Application of Particle Swarm Optimisation to Scheduling Problems

Particle swarm optimisation (PSO) has been first presented by Kennedy and Eberhart [68] in 1995. It was simulated as a social behaviour of bird flocking or fish schooling to be used as an optimiser.

PSO has been applied to a great variety such as optimisation problems, artificial neural network training, pattern recognition, fuzzy control [32] [120] [7], continuous nonlinear functions [68], nonlinear constrained optimisation problems [31] and some other fields.

Moreover, PSO has been growing rapidly with over 100 published papers every year.

21

hand, the number of the applications of PSO to scheduling problems is extremely low [78].

In 1989, Shaw and Whinston [110] proposed a PSO approach to scheduling of flexible manufacturing systems. Researchers, who have studied the nonlinear software problems adopting the PSO technique, usually believe that the parameters w, r1 and r2 are the key factors to affect the convergence of the PSO [18] [120] [92] and were r1 and r2 are chosen randomly, PSO cannot guarantee the optimisation‘s quality. Chuanwen and Bompard [18] provided a new method that introduces chaotic mapping with certainty, ergodicity and the stochastic property into particle swarm optimisation so as to improve the global convergence. This technique used to solve the short term generation scheduling of a hydro-system in a deregulated environment. The result introduced chaos mapping and an adaptive scaling term into the particle swarm optimisation algorithm, which increases its convergence rate.

Jerald et al [62] used four techniques GA, PSO, Mimetic Algorithm and Simulated Annealing for scheduling optimisation. They applied these techniques to Flexible Manufacturing Systems (FMS). The FMS contain five flexible machining cells each with two to six Computerised Numerical Control machines an independent and a self-sufficient tool magazine, one Automatic Tool Changer and one automatic pallet changer. Each cell is supported by one to three dedicated robots for intra-cell movement of materials between operations. The objective of the schedule is to minimise the machine ideal time and minimising the total penalty cost. Results of this works show that Particle swarm algorithm is found to be superior and gives the minimum combined objective function.

It is well known that the original PSO is designed as continuous technique. For solving discrete optimisation problems first, Kennedy and Eberhar [69] developed a discrete binary version of the PSO. There were two main differences the first is in the particle which was composed as binary variable while the second is in the velocity which is changed where it‘s probability having to be changed to give binary variable one value.

Consequently, many researches came to solve discrete optimisation problems. Few research concentrate on scheduling using discrete PSO. Liaoa et al [78] applied this

22

approach of discrete PSO in flow shop scheduling problems and found that the discrete PSO algorithm performs better than the continuous PSO algorithm of Tasgetiren et al.

[114]. Moreover, compared to GA, PSO gave better performance when it was allowed to run on the same time. In this work they used local search scheme to incorporate PSO.

They called it PSO-LS. The result shows how it is guided successfully by PSO.

Although it take more time than PSO, it performs better for some problems such as Max–Min Ant System and Posed Ant Colony Algorithm which are the latest versions of ant-colony algorithms which have been applied to the scheduling problem for nearly 10 years [78]. Another research used Discrete Particle Swarm Optimisation for scheduling made by Pana et al [96] . The researchers in this paper applied standard and Variable Neighbourhood Descent to the no-wait flowshop scheduling problems. 110 benchmark instances of Taillard [117] are treated as no-wait flowshop problems and applied on both algorithms. The computational results show that the proposed algorithms generated either competitive or better results than those by the descent and Travailing Salesman algorithms of Grabowski and Pempera [51]. Moreover, the variable neighbourhood local search in the PSO algorithm enhanced the solution quality significantly.

Sha and Lin [109] proposed PSO for multi objective optimisation problems. In this work, diversification strategy has been developed to prevent PSO sticking to local optimal solutions. This strategy has been applied based on the following set of rules.

 If the solution of the particle dominates the gbest solution, assign the particle solution to the gbest.

 If the solution of the particle equals to any solution in the non-dominated solution set, replace the non-dominated solution with the particle solution.

 If the solution of the particle is dominated by the worst non-dominated solution and not equal to any non-dominated solution, set the worst non-dominated solution equal to the particle solution

Also, in this work PSO was modified by changing representation of particle position, particle movement, and particle velocity using priority value for global best and local best. Moreover, using mutation operator PSO performance has been improved. This proposal has been tested by Taillard‘s benchmark [111]. The results show that

Multi-23

whose fitness function was weighted by sum of makespan, total tardiness, and total idle time of machines with random weights. In 22 of the 23 problems, the proposed PSO performed better for the solution considering the total tardiness. Finally, it can be said that the proposed algorithm was superior to the GA in solving the JSP with multiple objectives.

2.2.4 Application of Scheduling to Heating Treatment Operations