Solution Representation for Job Shop Scheduling Problems in Ant Colony Optimisation

Transcription

1 Solution Representation for Job Shop Scheuling Problems in Ant Colony Optimisation James Montgomery, Carole Faya 2, an Sana Petrovic 2 Faculty of Information & Communication Technologies, Swinburne University of Technology, Melbourne, Australia montgomery@ict.swin.eu.au 2 School of Computer Science & IT, University of Nottingham, Nottingham, UK {cxf,sxp}@cs.nott.ac.uk Abstract. Prouction scheuling problems such as the ob shop consist of a collection of operations (groupe into obs) that must be scheule for processing on ifferent machines. Typical ant colony optimisation applications for these problems generate solutions by constructing a permutation of the operations, from which a eterministic algorithm can generate the actual scheule. This paper consiers an alternative approach in which each machine is assigne a ispatching rule, which heuristically etermines the orer of operations on that machine. This representation creates a substantially smaller search space that likely contains goo solutions. The performance of both approaches is compare on a real-worl ob shop scheuling problem in which processing times an ob ue ates are moelle with fuzzy sets. Results inicate that the new approach prouces better solutions more quickly than the traitional approach. Keywors: Ant colony optimisation, fuzzy ob shop scheuling, solution representation. Introuction Ant colony optimisation (ACO) is a constructive metaheuristic in which, uring successive iterations of solution construction, a number of artificial ants buil solutions by probabilistically selecting from problem-specific solution components, influence by a parameterise moel of solutions (calle a pheromone moel in reference to ant trail pheromones). The parameters of this moel are upate at the en of each iteration using the solutions prouce so that, over time, the algorithm learns which solution components shoul be combine to prouce the best solutions. When aapting ACO to suit a problem an algorithm esigner must first ecie how solutions are to be represente an built (i.e., what base components are to be combine to form solutions) an then what characteristics of the chosen representation are to be moelle. Corresponing author

2 Prouction scheuling problems consist of a number of obs, mae up of a set of operations, each of which must be scheule for processing on one of a number of machines. Preceence constraints are impose on the operations of each ob. The maority of ACO algorithms for these problems represent solutions as permutations of the operations to be scheule (operations are the base components of solutions), which etermines the relative orer of operations that require the same machine (see, e.g., [, 2]). A eterministic algorithm can then prouce the best possible scheule given the preceence constraints establishe by the permutation. This approach is more generally referre to as the list scheuler algorithm []. An alternative approach is to assign ifferent heuristics to each machine which etermine the relative processing orer of operations, thereby searching the reuce space of scheules that can be prouce by ifferent combinations of the heuristics [3]. Builing solutions in this manner may offer an avantage by concentrating the search on heuristically goo solutions. This paper compares these two solution representations by using a real-worl ob shop scheuling problem (JSP). A formal escription of the JSP is given in Section 2, incluing further etails of the two solution construction approaches. Section 3 escribes the real-worl JSP instance to which both approaches are applie, in which processing times an ue ates are moelle by fuzzy sets to reflect the uncertain nature of these in inustrial settings. Details of the ACO algorithms evelope for the fuzzy JSP are given in Section 4, followe by analysis of their empirical performance in Section 5. Section 6 escribes the implications of the results for the future application of ACO to such problems. An extene version of this paper, incluing more extensive empirical analyses, is presente in [9]. 2 Job Shop Scheuling an Solution Construction The JSP examine in this stuy consists of a set of n obs J,..., J n, with associate release ates r,..., r n an ue ates,..., n. Each ob consists of a sequence of operations (etermine by the processing requirements of the ob) that must each be scheule for processing on one of m machines M,..., M m. Only one operation of a ob may be processe at any given time, only one operation may use a machine at any given time an operations may not be preempte. Two criteria have to be minimise simultaneously, the average tariness of obs C AT an the number of tary obs C NT, calculate as follows: C AT = n n T () where T = max{0, C } is the tariness of of ob J an C is the completion time of ob J. n C NT = u (2) = =

