Applications of particle swarm optimisation in integrated process planning and scheduling

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1 Robotics and Computer-Integrated Manufacturing ] (]]]]) ]]] ]]] Applications of particle swarm optimisation in integrated process planning and scheduling Y.W. Guo a,, W.D. Li b, A.R. Mileham a, G.W. Owen a a Department of Mechanical Engineering, University of Bath, Claverton Down, Bath, Avon BA2 7AY, UK b Department of Engineering and Manufacturing Management, Faculty of Engineering and Computing, Coventry University Priory Street, Coventry CV1 5FB, UK Received 17 July 2007; received in revised form 3 December 2007; accepted 11 December 2007 Abstract Integration of process planning and scheduling (IPPS) is an important research issue to achieve manufacturing planning optimisation. In both process planning and scheduling, vast search spaces and complex technical constraints are significant barriers to the effectiveness of the processes. In this paper, the IPPS problem has been developed as a combinatorial optimisation model, and a modern evolutionary algorithm, i.e., the particle swarm optimisation (PSO) algorithm, has been modified and applied to solve it effectively. Initial solutions are formed and encoded into particles of the PSO algorithm. The particles fly intelligently in the search space to achieve the best sequence according to the optimisation strategies of the PSO algorithm. Meanwhile, to explore the search space comprehensively and to avoid being trapped into local optima, several new operators have been developed to improve the particles movements to form a modified PSO algorithm. Case studies have been conducted to verify the performance and efficiency of the modified PSO algorithm. A comparison has been made between the result of the modified PSO algorithm and the previous results generated by the genetic algorithm (GA) and the simulated annealing (SA) algorithm, respectively, and the different characteristics of the three algorithms are indicated. Case studies show that the developed PSO can generate satisfactory results in both applications. r 2007 Elsevier Ltd. All rights reserved. Keywords: Particle swarm optimisation; Operation sequencing; Integrated process planning and scheduling; Genetic algorithm; Simulated annealing 1. Introduction Process planning, an essential component for linking design and downstream manufacturing processes, is the act of preparing detailed instructions to transform an engineering design to a final part [1]. One of core activities in process planning is to decide which manufacturing resources to select and in which sequence to use, mainly based on the objective of achieving the correct quality, the minimal manufacturing cost and ensuring good manufacturability. Scheduling is used to determine the most appropriate moment to execute each for the launched production orders, taking into account the due date of these orders, a minimum makespan, a balanced Corresponding author. Tel.: ; fax: address: y.guo@bath.ac.uk (Y.W. Guo). resource utilisation, etc., to obtain high productivity in the workshop [2,3]. In job shop and batch manufacturing, both process planning and scheduling are responsible for the effective allocation and utilisation of resources. A process plan is usually determined before the actual scheduling with no regard for the scheduling objectives and with the assumption that all the resources are available. However, if a process plan is prepared offline without due consideration of the actual shop floor status, it may become unfeasible due to changes or constraints in the manufacturing environment and heavily unbalanced resource assignments. Also, due to the different objectives of these two systems, it is difficult to produce a satisfactory result in the simple sequential execution of the two systems. The merit of integrated process planning and scheduling (IPPS) is to increase production feasibility and optimality by combining both the process planning and scheduling problems [4] /$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi: /j.rcim

2 2 ARTICLE IN PRESS Y.W. Guo et al. / Robotics and Computer-Integrated Manufacturing ] (]]]]) ]]] ]]] The most recent works related to the IPPS optimisation can be generally classified into two categories: the enumerative approach and the simultaneous approach [2]. In the enumerative approach [3,5 7], multiple alternative process plans are first generated for each part. A schedule can be determined by iteratively selecting a suitable process plan from alternative plans of each part to replace the current plan until a satisfactory performance is achieved. The simultaneous approach [2,8 12] is based on the idea of finding a solution from the combined solution space of process planning and scheduling. In this approach, the process planning and scheduling are both in dynamic adjustment until specific performance criteria can be satisfied. Although this approach is more effective and efficient in integrating the two functions, it also enlarges the solution search space significantly. To address the above two optimisation problems, some