Title: Dynamic scheduling of FMS using a real-time genetic algorithm.

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3 Title: Dynamic scheduling of FMS using a realtime genetic algorithm. Authors: Andrea Rossi, Gino Dini Affiliations: Dipartimento di Ingegneria della Produzione, Università di Pisa, via Bonanno Pisano 25/B, Pisa, Italy. Corresponding author: Gino Dini, Dipartimento di Ingegneria della Produzione, Università di Pisa, via Bonanno Pisano 25/B, Pisa, Italy, tel Fax , E mail: dini@itm.unipi.it. Abstract: The paper presents a genetic algorithm capable of generating optimised production plans in flexible manufacturing systems. The ability of the system to generate alternative plans following partflow changes and unforeseen situations is particularly stressed (dynamic scheduling). Two contrasting objectives represented by the reduction of machine idletimes, thanks to dynamic scheduling computation, and the reduction of the makespan are taken into account by the proposed system. The keypoint is the realtime response obtained by an optimised evolutionary strategy capable of minimising the number of genetic operations needed to reach the optimal schedule in complex manufacturing systems. Keywords: Dynamic Scheduling, Genetic Algorithms, Flexible Manufacturing System, Optimization

4 List of symbols. φιτ(χ) : fitness function of the chromosome Χ; ΓΧ(µγ, πσ, πχ, πµ) : genetic complexity (average number of operations) of a GA adopting the process parameters µγ, πσ, πχ and πµ; ΗΛΒ : κ : κ 1, κ 2 : Λ(µ 1,.., µ Μ ) : heuristic lower bound for the generalised jobshop scheduling problem; number of operations needed to evaluate the fitness function; parameters used by the genetic mutation operator; system layout, where µ η (with η=1,..,μ) represents the number of machine tools belonging to the η th machine category (e.g. lathes, machining centers, etc.); Μ : number of the categories of machine tools; µ : total number of machine tool categories; µ η : number of machines for the η th category; µακ(χ): makespan of the schedule chromosome Χ; µακ(χ,µγ,πσ,πχ,πµ): makespan of the schedule generated by the chromosome Χ having the best fitness and obtained by the module adopting the process parameters µγ,πσ,πχ,πµ; µγ : maximum number of generations; ν : Ν : Ο i : π,θ,ρ : πχ : number of jobs (or parts) to be rescheduled; set of the integers; ι th operation; positional indices in the operation queues of the chromosome; probability of crossover; πµ : probability of mutation; πσ : τ : ζ : population size (number of chromosomes); starting time of rescheduling; number of times that condition (1) is verified; τ ιη : processing time for the ι th job (ι=1,..,ν) on the machine category η; 2

5 τ : τ j : offline estimation of the scheduling computation time; time required to complete the operation processed at the instant τ on the j th machine tools, ϕ=1,..,µ; τ µ : σ : mean of the computation time; standard deviation of the computation time; 1. Introduction The problem of decreasing production costs through an appropriate management of available resources is fundamental in the field of industrial production. The performance of a flexible manufacturing system (FMS), not properly supported by an efficient resource assignment strategy, may be drastically limited, and the advantages derived from its flexibility in terms of production costs may suffer a sharp reduction. Furthermore, an FMS is composed of a large number of components, thus making the identification of a correct strategy for their management more difficult. It should also be emphasised that a static availability of resources (parts and machine tools) is never present in a real manufacturing scenario. Unforeseen situations have to be considered: delays in part availability, machine breakdowns, tool failures, changes in economic policies, etc. These events, the difficulties in predicting them and their effects on the system do not allow the use of a unreconfigurable production plan, and force manufacturers to develop flexible strategies capable of generating optimal plans for each situation. This problem, usually known as the dynamic scheduling of FMS, has been analysed in detail in Ramasesh (199) and in Suresh and Chaudhuri (1993). The main difference between dynamic scheduling and static or predictive scheduling lies in the robustness and the response reactivity of the algorithm to perturbations introduced into the manufacturing system. Two methods have usually been adopted in literature to address the problem of the dynamic scheduling of FMS. The first includes the ruleoriented methods which allow the identification of priority dispatching rules from a set of simple heuristic scheduling rules (e.g. SPT, LPT rules, etc.) with respect to a given set of jobs. This strategy has also been implemented through AI techniques such as neural networks (Wang, 1995), discrete event simulation (Ishii and Tavalage, 1991), fuzzy logic (Perrone et al., 1995), knowledgebased systems (O Kane et al., 1994) and hybrid systems (Rabelo et al., 1994). The second group includes joboriented methods which generate the schedule through the analysis of the most efficient alternatives in order to select the optimal (or the nearoptimal) solution. As this solution cannot usually be 3

