The increasing competitiveness and complexity of the market, the accelerated development of information and communication technologies have caused the decision-making process in many organizations to become of crucial and decisive importance. Decision-making involves human judgments and logic. Therefore, increasingly complex business conditions require a multi-criteria approach to solving business problems, which allows an objective comparison of several alternatives evaluated in a system of multiple heterogeneous and different criteria with other extremization requirements and different relative importance. Consequently, multi-criteria decision-making (MCDM) methods are useful for facilitation of decision-making process in situations when there are a number of often conflicting criteria (Karamaşa et al., 2020; Stanujkic et al., 2019). Hafezalkotob et al. (2019a) emphasizes three categories of the MCDM techniques, such as: Value Measurement Methods (the SAW method; the WASPAS method etc.); Goal or Reference Level Models (the VIKOR method, the TOPSIS method etc.); and Outranking Techniques (the ELECTRE method; the PROMETHEE method etc.).

Accelerated growth and the existence of numerous methods of multi-criteria decision-making can improve the decision-making process in all areas of life. Solving problems by utilizing MCDM is based on quantitative analyses and represent elegant solutions when making decisions between multiple alternatives based on multiple-criteria. Therefore, in due course of time, there are prominent and most common developed MCDM methods, among the dozens of approaches proposed over time for solving complex-decision making problems, such as the Maxmax method, the Maxmin method, the SAW method, the AHP method, the ELECTRE method, the PROMETHEE method, the TOPSIS method, the VIKOR method, the COPRAS method, the MACBETH method, the ANP method, the MOORA method, the MULTIMOORA method, and so forth (Zavadskas et al., 2020; Ulutaş et al., 2020; Jauković-Jocić et al., 2020).

The need to solve as wide a range of real-world problems has led to the creation of a new generation of MCDM methods and approaches, such as: the HEBIN method (Zavadskas et al., 2021); the MARCOS method (Stević et al., 2020); the CoCoSo method (Yazdani et al., 2019); the SECA method (Keshavarz-Ghorabaee et al., 2018); the FUCOM method (Pamučar et al., 2018); the ARCAS (Stanujkic et al., 2017a); the PIPRECIA method (Stanujkić et al., 2017b); the MABAC method (Pamučar and Ćirović, 2015); the EDAS method (Keshavarz Ghorabaee et al., 2015), and so forth. Some of the aforementioned methods were used for ranking of alternatives whereas some of them for the purpose of weight determination.

The MOOSRA method (Multi-Objective Optimization on the basis of Simple Ratio Analysis) belongs to the group of multi-objective optimization methods and is developed by Brauers (2004). The main difference between the MOOSRA method and the MOORA method is reflected in the negative performance scores that do not appear in the MOOSRA method, unlike the MOORA method. Besides, the MOOSRA method is less sensitive to the large variation in the values of the criteria (Adalı and Işık, 2017). So far, the MOOSRA method has been applied for solving of a various complex decision-making problems, such as: the laptop selection problem (Adalı and Işık, 2017); project-critical path selection (Dorfeshan et al., 2018); bio-medical waste disposal assessment (Narayanamoorthy et al., 2020); machine selection (Sarkar et al., 2015), and so forth. The simplicity of the calculation procedure of the MOOSRA method, which is based on the ratio between weighted ratings of beneficial and non-beneficial criteria, can be mentioned as an important characteristic of this method.

Brauers (2004) also developed the MOORA method (Multi-Objective on the basis of Ratio Analysis). Somewhat later, based on the ideas of the MOORA method, Brauers and Zavadskas (2010) have proposed the MULTIMOORA method (Multi-Objective Optimization by Ratio Analysis plus Full Multiplicative Form). Both methods have been proposed to cope with subjectivity problems. The usability of the MOORA method has been demonstrated in numerous cases, such as: credit evaluation model using MOORA method (İç, 2020); evaluation of the work performance (Fadli and Imtihan, 2019); decision-making in the production system (Attri and Grover, 2014); supplier selection (Karande and Chakraborty, 2012); decision-making in a manufacturing environment (Chakraborty, 2011); privatization in a transition economy (Brauers and Zavadskas, 2006), and so on. When it comes to the MULTIMOORA method, the same has also been applied in various and numerous cases, such as: personnel selection (Karabasevic et al., 2015; Baležentis et al., 2012); risk assessment (Liu et al., 2014); project management (Brauers and Zavadskas, 2010); strategy assessment (Fedajev et al., 2020); ranking of the renewable energy sources (Alkan and Albayrak, 2020); site selection (Rahimi et al., 2020); hybrid vehicle engine selection (Hafezalkotob et al., 2019b), and so on. Integration of several proven multicriteria approaches for ranking alternatives and the use of dominance theory for the final ranking of alternatives can be mentioned as an important characteristic of these methods, which is also proven in the above-mentioned articles.