optimisation approaches based on modern heuristics or evolutionary algorithms, such as the genetic algorithm (GA) (for sequencing problem [13 17]; for IPPS problem [9,10,12,18]), simulated annealing (SA) algorithm (for sequencing problem [19,20]; for IPPS problem [2,5]), Tabu search algorithm (for sequencing problem [20,21], for IPPS problem [11]) and agent-based approach (for IPPS problem [22]) have been developed in the last two decades and significant improvements have been achieved. However, for parts with complex structures and features and multiple parts involved, these two optimisation processes are well known as complicated decision problems. The major difficulties include: (1) both sequencing and IPPS problems are NP-hard (non-deterministic polynomial) combinatorial optimisation problems. The search space is usually very large especially for IPPS problem because it involves multiple parts scheduling, and many previously developed methods could not find optimised solutions effectively and efficiently, and (2) there are usually a number of precedence constraints in sequencing s and manufacturing resource utilisation constraints due to manufacturing practice and rules, which make the search more difficult. Therefore, it is necessary to develop efficient models for the sequencing and the IPPS optimisation problems and the optimisation algorithms need to be more agile and efficient to solve practical cases. Particle swarm optimisation (PSO) is a modern evolutionary computation technique based on a population mechanism [23]. It has been motivated by the simulation of the social behaviour of individuals (particles). This paper investigates the applications of this emerging optimisation algorithm into the intractable sequencing and the IPPS problems, and a newly developed PSO-based optimisation algorithm for them is elaborated. Firstly, the sequencing is defined and the representation of a solution for it by a particle is presented. Then, the representation model is expanded to represent the IPPS problem. The fitness functions of the solutions for these two problems are stated. Thirdly, the details of applying the PSO algorithm for them are described. Finally, case studies with computational experiments to test the algorithm are demonstrated, and a comparison between the result of the PSO algorithm and that of previous work is presented. 2. Representation of the process planning problem The PSO algorithm was inspired by the social behaviour of bird flocking and fish schooling [23]. Three aspects will be considered simultaneously when an individual fish or bird (particle) makes a decision about where to move: (1) its current moving direction (velocity) according to the inertia of the movement, (2) the best position that it has achieved so far, and (3) the best position that its neighbour particles have achieved so far. In the algorithm, the particles form a swarm and each particle can be used to represent a potential solution of a problem. In each iteration, the position and velocity of a particle can be adjusted by the algorithm that takes the above three considerations into account. After a number of iterations, the whole swarm will converge at an optimised position in the search space. To conduct process planning, parts are represented by manufacturing features. Fig. 1 shows a part composed of m features. Each feature can be manufactured by one or more machining s (n s in total for the part). Each can be executed by several alternative plans if different machines, cutting tools or set-up plans are chosen for this [28,29]. A set-up is usually defined as a group of s that are machined on a specified machine with the same fixture. Here, a set-up is equivalently defined as a group of s with the same tool approach direction (TAD) machined on a machine. For example, a through hole with two TADs is considered to be related to two set-ups. A process plan for a part consists of all the s needed to machine the part and their relevant machines, cutting tools, TADs, and sequences. A good process plan of a part is built up based on two elements: (1) the optimised selection of the machine, cutting tool and TAD for each ; and (2) the optimised sequence of the s of the part. Fig. 1. Representation of a process plan (particle).

3 Y.W. Guo et al. / Robotics and Computer-Integrated Manufacturing ] (]]]]) ]]] ]]] 3 Hence, the developed algorithm needs to address these two aspects. To apply the PSO algorithm to the process planning optimisation problem, two issues have to be handled first: (1) Encode a process plan to produce a particle. As shown in Fig. 1, each is modelled as a particle dimension that includes information on machines, cutting tools and TADs, and the details are listed in Table 1. Here a position variable and a velocity variable are used to represent the position and velocity of an, respectively. All the particle dimensions (s) executed to make the part form a particle (a process plan). In Table 2, an initialised particle with five ParticleDimensions is shown. (2) Decode the particle to get a sequenced process plan. In each iteration, when all the ParticleDimensions in a particle