, and the need for maintenance increases during the final periods of their technical life (usually based on safety criteria) (Gustavsson, Patriksson, Strömberg, Wojciechowski, & Önnheim, 2014). However, in some cases it may be economical to renew the component earlier than its technical life. There are several ways to determine the economic life of a system component. Wagner (1975) presented a dynamic programming formulation in which the state of the system is the time period. The decisions are no longer whether to keep or replace the asset, but rather, the number of periods to maintain the equipment. The Wagner formulation has been extended by researchers to deal with constraints and conditions such as demand, capital budgeting and technological advances (Oakford, Lohmann, & Salazar 1984; Bean, Lohmann, & Smith 1994; Hartman 2000). Recent studies in the area of the equipment replacement problem have considered continuous and discontinuous technological change ( Rogers & Hartman 2005; Hartman & Rogers 2006). Previous work on equipment replacement problems also examines capital replacement decisions (Hartman & Tan 2014; Jardine & Tsang 2013). They summarize the literature in the vast area of equipment replacement decisions, including technological change for finite and infinite planning horizons, variable utilization patterns and various uncertainties. Ke & Yao (2016) studied the block replacement policy with uncertain lifetimes to obtain the optimal scheduled replacement time. Block replacement policy is a type of a preventive replacement policy, in which the units are always replaced at failure or at a scheduled time periodically. A comprehensive review of asset replacement models and methods can be found in Yatsenko & Hritonenko (2016). These studies give relevant insight to our problem, but they do not consider maintenance activities, which clearly impact replacement schedules. These studies examine cases in which expected life-cycle costs over some horizon are estimated, and the ensuing problem is solved by defining a replacement schedule for the asset. Unlike previously published renewal-based studies, this paper determines the economic life (renewal time) of system components in multi-unit systems by finding the optimal balance between maintenance and replacement while considering their effect on each other. 2.3. Maintenance and renewal of multi-unit systems In the maintenance literature, the terms multi-unit and multi-components are sometimes used interchangeably. However, in this study, they have their own definitions. Multi-unit maintenance models are concerned with optimal maintenance policies for a system consisting of several multicomponent systems in series. One can observe an increasing interest in recent publications to the optimization of maintenance policies for multi-unit systems (Shafiee & Finkelstein, 2015). The selection of optimal maintenance policies for multi-unit systems is usually more complex than for single-unit systems. The reason for this complexity is that there often exists one or more types of dependence between the components in a multi-unit system (Nicolai & Dekker, 2008). Among these types of dependencies, the economic dependence has been addressed the most in the literature and is also used in this study. Moghaddam & Usher (2011a) declared that there are few works in the integration of preventive maintenance optimization and equipment replacement problem, especially for multicomponent systems. Here, we review the studies which considered preventive maintenance and replacement decisions simultaneously. Usher, Kamal, & Syed (1998) presented an optimization maintenance and replacement model for a single-component system. They determined an optimal preventive maintenance schedule for a new

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system subject to deterioration, increasing rate of occurrence of failure over time and the use of an improvement factor to provide for the case of imperfect maintenance actions. They also compared the computational results among random search, genetic algorithm, and branch and bound algorithms. Moghaddam & Usher (2011a) proposed a model to schedule preventive maintenance and replacement activities for a multi-component system. They assumed that each maintenance activity, effectively reduces the age of component and the failure rate becomes smaller than the previous cycle. They presented two non-linear models by considering several costs (such as failure cost, fixed cost, maintenance cost, and replacement cost), and solved the model with dynamic programming. By considering a fixed cost of “downtime”, the optimal time to perform maintenance/replacement activities on individual components is dependent upon the decision made for other components. There are also other works that studied the proposed problem by Moghaddam & Usher (2011a). Moghaddam & Usher (2011b) performed a sensitivity analysis for the same problem by investigating the effect of parameters on the structure of the solutions. Moghaddam & Usher (2011c) developed a multi-objective version of the same problem. Moghaddam & Usher (2010) modeled an improvement factor based on the ratio of maintenance and repair costs, and showed how it outperforms fixed improvement factor models by analyzing the effectiveness in terms of cost and reliability of the system. To summarize the literature review, most of the previous research deals with solution methodologies and algorithms developed for a particular context, such as those found in manufacturing or railways. Compared to the papers found in the literature, our research presents a more holistic view by developing an integrated scheduling approach for preventive maintenance and renewal planning of multi-unit systems. This is done by integrating resource utilization techniques such as grouping and balancing through proper scheduling. 3. Problem description In this section, the ideas behind grouping and balancing maintenance and renewal projects are discussed followed by illustrative examples. Later, the general assumptions of this study are explained. 3.1. Setup costs structure for group scheduling In this paper, we will speak of economies of scale by proposing a hierarchical (tree-like) setup structure for modeling maintenance of multi-unit systems. We include setup costs such as downtime cost, equipment preparation cost, and equipment installation cost as criteria for group scheduling. By considering these setup costs, renewal and maintenance projects can be combined if they are related to different components of the same unit (grouping in time) or the same type of components in several adjacent units (grouping in space). Three criteria for group scheduling are listed as follows: 1) Grouping with similar service interruption: There is a high fixed setup cost in each period for carrying out maintenance and renewal projects. This cost is known as downtime cost due to production loss and is not dependent on the type and the number of maintenance/renewal activities that are carried out in a given period. This setup cost creates an incentive to minimize the number of periods the system is down due to maintenance. 2) Grouping with identical unique maintenance/renewal machinery: There is a fixed setup cost in each period for preparing the required equipment for carrying out renewal/maintenance of each component. This cost is not dependent on the number of units that are maintained/renewed in that period. This setup cost creates an incentive to increase the number of activities which required the prepared equipment.