6 obtained by means of any heuristic rule, this second approach generates better results; nevertheless, they use timeconsuming algorithms, which can be critical when the system has to give the output in a very short time. A nearoptimal solution can be found, with reasonable computational efforts, in Adams et al. (1988), by using approximation methods for classic job shop scheduling, and in Brandimarte et al. (1995) by means of continuous flowbased relaxed models. This problem can also be effectively faced by means of one of the most promising AI techniques: genetic algorithms (GAs) (Holland, 1975). A genetic algorithm operates through an implicit parallelism (Booker et al. 1989), which may be considered as a pool of entities aiming to find the optimal solution. These entities are the chromosomes and their evolutionary strategy of mating. The chromosomes constitute the basic structure of a GA and, in scheduling problems, they represent the loading sequences of the parts on the machine tools. Each chromosome is associated to a fitness function which shows its closeness to a lower bound value previously calculated by a heuristic process of reasoning. A GA is a multiparametrical function. The parameters are the size of the chromosome population and its mechanism of initialisation, the number of generations used in the algorithm, the mechanism of reproduction (composition of mating pool, kind of genetic operators, etc.). From these initial considerations, it is clear that the research area of GAs is very extensive. Furthermore, in the field of FMS scheduling, the fitness indices which can be optimised are various (e.g. minimising the makespan, reducing the workinprocess, increasing the throughput rate, meeting order duedates, etc.). Interesting and very promising studies on enhancing the performance of genetic operators have been carried out by Syswerda (1991) and Park L. and Park C.H. (1995). Other papers have been published with the aim of detecting the optimal values of these operators (Fichera et al. 1995, Gen et al. 1994), their inprocess adjustments (Morikawa et al., 1994) and genetic strategies to optimise multiobjective fitness functions (Liang and Lewis 1994, Sridhar and Rajendran 1994). A hybrid genetic learningbased approach to minimising the makespan has been proposed in (Kim and Lee, 1995). In this context, the contribution of the present work consists mainly in the development of a scheduling system capable of giving a realtime response by a joboriented method using a genetic algorithm in order to determine the optimal solution. The implementation of a GA, without a strategy to accelerate the processing speed, is not sufficient to obtain such a system. This aspect has been partially studied in (Fang et al., 1993) through a dynamic sampling of convergence rates in different elements of the chromosome. The approach proposed in this paper is to realise a global optimisation of the parameters of a GA in order to minimise the number of genetic operations executed by the algorithm so as to obtain strategic advantages for the scheduling computingtime. 4

7 2. The problem of dynamic scheduling The dynamic scheduling of FMSs consists in the assignment and sequencing of a set of jobs among the machine tools and workstations in order to maintain an optimised schedule when unforeseen changes of production occur. In particular, a dynamic scheduler should allow the rescheduling of the system if one of the following events occurs: new batch to be processed by the system; temporary unavailability of parts to be machined (due to failure of feeding systems, presence of defects on workpieces, etc.); temporary breakdown of machine tools (due to unavailability of tools, maintenance, etc.). After an unforeseen event, some jobs present in the system are not included in the current schedule: the schedule is called unrepaired and a dynamic scheduling algorithm has to be used to repair the schedule. The complexity of the system and the quality of the required schedule may need a great amount of scheduling computingtime. As dynamic scheduling is performed in parallel to the production process, scheduling computingtime necessarily involves a slight drop in productive indices due to machine idletime problems and difficulty in obtaining a reschedule having the minimum makespan. The reduction in scheduling computation time by means of a realtime algorithm is the keypoint to avoid a drop in performance during dynamic scheduling. Methods to repair the schedule also depend on the period of time existing between two successive rescheduling. In Ishii and Tavalage (1991) the rescheduling is executed when an index for measuring the system performances is lower than a threshold value. In Duffie and Prabhu (1994) the rescheduling is carried out by a virtual system which continually generates production plans and updates the system when it finds a valuable schedule. Therefore, if a realtime dynamic scheduling algorithm is able to repair the schedule straight after an unforeseen event occurs, the schedule is maintained optimised. In this work, this eventdriven approach is used to build a dynamic scheduling problem solver. 3. Overview of the proposed dynamic scheduling system Figure 1 shows a schematic overview of the proposed system. As can be clearly seen, the system is fundamentally divided into the following modules: 1. module for the evaluation of rescheduling parameters; in particular, this module performs: 5

