Iterated Local Search for the Generator Maintenance Scheduling Problem

Transcription

1 MISTA 25 Iterated Local Search for the Generator Maintenance Scheduling Problem Ahmad Almakhlafi Joshua Knowles Abstract We consider the task of devising an effective metaheuristic for a variant of the preventive maintenance scheduling problem (PMSP) the (power) generator maintenance scheduling problem (GMSP). Recent research on metaheuristics for this problem has made progress on it, but the potential economic benefits of effective methods is significant in this area, and warrants further focused work. We propose here a solution method based on Iterated Local Search (ILS) following an earlier study by us on neighbourhood search for the same task. Several extensions to a basic ILS design are developed and analysed, including specialised operators and delta-evaluation, as well as restart and portfolio strategies. With these methods, we obtain a significant improvement in performance (in terms of solution quality, runtime and function evaluations) over recent techniques for real-world derived instances of the GMSP. We also provide a benchmark (and results on additional benchmark instances) for future studies of this problem. Keywords Iterated Local Search, Algorithm Portfolio, Maintenance Scheduling Problem, Generators, Delta Function, Variable Neighbourhood Descent, Hybrid, Evaluation Function, Restart Strategy, Run Length Distribution, Benchmarks. Introduction Preventive maintenance (PM) is a series of tasks, such as regular inspections or the replacement of aged parts, carried out to extend the life of a machine. PM is done preemptively before the machine has reached a point of critical wear and is about to fail [53], and hence it is a planned activity, dependent on scheduling. Due to the First Author School of Computer Science, University of Manchester, Kilburn Building, Oxford Road, Manchester, M3 9PL, United Kingdom almakhla@cs.man.ac.uk Second Author School of Computer Science, University of Manchester, Kilburn Building, Oxford Road, Manchester, M3 9PL, United Kingdom j.knowles@manchester.ac.uk

2 widespread use of PM in production and service industries, and its economic benefits in preventing damage to (often very expensive) equipment or infrastructure, some commentators have suggested that improving PM scheduling is one of the most significant problems faced by industry today [74]. Its economic importance is likely to easily rival more better-known planning problems such as production scheduling [43], nurse rostering [3] and timetabling [6]. One important form of the PMSP is the Generator Maintenance Scheduling Problem (GMSP) in the power generation industry, which concerns the maintenance of expensive infrastructure that can also be critical to the reliability of national power grids. This problem has been studied for a number of years in the OR literature (often using rather small benchmark instances) and several optimization methods, from exact methods to metaheuristics, have been developed and applied (e.g., see [74, 24, 4, 68]). We here tackle the GMSP using local search as our basis, following some earlier investigative work where we found it a promising approach to the problem [2]. Here we use the framework of Iterated Local Search (ILS), a simple but powerful metaheuristic framework for improving the performance of basic local search, which has been applied successfully to a number of problems including the travelling salesman problem [65], the permutation flowshop [22] and the quadratic assignment problem [64]. We aim here to achieve similarly good results with the GMSP. To pursue our goal, we use a staged development process. We first analyse the main components of the ILS algorithm using a set of in-house developed GMSP instances for the testing. Then we suggest several extensions of the proposed ILS: an ILS with restart strategy, an ILS with delta evaluation implementation, an ILS hybrid with Variable Neighbourhood Descent (VND) algorithm and a Portfolio of ILSs. These algorithms are developed carefully by looking at run-length distributions and other analysis tools. The performance of the proposed ILS and its variants are also tested on two GMSP instances from the existing literature, allowing us to compare directly with some previously proposed algorithms. The remainder of the paper is organized as follows: a description of the GMSP and its mathematical formulation is presented in Section 2. Section 3 details the different ILS operators proposed and the function of the run-length distributions used for analysis in this work. In Section 4 we suggest several extensions to the initial proposed ILS algorithm. In section 5, we present an analysis of the run-length behaviour of different configurations of the ILS. The proposed ILS algorithm and its variants are tested and compared with some results from the literature in Section 6. Finally, some concluding remarks are given in Section 7. 2 PMSP in power plants In this section, we introduce the generator maintenance scheduling problem, give the formulation we will consider in this paper, and briefly survey related work on the development of solution methods. 2. Overview Maintenance plays a crucial role in the power-generating industry s planning and operation. Modern power plants utilize large capacity units making a single outage cause