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3) Grouping with geographically adjacent units: There is a fixed setup cost in each period for installation of the required equipment for carrying out renewal/maintenance of each component in non-adjacent units. This cost is paid only once in carrying out renewal/maintenance of components in the adjacent units. Therefore, it creates an incentive to carry out activities in a long length of adjacent units. Example 1. Here, an illustrative example of a multi-unit system is considered to show the benefits of grouping projects. In this example, three geographically adjacent units with two components in each are considered. Figure 3 shows three different scenarios for carrying out the renewal of component 1 in all units and the maintenance of component 2 in the units 2 and 3, where r1 is the renewal of component 1 and m2 is the maintenance of component 2. All the cost data relevant for this example are included in Table 1. Table 1. The cost data set of Example 1.

Downtime cost of system (Euro/period) Preparation cost of equipment (Euro/period) Installation cost of equipment (Euro/unit) Processing cost (Euro/unit)

Renewal of component 1 (r1) 100 30 5 25

Maintenance of component 2 (m2) 100 20 5 10

Figure 2. An illustration of three different scenarios for grouping maintenance and renewal projects.

In Figure 2(a), there is no grouping and the system is down due to maintenance or renewal activities in all of the periods. As can be seen in Figure 2(b), one method to gain saving in costs is to group the activities in the adjacent units when the preparation cost of equipment (machinery cost) is paid. As a result, besides the equipment preparation cost, the installation cost of the equipment in non-adjacent units will be reduced. In Figure 2(c), all the maintenance and renewal activities are grouped in the first period, and so the interruption for customers is minimized, and the cost saving gained by combining activities is maximized. However, some components are maintained when they have passed only a fraction of their expected life. In Figure 3, a summary of the above-discussed setup costs is converted into a so-called maintenance tree. We have formulated the hierarchical structure of all possible setup properties. The root of this maintenance tree corresponds to the downtime of the overall system due to maintenance; the Leafs represent the preparatory works for renewal and maintenance activities.

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Figure 3. A maintenance tree for Example1. Setup costs are shown at the arcs; maintenance costs are shown in brackets at the corresponding nodes.

In this paper, we have also considered another criterion for group scheduling of renewal activities. In fact, in many real cases, there exist a special machinery for renewal of all component of a system together. We assume that the equipment preparation cost for special renewal machinery is less than the sum of preparation costs for renewal of each component. 3.2. Balancing between maintenance and renewal Balancing is novel technique to reduce PM costs by determining the economic life of a system component. In general, a system component deteriorates with age, and it is needed to increase the frequency of maintenance activities during final periods of its technical life; technical life is determined based on safety criteria (Gustavsson et al., 2014). By balancing, it may be economical to renew the component earlier than its technical life to prevent successive maintenance activities or to be able to combine its renewal with other renewal activities. Figure 4 represents the idea behind achieving a balance between maintenance and renewal in an example. In Figure 4, we have considered one component in one unit, and so there in no incentive to perform maintenance before its due date (no grouping). We assume that the latest possible time for carrying out the next maintenance relative to the previous one is known. These maximum time intervals between subsequent maintenance activities are known as PM intervals and may have a decreasing pattern with increasing the number of maintenance activities performed on the component. Here, we also assumed that component must be replaced after five maintenance due to safety problems.

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Figure 4. Maintenance schedules with different renewal times.

As can be seen in Figure 4 (a), the component is renewed at the end of its technical life, and so we should perform six maintenance activities over the planning horizon. However, as can be seen in Figure 4 (b), we reduce the required number of maintenance within the planning horizon to four activities by renewing the component on its economic lifetime. Consequently, maintenance costs can be reduced by finding the optimal balance between maintenance and renewal. Our proposed model is called an integrated optimization model because it not only combines the execution of maintenance and renewal projects but also determines the optimal renewal time. Therefore, the renewal of a component can be carried out before its technical lifetime for two reasons: (1) to group it with other activities; and (2) to perform it on its economic lifetime. 3.3. Assumptions The main assumptions of PMRSP are listed as follows: A1. All maintenance and renewal projects can be performed in discrete time periods on each unit. A2. It is assumed that the maintenance projects are imperfect and the deterioration rate of components can differ in different units. The deterioration rate of each component may increase with time and after each intervention. Therefore, maintenance activities could be applied with increasing frequency during the final phases of a component’s life. A3. The latest possible times between maintenance/renewal activities for each component are known. These time intervals may not be fixed and could have a decreasing order. A4. The maintenance and renewal of components can only be performed earlier to be combined with the maintenance and renewal of other components in other units. A5. There is a maximum allowed number of times that maintenance on each component can be performed. A6. System downtime cost, due to maintenance/renewal, is independent of component type but dependent on the location of the unit, its importance, and availability of an alternative. A7. All the parameters of the model are deterministic. It is assumed that the randomness has been already covered in a prior phase of this research paper. For example, failure modeling and

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reliability analysis are not covered in this study and have been included in the calculation of the latest possible times between maintenance/renewal activities. 4. Problem formulation In this section, we present a mathematical formulation of our problem. Problem parameters and decision variables will be introduced first. Then an integer programming model will be presented. Finally, numerical experiments with sensitivity analysis are carried out to analyze different features of the developed model. 4.1. Notation The sets, parameters, and variable that are used in developing the mathematical model are described below. Sets I K T