have been updated, the sequence can be determined by the relative positions of the Particle- Dimensions [30]. For example, in Table 2, the sequence of the particledimensions will be (3, 4, 5, 2, 1) according to the descending order of their position values. By using a number of iterations to update the positions and velocities of the particledimensions in each particle, an optimised sequence (i.e., an optimised process plan) can be achieved eventually. A feasible sequence must comply with precedence constraints that come from geometrical and technological considerations. The precedence constraints between s are usually classified into six types: (1) fixture interaction, (2) tool interaction, (3) datum interaction, (4) thin-wall interaction, (5) material-removal interaction, and (6) fixed order of machining s [21,31]. An optimised process plan needs to satisfy all of these constraints. Machining cost can be used to measure the quality of a process plan quantitively. The machining cost of a plan is comprised of machine utilisation costs, tool utilisation costs, machine change costs, set-up change costs, tool change costs and additional penalty cost. The formulas are given in [21,31]. The time of machine utilisation is not explicitly represented and handled here. However, it will be considered in the following scheduling and IPPS. 3. Representation of the IPPS problem The IPPS problem can be defined as: Given a set of n parts which are to be processed on m machines with s including alternative manufacturing resources (machines, tools and TADs), select suitable manufacturing resources and sequence the s so as to determine a schedule in which the precedence constraints among s can be satisfied and the corresponding objectives can be achieved. The problem can be modelled as an extension of the sequencing optimisation problem into one of multi-parts with scheduling objectives. To achieve this, the Table 1 Class definition of a particle dimension (an ) Class ParticleDimension: an Variable Description Operation_id Machine_id Tool_id TAD_id Machine_list[ ] Tool_list[ ] TAD_list[ ] Position Velocity Table 2 An initialised particle The id of the The id of a machine to execute the The id of a cutting tool to execute the The id of a TAD to apply the The candidate machine list for executing the The candidate tool list for executing the The candidate TAD list for applying the The position value of the The velocity value of the Operation_id Machine_id Tool_id TAD_id Position Velocity Relative position (sequence no.) Table 3 Class definition of a particle dimension (an ) Class ParticleDimension: an Variable Description Operation_id Part_id Machine_id Tool_id TAD_id Machine_list[ ] Tool_list[ ] TAD_list[ ] Mac_time Change_time Machine_s_time Machine_e_time Position Velocity The id of the The id of part to which the belongs The id of a machine to execute the The id of a cutting tool to execute the The id of a TAD to apply the The candidate machine list for executing the The candidate tool list for executing the The candidate TAD list for applying the The machining time for this The change time required for this including tool change, set-up change and machine change The start machining time of executing this The end machining time of executing this The position value of the The velocity value of the

4 4 ARTICLE IN PRESS Y.W. Guo et al. / Robotics and Computer-Integrated Manufacturing ] (]]]]) ]]] ]]] representations of the process plans and schedule need to be extended: (1) In encoding process: Compared to the representation of a process plan shown in Fig. 1, several new variables including Mac_time, Change_time, Machine_s_time and Machine_e_time are added to record and track the time related to the execution of the so as to determine the time allocation on the machines. Mac_time, Change_time, Machine_s_time and Machine_e_time are set as 0 initially. Table 3 shows the extended class definition of a particle dimension (an ). (2) In decoding process: To record the machine utilisation status (available time) and s being executed on every machine (including start time, time and end time for each ) at different times, a machine class is defined. As discussed in Section 2, when the sequence for all the s is generated Machine½jŠUtilisation ¼ Xn ðj ¼ 1; ::; mþ P m j¼1 i¼1 ðmachine½jšutilisationþ ðoperation½išmac_tþ, w ¼, (4) m vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ux m Utilisation_Level ¼ t ðmachine½jšutilisation wþ 2. j¼1 (5) A fixed penalty time (PT): applied to each violated constraint which type is described in Section 2. Thus APT ¼ XNVC PT, (6) where i¼2 ð3þ ( 1 The sequence of X before Y violates constraints; O 4 ðx; YÞ ¼ 0 The sequence of X before Y is in accordance to constraints: and the manufacturing resources are selected, the assignments of specific s and machines are determined and therefore the schedule is obtained. By using a number of iterations to update the positions and velocities of the particledimensions in each particle, an optimised sequence (i.e., an optimised solution) can be achieved eventually. When the process plans of all parts are generated and the manufacturing resources are specified, it is required to determine the schedule based on this information and calculate the makespan, total tardiness, etc. Here, three evaluation criteria of the IPPS problem can be calculated as follows: m Makespan : Makespan ¼ Max ðmachine½jš j¼1 Available_timeÞ. Total job tardiness: the due date of a part is denoted as DD, and the completion moment of the part is denoted as CM. Hence, 0 if DD is later than CM; Part_Tardiness ¼ CM DD Otherwise: Balanced level of machine utilisation: the standard deviation concept is introduced here to evaluate the balanced machine utilisation (assuming there are m machines, and each machine has n s). ð1þ (2) APT is the total additional penalty time and NVC is number of violating constraints. 