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A Linguistic Multi-Criteria Decision Making Approach

Based on Logical Reasoning

Shuwei Chena,

1

, Jun Liua, b, Hui Wanga, Yang Xub, Juan Carlos Augustoc

a School of Computing and Mathematics, University of Ulster at Jordanstown,

Newtownabbey, BT37 0QB, Northern Ireland, UK

b School of Mathematics, Southwest Jiaotong University, Chengdu 610031, Sichuan, China

c Department of Computer Science, School of Science and Technology, Middlesex University, London, UK

Abstract

In real decision making problems, it is always more natural for decision makers to use linguistic terms to express their

preferences/opinions in a qualitative way among alternatives than to provide quantitative values. Additionally, many of

these decision making problems are under uncertain environments with vague and imprecise information involved.

Following the idea of Computing with Words (CWW) methodology, we propose in this paper a linguistic valued

qualitative aggregation and reasoning framework for multi-criteria decision making problems, where a linguistic valued

algebraic structure is constructed for modelling the linguistic information involved in multi-criteria decision making

problems, and a linguistic valued logic based approximate reasoning method is developed to infer the final decision

making result. This method takes the advantage of handling the linguistic information, no matter totally ordered or

partially ordered, directly without numerical approximation, and having a non-classical logic as its formal foundation for

decision making process.

Keywords: Decision making, linguistic information, computing with words, lattice-valued logic, approximate reasoning,

lattice implication algebra

1. Introduction

Decision making, which, in many cases, can be seen as the process for choosing the most appropriate one among a set of

alternatives under provided criteria or preferences, is the crucial step in many real applications such as organization

management, financial planning, risk assessment, products evaluation and recommendation. Qualitative information is

frequently used in the area of decision-making, such as judgments/opinions from experts, which are always expressed by

linguistic terms in natural language. Linguistic terms, not like numerical ones whose value are crisp numbers, are always

vague and imprecise [13, 14, 25, 31, 34, 37]. For example, when we are evaluating the quality of a computer, which is

qualitative in nature, the evaluations are usually expressed as linguistic terms, such as “satisfied,” “acceptable,” or “good”.

Computing with Words (CWW) [39, 46] methodology has been the most popular one to model and manipulate linguistic

information to solve decision making problems under qualitative and uncertain environment.

The main inspiration of CWW comes from the human ability to deal with different uncertain, incomplete and imprecise

perceptions naturally without needing an explicit use of any measurements or computations. Therefore, CWW aims at

manipulating perceptions, or words and propositions from natural language, which contrasts to the usual sense of

1

Corresponding author. Tel: +44 28 9036 6095.

E-mail addresses: chensw915@gmail.com (S. Chen), j.liu@ulster.ac.uk (J. Liu), h.wang@ulster.ac.uk (H. Wang), xuyang@home.swjtu.edu.cn (Y.

Xu), j.augusto@mdx.ac.uk (J.C. Augusto)

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computing, i.e., manipulating numbers, and providing a methodology which can enhance computer’s ability to deal with

imprecision, uncertainty and partial truth which is pervasive in human daily recognition and decision processes [14, 31].

The conventional CWW methodology is based on fuzzy set theory [2, 35, 38, 39, 46], which generally converts the

linguistic information being processed into a fuzzy set (FS) and modelled by membership function or fuzzy number, and a

CWW engine, perceptual computer, then works to map these FSs into some other FS, after which the generated FS will be

transformed back into a word or proposition using linguistic approximation [33]. The process of converting words to FSs

and then FSs to words again, which is mainly based on the extension principle, is usually time consuming, computationally

complex, and involving loss of information [17]. Furthermore, it is not always an easy task to find the feasible FSs for

representing the corresponding words. Taking into account the fact that human beings manipulate the perceptions directly

without needing to convert them into numerical values, and the process is always accompanied with reasoning, it would be

more natural and reasonable to represent and reason about linguistic information in its original form, i.e., through the

symbolic way [42].

Symbolic approaches [1, 12, 30] use linguistic symbols (usually with indexes) to represent linguistic information

directly without the numerical approximation required by fuzzy set based methods, and aggregate or compute on the

indexes of these symbols to obtain the final result. One of the representative symbolic linguistic information processing

approaches, fuzzy ordinal linguistic approach [12, 18, 31], uses an ordered structure, linguistic labels with indexes, to

represent the set of linguistic terms, with the assumption that the terms under discussion is totally ordered [32]. The 2-tuple

linguistic representation (or computational) model [16, 17, 30], a continuous linguistic representation and computation

model, is one of most popular extended fuzzy ordinal linguistic approaches. In this model, the linguistic 2-tuple, a pair of

values, (Li, αi), is used to represent the linguistic information, where Li

S is a linguistic label and the number αi

[-0.5,

0.5) is called the symbolic translation, which supports “difference of information” between the result obtained after

aggregation and the closest one in the set of linguistic terms.

Although fuzzy ordinal linguistic approaches take the advantages of without loss of information and computational

simplicity by avoiding use of membership function [17, 30], it requires that the linguistic information is totally ordered and

can be manipulated by indexes. This limits the application of fuzzy ordinal linguistic approach to more general situations

where partially ordered information often involved.