Set of components, with its elements numbered for convenience from 1 to |I| Set of units, with its elements numbered for convenience from 1 to |K| Set of (time) periods, with its elements numbered for convenience from 1 to |T| Set of maintenance intervals belonging to component i, with its elements numbered for convenience from 1 to

Input parameters Maintenance cost of component i in each unit Renewal cost of component i in each unit System downtime cost due to maintenance or renewal activities in each period Equipment preparation cost for renewal of component i in each period Equipment preparation cost for maintenance of component i in each period Equipment installation cost for renewal of component i in each period Equipment installation cost for maintenance of component i in each period Equipment preparation cost for special renewal machinery (which can renew all components together) in each period Maintenance history; it indicates the number of maintenance activities performed since the previous renewal of component i in unit k before the planning horizon starts Maintenance history; it indicates the number of periods elapsed since the previous maintenance of component i in unit k before the planning horizon starts Latest possible time for carrying out hth maintenance for component i after the h-1th maintenance (where h=1, it means latest time to perform first maintenance after renewal) Decision variables Binary variable, 1, if renewal of component i in unit k is scheduled in period t, and 0 otherwise Binary variable, 1, if hth maintenance of component i in unit k is scheduled in period s after renewal in time t, and 0 otherwise When t=0, it means the renewal has been performed before the planning horizon starts Binary variable, 1, if the system is down in period t due to maintenance or renewal activities, and 0 otherwise Binary variable, 1, if renewal of component i is carried out in both units k and k-1 in period t, and 0 otherwise

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Binary variable, 1, if maintenance of component i is carried out in both units k and k-1 in period s, and 0 otherwise Binary variable, 1, if machinery for maintenance of component i is needed in period s, and 0 otherwise Binary variable, 1, if machinery for renewal of component i is needed in period t, and 0 otherwise Binary variable, 1, if special machinery is needed for renewal of all component together in period t, and 0 otherwise 4.2. Mathematical model The complete model of PMRSP can be written as follows: minimize (1.1) (1.2)

(1.3)

subject to (2) (3) (4) (5) (6) (7) (8) (9) (10)

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(11) (12) (13) The objective (1) is to minimize the sum of maintenance and renewal processing costs and their relevant setup costs such as possession cost, the cost of preparing and installing the machines. The first term, i.e. (1.1), calculates the total downtime cost of the asset for carrying out renewal and maintenance activities. The second term calculates the total machinery cost. The third term calculates both the installation cost of machines and the total processing cost of renewal and maintenance in all periods. Constraints (2) guarantees that in the first truncated interval, there must be a plan for at least one maintenance or renewal activity. Constraint set (3) ensures that the schedule will include a maintenance or renewal activity by the end of the first planning interval after the renewal. The planning intervals are the time intervals between sequential maintenance activities, which are given as an input. Considering an upper bound for t ensures that there is no necessity to plan a maintenance or renewal activity in the final planning interval. Constraint set (4) makes the next maintenance or renewal dependent on the previous maintenance. Constraint (5) ensures that a renewal will be carried out after reaching the end of each component’s technical life. Constraints (6) and (7) together with the objective function prevent the paying for an installation cost of components that are renewed/maintained in the adjacent units. Constraint sets (8) and (9) ensure that period t will be occupied for preventive renewal or preventive maintenance works if -and only if- for that period at least one project is planned. Constraint set (10) defines the type of machinery needed in each period for carrying out the maintenance projects. The relevant machinery for maintenance of component i is required in period t if -and only if- for that period, at least one activity is planned. Constraint set (11) defines the type of machinery needed in each period for carrying out the renewal activities. The value M in the constraints (8)-(11) is an upper bound on the left side of the constraints, and so M has to be large enough not to limit the left side of the constraints. To prevent the drawbacks of using one very large M for all the constraints, we have cut down M down to the size of each constraint. Constraint set (12) guarantees that when special machinery is used for renewal of a unit in each period, all the components are renewed together. Constraints (13) represent the conditions on the decision variables. 4.3. Numerical example with sensitivity analysis Example 2. This example aims to present and evaluate the results of a sensitivity analysis on the developed optimization model. The following experiments (scenarios) investigate the effect of the parameters on the structure of optimal preventive maintenance and renewal schedules in multi-unit systems. This example includes two components, three units with the number of allowed maintenance projects being equal to two. The data set relevant to this example is shown below in Table 2. Here, first, we consider different scenarios to illustrate the impact of parameters such as the length of planning horizon and the system downtime cost on the optimal schedule. Then, we consider the effect of balancing and grouping on the final solution. These illustrative examples are solved by root node processing and branch-and-cut methods using IBM ILOG CPLEX Optimization Studio 12.6. Table 2. Input parameter settings of test instances. Parameter

Value

Number of components Number of units Number of allowed maintenance

I=2 K=3 H=2

14

Number of time units Maintenance cost (Euro/unit) Renewal cost (Euro/unit) Downtime cost (Euro /period) Preparation cost of renewal equipment (Euro/period) Preparation cost of maintenance equipment (Euro/period) Installation cost of renewal equipment (Euro/unit) Installation cost of maintenance equipment (Euro/unit) Preparation cost of special equipment for renewal of all components together (Euro/period) Latest possible time for the next maintenance (year) Number of performed maintenance after the last renewal (year) Elapsed period after the last maintenance (year)

T= 8 or 16 = [40 60] = [300 600] = 500 or 1000 = [15 20] = [10 15] = [10 10] = [5 5] = 30

=[[3 1] [2 2]] =[[0 0 0] [0 0 0]] =[[2 1 1] [1 1 0]]

Figure 5 shows the maintenance history in each unit as an input parameter. For example, the last performed maintenance activity for component 1 in unit 1 was a renewal on two-time periods before the planning horizon. As can be seen in Table 2, the latest possible times between maintenance activities of each component are given in matrix . If the planning of each component of each unit is considered individually, then the best solution is to schedule the maintenance activities at the latest possible time (there is no basis for grouping); this is shown in Figure 5. In the solution shown in Figure 5, there is no optimization problem to be solved (i.e. without any grouping and balancing), and the schedule of maintenance and renewal activities is locally optimized for each component-unit combinations individually. As can be seen in Figure 5, some activities are combined in time anyway due to the structure of the problem, but still the downtime of the system is high (the whole system will be down in 14 time units).

