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[.]
2021 8th NAFOSTED Conference on Information and Computer Science (NICS) An Approach to Evaluate Alternatives in Multi-Attribute Decision Making Problems based on Linguistic Many-Valued Logic Phuong Le Anh, Hoai Nhan Tran Dinh Khang Tran Department of Computer Science, Hue University of Education, Hue University, Hue City, Vietnam {leanhphuong, tranhoainhan}@dhsphue.edu.vn 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 manyvalued 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, linguistic many-valued logic I I NTRODUCTION 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 complexity and uncertainty of the objective thing and the opacity 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 information (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 information, such as the methods based on making operations on the fuzzy numbers that support the semantics of the linguistic 978-1-6654-1001-4/21/$31.00 ©2021 IEEE 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 information, 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 manyvalued logic by extending 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 525 2021 8th NAFOSTED Conference on Information and Computer Science (NICS) II L INGUISTIC M ANY-VALUED L OGIC 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 h1 h2 hn c (n ≥ 0), with hi is called hedge, h1 h2 hn is 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 conditions [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 >H in H ∪{I} is very >H more >H I >H probably >H mol Then, in Mono-HA, hedges are “contextfree”, 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 + 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 , are the smallest, neutral, and biggest elements respectively in AX According to Nguyen and Wechsler [7]: + < x < W < y < for all x ∈ H(c− ), and ∀y ∈ H(c+ ), + hW = W , h1 = 1, h0 = 0, + −W = W , −1 = 0, −0 = Because AX is the linguistic truth-valued domain in finite monotonous hedge algebra and linear order [10, 13]; thus, we can rewrite it as: AX = { vi , i = 1, , n| v1 = 0, = 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 = ∧ 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 denote for “False” and “True”, respectively) with the linguistic truth-valued domain AX Ln is called a linguistic many-valued logic 526 2021 8th NAFOSTED Conference on Information and Computer Science (NICS) III K NOWLEDGE BASE AND L INGUISTIC R EASONING 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 elementary ones by means of logical connectives such as “and”, “or”, “if-then” and “not”, which are denoted correspondingly by ∧, ∨, → and ¬, called conjunction, disjunction, 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 conclusion (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: − − (hk hk−1 h1 ) δc = h− k h1 δc 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]: (p (x, hu) , δc) R1 ) RT1 : (p (x, u) , δhc) (p (x, u) , δc) R2 ) GRT2 : (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) (p(x,u),δc)→(q(y,v),σc),(p(x,u),δ ′ c) R4 ) EGMP: (Q,δ ′ c⊗(δc→σc)) EGMP is an extension of GMP; (p(x,¬u)→q(y,v),vi ),(p(x,u),vj ) R5 ) NGMP: (q(y,v),vi ⊗¬vj ) (p(x,¬u),v )→(q(y,v),v ),(¬q(x,u),vk ) i j R6 ) ENGMP: (q(y,v),(vi →vj )⊗¬vk ) ENGMP is an extension of NGMP , 4) Generalized modus ponens with linguistic modifiers: The rules of generalized Modus ponens with linguistic modifiers (GMPLM) have been introduced in [9, 16] Given α, β, δ, σ, θ, ∂, α′ , β ′ , δ ′ , θ′ , and ∂ ′ are the hedge strings; get α = h1 h2 hk , symbol α−1 = hk h2 h1 , the rules are: (p(x,δu)→q(y,∂v),αc),(p(x,δ ′ u),α′ c) R7 ) GMPLM: (q(y,∂v),αc⊗δ −1 (α′ δ ′−1 c)) (p(x,δu),αc)→(q(y,∂v),βc),(p(x,δ ′ u),α′ c) R8 ) EGMPLM: (q(y,∂ ′ v),(αδ−1 c→β∂ −1 c)⊗(α′ δ′−1 c)) (p(x,¬(δu))→q(y,∂v),αc),(p(x,δ ′ u),α′ c) R9 ) NGMPLM: (q(y,∂v),αc⊗¬(δ −1 (α′ δ ′−1 c))) (p(x,δu),αc)→(q(y,∂v),βc),(p(x,δ ′ u),α′ c) R10 ) ENGMPLM: (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, equivalent, and substitution of individual constant for an individual variable, which are proposed in [14, 16] (P ),δc) R11 ) RE: P ↔Q,(F (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 (P →Q,vi ),(P,vj ) , where vi , vj ∈ Ln R12 ) RMT: (Q,vi ⊗vj ) 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 manyvalued 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 truthvalued of formula P ) IV D ECISION M AKING USING L INGUISTIC M ANY-VALUED L OGIC 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 527 2021 8th NAFOSTED Conference on Information and Computer Science (NICS) Algorithm 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 RT1 and GRT2 rules to transform (fj (xi , uj ), tj ) to fj (xi , vj ), tj ; 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 ) on Fi , tj and 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 formula: 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 = { (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 following steps: Step 1: Apply RT1 and GRT rules to transform ′ (fj (xi , uj ), tj ) to fj (xi , vj ), tj , for all j = 1, 2, , n; Step 2: Compute the linguistic value ei based 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 Computational complexity: the computational complexity of the algorithm 