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Automatic design of scheduling policies for dynamic flexible job shop scheduling by multi-objective genetic programming based hyper-heuristic
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Procedia CIRP 00 (2017) 000–000 Procedia CIRP 79 (2019) 439–444
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12th 2018, 12thCIRP CIRPConference Conferenceon onIntelligent IntelligentComputation ComputationininManufacturing ManufacturingEngineering, Engineering,18-20 CIRPJuly ICME '18 Gulf of Naples, Italy 12th CIRP Conference on Intelligent Computation in Manufacturing Engineering, CIRP ICME '18 28th CIRP Design Conference, May 2018, Nantes, Automatic design of scheduling policies for dynamic flexible job France shop scheduling by multi-
Automatic design of scheduling policies for dynamic flexible job shop scheduling by multiobjective programming basedand hyper-heuristic A new methodology to genetic analyze the functional physical architecture of objective genetic programming based hyper-heuristic
existing products for an assembly oriented producta family identification a, Yong Zhou *, Jian-jun Yang a, Yong Zhou *, Alain Jian-jun Yanga PaulSchool Stief *, Jean-Yves Dantan, Etienne, Siadat of Mechanical Engineering and Automation, Beihang University,Ali Beijing, 100191,China a
a School of Mechanical Engineering and Automation, Beihang University, Beijing, 100191,China École Nationale Supérieure d’Arts et Métiers, Arts et Métiers ParisTech, LCFC EA 4495, 4 Rue Augustin Fresnel, Metz 57078, France * Corresponding author. Tel.: +86-10-82316783; E-mail address: [email protected] * Corresponding author. Tel.: +86-10-82316783; E-mail address: [email protected] * Corresponding author. Tel.: +33 3 87 37 54 30; E-mail address: [email protected]
Abstract Abstract Abstract This study proposes four multi-objective genetic programming based hyper-heuristic methods(MO-GPHH) for automated heuristic design to solvestudy the multi-objective flexible genetic job shopprogramming scheduling problem(MO-DFJSP). scheduling policy(SP)for used in the MO-DFJSP includes This proposes fourdynamic multi-objective based hyper-heuristicAmethods(MO-GPHH) automated heuristic design to Intwo today’s business the trend towards product variety andrule(MAR). customization is unbroken. thisindevelopment, the includes need of decision rules:environment, a jobdynamic sequencing rule(JSR) andmore a machine assignment These twopolicy(SP) rulesDue are to simultaneously evolved to solve solve the multi-objective flexible job shop scheduling problem(MO-DFJSP). A scheduling used the MO-DFJSP agile and reconfigurable systems emerged cope with variousand products and These product families. To design andthat optimize production threedecision scheduling objectives (mean weighted tardiness, maximum tardiness mean flow time). two The resultsare demonstrate the pareto front of two rules: a jobproduction sequencing rule(JSR) and ato machine assignment rule(MAR). rules simultaneously evolved to solve systems as well as to choose the optimal product matches, product analysis methods are needed. Indeed, most of the known methods aimSPs to the proposed methods dominate that of 320 human-made SPstardiness which are from literatures on demonstrate training set, that andthe thepareto evolved three scheduling objectives (mean weighted tardiness, maximum andselected mean flow time). The results front of analyze a product or SPs one product on level. Different product families,from however, may differ largely in terms theevolved number and outperform manual in 58/64family test scenarios. the proposed methods dominate that of the 320physical human-made SPs which are selected literatures on training set, andofthe SPs nature ofThe components. This fact by impedes anB.V. efficient comparison and choice of appropriate product family combinations for the production © 2018 Authors. Published outperform manual SPs in 58/64 testElsevier scenarios. system. AThe new methodology is proposed to analyze existing products in CIRP view of their functional and physical architecture. The aim is to cluster Peer-review under responsibility of the scientific committee of the 12th Conference on Intelligent Computation in Manufacturing © 2018 Authors. Published by Elsevier B.V. © 2019 The Authors. Publishedoriented by Elsevier B.V.families these products in new assembly product for the optimization ofConference existing assembly lines and the creation of future reconfigurable Engineering. Peer-review under responsibility of the scientific committee of the 12th CIRP on Intelligent Computation in Manufacturing Peer-review under responsibility of the scientific committee of the 12th CIRP Conference on Intelligent Computation in Manufacturing Engineering. assembly systems. Based on Datum Flow Chain, the physical structure of the products is analyzed. Functional subassemblies are identified, and Engineering. a Keywords: functionaldynamic analysisflexible is performed. Moreover,scheduling a hybrid policies; functional and physical architecture graph (HyFPAG) is the output which depicts the job shop scheduling; multi-objective genetic programming; cooperative coevolution similarity families by providing design support to both, production system planners and coevolution product designers. An illustrative Keywords:between dynamic product flexible job shop scheduling; scheduling policies; multi-objective genetic programming; cooperative example of a nail-clipper is used to explain the proposed methodology. An industrial case study on two product families of steering columns of thyssenkrupp Presta France is then carried out to give a first industrial evaluation of the proposed approach. ©1.2017 The Authors. Published by Elsevier B.V. implementation, satisfactory performance, low computational Introduction Peer-review under responsibility of the scientific committee of the 28th CIRP Design Conference 2018. toperformance, implementation, low computational 1. Introduction requirement, and satisfactory flexibility incorporate domain knowledge
Over the past decades, the static single-objective FJSP has
Keywords: Assembly; Design method; Family identification Over the past decades, single-objective FJSP has been extensively studied inthethestatic literature [1]. However, many
been extensively studied in the literature [1]. However, many real-world scheduling problems usually involve simultaneous real-world scheduling problems usually involve simultaneous optimization of several objectives which are in conflict to optimization of several objectives are inorconflict to some extent. And when we considerwhich a dynamic stochastic 1. Introduction some extent.environment And when we consider a dynamic or stochastic scheduling where the job arrival times are not scheduling wherefeatures the job arrival are not known in environment advance. These of thetimes MO-DFJSP Due to the fast These development domain of known in advance. features inof the MO-DFJSP significantly increase the complexity of the finding optimal communication and an ongoing trend of digitization and significantly increase the complexity optimal solutions[2]. Job shop scheduling problemofhasfinding been proven to digitalization, manufacturing enterprises arehas facing solutions[2]. Job shopasscheduling problem been important proven be NP-hard[3]. FJSP an extended problem, should also betoa challenges in today’s environments: a continuing be NP-hard[3]. FJSP as market an extended problem, should also be a NP-hard problem[4]. This type of scheduling problem tendency towards reduction of product development and NP-hard two problem[4]. This astype of scheduling problem involves subproblems follows: assignmenttimes of each shortened product lifecycles. In addition, there is an increasing involves two subproblems as follows: assignment of each operation to an appropriate machine (machine selection); demand of customization, beingmachine at the same time inselection); a global operation tooperations an appropriate (machine sequencing on each (job scheduling). competition with competitors all over the Thissubject, trend, sequencing on eachof machine (jobworld. scheduling). Due to aoperations large number references on this which from micro Due istoinducing a large the number of references on thistosubject, numerous techniques candevelopment be divided into the macro following types: numerous techniques can be divided into the types: heuristics,results meta-heuristic, hyper-heuristic andaugmenting artificial markets, in diminished lot sizes due following to heuristics, meta-heuristic, and rules artificial intelligence [5].(high-volume Heuristics hyper-heuristic named dispatching are product varieties to low-volume production) [1]. intelligence [5]. Heuristics named rules frequently in practice dueasdispatching to their ease of To cope with used this augmenting variety well as to be ableare to frequently used optimization in practice potentials due to intheir of identify possible the ease existing production system, it is important to have a precise knowledge 2212-8271 © 2017 The Authors. Published by Elsevier B.V.