Figure 5. Scenario 0- the sequence of maintenance and renewal projects without optimization.

One interesting aspect of this type of modeling is that one can analyze the effect of length of planning horizon and the effect of system downtime cost on the optimal sequence of maintenance and renewal activities. Two factorial design experiments are designed to find the effect of these parameters on the structure of the optimal schedule. We solve the above example using the proposed mathematical model while considering two different planning horizons (T=8 and T=16) and two different system downtime costs ( = 500 and = 1000). Therefore, we experiment four different scenarios to evaluate the impact of increasing the length of planning horizon and system downtime cost. Tables 3-6 represents the optimal solutions of the proposed scenarios. Table 3. Scenario 1- optimal maintenance and renewal schedule with

Unit 1

Year 1 -

2 3 m1 r1 r2 m2

4 5 6 7 8 - - - m1 - - m2 - -

15

= 500 and T=8.

2

-

r2

r1 m2

-

-

- m1 m2 -

-

3

-

-

r1 m2

-

-

- m1 m2 -

-

Table 4. Scenario 2- optimal maintenance and renewal schedule with

Unit

= 500 and T=16.

Year 1 -

2 3 m1 m1 m2 -

4 5 6 7 8 9 10 11 12 13 14 15 16 - r1 - - - m1 r1 - m1 - m2 - - r2 - m2 - m2 -

2

-

- m1 m2 -

-

r1 m2

-

-

- m1 r2 -

-

r1 m2

-

-

- m1 m2 -

-

3

-

- m1 m2 -

-

r1 m2

-

-

- m1 r2 -

-

r1 m2

-

-

- m1 m2 -

-

1

Table 5. Scenario 3- optimal maintenance and renewal schedule with

Unit

Year 1 -

2 3 4 m1 - r1 m2 - r2

5 6 7 8 - - m1 - - m2 -

2

-

m1 m2

-

r1 r2

-

-

m1 m2

-

3

-

m1 m2

-

r1 r2

-

-

m1 m2

-

1

= 1000 and T=8.

Table 6. Scenario 4- optimal maintenance and renewal schedule with

Unit

= 1000 and T=16.

Year 1 -

2 3 4 5 6 7 8 r1 - - r1 - - r1 m2 - - m2 - - r2

9 10 11 12 13 14 15 16 - r1 r1 - - m2 - m2 -

2

-

r1 m2

-

-

r1 m2

-

-

r1 r2

-

-

r1 m2

-

-

r1 m2

-

-

3

-

r1 m2

-

-

r1 m2

-

-

r1 r2

-

-

r1 m2

-

-

r1 m2

-

-

1

As can be seen in Tables 3-6, increasing system downtime cost (in general) leads to a decrease in the number of performed maintenance activities. Consequently, the frequency of renewal activities and their relevant costs will be increased. This increment in the number of renewals is because the time interval between maintenance activities could be increased after the renewal, and there would be less necessity to make the system down for carrying out maintenance activities. By increasing the number of periods (while downtime cost is fixed), the sequence of maintenance and renewal activities will be modified to minimize the total cost of the fixed time windows. It is worth to note that scenario 1 and scenario 3 have several optimal solutions so the maintenance manager can select the best-fitting solution. Table 7 shows the costs of scenarios 1-4 grouped by maintenance cost, renewal cost and their relevant setup costs.

16

Table 7. The cost of scenarios 1-4 for analyzing the effect of planning horizon (T) and downtime cost (

).

With grouping and balancing

Costs

T=8

T=16

= 500 520 2100 2125 4745 3

Maintenance cost Renewal cost Setup costs Total cost Computation time (sec)

= 1000 600 2700 3120 6420 2

= 500 1120 3600 4220 8940 3267

= 1000 720 6300 5230 12250 756

As can be seen in Table 7, the required runtime to solve these problems to optimality is highly dependent on the number of periods within the planning horizon. By increasing the number of periods from 8 to 16, the number of variables increases from 1025 to 3585 and the number of constraints is also increased from 787 to 3131. Therefore, the size of the problem dramatically increases with the length of planning horizon in a stepwise manner which is relevant to defining an upper bound for t and u indices in the constraints (3), (4) and (5). Here, two more experiments are conducted to show the significance of the main two features of the problem (i.e. grouping and balancing) on the schedule of preventive maintenance and renewal activities. Important feature presented in Table 8 is the effect of (just) balancing on the number and frequency of maintenance and renewal activities over the planning horizon. In obtaining this solution, it is assumed that maintenance and renewal of components cannot be advanced to be combined with the maintenance and renewal of other components in other units. Table 9 shows the effect of (just) grouping on the number and frequency of maintenance and renewal activities over the planning horizon. In obtaining the solution shown in Table 9, it is assumed that renewal of components cannot be carried out before their technical lifetime. Table 8. Scenario 5- optimal schedule while only balancing is considered (T=16 and

Unit

Year 1 -

2 r1 m2

2

-

- m1 m2 -

3

-

1

= 500).

