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A fuzzy rough number-based AHP-TOPSIS for design concept evaluation under uncertain environments

March 2020
Applied Soft Computing 91:106228

March 2020
91:106228

DOI:10.1016/j.asoc.2020.106228

Authors:

Guo-Niu Zhu

Fudan University

Jie Hu

Shanghai Jiao Tong University

Design concept evaluation in the early phase of product design plays a crucial role in new product development as it considerably determines the direction of subsequent design activities. However, it is a process involving uncertainty and subjectivity. The evaluation information mainly relies on expert’s subjective judgment, which is imprecise and uncertain. How to effectively and objectively evaluate the design concept under such subjective and uncertain environments remains an open question. To fill this gap, this paper proposes a fuzzy rough number-enhanced group decision-making framework for design concept evaluation by integrating a fuzzy rough number-based AHP (analytic hierarchy process) and a fuzzy rough number-based TOPSIS (technique for order preference by similarity to ideal solution). First of all, a fuzzy rough number is presented to aggregate personal risk assessment information and to manipulate the uncertainty and subjectivity during the decision-making. Then a fuzzy rough number-based AHP is developed to determine the criteria weights. A fuzzy rough number-based TOPSIS is proposed to conduct the alternative ranking. A practical case study is put forward to illustrate the applicability of the proposed decision-making framework. Experimental results and comparative studies demonstrate the superiority of the fuzzy rough number-based method in dealing with the uncertainty and subjectivity in design concept evaluation under group decision-making environment.

Comparison of criteria weighting

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Comparison of the ranking results from various methods

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Brief description of the evaluation criteria in the design concept evaluation of the heat exchanger

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Fundamental scales in fuzzy pairwise comparison (Liu et al., 2015)

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The separation values of the candidate design concepts

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A fuzzy rough number-based AHP-TOPSIS for design

concept evaluation under uncertain environments

Guo-Niu Zhua,∗, Jie Hub, Hongliang Renc

aSchool of Mechanical and Aerospace Engineering, Nanyang Technological University,

Singapore 639798, Singapore

bState Key Laboratory of Mechanical System and Vibration, School of Mechanical

Engineering, Shanghai Jiao Tong University, Shanghai 200240, China

cDepartment of Biomedical Engineering, National University of Singapore, Singapore

117583, Singapore

Abstract

Design concept evaluation in the early phase of product design plays a crucial

role in new product development as it considerably determines the direction

of subsequent design activities. However, it is a process involving uncertainty

and subjectivity. The evaluation information mainly relies on expert’s sub-

jective judgment, which is imprecise and uncertain. How to eﬀectively and

objectively evaluate the design concept under such subjective and uncertain

environments remains an open question. To ﬁll this gap, this paper proposes

a fuzzy rough number-enhanced group decision-making framework for design

concept evaluation by integrating a fuzzy rough number-based AHP (analytic

hierarchy process) and a fuzzy rough number-based TOPSIS (technique for

order preference by similarity to ideal solution). First of all, a fuzzy rough

number is presented to aggregate personal risk assessment information and

to manipulate the uncertainty and subjectivity during the decision-making.

Then a fuzzy rough number-based AHP is developed to determine the cri-

teria weights. A fuzzy rough number-based TOPSIS is proposed to conduct

the alternative ranking. A practical case study is put forward to illustrate

the applicability of the proposed decision-making framework. Experimen-

tal results and comparative studies demonstrate the superiority of the fuzzy

rough number-based method in dealing with the uncertainty and subjectivity

∗Corresponding author.

Email addresses: guoniu.zhu@ntu.edu.sg (Guo-Niu Zhu), hujie@sjtu.edu.cn (Jie

Hu), ren@nus.edu.sg (Hongliang Ren)

Preprint submitted to Applied Soft Computing February 4, 2020

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in design concept evaluation under group decision-making environment.

Keywords: Design concept evaluation, Fuzzy rough number, Uncertainty

and subjectivity, Group decision-making, AHP and TOPSIS

1. Introduction

Concept generation and evaluation are two crucial steps for obtaining an

optimal design concept at the conceptual design stage, while the former gen-

erates possible design concepts and the latter determines the ﬁnal selection

of candidate design alternatives (Huang et al.,2006). As a gatekeeper, design

concept evaluation has been considered to be extremely important because

of its impact on the novelty, feasibility, and quality of the ﬁnal product, as

well as its inﬂuence on the development cycle and cost (Zheng et al.,2018).

A proper selection of creative design schemes may lead to disruptive innova-

tion and great success while a poor concept selection may not only increase

the development cost and production postponement, but also result in ad-

ditional corrections, iterations, or even endanger the success of the entire

product development (Huang et al.,2013). Therefore, how to evaluate the

design concept at the early design stage is a key problem in new product

development (NPD). It is well recognized that up to 70-80% of the prod-

uct lifecycle cost is determined at the conceptual design stage (Saravi et al.,

2008). Besides, the design defects at this stage can hardly be corrected at

the downstream design activities.

However, design concept evaluation is a tough challenge. To select a desir-

able design concept, design groups need to take into account various factors,

ranging from customer needs, technical attributes to design constraints, and

develop a proper evaluation model to determine the risk priority ranking of

candidate design alternatives (Zhang and Chu,2009). Meanwhile, most of

the evaluation information at the conceptual design stage is derived from

expert’s personal judgment, which is imprecise, uncertain, and subjective

(Song et al.,2013). It varies from personal assessment to techniques that

aggregate, weight, and rank those risk factors and alternatives (Scott,2007).

To alleviate the inﬂuence of cognitive bias from respective decision maker,

group decision-making strategy is becoming more common in design concept

evaluation (Geng et al.,2010;Dong et al.,2015). Many experts are invited

to discuss the evaluation model, identify the evaluation criteria, give their

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assessment information, and determine the alternative ranking of design con-

cepts.

Due to the lack of knowledge, the abundance of alternatives, and the am-

biguity of the assessment information, there emerges a critical gap between

the objective evaluation and the uncertain and subjective environment. It is

too diﬃcult for the decision maker to present precise quantitative numbers

in design concept evaluation. Considering such a dilemma, various strategies

are exploited to deal with the uncertainty and ambiguity in decision-making,

such as fuzzy sets (Akay et al.,2011;Vinodh et al.,2014;Aikhuele,2017),

interval number (Sayadi et al.,2009;Yue,2011), rough number (Zhu et al.,

2017;Chatterjee et al.,2018;Roy et al.,2018), grey system theory (Golinska

et al.,2015;Li and Yuan,2017), and some other variants (Pires et al.,2011;

Pamuˇcar et al.,2018b). The basic idea of such solutions is to introduce in-

terval values to replace the precise numbers so that the uncertainty can be

represented by the intervals. However, the determination of the boundaries

in the interval values remains a challenge. For most of the uncertainty pro-

cessing methods, additional parameters need to be introduced to identify the

boundary values, which mainly depend on the expert’s subjective judgment.

Besides the uncertainty modeling in decision maker’s assessments, how to

eﬀectively and objectively aggregate these assessments from diﬀerent decision

makers in the group decision-making environment is another critical issue.

Several weighted averaging operators have been proposed to handle this prob-

lem, such as the weighted mean method (Zhang and Chu,2009), linguistic

arithmetic averaging operator (Xu,2006), and ordered weighted geometric

aggregation operator (Akay et al.,2011). However, the weighting strategy

needs to be predeﬁned, which involves additional subjective judgment. Each

decision maker may give a totally diﬀerent weighting parameter, which varies

from personal preferences to cognitive bias as well as randomness.

Although some research eﬀorts have been made in the uncertainty and

subjectivity manipulation during design concept evaluation, it remains a

challenging task. Most of these methods can only address part of them,

e.g. fuzzy sets, weighted averaging operators. Much of the uncertainty and

subjectivity are ignored, such as the determination of the weighted averag-

ing operators, the selection of the membership function, the cognitive bias

between diﬀerent decision makers, etc. Considering the shortcomings of the

weighted averaging operators, some rough number-based methods have been

proposed to further enhance the evaluation information aggregation in group

decision-making (Zhai et al.,2009;Pamuˇcar et al.,2017,2018b;Roy et al.,

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2018). Compared with the weighted averaging operators, the rough number

is an objective mathematical model. It is totally established on the original

evaluation dataset, without any predeﬁned parameters (Zhu et al.,2015).

Due to its objectivity in personal assessment information aggregation, some

other models such as crisp number (Chatterjee et al.,2018), interval number

(Pamuˇcar et al.,2019a), and fuzzy sets (Pamuˇcar et al.,2018a) have been

investigated to integrate with the rough number to handle the assessment

aggregation in group decision-making.

To enhance the uncertainty characterization and the subjectivity elimi-

nation in design concept evaluation, this paper proposes a systematic fuzzy

rough number-based group decision-making framework by integrating the

fuzzy rough number-based AHP (analytic hierarchy process) and the fuzzy

rough number-based TOPSIS (technique for order preference by similarity

to ideal solution). The purpose of this paper lies in three folds: 1) to develop

a novel multi-criteria decision-making (MCDM) framework to ﬁll the gap

that exists in design concept evaluation; 2) to present a fuzzy rough number

to deal with the uncertainty and subjectivity in group decision-making to

strengthen the objectivity of the evaluation results; 3) to propose a fuzzy

rough number-based AHP and a fuzzy rough number-based TOPSIS to en-

rich the MCDM algorithms and to emphasize the advantages of the fuzzy

rough number. The primary contributions of this paper include:

(1) An integrated fuzzy rough number-based group decision-making frame-

work is proposed for design concept evaluation. Based on the strength of

fuzzy sets in uncertainty representation, the beneﬁt of group decision-making

in cognitive bias elimination, and the advantage of fuzzy rough number in

subjective information aggregation, the proposed fuzzy rough number-based

decision-making framework provides an objective and eﬀective way in indi-

vidual assessments characterization and aggregation in design concept eval-

uation. The fuzzy rough number adopts a ﬂexible interval boundary and

is endowed with the ability to depict the uncertainty of the entire triangu-

lar fuzzy number and to reveal the vagueness of its individual components

including the lower bound, modal value, and the upper bound. It is just

based on the original evaluation dataset, without any predeﬁned parameters

or auxiliary information in the subsequent processing. It provides a way

to measure the uncertainty in fuzzy importance and linguistic variables in

design concept evaluation.

(2) A fuzzy rough number-based AHP (FRN-AHP) and a fuzzy rough

number-based TOPSIS (FRN-TOPSIS) are presented to enhance the MCDM

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algorithms in dealing with uncertainty and subjectivity under group decision-

making environments. To the best of our knowledge, this is the ﬁrst work

that proposes the AHP and TOPSIS with the fuzzy rough number.

(3) Comparative studies are conducted to demonstrate the superiority

of the proposed fuzzy rough number-based decision-making algorithm. The

subjectivity and uncertainty are eﬀectively characterized in design concept

evaluation.

The rest of this paper is structured as follows. Section 2outlines the

background. Section 3discusses the proposed fuzzy rough number. Sec-

tion 4presents the fuzzy rough number-based decision-making framework.

A real-world case study of the design concept evaluation for a bio-inspired

heat exchanger is provided in Section 5. The comparison and discussion are

presented in Section 6. The conclusion is drawn in Section 7.

2. Related work

2.1. Design concept evaluation

Design concept evaluation aims to select the optimal design concept for

subsequent design activities. As a typical MCDM problem, various decision-

making approaches have been introduced for evaluating the candidate design

concepts. Aya˘g (2016) presented an integrated ANP (analytic network pro-

cess) and modiﬁed TOPSIS for concept selection in a NPD. ANP was used to

calculate the criteria weights while a modiﬁed TOPSIS was adopted to deter-

mine the alternative ranking. Ahmad et al. (2017) studied the application of

AHP method in conceptual design selection for a manual wheelchair. Dong

et al. (2014) investigated the preference representation in concept selection.

A linguistic model was proposed to evaluate the dynamics of design team

priorities in the evaluation process. In addition, Goswami and Tiwari (2014)

suggested a predictive risk assessment method for modular product concept

selection. Bayesian network was introduced to describe the risk parameters

and overall enterprise risk index was developed to evaluate the candidate

design concepts. Based on data mining and domain ontology, Chang and

Chen (2015) studied the product concept evaluation under a crowdsourcing

environment. Hao et al. (2017) put forward a function-based computational

tool for evaluating design concepts. Some computational metrics were devel-

oped to evaluate the novelty, feasibility, and diversity based on the function

knowledge base and patent knowledge base. Tiwari et al. (2019) presented

a bijective soft set-based Shannon entropy and TOPSIS for design concept

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evaluation. Shannon entropy was used to calculate criteria weights while

TOPSIS was introduced to rank the design concepts. Similarly, Hayat et al.

(2020) developed an integrated Shannon entropy and TOPSIS based on soft

sets. The acceptable and satisfactory level requirements of customers were

taken into consideration in design concept evaluation. He et al. (2018) dis-

cussed the unascertained measure model for product environmental footprint

evaluation.

However, most of the evaluation values at the conceptual design stage

come from expert’s subjective judgment. The crisp number-based models are

unable to capture the uncertainty and vagueness during the evaluation pro-

cess. To enhance the design concept evaluation in uncertainty manipulation,

fuzzy sets were introduced and several fuzzy sets-based algorithms have been

developed. Aya˘g (2005) proposed an integrated fuzzy AHP and simulation

analysis for concept evaluation. A fuzzy AHP was developed to eliminate the

possible design concepts and a simulation analysis was adopted to evaluate

the remaining alternatives. Jenab et al. (2013) presented a multi-layer fuzzy

graph for conceptual design selection using conﬂict resolution. Huang et al.

