# Linking Scheduling Criteria to Shop Floor Performance in Permutation Flowships
## 3 Computational Experience
### Testbed Setting  
Although, in principle, a possible option to obtain flowshop instances to perform our research may be to extract these data from real-life settings, this option poses a number of difficulties. First, obtaining such data is a representative number is complicated. There are only few references publishing real data in the literature (see [45; 46]). It may be thus required to obtain such data from primary sources, which may be a research project itself. Second, processing time data are highly industry-dependent, and it is likely that a sector-by-sector analysis would be required, which in turn makes the analysis even more complicate and increases the need of obtaining additional data. Finally, extracting these data from industry would make processing times to be external (independent) variables in the analysis. Therefore, we generate these data according to test-bed generation methods available in the literature. For the flowshop layout in our research, this means establishing the problem size (number of jobs and machines) and processing times of each job on each machine. With respect to the values of the number of jobs \(n\) and \(m\) machines, we have chosen the following: \(n\in\{20,50,100,200\}\), and \(m\in\{10,20,50\}\). For each problem size, 30 instances have been generated. This number has been chosen so that the results have a relatively high statistical significance. Regarding the generation of the processing times, methods for generating processing times can be classified in random and correlated. In random methods, processing times are assumed to be independent from the jobs and the machines, i.e., they are generated by sampling them from a random interval using a uniform distribution [a, b]. The most usual values for this interval are [1; 99] (see, e.g., in [47; 48]), while in some other cases even wider intervals are employed (e.g., [49] uses [1,200]). Random methods intend to produce difficult problem instances, as it is known that, at least with respect to certain scheduling criteria, this generation method yields the most difficult problems [50; 51]. As foreseeable, random processing times are not found in practice [52]. Instead of random processing times, in real-life manufacturing environments it is encountered a mixture of job-correlation and machine-correlation for the processing times, as some surveys suggest (e.g., [53]). To model this correlation, several methods have been proposed, such as those of [54; 55; 56], or [57]. Among these, the latest method synthesises the others. This method allows obtaining problem instances with mixed correlation between jobs and machines. The amplitude of the interval from which the distribution means of the processing times are uniformly sampled depends on a parameter \(\alpha\in[0,1]\). For low values of \(\alpha\), differences among the processing times in the machines are small, while the opposite occurs for large values of \(\alpha\). For a detailed description of the implementation, the reader is referred to [57]. Finally, it is to note that several works claim the Erlang distribution to better capture the distribution of processing times (e.g., [4; 19], or [58]), yet these do not specify whether this has been confirmed in real-life settings. Therefore, we discard this approach. In [57], the processing times for each job on each machine \(p_{ij}\) are generated according to the following steps. 1. Set the upper and lower bounds of processing times, \(Dur_{LB}\) and \(Dur_{UB}\), respectively, and a factor \(\alpha\) controlling the correlation of the processing times. 2. Obtain the value \(Interval_{st}\) by drawing a uniform sample from the interval \([Dur_{LB},Dur_{UB}+Width_{eff}]\), where \(Width_{eff}=rint(\alpha\cdot(Dur_{UB}-Dur_{LB}))\). 3. For each machine \(j\), obtain \(D_{j}=[d_{j}^{lb},d_{j}^{ub}]=[\mu_{j}-d_{j}^{hw},\mu_{j}+d_{j}^{hw}]\), where \(\mu_{j}\) is sampled from the interval \([Interval_{st},Interval_{st}+Width_{eff}]\) and \(d_{j}^{hw}\) is uniformly sampled from the interval \([1,5]\). 4.