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 university 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 alternatives “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 “a2 is 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) a2 university teaching very very good is true and research very good is more true 528 2021 8th NAFOSTED Conference on Information and Computer Science (NICS) (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, a1 university can be selected TABLE I T HE INVERSE MAPPING OF 2-M ONO -HA Index 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 Linguistic truth-valued domain 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+ V− M− P− V c− M c− c− P c− P P c− P P c− P P c− P P c− P P c− P P c− P P c− P P c− P P c− W P P c+ P P c+ P P c+ P P c+ P P c+ P P c+ P P c+ P P c+ P P c+ P c+ c+ M c+ V c+ V V c− V V c− M V c− M V c− V c− M c− c− P c− P P c− P P c− P P c− P P c− P P c− W P P c+ P P c+ P P c+ P P c+ P P c+ P c+ c+ M c+ V c+ M V c+ M V c+ V V c+ V V c+ V V c− V V c− V V c− V V c− V V c− M V c− M V c− M V c− M V c− V c− M c− c− P c− W P c+ c+ M c+ V c+ M V c+ M V c+ M V c+ M V c+ V V c+ V V c+ V V c+ V V c+ V V c+ V C ONCLUSION In this paper, we proposed a novel method to create a synthesized 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 R EFERENCES 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”, “research”, and “university” The following steps illustrate the proposed algorithm for determining the best alternative Table I list the linguistic truthvalued 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 : 529 [1] P Taylor, L Mart´ınez, D Ruan and L Mart, “Computing with Words in Decision support Systems: An overview on Models and Applications,” International Journal of Computational Intelligence Systems, Vol 3, pp 382–395, 2010 [2] L Yang and Y Xu, “A decision method based on uncertainty reasoning of linguistic truth-valued concept lattice,” International Journal of General Systems, Vol 39, No 3, pp 235–253, 2010 [3] Z Xu, Linguistic decision making: Theory and methods, Springer, 2012 [4] F Herrera and E Herrera-Viedma, “Linguistic decision analysis: Steps for solving decision problems under linguistic information,” Fuzzy Sets and Systems, Vol 115, No 1, pp 67–82, 2000 [5] L A Zadeh, “The concept of a linguistic variable and its application to approximate reasoning-I,” Information Sciences Vol 8, No 3, pp 199–249, 1975 [6] S Chen, J Liu, H Wang, Y Xu and J C Augusto, “A linguistic multi-criteria decision making approach based on logical reasoning,” Information Sciences, Vol 258, pp 266–276, 2014 [7] N C Ho and W Wechler, “Hedge algebras: An algebraic approach to structure of sets of linguistic truth values,” Fuzzy sets and systems, Vol 35, No 3, pp 281–293, 1990 [8] V H Le, F Liu and D K Tran, “Fuzzy linguistic logic programming and its applications”, Theory and Practice of Logic Programming, Vol 9, No 3, pp 309–341, 2009 [9] L A Phuong and T D Khang, “Generalized If Then Else Inference Rules with Linguistic Modifiers 2021 8th NAFOSTED Conference on Information and Computer Science (NICS) [10] [11] [12] [13] for Approximate Reasoning,” International Journal of Computer Science Issues, Vol 9, No 6, pp 184–190, 2012 L A Phuong and T D Khang, “Linguistic reasoning based on generalized modus ponens with linguistic modifiers and hedge moving rules, ” In: Proc International conference on Fuzzy Theory and Its Applications (iFUZZY2012), pp 82–86, 2012 L A Phuong, T D Khang and N V Trung, “New approach to mining fuzzy association rule with linguistic threshold based on hedge algebras,” In: CEUR Workshop Proc., Vol 1494, No November, pp 0–10, 2015 L A Phuong and T D Khang, “A Resolution Method for Linguistic Many-valued Logic,” Applied Mathematics & Information Sciences, Vol 3, 2013 L A Phuong and T D Khang, “Generalized Modus Tollens with Linguistic Modifiers for Inverse Approxi- mate Reasoning,” International Journal of Computational Intelligence Systems, Vol 7, No 3, pp 556–564, 2014 [14] N C Ho, “A method in linguistic reasoning on a knowledge base representing by sentences with linguistic belief degree,” Fundamenta Informaticae, Vol 28, No (3–4), pp 247–259, 1996 [15] P V Hai, T D Khang and K Kamei, “Applications using hybrid intelligent decision support systems for selection of alternatives under uncertainty and risk,” International Journal of Innovative Computing, Information and Control, Vol 10, No 1, pp 39–56, 2014 [16] N C Ho, T D Khang, H Van Nam and N H Chau, “Hedge algebras, linguistic-valued logic and their application to fuzzy reasoning,” International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, Vol 7, No 4, pp 347–361, 1999 530 ... where x is a variable and a is a constant, and v ∈ Ln D Linguistic reasoning method based on linguistic manyvalued logic In the linguistic many- valued logic, an assertion is a pair A = (p(x,... valuation (linguistic truthvalued of formula P ) IV D ECISION M AKING USING L INGUISTIC M ANY -VALUED L OGIC In decision making, the decision maker needs to choose one or more alternatives based on the... Xu, Linguistic decision making: Theory and methods, Springer, 2012 [4] F Herrera and E Herrera-Viedma, ? ?Linguistic decision analysis: Steps for solving decision problems under linguistic information,”