(2013) investigated the application of fuzzy neural network, fuzzy weighted

average, and fuzzy compromise decision-making in design concept evalua-

tion. Furthermore, Vinodh et al. (2016) suggested an integrated fuzzy DE-

MATEL (decision making trial and evaluation laboratory), fuzzy ANP, and

fuzzy TOPSIS for agile concept selection. Fuzzy DEMATEL and fuzzy ANP

were developed to conduct the interdependency analysis within and among

the agile criteria. Fuzzy TOPSIS was adopted to evaluate the alternatives.

Shidpour et al. (2016) put forward a novel group MCDM framework by in-

tegrating rough sets and fuzzy sets. Criteria weights were calculated based

on extent analysis on fuzzy AHP and design concepts preference ranking was

determined by the interval-based relative closeness. Besides the fuzzy sets,

the interval-valued fuzzy sets were also introduced to further investigate the

uncertainty in decision-making. Pires et al. (2011) presented an AHP-based

fuzzy interval TOPSIS for sustainable expansion of the solid waste manage-

ment system. An interval-valued fuzzy TOPSIS was developed to determine

the ﬁnal ranking. Compared with the fuzzy sets, the interval-valued fuzzy

sets are enabled to reveal more uncertainty inherent in the decision-making.

However, the determination of the uncertainty degree involves additional

subjective judgment.

Despite the fact that the fuzzy and interval-valued fuzzy sets-based meth-

ods show powerful ability in uncertainty representation with the aid of mem-

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bership function, it only beneﬁts in individual assessment information charac-

terization. The aggregation of separate evaluation values within the decision-

making group remains a tricky problem. To resolve this issue, Geng et al.

(2010) proposed a vague sets-based integrated method. A vague sets-based

weighted least squares model was developed to aggregate personal assess-

ment values. A vague sets-based cross-entropy incorporated with a vague

sets-based TOPSIS was suggested to rank the conceptual design alterna-

tives. Based on the interval-valued fuzzy sets, Kuo and Liang (2012) put

forward an arithmetic averaging operator for evaluation values aggregation.

Moreover, Yan and Ma (2015) presented a three-stage fuzzy group decision-

making technique by combining a fuzzy weighted average method, a fuzzy

number weighted ordered weighted averaging operator, and a probability dis-

tribution model. Jing et al. (2018) developed a noncooperative-cooperative

game theory-based decision method with multiple interactive qualitative ob-

jectives. An arithmetic averaging strategy was adopted in the aggregation of

group evaluations. A noncooperative-cooperative game decision model was

suggested to evaluate the conceptual design schemes. However, the weighted

averaging operators need to be predeﬁned, which depends on intuitive judg-

ment and involves additional subjectivity.

Rather than the predeﬁned weighted averaging operators, some eﬀorts

have been devoted to the rough number. Zhai et al. (2009) proposed a rough-

grey analysis for design concept evaluation. Rough number was developed

to incorporate with grey relational analysis. Zhu et al. (2015) presented

an integrated rough number-based AHP and rough number-based VIKOR

(VIseKriterijumska Optimizacija I Kompromisno Resenje) to evaluate the

design concepts in NPD. A rough number-based AHP was developed to cal-

culate the criteria weights and a rough number-based VIKOR was proposed

to determine the priority ranking. Hu et al. (2015) studied the combina-

tion of rough number and information entropy theory. Furthermore, Zhang

et al. (2017) suggested a quantitative framework with data-driven perfor-

mance predictions. A rough number-based DEMATEL was put forward to

determine the criteria weights and VIKOR was introduced to identify the

priority ranking of design concepts. Chatterjee et al. (2018) discussed the in-

tegration of rough number, ANP, DEMATEL, and MAIRCA (multi-attribute

ideal-real comparative analysis). A hybrid rough DEMATEL-ANP and rough

MAIRCA method was developed to evaluate the performance of suppliers for

green supply chain implementation.

Among these approaches, the rough number is directly generated from the

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original evaluation values. Due to its objective characteristic, the rough num-

ber has shown strong potentials in evaluation information aggregation under

subjective environments. In view of the merit of the rough number in dealing

with subjectivity in decision-making, many extensions have been investigated

to integrate with the rough number to resolve the decision-making problems.

By integrating with the interval numbers and rough number, Pamuˇcar et al.

(2017) put forward a hybrid DEMATEL-ANP-MAIRCA method based on

interval rough numbers for group MCDM. Rough number is adopted to ag-

gregate the interval numbers in group decision-making. Likewise, Pamuˇcar

et al. (2018b) developed an integrated interval rough AHP and interval rough

MABAC (multi-attributive border approximation area comparison) method

to evaluate the university web pages. Pamuˇcar et al. (2019a) further com-

bined with interval rough number and BWM (best worst method), WASPAS

(weighted aggregated sum product assessment), MABAC to evaluate the

third-party logistics. Based on the fuzzy sets and rough number, Pamuˇcar

et al. (2018a) presented an interval-valued fuzzy-rough numbers (IVFRN)

and integrated this model with BWM and MABAC method. Generally, the

IVFRN proposed in this paper is the most similar to ours, but we diﬀer in the

uncertainty measurement mechanism and arithmetic operations. Similarly,

Pamuˇcar et al. (2019b) suggested the application of IVFRN in MAIRCA. Roy

et al. (2019) investigated the combination of the factor relationship (FARE)

and MABAC based on IVFRN.

Though traditional fuzzy sets and weighted averaging operators provided

useful techniques in uncertainty representation and subjectivity manipula-

tion, most of these methods can only address part of them. In view of the

prominent performance of fuzzy sets in uncertainty representation, the supe-

riority of group decision-making strategy in bias elimination, and the merit of

rough number in subjectivity manipulation, this paper proposes a systematic

fuzzy rough number-based MCDM framework to ﬁll the gap existed in de-

sign concept evaluation. Based on such assumptions, a fuzzy rough number is

proposed to incorporate the rough number in fuzzy sets. Then a fuzzy rough

number-based AHP is presented to determine the criteria weights. A fuzzy

rough number-based TOPSIS is proposed to identify the priority ranking of

the design concepts.

2.2. Fuzzy sets and fuzzy number

As a powerful technique in uncertainty manipulation, fuzzy sets have

been widely applied in information representation and quantiﬁcation under

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uncertain environments (Yager and Zadeh,2012;Wang et al.,2014;Olivas

et al.,2019). Basically, a fuzzy set e

Sis composed of a group of elements,

which each element has its own degree of membership µe

S(x). In past studies,

various types of membership function, including triangular, trapezoidal, and

Gaussian, have been incorporated with fuzzy logic (Karimi et al.,2020;Wu

et al.,2019;Jais et al.,2019). Due to the simpliﬁcation and easily calcula-

tion, the triangular membership function is extensively used for fuzzy sets

representation in real applications (Nguyen et al.,2019).

Furthermore, a fuzzy number is a generalization of the real number. It

is a special case of the fuzzy sets, whose membership is convex and nor-

malized. Given two positive triangular fuzzy numbers ep= (pl, pm, pu) and

eq= (ql, qm, qu), kis a positive real number, the basic arithmetic rules for

triangular fuzzy number are described as (Chen,2000):

ep⊕eq= (pl+ql, pm+qm, pu+qu) (1)

p⊖e

q= (pl−qu, pm−qm, pu−ql) (2)

ep⊗eq∼

=(pl×ql, pm×qm, pu×qu) (3)

ep⊘eq∼

=(pl

,pm

,pu

) (4)

ep⊗k∼

=(pl×k, pm×k, pu×k) (5)

ep−1∼

=(1

) (6)

3. Fuzzy rough number

Due to its eﬀectiveness in subjective information processing, rough num-

ber has been widely employed to combine with various decision-making meth-

ods and applied in many areas, such as quality function deployment (Zheng

et al.,2019), design concept evaluation (Song et al.,2013;Zhu et al.,2015),

failure mode and eﬀects analysis (Li et al.,2019), change mode and eﬀects

analysis (Zhu et al.,2017), and supply chain evaluation (Chatterjee et al.,

2018). Inspired by the manipulation strategy of the classical rough num-

ber in crisp number aggregation, this paper extends the application of rough

number to the manipulation of fuzzy sets and proposes a fuzzy rough number.

Basically, a fuzzy rough number shares the concept of lower limit, upper

limit, and rough boundary interval as conventional rough number. Suppose

Uis the universe, which is composed of the evaluation values collected from

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the decision maker. Generally, these values can be divided into nclasses,

which are ordered as e

C1≤e

C2≤ · · · ≤ e

Cn.e

Ci(1 ≤i≤n) is a triangular

fuzzy number, e

Ci= (Cil, Cim , Ciu). e

Ris the collection of {e

C1,e

C2,· · · ,e

Cn},

Yis an arbitrary element of the universe, then the lower approximation of

class e

Ciis deﬁned as:

Apr(Cil) = ∪{Y∈U/ e

R(Y)≤Cil}(7)

Apr(Cim) = ∪{Y∈U/ e

R(Y)≤Cim}(8)

Apr(Ciu) = ∪{Y∈U/ e

R(Y)≤Ciu}(9)

Correspondingly, the upper approximation of class e

Ciis described as:

Apr(Cil) = ∪{Y∈U/ e

R(Y)≥Cil}(10)

Apr(Cim) = ∪{Y∈U/ e

R(Y)≥Cim}(11)

Apr(Ciu) = ∪{Y∈U/ e

R(Y)≥Ciu}(12)

Then the lower limit of e

Ciis deﬁned as:

Lim(Cil) = 1

NLl

i=1

Y∈Apr(Cil) (13)

Lim(Cim) = 1

NLm

i=1

Y∈Apr(Cim) (14)

Lim(Ciu) = 1

NLu

i=1

Y∈Apr(Ciu) (15)

where NLl,NLm , and NLu are the number of elements in Apr(Cil), Apr(Cim),

and Apr(Ciu), respectively.

Likewise, the upper limit of e

Ciis represented as:

Lim(Cil) = 1

NUl

i=1

Y∈Apr(Cil) (16)

Lim(Cim) = 1

NUm

i=1

Y∈Apr(Cim) (17)

Lim(Ciu) = 1

NUu

i=1

Y∈Apr(Ciu) (18)

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where NUl,NU m , and NU u denote the number of elements in Apr(Cil),

Apr(Cim), and Apr(Ciu), respectively.

Then the fuzzy rough number of e

Cican be denoted as:

F RN (e

Ci)=(⌈Lim(Cil),Lim(Cil )⌋,⌈Lim(Cim), Lim(Cim )⌋,⌈Lim(Ciu), Lim(Ciu )⌋)(19)

The rough boundary interval of Cil,Cim,Ciu , and the whole e

Ciare deﬁned

as:

RBnd(Cil ) = Lim(Cil )−Lim(Cil) (20)

RBnd(Cim ) = Lim(Cim)−Lim(Cim) (21)

RBnd(Ciu ) = Lim(Ciu)−Lim(Ciu) (22)

RBnd(e

Ci) = Lim(Ciu)−Lim(Cil ) (23)

Generally, the rough boundary interval represents the vagueness of Cil,

Cim,Ciu , and e

Ci, respectively. A smaller interval usually means a better

precise and a larger one denotes vaguer. Therefore, it enables the fuzzy

rough number to measure the uncertainty in the triangular fuzzy number,

which can be further extended to fuzzy importance and linguistic variables.

The subjective risk assessment values can be aggregated and characterized

using the fuzzy rough number.

