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Rule-Evolutionary Approach for Single-Stage Multiproduct Scheduling with Parallel Units Yaohua He and Chi-Wai Hui* Chemical Engineering Department, Hong Kong UniVersity of Science and Technology, Clear Water Bay, Hong Kong, P. R. China
Process scheduling shows much more complexity than machine scheduling, and it has been widely studied mainly by using mathematic programming (MP). Due to the difficulties for MP to solve large-size problems, simple rule-base methods are often used in the industry. Metaheuristic methods, such as genetic algorithm and tabu search, combined with suitable heuristic rules, are effective to obtain near-optimal solution for largesize problems. The use of good heuristic rules is crucial to cut down the solution space. Traditionally, great simulation experiments are needed to select suitable rules for diverse scheduling objectives. This paper proposes a novel evolutionary approach to tackle rule selection, rule sequence, and subsequent rule combination for a certain scheduling objective. In our approach, the algorithm itself will automatically select the suitable rule/ rule sequence to synthesize an evolved order sequence into a high quality schedule. This approach is able to solve large-size scheduling problems. (1) Makespan (Cmax): The makespan, defined as,
Introduction From the literature and our research work,1-3 it is found that by using traditional mathematic programming (MP), formulating and modeling real-world problems is not an easy task; solving large complex problems is much more difficult. And, it is often impossible for MP to get an optimal solution or even a feasible solution within a reasonable time. Hence, in many cases, feasible solutions are more considered and practical, rather than optimal solutions. The aim of this study is to effectively solve largesize process scheduling problems through an evolutionary approach. Scheduling problems are typical combinatorial problems, the complexity of which can easily exceed today’s hardware and algorithm capacities, and no standard solution techniques are available. Researchers often use MP to solve scheduling problems. When the problem size increases linearly, the computation time of MP will increase exponentially. For largesize scheduling problems, MP shows limited search ability. The preferred approach to large-size scheduling problems in industry is the use of scheduling rules, such as the shortest processing time (SPT), earliest due date (EDD), etc. The scheduling rules change with the different types of scheduling problems, such as parallel machine scheduling (single-stage) and flow-shop scheduling (multistage). During the last 30 years, the performance of a large number of scheduling rules has been studied extensively by using simulation techniques. The research on the scheduling rules has shown that there is no single universal rule, and the effectiveness of a scheduling rule depends on the scheduling objective and the prevailing shop or plant conditions. Time-based scheduling objectives are commonly considered in the literature because cost-based objectives usually can be surrogated by time-based objectives.4 Common time-based objectives are described by Pinedo.5 Makespan, total tardiness, and total earliness are three typical scheduling objectives to be minimized. Assume that there are N jobs to be assigned to M machines. * To whom correspondence should be addressed. Tel.: +85223587137. Fax: +852-23580054. E-mail:
[email protected].
Cmax ) max{C1, C2, ..., CN}
(1)
is the completion time of the last job to leave the system, where Cj is the completion time of job j. A minimum makespan usually implies a high utilization of the machines. (2) Total tardiness (T): The total tardiness is the sum of the tardiness of all jobs, defined as, N
T)
Tj ∑ j)1
(2)
where Tj is the tardiness of job j, Tj ) max{Cj - dj, 0}, and dj is the due date of job j. (2) Total earliness (E): The total earliness is the sum of the earliness of all jobs, defined as, N
E)
Ej ∑ j)1
(3)
where Ej is the earliness of job j and Ej ) max{dj - Cj, 0}. Cheng and Sin6 presented a full review of parallel machine scheduling research, including a large number of scheduling rules. Park et al.7 investigated the significant scheduling rules for parallel machine scheduling. Rajendran,4 Holthaus,8 and their colleagues (from 1997 to 2004) have executed a lot of research on scheduling rules for job-shop and flow-shop scheduling. Besides studying extensively the performance of scheduling rules, dynamic selection of scheduling rules was proposed by Subramaniam et al.9 Subramaniam et al.10 proposed a fuzzy scheduler that used the prevailing conditions in the job shop to select dynamically the most appropriate scheduling rule from several candidate rules. Subramaniam et al.11 proposed a new two-stage scheduling approach: (stage 1) Apply a machine selection rule to identify a free machine to be scheduled. (stage 2) Apply a scheduling rule to the machine identified in stage 1. Different from the fact that scheduling rules decide the job sequence or its operation sequence on the machine, the purpose
10.1021/ie0512587 CCC: $33.50 © 2006 American Chemical Society Published on Web 05/18/2006
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Table 1. Changeover Times, Due Dates, and Process Times of Example 1 changeover times j1 i1 i2 i3 i4 i5 i6 i7 i8 i9 i10
0.78 1.66 0.95 0.72 1.24 1.96 0.97 1.07 1.54
process times
j2
j3
j4
j5
j6
j7
j8
j9
j10
due date