requirement, and flexibility to incorporate knowledge and expertise[6]. Due to these advantagesdomain mentioned above, and to these advantages mentioned above, they expertise[6]. remain a veryDue popular technique used by the operators in they remain popular technique used by theinclude operators in practice. Thea very major drawbacks of heuristics their practice. Thedependence major drawbacks of heuristics include their performance on the state of the system and none performance dependence on the state the system none of the rules that is superior to all theofothers for alland possible of the product range and characteristics manufactured and/or of the rules that is superior to all the others for all possible states[7]. assembled in this system. In this context, quite the main in states[7]. Meta-heuristics are able to perform wellchallenge and carry modelling and analysis is now not only to cope with single Meta-heuristics to perform and metacarry more knowledge ofare theable problem domain.quite Evenwell though products, a have limited range domain. or existing product more knowledge ofproduct the problem Even though heuristics attracted significantly attentions, it families, hasmetatwo but also to be ableresearchers to analyze and to compare products heuristics have attracted significantly it to hasdefine two disadvantages: have to attentions, design a specialized new product families. can behave observed that problem classical existing disadvantages: researchers to design a specialized representation for eachIt practical scheduling with the product familiesfor regrouped function clients or of features. representation each practical scheduling problem with the characteristics ofare problem. So in that meansofeach class metaHowever, assembly oriented product families are class hardlyof tometafind. characteristics of problem. So that means each heuristic doesn’t have a wide range of applications on shop On the doesn’t product family level, products differ two heuristic a wide range ofthe applications oninshop scheduling[2]. Thehave second drawback is cost mainly of prohibitive main characteristics: (i) the drawback number ofiscomponents (ii) the scheduling[2]. The second the cost of and prohibitive computational effort in real-time. computational effort in real-time. Inofthe context of (e.g. solving various types of shop scheduling type components mechanical, electrical, electronical). In the context oflearning-based solvingconsidering variousapproaches types of shop scheduling problems, many with a large Classical methodologies mainly single products problems, many learning-based approaches with a large number of their variants have been employed researchers. or solitary, already existing product familiesby analyze the number structure of their variants have been (components employed by level) researchers. These methods gaussian processes[8], imitation product oninclude a physical level which These difficulties methods include gaussian processes[8], imitation causes regarding an efficient definition and comparison of different product families. Addressing this
Peer-review the scientific committee 2212-8271 ©under 2017responsibility The Authors. of Published by Elsevier B.V.of the 11th CIRP Conference on Intelligent Computation in Manufacturing Engineering. Peer-review under responsibility of the scientific committee of the 11th CIRP Conference on Intelligent Computation in Manufacturing Engineering. 2212-8271©©2017 2019The The Authors. Published by Elsevier 2212-8271 Authors. Published by Elsevier B.V. B.V. Peer-reviewunder underresponsibility responsibility scientific committee of the CIRP Conference on 2018. Intelligent Computation in Manufacturing Engineering. Peer-review of of thethe scientific committee of the 28th12th CIRP Design Conference 10.1016/j.procir.2019.02.118
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learning [9], data mining[10], reinforcement learning[11], artificial neural-networks[12], fuzzy logic[13], ensemble learning[14], genetic programming[15] and artificial immune networks [16]. However, non-existence of any AI methods is superior to all the others for all possible states in shop scheduling. Besides, the training case need to be carefully designed because it has a great impact on the test performance in supervised learning. In recent years, the concept of hyper-heuristic has been proposed to solve the combinatorial optimization problem[17]. It refers to high-level iterative techniques, which guide low-level heuristics by using intelligent concepts to explore the search space of heuristics rather than search space of solutions[18]. In this study, we focus on MO-GPHH methods for heuristic generation to fabricate new SPs by combining various small system attributes. The motivation of this approach is to generate effective SPs offline and to use them online for fast application. The rest of paper is organized as follows. Section 2 provides a brief description of the shop scheduling problem. In Section 3, the proposed algorithm is illustrated in detail. The multi-objective performance measures for the algorithm are provided in Section 4. In Section 5 and 6, the nondominated SPs are then presented with a comparison to the 320 combinations of benchmark SPs on both training and test set. Finally, conclusions and directions for future research are drawn in Section 7. Nomenclature MO-DFJSP multi-objective dynamic flexible job-shop scheduling problem SP scheduling policy JSR job sequencing rule MAR machine assignment rule MO-GPHH multi-objective genetic programming based hyper-heuristic CCGP cooperative coevolution genetic programming with two populations TTGP genetic programming with single population that a GP individual contains two trees NSGAII nondominated sorting genetic algorithm II SPEA2 strength Pareto evolutionary algorithm 2 mean weighted tardiness WTmean maximum tardiness Tmax mean flow time Fmean 2. Problem Description The MO-DFJSP with functionally related machines is formulated as follows. 1) There are a set of independent jobs J {Ji }1i n indexed i be a set of n jobs to be scheduled. 2) Each job J i consists of a predetermined sequence of operations. Let Oi , j be operation j of J i . 3) Let M {M k }1k m indexed k be a set of m machines. 4) Each operation Oi , j is processed without interruption on one machine M k out of a set of given compatible
machines M i , j (for M k M i , j , M i , j M ). Therefore, we denote by Oi , j , k to be an operation j of J i that is processed on machine M k and Pi , j , k be its processing time on machine M k . It should be noted that the processing time Pi , j , k of each operation is machine dependent. If M i , j M for at least one operation, it is partial flexibility FJSP (P-FJSP); while M i , j =M for each operation, it is total flexibility FJSP (T-FJSP) [19]. The problem considered here is P-FJSP. 5) Let wi , ri , di , ci be the weight, release date, due date and completion date of job J i , respectively. The tardiness of this job can be calculated by the following formula: (1) Ti max{0, Ci di } The flowtime of job J i is calculated by the following formula: (2) F Ci ri i 6) Three objectives, namely, mean weighted tardiness, maximum tardiness and mean flow time are to be minimized, which are defined respectively as follows: WTmean 1 / n i 1 wi Ti n
(3)
Tmax max{ Ti | i 1,..., n}
(4)
Fmean 1/ n i 1 Fi
(5)
n
The discrete event simulation model of the DFJSP is used as a test-bed for the purpose of experimentation. The simulation runs for a sufficiently long period after the shop reaches the steady state. In each simulation replication, we begin with an empty shop. The interval from the beginning of the simulation until the arrival of the 500th job is considered as the warm-up time, and the statistics from the 500th job to the next completed 2500 jobs will be used to calculate the simulation performance. The simulation configurations and details are shown in Table 1. Because of the extremely high complexity of MO-DFJSP, some common assumptions are made in this study. Job arrivals follow Poisson distribution. Weights of jobs are assigned based on the 4:2:1 rule according to Pinedo’s study [20], which showed that 20% of the customers are particularly important, 60% of them are average importance and 20% of them are less importance. Three experimental factors that each factor has two levels are examined. They are allowance factor; utilization of the shop and the number of machines could be selected by each operation. Allowance factor is used to estimate the due date of a job. The Total Work Content (TWK) [21] method is used to assigns each job a delivery date that is a multiple of its total processing time, i.e. di ri c i 1 pi , j n
(6)
In equation (6), c denotes the tightness factor of the due date, pi , j denotes the mean processing time per operation.
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to O use one replication, and it is measured by the average value of the specific objective across all training scenarios. After all individuals have been evaluated, the archive A will be updated. To explore the pareto front of nondominated SPs, two multi-objective approaches are employed to assign ranks and crowding distance (NSGAII[23]) or pareto strength (SPEA2[24]). New subpopulations are generated by genetic operations and the algorithm starts a new generation if the maximum generation is not reached.
Table 1. Parameter setting of the training and testing scenarios. Parameter Weights of jobs Simulation Length Warmup Length Workstation Process number distribution Process time distribution Allowance factor Utilization Optional device number
Training Testing 4:2:1(20%:60%:20%) 2500 500 10 10, 20 Missing Missing, Full U[1,99] U[1,99], N(120,30) 2, 4 1, 3 70%, 90% 80%, 95% 1, U[1,3] U[1,2], U[1,4]
3. Proposed methods Simulation-based evaluation
MO-GPHH based policies generation Start
Decode individual
Start
Form new population Sub-population/tree of MAR
ct=ct+1
Y
Start dispatching
Performance index
Collaborate to construct complete SP
N
Stop N
New job arrived?
Sub-population/tree of JSR
N
Fitness evaluation
NSGAII(Assign ranks and crowding distance) or SPEA2(Assign pareto strength) Produce new population
max generation achieved? Y Stop and obtain Pareto set of SPs
N
Job needs to be processed ？ Y
Return fitness Build archive
N
Device idle?
