INTRODUCTION:
The core of Operations Research (OR) is the development of approaches for optimal decision making (Triantaphyllou, Shu, Sanchez, and Ray, 1998). In order to find the most appropriate solutions, Multi Criteria Decision Making (MCDM) methods are the translation system, which translate decision making problems, from less complex such as the daily decision-making problems to advanced decision-making problems, to the mathematical algorithms.
MCDM methods are developed to analyze alternatives against the criteria with the various algorithms to lead the Decision Maker(s) (DMs) to the optimal solutions for the decision-making problems (Zakeri, Cheikhrouhou, Konstantas, and Barabadi, 2022).
MCDM methods can be defined as structure frameworks that deal with the process of making decisions in the presence of multiple objectives. They are used to find the best opinion from all of the feasible alternatives in the presence of multiple, usually conflicting, decision criteria (Qu, Wan, Yang, and Lee, 2018). Since criteria are often conflicting there may be no solution satisfying all criteria simultaneously. Consequently, result of this process is often a compromise solution, which is obtained according to DMs preferences (Nenad and Zoran, 2017). In essence, MCDM involves four key components (Anand, Agarwal, and Aggrawal, 2022):
1. Alternatives to be ranked or chosen from.
2. Criteria by which the alternatives are evaluated and compared.
3. Weights representing the relative importance of the criteria.
4. DMs and potentially other stakeholders, whose preferences are to be represented.
The title of MCDM was used for the first time in a paper by Zeleny in 1975 (Zavadskas, Antucheviciene, Turskis, and Adeli, 2016), and is one of the most widely use decision methodologies in the sciences, business, and engineering worlds (Azadfallah, 2019).
According to Roszkowska (2011), the MCDM problems can be divided into two kinds. One is the classical MCDM set of problems among which the ratings and the weights of criteria are measured in crisp numbers. Another one is the MCDM set of problems where the ratings and the weights of criteria evaluated on incomplete information, imprecision, subjective judgement and vagueness are usually expressed by interval numbers, linguistic terms, fuzzy numbers or intuitive fuzzy numbers. However, there is no particular MCDM method that can be applied to all types of problems, since methods have been designed for specific cases, with their associated benefits and limitations (Sitorus, Cilliers, and Parada, 2019). But in reality, to a DM it is not always easy to specify the membership function or probability distribution in an inexact environment (Sayadi, Heydari, and Shahanaghi, 2009). Since, the interval number is a most commonly used mode (Zhang and Xu, 2006).
In decision making science, evaluation values are not always real numbers. Using interval numbers to represent evaluation values in the decision-making process more accurately represents the reality of uncertainty and is more consistent with the fuzzy human mind than using real numbers (Yao, Ye, Wang, and Hu, 2020). For instance, for locating gas stations and first aid stations on roads or locating a fire station or hospital in a city, an exact location cannot be usually candidate and these locations have to be considered as interval (Amiri, Nosratian, Jamshidi, and Kazemi, 2008). By definition, an interval number is a set of real numbers in which the members of the set lie between two limiting numbers (Hadiono, 2016). A MCDM problem with interval weight and data can be concisely expressed in format of one matrix as table1(Amiri, Zandieh, Vahdani, Yazdani, and Soltani, 2008).
Table1. MCDM problem with interval weight and data
|
Criteria Alternative |
C1 |
C2 |
Cn |
|
|
|
|
|
|
A1 |
|
|
|
|
A2 |
|
|
|
|
Am |
|
|
|
Where, is the weight of criterion Cj.
On the other side, ranking interval numbers is key in decision making approaches that use interval numbers to represent evaluation values (Yao, Ye, Wang, and Hu, 2020). However, several solutions for the above problem are possible (i.e., the acceptability index, minimax regret approach, etc.), but theoretically there is no reason to be restricted to these approaches. Therefore, in this paper, we propose the distance to ideal interval numbers model to evaluates the interval numbers based on the distance between each interval number and the ideal interval numbers. So, interval numbers are ranked according to proximity to ideal interval numbers.
