A Memetic Algorithm for the Maintenance Scheduling Problem Edmund K. Burke and Alistair J. Smith Automated Scheduling and Planning Group, University of Nottingham
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Abstract The maintenance scheduling problem has been previously tackled by various traditional optimisation techniques. While these methods can give an optimal solution to small scale problems, they are often inefficient when applied to larger scale problems. The memetic algorithm presented here is essentially a genetic algorithm with an element of local search. The effectiveness of the method is tested through its application to real scale problems.
2 The Memetic Algorithm 2.1 Introduction
Evolutionary algorithms are usually modelled on concepts observed in the propagation of life. However, it has been noted that ideas can also evolve [Mic92]. In a genetic algorithm [Hol75], possible solutions are created, selected according to some measure of fitness, combined with other solutions and possibly mutated. The solutions are made up of genes. As in a genetic algorithm, a population of ideas can be created. A good idea has a better chance of survival than a bad idea and ideas can be combined to form new ideas. It is also possible for an idea to be mutated if misinterpreted. However, an additional property of an idea is that it can be enhanced by 1 Introduction it’s owner. Memetic algorithms have as the unit of information The thermal generator maintenance scheduling prob- a meme [Daw76] instead of a gene. The memetic process can lem [Yam82] is concerned with scheduling essential main- be viewed as an attempt to model the processes which occur tenance over a fixed planning horizon for a number of thermal during the evolution of ideas. generator units while minimising the maintenance costs and providing enough capacity to meet the anticipated demand. 2.2 Implementation Several modern heuristic methods have been applied to the The problem as described in the previous section can be reduced ˇ problem. Examples are simulated annealing [C85, KGV83], to that of finding the optimal starting period of each generating stochastic evolution [SR91], genetic algorithms [Gol89] and unit such that the objective function is minimised. tabu search [Ree93]. This paper explores the application of a memetic algorithm to 2.2.1 Genetic Representation this problem. To represent a problem solution, a direct representation is used A brief description of the problem is given below. A more dein which the start period of each generator unit is encoded into a tailed description of the formulation is given in [BCS97]. binary string using gray-coding to reduce the impact of the difConsider a collection of generating units producing output over fering significances of individual bits which would be encouna fixed planning horizon. Each unit must be maintained for a tered using a base-2 encoding. An individual is the concatepredetermined number of contiguous periods during the hori- nation of each unit’s encoding. The encoding must be defined zon. The demand forecast throughout the planning horizon is such that there is an onto mapping between start periods and known, and also the anticipated fuel costs. encodings. In order to avoid random factors in the problem, such as unit In the instance where the number of periods is not a power of random outages, a reserve capacity variable proportional to the 2, then not all encodings are valid. To compensate for this, the demand is incorporated into the problem description. This prob- starting period is multiplied by a factor such that the resulting lem is classified as a deterministic cost-minimisation problem range of values is exactly a power of 2. The reverse is done and can be solved using an optimisation-based technique. during the decode operation. This will favour certain periods The objective function takes into account the running costs and in that more than one allelle might decode to produce the same maintenance costs of the plan, together with penalty functions period. However such periods are spread evenly throughout the to penalise infeasible or undesirable situations. An example of range. An alternative solution would be to take the modulus of an infeasible solution would be a plan which attempts to exceed the number of periods after the decoding but this would always the operating capacity of the generating units. favour the first few periods and this was felt to be unacceptable. In Proceedings of the International Conference on Neural Information Processing and Intelligent Information Systems Vol 1, pp469–473, Springer, ISBN: 981-3083-64-6
2.2.2 Initialisation For the problem of how to find a good initial population, various techniques were tried, including random generation, heuristic search and with suitable parameters, it was found that a sufficiently healthy initial population can be obtained through the application of local search to a randomly generated population. Experiments suggest that a good initial population is of less significance for a memetic algorithm than for a genetic algorithm. However, this is not particularly surprising since the fitness of each individual is greatly improved after the first application of local search.
