A new distance-based (DIBA) multi-attribute decision making method
چکیده

	Introduction
In recent years, decision making become one of the most fundamental tasks of managers, and the organizational goal achievement depends on its quality. Some experts believe that decision-making is the main essence of management (Simon, 1979). Many decisions have multiple options and multiple criteria. Increasing the number of indicators and options increases the complexity of the decision-making process. Due to these complexities, the problems and methods of multi-criteria decision-making (MCDM) have grown significantly in recent years (Nasrollahi, Fathi & Faghih, 2018) and have been used and paid attention by researchers and practitioners many times. Multiple-Criteria Decision-Making is a fundamental and multi-disciplinary field of Operations Research that helps managers make decisions based on multiple and even conflicting criteria. In organizational issues, where managers deal with various criteria with different scales for measurement, implementing MCDM methods is very useful. It includes the correct expression of objectives, determining different and possible solutions, evaluating their feasibility, assessing the consequences, and the results of implementing each solution, and finally, selecting and implementing the solution. Multiple Criteria Decision Making (MCDM) has been taught for several years at different educational levels in the fields of industrial engineering, management, and applied mathematics. Nowadays, a lot of articles, books, and dissertations have been published in this important and applied field of operations research, and scientific and research studies should continue in this regard.
MCDM methods support decision-makers in finding a compromise solution by considering various criteria (Ramezani, Sadraei & Nasrollahi, 2019). These methods include multi-objective decision-making (MODM) methods and multi-criteria decision-making (MADM) methods. MODM methods seek to find the best solution considering different objectives, while the decision maker in MADM methods wants to prioritize a certain number of decision alternatives. Since the application of MADM methods is more than MODM, Various MADM methods have been developed and utilized over the last recent decades. The focus of this research is on MADM. Many researchers use the two terms MADM and MCDM interchangeably.
Multi-Attribute Decision-Making includes screening and prioritizing several alternatives or choosing one alternative based on multiple and sometimes conflicting attributes with varying degrees of importance, and can provide a comprehensive view of alternatives (Diakoulaki, Mavrotas & Papayannakis, 1995). Each of the decision-making attributes gives different information to the decision-maker, and therefore it is not possible to evaluate the alternatives in general with one attribute. MADM methods provide a process that leads to rational, explainable and justifiable conclusions. In this type of problems, with the aim of evaluating various alternatives, a decision matrix is created as follows, which includes information (quantitative or qualitative) about the alternatives (Ai) considering the attributes (Cj). The importance of the attributes is also shown in the weight of the attributes (Hwang and Yoon, 1981).
1	D=■(&■(■(C_1&C_2 )&■(⋯&C_n ))@■(■(A_1@A_2 )@■(⋮@A_m ))&[■(■(x_11&x_12@x_21&x_22 )&⋯&■(x_1n@x_2n )@⋮&⋱&⋮@■(x_m1&x_m2 )&⋯&x_mn )] )
2	W=[■(■(w_1&w_2 )&■(⋯&w_n ))]
Where ■(■(A_1&A_2 )&■(⋯&A_m )) are possible alternatives, ■(■(C_1&C_2 )&■(⋯&C_n )) indicate the evaluation criteria of the alternatives, x_ij indicates the performance of alternative i is in criterion j, and finally w_j shows the importance (weight) of criterion j. In general, there are four main steps in the implementation of MADM methods: a) determination of decision alternatives and evaluation criteria; b) weighting the criteria; c) evaluation of each alternative in each criterion; and d) choosing the best alternative based on overall evaluation (Keshavarz-Ghorabaee et al., 2021). Various methods for implementing steps 2 and 3 have been introduced by researchers in the literature. Some methods for determining the weight of criteria are: Shannon Entropy (1948), AHP (Saaty, 1988), CRITIC (Diakoulaki, Mavrotas, & Papayannakis, 1995), ANP (Saaty, 2004), BWM (Rezaei, 2015), and SISMW (Nasrollahi et al., 2022). The Simple Additive Weighting (SAW) method is probably the simplest, most well-known and most common MADM method (Yamagishi et al., 2022). Although the final results of this method is not very reliable. However, this technique is useful when we assume that the preferences are independent or separate from each other. Some of the widely used methods for evaluating alternatives are TOPSIS (Hwang & Yoon, 1981), ELECTRE (Roy, 1968), PROMETHEE (Brans, Vincke & Mareschal, 1986), VIKOR, WASPAS, EDAS, COPRAS, and SECA. MADM methods are divided into two categories: non-compensatory methods and compensatory methods. The main disadvantage of the non-compensatory multi-attribute decision-making methods is that the weakness in one criterion is not compensated by the advantage in another