3 where u = if T > 0, 0 otherwise. It is common in ACO applications for the JSP an relate scheuling problems to generate a permutation of the operations, which implicitly etermines the relative processing orer of operations on each machine. These algorithms are restricte to creating permutations that respect the require processing orer of operations within each ob, which can consequently be calle feasible permutations. A eterministic algorithm transforms the relative processing orer into an actual scheule. Different solution construction approaches prouce ifferent search spaces. The space of feasible permutations of operations for a JSP is very large (a weak upper boun is O(k!), where k is the number of operations) an is certainly much larger than the space of actual solutions. This space also has a slight bias towars goo solutions, which can be exploite by some pheromone moels an proves isastrous for others [0]. Another notable feature of this search space is that while all solutions can be reache, solutions (scheules) are represente by iffering numbers of permutations. An alternative approach to builing solutions is to assign ifferent ispatching rules (i.e., orering heuristics) to each machine, which subsequently buil the actual scheule [3]. The search space then becomes the space of all possible combinations of rules assigne to machines, which is O( D m ) where D is the set of rules an m the number of machines. Given a small number of ispatching rules (this stuy uses four, escribe in Section 4) it is highly probable that this search space is a subset of the space of all feasible scheules. However, assuming the ispatching rules are iniviually likely to perform well it is expecte that this reuce space largely consists of goo quality scheules. The performance of these two approaches is compare on a real-worl JSP instance, escribe in the next section. 3 A Real-Worl JSP The ata set use has been provie by a printing company, Sherwoo Press, in Nottingham, Unite Kingom [5]. There are 8 machines in the shop floor, groupe within seven work centres: printing, cutting, foling, car-inserting, embossing an ebossing, gathering, stitching an trimming, an packaging. Due to both machine an human factors, processing times of obs are uncertain an ue ates are not fixe but promise instea. Therefore, fuzzy sets are use to moel these uncertain values. A triangular membership function µ pi (t) = (p i, p2 i, p3 i ) is use to moel the fuzzy processing time p i of ob J on machine M i, i =,..., m, =,..., n, where p i an p3 i are lower an upper bouns of the processing time, while p 2 i is the so-calle moal point [7]. An example of fuzzy processing time is shown in Fig. (a). A trapezoial fuzzy set (, 2 ) is use to moel the ue ate of each ob, where is the crisp ue ate an the upper boun 2 of the trapezoi excees by 0%, following the policy of the company. An example of a fuzzy ue ate is given in Fig. (b).

4 µ ~ p i ( t) ~ p i µ ~ ( t) ~ p i p i p i t 0 2 t (a) (b) Fig.. Fuzzy (a) processing time an (b) ue ate µ(t) 0 ~ π ~ ( ) ~ C C ~ 2 (a) t µ(t) 0 ~ ~ ~ C I C ~ 2 (b) t Fig. 2. Satisfaction grae of tariness using (a) possibility measure an (b) area of intersection The obective function takes into account both the average tariness of obs an the number of tary obs. As these are measure in ifferent units they are mappe onto satisfaction graes in the range [0, ], which are then combine in an overall satisfaction grae. Two approaches use to measure tariness in [5] are investigate:. The possibility measure π C ( ), use by Itoh an Ishii [6] to hanle tary obs in a JSP, measures the satisfaction grae of a fuzzy completion time SG T ( C ) of ob J by evaluating the possibility of a fuzzy event C occurring within the fuzzy set [6] (illustrate in Fig. 2(a)): SG T ( C ) = π C ( ) = sup min{µ C (t), µ (t)} =,..., n (3) where µ C (t) an µ (t) are the membership functions of fuzzy sets C an respectively. This measure is referre to as poss hereafter. 2. The area of intersection measure (enote area hereafter), introuce by Sakawa an Kubota [], measures the proportion of C that is complete by the ue ate (illustrate in Fig. 2(b)): SG T ( C ) = (area C )/(area C ) (4) The satisfaction graes of tariness efine in (3) an (4) are use in two obectives:

