**1. Introduction**

While conducting their activities, manufacturing enterprises establish a range of various goals. Certainly, one of the common strategic business objectives is to strengthen the market position. An enterprise that aims to broaden the group of clients, as well as foster the already existing business relations, must first and foremost be reliable and deliver quality goods within contractual deadlines [1,2]. Therefore, proper planning of works becomes central to sound execution of production processes. Production scheduling is the solution that can boost the capacity of manufacturers, hence there are numerous scientific publications in the field [3]. Researchers are still taking active efforts to optimise the effectiveness of production jobs scheduling in order to streamline the production planning process [2,4].

Unfortunately, most of the proposed solutions display numerous limitations [5]. It is common practice that the job scheduling algorithms build schedules for idealized production environments, i.e., assuming a static and stable production flow [5,6]. Thereby, a number of disruptive factors are excluded, which would bring the production to a halt in case they occur [7,8]. As a consequence, contractual deadlines would be missed, penalties would be imposed and the manufacturer's credibility would diminish. Among the various uncertainty factors, we can highlight the following [5,9]:

• disruptions of resource availability (machine or robot failure)


Scheduling production in real manufacturing systems cannot afford to pretend to be disruption-free. It is, therefore, of the essence that scheduling endeavors should consider production problems under uncertainty, which is capable of having a colossal effect on the timeliness of production [10,11]. Given that the more the process changes, the greater its disorganization, the scientific literature in the field of scheduling has recently turned towards robust scheduling [5,9].

The predictive scheduling method proposed in this work employs Markov chains and ARIMA (auto-regressive integrated moving average) models whose combination enables determining the values of the machine failure parameters (time to failure and repair time of the machine). In the next step time buffers are directly integrated into the scheduling process and determine the completion time of the production, which corresponds to the delivery date agreed with the customer.

Section 2 summarizes the essential information regarding robust production scheduling and reviews existing literature. The new methodology for scheduling under machine failure and failure prediction is described in Section 3, and the proposed solutions and results are discussed in the subsequent section. Conclusions and plans for further research work are presented in the last section of the work.

#### **2. Existing Work on Robust Production Scheduling**

#### *2.1. Essentials of Robust Scheduling*

The purpose of a robust production schedule is predominantly to absorb potential disruptions, by allowing variability to the production system parameters.

Two phases of scheduling are distinguished [9,12]:


A well-executed process of scheduling production jobs must pertain to the first of the phases (also referred to as the offline phase) when the available production data give the foundation for creating [3]:


Reasonable scheduling in this phase requires not only implementing appropriate tools but also suitable methods for determining uncertainty factors [6]. Unfortunately, there is a distinct paucity of solutions that consider the impact of process disruptions [5,6].

#### *2.2. Existing Literature on Robust Production Scheduling*

Due to the practical nature of the problem, robust production scheduling solutions are mainly developed for flow-shop and job-shop systems, which are the prevailing forms of organization in real production systems.

Although robust task scheduling in a flow-shop environment is rather neglected in the literature, a certain number of publications on this issue can be found [13–15]. Various approaches have been applied to building robust schedules in a flow-shop environment—from classical local search algorithms [16] to genetic algorithms [17], integer programming applications [18] and dynamic programming [19]. However, in an overwhelming majority, the publications are concerned with

predictive-reactive scheduling, thus, tend to focus on the investigation of effective re-scheduling methods [19], and not on the analysis of uncertainty factors itself.

The second most-investigated scheduling problem is robust production scheduling in job-shop systems [20]. This system is a close reflection of a typical production environment, where the order of operations is imposed by the technological routes of their jobs. Researchers have long highlighted that the real job-shop problem requires a distinctly different approach than the shown by the prevailing theoretical tendencies [21], however, to date no clear trend has emerged. Robust scheduling solutions proposed in the aspects of job-shop production processes resemble the solutions for flow-shop systems inasmuch as they are mainly dedicated to the predictive-reactive approach. Standard approaches are shown to draw from various methods, such as genetic algorithms and their hybrids [20,22], immunological algorithms [23,24] and stochastic programming [25]. Other authors propose robust scheduling methods using expert systems [26].

### *2.3. Machine Failure as the Major Uncertainty Factor*

Although many uncertainty factors can be named, the failure of technological machinery is still considered to be the central problem in manufacturing. This disturbance is regarded to have the greatest impact on performed processes. Failure will not only halt the production but its consequences will linger throughout the remaining production process [5,7].

From the analyzed scientific papers dealing with the topic of machine failure in scheduling, it can be seen that the search for methods that will enable approaching the problem of machine failure and predicting its occurrence are very much in place. Developing effective prediction methods is extremely important from the perspective of robust scheduling [18,19,26].

To this end, a probability distribution is among the most widely used approaches in the field of failure analysis. Researchers employ typical distributions and their combinations. The failure description proposed by Jensen [27] applies a uniform distribution. A similar solution is proposed by Al-Hinai and ElMekkawy [28], who, however, assume that the probability of failure is constant. In contrast, in their description of production process disturbances, Davenport et al. [29] implement a normal distribution, while Mehta and Uzsoy [21] utilize an exponential distribution. The authors propose the use of interesting approaches, such as the methods based on combinations of various distributions. The latter is used by Gürel et al. [4], who combine normal, triangular and exponential distributions.

Recently, researchers have also investigated the application of typical key performance indicators (KPIs) used in maintenance, e.g., MTTF (mean time to failure), MTBF (mean time between failures) and MTTR (mean time to repair). In their direct application of the indicators, Deepu [9] and Gao [5] analyze specially prepared scenarios that assume a certain frequency of machine failure, i.e., high, medium or low, to study the consequences of machine downtime and propose solutions to absorb the emerging disruptions to the schedule. With respect to the indirect use of the indicators, Kempa et al. [30,31] propose the use of the aforementioned reliability indicators indirectly for the purpose of estimating Weibull distribution parameters, while Rosmaini and Shahrul [32] in their study, couple the said indicators with statistical methods. These studies, however, suffer from the major drawback—the acquisition and use of the respective quantities is treated quite theoretically and lacks practical verification on real data of machine failure rates [9,31].

In addition to the methods referenced in the preceding paragraphs, a range of alternative failure prediction methods can be found in the literature. Jian et al. [20] propose accumulating failures to a single occurrence, describing it by means of the MTTR parameter and their original indicator, MBL (machine breakdown level). In turn, Rawat and Lad [33] determine failure rates from the analysis of machine load time distributions, and Baptista et al. in [34] use artificial neural networks for failure analysis.

Although constituting an interesting and important voice in the robust production scheduling studies, these models are associated with certain limitations. Their verification is often carried out on test data, which may not be the most accurate representation of actual problems in manufacturing systems. Secondly, the questions arise as to insufficient argumentation regarding the selection of the solutions. Consequently, the key aspect of implementing historical data in studies of the failure rate of machines is omitted.

The need to use real data on uncertainty factors is also emphasized by Davenport et al. [29] and Kalinowski et al. [35]. Only real knowledge on process disruptions can actually solve actual the problems that result from their occurrence. The issue was addressed in our previous work [36], where a model for the prediction of technological operation times in the framework of an intelligent job scheduling system was conceptualized. The study in question considered the impact of real processing time uncertainty on the production schedule and the developed intelligent module also implemented ARMA/ARIMA time series models, however, a problem of a different size was concerned and the verification was carried out for different production data. While such solutions can be found in the literature, the body of knowledge in the field still appears to lack proper depth [6].