3 Table : List of parameters used to formulate the GMSP. i Index of generating units I Total number of units t Index of intervals T Total number of intervals C it Generating capacity of C s Generating capacity of the system unit i in interval t M t Available manpower at interval t m it Manpower needed to maintain R t Nett reserve for interval t unit i in interval t d i Duration of maintenance for unit i R m Reserve safety margin P t Predicted load for interval t both a possible large loss of generation capacity and increased cost of corrective maintenance [5]. Hence, the effective scheduling of preventive maintenance activities, where units need to be taken off-line for maintenance, allows the power system to perform its function reliably and achieve considerable savings. The preventive maintenance scheduling problem of power plants is dealt with as a long-term planning problem in which the power and energy resources are utilized and maintained during a time horizon from several weeks to several months or years into the future, discretized into intervals [5, 39]. Usually, the maintenance outages are planned on a yearly time horizon with scheduling intervals being set at one week [4,3]. The starting times of these outages have to be determined taking into account organizational objectives, system constraints and consumers power demands [5, 39]. The GMSP is a constrained optimization problem [38,73], and is NP-hard in most forms [53,2,,6]. 2.2 Problem formulation Consider a collection of I Z + generating units which can generate a total power of C s over a scheduling horizon of T Z + intervals. The scheduling horizon is organized in weeks and remains constant for all units. Each i =,...I has a generating capacity C i and must be maintained exactly once for d i consecutive weeks during the period T. The maintenance tasks are performed by a workforce of size m it, where i indexes the unit under maintenance and t,...,t is the maintenance interval. However, the total size of the workforce maintaining generators at an interval t cannot exceed the maximum manpower available M t at that interval. The nett reserve of the system R t at any interval t is the power remaining after subtracting the generation loss due to meeting demand and scheduled maintenance outages for that interval. At any interval t, nett reserve must be greater than or equal to the minimum reserve, which is the total of the predicted load P t for interval t and any reserve safety margin R m. The problem parameters are summarized in Table. The objective function aims to level the reserve load over the planning horizon by minimizing the sum of the squares of the reserve loads (SSR). In real-world terms, the SSR value measures the reliability of the power system. The constraints present in this problem consist of the system meeting the minimum reserve, the availability of the maintenance workforce, the maintenance window and the duration for each unit to be offline for maintenance, as well as any exclusion constraints to prevent the simultaneous maintenance of certain combinations of generators. In our formulation, the pseudoboolean solution vector X is two-dimensional, and its element X it represents the status of unit i in interval t as follows:

4 { if unit i is under maintenance during interval t X it = otherwise. () The optimization problem, which has the goal to minimize the sum of squares of the reserve (SSR) generation, can be written as Minimize T ( SSR = C s P t t= subject to the minimum reserve constraints I i= ) 2 C it X it (2) I C it ( X it ) P t +R m for all t =,...,T, (3) i= the manpower constraints I X it m it M t for all t =,...,T, (4) i= and the duration constraints s i+d i t=s i X it = d i for all i =,...,I (5) where s i represents the interval in which the maintenance of unit i starts. In addition, this expression ensures the continuous maintenance requirements of units. In general, a GMSP may include alternative or additional constraints. Although this formulation seems to be pure and simple, it actually reflects the problem settings in the power company described in [2]. 2.3 GMSP Optimization Methods The initial formulation of the GMSP was presented in [6,34,33]. Later, the cost function of the problem formulation was improved by Yamayee and Sidenblad [75]. In literature, GMSP has been optimized by many different techniques and each technique has its own difficulties. They can be classified into two groups: exact optimization methods, and metaheuristic methods. The exact methods, such as integer programming[4, 44], dynamic programming[78, 7] and the branch and bound method [23,24,5,69], have generally been applied to small GMSPs. Although these methods have the capability of finding the optimal solution, they cannot generally be applied to large-scale problems, because the size of the solution space increases exponentially with the increasing number of generator units, increasing the computational time of these algorithms accordingly. To overcome these disadvantages, metaheuristics have been applied to solve the GMSP. These methods include the application of GAs [73,8,2], Simulated Annealing [56, 58] and Tabu Search [4, 25]. Although metaheuristics can find (near-) optimal

5 solutions for large problem sizes better than the exact methods with a reasonable computational time, finding such solutions is not guaranteed. By exploiting individual advantages of the exact and metaheuristic techniques, they can be combined to form hybridized algorithms with good overall performance. Variants of such algorithms which have been proposed in recent years are reported to be superior to their pure counterparts in terms of solution quality and computational time, especially when optimizing real-world NP-hard problems [54]. These approaches are commonly referred to as hybrid algorithms. An emerging optimization method is algorithm portfolios [32]. This method uses collections of multiple algorithms, either different copies of the same algorithm or different algorithms running in a parallel or interleaved in time [59], to optimize a problem. Compared to a single algorithm, a portfolio can offer better performance when optimizing many instances or even on a single instance [3, 32]. An application of this method to optimize GMSP can be found in [3]. Other approaches to solve the maintenance scheduling problem in the power-generating industry include Knowledgebased [5], fuzzy logic [26] and expert systems [46]. Algorithm : Iterated Local Search s GenerateInitialSolution ; 2 s LocalSearch(s ) ; 3 repeat 4 s Perturbation(s,history) ; 5 s LocalSearch(s,history) ; 6 s AcceptanceCriterion(s,s,history) ; 7 until termination condition met; 8 return s 3 ILS for GMSP Iterated local search [47, 48] is a simple yet powerful framework which can be implemented in any type of local search algorithm to improve its performance. The algorithm walks randomly in the space of the local optimum performing a stochastic greedy search. The search strategy of ILS consists of utilizing the local search algorithm to find the local optimum in the defined neighbourhood and then applying small perturbations on the local optimum to escape the basin of attraction of the current local optimum. The local search is then restarted from the perturbed solution. Despite its simplicity, ILS has been applied successfully to several combinatorial optimization problems like the travelling salesman problem (TSP) [65, 49] and various scheduling problems [7, 27, 42]. Its performance is comparable with several state-ofthe-art metaheuristics, such as Simulated Annealing, Tabu Search, Genetic Algorithm and Ant-Colony Optimization [22]. In order to apply ILS to the GMSP, its building components have to be defined: initial solution generation, to generate an initial solution; local search, to find an improved solution; perturbation, to perturb a solution; and acceptance criterion, to decide from which solution to continue the search. The architecture of ILS is given by Algorithm [48].

