# Service Station Evaluation Problem in Catering Service of High-Speed Railway: A Fuzzy QFD Approach Based on Evidence Theory

Authors: Xin Wu; Lei Nie; Meng Xu
Journal: Mathematical Problems in Engineering
 (2015)

---

## Abstract

Catering Service of High-Speed Railway (CSHR) starts at suppliers, includes distribution centers and service stations in cities, and ends at cabinets in high-speed trains. In Distribution System Design (DSD) Problem for CSHR, it is critical to evaluate the alternatives of service stations, which is termed as Service Station Evaluation Problem in Catering Service of High-speed Railway (SSEP-CSHR). As a preparation work for DSD, SSEP-CSHR needs to be solved without detailed information and being accompanied with uncertainty. Fuzzy Quality Function Deployment (F-QFD) has been given in the literatures to deal with vagueness in Facility Location Evaluation (FLE). However, SSEP-CSHR that includes identifying and evaluating stations requires not only dealing with the vague nature of assessments but also confirming them. Based on evidence theory, this paper introduces the framework to give the truth of proposition “x is A.” Then it is incorporated into a two-phase F-QFD with an approximate reasoning to enable the truth of the decisions to be measured. A case study that refers to 85 alternative stations on Chinese high-speed railway will be carried out to verify the proposed method. Analysis shows that the proposed evaluation method enhances scientific credibility of FLE and allows decision makers to express how much is known.


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## Body

## 1. Introduction

In China, the significance of Catering Service of High-Speed Railway (CSHR) has emerged with the rapid development of high-speed passenger rail line’s construction. The process of the service starts at suppliers, includes distribution centers and service stations in various cities, and ends at cabinets in high-speed trains (see Figure1). Service stations, that is, places to catering foods, play an essential role as link-ups between suppliers and trains in food distribution system. The performance of each station for catering is extremely important for CSHR to maintain a continuous supply and rapid distribution. Hence for a new institute to establish a distribution system to support CSHR, it is an inevitable decision to evaluate alternative sites of service station as a preliminary work for other further decisions in CSHR. Service station Evaluation Problem (SSEP-CSHR) thus becomes a critical issue in CSHR.Figure 1
The process of catering service for High-Speed Railway (CSHR).The evaluation of alternative locations is a Multi-Criteria Decision-Making (MCDM) problem referring to qualitative criteria. Under this situation, the values referred in MCDM are often imprecisely defined for the decision-makers. So a fuzzy perspective is adopted naturally in related literatures to deal with above uncertainty. Kuo et al. [1] develop a decision support system in basis of Fuzzy Analytic Hierarchy Process (F-AHP) to assess new stores. Chen [2] proposes a MCDM approach to solve a distribution center evaluation problem under fuzzy environment. Kahraman et al. [3] solve Facility Location Evaluation (FLE) problem by four different fuzzy group decision-making methods, including a fuzzy model proposed by Blin [4], fuzzy synthetic evaluation, weighted goals method [5], and Extent Analysis by Chang [6]. Kulak et al. [7] develop an MCDM based on axiomatic design and Analytic Hierarchy Process (AHP) to select transportation companies. Guo [8] proposes a Fuzzy Data Envelopment Analysis (F-DEA) model for a case involving a new restaurant location. Like other MCDM problems, SSEP-CSHR is also required to be done without detailed information and thus is always accompanied with different types of uncertainty.Service stations in CSHR are also required to satisfy some Basic Requirements, for example, accessibility to trains and, meanwhile, enough time for catering food. Quality Function Deployment (QFD) is one of coherent systems widely used to translate Basic Requirements into actionable alternatives [9]. A typical QFD contains info about “WHATs” and “HOWs,” the relations between “WHATs” and “HOWs,” and the correlations between “WHATs”/“HOWs” themselves. All above information can be presented by Houses of Quality (HOQ) [10].Jamalnia et al. [11] have attempted to develop a Fuzzy Quality Function Deployment (F-QFD) approach for FLE. they proposed some facility locations requirements and then use alternative location to realize them. It follows that F-QFD methodology is capable of solving SSEP-CSHR as a special case of FLE problems. But one of challenges that FLE-F-QFD is encountering when it is utilized in SSEP-CSHR is accountability. Because expensive cost will be spent if a decision-maker plans to establish a cold chain for CSHR and SSEP-CSHR that involves identifying, evaluating alternative service stations requires not only handling the vague nature of assessments but also confirming them, that is, professing what decision makers do know and what they do not know [12]; It means that decision makers lead the truth of the outputs to be measured. Furthermore CSHR is a new field in logistic management, experts are far from having general agreement about the importance of certain Basic Requirements. So the results provided by FLE-F-QFD are indispensable to be crosschecked through various experts to ensure its credibility.It leads to some important problems in related studies about uncertainty management in F-QFD. Kim et al. [13] conclude the uncertainty in QFD mainly arising from the following sources,Fuzziness,Fluctuation, andHeterogeneity.(1)Fuzziness accounts for the imprecise nature or vagueness of the linguistic terms used in QFD. In fact, QFD teams usually do not have sufficient knowledge about the values taken in a HOQ due to the lack of clarity. It makes the application of fuzzy sets theory in QFD (i.e., F-QFD) significant in many problems; fuzzy sets are used to describe natural language-based criteria. F-QFD is one of extended version of typical QFD approaches, which have been widely used in many field [11, 14–20]. In F-QFD, (Fuzzy) AHP approaches [6, 21–25] are always used to generate the inputs.(2)Fluctuation is associated with the change of Basic Requirements over time. SSEP-CSHR is placed on the early stage of SCD problem, so the Fluctuation of the requirements is inevitable. The framework of Robust QFD was presented by Kim et al. [13, 26] to accommodate this uncertainty and to prevent QFD teams obtaining misleading outputs.(3)Heterogeneity comes from the different viewpoints of multiple experts. Hence it is required to justify the outputs obtained by a QFD team by other experts’ opinions.To achieve justifications to cope with theHeterogeneity, evidences from questionnaires must be referred to. Evidences from questionnaires are generally related to the occurrence of deterministic or stochastic events and treat crisp values or probability distributions as their paradigm. However sometimes the evidences involve nonrandom events, such as “an estimated strength of one passenger demand,” and thus are biased by perception or even guess [12]. SSEP-CSHR apparently refers to subjective experience or uncertain considerations. In this case, evidences are represented by possibility distributions, or mixed evidences consist of both probability and possibility distribution. In regard to the above-mentioned status, FLE-F-QFD should be improved to adapt above complicated evidences in order to solve SSEP-CSHR.Recently, great advances have been made in the field of evidence theory. Then complicated empirical evidence is utilized to confirm propositions “x is A” in evidence theory. Various types of proofs are thus defined by the way of an evidence x proving its membership in any set A [27]. The objective of this paper is to introduce evidence theory to make an approximating reasoning possible in a modified QFD to dampen the effects of bothFuzziness andHeterogeneity.Figure2 concludes the relations among application background, motivations, existing versions of QFD, and our proposed method. In the paper, we firstly present the essence of evidence theory including its concepts and formulas to measure the truth of “x is A.” Meanwhile we propose two measures which will be used in our modified F-DFD. Further two-phase F-QFD is introduced. Based on the two-phase F-QFD method, we devote ourselves to incorporate evidence theory into SSEP-CSHR by a procedure of approximate reasoning and form an effective approach to determine alternatives in CSHR with respect to certain qualitative criteria.Figure 2
The precedence relationship among background, motivations, various QFDs, and our method.This paper is organized as follows. In Section2, a framework for determining the truth of “x is A” in basis of evidence theory is introduced. In Section 3.2, a two-phase F-QFD methodology will be described briefly and in Section 3.2 HOQs in the context of SSEP-CSHR will be discussed in detail. Further, in Section 3.3, a procedure of approximate reasoning based on evidence theory is incorporated into FLE-F-QFD for SSEP-CSHR. In Section 4, a case study in the context of CSHR will be done to verify our methodology. The analysis shows that the proposed method enhances scientific credibility of FLE-F-QFD in CSHR and allows decision makers to express how much is known. Figure 3 illustrates the precedence relationship between our sections.Figure 3
The precedence relationship between sections.


## 2. Framework for Determining the Truth of “x Is A” Based on Evidence Theory

### 2.1. Proposition

The canonical form of a proposition is “x is A,” where subject x refers to the condition of an object and A is a set that represents a predicate of x. The following are examples [12].Example 1.
Constructions Costs (CC) are very important.x
= Constructions Costs (CC);A = very important.Example 2.
Proximity to trains (PT) is quite important.x = proximity to trains (PT); A = quite important.
A is represented by fuzzy sets. Truth of a proposition is evaluated by the degree of x supporting A.


### 2.2. SetA in “x Is A”

The predicateA is a description of a situation and is treated as either crisp or fuzzy sets.GivenA~=(x,A~(x))∣x∈U is a fuzzy set, where A~(x) denotes normalized membership function, U is a universal set. According to the first decomposition theorem:(1)A~=⋃α∈0,1α·Aα.And sets called α-cut of the fuzzy set are defined as follows:(2)Aα=x∈U∣A~x≥α=AαL,AαU∀α∈0,1.We setα∈{1/K,2/K,…,1}. K is the number of α-cut considered in [0,1]. For brevity, in this paper, we use Triangular Fuzzy Numbers (TFNs) a/b/c to denote a fuzzy number on an x-α graph; say α=1 with x=b, and say α=0 with x in [a,c] [27].A membership function is a graphical expression about decision makers’ opinions. For example, a group of decision makers is asked to express “the strength degree of the relation between CP and PT” by fuzzy scores. Each decision maker draws two lines on anx-y graph to describe the feeling about the strength, x-axis being the score, and y-axis being the membership level between 0 and 1 (like a triangle or a trapezoid). As shown in Figure 4(a), two lines will both start from y=1 at x=5, which means that “the strength of the relation between CP and PT is 5” is with the most certainty, and end at two points, respectively, on left and right side of score 5, where the scores are totally uncertain for the decision maker (thus y=0 at the points). Many similar lines are drawn by different decision makers and a membership function can be obtained by combining them as shown in Figure 4(a). In Figure 4(b), we take K=5; α-cuts of the fuzzy set are used to fit the depicted membership function. The larger the parameter K is, the better the α-cuts fit the function.Figure 4
An illustration of the development of predicateA.
(a)
(b)


### 2.3. Evidence aboutx

Let a universal set beU. The family of all subsets of a given set A is called the power set of A, and it is denoted by P(A). The following function is called basic probability assignment of evidences:(3)m:PU⟶0,1 such that m(⌀)=0 and(4)∑A∈PUmA=1.Every set A in P(U) for which m(A)>0 is usually named a focal element of m that is a subset of U on which the available evidence focuses. The pair (F,m) is termed as a body of evidence (BOE), where F denotes a set of focal elements and m denotes the associated basic assignment. Moreover, because m can be fully characterized by a list of focal elements F={f1,f2,…,fM}; therefore in the paper F is used instead of (F,m) for short [27].According to the properties of a BOE, the evidence aboutx is divided into three types.(
1) Conflicting Evidence (C-BOE). The BOE represents a random situation that each focal element is conflicting with one another, that is, for each i≠j∈{1,2,…,M}: (5)fi∩fj=⌀.Each event yields a piece of evidence pointing to distinct focal elements. Based on the frequency at which pieces of evidence focuse on each focal element, a probability distribution is developed.(
2) Nested Evidence (N-BOE). In this type of BOE, pieces of evidence point to a range of nested sets. Nested sets indicate the focal elements in F which can be sorted as follows:(6)f1⊆f2⋯⊆fM.This is a situation that one faces in the data that describes estimated quantities. The distribution of this type is termed the possibility distribution.(
3) Mixed Evidence (M-BOE). This type is a generalization of C-BOE and N-BOE. In this case, some pieces of evidence point to sets that are nested and some to mutually exclusive sets.By Venn Diagram, above types of BOEs are illustrated in Figure5. Figure 5(a) is a C-BOE. Figure 5(b) is a N-BOE. Figure 5(c) is a M-BOE. All the BOEs satisfy mf1+mf2+mf3+mf4+m(f5)+m(f6)=1.Figure 5
Three illustrative examples of body of evidence (BOE).A further question is how to develop the BOEs in SSEP-CSHR context. For example, a statement proposed by decision makers is “Cost performance (CP) is important.” How do we generate a BOE to support the statement?A C-BOE is often developed by choice questions with mutually exclusive options in questionnaires. Each interviewee is asked to select only one option from options. The accumulative frequency of each option gives a probability distribution. For example, in Figure5(d) each interviewee is asked to select a score from the set {1,2,3,4,5,6,7,8}.An approach to develop N-BOE or M-BOE is to ask each interviewee to indicate the range of her agreeable score, and the enquirer draws a horizontal line corresponding to the range, and finally stacks the responses. IfM people are asked during our survey, the height of one horizontal line is 1/M. If a representation of the set of opinions is in general agreement, the evidence can be viewed as a possibility distribution (i.e., N-BOE) [12]. The differences among the opinions are only boundaries of the agreeable ranges. For instance, in Figure 5(e), we assume an “ideal” score ranging from 4 to 4.7 exists, which every interviewee agrees on, and interviewees give their acceptable closeness to the “ideal” score range. Sometimes evidences from questionnaires are without any general agreement. Then the evidence is an M-BOE. As shown in Figure 5(f), M-BOE needs no requirements of an “ideal” score range, so M-BOE can be used to describe the opinions without any consensus. Owing to info deficiency during solving SSEP-CSHR, sometimes SSEP-CSHR is required to deal with M-BOE.


