Fuzzy Approach to Analysis of Investment Alternatives

Abstract

With significant market unsureness, “static” methods fail to account for economic uncertainty, may be less precise and, accordingly, less helpful when selecting investment alternatives. Methods that take into account the current economic situation and allow for adapting the alternative selection to external uncertainty are becoming more relevant. One of such methods is the fuzzy set theory. This article addresses the mathematical framework of such an approach for the economic analysis of investment project selection. A step-by-step scheme for implementing the fuzzy set method for investment projects is presented. Studies performed on the example of three investment alternatives give grounds for asserting the compatibility and feasibility of using two methods (the fuzzy set method may be partly based on the results of pairwise comparisons of experts according to the Saaty method) and confirmation or refutation of previous intuitive decisions of investors based on a comprehensive analysis of the criterion composition and the use of mathematical grounded technique.

1. Introduction

The unstable economic environment, changing legal conditions, exchange rate fluctuations, and rapid technological development significantly complicate investment decision-making, particularly in the context of startup investments. Under such uncertainty, traditional “static” evaluation methods often fail to reflect dynamic market conditions, leading to less accurate and less reliable investment choices.

In such circumstances, decision-making approaches that incorporate uncertainty and allow flexibility in evaluating alternatives are gaining importance. In particular, methods based on fuzzy logic provide a structured way to model imprecision and incomplete information, which are inherent in real-world investment scenarios.

The groundbreaking studies by Saaty (1972, 1975, 1977, 1980, 1988) on the analytic hierarchy process and by Zadeh (1965, 1975), R. Bellman and Giertz (1973) and R. E. Bellman and Zadeh (1970, 1977) have led the foundations of using fuzzy sets in decision making when alternatives (goals and constraints) are stated imprecisely. Fuzzy sets and fuzzy logic are powerful mathematical tools for decision making in the absence of complete and precise information.

This approach has proven versatile in addressing complex decision-making problems across multiple domains. In financial and investment contexts, it has been used for capital budgeting (Graham & Harvey, 2001), portfolio selection (Belabbes et al., 2025; Derya et al., 2025; Mohseny-Tonekabony et al., 2025; Șerban, 2025; Watada, 2001), long-term savings decisions (Mugerman et al., 2014), and evaluating strengths and risks in investment analyses (Kaplan & Strömberg, 2004), optimal investment and employment (Bloom, 2009). In organizational and human-resource management, it has supported the measurement of organizational capital (Bozbura & Beskese, 2007), faculty selection (Smith & Bayazit, 2024) and candidate evaluation (Paudel et al., 2022). In engineering and technical applications, the method has guided sector selection (Akkaya et al., 2015), power engineering problem-solving (P. Y. Ekel et al., 2007), and programming task prioritization (Nishad & Singh, 2015). It has also been applied in service quality and infrastructure evaluation, including healthcare services (Büyüközkan et al., 2011), irrigation projects (Anagnostopoulos & Petalas, 2011), and general service assessment (Mikhailov & Tsvetinov, 2004). Finally, its scope extends to specialized areas such as weapon system evaluation (Cheng & Mon, 1994), higher education investment (Kahraman et al., 2013), banking product selection (Ishizaka & Nguyen, 2013), and sports analytics (Ballı & Korukoğlu, 2014), demonstrating its broad adaptability in complex, multi-criteria decision-making scenarios.

Several books are devoted to different applications of fuzzy sets approach to decision making (Aliev, 2018; Aliev & Huseynov, 2014; Bhushan & Rai, 2004; Borisov et al., 1990; P. Ekel et al., 2020; Kitainik, 1993; Saaty & Vargas, 2012); see also the review papers (Aliev et al., 2016; Dubois, 2011; Dymova et al., 2021; Kubler et al., 2016; Liu et al., 2020; Mardani et al., 2015; Vaidya & Kumar, 2006), where mathematical tools, methodologies and techniques of the decision-making practice are comprehensively discussed.

Analysis of priority vector based on pairwise comparison matrix with fuzzy elements still attracts the interest of researchers (Borisov et al., 1990; Greblikaitė & Daugėlienė, 2009; Innan & Inuiguchi, 2024, 2025; Kubler et al., 2018; Ramik, 2021, 2023; Siebert, 2019).

