## 5 Heuristic Approaches
### Dynamic Island Model
#### 5.2.1 Principle
We gratefully acknowledge and express our appreciation to the Editor and the anonymous reviewers who provided feedback. The authors declare no conflicts of interest. The following abbreviations are used in this manuscript:

OAS Order Acceptance Scheduling

TOU Time-of-Use

FR Fix-and-Relax

DIM Dynamic Island Model

GA Genetic Algorithm

EA Evolutionary Algorithm

GHG Greenhouse Gas

MILP Mixed Integer Linear Program

ATI Arc-Time indexed

TI Time-Indexed

TEC Total Energy Cost Consumption

FW Frozen Window

AW Approximation Window

OW Observation Window

BKS Best Known Solution

## Appendix A Experimental Design

The Taguchi method is a fractional factorial design of experiments developed by Dr. [PERSON] in the late 1940s [58]. It limits the number of parameter settings to be tested to estimate the optimal one, when testing all of them would take a tremendous amount of time. To put things into perspective, when five parameters taking two values, a brute-force approach would consist in testing \(2^{5}\) settings on a sample of 15 instances. A high estimate of 3600 s per instance gives a total of 1.7 million seconds to carry out the tests, which is not reasonable. To avoid such thing, [PERSON] designs orthogonal arrays, which consider a reduced amount of settings to run. The results of each experiment are converted to a Signal-to-Noise (S/N) ratio in order to estimate the effects of control factors on the data mean and variation. The S/N ratio response table provides the best tuning for the process. Using Minitab 19 [59], the procedure involves the following steps:

1. Identify factors
2. Characterize levels of factors
3. Select an orthogonal array
4. Run experiments and collect the response data
5. Analyze the experimental data
6. Identify the optimal levels of factors
7. Validate experiment

First, control factors with their range of values are selected. A large catalog of designs are available, as required. For instance, with 4 factors and 2 levels by factor, the proposed design referrers as 24 or \(L_{8}\), which means a \(P=8\) runs design. The resulting orthogonal array must be judiciously selected, particularly if factors are dependent. In step 3, the default orthogonal array given by Minitab is chosen. During step 4, experiments are carried out with the proper settings described by the orthogonal array. In each experiment, objective values are collected. As the problem aims at determining factor levels that result in the largest means, the S/N ratio of each experiment \(e=1,\ldots,P\) is calculated with the _Larger-is-Better_ formula as follows, with \(y_{i}\) the \(i\)th objective value and \(M\) the sample size:

\[\text{S/N}(e)=-10\log\frac{1}{M}\sum_{i=1}^{M}\frac{1}{y_{i}^{2}}\qquad e=1, \ldots,P \tag{1}\]

Adjustment have been made to the collected response values in order to properly use the formula described in (1) and remove statistical biases. First, response data have been smoothed using min-max normalization. Second, as the S/N function takes only strictly positive values, normalized values have been scaled into the interval \([1,2]\). ### Fr Heuristics

For the tests, two values for the setup times are used, as shown in Table 1.

The FR heuristic depends on parameters such as the decision window size \(\sigma\) and the number of overlapping periods \(\delta\). The impact of these parameters is studied with \(\sigma=2p_{\text{max}}\) and \(\sigma=3p_{\text{max}}\) (\(p_{\text{max}}=\max\limits_{j=1,\ldots,n}p_{j}\)) and two values of \(\delta\) representing 25% and 50% of the length of the decision window \(\sigma\). For the FR TI Pulse and the TI On/Off formulations, 3 factors and 2 levels are chosen in the design of experiment. Table 2 presents factors A, B and C and their corresponding description and levels (set of possible values).