The paper is organized as follows. The second section briefly describes the method considered in this study (i.e., interval number, distance of interval numbers, and the acceptability index [or Sengupta and Pal approach]). In third, fourth, fifth, and sixth sections, prior work, research design, propose procedure, and numerical example are discussed, respectively. Discussion is provided in the next section. The paper is concluded in the eight and the last section.
Interval Number:
According to Zhang and Xu (2006), because of the complexity of objective affairs and the restriction of mankind’s knowledge and cognition, in real decision-making problems, sometimes the decision information is expressed by interval numbers but not numerical points. Nevertheless, in continuation, we describe the basic definition and operations of interval numbers.
Definition1.
Let, then a is called a nonnegative interval
number. Especially, a is a nonnegative real number, if 
.
For convenience, throughout this chapter, all the
interval arguments are nonnegative interval numbers, and let Ω be
the set of all interval numbers,
.
Definition2.
Let, then:
(1)
(2)
. Especially,. (3)
Definition 3.
If 
is a crisp weight vector, such that:
(4)
Then W is called normalized weight vector.
Definition4. Let be non-normalized interval weights, then:
,(5)
Are called normalized interval weights of, (Yue, 2013; Azadfallah, 2017).
The Distance of Interval Numbers:
The distance of interval numbers between is (Jiang, Ren, and Wang, 2022):
.(6)
The Acceptability Index:
Let < be an extended relation between the
intervals on the real line, then for, we construct a premise 
,
which implies that A is inferior to B (or B is superior to
A). here, the term ‘inferior to’ (‘superior to’) is analogous to ‘less
than’ (‘greater than’).
Definition1. Let I be the set of all closed
intervals on the real line
.
Here, we further define an acceptability function 
such
that (Eq. 6):
, (7)
Where. may be interpreted as the grade of acceptability of the ‘first interval to be inferior to the second interval’.
The grade of acceptability of may be classified and interpreted further on the basis of comparative position of mean and width of interval B with respect to those of interval A as follows (Eq. 7):
(8)
The classification of the acceptability grades is interpreted as follows:
1. If, then the premise ‘A is inferior to B’ is not accepted.
2. If, then the interpreter accepts the premise with different grades of satisfaction ranging from zero to one (excluding zero and one).
3. If, the interpreter is absolutely satisfied with the premise or, in other words, he accepts that is true (Sengupta, Pal, and Chakraborty,2001; Sengupta and Pal, 2009).
Prior Work:
Ranking decision for interval data is a very important issue in decision making analysis (Song, Liang, and Qian, 2012). Hence, the ranking of interval numbers was a subject of many papers. For instance, Edelsbrunner, Overmars, Welzl, Hartman, and Feldman (1990) developed a ranking interval model under visibility constraints. Zhang, Fan, and Pan (1999) introduced a dominance and possibilities-based ranking model for interval numbers in MCDM problems under uncertain conditions. Liu (2001) proposed an axiomization definition for the ranking of interval number from the mathematical and practical angle. then, two families of ordering relations are proposed and some rationality properties related to them are investigated. Xu (2003) extended a possibility degree procedure for a comparison between two interval numbers. Sevastianov (2007) applied the Dempster-Shafer theory of evidence with its probabilistic interpretation to justify and construct the method which provides the result of comparison in the form of an interval or fuzzy number. Wang (2008) proposed a method for ranking and applying interval numbers based on partial connection numbers. Fan and Liu (2010) investigated how the ranking order of alternatives is determined on preference information of ordinal interval numbers in group decision making problems. Song, Liang, and Qian (2012) proposed a two-stage approach to ranking interval data. In this approach they keep the ranking result induced by the entire dominance degree in the first stage, and then refine the objects that cannot be ranked through introducing a so-called entire directional distance index. Ye, Yao, Wang, and Wang (2016) presented a model of ranking interval values based on degree for MADM (Multiple Attribute Decision Making; also, often called MCDM). Li, Zeng, and Yin (2018) suggested an improved ranking algorithm for a set of interval numbers based on the Boolean matrix. Yue (2018) introduced a new model for comparing two interval based on a two-dimensional geometric interpretation. Zhao and Ping (2020) proposed a possibility degree approach for ranking intervals scores under non-uniform distribution. Hosseinzadeh and ghasemi (2021) extended a ranking function for ordering interval numbers, and explain some features of this function. Luo, Ye, Yao, and Wang (2021) developed an interval number ranking method based on multiple decision attitudes. Jiang, Ren, and Wang (2022) presented interval data MADM method based on TOPSIS. Then, the average of the ranking number based on different values for α is used to reflect the actual situation better. Li and Li (2023) established a ranking model based on the generalized greyness of interval number for interval grey numbers. In addition, Romanenkov, Mukhin, Kosenko, Revenko, Lobach, Kosenko, and Yakovleva (2024) improved the efficiency of decision-making based on interval expert data under conditions of uncertainty and risk by developing a criterion for the preferences of interval evaluations of overlapping alternatives. Nevertheless, in this paper we focus our main attention to on the distance of interval numbers from the ideal interval numbers, and its uses in the interval numbers ranking.