4. The modified PSO algorithm A traditional PSO algorithm can be applied to optimise the IPPS in the following steps: (1) Initialisation: Set the size of a swarm, e.g., the number of particles Swarm_Size and the max number of iterations Iter_Num. Initialise all the particles in the method introduced in Sections 2 and 3. Decode every particle (solution) to get the schedule of the particle and then calculate the corresponding criteria of particle (the result is called fitness here). Set the local best particle and the global best particle with the best fitness. (2) Iterate the following steps until Iter_Num is reached: For each particle in the swarm, and each ParticleDimension (i.e., in particle), update ParticleDimension s velocity and position values. Decode the particle into a solution in terms of new position values and calculate the fitness of the particle. Update the local best particle and the global best particle if a lower fitness is achieved. (3) Decode global best particle to get the optimised solution. However, the traditional PSO algorithm introduced above is still not effective in resolving the

5 Y.W. Guo et al. / Robotics and Computer-Integrated Manufacturing ] (]]]]) ]]] ]]] 5 sequencing problem. There are two major reasons for it: (1) Due to the inherent mathematical operators, it is difficult for the traditional PSO algorithm to consider the different arrangements of machines, tools and TADs for each, and therefore the particle is unable to fully explore the entire search space. (2) The traditional algorithm usually works well in finding solutions at the early stage of the search process (the optimisation result improves fast), but is less efficient during the final stage. Due to the loss of diversity in the population, the particles move quite slowly with low or even zero velocities and this makes it hard to reach the global best solution [32]. Therefore, the entire swarm is prone to be trapped in a local optimum from which it is difficult to escape. To solve these two problems and enhance the ability of the traditional PSO algorithm to find the global optimum, new s, including mutation, crossover and shift, have been developed and incorporated in a modified PSO algorithm. Meanwhile, considering the characteristics of the algorithm, the initial values of the particles have been well planned. Some modification details are depicted below. (1) New operators in the algorithm Mutation: In this strategy, an is first randomly selected in a particle. From its candidate machining resources (Machine_list[], Tool_list[] and TAD_list[]), an alternative set (machine, tool, TAD) is then randomly chosen to replace the current machining resource in the. The probability of applying this strategy is defined as P m. Crossover: Two particles in the swarm are chosen as parent particles for a crossover. In the crossover, a cutting point is randomly determined, and each parent particle is separated as left and right parts of the cutting point. The positions and velocities of the left part of Parent 1 and the right part of Parent 2 are reorganised to form Child 1. The positions and velocities of the left part of Parent 2 and the right part of Parent 1 are reorganised to form Child 2. The probability of applying the crossover is defined as P c. Shift: This operator is used to exchange the positions and velocities of two s in a particle so as to change their relative positions in the particle. The probability of applying the shift is defined as P s. (2) Escape method During the optimisation process, if the iteration number of obtaining the same best fitness is more than 10, then the mutation and shift s are applied to the best particle to try to escape from the local optima. The workflow of the modified PSO algorithm is shown in Fig Case study and discussions Two experiments are used here to verify the efficiency of the PSO algorithm for the sequencing and IPPS problems. The first experiment is designed to compare the efficiencies of the PSO, GA and SA algorithms in the application of sequencing optimisation. The second experiment is used to further compare them for the IPPS optimisation. For simplification, the parameters of the PSO algorithm recommended in Guo et al. s work [31] are used in the PSO algorithm for experiments in this paper (Swarm_Size are set as 5000, Iter_Num as 200; P m ¼ 0.65, P c ¼ 0.2, and P s ¼ 0.3) Experiment 