-

3 -

m1 m2

4 5 6 - - m1 - m2 -

7 8 - r1 - r2

9 -

10 11 12 13 14 - m1 r1 - m2 - m2

-

r1 m2

-

-

r2

m1 -

-

r1 m2

-

-

-

r1 -

m2

-

-

m1 r2

-

r1 -

m2

-

15 -

- m1 m2 -

m1 m2

16 -

Table 9. Scenario 6- optimal schedule while only grouping is considered (T=16 and

Unit

Year 1 -

2 m1 m2

3 -

4 5 6 7 8 m1 r1 - - - m2 - - r2

9 10 11 12 13 14 m1 - m1 - r1 - m2 - m2

2

-

m1 m2

-

m1 r1 - m2

-

-

r2

m1 -

-

m1 m2

-

r1 -

3

-

m1 m2

-

m1 r1 - m2

-

-

r2

m1 -

-

m1 m2

-

r1 -

1

= 500).

15 -

16 -

m2

-

-

m2

-

-

The results of scenarios 5 and 6 show that the downtime of the system is increased if only balancing is considered. Likewise, the number of maintenance activities is increased if only grouping is considered. Table 10 illustrates the costs of different scenarios with integrated and separated balancing and 17

grouping while T=16 and = 500. The costs are grouped by maintenance cost, renewal cost and their relevant setup costs. Figure 6 below demonstrates the effect of different setup costs in the scenarios with integrated and separated balancing and grouping. Table 10. The cost of scenarios 0, 2, 5, and 6 for analyzing the significance of balancing and grouping.

Costs Maintenance cost Renewal cost Setup costs Total cost Computation time (sec)

No optimization Without grouping & balancing (Scenario 0) 1200 3600 7440 12240 0

Separated optimization Just Just balancing grouping (Scenario 5) (Scenario 6) 1040 1200 3900 3600 5420 4220 10360 9020 2 35

Integrated optimization With grouping & balancing (Scenario 2) 1120 3600 4220 8940 3267 a

a

This solution was obtained for the first time after 4 seconds while the gap was 60%. However, it takes 3263 seconds to prove the optimality of this solution.

Figure 6. Detailed view of setup costs for different scenarios with integrated and separated balancing and grouping while T=16 and = 500.

The cost effect of integrated optimization can be seen in Table 10 and Figure 6. As can be seen in Figure 6, the setup costs are the same in scenarios 2 and 6. Nevertheless, the maintenance cost of scenario 6 is higher than scenario 2 due to two extra maintenance activities for component 1 in units 2 and 3. Experimental results show that the proposed techniques (grouping and balancing) perform well in reducing setup costs and eliminate the need for setups through proper scheduling. The reduction in the costs as a consequence of using our developed model is based on reduced amount of infrastructure possession (system downtime) for carrying out maintenance activities and the efficient use of resources and machines. 5. Numerical results and discussion This section explains our solution approach and numerical experiments. The PMRSP is implemented using the IBM ILOG CPLEX Optimization Studio 12.6 in an Intel Core i7 [email protected] gigahertz, with 16 gigabytes RAM running 64-bit Windows.

18

5.1. Solution approach In this section, we present a solution approach based on decomposition technique. The following heuristics are developed based on the structure of the problem. These heuristic methods decompose the proposed problem into less complex sub-problems and then solve them sequentially. In the literature, the idea of these heuristics is sometimes called decomposition approach. The focus in the first step of the heuristics is in the individual plan of each component in each unit. That is, we find the best sequence of actions for a component regardless of the actions taken to the other components and units. This would result in independent optimization problems. As a result, the grouping of maintenance and renewal activities is not considered. The grouping of projects is added later to the schedule in the next steps of the developed heuristics. Heuristic 1: bounding the first renewal time Step 1: Determine the optimal renewal time for each component of each unit individually. Consequently, the grouping of maintenance and renewal activities are not considered. Step 2: Add a new constraint for bounding the first renewal time in each component-unit combination. For this aim, the first obtained renewal time from the previous step is considered as the latest possible time for performing the first renewal. Step 3: Solve the general model with considering the obtained constraints in Step 2. Totally, constraints will be added to the general model. Heuristic 2: warm start Step 1: Determine the optimal renewal time for each component of each unit individually. Consequently, the grouping of maintenance and renewal activities are not considered. Step 2: Solve the general model with providing the obtained solution from the previous step as a starting search point for MIP optimization. To improve the performance of the PMRSP, we added to the model several cuts based on certain logical aspects of the model. These constraints are listed below: (14) (15) (16) Constraints (14) ensure that both maintenance and renewal of a component cannot be performed together. Constraints (15) and (16) guarantee that each maintenance position is performed in each period once at most. More details about performances are presented in Section 5.3. 5.2. Test instances We define three problem sets (PS): PS1, PS2, and PS3. Table1 lists the settings used to modify the structure of multi-unit systems for each of the three sets. The generated values for different parameters of the problems are shown in Table 11. We generated nine problems by considering three different system downtime costs ( = 100 or 500 or 1000) and three different periods (T=8 or 16 or 24) for each problem set. We tackled each instance three times to yield more reliable information.