Given two positive fuzzy rough numbers F RN (eα) = (⌈Lim(αl), Lim(αl)⌋,

⌈Lim(αm), Lim(αm)⌋,⌈Lim(αu), Lim(αu)⌋)and F RN (e

β) = (⌈Lim(βl), Lim(βl)⌋,

⌈Lim(βm), Lim(βm)⌋,⌈Lim(βu), Lim(βu)⌋),µis a positive real number, the

arithmetic operations for the fuzzy rough number are described as (Chen,

2000):

F RN (eα)⊕F RN (e

β) =(⌈Lim(αl), Lim(αl)⌋,⌈Lim(αm), Lim(αm)⌋,⌈Lim(αu), Lim(αu)⌋)⊕

(⌈Lim(βl), Lim(βl)⌋,⌈Lim(βm), Lim(βm)⌋,⌈Lim(βu), Lim(βu)⌋)

=(⌈Lim(αl) + Lim(βl), Lim(αl) + Lim(βl)⌋,⌈Lim(αm) + Lim(βm),

Lim(αm) + Lim(βm)⌋,⌈Lim(αu) + Lim(βu), Lim(αu) + Lim(βu)⌋)

(24)

F RN (eα)⊖F RN (e

β) =(⌈Lim(αl), Lim(αl)⌋,⌈Lim(αm), Lim(αm)⌋,⌈Lim(αu), Lim(αu)⌋)⊖

(⌈Lim(βl), Lim(βl)⌋,⌈Lim(βm), Lim(βm)⌋,⌈Lim(βu), Lim(βu)⌋)

=(⌈Lim(αl)−Lim(βu), Lim(αl)−Lim(βu)⌋,⌈Lim(αm)−Lim(βm),

Lim(αm)−Lim(βm)⌋,⌈Lim(αu)−Lim(βl), Lim(αu)−Lim(βl)⌋)

(25)

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F RN (eα)⊗F RN (e

β) =(⌈Lim(αl), Lim(αl)⌋,⌈Lim(αm), Lim(αm)⌋,⌈Lim(αu), Lim(αu)⌋)⊗

(⌈Lim(βl), Lim(βl)⌋,⌈Lim(βm), Lim(βm)⌋,⌈Lim(βu), Lim(βu)⌋)

=(⌈Lim(αl)×Lim(βl), Lim(αl)×Lim(βl)⌋,⌈Lim(αm)×Lim(βm),

Lim(αm)×Lim(βm)⌋,⌈Lim(αu)×Lim(βu), Lim(αu)×Lim(βu)⌋)

(26)

F RN (eα)⊘F RN (e

β) =(⌈Lim(αl), Lim(αl)⌋,⌈Lim(αm), Lim(αm)⌋,⌈Lim(αu), Lim(αu)⌋)⊘

(⌈Lim(βl), Lim(βl)⌋,⌈Lim(βm), Lim(βm)⌋,⌈Lim(βu), Lim(βu)⌋)

=(⌈Lim(αl)÷Lim(βu), Lim(αl)÷Lim(βu)⌋,⌈Lim(αm)÷Lim(βm),

Lim(αm)÷Lim(βm)⌋,⌈Lim(αu)÷Lim(βl), Lim(αu)÷Lim(βl)⌋)

(27)

F RN (eα)⊗µ=(⌈Lim(αl), Lim(αl)⌋,⌈Lim(αm), Lim(αm)⌋,⌈Lim(αu), Lim(αu)⌋)⊗µ

=(⌈Lim(αl)×µ, Lim(αl)×µ⌋,⌈Lim(αm)×µ, Lim(αm)×µ⌋,

⌈Lim(αu)×µ, Lim(αu)×µ⌋)

(28)

From above deﬁnitions, it can be concluded that the fuzzy rough number

is only determined by the original dataset. It does not need any auxiliary in-

formation. Thus, it is an objective model, which can better capture the deci-

sion maker’s true perception and strengthen the objectivity under subjective

environments. Original fuzzy numbers are aggregated and transformed into

the fuzzy rough numbers. In view of the potential possibility in subjective

information manipulation, the fuzzy rough number is presented to combine

with the group decision-making framework for design concept evaluation.

Compared with the IVFRN which is proposed in Pamuˇcar et al. (2018a),

we diﬀer in two folds. First of all, we use diﬀerent uncertainty measure-

ment mechanisms. In IVFRN, the rough boundary interval of I(a1)qis

deﬁned as RB(I∗(a1)q) = Lim(I∗(a1)uq )−Lim(I∗(a2)lq ). It is not clear

what is the rough boundary interval of I(a2)q,I(a3)q, and the whole fuzzy

number Aq= (a1q, a2q, a3q). According to the deﬁnition of Lim(I∗(a1)lq)≤

Lim(I∗(a1)uq)≤Lim(I∗(a2)lq)≤ · · · ≤ Lim(I∗(a3)uq ), then RB(I∗(a1)q)≤

0. By contrast, the fuzzy rough number identiﬁed the rough boundary inter-

val of individual components including the lower bound (RBnd(Cil )), modal

value (RBnd(Cim)), and the upper bound (RBnd(Ciu)), as well as the rough

boundary interval of the whole triangular fuzzy number (RBnd(e

Ci)). The

series of deﬁnitions of the rough boundary interval enables the fuzzy rough

number to reveal both the vagueness of the whole triangular fuzzy num-

ber ( e

Ci) and its individual components (Cil,Cim , and Ciu ). The rough

Journal Pre-proof

boundary interval is always a nonnegative value in the fuzzy rough num-

ber. It provides a way to measure the uncertainty in fuzzy importance and

linguistic variables. Thus, the uncertainty measurement mechanism is well

enhanced in the fuzzy rough number. Second, we adopt diﬀerent arithmetic

operations. Take the subtraction operation for example, let F RN (eα) =

(⌈2.2,3⌋,⌈4.2,5⌋,⌈6.2,7⌋) and F RN (e

β)=(⌈1,2.2⌋,⌈3,4.2⌋,⌈5,6.2⌋), then it

is calculated as F RN (eα)−F RN(e

β) = ([2.2−1,3−2.2],[4.2−3,5−4.2],[6.2−

5,7−6.2]) = ([1.2,0.8],[1.2,0.8],[1.2,0.8]) in Pamuˇcar et al. (2018a) while

F RN (eα)⊖F RN(e

β) = (⌈2.2−6.2,3−5⌋,⌈4.2−4.2,5−3⌋,⌈6.2−2.2,7−1⌋) =

(⌈−4,−2⌋,⌈0,2⌋,⌈4,6⌋) in our method. Obviously, the result of Pamuˇcar

et al. (2018a) looks so confusing. The lower limit is even bigger than the

upper limit. On the other hand, the result is consistent with the deﬁnition

in our method. The lower limit is always smaller than the upper limit. Sim-

ilarly, the fuzzy rough number of Cil is always smaller than the fuzzy rough

number of Cim and then the Ciu. It is more clear and stable in our method.

Similar situations can be also found in the division operation.

4. Fuzzy rough number-based group decision-making for design

concept evaluation

4.1. Framework of the proposed method

To enhance the objectivity in design concept evaluation, this paper pro-

poses a fuzzy rough number-enhanced group decision-making technique by

combining a fuzzy rough number-based AHP and a fuzzy rough number-

based TOPSIS. The evaluation process is divided into three phases: 1) de-

termine the evaluation components, including the evaluation objective, eval-

uation criteria, and evaluation alternatives; 2) relative weight calculation by

fuzzy rough number-based AHP; 3) evaluation alternative ranking by fuzzy

rough number-based TOPSIS. First of all, a group of experts are invited

as the decision makers. Expert interviews are conducted to determine the

evaluation components and to collect the experts’ assessments about the

evaluation criteria and alternatives. Then a fuzzy rough number is presented

to aggregate individual risk ratings. A fuzzy rough number-based AHP is

developed to determine the criteria weights. A fuzzy rough number-based

TOPSIS is proposed to identify the risk priority ranking of the candidate

design concepts. The framework of the proposed fuzzy rough number-based

design concept evaluation is illustrated in Fig. 1.

Journal Pre-proof

Phase 3: Fuzzy rough numberbased

TOPSIS for alternative ranking

IdentifyWKH

evaluation

objective,

evaluation

alternatives,

and

evaluation

criteria

Collect pairwise comparison data and construct

an integrated pairwise comparison matrix

Generate fuzzy rough numbers and construct a

fuzzy rough pairwise comparison matrix

Calculate WKHrelative weights for evaluation

criteria EDVHGRQWKH fuzzy rough number

Collect evaluation data and construct a fuzzy

rough decision matrix

Decision matrix normalization and weighted

normalized decision matrix construction

Identify WKHpositiYH ideal solution (PIS)

and negative ideal solution (NIS)

Determine WKHseparation

CalculateWKHrelative closeness and determine

the alternative ranking

Phase 2: Fuzzy rough

numberbased AHP for

criteria weighting

Phase 1: Evaluation

components

determination

Fig. 1. Framework of the proposed fuzzy rough number-based design concept evaluation

Journal Pre-proof

4.2. Fuzzy rough number-based AHP for criteria weighting

At the beginning of design concept evaluation, a group of experts are

interviewed to identify the evaluation components. Considering the power-

ful capabilities of AHP methods in analyzing complex decisions, handling

multiple decision makers, conducting consistency measurement, and manip-

ulating tangible and non-tangible criteria, a fuzzy rough number-based AHP

is presented to calculate the criteria weights, which is performed as follows.

Step 1: Data collection and integrated pairwise comparison matrix con-

struction.

Conduct AHP surveys and ask the experts to give their respective judg-

ment about the evaluation criteria. Collect risk ratings from each expert and

build corresponding fuzzy pairwise comparison matrix. The fuzzy pairwise

comparison matrix given by expert eis expressed as:

Me=







1exe

12 · · · exe

exe

21 1· · · exe

exe

p1exe

p2· · · 1





(29)

where exe

ij = (xe

ijl , xe

ijm , xe

iju ),1≤i, j ≤pis the assessment value of criterion

ion criterion j,pis the number of the evaluation criteria.

After the construction of f

Me, a consistency test is implemented to each

of the fuzzy pairwise comparison matrices. The consistency ratio (C R) is

deﬁned as:

CR =CI

RI (30)

where RI is the random consistency index as listed in Table 1,CI is the

consistency index, and

CI =λe

max −p

p−1(31)

where λe

max is the maximum eigenvalue of f

Me,pis the number of the evalu-

ation criteria.

If CR < 0.1, the pairwise comparison matrix is acceptable. Otherwise,

the risk ratings should be adjusted until CR < 0.1.

Journal Pre-proof

Table 1. Random consistency index (RI) (Satty,1980)

p 3 4 5 6 7 8 9 10 11

RI 0.52 0.89 1.11 1.25 1.35 1.40 1.45 1.49 1.51

For the sake of simplicity, the matrix Me= [xe

ijm ]p×pis used in the con-

sistency test to replace the original matrix f

Me= [exe

ij ]p×p. If Meis consistent,

then f

Meis acceptable (Lin,2010).

Integrate all the pairwise comparison matrices and construct an inte-

grated pairwise comparison matrix, which is represented as:







ex12 · · · e

ex1p

x21 1· · · e

x2p

exp1e

exp2· · · 1





(32)

where e

exij ={ex1

ij ,ex2

ij ,· · · ,exk

ij }, 1 ≤i, j ≤pis the integrated pairwise com-

parison value of criterion ion criterion j,kis the number of the decision

makers. Alternatively, e

exij can be also written as e

exij = (exijl ,exij m,exij u),

where exijl ={x1

ijl , x2

ijl ,· · · , xk

ijl },exijm ={x1

ijm , x2

ijm ,· · · , xk

ijm }, and exiju =

{x1

iju , x2

iju ,· · · , xk

iju }.

Step 2: Fuzzy rough number generation and fuzzy rough pairwise com-

parison matrix construction.

Given the element e

exij in the integrated pairwise comparison matrix f

the component exe

ij can be transformed to a fuzzy rough number using Eqs.

(7)-(19):

F RN (exe

ij )=(⌈Lim(xe

ijl ), Lim(xe

ijl )⌋,⌈Lim(xe

ijm ), Lim(xe

ijm )⌋,⌈Lim(xe

iju ), Lim(xe

iju )⌋) (33)

Then e

exij is represented as a sequence of the fuzzy rough numbers.

F RN (e

exij ) = {F RN (ex1

ij ), F RN (ex2

ij ),··· , F RN (exk

ij )}

={(⌈Lim(x1

ijl ), Lim(x1

ijl )⌋,⌈Lim(x1

ijm ), Lim(x1

ijm )⌋,⌈Lim(x1

iju ), Lim(x1

iju )⌋),

(⌈Lim(x2

ijl ), Lim(x2

ijl )⌋,⌈Lim(x2

ijm ), Lim(x2

ijm )⌋,⌈Lim(x2

iju ), Lim(x2

iju )⌋),

··· ,

(⌈Lim(xk

ijl ), Lim(xk

ijl )⌋,⌈Lim(xk

ijm ), Lim(xk

ijm )⌋,⌈Lim(xk

iju ), Lim(xk

iju )⌋)}

(34)

Journal Pre-proof

It is further translated into a fuzzy rough number, which is deﬁned as the

average of the sequence.

F RN (e

exij ) = (⌈xL

ijl , xU

ijl ⌋,⌈xL

ijm , xU

ijm ⌋,⌈xL

iju , xU

iju ⌋) (35)

where

ijl =1

e=1

Lim(xe

ijl ) (36)

ijl =1

e=1

Lim(xe

ijl ) (37)

ijm =1

e=1

Lim(xe

ijm ) (38)

ijm =1

e=1

Lim(xe

ijm ) (39)

iju =1

e=1

Lim(xe

iju ) (40)

iju =1

e=1

Lim(xe

iju ) (41)

where xL

ijl ,xL

ijm , and xL

iju are the lower limit of corresponding exij l,exij m, and

exiju , respectively; while xU

ijl ,xU

ijm , and xU

iju are the upper limit of correspond-

ing exijl ,exijm , and exij u, respectively.

Then a fuzzy rough pairwise comparison matrix is obtained as:







(⌈1,1⌋,⌈1,1⌋,⌈1,1⌋)··· (⌈xL

1pl, xU

1pl⌋,⌈xL

1pm, xU

1pm⌋,⌈xL

1pu, xU

1pu⌋)

(⌈xL

21l, xU

21l⌋,⌈xL

21m, xU

21m⌋,⌈xL

21u, xU

21u⌋)··· (⌈xL

2pl, xU

2pl⌋,⌈xL

2pm, xU

2pm⌋,⌈xL

2pu, xU

2pu⌋)

(⌈xL

p1l, xU

p1l⌋,⌈xL

p1m, xU

p1m⌋,⌈xL

p1u, xU

p1u⌋)··· (⌈1,1⌋,⌈1,1⌋,⌈1,1⌋)





(42)

Step 3: Criteria weights calculation.

Based on the fuzzy rough pairwise comparison matrix c

M, the fuzzy rough

weight for each evaluation criterion is calculated as:

bwi= (⌈wL

il , wU

il ⌋,⌈wL

im, wU

im⌋,⌈wL

iu, wU

iu⌋)

= (⌈p

j=1

ijl ,p

j=1

ijl ⌋,⌈p

j=1

ijm ,p

j=1

ijm ⌋,⌈p

j=1

iju ,p

j=1

iju ⌋)(43)

Journal Pre-proof

It can be further normalized as:

bw′

i=bwi/max(wU

iu) (44)

Thus, the criteria weights are determined.