Partially ordered information is ubiquitous in our daily decision making problems because of the uncertain and

dynamic environment [29]. For example, one alternative is better in one aspect but may be worse in another, and we

always find it difficult to make a decision when multiple criteria are involved where conflicting opinions always exist.

Partial orders are more flexible than total orders to represent incomplete, uncertain and imprecise knowledge. Moreover,

they avoid comparing unrelated pieces of information which is required by total order based approaches.

Lattice, a special kind of partially ordered structure, has been shown to be an appropriate and efficient structure for

representing ordinal qualitative information in the real world due to its additional operations and better properties [6, 23].

Mainly from the algebraic point of view, Ho et al. [19-22] proposed an algebraic structure for modelling linguistic terms,

Hedge algebra. Hedge algebra consists of two parts: linguistic hedges and prime terms (also called generators), where

linguistic hedges are actually some kind of linguistic modifiers, e.g., “more or less,” “quite,” “highly,” on the prime terms

such as “true and false,” “high and low”. These linguistic hedges take the role in strengthening or weakening the meaning

of prime terms. By applying the set of hedges to the prime terms, Hedge algebra is then constructed which is essentially a

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partially ordered structure according to the natural meanings of the represented linguistic terms, and generally a lattice as

Fig. 1.1 shows.

Fig. 1.1 The poset of a Hedge algebra

V=very, M=more, A=approximately, P=possibly, ML=more or less, S=less.

Although Hedge algebra is able to reflect the semantic ordering relation, partial order in many cases, among the

considered linguistic terms [22], and has a close relation with logic systems and approximate reasoning methods due to the

fact that it is based on lattice structure, there is no logic system and the corresponding approximate reasoning methods has

been built based on Hedge algebras [9].

In our view, the process of decision making can be essentially interpreted as a reasoning process from provided

information or domain knowledge to some conclusion, and decision making under qualitative and uncertain environments

is essentially an approximate reasoning process. From the viewpoint of symbolism, the confidence and rationality of

certain reasoning is rooted in the solid classical logic foundation. Accordingly, the confidence and rationality of

approximate reasoning then relies on non-classical logics [7, 36], which are extensions of the classical logic. Therefore, it

is important and necessary to study the rational logic-based approximate reasoning approach for decision making problems

under uncertain environment [9, 42].

Logic generally can be used for modelling decision making problems in two different ways: syntactic and semantic [9].

From the syntactic point of view, logic uses formulas and propositions to represent judgments from decision makers. For

example, the judgments of a set of decision makers among a set of alternatives can be represented by the propositions of a

logic system, e.g., p1, p2. Such as, p1 means that alternative 1 performs well in some specified property. The composite

propositions, which are composed by the primitive propositions p1, p2, etc. with logical connectives (not), ⋀ (and), ⋁ (or),

→ (if-then) and ↔ (if and only if), can be used for modelling more complex judgments. Then different logical reasoning

methods, such as fuzzy Modus Ponens (MP) rule and fuzzy Compositional Rule of Inference (CRI) [46], are applied to

reach the collective evaluation. From the semantic side, the truth-value field of logic system, such as {0, 1} for classical

logic, or [0, 1] for fuzzy logic, is used for modelling the set of evaluations on the alternatives. Take Fig. 1.1 as a truth-

VTrue

MTrue

True

(

A P ML

)True

ATrue

PTrue

MLTrue

(

A P ML

)True

STrue

SFalse

VFalse

MFalse

False

AFalse

PFalse

MLFalse

(

A P ML

)False

(

A P ML

)False

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value field example, the truth-value of p1 is ATrue means that the judgment of the decision maker on the first alternative is

approximately true. This kind of truth-value will change accordingly along with the syntactic inference process.

For the method mainly from the syntactic representation point of view, Das [11] developed a formal logic for reasoning

about qualitative preferences by representing preference through the binary relation R among propositional formulae which

represent the considered alternatives or actions. For example, R(a, b) is interpreted as the alternative a is preferred to b.

While most of the methods for modelling decision making problems in logic framework combine syntactic and semantic

parts together. Among them, Benferhat et al. [5] proposed some reasoning methods with partial information by using

extended possibilistic distribution in the framework of possibilistic logic. They [3, 4] also extended the possibilistic logic

by defining new combination rules to aggregate multiple-source information, which provides a coherent way to represent

and reason uncertain information from different sources.

By combining lattice and implication algebra, Xu et al. [44] established a logical algebraic structure, lattice implication

algebra (LIA), which is not only an efficient algebraic representation structure, but also a bridge to link with the

corresponding logic systems, lattice-valued logic. Inspired by the idea of Hedge algebra and by analysing the underlying

semantic meaning of linguistic hedges, based on the extensive study of LIA and lattice-valued logic, they further

constructed linguistic truth-valued LIAs [26, 40, 43] for modelling linguistic information, and discussed the linguistic

truth-valued logic system [24, 27, 41] and the corresponding approximate reasoning and automated reasoning approaches

[10, 43]. Lu et al. [28] proposed a linguistic truth-valued logic based reasoning system with several temporal predicates for

modelling and reasoning information under dynamic qualitative situations, and applied this reasoning mechanism to design

an intelligent environment, e.g., Smart Home. Liu et al. [26] laid some basic views on lattice ordered linguistic-valued

decision making, and constructed some lattice structures for modelling the linguistic information involved in decision

making process. These works are the bases of the linguistic multi-criteria decision making approach proposed in this paper.