u1
u2
u3
1.02
0.89 1.86
0.80 1.96 1.69
0.98 1.69 1.88 1.30
1.24 1.97 0.81 1.02 1.26
1.83 1.37 1.43 0.71 1.17 1.11
0.84 1.03 0.78 1.45 1.74 1.39 1.83
1.58 0.96 1.49 1.26 0.77 0.96 1.34 0.66
1.02 1.66 0.63 0.71 0.74 0.65 0.75 1.77 0.54
31 39 22 34 55 28 55 29 26 42
14.07 15.64 16.49 10.70 15.53 8.20 14.41 6.83 5.08 8.60
12.20 8.95 14.29 16.18 5.13 10.68 17.24 9.70 6.43 13.11
5.40 19.41 8.56 5.97 10.68 7.19 5.73 8.02 10.41 14.12
1.87 1.41 1.44 1.60 1.99 1.78 1.81 1.76
1.06 1.68 1.68 1.85 1.32 0.71 1.27
1.81 1.26 0.81 0.67 1.57 1.46
1.37 0.70 1.22 0.78 1.51
1.13 1.89 1.83 0.83
0.99 1.34 1.19
of machine selection rules is to identify an appropriate machine to be scheduled. Metaheuristics, such as genetic algorithm (GA), simulated annealing (SA), and tabu search (TS), and their hybrids have shown their potentials to solve large-size complex scheduling problems in discrete manufacturing during recent years. Tanev12 developed a hybrid evolutionary algorithm that was able to evolve a schedule of 400 customers’ orders. Nowicki13 proposed an algorithm based on a tabu search technique to solve the permutation flow-shop problem (up to 500 jobs and 20 machines). Grabowsk and Wodecki14 proposed a new very fast local search procedure based on a tabu search approach for the permutation flow-shop problem with the makespan criterion. They solved the 500-job/20-machine flow-shop instance with high accuracy in a very short time. Due to the complexity and high constraints of process scheduling, some authors believe that metaheuristics are not suitable for process scheduling problems, especially large-size complex problems. Nevertheless, our research1-3 has shown that metaheuristic methods, combined with suitable heuristic rules, are effective to search near-optimal solutions for large-size process scheduling problems. In the research on process scheduling, heuristic rules are often combined into the mixed-integer linear programming (MILP) model with the purpose of cutting down the size of the model.15-19 To sum up, heuristic rules (scheduling rules and machine selection rules) are human intelligent knowledge for solving scheduling problems that, if properly selected and combined into solution techniques (MILP or metaheuristics), will greatly cut down the solution space to search, greatly reduce the search time, and raise the solution quality. Traditionally, great simulation experiments are needed to select suitable rules for diverse scheduling objectives. The knowledge-based scheduling approach4 relies on expert system technology mainly based on if-then rules. Our research proposes a novel evolutionary approach to tackle the selection of rules and rule sequences. Heuristic rules for particular scheduling objective are coded into an evolutionary algorithm, and the algorithm will automatically select the suitable rule/ rule sequence to assign the orders to the production units. The algorithm is easily implemented. Our evolutionary approach not only searches the near-optimal solutions to the large-size problem but also finds out the suitable knowledge for solving the problem. The new approach can also be applied to discrete scheduling, project scheduling, and other combinatorial problems. Single-stage multiproduct scheduling with parallel units (SMSP) is used to illustrate our rule evolutionary approach. SMSP has been widely studied by process scheduling researchers, such as Cerda et al.,16 Hui and Gupta,20 and Chen et al.19 In our future work, the proposed approach will be applied to
1.43 0.51
1.14
other process scheduling problems, e.g., a multistage multiproduct scheduling problem (MMSP) in a batch plant. The algorithms in this paper were implemented in the C language, and the computation tests were run on computer with an Intel Pentium M 1500 MHz CPU and 768 MB of memory. GA is just for illustration; it is not the only choice. Other metaheuristics, such as tabu search and hybrid algorithms, will be more effective to solve large-size problems. 2. Problem Definition and Heuristic Rules 2.1. Problem Definition. In SMSP, a fixed number of production units (forming a set of units: U) are available to process all customer orders (forming a set of orders: O). Assume the number of production units is M and the number of customer orders is N. Each order involves a single product, requiring a single processing stage, has a predetermined due date, and can only be processed in a subset of the units available. The production capacity of a unit depends on the order processed. The size of an order may be larger than the size of a batch, requiring several batches to satisfy an order. Batches of the same order are processed consecutively in the same unit. A production unit processes only one batch at a time. The batch-time of an order is fixed and unit-dependent. When one order changes over to another order, time is required for the preparation of the unit for the changeover. The changeover time is sequence-dependent. Forbidden changeovers and processes may exist in the problem, named as CP constraints. The scheduling objective is to minimize the makespan Cmax or total tardiness T, depending on the due dates of the orders. If it is obvious that all the orders can be completed before their due dates, it makes no sense to minimize the total tardiness. Just as was done in the literature,16,20 when minimizing the makespan of the problem, the due dates of the orders are not considered as constrains. Example 1 will be used to illustrate our approach. Example 1 is a random SMSP in which 10 orders (N ) 10) are to be assigned to 3 units (M ) 3). Each order involves only one batch and is required to be completed before the due date di. The due date di is produced randomly, di ∈ (20, 60). All the order release times and unit release times are null. Changeover time cij is produced randomly, cij ∈ (0.50, 2.00), and process time piu is also produced randomly, piu ∈ (5.00, 20.00). Neither a forbidden changeover nor a forbidden process exists in this example, that is to say, no CP constraint exists. The data for example 1 is presented in Table 1. 