4.3. Fuzzy rough number-based TOPSIS for alternative ranking

Based on the relative weights calculated by the fuzzy rough number-

based AHP, a fuzzy rough number-based TOPSIS is proposed to rank the

risk priority of candidate design concepts, which is described as follows.

Step 1: Data collection and fuzzy rough decision matrix construction.

Conduct expert interviews and ask the decision makers to give their inde-

pendent assessments about evaluation alternatives with respect to the eval-

uation criteria. Collect personal evaluation values, convert them to fuzzy

numbers, and construct corresponding decision matrix. The fuzzy decision

matrix given by expert eis represented as:

De=







11 e

12 · · · e

21 e

22 · · · e

q1e

q2· · · e





(45)

where e

ij = (fe

ijl , f e

ijm , f e

iju ),1≤i≤q, 1≤j≤pis the evaluation value of

alternative ion criterion j,qis the number of the evaluation alternatives, p

is the number of the evaluation criteria.

Integrate all the fuzzy decision matrices and transform the elements into

the corresponding fuzzy rough numbers. Then a fuzzy rough decision matrix

is constructed as:







(⌈fL

11l, f U

11l⌋,⌈fL

11m, f U

11m⌋,⌈fL

11u, f U

11u⌋)··· (⌈fL

1pl, f U

1pl⌋,⌈fL

1pm, f U

1pm⌋,⌈fL

1pu, f U

1pu⌋)

(⌈fL

21l, f U

21l⌋,⌈fL

21m, f U

21m⌋,⌈fL

21u, f U

21u⌋)··· (⌈fL

2pl, f U

2pl⌋,⌈fL

2pm, f U

2pm⌋,⌈fL

2pu, f U

2pu⌋)

(⌈fL

q1l, f U

q1l⌋,⌈fL

q1m, f U

q1m⌋,⌈fL

q1u, f U

q1u⌋)··· (⌈fL

qpl, f U

qpl⌋,⌈fL

qpm, f U

qpm⌋,⌈fL

qpu, f U

qpu⌋)





(46)

where fL

ijl ,fL

ijm , and fL

iju , 1 ≤i≤q, 1≤j≤pare the lower limit of

corresponding e

fijl ,e

fijm , and e

fiju , respectively; while fU

ijl ,fU

ijm , and fU

iju are

the upper limit of corresponding e

fijl ,e

fijm , and e

fiju , respectively.

Step 2: Decision matrix normalization and weighted normalized decision

matrix construction.

Journal Pre-proof

The fuzzy rough decision matrix b

Dis normalized as (Chen,2000;Ashtiani

et al.,2009):

brij = (⌈fL

ijl

f∗

,fU

ijl

f∗

⌋,⌈fL

ijm

f∗

,fU

ijm

f∗

⌋,⌈fL

iju

f∗

,fU

iju

f∗

⌋), j ∈B(47)

brij = (⌈f−

iju

,f−

iju

⌋,⌈f−

ijm

,f−

ijm

⌋,⌈f−

ijl

,f−

ijl

⌋), j ∈C(48)

where Bis the beneﬁt criterion, Cis the cost criterion, and

f∗

j= max

ifU

iju , j ∈B(49)

f−

j= min

ifL

ijl , j ∈C(50)

Thus, the normalized fuzzy rough decision matrix b

R= [brij ]q×pis ob-

tained. Furthermore, the weighted normalized decision matrix b

V= [bvij ]q×p

is determined as:

bvij =brij ⊗bw′

i(51)

where bw′

iis the criterion weight determined by the fuzzy rough number-based

AHP in Section 4.2.

Step 3: Positive ideal solution (PIS) and negative ideal solution (NIS)

identiﬁcation.

As all the evaluation criteria have been transformed to the beneﬁt crite-

rion after normalization, then the PIS and NIS are determined as:

V∗={bv∗

1,bv∗

2,· · · ,bv∗

p}(52)

V−={bv−

1,bv−

2,· · · ,bv−

p}(53)

where bv∗

j= (⌈1,1⌋,⌈1,1⌋,⌈1,1⌋) and bv−

j= (⌈0,0⌋,⌈0,0⌋,⌈0,0⌋), j= 1,2,· · · , p.

Step 4: Separation determination.

The separation of each evaluation alternative to the PIS is calculated as:

d∗

i=d(b

Vi,b

V∗)

j=1

d(bvij ,bv∗

j=1 s1

6[(vL

ijl −v∗L

jl )2+ (vU

ijl −v∗U

jl )2+ (vL

ijm −v∗L

jm )2+ (vU

ijm −v∗U

jm )2+ (vL

iju −v∗L

ju )2+ (vU

iju −v∗U

ju )2]

j=1 s1

6[(vL

ijl −1)2+ (vU

ijl −1)2+ (vL

ijm −1)2+ (vU

ijm −1)2+ (vL

iju −1)2+ (vU

iju −1)2]

(54)

Journal Pre-proof

Similarly, the separation of each evaluation alternative to the NIS is de-

ﬁned as:

d−

i=d(b

Vi,b

V−)

j=1

d(bvij ,bv−

j=1 s1

6[(vL

ijl −v−L

jl )2+ (vU

ijl −v−U

jl )2+ (vL

ijm −v−L

jm )2+ (vU

ijm −v−U

jm )2+ (vL

iju −v−L

ju )2+ (vU

iju −v−U

ju )2]

j=1 s1

6[(vL

ijl −0)2+ (vU

ijl −0)2+ (vL

ijm −0)2+ (vU

ijm −0)2+ (vL

iju −0)2+ (vU

iju −0)2]

(55)

Step 5: Relative closeness calculation and risk priority ranking.

The relative closeness of each evaluation alternative is determined as:

CCi=d−

d−

i+d∗

(56)

Then the risk priority ranking can be determined according to the relative

closeness CCi, in the ascending order.

5. Case study

In this section, a real-world case study of the design concept evalua-

tion for the heat exchanger of the automotive air-conditioning is presented

to illustrate the proposed fuzzy rough number-based group decision-making

method. In general, the heat exchanger is a key component in the automotive

air-conditioning system, which aims to exchange heat between the refrigerant

and the interior so that the vehicle is enabled to provide a comfortable tem-

perature for the passengers. Thus, the heat transfer performance of the heat

exchanger is crucial, which has a direct impact on the overall automotive

air-conditioning system.

With the continuous upgrading in the demand for energy conservation

and emission reduction, the increasing competition, and the rapid advances

in technology, the product development of the heat exchanger is updated

frequently. In a professional heat exchanger manufacturer, a design group

has been devoted to developing a new version of the heat exchanger based on

the bio-inspired design. Various heat exchange mechanisms from the natural

phenomena are investigated to explore the possibility to inspire the design

Journal Pre-proof

of the engineering heat exchanger. During the concept generation stage of

the conceptual design, nine candidate design concepts are obtained by the

design group, namely A1,A2,A3,A4,A5,A6,A7,A8, and A9.

As illustrated in Table 2, these nine design concepts are derived from

the bio-inspired design philosophy, which takes inspiration from the natural

phenomena. A lot of biological structures or mechanisms are identiﬁed as

the inspirations, such as the structure of cardiac valves in human beings, the

ﬁne pipeline transmission mechanism from the ginkgo leaves, the fractal con-

nection network in mycorrhizal fungi, and the multi-dimensional bifurcation

structure in blood vessels. Therefore, the design concept evaluation of the

heat exchanger is to determine the optimal design concepts from the nine

candidates.

Corresponding to the group decision-making framework, ﬁve experts from

the research and development (R & D) department are invited to act as the

decision makers, which are composed of the design engineer, manufacturing

engineer, bionic advisor, refrigeration engineer, and system engineer. They

are experienced senior directors, who have a strong background in the heat

exchanger and the bio-inspired design, especially in the R & D of the bio-

inspired heat exchanger. During the evaluation process, each expert is asked

to give their independent judgment towards the evaluation criteria and eval-

uation alternatives.

In view of the experts’ opinions and the technical characteristics of the

heat exchanger, eleven indexes are identiﬁed as the evaluation criteria, which

are comprised of the eﬃciency (C1), strength (C2), reliability (C3), pressure

(C4), seal (C5), temperature diﬀerence (C6), corrosion resistance (C7), man-

ufacturing cost (C8), maintainability (C9), power consumption (C10), and

noise (C11). As outlined in Table 3, the criteria of manufacturing cost, power

consumption, and noise are categorized as the cost criteria (the-smaller-the-

better), while the other nine are classiﬁed as the beneﬁt criteria (the-bigger-

the-better). In addition, the criteria of eﬃciency, pressure, and tempera-

ture diﬀerence determine the quality of the heat exchange. The criteria of

strength, reliability, and seal reveal the structural property. Likewise, the

criteria of corrosion resistance and maintainability denote the maintenance

characteristics while the power consumption, manufacturing cost, and noise

are the negative criteria.

After the determination of decision makers, evaluation criteria, and eval-

uation alternatives, the design concept evaluation of the heat exchanger can

be carried out according to the framework shown in Fig. 1. First of all, each

Journal Pre-proof

Table 2. Brief description of the evaluation alternatives in the design concept evaluation

of the heat exchanger

Alternatives Description

A1A1is a micro-channel heat exchanger with a counter ﬂow con-

ﬁguration and a valve-based regulatory mechanism, which is

inspired by the structure of cardiac valves as existed in human

and some animals’ heart.

A2A2is a micro-channel heat exchanger with a parallel ﬂow con-

ﬁguration, which is developed based on the inspiration from

the ﬁne pipeline transmission mechanism as widely existed in

sacred ﬁg leaves, xylem, and ginkgo leaves.

A3A3is a micro-channel heat exchanger with multi-dimensional,

variable diameters, fractal, and staggered structure conﬁgura-

tion. The design inspiration comes from the structure of the

fractal connection network in mycorrhizal fungi.

A4A4is a micro-channel parallel ﬂow heat exchanger with a helical

structure conﬁguration in the internal surface of the pipeline,

which is inspired by the helical structure of the inner wall in

the cuttlebone of the cuttleﬁsh.

A5A5is a micro-channel heat exchanger with a counter ﬂow con-

ﬁguration, which is developed on the inspiration from the multi-

dimensional connection and countercurrent exchange mecha-

nism as widely existed in mammals’ blood vessels and ﬁsh gills.

A6A6is a micro-channel heat exchanger with a multi-dimensional

bifurcation structure conﬁguration, which is inspired by the

structure of blood vessels.

A7A7is a micro-channel heat exchanger with a counter ﬂow and

porous liquid separation baﬄes conﬁguration, which comes from

the inspiration of the porous structure in the pit membrane as

existed in conifers.

A8A8is a micro-channel heat exchanger with a fractal pipeline

network conﬁguration, which is inspired by the fractal layout

of the leaf vein.

A9A9is a micro-channel heat exchanger with a counter ﬂow and

helical structure in the inner wall of the pipeline, which is de-

veloped on the inspiration of ﬁsh gills and internal structure in

the cuttlebone of the cuttleﬁsh.

Journal Pre-proof

Table 3. Brief description of the evaluation criteria in the design concept evaluation of the

heat exchanger

Evaluation criteria Description Category

C1Eﬃciency Beneﬁt

C2Strength Beneﬁt

C3Reliability Beneﬁt

C4Pressure Beneﬁt

C5Seal Beneﬁt

C6Temperature diﬀerence Beneﬁt

C7Corrosion resistance Beneﬁt

C8Manufacturing cost Cost

C9Maintainability Beneﬁt

C10 Power consumption Cost

C11 Noise Cost

decision maker is requested to give his individual pairwise comparison values

of the evaluation criteria and the assessments of the evaluation alternatives

with regard to these criteria. Then the criteria weighting and the alternative

ranking are conducted by the fuzzy rough number-based AHP and the fuzzy

rough number-based TOPSIS, respectively.

5.1. Criteria weighting by fuzzy rough number-based AHP

Once the evaluation criteria are identiﬁed, the fuzzy rough number-based

AHP is presented to determine the criteria weights.

Step 1: Collect pairwise comparison information and construct an inte-

grated pairwise comparison matrix.

Fundamental fuzzy scales are introduced to depict the fuzzy assessments

in the pairwise comparison, which are corresponding to the fundamental

scales used in traditional AHP (Saaty and Vargas,2012). As listed in Table

4, the fuzzy scales are further quantiﬁed to triangular fuzzy numbers.