8 the generation of the batch of ν jobs to reschedule; the recognition of machine breakdowns or other failures; the evaluation of the deadlines τ + τ ϕ ; being τ ϕ the time required to complete the job processed at the time τ on the ϕ th machine; 2. the REaltime Genetic ALgorithm () module, which represents the kernel of the system; 3. the module for downloading the plan to production level. [Insert figure 1 about here] The system also includes a database containing the estimation of the GA process parameters, performed by the gradientbased technique (described in section 4.6), and the estimation of the scheduling computation time τ. The computation time τ is an approximation of the computation time of the module for typical values of ν and appropriate configurations of the system layout Λ(µ 1,.., µ Μ ), where Μ is the number of the different categories of machine tool and µ η, η=1,..,μ, is the number of machine tools belonging to the category η. Dynamic scheduling starts whenever an unforeseen event occurs, and in order to repair the schedule, the system performs the following steps: a) the old plan of each machine tool ϕ is stopped at the time (τ + τ ϕ ); b) the first module evaluates of the rescheduling parameters and the intervals of time µαξ{, ( τ ϕ τ)} needed to synchronise the old and the new plan; c) generates the new plan; d) the new plan is downloaded to the production level; e) the new plan of each machine ϕ is activated at the time µαξ{(τ + τ), (τ + τ ϕ )}. As can be deduced from the previous steps, the proposed dynamic scheduling system makes it possible to link old and new plans so as to avoid initial leadtimes and illegal schedules. As mentioned before, the use of a realtime algorithm is the keypoint for an efficient solution to jobshop dynamic scheduling. The condition of a realtime response can now be precisely expressed by the following condition concerning the computation time: τ τ ϕ for each no idletime machine ϕ at the time τ. (1) 6

9 4. The module. GAs are very promising to solve scheduling problems since they use an implicit parallel strategy of searching for the optimal solution within the set of all the feasible job assignments on the machines. Nevertheless, in several manufacturing systems, the complexity of inputs (different categories of machine tools, number of machines, number of jobs, different products, etc.) and the lack of lowcost highperformance computers, do not allow the implementation of a realtime response system. This section describes a GA developed for realtime scheduling of an FMS, and developed in accordance with a joboriented approach and running on a 2 MHz PC. 4.1 Definition and limits of the problem The main goal of this study is to develop a flexible algorithm capable of solving a model of a real manufacturing environment represented by generalised jobshop scheduling (Papadimitriou and Steiglitz, 1983). This problem is more precisely defined by the following constraints and features: 1. one or more machines can be present for each category of machines (i.e. lathes, milling machines, etc.); 2. each job (or part) can be processed at the most once on each category of machine; 3. no machine may process more than one job at a time; 4. jobs can follow different routings among categories of machines. The model simulates the scheduling in a real manufacturing environment because both the assignment constraints (denoted by constraints 1 and 2) and the sequencing constraint (denoted by constraint 3) are taken into account (Shaw 1986, Brandimarte and Calderini 1995). This feature is schematically shown in figure 2.a. The production cycle contains a set of 1 parts to be scheduled for an FMS including two categories of machine tools. Each operation is represented by a twocomponent array: the first component is the operation name and the second component represents the processing time (i.e. the sum of the time needed to load and to unload the part on the machine tool and to execute the operation). These operations are assigned to a particular machine tool among the µ η present for each category η, according to constraints 1 and 2, and for each machine the operations are ordered to verify constraint 3. The structure which supports the assignment and sequencing of operations to the machines is the chromosome used in the module (described in section 4.2). Feature 4 highlights the fact that the module performs the scheduling with unconstrained partroutings. 7

10 [Insert figure 2 about here] 4.2 Chromosome coding Chromosome coding is mainly based on the typical structure for ordering problems used in other applications concerning genetic scheduling (Syswerda 1991, Forrest 1993, Park and Park 1995). The coding used in these applications is commonly known as a jobordered list on each machine, where the components represent the jobs to be scheduled for each machine. With respect to previous coding, the proposed representation has the following peculiar aspects: the chromosome is formed by µ queues of operations (where µ is the number of machines included in the system), each of them assigned to a specific machine; all the µ η queues assigned to a machine category η, is a permutation among the operations to be processed on that machine category; this permutation is called a gene. The second aspect verifies constraints 1 and 2 (assignment constraints) and, for this reason, the proposed chromosome coding can be termed a jobordered list on each machine category (with respect to the jobordered list on each machine mentioned before, where the assignment constraints are not taken into account). Figure 2.b shows how chromosome coding and partroutings generate the production plan. Starting from the time µαξ{, ( τ ϕ τ)}, the operation are successively processed following the order in which they appear in queue of the machine tool ϕ and in the partroutings. A chromosome is feasible if this procedure locates a legal schedule, i.e. a production plan including all operations present in the chromosome. The procedure allows the evaluation of the production indices associated with a feasible chromosome. In this study, the minimum makespan is used to determine the efficiency of a plan included in a given chromosome. The makespan µακ(χ) obtained from a chromosome Χ, is compared with a heuristic lower bound ΗΛΒ (described in section 4.6.3) in order to obtain a general value of efficiency in the range of [,1], represented by the fitness function of the chromosome. In this paper, this is expressed by: & ( ) ( = (2) 8