Inspired by the idea of providing a strict logical foundation for decision making under qualitative and uncertain

environment, this paper proposes a logical reasoning framework, based on a lattice ordered linguistic truth-valued logic,

for linguistic multi-criteria decision making problems. In this framework, an algebraic structure is constructed for

representing the linguistic information involved in multi-criteria decision making problems, and a logic-based approximate

reasoning method is developed to reach the decision making result based on provided information. This logical reasoning

based decision making approach is capable of handling both totally ordered and partially ordered linguistic information

directly without numerical approximation, and has a non-classical logic as its formal theoretical foundation.

The rest of this paper is structured as follows. Section 2 constructs an algebraic structure for representing the linguistic

information involved in multi-criteria decision making problems. Then, a logic based approximate reasoning approach is

proposed in Section 3 for linguistic multi-criteria decision making problems. An example is given in Section 4 to illustrate

the proposed method. Section 5 comes to the concluding remarks.

2. Linguistic Truth-Valued LIA for Representing Linguistic Information

Generally, the first stage of decision making under qualitative environment is to find suitable structure for representing the

linguistic information involved, which is known as information representation. Then the suitable aggregation algorithm or

reasoning mechanism needs to be chosen to aggregate the alternatives according to the provided linguistic information.

The next step is to decide the “best” alternative, which normally consists of two phases: (a) The aggregation phase for

combining linguistic evaluations to reach a collective preference value for each alternative, and (b) the exploitation phase

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that establishes a rank ordering among the alternatives according to their collective preference values and then chooses the

most appropriate one [15, 38]. This section constructs an algebraic structure, linguistic truth-valued LIA, for representing

the linguistic information involved in multi-criteria decision making problems.

LIA [44] is an important kind of logical algebra, which takes both the advantage of lattice as an efficient algebraic

representation model, and additional operations, e.g., “implication,” to link with the corresponding logic systems, lattice-

valued logic. Some preliminary knowledge about LIA is provided as follows.

Definition 2.1 Let (L, , , O, I ) be a bounded lattice, where O and I are the smallest element and greatest element of

L respectively, with an order-reversing involution “ ”, : LLL be a mapping. (L, , , , ) is a lattice implication

algebra (LIA) if it satisfies the following conditions for any x, y, zL:

(I1) x(yz)=y(xz) (exchange property)

(I2 ) xx=I (identity)

(I3) xy=yx (contraposition)

(I4) xy=yx=I implies x=y

(I5) (xy)y=(yx)x

(l1) (xy)z=(xz)(yz)

(l2) (xy)z=(xz)(yz).

Example 2.1 (Łukasiewicz implication algebra on a finite chain). Let L be a finite chain, L={ai | 1in} and

O=a1<a2< ... <an=I, define for any ai, ajL,

aiaj=amax(i,j), ai aj=amin(i ,j), (ai)=an-i+1, aiaj=amin(n-i+j,n).

Then (L,

,

, ',

) is an LIA, called as Łukasiewicz chain and denoted by Ln.

Some properties of these operations on LIA which will be used in the proposed linguistic decision making method are

given as follows, and the readers may refer to [43, 44] for details.

Proposition 2.1 Let (L, ,

, , ) be an LIA. For any x, y, z

L,

(1) xy=(xy) y, xy=(xy);

(2) xO=x, Ox=I, xI=I, Ix=x; (extreme properties)

(3) xy=O iff x=I and y=O;

(4) xy(yz) (xz); (transitivity)

(5) xy x y;

(6) If xy, then xz yz;

(7) If yz, then xy xz; (order preserving)

(8) xy if and only if xy=I.

LIA is a general type of algebraic structure where all LIAs constitute a proper class [43, 44]. In order to model the

linguistic terms involved in the decision making and evaluation process through symbolic way, [26, 27, 40] constructed a

special kind of LIA, called linguistic truth-valued LIA, and denoted as L-LIA.

The generation of L-LIA takes the similar idea with Hedge algebra, i.e., the set of linguistic terms involved in decision

making problems constitute a partially ordered set whose elements are ordered according to their intuitive semantic

meanings and also can be regarded as an algebraically generated set by applying a set of linguistic modifiers, e.g., “less,”

“quite,” “more,” to the prime terms (generators) such as “dissatisfied and satisfied,” “high and low” as illustrated in Fig.

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2.1. We choose to use “dissatisfied and satisfied” as the prime terms in this paper because the developed approach is

mainly about linguistic decision making problems.

Fig. 2.1. A lattice structure of linguistic information

I=more satisfied, b= satisfied, c=less satisfied, a=less dissatisfied, d= dissatisfied, O=more dissatisfied.