criterion. Each criterion, apart from other criteria, is the basis for evaluating competing alternatives. On the other hand, in compensatory models, the decision maker is willing to trade off between criteria. This means that the difference in the values of one criterion can be compensated by the values of other criteria. A change in one attribute is offset by an opposite change in the other attribute or attributes. Most of the methods used by researchers are in the category of compensatory methods. Each of the compensatory methods has advantages and disadvantages and can be efficient in different situations. In this study, we do not aim to examine the advantages and disadvantages of these methods; but it can be said that the efficiency of the evaluation method is very important to determine the importance of alternatives.
Given that the applications of MADM are mostly more than MODM, wide various techniques have been developed for MADM by researchers over the last 60 years. The problems related to Multiple Attribute Decision Making (MADM) include components such as objective or a set of objectives, decision-maker or a group of decision-makers, a set of evaluation attributes, a set of decision alternatives, a set of unknown variables, or decision variables and a set of results obtained from any pair of alternatives. The criterion for the central element of this structure is a decision matrix such as a set of rows and columns expressing the decision outcomes for a set of alternatives and evaluation attributes.
The VIKOR method was developed for multi-criteria optimization of complex systems. It determines the compromise ranking-list, the compromise solution, and the weight stability intervals for preference stability of the compromise solution obtained with the initial (given) weights. This method focuses on ranking and selecting from a set of alternatives in the presence of conflicting criteria. It introduces the multi-criteria ranking index based on the particular measure of ‘‘closeness’’ to the ‘‘ideal’’ solution (Opricovic, 2007).
TOPSIS (technique for order preference by similarity to an ideal solution), one of the known classical MCDM method, was first developed by Hwang and Yoon (1981). The basic principle is that the chosen alternative should have the shortest distance from the ideal solution and the farthest distance from the negative-ideal solution. In the process of TOPSIS, the performance ratings and the weights of the criteria are given as crisp values.
The Simple Multi-Attribute Rating Technique (SMART) was introduced by Winterfeldt and Edwards in 1986 (Edwards & Barron, 1994; Alinezhad & Khalili, 2019), in which a limited number of alternatives are examined based on a limited number of attributes. The present method aimed to rank the alternatives by a combination of quantitative and qualitative attributes. The SMART method with the feature of compensatory attributes as well as the lack of need for the dependence or independence of attributes, can be considered as one of the appropriate techniques for choosing the best alternative according to the different attributes.
The REGIME method, initially introduced by Hinloopen, Nijkamp, and Rietveld in 1983 (Hinloopen & Nijkamp, 1986; Hinloopen & Nijkamp, 1990) is a multiple attribute qualitative method which solves the problem using the REGIME matrix, and a final ranking of the alternatives is done. In the final ranking, the weight of attributes, introduced by the decision maker, is important and can influence the results. This technique is used for ranking the sawability of ornamental and building stones (Kamran et al., 2017) and evaluation and ranking of coastal areas (Hinloopen, Nijkamp, & Rietveld, 1983) due to its features. The strength point of this method is the REGIME matrix formation, which is a combination of quantitative and qualitative attributes at penultimate stage. It allows decision makers to use this technique in many cases without any need to convert the qualitative attributes into quantitative attributes.
In general, many scholars have developed techniques for prioritizing alternatives, which emphasizes the importance of having robust methods for ranking alternatives. Each method has its advantages as well as disadvantages. The impact of selecting the correct alternative on the success or failure of a project or a decision-making situation is very high. It is a direct relationship between ranks of alternatives and the cost and/or benefit of a decision. In view of this, this work proposes a new technique for prioritizing alternatives. This method is named a new distance-based (DIBA) multi-attribute decision making method, and has some features that have not been considered in the other MCDM methods.
The remainder of this paper is structured as follows: In Section 2, a new distance-based (DIBA) multi-attribute decision making method is presented in detail. In Section 3, we use numerical example to illustrate the process of the DIBA method. In Section 4, a comparative sensitivity analysis is made to demonstrate the efficiency of the proposed method. Section 5 concludes this paper offering a discussion of the findings and conclusions.

