Simulation optimisation

The term simulation optimisation is an umbrella term for techniques used to optimise stochastic simulation problems (Amaran et al., 2014) or simply SO problems. The

term SO problem is used here to refer to optimisation problems solved with computer simulation for reasons mentioned in Section1.1.

In their work, Fu et al. (2000) distinguished between two kinds of approaches for solving SO problems: one where a constraint set (possibly unbounded and uncountable) is provided, over which an algorithm seeks improved solutions, and another where a fixed set of alternatives is provided a priori and the so called ranking and selection (R&S) procedures are used to determine the best alternative. According to Fu et al.

(2000), the focus in the first approach is on the searching mechanism, whereas in the second approach, statistical considerations are paramount.

In a similar way, Yoon & Bekker (2017) have also distinguished between SO prob-lems based on their solution space size which, in the words of the researchers, determines the fundamental approaches needed to solve them. They have categorised, on one hand, SO problems with a relatively small solution space (small-scale SO problems) for which R&S procedures are sufficient to find the best solutions and, on the other hand, SO problems with a large solution space (large-scale SO problems) for which intelligent search mechanisms, with or without the partnership of R&S procedures, are needed for seeking the optimal or near-optimal solutions.

Both researches are in agreement regarding how to approach SO problems. It is clear that the size of the solution space matters.

2.2.1 Decision variables and solution space size

Given that potential solutions to an SO problem are not definitive nor known in ad-vance, it is important to study the size of the solution space of the problem at hand in order to solve it accordingly. The size of a solution space can be determined by the nature of the decision variables of interest; that is, whether the decision variables are discrete, continuous or mixed; as well as by the boundaries over which the decision variable values are allowed to be selected. Decision variables that can be defined in this manner are often referred to as quantitative decision variables. Besides them, an-other type also exists that is sometimes referred to as categorical or qualitative (Law

& Kelton,2000) (see for example the problem in Section3.1).

SO problems with qualitative decision variables are generally small in scale (i.e.

the size of their solution space is generally small). SO problems with quantitative decision variables, on the other hand, can be either small or large in scale. When the

potential solutions to be evaluated are known in advance and no searching mechanism is needed, then the problem can again be treated as a small-scale problem, despite having quantitative decision variables. When none of the previous applies, then the problem should be treated as a large-scale problem if “actual” optimality is to be attained or approximated.

Simulation problems (i.e. stochastic problems solved with discrete-event simula-tion) are generally treated as small-scale problems in simulation studies. Optimisation in this case is reduced to the identification or selection of the best solution(s) out of all potential solutions being considered. But unless such problems are truly small-scale problems, then the solutions found are not “truly” optimal. In effect, when a problem that should be treated as a large-scale problem is reduced to a small-scale one, the approach being taken for the problem is fundamentally wrong. Hence, large-scale and small-scale problems must be differentiated and solved accordingly.

2.2.2 Solution approaches for SO problems in the literature

It is important to mention that in a large portion of the literature on SO, those specif-ically on large-scale SO, there is a clear separation between solution approaches (or algorithms) used when decision variables are continuous and when they are discrete.

In other words, after the size of the space has been determined as being large, it is the nature of the decision variables that dictates which approach (i.e. search mechanism) is to be used to solve the problem.

Hong & Nelson (2009a) actually divide SO problems into three categories rather than simply two because of this, with each category requiring distinctive solution ap-proaches. In the first category, the solution space has a small number of solutions (often less than 100, according to the researchers) and the decision variables are numerical or categorical. (This category is identical to the small-scale SO category described earlier.) In the second and third categories, the solution space is large. In the second category in particular, decision variables are exclusively continuous. Problems in this category are also referred to as continuous optimisation via simulation (COvS) prob-lems. (Optimisation via simulation (OvS) is another term for simulation optimisation in the literature.) Finally, in the third category, decision variables are exclusively dis-crete. Problems in this category are also known as discrete optimisation via simulation (DOvS) problems. As mentioned already, for each of these categories, the researchers

present in their work a number of solution approaches that are distinctively different from each other (some of them will be discussed in Section2.5.2). An earlier and sim-ilar work in the literature byAndradottir(1998) also presents a review of methods for solving SO problems by distinguishing them as done byHong & Nelson (2009a).

In this study, however, the author is interested in a class of search mechanisms that is not limited by the nature of decision variables. In other words, the algorithms in this class can be used for both discrete and continuous large-scale problems without the need to be distinctively different for each case. The reason for this choice will be made known as the study progresses. The earlier distinction of SO problems as simply being small or large (in solution space scale) in order to determine the solution approaches to be used to solve them is therefore, somewhat, justified for the purpose of this work.

Before discussing small-scale and large-scale SO problems further, SO problems with multi-objectives are first introduced and discussed next.