The main motivation of this research is to develop a new simpler and much more reliable MCDM approach for selecting alternatives. Accordingly, the paper aims to propose a new MCDM-based technique that is based in some segments on previous well-known MCDM techniques (MOOSRA, MOORA, MULTIMOORA), and as a novelty also includes the logarithmic approximation (LA) approach. Therefore, it is also important to state that there are four arithmetic operations in the proposed MULTIMOOSRAL method. These are as follows: addition, subtraction, multiplication, and division. In addition to these arithmetic operations, a fifth evaluation technique, which is the logarithmic approach has been added to the MULTIMOOSRAL method. Thus, unlike other methods (MULTIMOORA, MOORA, and MOOSRA), the MULTIMOOSRAL method was tried to reach a stronger result with five different techniques. For example, MULTIMOORA method uses only three arithmetic operations, which are subtraction, division, and multiplication. On the other hand, MOORA and MOOSRA use only two arithmetic operations. However, the MULTIMOOSRAL method uses all arithmetic operations and a logarithmic approach to reach much more valid and rigorous results.

Further, highlights of the logarithmic normalization are emphasized by Zavadskas and Turskis (2008). Therefore, the newly-developed, so-called MULTIMOOSRAL method integrates five approaches for ranking alternatives. In order to apply and test the new approach, an illustrative case study of the supplier selection is conducted. Accordingly, the paper is structured as follows: in Section 1, the introductory consideration are given. In Section 2, the MOOSRA method, the MOORA method and the MULTIMOORA method are presented. The new MULTIMOOSRAL approach is presented in Section 3, whereas in Section 4, a conducted case study is demonstrated. Finally, at the end of the article, conclusions are given.

The newly proposed MULTIMOOSRAL method integrates five approaches for ranking alternatives. In addition to the previously discussed approaches, applied in MOOSRA, MOORA and MULTIMOORA methods, the MULTIMOOSRAL method also includes the LA approach. In addition, an important characteristic of MULTIMOOSRAL method can be mentioned, that is, it uses a new approach for determining the final ranking order of alternatives, which is not based on the dominance theory.

The computational procedure of the MULTIMOOSRAL method, presented in Fig. 2, can be precisely presented using the following steps:

Step 1. Forming the initial decision matrix and determining criteria weights.

Step 2. Forming the normalized decision matrix. The normalized decision matrix is formed using Eq. (7), which is: (7) rij=xij∑i=1n(xij)2.

Step 3. Calculating the normalized overall utilities of alternatives based on the five approaches included in the MULTIMOOSRAL method, as follows:

Step 3.1. Determining the utility of alternatives based on the RS approach by applying the following substeps:

Substep 3.1.1. Calculating the overall importance of considered alternatives is performed using Eq. (8), which is: (8) yi=∑j∈θmaxwjrij−∑j∈θminwjrij.

Substep 3.1.2. Calculating the overall utility of considered alternatives as follows: (9) mi=yi,maxi(yi)>0,yi+1,maxi(yi)=0,−1/yi,maxi(yi)<0, 0,\\ {} {y_{i}}+1,\hspace{1em}& {\max _{i}}({y_{i}})=0,\\ {} -1/{y_{i}},\hspace{1em}& {\max _{i}}({y_{i}})<0,\end{array}\right.\]]]> where mi denotes overall utility of alternative i obtained on the basis of RS approach.

Substep 3.1.3. Normalizing the overall utilities obtained on the basis of RS approach as follows: (10) mi′=mi−min(mi)max(mi)−min(mi), where mi′ denotes normalized overall utility of alternative i obtained on the basis of RS approach.

Step 3.2. Determining the utility of alternatives based on the RP approach by applying the following substeps:

Substep 3.2.1. Determining reference point r∗ as follows: (11) r∗=(r1∗,r2,∗,…,rn∗)={maxirij|j∈θmax,minirij|j∈θmin}.

Substep 3.2.2. Calculating the maximal distance between each alternative and the reference point using Eq. (12), which is: (12) ti=maxj(wj|rj∗−rij|).

Substep 3.2.3. Normalizing maximal distances as follows as follows: (13) ti′=max(ti)−timax(ti)−min(ti), where ti′ denotes normalized overall utility of alternative i obtained on the basis of RP approach.

Step 3.3. Determining the utility of alternatives based on the FMF approach by applying the following substeps:

Substep 3.3.1. Calculating the overall utility of the alternatives using Eq. (14), which is: (14) ui=∏j∈θmaxwjrij∏j∈θminwjrij.

Substep 3.3.2. Normalizing the overall utilities obtained on the basis of FMF approach as follows: (15) ui′=ui−min(ui)max(ui)−min(ui), where ui′ denotes normalized overall utility of alternative i obtained on the basis of FMF approach.

Step 3.4. Determining the utility of alternatives based on an addition form (AF) approach by applying the following substeps:

Substep 3.4.1. Calculating the overall utility of each alternative using Eq. (16), which is: (16) vi=∑j∈θmaxwjrij∑j∈θminwjrij.

Substep 3.3.2. Normalizing the overall utilities obtained on the basis of AF approach as follows: (17) vi′=vi−min(vi)max(vi)−min(vi), where vi′ denotes normalized overall utility of alternative i obtained on the basis of AF approach.