### 2.4. Measuring the Truth of “x Is A”

How BOE and setA are combined dictates the framework for determining the truth of “x is A.”Given a fuzzy setA~(x) and a BOE (F,m) on U, where F={f1,…,fM}, a framework to determine the truth of “x is A” is based on two dual measures: Belief (Bel) and Plausibility (Pl) as follows:(7)BelF,A~=1K∑i=1Mmfi∑α∈0,1Ifi⊂AαL,AαU,(8)PlF,A~=1K∑i=1Mmfi∑α∈0,1Ifi∩AαL,AαU≠⌀such that if A~=⌀, then Bel(F,A~)=Pl(F,A~)=0 and if A~=U, Bel(F,A~)=Pl(F,A~)=1. In addition,(9)BelF,A~=1-PlF,A~¯;PlF,A~=1-BelF,A~¯.In (7) and (8), K is the number of α-cuts considered in [0,1]; Ip is a Boolean function.While, for everyi∈{1,2,…,M}, m(fi) characterizes the frequency of pieces of evidence focusing on exactly focal elements fi alone, equation (7) represents the total evidence that each focal element belongs to every α-cut of the fuzzy set. Equation (8) represents not only the evidence that each focal element belongs to every α-cut, but also the additional evidence associating with the focal element overlapping every α-cut. Hence, the Bel is a conservative measure and the Pl is an optimistic one. Then the truth of “x is A” can be defined by following weighted summation:(10)TF,A~=w1BelF,A~+w2PlF,A~.Based on the above discussion, the types of BOE are divided into C-BOE, N-BOE, and M-BOE. The types of set forA that we are mainly concerned with are the crisp set and the fuzzy set. Hence there are six cases of the proposition, “x is A,” corresponding to the combination of the type of BOE and the type of set A. We give an illustrative example to illustrate the two measures.Letw1=w2=0.5 and K=5. Let m(f1)=0.05; m(f2)=0.15; m(f3)=0.3; m(f4)=0.3; m(f5)=0.15; m(f6)=0.05 (the same as the values in Figure 5). All the six cases are illustrated in Figure 6 and the truth of each case is shown in Table 1.Table 1
The truth of each case in illustrative example in Figure5.
BOE
A
Bel
(
F
,
A
)
Pl
(
F
,
A)
T
(
F
,
A
)
Example
C-BOE
Crisp
=
0.2
×
(
0.3
+
0.3
+
0.15
+
0.05
)
×
5
=
0.8
=
0.3
+
0.3
+
0.15
+
0.05
=
0.8
0.8
Figure6(a)
C-BOE
Fuzzy
=
0.2
×
(
0.3
×
1
+
0.3
×
3
+
0.15
×
4
+
0.05
×
5
)
=
0.41
=
0.2
×
(
0.3
×
1
+
0.3
×
3
+
0.15
×
4
+
0.05
×
5
)
=
0.41
0.41
Figure6(b)
N-BOE
Crisp
=
0.2
×
(
0.05
+
0.15
)
×
5
=
0.2
=
0.2
×
(
0.05
+
0.15
+
0.3
+
0.3
+
0.15
+
0.05
)
×
5
=
1
0.6
Figure6(c)
N-BOE
Fuzzy
=
0
=
0.2
×
(
0.3
+
0.15
+
0.05
)
×
5
=
0.5
0.25
Figure6(d)
N-BOE
Fuzzy
=
0.2
×
(
0.05
+
0.15
)
=
0.04 (Equation (7))
=
(
0.05
×
5
+
0.15
×
5
+
0.3
×
4
+
0.3
×
3
+
0.15
×
2
+
0.05
×
2
)
=
0.7
0.37
Figure6(e)
=
1
-
0.2
×
(
0.05
×
5
+
0.15
×
5
+
0.3
×
5
+
0.3
×
5
+
0.15
×
4
+
0.05
×
4
)
=
0.04 (Equation (9))
Figure6(f)
M-BOE
Crisp
=
0.2
×
(
0.05
+
0.15
)
×
5
=
0.2
=
0.2
×
(
0.05
+
0.15
+
0.3
+
0.3
+
0.15
+
0.05
)
×
5
=
1
0.6
Figure6(g)
M-BOE
Fuzzy
=
0.2
×
(
0.05
+
0.15
)
=
0.04
=
0.2
×
(
0.05
×
5
+
0.15
×
5
+
0.3
×
2
+
0.3
+
0.15
×
2
+
0.05
×
2
)
=
0.46
0.25
Figure6(h)Figure 6
Illustrative example of Belief and Plausibility measures.
(a)
C-BOE + crisp set
(b)
C-BOE + fuzzy set
(c)
N-BOE + crisp set: the core ofg(x) is contained in A
(d)
N-BOE + crisp set: the core ofg(x) is not contained in A
(e)
N-BOE + fuzzy set: using (7) and (8)
(f)
N-BOE + fuzzy set: using (9)
(g)
M-BOE+ crisp set
(h)
M-BOE fuzzy setTo sum up, Figure7 depicts different frameworks corresponding to different cases (Cases 1–6).Figure 7
Different theory corresponding to the combination of the type of BOE and the type of setA.Case 1.
C-BOE + crisp set.Case 2.
C-BOE + fuzzy set.Case 3.
N-BOE + crisp set.Case 4.
N-BOE + fuzzy set.Case 5.
M-BOE + crisp set.Case 6.
M-BOE + fuzzy set.Firstly, Figure7 displays that Cases 1-2 are traditionally expressed based on probability theory, while possibility theory is used to deal with Cases 3-4; secondly, Cases 1–4 are special cases of Cases 5-6, in that M-BOE is a mix of C-BOE and N-BOE. Evidence theory is a generalized form to measure the truth of “x is A” which suits for Cases 1–6 all [27].


## 2.1. Proposition

The canonical form of a proposition is “x is A,” where subject x refers to the condition of an object and A is a set that represents a predicate of x. The following are examples [12].Example 1.
Constructions Costs (CC) are very important.x
= Constructions Costs (CC);A = very important.Example 2.
Proximity to trains (PT) is quite important.x = proximity to trains (PT); A = quite important.
A is represented by fuzzy sets. Truth of a proposition is evaluated by the degree of x supporting A.


## 2.2. SetA in “x Is A”

The predicateA is a description of a situation and is treated as either crisp or fuzzy sets.GivenA~=(x,A~(x))∣x∈U is a fuzzy set, where A~(x) denotes normalized membership function, U is a universal set. According to the first decomposition theorem:(1)A~=⋃α∈0,1α·Aα.And sets called α-cut of the fuzzy set are defined as follows:(2)Aα=x∈U∣A~x≥α=AαL,AαU∀α∈0,1.We setα∈{1/K,2/K,…,1}. K is the number of α-cut considered in [0,1]. For brevity, in this paper, we use Triangular Fuzzy Numbers (TFNs) a/b/c to denote a fuzzy number on an x-α graph; say α=1 with x=b, and say α=0 with x in [a,c] [27].A membership function is a graphical expression about decision makers’ opinions. For example, a group of decision makers is asked to express “the strength degree of the relation between CP and PT” by fuzzy scores. Each decision maker draws two lines on anx-y graph to describe the feeling about the strength, x-axis being the score, and y-axis being the membership level between 0 and 1 (like a triangle or a trapezoid). As shown in Figure 4(a), two lines will both start from y=1 at x=5, which means that “the strength of the relation between CP and PT is 5” is with the most certainty, and end at two points, respectively, on left and right side of score 5, where the scores are totally uncertain for the decision maker (thus y=0 at the points). Many similar lines are drawn by different decision makers and a membership function can be obtained by combining them as shown in Figure 4(a). In Figure 4(b), we take K=5; α-cuts of the fuzzy set are used to fit the depicted membership function. The larger the parameter K is, the better the α-cuts fit the function.Figure 4
An illustration of the development of predicateA.
(a)
(b)


## 2.3. Evidence aboutx

Let a universal set beU. The family of all subsets of a given set A is called the power set of A, and it is denoted by P(A). The following function is called basic probability assignment of evidences:(3)m:PU⟶0,1 such that m(⌀)=0 and(4)∑A∈PUmA=1.Every set A in P(U) for which m(A)>0 is usually named a focal element of m that is a subset of U on which the available evidence focuses. The pair (F,m) is termed as a body of evidence (BOE), where F denotes a set of focal elements and m denotes the associated basic assignment. Moreover, because m can be fully characterized by a list of focal elements F={f1,f2,…,fM}; therefore in the paper F is used instead of (F,m) for short [27].According to the properties of a BOE, the evidence aboutx is divided into three types.(
1) Conflicting Evidence (C-BOE). The BOE represents a random situation that each focal element is conflicting with one another, that is, for each i≠j∈{1,2,…,M}: (5)fi∩fj=⌀.Each event yields a piece of evidence pointing to distinct focal elements. Based on the frequency at which pieces of evidence focuse on each focal element, a probability distribution is developed.(
2) Nested Evidence (N-BOE). In this type of BOE, pieces of evidence point to a range of nested sets. Nested sets indicate the focal elements in F which can be sorted as follows:(6)f1⊆f2⋯⊆fM.This is a situation that one faces in the data that describes estimated quantities. The distribution of this type is termed the possibility distribution.(
3) Mixed Evidence (M-BOE). This type is a generalization of C-BOE and N-BOE. In this case, some pieces of evidence point to sets that are nested and some to mutually exclusive sets.By Venn Diagram, above types of BOEs are illustrated in Figure5. Figure 5(a) is a C-BOE. Figure 5(b) is a N-BOE. Figure 5(c) is a M-BOE. All the BOEs satisfy mf1+mf2+mf3+mf4+m(f5)+m(f6)=1.Figure 5
Three illustrative examples of body of evidence (BOE).A further question is how to develop the BOEs in SSEP-CSHR context. For example, a statement proposed by decision makers is “Cost performance (CP) is important.” How do we generate a BOE to support the statement?A C-BOE is often developed by choice questions with mutually exclusive options in questionnaires. Each interviewee is asked to select only one option from options. The accumulative frequency of each option gives a probability distribution. For example, in Figure5(d) each interviewee is asked to select a score from the set {1,2,3,4,5,6,7,8}.An approach to develop N-BOE or M-BOE is to ask each interviewee to indicate the range of her agreeable score, and the enquirer draws a horizontal line corresponding to the range, and finally stacks the responses. IfM people are asked during our survey, the height of one horizontal line is 1/M. If a representation of the set of opinions is in general agreement, the evidence can be viewed as a possibility distribution (i.e., N-BOE) [12]. The differences among the opinions are only boundaries of the agreeable ranges. For instance, in Figure 5(e), we assume an “ideal” score ranging from 4 to 4.7 exists, which every interviewee agrees on, and interviewees give their acceptable closeness to the “ideal” score range. Sometimes evidences from questionnaires are without any general agreement. Then the evidence is an M-BOE. As shown in Figure 5(f), M-BOE needs no requirements of an “ideal” score range, so M-BOE can be used to describe the opinions without any consensus. Owing to info deficiency during solving SSEP-CSHR, sometimes SSEP-CSHR is required to deal with M-BOE.


