# Uncertainty

**The term uncertainty has different definitions and taxonomies in different research fields. Here, we briefly introduce uncertainty in three contexts: predictive modelling, optimization problem and scheduling research.**

## Uncertainty in predictions

In this region of predictive modelling, uncertainty can be defined as the incompleteness in knowledge (either in information or context) that causes model-based predictions to differ from reality in a manner described by some distribution function. There are different possibilities to quantify the uncertainties mathematically.

- The
*deterministic type*defines the crisp possibility of whether a state of an uncertain variable is possible or not. - The
*probabilistic type*defines probability measures describing the likelihood by which a certain event occurs. - The
*fuzzy type*defines fuzzy measures describing the possibility or membership grade by which a certain event can be plausible or believable.

There are numerous taxonomies in literature to address uncertainty classification. Especially in the engineering field, uncertainty is characterized in two types: epistemic uncertainty and aleatory uncertainty [1] [2]. Epistemic uncertainty is the uncertainty that is due to incomplete or inadequate information. Aleatory uncertainty, on the other hand, is used to describe uncertainties within a system or model that have an intrinsic stochastic nature.

Epistemic uncertainties reflect the lack of insufficient knowledge. This kind of uncertainty, in principle, could be reduced by increased efforts, like for example gathering more data or refining models. Epistemic uncertainties include uncertainties about the model used to describe the reality, its boundary and operation conditions, also referred to as model form errors, and also the errors introduced by the numerical solution methods used (e.g., discretization error, approximation error, convergence problems). Such uncertainties can be modeled by type 1 (deterministic type) and 3 (fuzzy type) techniques.

Aleatory uncertainties are from the inherent stochastic nature of the degradation and failure phenomena. That is, these kinds of uncertainties are of physical nature, e.g., the noise in electrical devices, seismic and wind load, humidity, temperature, server load, material parameters (steel yielding strength, stiffness, conductivity), stochastic imperfections in the design of mesostructures of materials, and so on. These uncertainties cannot be removed. The optimization has to “live with them” and to optimize the design according to this reality. Due to the probabilistic nature, probability distributions are the adequate means for the mathematical description of these uncertainties. Thus aleatory uncertainties basically belong to item 2 (probabilistic type) uncertainties.

## Uncertainty in optimization problems

For optimization problems, the terms uncertainty (or noise) refer to behavior in any part of the optimization model that cannot (fully) be predicted or controlled, or that is subject to vagueness. Classifying the uncertainty that arises in optimization problems has been made in literature, e.g., [3], [4]. According to the sources of uncertainties, uncertainty can be distinguished to five types [5]:

- Uncertainties in the design variables
- Uncertainties in the environmental parameters
- Uncertainties in the output
- Vagueness in the constraints
- Preference uncertainty in the objectives

Uncertainty leads to the question about robust design, an optimization approach which tries to account for uncertainties. Robust optimization formulates the optimization problem to search for the robust feasible optimal solution. If a variable is subject to perturbations, a common requirement is that a solution should still work satisfactorily. Such solutions are termed robust solutions. A robust optimal solution is not necessarily an optimum, but there is usually a tradeoff between the quality and robustness of the solution. In [6], search for robust solutions is treated as a three-objective optimization problem, where the fitness of a solution, the mean and the standard deviation of the fitness are used as the objectives. The mean and the standard deviation have been obtained by sampling a number of points in the neighborhood. In [7], fitness and a robustness measure are optimized simultaneously. The robustness measure is defined as the ratio between the standard deviation of the fitness and that of the design variables. The deviation of both the performance and the design variables are calculated using the neighboring solutions in the current population.

## Uncertainty in scheduling research

In the domain of scheduling, the literature that discusses uncertainty mainly focuses on the aspect of uncertainty related to the dynamic nature of scheduling, for instance because machines might fail, unexpected jobs might arrive or because there are changes in processing time.