Following the above remarks, construction of an appropriate set of linguistic values for an application can be carried

out step by step. The set of linguistic modifiers usually consists of two kinds of modifiers, e.g., H+={highly, very, more},

H-={approximately, possibly, more or less, slightly}, where H+ consists of modifiers which strengthen the meanings of

“satisfied” and the modifiers in H- weaken it. Put H=H+H-. H+, H- can be ordered by the degree of strengthening or

weakening. For example, one may assume that very>more, slightly>approximately, possibly, and more or less, and

“approximately,” “possibly,” “more or less” are incomparable.

Definition 2.2 Let a and b be two linguistic modifiers. We say that ab if and only if a(satisfied)b(satisfied) in

natural language.

Applying the linguistic modifiers of H to the primary terms, “satisfied” and “dissatisfied”, we obtain a partially ordered

set or lattice. We add four special elements I, E, F, O as “absolutely satisfied,” “exactly satisfied,” “exactly dissatisfied,”

and “absolutely dissatisfied” to the obtained set so that they have a natural ordering relationship with linguistic terms. The

set of linguistic terms obtained by the above procedure is a lattice with a boundary. One can further define , ,

implication “” and complement operations “” on this lattice according to the product LIA structure defined as follows.

Definition 2.3 Let Lm, Ln be two LIAs and Lm={a1,…, am}: a1,…am, Ln={b1,…, bn}: b1,…bn, ai aj = am(m-i+j),

ai= ai a1, bk bl = bn(n-k+l), bk= bk b1. Define the product of Lm and Ln as follows: LmLn ={(a, b) | aLm, b Ln}.

The operations on LmLn are defined respectively as follows:

(ai, bj) =(ai, bj), (ai, bk) (aj, bl) = (ai aj, bk bl). (1)

Then (Lmn, , , , , (a1, b1), (am, bn)) is a LIA, denoted by Lmn.

Based on Definition 2.3, L-LIA, is generated by the product of two Łukasiewicz chains Ls2=LsL2, where L2={b1, b2}

(b1<b2) is the set of the prime linguistic terms, e.g., {b1, b2}={Dissatisfied (Ds for short), Satisfied (Sa)}, and Ls={a1, a2, …,

as} (a1<a2<…<as) is the set of the linguistic modifiers. According to the natural requirements, the number of linguistic

modifiers is usually set to odd number, 3, 5, 7 or 9, and the structure of L-LIA corresponding to 3 modifiers can be

illustrated as Fig. 2.1.

In the following, we use L-LIA with 9 modifiers for modelling the linguistic terms under consideration, which can

meet the general requirements of real decision making and evaluation problems [40]. The linguistic modifiers of L-LIA

with 9 modifiers are L9={Slightly (Sl for short), Somewhat (So), Rather (Ra), Almost (Al), Exactly (Ex), Quite (Qu), Very

O

b

a

d

I

c

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(Ve), Highly (Hi), Absolutely (Ab)} with the semantic ordering relationship Sl < So < Ra < Al < Ex < Qu < Ve < Hi < Ab,

and the prime linguistic terms are L2={Dissatisfied (Ds for short), Satisfied (Sa)}with Ds < Sa.

Let a mapping f: L-LIA → L92 be defined as follows (L92 is defined as in Definition 2.3).

f(Ab, Sa)=(a9, b2), f(Hi, Sa)=(a8, b2), f(Ve, Sa)=(a7, b2), f(Qu, Sa)=(a6, b2), f(Ex, Sa)=(a5, b2), f(Al, Sa)=(a4, b2), f(Ra,

Sa)=(a3, b2), f(So, Sa)=(a2, b2), f(Sl, Sa)=(a1, b2), f(Sl, Ds)=(a9, b1), f(So, Ds)=(a8, b1), f(Ra, Ds)=(a7, b1), f(Al, Ds)=(a6, b1),

f(Ex, Ds)=(a5, b1), f(Qu, Ds)=(a4, b1), f(Ve, Ds)=(a3, b1), f(Hi, Ds)=(a2, b1), f(Ab, Ds)=(a1, b1).

Then f is a bijection. Denote its inverse mapping as f-1. In addition, if for any x, yL,

xy= f -1( f (x) f (y)), xy= f -1( f (x) f (y)),

xy= f -1( f (x) f (y)), x=f -1(( f (x))),

then it is easy to prove that (L-LIA, , , , ) is a LIA, and L-LIA is isomorphic to L92, i.e., f is an isomorphic mapping

from L-LIA onto L92, so we still use L92 for denoting the L-LIA with 9 modifiers in the subsequent sections.

The Hasse diagram of L-LIA L92 is depicted in Fig. 2.2, where I=(a9, b2), A=(a8, b2), B=(a7, b2), C=(a6, b2), D=( a5, b2),

E=(a4, b2), F=(a3, b2), G=(a2, b2), H=(a1, b2), R=(a9, b1), J=(a8, b1), K=(a7, b1), S=(a6, b1), M=(a5, b1), N=(a4, b1), P=(a3, b1),

Q=(a2, b1), O=(a1, b1), which shows the ordinal relation between these linguistic terms. Note that, some linguistic terms are

incomparable according to the Hasse diagram in Fig. 2.2. For example, A // R (// means incomparable) according to the

Hasse diagram in Fig. 2.2, so f(Hi, Sa)=(a8, b2)=A and f(Sl, Ds)=(a9, b1)=R are incomparable, i.e., Highly Satisfied //

Slightly Dissatisfied, which intuitively is true in terms of their meanings in natural language.The basic operations of L92

can be defined easily based on Definition 2.3.

