Algorithms for interval-valued fuzzy soft sets in emergency decision making based on WDBA and CODAS with new information measure.

This paper aims to give deeper insights into decision making problem based on interval-valued fuzzy soft set (IVFSS). Firstly, a new score function for interval-valued fuzzy number is proposed for tackling the comparison problem. Subsequently, the formulae of information measures (distance measure, similarity measure and entropy) are introduced and their transformation relations are pioneered. Then, the objective weights of various parameters are determined via new entropy method, meanwhile, we develop the combined weights, which can show both the subjective information and the objective information. Moreover, we propose three algorithms to solve interval-valued fuzzy soft decision making problem by Weighted Distance Based Approximation (WDBA), COmbinative Distance-based ASsessment (CODAS) and similarity measure. Finally, the effectiveness and feasibility of approaches are demonstrated by a mine emergency decision making problem. The salient features of the proposed methods, compared to the existing interval-valued fuzzy soft decision making methods, are (1) it can obtain the optimal alternative without counterintuitive phenomena; (2) it has a great power in distinguishing the optimal alternative; and (3) it can avoid the parameter selection problems.

Introduction

During the process of economic and social globalization in the 21st century, the world is undergoing an unheard-of period of crisis so far. A substantial emergency events arose, such as September 11 attacks in 2001, severe acute respiratory syndrome (SARS) in 2003, Wenchuan earthquake in 2008, Tianjin explosion accident in 2015, Jiuzhai Valley earthquake in 2017, and so on, which demolished the country or the society involving massive humans, material resources, economic, or local environment. The influences of the emergency incidents ordinarily beyond the ability of the affected country or society to dispose of its own resources. As a consequence, the topic about emergency response has touched off the heated discussions in recent times (Ren, Xu, & Hao, 2017). Emergency decision making, one of the significant domains in the emergency response, is urgent to be explored by building the decision making techniques to polish up the efficiency and productiveness of the emergency response, which can enormously reduce the casualties, environment disruption and the economic losses. Every barber knows that one of the momentous characters of the emergency response is timeliness, which implies that the rescue forces are required for taking impactful actions in the short time. Meanwhile, for the time-starved in the emergency response, there exists much uncertain and unknown information. Under such circumstance, the experts make decisions with a great power in distinguishing the optimal alternative rather than in low differentiated alternatives. Furthermore, the indeterminacies of the emergency circumstance determine that the experts can’t offer the precise decision making information when they confront diverse choices. While a wide diverse of the existing theories such as theory of fuzzy set (Zadeh, 1965), theory of probability, and theory of rough set (Pawlak, 1982) have been initiated to simulate vagueness. Each of these theories has its inherent insufficiencies as pointed out in (Molodtsov, 1999) nevertheless. The theory of soft set (SS), poineered by Molodtsov (Molodtsov, 1999), has come fresh from the deficiency of the parameterization tools of those groundbreaking theories (Pawlak, 1982, Zadeh, 1965). It has potential applications in diverse fields such as rule mining (Feng, Cho, Pedrycz, Fujita, & Herawan, 2016), game theory (Deli & Çağman, 2016), feature selection (Yang, Xiao, Xu, Wang, & Pan, 2016), decision making (Aktaş & Çağman, 2016), parameter reduction (Xie, 2016).

By combining soft sets with other mathematical models, diverse extensions of soft set with their applications have been presented such as fuzzy soft set (FSS) (Alcantud, 2016a, Alcantud, 2016b, Alcantud and Mathew, 2017, Maji et al., 2001), Pythagorean fuzzy soft set (PFSS) (Peng, Yang, Song, & Jiang, 2015), intuitionistic fuzzy soft set (IFSS) (Maji, Biswas, & Roy, 2001), neutrosophic soft set (NSS) (Peng & Liu, 2017), rough soft set (Zhan, Ali et al., 2017, Zhan, Liu et al., 2017, Zhan, Zhou et al., 2017), hesitant fuzzy soft set (Peng and Dai, 2017, Wang et al., 2014, Wang et al., 2015).

The above extension models are all proposed to deal with uncertainties by taking advantages of soft set. In those extensions of SSs, the value of membership is either a fuzzy value or rough value. But in fact, the membership degree may be an interval values in a SS, so combining the interval-valued fuzzy set with soft set, Son (2007) proposed interval-valued fuzzy soft set (IVFSS). Yang, Lin, and Yang (2009) proposed choice values method for interval-valued fuzzy soft set method. Moreover, there have been some applications of interval-valued fuzzy soft sets such as decision making (Chen and Zou, 2017, Feng et al., 2010, Mukherjee and Sarkar, 2014, Peng et al., 2017, Peng and Yang, 2017, Xiao et al., 2013, Yang and Peng, 2017, Yuan and Hu, 2012), information measure (Jiang et al., 2013, Mukherjee and Sarkar, 2014, Peng et al., 2017, Peng and Yang, 2015), matrix theory (Rajarajeswari & Dhanalakshmi, 2014), algebraic structure (Liu, Feng, Yager et al., 2014, Liu, Feng and Zhang, 2014), parameter reduction (Ma, Qin, Sulaiman, Herawan, & Abawajy, 2014), medical diagnosis (Chetia & Das, 2010).

Due to the drawbacks (counterintuitive phenomena (Peng et al., 2017), discrimination problem (Feng et al., 2010, Peng et al., 2017, Yang et al., 2009, Yuan and Hu, 2012), parameter selection problems (Feng et al., 2010)) of the existing decision making methods for IVFSSs, it may be difficult for decision makers to select optimal or convincible alternatives. Therefore, the aim of this paper is to tackle the three challenges mentioned above by developing three MCDM approaches to managing evaluation information for IVFSSs, which not only have a great power in distinguishing the optimal alternative, but also can obtain an optimal alternative out of counterintuitive phenomena and parameter selection problems. The decision making methods are shown in the following.

Weighted Distance Based Approximation (WDBA), poineered by Rao and Singh (2012), measures the distance from the optimal point (best value of alternatives) and non-optimal point (worst value of alternatives). Finally, the alternatives are ranked by their suitability index (SI). The alternative having the least value of SI is ranked at first position and with the maximum value at last. WDBA approach has already been applied in many fields such as E-learning websites selection (Garg and Jain, 2017, Jain et al., 2016), COTS component selection (Garg, Sharma, & Sharma, 2016), selection of software effort estimation (Bansal, Kumar, & Garg, 2017).