11 The efficiency of a nonfeasible chromosome is penalised by associating a zero value to its fitness function. 4.3 The evolutionary strategy It is wellknown that a GA works through successive chromosome populations (called generations) which have the peculiarity of covering, in the space of all the possible solutions, regions with increasing fitness (Goldberg, 1989). A generation includes πσ individuals or chromosomes; it is obtained from the previous one through the application of genetic operators, such as selection of a mating pool, crossover to pairs of chromosomes included in the mating pool, and genetic mutation of a few chromosomes. Random values πχ and πµ (with πχ and πµ ranging between and 1) are generated in order to set, respectively, the number of crossover operations and the number of mutation operations. In any case, the algorithm ends within a finite number of generations µγ, whether it reaches the final goal or not. The final goal is represented by a chromosome Χ such that: φιτ(χ ) 1ε, (3) where ε is an appropriate positive number near to. If the final goal is not reached, the algorithm gives the solution having the best fitness. 4.4 Initial population To improve the performance of, all chromosomes of the initial population are chosen in order to represent feasible schedules. In this regard, the main concept is the layer. The layer is the basic structure of the gene and includes all the operations with the same precedence in the partrouting (figure 3). Different layers are characterised by different precedences of operations. Each layer is formed by a sequence of blocks ; each of these is then included in a queue of operations. [Insert figure 3 about here] 9

12 The blocks are assembled in accordance with the different precedences of the layer. No intersection takes place among the blocks. Schedule feasibility results from the ordering of precedencebased partrouting imposed among the layers. Figure 3 also shows the relations between genes, layers and blocks and the procedure for building an initial chromosome. 4.5 Genetic operators The following genetic operators have been used in the system: reproduction operator: the reproduction of chromosomes is performed by means of a roulette wheel strategy (Goldberg, 1989). This strategy selects a chromosome and includes it in a mating pool with a probability equal to its fitness function, assuring the heredity and the improvement of highfitness strings of operations in the chromosome. crossover operator: it operates on the corresponding layers of two chromosomes (parents), selecting a random number of crossover sites which represent the elements to be changed. In both parents, the crossover sites are replaced by arranging the elements according to the order of the other parent (thus obtaining two offsprings) (figure 4). In one offspring, the average number of changed elements (which corresponds to the number of genetic operations) is equal to half its elements (ν/2); [Insert figure 4 about here] mutation operator: the operator used for genetic mutation includes two suboperators which respectively perform two procedures: the Layer Subdivision procedure (LS) and the Limited Mixing procedure (LM). The first procedure selects a layer with a probability of κ 1 /Μ 2. If the selected layer belongs to gene η*, the LS procedure generates µ η* blocks with a random size within the layer, replacing the original structure of the blocks (figure 5.a). The LM procedure selects a queue of operations in the chromosome, with a probability of κ 2 /µ, and it operates according to the following steps (figure 5.b): 1. for each block in the different layers, it computes the middle position π in the queue; 2. it selects a random element in the block (let θ be its position in the queue); 3. it fixes an element whose position in the queue is found through the following steps: 1

13 a) toward left or right from the position θ with an equal probability; b) along ρ positions with a probability given by a normal density with parameters (π, π /2) in (πρ); 4. it exchanges elements between position θ and the element fixed in step 3 [Insert figure 5 about here] As the optimal chromosome does not necessarily involve ordering with no intersection among the blocks in a selected queue of operations, the mutation operator does not maintain the initial subdivision among the blocks of different layers. Nevertheless, as only works with feasible chromosomes, if a mutated chromosome involves an illegal schedule, it is replaced by the original chromosome. 4.6 Method of increasing the convergence rate The problem of dynamic scheduling is characterised by two contrasting objectives: 1. reducing the scheduling computation time in order to limit the idletimes of machines; 2. reducing makespan (or other production indices). In the proposed system, the former objective is obtained by minimising the genetic complexity (ΓΧ) of the algorithm, i.e. the average number of the operations performed by the algorithm. The second objective is fulfilled by means of an approximate minimum makespan with respect to a heuristic lower bound. Experimental results seem to show that a GA generates a chromosome with nearoptimal fitness after a certain number of genetic operations have been performed. Therefore a particular configuration of process parameters (µγ, πσ, πµ, πχ) can be set in order to reduce the computation time needed to obtain the final goal Χ. In this regard, the present work proposes an offline optimisation of the genetic complexity in the multidimensional space of process parameters, with the goal of reaching a schedule with a nearoptimal makespan. The following sections describe the evaluation of the average number of operations performed by the module, the guidelines for increasing the convergence rate and the method used to estimate the heuristic lower bound. 11

