Abstract
In this article, the gained and lost dominance score (GLDS) method is extended into the 2-tuple linguistic neutrosophic environment, which also combined the power aggregation operator with the evaluation information to deal with the multi-attribute group decision-making problem. Since the power aggregation operator can eliminate the effects of extreme evaluating data from some experts with prejudice, this paper further proposes the 2-tuple linguistic neutrosophic numbers power-weighted average operator and 2-tuple linguistic neutrosophic numbers power-weighted geometric operator to aggregate the decision makers’ evaluation. Moreover, a model based on the score function and distance measure of 2-tuple linguistic neutrosophic numbers (2TLNNs) is developed to get the criteria weights. Combing the GLDS method with 2-tuple linguistic neutrosophic numbers and developing a 2TLNN-GLDS method for multiple attribute group decision making, it can express complex fuzzy information more conveniently in a qualitative environment and also consider the dominance relations between alternatives which can get more effective results in real decision-making problems. Finally, an applicable example of selecting the optimal low-carbon logistics park site is given. The comparing results show that the proposed method outperforms the other existing methods, as it can get more reasonable results than others and it is more convenient and effective to express uncertain information in solving realistic decision-making problems.
Keywords
Introduction
Due to the uncertainty and complexity of the decision-making environment, it is difficult to express a wealth of information with precise numbers, which will result in the loss of information. Zadeh [1] defined the fuzzy set theory which used the membership function to describe the vague information. Atanassov [2] proposed the intuitionistic fuzzy set (IFS) which are characterized by its membership function and non-membership function. Considering the uncertainty and contradictions of people’s subjective cognition, and the information given by decision makers is full of indeterminacy and inconsistent, Smarandache [3] further introduced the neutrosophic set (NS) which are characterized by its truth membership function, indeterminacy membership function and falsity membership function. The NS can be better express uncertain information by adding an indeterminacy membership function. Long [4] proposed a fuzzy clustering algorithm through neutrosophic association matrix, which are more effective to solve the benchmark datasets using different clustering criteria. Jha [5] extended the NS into image segmentation field, then combined the Dice’s Coefficients with NS to ensure proper evaluation of uncertainty of the missing data and their indeterminacy for image segmentation. Son [6] proposed granular representation of single valued triangular neutrosophic numbers, defined the calculus properties of neutrosophic gr-integral and the neutrosophic gr-derivative. Abdel-Basset [7] integrated the internet of things (IOT) system and neutrosophic multicriteria decision making (N-MCDM) technique together for detecting, monitoring and controlling heart failure with minimum cost and time. Dey [8] proposed the minimum spanning tree problem with undirected connected weighted interval type 2 fuzzy graph (FMST-IT2FS), then introduced an agenetic algorithm to solve the FMST-IT2FS problems. Thong [9] extended NS into dynamic multi-criteria decision-making (DMCDM) environment, proposed generalized dynamic internal-valued neutrosophic sets which can solve the problems better in the education domain. Can [10] extended NS to classify harmful domain names in domain generation algorithms (DGA) botnet detection. Patro [11] developed a knowledge-based preference learning (KBPL) system combined content-based filtering (CBF) and collaborative filtering (CF), then through the adaptive neuro-fuzzy inference system (ANFIS) to predict the output in recommender system. Garg [12] introduced neutrality addition and scalar multiplication for single-valued neutrosophic numbers which can take the neutral characters of the decision-maker’s preferences into account. Garg [13] defined some new sine-trigonometric (ST) operations laws (STOLs) corresponding to SVNSs to solve the group decision-making, then developed the ST weighted average operator and ST weighted average operator to aggregate the single-valued neutrosophic information. Zulqarnain [14] introduced the Pythagorean fuzzy soft weighted average (PFSWA) and Pythagorean fuzzy soft weighted geometric (PFSWG) operators and applied it in green supplier selection. Ji [15] depicted the Frank aggregation operations for aggregating SVNS information to deal with the selection of third-party logistics (TPL) providers.