Combinative Distance-based Assessment (CODAS), presented by Ghorabaee, Zavadskas, Turskis, and Antucheviciene (2016), measures the overall performance of an alternative by Euclidean and Hamming distances from the negative-ideal point. The CODAS utilizes the Euclidean distance as the primary measure of assessment. If the Euclidean distance of two alternatives is very close to each other, the Hamming distance is utilized to compare them. The degree of closeness of Euclidean distance is set by a threshold parameter. The Euclidean distance and Hamming distance are measured for -norm and -norm indifference spaces, respectively (Yoon, 1987). Therefore, in the CODAS method, we first assess the alternatives in an -norm indifference space. If the alternatives are not comparable in this space, we go to an -norm indifference space. To perform this process, we should compare each pair of alternatives. CODAS approach has already been applied in market segment evaluation (Ghorabaee, Amiri, Zavadskas, Hooshmand, & Antuchevic̆ienė, 2017).

In order to compute the information measure (distance measure, similarity measure and entropy) of two interval-valued fuzzy soft sets, we propose a new method for constructing of distance measure and similarity measure. Meanwhile, some special properties are discussed in detail. Later, we apply the proposed similarity measure for decision making.

Considering that different parameters’ weights will have an effect in the ranking results of alternatives, inspired by Peng et al., 2017, Peng and Yang, 2017, we also develop a novel method to determine the parameter weights by combining the subjective elements with the objective ones. This model is not the same from the existing interval-valued fuzzy soft weight determining methods, which can be divided into two aspects: one is the subjective weighting evaluation methods and the other is the objective weighting determine methods, which can be computed by the new proposed entropy determining method. The subjective weighting methods concentrate on the preference information of the experts (Feng et al., 2010), while they overlook the objective information. The objective weighting determine methods do not take the preference of the decision maker into consideration, in other words, these methods fail to take the risk attitude of the decision maker into account (Xiao et al., 2013). The feature of our weighting model can show both the subjective information and the objective information. Consequently, combining objective weights with subjective weights, a combined model to achieve parameters’ weights is presented. Moreover, our weight determining method has keep in accordance with the references (Peng et al., 2017, Peng and Yang, 2017) when using the corresponding decision making method for achieving the final decision results.

To achieve these goals, the main contributions are as follows:

A new weight determining model is proposed for avoiding the influence of subjective factor and objective factor, respectively.

The new information measure (similarity measure, distance measure, entropy) for interval-valued fuzzy soft set is presented and the new entropy is used for determining the weight.

A new score function is examined for tackling the comparison problem.

Three proposed algorithms with some existing algorithms (Feng et al., 2010, Peng et al., 2017, Yang et al., 2009, Yuan and Hu, 2012) are compared by some examples. Their objective evaluation is carried out, and the methods which maintain consistency of their results are chosen.

The remainder of this paper is organized as follows: In Section 2, we review some fundamental conceptions of soft sets and interval-valued fuzzy sets. In Section 3, new information measure (similarity measure, distance measure, entropy) is conceived and their inner transformation relation is studied. In Section 4, three interval-valued fuzzy soft decision making approaches based on WDBA, CODAS and similarity measure are shown and a numerical example is given to illustrate the proposed methods. In Section 5, we compare the novel proposed approaches with the existing interval-valued fuzzy soft set decision making approaches, and show the effectiveness of the proposed approaches. The paper is concluded in Section 6.

Preliminaries

In this section, we will briefly recall the basic concepts of interval-valued fuzzy sets, interval-valued fuzzy soft sets and score function.

Zadeh, 1975

An interval-valued fuzzy set (IVFS) I in U is given bywhere . For simplicity, we call an interval-valued fuzzy number (IVFN) denoted by .

Xu and Da, 2002

Let and be two IVFNs, , then their operational laws are defined as follows:

;

;

;

;

iff and ;

;

.

From (1), Peng and Yang (2017) see that it may be not satisfied the definition of IVFN when , but it has fundamentality of computation. Moreover, for (7), although it can be easily known that it will be out of positive number range, it also has fundamentality of computation. That is to say, it can see IVFN as a special interval number which can satisfy the above cases.

Son, 2007

Let U be an initial universe and A be a set of parameters, a pair is called an interval-valued fuzzy soft set over , where F is a mapping given by . denotes the set of all interval-valued fuzzy subsets of U.

is an interval-valued fuzzy subset of U, and it is called an interval-valued fuzzy value set of parameter . Let denote the membership value that object x holds parameter , then can be written as an interval-valued fuzzy set that .

The complement of an interval-valued fuzzy soft set , is denoted by and is defined by . .

In the following, Peng and Yang (2017) give a comparative law based on interval-valued fuzzy soft sets by score function and deviation function.

Peng and Yang, 2017

Let be an interval-valued fuzzy soft set, then the score function of is defined as follows:

Let be an interval-valued fuzzy soft set, then the deviation function of is defined as follows:

Based on above score function and deviation function, Peng and Yang (2017) derive the following comparative law.

For two interval-value fuzzy soft sets and , then

if , then is superior to , denoted by ;

if , then is superior to , denoted by ;

if , then

if , then is superior to , denoted by ;

if , then is equivalent to , denoted by ;

if , then is superior to , denoted by .

For simplicity, we only take into consideration for computing the score function of two IVFNs. The deviation function has no reference value due to . Hence, the above score function can be rewritten as follows:

Let be an IVFN, then the score function of is defined as follows:

Suppose that and . If we use the above score function to select the desirable alternative, we can get . Hence, we can’t make a distinction between and , which means that this score function can’t achieve the order in this case.

Notice the deficiencies of the score function proposed by Peng and Yang (2017), in the following, we give a superior score function which can identify the difference of two IVFNs.

For an IVFN , its score function is defined as follows:

For an IVFN monotonically increases with respect to and , respectively.

According to Eq. (5), we get the first partial derivative of with ,

Similarly, we can obtain the first partial derivative of with ,

Hence, we can have monotonically increases with respect to and , respectively. □

For an IVFN , a new score function satisfies:

;

iff ;

iff .

According to Theorem 2.12, we can know that if we only take or into consideration, obtain the min-value or max-value when or . That is to say, and . Therefore, . □

Let and be two IVFNs, if and , then .

According to Theorem 2.12, we can know that monotonically increases with respect to or .

Hence, if and , then . □

To test the feasibility of the proposed score function for ranking IVFNs, Table 1 shows the comparison between the results achieved by the proposed score function and the cases of the score function obtained by Peng and Yang (2017).

Table 1

The comparison of score functions.

