An approach to evaluate alternatives in multi attribute decision making problems based on linguistic many valued logic

An Approach to Evaluate Alternatives in Multi Attribute Decision Making Problems Based on Linguistic Many Valued Logic An Approach to Evaluate Alternatives in Multi Attribute Decision Making Problems[.]

An Approach to Evaluate Alternatives in

Multi-Attribute Decision Making Problems

based on Linguistic Many-Valued Logic

Phuong Le Anh, Hoai Nhan Tran

Department of Computer Science, Hue University of Education, Hue University,

Hue City, Vietnam {leanhphuong, tranhoainhan}@dhsphue.edu.vn

Dinh Khang Tran

Department of Information Systems Hanoi University of Science and Technology,

Hanoi, Vietnam khangtd@soict.hust.edu.vn

Abstract—Humans are often faced with decisions, that is, have

to choose between different alternatives every day Many of these

decision problems are under uncertain environments with vague

and imprecise information Thus, linguistic decision making

problem is important research topic Besides, decision making

is widely applied in fields such as society, economy, medicine,

management, and military affairs, etc In the framework of the

linguistic many-valued logic, both comparable and incomparable

truth-values can be expressed This paper further studies the

applicability of linguistic many-valued logic for dealing with

the decision making problem and proposes a linguistic

many-valued reasoning approach for the decision making model This

approach can also address the reasoning of decision making

directly on linguistic truth values

Index Terms—Linguistic decision making, Hedge algebra,

lin-guistic many-valued logic

I INTRODUCTION Decision making is inherent to humans, as people have

to choose between different alternatives every day [1] Many

of these decision problems are under uncertain environments

with vague and imprecise information [2, 3] The

complex-ity and uncertainty of the objective thing and the opaccomplex-ity

of human thought lead to decision making with linguistic

information [3] For example, when evaluating the “comfort”

or “design” of a car, people usually used linguistic labels like

“good”, “fair” and “poor”

There are decision situations where information cannot be

accurately assessed in a quantitative form but can be in a

qualitative form, and therefore the use of a linguistic approach

is necessary [4] The linguistic approach is an approximation

technique for expressing qualitative aspects such as linguistic

values by linguistic variables [5], i.e variables whose values

are not numbers but words or sentences in natural language

Therefore, decision making problem under linguistic

informa-tion (linguistic decision making problem) is an interesting and

important research topic that has been attracting more and

more attention in recent years

There are many methods for dealing with linguistic

infor-mation, such as the methods based on making operations on

the fuzzy numbers that support the semantics of the linguistic

labels, the methods based on making computations on the indexes of the linguistic terms, the methods based on fuzzy linguistic representation model, the methods which compute with words directly The methods which compute with words directly can not only avoid the loss of any linguistic informa-tion, but also simple and very convenient in calculation [3] The decision making process can basically be understood

as a process of reasoning from provided information or knowledge base to some conclusion, and decision making under qualitative and uncertain is an approximate reasoning process [6]

The theory of hedge algebras was introduced by Nguyen and Wechler in [7] which is an algebraic approach to linguistic hedges in Zadeh’s fuzzy logic The types of hedge algebras and methods of computing with words have been developed

in [8–12] Le and Tran have proposed L-Mono-HA as a form

of linear hedge algebra and limited to the length of the hedge string [12] In [12], the authors proposed linguistic many-valued logic by extending many-many-valued logic and choosing Lukasiewicz operators ∨, ∧, ⊗, ⊕, ¬, → for linguistic-valued domain, which represent linguistic information

These theories are well suited for linguistic reasoning and

we will continue to use them for this work We will study how

to apply “if then” inference rules with linguistic modifiers for evaluating of alternatives, which is an important step in linguistic decision making problems Then we propose a novel method for dealing linguistic decision making problem In the proposed method, the string of hedges with more than one hedge can be address