However, there are vagueness and limitations in human thinking, so that people are more inclined to use language words to express evaluation values. Garg [16] introduced the power average, weighted average, ordered weighted average, hybrid average and geometric power aggregation operators for linguistic single-valued neutrosophic set (LSVNS) to deal with the group decision-making problems. Subsequently, Garg [17] proposed prioritized ordered weighted average operator, prioritized ordered weighted geometric operator of linguistic single-valued neutrosophic numbers and discussed their properties. Dat [18] introduced the single-valued linguistic complex neutrosophic set (SVLCNS-2) and interval linguistic complex neutrosophic set (ILCNS-2), and applied it to lecturer selection of the University to verify the usefulness and efficiency. Luo [19] extended a new similarity measure of LNNs based on consistency degree and used it for the selection of mine development plan. Garg and Harish [20] introduced the concept of the possibility linguistic single-valued neutrosophic set (PLSVNS), and defined the weighted averaging operators and weighted geometric operators of PLSVNS. Li [21] developed an approach combined the evaluation based on distance from average solution (EDAS)with LNSs, and presented power aggregation (PA) operator of LNSs to select the suitable property company. Wang [22] proposed the hesitant interval neutrosophic uncertain linguistic sets (HINULSs) and aggregated it with prioritized weighted averaging average (LPWA) operator, prioritized weighted geometric (LPWG) operator, and generalized prioritized weighted aggregation (LPWA) to solve the problems in investment.
Wang [23] proposed the 2-tuple linguistic neutrosophic sets (2TLNS) and defined some bonferroni mean (BM) operator for 2TLNSs to find the desirable green supplier in green supply chain management. Ju [24] expanded the maclaurin symmetric mean (MSM) operator into single-valued neutrosophic interval 2-tuple linguistic environment and proposed the single-valued neutrosophic interval 2-tuple linguistic maclaurin symmetric mean (SVN-ITLMSM) operator, the single-valued neutrosophic interval 2-tuple linguistic weighted average (SVN-ITLWA) operator. Wu [25] presented dombi bonferroni mean (DBM) operators and dombi geometric Bonferroni mean (DGBM) operators of 2TLNNs based on the operations of dombi t-norm and t-conorm. Wang [26] developed a combinative distance-based assessment (CODAS) method for safety assessment of construction project with the 2-Tuple Linguistic Neutrosophic information. Wang [27] presented the 2-tuple linguistic neutrosophic MABAC model. Wang [28] combined the VIKOR method with 2TLNNs which are more reasonable and scientific for considering the conflicting criteria.
The 2TLNS can effectively describe the uncertain information presented in complex decision-making problems. However, there are many shortcomings in the existing research on 2TLNS. Such as VIKOR and TOPSIS, their group utility value can be fully compensated by some other attributes, which will affect the accuracy of the final ranks of all alternatives. Inspired by the GLDS method introduced by Liao [29] which considers both the gain and loss in the uncertain circumstances, the bad performances of some criteria cannot compensated by good performances of other criteria, it can avoid choosing an alternatives which have poor performance under certain attributes. Information operators is useful tool to aggregate the evaluation received by decision makers. However, the operators of existing 2TLNS ignore the influence of unreasonable information biased by decision makers. Such as the BM operator and MSM operator. But in some realistic decision-making problems, the evaluation information given by experts based on different cognitive levels will eventually bias the results. Thus, the evaluation cannot using the above mentioned operators. We find that the PA operator is simple and powerful, and suitable for processing 2-tuple language information. The PA operator can manage information about the interrelationship of aggregated values, and they enable the values to be mutually reinforced. When use the PA operator to aggregate the evaluation information, the influences of unreasonable information received by biased experts can be relieved. In order to better deal with uncertain information, and overcome the limits of the existing methods, an effective and convenient approach based on PA operator and GLDS method is developed, which can get more reasonable outputs in solving realistic decision-making problems.
The arrangement of this article is as follows. Section 2 gives a brief review of 2TLNSs. Section 3 presents the power average operator and the power average operator with 2TLNNs. Section 4 introduces the GLDS method. Section 5 presents a model to derive the weights of attributes, and introduces the steps of 2TLNN-GLDS method. Section 6 illustrates a case of site selection of a low-carbon logistics park to verify the feasibility of proposed approach. In addition, there are some comparisons with other methods to further demonstrate the merit of the method proposed. Section 7 concludes the manuscript and points out future research directions.