The present article addresses the mathematical framework of this approach for the economic analysis of investment project selection. A step-by-step scheme for implementing the fuzzy set method for investment projects is carried out. Studies performed on the example of three investment alternatives give grounds for asserting the compatibility and feasibility of using two methods (the fuzzy set method may be partly based on the results of pairwise comparisons of experts according to the Saaty method) and confirmation or refutation previous intuitive decisions of investors based on a comprehensive analysis of the criterion composition and the use of mathematical grounded technique.

2. Economic-Mathematical Model

The decision to invest in modern conditions is based on the choice of such an alternative that satisfies the conditions of fuzzy goals and fuzzy restrictions. Thus, the goals and limitations will be symmetric with respect to the solution, which allows it to be presented as a fusion of fuzzy goals and constraints. It is also necessary to distinguish between the notion of randomness and fuzziness. Randomness is associated with uncertainty as to the affiliation or non-existence of a particular object to a fuzzy set, while fuzziness (vagueness, blurriness) can be interpreted as belonging to classes that have different gradations of the degree of affiliation intermediate between complete affiliation and complete inactivity to this class. But the main difference in this approach is the mathematical tool used to solve such problems. The probability theory deals with the investigation of randomness; instead, to solve the problems of the theory of fuzzy sets, the apparatus of the theory of fuzzy sets should be used.

Simply put, in our case, we will deal with vague goals (the choice of investment project) and vague terms (choice according to individual criteria). The essence of the proposed fuzzy set theory can be applied as follows. Let $X=\{x_1,x_2,\dots x_k\}$ be the set of investment projects/startups and $G=\{g_1,g_2,\dots g_n\}$ be the set of criteria under which investment projects will be evaluated. The fuzzyness of sets is described by eigenvectors of the matrix of pairwise comparisons.

In the previous paper (Kyrylych & Povstenko, 2023), three investment alternatives were studied (production of LED traffic lights—the first investment alternative, manufacture of information–reference electronic terminals—the second investment alternative, and manufacture of rotor-reactive turbo-rotational heaters of liquids—the third investment alternative). In the process of evaluating investment alternatives, each of them is analyzed in terms of the degree to which particular groups of economic criteria are met. The following group of criteria were defined for choice of invetment alternatives: financial strength; product/service potential; marketing capability; organization potential; scientific and technical possibility; staff competence; evaluation of the governmental, international, economic and political situation; time potential; autonomy; ecological strength; social potential; information capability.

The selection of a team of experts for choosing the best investment alternative should be purposeful, balanced, and as small as possible, while still covering key perspectives: financial, market, risk, and operational. In practice, a team of 5–7 experts works best—this size helps avoid decision-making chaos while still providing a broad view of the situation. In the proposed example, synthetic data were used. To conduct such an analysis, the suggested composition of the expert team could include: 1. Financial/investment analyst. 2. Market and strategy expert (e.g., strategic consultant). 3. Risk specialist (risk manager). 4. Industry expert (domain expert). 5. Legal and regulatory expert. 6. Operational/technological expert. 7. Capital allocation expert/practitioner investor.

Experts are usually recruited from consulting and advisory firms, financial and investment institutions, academia (experts from universities in finance, economics, and technology), and independent industry experts (practitioners with operational experience). For collecting the expert opinions, the Delphi method is recommended.

The mathematical apparatus for selecting optimal investment alternative based on the fuzzy set method can be presented in the form of the diagram shown in Figure 1.

3. Implementation of the Fuzzy Set Method

Relative importance coefficients necessary for decision making under uncertainty are determined based on the paired comparison procedure. Due to estimations of expert council of potential investors, the pairwise comparisons of investment alternatives have been made in each group of criteria (Kyrylych & Povstenko, 2023).

Table 1. The matrix of pairwise comparisons for the group of criteria “Financial strength”.

	The First	The Second	The Third
Startup	Investment	Investment	Investment
	Alternative	Alternative	Alternative
The first investment
alternative	1	1/3	1
The second investment
alternative	3	1	2
The third investment
alternative	1	1/2	1

shows such comparison for the first group of criteria “Financial strength”.