Fig. 2.2. Hasse diagram of L-LIA L92.

The process of constructing L-LIA is similar to that for Hedge algebra by applying the set of linguistic modifiers to the

prime terms, but L-LIA is more powerful in logical connection due to its additional operations. In other words, the

constructed L-LIA not only is a practical linguistic algebraic structure for modelling linguistic terms, but also provides

some logical operations between the linguistic terms which can bridge the gap between real problem and logical system.

Accordingly, a strict theoretical foundation as well as a practical and effective approximate reasoning mechanism can be

built for linguistic decision making and other intelligent information processing problems.

R

O

A

J

K

I

B

C

D

E

F

G

H

S

M

N

P

Q

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There are some additional operations, and , in L-LIA given in the following definition, which play an important

role in the corresponding linguistic decision making approach. Intuitively, operation takes the similar role with the

arithmetic operation and algebraic operation , and has the similar meaning with + and respectively.

Definition 2.4 For any ai, ak Ls, bj, bm L2,

)),(),((),(),(

mkjimkji babababa

(2)

),(),(),(),( mkjimkji babababa

(3)

Some properties of the two operations are given in Proposition 2.2, which will be used in the reasoning process for the

decision making approach.

Proposition 2.2 For any (ai, bj), (ak, bm), (al, bn)

L-LIA, the following statements hold.

(1) (ai, bj) (ak, bm) = (ai ak, bj bm), (ai, bj) (ak, bm) = (ai ak, bj bm);

(2) (ai, bj) (ak, bm) = (ak, bm) (ai, bj), (ai, bj) (ak, bm) = (ak, bm) (ai, bj);

(3) ((ai, bj) (ak, bm)) (al, bn) = (ak, bm) ((ai, bj) (al, bn));

(4) ((ai, bj) (ak, bm)) (al, bn) = (ak, bm) ((ai, bj) (al, bn));

(5) ((ai, bj) (ak, bm))= (ai, bj) (ak, bm), ((ai, bj) (ak, bm))= (ak, bm) (ai, bj);

(6) (ai, bj) (ak, bm) (ai, bj) (ak, bm), (ai, bj) (ak, bm) (ai, bj) (ak, bm);

(7) (a1, b1) (ai, bj) = (a1, b1), (aN, bM) (ai, bj) = (ai, bj), (ai, bj) (ai, bj) = (a1, b1);

(8) (a1, b1) (ai, bj) = (ai, bj), (aN, bM) (ai, bj) = (aN, bM), (ai, bj) (ai, bj) = (aN, bM);

(9) (ai, bj) ((ai, bj) (ak, bm)) = (ai, bj) (ak, bm) = ((ai, bj) (ak, bm)) (ak, bm);

(10) ((ai, bj) (ak, bm)) (ai, bj) = (ai, bj) (ak, bm);

where (a1, b1), (aN, bM) are the smallest and greatest element of L-LIA respectively.

The linguistic multi-criteria decision making problem based on the linguistic algebraic structure, L92, can now be

described as follows. Suppose that there are a set of alternatives A={x1, x2, …, xn}, which are under evaluation according to

several criteria or attributes: F={f1, f2, …, fm} associated with a set of weights W={ω1, ω2, …, ωm}. The evaluation terms

from the experts to evaluate the alternatives are linguistic terms taken from the set E=L92, and the weights W={ω1, ω2, …,

ωm} may also be linguistic terms in the set of modifiers L9, which can be interpreted as the linguistic importance degrees of

different criteria. We denote the evaluation results obtained from the experts as R={r1, r2, …, rm},

92

( , )

i i i

r E L

.

The linguistic multi-criteria decision making problem is then to find a suitable aggregation or reasoning mechanism for

reaching a reasonable comprehensive evaluation based on provided evaluations from the experts. For example, when

evaluating several cars according to four criteria (attributes): safety (f1), price (f2), comfort (f3) and fuel economy (f4), one

may express his opinion about Passat as “Its safety is very satisfied, price is somewhat satisfied, comfort is quite satisfied,

and fuel economy is rather dissatisfied”, which can be denoted as r1=(a7, b2), r2=(a2, b2), r3=(a6, b2), r4=(a7, b1). Suppose

that the weights of four criteria are highly, very, quite and quite, which denotes the importance degrees of the four criteria

contributing to the comprehensive evaluation about the car. The logical reasoning based approach is then proposed in the

following section to infer a reasonable comprehensive evaluation based on the provided evaluations.

3. Logical Reasoning based Linguistic Decision Making Approach

After the representation of linguistic information involved in decision making, the next step is to apply aggregation

algorithm or inference mechanism to obtain the final decision result [15]. Current linguistic decision making methods

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usually transform the linguistic terms into fuzzy sets or some other numerical models, and then different aggregation

operators are applied to obtain a final ordering or evaluation result. Inspired by the idea that the underlying mental process

of human decision making is reasoning and logic is the foundation and standard for justifying the soundness and

consistency of the methods [7], we propose in this section a logical reasoning based linguistic multi-criteria decision

making approach.