According to the fundamental fuzzy scales deﬁned in Table 4, AHP survey

Journal Pre-proof

Table 4. Fundamental scales in fuzzy pairwise comparison (Liu et al.,2015)

Fuzzy importance Deﬁnition Triangular fuzzy number

1 Equally important (1,1,3)

3 Moderately important (1,3,5)

5 Strongly important (3,5,7)

7 Very strongly important (5,7,9)

9 Extremely important (7,9,9)

is conducted and a group of pairwise comparison matrices are built as:

M1=







5−11e

1−1e

5−1e

3−1e

5−1e

3−1e

5−1e

1 1 e

5−1e

3−1e

5−1e

3−1e

5 1 e

1−1e

5−1e

3−11e

3−1e

3 1 e

7−1e

5−1e

7−1e

5−1e

7−11e

5−1e

3−1e

1−1

5−1e

1−1e

5−1e

5 1 e

7−1e

3−1e

7−1e

3−1e

7−1e

3−11e

5−1e

3−1e

5−1e

3−1e

5−1e

3−1e

3−11e

7−1e

5−1e

7−1e

5−1e

7−1e

5−1e

3−11







M2=







5−11e

3−1e

5−1e

3−1e

5−1e

3−1e

7−1e

3 1 e

5−1e

3−1e

5−1e

3−1e

5 1 e

3−1e

5−1e

3−11e

3−1e

3 1 e

7−1e

3−1e

5−1e

7−1e

5−1e

7−11e

5−1e

3−1e

3−1

5−1e

3−1e

5 1 e

7−1e

3−1e

5−1e

3−1e

5−1e

3−11e

5−1e

3−1e

1−11e

7−1e

3−1e

7−1e

3−1e

7−1e

5−1e

3−1e

1−11







Journal Pre-proof

M3=







5−11e

1−1e

5−1e

3−1e

5−1e

1−1e

5−1e

1 1 e

5−1e

3−1e

5−1e

3−1e

5 1 e

5−1e

5−11e

3−1e

3 1 e

7−1e

3−1e

5−1e

7−1e

5−1e

7−11e

5−1e

3−1e

3−1

5−1e

1−1e

5−1e

5 1 e

7−1e

3−1e

5−1e

3−11e

5−1e

3−1e

5−1e

3−1e

5−1e

3−1e

3−11e

7−1e

3−1e

7−1e

5−1e

7−1e

5−1e

3−11







M4=







5−11e

5−1e

3−1e

5−1e

1−1e

5−1e

3−11e

5−1e

3−1e

5−1e

3−1e

1−1e

5 1 e

3−1e

3−11e

3−1e

1−1e

3 1 e

9−1e

5−1e

7−1e

5−1e

7−11e

5−1e

3−1e

5−1e

3−1

5−1e

1−1e

5−1e

5 1 e

7−1e

3−1e

1−1e

7−1e

3−1e

7−1e

3−11e

5−1e

3−1e

1−1e

5−1e

3−1e

5−1e

1−1e

1−11e

7−1e

5−1e

3−1e

7−1e

5−1e

7−1e

3−1e

3−11







M5=







5−11e

5−1e

1−1e

5−1e

1−1e

7−1e

3−11e

5−1e

1−1e

5−1e

3−1e

1−1e

5 1 e

5−1e

5−11e

5−1e

1−1e

5 1 e

9−1e

5−1e

9−1e

5−1e

9−11e

5−1e

3−1e

5−1e

3−1

5−1e

1−1e

5−1e

5 1 e

3−1e

7−1e

3−1e

1−1e

7−1e

3−1e

7−1e

3−11e

3−1e

5−1e

3−1e

5−1e

1−1e

5−1e

3 1 e

9−1e

5−1e

3−1e

9−1e

5−1e

9−1e

3−1e

3−11







Journal Pre-proof

Based on Eq. (30), the consistency ratio is obtained as:

CR1= 0.0791, CR2= 0.0809, CR3= 0.0907, C R4= 0.0504, CR5= 0.0544

Obviously, CRe<0.1(e= 1,2,· · · ,5). All the fuzzy pairwise comparison

matrices are acceptable.

Corresponding to Eq. (32), the integrated pairwise comparison matrix is

constructed as:







5,e

5··· e

7,e

5−1,e

5−11··· e

5,e

3,e

5,e

5−1,e

7−1,e

5−1,e

7−1e

1,e

3,e

1,e

3−1,e

3−1··· e

5,e

3,e

3−1,e

1−1,e

1−1e

5,e

5··· e

7,e

5−1,e

3−1,e

5−1e

3,e

1··· e

5,e

3,e

5,e

3−1,e

1−1,e

1−1e

5,e

5··· e

7,e

7−1,e

9−1,e

9−1e

5−1,e

3−1,e

5−1,e

5−1··· e

1−1,e

3−1,e

3−1

5−1,e

5−1e

3,e

1,e

1··· e

5,e

3,e

7−1,e

7−1e

3−1,e

3−1··· e

5,e

3,e

5,e

3,e

5−1,e

5−1e

3−1,e

3−1··· e

3,e

1,e

3,e

7−1,e

9−1e

5−1,e

3−1,e

5−1,e

5−1··· 1







Step 2: Translate the elements in f

Minto fuzzy rough numbers and con-

struct a fuzzy rough pairwise comparison matrix.

Take the element e

ex2,11 ={e

5,e

3,e

5,e

5}as an example, according to the

fundamental scales deﬁned in Table 4:

ex1

2,11 =ex4

2,11 =ex5

2,11 =e

5 = (3,5,7)

ex2

2,11 =ex3

2,11 =e

3 = (1,3,5)

Based on Eqs. (7)-(18):

Lim(e

5l) = 1

5×(3+1+1+3+3)=2.2

Lim(e

5l) = 1

3×(3 + 3 + 3) = 3

Lim(e

5m) = 1

5×(5+3+3+5+5)=4.2

Lim(e

5m) = 1

3×(5 + 5 + 5) = 5

Lim(e

5u) = 1

5×(7+5+5+7+7)=6.2

Lim(e

5u) = 1

3×(7 + 7 + 7) = 7

Journal Pre-proof

Then ex1

2,11 is translated into a fuzzy rough number by Eq. (19):

F RN (ex1

2,11) = F RN (ex4

2,11) = F RN (ex5

2,11) = F RN (e

5) = (⌈2.2,3⌋,⌈4.2,5⌋,⌈6.2,7⌋)

Similarly, ex2

2,11 is obtained as:

F RN (ex2

2,11) = F RN (ex3

2,11) = F RN (e

3) = (⌈1,2.2⌋,⌈3,4.2⌋,⌈5,6.2⌋)

Finally, e

ex2,11 can be transformed using Eqs. (35)-(41):

F RN (e

ex2,11) = (⌈1.72,2.68⌋,⌈3.72,4.68⌋,⌈5.72,6.68⌋)

As a result, the fuzzy sequence e

ex2,11 in f

Mis translated into a fuzzy rough

number. Other elements can be operated in the same way.

After the fuzzy rough number transformation of each element in the in-

tegrated pairwise comparison matrix f

M, a fuzzy rough pairwise comparison

matrix is generated as:







(⌈1.00,1.00⌋,⌈1.00,1.00⌋,⌈1.00,1.00⌋)··· (⌈5.08,5.72⌋,⌈7.08,7.72⌋,⌈9.00,9.00⌋)

(⌈0.14,0.14⌋,⌈0.20,0.20⌋,⌈0.33,0.33⌋)··· (⌈1.72,2.68⌋,⌈3.72,4.68⌋,⌈5.72,6.68⌋)

(⌈0.12,0.14⌋,⌈0.16,0.19⌋,⌈0.25,0.31⌋)··· (⌈1.08,1.72⌋,⌈3.08,3.72⌋,⌈5.08,5.72⌋)

(⌈0.22,0.29⌋,⌈0.44,0.76⌋,⌈1.00,1.00⌋)··· (⌈5.08,5.72⌋,⌈7.08,7.72⌋,⌈9.00,9.00⌋)

(⌈0.15,0.16⌋,⌈0.21,0.25⌋,⌈0.36,0.57⌋)··· (⌈2.28,2.92⌋,⌈4.28,4.92⌋,⌈6.28,6.92⌋)

(⌈0.22,0.29⌋,⌈0.40,0.76⌋,⌈1.00,1.00⌋)··· (⌈5.08,5.72⌋,⌈7.08,7.72⌋,⌈9.00,9.00⌋)

(⌈0.11,0.11⌋,⌈0.12,0.14⌋,⌈0.16,0.19⌋)··· (⌈0.21,0.25⌋,⌈0.36,0.57⌋,⌈1.00,1.00⌋)

(⌈0.14,0.14⌋,⌈0.20,0.20⌋,⌈0.33,0.33⌋)··· (⌈1.72,2.68⌋,⌈3.72,4.68⌋,⌈5.72,6.68⌋)

(⌈0.11,0.11⌋,⌈0.14,0.14⌋,⌈0.20,0.20⌋)··· (⌈1.32,2.28⌋,⌈3.32,4.28⌋,⌈5.32,6.28⌋)

(⌈0.14,0.14⌋,⌈0.20,0.20⌋,⌈0.33,0.33⌋)··· (⌈1.00,1.00⌋,⌈2.28,2.92⌋,⌈4.28,4.92⌋)

(⌈0.11,0.11⌋,⌈0.13,0.14⌋,⌈0.18,0.20⌋)··· (⌈1.00,1.00⌋,⌈1.00,1.00⌋,⌈1.00,1.00⌋)







Step 3: Calculate the criteria weights.

Based on the fuzzy rough pairwise comparison matrix c

M, the relative

weights are determined by Eqs. (43)-(44), which are given in Table 5.

Therefore, all the criteria weights are obtained by the fuzzy rough number-

based AHP.

5.2. Design concept priority ranking by fuzzy rough number-based TOPSIS

After the determination of criteria weights in Section 5.1, the fuzzy rough

number-based TOPSIS is developed to identify the priority ranking of the

evaluation alternatives.

Step 1: Collect evaluation information and construct a fuzzy rough deci-

sion matrix.

Journal Pre-proof

Table 5. Relative weights of the evaluation criteria

bwbw′

C1(⌈2.598,2.728⌋,⌈3.886,4.415⌋,⌈5.567,5.921⌋) (⌈0.439,0.461⌋,⌈0.656,0.746⌋,⌈0.940,1.000⌋)

C2(⌈0.452,0.542⌋,⌈0.869,1.058⌋,⌈1.422,1.570⌋) (⌈0.076,0.092⌋,⌈0.147,0.179⌋,⌈0.240,0.265⌋)

C3(⌈0.437,0.509⌋,⌈0.695,0.905⌋,⌈1.299,1.525⌋) (⌈0.074,0.086⌋,⌈0.117,0.153⌋,⌈0.219,0.258⌋)

C4(⌈1.809,2.238⌋,⌈2.968,3.613⌋,⌈4.369,4.868⌋) (⌈0.306,0.378⌋,⌈0.501,0.610⌋,⌈0.738,0.822⌋)

C5(⌈0.709,0.802⌋,⌈1.222,1.443⌋,⌈2.213,2.661⌋) (⌈0.120,0.135⌋,⌈0.206,0.244⌋,⌈0.374,0.449⌋)

C6(⌈1.726,2.151⌋,⌈2.948,3.568⌋,⌈4.253,4.775⌋) (⌈0.292,0.363⌋,⌈0.498,0.603⌋,⌈0.718,0.806⌋)

C7(⌈0.171,0.180⌋,⌈0.232,0.260⌋,⌈0.404,0.458⌋) (⌈0.029,0.030⌋,⌈0.039,0.044⌋,⌈0.068,0.077⌋)

C8(⌈0.607,0.686⌋,⌈1.106,1.354⌋,⌈1.751,2.209⌋) (⌈0.103,0.116⌋,⌈0.187,0.229⌋,⌈0.296,0.373⌋)

C9(⌈0.308,0.350⌋,⌈0.497,0.601⌋,⌈0.937,1.074⌋) (⌈0.052,0.059⌋,⌈0.084,0.102⌋,⌈0.158,0.181⌋)

C10 (⌈0.312,0.423⌋,⌈0.545,0.830⌋,⌈1.039,1.422⌋) (⌈0.053,0.071⌋,⌈0.092,0.140⌋,⌈0.175,0.240⌋)

C11 (⌈0.205,0.222⌋,⌈0.292,0.351⌋,⌈0.520,0.665⌋) (⌈0.035,0.037⌋,⌈0.049,0.059⌋,⌈0.088,0.112⌋)

Similar as conducted in AHP surveys, linguistic variables are introduced

to represent the fuzzy assessments in the alternative ranking. Triangular

fuzzy numbers are adopted in linguistic variables quantiﬁcation, which are

illustrated in Table 6.

Table 6. Linguistic variables and corresponding values in risk rating(Kutlu and Ek-

mek¸cio˘glu,2012)

Linguistic variable Triangular fuzzy number

Very low (VL) (0,0,1)

Low (L) (0,1,3)

Medium low (ML) (1,3,5)

Medium (M) (3,5,7)

Medium high (MH) (5,7,9)

High (H) (7,9,10)

Very high (VH) (9,10,10)

Based on the linguistic variables deﬁned in Table 6, the evaluation values

are collected during the expert interview, which are listed in Table 7.