## 2.4. Measuring the Truth of “x Is A”

How BOE and setA are combined dictates the framework for determining the truth of “x is A.”Given a fuzzy setA~(x) and a BOE (F,m) on U, where F={f1,…,fM}, a framework to determine the truth of “x is A” is based on two dual measures: Belief (Bel) and Plausibility (Pl) as follows:(7)BelF,A~=1K∑i=1Mmfi∑α∈0,1Ifi⊂AαL,AαU,(8)PlF,A~=1K∑i=1Mmfi∑α∈0,1Ifi∩AαL,AαU≠⌀such that if A~=⌀, then Bel(F,A~)=Pl(F,A~)=0 and if A~=U, Bel(F,A~)=Pl(F,A~)=1. In addition,(9)BelF,A~=1-PlF,A~¯;PlF,A~=1-BelF,A~¯.In (7) and (8), K is the number of α-cuts considered in [0,1]; Ip is a Boolean function.While, for everyi∈{1,2,…,M}, m(fi) characterizes the frequency of pieces of evidence focusing on exactly focal elements fi alone, equation (7) represents the total evidence that each focal element belongs to every α-cut of the fuzzy set. Equation (8) represents not only the evidence that each focal element belongs to every α-cut, but also the additional evidence associating with the focal element overlapping every α-cut. Hence, the Bel is a conservative measure and the Pl is an optimistic one. Then the truth of “x is A” can be defined by following weighted summation:(10)TF,A~=w1BelF,A~+w2PlF,A~.Based on the above discussion, the types of BOE are divided into C-BOE, N-BOE, and M-BOE. The types of set forA that we are mainly concerned with are the crisp set and the fuzzy set. Hence there are six cases of the proposition, “x is A,” corresponding to the combination of the type of BOE and the type of set A. We give an illustrative example to illustrate the two measures.Letw1=w2=0.5 and K=5. Let m(f1)=0.05; m(f2)=0.15; m(f3)=0.3; m(f4)=0.3; m(f5)=0.15; m(f6)=0.05 (the same as the values in Figure 5). All the six cases are illustrated in Figure 6 and the truth of each case is shown in Table 1.Table 1
The truth of each case in illustrative example in Figure5.
BOE
A
Bel
(
F
,
A
)
Pl
(
F
,
A)
T
(
F
,
A
)
Example
C-BOE
Crisp
=
0.2
×
(
0.3
+
0.3
+
0.15
+
0.05
)
×
5
=
0.8
=
0.3
+
0.3
+
0.15
+
0.05
=
0.8
0.8
Figure6(a)
C-BOE
Fuzzy
=
0.2
×
(
0.3
×
1
+
0.3
×
3
+
0.15
×
4
+
0.05
×
5
)
=
0.41
=
0.2
×
(
0.3
×
1
+
0.3
×
3
+
0.15
×
4
+
0.05
×
5
)
=
0.41
0.41
Figure6(b)
N-BOE
Crisp
=
0.2
×
(
0.05
+
0.15
)
×
5
=
0.2
=
0.2
×
(
0.05
+
0.15
+
0.3
+
0.3
+
0.15
+
0.05
)
×
5
=
1
0.6
Figure6(c)
N-BOE
Fuzzy
=
0
=
0.2
×
(
0.3
+
0.15
+
0.05
)
×
5
=
0.5
0.25
Figure6(d)
N-BOE
Fuzzy
=
0.2
×
(
0.05
+
0.15
)
=
0.04 (Equation (7))
=
(
0.05
×
5
+
0.15
×
5
+
0.3
×
4
+
0.3
×
3
+
0.15
×
2
+
0.05
×
2
)
=
0.7
0.37
Figure6(e)
=
1
-
0.2
×
(
0.05
×
5
+
0.15
×
5
+
0.3
×
5
+
0.3
×
5
+
0.15
×
4
+
0.05
×
4
)
=
0.04 (Equation (9))
Figure6(f)
M-BOE
Crisp
=
0.2
×
(
0.05
+
0.15
)
×
5
=
0.2
=
0.2
×
(
0.05
+
0.15
+
0.3
+
0.3
+
0.15
+
0.05
)
×
5
=
1
0.6
Figure6(g)
M-BOE
Fuzzy
=
0.2
×
(
0.05
+
0.15
)
=
0.04
=
0.2
×
(
0.05
×
5
+
0.15
×
5
+
0.3
×
2
+
0.3
+
0.15
×
2
+
0.05
×
2
)
=
0.46
0.25
Figure6(h)Figure 6
Illustrative example of Belief and Plausibility measures.
(a)
C-BOE + crisp set
(b)
C-BOE + fuzzy set
(c)
N-BOE + crisp set: the core ofg(x) is contained in A
(d)
N-BOE + crisp set: the core ofg(x) is not contained in A
(e)
N-BOE + fuzzy set: using (7) and (8)
(f)
N-BOE + fuzzy set: using (9)
(g)
M-BOE+ crisp set
(h)
M-BOE fuzzy setTo sum up, Figure7 depicts different frameworks corresponding to different cases (Cases 1–6).Figure 7
Different theory corresponding to the combination of the type of BOE and the type of setA.Case 1.
C-BOE + crisp set.Case 2.
C-BOE + fuzzy set.Case 3.
N-BOE + crisp set.Case 4.
N-BOE + fuzzy set.Case 5.
M-BOE + crisp set.Case 6.
M-BOE + fuzzy set.Firstly, Figure7 displays that Cases 1-2 are traditionally expressed based on probability theory, while possibility theory is used to deal with Cases 3-4; secondly, Cases 1–4 are special cases of Cases 5-6, in that M-BOE is a mix of C-BOE and N-BOE. Evidence theory is a generalized form to measure the truth of “x is A” which suits for Cases 1–6 all [27].


## 3. An Integration of Evidence Theory and SSEP-CSHR

### 3.1. The Two-Phase F-QFD

In practice, series of HOQs can be integrated into a multiphase QFD. Each phase is closely correlated in a multiphase QFD since the inputs of one phase needs to apply the outputs of the previous phase [17, 18]. In the paper, we focused on QFD whose procedure has been thoroughly described by Mazur [10]. This approach consists of solving some successive HOQs. Among these HOQs, two HOQs named as Quality Deployment (QD) and Functional Deployment (FD) which is of the most essential are applied here to develop our MCDM model for SSEP-CSHR.Quality Deployment (QD). It translates some Basic Requirements to certain quality attributes.Functional Deployment (FD). It realized the quality attributes by specific alternatives.At the beginning of the process, Demanded Qualities (DQs) have to be identified. The elements are gathered from surveys or experts’ interviews. They are “WHATs” in QD phase:(11)L1=R1,…,Ri,…,RN1.We determine the fuzzy importance ratings of DQs (L1). The fuzzy importance ratings are usually obtained by subjective method discussed in Section 2.2. F-AHP, for example, Extent Analysis (EA) or Lambda-Max Method (LMM) can also be used to capture the fuzzy ratings [6, 25]: (12)W~L1=W~1L1,…,W~iL1,…,W~N1L1. DQs are translated into “Quality Attributes” (QAs). QAs (L2) are assessment criteria that are related to the accomplishment of above DQs. Then QAs are “HOWs” in QD phase. And QFD team then generates the QAs as follows:(13)L2=C1,…,Ci,…,CN2.Similarly, some Functional Techniques (FTs) (L3), that is, our alternatives to realize above QAs, are further defined by panel of experts. Here QAs are “WHATs” for FD and FTs are “HOWs” for FD:(14)L3=F1,…,Fi,…,FN3.The core elements of F-QFD are “relationships matrices” in QD and FD, respectively:(1)
The relationship matrix between DQs (L1) and QAs (L2) to express how a quality attribute to meet a requirement;(2)
The relationship matrix between QAs (L2) and FTs (L3) to express how an alternative to realize an attribute;(3)
The correlation matrix between DQs (L1), QAs (L2), and FTs (L3) themselves.To sum up, Table2 defines all fuzzified inputs and outputs in the two-phase F-QFD.Table 2
Inputs and outputs in HOQs of the two-phase F-QFD.
Inputs
Descriptions
W
~
i
L
1
The fuzzy important ratings ofRi where i=1,2,…,N1
B
~
i
j
L
2
The fuzzy relationship score betweenRi and Cj, where i=1,2,…,N1; j=1,2,…,N2
B
~
i
j
L
3
The fuzzy relationship score betweenCi and Fj, where i=1,2,…,N2; j=1,2,…,N3
A
~
i
j
L
1
The fuzzy correlation score betweenRi and Rj, where i=1,2,…,N1; j=1,2,…,N1
A
~
i
j
L
2
The fuzzy correlation score betweenCi and Cj, where i=1,2,…,N2; j=1,2,…,N2
A
~
i
j
L
3
The fuzzy correlation score betweenFi and Fj, where i=1,2,…,N3; j=1,2,…,N3
Outputs
Descriptions
W
~
i
L
2
The fuzzy important ratings ofCi where i=1,2,…,N2
W
~
i
L
3
The fuzzy important ratings ofFi where i=1,2,…,N3All inputs and outputs are described by linguistic terms and defined as the fuzzy sets. A fuzzy set is fully and uniquely represented by itsα-cuts (see Table 3).Table 3
Theα-cuts of the fuzzy sets in the two-phase F-QFD.
Notations
Descriptions
W
~
i
α
L
1
=
W
i
L
1
α
L
,
W
i
L
1
α
U
Theα-cuts of fuzzy important ratings W~iL1
B
~
i
j
α
L
2
=
B
i
j
L
2
α
L
,
B
i
j
L
2
α
U
Theα-cuts of fuzzy relationship B~ijL2
B
~
i
j
α
L
3
=
B
i
j
L
3
α
L
,
B
i
j
L
3
α
U
Theα-cuts of fuzzy relationship B~ijL3
A
~
i
j
α
L
1
=
A
i
j
L
1
α
L
,
A
i
j
L
1
α
U
Theα-cuts of fuzzy correlations A~ijL1
A
~
i
j
α
L
2
=
A
i
j
L
2
α
L
,
A
i
j
L
2
α
U
Theα-cuts of fuzzy correlation A~ijL2
A
~
i
j
α
L
3
=
A
i
j
L
3
α
L
,
A
i
j
L
3
α
U
Theα-cuts of fuzzy correlation A~ijL3
W
~
i
α
L
2
=
W
i
L
2
α
L
,
W
i
L
2
α
U
Theα-cuts of fuzzy important ratings W~iαL2
W
~
i
α
L
3
=
W
i
L
3
α
L
,
W
i
L
3
α
U
Theα-cuts of fuzzy important ratings W~iαL3Chen and Ko [17, 18] gave a formulation to implement multiphase F-QFD. Wang et al. [20, 28] have demonstrated that the method is not proper and provided a set of nonlinear programming (NLP) models to do F-QFD.LetA~ and B~ be two fuzzy numbers; operations on two fuzzy numbers are defined as(15)A~∗B~=⋃α∈0,1αA∗Bα=⋃α∈0,1αAα∗Bα=⋃α∈0,1αff∗∈gAα,g∈Bα.Let ∗ denote any of the four basic arithmetic operations. Aα∗Bα follows arithmetic operations on closed intervals [27]. Then based on Chen et al. [17, 18], and Wang and Chin [20], an algorithm to solve the two-phase F-QFD is introduced.Algorithm 3 (the algorithm to implement the two-phase F-QFD).
Consider the following.
Step
1. Set α-cuts of the fuzzy set equal to 1/K,…, and 1 respectively. K is the number of α-cut in [0,1]. Then the fuzzy importance ratings of FTs are obtained by following substeps.
Step
1.1. Let k=1.
Step
1.2. The fuzzy correlations between Lk can be normalized by formula:(16)ω~iLk=∑n=1NkW~nLk⊗A~niLk∑n=1Nk∑m=1NkW~nLk⊗A~nmLk∀i∈1,2,…,Nk.
The formula can be achieved by the following Nonlinear Programming (NLP) models.
For alli∈{1,2,…,Nk}, α∈[0,1], determine interval [(ωiLk)αL,(ωiLk)αU] by models:
(Model 1)(17)ωiLkαL=min⁡∑n=1NkϖnLkaniLk∑n=1Nk∑m=1NkϖnLkanmLk,ωiLkαU=max⁡∑n=1NkϖnLkaniLk∑n=1Nk∑m=1NkϖnLkanmLk,s.tWiLkαL≤ϖiLk≤WiLkαU∀i∈1,2,…,Nk,AijLkαL≤aijLk≤AijLkαU∀i,j∈1,2,…,Nk.
Step
1.3. Furthermore, the fuzzy relationship between Lk and Lk+1 can be defined by(18)Y~iLk+1=∑η=1Nk+1B~iηLk+1⊗A~ηjLk+1∑l=1Nk+1∑η=1Nk+1B~iηLk+1⊗A~ηlLk+1∀i∈1,2,…,Nk.
Then we calculate the importance rating ofLk+1 as the fuzzy weighted average of the normalized fuzzy relationships:(19)W~jLk+1=∑i=1NkW~iLk⊗YijLk+1∑i=1NkW~iLk∀j∈1,2,…,Nk+1.
The above two formulas can be achieved by following NLP models.
For alli∈{1,2,…,Nk}, α∈[0,1], we can determine interval:
(Model 2)(20)WjLk+1αL=min⁡∑η=1NkwiLk∑η=1Nk+1biηLk+1aηjLk+1/∑l=1Nk+1∑η=1Nk+1biηLk+1aηlLk+1∑i=1NkwiLkWjLk+1αU=max⁡∑i=1NkwiLk∑η=1Nk+1biηLk+1aηjLk+1/∑l=1Nk+1∑η=1Nk+1biηLk+1aηlLk+1∑i=1NkwiLk(21)s.tAijLk+1αL≤aijLk+1≤AijLk+1αU∀i,j∈1,2,…,Nk+1(22)ωiLkαL≤wiLk≤ωiLkαU∀i∈1,2,…,Nk(23)BijLk+1αL≤bijLk+1≤BijLk+1αU∀i∈1,2,…,Nkj∈1,2,…,Nk+1.
Step
1.3. By (1), the fuzzy importance ratings of Lk+1 are obtained:(24)W~Lk+1=W~1Lk+1,…,W~iLk+1,…,W~Nk+1Lk+1.
Step
1.4. If k+1=3, go to Step 2; else k=k+1 return to Step 1.2.
Step
2. Fuzzy importance rating can be ranked by different methods. In the paper, the fuzzy importance ratings are compared with each other by method developed in Wang [29]:(25)WiLk+1=1K∑α∈0,1WiLk+1αL+WiLk+1αU2∀i∈1,2,…,Nk.
Step
3. ENDBased on the method proposed in Wang and Chin [20], Models 1 and 2 can be linearized and solved optimally.