	New DIstance-BAsed (DIBA) multi-attribute decision making method 
In this section, we propose a new distance-based multi-attribute decision-making method called DIBA. This method is very useful and fast when we have conflicting criteria. In the compromise MCDM methods such as VIKOR and TOPSIS (Opricovic and Tzeng, 2004), the best alternative is obtained by calculating the distance from ideal and nadir solutions. The desirable alternative has a lower distance from the ideal solution and a higher distance from the nadir solution in these MCDM methods. In this method, we don’t need to calculate the ideal and the nadir solution. Instead, we make some pairwise comparisons between alternatives. The alternatives are evaluated according to their distance from each other. The steps for using the DIBA method are presented as follows:
Step 1. Construct the decision-making matrix (D). The decision-making matrix shown as follows:
1	D=■(&■(■(C_1&C_2 )&■(⋯&C_n ))@■(■(A_1@A_2 )@■(⋮@A_m ))&[■(■(x_11&x_12@x_21&x_22 )&⋯&■(x_1n@x_2n )@⋮&⋱&⋮@■(x_m1&x_m2 )&⋯&x_mn )] )
2	W=[■(■(w_1&w_2 )&■(⋯&w_n ))]
Where x_ij (x_ij≥0) denotes the performance value of ith alternative on jth attribute, and W is the weight of attributes (i∈{1,2,…,m}) and (j∈{1,2,…,n}).
Step 2. Calculate the normalized decision matrix. We use Euclidean normalization of performance values as follows:
3	n_ij=x_ij/√(∑_(i=1)^m▒x_ij^2 )
Obviously, n_ij indicates the normalized amount of the decision matrix of i^th alternative in the j^th attribute.
Step 3. Calculate the weighted normalized decision matrix. The weighted normalized performance values are calculated as follows:
4	r_ij=w_j n_ij
5	■(W=[■(■(w_1&w_2 )&■(⋯&w_n ))]&and&∑_(j=1)^n▒〖w_j=1〗)
Where (0<w_j<1) denotes the weight of j^th criterion.
Step 4. Pairwise comparisons: The Euclidean distance between the pair of alternatives is computed. In this step, the distance of alternative a_i compared to alternative a_k is calculated in the criteria where a_i is better than a_k. The distance of alternative a_k compared to alternative a_i is also calculated in criteria where a_k is better than a_i.
6	d_j (a_i,a_k )=r_j (a_i )-r_j (a_k)
d_j (a_i,a_k ) is the difference between the evaluations of two alternatives for criterion C_j. Of course, these differences depend on the measurement scales used and are not always easy to compare for the decision maker.
Step 5. Calculating preference flows: Each alternative a is facing (m-1) other alternatives in d. We define positive (leaving) and negative (entering) preference flows. The sum of the values in each row shows the superiority or strength of an alternative over other alternatives and is called the positive (leaving) flow. The sum of the values of each column also shows the superiority of other alternatives to one alternative and is called negative (entering) flow.
7	Φ^+= ∑_(j=1)^n▒〖d_j (a,a_i ) 〗,i={1,2,…,m}

8	Φ^-= ∑_(j=1)^n▒〖d_j (a_i,a)〗,i={1,2,…,m}

Φ^+ is the positive flow and it checks the preference of a over m-1 other alternatives. This is actually the strength of alternative a. The higher the strength of Φ^+, the better alternative a is. Φ^- is a negative flow and checks the preference of m-1 other alternatives over alternative a. This is actually the weakness of alternative a. The lower the power of Φ^-, the better alternative a is.
Step 6. Net outranking flow: To calculate the net outranking flow, the positive outranking flow should be subtracted from the negative outranking flow.
9	Φ_i^n=Φ_i^+-Φ_i^-  ,i={1,2,…,m}

Φ_i^n is net flow and it represents positive and negative flow values simultaneously. The higher the strength of Φ_i^n, the better Alternative a is.

	Computational Analysis
In this section, a computational analysis is made to validate the results of the proposed method. This analysis is presented in the two following sub-sections. Firstly, the overall performance of proposed method is analyzed by solving a numerical example and secondly, we make a comparative analysis based on some existing MCDM methods. 
	Validation of the proposed method
To make the validation of the proposed method, a numerical example is borrowed from the study of Asgharizadeh and Nasrollahi (2007). Based on step 1 the decision-matrix of the example along with the weight of each criterion is shown in Table 1 (step 1).
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