Research Design:
According to Yue (2013), in many cases, the attributes values are not precisely known but value ranges can be obtained. The information provided by DMs is usually uncertain, due to time pressure, lack of knowledge, and the DM’s limited expertise related with problem domain. In these cases, the use of interval data instead of precise numerical values seems to be more adequate.
Historically, since interval numbers were first proposed by Dwyer [in 1951] (Yao, Ye, Wang, and Hu, 2020), a difficulty to solve the interval decision making problems is the ranking method of interval numbers, because among the interval numbers exist cross, it’s difficult to rank the alternatives directly (Zhang and Xu, 2006). however, several solutions (i.e., the acceptability index, minimax regret approach, etc.) have been proposed to deal with these problems, but theoretically there is no reason to be limited to these approaches. Therefore, in this paper, we propose the distance to ideal interval numbers model to evaluates the interval numbers based on the distance between each interval number and the ideal interval numbers.
Proposed Procedure:
In the proposed procedure, non-negative interval numbers ranking process has three simple steps, as follows.
Step 1. Start by setting a reference point (or ideal interval number) for comparison.
Start by setting a reference point that may be established subjectively (in other words, by DM) /objectively (in other words, by mathematical calculation).
Consider two intervals. where be the reference point (Eq.9).
(9)
Step 2. Calculate the difference between the interval numbers and reference point.
Distances of each interval number (or A) to ideal interval number (or AI) is calculated based on equation (6). So, in the proposed model, then, we have (Eq. 10):
. (10)
Step 3. Rank the interval numbers.
Rank the interval numbers based on the distance between two intervals (i.e., A and AI). further, it ranks interval numbers based on their proximity to an ideal interval numbers.
Numerical Example:
In this section, let us take two examples to understand how to implement proposed approach to calculate distance between two interval numbers.
Example 1:
Consider five alternatives / intervals
distance to the ideal interval numbers is calculated as follows.
By using step1:
Set a reference point (or ideal interval numbers) for comparison, by using equation (9), as follows.
here (objectively):
By using step 2:
Calculate the difference between the interval numbers and reference point, by using equation (10), as follows.
;
for instance, for A1, we have:
Similarly:
By using step3:
Rank the interval numbers (step2) based on their proximity to the ideal interval numbers (in other words, the smaller the distance, the higher the interval numbers), as follows.
Therefore, the best interval numbers by the ranking order is A4.
Example2
For intervals
and again
the distance is:
then; the ranking order of the interval numbers is:
Therefore, the best alternative by the ranking order is A1.
In this section, more studies have been done. In order to compare this result with the existing approach (particularly, the acceptability index or Sengupta and Pal approach, Eqs. 7-8), we will use the same numerical example.
Here, with no intention to describe the whole procedure, we shall only point to the final results (table2).
Table 2. Comparison of interval ranked by different approach
|
Example |
Interval Numbers |
Priority |
|
1 |
Proposed approach |
|
|
Existing approach |
|
|
|
2 |
Proposed approach |
|
|
Existing approach |
|
According to the results of table 2, we can find the priority is and, for 2 examples, respectively.