### 3.2. Service Station Evaluation Problem for CSHR

SSEP-CSHR is based on a reinterpretation of above two-phase F-QFD. In SSEP-CSHR, DQs in terms of Basic Requirements (L1) are translated into QAs in terms of Assessment Criteria (L2). And then service stations (L3), that is, specific alternatives to realize above Assessment Criteria, are served as FTs. Figure 8 gives the corresponding relationship between the two-phase F-QFD and SSEP-CSHR.Figure 8
Corresponding relationship between the two-phase F-QFD and SSEP-CSHP.Basic Requirements (L1) in SSEP-CSHR are comprised of two aspects of requirements: Firstly, it contains some requirements used in FLE-F-QFD. These demands have been extensively described by the literatures about facility location and supply chain management [11, 19]. Moreover, it involves some distinct requirements because of special CSHR background. For the sake of clarity, all those requirements are shown in Table 4.Table 4
List ofL1 for SSEP-CSHR.
Basic requirements
Description
Proximity to trains (PT)
Proximity to demands; number of trains that can be served; cost of serving trains
Sound catering condition (SC)
Space and time condition for delivery
Labor availability (LA)
Availability of required professional and nonprofessional workforce; wage rates; quality of workforce; and so forth
Availability of foods (AF)
Closeness to food suppliers or distributional centers; freight costs of transportation
Government incentives (GI)
Convenience provided by government or railway authorities
Constructions costs (CC)
Construction costs of building cold stores at the station and also availability of space for future expansionBasic Requirements (L1) are some requirements described by broad nature language. However, the set of Assessment Criteria (L2) usually use more specific descriptions. Similarly Assessment Criteria (L2) also consist of two parts: some criteria that are derived from relevant literatures [11, 19] and other ones that are gathered from experts’ comments under certain background. All the criteria are presented in Table 5.Table 5
List ofL2 for SSEP-CSHR.
Assessment criteria
Description
Quality and location of neighboring suppliers  (QL)
Proximity to suppliers; alternative suppliers; and speed and responsiveness of suppliers
Accessibility to trains  (AT)
Number of train that can be served  (number of trains departing from the city; number of trains stopping at the city, the ratio of the above two)
Available catering time (AC)
Average available catering time
Physical catering conditions  (PC)
The platform area
Human resource conditions  (HR)
Labor costs; attitudes toward works and labor turnover, economical growths, and declines effects
Economics related factors (EF)
Urban economic development, tax incentives,and business climate; interest rates
Political issues (PI)
Availability and size of government; aids attitude of government to inward investmentBy solving the steps of the two-phase F-QFD (Algorithm3), the importance ratings of alternative Service stations satisfying above requirements and criteria can be determined. Figure 9 provides the basic structure of two essential HOQs (QD and FD) used in SSEP-CSHR.Figure 9
Basic structure of HOQs for SSEP-CSHR.


### 3.3. An Approximate Reasoning Procedure in SSEP-CSHR

Approximate reasoning is related to the appropriate processing of uncertain knowledge, given in a rule based manner. Deductive inference can be performed by means of classical logic modus ponens (MP). For using linguistic variables expressed fuzzy sets, a generalized MP has been proposed by Zadeh [30]. It enables an approximate deductive reasoning with the following general scheme:(26)
F
a
c
t
s
:
x
i
s
A
~
′
A
N
D
y
i
s
B
~
′
R
u
l
e
1
:
I
F
x
i
s
A
~
1
A
N
D
y
i
s
B
~
1
T
H
E
N
z
i
s
C
~
1
R
u
l
e
2
:
I
F
x
i
s
A
~
2
A
N
D
y
i
s
B
~
2
T
H
E
N
z
i
s
C
~
2
C
o
n
c
l
u
s
i
o
n
s
:
z
i
s
C
~
′
.Let us proceed one step further and provide a generalization of the two-phase F-QFD based on the generalized MP. We find that the situation of “WHATs” in QD and “HOWs” in FD can all be described by propositions “x is A.” The fundamental idea in our paper is to obtain the truth of the propositions about Basic Requirements by evidence theory to infer the truth of the propositions about service stations by implication “if x is A, then y is B.”The methods to generate the evidencesx and the linguistic terms of predicates in SSEP-CSHR are analyzed in Section 3.3.1. In Section 3.3.2, the expression of anif-then production rules and a general scheme of a multiconditional approximate reasoning in SSEP-CSHR are discussed.


#### 3.3.1. Methods to Generate Evidence and Predicates

Figure9 has provided the basic structure of two essential HOQs (QD and FD) used in SSEP-CSHR. Here we propose a method to generate the predicates of propositions incorporated in HOQs.(
1) Propositions about Basic Requirements
L
1. Column vector W~iL1 in Figure 9 in QD shows the importance ratings of each Ri, that is, the following.Proposition 4.
R
i is W~iL2∀i∈{1,2,…,N1}.Proposition4 can be truth-qualified by a canonical form “x is A isT,” where T is a truth qualifier represented by crisp values. We should do the following analysis on truth-qualified proposition “x is A is T” [27] to capture the truth of “x is A.”Algorithm 5 (the Algorithm to capture the truth of “x is A”).
Consider the following
Step
1. The evidences about Ri (that might be C-BOE, N-BOE, or M-BOE) are collected from many experts by questionnaires (the method to generate BOE has been discussed in Section 2.3).
For eachi∈{1,2,…,N1}, a list of focal elements with basic probability distribution is then generated from the questionnaires (the list of focal elements is expressed by (FiL1,miL1)).
Step
2. The predicates in Proposition 4 are the same as the fuzzy importance ratings of L1 obtained by the two-phase F-QFD (see Algorithm 3 Step 2).
Step
3. Then the truth of Proposition 4 is calculated by the following formulas:(27)TFiL1,W~iL1=w1BelFiL1,W~iL1+w2PlFiL1,W~iL1∀i∈1,2,…,N1,where T(FiL1,W~iL1) is the truth qualifier of proposition “Ri is W~iL2” that is; “Ri is W~iL2” is T(FiL1,W~iL1).
Step
4. End.Table6 shows the steps to analyze a truth-qualified proposition.Table 6
An analysis on a truth-qualified proposition.
Column 1
Column 2
Column 3
Canonical form
x
is
A
is
T
Example
Construction cost
is
important
is
true
Description
BOE expressing evidence aboutx
Fuzzy sets expressing predicates
The truth of “x is A”
Math expression
(
F
i
L
1
,
m
i
L
1
)
W
~
i
L
2
T
(
F
i
L
1
,
W
~
i
L
1
)
Methods
By organizing questionnaires from experts
By estimation from QFD team (see Algorithm3)
By (10) in evidence theory
Analysis step
Step  1 in Algorithm5
Step  2 in Algorithm5
Step  3 in Algorithm5“‘Construction Cost is important’ is true”:(i)
The predicate “important” is given by a fuzzy setW~iL2 (see Column 2).(ii)
The evidences aboutx are generated from questionnaires (see Column 1).(iii)
The predicate “true” is expressed by a crisp number onT(FiL1,W~iL1)∈[0,1] (see Column 3).(
2) Propositions about Service Stations
(
L
3
). Row vector W~iL2 in FD of Figure 9 shows the importance ratings of each Fi (i.e., each station).Proposition 6.
F
i is W~iL3∀i∈{1,2,…,N3}.Here an approximate reasoning is demanded to infer the truth of Proposition6 based on the truth of Proposition 4 (see Section 3.3.2).


#### 3.3.2. Evidence-Based Inference Scheme in SSEP-CSHR

The evidence-based inference schema of multiconditional approximate reasoning in the two-phase F-QFD has the following form:(28)
Facts:
R
1
is
F
1
L
1
AND
R
2
is
F
2
L
1
AND
…
R
N
1
is
F
N
1
L
1
Rule
1
:  IF
R
1
is
W
~
11
L
1
AND
R
2
is
W
~
12
L
1
AND
…
R
N
1
is
W
~
1
N
1
L
1
THEN
F
1
is
W
~
11
L
3
AND
F
2
is
W
~
12
L
3
AND
…
F
N
3
is
W
~
1
N
3
L
3
Rule
2
:  IF
R
1
is
W
~
21
L
1
AND
R
2
is
W
~
21
L
1
AND
…
R
N
1
is
W
~
2
N
1
L
1
THEN
F
1
is
W
~
21
L
3
AND
F
2
is
W
~
22
L
3
AND
…
F
N
3
is
W
~
2
N
3
L
3
⋮
Rule
N
:  IF
R
1
is
W
~
N
1
L
1
AND
R
2
is
W
~
N
1
L
1
AND
…
R
N
1
is
W
~
N
N
1
L
1
THEN
F
1
is
W
~
N
1
L
3
AND
F
2
is
W
~
N
2
L
3
AND
…
F
N
3
is
W
~
N
N
3
L
3
Conclusions:
F
1
is
W
~
1
L
3
′
AND
F
2
is
W
~
2
L
3
′
AND
…
F
N
3
is
W
~
N
3
L
3
′
.Above inference schema developed to infer the truth of Proposition6 consists of the following parts:(1)
The IF part of a rule consists of (i) some expressions (Proposition4) linked by AND, and (ii) the fuzzy sets expressing the importance of the Basic Requirements in Rule n, for example:W~niL1 is the predicates describing the situation of Ri by Rule n,(2)
the THEN part of a rule consists of (i) some expressions linked by AND, and (ii) the fuzzy sets corresponding to the importance of the service stations in Rulen, for example,W~niL3 is the predicates describing the situation of Fi by Rule n,(3)
givenNif-then rule, rules 1 though N, and Facts, Conclusions will be obtained,FiL1is the evidences (BOE) describing the score range of Ri obtained by questionnaires,W~iL3′ is a new predicate for Fi, which is modified by the truth of Proposition 6.Figure10 explicates the corresponding relationship between SSEP-CSHR and the above reasoning.Figure 10
Corresponding relationship between SSEP-CSHR and proposed reasoning.The most common way to determine conclusions in the above reasoning is referred to as a method ofinterpolation [27]. Theinterpolation method is actually a special case of the max-min compositional rule of inference that uses operations:(1)
intersection operation: minimization;(2)
implication operation: minimization;(3)
aggregations operation: maximization.Based on the operations, we propose an algorithm as follows.Algorithm 7 (the Algorithm to do approximate reasoning in SSEP-CSHR).
Consider the following.
Step
1. Use Algorithm 3 to implement the two-phrase F-QFD in SSEP-CSHR.
Step
2. For each i∈{1,2,…,N1}, develop a BOE for Ri by the method proposed in Section 2.3.
Step
3. For each i∈{1,2,…,N1}, k∈{1,2,…,n}, by Algorithm 5, we calculate the truth of propositions:(29)TFiL1,W~niL1=w1BelFiL1,W~niL1+w2PlFiL1,W~niL1∀i∈1,2,…,N1,n∈1,2,…,N.The truth value here is a generalization of the concept ofthe degree of consistency in theinterpolation method of multiconditional approximate reasoning [27].
Step
4. Calculate the degree of truth between the given facts and the antecedent of eachif-then rule n in terms of the fuzzyintersection operation:(30)Tn=mini∈1,2,…,N1⁡TFiL1,W~niL1∀n∈1,2,…,N.
Step
5. Byimplication andaggregation operations above, we can calculate the Conclusions. That is,(31)W~jL3x′=supn∈1,2,…,N⁡min⁡Tn,W~njL3x∀j∈1,2,…,N3.for all x∈U.
Step
6. End.An self-explanatory example of above approximate reasoning for twoif-then rules with two Basic Requirements (R1,R2) and a service station (F1) is given in Figure 11. In Figure 11 the fuzzy set, crisp set, and evidences about both R1 and R2 (a M-BOE and a N-BOE) have all been discussed in Section 2.4.Figure 11
Illustration of approximate reasoning procedure in SSEP-CSHR.When BOE for eachi in {1,2,…,N1} is a crisp number ai,(32)Tai,W~niL1=W~niL1ai∀i∈1,2,…,N1,n∈1,2,…,N.Then Algorithm7 becomes a normalinterpolation for a multiconditional approximate reasoning [27]. Figure 12 manifests this vestigial form.Figure 12
Illustration of normalinterpolation for multiconditional approximate reasoning procedure.The general steps of our integrated approach for SSEP-CSHR are summarized in Figure13.Figure 13
General steps of proposed approach.It is noted that not only Proposition4 but also any other propositions (of course containing the relation propositions abstract from Relation Matrices and Correlation Matrices in Figure 9) might serve as “cause” to affect the predicates of Proposition 6. However, in the above approximate reasoning we did not consider these “causes” for the following two reasons: Firstly, adding more “causes” did not influence the basic schema of above evidence-based reasoning. Secondly, the relationship between a “WHAT” and a “HOW” and the correlations of “WHATs” and “HOWs” are usually more “objective” (and thus truer) than the importance ratings of “WHATs.” It means that the corresponding propositions of them are usually accompanied with higher truth value. If standard fuzzy operations are used (as in Algorithm 7), these propositions only have little impact on the predicates of Conclusions.