Step 3.5. Determining the utility of alternatives based on the LA approach by applying the following substeps:

Substep 3.5.1. Calculating the overall utility of alternatives based on the LA approach ki as follows: (18) ki=∑j∈θmaxln(1+wjrij)+1∑j∈θminln(1+wjrij).

Substep 3.5.2. Normalizing the overall utilities obtained on the basis of AF approach as follows: (19) ki′=ki−min(ki)max(ki)−min(ki), where ki′ denotes normalized overall utility of alternative i obtained on the basis of AF approach.

Step 4. Determining the final ranking orders of alternatives. The final ranking of alternatives is determined based on their total utility Si , which is calculated as follows: (20) Si=mi′+ti′+ui′+vi′+ki′.

After that, the alternatives are ranked on the basis of values of Si in descending order and the alternative with higher value of Si is the most preferable.

In this study the usability of the MULTIMOOSRAL method was demonstrated on supplier selection problem for a textile company. All data were collected from three managers of the company and the actual data of company. Managers of the company evaluated the criteria (indicated in Table 1) to obtain criteria weights. The evaluation criteria, as well as their weights (obtained by using the SWARA method, Keršuliene et al., 2010), are shown in Table 1.

The data of the first three criteria are actual data and the data of the other criteria are obtained from managers of the company. The ratings of suppliers and normalized decision matrix are shown in Table 2 and Table 3.

The overall importance of alternatives, overall utilities and normalized overall utilities obtained on the basis of RS approach, calculated by Eqs. (8), (9) and (10), are shown in Table 4.

The reference point, obtained using Eq. (11) and data from Table 3, is shown in Table 5.

After that, the maximal distances and normalized maximal distances are calculated, using Eq. (12) and Eq. (13), as shown in Table 6.

The overall utility and normalized overall utility of alternatives obtained on the basis of FMF approach, using Eqs. (14) and (15), are shown in Table 7.

The overall utility and normalized overall utility of alternatives obtained on the basis of AF approach, using Eqs. (16) and (17), are shown in Table 8.

The overall utility and normalized overall utility of alternatives obtained on the basis of LA approach, using Eqs. (18) and (19), are shown in Table 9.

Finally, in Table 10 results obtained using five approaches integrated in MULTIMOOSRAL method, overall utility of considered alternatives, calculated using Eq. (20), and ranking order of alternatives are presented.

As it can be seen from Table 10, the best supplier selected using the MULTIMOOSRAL method is supplier denoted as “Supplier 1”.

In order to further verify the obtained results, a comparison with the results obtained using the MOOSRA, MOORA and MULTIMOORA methods was performed below. The results obtained using the above methods are shown in Tables 11, 12 and 13.

From Tables 11, 12 and 13 it can be seen that the ranking results obtained using the MULTIMOOSRAL method are identical to the results obtained using the MOOSRA, MOORA and MULTIMOORA methods.

In order to finally verify the results obtained using the MULTIMOOSRAL method, a comparative analysis was performed with several well-known MCDM methods, such as TOPSIS, MULTIMOORA, and CoCoSo methods, as shown in Table 14.

It can be observed from Table 14, that the MULTIMOOSRAL method gives the same ranking orders as the TOPSIS and MULTIMOORA methods. Some discrepancy in the rank of the alternative can be observed in the case of the use of the CoCoSo method, which refers to the fourth and fifth-ranked alternatives. However, such phenomena are expected because the newly proposed MULTIMOOSRAL method integrates more ranking approaches and because of that, it should allow a more accurate ranking of the alternatives.

This paper proposes a new MCDM technique called MULTIMOOSRAL that is based on the approaches in MOOSRA, MOORA, and MULTIMOORA methods and LA approach for the facilitation of a decision-making process. The main incentive for proposing the new method reflects the desire to develop such an approach that will contribute to the increasing of the credibility of the obtained results. In this case, by involving five approaches (RS, RP, FMF, AF, and LA) the reliability of the final ranking order as well as its stability is raised to a higher level.

In order to demonstrate the applicability and usefulness of the proposed method, the illustrative case study pointed to the selection of the adequate supplier of the textile company is presented. The evaluation process, entrusted to 3 managers, is based on the 7 criteria and 5 alternatives. The gained result revealed that supplier 1 is the most suitable to work within the present conditions, while supplier 3 is the worst choice according to the given performances.

To acknowledge the reliability of the obtained ranking order, the MOOSRA, MOORA, and MULTIMOORA methods are applied. The results from all three used techniques confirmed that one was obtained by using the newly proposed MULTIMOOSRAL method. Namely, in all three observations, supplier 1 is ranked as the best alternative, while supplier 3 is ranked as the last and worst option. In this way, the stability of the proposed MCDM method is verified as well as its suitability for applying in the decision-making process.

Although this novel method contributes to the reliability of the performed decision process by involving more approaches, the same thing could be considered as its main deficiency, too. Namely, the computational procedure could be considered as complex for application by the users that are not familiar with the MCDM field. Additionally, in order to better incorporate the uncertainty, this model could be extended by involving the fuzzy, grey, or neutrosophic numbers. But, despite the mentioned imperfections, the MULTIMOOSRAL method proved its efficiency in enhancing the decision-making process and its possibilities should be further examined.