2.2. Scheduling by Heuristic Rules. To minimize the makespan of SMSP, let us enumerate the following heuristic rules (forming a candidate rule base, or a set of rules: SR) for assignment of the orders to units. (Rule 1) Assign the order on the unit that makes the order’s completion time as early as possible (earliest completion time, ECT).
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Figure 1. Schedule synthesis by one rule. Table 2. Random Order Sequence Scheduled by Different Rules order sequence π rule used makespan Cmax total tardiness T
3 rule 1 30.14 0.09
2
7 rule 2 38.24 10.46
6
4
rule 3 37.62 6.62
5 rule 4 58.86 47.86
9
10 rule 5 58.37 53.64
1
8
rule 6 30.72 1.72
(Rule 2) Assign the order on the unit that makes the order’s start time as early as possible (earliest start time, EST). (Rule 3) Assign the order on the unit that makes the order’s process time on the unit the shortest (shortest process time, SPT). (Rule 4) Assign the order on the unit that makes the order’s changeoVer time on the unit the shortest (shortest changeoVer time, SCT). (Rule 5) Assign the order on the units by random (random rule, RR). (Rule 6) Assign the order on the unit that makes the sum of the order’s changeoVer time and process time on the unit the shortest (shortest changeoVer + process time, SCPT). Natural numbers 1, 2, 3, ..., N are used to denote the N orders. An order sequence π ) (π1, π2, π3, ..., πN) is produced randomly, πi ∈ {1, 2, 3, ..., N}, and i ) 1, 2, 3, ..., N. And then, from π1 to πN, one by one, each order will be assigned over the units according to a certain heuristic rule above. As a result, a schedule is formed with an objective value, f(π). The value of f(π) can be calculated by the function of eqs 1-3. Figure 1 is the procedure to synthesize an order sequence into a schedule according to one selected rule. It is easy to understand the procedure. Assume that a random order sequence is π ) (3, 2, 7, 6, 4, 5, 9, 10, 1, 8) in example 1. We have used the above six rules respectively to assign all the orders in the order sequence over the units and formed different schedules with various makespan and total tardiness; see Table 2 and Figure 2. From the values of Cmax and T in Table 2 and through analyzing the schedule charts in Figure 2, it has been seen that rules 1 and 6 are the
best, rules 3 and 2 are the better, but rules 4 and 5 are the worst. Rule 1 assigns the orders over the units averagely. Therefore, for the same order sequence, if different rules are used for assignment, different quality schedules are obtained. 2.3. Heuristic Rules for Different Scheduling Objectives. In general, for different scheduling objectives, there are different heuristic rules that form a rule base. It is easy to understand that the heuristic rules for minimization of the total tardiness are the same or similar as those rules for minimization of the makespan. However, heuristic rules for minimization the total earliness might be different, for instance, rules 1 and 2 should be changed into the following: (Rule 1) Assign the order on the unit that makes the order’s completion time as near as possible to the due date. (Rule 2) Assign the order on the unit that makes the order’s start time as near as possible to the due date. The other rules in SR can be still used for minimization of the total earliness. In this paper, the rule-evolutionary approach is applied to the minimization of the makespan and total tardiness. Nevertheless, the proposed approach can be easily used for other scheduling objectives, with only a little change to the subroutines for specific rules. 2.4. Criteria to Evaluate the Algorithms. The following criteria are used to evaluate the performance of the algorithms: (1) the best solution that the algorithm obtains; (2) the computational time of the algorithm; (3) the mean objective value of the computational tests; (4) the deviation from the best solution obtained up to date. For large-size problems, we cannot expect to find globally optimal solutions within a reasonable time. Therefore, relative deviations from the best solutions are used as the criteria for evaluation.21,22 Relative deviation is calculated with respect to the best (or optimal) solution obtained up to now. For makespan, we have
Dev from best (%) ) 100[(algorithm makespan best makespan)/best makespan] (4) 3. Genetic Algorithm with a Prefixed Rule GA is inspired by Darwin’s theory of evolution.23 GA has experienced increasing application for numerical optimization and combinatorial optimization and has shown great promise for performance in many industrial engineering areas.24 In GA for scheduling, solutions to the problem are represented by chromosomes. Each chromosome is evaluated through synthesizing it into a schedule according to a certain heuristic
Figure 2. Different schedules for the same order sequence by different rules: (a) rule 1; (b) rule 2; (c) rule 3; (d) rule 4; (e) rule 5; (f) rule 6.