Job needs to be arranged ？ Y
Y
Apply JSR
Apply MAR
Determine which job to process on the device
Determine which device to select for the current process
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Fig. 1. Framework of the MO-GPHH based policies generation and the simulation-based fitness evaluation.
A scheduling policy which is used in the MO-DFJSP includes two decision rules: a job sequencing rule(JSR) and a machine assignment rule(MAR). As shown in Fig. 1, the framework of the automated heuristics design approach contains two parts: the MO-GPHH based SPs generation and the simulation-based fitness evaluation. In the evolutionary stage, GP[22] is employed as the learning mechanism to evolve SPs. Different from other evolution algorithms, each GP individual is not represented by a fixed length string of genes but usually represent in a tree form of various lengths. Because a GP individual is an expression of computer program for solving a specific task. In the evaluation stage, a GP individual is decoded into decision rules and then applied to the relevant decision points in the discrete-event simulation model. In the evolution stage, the results are collected and returned to assign fitness to the GP individual. The MO-GPHH algorithms include two collaborative models. One is MO-CCGP method which evolves two decision rules in two separate populations, and the other is MO-TTGP method that have single population and a GP individual includes two trees for two decision rules. The MOCCGP method starts with two randomly generated populations, one is for JSR and the other is for MAR. The algorithm begins with an initial population Pop for JSR and an initial population P ws for MAR. In the training stage, eight simulation scenarios (more details are shown in Table 1) are loaded to evaluate the performance of an evolved SP. The individual R iop from the population P op is paired with the individual Rjws from the population Pws using random shuffling. Rirep is the complete SP that is formed by the combination of (Riop, Rjws). The fitness of (Riop, Rjws) is obtained by applying Rirep
Algorithm: MO-CCGP Inputs: simulation model O Outputs: the pareto front of nondominated SPs F initialize population Pop, Pws at random, Pop←{R1op, R2op…Rnop}, Pws←{R1ws,R2ws …Rnws} generation←0 archive A← {} while generation ≤max Generation do pair up the Riop, Rjws using random shuffling for all Riop∈Pop do Rkrep ←collaborate (Riop, Rjws) evaluate f(Rkrep) by applying Rkrep to O use 1 replication f(Riop), f(Rjws) ← f(Rkrep) end for build archive A← update { A Pop Pws } assign ranks and crowding distance(NSGAII) or pareto strength (SPEA2) apply genetic operations to archive A to generate new population generation ←generation + 1 end while apply fast-nondominated-sort to archive A to obtain pareto front F return F
The MO-TTGP method is simpler than the MO-CCGP algorithm. The difference is that an individual in MO-TTGP contains two sub-trees for two decision rules when decoding for fitness evaluation. In this case, each individual is equivalent to a complete SP. Table 2 shows the parameter settings of the proposed algorithms. The function set consists of basic operators (+, -, ×, /, max, min). The function ‘/’ is the protected division, which returns 1 if the denominator is 0. To make a relative fair comparison, all MO-GPHH methods use the same number of function evaluations (NFEs). For CCGP-NSGAII and TTGP-NSGAII methods, generation is set to 50 and archive size is fixed to 200. Therefore, the number of fitness evaluations per generation is 400, and the total number of fitness evaluations is 50×400=20000. For CCGP-SPEA2 and TTGP-SPEA2 methods, generation is set to 100 and the archive size is fixed to 100. The number of fitness evaluations per generation is 200. Hence, the total number of fitness evaluations is 100×200=20000. These settings are applied to achieve the same NFEs for each algorithm. Since two kinds of rules are evolved by GP, the terminal set for constructing JSR and MAR are presented in Table 3 and Table 4, respectively. Table 2. Parameter settings of the proposed method. Parameter Population Number Population Size Generation Crossover/mutation rates Initialization Selection Parsimony Pressure Maximum depth Operators Terminals
CCGP TTGP 2 1 200 200 NSGAII: 50; SPEA2: 100 90%/10% ramped half-and-half (depth 2~6) tournament selection (size 5) double tournament selection (t1=7, t2=2) 8 +, −, ×, /, max, min Shown in Table 3 and Table 4
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convergence performance. Spacing: This indicator measures the distance variance of neighboring vectors in PFkonwn . A lower Spacing value
Table 3. Terminal set for JSR. Terminals
Description
PT
processing time of operation Oi , j .
OL
number of operations left for job
RPT
work remaining of job
indicates a good distribution of solutions along PFkonwn .
Ji
Ji Ji
(current time-
TIS
time spend in system of job
TIQ
time spent in the device queue of job
TDD
time to the due date of job ( d i -current time)
SLACK TOD W ERC
Spacing
ri )
Ji
the difference between job’s remaining delivery time and its remaining work time=TDD-RPT time to operation’s due date weight of job J i
NPT
Random number generating from (0, 1.0] processing time of the next operation Oi , j +1
WINQ
total work time in the next queue.
Table 4. Terminal set for MAR. Terminals WQT WQS WQD WD WPT WTW WRL ERC
Description total work time of the job in the device queue queue size of the device total queueing time of the device total idle time of the device processing time of the job on the device total work time of the device mean queue size of the device Random number generating from (0, 1.0]
4. Performance measures Three popular metrics are employed to evaluate the performances of the proposed methods: hypervolume ratio (HVR)[25], Inverted Generational Distance (IGD)[26], and Spacing[27] . They can be expressed as follows: Hypervolume ratio (HVR): hypervolume is used to measures the size of the objective space dominated by the obtained non-dominated front PFkonwn . A higher HV value is desirable and denotes a good dominate performance. nPF
HVR volume(
i 1
vi )
(7)
1 nPF
1
nPF i 1
(d d i )
member and its nearest member in PFkonwn , d is the average value of all d i .
PFref is normally the true Pareto front, which is extracted from all SPs found by the four MO-GPHH methods in all independent runs. Each experiment is conducted 30 independent run times for each algorithm. In summary, the evolved SPs from 4 methods × 30 runs =120 Pareto fronts are combined into a common pool, and the nondominated sorting technique is used to extract the Pareto fronts from this pool[28]. The experiments are implemented in Java 8.0 and run on a computer with Intel Core i5-4590 3.30 GHz, 8 GB RAM. For each performance metric, a Wilcoxon signed-rank test with the significance level of 0.05 [29] is carried out on the results obtained by 30 independent runs of each method. Figure 2 shows that the metrics HVR and IGD produced by MO-CCGP are significantly better than that of MO-TTGP methods. However, in MO-CCGP or MO-TTGP methods, there is no significant difference between NSGAII and SPEA2 based approach. In terms of Spacing, the value obtained by NSGA-II based method is significantly better than those of SPEA2 based method. In addition, there is no significant difference between MO-CCGP and MO-TTGP methods. In summary, collaborative patterns(CCGP/TTGP) pose a great impact on the performance indicators of HVR and IGD, and pareto approaches(NSGAII/SPEA2) significantly affect the indicator of Spacing. Overall, CCGP-NSGAII is the most competitive method among the four proposed methods.
HV of the reference Pareto front PFref .
HV ( PFkonwn ) HV ( PFref )
(8)
Inverted Generational Distance(IGD): This is a variant of the Generational Distance (GD) but represents a combined or comprehensive indicator. It measures the average distance from the reference Pareto front PFref to Pareto
(a)
front PFkonwn obtained by the algorithm.