1 Three parts taken from the works of Li and MacMahon [2] and Zhang et al. [16] are used here as example parts. The relevant information of features, s, and precedence constraints for each part is given in Table 4 briefly. The GA and SA algorithms developed by Li et al. [21,33] have been used to compare their performance with this developed PSO algorithm. As shown in Fig. 3, at the initial optimisation stage, the GA optimises faster than the SA Part 1 Part p Operation 1 Operation 2 Operation n ParticleDimension 1 Applicable machines Applicable tools Applicable TADs Machining time Machine change time Machining start time Machining end time Position Velocity Particle ParticleDimension 2 ParticleDimension n Applicable machines Applicable tools Applicable TADs Machining time Machine change time Machining start time Machining end time Position Velocity Fig. 2. Representation of a solution for the IPPS problem. Table 4 The technical specifications for the part in Group 1 (Li and McMahon [2]) Part Number of s (with numbers of alternative machining plans for each ) 1 20 (9, 9, 9, 8, 12, 12, 6, 12, 3, 4, 12, 12, 3, 4, 4, 58 3, 6, 12, 3, 4) 2 16 (8, 12, 9, 9, 18, 8, 6, 8, 9, 4, 9, 9, 8, 18, 4, 10 60) 3 14 (9, 9, 36, 16, 36, 24, 27, 36, 24, 6, 8, 8, 6, 8) 51 Number of constraints

6 6 Y.W. Guo et al. / Robotics and Computer-Integrated Manufacturing ] (]]]]) ]]] ]]] Initialise Swarm_Size and Iter_Num, Initialise the particle swarm set the Pi as particle itself Calculate the fitness of particles, Initialise Pg, set N=0 Swarm_Size: the population of particles, Iter_Num: the max iteration number of PSO N: Iteration index number For each iteration Select particles randomly to do mutation, crossover and shift s, set M=0 M: Particle index number For each particle in swarm 1. Update Mth particle s position and velocity 2. Calculate the fitness of Mth particle Fig. 4. Comparisons of PSO, GA and SA for Part 3 in the process planning optimisation. N and the PSO (this is shown by a more rapid fall in Fig. 4). However, at the middle and late stages, the GA converges while the SA and the PSO continue to decline to give better results. From Table 5, it can be observed that the SA and PSO algorithms outperform the GA in all the experiments of all three parts and both the SA and PSO can achieve results that are nearer the optimum Experiment 2 If current fitness < fitness of Pi? Y Update the Pi with the current position N If fitness of Pi < fitness of Pg? N N Y Update the Pg with Pi M=M+1 M>= Swarm_Size? Y N=N+1 N>=Iter_Num? Y Output current Pg and the fitness Fig. 3. Workflow of the PSO algorithm. Two criteria are used here as the optimising direction for IPPS problem, i.e., the makespan and the balanced machine utilisation. The example parts and manufacturing resources from Li and McMahon [2] are used here to verify the efficiencies of the PSO. Eight parts have been used to test the algorithm under more complex conditions. The relevant specifications of the parts are given in Table 6. The population of the GA and the PSO are both set as 200. It can be found that the PSO can optimise the Makespan after nearly 4000 iterations and the balanced machine utilisation after 3000 iterations Makespan As shown in Table 7 and Fig. 5, with the same time period, the PSO and the SA can achieve better results than the GA. But for 20 random consecutive trials, the SA can only proceed with optimisation successfully in six trials, the PSO and the GA can proceed with optimisation successfully in all 20 trials Balanced machine utilisation From Table 8 and Fig. 6, it can be observed that all of the algorithms can reach good results, while different characteristics are shown due to the inherent mechanisms of the algorithms. The SA is much sharper to find optimised solutions than the GA and the PSO. The SA can achieve better results than the GA and the PSO. However, in 20 trials, the SA can only proceed with optimisation successfully in six trials but the GA and the PSO can proceed with optimisation successfully in all 20 trials Summary of GA, SA and PSO algorithms As discussed in above sections, the GA, SA and PSO algorithms are used to optimise the IPPS problem. All of them can yield good results, but they have different characteristics. It is sometimes difficult to make sure that the algorithms converge already as they can go to another stage after staying at one stage for a while. Therefore, a single time is used for all of them to unify the comparison standards. The GA and the PSO are both population-based algorithms but the SA is not. So the optimising processes of the GA and the PSO take a longer time than that of the SA [31]. It can also be observed that the PSO needs to adjust the particledimensions by updating the velocities and positions of them due to its intrinsic mechanism so that it needs more computation time than the GA. For the

