Creating dispatching rules by simple ensemble combination

Abstract

Dispatching rules are often the method of choice for solving scheduling problems since they are fast, simple, and adaptive approaches. In recent years genetic programming has increasingly been used to automatically create dispatching rules for various scheduling problems. Since genetic programming is a stochastic approach, it needs to be executed several times to ascertain that good dispatching rules were obtained. This paper analyses whether combining several dispatching rules into an ensemble leads to performance improvements over the individual dispatching rules. Two methods for creating ensembles of dispatching rules, based on the sum and vote methods applied in machine learning, are used and their effectiveness is analysed with regards to the size of the ensemble, the genetic programming method used to generate the dispatching rules, the size of the evolved dispatching rules, and the method used for creating the ensembles. The results demonstrate that the generated ensembles achieve significant improvements over individual automatically generated dispatching rules.

Appendices

Appendix

A Problem instance details

The scheduling problem which is solved in this paper can be classified as $Rm|r_j|\gamma$, where $\gamma$ represents one of the four scheduling criteria that are considered. In the unrelated machines scheduling problem each job needs to be scheduled on a single machine. When the job is scheduled on a certain machine, the machine needs to execute this job until it is finished before it can start executing another job (thus preemption is not permitted). At each moment in time each machine can execute at most one job. However, all machines work in parallel, which means that each of them is executing the job assigned to it, independently from the other machines. The unrelated machines environment is a single stage environment, which means that each job needs to be executed on only a single machine to be completed. Furthermore, the specificness of this environment is that each job has a different execution time on each of the machines, thus selecting the appropriate machine for each job constitutes an important part in this problem. Except for the release times, no additional constraints, like breakdowns or set-up times, are considered in these problems. However, the proposed SEC method should work with any of those without any changes, if the DRs are adapted for these additional constraints. The way in which the problem instances are generated, the system will have a high utilisation most of the time during its execution.

The processing times of jobs are generated from the interval

$$\begin{aligned} p_{ij} \in [0,100], \end{aligned}$$

by using one of the following three probabilistic distributions: uniform, normal (Gaussian), and quasi-bimodal. Which of the aforementioned three distributions will be used for generating the processing times is chosen randomly for each job (with all three distributions having the same probability of being chosen). The motivation behind the use of three distributions for generating processing times is to make the evolved priority functions more resilient, since in real conditions jobs could be received from different sources. All job weights are generated uniformly from the interval

$$\begin{aligned} w_{T_j}\in <0,1]. \end{aligned}$$

The release times of jobs are generated by a uniform distribution from the interval

$$\begin{aligned} r_j\in \left[ 0,\frac{\hat{p}}{2}\right] , \end{aligned}$$

where $\hat{p}$ is defined as

$$\begin{aligned} \hat{p}=\frac{\sum _{j=1}^{n}\sum _{i=1}^{m}p_{ij}}{m^2}, \end{aligned}$$

and $p_{ij}$ denotes the processing time of job j on machine i, while m denotes the total number of machines. The due dates of jobs are also defined using a uniform distribution from the interval

$$\begin{aligned} d_j \in \left[ r_j + (\hat{p}-r_j)*\left( 1-T-\frac{R}{2}\right) ,r_j +(\hat{p}-r_j)*\left( 1-T+\frac{R}{2}\right) \right] , \end{aligned}$$

where T represents the due date tightness parameter, while R represents the due date range parameter. The due date range parameter defines the dispersion of the due date values, while the due date tightness adjusts the amount of jobs that will be late. While generating the problem set, both of those parameters assumed values of 0.2, 0.4, 0.6, 0.8, and 1 in various combinations.

Because some problem instances have significantly different characteristics, and will therefore also have significantly different objective values. This leads to a problem in which smaller instances have little or no influence in the total fitness value, and thus the GP procedure would focus less on optimising these instances. To avoid this problem all the objective values were normalised in order for the problem instances with different characteristics to have similar objective values. Therefore, the normalised objective functions for the problem instance with the index i are defined as follows:

• for the weighted tardiness criterion $f_i=\frac{\sum _{j=1}^{n}w_jT_j}{n\bar{w}\bar{p}}$

• for the weighted number of tardy jobs criterion $f_i=\frac{\sum _{j=1}^{n}w_jU_j}{n\bar{w}}$

• for the flowtime criterion $f_i=\frac{\sum _{j=1}^{n}F_j}{n\bar{p}}$

• for the makespan criterion $f_i=\frac{\max \{C_j\}}{n\bar{p}},$

where n denotes the number of jobs in the problem instance, $\bar{w}$ the average weight of the jobs and $\bar{p}$ the average job processing duration. The total objective function is then calculated as the sum of the objective functions of the individual problem instances.