solution would make it too slow to be practical. Figure1 shows that tabu search improves rapidly at first and once the first local optimum is found, many iterations are required for a further significant improvement. The number of iterations for an initial gain can give us an indication of the number of iterations that would yield a faster but worthwhile application of tabu search to improve the solution. 1700000 Objective function 1680000
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2.2.3 Selection 1640000
At first, roulette wheel selection was used to select pairs of in1620000 dividuals for reproduction. However extensive testing led us to abandon this in favour of tournament selection. After this 1600000 change, the performance was increased to the point that the best solution was often better than that reached by the previous best 1580000 performing algorithm, tabu search [BCS97]. 1560000 Since this change had a dramatic impact on the algorithm’s per0 10 20 30 40 50 60 70 formance, it suggested that the use of elitism may also be beneficial. Automatically selecting the best individual from the preFigure 1: Progress of tabu search optimisation vious generation actually showed no improvement in the results but the number of generations necessary to attain the best result was reduced. 2.2.7 Neighbourhood search 2.2.4 Recombination In our representation, single-point crossover is used and has the effect of taking the starting period for the first few units from the first chromosome and the remainder from the second chromosome. 2.2.5 Mutation Two mutation operators are applied to each new chromosome, light mutation and heavy mutation. The light mutation operator probabilistically flips bits in the chromosome. With a sufficiently small probability, this has the effect of increasing diversity while still retaining the potentially beneficial effects of the crossover operator. This is followed by heavy mutation which attempts to target periods in which a penalty function is incurred due to capacity being below the desired level. Each unit which is being maintained during these periods, it is rescheduled to a random starting period. 2.2.6 Local Search Local search can be thought of as the process of an individual improving its idea of the solution. Since the tabu search algorithm has been observed [BCS97] to be an extremely efficient algorithm for this problem, we have used it to effect the local search. One of the most important characteristics of the local search is that it should be fast. Clearly using tabu search with the same parameters that enables the algorithm to find a very good
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During each iteration the solutions in the neighbourhood of the current solution are considered as candidates for the next solution. Two different search functions are considered for use by the algorithm. The first search function returns solutions where for each generating unit the start period is moved to each possible starting period. In each case, only one start period is moved. (J ? 1) I solutions are considered, where the number of units is denoted by I , and the number of periods is denoted by J . The second search function is similar to the first, except for each unit only the periods adjoining the current start period are considered. So there are 2 I considered solutions. For each function, only one generating unit is selected and its maintenance start period is changed. Thus it is possible to implement the cost function calculation by means of computing the change caused by the move rather than by evaluating the entire cost function. Random move functions were used in an initial implementation but it was found that the use of these search methods improved the effectiveness of the algorithms. 2.2.8 Tabu criteria Two different methods of tabu classification were implemented. The first method compares state of the trial solution to that of the last N solutions, where N denotes the size of the tabu list. Thus the algorithm cannot re–visit a previous solution for N iterations. This is referred to as “solution tabu”. The second method compares the move that takes the current
In Proceedings of the International Conference on Neural Information Processing and Intelligent Information Systems Vol 1, pp469–473, Springer, ISBN: 981-3083-64-6
solution to the trial solution with those moves made in the last N iterations. Once a solution is accepted, the inverse move is entered into the tabu list, thus the algorithm is forbidden to visit the previous solution. This is referred to as “move tabu”. 2.2.9 Termination criteria
System I J 15 25 30 40 60 52 System I J 15 25 30 40 60 52
MA1 Cost Time 249325 402 742720 787 2523219 1604 TS1 Cost Time 249325 56 742732 111 2523523 320
MA2 Cost Time 249325 125 742720 451 2523219 1018 TS2 Cost Time 249448 14 744426 19 2527211 45