Examples	s_pPeng and Yang (2017)	Ranking	s_pxd	Ranking
F(ε)(x)=[0.5,0.6]	s_p(F(ε)(x))=0.05		s_pxd(F(ε)(x))=1.0047
		F(ε)(x)=G(ε)(x)		F(ε)(x)>G(ε)(x)
G(ε)(x)=[0.45,0.65]	s_p(G(ε)(x))=0.05		s_pxd(G(ε)(x))=0.9210
F(ε)(x)=[0.6,0.7]	s_p(F(ε)(x))=0.15		s_pxd(F(ε)(x))=1.2271
		F(ε)(x)=G(ε)(x)		F(ε)(x)>G(ε)(x)
G(ε)(x)=[0.5,0.8]	s_p(G(ε)(x))=0.15		s_pxd(G(ε)(x))=1.0384
F(ε)(x)=[0.4,0.8]	s_p(F(ε)(x))=0.1		s_pxd(F(ε)(x))=0.8724
		F(ε)(x)=G(ε)(x)		F(ε)(x)<G(ε)(x)
G(ε)(x)=[0.5,0.7]	s_p(G(ε)(x))=0.1		s_pxd(G(ε)(x))=1.0178
F(ε)(x)=[0.2,0.6]	s_p(F(ε)(x))=-0.1		s_pxd(F(ε)(x))=0.5848
		F(ε)(x)=G(ε)(x)		F(ε)(x)<G(ε)(x)
G(ε)(x)=[0.3,0.5]	s_p(G(ε)(x))=-0.1		s_pxd(G(ε)(x))=0.6823

From Table 1, we can conclude that the proposed score function can effectively solve the deficiencies of proposed by Peng and Yang (2017). That is to say, the proposed score function can distinguish the difference of alternatives when the existing score function cannot deal with.

New information measure for interval-valued fuzzy sets

Liu, 1992

Let and be three interval-valued fuzzy sets (IVFSs) on . A distance measure is a mapping , possessing the following properties:

;

iff is a crisp set;

iff ;

;

If , then and .

Let and be two IVFSs, then is a distance measure. where t identifies the level of uncertainty and p is parameter-varying.

Let and be two IVFS.

We can have the following equations for any i:

It can be easily known that , , , .

Hence, we can have the following inequalities:then we have

That is,

Furthermore,

Similarly, we have

Finally, we have

Consequently,

Hence, is a crisp set.

Let and be two IVFSs, if , then , , , .

Therefore, the distance .

Let and be two IVFSs.

We can have the following equations for any i:

According to the definition of absolute value, we have.

Therefore, we can have .

Let , and be three IVFSs. The distance measures between and , and and are given as follows:

We can give the following equations:

If , then and .

Hence, we have the following inequalities:

So it is easily known that:

Furthermore, we can finally have the inequalities: and .

We can conclude that is a distance measure between IVFS and since it satisfies . □

is a weighted distance measure between two IVFSs and . where t identifies the level of uncertainty, p is parameter-varying, and is the weights of the attributes with .

The proof is similar to Theorem 3.2. □

Let and be three interval-valued fuzzy sets (IVFSs) on . A similarity measure is a mapping , possessing the following properties:

;

iff is a crisp set;

iff ;

;

If , then and .

Let and be two IVFSs, then is a similarity measure. where t identifies the level of uncertainty and p is parameter-varying.

It is similar to Theorem 3.2. □

is a weighted similarity measure between two IVFSs and . where t identifies the level of uncertainty, p is the norm, and is the weights of the attributes with .

The proof is similar to Theorem 3.2. □

Zeng and Li, 2006

Let be an interval-valued fuzzy set on . An entropy measure is a mapping , possessing the following properties:

iff is crisp set;

iff for any i;

for all ;

iff and for or and for .

Let be an IVFS, then is an entropy measure. where p is parameter-varying.

(E1) for any i or or .

Hence, is a crisp set.

(E2) and (E3) is straightforward.

(E4) If , then.

and for or and for . □

Let be an IVFS, then .

It is straightforward by three equations of similarity measure, distance measure and entropy measure. □

Three algorithms for interval-valued fuzzy soft set based decision making

Problem description

Let be a limited set of m alternatives, be a finite set of n parameters, and the weight of parameter is , , . is interval-valued fuzzy soft set which can be shown in Table 2 . denotes the possible IVFN of the ith alternative satisfying the jth parameter given by the decision maker.

Table 2

The tabular representation of interval-valued fuzzy soft set .

	ε_1	ε_2	⋯	ε_n
x_1	F(ε_1)(x_1)	F(ε_2)(x_1)	⋯	F(ε_n)(x_1)
x_2	F(ε_1)(x_2)	F(ε_2)(x_2)	⋯	F(ε_n)(x_2)
⋮	⋮	⋮	⋱	⋮
x_m	F(ε_1)(x_m)	F(ε_2)(x_m)	⋯	F(ε_n)(x_m)

The method of determining the combined weights

Determining the objective weights: the entropy method

Shannon entropy (Shannon, 1948) evaluates the expected information content of a certain message. The degree of uncertainty in information can be measured using the entropy concept. Information entropy idea can regulate decision making process because it is able to measure existent contrasts between sets of data and thus clarify the intrinsic information for decision maker. Inspired by Shannon, we propose a new entropy method (Theorem 3.8) to compute objective weights.

The interval-valued fuzzy entropy of jth parameter is calculated as follows:

The weight of jth parameter can be computed as follows:

When the decision information of parameter with each alternative is interval-valued fuzzy number, from the perspective of the original decision information, the parameter information increasingly blurred, more uncertain, the property for the amount of information available for the less, the greater entropy should be given less weight, and vice verse. Hence, the use of interval-valued fuzzy entropy to determine the weights of parameters, both to reduce the loss of information evaluation, and can reflect the willingness of decision makers.

Determining the combined weights: the linear weighted comprehensive method

Suppose that the vector of the subjective weight, given by the decision makers directly, is , where , . The vector of the objective weight, computed by Eq. (12) directly, is , where , .

Therefore, the vector of the combined weight can be defined as follows:where , .

The subjective weight and objective weight are assembled by non-linear weighted comprehensive method. Based on the multiplier effect, the greater the value of the objective weight and subjective weight are, the greater the combined weight is, or vice versa. Furthermore, we can know that the Eq. (13) breaks through the limitation of only considering either subjective or objective influence. The superiority of Eq. (13) is that the rankings of alternatives and the parameter weights can show both the objective information and the subjective information.