DISCUSSION:
According to the results (table2), we can find the priority is, and. These priorities are slightly different from the results obtained from the traditional method (or Sengupta and Pal approach) [i.e., and], for two examples, respectively. Because the distance to an ideal alternative/interval number is considered in the proposed approach. in this situation, the two priority alternatives (A4>A5, and A1>A5, for two examples, respectively) remain unchanged, but the other alternatives (i.e., A2>A1>A3 and A4>A2>A3, to A3>A2>A1 and A3>A4>A2, for example 1 and example 2, respectively) are shifted. In general, these analyses show that the proposed approach is efficient and the result is consistent.
CONCLUSION:
In this paper, we propose the distance to ideal interval number/reference point approach to ranking the interval numbers. So, it ranks the interval numbers based on their proximity to the ideal alternative. In continuation, the results compared with the existing approaches (i.e., the acceptability index or Sengupta and Pal method). According to the results (table2), we can find the priority is, and. These priorities are slightly different from the results obtained from the traditional method [i.e., and], for two examples respectively. Because the distance to an ideal alternative/interval number is considered in the proposed approach. Generally, compared results show that the proposed procedure is effective and feasible.
The merits and advantages of the proposed approach are shown as follows.
· Better use of resources,
· Very simple computation process,
· Also the proposed approach improves the quality and degree of acceptance of decisions.
In sum, the finding in this paper confirms the effectiveness of proposed method. Therefore, we believe that the mechanisms of proposed approaches are reasonable. In final, it is expected that the models introduced in this paper may have more potential applications in the future work. Further, compare it with another existing models (i.e., except for Sengupta and Pal model, which was reviewed and compared here), or extend it for support decision situation where the information is in non-positive forms.
REFERENCES:
1. Amiri, M., Nosratian, N. E., Jamshidi, A., and Kazemi, A. Developing a new ELECTRE method with interval data in multi attribute decision making problems. Journal of Applied Sciences. 2008; 8(22): 4017-4028.
2. Amiri, M., Zandian, M., Vahdani, B., Yazdani, M., and Soltani, R. A hybrid MCDM model with interval weights and data for convention site selection. Journal of Applied Sciences. 2008; 8(15): 2678-2686.
3. Anand, A. Agarawal, M. and Aggrawal, D. Multiple Criteria Decision Making methods: Application for Managerial Discretion. 2022 Walter de Gruyter GmbH, Berlin/Boston. 2022.
4. Azadfallah, M. Supplier selection with interval SAW for a group of decision makers when a group cannot reach to consensus. Journal of Supply Chain Management System. 2017; 6(3): 25-34.
5. Azadfallah, M. A new MCDM approach for ranking of candidates in voting systems. Int. J. Society Systems Sciences. 2019; 11(2): 119-133.
6. Fan, Z. P. and Liu, Y. An approach to solve group decision making problems with ordinal interval numbers. Cybern B Cybern. 2010; 40(5): 1413-23.
7. Hadiono, K. A practical implementation of interval arithmetic in AHP. IJAHP. 2016; 8(2).
8. Hosseinzadeh, A. and Ghasemi, M. V. Solving fully interval linear programming problems using ranking interval numbers. International Journal Industrial Mathematics. 2021; 13(3): 1-12.
9. Jiang, J., Ren, M., and Wang, J. Interval number Multi Attribute Decision making method based on TOPSIS. Alexandria Engineering Journal. 2022; 61(7): 5059-5064.
10. Liu, J. S. The ranking of interval numbers. Chines Journal of Engineering Mathematics. 2001; 18(4): 103-109.
11. Li, D., Zeng, W., and Yin, Q. Novel ranking method of interval numbers based on the Boolean matrix. Soft Computing. 2018; 22: 4113-4122.
12. Nenad, M. and Zoran, A. Multi-Criteria Decision Making method comparative analysis of PROMETHEE and VIKOR. XVII International Scientific Conference on Industrial Systems (IS’17), Novi Sad, Serbia. 2017: 284-287.