5 . To maximise the satisfaction grae of average tariness S AT : S AT = n n SG T ( C ) (5) = 2. To maximise the satisfaction grae of number of tary obs S NT : A parameter λ is introuce such that a ob J, =,..., n, is consiere to be tary if SG T ( C ) λ, λ [0, ]. After calculating the number of tary obs nt ary, the satisfaction grae S NT is calculate as: if nt ary = 0 S NT = (n nt ary)/n if 0 < nt ary < n (6) 0 if nt ary > n where n = 5% of n, where n is the number of obs. Two ifferent aggregation operators, average an minimum (enote average an min hereafter), were investigate for combining the satisfaction graes of the obectives. 4 ACO for a Fuzzy JSP. Blum an Sampels [] take the minimum of the relevant pheromone values. Thus, the probability of selecting an available operation o to a to the partial permutation p is given by Two ACO algorithms were evelope base on the MAX MIN Ant System (MMAS), which has been foun to work well in practice [2]. The first of these, enote MMAS perm, constructs solutions as permutations of the operations, while the secon, enote MMAS rules, assigns ispatching rules to machines. The set of ispatching rules D consists of the following four rules: Early Due Date First, Shortest Processing Time First, Longest Processing Time First an Longest Remaining Processing Time First. The two solution representations require ifferent pheromone moels. The moels chosen have been foun to prouce the best performance for their respective solution representations [8]. For MMAS perm, a pheromone value, enote τ(o i, o ), exists for each irecte pair of operations that use the same machine, an represents the learne utility of operation o i preceing operation o []. At each step of solution construction, the set of unscheule operations that require the same machine as a caniate operation o is enote by O rel o P (o, p) = min or O rel o o p min o r O rel τ(o, o r ) o τ(o, o r ). (7) The last operation on each machine is scheule as soon as it becomes available. For MMAS rules, a pheromone value τ(m k, ) is associate with each combination of machine an ispatching rule (M k, ) M D, where M is the set

6 of machines. At each step of solution construction, a machine is assigne a ispatching rule. The probability of assigning a ispatching rule D to machine M k is given by τ(m k, ) P (M k, ) = D\{} τ(m k, ). (8) In both algorithms, after each iteration all pheromone values are reuce in proportion to (ρ ) while those for the iteration best solution s ib are increase by ρ F (s ib ), where ρ is the pheromone evaporation rate an F is the overall satisfaction grae of s ib (given by either the average or min aggregation operator). 5 Computational Results The performance of the algorithms was compare on one month s ata collecte from Sherwoo Press (the March set use by Faya an Petrovic [5]). The resulting JSP instance consists of 549 operations partitione into 59 obs. The algorithms were implemente in the C language an execute uner Linux on a 2.6GHz Pentium 4 with 52Mb of RAM. The MMAS control parameters use were: 0 ants per iteration; 3000 iterations; ρ = 0.; τ max = ; τ min = 0 3 in MMAS rules an τ min = 0 4 in MMAS perm. The values of τ min an τ max were chosen to approximate those suggeste by Stützle an Hoos [2] base on the size of the solution representation an pheromone upate. Both algorithms were execute with ifferent combinations of parameter values for solution evaluation: poss an area tariness measures, an average an min aggregation operators. The value of λ was fixe at 0.7. Each combination was run with 0 ifferent ranom sees. 5. Solution quality The results reveale that when using the min aggregation operator, MMAS perm is unable to fin a solution with a non-zero obective value. This is because the algorithm, facing a large number of solutions with S NT = 0, searches ranomly until a subset of pheromone values is upate. Further testing confirme that a ranom search of permutations is unlikely to prouce solutions with S NT > 0. A secon version of the algorithm, name MMAS min perm, was evelope in which the pheromone upate was moifie such that, if all solutions in an iteration have an obective value of zero, the best solution in terms of S AT is use to upate pheromone values using the average aggregation operator. Such a moification was not necessary for MMAS rules as ranom assignments of ispatching rules to machines typically prouce solutions with S NT > 0. Table summarises the satisfaction graes of tariness measures accoring to the aggregation operator use for each algorithm. It is evient that MMAS min perm is much more successful than its original form when using the min aggregation