## 3.1. The Two-Phase F-QFD

In practice, series of HOQs can be integrated into a multiphase QFD. Each phase is closely correlated in a multiphase QFD since the inputs of one phase needs to apply the outputs of the previous phase [17, 18]. In the paper, we focused on QFD whose procedure has been thoroughly described by Mazur [10]. This approach consists of solving some successive HOQs. Among these HOQs, two HOQs named as Quality Deployment (QD) and Functional Deployment (FD) which is of the most essential are applied here to develop our MCDM model for SSEP-CSHR.Quality Deployment (QD). It translates some Basic Requirements to certain quality attributes.Functional Deployment (FD). It realized the quality attributes by specific alternatives.At the beginning of the process, Demanded Qualities (DQs) have to be identified. The elements are gathered from surveys or experts’ interviews. They are “WHATs” in QD phase:(11)L1=R1,…,Ri,…,RN1.We determine the fuzzy importance ratings of DQs (L1). The fuzzy importance ratings are usually obtained by subjective method discussed in Section 2.2. F-AHP, for example, Extent Analysis (EA) or Lambda-Max Method (LMM) can also be used to capture the fuzzy ratings [6, 25]: (12)W~L1=W~1L1,…,W~iL1,…,W~N1L1. DQs are translated into “Quality Attributes” (QAs). QAs (L2) are assessment criteria that are related to the accomplishment of above DQs. Then QAs are “HOWs” in QD phase. And QFD team then generates the QAs as follows:(13)L2=C1,…,Ci,…,CN2.Similarly, some Functional Techniques (FTs) (L3), that is, our alternatives to realize above QAs, are further defined by panel of experts. Here QAs are “WHATs” for FD and FTs are “HOWs” for FD:(14)L3=F1,…,Fi,…,FN3.The core elements of F-QFD are “relationships matrices” in QD and FD, respectively:(1)
The relationship matrix between DQs (L1) and QAs (L2) to express how a quality attribute to meet a requirement;(2)
The relationship matrix between QAs (L2) and FTs (L3) to express how an alternative to realize an attribute;(3)
The correlation matrix between DQs (L1), QAs (L2), and FTs (L3) themselves.To sum up, Table2 defines all fuzzified inputs and outputs in the two-phase F-QFD.Table 2
Inputs and outputs in HOQs of the two-phase F-QFD.
Inputs
Descriptions
W
~
i
L
1
The fuzzy important ratings ofRi where i=1,2,…,N1
B
~
i
j
L
2
The fuzzy relationship score betweenRi and Cj, where i=1,2,…,N1; j=1,2,…,N2
B
~
i
j
L
3
The fuzzy relationship score betweenCi and Fj, where i=1,2,…,N2; j=1,2,…,N3
A
~
i
j
L
1
The fuzzy correlation score betweenRi and Rj, where i=1,2,…,N1; j=1,2,…,N1
A
~
i
j
L
2
The fuzzy correlation score betweenCi and Cj, where i=1,2,…,N2; j=1,2,…,N2
A
~
i
j
L
3
The fuzzy correlation score betweenFi and Fj, where i=1,2,…,N3; j=1,2,…,N3
Outputs
Descriptions
W
~
i
L
2
The fuzzy important ratings ofCi where i=1,2,…,N2
W
~
i
L
3
The fuzzy important ratings ofFi where i=1,2,…,N3All inputs and outputs are described by linguistic terms and defined as the fuzzy sets. A fuzzy set is fully and uniquely represented by itsα-cuts (see Table 3).Table 3
Theα-cuts of the fuzzy sets in the two-phase F-QFD.
Notations
Descriptions
W
~
i
α
L
1
=
W
i
L
1
α
L
,
W
i
L
1
α
U
Theα-cuts of fuzzy important ratings W~iL1
B
~
i
j
α
L
2
=
B
i
j
L
2
α
L
,
B
i
j
L
2
α
U
Theα-cuts of fuzzy relationship B~ijL2
B
~
i
j
α
L
3
=
B
i
j
L
3
α
L
,
B
i
j
L
3
α
U
Theα-cuts of fuzzy relationship B~ijL3
A
~
i
j
α
L
1
=
A
i
j
L
1
α
L
,
A
i
j
L
1
α
U
Theα-cuts of fuzzy correlations A~ijL1
A
~
i
j
α
L
2
=
A
i
j
L
2
α
L
,
A
i
j
L
2
α
U
Theα-cuts of fuzzy correlation A~ijL2
A
~
i
j
α
L
3
=
A
i
j
L
3
α
L
,
A
i
j
L
3
α
U
Theα-cuts of fuzzy correlation A~ijL3
W
~
i
α
L
2
=
W
i
L
2
α
L
,
W
i
L
2
α
U
Theα-cuts of fuzzy important ratings W~iαL2
W
~
i
α
L
3
=
W
i
L
3
α
L
,
W
i
L
3
α
U
Theα-cuts of fuzzy important ratings W~iαL3Chen and Ko [17, 18] gave a formulation to implement multiphase F-QFD. Wang et al. [20, 28] have demonstrated that the method is not proper and provided a set of nonlinear programming (NLP) models to do F-QFD.LetA~ and B~ be two fuzzy numbers; operations on two fuzzy numbers are defined as(15)A~∗B~=⋃α∈0,1αA∗Bα=⋃α∈0,1αAα∗Bα=⋃α∈0,1αff∗∈gAα,g∈Bα.Let ∗ denote any of the four basic arithmetic operations. Aα∗Bα follows arithmetic operations on closed intervals [27]. Then based on Chen et al. [17, 18], and Wang and Chin [20], an algorithm to solve the two-phase F-QFD is introduced.Algorithm 3 (the algorithm to implement the two-phase F-QFD).
Consider the following.
Step
1. Set α-cuts of the fuzzy set equal to 1/K,…, and 1 respectively. K is the number of α-cut in [0,1]. Then the fuzzy importance ratings of FTs are obtained by following substeps.
Step
1.1. Let k=1.
Step
1.2. The fuzzy correlations between Lk can be normalized by formula:(16)ω~iLk=∑n=1NkW~nLk⊗A~niLk∑n=1Nk∑m=1NkW~nLk⊗A~nmLk∀i∈1,2,…,Nk.
The formula can be achieved by the following Nonlinear Programming (NLP) models.
For alli∈{1,2,…,Nk}, α∈[0,1], determine interval [(ωiLk)αL,(ωiLk)αU] by models:
(Model 1)(17)ωiLkαL=min⁡∑n=1NkϖnLkaniLk∑n=1Nk∑m=1NkϖnLkanmLk,ωiLkαU=max⁡∑n=1NkϖnLkaniLk∑n=1Nk∑m=1NkϖnLkanmLk,s.tWiLkαL≤ϖiLk≤WiLkαU∀i∈1,2,…,Nk,AijLkαL≤aijLk≤AijLkαU∀i,j∈1,2,…,Nk.
Step
1.3. Furthermore, the fuzzy relationship between Lk and Lk+1 can be defined by(18)Y~iLk+1=∑η=1Nk+1B~iηLk+1⊗A~ηjLk+1∑l=1Nk+1∑η=1Nk+1B~iηLk+1⊗A~ηlLk+1∀i∈1,2,…,Nk.
Then we calculate the importance rating ofLk+1 as the fuzzy weighted average of the normalized fuzzy relationships:(19)W~jLk+1=∑i=1NkW~iLk⊗YijLk+1∑i=1NkW~iLk∀j∈1,2,…,Nk+1.
The above two formulas can be achieved by following NLP models.
For alli∈{1,2,…,Nk}, α∈[0,1], we can determine interval:
(Model 2)(20)WjLk+1αL=min⁡∑η=1NkwiLk∑η=1Nk+1biηLk+1aηjLk+1/∑l=1Nk+1∑η=1Nk+1biηLk+1aηlLk+1∑i=1NkwiLkWjLk+1αU=max⁡∑i=1NkwiLk∑η=1Nk+1biηLk+1aηjLk+1/∑l=1Nk+1∑η=1Nk+1biηLk+1aηlLk+1∑i=1NkwiLk(21)s.tAijLk+1αL≤aijLk+1≤AijLk+1αU∀i,j∈1,2,…,Nk+1(22)ωiLkαL≤wiLk≤ωiLkαU∀i∈1,2,…,Nk(23)BijLk+1αL≤bijLk+1≤BijLk+1αU∀i∈1,2,…,Nkj∈1,2,…,Nk+1.
Step
1.3. By (1), the fuzzy importance ratings of Lk+1 are obtained:(24)W~Lk+1=W~1Lk+1,…,W~iLk+1,…,W~Nk+1Lk+1.
Step
1.4. If k+1=3, go to Step 2; else k=k+1 return to Step 1.2.
Step
2. Fuzzy importance rating can be ranked by different methods. In the paper, the fuzzy importance ratings are compared with each other by method developed in Wang [29]:(25)WiLk+1=1K∑α∈0,1WiLk+1αL+WiLk+1αU2∀i∈1,2,…,Nk.
Step
3. ENDBased on the method proposed in Wang and Chin [20], Models 1 and 2 can be linearized and solved optimally.