Understanding scheduling robustness through the explicit consideration of uncertain information is a direction in scheduling research [8]. Dynamic scheduling methods have been developed to achieve good performance by dispatching jobs dynamically to account for random disruptions as they occur. Another line of research focuses on revising the precomputed schedule by a partial or complete updating in the event of disruptions.

For flexible job shop scheduling problems (FJSPs), a drawback of precomputed schedules is that after they are released for execution, continual updating is required due to the random nature of shop floor conditions. Uncertainty in FJSPs usually refers to dynamic elements that are to be taken into account. It is therefore desirable to generate schedules that are robust within a reasonable range of disruptions such as machine breakdowns and processing time variability. According to the literature in robust scheduling methodologies, robustness is mainly grouped into quality robustness and solution robustness [9]. The quality robustness refers to the insensitivity of the scheduling performance such as makespan and total tardiness in the presence of uncertainty. When there is a machine breakdown, actual scheduling may be shifted away from the baseline schedule. The property that the start and completion of each job should be as close as possible to its baseline schedule is known as the solution robustness and is usually considered as a “stability” measurement of the schedule.

Numerous studies have been conducted in real production environments where scheduling is often influenced by uncertain or stochastic factors. [10] addresses the problem of finding robust and stable solutions for the flexible job shop scheduling problem with random machine breakdowns. Several measures of stability are suggested and linearly combined (using weighted sum) with the primary objective function (makespan) to form a bi-objective function to guide the genetic algorithm search procedure. The breakdown machine, the breakdown time, and the breakdown duration are generated using the uniform distributions. [11] and [12] solve similar dynamic problems as well.

## References

[1] M Elisabeth Paté-Cornell. Uncertainties in risk analysis: Six levels of treatment. Reliability Engineering & System Safety, 54(2-3):95–111, 1996.

[2] Armen Der Kiureghian and Ove Ditlevsen. Aleatory or epistemic? does it matter? Structural Safety, 31(2):105–112, 2009.

[3] Hans-Georg Beyer and Bernhard Sendhoff. Robust optimization–a comprehensive survey. Computer methods in applied mechanics and engineering, 196(33-34):3190–3218, 2007.

[4] Yaochu Jin and Jürgen Branke. Evolutionary optimization in uncertain environments-a survey. IEEE Transactions on evolutionary computation, 9(3):303–317, 2005.

[5] Johannes Willem Kruisselbrink. Evolution strategies for robust optimization. Leiden Institute of Advanced Computer Science (LIACS), 2012.

[6] Tapabrata Ray. Constrained robust optimal design using a multiobjective evolutionary algorithm. In wcci, pages 419–424. IEEE, 2002.

[7] Yaochu Jin and Bernhard Sendhoff. Trade-off between performance and robustness: an evolutionary multiobjective approach. In international conference on Evolutionary Multi-Criterion Optimization, pages 237–251. Springer, 2003.

[8] Stephen C Graves. A review of production scheduling. Operations research, 29(4):646–675, 1981.

[9] Willy Herroelen and Roel Leus. Project scheduling under uncertainty: Survey and research potentials. European journal of operational research, 165(2):289–306, 2005.

[10] Nasr Al-Hinai and Tarek Y ElMekkawy. Robust and stable flexible job shop scheduling with ran- dom machine breakdowns using a hybrid genetic algorithm. International Journal of Production Economics, 132(2):279–291, 2011.

[11] Ehsan Ahmadi, Mostafa Zandieh, Mojtaba Farrokh, and Seyed Mohammad Emami. A multi objec- tive optimization approach for flexible job shop scheduling problem under random machine break- down by evolutionary algorithms. Computers & Operations Research, 73:56–66, 2016.

[12] Jian Xiong, Li-ning Xing, and Ying-wu Chen. Robust scheduling for multi-objective flexible job- shop problems with random machine breakdowns. International Journal of Production Economics, 141(1):112–126, 2013.