7 Table. Performance of the algorithms. The best result for each measure is given with the mean value in parentheses. Bol items are best within each solution quality measure Algorithm F S AT S NT C NT Using poss an average MMAS perm 0.69 (0.62) 0.9 (0.9) 0.46 (0.34) 3 (5.9) MMAS rules 0.73 (0.73) 0.93 (0.93) 0.54 (0.53) (.2) Using poss an min MMAS perm (0.7) 0 48 (5.2) MMAS min perm 0.42 (0.35) 0.89 (0.88) 0.42 (0.35) 4 (5.5) MMAS rules 0.54 (0.53) 0.93 (0.93) 0.54 (0.53) (.3) Using area an average MMAS perm 0.62 (0.59) 0.90 (0.90) 0.33 (0.28) 6 (7.3) MMAS rules 0.7 (0.70) 0.93 (0.93) 0.50 (0.48) 2 (2.5) Using area an min MMAS perm (0.69) 0 49 (52.) MMAS min perm 0.42 (0.32) 0.88 (0.87) 0.42 (0.32) 4 (6.4) MMAS rules 0.50 (0.48) 0.93 (0.92) 0.50 (0.48) 2 (2.5) operator. Further investigation reveale that it require the use of the average aggregation operator in up to 33% of iterations. Across solution evaluation measures, MMAS rules clearly outperforms MMAS perm. 5.2 CPU time An orer of magnitue ifference was observe between the CPU time of the two algorithms, with MMAS perm taking more than 400 secons compare to approximately 00 secons for MMAS rules. This is to be expecte given the respective number of components each must consier at each constructive step; MMAS perm consiers approximately 40 operations on average, while MMAS rules consiers only four. Moreover, MMAS rules fins its best solutions very early in each run (often within secon) while MMAS perm oes not converge until quite late. 6 Conclusions Typical ACO algorithms for prouction scheuling problems such as the JSP buil solutions as permutations of the operations to be scheule, from which actual scheules are generate eterministically. An alternative approach when the problem in question has multiple machines an various criteria upon which to uge the urgency of competing operations is to assign ifferent ispatching rules to each machine. The chosen ispatching rules are then responsible for etermining the relative processing orer of operations on each machine. This paper compare both approaches on a multi-obective real-worl JSP, moelle

8 with fuzzy operation processing times an ob ue ates. The results show that assigning ispatching rules to machines prouces higher quality solutions in far less time than builing a permutation of the operations. This supports the claim that the assignment of ispatching rules restricts the search space to an area of goo quality solutions. As this stuy focuse on a single, real-worl JSP instance (albeit using a variety of solution quality measures) future work is require to etermine if these results hol for other prouction scheuling instances. Aitionally, it is now common practice in most ACO algorithms to use a local search proceure to improve the solutions prouce, something not one in this stuy so that ifferences between the two solution construction approaches coul be observe. While the aition of local search to a permutation-base ACO algorithm for these problems may allow it to perform better, it is potentially more useful in the new approach, where it can explore solutions that combinations of ispatching rules woul otherwise never prouce. References [] C. Blum an M. Sampels. An ant colony optimization algorithm for shop scheuling problems. J. Math. Moel. Algorithms, 3(3): , [2] A. Colorni, M. Dorigo, V. Maniezzo, an M. Trubian. Ant system for ob-shop scheuling. JORBEL, 34():39 53, 994. [3] U. Dornorf an E. Pesch. Evolution base learning in a ob shop scheuling environment. Comput. Oper. Res., 22:25 44, 995. [4] D. Dubois an P. H. Possibility theory: An approach to computerize processing of uncertainty. Kluwer Acaemic, New York, 988. [5] C. Faya an S. Petrovic. A fuzzy genetic algorithm for real-worl ob shop scheuling. In M. Ali an F. Esposito, eitors, Innovations in Applie Artificial Intelligence, volume 3533 of LNAI, pages , Springer-Verlag. [6] T. Itoh an H. Ishii. Fuzzy ue-ate scheuling problem with fuzzy processing time. Int. Trans. Oper. Res., 6: , 999. [7] G. Klir an T. Folger. Fuzzy sets, uncertainty an information. Prentice Hall, New Jersey, 988. [8] E. J. Montgomery. Solution biases an pheromone representation selection in ant colony optimisation. PhD thesis, Bon University, [9] J. Montgomery, C. Faya an S. Petrovic. Representation of solutions to ob shop scheuling problems in ant colony optimisation. Technical Report SUTICT- TR , Faculty of Information & Communication Technologies, Swinburne University of Technology, [0] J. Montgomery, M. Ranall, an T. Hentlass. Structural avantages for ant colony optimisation inherent in permutation scheuling problems. In M. Ali an F. Esposito, eitors, Innovations in Applie Artificial Intelligence, volume 3533 of LNAI, pages , Springer-Verlag. [] M. Sakawa an R. Kubota. Fuzzy programming for multiobective ob shop scheuling with fuzzy processing time an fuzzy ueate through genetic algorithms. Eur. J. Oper. Res., 20(2): , [2] T. Stützle an H. Hoos. MAX MIN ant system. Future Gen. Comp. Sys., 6:889 94, 2000.