14 4.6.1 Genetic complexity of For each generation, processes a total number of πχ πσ chromosomes with crossover operations and a total number of πµ πσ chromosomes with mutation operations. Considering that the maximum number of generations is µγ, the total number of chromosomes evaluated by is: (4) For each chromosome, κ operations are produced in order to evaluate its fitness function. Furthermore, the crossover operator performs ν/2 changes of operations in a chromosome. On the other hand, as a result of the considerations explained in section 4.5, the mutation operator generates: κ 1 layers, which correspond to κ 1 m blocks modified in a chromosome by the LS procedure; κ 2 queues of operations, which correspond to κ 2 Μ 2 blocks modified in a chromosome, by the LM procedure. Integrating the number of operations performed by a single operator, the total number of these, defined as the genetic complexity ΓΧ of the algorithm, can be expressed by: (,,, ) = & + (& + & ) (5) In this work the condition κ 1 =κ 2 =2 has been adopted on the basis of the results obtained in preliminary tests Gradientbased parameter optimisation Selecting the heuristic lower bound as a proper upper bound for the minimum makespan (thus considering ε=), from the expressions (2) and (3) it derives that adopting the process parameters µγ, πσ, πχ and πµ, converges to an optimal solution if the best chromosome Χ verifies the following condition: µακ(χ,µγ, πσ, πχ, πµ) ΗΛΒ (6) 12

15 Condition (6) is used as a constraint for the problem of minimising the genetic complexity of, obtaining the following problem of optimisation of the process parameters:,,, subject to &,,,,, defined by (7) The approach used in searching for the optimal values of µγ, πσ, πµ and πχ can be considered to be a variant of the gradient method in the simplex algorithm (Knout and Quandt, 1963). The peculiarity of this technique is the subdivision of the space of searching into two halfspaces. By the following iterative procedure, in the first halfspace, ΝxΝ, is defined the main direction of searching, which corresponds to the maximum gradient of ΓΧ(µγ, πσ, πχ, πµ) with respect to µγ and πσ: ϑ ϑ ϑ ϑ ϑ ϑ Analogously, in the second halfspace [,1]x[,1], a secondary searching direction follows the direction of the maximum gradient of ΓΧ(µγ, πσ, πχ, πµ) with respect to πµ and πχ, restricted to the points µγ=µγ ι, πσ=πσ ι, ι=,1,.. of the main direction. The secondary searching direction is defined by the following iterative procedure: & & ϑ & & ϑ & ϑ ϑ & & & & & & & 13

16 The gradient method is stopped when the following condition is verified for the ζ th time: µακ(χ,µγ ι,πσ ι,πχ ϕ,πµ ϕ ) ΗΛΒ, ι,ϕ Ν (1) obtaining the optimal configuration of the parameters µγ*, πσ*, πχ* and πµ*. In this work the condition ζ=2 has been adopted. In order to calculate the previous parameters µγ*, πσ*, πχ* and πµ*, random sequences of partroutings and processing times have been used. The presented method of searching is a compromise between a greater flexibility of the technique compared with searching on one direction and a lesser complexity compared with searching in a number of directions higher than Heuristic lower bound definition It is clear at this point that the keypoint of the method is the definition of the ΗΛΒ. In order to adapt the gradientbased convergence method, the ΗΛΒ must be an appropriate upper bound of the minimum makespan. For this purpose, if µ η >1 for each η=1,..,μ and ν»µ, can be adopted the following lower bound: = ( ) = 1,.., + τ τ = (11) where τ ιη is the processing time for the job ι=1,..,ν on the machine category η. The effectiveness of expression (11) is subjected to the condition expressed in the previous section, i. e. the sequences of machine categories included in the routing of the parts must be uniformly distributed on all the possible combinations. 5. Results and applications 14

17 This paragraph reports the results obtained in the use of the system and concerning the following two aspects: 1. analysis of the performance of the proposed algorithm compared with other scheduling algorithms (a Pentium MMX 2 MHz is used); 2. applications to dynamic scheduling of a flexible manufacturing cell. 5.1 Analysis of the algorithm performance Three kinds of scheduling algorithms have been compared with respect to instances 1 and 2 of section 4.6: 1. RuleOriented Algorithm (ROA); 2. Generic GA (); 3. the algorithm proposed in this work (). Algorithm 1 is a standard dispatching rule which selects the best schedule among those ones given by FIFO, LIFO, SPT and LPT dispatching. In accordance with a specific rule, operation queues are generated dynamically when each part is available for processing in accordance with its partrouting. Algorithm 2 is similar to the module except that the gradientbased optimisation technique has not been applied. The following process parameters have been used in this algorithm: µγ=1, πσ=5, πχ=.4 and πµ=.15. Algorithm comparison is executed for the scheduling of 5, 75, 12, 2 and 36 jobs on different layouts of FMS, each of these with the overall number of machines m between 12 and 21. Table 1 and table 2 show the results subdivided for different layout configurations of manufacturing plants indicated by the symbol Λ(µ 1,.., µ Μ ). For each category of data, 1 different instances are executed. Each instance corresponds to the execution of the scheduling algorithm 1, 2 and 3 on the same data input characterised by a random generation of production cycles (processing times and partroutings). The computation time τ shown in the tables is computed by ( τ µ +3 σ), where τ µ and σ are, respectively, the mean and the standard deviation of the computation time. [Insert table 1 and table 2 about here] 15