In the following, we will apply the WDBA, CODAS and similarity measure methods to interval-valued fuzzy soft sets.

The interval-valued fuzzy soft decision making approach based WDBA

To solve the MCDM problem with interval-valued fuzzy soft information, we try to present an interval-valued fuzzy soft WDBA approach, which is measured the distance from the optimal point (best value of alternatives) and non-optimal point (worst value of alternatives) and ranked the alternatives according to their suitability index.

At the beginning, we consider to normalize evaluation information because there may be some benefit parameters and cost parameters in decision matrix. The above two kinds of parameters react oppositely, in other words, the larger value reflects the better performance of a benefit parameter but means the worse performance of a cost parameter. Therefore, in order to ensure all parameters are compatible, we make further efforts to convert the cost parameters into benefit parameters by the following formula:According to Eq. (15), we can achieve the normalized interval-valued fuzzy soft matrix .

Next, we calculate the score function of by Eq. (16),

In order to show the standardized matrix visually, we express it in an average value matrix and revised deviation matrix . The concrete process is shown as follows:

The developed WDBA method is based on the concept that the selected alternative (optimum) should have the shortest distance from the ideal solution (optimal possible alternative) and be farthest from the anti-ideal solution (worst possible alternative). The ideal points are the set of parameter values ideally (most) desired. The anti-ideal points are the set of parameter values ideally not desired at all or least desirable. The ideal points, denoted by and anti-ideal points, denoted by are established from standardized decision matrix, which can be defined as follows: where .

This mechanism guarantees that the top ranked alternative is closest to the ideal solution and farthest from the anti-ideal solution. Euclidean distance is the shortest distance among two points. The overall performance index score of an alternative is ascertained by its Euclidean distance between ideal solution and anti-ideal solutions. This distance is interrelated with the parameters’ weights and should be united in the distance measurement. This is reason that all alternatives are compared with ideal and anti-ideal solutions, rather than directly among themselves. Therefore, weighted Euclidean distances are considered in the developed method.

Fig. 1 shows Euclidean distance of an alternative to ideal and anti-ideal solutions in 2D space in case of two parameters and . The rectangular box means the real domain. The Euclidean distance between points P and Q in n dimensional space is the length of the line segment, PQ. In Cartesian coordinates, if and are two points in Euclidean n-space, so the distance from P to Q is given by .

Fig. 1

Euclidean distance of an alternative to ideal and anti-ideal points in 2D space (case of two parameters).

Weighted Euclidean distance (WED) between an alternative and ideal point is expressed as and between an alternative and anti-ideal point is expressed as . where .

The suitability index (SI) denotes the relative closeness of an alternative to the ideal solution. The higher the index suitability scores of an alternative, the closer the alternative to the ideal solution. The alternative for which the value of suitability index is the highest is the optimal choice for the considered decision making problem. The higher the suitability index, the higher the ranking of the alternative. The suitability index is computed as follows:

A final decision can be achieved by taking into account its real considerations. All possible constraints likely to be experienced by the decision maker have to be observed and studied at the moment. These include management, social, availability, economic and political constraints, among others. However, compromise may be obtained in favor of an alternative with a higher value of SI.

In general, interval-valued fuzzy soft WDBA approach involves the following steps:

WDBA

It is indispensable to illustrate that in the interval-valued fuzzy soft WDBA approach, the information in decision matrix is expressed by IVFNs to represent the DMs’ opinions. Expressed by the membership degree with a possible interval values, the IVFNs are quite efficacious for capturing the uncertainty and imprecision of the DMs in the MCDM problem. In addition, the interval-valued fuzzy soft WDBA approach is an invaluable tool to manage the decision making problems with IVFNs possessing a great power in distinguishing the optimal alternative and obtaining the optimal alternative without counterintuitive phenomena. But other approaches (Feng et al., 2010, Peng et al., 2017, Yang et al., 2009, Yuan and Hu, 2012) to solving the decision making problems with IVFNs do not have this desirable characteristic.

The interval-valued fuzzy soft decision making approach based CODAS

To solve the MCDM problem with interval-valued fuzzy soft information, we try to present an interval-valued fuzzy soft CODAS approach. It is a new and efficient MCDM method which introduced by Ghorabaee et al. (2016) recently. The desirability of alternatives in the CODAS method is determined according to -norm and -norm indifference spaces for criteria. According to these spaces, in the procedure of this method, a combinative form of the Euclidean and Hamming distances is used for the calculation of the assessment score of alternatives. However, the Euclidean and Hamming distances are defined in a crisp environment and we can’t utilize them in interval-valued fuzzy soft set problems. The aim of this study is to develop an interval-valued fuzzy soft CODAS method. In order to achieve this aim, we use the fuzzy weighted Euclidean distance and fuzzy weighted Hamming distance, instead of the crisp distances.

At the beginning, we consider to normalize information by Eq. (14) and calculate the score function of by Eq. (15).

In order to compute the weighted normalized decision matrix , we express it in weighted normalized performance values, shown as follows:where denotes the combined weight of jth parameter, and .

The proposed CODAS method is based on the negative-ideal solution (NIS). We define a negative-ideal solution as follows: where .

Later, we compute the Euclidean distance measure and Hamming distance of alternatives from negative-ideal solution as follows:

Based on above two distances, we can construct the relative assessment matrix, shown as follows: where and denotes a threshold function to recognize the equality of the Euclidean distance of two alternatives, and is shown as follows:

In this function, is the threshold parameter that can be determined by decision-maker. It is suggested to determine this parameter at a value between 0.01 and 0.05 (a value close to 0). If the difference between Euclidean distance of two alternatives is more than , these two alternatives would be compared by the Hamming distance. It has been verified that when  = 0.02, the outcomes are consistent with the original data (Ghorabaee et al., 2017).

Now, we can collect the assessment score of each alternative. The alternative for which the value of assessment score is the highest is the optimal choice for the considered decision making problem shown as follows:

To describe the proposed method, Ghorabaee et al. (2017) utilized a simple situation with seven alternatives and two parameters . Suppose that weighted normalized performance values have been calculated. These values are dimensionless and between 0 and 1. Fig. 2 (Ghorabaee et al., 2017) shows the position of all alternatives according to these values.

Fig. 2

A simple graphical example with two parameters.

In general, interval-valued fuzzy soft CODAS approach involves the following steps:

CODAS

The interval-valued fuzzy soft CODAS approach is an invaluable tool to manage the decision making problems with IVFNs obtaining the optimal alternative without counterintuitive phenomena. But other approaches (Feng et al., 2010, Peng et al., 2017, Yang et al., 2009, Yuan and Hu, 2012) to solving the decision making problems with IVFNs do not have this desirable characteristic.