IGD
p
( i 1 di )1/ p n
(9) n Where n is the number of all elements in PFref , p is set to 2 in this study, d i is the Euclidean distance between the member i in PFkonwn and its nearest member in PFref . Fronts with a lower IGD value are desirable and denotes a good
(10)
Where nPF is the number of members in the obtained Pareto front PFkonwn , d i is the minimum distance between the
Where nPF is the number of members in the obtained nondominated front PFkonwn , vi is the hypercube constructed with a reference point and the member i as the diagonal of the hypercube [23]. HVR is the ratio of the HV of PFkonwn and the
HVR
2
(b)
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(c) Fig. 2 Performances of the proposed methods on training scenarios. (HVR to be maximized and Spacing and IGD to be minimized). (a)HVR (b)IGD (c)Spacing
5. Comparison of nondominated SPs to existing SPs In this study, 320 benchmark SPs found in the literature [6][21] are used to verify the effectiveness of the evolved SPs. The benchmark SPs are made up of 30 well-known JSRs and 10 MARs, which are shown in Table 5 and Table 6. Because the ATC rule contains parameter k, and three parameter configurations with k=1.0, 2.0, 3.0 are used; therefore, there are 32×10=320 combinations of benchmark SPs. Table 5. Benchmark rules for JSR. No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Benchmark Rule ATC CR CR+SPT EDD FASFS FCFS LCFS LPT LRPT MDD MOD ODD RND SI SL SLK SPT SRN SRPT SRPT/PT SRPT/SLK PT+WINQ PT+WINQ+Slack 2PT+WINQ+NPT WINQ NPT LW WMDD WMOD WSPT
Description apparent tardiness cost critical ratio critical ratio plus SPT earliest due date first arrival at shop first served first come first served last come first served longest processing time longest remaining processing time modified due date modified operation due date operational due date random rule a truncated version of SPT negative minimum slack minimum slack shortest processing time shortest remaining operation number shortest remaining processing time remaining processing time per processing time remaining processing time per slack processing time plus WINQ processing time plus WINQ and slack double processing time plus WINQ and NPT work in the next queue processing time of next operation largest weight weighted modified due date weighted modified operational due date weighted shortest processing time
Table 6. Benchmark rules for MAR. No. 1 2 3 4 5 6 7 8 9 10
Terminals WQT WQS WQD WD WPT WTW WRL WTN WU WRD
Description total process time of the job in the device queue queue size of the device total waiting time of the job in the device queue total idle time of the device process time of the job on the device total work hours of the device average queue size of the device in one minute total work numbers of the device device utilization average queueing time of the device
Fig. 3. Reference Pareto front of the evolved SPs and the results of the existing SPs on training scenarios (‘*’ and ‘o’ respectively represent evolved and existing SPs)
As mentioned earlier, 320 benchmark SPs are applied to 8 training scenarios (see Table 1), and 100 simulation replications are performed for each scenario. Therefore, we perform 8×100=800 simulation replications to test the performance of each manual SP. The results generated by the manual SPs are recorded to compare with the results of the evolved SPs ( PFref )on the training set. As shown in Fig.3, the nondominated SPs evolved by MO-GPHH dominate all manual SPs found in the literature under any objective. 6. Testing performance To evaluate the generalization of the evolved SPs, we applied a design of experiments (DOEs) approach to design the test set. In the DOE, we consider 5 factors (see Table 1), each with 2 levels. The full factorial design (including 25 =64 combinations) is adopted in this case. Since many SPs have been obtained from the training stage, we choose three SPs that perform well on the specific objective (see Fig.4) from the nondominated SPs generated by CCGP-NSGAII. The three selected evolutionary SPs and the 320 kinds of benchmark SPs are simultaneously applied to 64 test scenarios (see Table 1). Meanwhile, 100 simulation replications are performed for each scenario. Therefore, we perform 64×100=6400 simulation replications to test the performance of each SP. Average value of the specific objective under certain scenario obtained from 100 independent simulation replications are recorded as the performance metric of the test SP. The relative deviation between the best performance obtained by three evolved SPs with the best performance obtained by 320 manual SPs on the same instance under the same target is recorded to verify the generalization of the evolved SPs. The number of test scenarios that the evolved SPs perform better than the 320 types of manual SPs under each objective WTmean , Tmax , Fmean is 62, 58 and 64, respectively. There are 6 scenarios with relatively poor performance Tmax . Because Tmax is a maximum target rather than other average targets, it may change drastically with the change of the random seed. Besides, the evolved SPs perform better than the 320 types of manual SPs in all test scenarios with the indicator Fmean .
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Fig. 4. Pareto front and selected evolved SPs generated by CCGP-NSGA2 (‘*’ denotes the nondominated evolved SPs and ‘o’ denotes the selected evolved SPs)
7. Conclusion and future work This study proposes four MO-GPHH algorithms to automatically design SPs for MO-DFJSP, including JSR and MAR. The training performances show that CCGP-NSGAII is the most competitive approach among the proposed methods for evolving efficient non-dominated SPs for MO-DFJSP. Statistical tests indicate that it has the best overall performances in three popular metrics (HVR, IGD, Spacing). Then, the evolved SPs which were extracted from all SPs found by the four MO-GPHH methods are compared with the 320 types of benchmark SPs found in the literature on training set. The results reveal that the performances of the evolved SPs dominate those of all manual SPs under any objective. To evaluate the generalization of the evolved SPs, 64 simulation scenarios are designed for the test set. The results show that the evolved SPs outperform the 320 types of manual SPs in 58/64 test scenarios. It also demonstrates that the evolved SPs have a strong generalisation ability to be reused on new unobserved scheduling scenario. The results show that the evolved SPs can effectively solve the MO-DFJSP and obtained trade-offs among different objectives. However, dispatching rules made by human experts are still widely used in many practical scheduling systems. Therefore, we plan to apply the proposed MOGPHH methods to automatically evolve SPs based on the real case. And then, the appropriate SP is selected to replace the artificial SPs designed by experts in the scheduling system. Acknowledgements This work was supported by Beijing Municipal Education Commission (Build a Project) and the Key Laboratory Development Projects of Beijing, China. References [1] Nouiri M, Bekrar A, Jemai A, Niar S, Ammari AC. An effective and distributed particle swarm optimization algorithm for flexible job-shop scheduling problem. J Intell Manuf 2018; 29:603-15. [2] Gen M, Lin L. Multiobjective evolutionary algorithm for manufacturing scheduling problems: state-of-the-art survey. J Intell Manuf 2014; 25:84966. [3] Garey M R, Johnson D S, Sethi R. The Complexity of Flowshop and Jobshop Scheduling. Math Method Oper Res 1976; 1:117-129. [4] Mati Y, Xie X. The complexity of two-job shop problems with multipurpose unrelated machines. Eur J Oper Res 2004; 152:159-169.