18 The parameters are reported in table 3 for each layout. It can be noticed that the number of generations and the number of chromosomes are equal, due to (8). Furthermore, it can be seen that the parameters πχ and πµ range respectively between and For each example of layout, the makespan obtained by is from 1% to 7% lower than the makespan obtained by standard dispatching rules in layout with two machine categories, and from 8% to 19% in layouts with three or more categories. To a lesser extent this index improves compared with the one obtained by (from 1% to 3%). Consequently, the system throughput and machine utilisation obtained by the standard ruleoriented algorithms is considerably improved by. gives good performances also with respect to the computation time, being very close to the realtime response of the standard ruleoriented algorithm. Some seconds are sufficient for to converge to ΗΛΒ in layouts with different values of µ ι (e.g. Λ(15,6), Λ(8,5,3), etc.). Instead in a symmetrical layout (e.g. Λ(8,8), Λ(4,4,4), etc.) the greater competition among the machine categories for minimising makespan involves a greater complexity of the scheduling problem and a longer computation time which takes up to 2 minutes for the scheduling with ν 12 jobs. The computational time obtained by is always higher than (from 2 to 4 times). A consideration can be also made for the values of ΗΛΒ obtained using (11). From the results of table 1 and table 2, the ΗΛΒ is lower than the makespan obtained by ROA and greater than those ones obtained by (except in few cases where ν 75, being the condition ν»µ not properly verified). This practical result demonstrates that in the generalised jobshop scheduling problem, ΗΛΒ is a good lower bound for the standard dispatching algorithms and must be appropriately utilised to define the fitness function for a GA. [Insert table 3 about here] 5.2 Application to dynamic scheduling of a flexible manufacturing system. As an example of a practical application, the system was used for the batchoriented scheduling of the flexible manufacturing system shown in figure 6. The FMS includes 8 CNC machining centers and 8 CNC lathes. A global feeder receives batches of raw parts from an input of the FMS; it moves the parts to the local buffers in accordance with the scheduling planned by the software system. 16

19 [Insert figure 6 about here] Table 4 shows the production plans of the 14 different parts used in this application for the scheduling of 12 operations in a batch of 7 parts (5 objects for each different part). performs a realtime scheduling in.8 minutes. The best schedule corresponding to the chromosome having the best fitness is showed in the Gantt diagram of figure 7. Being τ=1 minute, for each machine ϕ the plan is actived at µαξ{1, τ ϕ } (supposing that the start of rescheduling occurs at the time τ =), obtaining a realtime response in accordance with the expression (1) and a makespan of 49 minutes (4.5 minutes less than ΗΛΒ). [Insert figure 7 about here] In order to evaluate the performance of the proposed dynamic scheduling system in this application, the plan was activated at the time µαξ{, τ ϕ }= τ ϕ (i.e. forcing τ=). This also gave a makespan of 49 minutes, leading to the conclusion that the value τ=1 does not increase the makespan. For this application, the algorithms ROA and perform a production plan with makespan of 56 and 57 min. respectively. It may be noticed that neither of the schedules performed by the algorithms are optimal and the differences with respect to ΗΛΒ are, respectively, 2.5 min. and 3.5 min.. Moreover, the computation time of is equal to the estimated time τ=4.5 minutes. Being µιν ϕ=1,..,16 { τ τ ϕ }=.5 min., a machine idletime of at least.5 min. is bound to be obtained. [Insert table 4 about here] 6. Conclusions The present work represents a contribution to the development of dynamic joboriented schedulers for FMS using realtime genetic algorithms. In particular, a technique based on the optimisation of genetic complexity is proposed in order to drastically reduce the time required to generate a new schedule. 17