Therefore, the individual decision matrices are constructed as:

D1=







M MH MH M MH H MH M H ML ML

MH VH VH VH H MH H ML VH M ML

H ML ML L L M L VH VL M M

MH H H M MH MH M MH ML ML M

VH VH VH VH H H H ML VH L ML

MH M M M ML M ML H L M M

VH VH VH VH MH H MH ML H VL ML

H ML ML ML L M ML H L M M

VH H H M MH H M MH ML VL M







,· · ·

Journal Pre-proof

Table 7. Evaluation data for alternative ranking

Alternatives Experts Evaluation criteria

C1C2C3C4C5C6C7C8C9C10 C11

1 H MH MH M MH H MH M H ML ML

2 H M M M M MH M M MH ML M

3 VH MH MH MH MH H MH ML H L ML

4 H H MH H M H H M MH L L

5 VH H H MH M MH MH ML MH L ML

1 MH VH VH VH H MH H ML VH M ML

2 M H H H H M H L H M L

3 M H H H H ML H L VH M L

4 MH H H H H M H L H ML VL

5 MH VH H H H M MH L H M L

1 H ML ML L L M L VH VL M M

2 MH L L L ML ML L H VL MH M

3 MH ML ML ML M M M H ML M ML

4 M M M M ML M MH H M ML L

5 M M ML M L M M VH L M ML

1 MH H H M MH MH M MH ML ML M

2 M M M M M MH ML H L ML M

3 MH MH MH MH H M M MH M ML ML

4 MHHHHHMHHHL L

5 MH H H MH H M MH MH H ML ML

1 VH VH VH VH H H H ML VH L ML

2 HHHHHHHL HMLL

3 MH H H H H H H L VH ML L

4 HHHHHHHL HL VL

5 H VH H H H MH MH L H ML L

1 MH M M M ML M ML H L M M

2 M L L ML ML M L H VL MH M

3 MH ML ML ML M M M H ML M ML

4 M M M M ML M MH H M ML L

5 M M M M L M M H L M ML

1 VH VH VH VH MH H MH ML H VL ML

2 H H H H M H MH L H ML L

3 VH H H H H H MH L VH L L

4 HHHHHHHL HL VL

5 VH VH H H H MH MH L H L L

1 H ML ML ML L M ML H L M M

2 MH L L ML ML M L H VL M M

3 MH ML ML ML M M M H ML M ML

4 MH M M M ML M MH H M ML L

5 M ML ML M L M M H L M ML

1 VH H H M MH H M MH ML VL M

2 H M M M M H ML H L ML M

3 H MH MH MH H H M MH M L ML

4 HHHHHHHHHL L

5 H H H MH H MH MH MH H L ML

Journal Pre-proof

As illustrated in Section 5.1, integrate all the decision matrices and trans-

form the elements into the fuzzy rough numbers. Then the fuzzy rough

decision matrix is obtained as:







(⌈7.32,8.22⌋,⌈9.16,9.64⌋,⌈10.00,10.00⌋)··· (⌈0.69,1.74⌋,⌈2.30,3.70⌋,⌈4.30,5.70⌋)

(⌈3.72,4.68⌋,⌈5.72,6.68⌋,⌈7.72,8.68⌋)··· (⌈0.04,0.36⌋,⌈0.69,1.74⌋,⌈2.30,3.70⌋)

(⌈3.72,5.51⌋,⌈5.72,7.51⌋,⌈7.68,9.09⌋)··· (⌈0.91,2.32⌋,⌈2.49,4.28⌋,⌈4.49,6.28⌋)

(⌈4.28,4.92⌋,⌈6.28,6.92⌋,⌈8.28,8.92⌋)··· (⌈0.91,2.32⌋,⌈2.49,4.28⌋,⌈4.49,6.28⌋)

(⌈6.30,7.70⌋,⌈8.26,9.31⌋,⌈9.64,9.96⌋)··· (⌈0.04,0.36⌋,⌈0.69,1.74⌋,⌈2.30,3.70⌋)

(⌈3.32,4.28⌋,⌈5.32,6.28⌋,⌈7.32,8.28⌋)··· (⌈0.91,2.32⌋,⌈2.49,4.28⌋,⌈4.49,6.28⌋)

(⌈7.72,8.68⌋,⌈9.36,9.84⌋,⌈10.00,10.00⌋)··· (⌈0.04,0.36⌋,⌈0.69,1.74⌋,⌈2.30,3.70⌋)

(⌈4.30,5.70⌋,⌈6.30,7.70⌋,⌈8.26,9.31⌋)··· (⌈0.91,2.32⌋,⌈2.49,4.28⌋,⌈4.49,6.28⌋)

(⌈7.08,7.72⌋,⌈9.04,9.36⌋,⌈10.00,10.00⌋)··· (⌈0.91,2.32⌋,⌈2.49,4.28⌋,⌈4.49,6.28⌋)







Step 2: Construct the weighted normalized decision matrix.

According to Eqs. (47)-(50), the normalized decision matrix is determined

as:







(⌈0.73,0.83⌋,⌈0.92,0.96⌋,⌈1.00,1.00⌋)· · · (⌈0.01,0.01⌋,⌈0.01,0.02⌋,⌈0.02,0.06⌋)

(⌈0.37,0.47⌋,⌈0.57,0.67⌋,⌈0.77,0.87⌋)· · · (⌈0.01,0.02⌋,⌈0.02,0.06⌋,⌈0.11,1.00⌋)

(⌈0.37,0.55⌋,⌈0.57,0.75⌋,⌈0.77,0.91⌋)· · · (⌈0.01,0.01⌋,⌈0.01,0.02⌋,⌈0.02,0.04⌋)

(⌈0.43,0.49⌋,⌈0.63,0.69⌋,⌈0.83,0.89⌋)· · · (⌈0.01,0.01⌋,⌈0.01,0.02⌋,⌈0.02,0.04⌋)

(⌈0.63,0.77⌋,⌈0.83,0.93⌋,⌈0.96,1.00⌋)· · · (⌈0.01,0.02⌋,⌈0.02,0.06⌋,⌈0.11,1.00⌋)

(⌈0.33,0.43⌋,⌈0.53,0.63⌋,⌈0.73,0.83⌋)· · · (⌈0.01,0.01⌋,⌈0.01,0.02⌋,⌈0.02,0.04⌋)

(⌈0.77,0.87⌋,⌈0.94,0.98⌋,⌈1.00,1.00⌋)· · · (⌈0.01,0.02⌋,⌈0.02,0.06⌋,⌈0.11,1.00⌋)

(⌈0.43,0.57⌋,⌈0.63,0.77⌋,⌈0.83,0.93⌋)· · · (⌈0.01,0.01⌋,⌈0.01,0.02⌋,⌈0.02,0.04⌋)

(⌈0.71,0.77⌋,⌈0.90,0.94⌋,⌈1.00,1.00⌋)· · · (⌈0.01,0.01⌋,⌈0.01,0.02⌋,⌈0.02,0.04⌋)







Then the weighted normalized decision matrix is obtained by Eq. (51):







(⌈0.32,0.38⌋,⌈0.60,0.72⌋,⌈0.94,1.00⌋)· · · (⌈0,0⌋,⌈0,0⌋,⌈0,0.01⌋)

(⌈0.16,0.22⌋,⌈0.37,0.50⌋,⌈0.72,0.87⌋)· · · (⌈0,0⌋,⌈0,0⌋,⌈0.01,0.11⌋)

(⌈0.16,0.25⌋,⌈0.37,0.56⌋,⌈0.72,0.91⌋)· · · (⌈0,0⌋,⌈0,0⌋,⌈0,0⌋)

(⌈0.19,0.23⌋,⌈0.41,0.51⌋,⌈0.78,0.89⌋)· · · (⌈0,0⌋,⌈0,0⌋,⌈0,0⌋)

(⌈0.28,0.35⌋,⌈0.54,0.69⌋,⌈0.90,1.00⌋)· · · (⌈0,0⌋,⌈0,0⌋,⌈0.01,0.11⌋)

(⌈0.14,0.20⌋,⌈0.35,0.47⌋,⌈0.69,0.83⌋)· · · (⌈0,0⌋,⌈0,0⌋,⌈0,0⌋)

(⌈0.34,0.40⌋,⌈0.62,0.73⌋,⌈0.94,1.00⌋)· · · (⌈0,0⌋,⌈0,0⌋,⌈0.01,0.11⌋)

(⌈0.19,0.26⌋,⌈0.41,0.57⌋,⌈0.78,0.93⌋)· · · (⌈0,0⌋,⌈0,0⌋,⌈0,0⌋)

(⌈0.31,0.35⌋,⌈0.59,0.70⌋,⌈0.94,1.00⌋)· · · (⌈0,0⌋,⌈0,0⌋,⌈0,0⌋)







Step 3: As all the evaluation criteria are transformed to the beneﬁt cri-

teria, the PIS and NIS values are identiﬁed based on Eqs. (52)-(53):

V∗={bv∗

1,bv∗

2,· · · ,bv∗

11}

V−={bv−

1,bv−

2,· · · ,bv−

11}

where bv∗

j= (⌈1,1⌋,⌈1,1⌋,⌈1,1⌋) and bv−

j= (⌈0,0⌋,⌈0,0⌋,⌈0,0⌋), j= 1,2,· · · ,11.

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Table 8. The separation values of the candidate design concepts

d∗

id−

A12.814 1.035

A22.790 0.950

A33.003 0.725

A42.857 0.896

A52.727 1.109

A62.984 0.723

A72.716 1.131

A82.987 0.766

A92.797 1.051

Step 4: Determine the separation values according to Eqs. (54)-(55),

which are given in Table 8.

Step 5: Calculate the relative closeness by Eq. (56), which are listed in

Table 9.

Table 9. The relative closeness and risk priority ranking of the design concepts

CCiRank

A10.269 4

A20.254 5

A30.194 9

A40.239 6

A50.289 2

A60.195 8

A70.294 1

A80.204 7

A90.273 3

As shown in Table 9, the relative closeness is ranked as CC3< CC6<

CC8< CC4< CC2< CC1< CC9< CC5< CC7. Accordingly, the

alternative ranking is obtained as A3< A6< A8< A4< A2< A1< A9<

A5< A7. Among the nine candidate design concepts, A7is the best while A3

is the worst. Thus, the priority ranking of the nine candidate design concepts

are determined by the fuzzy rough number-based TOPSIS.

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5.3. Sensitivity analysis

To further investigate the evaluation process of the fuzzy rough number-

based AHP-TOPSIS, a sensitivity analysis is conducted by changing the

criteria weights. Speciﬁcally, the criteria weights obtained from the fuzzy

rough number-based AHP are used in the changing operation, namely the

bw′

i(i= 1,2,· · · ,11) as listed in Table 5. In each round of the sensitivity anal-

ysis, the weights of two evaluation criteria are exchanged, while the others

remain consistent. Then the new criteria weights are adopted to integrate

with the fuzzy rough number-based TOPSIS to conduct the alternative rank-

ing.

For the sake of simplicity, only the weight of the ﬁrst evaluation criterion

C1is used to exchange sequentially with other criteria. In the ﬁrst round

(R1), the original weights of bw′

1,bw′

2,· · · ,bw′

10,bw′

11 are used as the benchmark.

In the second round (R2), the weights of criteria C1and C2are exchanged.

Then the criteria weights are changed to bw′

2,bw′

1,· · · ,bw′

10,bw′

11. Likewise, in

the eleventh round (R11), the weights of criteria C1and C11 are exchanged.

The new criteria weights are obtained as bw′

11,bw′

2,· · · ,bw′

10,bw′

1. Based on

such assumptions, the results of detailed sensitivity analysis are illustrated

in Table 10 and Fig. 2, respectively.

Table 10. Results of the sensitivity analysis

No Variables Alternatives

A1A2A3A4A5A6A7A8A9

R1CCi0.269 0.254 0.194 0.239 0.289 0.195 0.294 0.204 0.273

Rank 4 5 9 6 2 8 1 7 3

R2CCi0.257 0.274 0.162 0.247 0.292 0.176 0.293 0.165 0.264

Rank 5 3 9 6 2 7 1 8 4

R3CCi0.254 0.274 0.158 0.247 0.291 0.175 0.292 0.164 0.264

Rank 5 3 9 6 2 7 1 8 4

R4CCi0.2619 0.2618 0.183 0.238 0.290 0.189 0.293 0.193 0.267

Rank 4 5 9 6 2 8 1 7 3

R5CCi0.248 0.271 0.160 0.246 0.290 0.171 0.285 0.167 0.265

Rank 5 3 9 6 1 7 2 8 4

R6CCi0.266 0.251 0.186 0.237 0.289 0.192 0.292 0.197 0.272

Rank 4 5 9 6 2 8 1 7 3

R7CCi0.253 0.272 0.165 0.232 0.289 0.179 0.282 0.175 0.250

Rank 4 3 9 6 1 7 2 8 5

R8CCi0.214 0.238 0.140 0.198 0.260 0.151 0.261 0.148 0.222

Rank 5 3 9 6 2 7 1 8 4

R9CCi0.259 0.276 0.147 0.229 0.293 0.160 0.292 0.156 0.248

Rank 4 3 9 6 1 7 2 8 5

R10 CCi0.210 0.216 0.134 0.194 0.239 0.146 0.258 0.142 0.237

Rank 5 4 9 6 2 7 1 8 3

R11 CCi0.206 0.236 0.130 0.191 0.2559 0.143 0.2561 0.138 0.213

Rank 5 3 9 6 2 7 1 8 4

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Fig. 2. Ranking results of the sensitivity analysis

From Table 10 and Fig. 2, the results show that A7and A5are ranked

with the highest relative closeness (CCi). But the values of these two alterna-

tives are very close in most cases. By changing the weights of two evaluation

criteria, the changes of the ranking results are not drastic. Therefore, the

proposed fuzzy rough number-based AHP-TOPSIS is stable and credible for

design concept evaluation.

Alternatively, there are many diﬀerent changing strategies in sensitivity

analysis. Due to the space limitation, we just list the exchange of criterion

C1with other criteria. More extensions can be introduced to further enhance

the sensitivity analysis.

6. Comparison and discussion

In view of the evaluation process of the fuzzy rough number-based decision-

making framework, the conventional crisp model, fuzzy sets, rough number,

and interval-valued fuzzy sets are introduced to evaluate the performance of

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the proposed method. Accordingly, crisp AHP and crisp TOPSIS (Akg¨un

and Erdal,2019), fuzzy AHP and fuzzy TOPSIS (Chou et al.,2019), rough

AHP and rough TOPSIS (Song et al.,2013), interval-valued fuzzy AHP and

interval-valued fuzzy TOPSIS (Pires et al.,2011) are executed in the com-

parison. All the experiments are implemented under group decision-making

framework. Corresponding to the evaluation process of design concept evalu-

ation, the comparative studies are divided into two parts: criteria weighting

and alternative ranking.