The rest of this paper is structured as follows Section II and Section III introduce the linguistic many-valued logic and the knowledge base and linguistic reasoning, respectively An approach based on linguistic many-valued logic is proposed in Section IV for linguistic decision making problems Section V comes to the concluding

II LINGUISTICMANY-VALUEDLOGIC

A Hedge and its properties

Let X be the set of linguistic values of a language variable,

and each element in X is of the form h1h2 hnc (n ≥ 0),

with hiis called hedge, h1h2 hnis called hedge string, and

c is called generator, respectively Thus, we consider that X is

generated from a set of generators (denoted G) using hedges

(denoted H) as unary operations

There exists a natural ordering among these values, with

a ≤ b meaning that meaning that a indicates a degree less

than or equal to b, where a, b ∈ X, a < b iff a ≤ b and a ̸= b

The relation ≤ is called the semantically ordering relation on

X

There are natural semantic properties of linguistic terms and

hedges Given two hedges h, k, it can be said that: [9, 13]

i) h and k are converse if x ∈ X: hx > x iff kx < x;

ii) h and k are compatible if x ∈ X: hx > x iff kx > x;

iii) h ≥ k, if x ∈ X: (hx ≤ kx ≤ x) or (hx ≥ kx ≥ x);

iv) h > k if h ≥ k and h ̸= k;

v) h is positive w.r.t k if x ∈ X: (hkx < kx < x) or

(hkx > kx > x);

vi) h is negative w.r.t k if x ∈ X: (kx < hkx < x) or

(kx > hkx > x)

B Linear symmetric hedge algebra

An abstract algebra (X, G, H, ≤), where X is a term

domain, G is a set of generators, H is a set of hedges, H

is decomposed in to H+ and H−, and ≤ is an semantically

ordering relation on X, is called a linear symmetric hedge

algebra (HA in short) if it satisfies the following

condi-tions [8, 9]:

i) For all h ∈ H+ and k ∈ H−, h and k are converse;

ii) The sets H+∪ {I} and H−∪ {I} are linearly ordered

with the least element I;

iii) For each pair h, k ∈ H, either h is positive or negative

w.r.t k;

iv) If h ̸= k and hx < kx then h′hx < k′kx, for all h, k,

h′, k′ ∈ H and x ∈ X;

v) If u ∈ H(v) and u < v (u > v) then u < hv (u > hv),

respectively, for any h ∈ H

For example, consider a HA (X, {T, F }, H, ≤), where H =

{very, more, probably, mol} (“mol” stands for “more-or-less”),

“T” is “true”, “F” is “false” H is decomposed into H+ =

{very, more} and H− = {probably, mol} In H+∪ {I} we

have very > more > I, whereas in H−∪ {I} we have mol >

probably> I

• “very” and “more” are positive w.r.t “very” and “more”,

negative w.r.t “probably” and “mol”;

• “probably” and “mol” are negative w.r.t “very” and

“more”, positive w.r.t “probably” and “mol”

C Monotonous hedge algebra

A linear symmetric HA (X, G, H, ≤) is called monotonic,

denoted Mono-HA, if each h ∈ H is positive w.r.t all k ∈

H+(H−), and negative w.r.t all h ∈ H−(H+) [9]

As defined, both sets H+∪ {I} and H−∪ {I} are linearly ordered, however, H = H+∪ H−∪ {I} is not For example, very∈ H+and mol ∈ H− are not comparable Let us extend the order relation on H+∪ {I} and H−∪ {I} to one on

H ∪ {I} as follows [9]

Given h, k ∈ H ∪ {I} iff i) h ∈ H+, k ∈ H−; or ii) h, k ∈ H+∪ {I} and h ≥ k; or iii) h, k ∈ H− ∪ {I} and h ≥ k, h >h k iff h ≥h k and

h ̸= k

The order relation >Hin H ∪ {I} is very >Hmore>H I >H

probably >H mol Then, in Mono-HA, hedges are “context-free”, i.e., a hedge modifies the meaning of a linguistic value independently of preceding hedges in the hedge chain [9, 13]