The interval-valued fuzzy soft decision making approach based similarity measure

In this subsection, we introduce a method for the MCDM problem by the proposed similarity measure between IVNSs. The concept of ideal point has been applied to help determine the best alternative in the decision process. Although the ideal alternative does not exist in actual cases, it does offer an invaluable theoretical construct against which to appraise alternatives. Hence, we define the ideal alternative as the IVFN for .

Consequently, according to Eq. (8), the proposed similarity measure S between an alternative and the ideal alternative presented by the IVFSs is defined as follows:

In general, interval-valued fuzzy soft similarity measure approach involves the following steps:

Similarity measure

A case study in mine emergency decision making

We will consider the emergency decision making problems of mine accidents employing the proposed decision making algorithms based on WDBA, CODAS and similarity measure. The mine explosion is one the most hazardous dangers in mine accidents. The mine explosion enormously threatens the safety of work and life and imperils the safety production of mine. Since the explosion accidents often occur unexpectedly and suddenly, it is not easy to predict the accident to a crumb and have enough preparations and emergency actions ahead of time. Therefore, the emergency response plans and the simulations of the accidents are a requisite approach in disaster preparedness and appropriate responses. The high quality and feasibility of the emergency plans will directly influence the later emergency actions, and affect the evolution of disasters. Consequently, the evaluation and decision of the given emergency plans with simulations is considered essential for the disaster management of mine accidents (Hao, Xu, Zhao, & Fujita, 2017).

Assume that there are five emergency plans to be considered for an explosion accident in the coal mine. The expert chooses the decision parameters set to be the noxious gas concentration (denoted as gas), reducing casualty of current events (denoted as casualty), the smoke and the dust level (denoted as smoke), the feasibility of rescue operations (denoted as feasibility) and repairing facility damages caused by the emergency (denoted as facility). Based on the general evolving principle and the characteristics of the mine accidents, we can determine that all parameters are benefit parameters. Suppose that the expert has the following prior weight set given by his/her prior experience or preference: . The assessments for emergency plans arising from questionnaire investigation to the expert and constructing an interval-valued fuzzy soft set with its tabular form given by Table 3 .

Table 3

The tabular form of interval-valued fuzzy soft set in Example 5.3.

	ε_1	ε_2	ε_3	ε_4	ε_5
x_1	[0.55,0.65]	[0.41,0.62]	[0.2,0.3]	[0.4,0.6]	[0.7,0.8]
x_2	[0.58,0.67]	[0.51,0.62]	[0.2,0.3]	[0.4,0.5]	[0.6,0.8]
x_3	[0.61,0.71]	[0.55,0.65]	[0.3,0.4]	[0.5,0.6]	[0.7,0.8]
x_4	[0.54,0.65]	[0.61,0.68]	[0.2,0.3]	[0.4,0.5]	[0.6,0.7]
x_5	[0.53,0.61]	[0.43,0.49]	[0.3,0.4]	[0.4,0.6]	[0.4,0.6]

In what follows, we utilize the algorithms proposed above to select emergency plans under interval-valued fuzzy soft information.

According to Algorithm 1, Algorithm 2, Algorithm 3 shown in Table 4 , we can conclude that the final decision results are the same, i.e., is the most desirable emergency plan. Hence, the three approaches proposed above are effective and available.

Table 4

Final results and ranking.

Algorithms	Results	Ranking	Optimal alternative
Algorithm 1: WDBA	SI_1=0.4552, SI_2=0.4708, SI_3=0.8745, SI_4=0.4319, SI_5=0.2953	x_3≻x_2≻x_1≻x_4≻x_5	x_3
Algorithm 2: CODAS	RA_1=0.3385, RA_2=-0.2215, RA_3=0.6974, RA_4=-0.2115, RA_5=-0.6030	x_3≻x_1≻x_4≻x_2≻x_5	x_3
Algorithm 3: Similarity measure	S_1=0.6941, S_2=0.6955, S_3=0.7349, S_4=0.6935, S_5=0.6872	x_3≻x_2≻x_1≻x_4≻x_5	x_3

Comparison of the newly proposed approaches with the other approaches to interval-valued fuzzy soft set based decision making

Comparison of the newly proposed three approaches with their own

Comparison of computational complexity

It is easily known that Algorithm 1, Algorithm 2 will require more computational complexity than Algorithm 3. Hence, if we take the computational complexity into consideration, the Algorithm 3 is given priority to make decision.

Comparison of applied situation

When the decision maker takes ideal point into consideration in the decision process, the Algorithm 2, Algorithm 3 are given priority to make decision.

Comparison of discrimination

Comparing the results in Algorithm 3 with Algorithm 1, Algorithm 2, we can see that the results of Algorithm 3 are quite close and vary from 0.6872 to 0.7349. These result of decision values can’t clearly distinguish, that is to say, the results obtained from Algorithm 3 are inconvincible (or at least not applicable). On the contrary, the Algorithm 1, Algorithm 2 have a clearly distinguish. Hence, if we take the discrimination into consideration, the Algorithm 1, Algorithm 2 are given priority to make decision. For better understanding, we give the figure form shown in Fig. 3 .

Fig. 3

The discrimination of the proposed three algorithms.

Comparison of the newly proposed three methods with other approaches

The MABAC method (Peng et al., 2017) and its limitation

MABAC (Peng et al., 2017)

However, we can find that the above algorithm will meet the counterintuitive phenomena when or is 0. That is to say, it only determines by one of them to make decision and decision information of others can be omitted.

The similarity method (Peng et al., 2017) and its limitation

Similarity measure (Peng et al., 2017)

According to above algorithm, one can establish an approach to make decision. However, the limiting condition that the probability of equality of final decision results is great. That is to say, we can hardly make decision.

Suppose that we invite another expert to evaluate the coal mine in Example 5.3. The weight information given by expert has the prior weight set given by his/her prior experience or preference: . The assessments for emergency plans arising from questionnaire investigation to the expert are different, which is shown in Table 5 .

Table 5

The tabular form of interval-valued fuzzy soft set in Example 5.3.