6 Start algorithm Is the solution fesabile? No Yes Scan all intervals for intervals with maximum violations Scan all intervals for intervals with minimum nett reserve Start algorithm Scan scheduling period for the worst and best intervals Is there more than one interval with the same worst value? No Yes Select one interval randomly to betheworst interval Scan interval with violations/minimum nett reserve Is there more than one interval with the same value? No Move units until First Improved solution is found Yes Scan intervals for the interval with the maximum violations/ minimum nett reserve to locate units Move units until Best Improved solution is found Select one interval randomly to be the worst interval Alocal optimum is found Yes Is the empty? No Is the Yes of moved? No Moveunittothebestinterval, Yes Is a better solution found? No A local optimum is found Yes Does the new solution dominate the current one? No (a) FILS and BILS (b) WBLS Fig. : Flowcharts of the proposed local search operators. Flowchart (a) is similar for BILS and FILS algorithms except for the moving units part. The WBLS algorithm is presented in Flowchart (b). Making the best possible choice of ILS components is essential to achieve the best overall performance of the algorithm for a particular problem. In the following subsections we provide further details of the components used in this work. 3. Initial solution generation We use a heuristic to generate the initial solution. It is encoded using a two-dimensional binary string representation of (T I) bits such that means that the unit is under maintenance (offline), whilst means that it is operational (online). The heuristic assigns each unit exactly one job in the scheduling horizon and determines the position of this job uniformly at random. The schedule is regenerated if a maintenance duration collides with the end of the scheduling period.

7 3.2 Local search A local search algorithm searches for an improved solution by exploring the neighbourhood of a given initial one. If a better solution is found, it replaces the current solution and the search is continued until a local optimum is found. One of the main ingredients of a local search algorithm is defining the neighbourhood structure. A proper neighbourhood structure definition allows for an efficient move from one solution to another. For the local search component of the ILS algorithm we considered three algorithms (FILS, BILS and WBLS, described next) with different neighbourhood definitions. These algorithms were designed to guide the search to the feasible region of the search space and maintain feasibility afterwards. Hence, they would not move to an infeasible solution when the current one was feasible. First Improvement Local Search (FILS): This algorithm has an advanced version of the intelligent mutation operator described in [2] and is shown in Figure (a). The algorithm starts by checking the feasibility of the solution. When the solution is infeasible, it scans all intervals in the scheduling horizon to identify the interval(s) with the maximum violations. If a solution is feasible, it scans all intervals to find one(s) with the minimum nett reserve. In acase where there is more than one interval with the same maximum violations / minimum nett reserve, one interval is selected uniformly at random as the worst interval. The worst interval is then scanned to locate the offline units (under maintenance). Each of these is tried in each of the intervals in sequence, starting from the first week of the scheduling horizon, until an improvement is found and this is moved to. If the end of the scheduling period is reached without finding a fitter solution, the next unit is considered until a better neighbourhood is found. The algorithm continues until no improved solution is found and terminates with the current best when there are no more units to be moved. Best Improvement Local Search (BILS): BILS has the same mutation operator as that used in FILS, except that it evaluates all possible solutions resulting from moving units from the targeted interval. The best solution among the possible solutions is then chosen. The FILS and BILS algorithms are enhanced versions of the hill climbers (FIHC and BIHC) we presented previously in [2]. The mutation operator in FIHC and BIHC scans all intervals in the scheduling horizon to identify the one with minimum nett reserve, while in FILS and BILS it scans first for the intervals with high violations. When the solution is feasible, it scans for the intervals with minimum reserve. Figure (a) illustrates a flowchart of the FILS and BILS algorithms. Worst Best Local Search (WBLS): WBLS incorporates a tracking heuristic and a memory structure. The heuristic tracks changes in the constraint violations and in the objective function value and locates the worst and best intervals. When a solution is infeasible, the worst interval is the interval with the highest number of violations. If these violations are caused by different constraints, the heuristic checks for which of them contributes more of the violations. The best interval is then determined such that moving the units from the worst interval will reduce the total number of violations for the solution. When the solution becomes feasible, the worst and best intervals are the intervals with minimum and maximum nett reserve respectively. The heuristic makes sure that the search is restricted to the feasible region.

8 Weeks Weeks New intervals Old intervals New intervals Old intervals (a) Perturbation INSERT (b) Perturbation SWAP Fig. 2: Two perturbation operators are illustrated. Figure (a) shows the principle of the INSERT operator where units are relocated randomly in other intervals. The SWAP operator shown in Figure (b) where units exchange their maintenance starting intervals. The memory structure tracks the units moved during the search. Units which are scheduled to be maintained on the worst interval are added to the unitstobemoved list, from where they are selected one by one and moved to the best interval. When a unit is moved, it is added to the movedunits list to prevent it from moving again. An attempt is also made at this juncture to move a different unit in order to bias the search towards promising areas of the search space. Solutions are compared on the basis of a constraint violation test and objective function value. In cases where there is more than one worst interval, the algorithm utilizes a similar mechanism to that described for the FILS and BILS algorithms. When there are no more units to be moved, the search is considered to have reached equilibrium and the current solution is reported as a local optimum. Figure (b) is a flowchart of the WBLS algorithm. 3.3 Perturbation With a local search algorithm, progress is made by moving only to better solutions and the search terminates in a local optimum. The aim of the perturbation phase is to modify a current local optimum so that it can be effectively escaped and to provide a good starting solution for the local search. The perturbation should be strong enough toallow the local search toescape the local optimum, but also weak enough tomaintain some features of the current solution. In this context, we identify two concepts related to perturbation: perturbation strength and perturbation nature [48]. Perturbation Strength: In this work, perturbation strength refers to the number of units that are moved from their current intervals to some other ones. We examine two types of perturbation strengths, deterministic and adaptive. The deterministic perturbation strengths are fixed for all of the instances where 2 and 3 units are considered for perturbation for any instance. The adaptive perturbation strengths are determined by the size of the instance as a percentage of the number of units in an instance (25%, 5% and 75%). Type of Perturbation: We considered two different types of perturbation for the GMSP described previously: INSERT and SWAP. The INSERT perturbation modifies a solution by moving a uniformly randomly selected unit from its interval(s) to another