When no improving solution has been found over the last n iterations the algorithm terminates. Typically this value is between 200 and 1000 depending on the desired quality of the solution Table 1: Comparison of the algorithms and the size of the problem since for a larger problem, improving moves are likely to be found with a lower frequency. An alternative termination criterion was considered: the algorithm stops after computing n iterations. This was found to be the solutions obtained by using the three algorithms were optimal. However the memetic algorithm was somewhat slower. more dependant upon the problem size. The real advantage of using a memetic algorithm is when large 2.3 Experimental results problems are considered. Results for the application of a genetic algorithm to this prob- In [SN91], three problems of varying complexity are given and lem [BCS97] are relatively poor. However, it is suggested that a comparison performed between simulated annealing and intethe optimum mutation parameter is around 0:04 and that the ger programming. [BCS97] compares the same problems with optimum crossover parameter is in the neighbourhood of 0:1. simulated annealing, a genetic algorithm, tabu search and a hyTabu search was also explored, and it was found that a large brid algorithm composed with elements of simulated annealing search space combined with forbidding previous solution states and tabu search. Table 1 shows the results obtained from the memetic algorithm, together with those of tabu search. The ‘1’ achieved good results. suffix denotes that a large neighbourhood search space was used Although the memetic algorithm is a genetic algorithm comin combination with ‘solution tabu’ in which recent solution bined with tabu search, it is not necessarily the case that the states are in tabu. ‘2’ denotes that a smaller search space was parameters discussed above are the most ideal for a memetic used together with ‘move tabu’ in which recently performed algorithm. Extensive testing suggests that optimum parameters moves are in tabu. for this problem are a smaller population, combined with a reduced tabu list length and a smaller number of iterations when For the small problem, tabu search can be seen to be the most encountering non-improving solutions during the local search. effective algorithm as the same solution is achieved in a shorter The characteristics of tabu search are that it climbs rapidly to time. But for larger scale problems the memetic algorithm can the nearest optima and then (subject to the tabu list constraints) be seen to outperform tabu search. The running time of both algorithms can be reduced by using the second search algosearches for other optima in the region. rithm since less of the neighbourhood is searched on each iterIntuitively, the large neighbourhood space directs the tabu ation. The solution quality of tabu search is severly reduced by search to search more of the neighbourhood of the first local this choice whereas results suggest that the memetic algorithm optima. The change in the size of the tabu list means the overis unaffected. It is suggested that this occurs since the local heads of operating the list are reduced while the effectiveness search is a subsidiary part of the algorithm and a reduction in is maintained since the number of iterations is also reduced. its effectiveness is compensated through the advantages of the The resultant abbreviated tabu search encounters the first local underlying genetic algorithm. optima and performs a cursory search around the neighbourIn conclusion, the experiments show that the proposed memetic hood before terminating. If a better set of improving solutions method produces results that indicate that it can be usefully apis found then the search follows these until no more increasingly plied to real problems. optimal solutions are found in the neighbourhood. While it might appear that there is little value in the genetic operators, replacing the process in which new individuals are References created from the previous generation by one in which new indi[BCS97] E. K. Burke, J. A. Clark, and A. J. Smith. Four viduals are created randomly reduces the utility of the algorithm methods for maintenance scheduling. In Proceedto the point where its effectiveness is not as good as tabu search ings of the International Conference on Artificial Neualone. ral Networks and Genetic Algorithms, pages 264–269. Springer, 1997. 3 Comparison For very small problems it is the case that the memetic algorithm, finds the same solution as simulated annealing and tabu [Daw76] R. Dawkins. The Selfish Gene. Oxford University search. By enumeration the same solution was found, therefore Press, 1976. In Proceedings of the International Conference on Neural Information Processing and Intelligent Information Systems Vol 1, pp469–473, Springer, ISBN: 981-3083-64-6
[Gol89] David E. Goldberg. Genetic algorithms in search, optimization and machine learning. Addison-Wesley, 1989. [Hol75] J. H. Holland. Adaptation in natural and artificial systems. University of Michigan, 1975. [KGV83] S. Kirkpatrick, C. D. Gelatt, Jr., and M. P. Vecchi. Optimisation by simulated annealing. Science, 220:671–680, 1983. [Mic92] Z. Michalewicz. Genetic Algorithms + Data Structures = Evolution Programs. Springer-Verlag, Heidelberg, 1992. [Ree93] C. Reeves. Modern Heuristic Techniques for Combinatorial Problems. Blackwell, 1993. [SN91] T. Satoh and K. Nara. Maintenance scheduling by using the simulated annealing method. IEEE Transactions on Power Systems, 6:850–857, 1991. [SR91] Youssef G. Saab and Vasant B. Rao. Combinatorial optimisation by stochastic evolution. IEEE Transactions on Computer-Aided Design, 10:525–535, 1991. ˇ ˇ [C85] V. Cerny. Thermodynamic approach to the travelling salesman problem: An efficient simulation algorithm. Journal of Optimization Theory and Applications, 45:41–52, 1985. [Yam82] Z. A. Yamayee. Maintanance scheduling: Description, literature survey and interface with overall operations scheduling. IEEE Transactions on Power Apparatus and Systems, PAS-101(8):2770–2779, 1982.
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