20 The results obtained from the tests have demonstrated the ability to give fast, optimal responses, and thus to prevent the problems related to high computing times, such as machine idletimes and reduction in performance for dynamic scheduling applications. Besides, the proposed method greatly reduces the makespan of the schedule produced by the standard dispatching rules in complex cell layouts. Further investigation is needed to use these optimised evolutionary strategies for realtime response in very complex FMS. The main goal is to obtain a low increase in genetic complexity when the variables ν, µ and Μ assume high values. New developments in gradient method or other optimisation algorithms of the GA process parameters may lead to optimal results in generalised jobshop scheduling. References Adams, J., Balas, E., and Zawack, D., 1988, The shifting bottleneck procedure for job shop scheduling. Management Science, 34, Booker, L.B., Goldberg, D.E. and Holland, J.H., 1989, Classifier systems and genetic algorithms. Artificial Intelligence, 4, Brandimarte, P., and Calderini, M., 1995, A hierarchical bicriterion approach to integrated process plan selection and job shop scheduling. International Journal of Production Research, 33, Brandimarte, P., Ukovich, W., and Villa, A., 1995, Continuous flow models for batch manufacturing: a basis for a hierarchical approach. International Journal of Production Research, 33, Duffie, N. A. and Prabhu, V. V., 1994, Realtime distributed scheduling of heterarchical manufacturing systems. Journal of Manufacturing Systems, 13, Fang, H.L., Ross, P., and Corne, D., 1993, A Promising Genetic Algorithm Approach to JobShop Scheduling, Rescheduling and OpenShop Scheduling Problems. Proceedings of the 5th Conference on Genetic Algorithms, Fichera, V., Grasso, S., and Lombardo, A., 1995, Genetic Algorithms Efficiency in Flow Shop Scheduling. 1th International Conference on Application of AI in Engineering, Forrest, S., 1993, Genetic Algorithms: Principles of Natural Selection Applied to Computation. Science, 261,

21 Gen, M., Tsujimura, Y., and Kubota, E., 1994, Solving JobShop Scheduling Problems by Genetic Algorithm. Proceedings of the IEEE International Conference on Systems Man and Cybernetics, 2, Goldberg, D.E., 1989, Genetic Algorithms in Search, Optimization and Machine Learning (Reading, MA: Addison Wesley). Holland, J.H, 1975, Adaptation in Natural and Artificial Systems. Univ. of Michigan Press, Ann Arbor, MI. Ishii, N., and Talavage, J. J., 1991, A TransientBased RealTime Scheduling Algorithm in FMS. International Journal of Production Research, 29, Kim, G.H., and Lee, C.S.G., 1995, Genetic Reinforcement Learning Approach to the Machine Scheduling Problem. Proceedings of the IEEE International Conference on Robotics and Automation, 1, Kuhn, H. W., and Quandt, R.E., 1963, An Experimental Study of Simplex Method. Proceedings of Symposia on Applied Mathematics, 25, Liang, S.J., and Lewis, J.M., 1994, JobShop Scheduling Using Multiple Criteria. Proceedings of the Joint HungarianBritish International Mechatronics Conference, Morikawa, K., Furuhashi, T., and Uchikawa, Y., 1994, Evolution of CIM System with Genetic Algorithm. Proceedings of the 1st IEEE Conference on Evolutionary Computation, 2, O Kane, J.F., Harrison, D.K., and Gentili, E., 1994, The Analysis of Reactive Scheduling Issues in a FMS using a Dynamic KnowledgeBased System Approach. Proceedings of the 1th International Conference on Computer Aided Production Engineering, Palermo, Papadimitriou, C.H., and Steiglitz, K., 1982, Combinatorial Optimization: Algorithms and Complexity (Englewood Cliffs, NJ: Prentice Hall Inc.), pp Park, L., and Park, C.H., 1995, Application of Genetic Algorithm to JobShop Scheduling Problems with Active Schedule Constructive Crossover. Proceedings of the IEEE International Conference on Systems, Man and Cybernetics, 1, Perrone, G., La Commare, U., Lo Nigro, G., and Nuccio, C., 1995, Dynamic Scheduling in a Multiple Objective Production Environment using a Fuzzy Adaptive Controller. Proceedings of the 11th International Conference on Computer Aided Production Engineering, London, Rabelo, L.C., Jones, A., and Yih, Y., 1994, Development of a RealTime Learning Scheduler Using Reinforcement Learning Concepts. Proceedings of the IEEE International Symposium on Intelligent Control, Ramasesh, R., 199, Dynamic Job Shop Scheduling: A Survey of Simulation Research. International Journal of Management Science, 18,

22 Shaw, M.J., 1986, Dynamic scheduling in cellular manufacturing systems: a framework for networked decision making. Journal of Manufacturing Systems, 7, Sridhar, J., and Rajendran, C., 1994, Genetic algorithm for family and shop scheduling in a flowlinebased manufacturing cell. Computers & Industrial Engineering, 27, Suresh, V., and Chaudhuri, D., 1993, Dynamic scheduling A survey of research. International Journal of Production Economic, 32, Syswerda, G., 1991, Schedule optimization using genetic algorithms. In Davis, L., Handbook of Genetic Algorithms (New York: Van Nostrand Reinhold), chapter 21, pp Wang, L.C., 1995, Intelligent scheduling of FMSs with inductive learning capability using neural networks. International Journal of Flexible Manufacturing Systems, 7,