19

Table 11. Parameters settings or sampling range for PS1, PS2, and PS3. Parameter

Setting PS1/PS2

Number of components Number of units Number of allowed maintenance Maintenance cost (Euro/unit) Renewal cost (Euro/unit) Preparation cost of renewal equipment (Euro/period) Preparation cost of maintenance equipment (Euro/period) Installation cost of renewal equipment (Euro/unit) Installation cost of maintenance equipment (Euro/unit) Preparation cost of special equipment for renewal of all components together (Euro/period) Latest possible time for the next maintenance (month) Number of performed maintenance after the last renewal (month) Elapsed period after the last maintenance (month)

I K H

PS3

a

3 3 2 [20, 80] [200, 600] [10, 40] [10, 20] [5, 15] [3, 10] 30

4 6 4 [150, 300] [400, 800] [200, 400] [100, 300] [30, 70] [15, 40] 1000

[1, 5] [0, H]

[1, 5] [0, H]

[0,

]

[0,

]

a

For PS1 I is equal to 2. The real-size number of components is equal to 3 for the application areas of the developed model.

5.3. Results for the proposed algorithms The performance of the solution methods and the integrated optimization model are demonstrated in Table 12. The objective function values, after running the model for medium and large size problem instances proposed in Section 5.2, are shown in Table 12. We have also considered several criteria to compare the developed heuristics (See Table 13). We applied a time limit of 60 minutes to compare the standard CPLEX and the heuristic methods; this resulted after testing on some candidate values and analyzing the objective function improvement by increasing the running time. As it was expected, it can be seen that the optimal solutions of the integrated planning problem are never worse than solutions of the separated planning problem. Table 12. The objective values of the integrated optimization model and the heuristic methods against the separated planning. The best solutions are in bold, and the solutions marked with an asterisk indicate non-zero gap1. Problem set

Parameter

T=8

PS1

T=16

Separated optimization Just Just balancing grouping 3325 3245

CPLEX

Heuristic1

Heuristic2

3145

3145

3145

5820

5740

5740

5740

10495

9780

*

9665

*

9665

9665*

5325

4845

4745

4745

4745

10360

9020

8940 14445

8940 14365*

8940 14365*

6420

6420

6345

16910 T=24

1

Integrated optimization

14580

*

7825

6420

6420

15360

13020

12250

12250

12250

24910

20445

19460*

19460*

19465*

Gap= |bestbound-bestinteger|/ |bestinteger|

20

T=8

PS2

T=16

T=24

T=8

PS3

T=16

T=24 Total average

6080 11910 18445 9055 17225 26055 12555 23725 35555

5740 11770 17700 7640 15470 23235 9640 19970 30370

5495 11095* 17350* 7095 16220* 22500* 8885 18875* 28905*

5495 11075* 17190*

5495 11075* 17340*

7095 14480* 24200*

7095 14400* 22140*

8885 17865* 27885*

8885 17925* 28785*

7450 15730 24130 12660 22930 34030

6380 17200 24990 9080 22380 32190

6050 15760* 23940* 8750 17200*

6050 14960* 24130*

6050 14960* 22850*

8750 18560*

8750 17200*

32660 54930

21080 42380

78030

62540

30870* 18730 32950 48180*

28180* 18730 33050* 48180*

28010* 18730 32950 50380*

20148

17293

15691

15536

15472

The average runtime for solving the separated optimization with balancing is about 2 seconds while the average runtime for solving the separated optimization with grouping is about 1438 seconds. This shows that grouping is more time consuming than balancing. In fact, when the number of periods is equal to 24 even in the separated optimization problems with grouping, the optimality of the obtained solutions cannot be proven within the one-hour time limit. In Table 13, we show for each optimization method, the average (sub-optimal) cost of each method as compared with the best solution among the methods (error). We also show what percentage of the instances each algorithm gives the best (or equal to the best) solution among the others. Also, the average runtime, gap and the first time the solver reached the best-known solution of each method are given for each problem set in Table 13 below. Since the results of problems with T=8 are obtained within a few seconds, we do not consider them in the calculation of the average runtime and the average first-time that solver reaches the best-obtained solution. Table 13. Average total cost error for each optimization method. The percentage number of instances in which each method gives the best solution. The difference between the best integer objective and the objective of the best node remaining. The first time the solver reaches the best-known solution. The runtime shows the computation time required to solve each problem given the time limit of 3600 seconds. Problem set

PS1

Average error (%) Best solution (%) Average gap (%) Average first-time (sec) Average runtime (sec)

CPLEX 0.0006 88.89 27.35 532.36 2898.13

Heuristic1 0.0000 100 14.85 793.33 2375.67

Heuristic2 0.0000 88.89 16.89 539.84 2382.16

PS2

Average error (%) Best solution (%) Average gap (%) Average first-time (sec) Average runtime (sec)

0.0274 33.33 38.25 1792.83 3600

0.0110 77.78 26.37 1869 3600

0.0049 66.67 28.97 2047.50 3600

21

PS3

Average error (%) Best solution (%) Average gap (%) Average first-time (sec) Average runtime (sec)

0.0226 66.67 34.30 2228.25 3151.89

0.0160 55.56 25.12 2412.50 3600

0.0051 88.89 26.53 1802.50 3330.17

Total average

Average error (%) Best solution (%) Average gap (%) Average first-time (sec) Average runtime (sec)