## 3.2. Service Station Evaluation Problem for CSHR

SSEP-CSHR is based on a reinterpretation of above two-phase F-QFD. In SSEP-CSHR, DQs in terms of Basic Requirements (L1) are translated into QAs in terms of Assessment Criteria (L2). And then service stations (L3), that is, specific alternatives to realize above Assessment Criteria, are served as FTs. Figure 8 gives the corresponding relationship between the two-phase F-QFD and SSEP-CSHR.Figure 8
Corresponding relationship between the two-phase F-QFD and SSEP-CSHP.Basic Requirements (L1) in SSEP-CSHR are comprised of two aspects of requirements: Firstly, it contains some requirements used in FLE-F-QFD. These demands have been extensively described by the literatures about facility location and supply chain management [11, 19]. Moreover, it involves some distinct requirements because of special CSHR background. For the sake of clarity, all those requirements are shown in Table 4.Table 4
List ofL1 for SSEP-CSHR.
Basic requirements
Description
Proximity to trains (PT)
Proximity to demands; number of trains that can be served; cost of serving trains
Sound catering condition (SC)
Space and time condition for delivery
Labor availability (LA)
Availability of required professional and nonprofessional workforce; wage rates; quality of workforce; and so forth
Availability of foods (AF)
Closeness to food suppliers or distributional centers; freight costs of transportation
Government incentives (GI)
Convenience provided by government or railway authorities
Constructions costs (CC)
Construction costs of building cold stores at the station and also availability of space for future expansionBasic Requirements (L1) are some requirements described by broad nature language. However, the set of Assessment Criteria (L2) usually use more specific descriptions. Similarly Assessment Criteria (L2) also consist of two parts: some criteria that are derived from relevant literatures [11, 19] and other ones that are gathered from experts’ comments under certain background. All the criteria are presented in Table 5.Table 5
List ofL2 for SSEP-CSHR.
Assessment criteria
Description
Quality and location of neighboring suppliers  (QL)
Proximity to suppliers; alternative suppliers; and speed and responsiveness of suppliers
Accessibility to trains  (AT)
Number of train that can be served  (number of trains departing from the city; number of trains stopping at the city, the ratio of the above two)
Available catering time (AC)
Average available catering time
Physical catering conditions  (PC)
The platform area
Human resource conditions  (HR)
Labor costs; attitudes toward works and labor turnover, economical growths, and declines effects
Economics related factors (EF)
Urban economic development, tax incentives,and business climate; interest rates
Political issues (PI)
Availability and size of government; aids attitude of government to inward investmentBy solving the steps of the two-phase F-QFD (Algorithm3), the importance ratings of alternative Service stations satisfying above requirements and criteria can be determined. Figure 9 provides the basic structure of two essential HOQs (QD and FD) used in SSEP-CSHR.Figure 9
Basic structure of HOQs for SSEP-CSHR.


## 3.3. An Approximate Reasoning Procedure in SSEP-CSHR

Approximate reasoning is related to the appropriate processing of uncertain knowledge, given in a rule based manner. Deductive inference can be performed by means of classical logic modus ponens (MP). For using linguistic variables expressed fuzzy sets, a generalized MP has been proposed by Zadeh [30]. It enables an approximate deductive reasoning with the following general scheme:(26)
F
a
c
t
s
:
x
i
s
A
~
′
A
N
D
y
i
s
B
~
′
R
u
l
e
1
:
I
F
x
i
s
A
~
1
A
N
D
y
i
s
B
~
1
T
H
E
N
z
i
s
C
~
1
R
u
l
e
2
:
I
F
x
i
s
A
~
2
A
N
D
y
i
s
B
~
2
T
H
E
N
z
i
s
C
~
2
C
o
n
c
l
u
s
i
o
n
s
:
z
i
s
C
~
′
.Let us proceed one step further and provide a generalization of the two-phase F-QFD based on the generalized MP. We find that the situation of “WHATs” in QD and “HOWs” in FD can all be described by propositions “x is A.” The fundamental idea in our paper is to obtain the truth of the propositions about Basic Requirements by evidence theory to infer the truth of the propositions about service stations by implication “if x is A, then y is B.”The methods to generate the evidencesx and the linguistic terms of predicates in SSEP-CSHR are analyzed in Section 3.3.1. In Section 3.3.2, the expression of anif-then production rules and a general scheme of a multiconditional approximate reasoning in SSEP-CSHR are discussed.


### 3.3.1. Methods to Generate Evidence and Predicates

Figure9 has provided the basic structure of two essential HOQs (QD and FD) used in SSEP-CSHR. Here we propose a method to generate the predicates of propositions incorporated in HOQs.(
1) Propositions about Basic Requirements
L
1. Column vector W~iL1 in Figure 9 in QD shows the importance ratings of each Ri, that is, the following.Proposition 4.
R
i is W~iL2∀i∈{1,2,…,N1}.Proposition4 can be truth-qualified by a canonical form “x is A isT,” where T is a truth qualifier represented by crisp values. We should do the following analysis on truth-qualified proposition “x is A is T” [27] to capture the truth of “x is A.”Algorithm 5 (the Algorithm to capture the truth of “x is A”).
Consider the following
Step
1. The evidences about Ri (that might be C-BOE, N-BOE, or M-BOE) are collected from many experts by questionnaires (the method to generate BOE has been discussed in Section 2.3).
For eachi∈{1,2,…,N1}, a list of focal elements with basic probability distribution is then generated from the questionnaires (the list of focal elements is expressed by (FiL1,miL1)).
Step
2. The predicates in Proposition 4 are the same as the fuzzy importance ratings of L1 obtained by the two-phase F-QFD (see Algorithm 3 Step 2).
Step
3. Then the truth of Proposition 4 is calculated by the following formulas:(27)TFiL1,W~iL1=w1BelFiL1,W~iL1+w2PlFiL1,W~iL1∀i∈1,2,…,N1,where T(FiL1,W~iL1) is the truth qualifier of proposition “Ri is W~iL2” that is; “Ri is W~iL2” is T(FiL1,W~iL1).
Step
4. End.Table6 shows the steps to analyze a truth-qualified proposition.Table 6
An analysis on a truth-qualified proposition.
Column 1
Column 2
Column 3
Canonical form
x
is
A
is
T
Example
Construction cost
is
important
is
true
Description
BOE expressing evidence aboutx
Fuzzy sets expressing predicates
The truth of “x is A”
Math expression
(
F
i
L
1
,
m
i
L
1
)
W
~
i
L
2
T
(
F
i
L
1
,
W
~
i
L
1
)
Methods
By organizing questionnaires from experts
By estimation from QFD team (see Algorithm3)
By (10) in evidence theory
Analysis step
Step  1 in Algorithm5
Step  2 in Algorithm5
Step  3 in Algorithm5“‘Construction Cost is important’ is true”:(i)
The predicate “important” is given by a fuzzy setW~iL2 (see Column 2).(ii)
The evidences aboutx are generated from questionnaires (see Column 1).(iii)
The predicate “true” is expressed by a crisp number onT(FiL1,W~iL1)∈[0,1] (see Column 3).(
2) Propositions about Service Stations
(
L
3
). Row vector W~iL2 in FD of Figure 9 shows the importance ratings of each Fi (i.e., each station).Proposition 6.
F
i is W~iL3∀i∈{1,2,…,N3}.Here an approximate reasoning is demanded to infer the truth of Proposition6 based on the truth of Proposition 4 (see Section 3.3.2).


### 3.3.2. Evidence-Based Inference Scheme in SSEP-CSHR

The evidence-based inference schema of multiconditional approximate reasoning in the two-phase F-QFD has the following form:(28)
Facts:
R
1
is
F
1
L
1
AND
R
2
is
F
2
L
1
AND
…
R
N
1
is
F
N
1
L
1
Rule
1
:  IF
R
1
is
W
~
11
L
1
AND
R
2
is
W
~
12
L
1
AND
…
R
N
1
is
W
~
1
N
1
L
1
THEN
F
1
is
W
~
11
L
3
AND
F
2
is
W
~
12
L
3
AND
…
F
N
3
is
W
~
1
N
3
L
3
Rule
2
:  IF
R
1
is
W
~
21
L
1
AND
R
2
is
W
~
21
L
1
AND
…
R
N
1
is
W
~
2
N
1
L
1
THEN
F
1
is
W
~
21
L
3
AND
F
2
is
W
~
22
L
3
AND
…
F
N
3
is
W
~
2
N
3
L
3
⋮
Rule
N
:  IF
R
1
is
W
~
N
1
L
1
AND
R
2
is
W
~
N
1
L
1
AND
…
R
N
1
is
W
~
N
N
1
L
1
THEN
F
1
is
W
~
N
1
L
3
AND
F
2
is
W
~
N
2
L
3
AND
…
F
N
3
is
W
~
N
N
3
L
3
Conclusions:
F
1
is
W
~
1
L
3
′
AND
F
2
is
W
~
2
L
3
′
AND
…
F
N
3
is
W
~
N
3
L
3
′
.Above inference schema developed to infer the truth of Proposition6 consists of the following parts:(1)
The IF part of a rule consists of (i) some expressions (Proposition4) linked by AND, and (ii) the fuzzy sets expressing the importance of the Basic Requirements in Rule n, for example:W~niL1 is the predicates describing the situation of Ri by Rule n,(2)
the THEN part of a rule consists of (i) some expressions linked by AND, and (ii) the fuzzy sets corresponding to the importance of the service stations in Rulen, for example,W~niL3 is the predicates describing the situation of Fi by Rule n,(3)
givenNif-then rule, rules 1 though N, and Facts, Conclusions will be obtained,FiL1is the evidences (BOE) describing the score range of Ri obtained by questionnaires,W~iL3′ is a new predicate for Fi, which is modified by the truth of Proposition 6.Figure10 explicates the corresponding relationship between SSEP-CSHR and the above reasoning.Figure 10
Corresponding relationship between SSEP-CSHR and proposed reasoning.The most common way to determine conclusions in the above reasoning is referred to as a method ofinterpolation [27]. Theinterpolation method is actually a special case of the max-min compositional rule of inference that uses operations:(1)
intersection operation: minimization;(2)
implication operation: minimization;(3)
aggregations operation: maximization.Based on the operations, we propose an algorithm as follows.Algorithm 7 (the Algorithm to do approximate reasoning in SSEP-CSHR).
Consider the following.
Step
1. Use Algorithm 3 to implement the two-phrase F-QFD in SSEP-CSHR.
Step
2. For each i∈{1,2,…,N1}, develop a BOE for Ri by the method proposed in Section 2.3.
Step
3. For each i∈{1,2,…,N1}, k∈{1,2,…,n}, by Algorithm 5, we calculate the truth of propositions:(29)TFiL1,W~niL1=w1BelFiL1,W~niL1+w2PlFiL1,W~niL1∀i∈1,2,…,N1,n∈1,2,…,N.The truth value here is a generalization of the concept ofthe degree of consistency in theinterpolation method of multiconditional approximate reasoning [27].
Step
4. Calculate the degree of truth between the given facts and the antecedent of eachif-then rule n in terms of the fuzzyintersection operation:(30)Tn=mini∈1,2,…,N1⁡TFiL1,W~niL1∀n∈1,2,…,N.
Step
5. Byimplication andaggregation operations above, we can calculate the Conclusions. That is,(31)W~jL3x′=supn∈1,2,…,N⁡min⁡Tn,W~njL3x∀j∈1,2,…,N3.for all x∈U.
Step
6. End.An self-explanatory example of above approximate reasoning for twoif-then rules with two Basic Requirements (R1,R2) and a service station (F1) is given in Figure 11. In Figure 11 the fuzzy set, crisp set, and evidences about both R1 and R2 (a M-BOE and a N-BOE) have all been discussed in Section 2.4.Figure 11
Illustration of approximate reasoning procedure in SSEP-CSHR.When BOE for eachi in {1,2,…,N1} is a crisp number ai,(32)Tai,W~niL1=W~niL1ai∀i∈1,2,…,N1,n∈1,2,…,N.Then Algorithm7 becomes a normalinterpolation for a multiconditional approximate reasoning [27]. Figure 12 manifests this vestigial form.Figure 12
Illustration of normalinterpolation for multiconditional approximate reasoning procedure.The general steps of our integrated approach for SSEP-CSHR are summarized in Figure13.Figure 13
General steps of proposed approach.It is noted that not only Proposition4 but also any other propositions (of course containing the relation propositions abstract from Relation Matrices and Correlation Matrices in Figure 9) might serve as “cause” to affect the predicates of Proposition 6. However, in the above approximate reasoning we did not consider these “causes” for the following two reasons: Firstly, adding more “causes” did not influence the basic schema of above evidence-based reasoning. Secondly, the relationship between a “WHAT” and a “HOW” and the correlations of “WHATs” and “HOWs” are usually more “objective” (and thus truer) than the importance ratings of “WHATs.” It means that the corresponding propositions of them are usually accompanied with higher truth value. If standard fuzzy operations are used (as in Algorithm 7), these propositions only have little impact on the predicates of Conclusions.