Preliminaries
2-Tuple Linguistic Neutrosophic Set
Wang [32] was the first to put forward the 2TLNS, which used the combination of 2-tuple linguistic sets (2TLSs) and SVNSs to solve the multi-attribute decision-making problems and reduce the deviation caused by the traditional LNNs in solving real life problems. The definition of 2TLNSs can be expressed as follows:
The score function and the accuracy function can intuitively compare the size of two 2TLNNs.
if S (φ1) < S (φ2), then φ1 < φ2; if S (φ1) > S (φ2), then φ1 > φ2; if S (φ1) = S (φ2) and H (φ1) < H (φ2), then φ1 < φ2; if S (φ1) = S (φ2) and H (φ1) > H (φ2), then φ1 > φ2; if S (φ1) = S (φ2) and H (φ1) = H (φ2), then φ1 = φ2.
(1)
(2)
(3)
(4)
The power operator considers the relationship between the associated data which are the support degree φ, and can reduce the effects of extreme evaluating data from some experts with prejudice. Therefore, it has more practical applicability in solving realistic decision-making problems.
Here,
sup(φ
i
, φ
j
) = sup(φ
j
, φ
i
) sup(φ
i
, φ
j
) ∈ [0, 1] If |φ
i
- φ
j
| ⩽ |φ
m
- φ
n
|, then sup(φ
i
, φ
j
) ⩾ sup (φ
m
, φ
n
).
Here T (φ
i
) satisfies
Obviously, the degree of support is similar to the similarity index. The higher the degree of similarity, the closer the two values are, the higher the degree of support between each other.
PA operator introduced by Yager [36], the advantage of the PA operator is taking the relationship between the evaluation values into consideration and the figures which are too large or too small can be handled well through different evaluation weights. In this section, the power operator is extended as 2-tuple linguistic neutrosophic numbers power-weighted average (2TLNNPWA) operator and 2-tuple linguistic neutrosophic numbers power-weighted geometric (2TLNNPWG) operator to solve the problem of multi-attribute group decision-making in which the attribute values is represented by 2TLNNs.
2TLNNPWA Operator
Where
sup(φ
i
, φ
j
) = sup(φ
j
, φ
i
) sup(φ
i
, φ
j
) ∈ [0, 1] If d (φ
i
, φ
j
) ⩽ d (φ
m
, φ
n
), then sup(φ
i
, φ
j
)⩾ sup(φ
m
, φ
n
), where d (φ
i
, φ
j
) is the normalized hamming distance between φ
i
and φ
j
.
Where
when n = 1, the Equation (7) is easy to know correctly. Assume that when n = t, the Equation (7) is true, so
Then, when n = t+1, we have
So when n = t+1, Equation (7) is true.
According to (1) and (2), we can know that Eq. (7) is still correct for any n.
Thus, φ- ⩽ 2TLNNPWA (φ1, φ2, . . . , φ n ) ⩽ φ+.
Where
sup(φ
i
, φ
j
) = sup(φ
j
, φ
i
) sup(φ
i
, φ
j
) ∈ [0, 1] If d (φ
i
, φ
j
) ⩽ d (φ
m
, φ
n
), then sup(φ
i
, φ
j
)⩾ sup(φ
m
, φ
n
), where d (φ
i
, φ
j
) is the normalized hamming distance between φ
i
and φ
j
.
Where
Obviously, the 2TLNNPWG operates also have the idempotency and boundedness, the proof are omitted here.
The GLDS method, which was first proposed by liao [29], uses both the gained and lost dominance relations between alternatives to derive the ranking orders of alternatives and select the most desirable alternatives. Then the specific steps of the GLDS method are expressed as follows:
Step 1. Define the dominance flow of alternative a
i
over alternative a
v
based on the distance measure, shown as:
Where τ (r
ij
) denotes the function of turning the 2-tuple linguistic neutrosophic fuzzy variables represented by alternative a
i
under criterion j into the real numbers. J1 and J2 are the set of benefit and cost criteria. Considering the bias caused by the dominance flows, then the normalized dominance flows are represent.
Step 2. According to the normalized dominance flows, the overall gained dominance score of a
i
under c
j
can be calculated by Equation (12), and get the subordinate order set in descending order as R1 = { r1 (a1) , r1 (a2) , . . . , r1 (a
m
) } (i = 1, 2, . . . , m).
Step 3. The overall lost dominance score of a
i
under c
j
can be calculated by Equation (13), and derive a subordinate order set in ascending order as R2 = { r2 (a1) , r2 (a2) , . . . , r2 (a
m
) } (i = 1, 2, . . . , m).
