#### 3.1. The Application of Multi-Criteria Decision-Making Methods in Developing Modern Portfolio Theory

Project portfolio selection, which involves computing the proportion of the initial financial resources to be allocated among the existing projects, is essential in the field of project portfolio management [

28,

39]. A fundamental answer to this problem was given by Markowitz [

38], who proposed the mean-variance model laying the basis of modern portfolio theory [

40,

41,

42,

43]. Markowitz defined the problem as a trade-off between the expected return and the expected risk of a portfolio [

44].

Although mean-variance has been the prevalent model in portfolio selection for over sixty years, there have been attempts to extend its scope [

42]. The initial Markowitz’s [

38] mean-variance model was in principle elaborated in three directions [

41,

45,

46] (see

Figure 1).

This paper will not address the first two directions and will only focus on the third development direction, even though it is interwoven with the second one, namely the introduction of alternative measures of risk. Lately, more and more authors come to realize the benefit of incorporating an additional criterion and/or constraint in project selection [

41,

42,

43,

44,

45,

47]. The analysis of this direction demonstrates mean-risk-third parameter models [

41,

42,

46,

48,

49,

50,

51,

52,

53,

54]. In addition to the mean and risk, such models also offer a third parameter, that of the project value (e.g., liquidity [

55], investment portfolio size [

50], transaction costs [

49,

56] and so on). However, most recently, the additional criterion that seems to be receiving the most consideration is social responsibility [

42,

57]. Over the past few years numerous papers on social responsibility in portfolio selection have been published [

51,

52,

54,

58,

59,

60,

61,

62,

63,

64,

65,

66,

67,

68,

69]. One of the most recent works on this subject comes from Utz

et al. [

52] who extended the Markowitz model by complementing it with a social responsibility objective, in addition to the portfolio return and variance, thereby making the traditional efficient frontier a surface. It is noteworthy that the terms “social responsibility” and “sustainability” are often used interchangeably.

Thus, with new conflicting criteria being added to the traditional conflict between risk and return, portfolio selection and resource allocation have become even more complex. In this context, the application of multi-criteria decision-making methods is very useful. MCDM methods can be used to support decision-makers in circumstances where multiple conflicting decision criteria or alternatives have to be considered simultaneously [

63]. In other words, MCDM techniques intend to find compromises in the decision-making process [

70].

MCDM models and methods can be divided into two groups [

71,

72,

73,

74]:

Multi-Objective Decision-Making (MODM) methods: An optimization problem is solved with an objective function while evaluating certain constraints. These methods are directly applied to solve a portfolio optimization problem.

Multiple-Attribute Decision-Making (MADM) methods: Decision-making is intended for discrete comparison of alternatives. In scholarly works, methods falling within this group are usually used to rate assets to be included in a portfolio.

MCDM methods are widely used in solving portfolio selection and resource allocation problems. When classifying scholarly articles based on the field of application of MCDM methods (between 2002 and 2014), Zopounidis

et al. [

75] established that 40 percent of scholarly articles addressed portfolio selection. They also established that for the purpose of portfolio selection multi-objective optimization was used in most cases (72 percent). Steuer and Na [

76] also considered portfolio analysis to be the prevailing research field in the period before 2002. The popularity of portfolio optimization can be explained by a number of reasons. It is a multifaceted problem that presents a number of algorithmic and modeling challenges (e.g., dynamic nature, data of various types, risk modeling,

etc.). Most multi-objective optimization and goal programming models used for the purpose of portfolio optimization are based on the combination of multiple risk measures (e.g., skewness/kurtosis, value-at-risk measures,

etc.), usually with further consideration of some additional goals and objectives (e.g., liquidity, dividends,

etc.). For that matter, evolutionary algorithms (EA) and metaheuristics (MH) have also been very popular, especially with regard to non-convex portfolio selection criteria and models or when additional real features, such as cardinality constraints, are used in the analysis [

75,

77].

Given that multi-objective optimization is most commonly used for solving a portfolio selection problem [

75], a short overview of multi-objective optimization is presented below. Multi-objective optimization (also known as multi-objective programming, vector optimization, multicriteria optimization, multiattribute optimization or Pareto optimization) is an area of dealing with a mathematical optimization problem where more than one objective function has to be optimized simultaneously. In the case of multi-objective optimization with conflicting objective functions, a set of Pareto optimal solutions is derived rather than one optimal solution because none of these solutions can be regarded as being better than any other with respect to all objective functions.

A multi-objective optimization problem can be defined as follows [

41]:

Subject to:

where

$x=\left({x}_{1},\dots ,{x}_{n}\right)$ is the vector of decision variables and

X is the set of feasible solutions. The objective function vector

$F\left(x\right)$ which contains the values of

k objectives maps the feasible set

X into the set

F which presents all possible values of the objective function. The objective functions may all be maximized, minimized or be in a mixed form. The usual process in multi-objective optimization is to find all non-dominated or Pareto optimal solutions of the problem.

Since all the Pareto optimal solutions are equally good from a mathematical point of view, they can be regarded as equally valid compromise solutions of the problem. Thus, there is no trivial mathematical tool in order to find the best solution in the Pareto optimal set because vectors cannot be ordered completely. For this reason some additional information is required in the decision-making process. Decision-making can take place before or during the multi-objective optimization process. Usually, a human decision-maker is necessary to make tough trade-offs between conflicting objectives as the decision-maker is considered to have a better insight into the problem and can therefore express preference relations between different solutions. Thus, for example, it can be useful for the decision-maker to know the ranges of the objective function values (ideal and nadir points) in the Pareto optimal set. The decision-maker can participate in the solution process, and, in some way, determine which of the obtained Pareto optimal solutions is the most suitable as the final solution.

Depending on the participation of the decision-maker in the problem-solving process, methods are classified as follows [

78,

79,

80].

In no-preference methods, no input from the decision-maker concerning the importance of criteria is used, which results in one Pareto optimal solution. When using posteriori methods (also called Methods for Generating Pareto Optimal Solutions), a set of Pareto optimal solutions is derived, which is presented to the decision-maker who then chooses the most satisfying solution. In priori methods, the decision-maker has to indicate his priorities and objectives before the solution process [

81]. If the obtained solution satisfies the decision-maker’s requirements, the decision-maker does not have to spend much time for the solution process. However, the problem is that the decision-maker does not necessarily know before the solution process what outcome he could expect and how viable his goals are. As a consequence, when the decision-maker is not satisfied with the solution, he has to change priorities and adjust his goals. Thus, it is very important to involve all stakeholders in decision-making from the outset so as to reach consensus on priorities and targets and thereby avoid their frequent adjustments [

81]. In interactive methods, the decision-maker takes active part in the problem-solving process. The specificity of these methods is that, owing to the complexity of the problem concerned, the decision-maker cannot properly identify priorities and goals before the solution process. However, the decision-maker can adjust his priorities and goals during the solution process as the solution process is interactive. Interactive methods are most time-consuming for the decision-maker, as compared to the other methods, yet, they allow solving complex problems with multiple criteria and constraints [

82].

The above-given concise overview of the classical portfolio evolution suggests that scholars more and more often conclude that it would be appropriate to include, in addition to return and risk, additional parameters (one of which is social responsibility, or sustainability) in a portfolio selection and resource allocation problem and that the inclusion of new variables in the traditional model has been made possible by multi-criteria decision-making theory.