[5] Çaliş B, Bulkan S. A research survey: review of AI solution strategies of job shop scheduling problem. J Intell Manuf 2015; 26:961-973. [6] Chiang T C. Enhancing rule-based scheduling in wafer fabrication facilities by evolutionary algorithms: Review and opportunity. Comput Ind Eng 2013; 64:524-535. [7] Vinod V, Sridharan R. Simulation modeling and analysis of due-date assignment methods and scheduling decision rules in a dynamic job shop production system. Int J Prod Econ 2011; 129:127-146. [8] Heger J, Branke J, Hildebrandt T, Scholz-Reiter B. Dynamic adjustment of dispatching rule parameters in flow shops with sequence-dependent set-up times. Int J Prod Res 2016; 54: 6812-6824. [9] Ingimundardottir H, Runarsson T P. Discovering dispatching rules from data using imitation learning: A case study for the job-shop problem. J Sched 2017; 12:1-16. [10] Shahzad A, Mebarki N. Data mining based job dispatching using hybrid simulation-optimization approach for shop scheduling problem. Eng Appl Artif Intel 2012; 25:1173-1181. [11] Zhang Z, Zheng L, Hou F, Li N. Semiconductor final test scheduling with Sarsa (λ, k) algorithm. Eur J Oper Res 2011; 215:446-458. [12] Golmohammadi D. A neural network decision-making model for jobshop scheduling. Int J Prod Res 2013; 51:5142-5157. [13] Abdullah S, Abdolrazzagh-Nezhad M. Fuzzy job-shop scheduling problems: A review. Inf Sci 2014; 278:380-407. [14] Ɖurasević M, Jakobović D. Comparison of ensemble learning methods for creating ensembles of dispatching rules for the unrelated machines environment. Genet Program Evol M 2018; 19:53-92. [15] Branke J, Hildebrandt T, Scholz-Reiter B. Hyper-heuristic evolution of dispatching rules: A comparison of rule representations. Evol Comput 2015; 23:249-277. [16] Xiong W, Fu D. A new immune multi-agent system for the flexible job shop scheduling problem. J Intell Manuf 2018; 29:857-73. [17] Burke E K, Gendreau M, Hyde M, Kendall M, Ochoa G, Özcan E. Hyper-heuristics: A survey of the state of the art. J Oper Res Soc 2013; 64: 1695-1724. [18] Nguyen S, Mei Y, Zhang M. Genetic programming for production scheduling: a survey with a unified framework. Complex & Intelligent Systems 2017; 3:41-66. [19] Kacem I, Hammadi S, Borne P. Approach by localization and multiobjective evolutionary optimization for flexible job-shop scheduling problems. IEEE Transactions on Systems, Man, and Cybernetics, Part C (Applications and Reviews) 2002; 32:1-13. [20] Pinedo M, Singer M. A shifting bottleneck heuristic for minimizing the total weighted tardiness in a job shop. Naval Research Logistics 1999; 46:1-17. [21] Baker K R. Sequencing Rules and Due-Date Assignments in a Job Shop. Management Science 1984; 30:1093-1104. [22] Koza J R. Genetic programming as a means for programming computers by natural selection. Statistics and computing 1994; 4:87-112. [23] Deb K, Pratap A, Agarwal S, Meyarivan T. A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Trans. Evol. Comput 2002; 6:182-197. [24] Zitzler E, Laumanns M, Thiele L. SPEA2: Improving the Strength Pareto Evolutionary Algorithm for Multi-objective Optimization. Evolutionary Methods for Design, Optimization and Control with Applications To Industrial Problems. Proceedings of the Eurogen 2001. [25] Van Veldhuizen D A, Lamont G B. Multi-objective evolutionary algorithm test suites. ACM Symp. Appl. Comput 1999; 8:351–357. [26] Coello C A C, Cortés N C. Solving multi-objective optimization problems using an artificial immune system. Genet Program Evol M 2005; 6: 163-190. [27] Schott J R. Fault Tolerant Design Using Single and Multicriteria Genetic Algorithm Optimization. Master's Thesis, Massachusetts Institute of Technology 1995. [28] Nguyen S, Zhang M, Johnston M, Tan K C. Automatic design of scheduling policies for dynamic multi-objective job shop scheduling via cooperative coevolution genetic programming. IEEE T Evolut Comput 2014; 18: 193-208 [29] Demšar J. Statistical comparisons of classifiers over multiple data sets. J. Machine Learning Res 2006; 7: 1-30.
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12th 2018, 12thCIRP CIRPConference Conferenceon onIntelligent IntelligentComputation ComputationininManufacturing ManufacturingEngineering, Engineering,18-20 CIRPJuly ICME '18 Gulf of Naples, Italy 12th CIRP Conference on Intelligent Computation in Manufacturing Engineering, CIRP ICME '18 28th CIRP Design Conference, May 2018, Nantes, Automatic design of scheduling policies for dynamic flexible job France shop scheduling by multi-
Automatic design of scheduling policies for dynamic flexible job shop scheduling by multiobjective programming basedand hyper-heuristic A new methodology to genetic analyze the functional physical architecture of objective genetic programming based hyper-heuristic
existing products for an assembly oriented producta family identification a, Yong Zhou *, Jian-jun Yang a, Yong Zhou *, Alain Jian-jun Yanga PaulSchool Stief *, Jean-Yves Dantan, Etienne, Siadat of Mechanical Engineering and Automation, Beihang University,Ali Beijing, 100191,China a
a School of Mechanical Engineering and Automation, Beihang University, Beijing, 100191,China École Nationale Supérieure d’Arts et Métiers, Arts et Métiers ParisTech, LCFC EA 4495, 4 Rue Augustin Fresnel, Metz 57078, France * Corresponding author. Tel.: +86-10-82316783; E-mail address: [email protected] * Corresponding author. Tel.: +86-10-82316783; E-mail address: [email protected] * Corresponding author. Tel.: +33 3 87 37 54 30; E-mail address: [email protected]
Abstract Abstract Abstract This study proposes four multi-objective genetic programming based hyper-heuristic methods(MO-GPHH) for automated heuristic design to solvestudy the multi-objective flexible genetic job shopprogramming scheduling problem(MO-DFJSP). scheduling policy(SP)for used in the MO-DFJSP includes This proposes fourdynamic multi-objective based hyper-heuristicAmethods(MO-GPHH) automated heuristic design to Intwo today’s business the trend towards product variety andrule(MAR). customization is unbroken. thisindevelopment, the includes need of decision rules:environment, a jobdynamic sequencing rule(JSR) andmore a machine assignment These twopolicy(SP) rulesDue are to simultaneously evolved to solve solve the multi-objective flexible job shop scheduling problem(MO-DFJSP). A scheduling used the MO-DFJSP agile and reconfigurable systems emerged cope with variousand products and These product families. To design andthat optimize production threedecision scheduling objectives (mean weighted tardiness, maximum tardiness mean flow time). two The resultsare demonstrate the pareto front of two rules: a jobproduction sequencing rule(JSR) and ato machine assignment rule(MAR). rules simultaneously evolved to solve systems as well as to choose the optimal product matches, product analysis methods are needed. Indeed, most of the known methods aimSPs to the proposed methods dominate that of 320 human-made SPstardiness which are from literatures on demonstrate training set, that andthe thepareto evolved three scheduling objectives (mean weighted tardiness, maximum andselected mean flow time). The results front of analyze a product or SPs one product on level. Different product families,from however, may differ largely in terms theevolved number and outperform manual in 58/64family test scenarios. the proposed methods dominate that of the 320physical human-made SPs which are selected literatures on training set, andofthe SPs nature ofThe components. This fact by impedes anB.V. efficient comparison and choice of appropriate product family combinations for the production © 2018 Authors. Published outperform manual SPs in 58/64 testElsevier scenarios. system. AThe new methodology is proposed to analyze existing products in CIRP view of their functional and physical architecture. The aim is to cluster Peer-review under responsibility of the scientific committee of the 12th Conference on Intelligent Computation in Manufacturing © 2018 Authors. Published by Elsevier B.V. © 2019 The Authors. Publishedoriented by Elsevier B.V.families these products in new assembly product for the optimization ofConference existing assembly lines and the creation of future reconfigurable Engineering. Peer-review under responsibility of the scientific committee of the 12th CIRP on Intelligent Computation in Manufacturing Peer-review under responsibility of the scientific committee of the 12th CIRP Conference on Intelligent Computation in Manufacturing Engineering. assembly systems. Based on Datum Flow Chain, the physical structure of the products is analyzed. Functional subassemblies are identified, and Engineering. a Keywords: functionaldynamic analysisflexible is performed. Moreover,scheduling a hybrid policies; functional and physical architecture graph (HyFPAG) is the output which depicts the job shop scheduling; multi-objective genetic programming; cooperative coevolution similarity families by providing design support to both, production system planners and coevolution product designers. An illustrative Keywords:between dynamic product flexible job shop scheduling; scheduling policies; multi-objective genetic programming; cooperative example of a nail-clipper is used to explain the proposed methodology. An industrial case study on two product families of steering columns of thyssenkrupp Presta France is then carried out to give a first industrial evaluation of the proposed approach. ©1.2017 The Authors. Published by Elsevier B.V. implementation, satisfactory performance, low computational Introduction Peer-review under responsibility of the scientific committee of the 28th CIRP Design Conference 2018. toperformance, implementation, low computational 1. Introduction requirement, and satisfactory flexibility incorporate domain knowledge
Over the past decades, the static single-objective FJSP has
Keywords: Assembly; Design method; Family identification Over the past decades, single-objective FJSP has been extensively studied inthethestatic literature [1]. However, many
been extensively studied in the literature [1]. However, many real-world scheduling problems usually involve simultaneous real-world scheduling problems usually involve simultaneous optimization of several objectives which are in conflict to optimization of several objectives are inorconflict to some extent. And when we considerwhich a dynamic stochastic 1. Introduction some extent.environment And when we consider a dynamic or stochastic scheduling where the job arrival times are not scheduling wherefeatures the job arrival are not known in environment advance. These of thetimes MO-DFJSP Due to the fast These development domain of known in advance. features inof the MO-DFJSP significantly increase the complexity of the finding optimal communication and an ongoing trend of digitization and significantly increase the complexity optimal solutions[2]. Job shop scheduling problemofhasfinding been proven to digitalization, manufacturing enterprises arehas facing solutions[2]. Job shopasscheduling problem been important proven be NP-hard[3]. FJSP an extended problem, should also betoa challenges in today’s environments: a continuing be NP-hard[3]. FJSP as market an extended problem, should also be a NP-hard problem[4]. This type of scheduling problem tendency towards reduction of product development and NP-hard two problem[4]. This astype of scheduling problem involves subproblems follows: assignmenttimes of each shortened product lifecycles. In addition, there is an increasing involves two subproblems as follows: assignment of each operation to an appropriate machine (machine selection); demand of customization, beingmachine at the same time inselection); a global operation tooperations an appropriate (machine sequencing on each (job scheduling). competition with competitors all over the Thissubject, trend, sequencing on eachof machine (jobworld. scheduling). Due to aoperations large number references on this which from micro Due istoinducing a large the number of references on thistosubject, numerous techniques candevelopment be divided into the macro following types: numerous techniques can be divided into the types: heuristics,results meta-heuristic, hyper-heuristic andaugmenting artificial markets, in diminished lot sizes due following to heuristics, meta-heuristic, and rules artificial intelligence [5].(high-volume Heuristics hyper-heuristic named dispatching are product varieties to low-volume production) [1]. intelligence [5]. Heuristics named rules frequently in practice dueasdispatching to their ease of To cope with used this augmenting variety well as to be ableare to frequently used optimization in practice potentials due to intheir of identify possible the ease existing production system, it is important to have a precise knowledge 2212-8271 © 2017 The Authors. Published by Elsevier B.V.
