# Minimum-Energy Scheduling of Flexible Job-Shop Through Optimization and Comprehensive Heuristic
## 1 Introduction
### Literature Review  
Before introducing the minimum-energy flexible job-shop scheduling problem with the due times considered in this study, an overview of previous instances of job-shop scheduling problems is presented here. The flexible job-shop scheduling problem, at first considered as a mere extension of the job-shop scheduling problem, where it is alsonecessary to assign machines to the operations of different jobs, has turned out to be, by its added complexity and its adequacy with current industrial practice (Industry 4.0), \"a job shop scheduling problem of its own\" [7]. In recent years, different reviews covering assumptions, models, and solution techniques for the flexible job-shop scheduling problem have been published [7; 8; 9; 10; 11]. Most of the formulations adopted have resulted in Mixed-Integer Linear Programming (MILP) optimization problems with sizes and complexity that often limit the use of exact methods. This has led to the proliferation of heuristics and meta-heuristics, notably nature-inspired ones, which proved rather effective [6; 12]. In many studies, the exercise carried out when solving the scheduling problem boils down to delivering a mathematical solution without considering the conditions of its implementation. However, some authors have considered the real-life issues of their implementation by integrating them into the formulation of the scheduling problem to have more effective scheduling: the maintenance of machine condition [13], the availability of machines during production [14], and processing time uncertainties [15] are some of the covered topics. The real environment of production systems is dynamic and is subject to unexpected events such as sudden machine breakdowns, power outages, shortages of materials, the unavailability of operators, and unusual activity durations. Dynamic job-shop scheduling techniques were developed to cope with this reality [16; 17]. These techniques are based on three different scheduling policies: predictive, reactive, and predictive-reactive [18]. Predictive approaches build a schedule that is maintained during execution, but when disturbances may appear, the maintenance of its feasibility has a time cost (slack times incorporated into the duration of the operations to absorb delays) and an equipment cost (spare machines to face breakdowns). The scheduling is recomputed with purely reactive approaches after each notable disturbance. Predictive-reactive approaches are a mix of the two previous approaches, where a pre-existent schedule is adapted online during its execution. These techniques increasingly involve tools from artificial intelligence (AI) [19]. When considering energy consumption in a job shop, it is composed of the processing energy consumed by machines; of the transfer energy of raw, semi-finished, or finished products within and around the job shop; of the idle energy consumed by machines during the time intervals between consecutive operations; of the set-up energy of machines to enable the processing of operations; and of the common energy consumed for maintaining acceptable operating conditions in the workshop (lighting, air conditioning, and heating). In flexible job shops, the energy-saving opportunities are not limited to those of job shops, i.e., turning off idle machines, slowing down machine speed, and producing during off-peak periods [20]; they also include the allocation of operations to more energy-efficient machines and the choice of energy-efficient means of transport between machines. This newer and more specialized area of the flexible job shop has been the subject of far fewer survey articles [4; 21], even though articles on this subject are published regularly. In these studies, the classical concern has been about minimizing the make-span shifts toward the minimization of energy.