13. Qu, Z., Wan, C., Yang, Z., and Lee, P. T. W. A discourse of Multi Criteria Decision making (MCDM) approaches, Ch.2 [pp.8], in: P. T. W. Lee, Z. Yang (eds.), Multi Criteria Decision making, in marine studies and logistics, Springer International Publishing AG 2018.
14. Romanenkov, Y. Criterion for ranking interval alternatives in a decision making task. I. J. Modern Education and Computer Science. 2024; 2: 72-82.
15. Roszkowska, E. Multi-Criteria Decision making models by applying the TOPSIS method to crisp and interval data. Multi-Criteria Decision making/University of Economics in Katowice. 2011; 6: 200-230.
16. Sayadi, M. K., Heydari, M. and Shahanaghi, K. Extension of VIKOR method for decision making problem with interval numbers. Applied Mathematical Modelling. 2009; 33: 2257-2262.
17. Sengupta, A. and Pal, T. K. Fuzzy preference ordering of interval numbers in decision problems. Springer-Verlag Berlin Heidelberg 2009.
18. Sengupta, A., Pal, T. K., and Chakraborty, D. Interpretation of inequality constraints involving interval coefficients and a solution to interval linear programming. Fuzzy Sets and Systems. 2001; 119: 129-138.
19. Sevastianov, P. Numerical methods for interval and fuzzy number comparison based on the probabilistic approach and Dempster-Shafer theory. Information Sciences. 2007; 177(21): 4645-4661.
20. Sitorus, F., Cilliers, J. J., and Parada, P. R. B. Multi-Criteria Decision Making for the choice problem in mining and mineral processing: applications and trends. Expert Systems with Applications. 2019; 121: 393-417.
21. Song, P., Liang, J., and Qian, Y. A two-grade approach to ranking interval data. Knowledge-Based Systems. 2012; 27: 234-244.
22. Triantaphyllou, E., Shu, B., Sanchez, S. N., and Ray, T. Multi-Criteria Decision making: An operations research approach. Encyclopedia of Electrical and Electronics Engineering, (J. G. Webster, Ed.), John Wiley and Sons, New York, NY. 1998; 15: 175-186.
23. Wang, W. J. Ranking and application of interval numbers based on partial connection numbers. Journal of Gansu Education College. 2008; 1: 48-50.
24. Yao, N., Ye, Y., Wang, Q., and Hu, N. Interval number ranking method considering multiple decision. Iranian Journal of Fuzzy Systems. 2020; 17(2): 115-127.
25. Yue, C. A novel approach to interval comparison and application to software quality evaluation. Journal of Experimental and Theoretical Artificial Intelligence. 2018; 30(5): 583-602.
26. Yue Z. Group decision making with Multi attribute interval data. Information Fusion. 2013; 14: 551-561.
27. Zakeri, S., Cheikhrouhou, N., Konstantas, D., and Barabadi, F. S. (2022). A grey approach for the computation of interactions between two groups of irrelevant variables of decision matrices. Ch. 10[pp. 193-222],
28. A. J. Kulkarni [Editor]: Multi Criteria Decision making; techniques, analysis and applications, Studies in Systems, Decision and Control 407, Springer Nature Singapore Ptd Ltd.
29. Zavadskas, E. K., Antucheviciene, J., Turskis, Z., Adeli, H. Hybrid Multi-Criteria Decision Making methods: A review of applications in engineering, Scientia Iranica. 2016; 23(1): 1-20.
30. Zhang, Y. and Xu, J. The ELECTRE method based on interval numbers and its application to the selection of leather manufacture alternatives. World Journal of Modelling and Simulation. 2006; 2(2): 119-128.
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Received on 09.07.2025 Revised on 07.08.2025 Accepted on 01.09.2025 Published on 07.11.2025 Available online from November 17, 2025 Asian Journal of Management. 2025;16(4):311-316. DOI: 10.52711/2321-5763.2025.00047 ©AandV Publications All right reserved
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