requirement, and flexibility to incorporate knowledge and expertise[6]. Due to these advantagesdomain mentioned above, and to these advantages mentioned above, they expertise[6]. remain a veryDue popular technique used by the operators in they remain popular technique used by theinclude operators in practice. Thea very major drawbacks of heuristics their practice. Thedependence major drawbacks of heuristics include their performance on the state of the system and none performance dependence on the state the system none of the rules that is superior to all theofothers for alland possible of the product range and characteristics manufactured and/or of the rules that is superior to all the others for all possible states[7]. assembled in this system. In this context, quite the main in states[7]. Meta-heuristics are able to perform wellchallenge and carry modelling and analysis is now not only to cope with single Meta-heuristics to perform and metacarry more knowledge ofare theable problem domain.quite Evenwell though products, a have limited range domain. or existing product more knowledge ofproduct the problem Even though heuristics attracted significantly attentions, it families, hasmetatwo but also to be ableresearchers to analyze and to compare products heuristics have attracted significantly it to hasdefine two disadvantages: have to attentions, design a specialized new product families. can behave observed that problem classical existing disadvantages: researchers to design a specialized representation for eachIt practical scheduling with the product familiesfor regrouped function clients or of features. representation each practical scheduling problem with the characteristics ofare problem. So in that meansofeach class metaHowever, assembly oriented product families are class hardlyof tometafind. characteristics of problem. So that means each heuristic doesn’t have a wide range of applications on shop On the doesn’t product family level, products differ two heuristic a wide range ofthe applications oninshop scheduling[2]. Thehave second drawback is cost mainly of prohibitive main characteristics: (i) the drawback number ofiscomponents (ii) the scheduling[2]. The second the cost of and prohibitive computational effort in real-time. computational effort in real-time. Inofthe context of (e.g. solving various types of shop scheduling type components mechanical, electrical, electronical). In the context oflearning-based solvingconsidering variousapproaches types of shop scheduling problems, many with a large Classical methodologies mainly single products problems, many learning-based approaches with a large number of their variants have been employed researchers. or solitary, already existing product familiesby analyze the number structure of their variants have been (components employed by level) researchers. These methods gaussian processes[8], imitation product oninclude a physical level which These difficulties methods include gaussian processes[8], imitation causes regarding an efficient definition and comparison of different product families. Addressing this
Peer-review the scientific committee 2212-8271 ©under 2017responsibility The Authors. of Published by Elsevier B.V.of the 11th CIRP Conference on Intelligent Computation in Manufacturing Engineering. Peer-review under responsibility of the scientific committee of the 11th CIRP Conference on Intelligent Computation in Manufacturing Engineering. 2212-8271©©2017 2019The The Authors. Published by Elsevier 2212-8271 Authors. Published by Elsevier B.V. B.V. Peer-reviewunder underresponsibility responsibility scientific committee of the CIRP Conference on 2018. Intelligent Computation in Manufacturing Engineering. Peer-review of of thethe scientific committee of the 28th12th CIRP Design Conference 10.1016/j.procir.2019.02.118
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learning [9], data mining[10], reinforcement learning[11], artificial neural-networks[12], fuzzy logic[13], ensemble learning[14], genetic programming[15] and artificial immune networks [16]. However, non-existence of any AI methods is superior to all the others for all possible states in shop scheduling. Besides, the training case need to be carefully designed because it has a great impact on the test performance in supervised learning. In recent years, the concept of hyper-heuristic has been proposed to solve the combinatorial optimization problem[17]. It refers to high-level iterative techniques, which guide low-level heuristics by using intelligent concepts to explore the search space of heuristics rather than search space of solutions[18]. In this study, we focus on MO-GPHH methods for heuristic generation to fabricate new SPs by combining various small system attributes. The motivation of this approach is to generate effective SPs offline and to use them online for fast application. The rest of paper is organized as follows. Section 2 provides a brief description of the shop scheduling problem. In Section 3, the proposed algorithm is illustrated in detail. The multi-objective performance measures for the algorithm are provided in Section 4. In Section 5 and 6, the nondominated SPs are then presented with a comparison to the 320 combinations of benchmark SPs on both training and test set. Finally, conclusions and directions for future research are drawn in Section 7. Nomenclature MO-DFJSP multi-objective dynamic flexible job-shop scheduling problem SP scheduling policy JSR job sequencing rule MAR machine assignment rule MO-GPHH multi-objective genetic programming based hyper-heuristic CCGP cooperative coevolution genetic programming with two populations TTGP genetic programming with single population that a GP individual contains two trees NSGAII nondominated sorting genetic algorithm II SPEA2 strength Pareto evolutionary algorithm 2 mean weighted tardiness WTmean maximum tardiness Tmax mean flow time Fmean 2. Problem Description The MO-DFJSP with functionally related machines is formulated as follows. 1) There are a set of independent jobs J {Ji }1i n indexed i be a set of n jobs to be scheduled. 2) Each job J i consists of a predetermined sequence of operations. Let Oi , j be operation j of J i . 3) Let M {M k }1k m indexed k be a set of m machines. 4) Each operation Oi , j is processed without interruption on one machine M k out of a set of given compatible
machines M i , j (for M k M i , j , M i , j M ). Therefore, we denote by Oi , j , k to be an operation j of J i that is processed on machine M k and Pi , j , k be its processing time on machine M k . It should be noted that the processing time Pi , j , k of each operation is machine dependent. If M i , j M for at least one operation, it is partial flexibility FJSP (P-FJSP); while M i , j =M for each operation, it is total flexibility FJSP (T-FJSP) [19]. The problem considered here is P-FJSP. 5) Let wi , ri , di , ci be the weight, release date, due date and completion date of job J i , respectively. The tardiness of this job can be calculated by the following formula: (1) Ti max{0, Ci di } The flowtime of job J i is calculated by the following formula: (2) F Ci ri i 6) Three objectives, namely, mean weighted tardiness, maximum tardiness and mean flow time are to be minimized, which are defined respectively as follows: WTmean 1 / n i 1 wi Ti n
(3)
Tmax max{ Ti | i 1,..., n}
(4)
Fmean 1/ n i 1 Fi
(5)
n
The discrete event simulation model of the DFJSP is used as a test-bed for the purpose of experimentation. The simulation runs for a sufficiently long period after the shop reaches the steady state. In each simulation replication, we begin with an empty shop. The interval from the beginning of the simulation until the arrival of the 500th job is considered as the warm-up time, and the statistics from the 500th job to the next completed 2500 jobs will be used to calculate the simulation performance. The simulation configurations and details are shown in Table 1. Because of the extremely high complexity of MO-DFJSP, some common assumptions are made in this study. Job arrivals follow Poisson distribution. Weights of jobs are assigned based on the 4:2:1 rule according to Pinedo’s study [20], which showed that 20% of the customers are particularly important, 60% of them are average importance and 20% of them are less importance. Three experimental factors that each factor has two levels are examined. They are allowance factor; utilization of the shop and the number of machines could be selected by each operation. Allowance factor is used to estimate the due date of a job. The Total Work Content (TWK) [21] method is used to assigns each job a delivery date that is a multiple of its total processing time, i.e. di ri c i 1 pi , j n
(6)
In equation (6), c denotes the tightness factor of the due date, pi , j denotes the mean processing time per operation.