23 Acknowledgements The authors would like to acknowledge Prof. Francesco Giusti, Prof. Marco Santochi and the technical staff of the Department of Production Engineering of the University of Pisa for their contribution to this work. 21

24 Tab. 1 Comparison of algorithms for scheduling of ν=5, 75, 12, 2 and 36 jobs and for the layouts Λ(8,8), Λ(15,6) and Λ(4,4,4) ( & = makespan obtained by the algorithms & & obtained by ; % = & obtained by 1 ) Case Λ(µ 1,..,µ Μ ) ν ΗΛΒ αλγοριτηµ µακ ιφφ% τ (min.) 1 Λ(8,8) ROA Λ(8,8) ROA Λ(8,8) ROA Λ(8,8) ROA Λ(8,8) ROA Λ(15,6) ROA Λ(15,6) ROA Λ(15,6) ROA Λ(15,6) ROA Λ(15,6) ROA Λ(4,4,4) ROA Λ(4,4,4) ROA Λ(4,4,4) ROA Λ(4,4,4) ROA Λ(4,4,4) ROA

25 Tab. 2 Comparison of algorithms for scheduling of ν=5, 75, 12, 2 and 36 jobs and for the layouts Λ(8,5,3), Λ(3,3,3,3) and Λ(4,4,3,2). Case Λ(µ 1,..,µ Μ ) ν ΗΛΒ αλγοριτηµ µακ ιφφ% τ (min.) 1 Λ(8,5,3) ROA Λ(8,5,3) ROA Λ(8,5,3) ROA Λ(8,5,3) ROA Λ(8,5,3) ROA Λ(3,3,3,3) ROA Λ(3,3,3,3) ROA Λ(3,3,3,3) ROA Λ(3,3,3,3) ROA Λ(3,3,3,3) ROA Λ(4,4,3,2) ROA Λ(4,4,3,2) ROA Λ(4,4,3,2) ROA Λ(4,4,3,2) ROA Λ(4,4,3,2) ROA

26 Tab. 3 Process parameters of derived by the gradientbased optimisation. Λ(µ 1,..,µ Μ ) ν Maximum number of generation (µγ) Number of chromosomes (πσ) Probability of crossover (πχ) Probability of mutation (πµ) Λ(8,8) Λ(15,6) Λ(4,4,4) Λ(8,5,3) Λ(3,3,3,3) Λ(4,4,3,2)

27 Tab. 4 Production cycles and partroutings of the 7 parts taken as an example of application (L lathe, MC machining center). Parts Operation for lathes Operation for machining center Partroutings Part 1 Part 5 (O 2, 4) (O 1, 7) (O 1, MC), (O 2, L) Part 6 Part 1 (O 3, 3) (O 3, L) Part 11 Part 15 (O 4, 3) (O 5, 6) (O 4, L), (O 5, MC) Part 16 Part 2 (O 6, 3) (O 6, L) Part 21 Part 25 (O 7, 9) (O 8, 9) (O 7, L), (O 8, MC) Part 26 Part 3 (O 9, 3) (O 9, L) Part 31 Part 35 (O 11, 5) (O 1, 9) (O 1, MC), (O 11, L) Part 36 Part 4 (O 12, 3) (O 12, MC) Part 41 Part 45 (O 14, 2) (O 13, 5) (O 13, MC), (O 14, L) Part 46 Part 5 (O 15, 3) (O 15, L) Part 51 Part 55 (O 16, 3) (O 16, MC) Part 56 Part 6 (O 17, 3) (O 18, 7) (O 17, L), (O 18, MC) Part 61 Part 65 (O 19, 9) (O 2, 4) (O 19, L), (O 2, MC) Part 66 Part 7 (O 21, 6) (O 22, 2) (O 21, L), (O 22, MC) 25

28 Captions to illustrations. Fig. 1 Schematic overview of the proposed system. Fig. 2. Chromosome coding: (a) assignment and sequencing of jobs to the machines; (b) generation of the schedule from the chromosome and the partroutings. Fig. 3 Procedure used to obtain the chromosomes of the initial population. Fig. 4 Adopted positionbased crossover operator. Fig. 5 Genetic mutation operator: (a) Layer Subdivision procedure; (b) Limited Mixing procedure. Fig. 6 Example of FMS layout Λ(8,8); L lathe; MC machining center. Fig. 7 Gantt diagram of the schedule generated by for the FMS of Fig.6 using production cycles described in Tab. 4. Operations are indicated by means of the respective part. 26