9 interval or intervals selected uniformly at random, while the SWAP perturbation modifies a solution by exchanging the starting intervals of two uniformly random selected units (without replacement), as Figure 2 shows. 3.4 Acceptance criterion The acceptance criterion defines the conditions of transition from the current solution toanewlygenerated one. Ifthe newsolution is accepted, it will be perturbedand serves as an initial solution for the next iteration of the search. We consider two acceptance criteria: BETTER and PROBABILITY. BETTER accepts the new solution if it is better than the current solution. This criterion, which is similar to one described in [64], is defined as: { BETTER(s,s s if f(s ) < f(s ) ) = s (6) otherwise where f(s ) and f(s ) are the objective functions for the new and current solutions respectively. Such a criterion may result in a very strong intensification of the search. To introduce some diversification of the search, PROBABILITY can accept a worse solution with a probability p. We used three values for p: %, 3% and 5%. PROBABILITY can be defined as: s if f(s ) < f(s ) PROBABILITY(s,s ) = s if r p < p (7) s otherwise where r p is a randomly generated number within the range [,]. This criterion can be used to control the balance between intensification and diversification of the search. Note that PROBABILITY tends to make the search more intensive when p is smaller and to make the search more diverse if it is larger. 3.5 Run-length distributions for evaluating ILS We follow visualization methods introduced in [62] and [36] to support the analysis of our ILS algorithms. These methods, called run-length distributions (RLDs), are useful to see how the solution quality grows with time (or, here, function evaluations) while properly accounting for the stochastic nature of an ILS or other metaheuristic. The RLDs in Figure 3 (a) show the best solution found and various bounds on the solution quality given as the percentage deviation from the best found solution; these are plotted against the run length (number of function evaluations) for a large number of independent algorithm runs. As well as basic visualization of an algorithm s performance, RLDs can be used to compare algorithms in detail [36, 62], which can facilitate the design of algorithm portfolios that perform better than any of the constituent algorithms (see Section 4.4). Figure 3 (b) shows the RLDs for two ILS algorithms, A and B, and the crossing over of their RLD curves indicates that a portfolio based on these two could be beneficial. Finally, RLDs may also be used to consider restart strategies. By fitting an exponential distribution to an RLD, one can detect stagnation of an optimizer. An example of this method is given in Figure 3 (c), where the run-length distributions of an ILS

10 Empirical Solution Probability best.25%.5% % 3% 5% (a) Empirical Solution Probability A B (b) Crossing of RLDs Empirical Solution Probability s f(x) s 2 g(x) (c) Optimal cutoff Fig. 3: The empirical run-length distributions for GMSP instances across independent runs of different ILS algorithms in semi-log plots. The x-axis gives the number of evaluations used and the y-axis represents the empirical probability to find a solution. (a) The RLD for an ILS algorithm that is allowed to run for 2 evaluations. The RLDs are for the best solution found and various bounds on the solution quality given as the percentage deviation from the best found solution. (b) RLDs for two ILS algorithms, A and B. The algorithm allowed to run for evaluations per each run. (c) The RLDs of an ILS algorithm measured on two GMSP instances, s and s 2. The exponential approximations of the ILS are indicted by f(x) for instance s and g(x) for instance s 2. algorithm measured on two GMSP instances are plotted. For instance s, the run length distribution is well approximated by the exponential distribution f(x). This indicates that the restart strategy would not improve the algorithm performance on this instance. The instance s 2, on the hand, develops significantly below the exponential curve g(x) from the tangential point on the empirical RLD. Hence, the algorithm suffers from a stagnation behaviour when optimizing this instance and the tangential point (here at about 8 evaluations) is hypothesised to be the optimum cut-off value [64]. However, an optimum value of a cut-off may depends on the particular GMSP instance. Hence, Stützle suggested using soft restarts instead of applying fixed cut-offs for restarting an ILS algorithm [63, 65, 64]. We refer the interested reader to the extensive discussion provided by Hoos [36] and Stützle [62] in their PhD thesis where they discussed the RLDs theoretically and practically. 4 ILS-extended algorithms In the previous section, we have presented different basic ILS operators for the GMSP. In the following, we discuss four possible extensions to the base method: using a restart strategy, use an objective function with delta evaluation, a hybrid with a VND algorithm, and an algorithm portfolio of ILSs. 4. ILS with restart strategy Stagnation, where an algorithm fails to find a better solution for an extensive number of evaluations, can afflict metaheuristics such as ILS, reducing their effectiveness. The