## 3.3.1. Methods to Generate Evidence and Predicates

Figure9 has provided the basic structure of two essential HOQs (QD and FD) used in SSEP-CSHR. Here we propose a method to generate the predicates of propositions incorporated in HOQs.(
1) Propositions about Basic Requirements
L
1. Column vector W~iL1 in Figure 9 in QD shows the importance ratings of each Ri, that is, the following.Proposition 4.
R
i is W~iL2∀i∈{1,2,…,N1}.Proposition4 can be truth-qualified by a canonical form “x is A isT,” where T is a truth qualifier represented by crisp values. We should do the following analysis on truth-qualified proposition “x is A is T” [27] to capture the truth of “x is A.”Algorithm 5 (the Algorithm to capture the truth of “x is A”).
Consider the following
Step
1. The evidences about Ri (that might be C-BOE, N-BOE, or M-BOE) are collected from many experts by questionnaires (the method to generate BOE has been discussed in Section 2.3).
For eachi∈{1,2,…,N1}, a list of focal elements with basic probability distribution is then generated from the questionnaires (the list of focal elements is expressed by (FiL1,miL1)).
Step
2. The predicates in Proposition 4 are the same as the fuzzy importance ratings of L1 obtained by the two-phase F-QFD (see Algorithm 3 Step 2).
Step
3. Then the truth of Proposition 4 is calculated by the following formulas:(27)TFiL1,W~iL1=w1BelFiL1,W~iL1+w2PlFiL1,W~iL1∀i∈1,2,…,N1,where T(FiL1,W~iL1) is the truth qualifier of proposition “Ri is W~iL2” that is; “Ri is W~iL2” is T(FiL1,W~iL1).
Step
4. End.Table6 shows the steps to analyze a truth-qualified proposition.Table 6
An analysis on a truth-qualified proposition.
Column 1
Column 2
Column 3
Canonical form
x
is
A
is
T
Example
Construction cost
is
important
is
true
Description
BOE expressing evidence aboutx
Fuzzy sets expressing predicates
The truth of “x is A”
Math expression
(
F
i
L
1
,
m
i
L
1
)
W
~
i
L
2
T
(
F
i
L
1
,
W
~
i
L
1
)
Methods
By organizing questionnaires from experts
By estimation from QFD team (see Algorithm3)
By (10) in evidence theory
Analysis step
Step  1 in Algorithm5
Step  2 in Algorithm5
Step  3 in Algorithm5“‘Construction Cost is important’ is true”:(i)
The predicate “important” is given by a fuzzy setW~iL2 (see Column 2).(ii)
The evidences aboutx are generated from questionnaires (see Column 1).(iii)
The predicate “true” is expressed by a crisp number onT(FiL1,W~iL1)∈[0,1] (see Column 3).(
2) Propositions about Service Stations
(
L
3
). Row vector W~iL2 in FD of Figure 9 shows the importance ratings of each Fi (i.e., each station).Proposition 6.
F
i is W~iL3∀i∈{1,2,…,N3}.Here an approximate reasoning is demanded to infer the truth of Proposition6 based on the truth of Proposition 4 (see Section 3.3.2).


## 3.3.2. Evidence-Based Inference Scheme in SSEP-CSHR

The evidence-based inference schema of multiconditional approximate reasoning in the two-phase F-QFD has the following form:(28)
Facts:
R
1
is
F
1
L
1
AND
R
2
is
F
2
L
1
AND
…
R
N
1
is
F
N
1
L
1
Rule
1
:  IF
R
1
is
W
~
11
L
1
AND
R
2
is
W
~
12
L
1
AND
…
R
N
1
is
W
~
1
N
1
L
1
THEN
F
1
is
W
~
11
L
3
AND
F
2
is
W
~
12
L
3
AND
…
F
N
3
is
W
~
1
N
3
L
3
Rule
2
:  IF
R
1
is
W
~
21
L
1
AND
R
2
is
W
~
21
L
1
AND
…
R
N
1
is
W
~
2
N
1
L
1
THEN
F
1
is
W
~
21
L
3
AND
F
2
is
W
~
22
L
3
AND
…
F
N
3
is
W
~
2
N
3
L
3
⋮
Rule
N
:  IF
R
1
is
W
~
N
1
L
1
AND
R
2
is
W
~
N
1
L
1
AND
…
R
N
1
is
W
~
N
N
1
L
1
THEN
F
1
is
W
~
N
1
L
3
AND
F
2
is
W
~
N
2
L
3
AND
…
F
N
3
is
W
~
N
N
3
L
3
Conclusions:
F
1
is
W
~
1
L
3
′
AND
F
2
is
W
~
2
L
3
′
AND
…
F
N
3
is
W
~
N
3
L
3
′
.Above inference schema developed to infer the truth of Proposition6 consists of the following parts:(1)
The IF part of a rule consists of (i) some expressions (Proposition4) linked by AND, and (ii) the fuzzy sets expressing the importance of the Basic Requirements in Rule n, for example:W~niL1 is the predicates describing the situation of Ri by Rule n,(2)
the THEN part of a rule consists of (i) some expressions linked by AND, and (ii) the fuzzy sets corresponding to the importance of the service stations in Rulen, for example,W~niL3 is the predicates describing the situation of Fi by Rule n,(3)
givenNif-then rule, rules 1 though N, and Facts, Conclusions will be obtained,FiL1is the evidences (BOE) describing the score range of Ri obtained by questionnaires,W~iL3′ is a new predicate for Fi, which is modified by the truth of Proposition 6.Figure10 explicates the corresponding relationship between SSEP-CSHR and the above reasoning.Figure 10
Corresponding relationship between SSEP-CSHR and proposed reasoning.The most common way to determine conclusions in the above reasoning is referred to as a method ofinterpolation [27]. Theinterpolation method is actually a special case of the max-min compositional rule of inference that uses operations:(1)
intersection operation: minimization;(2)
implication operation: minimization;(3)
aggregations operation: maximization.Based on the operations, we propose an algorithm as follows.Algorithm 7 (the Algorithm to do approximate reasoning in SSEP-CSHR).
Consider the following.
Step
1. Use Algorithm 3 to implement the two-phrase F-QFD in SSEP-CSHR.
Step
2. For each i∈{1,2,…,N1}, develop a BOE for Ri by the method proposed in Section 2.3.
Step
3. For each i∈{1,2,…,N1}, k∈{1,2,…,n}, by Algorithm 5, we calculate the truth of propositions:(29)TFiL1,W~niL1=w1BelFiL1,W~niL1+w2PlFiL1,W~niL1∀i∈1,2,…,N1,n∈1,2,…,N.The truth value here is a generalization of the concept ofthe degree of consistency in theinterpolation method of multiconditional approximate reasoning [27].
Step
4. Calculate the degree of truth between the given facts and the antecedent of eachif-then rule n in terms of the fuzzyintersection operation:(30)Tn=mini∈1,2,…,N1⁡TFiL1,W~niL1∀n∈1,2,…,N.
Step
5. Byimplication andaggregation operations above, we can calculate the Conclusions. That is,(31)W~jL3x′=supn∈1,2,…,N⁡min⁡Tn,W~njL3x∀j∈1,2,…,N3.for all x∈U.
Step
6. End.An self-explanatory example of above approximate reasoning for twoif-then rules with two Basic Requirements (R1,R2) and a service station (F1) is given in Figure 11. In Figure 11 the fuzzy set, crisp set, and evidences about both R1 and R2 (a M-BOE and a N-BOE) have all been discussed in Section 2.4.Figure 11
Illustration of approximate reasoning procedure in SSEP-CSHR.When BOE for eachi in {1,2,…,N1} is a crisp number ai,(32)Tai,W~niL1=W~niL1ai∀i∈1,2,…,N1,n∈1,2,…,N.Then Algorithm7 becomes a normalinterpolation for a multiconditional approximate reasoning [27]. Figure 12 manifests this vestigial form.Figure 12
Illustration of normalinterpolation for multiconditional approximate reasoning procedure.The general steps of our integrated approach for SSEP-CSHR are summarized in Figure13.Figure 13
General steps of proposed approach.It is noted that not only Proposition4 but also any other propositions (of course containing the relation propositions abstract from Relation Matrices and Correlation Matrices in Figure 9) might serve as “cause” to affect the predicates of Proposition 6. However, in the above approximate reasoning we did not consider these “causes” for the following two reasons: Firstly, adding more “causes” did not influence the basic schema of above evidence-based reasoning. Secondly, the relationship between a “WHAT” and a “HOW” and the correlations of “WHATs” and “HOWs” are usually more “objective” (and thus truer) than the importance ratings of “WHATs.” It means that the corresponding propositions of them are usually accompanied with higher truth value. If standard fuzzy operations are used (as in Algorithm 7), these propositions only have little impact on the predicates of Conclusions.