Step 4. Calculate the Normalized OGDS and Normalized OLDS based on the Equation (14) and Equation (15).
Step 5. The final ranks of alternatives should take the normalized gained score and the normalized lost score as well as two subordinate order sets (R1 and R2) into account, so we can get the final score CS
i
by the following equation.
In this section, the model based on the distance and score function of deriving the criteria weights are constructed. By comparison between the two alternatives to get the priority degree of criterion, the weight function and normalization of weights can eliminate the bias in attribute weights distribution. Subsequently, the GLDS and the PA operator are combined to effectively manage the 2TLNNs, then a new model of 2-Tuple linguistic neutrosophic set is developed. The PA operator can very well gather the evaluation information of experts, while the GLDS obtains “group utility” values and “individual regret” values through the comparison of dominance flow between alternatives, and then comprehensively considers them for selecting the best solution. The flowchart of 2TLNN-GLDS method is shown as Fig. 1.
The flowchart of 2TLNN-GLDS method.
The determination of attribute weights is an important part of multi-attribute decision-making problems. The objective weight determination method based on the score function of HFLE proposed by Fu [37] is an effective way to distinguish the importance of each attributes. Then it can be extended to the 2TLNNs environment.
Assume that R = [r
ij
] m×n (i = 1, 2, . . . , m ; j = 1, 2, . . , n) is a 2-Tuple linguistic neutrosophic decision matrix, where r
ij
= 〈 (s
T
1
, α1) , (s
I
1
, β1) , (s
F
1
, γ1)〉 is expressed in the form of 2TLNNs. ω
j
(j = 1, 2, . . . , n) is weight of j - th criterion. Then, ω
j
⩾ 0,
Where d (r ij , r vj ) is the distance between a i and a j under the criterion c j , S (φ ij ) is the score function of a i under the criterion c j . p ij is the priority degree of criterion c j , p ij ∈ (0, 1).
The lager the priority degree among all the alternatives, the higher weight should be assigned to highlight the importance of other criteria. Otherwise it should be given a smaller weight to weaken its importance to alternatives. Therefore, a following weight function
Where η1, η2 are the optimized parameters to ensure that the weight function range is within [0,1] and the value of η1, η2 are determined according to the actual situation. The above result fail to distinguish the deviation of some attributes, so the normalized weight model is proposed to derive the final weight information and the criteria weight is ω
j
.
Suppose there are m alternatives {a1, a2, . . . , a
m
}, n attributes {c1, c2, . . . , c
n
} and e
t
experts {e1, e2, . . . , e
k
} with an expert’s weighting vector of ω
t
(t = 1, 2, . . . , k), which satisfies 0 ⩽ ω
t
⩽ 1,
Step 1. Organize the assessment information expressed using 2TLNNSs that are given by experts e
t
(t = 1, 2, . . . , k) and construct the decision-making evaluation matrix
Step 2. Based on the decision-making evaluation matrix
Step 3. Use the 2TLNNPWA operator or the 2TLNNPWG operator to aggregate the assessment information for the comprehensive evaluation matrix R = (r ij ) m×n.
Step 4. According to the comprehensive evaluation matrix R = (r ij ) m×n, and obtain the attribute weights using the Eqs. (17–19).
Step 5. Obtain the dominance flow of alternative a i over alternative a v according to Eq. (10) and get the normalized dominance flow using Eq. (11).
Step 6. Obtain the OGD and OLD scores according to Eqs. (12) and (13), then derive the subordinate order set R1 and R2.
Step 7. Normalize the OGD and OLD scores using the Eqs. (14) and (15).
Step 8. Obtain the CS according to Eq. (16), then rank all the alternatives.