11 simplest solution to this problem is to restart the algorithm from a new initial solution after a predefined number of evaluations (cut-off value). As stated in Section 3.5, an empirical RLD can be approximated by an exponential distribution, while departure from that distribution can serve to detect stagnation and help to identify the most appropriate cut-off value for the restart. Based on a well-known result from probability theory, if a given algorithm has an exponential RLD and it is allowed to run k runs for e evaluations, the probability of finding a target solution using such configurations is the very same as when running the algorithms once for evaluations k.e. Hence, using a restart strategy for an exponentially distributed ILS algorithm will not affect the probability of obtaining the target solution [64, 65]. Restarting an ILS algorithm will enhance its performance only if the empirical run-length distribution falls below the plotted exponential. 4.2 ILS with delta evaluation Delta evaluation is an important technique to reduce the amount of computational work performed by local search algorithms [9, ]. In this approach, any solution is partially evaluated by computing only the cost difference between that solution and its neighbouring solution. When scheduling the generator maintenance tasks, it considers only the fitness function contribution of intervals that are not common between the two schedules, which reduces the repeated evaluations. Let us demonstrate the usefulness of this method using an example of GMSP. Consider the case where a unit i with a duration of d i needs to be moved from its current position within a range of total intervals of T. Usually, the solution is evaluated by computing the change of the nett reserve on each interval, which is equal to T times. Using delta evaluation, only the intervals that the unit i has moved from and to are computed and these will be in the range between d+ and d 2 intervals. Hence, the cost of moving the unit i evaluated using the delta function compared to using a full evaluation would be in the range between (d+)/t and (d 2)/T to. For instance, for a unit with a duration of d i = 6 and scheduling horizon of T = 52, the delta evaluation would be between.35 and.23 evaluations. This method is more effective for schedules with many units having a small number of durations. In this ILS variant, the algorithm has the same components as a standard ILS, but a different fitness function. Delta evaluation is utilized and the new fitness is calculated using the function: SSR new = F delta (solution old,solution new,ssr old ). This function uses the information from the fitness and content of an existing solution in order to calculate a new neighbouring solution s fitness much more rapidly. Algorithm 2: Hybrid ILS/VND s GenerateInitialSolution; 2 s VND(s ); 3 repeat 4 s Perturbation(s,history); 5 s VND(s ); 6 s AcceptanceCriterion(s,s,history); 7 until termination condition met; 8 return s

12 4.3 ILS/VND hybrid Each of the various exact methods and metaheuristics has its assets and shortcomings. By exploiting individual advantages, algorithms can be combined to form hybridized algorithms with better overall performance. Hybrid metaheuristics tend to be superior to their pure counterparts in terms of solution quality and computational time, especially when optimizing larger-scale instances [7]. Several ILS and VND [35] hybrid algorithms are proposed for several problems, such as the Vehicle Routing Problem [67], the attribute reduction in rough set theory [4] and the TSP [55]. In this section, we propose a new hybrid ILS and VND algorithm for solving the GMSP. This hybrid integrates a VND procedure into the framework of an ILS algorithm. Algorithm 3: VND Procedure N: Set of neighborhood structures ; 2 k: Index of the current neighbourhood structure ; 3 Set k = ; 4 while k N do 5 s LocalSearch(s,N k ) ; 6 if s dominates s then 7 s s ; 8 end 9 k = k + ; end return s Variable neighbourhood descent is a variant of Variable Neighbourhood Search (VNS) [35], where there is no shaking step and the change of neighbourhood is performed in a deterministic way, in the descent to local optimum. More precisely, the idea behind VND is to alternate between different neighbourhood structures where the local minimum found by a local search algorithm on one neighbourhood is the starting point of the algorithm within the next neighbourhood. The framework of the proposed algorithm is presented in Algorithm 2 and explained as follows: Initial solution generation: The initial solution is generated using the same heuristic described in section 3. Each unit is assigned exactly one job in the scheduling horizon and the outage period of this job is determined uniformly at random. Base ILS algorithm: The ILS considered here utilizes an INSERT(3) perturbation operator, BILS as its local search and the BETTER acceptance criterion. These ILS components have been described earlier in Section 3. Variable Neighbourhood Descent: In GMSP, an incumbent solution can often be improved by a valid movement of the units from their current intervals to others within the scheduling horizon. How many units should move determines a neighbourhood s structure and size. This moving process is essential to the definition of the neighbourhood structures used in the proposed algorithm, which defines two neighbourhoods:

13 N : This is a large neighbourhood structure where the effective search space can be as large as T I. An INSERT operator is used to remove a unit from interval i and reinsert it in interval j, anywhere in the scheduling horizon. If the maintenance period has multiple intervals, i and j represent the first interval of the maintenance period. N 2 : This is a small neighbourhood search structure where the effective search space can be approximatly 2I. An INSERToperator is also used toremove a unit, but the movement is limited to one interval earlier and one interval later than the current first interval of the maintenance period. The VND procedure, described by Algorithm 3, is used to search the neighbourhood defined by the operator N k of the current solution using the local search component. The procedure starts by exploring in the large neighbourhood N until a local optimum is reached, then it performs a small neighbourhood search for a better solution than the current one in N 2. The best solution found by the procedure is returned. Stopping criterion: Similar to the other ILS variants, the hybrid algorithm stops when it reaches the maximum number of evaluations allowed. The performance of the hybrid algorithm is tested and compared with other ILS algorithms presented here and some metaheuristic results from the literature in Section A portfolio of ILS algorithms The term algorithm portfolio was introduced originally by Huberman et al [37] and was also studied by Gomes and Selman [32]. Algorithm portfolio can be defined as A collection of different algorithms (heterogeneous portfolio) and/or different copies of the same algorithm (homogeneous portfolio) running on different processors [59]. The motivation behind studying algorithm portfolios arose from the fact that there is not a single algorithm which can guarantee to be able to find an optimum or near-optimum solution for a specific type of problem with varying dimensions and complexities where its performance varies based on the problem instance. Portfolios comprising different algorithms, or differently configured instances of an algorithm, can offer better performance even on a single instance, since switching between algorithms is likely to match each phase of the search to a good algorithm, or search can be terminated earlier when one constituent algorithm finds a good solution. Such portfolios can probably be further enhanced by exchanging of information between constituent algorithms. For instance, Peng et. al. [52] have used different strategies to communicate adaptively between consistent algorithms and reported an improvement in performance. In a well-regarded technique called AMALGAM [7, 72], multiple different search algorithms are run simultaneously and learn from each other through information exchange using a common population of points. The computational effort of each one is adapted continuously during the course of the optimization, in order to favour individual algorithms that exhibit the highest reproductive success during the search. Results from experiments show that AMALGAM significantly improves the efficiency of evolutionary search.