0.0169 62.97 33.14 1517.81 3216.67

0.0090 77.78 22.11 1691.61 3191.88

0.0034 81.49 24.13 1463.28 3104.11

The results, as shown in Table 13, indicate that Heuristic 1 outperforms Heuristic 2 in PS1 and PS2. Although in this problems Heuristic 1 has a greater average error, it reaches the best solution in a larger number of instances. In PS3, Heuristic 2 not only gives the best solution in 88.89% of the problems but also has the least average error. The results show that the best solutions are reached in 81.49% of the problems by using Heuristic 2 in comparison with Heuristic 1 which gives the best solution in 77.78% of the problems. We cannot draw a specific conclusion under which condition each of these optimization methods performs best. However, Heuristic 1 performs best in the instances with a large number of periods and high system downtime cost. In Table 13 above, we show the total average gap of CPLEX was 33% after the 60 minutes time limit, and the total average gap of Heuristic 1 and Heuristic 2 were respectively 22.11% and 24.13%. To check whether there is any better solution for these problems, we extend the time limit to 5 hours on PS2. By increasing the time limit, the average gaps of the heuristics decreased below 10% without finding better solutions (detailed results for those tests are not depicted here). It is worth to note that the best-found solutions of Heuristic 1 and Heuristic 2 were obtained, respectively, after 31 and 34 minutes. The analysis shows that in a significant proportion of the solution time, solver tries to prove the optimality of the best-obtained solution by increasing the bound of the objective function and decreasing the gap. The problem studied in this paper is NP-hard, and it is in general not time efficient to solve this type of problems using a MIP-solver. However, the PMRSP is not a daily problem. In fact, the application of our proposed model is at the strategic level with time horizons of one to several years. Therefore, it is acceptable to reach (near) optimal solution within several hours runtime. As explained in Section 5.1, our developed heuristics decompose the proposed problem into less complex sub-problems and then solve them sequentially by using the branch and bound method. The results, as shown in Table 13, indicate that the average runtime of the heuristics is less than the standard CPLEX. Furthermore, heuristics reaches to better solutions with less gaps in the proposed time. In summary, as we discussed before, both the developed heuristic methods are possible options to optimize the problem and for problems with a large number of periods and high system downtime cost, using Heuristic 1 is highly recommended. In addition, in many real cases, there is no integrated optimization tool for grouping and balancing decisions. In these cases, making maintenance grouping decisions and finding the optimal renewal times are not taken simultaneously, and maintenance managers just evaluate a limited number of simple scenarios and select the best. 5.4. Cost saving of the integrated optimization In this section, we show how the maintenance grouping and the optimal renewal decisions (balancing) affect the optimization results. The developed integrated optimization approach is compared with the separated optimization problems, where grouping and balancing are done separately. The cost savings of integrated optimization are shown in Table 14.

22

Table 14. Average and maximum integrated optimization cost saving compared to the separated planning (i.e. just balancing, and just grouping). Problem set PS1 PS2 PS3

Integrated optimization V.S. just balancing Average cost saving Maximum cost (%) saving (%) 13.62 21.86 16.49 29.23 24.02 42.65

Integrated optimization V.S. just grouping Average cost saving Maximum cost (%) saving (%) 2.31 5.91 6.03 10.24 12.78 23.15

As can be seen in Table 14, the cost savings gained from grouping are significantly greater than the savings from balancing. In addition, cost savings increase considerably when the problem size increases. The significance of the integrated optimization model in cost reduction can be explicitly seen in cases which the cost of preparatory works are high (high setup costs). 5.5. Case study on railway track Railway track as a multi-unit system includes several adjacent units that are called segments. A segment can be defined as a specific length of track where the traffic, track conditions, and component types are all uniform. Therefore, segments can have non-fixed lengths. Rail, ballast, and sleepers are the three major components in conventional railway tracks. In general, the components deteriorate at different rates and the criteria for their maintenance and renewal are different. The main activities in the maintenance of rail and ballast are called, respectively, rail grinding and ballast tamping, and there is no maintenance activity relevant to sleepers. Rail grinding operation extends the service life of rail by removing the rolling contact fatigue layer on the running surface of the rail and preventing the shelling damage. Ballast tamping is conducted to correct the longitudinal profile, the cross-level and the alignment of the track. In general, track maintenance works are planned well in advance and, when possible; railway infrastructure managers try to schedule these interventions during periods of low activity or no activity. When this is not possible, the train schedule needs to be adapted, and this leads to high system downtime cost (Vansteenwegen, Dewilde, Burggraeve, & Cattrysse, 2016). As shown by Nicolai & Dekker (2008), case studies are not well represented in maintenance optimization literature, although maintenance is something that should be done in practice and not in theory. Case studies are often only used to demonstrate the applicability of a developed model, rather than finding an optimal solution to a specific problem of interest to a practitioner (Van Horenbeek et al., 2010). In conducting our research, we had an opportunity to collaborate with the maintenance contractors of railway infrastructures through our research, and discuss our assumptions, analyses, and results with them. As discussed in Section 2.2, the scheduling approach is typically used in stationary deterministic models whereas the rolling horizon approach is typically used in dynamic stochastic models. In this study, we chose scheduling approach because of its practicality and the availability of the required data. We skip focusing on the rolling horizon approach because infrastructure managers have little flexibility in readapting the maintenance plan and rebalancing time for operation and maintenance. For the case study, a planning horizon of 10 years is used for scheduling the maintenance and renewal projects of 10 kilometers of railway track which includes rails (component 1), ballast (component 2), and sleepers (component 3). The considered section of track is divided into ten segments with variant lengths. Since the maintenance manager is interested in planning these maintenance activities in a threemonth period, we considered forty time periods in our planning time window (40×3=120 months≈10 years). Most of the parameters are based on the data of the company. However, we have to estimate the time intervals between the maintenance activities of different components. The input parameters which includes different costs involved in maintenance and renewal planning of railway tracks are not reported because of confidentiality restrictions. 23