D Mono-HA with limited length of the hedge string Linguistic truth-valued domain

Mono-HA with a limited length of the hedge string is proposed in [12] to generate a linguistic truth-valued domain with finite hedge string

L-Mono-HA, L is a natural number, is a Mono-HA with standard presentation of all elements having the length not exceed L + 1

A linguistic truth-valued domain AX taken from a L-Mono-HA = (X; {c+, c−}; H; ≤) is defined as AX =

X ∪ {0, W, 1} of which 0, W , 1 are the smallest, neutral, and biggest elements respectively in AX

According to Nguyen and Wechsler [7]:

+ 0 < x < W < y < 1 for all x ∈ H(c−), and ∀y ∈ H(c+),

+ hW = W , h1 = 1, h0 = 0, + −W = W , −1 = 0, −0 = 1

Because AX is the linguistic truth-valued domain in fi-nite monotonous hedge algebra and linear order [10, 13]; thus, we can rewrite it as: AX = { vi, i = 1, , n|

v1= 0, vn= 1; vi ≤ vj⇔ i ≤ j, ∀i, j = 1, , n}

E Linguistic many-valued logic Many-valued logic is a generalization of Boolean logic It provides truth values that are intermediate between “True’’ and

“False”, n is the number of truth degrees in many-valued logic

In linguistic many-valued logic, the truth degree of proposition

is vi ∈ AX

Let Ln= (AX, ∨, ∧, ⊗, ⊕, ¬, →, 0, 1), where operators ∨,

∧, ⊗, ⊕, ¬, → are defined as follows, for all vi, vj∈ AX i) vi∨ vj = vmax(i,j);

ii) vi∧ vj = vmin(i,j); iii) vi⊗ vj= v1∨ vi+j−n; iv) vi⊕ vj= vn∧ vi+j; v) ¬vi= vn−i+1; vi) vi→ vj= vmin(n,n−i+j); Then Ln = (AX, ∨, ∧, ⊗, ⊕, ¬, →, 0, 1) (Ln in short) is an extension of many-valued logic by replacing truth domain in [0, 1] (0 and 1 denote for “False” and “True”, respectively) with the linguistic truth-valued domain AX Ln is called a linguistic many-valued logic

III KNOWLEDGEBASE ANDLINGUISTICREASONING

A Vague sentences and linguistic valuation

An assertion is a pair A = (p(x, u), δc), where u is a vague

concept, δ is a string of hedges, p(x, u) is a vague sentence,

δc is a linguistic truth value to valuate for the vague sentence

p(x, u) We can rewrite an assertion as a pair (P, t), where

P = p(x, u) and t = δc For example, the sentence “John is

faithful” in a certain context may be assigned to a truth degree

“very true” [14]

The composed vague sentences are formed from

elemen-tary ones by means of logical connectives such as “and”,

“or”, “if-then” and “not”, which are denoted

correspond-ingly by ∧, ∨, → and ¬, called conjunction,

disjunc-tion, implication and negation An example for composed

fuzzy sentences is “if a person is old then he or she can

not run quickly” and this sentence can be expressed by

(AGE(x, old) → ¬(RU N (x), quickly))

B Knowledge base

One knowledge base K is a finite set of assertions From the

given knowledge base K, we can deduce new assertions by

using on derived rules The deduction method is derived from

knowledge base K using the above rules to deduce the

con-clusion (P, v), we can write K ⊢ (P, v) Let C(K) denote the

set of all possible conclusions: C(K) = {(P, v) : K ⊢ (P, v)}

A knowledge base K is called consistent if , from K, we can

not deduce two assertions (P, v) and (¬P, v) [14]

C Linguistic reasoning

1) Inverse mapping of hedge: A mapping h−: AX → AX

is called inverse mapping of h if it meets the following

conditions [10, 13]:

i) h−(δhc) = δc of which c ∈ C, δ ∈ H∗;

ii) x ≤ y ⇒ h−(x) ≤ h−(y) of which x, y ∈ AX;