14 Algorithm 4: ILS Portfolio Define: 2 A : The total number of constituent algorithms; 3 P : The main population; 4 ϕ a : Algorithm a contribution to overall improvements in objective function where a =,...,A; 5 N a: Number of individuals in Algorithm a to be migrated from its sub-population P a to P; 6 P s : Size of population and sub-populations; 7 e pm : Maximum evaluations for the portfolio algorithm; 8 e im : Maximum evaluations for each iteration of optimizing the main population; 9 ν a : Minimum contribution of solution to be migrated for each algorithm a; Set e p = /* Portfolio evaluations counter */ Generate initial solutions in population P; 2 Calculate fitness and constraints violations for all individuals in P; 3 while e p e pm do 4 Set e i = /* Iteration evaluations counter */ 5 while e i e im do 6 for a = to A do 7 Algorithm a uses each individual in P as s to generate a sub-population P a of size P s; 8 Calculate fitness and constraints violations for all individuals in sub-population P a; 9 end 2 end 2 Clear population P; 22 Calculate ϕ a where a =,...,A; 23 Update N = N a,...,n A according to equation 8; 24 for a = to A do 25 for n = to N A do 26 while solution does not exist in P do 27 Move N a solution from sub-population P a to population P; 28 end 29 end 3 end 3 Update e p; 32 end 33 return Best solution For the ILS algorithms studied here, several observed phenomena suggest that the portfolio approach would be beneficial for optimizing the GMSP. First, no single algorithm has dominated the others and we occasionally observed the crossing of the RLDs of different algorithms. Secondly, we noticed that for different instances, different ILS algorithms showed the best performance. Thus, to improve performance over a broad range of instances, it would be advantageous to combine several of the best-performing ILS algorithms mentioned previously into a portfolio. The approach has proved ef-

15 fective for solving many problems such as satisfiability, planning and scheduling [6]. Although algorithm portfolios have been applied to different scheduling problems [59, 5], they appear to have limited application to the GMSP. In fact, to the best of our knowledge, it has been applied only as reported in [3], where constituent algorithms in a portfolio were run in parallel (all algorithms being executed concurrently), with no communication between algorithms. A similar application was used here, except that the constituent algorithms were able to exchange information, based on the AMAL- GAM approach. The pseudocode of the portfolio algorithm is given in Algorithm 4. It employs a population-based elitism search strategy to find the best solution of a GMSP using the predefined objective function. The algorithm is initiated by creating a main population P of randomly generated solutions using the procedure GenerateInitialSolution described in section 3. The fitness value and number of constraint violations are then computed for each solution in the population. Each algorithm a creates a sub-population P a of the size of the main population, where each individual in P is optimized and the solution achieved is added to P a. The solutions in all sub-populations are evaluated and their constraint violations are recorded. A defined number of solutions N = {N,...,N A } is migrated from each sub-population P a for a =,...,A. This number is different for each algorithm and depends on its performance. A twostep procedure is used to update the N in each iteration. First, we measure for each algorithm a its contribution to the overall improvement in objective function, denoted as ϕ a. The second step is to update N for each algorithm a according to this equation: ϕ a N a = (P s (ν a A)) A. (8) i ϕa Based on the calculated values of N, the best solutions are migrated from the subpopulations to the main population. Whatever the values of N, there is always a minimum number of solutions that each algorithm needs to contribute to the main population to avoid the possibility of disabling an algorithm which may provide better solutions in the future with higher evaluations. This learning mechanism ensured that the algorithms with the best performance were rewarded in the next iteration by allowing them to contribute more solutions. As a result, a faster convergence may have been achieved. However, to preserve population diversity and prevent too early convergence of the portfolio algorithm, we took three precautions:. A diverse set of ILS algorithms was considered as constituent algorithms for the portfolio. The best performing ILSs using BILS and WBLS were selected; these were A5 (BILS, INSERT(3), BETTER) and A69 (WBLS, INSERT(3), BETTER) respectively, see section 5.2. In the same section, it is shown that ILSs with these two local searches may achieve better performance when optimizing a GMSP cooperatively. In addition, the ILS variants mentioned in this section were considered, as they have different approaches to performing optimization. Two algorithms of each variant were added, one with BILS and the other with WBLS. Choosing constituent algorithms that exhibited different behaviours on the GMSP problem to be optimized may have led to better performance. 2. When migrating a solution from a sub-population to the main population, it was compared to the solutions already in the population, based on the Hamming distance. If the Hamming distance of the solution to be migrated was equal to zero