In the proposed method, we rely on optimization to build the schedule, which can easily adapt when new constraints are added. Since in practice there exist no maintenance activity for component 3, the following constraints should be added to the proposed problem just to schedule the renewal of this component. (17) (18) The proposed approach is now demonstrated on a case study. We show the cost of the integrated planning as compared with a spreadsheet-based approach used currently at the operation. By using the spreadsheet-based approach, a maintenance manager can just evaluate a limited number of simple scenarios for group scheduling and then select the best alternative. Furthermore, there is no sophisticated tool in the spreadsheet based approach for doing the balancing and the renewal times are determined based on expert’s judgments. The results of solving the case study with heuristic methods and the spreadsheet based approach are summarized in Table 15. The same analysis can be done for other scenarios in the problem to get more managerial insights for real problems. We have considered two extra scenarios in Table 15, which are: i) due to the technology change and technical issues, one component should be renewed for the whole section at the end of year 5 (period 20), and ii) maintenance activities of different components cannot be performed at the same time. Under this scenario, we assume that different contractors are responsible for the maintenance of each component. As such, they cannot use the benefits of system downtime in setting up the preparatory works on site. Table 15. Results of comparison for different scenarios relevant to the case study.

Maintenance cost (€) Renewal cost (€) Setup cost (€) Total cost (€)

Spreadsheet-based approach used in practice 2171951 728131 880000

Integrated optimization (Heuristic 1) 443563 2141951 408000

Integrated optimization (Heuristic 2) 457095 2171951 457300

3140082

2695514

2749046

Scenario i

Scenario ii

622604 2285054 788500

627989 4044132 888700

3117658

4942121

In the integrated optimization of the case study, we run the problem for 6 hours, and the best solution of the heuristic methods was obtained after 2 hours while the gaps were about 25%. The results of the analysis show that the obtained solutions by these heuristics result in a fast (near) optimal solution, and it requires a large amount of time to verify the optimality of the obtained solution. Therefore, it would be of interest to investigate proving the optimality of the induced solutions for future research. Another direction for future research is to develop other efficient heuristic or meta-heuristic algorithms for this problem. All in all, the presented optimization method is very appropriate to use for applications in practice. The heuristic algorithms result in a solution with much lower total cost in comparison with the approach used in practice; it results in 14 % reduction in cost. 6. Conclusions In this paper, we have discussed preventive maintenance and renewal scheduling in multi-unit systems and defined a general configuration for modeling it by proposing a hierarchical setup structure. The economic life of the system components are determined by finding the optimal balance between maintenance and renewal while considering their effect on each other. We have presented an integer 24

programming model to find the optimal maintenance and renewal times while considering the economic dependency between components. The proposed model considers a finite and discretized planning horizon in which three possible actions must be planned for each component in each unit, namely maintenance, renewal, or do nothing. The solution aims to find a schedule that minimizes the cost of maintenance and renewal activities and preparatory works (i.e. system downtime, machinery) by grouping them while optimizing their balance. This integration of grouping and balancing techniques is the main contribution of this paper. The experimental results show that the cost savings gained from grouping are significantly greater than the savings from balancing, and that using an integrated approach can reduce the system downtime cost without reducing the level of maintenance. The proposed model can be applied to a wide variety of infrastructures such as railways, roadways, electricity distribution networks, and distributed pipelines. Due to the complexity of the problem, we have developed two heuristic algorithms, which decompose the proposed problem into less complex sub-problems and solves them sequentially. Computational experiments were performed on various test instances to show the performance of the heuristics. In particular, a case study took place using a real instance, and a significant degree of improvement was found over the conventional method. These results show that the proposed techniques reduce costs through elimination of setup costs and proper scheduling. The results of this research can be used to find the optimal or near optimal preventive maintenance and renewal schedule of large-scale multi-unit systems or even any single-unit system with multi-components. As the focus of this study was on long-term preventive maintenance scheduling, topics such as unexpected failures and corrective maintenance decisions are not included in this study. It is assumed that analyses such as reliability analysis and failure modeling have been performed in a prior planning phase to calculate the latest possible times between maintenance/renewal activities. The dependency between the components of the multi-unit system is limited to economic dependency. An extension of this model could consider other criteria such as system reliability, availability and demand satisfaction, which would make the model more practical but also more complex. As system downtime costs may differ due to demand fluctuation in weekdays, nights or weekends, determining the exact execution time of projects is another possible elaboration of the model. In long-term maintenance planning, one could also account for the inflation rate. While these additions could make the model more accurate, this research has avoided these potential refinements for the sake of simplicity. It would also be interesting to see how the proposed model could be applied to other industrial sectors such as maintenance of roadways, pipelines and electricity networks. Another direction for future research is to further develop the heuristic or meta-heuristic algorithms for solving the proposed problem. Acknowledgements The authors gratefully acknowledge the support of Case Corporation in providing the data used for this study. We also thank the anonymous reviewers for their constructive remarks during the finalization of this paper. A part of this research has been presented at the 2015 ESREL Conference (Pargar, 2015).

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Highlights

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We study preventive maintenance and renewal scheduling problem An integrated approach with grouping and balancing is proposed for multi-unit systems A pure integer programming model is developed to formulate the considered problem Two heuristic algorithms based on decomposition approach are developed We examine our heuristics efficiency via generating and solving numerical examples A case study on railway is studied and resulted in 14% reduction in maintenance cost

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