7 Y.W. Guo et al. / Robotics and Computer-Integrated Manufacturing ] (]]]]) ]]] ]]] 7 Table 5 The comparisons of GA, SA and PSO in the process planning optimisation Algorithm Part 1 Part 2 Part 3 Best cost achieved Mean cost of 10 trials Best cost achieved Mean cost of 10 trials Best cost achieved Mean cost of 10 trials GA SA PSO Table 6 The technical specifications for the part in Group 2 (Li and McMahon [2]) Part Number of s (with numbers of alternative machining plans for each ) constraints 1 7 (9, 9, 27, 8, 8, 9, 36) (9, 9, 36, 18, 27, 8, 27, 18) (9, 9, 36, 36, 18, 6, 6) (9, 9, 27, 6, 36, 36, 6, 18, 18) (9, 9, 36, 36, 36, 18, 6) (9, 9, 36, 27, 18, 6, 27, 6, 18) (9, 27, 27, 18, 9) (9, 9, 27, 36, 36, 6, 6) 13 Number of Table 8 The comparisons of GA, SA and PSO of balance machine utilisation for Group 1 Algorithm Time for 5000 iterations GA 16 min 45 s 20 SA 22 min 6 PSO 7 min 30 s 20 Robustness (successful optimisation trials out of 20 trials) Table 7 The comparisons of GA, SA and PSO of Makespan Algorithm Time for 5000 iterations Robustness (successful optimisation trials out of 20 trials) GA 16 min 45 s 20 SA 45 min 6 PSO 7 min 20 Fig. 6. Comparisons of PSO, GA and SA for Group 2. Fig. 5. Comparisons of PSO, GA and SA for Group 2 (in 7 min). optimisation results, the SA and the PSO both outperform the GA in all the above case studies. As the complexity of the problem increases (for example when optimising IPPS problems), the SA can achieve better results than the GA and the PSO in the case studies described above. But as the complexity of the problem increases, the SA is not as robust as the GA and the PSO. Also as the complexity of the problem increases, it can be seen that the optimisation speed advantages of the GA and the SA over the PSO diminish. It is well known that simple mathematic s run much faster than other position changing s. This can probably be attributed to the fact that each iteration of the PSO algorithm uses mainly simple mathematical operators that can be finished in a shorter time than for the GA and the SA algorithms with mainly complex position changing operators. In constraints handling, the GA and the SA can use the adjust method developed by Li et al. [33] that keep the plan feasible, but the PSO can only use the penalty method to enable the results to comply with the constraints due to its intrinsic mechanism. The above discussion is illustrated in Table 9. For above two experiments, it only takes a few minutes to carry out a simulation for all the algorithms. However, for more complex problems with thousand parameters in the real world, the difference of the algorithms in terms of computation efficiency and robustness will be more

8 8 ARTICLE IN PRESS Y.W. Guo et al. / Robotics and Computer-Integrated Manufacturing ] (]]]]) ]]] ]]] Table 9 The comparison of GA, SA and PSO algorithms Algorithm Population based Optimisation result (out of 10) Optimisation speed Constraints handling Robustness GA Yes 6 Fast but get slow when Adjust penalty Robust complexity of problems increases SA No 9 Faster but get slow when complexity of problems increases Adjust penalty Not robust when complexity of problems increases PSO Yes 8 Fast Penalty Robust significant. On the other hand, it is also an important research issue to compare the performance of the algorithms to understand their mechanisms. 6. Conclusions In this research, the sequencing and the IPPS problems have been modelled and a modified PSO algorithm has been used to optimise them. Solutions to the sequencing and the IPPS problems are encoded into PSO particles to intelligently search for the best sequence of the s through leveraging the optimisation strategies of the PSO algorithm. To explore the search space more effectively, new operators, i.e., mutation, crossover and shift have been developed and incorporated to produce a modified PSO algorithm with improved performance. The GA and SA algorithms developed by Li et al. [21,33] and Li and McMahon [2] have been used to verify and compare the performance of the modified PSO algorithm with experiments of two groups parts. It is shown that the PSO algorithm can obtain a satisfactory optimisation result for these two problems. The characteristics of the GA, SA and PSO algorithms have been given, and for these cases the PSO algorithm has been shown to outperform both the GA and SA in the majority of applications by consideration of the computation efficiency, optimality and robustness. At this point in time the conclusions are limited by this computational experience, and more theoretical analysis needs to be made in future. However, there is still potential for further improvement in computation efficiency and optimality if introducing new operators and characteristics of other algorithms (For example, SA or Ant Colony Optimisation algorithm). And also with the population-based characteristics, a bounded rationality mechanism which is used in social science and economics can also be applied to the PSO algorithm to improve the performance further. Acknowledgements This work is funded by the Innovative design & Manufacturing Research Centre (IdMRC) and the Department of Mechanical Engineering at the University of Bath. References [1] Chang TC, Wysk RA. An introduction to automated process planning systems. Englewood Cliffs, NJ, USA: Prentice-Hall Inc.; [2] Li WD, Mcmahon CA. 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