16 Table 2: List of the components of the best ILS algorithms. ILS Local Search Perturbation Acceptance Criterion ILS Local Search Perturbation Acceptance Criterion A5 BILS INSERT(3) BETTER A65 WBLS INSERT(2) BETTER A69 WBLS INSERT(3) BETTER A3 BILS INSERT(.5) BETTER A BILS INSERT(2) BETTER A33 FILS INSERT(2) BETTER A9 BILS INSERT(.25) BETTER A37 FILS INSERT(3) BETTER A73 WBLS INSERT(.25) BETTER compared with any solution in P, the next better solution was selected, even if its quality was lower than that of the first chosen solution. 3. To introduce new solutions during the search process, we used two ILS algorithms which both had INSERT perturbation and the PROBABILITY acceptance criterion, but different local search algorithms: BILS and WBLS. For both algorithms, the perturbation strength and probability of accepting worsening moves were set high (.75). New solutions which improved upon the current best solution could then be found during the search. The portfolio algorithm is stopped when it reached the maximum evaluations allowed. The portfolio seems to be a promising approach to optimize different large, complex and dynamic problems such as the GMSP. Just as for the other ILS variants suggested, the performance of the proposed ILS-based portfolio was investigated on two GMSPs from the literature, as reported in section 6. 5 Selecting ILS best components In this section, we investigate the effectiveness of different ILS components (in all, 96 ILS algorithm with different components and parameters) using run-length distributions over a set of problem instances. The more advanced variants of the ILS, using delta-evaluation, restarts or a portfolio, were formed from the best of the ILS components tested here; these are evaluated in Section Experimental Settings A set of 96 ILS algorithms was constructed by combining the components described in section 3. To test the performance of these algorithms, we ran them on a set of instances from our own GMSP instance generator, based on the problem model described in section 2. The instances were hand-crafted to represent different sizes and characteristics of generator scheduling problems, with the number of units from 4 to 29. The scheduling horizon for all instances was 52 weeks. Each algorithm optimized each instance for runs and the mean Percentage Deviation (PD) from the best known solution for that instance was calculated using theformulapd = (f(s) f(s best ))/f(s best ). Foreachalgorithm we alsomeasured the Infeasibility Ratio (IR) (IR = number of infeasible solutions/ number of runs ). All algorithms were allowed to run to their maximum evaluations, calculated by the formula maxevaluation = αi + β, where α = β = 2 and I is the total number of units for an instance (Table ). A list of the ILSs tested, showing the their components and the instance details, is provided at [], including instances and results.

17 Timing information is given below; these results are from an implementation in Java run on a server with 2 cores (2x6 XEON GHz) using 32 GB of RAM. 5.2 Results and findings For the sake of making this paper short, a detailed discussion of the results is provided in the supplementary section S while in this section we wish to identify some overall best ILS algorithms. To do so, we ranked the PD and IR values obtained by all algorithms for each instance. In both cases, these rankings start from zero and the lower the rank, the better the performance of the algorithm. Each algorithm s ranks were then summed for all instances and plotted in Figure 4. The left plot of the figure shows thesumoftheranksfor all thealgorithms dividedintothree groups, basedonthe local search algorithm used. It can be observed from the plot that the ILS algorithms using BILS tended in general to perform better than the other ILSs and to have a spread of data points towards the bottom left. On the other hand, the ILS algorithms with FILS seem to have varied in performance, indicating that they struggled to find feasible and good quality solutions for all instances. The best performing ILS algorithm with FILS was A33. The ILSs with WBLS showed less variation in performance than those with FILS, but more variation than the ILSs with BILS. The seven best ranked ILSs are shown in the right plot of the figure and listed in Table S, while Table 2 lists their components and those of algorithms A33 and A37. All nine of these ILSs managed to obtain feasible solutions for all runs on all instances; IR =. The best ranked ILS algorithm was A5, with the components BILS, INSERT(3) perturbation and BETTER acceptance criterion. This choice of configurations seems to have allowed the ILS algorithm to achieve good solutions for all GMSP instances. Indeed, it was the best solver for many of them. Algorithm A69 also showed solid performance over all instances. As an outcome of this analysis, we propose the ILS algorithms A5 and A69 as effective algorithms for GMSPs such as the one modelled here. The implementation and behaviour analysis of these algorithms showed them to be easily adaptable alternatives to other more complex metaheuristics, as they showed excellent performance after some straightforward optimizations. The performance of the proposed ILSs were tested on two benchmark GMSPs from the literature and compared to the results for other metaheuristics on the same instances. Finally, there seems to have been a strong positive correlation between the IR and PD sums of ranks of the ILS algorithms, as can be seen from the scatter plot in Figure 4 (Spearman rank correlation coefficient ρ =.976). This suggests that the ability to obtain feasible solutions was strongly linked to the ability to find high quality solutions for these instances. 5.3 Overall benchmark results Results for runs of all 96 algorithms across the full set of instances are presented in Table 3. It should be noted that every one of the algorithms managed to find a feasible solution for instance s during all runs. On the other hand, a feasible solution for instance s 9 proved difficult to find on more than half of the optimization runs.

18 9 A3 Sum of Percentage Deviation Ranks BIHC FIHC WBHC Sum of Infeasibility Ratio Ranks Sum of Percentage Deviation Ranks A65 A73 A9 A A69 A5 BIHC FIHC WBHC Sum of Infeasibility Ratio Ranks Fig. 4: The performance of different ILS algorithms represented as the sum of the ranks of the algorithms percentage deviation against the sum of the ranks of their infeasibility ratio. For each algorithm optimizes an instance, the percentage deviation from the best know solution for that instances is calculated for all runs and then averaged over them while the infeasibility ratio is calculated according to the equation in Section 5.. The percentage deviation and infeasibility ratio ranking starts from zero. The lower the x-axis and y-axis values the better. The last column shows the number of times the best solution was found for an instance among all optimizations. The highest number of runs resulting in a best solution was obtained on instance s, yet it represented only about 29% of the optimization runs. For the instances s 7 to s, which were the largest instances in the set, only once was the best solution achieved, and in general they seem to have been the hardest among the set. As we belive that this set of instances are likely to be valuable as benchamrks for the GMSP, due to the different sizes and characteristics of the instances, we have made it available to the optimization community and it can be downloaded from []. 6 Comparative study This section reports the experimental evaluation of the performance of two standard ILS algorithms and the ILS variant algorithms proposed in section 4 on the test systems described in section 6.. The standard ILS algorithms are those proposed in section 5.2 based on their overall performance: A5 and A69, referred to here as ILS-BILS and ILS-WBLS respectively. The ILS with the restart strategy is referred to as Restart and that with delta evaluation as Delta. The ILS/VND hybrid is referred to as Hybrid. Finally, we refer to the portfolio of different ILS-based algorithms simply as Portfolio. For each problem, all algorithms were run for independent trials and were given the same number of evaluations per optimization run. In order to facilitate fair comparisons for the 2-unit system, we used the same number of evaluations (3) as were used in previous studies which investigated this problem. As the 32-unit system had more units, the algorithms were given 4 evaluations per run to optimize the problem. All experiments were implemented in Java and run on the same hardware configuration