2-tuple linguistic neutrosophic number decision matrix (e1)
2-tuple linguistic neutrosophic number decision matrix (e1)
2-tuple linguistic neutrosophic number decision matrix (e2)
2-tuple linguistic neutrosophic number decision matrix (e3)
The aggregating results by 2TLNNWPA operator
Numerical example of low-carbon logistics park site selection
With the advancement of the national low-carbon economic industry construction pilot, the logistics industry, which is the foundation of national economic development, must also find a new model of low-carbon operation and sustainable development. Building a low-carbon logistics park is the primary task of a low-carbon logistics system, and the first step in building a logistics park is to solve the problem of logistics park location from a low-carbon perspective. It is now preparing to build logistics parks in the following five areas a i = (i = 1, 2, 3, 4, 5). In order to select the most desirable logistics park location, the company has convened three experts from different fields e1, e2 and e3 to evaluate the plans. After negotiation and discussion, four vital criteria were selected: the growth rate of corporate income c1, land use c2, pollutant treatment c3, and regional cargo turnover c4. The five feasible areas a i = (i = 1, 2, 3, 4, 5) were evaluated by the decision-maker in terms of the above four attributes using the 2TLNN information (the weight vectors of the three experts was considered as ω = (0.41, 0.26, 0.33). The evaluation results are shown in Tables 1–3, respectively.
Step 1. The evaluation information expressed using2TLNNSs of each experts e
t
(t = 1, 2, . . . , k) were gathered and shown as a decision-making evaluation matrix
Step 2. According to the expert’s weight and decision-making evaluation matrix, we utilize the 2TLNNWPA operator to derive the Supports T ijt and the Weight coefficient φ ijt .
Step 3. Through the Supports T ijt and the Weight coefficient φ ijt , the assessment information can be aggregated to obtain the comprehensive evaluation matrix shown in Table 4.
The gained dominance scores of each alternatives
The gained dominance scores of each alternatives
The lost dominance scores of each alternatives
Step 4. Calculate the score of each 2TLNNs in the comprehensive evaluation matrix, then based on the score matrix to derive the weights of criteria.
The attribute’s weight can derived by Eqs. (17–19) based on the score function and distance measure. By calculation, the priority degree p of criteria p = (0.261, 0.249, 0.382, 0.207). Suppose η1 = 4, η2 = 2 and the attribute’s weight ω j can be obtained as ω1 = 0.257, ω2 = 0.249, ω3 = 0.274, ω4 = 0.219.
Step 5. Calculate the dominance flow of a i over others corresponding to experts e k and the results of the normalized dominance flow are shown as following:
Step 6. The overall gained dominance (OGD) scores with a1, a2, a3, a4, a5are 0.053, 1.195, 0.505, 0.568, 0.385, respectively. And the overall lost dominance (OLD) scores are 0.053, 0.133, 0.130, 0.132 and 0.107, respectively. The detail presents in the Table 5 and 6. From the table, the gained dominance scores and the lost dominance scores through calculating the dominance flow between an alternative and others are presented, it can reflect the group utility of the alternatives, while the lost dominance scores are represented the individual regret. The OGD scores and OLD scores take the attributes weight into account, which considers the importance of attributes on the basis of overall group utility and individual regret value.
Step 7. Normalize the OGD and OLD scores using the Eqs. (13) and (14). The normalization can eliminate different metrics between data. And the normalized overall gained dominance (NOGD) scores with a1, a2, a3, a4, a5 are 0.036, 0.814, 0.343, 0.387, 0.262, respectively, the normalized overall lost dominance (NOLD) scores are 0.206, 0.518, 0.506, 0.514, 0.416, respectively.
Step 8. Based on the collective scores of all alternatives as CS1 = 0.011, CS2 = 0.291, CS3 = 0.122, CS4 = 0.140, CS5 = 0.100 and rank in a decrease order of a2 ≻ a4 ≻ a3 ≻ a5 ≻ a1. Obviously, a2 is the best logistics park site.
To test the rationality and effectiveness of the proposed method, it is indispensable to compare with other methods. In this part, we compare our proposed method with other existing methods, including the 2-tuple linguistic neutrosophic numbers weighted average (2TLNNWA) operator and 2-tuple linguistic neutrosophic numbers weighted geometric (2TLNNWG) operator presented by Wu [32] and the 2TLNN-VIKOR method presented by Wang [28]. The analysis process is shown as below.
1) Comparing with 2TLNNWA operator and 2TLNNWG operator
Based on the comprehensive evaluation matrix presented in Table 4. The alternative score results derived by using 2TLNNWA and 2TLNNWG operators which also combined with the score function of 2TLNNs.
The fused results by 2TLNNWA were
The fused results by 2TLNNWG were
2) Comparing with 2TLNN-VIKOR method
The VIKOR method is an effective ranking method which can adjust the range of “group utility” value and “individual regret” value by the strategic weight ρ. Using the VIKOR method to handle the comprehensive evaluation matrix presented in Table 4 and the value of all the alternatives are presented below. In this case, the strategic weight ρ is 0.4, the alternative with minimum value is the best logistics park location.