	ε_1	ε_2	ε_3	ε_4	ε_5
x_1	[0.0,0.1]	[0.8,1.0]	[0.8,0.9]	[0.8,0.9]	[0.9,0.9]
x_2	[0.58,0.87]	[0.51,0.62]	[0.5,0.6]	[0.5,0.6]	[0.6,0.8]
x_3	[0.61,0.81]	[0.55,0.65]	[0.3,0.4]	[0.4,0.6]	[0.7,0.8]
x_4	[0.14,0.15]	[0.61,0.68]	[0.2,0.3]	[0.4,0.5]	[0.6,0.7]
x_5	[0.13,0.15]	[0.43,0.49]	[0.3,0.4]	[0.4,0.6]	[0.4,0.6]

The final decision results and ranking are shown in Table 6 . Based on the above discussion in Remarks 6.1 and 6.2, we know that the unreasonable results for MABAC method (Peng et al., 2017) owe to the “0” problem.

Table 6

Final results and ranking in Example 5.3.

Algorithms	Results	Ranking	Optimal alternative
Algorithm 1: WDBA	SI_1=0.5853, SI_2=0.4894, SI_3=0.5060, SI_4=0.2110, SI_5=0.0761	x_1≻x_3≻x_2≻x_4≻x_5	x_1
Algorithm 2: CODAS	RA_1=1.5076, RA_2=0.9874, RA_3=1.2165, RA_4=-1.6670, RA_5=-2.0444	x_1≻x_3≻x_2≻x_4≻x_5	x_1
Algorithm 3: Similarity measure	S_1=0.6677, S_2=0.5839, S_3=0.6363, S_4=0.2882, S_5=0.2803	x_1≻x_3≻x_2≻x_4≻x_5	x_1
Algorithm 4: MABAC (Yang & Peng, 2017)	S_1=0.1682, S_2=0.2129, S_3=0.1828, S_4=-0.0023, S_5=-0.0385	x_2≻x_3≻x_1≻x_4≻x_5	x_2
Algorithm 5: Similarity measure (Yang & Peng, 2017)	S_1=0.1948, S_2=0.2826, S_3=0.2240, S_4=-0.2308, S_5=-0.3033	x_2≻x_3≻x_1≻x_4≻x_5	x_2

“Bold” denotes the unreasonable results.

From Fig. 4 , we observe that the proposed algorithms (WDBA, CODAS and similarity measure) are moderately correlated with similarity measure (Peng et al., 2017) and MABAC (Peng et al., 2017) . The reason for such variation is implicit from the nature of formulation used by each of these methods. To further demonstrate the strength of our algorithms, we compare them to existing methods (Peng et al., 2017). Fig. 5 shows the comparison of these five methods which clarifies the power of proposed algorithms better in distinguishing the optimal alternative and obtain the optimal alternative without counterintuitive phenomena. In Fig. 6 , we can find that the proposed weight determining model can effectively reflect the subjective information and objective information, but the combined weight proposed by Peng et al. (2017) can’t reveal the difference compared with the given weight information. That is to say, our weight determining model can illustrate the data law given by expert, not limit to the known subjective weight.

Fig. 4

Correlation inference.

Fig. 5

The comparison of the proposed three algorithms with the existing algorithms.

Fig. 6

The comparison of the weight information.

The choice values method (Yang et al., 2009) and its limitation

choice values (Yang et al., 2009)

The primary problem for choice values method originally developed in Yang et al. (2009) comes from the use of interval fuzzy choice values. As it was pointed out by Feng et al. (2010), the “interval fuzzy choice value” presented by Yang et al. (2009) is just the sum of membership degrees and non-membership of an alternative with respect to all parameters, interval fuzzy choice value can’t be interpreted as the number of the parameter satisfied by the alternative. It can be seen as a synthesized measure to size up each alternative by a fusion of all parameters. However, this direct addition of all the membership values and non-membership values with respect to diverse parameters is not always rational. In fact, it just like the addition of weight and height.

The following example will illustrate that algorithm in Yang et al. (2009) may not be successfully applied to some soft decision making problems.

Suppose that is an interval-valued fuzzy soft set and Table 7 gives its tabular representation with choice values and scores.

Table 7

The tabular form of interval-valued fuzzy soft set in Example 5.5.

	ε_1	ε_2	ε_3	ε_4	Choice value(c_i)	Score(r_i)
x_1	[0.5,0.6]	[0.6,0.8]	[0.2,0.4]	[0.3,0.6]	[1.6,2.4]	0
x_2	[0.4,0.6]	[0.5,0.6]	[0.4,0.5]	[0.3,0.7]	[1.6,2.4]	0
x_3	[0.3,0.5]	[0.6,0.7]	[0.3,0.4]	[0.4,0.8]	[1.6,2.4]	0
x_4	[0.5,0.6]	[0.4,0.6]	[0.3,0.5]	[0.4,0.7]	[1.6,2.4]	0

From Table 7, we can find that the scores of all alternatives are identical, in other words, all of them could be selected as the best choice by algorithm in Yang et al. (2009). Hence, it is unreasonable in current case. But in fact, we can achieve an optimal alternative by our pioneered algorithms and the existing algorithms (Peng et al., 2017). Suppose that we use Eq. (13) to determine the objective weight instead of combined weight by our algorithms and use Eq. (7) in Peng et al. (2017) to determine the objective weight instead of combined weight by Peng and Dai’s algorithms Peng et al. (2017). Meanwhile, assume that there is no discriminative power of cost parameters and benefit parameters, i.e., we think that the parameters are benefit parameters. Therefore, we can have the decision results by our algorithms and the existing algorithms (Peng et al., 2017), which is presented in Table 8 .

Table 8

A comparison study with some existing methods in Example 5.5.

Algorithms	Ranking	Optimal alternative
Algorithm 6:choice values (Peng & Dai, 2017)	x_1∼x_2∼x_3∼x_4	∗
Algorithm 1: WDBA	x_2≻x_3≻x_1≻x_4	x_2
Algorithm 2: CODAS	x_2≻x_1≻x_3≻x_4	x_2
Algorithm 3: Similarity measure	x_2≻x_3≻x_4≻x_1	x_2
Algorithm 4: MABAC (Yang & Peng, 2017)	x_2≻x_1≻x_4≻x_3	x_2
Algorithm 5: Similarity measure (Yang & Peng, 2017)	x_2≻x_1≻x_4≻x_3	x_2

“∗” denotes that there is no unified alternative to selected.