19 Table 3: Computational times required to find the best solution (averaged over runs and algorithms), percentage of feasible solutions (of all runs and algorithms) and the number where the best fitness was achieved (among 96 optimizations). Instance Number Best Avg Best Min Avg Max % Feasible NBSA of units Solution Solution Time (s) Time (s) Time (s) Solutions s s s s s s s s s s s mentioned in section 5.. The resultant objective functions (SSR) for all runs of all algorithms are available to be downloaded at []. 6. Test systems The case studies considered in this research as benchmarks to investigate the effectiveness of the proposed algorithms are two GMSP test systems obtained from the literature in which reliability is considered as the optimality criterion. The objective of both problems is to minimise the sum of the squares of the reserve (SSR) over a 52-week scheduling horizon while meeting the problem constraints. The first test system comprised 2 units. It was loosely derived from the problem in [76] and presented with some simplifications and additional constraints in [2]. Each unit must be maintained continuously for a given duration within a specified window, either in the first or second half of a year s scheduling horizon, while meeting problem constraints such as the load demand and the availability of maintenance crew. The details of the problem including the unit capacities, maintenance allowed periods, duration of maintenance and manpower needed are available in [2, 9]. Several metaheuristics have been applied to this test system. They are GAs in [2, 8, 2], Simulated Annealing (SA) in [2] and hybrid algorithms (GA,SA and a heuristic) in [9]. The best results were obtained by Foong [29] using the Ant Colony Optimisation (ACO) algorithm. Recently, Schlünz and Vuuren [57, 58] managed to match the best known objective function found by Foong using an SA algorithm and an SA/Heuristic hybrid. However, they were unable to match or improve the best known average incumbent solution quality. The second test system was the 32-unit scheduling problem first introduced in [58]. This is a modified version of the system presented in the 979 IEEE Reliability Test System [66], with parameter values and constraints added. It has additional exclusion constraints compared to the 2-unit problem, to prevent certain units from being in a state of simultaneous maintenance. The problem has a safety reserve to be considered during scheduling and is assumed to be highly constrained by both its maintenance crew and power demand constraint sets. The details of the test system are available in [57,58].

20 6.2 Algorithm parameters All ILS variants were extensions to the standard ILS-BILS algorithm. The Restart algorithm required a single parameter ( ) to be set in order to calculate the cut-off evaluations. If the objective function of a solution has not improved for e r = I. evaluations, then the algorithm will start with a new random solution where = 2. The Delta and Hybrid algorithms have no parameters to be set. Finally, the Portfolio algorithm starts by creating a P of size 5. The same size was chosen for the subpopulation of the algorithms. The e im was set according to the formula (e im = αi+β, where α = and β = 9). The minimum number of solutions that an algorithm a needs to contribute was set to ν a = 2. All the parameter settings were based on preliminary experiments performed to check the effects of different settings on the performance of the algorithms. 6.3 Performance of ILS variants Similar to section 5.2, we provide an extensive results and detailed discussion of them in the supplementary section S2. Here, we can summarize the above observations from Figures S6 and S7 concerning the performance of the ILS standard and variant algorithms as follows: Keeping in mind that ILS-BILS and ILS-WBLS were standard ILS algorithms that had not been modified or tuned on the test problems, they showed good performance, especially ILS-WBLS. This difference can be credited to the advanced local search in WBLS, as the two algorithms were similar in terms of their other components. The restart strategy was not effective for the ILS algorithms, indicating that they did not suffer from stagnation behaviour when optimizing the GMSP modelled in this work. Implementing the VND within the ILS framework as a hybrid algorithm significantly improved the performance of the ILS algorithm on both objective values achieved and computation evaluations required. As shown by the plots, the principle of using a portfolio of ILS algorithms that can communicate proved to be efficient and effective in optimizing the test instances in a short time and on finding excellent solutions. These findings encourage the use of such a method for larger and harder problems, such as those ocurring in the industry. We validated these outcomes by a further exploration of the results using applicable statistical tests (see supplementary section S3). These tests confirmed the conclusions mentioned earlier, as follows: There were statistically significant differences in performance between the ILS algorithms with BILS and WBLS for both test cases. Both median and mean values showed better performance for ILS with the WBLS algorithm. The t-test(2-unit problem) and Mann-Whitney test (32-unit problem) revealed no significant difference in the SSR results of the standard ILS algorithm (ILS-BILS) and its variant with the restart strategy (Restart). This indicates that the ILS algorithms did not suffer stagnation behaviour on the GMSP instances studied here, making the restart strategy superfluous.