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to O use one replication, and it is measured by the average value of the specific objective across all training scenarios. After all individuals have been evaluated, the archive A will be updated. To explore the pareto front of nondominated SPs, two multi-objective approaches are employed to assign ranks and crowding distance (NSGAII[23]) or pareto strength (SPEA2[24]). New subpopulations are generated by genetic operations and the algorithm starts a new generation if the maximum generation is not reached.
Table 1. Parameter setting of the training and testing scenarios. Parameter Weights of jobs Simulation Length Warmup Length Workstation Process number distribution Process time distribution Allowance factor Utilization Optional device number
Training Testing 4:2:1(20%:60%:20%) 2500 500 10 10, 20 Missing Missing, Full U[1,99] U[1,99], N(120,30) 2, 4 1, 3 70%, 90% 80%, 95% 1, U[1,3] U[1,2], U[1,4]
3. Proposed methods Simulation-based evaluation
MO-GPHH based policies generation Start
Decode individual
Start
Form new population Sub-population/tree of MAR
ct=ct+1
Y
Start dispatching
Performance index
Collaborate to construct complete SP
N
Stop N
New job arrived?
Sub-population/tree of JSR
N
Fitness evaluation
NSGAII(Assign ranks and crowding distance) or SPEA2(Assign pareto strength) Produce new population
max generation achieved? Y Stop and obtain Pareto set of SPs
N
Job needs to be processed ？ Y
Return fitness Build archive
N
Device idle?
Job needs to be arranged ？ Y
Y
Apply JSR
Apply MAR
Determine which job to process on the device
Determine which device to select for the current process
441
Fig. 1. Framework of the MO-GPHH based policies generation and the simulation-based fitness evaluation.
A scheduling policy which is used in the MO-DFJSP includes two decision rules: a job sequencing rule(JSR) and a machine assignment rule(MAR). As shown in Fig. 1, the framework of the automated heuristics design approach contains two parts: the MO-GPHH based SPs generation and the simulation-based fitness evaluation. In the evolutionary stage, GP[22] is employed as the learning mechanism to evolve SPs. Different from other evolution algorithms, each GP individual is not represented by a fixed length string of genes but usually represent in a tree form of various lengths. Because a GP individual is an expression of computer program for solving a specific task. In the evaluation stage, a GP individual is decoded into decision rules and then applied to the relevant decision points in the discrete-event simulation model. In the evolution stage, the results are collected and returned to assign fitness to the GP individual. The MO-GPHH algorithms include two collaborative models. One is MO-CCGP method which evolves two decision rules in two separate populations, and the other is MO-TTGP method that have single population and a GP individual includes two trees for two decision rules. The MOCCGP method starts with two randomly generated populations, one is for JSR and the other is for MAR. The algorithm begins with an initial population Pop for JSR and an initial population P ws for MAR. In the training stage, eight simulation scenarios (more details are shown in Table 1) are loaded to evaluate the performance of an evolved SP. The individual R iop from the population P op is paired with the individual Rjws from the population Pws using random shuffling. Rirep is the complete SP that is formed by the combination of (Riop, Rjws). The fitness of (Riop, Rjws) is obtained by applying Rirep
Algorithm: MO-CCGP Inputs: simulation model O Outputs: the pareto front of nondominated SPs F initialize population Pop, Pws at random, Pop←{R1op, R2op…Rnop}, Pws←{R1ws,R2ws …Rnws} generation←0 archive A← {} while generation ≤max Generation do pair up the Riop, Rjws using random shuffling for all Riop∈Pop do Rkrep ←collaborate (Riop, Rjws) evaluate f(Rkrep) by applying Rkrep to O use 1 replication f(Riop), f(Rjws) ← f(Rkrep) end for build archive A← update { A Pop Pws } assign ranks and crowding distance(NSGAII) or pareto strength (SPEA2) apply genetic operations to archive A to generate new population generation ←generation + 1 end while apply fast-nondominated-sort to archive A to obtain pareto front F return F
The MO-TTGP method is simpler than the MO-CCGP algorithm. The difference is that an individual in MO-TTGP contains two sub-trees for two decision rules when decoding for fitness evaluation. In this case, each individual is equivalent to a complete SP. Table 2 shows the parameter settings of the proposed algorithms. The function set consists of basic operators (+, -, ×, /, max, min). The function ‘/’ is the protected division, which returns 1 if the denominator is 0. To make a relative fair comparison, all MO-GPHH methods use the same number of function evaluations (NFEs). For CCGP-NSGAII and TTGP-NSGAII methods, generation is set to 50 and archive size is fixed to 200. Therefore, the number of fitness evaluations per generation is 400, and the total number of fitness evaluations is 50×400=20000. For CCGP-SPEA2 and TTGP-SPEA2 methods, generation is set to 100 and the archive size is fixed to 100. The number of fitness evaluations per generation is 200. Hence, the total number of fitness evaluations is 100×200=20000. These settings are applied to achieve the same NFEs for each algorithm. Since two kinds of rules are evolved by GP, the terminal set for constructing JSR and MAR are presented in Table 3 and Table 4, respectively. Table 2. Parameter settings of the proposed method. Parameter Population Number Population Size Generation Crossover/mutation rates Initialization Selection Parsimony Pressure Maximum depth Operators Terminals
CCGP TTGP 2 1 200 200 NSGAII: 50; SPEA2: 100 90%/10% ramped half-and-half (depth 2~6) tournament selection (size 5) double tournament selection (t1=7, t2=2) 8 +, −, ×, /, max, min Shown in Table 3 and Table 4
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convergence performance. Spacing: This indicator measures the distance variance of neighboring vectors in PFkonwn . A lower Spacing value
Table 3. Terminal set for JSR. Terminals
Description
PT
processing time of operation Oi , j .
OL
number of operations left for job
RPT
work remaining of job
indicates a good distribution of solutions along PFkonwn .
Ji
Ji Ji
(current time-
TIS
time spend in system of job
TIQ
time spent in the device queue of job
TDD
time to the due date of job ( d i -current time)
SLACK TOD W ERC
Spacing
ri )
Ji
the difference between job’s remaining delivery time and its remaining work time=TDD-RPT time to operation’s due date weight of job J i
NPT
Random number generating from (0, 1.0] processing time of the next operation Oi , j +1
WINQ
total work time in the next queue.
Table 4. Terminal set for MAR. Terminals WQT WQS WQD WD WPT WTW WRL ERC
Description total work time of the job in the device queue queue size of the device total queueing time of the device total idle time of the device processing time of the job on the device total work time of the device mean queue size of the device Random number generating from (0, 1.0]
4. Performance measures Three popular metrics are employed to evaluate the performances of the proposed methods: hypervolume ratio (HVR)[25], Inverted Generational Distance (IGD)[26], and Spacing[27] . They can be expressed as follows: Hypervolume ratio (HVR): hypervolume is used to measures the size of the objective space dominated by the obtained non-dominated front PFkonwn . A higher HV value is desirable and denotes a good dominate performance. nPF
HVR volume(
i 1
vi )
(7)
1 nPF
1
nPF i 1
(d d i )
member and its nearest member in PFkonwn , d is the average value of all d i .
