Literature study
2.1 Multi-objective optimisation
In general, a multi-objective optimisation problem is formulated as follows, without loss of generality:
Minimise f(x) = [f1(x), f2(x), ..., fk(x)]T (2.1) Subject to
gj(x) ≤ 0, j = 1, 2, ..., Ng, (2.2) hi(x) = 0, i = 1, 2, ..., Nh (2.3) where k is the number of conflicting objective functions, Ng is the number of in-equality constraints, and Nh is the number of equality constraints. x ∈ X is a vector of decision variables and X is the feasible decision or solution space formally defined as {x | gj(x) ≤ 0, j = 1, 2, ...Ng and hi(x) = 0, i = 1, 2, ...Nh}. Similarly, f (x) ∈ Y is a vector of objective functions and Y is the feasible objective space formally defined as {f (x) | x ∈ X}. For each element in X, there exists an equivalent element in Y (Deb, 2005).
Though (2.1) says “Minimise (or Maximise)” f (x), not all components of f (x) fol-low, necessarily, the same optimisation direction. In effect, the example presented in Section1.1, showed that the performance and cost objectives had opposite optimisation directions (i.e. performance was maximised while cost was minimised). Nonetheless, it is possible through the duality principle (Deb,2005), to use the same optimisation direction for all the objectives in f (x). According to this principle, if one desires to solve, say, the example in Section 1.1 by using a technique that uses a minimisation approach, one must multiply the performance objective by −1. The objective must then, of course, be converted back to its original form once the problem is solved.
Multi-objective optimisation problems as described here have more than one optimal solution. These are often referred to as Pareto optimal solutions. The reason why this is the case is due to the existing conflict between the objectives, causing the candidate solutions (i.e. the decision vectors) to “score” unevenly on the different objectives.
It becomes, therefore, difficult to declare one single solution as the ultimate best (see Section1.1) but rather a set, the Pareto optimal set. The set of Pareto optimal solutions (or Pareto set for short), consequently, consists of all decision vectors for which the corresponding objective vectors cannot be improved in a given dimension (i.e. objective function) without worsening another. In other words, they form a set of trade-offs (Chankong & Haimes,1983).
Throughout this study, the terms system design (or simply design) as well as sce-nario will be used interchangeably in addition to decision vector and solution, to refer to x.
Following are definitions fromCoello Coello(2009) that formally describe the Pareto optimality (minimisation) concept in a deterministic context:
Definition 2.1: Given two vectors u = (u1, u2, ..., uk)T, v = (v1, v2, ..., vk)T ∈ Y it is said that u ≤ v if ui≤ vi for i = 1, 2, ..., k, and that u < v if u ≤ v and u 6= v.
Definition 2.2: Given two vectors u, v ∈ Y , it is said that u dominates v (denoted by u ≺ v) if u < v.
Definition 2.3: It is said that a vector of decision variables x∗ ∈ X is Pareto optimal if there does not exist another x ∈ X such that f (x) ≺ f (x∗).
Definition 2.4: The Pareto optimal set Sp is defined by: Sp = {x ∈ X | x = x∗}.
Definition 2.5: The Pareto front Spf , which is the set of all Pareto optimal solutions’ equivalents in the objective space, is defined by: Spf = {f (x) ∈ Y | x ∈ Sp}.
The decision vectors in Sp are called non-dominated and there is no x in X such that f (x) dominates f (x∗). The dominance concept it illustrated in Figure 2.1, where the red solutions are considered to be non-dominated and the blue ones dominated.
The red solutions form, therefore, the Pareto front.
Obj. 2
Figure 2.1: An example of Pareto optimal solutions for two minimised objectives.
The goal, when solving a MOO problem, is therefore to obtain for (2.1) the Pareto optimal set Spby identifying in X all the decision vectors x∗that satisfy the constraints
(2.2) and (2.3), if they exist.
Goldberg(1989) developed a Pareto ranking algorithm that finds Sp with respect to a user-specified threshold th, when given a set of N decision vectors xi (i = 1, 2, ..., N ) and their respective f (x) values. th is an integer value that allows the algorithm to include in Sp, x /∈ Sp that are dominated by, at most, th x(s) (x ∈ Sp). Now consider W, a matrix with N rows and n + m + 1 columns, where n is the number of decision variables in x and m is the number of objective functions (m > 1). Goldberg (1989)’s algorithm, thus, is as presented in Algorithm1.
Algorithm 1 Pareto ranking algorithm (minimisation)
1: Input: W and th.
2: Set j = n + 1.
3: Sort the working matrix W with the values in column j in descending order.
4: Set rp = 1. not exceeding th as the non-dominated members of Sp.
The reality in practice, however, is that Sp can only be approximated as in many cases it is hard to know with certainty whether the true set was obtained. In effect, many real-world problems are such that X is very large and cannot be fully explored practically. Moreover, the problems are often subject to stochastic elements, meaning that the true values of f (x) ∈ Y can only be estimated.
Although this work focuses on methods for approximating the entire Pareto set, it is important to state that in some cases this may not be necessary. There exist situations in practice where the decision-maker does already have particular preferences for some objectives over others prior to the problem being solved. For example, a decision-maker in the example considered in Section1.1, may desire a solution whereby performance maximisation is given more importance or more “weight” relative to cost
minimisation. While the Pareto set would, in principle, have such a solution as one of the trade-offs, computational effort could be reduced significantly by focusing solely on finding the unique solution that matches the “preference” of the decision-maker via an appropriate method. Literature is not short of methods for solving MOO problems in this way. In particular, these methods are generally classified into two main groups often referred to as Scalarisation and Constraint methods. The interested reader can refer toMarler & Arora(2004), where a comprehensive survey on different methods for solving multi-objective optimisation problems is presented. Nevertheless, according to Li et al. (2015), it is not always easy to assign fair weights to various objectives, that truly reflect the decision-maker’s bias. Moreover, the complexity of some problems may not allow these methods to work correctly (more detail about this will be provided in Chapter3). So though these methods may be effective in certain cases, using techniques that attempt to find the entire Pareto set is ultimately the ideal approach. In this study, the author refers to such techniques as Pareto approach methods/techniques or MOO methods/techniques that use the Pareto approach. This is done to distinguish them from MOO methods that focus on finding single optimal solutions.
So far in this chapter, most of the discussion has been limited to the deterministic context. This is a context whereby it is assumed that there is no random, or stochastic element affecting the correct analysis of a problem. In the next section, simulation optimisation is introduced. The simulation optimisation field is concerned with meth-ods for solving stochastic optimisation problems using simulation (i.e. discrete-event simulation, for the purpose of this study).
The simulation optimisation field is vast and has been researched very actively over many years. The oldest contribution towards the SO field in this literature study dates back as far as the year 1954, while the newest contribution is from 2018. It is in this particular field that some of the most significant advances in solution approaches for real-world optimisation problems are being developed.