This linguistic multi-criteria decision making approach is based on the linguistic-valued approximate reasoning in a

linguistic truth-valued logic system which is a lattice-valued logic system with truth-value in L-LIA. It will be based on

the direct reasoning on linguistic variables, which offers the advantage of not requiring the linguistic approximation step.

Furthermore, it does not require the definition of the membership functions associated with the linguistic terms as in

conventional fuzzy set based methods [45]; this is usually a burdensome step requiring the investigation of human factors,

semantics of the linguistic terms, subjective beliefs etc. Instead, we use a finite set of linguistic terms with a rich lattice

ordering algebraic structure. This proposed procedure has another advantage, i.e., the handling of incomparable linguistic

terms. In general, we conjecture that the domain of a linguistic-valued algebra (LA) can be represented as a lattice. Thus, a

linguistic-valued logic is a logic in which the truth degree of an assertion is a linguistic value in LA. For example, the idea

is meaningful because in daily life, when being asked to assess the degree of a person being Young, it is usually easier to

give a verbal answer, e.g., “Very Young” or “Quite Young,” rather than to give a numerical answer, such as, 0.5 or 0.7. A

key insight behind the linguistic-valued logic scheme is that we can use natural language to express a logic in which the

truth values of propositions are expressed as linguistic values in natural language terms such as “satisfied,” “very satisfied,”

“less satisfied,” “very dissatisfied,” “dissatisfied,” etc., instead of a numerical scale. It is expected that such an approach

could reduce numerical approximation errors that could be caused in estimating membership functions and also will treat

vague information in its true format.

3.1 Some concepts in lattice-valued logic based on LIA

Some related concepts are given in this subsection, and the readers may refer to [27, 43, 44] for more details. In the

following, we always assume that (L, , , , , O, I ) is an LIA, in short L.

Definition 3.1 Let X be a set of propositional variables, X={p, q, r, …}, T=L{, } be a type with ar(

) =1,

ar()=2 and ar(a)=0 for every aL. The propositional algebra of the lattice-valued propositional calculus on the set X of

propositional variables is the free T algebra on X and is denoted by LP(X).

Proposition 3.1 LP(X) is the minimal set Y which satisfies:

(1) XLY.

(2) If p, qY, then p, pqY.

Note that L and LP(X) are the algebras with the same type T, where T=L{, }.

Definition 3.2 A valuation of LP(X) is a propositional algebra T-homomorphism v : LP(X)L.

If v is a valuation of LP(X), we have v(

)=

for every

L.

In what follows, for convenience, pLP(X), stands for p is a lattice-valued propositional logical formula of LP(X).

Definition 3.3 Let pLP(X),

L. If there exists a valuation v of LP(X) such that v(p)

, p is satisfiable by a truth-

10

value level

, in short,

-satisfiable; if v(p)

for every valuation v of LP(X), p is valid by the truth-value level

, in short,

-valid. If

=I, then p is valid simply.

As stated before, logic uses formulas and propositions to represent judgments or evaluations from decision makers

among a set of alternatives in decision making problems. The composite propositions, which are composed by the

primitive propositions with logical connectives (not), ⋀ (and), ⋁ (or), and → (if-then), can be used for modelling more

complex judgments. The truth-value level of the logical formula is used correspondingly for denoting the truth-value level

of the judgment. Take the car evaluation problem mentioned in Section 2 as an example, we can use p to denote the

evaluation “Passat is comfortable” and the truth-value level of p is

=quite true. It means that “Passat is quite comfortable”

by combining these two parts together.

3.2 Linguistic decision making approach based on logical reasoning

The weights associated with different criteria are actually the consensus knowledge reached by a group or panel, which

reflect the importance degrees of the criteria contributing to the comprehensive evaluation result. In the present work, we

interpret the combination process of the individual evaluation results and the weights as the inference engine based on the

sound and complete logic-based approximate reasoning scheme [27, 43, 44]. Therefore, we have m inference rules

corresponding to m criteria as follows, which are then used to infer the decision result.

If the evaluation result corresponding to criterion fi is ri=(αi, βi), then the result ci taking into account the weight ωi with

respect to criterion fi is (ωiαi, βi), i=1, …, m, where ri=(αi, βi)L92, ωiL9.

The above rules can be denoted in a simpler form as follows.

Ri: If fi is ri=(αi, βi), then ci is (ωiαi, βi), i=1, …, m.

Suppose now that there are currently some evaluations H={h1, h2, …, hm} (hi=(pi, qi)L92) from the experts, without

regarding the weights, with respect to the criteria F={f1, f2, …, fm}, which are then fed to the above pre-established rules to

infer the comprehensive evaluation result.

The process of obtaining the evaluation ei=(di, si) corresponding to the ith criterion is as follows, which is based on a

kind of approximate reasoning in the framework of a non-classical logic with truth values in a linguistic LIA [27, 40].

)),(),((),()(),( iiiiiiiiiiii qpcrhsde

. (4)

The logical reasoning process stated in Eq. (4) is actually based on the following approximate reasoning model based

on lattice-valued logic LP(X).