The inverse mapping of a hedge string is determined via the

inverse mapping of single hedges as follows:

(hkhk−1 h1)−δc = h−k h−1δc

In [8, 15], it is shown that inverse mapping of hedge always

exists and it is not unique

2) Hedge moving rules: Given h is hedge, δ is the hedge

strings, the hedge moving rules are set [7, 10, 12]:

R1) RT1: (p (x, hu) , δc)

(p (x, u) , δhc)

R2) GRT2: (p (x, u) , δc)

(p (x, hu) , h−(δc)) 3) Generalized modus ponens: Given δ, σ, and δ′ are the

hedge strings, in [8, 15, 16], the generalized modus ponens

(GMP) was proposed:

R3) GMP: (p(x,u)→q(y,v),δc),(p(x,u),σc)(q(y,v),δc⊗σc)

R4) EGMP: (p(x,u),δc)→(q(y,v),σc),(p(x,u),δ

′ c)

(Q,δ ′ c⊗(δc→σc))

EGMP is an extension of GMP;

R5) NGMP: (p(x,¬u)→q(y,v),vi ),(p(x,u),vj)

(q(y,v),v i ⊗¬v j )

R6) ENGMP: (p(x,¬u),vi )→(q(y,v),v j ),(¬q(x,u),v k )

(q(y,v),(vi→v j )⊗¬vk) , ENGMP is an extension of NGMP

4) Generalized modus ponens with linguistic modifiers: The rules of generalized Modus ponens with linguistic modi-fiers (GMPLM) have been introduced in [9, 16] Given α, β,

δ, σ, θ, ∂, α′, β′, δ′, θ′, and ∂′ are the hedge strings; get

α = h1h2 hk, symbol α−1 = hk h2h1, the rules are:

R7) GMPLM: (p(x,δu)→q(y,∂v),αc),(p(x,δ

′ u),α′c)

(q(y,∂v),αc⊗δ −1 (α ′ δ ′−1 c))

R8) EGMPLM: (p(x,δu),αc)→(q(y,∂v),βc),(p(x,δ

′ u),α′c)

(q(y,∂ ′ v),(αδ −1 c→β∂ −1 c)⊗(α ′ δ ′−1 c))

R9) NGMPLM: (p(x,¬(δu))→q(y,∂v),αc),(p(x,δ

′ u),α ′ c)

(q(y,∂v),αc⊗¬(δ −1 (α ′ δ ′−1 c)))

R10) ENGMPLM: (p(x,δu),αc)→(q(y,∂v),βc),(p(x,δ

′ u),α′c)

(q(y,∂ ′ v),(αδ −1 c→β∂ −1 c)⊗¬(α ′ δ ′−1 c))

5) Rule of proportion implication: Given F , P , Q, and G are formulas The following rules are modus ponens, equiva-lent, and substitution of individual constant for an individual variable, which are proposed in [14, 16]

R11) RE: P ↔Q,(F (P ),δc)(F (P /Q),δc) ; where F (P ) is a formula containing P as a subexpression, and F (Q/P ) denotes the formula obtained from F by replacing all occurrences

of P in F with Q

R12) RMT: (P →Q,vi ),(P,vj)

(Q,v i ⊗v j ) , where vi, vj ∈ Ln

R13) RS: (p(x,u),v)(p(a,u),v), where x is a variable and a is a constant, and v ∈ Ln

D Linguistic reasoning method based on linguistic many-valued logic

In the linguistic many-valued logic, an assertion is a pair

A = (p(x, u), δc), where u is a vague concept, δ is a string

of hedges, p(x, u) is a vague sentence, δc is a linguistic truth value We can rewrite an assertion as a pair (P, t), where P

is a formula P = p(x, u) and t is a valuation (linguistic truth-valued of formula P )

IV DECISIONMAKING USINGLINGUISTIC

MANY-VALUEDLOGIC

In decision making, the decision maker needs to choose one

or more alternatives based on the evaluation for alternatives Assume that the value used to evaluate the alternative is the linguistic value and the decision maker needs to choose a reasonable alternative based on a hypothesis, for example:

“A good university should be one with very high quality

of teaching and high quality of research” There are many alternative evaluation methods, in which, the method based

on logical reasoning has many advantages, one of which is accuracy because this is one of the methods of calculating directly on words

In this section, we shall describe our methodology, which computes evaluation for a given set of alternatives

A Methodology

We propose a decision making model for solving a decision problem based on the following two ideas Firstly, we use the linguistic algebraic structure 2-Mono-HA for the decision making problem Secondly, applying the deduction method

is derived from knowledge base K using the above rules

to deduce the conclusion (P, v) The model is described as follows

Algorithm 1 Determining the best alternative

Input:

+ The finite set of alternatives, X

+ The finite set of assertions, A

+ The finite set of reasoning rules, R, as described in Section III;

+ The hypothesis, H

Output: x∗∈ X is the best alternative

Begin

1: label Fi for fj(wj, vj), for all j = 1, 2, , n;

2: for each xi∈ X

3: assign H′= H;

4: for i = 1, , n

5: apply RT1and GRT2 rules to transform (fj(xi, uj), tj) tofj(xi, vj), t′j;

6: remove the outside left hedge in ti or tj if hedge string of ti or tj is greater than L;

/* the outside left hedge of hedge string make little change to the semantics of linguistic truth value */

7: label Fi for fj(xi, vj);

8: apply RMT (R12) onFi, t′jand H′;

9: end for

10: end for each

11: select xi with the highest ei evaluation value;

End

We form the hypothesis according to the following

for-mula:

H = (F1∧ F2∧ · · · ∧ Fn −→ F, t) ,

where t ∈ AX is a linguistic truth value; Fj =

fj(wj, vj), fj is a vague sentence, wj is a language

variable, vj ∈ AX is a vague concept; F = f (w, v),

f is a vague sentence, w is a language variable, v ∈ AX

is a vague concept

The hypothesis is the opinion of the decision maker

• Let X = {x1, x2, , xm} be a finite set of alternatives

• Let ai = { (Pj, tj)| j = 1, 2, , n} be a finite set of

assertions corresponding to xi alternative, where, tj ∈

AX, Pj = fj(xi, uj), uj ∈ AX is a vague concept,

(j = 1, 2, , n) The evaluation term tj from the experts

is used to evaluate the alternatives

• The knowledge base K consists of hypothesis H and set

of assertions A = {ai| i = 1, , m}, denoted K =

⟨H, A⟩

• Let R be a finite set of rules;

• Assume that, K is consistent It is necessary to evaluate

each alternative xi and select the one with the highest

evaluation That is, it needs to be done K ⊢ (f (xi, v), ei)

for every i = 1, , m, and select xi with the highest ei

The decision making process can be carried out in the

follow-ing steps:

Step 1: Apply RT1 and GRT2 rules to transform

(fj(xi, uj), tj) tofj(xi, vj), t′j, for all j = 1, 2, , n;

Step 2: Compute the linguistic value eibased on rule set

R and the hypothesis, for all i = 1, 2, , m;

Step 3: Select the alternative with the highest evaluation

B Algorithm for determining the best alternative According to the model proposed in subsection IV-A, we propose an algorithm to determine the best alternative, as described in Algorithm 1

Computational complexity: the computational complexity

of the algorithm 1 is O(mnk), where k is the maximum complexity of computing RT1 and GRT2

C Illustrative example

In this section, we provide an example of univer-sity evaluation Given finite monotonous hedge algebra

as follow: 2-Mono-HA = (X, {c+, c−} , H, ≤), H = {V = very, M = more, P = possibly}, c+ can be “good” or

“true”; c− can be “poor” or “false”