From the above comparative analysis and the final order shown in Table 7, we can find that a2 is the best logistics park location, which verifies the effectiveness of the 2TLNN-GLDS method. However, the ranking result obtained in the 2TLNNWA operator is a4 ≈ a2 ≻ a5 ≻ a3 ≻ a1, it means that the optimal alternatives are a4 and a2. This is because the method uses a relatively simple weighted average operator, it can only handle the situation where the attribute indicators are mutually independent, and when the evaluation value of some attributes is too high or too low, the weighted average operator cannot get accurate results, which will bias the final result. Comparing the results with 2TLNNWG operator, it has the same optimal alternative a2 as well as the worst one a1, but there have a little bit differences across the final result. The reason is that the weighted geometric operator is an incompletely compensated model, under some certain attributes, the inferior performance of alternatives cannot be fully compensated by the superior performance of other attributes. For example, the final value of a4 is not so good compared with other methods, since it performs badly under the attribute c2. It is worth mentioning that these two methods ignore the process of standardization and do not consider the relationship between attributes. These two points are very important in the decision-making process because different attributes are handled differently.
Ordering of the logistics park location
Ordering of the logistics park location
The VIKOR method is an ideal point-based method, which considers the relative importance of the distance from the alternative to the positive ideal solution and the distance to the negative ideal solution. When the strategic weight ρ > 0.5, it depicts “the maximum group utility”, then when ρ = 0.5 depicts equality and ρ < 0.5 depicts the minimum regret. In this case, the strategic weight between the group utility and individual regret value be 0.4, when considering the index of R, a3 is superior to a5, but the final ranking of a3 is not as good as that of a5. That is because a3 has a lower S value than a5. Compared with the GLDS, the VIKOR fails to make a reasonable compromise between the “group utility” and the “individual regret” values, and it also ignore the process of normalization.
In the 2TLNN-GLDS method, the alternatives compared in pairs, calculates the dominance score obtained through the dominance flow between alternatives and others, and then the gained dominance score derived after integration represents group utility, while the lost advantage score represents personal regret, after the normalization process of it and finally get the ranks by considering the group utility and the personal regret comprehensively. Since the GLDS does not need to determine the additional parameters like the VIKOR, it is more convenient in the actual decision-making process, and the results obtained are more in line with reality.
All the four methods can handle imprecise information under 2-Tuple linguistic neutrosophic environment, but the proposed method has less information loss than others which can get more reasonable result, and it also can decrease the complexity of calculation using language information. What’s more, in the process of processing information with the PA operator, the relationship between attributes is also considered, making the final result more reliable and accurate.
In this paper, we develop the 2TLNNs with the power average operator and the power geometric operator to deal with the MAGDM problems expressed by language variables. Then, define a model based on score function and distance measure of 2TLNNs to derive criteria weights. Next, the GLDS approach which considers the relationship between alternatives over attributes is combined with 2TLNNs to obtain the best alternatives. Besides, through a numerical example of low-carbon logistics park site selection and comparative analyses with other existing method, it can be found that the proposed method has a better degree of discrimination of alternatives, and also has better reliability and wider application space than others. The highlights of this paper can be summarized as follows: The 2TLNNs can express complex fuzzy information more conveniently in a qualitative environment, and reduce the loss of language information, which makes the decision result more reasonable and effective. The model of attribute’s weights considers the priority degree of criterion and the relationship of evaluation information, which can distinguish the importance of attributes well. It provides a reliable and accurate guarantee for the final result. By comparing with other methods, it can be found that the method takes the “group utility” and “individual regret” values into account which are more in line with realistic results. It also has a better discrimination than other methods and can select the best solution. Establish an information evaluation method based on 2TLNNs, and apply it to multi-attribute group decision making, which enrich the existing knowledge of decision-making theory and providing meaningful reference suggestions for realistic decision-making problems.
In the future, it would be interesting to explore more methods of 2TLNNs in the field of multi-attribute group decision making to solve problems better in real life. And the 2TLNN-GLDS model could be applied to many other fields such as risk analysis [38–40], pattern recognition [41] and other uncertain environments [42].