),(

),(

),(

21

2

1

g

f

gf

,

where f, gLP(X) are propositional logical formulas to represent the statements about the evaluations, τ1, τ2 L-LIA are

truth-values of the corresponding propositional formulas, and τ1= rici, and τ2= hi here. We can also say that Eq. (4) is the

simplified form, or the semantic part of the logical reasoning process.

The reasoning process in Eq. (4) can be further elaborated, according to Proposition 2.2, as:

11

))),(((),( 2

bqpsde iiiiiiii

))),((( iiiii qp

( ( ), )

i i i i

pq

,

where the operations and are not specified to be defined in L92 or L9, but can be distinguished according to their

positions easily.

The comprehensive evaluation result can then be obtained according to the real logical relation between the m

inference rules, or the m criteria. If the logical relation between the rules is conjunction (and), then the comprehensive

evaluation is

i

m

ie

1

or

i

m

ie

1

. If the logical relation is disjunction (or), then the comprehensive evaluation will be

i

m

ie

1

or

i

m

ie

1

.

The overall procedure of the linguistic multi-criteria decision making approach based on logical reasoning is

summarized as follows.

Step 1. Construct the algebraic structure for modelling the linguistic terms involved in the decision making problem,

which is isomorphic to the product algebra of two Łukasiewicz chains LsL2, and we will usually use L-LIA L92 as stated

in Section 2.

Step 2. Generate the rule base according to the prior domain knowledge, mainly about the weights associated with

different criteria.

Step 3. Ask the decision makers to provide evaluations about the alternatives, and then infer the m evaluations by

taking account into the weights corresponding to m criteria based on the logical reasoning process stated in Section 3.2.

Step 4. Elaborate the comprehensive evaluation by aggregating the m evaluations obtained in Step 3 according to the

logical relationship between the m inference rules, and then rank the alternatives according to their comprehensive

evaluations if necessary.

3.3 Some properties of the logical reasoning based linguistic decision making approach

There are some properties of the proposed logical reasoning based linguistic multi-criteria decision making approach that

can be obtained easily from the corresponding L-LIA and the logic system based on it [27, 41].

Property 3.1 MP rule holds for the logical reasoning based decision making method, that is:

For rule Ri: If fi is ri=(αi, βi), then ci is (ωiαi, βi), i=1, …, m, if the new evaluations H={h1, h2, …, hm} (hi=(pi, qi))

corresponding to the criteria F={f1, f2, …, fm} are exactly R={r1, r2, …, rm}, then the inferred evaluations are

iiii ccrr )(

.

Property 3.2 If there are two sets of new evaluations H(j)={h1(j), h2(j), …, hm(j)} (hi(j)=(pi(j), qi(j))), j=1, 2,

corresponding to the criteria F={f1, f2, …, fm}, then

(1) (hi(1) hi(2)) (ri ci) = (hi(1) (ri ci)) (hi(2) (ri ci));

(2) (hi(1) hi(2)) (ri ci) = (hi(1) (ri ci)) (hi(2) (ri ci)), i=1, …, m.

12

According to these two equations, it can be concluded that the logical reasoning based decision making method is

monotonously increasing, i.e., if there are two new evaluations which satisfy (pi, qi)

(ti, xi), then the corresponding

evaluations inferred from the rule base satisfy

)),(),((),()),(),((),( iiiiiiiiiiii xtqp

.

Property 3.2 shows that the proposed logical reasoning based linguistic decision making approach can preserve the

ordering relation of the original evaluations. The following property can be obtained easily from the Property 3.2.

Property 3.3 If the new evaluations H={h1, h2, …, hm} corresponding to the criteria F={f1, f2, …, fm} are H={(a1, b1),

(a1, b1), …, (a1, b1)}, then the inferred evaluations by taking into account the weights are

),()( 11 bacrhe iiii

,

i=1, …, m.

It can be inferred from Property 3.3 that the final evaluation is still the worst, (a1, b1), if all the original evaluations are

the worst ones. Similarly, the following property shows that the final evaluation is decided by the corresponding weight if

the original evaluations are all the best ones, (a9, b2).

Property 3.4 If the new evaluations H={h1, h2, …, hm} corresponding to the criteria F={f1, f2, …, fm} are H={(a9, b2),

(a9, b2), …, (a9, b2)}, then the inferred evaluations by taking into account the weights are

),)(( 2

be iii

, i=1, …, m.

In fact,

),)(()),(()()( 22 bbcrcrhe iiiiiiiiiii

.

Property 3.5 If the new evaluations H={h1, h2, …, hm} corresponding to the criteria F={f1, f2, …, fm}, then the inferred

evaluation results

)()( iiiiiii crhcrhe

.

We can see from this property that the proposed linguistic decision making approach takes the pessimistic viewpoint.

There are also some other properties of the proposed linguistic decision making approach which may be useful to show

different aspects of the proposed logical reasoning process.

Property 3.6 If the new evaluations H={h1, h2, …, hm} corresponding to the criteria F={f1, f2, …, fm}, then for any

ziL-LIA, the inferred evaluations

iiiii zcrhe )(

iff

iiii zcrh )(

, i=1, …, m.

