VIÑAS ASOCIADAS

Three main criteria are used to evaluate a heuristic for adaptive production scheduling and control (Li et al., 2011a): optimality, computational complexity, and flexibility. Usually optimality is used to evaluate a heuristic for offline production scheduling. However, when adaptive production control is taken into consideration, the computational complexity becomes critical. That is why some heuristics based on artificial intelligence are not suitable for adaptive production control, although they can get better solutions. Another criterion is

Production Scheduling 120

the flexibility, that is, whether a heuristic can deal with a disturbance. Of course, different situations have different requirements for optimality, computational complexity, and flexibility of a heuristic. There is inevitably a trade-off among these criteria, and the selection of heuristics for production scheduling and control depends on specifics of different situations, such as the value of optimality as compared to near optimal scheduling, as well as the type and volume of disturbances that underlies the requirements of response time. 2.2.2 Summary of existing heuristics for adaptive production control

For optimality, heuristics in phase 3 can get better solutions than heuristics in phases 1 and 2. However, for computational complexity, they take much longer time. For example, an adaptive learning approach (ALA) heuristic is in phase 3 for Fm/prmu/Cmax problems

(Agarwal et al., 2006). The deviation of ALA heuristic is only 1.74% for Taillard’s benchmarks (Taillard, 1993), much better than 3.56% of NEH heuristic. However, for the largest instance in Taillard’s benchmarks, i.e. 500 jobs and 20 machines, it takes more than 19 hours for ALA heuristic to get a solution, more than 20 hours for Simulated Annealing, and more than 30 hours for Tabu search (Agarwal et al., 2006). Even for the recent IG_RS heuristic, it takes 300 seconds to get a solution to a 500-job 20-machine instance. For flexibility, we need to see if a heuristic can deal with a disturbance. According to Pinedo (2002), there are three types of disturbances in general for flow shop production, job insertion or cancellation, operator absence or machine breakdown, and variation in processing times. The perfect production information in OKP is available only after the production (Wortmann, 1992). Therefore, if a heuristic operates the known processing time only, it cannot deal with variation in processing times.

The performance of first eight of 19 heuristics is summarized in Table 2.1, and the optimality of each heuristic is quoted from Ruiz and Maroto (2005). However, there is a discrepancy of optimality of heuristics in the literature, because the optimality is evaluated by the deviation from the best known upper bounds that are under continuous improvement. For example, the deviation of 3.33% is for NEH and 9.96% for CDS in Ruiz and Maroto (2005), but 3.56% for NEH and 10.22% for CDS in Agarwal et al. (2006), and 3.59% for NEH and 11.28% for CDS in our case study. In the table, the column “Opt.” means the optimality on Taillard’s benchmarks for Fm/prmu/Cmax problems, “I/C” means the job insertion or cancellation,

“OA/MB” means the operator absence or machine breakdown, and “Var.” means the variation in processing times. The mark of “Yes§_{” means a heuristic can deal with a}

disturbance only with a modification of processing times, e.g. in Botta-Genoulaz (2000).

Computational Complexity Flexibility

Opt. Note I/C OA/MB Var.

NEH 3.33% O(mn2_{) O(mn}3_{) Yes Yes}§ _No

Suliman 6.21% Intractable CDS first, then swap job pairs Yes Yes§ _No

RAES 7.43% Intractable RA first, then swap jobs Yes Yes§ _No

HoCha 8.06% Intractable CDS first, then swap job pairs Yes Yes§ _No

RACS 9.17% Intractable RA first, then swap jobs Yes Yes§ _No

Koula 9.22% O(m2_n2_{) JA first, then job passing Yes Yes}§ _No

HunRa 9.69% O(mn+nlogn) 3 × Palmer's slope index Yes Yes§ _No

CDS 9.96% O(m2_{n+mnlogn) JA Yes Yes}§ _No

It is self-illustrative for optimality and flexibility of each heuristic in the above table. We only discuss the computational complexity in the following. NEH heuristic, in its original version, has a computational complexity of O(mn3_{), but, by calculating the performance of all partial}

sequences in a single step, its complexity is reduced to O(mn2_{) (Taillard, 1990). Both Suliman}

(Suliman, 2000) and HoCha (Ho & Chang, 1991) heuristics use CDS heuristic to generate an initial sequence, and then exchange job pairs to improve the performance, but they use different mechanisms for job pair swaps. Because the number of job pair swaps depends on the calculation of performance of each job pair, the computational complexities of Suliman and HoCha heuristics are intractable. Job swaps are also involved in RACS and RAES heuristics (Dannenbring, 1977), and their computational complexities are intractable too. These two heuristics are based on a rapid access (RA) heuristic (Dannenbring, 1977), which is a mixture of JA and Palmer’s slope index (Plamer, 1965). Koula heuristic (Koulamas, 1998) is not purely for permutation flow shop problems. The job passing is allowed in Koula heuristic, because Potts et al. (1991) point out that a permutation schedule is not necessarily optimal for all n-job m- machine flow shop problems. Koula heuristic extensively uses JA to generate initial sequences, and then job passing is allowed to make further improvement. The overall computational complexity of Koula heuristic is O(m2_n2_{). HunRa heuristic (Hundal & Rajgopal, 1988) is a}

simple extension of Palmer’s slope index. HunRa heuristic generates three sequences, one by Palmer’s slope index, the other two by calculating indices differently. Therefore, the HunRa heuristic has the same computational complexity as Palmer’s slope index, O(mn+nlogn). Usually, the number of jobs n is much larger than the number of machines m, thus, the computational complexity of O(m2_{n+mnlogn) for CDS heuristic is comparable with that of}

O(mn+nlogn) for HunRa heuristic.

For an industrial instance in Gienow with 1396 jobs and 5 machines, it takes NEH heuristic more than 70 seconds to generate a sequence, which is too slow to keep up with the production pace in Gienow. Therefore, NEH and other five heuristics, with computational complexity higher than O(mn2_{), are not suitable for adaptive production scheduling and}

control in Gienow. It takes less than one second for CDS or HunRa heuristics to generate a sequence for the same industrial instance. However, their performance is not good from the optimality perspective, with more than 9% deviation on Taillard’s benchmarks.