The level soft set method (Feng et al., 2010) and its limitation

Feng et al., 2010

Let and . The vector is called an opinion weighting vector. The fuzzy soft set over U such that is called the weighted reduct fuzzy soft set (WRFS) of the interval-valued fuzzy soft set with respect to the opinion weighting vector W.

left soft set (Feng et al., 2010)

The level soft set method (Feng et al., 2010) is in fact an adjustable algorithm which seizes a key feature for decision making in a vague circumstances: some of these issues are essentially humanistic and thus subjective in nature; there does not exist a uniform criterion for evaluating the alternatives. Apparently, the level soft set algorithm in Feng et al. (2010) can’t be applied to IVFSS based decision making in some special cases.

Suppose that is an interval-valued fuzzy soft set. For obtaining the choice value , we set (Step 2), (Step 3) in Feng et al. (2010). Then Table 9 gives its tabular representation with choice values.

Table 9

The tabular form of interval-valued fuzzy soft set in Example 5.8.

	ε_1	ε_2	ε_3	ε_4	Choice value(c_i)
x_1	[0.55,0.70]	[0.3,0.5]	[0.5,0.6]	[0.49,0.51]	1
x_2	[0.55,0.65]	[0.34,0.47]	[0.36,0.64]	[0.36,0.64]	1
x_3	[0.45,0.80]	[0.37,0.43]	[0.35,0.65]	[0.46,0.54]	1
x_4	[0.53,0.67]	[0.36,0.46]	[0.45,0.65]	[0.38,0.62]	1

From Table 9, we can find that the choice values of all alternatives are identical, in other words, all of them could be selected as the best choice by algorithm in Feng et al. (2010). Hence, it is unreasonable in current case. But in fact, we can achieve an optimal alternative by our pioneered algorithms and the existing algorithms (Peng et al., 2017). Suppose that we use Eq. (13) to determine the objective weight instead of combined weight by our algorithms and use Eq. (7) in Peng et al. (2017) to determine the objective weight instead of combined weight by Peng and Dai’s algorithms (Peng et al., 2017). Meanwhile, assume that there is no discriminative power of cost parameters and benefit parameters, i.e., we think that the parameters are benefit parameters. Therefore, we can have the decision results by our algorithms and the existing algorithms (Feng et al., 2010, Peng et al., 2017), which is presented in Table 10 .

Table 10

A comparison study with some existing methods in Example 5.8.

Algorithms	Ranking	Optimal alternative
Algorithm 7: left soft set (Xiao et al., 2013)	x_1∼x_2∼x_3∼x_4	∗
Algorithm 1: WDBA	x_1≻x_4≻x_3≻x_2	x_1
Algorithm 2: CODAS	x_1≻x_4≻x_3≻x_2	x_1
Algorithm 3: Similarity measure	x_1≻x_4≻x_3≻x_2	x_1
Algorithm 4: MABAC (Yang & Peng, 2017)	x_1≻x_4≻x_3≻x_2	x_1
Algorithm 5: Similarity measure (Yang & Peng, 2017)	x_1≻x_4≻x_3≻x_2	x_1

“∗” denotes that there is no unified alternative to selected.

The comparison table method (Yuan & Hu, 2012) and its limitation

Yuan and Hu, 2012

Let U be an initial unverse, E be a parameter set, is an interval-valued fuzzy soft set on U. If , then we say the lower membership of related to is less than the lower membership of , its corresponding characteristic function is given as follows:

Let U be an initial unverse, E be a parameter set, is an interval-valued fuzzy soft set on U. If , then we say the upper membership of related to is less than the upper membership of , its corresponding characteristic function is given as follows:

The comparison table of interval-valued fuzzy soft set means that row numbers and column numbers are equal. is given as follows: j is the number of parameters. Obviously, , and .

comparison table (Yuan & Hu, 2012)

As stated in Yuan and Hu (2012), the given approach is in fact a comparable table method. Some of these problems are essentially humanistic and thus subjective in nature; there does not exist a uniform criterion for evaluating the alternatives. Apparently, the comparable table proposed by Yuan and Hu (2012) cannot be applied to IVFSS based decision making in some cases.

Peng et al., 2017

Assume that is an interval-valued fuzzy soft set. Then Table 11 gives its tabular representation and their decision results.

Table 11

The tabular form of interval-valued fuzzy soft set in Example 5.13.

	ε_1	ε_2	ε_3	ε_4	P_i	Q_i	S_i
x_1	[0.4,0.6]	[0.5,0.7]	[0.6,0.8]	[0.7,0.9]	12	12	0
x_2	[0.5,0.55]	[0.6,0.65]	[0.67,0.89]	[0.6,0.8]	12	12	0

From Table 11, we can find that the score values of all alternatives are identical, in other words, all of them could be selected as the best choice by algorithm in Yuan and Hu (2012). Hence, it is unreasonable in current case. But in fact, we can achieve an optimal alternative by our proposed algorithms and the existing algorithms (Peng et al., 2017). Suppose that we use Eq. (13) to determine the objective weight instead of combined weight by our algorithms and use Eq. (7) in Peng et al. (2017) to determine the objective weight instead of combined weight by Peng and Dai’s algorithms Peng et al. (2017). Meanwhile, assume that there is no discriminative power of cost parameters and benefit parameters, i.e., we think that the parameters are benefit parameters. Therefore, we can have the decision results by our algorithms and the existing algorithms (Peng et al., 2017, Yuan and Hu, 2012), which is presented in Table 12 .

Table 12

A comparison study with some existing methods in Example 5.13.

Algorithms	Ranking	Optimal alternative
Algorithm 8: comparison table (Mukherjee & Sarkar, 2014)	x_1∼x_2	∗
Algorithm 1: WDBA	x_2≻x_1	x_2
Algorithm 2: CODAS	x_2≻x_1	x_2
Algorithm 3: Similarity measure	x_2≻x_1	x_2
Algorithm 4: MABAC (Yang & Peng, 2017)	x_2≻x_1	x_2
Algorithm 5: Similarity measure (Yang & Peng, 2017)	x_2≻x_1	x_2

“∗” denotes that there is no unified alternative to selected.

Conclusion

The major contributions in this paper can be summarized as follows:

We construct a new score function for interval-valued fuzzy number. Comparing with the existing literature (Peng et al., 2017, Peng and Yang, 2017), it can distinguish the difference of alternatives when the existing score functions (Peng et al., 2017, Peng and Yang, 2017) cannot work.

A new information measure (distance measure, similarity measure and entropy) are introduced and their transformation relations are pioneered.

We gives new entropy method for obtaining the objective weight, meanwhile, the combined weights are developed, which can effectively reflect the subjective information and objective information, but the combined weight proposed by Peng et al. (2017) can’t reveal the difference compared with the given weight information.