Property 3.7 If the new evaluations H={h1, h2, …, hm} corresponding to the criteria F={f1, f2, …, fm}, then

)())(( iiiiiii crhcrhh

, i=1, …, m.

Property 3.8 If the new evaluations H={h1, h2, …, hm} corresponding to the criteria F={f1, f2, …, fm}, then

)())((

iiiiii crhcrh

,

and inversely,

))(()(

iiiiii crhcrh

, i=1, …, m.

Property 3.9 If the new evaluations H={h1, h2, …, hm} corresponding to the criteria F={f1, f2, …, fm}, then

)()( iiiiii chrcrh

,

))(())(()( iiiiiiiii crhcrhcrh

, i=1, …, m.

13

4. Illustrative Example

In this section, we provide an example of car evaluation, which is adapted from [8], to show how the proposed logical

reasoning based linguistic decision making approach works.

Suppose that there are three kinds of cars: BMW (x1), Hyundai (x2) and Passat (x3) are under evaluation according to

four criteria (attributes): safety (f1), price (f2), comfort (f3) and fuel economy (f4). The linguistic judgments for evaluating

the cars are chosen from L-LIA L92 as in Section 2, and the weights associated with the four criteria are supposed to be

a9=absolutely, a7=very, a4=almost, and a6=quite. The judgment for each kind of car with respect to different criteria is

given in Table 4.1, by taking a simple normalization of these naturally expressed evaluations. For example, the value of

f3(x3) in Table 4.1 expresses the evaluation “the comfort of Passat is quite satisfied”, which naturally means “Passat is

quite comfortable”.

Table 4.1. Evaluations about cars

f1

f2

f3

f4

x1

(a7, b2)

(a5, b1)

(a8, b2)

(a2, b2)

x2

(a8, b1)

(a6, b2)

(a2, b2)

(a5, b2)

x3

(a7, b2)

(a2, b2)

(a6, b2)

(a7, b1)

The rule base comes from our daily experience: “If the car is highly safe, rather cheap, very comfortable and with quite

satisfied fuel economy, then the car is highly satisfied”, which can be expressed as:

R1: If f1 is (a8, b2), then c1 is (a9a8, b2),

R2: If f2 is (a3, b2), then c2 is (a7a3, b2),

R3: If f3 is (a7, b2), then c3 is (a4a7, b2),

R4: If f4 is (a6, b2), then c4 is (a6a6, b2).

Based on Eq. (4), we can obtain the evaluations, as shown in Table 4.2, after taking the weights into account. We only

take the top left one as an example to show the reasoning process, and the others can be done similarly.

11 1 1 7 2 8 2 9 8 2

( ) ( , ) (( , ) ( , ))e f x a b a b a a b

7 8 9 2 7 9 2 7 2

( ( ), ) ( , ) ( , )a a a b a a b a b

.

Table 4.2. Inferred evaluations

f1

f2

f3

f4

x1

(a7, b2)

(a5, b1)

(a4, b2)

(a2, b2)

x2

(a8, b1)

(a6, b2)

(a2, b2)

(a5, b2)

x3

(a7, b2)

(a2, b2)

(a4, b2)

(a6, b1)

The comprehensive evaluations with respect to different cars are then obtained, with the assumption that the logical

relation between the criteria is disjunction (or), as e1 is (a7, b2), e2 is (a8, b2), e3 is (a7, b2), which can be interpreted in

natural language as “BMW is very satisfied, Hyundai is highly satisfied and Passat is very satisfied”. Because the results

follows the logic-based approximate reasoning process, not simple aggregation without theoretical foundation, the results

are more reliable and the results obtained are intuitively reasonable according to our common sense and perception.

5. Conclusions

Decision making under qualitative and uncertain environment is pervasive in reality. This paper proposed a framework for

linguistic multi-criteria decision making problems based on logical reasoning, where an algebraic structure is constructed

14

for modelling the linguistic information involved in multi-criteria decision making problems, and a logic-based

approximate reasoning method is developed to infer the comprehensive decision making result. This method is able to

handle the linguistic information directly without underlying numerical approximation which is necessary for some other

methods, and have a non-classical logic as the formal foundation for decision making process. Some properties of the

proposed logical reasoning linguistic multi-criteria decision making approach are discussed to show its intuitive rationality.

A car evaluation example was given to illustrate how the proposed logical reasoning based decision making approach

works.

This work supports linguistic decision making for CWW. Most of the computational approaches in the literature for

CWW assume that the set of linguistic terms can be totally ordered, while they may not be strictly linearly ordered due to

the real complex and dynamic environment. The present work can directly focus on CWW based on lattice-ordered

linguistic terms, i.e., modeling the term-domain of linguistic variables by a lattice structure, and implementing CWW

based on lattice-valued logic and reasoning, i.e., reasoning with words. Future work will focus on how to make the

proposed approach more applicable. Furthermore, the proposed approach can be extended to group decision making

problems under qualitative and uncertain environment.

Acknowledgments

The authors are grateful to the editors and anonymous referees for their valuable comments and suggestions that led to an

improved version of this paper. This contribution has been partially supported by the VCRS scholarship from University

of Ulster and the research project TIN2012-31263.

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