6.1. Comparison of criteria weighting

First of all, the classical crisp AHP (Akg¨un and Erdal,2019), fuzzy AHP

(Chou et al.,2019), rough number-based AHP (rough AHP) (Song et al.,

2013), and interval-valued fuzzy AHP (IVF-AHP) (Pires et al.,2011) are

performed to evaluate the proposed fuzzy rough number-based AHP (FRN-

AHP) in criteria weighting, which are shown in Fig. 3.

Crisp AHP

Rough AHP

Fuzzy AHP

IVF-AHP

FRN-AHP

C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11

0.0

0.2

0.4

0.6

0.8

1.0

Criteria weights

Evaluation criteria

Fig. 3. Comparison of criteria weighting

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In crisp AHP, the crisp number comes from the fuzzy importance as

deﬁned in Table 4, and the arithmetic averaging operator is introduced to

aggregate individual crisp numbers. For rough AHP, the calculation is based

on the crisp number and AHP method as presented in Zhu et al. (2015). In

fuzzy AHP, the arithmetic averaging operator is used to aggregate the three

components in the triangular fuzzy number. Likewise, the same arithmetic

averaging operator-based aggregation method is adopted in IVF-AHP. Mean-

while, an uncertainty degree of 5% is introduced to determine the interval

boundary of the triangular fuzzy numbers (Pires et al.,2011).

From Fig. 3, all the ﬁve algorithms generate the relative weights with the

identical order (C7< C11 < C9< C10 < C3< C2< C8< C5< C6< C4<

C1), but diﬀer in size. In most cases, the interval size of the ﬁve algorithms

is ranked as crisp AHP, rough AHP, fuzzy AHP, IVF-AHP, and FRN-AHP,

in ascending order. In other words, the criteria weights obtained from crisp

AHP are contained in the interval of rough AHP while the weights gained

from rough AHP fall into the interval of fuzzy AHP. Similarly, in fuzzy AHP,

IVF-AHP, and FRN-AHP, the interval size of the criteria weights is getting

bigger. In addition, the interval size of the fuzzy sets-based methods (fuzzy

AHP, IVF-AHP, and FRN-AHP) is much bigger than the crisp number-based

methods (crisp AHP and rough AHP). The lower and upper bounds of fuzzy

AHP are contained in the corresponding intervals of IVF-AHP, which are

further contained in the relevant intervals of FRN-AHP. Besides the lower and

upper bound intervals, the FRN-AHP is endowed with an additional modal

value interval. All the three components of the triangular fuzzy number in

fuzzy AHP are fully contained in the corresponding intervals of FRN-AHP.

Diﬀerent size in criteria weights reﬂects diﬀerent manipulation strategy

used in criteria weighting. The crisp AHP and rough AHP use crisp number

in criteria weights calculation. Even though the rough number provides an

objective tool in evaluation information aggregation, it is still a tough chal-

lenge for experts to give precise judgments at the early stage of NPD. By

contrast, the fuzzy AHP is more helpful to depict the uncertainty in decision

maker’s judgment. It avoids the precise assessment of risk ratings, which

makes the decision-making much easier. The IVF-AHP is further enabled

to depict parts of the uncertainty inherent in the triangular fuzzy numbers.

However, additional parameters need to be introduced in the determination

of the interval-valued fuzzy number, such as the uncertainty degree. Mean-

while, the subjectivity of the decision maker is not taken into consideration.

On the contrary, the FRN-AHP takes advantage of both the fuzzy sets and

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the rough number. With the generation of the fuzzy rough number, the

fuzzy numbers are aggregated automatically. No additional parameters are

involved during the transformation and aggregation. The objectivity is sub-

stantially enhanced during the determination of criteria weights.

6.2. Comparison of alternative ranking

Based on the relative weights determined by crisp AHP, fuzzy AHP, rough

AHP, and IVF-AHP, the corresponding crisp TOPSIS (Akg¨un and Erdal,

2019), fuzzy TOPSIS (Chou et al.,2019), rough TOPSIS (Song et al.,2013),

and interval-valued fuzzy TOPSIS (Pires et al.,2011) are adopted to con-

duct the risk priority ranking of candidate design concepts. Correspond-

ingly, the crisp AHP and crisp TOPSIS (crisp AHP-TOPSIS), fuzzy AHP

and fuzzy TOPSIS (fuzzy AHP-TOPSIS), rough AHP and rough TOPSIS

(rough AHP-TOPSIS), interval-valued fuzzy AHP and interval-valued fuzzy

TOPSIS (IVF-AHP-TOPSIS) are carried out to evaluate the proposed fuzzy

rough number-based AHP and fuzzy rough number-based TOPSIS (FRN-

AHP-TOPSIS) in alternative ranking, which are depicted in Fig. 4.

From Fig. 4, the ranking results for fuzzy AHP-TOPSIS, IVF-AHP-

TOPSIS, and FRN-AHP-TOPSIS are obtained as A3< A6< A8< A4<

A2< A1< A9< A5< A7. In crisp AHP-TOPSIS, the ranking orders are

A3=A6< A8< A4< A2< A1< A9< A5< A7. For rough AHP-TOPSIS,

the ranking results are ordered as A3< A8< A6< A4< A9< A1< A2<

A5< A7. In all the ﬁve methods, A7is the best design concept while A3

is the worst one. The ranking results are very similar in most of the ﬁve

methods.

In crisp AHP-TOPSIS and rough AHP-TOPSIS, modal values of the

evaluation data in Table 7are extracted as the evaluation values for alter-

native ranking. However, only the modal value is unable to characterize the

whole fuzzy number smoothly, as some of the values are asymmetric, like

(0,0,1) and (9,10,10). Some key information is lost during the simpliﬁ-

cation process. Moreover, in crisp AHP-TOPSIS and fuzzy AHP-TOPSIS,

the arithmetic averaging operator is introduced to aggregate individual as-

sessment information. Although it provides a compromise strategy in group

information aggregation, it is incapable of revealing the uncertainty and sub-

jectivity in teams’ perceptions and preferences under group decision-making

environment.

Compared with the crisp number-based method, the rough number, fuzzy

sets, and interval-valued fuzzy sets are enabled to characterize parts of the

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Fig. 4. Comparison of alternative ranking

uncertainty in the evaluation process. However, the proposed fuzzy rough

number adopts a ﬂexible interval boundary, which is endowed with the ability

to depict the uncertainty of the entire fuzzy number and to reveal the vague-

ness of the individual components including the lower bound, modal value,

and the upper bound. It provides a possibility to measure the uncertainty

in fuzzy importance scales and linguistic variables. The fuzzy rough num-

ber just depends on the original risk assessment values given by the decision

maker in the expert interview. It does not need any subjective parameters.

By contrast, it aﬀords a natural way to aggregate personal assessment val-

ues to reveal the decision maker’s true perception in group decision-making.

Hence, the fuzzy rough number-based decision-making framework integrates

the strength of fuzzy logic in uncertain information characterization, the

beneﬁt of group decision-making in cognitive bias elimination, and the ad-

vantage of fuzzy rough number in subjective information aggregation. Not

only the uncertainty inherent in individual assessment information, but also

the subjectivity within the decision-making group are taken into considera-

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tion. Thus, the proposed fuzzy rough number-based group decision-making

can eﬀectively manipulate the uncertainty and subjectivity in design concept

evaluation. The objectivity of the evaluation process is well enhanced.

Furthermore, compared with the interval-valued fuzzy sets (Pires et al.,

2011) which look formally similar to the fuzzy rough number, they are quite

diﬀerent. Some of the diﬀerences are brieﬂy summarized as:

1) The interval-valued fuzzy sets is presented for one single MCDM.

Although it has potentials to extend to group decision-making, additional

weighted averaging operators or aggregating strategies should be introduced,

which involves additional subjective judgment. On the contrast, the fuzzy

rough number is proposed for group decision-making. No additional operator

is required in the fuzzy rough number-based decision-making.

2) The interval-valued fuzzy sets are generated by changing the uncer-

tainty degree of the corresponding triangular fuzzy numbers. However, a

diﬀerent uncertainty degree changing strategy may produce a totally dif-

ferent interval-valued fuzzy number. Meanwhile, the determination of the

changing strategy relies on subjective judgment. Conversely, the fuzzy rough

number is generated from the original triangular fuzzy numbers. It does not

need to predeﬁne any parameter. It is a consistent value.

3) The interval-valued fuzzy sets use a symmetrical interval boundary,

which is generated by changing the uncertainty degree. However, the fuzzy

rough number adopts a ﬂexible interval boundary, which is determined by

the original fuzzy numbers.

4) The interval-valued fuzzy sets only illustrate the uncertainty degree of

the lower bound and upper bound of the triangular fuzzy number, namely

the land uin fuzzy number (l, m, u). The original triangular fuzzy number

(l, m, u) is transformed into ([ll, lu], m, [ul, uu]) in the interval-valued fuzzy

number. On the other hand, the uncertainty of all the components in the

triangular fuzzy number (l, m, u) can be fully characterized in fuzzy rough

number, including the lower bound (l), modal value (m), and the upper

bound (u). After the transformation, the triangular fuzzy number (l, m, u)

is translated to (⌈ll, lu⌋,⌈ml, mu⌋,⌈ul, uu⌋) in the fuzzy rough number.

Therefore, the fuzzy rough number has overwhelming advantages over

the interval-valued fuzzy sets in uncertainty representation and subjectivity

manipulation, especially in the group decision-making environment.

Besides the comparison of AHP-TOPSIS in various uncertain environ-

ments, some other MCDM methods including VIKOR (Liang et al.,2019),

COPRAS (compressed proportional assessment) (B¨uy¨uk¨ozkan and G¨o¸cer,

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2019), and MABAC (Roy et al.,2019) are introduced and modiﬁed by

the fuzzy rough number to further evaluate the eﬀectiveness of the pro-

posed FRN-AHP-TOPSIS. Speciﬁcally, the fuzzy rough number-based AHP

and fuzzy rough number-based VIKOR (FRN-AHP-VIKOR), fuzzy rough

number-based AHP and fuzzy rough number-based COPRAS (FRN-AHP-

COPRAS), fuzzy rough number-based AHP and fuzzy rough number-based

MABAC (FRN-AHP-MABAC) are developed to compare with the FRN-

AHP-TOPSIS, which are illustrated in Fig. 5.

Fig. 5. Comparison of the ranking results from various methods

From Fig. 5, the ranking results for FRN-AHP-VIKOR are arranged as

A6< A3< A8< A4< A2< A9=A1=A5=A7. For FRN-AHP-COPRAS

and FRN-AHP-MABAC, the ranking orders are A3< A6< A8< A4<

A2< A9< A1< A5< A7. In FRN-AHP-TOPSIS, the ranking results are

ordered as A3< A6< A8< A4< A2< A1< A9< A5< A7. In all the four

methods of FRN-AHP-VIKOR, FRN-AHP-COPRAS, FRN-AHP-MABAC,

and FRN-AHP-TOPSIS, A7is the best alternative. Moreover, the ranking

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results of FRN-AHP-COPRAS and FRN-AHP-MABAC are very similar to

the proposed FRN-AHP-TOPSIS, which only diﬀer in the ranking orders of

A9and A1. On the basis of the Qvalue (µ= 0.5), the ranking orders of the

FRN-AHP-VIKOR are A6< A3< A8< A4< A2< A9< A1< A5< A7.

However, according to the ranking rules of the VIKOR (Zhu et al.,2015), a

series of compromise solutions are selected in FRN-AHP-VIKOR, namely A7,

A5,A1, and A9. Therefore, the FRN-AHP-TOPSIS, FRN-AHP-COPRAS,

and FRN-AHP-MABAC methods have better capabilities of discrimination

than the FRN-AHP-VIKOR. The ranking results are more distinguishable

in the former three methods.

In addition to the theoretical implication, there are three main manage-

rial applications of this study. First, the evaluation criteria, alternatives, and

the analysis procedures developed in this study help the manufacturer of the

heat exchanger to establish the systematic approach to select the optimal

design alternatives at the conceptual design stage and assists to analyze the

candidate design concepts in a structured manner. Second, the fuzzy rough

number-based group decision-making framework provides a practical quanti-

tative modeling technique towards the analysis of complex decisions in design

concept evaluation. It is a versatile framework for analyzing and evaluating

design concepts in various new product development. Third, the uncertainty

quantiﬁcation and subjectivity elimination technique are not only beneﬁting

for analyzing and discussing in design concept evaluation. It is a general

model, which can be further extended to other expert interview or informa-

tion aggregation areas, such as the risk assessment in NPD and the customer

requirement identiﬁcation in market analysis.

7. Conclusion

This paper proposed a fuzzy rough number-based group decision-making

technique for design concept evaluation under uncertain and subjective en-

vironments, which contains a fuzzy rough number-based AHP and a fuzzy

rough number-based TOPSIS. Inspired by the classical rough number in crisp

number aggregation, a novel fuzzy rough number was presented to deal with

the uncertainty and subjectivity in fuzzy sets. During the generation of the

fuzzy rough number, the personal assessment information was aggregated,

both in criteria weighting and in alternative ranking. Meanwhile, the respec-

tive fuzzy number was transformed into the fuzzy rough number. Then the

criteria weighting was determined by the fuzzy rough number-based AHP

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and the alternative ranking was conducted by the fuzzy rough number-based

TOPSIS. A real-world case study and comparative analysis demonstrated the

superiority of the proposed decision-making technique. The uncertainty and

subjectivity manipulation in design concept evaluation under group decision-

making environment were well addressed. Both the uncertainty inherent in

personal evaluation information and the subjectivity within the decision-

making group were taken into consideration. The objectivity and credibility

of the ranking results were well enhanced.