## 4. Model Implementation

In this section, the integrated evidence theory and F-QFD for SSEP-CSHR will be applied to a practical case study, which refers to 85 alternative service stations on Chinese high-speed rail network. The set of service stations (L3) is generated based on the real timetables of high-speed trains on April in 2014 in China (see Figure 14). The stations are selected from the rail stations with plenty of starting train lines and stations at which trains usually stop. Table 7 presents names of the stations and their indexes.Table 7
Alterative service rail stations and their IDs in example network.
ID
Station
1
Bengbu south
2
Baoji south
3
Beidaihe
4
Beijing
5
Beijing south
6
Beijing west
7
Cangnan
8
Changzhou
9
Chengdu
10
Chengdu east
11
Dalian
12
Dalian north
13
Fuding
14
Fuzhou
15
Fuzhou south
16
Guangzhou
17
Guangzhou east
18
Guangzhou south
19
Guilin
20
Harbin west
21
Handan east
22
Hankou
23
Hangzhou
24
Hangzhou east
25
Hefe
26
Huainan east
27
Jilin
28
Jinan
29
Jinan west
30
Jiangshan
31
Jinhua west
32
Jingzhou
33
Jiujiang
34
Lichuan
35
Lingbao west
36
Liu‘an
37
Longyan
38
Luoyang Longmen
39
Nanchang
40
Nanchang west
41
Nanjing
42
Nanjing south
43
Nanning
44
Ningbo
45
Qinhuangdao
46
Qingdao
47
Qingdao north
48
Sanming north
49
Xiamen north
50
Shangqiu
51
Shanghai
52
Shanghai Hongqiao
53
Shenzhen
54
Shenzhen north
55
Shenyang
56
Shenyang north
57
Shiyan
58
Shijiazhuang
59
Suzhou
60
Taiyuan
61
Taining
62
Tianjin
63
Tianjin south
64
Tianjin west
65
Wenzhou south
66
Wuchang
67
Wuhan
68
Xi’an north
69
Xiangyang
70
Xinxiang east
71
Xinyang east
72
Xuzhou
73
Xuzhou east
74
Yan’an
75
Yichang east
76
Yiwu
77
YueYangDong
78
Zhangzhou
79
Changchun
80
Changchun west
81
Changsha
82
Changsha south
83
Zhengzhou
84
Zhengzhou east
85
Chongqing northFigure 14
Alternative service stations considered in our case study.Tables4 and 5 have presented L1 and L2 by details. Further QFD team expresses the opinions about the relationship between L1 and L2 in Table 8 (i.e., the relationship matrix in QD). The relationship between L2 and L3 is expressed in Appendix ((i.e., the relationship matrix in FD)). Correlation degrees between L1, L2, and L3 themselves are not considered in this study for simplification. In all the following matrices, a/b/c means a triangle fuzzy number on an x-α graph; say α=1 with x=b, and say α=0 with x∈[a,c].Table 8
Relation matrices of QD (betweenL1 and L2).
QL
AT
AC
PC
HR
EF
PI
PT
0.08/0.09/0.1
0.29/0.29/0.29
0.1/0.12/0.14
0.11/0.12/0.15
0.08/0.09/0.1
0.11/0.12/0.15
0.16/0.17/0.19
SC
0.07/0.08/0.08
0.07/0.08/0.08
0.24/0.25/0.25
0.24/0.25/0.26
0.07/0.08/0.09
0.13/0.14/0.15
0.13/0.14/0.15
LA
0.08/0.09/0.11
0.11/0.14/0.16
0.08/0.09/0.12
0.09/0.09/0.12
0.31/0.31/0.31
0.11/0.14/0.16
0.12/0.14/0.17
AF
0.3/0.3/0.3
0.08/0.09/0.11
0.1/0.13/0.15
0.11/0.13/0.16
0.08/0.09/0.11
0.11/0.13/0.16
0.11/0.13/0.17
GI
0.13/0.15/0.17
0.13/0.15/0.18
0.08/0.09/0.11
0.08/0.09/0.12
0.06/0.07/0.08
0.26/0.26/0.26
0.18/0.2/0.23
CC
0.15/0.17/0.18
0.15/0.17/0.19
0.1/0.12/0.13
0.1/0.12/0.14
0.08/0.08/0.09
0.22/0.23/0.23
0.11/0.12/0.14Table9 is a pairwise comparison matrix of F-AHP used here to give the importance ratings of Basic Requirements. Three F-AHPs are used to determine the importance rating of Basic Requirements L1 including Extent Analysis (EA) [6], Modified Extent Analysis (MEA) [24], and Lambda-Max Method (LMM) [25]. Different QFD teams might generate different comparison matrix of F-AHP and thus generate different production rules in approximate reasoning. In this case, only one QFD team has made our decision.Table 9
Pairwise comparison matrix of F-AHP.
PT
SC
LA
AF
GI
CC
PT
1/1/1
0.5/1/1.5
1/1.5/2
1.5/2/2.5
1.5/2/2.5
1/1.5/2
SC
0.67/1/2
1/1/1
1/1.5/2
1.5/2/2.5
1.5/2/2.5
1/1.5/2
LA
0.5/0.67/1
0.5/0.67/1
1/1/1
1/1.5/2
1/1.5/2
0.5/1/1.5
AF
0.4/0.5/0.67
0.4/0.5/0.67
0.5/0.67/1
1/1/1
0.5/1/1.5
0.5/0.67/1
GI
0.4/0.5/0.67
0.4/0.5/0.67
0.5/0.67/1
0.67/1/2
1/1/1
0.5/0.67/1
CC
0.5/0.67/1
0.5/0.67/1
0.67/1/2
1/1.5/2
1/1.5/2
1/1/1Table10 shows the fuzzy importance ratings of requirements in L1 obtained by EA, MEA, and LMM. By using (25) to defuzzify the fuzzy importance ratings, all three methods obtain the crisp results that signify “SC>PT>CC>LA>GI>AF”. We also find that the fuzzy importance ratings obtained by LMM are “narrower” than the ratings from MEA and EA. Klir and Yuan [27] characterize this nature as nonspecificity (Non). Then we get NonEA>NonMEA>NonLMM.Table 10
Fuzzy importance ratings ofL1 obtained from EA, MEA, and LMM.
Importance ratings ofL1 by different F-AHP
EA
Equation (25)
MEA
Equation (25)
LMM
Equation (25)
PT
0.12/0.23/0.4
0.25
0.13/0.23/0.34
0.23
0.21/0.23/0.23
0.223
SC
0.13/0.23/0.41
0.26
0.14/0.23/0.35
0.24
0.22/0.23/0.24
0.230
LA
0.08/0.16//0.29
0.18
0.09/0.16/0.26
0.17
0.14/0.15//0.17
0.157
AF
0.06/0.11/0.2
0.12
0.07/0.11/0.18
0.12
0.11/0.11/0.12
0.111
GI
0.07/0.11/0.22
0.13
0.07/0.11/0.20
0.13
0.11/0.11/0.12
0.114
CC
0.09/0.16/0.31
0.19
0.10/0.16/0.27
0.18
0.15/0.16/0.18
0.161N-BOEs are generated randomly with 20,000 samples corresponding to eachRi in L1. Based on the N-BOE and the fuzzy importance ratings in Table 10, we can measure the truth between the evidences about Basic Requirements and the predicates of Proposition 4 (which are obtained by fuzzy importance ratings of L1). In our experiment, N-BOEs are used instead of M-BOE for the purpose of easy presentation. Table 11 shows that fuzzy importance ratings calculated by LMM have lower truth than the results by EA and MEA, that is TruthEA>TruthMEA>TruthLMM. It is just the opposite of the state of nonspecificity. It follows that vagueness usually leads propositions more difficult to be falsified.Table 11
Comparison analysis between truth and nonspecificity.
L
1
Nonspecificity
by different F-AHPs
L
1
Truth of L1 by different F-AHPs
EA
MEA
LMM
EA
MEA
LMM
PT
1.5
1.11
0.09
PT
0.88
0.80
0.5
SC
1.58
1.15
0.1
SC
0.54
0.45
0.25
LA
1.14
0.91
0.11
LA
0.6
0.51
0.30
AF
0.76
0.66
0.04
AF
0.44
0.42
0.35
GI
0.84
0.71
0.05
GI
0.49
0.48
0.28
CC
1.22
0.96
0.12
CC
0.61
0.48
0.5
Max nonspecificity
1.58
1.15
0.18
Min truth
0.44
0.42
0.25Figure15 explicitly illustrates the relationship between the fuzzy important ratings (or the predicates of Proposition 4) obtained by EA, MEA, and LMM and the N-BOE of PT. Apparently, the predicates obtained by LMM are too strict to be proved by our evidences (Bel measure is 0 and Pl measure is 1). The fuzzy important rate of PT from EA although hardly provides any precision information, in this case, but the proposition “R1 is W~1L1” is truer from the view of the N-BOE. The values of “Min Truth” in Table 11 are important for that they determine the truth of our results of SSEP-CSHR when we only have one Rule in evidence-based inference schema.Figure 15
Fuzzy important rate of PT inL1 by three different methods.Figure16 displays the trade-off between nonspecificity and truth of fuzzy important ratings in F-QFD by a fitting curve based on all the fuzzy important ratings obtained by the above three methods. It actually shows an inclination that truer propositions are usually accompanied with more Fuzziness. It is noted that Klir and Yuan [27] treat fuzziness and nonspecificity as different conceptions; however, in many researches, nonspecificity is actually used the same as Fuzziness, such as Csutora and Buckley [25], and in fact, in general case nonspecificity is normally directly proportional to fuzziness.Figure 16
Trade-off betweenimprecision and truth of fuzzy important ratings.It leads to some interesting conclusions in the field of “uncertainty management.” In many existing literatures, minimizing Fuzziness usually serves as a basic principle in MCDM problems. For example, Csutora and Buckley [25] designed LMM in basis of the principle of minimizing Fuzziness. Wang et al. [20, 24] criticized the normalization methods in Chang’s [6] EA and Chen et al.'s (2011) fuzzy linear programming because they did not generate the narrowest interval and fuzzy weights.Our experiments partly justify a diversity of “uncertainty management,” just as Klir and Yuan discussed in their book [27]. In F-QFD, when we use precise predicates to describe the situation of L1, it is possible (not necessary) to result in less truth Conclusions in reasoning and thus decrease the credibility of our evaluation.Using the results of LMM as inputs, then we execute our two-phase F-QFD methods. LetK be the number of α-cut in [0,1]. When solving the NLPs, that is, (15)–(19), we define K=10. Above algorithms are all coded in MATLAB and executed by using a desktop with CPU Intel-Core i7-4700MQ and 8 G RAM. Finally using about 15 min we get the fuzzy important ratings of 85 stations.


### 4.1. Small Scale Example

We use the data in Appendix  (ID: 1, 5, 9, 18, 22, 42, 52, 68) to construct a7∗8 relationship matrix between QAs and these eight service stations. Then by Algorithm 3, based on the outputs of phase 1, we determine the fuzzy important rating of stations. Due to large amount of outputs, only eight stations’ important ratings at certain α-cuts (α=0 and α=1) are displayed in Table 12. The fuzzy sets which are used to express the fuzzy important ratings are shown in Figure 17(a). Figure 17(b) shows the Conclusions obtained by proposed approximate reasoning.Table 12
Fuzzy important ratings of some service stations.
α-cut
Bengbu south
Beijing south
Chengdu
Guangzhousouth
Hankou
Nanjingsouth
ShanghaiHongqiao
Xi’annorth
(
A
~
)
0
L
0.054
0.114
0.069
0.118
0.110
0.123
0.142
0.078
(
A
~
)
1
L
0.060
0.152
0.081
0.152
0.136
0.152
0.173
0.093
(
A
~
)
0
U
0.070
0.180
0.099
0.182
0.170
0.191
0.212
0.119Figure 17
Fuzzy important ratings of the eight service stations and the conclusions after reasoning.Figure18 gives all fuzzy important ratings of above eight service stations (with lower and upper bounds when α=0, as well as the core of the fuzzy sets) based on all the above three types of F-AHPs (i.e., EA, MEA, and LMM). We find that the Conclusions obtained based on LMM provide the most precision information, but with only “truth value = 0.25.” The Conclusions obtained from EA provide the most vague outputs, however, with “truth value = 0.48.”Figure 18
Fuzzy important ratings of service stations based on different F-AHPs.


### 4.2. Large Scale Example

Figure19 utilizes a boxplot to display all the fuzzy important ratings of above 85 stations (L3) (a/b/c, the minimum of a box is a in fuzzy triangle number a/b/c; the median of a box is b in fuzzy triangle number a/b/c; the maximum of a box is c in fuzzy triangle number a/b/c). Based on LMM inputs and the approximate reasoning procedure in SSEP-CSHR, we know that the outputs only have truth value 0.25 (see Table 11).Figure 19
Boxplots to express the fuzzy important ratings for service stations.


## 4.1. Small Scale Example

We use the data in Appendix  (ID: 1, 5, 9, 18, 22, 42, 52, 68) to construct a7∗8 relationship matrix between QAs and these eight service stations. Then by Algorithm 3, based on the outputs of phase 1, we determine the fuzzy important rating of stations. Due to large amount of outputs, only eight stations’ important ratings at certain α-cuts (α=0 and α=1) are displayed in Table 12. The fuzzy sets which are used to express the fuzzy important ratings are shown in Figure 17(a). Figure 17(b) shows the Conclusions obtained by proposed approximate reasoning.Table 12
Fuzzy important ratings of some service stations.
α-cut
Bengbu south
Beijing south
Chengdu
Guangzhousouth
Hankou
Nanjingsouth
ShanghaiHongqiao
Xi’annorth
(
A
~
)
0
L
0.054
0.114
0.069
0.118
0.110
0.123
0.142
0.078
(
A
~
)
1
L
0.060
0.152
0.081
0.152
0.136
0.152
0.173
0.093
(
A
~
)
0
U
0.070
0.180
0.099
0.182
0.170
0.191
0.212
0.119Figure 17
Fuzzy important ratings of the eight service stations and the conclusions after reasoning.Figure18 gives all fuzzy important ratings of above eight service stations (with lower and upper bounds when α=0, as well as the core of the fuzzy sets) based on all the above three types of F-AHPs (i.e., EA, MEA, and LMM). We find that the Conclusions obtained based on LMM provide the most precision information, but with only “truth value = 0.25.” The Conclusions obtained from EA provide the most vague outputs, however, with “truth value = 0.48.”Figure 18
Fuzzy important ratings of service stations based on different F-AHPs.


## 4.2. Large Scale Example

Figure19 utilizes a boxplot to display all the fuzzy important ratings of above 85 stations (L3) (a/b/c, the minimum of a box is a in fuzzy triangle number a/b/c; the median of a box is b in fuzzy triangle number a/b/c; the maximum of a box is c in fuzzy triangle number a/b/c). Based on LMM inputs and the approximate reasoning procedure in SSEP-CSHR, we know that the outputs only have truth value 0.25 (see Table 11).Figure 19
Boxplots to express the fuzzy important ratings for service stations.


## 5. Conclusions

QFD is a method to specify the customer wants and needs and then to evaluate each alternative in terms of its impact on meeting those needs. This paper is an effort to improve the scientific credibility and practicability of F-QFD and to offer a qualitative approach for SSEP-CSHR based on the two-phase F-QFD with an approximate procedure based on evidence theory. Belief and Plausibility are two types of measures in the view of evidence theory. Using the two measures, a framework to determine the truth of “x is A” is investigated in Section 2 and then is used to deal with different types of BOE that might be met in CSHR’s survey. The contribution of Section 3 is twofold. Firstly, this paper offers a MCDM approach for SSEP-CSHR based on the two-phase F-QFD to select potential service stations for CSHR with regard to series of practical criteria and Basic Requirements in context of CSHR. Secondly, this paper illustrates that the usage of evidence theory allows an approximate reasoning based on various types of BOE to be incorporated into F-QFD. In our case study, the proposed method is implemented in a high-speed rail network of China referring to 85 railway stations. The application, on the one hand, presents the fuzzy importance ratings for 85 service stations; on the other hand, it shows the truth of our Conclusions to be assessed by using proposed evidence-based inference schema. It demonstrated that the proposed method enhances scientific credibility of F-QFD in SSEP-CSHR and allows decision makers to express how much they know. By deep analysis, we realize an interesting trade-off between nonspecificity and truth in F-QFD. Hence our experiments partly justify a diversity of “uncertainty management.”



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*Source: 404926-2015-09-28.xml*