We propose three algorithms to solve interval-valued fuzzy soft decision making problem by Weighted Distance Based Approximation (WDBA), COmbinative Distance-based ASsessment (CODAS) and similarity measure. Compared to the existing interval-valued fuzzy soft decision making methods, are (i) it can obtain the optimal alternative without counterintuitive phenomena (Peng et al., 2017); (ii) it has a great power in distinguishing the optimal alternative (Feng et al., 2010, Peng et al., 2017, Yang et al., 2009, Yuan and Hu, 2012); (iii) it can avoid the parameter selection problems (Feng et al., 2010).

In the future, we shall apply more groundbreaking theories into interval-valued fuzzy soft set and also take different fuzzy environment (Li and Wang, 2017, Nie et al., 2017, Peng and Dai, 2017, Peng et al., 2017, Peng and Yang, 2015, Yang and Wang, 2018, Zhou et al., 2018) into consideration by proposed algorithms (WDBA and CODAS) for solving more emergency decision-making problems.

1:	Input the interval-valued fuzzy soft decision matrix I=(F(ε_j)(x_i))_n×m(i=1,2,…,m;j=1,2,…,n).
1:	Transform the matrix I=(F(ε_j)(x_i))_m×ninto a normalized interval-valued fuzzy soft matrix I^′=(F∼(ε_j)(x_i))_n×mby Eq. (14).
3:	Calculate the combined weight ϖ by Eq. (13).
4:	Compute the score matrix T=(t_ij)_m×n of I^′=(F∼(ε_j)(x_i))_n×m by Eq. (15).
5:	Form the standardized matrix SM=(SM_ij)_m×n by Eq. (16).
6:	Compute the ideal points SM^+ and anti-ideal points SM^- by Eqs. (19), (20).
7:	Compute the WEDi+ and WEDi- by Eqs. (21), (22).
8:	Derive the suitability index value SI_i of each alternative A_i by Eq. (23).
9:	Determine the ranking of the alternatives according to the suitability index values SI.

1:	It is similar to 1–3 in Algorithm 1.
2:	Compute the score matrix T=(t_ij)_m×n of I^′=(F∼(ε_j)(x_i))_n×m by Eq. (15).
3:	Calculate the weighted normalized decision matrix r_ij by Eq. (24).
4:	Determine the negative-ideal solution NIS by Eq. (25).
5:	Compute the Euclidean distance E and Hamming distance T from the NIS by Eqs. (27), (28).
6:	Construct the relative assessment matrix RA by Eq. (29).
7:	Calculate the assessment score of each alternative RA_i by Eq. (32).
8:	Rank the alternatives according to the decreasing values of assessment RA.

1:	It is similar to 1–3 in Algorithm 1.
2:	Calculate the similarity measure S(x_i,x^∗)(i=1,2,…,m) by Eq. (34) (p=2,t=2).
3:	Rank the alternatives according to the decreasing values of similarity measure S(x_i,x^∗).

1:	It is similar to 1–3 in Algorithm 1.
2:	Compute the weighted matrix T=(t_ij)_m×n by Eq. (35),
	(34)t_ij=[tij-,tij+]=ϖ_j[F∼^-(ε_j)(x_i),F∼^+(ε_j)(x_i)]=[ϖ_jF∼^-(ε_j)(x_i),ϖ_jF∼^+(ε_j)(x_i)].
3:	Compute the border approximation area (BAA) matrix G=(g_j)_1×n. The BAA for each parameter is obtained by Eq. (36).
	(35)g_j=∏i=1m(t_ij)^1/m=∏i=1m(tij-)^1/m,∏i=1m(tij+)^1/m.
4:	Compute the distance matrix D=(d_ij)_m×n by Eq. (37),
	(36)d_ij=d_e(t_ij,g_j),ifs(t_ij)>s(g_j),0,ifs(t_ij)=s(g_j),-d_e(t_ij,g_j),ifs(t_ij)<s(g_j),
	where distance measure d_e is defined in Eq. (6) (t=0,p=1 as (Peng and Yang, 2015)), and s(t_ij),s(g_j) are score function (Peng et al., 2017) of t_ij and g_j, respectively.
5:	Rank the alternatives by Q_i(i=1,2,…,m). The most desired alternative is the one with the biggest value of Q_i.
	(37)Q_i=∑j=1nd_ij,i=1,2,…,m;j=1,2,…,n.

1:	It is similar to 1–3 in Algorithm 1.
2:	Calculate the similarity measure S(x_i,x^∗)(i=1,2,…,m) by Eq. (39).
	(38)S^∇(x_i,x^∗)=∑j=1nϖ_jF∼^-(ε_j)(x_i),∑j=1nϖ_jF∼^+(ε_j)(x_i)
3:	Compute the each alternative of score function s_p(S(x_i,x^∗)) by Eq. (4).
4:	Rank the alternatives according to the decreasing values of similarity measure s(S^∇(x_i,x^∗)). The most desired alternative is the one with the biggest value of x_i.

1:	It is similar to Steps 1–2 in Algorithm 1.
2:	Compute the interval fuzzy choice value c_i for each alternative x_i such that c_i=[ci-,ci+]=∑j=1nF∼^-(ε_j)(x_i),∑j=1nF∼^+(ε_j)(x_i).
3:	Compute the score r_i of x_i such that r_i=∑j=1n(ci--cj-)+(ci+-cj+)
4:	The optimal alternative is to choose any one of the alternative x_k∈U such that r_k=max_x_i∈U{r_i}.

1:	It is similar to Steps 1–2 in Algorithm 1.
2:	Input an opinion weighting vector W=(α,β) and calculate the WRFS ϒ_W=(F_W,A) of the interval-valued fuzzy soft set (F,A) with respect to W.
3:	Input a threshold value t∈[0,1].
4:	Compute the t-level soft set L(ϒ_W;t).
5:	Present the level soft set L(ϒ_W;t) in tabular form and compute the choice value c_i of x_i for ∀x_i∈U.
6:	The optimal alternative is to select x_k if c_k=max_x_i∈Uc_i.
7:	If k has more than one value, then any one of x_k may be chosen.

1:	It is similar to Steps 1–2 in Algorithm 1.
2:	Compute the comparison table by Eqs. (40), (41).
3:	Compute the P_i=∑i=1mC_ik and Q_i=∑k=1mC_ik.
4:	Compute the score S_i=P_i-Q_i.
5:	The optimal alternative is to select x_i if S_k=max_x_i∈US_i.