Table 14 Performance of the ensemble construction methods on the Nwt criterion with the sum combination method

Fig. 7

Box-plot representation of results for the ensemble construction approaches and the sum combination method on the Nwt criterion

B Results for the ensemble construction methods on different criteria

In this section the results of the ensemble construction methods will be presented on the remaining three scheduling criteria. Table 14 represents the results obtained by the ensemble construction methods for the Nwt criterion when using the sum combination method. The box-plot representation of these results is given in Fig. 7. The results denote that the instance based method is the only one which consistently performed better on average than the best DR. The other methods performed well only when the ensemble size of 7 DRs was used, while in other cases the average results that were obtained were worse than that of the best DR. The box-plots show that for the smallest ensemble size the performance of the ensembles is usually much worse than that of the best individual DRs. However, for the largest ensemble size it is evident that for certain experiments most of the ensembles perform better than the best DR.

Table 15 denotes the same results using the vote combination method. Figure 8 represents the results presented by using box-plots. In this case all the ensemble combination methods obtained average results which are better than the result obtained by the best DR. The results are quite similar, and between most of them there is no statistically significant difference. The box-plot shows that most of the ensembles which were generated by SEC actually achieve a better performance than the best individual DR.

Table 15 Performance of the ensemble construction methods on the Nwt criterion with the vote combination method

Fig. 8

Box-plot representation of results for the ensemble construction approaches and the vote combination method on the Nwt criterion

Table 16 represents the results obtained for the Ft criterion when using the sum combination method. The box-plot representations of the results are presented in Fig. 9. The table shows some interesting results for this criterion. Namely, the results depend heavily on the size of the constructed ensemble. For the two smaller ensemble sizes the SEC method was unable to obtain better results on average than those obtained by the best DR. However, for the ensemble size 7 almost all tested ensemble construction methods obtained better results than the best DR. The best construction method seems to be the random selection method in this case.

Table 16 Performance of the ensemble construction methods on the Ft criterion with the sum combination method

Table 17 shows the results obtained when optimising the Ft criterion and using the vote combination method. The box-plots of the results are presented in Fig. 10. Similarly as with the Nwt criterion, all of the results obtained by the ensemble combination methods are better than those obtained by the best DR. The box-plots denote that the different ensemble combination methods achieve results with no significant difference between them. The constructed ensembles were once again better in most cases than the best individual DR.

Table 18 shows the results obtained with the sum combination method when optimising the $C_{max}$ criterion. Figure 11 shows the box-plot representation of the results. Most of the results obtained by the SEC method are not better than those obtained by individual DRs. Only in very few cases the ensembles were actually better on average than the best DR.

Table 19 shows the results obtained when optimising the $C_{max}$ criterion and when using the vote combination method. Figure 12 represent the results in box-plots. All ensemble combination methods obtain better results on average than the individual DRs. The only exception is the instance based method which constantly achieved quite bad results in comparison with the individual DRs. The differences between the different ensemble construction methods are usually small and not significantly different. The box-plots show that for ensemble sizes 5 and 7 the SEC method has once again proven to achieve better results when it is used with the vote combination method. In addition, the box-plots also show that most of the constructed ensembles achieve a better performance than the best individual DR.

Fig. 9

Box-plot representation of results for the ensemble construction approaches and the sum combination method on the Ft criterion

Table 17 Performance of the ensemble construction methods on the Ft criterion with the vote combination method

Fig. 10

Box-plot representation of results for the ensemble construction approaches and the vote combination method on the Ft criterion

Table 18 Performance of the ensemble construction methods on the $C_{max}$ criterion with the sum combination method

Fig. 11

Box-plot representation of results for the ensemble construction approaches and the sum combination method on the $C_{max}$ criterion

Table 19 Performance of the ensemble construction methods on the $C_{max}$ criterion with the vote combination method

Fig. 12

Box-plot representation of results for the ensemble construction approaches and the vote combination method on the $C_{max}$ criterion