PFref is normally the true Pareto front, which is extracted from all SPs found by the four MO-GPHH methods in all independent runs. Each experiment is conducted 30 independent run times for each algorithm. In summary, the evolved SPs from 4 methods × 30 runs =120 Pareto fronts are combined into a common pool, and the nondominated sorting technique is used to extract the Pareto fronts from this pool[28]. The experiments are implemented in Java 8.0 and run on a computer with Intel Core i5-4590 3.30 GHz, 8 GB RAM. For each performance metric, a Wilcoxon signed-rank test with the significance level of 0.05 [29] is carried out on the results obtained by 30 independent runs of each method. Figure 2 shows that the metrics HVR and IGD produced by MO-CCGP are significantly better than that of MO-TTGP methods. However, in MO-CCGP or MO-TTGP methods, there is no significant difference between NSGAII and SPEA2 based approach. In terms of Spacing, the value obtained by NSGA-II based method is significantly better than those of SPEA2 based method. In addition, there is no significant difference between MO-CCGP and MO-TTGP methods. In summary, collaborative patterns(CCGP/TTGP) pose a great impact on the performance indicators of HVR and IGD, and pareto approaches(NSGAII/SPEA2) significantly affect the indicator of Spacing. Overall, CCGP-NSGAII is the most competitive method among the four proposed methods.
HV of the reference Pareto front PFref .
HV ( PFkonwn ) HV ( PFref )
(8)
Inverted Generational Distance(IGD): This is a variant of the Generational Distance (GD) but represents a combined or comprehensive indicator. It measures the average distance from the reference Pareto front PFref to Pareto
(a)
front PFkonwn obtained by the algorithm.
IGD
p
( i 1 di )1/ p n
(9) n Where n is the number of all elements in PFref , p is set to 2 in this study, d i is the Euclidean distance between the member i in PFkonwn and its nearest member in PFref . Fronts with a lower IGD value are desirable and denotes a good
(10)
Where nPF is the number of members in the obtained Pareto front PFkonwn , d i is the minimum distance between the
Where nPF is the number of members in the obtained nondominated front PFkonwn , vi is the hypercube constructed with a reference point and the member i as the diagonal of the hypercube [23]. HVR is the ratio of the HV of PFkonwn and the
HVR
2
(b)
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(c) Fig. 2 Performances of the proposed methods on training scenarios. (HVR to be maximized and Spacing and IGD to be minimized). (a)HVR (b)IGD (c)Spacing
5. Comparison of nondominated SPs to existing SPs In this study, 320 benchmark SPs found in the literature [6][21] are used to verify the effectiveness of the evolved SPs. The benchmark SPs are made up of 30 well-known JSRs and 10 MARs, which are shown in Table 5 and Table 6. Because the ATC rule contains parameter k, and three parameter configurations with k=1.0, 2.0, 3.0 are used; therefore, there are 32×10=320 combinations of benchmark SPs. Table 5. Benchmark rules for JSR. No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Benchmark Rule ATC CR CR+SPT EDD FASFS FCFS LCFS LPT LRPT MDD MOD ODD RND SI SL SLK SPT SRN SRPT SRPT/PT SRPT/SLK PT+WINQ PT+WINQ+Slack 2PT+WINQ+NPT WINQ NPT LW WMDD WMOD WSPT
Description apparent tardiness cost critical ratio critical ratio plus SPT earliest due date first arrival at shop first served first come first served last come first served longest processing time longest remaining processing time modified due date modified operation due date operational due date random rule a truncated version of SPT negative minimum slack minimum slack shortest processing time shortest remaining operation number shortest remaining processing time remaining processing time per processing time remaining processing time per slack processing time plus WINQ processing time plus WINQ and slack double processing time plus WINQ and NPT work in the next queue processing time of next operation largest weight weighted modified due date weighted modified operational due date weighted shortest processing time
Table 6. Benchmark rules for MAR. No. 1 2 3 4 5 6 7 8 9 10
Terminals WQT WQS WQD WD WPT WTW WRL WTN WU WRD
Description total process time of the job in the device queue queue size of the device total waiting time of the job in the device queue total idle time of the device process time of the job on the device total work hours of the device average queue size of the device in one minute total work numbers of the device device utilization average queueing time of the device
Fig. 3. Reference Pareto front of the evolved SPs and the results of the existing SPs on training scenarios (‘*’ and ‘o’ respectively represent evolved and existing SPs)
As mentioned earlier, 320 benchmark SPs are applied to 8 training scenarios (see Table 1), and 100 simulation replications are performed for each scenario. Therefore, we perform 8×100=800 simulation replications to test the performance of each manual SP. The results generated by the manual SPs are recorded to compare with the results of the evolved SPs ( PFref )on the training set. As shown in Fig.3, the nondominated SPs evolved by MO-GPHH dominate all manual SPs found in the literature under any objective. 6. Testing performance To evaluate the generalization of the evolved SPs, we applied a design of experiments (DOEs) approach to design the test set. In the DOE, we consider 5 factors (see Table 1), each with 2 levels. The full factorial design (including 25 =64 combinations) is adopted in this case. Since many SPs have been obtained from the training stage, we choose three SPs that perform well on the specific objective (see Fig.4) from the nondominated SPs generated by CCGP-NSGAII. The three selected evolutionary SPs and the 320 kinds of benchmark SPs are simultaneously applied to 64 test scenarios (see Table 1). Meanwhile, 100 simulation replications are performed for each scenario. Therefore, we perform 64×100=6400 simulation replications to test the performance of each SP. Average value of the specific objective under certain scenario obtained from 100 independent simulation replications are recorded as the performance metric of the test SP. The relative deviation between the best performance obtained by three evolved SPs with the best performance obtained by 320 manual SPs on the same instance under the same target is recorded to verify the generalization of the evolved SPs. The number of test scenarios that the evolved SPs perform better than the 320 types of manual SPs under each objective WTmean , Tmax , Fmean is 62, 58 and 64, respectively. There are 6 scenarios with relatively poor performance Tmax . Because Tmax is a maximum target rather than other average targets, it may change drastically with the change of the random seed. Besides, the evolved SPs perform better than the 320 types of manual SPs in all test scenarios with the indicator Fmean .
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Fig. 4. Pareto front and selected evolved SPs generated by CCGP-NSGA2 (‘*’ denotes the nondominated evolved SPs and ‘o’ denotes the selected evolved SPs)
7. Conclusion and future work This study proposes four MO-GPHH algorithms to automatically design SPs for MO-DFJSP, including JSR and MAR. The training performances show that CCGP-NSGAII is the most competitive approach among the proposed methods for evolving efficient non-dominated SPs for MO-DFJSP. Statistical tests indicate that it has the best overall performances in three popular metrics (HVR, IGD, Spacing). Then, the evolved SPs which were extracted from all SPs found by the four MO-GPHH methods are compared with the 320 types of benchmark SPs found in the literature on training set. The results reveal that the performances of the evolved SPs dominate those of all manual SPs under any objective. To evaluate the generalization of the evolved SPs, 64 simulation scenarios are designed for the test set. The results show that the evolved SPs outperform the 320 types of manual SPs in 58/64 test scenarios. It also demonstrates that the evolved SPs have a strong generalisation ability to be reused on new unobserved scheduling scenario. The results show that the evolved SPs can effectively solve the MO-DFJSP and obtained trade-offs among different objectives. However, dispatching rules made by human experts are still widely used in many practical scheduling systems. Therefore, we plan to apply the proposed MOGPHH methods to automatically evolve SPs based on the real case. And then, the appropriate SP is selected to replace the artificial SPs designed by experts in the scheduling system. Acknowledgements This work was supported by Beijing Municipal Education Commission (Build a Project) and the Key Laboratory Development Projects of Beijing, China. References [1] Nouiri M, Bekrar A, Jemai A, Niar S, Ammari AC. An effective and distributed particle swarm optimization algorithm for flexible job-shop scheduling problem. J Intell Manuf 2018; 29:603-15. [2] Gen M, Lin L. Multiobjective evolutionary algorithm for manufacturing scheduling problems: state-of-the-art survey. J Intell Manuf 2014; 25:84966. [3] Garey M R, Johnson D S, Sethi R. The Complexity of Flowshop and Jobshop Scheduling. Math Method Oper Res 1976; 1:117-129. [4] Mati Y, Xie X. The complexity of two-job shop problems with multipurpose unrelated machines. Eur J Oper Res 2004; 152:159-169.
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