For the matrix in Table 1, we calculate the eigenvectors $\mathbf{A}·\mathbf{w}=\lambda \mathbf{w}$:

$$\det \left[\begin{array}{ccc}1-\lambda & 1/3& 1\\ 3& 1-\lambda & 2\\ 1& 1/2& 1-\lambda \end{array}\right]=-{\lambda }^3+3{\lambda }^2+1/6=0,$$

(1)

thus obtaining ${\lambda }_1=-0.01+0.23i$, ${\lambda }_2=-0.01-0.23i$, ${\lambda }_3=3.02$, and choose the maximum value ${\lambda }_{\max }=3.02$.

Next, substituting the value ${\lambda }_{\max }$ into the main diagonal of the matrix we obtain a system of linear equations

$$\left[\begin{array}{ccc}-2.02& 1/3& 1\\ 3& -2.02& 2\\ 1& 1/2& -2.02\end{array}\right]\left[\begin{array}{c}{w}_1\\ w_2\\ w_3\end{array}\right]=0,$$

(2)

under the constraint that (Borisov et al., 1990)

$$w_1+w_2+w_3=1.$$

(3)

The solution has the form:

$$w_1=0.21,w_2=0.55,w_3=0.24.$$

(4)

Similar calculations have been made for all other groups of criteria (see Appendix A). It should be emphasized that the inequality ${\lambda }_{\max }\ge k$ is always satisfied. The deviation of ${\lambda }_{\max }$ from k (${\lambda }_{\max }-k$) can serve as a measure of coherence between judgments of experts. The closer the ${\lambda }_{\max }$ to k, the judgments of experts are more agreed. To check consistency of expert judgments, the consistency ratio (CR)^p was introduced (Saaty, 1980; Saaty & Vargas, 2012). This ratio is defined as the ratio (CI)^p/(RI)^p, where the consistency index (CI)^p of matrices of alternatives is calculated as

$${\left(\mathrm{CI}\right)}^p={\displaystyle \frac{{\lambda }_{\max }^{\left(p\right)}-k}{k-1}},p=1,2,...,n,$$

(5)

and (RI)^p is the random consistency index. According to (Saaty, 1980; Saaty & Vargas, 2012), (RI)^p = 0.52 for $k=3$. The consistency ratio should not exceed 0.1; when it is more than 0.2, the opinions of experts should be reviewed and, if necessary, experts should be changed.

The obtained results are summarized in

Table 2. Consistency indeces and consistency ratios of matrices of pairwise comparisons.

Group of Criteria	(CI)(p)	(CR)(p)
1	0.0091	0.0175
2	0	0
3	0	0
4	0	0
5	0.0046	0.0088
6	0.0268	0.0515
7	0	0
8	0.0268	0.0515
9	0.0046	0.0088
10	0	0
11	0	0
12	0.0268	0.0515

.

In

Table 3. Eigenvectors of matrices of pairwise comparisons.

Group of Criteria	${\mathit{w}}_1$	${\mathit{w}}_2$	${\mathit{w}}_3$
1	0.21	0.55	0.24
2	0.20	0.40	0.40
3	0.20	0.40	0.40
4	0.25	0.50	0.25
5	0.21	0.55	0.24
6	0.20	0.31	0.49
7	0.25	0.25	0.50
8	0.59	0.16	0.25
9	0.16	0.30	0.54
10	0.25	0.50	0.25
11	1/3	1/3	1/3
12	0.33	0.14	0.53
MIN	0.16	0.14	0.24

, we present the eigenvectors of all twelve matrices of pairwise comparisons for three investment alternatives.

In each column, the minimum value is choosen: 0.16; 0.14; 0.24. The best investment alternative (manufacture of rotor-reactive turbo-rotational heaters of liquids) corresponds to the maximum value of these options: 0.24. On the second place is the first alternative (production of LED traffic lights), and on the third place is the second alternative (manufacture of information–reference electronic terminals). These calculations were carried out under assumption that all groups of criteria have the same weight.