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rule. In the beginning of GA, an initial generation of chromosomes is produced randomly. Throughout the genetic evolution, because of the mechanism of selection, crossover, and mutation, good-quality offspring are born from the previous generation (parents). Generation by generation, the stronger chromosomes are the survivors in a competitive environment. At the end of GA, near-optimal solutions can be achieved. 3.1. Components of Genetic Algorithm. 3.1.1. Representation and evaluation. To solve scheduling problems by using GA, the first task is to represent a solution of the problem as a chromosome. A permutation-based representation is adopted in this paper, which is a kind of integer coding. As stated in section 2, natural numbers 1, 2, 3, ..., N are used to denote N orders. A random order sequence π ) (π1, π2, π3, ..., πN) is produced and πi ∈ {1, 2, 3, ..., N}. In GA, π is called a chromosome. For example, π ) (6, 3, 5, 4, 8, 2, 7, 1, 9, 10) is a sample chromosome of the 10-order problem. The evaluation of a chromosome is a process to synthesize the chromosome (according to the heuristic rule) into a schedule with an objective value, as shown in Figure 1. Traditionally, only one rule is applied for the evaluation of all the chromosomes in the GA process. 3.1.2. Generation. At the beginning of GA, an initial generation of chromosomes are randomly generated. Assume the number of chromosomes in the initial generation is the popsize, which depends on the problem size and is an important parameter in GA for controlling the solution quality. In all the genetic algorithms in this paper, popsize ) 200. At every iteration of GA, a new generation will be produced through crossover, mutation, and selection and popsize will remain constant. 3.1.3. Selection. Selection is the process to select most of the better chromosomes in each generation to crossover and the rest or some of the rest to mutate. Common selection methods include roulette-wheel selection25 and the tournament method.26 Although roulette-wheel selection is one commonly used technique, the tournament method is adopted in our GA. The reason is that in the tournament method the objective value of a chromosome can be used as the selection criterion but in roulette-wheel selection the fitness of a chromosome is always used as the selection criterion. Assume that the number of chromosomes that are selected to crossover is xsize and that the number of chromosomes that are selected to mutate is msize. We can let xsize + msize ) popsize. The ratio Cr ) xsize/popsize is called crossover rate and is often ∈ [0.5,0.9]; the ratio Mr ) msize/popsize called mutation rate, and often, Mr ∈ [0.1,0.3]. So, Cr + Mr ) 1. Cr and Mr are two important parameters that influence the convergent performance of GA. In general, if Mr increases, GA will be convergent to the final solution slowly, so that GA has more chances to find out better solutions. But if Mr is too large, GA tends to be like random search. In all the genetic algorithms in this paper, we let Cr ) 0.8 and Mr ) 0.2. Due to popsize ) 200, we get xsize ) 160 and msize ) 40. 3.1.4. Crossover. Crossover is the process during which two parents generate two offspring, such that the children inherit a set of building blocks from each parent. However, genetic operators are related to representation schemes. Poon and Carter27 presented a survey of crossover operators for ordering applications. Regarding the permutation-based representation, the following crossover operators have been widely used: partially matched crossover (PMX), intended to keep the absolute positions of elements, and linear order crossover (LOX), intended to respect relative positions. PMX is adopted in our GA.
Figure 3. Flowchart of GA. Table 3. Results of Example 1 by GA Combined with Different Rules (min Cmax)
method
best
GA_R1 (ECT)a GA_R2 (EST) GA_R3 (SPT) GA_R4 (SCT) GA_R5 (RR) GA_R6 (SCPT)
28.31 28.31 36.31 28.31 28.83 30.53
a
mean dev mean from CPU mean tests of rule optimal time(s) iterations computation quality 0 1.05 28.26 5.74 6.69 7.84