We have the linguistic truth-valued domain: AX = {0,

V V c−, M V c−, V c−, P V c−, V M c−, M M c−, M c−,

P M c−, c−, V P c−, M P c−, P c−, P P c−, W, P P c+, P c+,

M P c+, V P c+, c+, P M c+, M c+, M M c+, V M c+, P V c+,

V c+, M V c+, V V c+, 1} Suppose that we have two alter-natives “a1 university” and “a2 university” respectively We make a decision to select one of a1 or a2 by computing linguistic value for sentences “a1 is a good university” and

“a2is a good university”

The hypothesis that “If a university teaching is good and research is very good, then it will be a good university” is more very true

Assume that, we have the following assertions:

(1) a1 university teaching very good is possibly true and research very good is more very true

(2) a2university teaching very very good is true and research very good is more true

TABLE I

T HE INVERSE MAPPING OF 2-M ONO -HA

truth-valued domain

2 V V c− V c− V V c− V V c−

3 M V c − M c − V V c − V V c −

5 P V c− P c− M V c− V V c−

6 V M c − P P c − V c − V V c −

7 M M c− P P c− M c− M V c−

9 P M c− P P c− P c− M V c−

10 c− P P c− P P c− M V c−

11 V P c− P P c− P P c− V c−

12 M P c− P P c− P P c− M c−

13 P c − P P c − P P c − c −

14 P P c− P P c− P P c− P c−

16 P P c + P P c + P P c + P c +

17 P c + P P c + P P c + c +

18 M P c + P P c + P P c + M c +

19 V P c + P P c + P P c + V c +

20 c + P P c + P P c + M V c +

21 P M c + P P c + P c + M V c +

23 M M c+ P P c+ M c+ M V c+

24 V M c + P P c + V c + V V c +

25 P V c + P c + M V c + V V c +

27 M V c + M c + V V c + V V c +

28 V V c + V c + V V c + V V c +

Based on the hypothesis, we rewrite it as follows:

teach(w, c+) ∧ res(w, V c+) → uni(w, c+), M V c+

where, “teach”, “res”, and “uni” stand for “teaching”,

“re-search”, and “university”

The following steps illustrate the proposed algorithm for

determining the best alternative Table I list the linguistic

truth-valued domain and the inverse mapping of hedge, which is

used for lookup when applying the GRT2 rule

(1) (teach(a1, V+), P c+) ∧ (res(a1, V+), M V c+)

1) Denote A = (teach(w, c+); B = res(w, V c+); C =

uni(w, c+);

2) Apply RT1:

(teach(a1, V c+), P c+) ≡ (teach(a1, c+), P V c+)

3) ((A ∧ B) → C) , v27)

4) (A, v25)

5) (B, v27)

6) Apply rule RMT on 3, 4: (B → C, v25⊗ v27)

(6a) (B → C, v23)

7) Apply rule RR on 5, 6a: (C, v27⊗ v23)

(7a) (C, v21) ≡ (C, P M c+)

(2) (teach(a2, V V+), c+) ∧ res(a2, V+), P c+)

1) Denote A = (teach(x, c+); B = res(x, V c+); C =

uni(x, c+);

2) Apply RT1:

(teach(a2, V V c+), c+) ≡ (teach(a2, c+), V V c+) 3) ((A ∧ B) → C) , v27)

4) (A, v28) 5) (B, v22) 6) Apply rule RMT on 3, 4: (B → C, v27⊗ v28) (6a) (B → C, v26)

7) Apply rule RR on 5, 6a: (C, v22⊗ v26) (7a) (C, v19) ≡ (C, V P c+)

Thus, the truth value of the sentence “a1 university teaching very good is possibly true and research very good is more very true” is P M c+ and the truth value of the sentence “a2

university teaching very very good is true and research very good is more true” is V P c+ Therefore, a1university can be selected

V CONCLUSION

In this paper, we proposed a novel method to create a syn-thesized evaluation for the alternatives This method generates the final evaluation of the alternatives from the point of view of the decision maker This method can be applied to linguistic multi-attribute decision making problems The advantage of this method is that it can be used directly on language values without any transformation

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