Although the proposed method provides an objective and quantitative

modeling approach towards the design concept evaluation in practical case

studies, there are some limitations of this study. The arithmetic operation

of the fuzzy rough number is more complex than conventional crisp number

and fuzzy sets, especially for the fuzzy rough number-based group decision-

making algorithms. Some computational programs should be developed to

alleviate the workload of the decision maker. Besides, the potential conﬂict

within and among the evaluation criteria and evaluation alternatives have

not been taken into consideration. To further strengthen the rationality of

the design concept evaluation, conﬂict resolution should be incorporated to

reveal the relationship within the evaluation components.

Moreover, the fuzzy rough number generation method is not only applica-

ble to the fuzzy number and linguistic variables but also applicable to interval

numbers. More practical case studies will be investigated to evaluate the gen-

eralization of the proposed fuzzy rough number-based group decision-making

technique. Besides, some other algorithms like ANP, BWM, FUCOM (full

consistency method), and DEMATEL will be exploited to combine with the

fuzzy rough number to develop more decision-making models for subjective

and uncertain environments.

Acknowledgments

This work is partly supported by the National Natural Science Foundation

of China (No. 51775332).

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Full-text available

May 2024

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ENVIRON SCI POLLUT R

The escalating volume of healthcare waste (HCW) generated by healthcare facilities poses a pressing challenge for all nations. Adequate management and disposal of this waste are imperative to mitigate its adverse impact on human lives, wildlife, and the environment. Addressing this issue in Bosnia and Herzegovina involves the establishment of a regional center dedicated to HCW management. In practice, there are various treatments available for HCW management. Therefore, it is necessary to determine the priority for procuring different treatments during the formation of this center. To assess these treatment devices, expert decision-making employed the fuzzy-rough approach. By leveraging extended sustainability criteria, experts initially evaluated the significance of these criteria and subsequently assessed the devices for HCW treatment. Employing the fuzzy-rough LMAW (Logarithm Methodology of Additive Weights), the study determined the importance of criteria, highlighting "Air emissions" and "Annual usage costs" as the most critical factors. Utilizing the fuzzy-rough CoCoSo (the Combined Compromise Solution) method, six devices employing incineration or sterilization for HCW treatment were ranked. The findings indicated that the "Rotary kiln" and "Steam disinfection" emerged as the most favorable devices for HCW treatment based on this research. This conclusion was validated through comparative and sensitivity analyses. This research contributes by proposing a solution to address Bosnia and Herzegovina's HCW challenge through the establishment of a regional center dedicated to HCW management.

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ANN OPER RES

In today’s world, industries are facing massive pressure to integrate sustainability issues for efficient and successful supply chain management (SCM). Hence, worldwide it has become critically important to make economic operational balance satisfying environment protection norms and social welfare perspectives. Consequently, the industries are investigating their SCM structures in association with a third party logistics (3PL) service provider adopting the triple bottom line framework for improving the overall supply chain performance. Therefore, selection of the right 3PL provider for the sustainable alliance is supremely important for broader perspective of greater business value. Thus, the main objective of this research work is the selection of most appropriate 3PL provider for a food manufacturing company (FMC) after systematic evaluation of six different feasible logistic providers serving over a decade in India. Selection of optimal alternative 3PL provider is very complex and challenging because of the qualitative description of service provider performances and the inherent uncertainty due to subjectivity. The concept of interval-valued fuzzy-rough number (IVFRN) offers perfect treatment of such uncertainty. In this paper, we develop a multi criteria decision making (MCDM) model combining the factor relationship (FARE) and multi-attributive border approximation area comparison (MABAC) models based on IVFRN. The proposed model is tested and validated on a case study where the optimal selection of 3PL providers is performed for an Indian FMC. Based on the results obtained in sensitivity analysis, it was shown that the proposed IVFRN based FARE-MABAC model produces stable/consistent solutions. Through the research presented in this paper, it is shown that the new hybrid MCDM method is a useful and reliable tool for rational decision-making.

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Angle-of-arrival (AOA) estimation is an important factor in various wireless sensing applications, especially localization systems. This paper proposes a new type of AOA estimation sensor node, known as AOA-estimation system (AES) where the received signal strength indication (RSSI) from multiple radiation pattern reconfigurable (RPR) antennas are used to calculate the AOA. In the proposed framework, three sets of RPR antennas have been used to provide a coverage of 15 regions of radiation patterns at different angles. The salient feature of this RPR-based AOA estimation is the use of Fuzzy Inferences System (FIS) to further enhance the number of estimation points. The introduction of a modified FIS membership function (MF) based on Gaussian function resulted in an improved 85% FIS aggregation percentage between the fuzzy input and output. This later resulted in a low AOA error (of less than 5%) and root-mean-square error (of less than 8°).

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Article

Full-text available

May 2019

This paper presents a new approach for the treatment of uncertainty and imprecision based on interval-valued fuzzy-rough numbers (IVFRNs). IVFRNs make a decision making possible using only the internal knowledge from the data, using objective indeterminacy without the need to rely on models of any assumption. Namely, instead of subjectively entering external uncertainties, the structure of the given data is used. Taking into account the given assumptions, we developed an original multi-criteria model based upon the IVFR approach. In the multi-criteria model the traditional MAIRCA (Multi-Attribute Ideal-Real Comparative Analysis) method was modified. The model was tested and validated on a case study, considering selection of the optimal landing operations point for overcoming water obstacles. The sensitivity analysis of the IVFRN MAIRCA model was carried out through 24 scenarios which showed that our results are of a high stability degree.

Design concept evaluation using soft sets based on acceptable and satisfactory levels: An integrated TOPSIS and Shannon entropy

Article

Full-text available

Feb 2020
SOFT COMPUT

As among several phases of new product development (NPD), concept selection is the most crucial activity, and it gives perfection to further progress of a product. Customer‘s ideas and linguistic requirements are often substantial in concept specifications to assess quantitative criteria, which gives satisfaction for a product to progress in the markets. This work aggregates concept selection on design parameters values by merging acceptable and satisfactory level needs of the customers. A promising framework is developed based on soft sets, TOPSIS and the Shannon entropy. Customer‘s preferences on incorporating design values are identified based on acceptable and satisfactory level needs, these preferences are weighted through Shannon entropy. By performing AND operation on the soft set of level requirements of one customer with the soft set of requirements of another customer, several weighted tables of soft sets are obtained on the pair of design parameters values. To obtain the best concept on different levels of requirements, TOPSIS is performed which provides several integrated evaluations. An illustration is considered for the demonstration of the method, brings the best concept for two customers which is acceptable for both of the customers, satisfactory for both of the customers and vise-versa. Finally, the comparisons are presented with recent major existing methods.

Assessing the Human Resource in Science and Technology for Asian Countries: Application of Fuzzy AHP and Fuzzy TOPSIS

Article

Full-text available

Feb 2019

The fuzzy analytic hierarchy process (AHP) and fuzzy Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) are extremely beneficial when a decision-making process is complex. The reason is that AHP and TOPSIS can prioritize multiple-choice criteria into a hierarchy by assessing the relative importance of criteria and can thus generate an overall ranking of the alternatives. This study uses fuzzy AHP and fuzzy TOPSIS to evaluate the human resource in science and technology (HRST) performance of Southeast Asian countries. The fuzzy TOPSIS analysis indicates that Singapore, South Korea, and Taiwan have similarities in their desired levels of HRST performance. That is, these three countries have better HRST performances than other Southeast Asian countries.

A fully fuzzy best-worst multi attribute decision making method with triangular fuzzy number: A case study of maintenance assessment in the hospitals

Article

Oct 2019
APPL SOFT COMPUT

Abstract This paper introduces the best–worst method to solve multi-attribute decision-making (MADM) problems in the fuzzy environment. In the proposed method, there is no need to do all the possible pairwise comparisons. In other words, only reference comparisons should be done. Reference comparisons consist of assessing the relative fuzzy preference of the best criterion (alternative) over others and all the criteria (alternatives) over the worst one. Afterward, a fully fuzzy linear mathematical model will be formulated and solved to determine the weight of the criteria. The same action will be performed to find the score of alternatives. This method has some interesting and valuable characteristics: (a) less required data for pairwise comparison, (b) high ability to provide a reliable solution, (c) it is an autonomous method along with its high capability to accompany another method. To evaluate the performance, it is compared with another fuzzy MADM method in an example. Furthermore, we apply this method for the maintenance evaluation of hospitals in Bojnord. The computational study confirms the high efficiency and satisfactory performance of the method, and results are validated by a low consistency ratio. Furthermore, the suggested methodology outperforms fuzzy AHP and well verified in the test instance.

A Novel Approach Integrating AHP and COPRAS Under Pythagorean Fuzzy Sets for Digital Supply Chain Partner Selection

Article

May 2019

Digital supply chains (DSCs) can be useful in unlocking supply chains to gain competitive advantage as a driver of growth, while enabling rapid response, supporting novel technology-driven approaches, and creating innovative products and services. Partner selection (PS) is one of the crucial tasks in the successful digitalization of supply chain. However, the selection of a suitable partner is not an easygoing process and is mostly associated with complexity. Various criteria are considered during the partner evaluation process. This paper delivers an efficient evaluation process to assess alternative DSC partners. The proposed approach integrates the Pythagorean fuzzy sets (PFSs), analytic hierarchy process, and complex proportional assessment under a group decision making environment for the first time in the literature. PFSs can depict experts’ evaluations with a richer structure, allowing for a more representative decision making. A case study from Turkey is conducted to validate proposed approach. This study contributes to the existing literature by developing a new evaluation model to improve DSC PS process and by proposing a new hybrid PFSs-based framework. This paper can be useful to practitioners and researchers to better understand DSC PS problem and designing effective DSC partner evaluation systems.

Pythagorean fuzzy VIKOR approaches based on TODIM for evaluating internet banking website quality of Ghanaian banking industry

Article

Mar 2019
APPL SOFT COMPUT

With the rapid development of Internet banking technology in Ghana, the website quality evaluation is the essential core of the customer, which is regarded as a multi-criteria decision making (MCDM) problem. Due to the uncertainty of Internet banking, the evaluation of criteria may be measured by Pythagorean fuzzy numbers (PFNs). In addition, the customer usually does not exhibit complete rationality in the decision procedure. Based on the traditional VIKOR (Vise Kriterijumska Optimizacija I Kompromisno Resenje) method of MCDM, this paper provides a new perspective of a compromised solution, which can handle the decision maker's psychological behavior by inducing TODIM (a Portuguese acronym meaning Interactive Multi-Criteria Decision Making). By defining Pythagorean fuzzy entropy and cross-entropy measures, we study the determination of the weights of the criteria in advance. Then, considering the psychological behavior of the customer, we design two types of strategies for the combination between TODIM and VIKOR. Meanwhile, the corresponding methods have been developed, i.e., Approaches I and II. After that, a simulated example of ranking Internet banking websites in the Ghanaian banking industry is given to illustrate the validity and applicability of our proposed approaches. Finally, we utilize the Wilcoxon signed-rank test and then discuss the differences among VIKOR, TODIM and our proposed methods.

Solving an Ammunition Distribution Network Design Problem Using Multi-Objective Mathematical Modeling, Combined AHP-TOPSIS, and GIS

Article

Feb 2019
COMPUT IND ENG

We study a strategic-level ammunution distribution network design problem (ADNDP) where the purpose is to determine the locations and the service assignments of main, regional, and local depots in order to meet the ammunition needs of military units considering several factors, e.g., stock levels at the depots, costs, and risk levels of depot locations. ADNDP is a real-world and large-scale problem for which scientific decision making methods do not exist. We propose a methodology that uses multiobjective mathematical modeling, Analytic Hierarchy Process (AHP), The Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS), and Geographic Information System (GIS) to solve the problem. The multi-objective mathematical model determines the locations and the service assignments of depots considering two objectives, namely, to minimize transportation costs and to minimize risk scores of main depot locations. The risk score of a depot location indicates how vulnerable the location is to disruptions and is determined by a combined AHP-TOPSIS analysis where TOPSIS is used to compute the risk scores and AHP is used to compute the weights needed by TOPSIS for the identified risk attributes. The GIS analysis is conducted to determine the potential depot locations using map layers based on spatial criteria. We have applied the proposed methodology in designing and evaluating a real ammunition distribution network under different scenarios in collaboration and cooperation with the area experts. We have employed the weighted-sum method to find non-dominated solutions for each scenario and discussed their tradeoffs with the area experts. The purpose of this paper is to present the proposed methodology, findings, and insights.

Interval type-2 fuzzy logic for dynamic parameter adaptation in a modified gravitational search algorithm

Article

Oct 2018
INFORM SCIENCES

In this paper a method for dynamically adjusting parameters in meta-heuristics based on interval type-2 fuzzy logic is proposed. Nowadays meta-heuristic algorithms have become a powerful choice in solving complex optimization problems. The gravitational search algorithm (GSA) based on the Newton laws of gravity and acceleration can be used to solve optimization problems achieving good results, however like in other optimization algorithms a critical issue is an appropriate adjustment of its parameters depending on the type of problem. In this paper the main contribution is a proposed method aimed at dynamic parameter adjustment in GSA with the help of type-2 fuzzy logic. Simulation results on benchmark problems show the advantages of the proposed approach.

A fuzzy rough number-based AHP-TOPSIS for design concept evaluation under uncertain environments

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