In the previous paper (Kyrylych & Povstenko, 2023), we have calculated the weights of groups of criteria (normalized vectors of geometric means) according to the formula

$${\mu }_i={\displaystyle \frac{\sqrt[n]{{\displaystyle \prod _{j=1}^n}{b}_{ij}}}{{\displaystyle \sum _{i=1}^n}\sqrt[n]{{\displaystyle \prod _{j=1}^n}{b}_{ij}}}},i=1,2,...,n,$$

(6)

where $b_{ij}$ are the elements of the matrix of pairwise comparisons for the complet of all groups of criteria, while

$$\sum _{i=1}^n{\mu }_i=1.$$

(7)

If we assume that different groups of criteria have different weights, then the results given in Table 3 will be slightly changed taking into account the weight of each group of criteria. The weighted results are presented in

Table 4. Weighted eigenvectors of matrices of pairwise comparisons.

Group of Criteria	${\mathit{\mu }}_{\mathit{i}}$	Weighted ${\mathit{w}}_1$	Weighted ${\mathit{w}}_2$	Weighted ${\mathit{w}}_3$
1	0.1893	0.0398	0.1041	0.0454
2	0.0800	0.0160	0.0320	0.0320
3	0.0911	0.0182	0.0364	0.0364
4	0.0490	0.0122	0.0245	0.0122
5	0.1058	0.0222	0.0582	0.0254
6	0.1262	0.0256	0.0397	0.0628
7	0.0666	0.0167	0.0167	0.0333
8	0.0942	0.0556	0.0151	0.0236
9	0.0483	0.0077	0.0145	0.0261
10	0.0422	0.0106	0.0211	0.0106
11	0.0511	0.0170	0.0170	0.0170
12	0.0542	0.0179	0.0076	0.0287
−	MIN	0.0077	0.0076	0.0106

.

In each column, the minimum value is choosen: 0.0077; 0.0076; 0.0106. The best investment alternative (manufacture of rotor-reactive turbo-rotational heaters of liquids) corresponds to the maximum value of these options: 0.0106. In the considered case, accounting for the weights of groups of criteria does not change the final choice.

4. Concluding Remarks

The fuzzy set method is a key tool for multi-criteria decision support under uncertainty based on the concept of fuzzy decision objectives. Its distinctive feature is its ability to model the vague, subjective, and qualitative assessments that dominate real-world economic and investment problems. Unlike classical decision-making methods, which assume precise input data and clearly defined preferences, the fuzzy set approach allows for the formal consideration of market uncertainty, forecast variability, and expert linguistic judgments. The selection of a compromise solution minimizes the risk of extremely unfavorable decision-making outcomes.

This approach is particularly important in investment analysis, where even a single, poorly fulfilled criterion can significantly reduce the attractiveness of the entire investment. This makes it an effective analytical tool in situations where financial data are incomplete and decisions must be made based on estimates, forecasts, and subjective preferences. Due to its flexibility, ability to model information ambiguity and emphasis on decision security, the fuzzy set method is widely used in the evaluation of investment alternatives, constituting a valuable complement to classical optimization methods.

The calculations made according to the fuzzy set method show that the best is the third investment project, on the second place—project number one, and on the third place—project number two. Comparing them with the results of (Kyrylych & Povstenko, 2023) obtained using the AHP Saaty method, it is easy to notice that in the AHP method we obtained a slightly different ranking: investment project 2, project 3 and project 1. The difference in the results obtained can be explained by the peculiarities of the methods used.

The fuzzy set method describes each group of criteria with a membership function, defining the degree to which a given alternative satisfies a given group of criteria. The summary decision is made by maximizing the minimum degree of satisfaction of all the groups of criteria. A significant advantage of this method is its good modeling of uncertainty and data fuzziness, and it is more dynamic and responsive to environmental changes. Each method excels at addressing a large number of specific criteria, confirming their reliability. The fuzzy set method seeks a “worst-case” compromise, while the AHP method aggregates weighted preferences. In a preference analysis conducted using the AHP method, a high-return alternative can be selected despite a high degree of risk, while the fuzzy set method is more likely to indicate a more stable but less profitable investment, thus protecting the investor from an extremely negative scenario. A distinctive feature of our method in practice is that it takes into account groups of criteria rather than only individual indicators, as is typically done in standard methods for evaluating investment alternatives.

The fuzzy approach is especially useful when data is imprecise, incomplete, or based on expert judgment. It effectively aggregates subjective opinions from multiple experts, making it suitable for complex investment decisions where quantitative data alone is insufficient providing a more holistic assessment. The method can be easily adjusted to different industries, project types, or strategic goals without requiring strict data conditions.