Neutrosophic Sets and Systems, Vol. 38, 2020 University of New Mexico The Score, Accuracy, and Certainty Functions determine a Total Order on the Set of Neutrosophic Triplets (T, I, F) Florentin Smarandache 1* 1 Division of Mathematics, Physical and Natural Sciences, University of New Mexico, 705 Gurley Ave., Gallup, NM 87301, USA * Correspondence: [email protected] Abstract: In this paper we prove that the Single-Valued (and respectively Interval-Valued, as well as Subset-Valued) Score, Accuracy, and Certainty Functions determine a total order on the set of neutrosophic triplets (T, I, F). This total order is needed in the neutrosophic decision-making applications. Keywords: single-valued neutrosophic triplet numbers; single-valued neutrosophic score function; single-valued neutrosophic accuracy function; single-valued neutrosophic certainty function. 1. Introduction We reveal the easiest to use single-valued neutrosophic score, accuracy, and certainty functions that exist in the literature and the algorithm how to use them all together. We present Xu and Da’s Possibility Degree that an interval is greater than or equal to another interval, and we prove that this method is equivalent to the intervals’ midpoints comparison. Also, Hong-yu Zhang et al.’s interval- valued neutrosophic score, accuracy, and certainty functions are listed, that we simplify these functions. Numerical examples are provided. 2. Single-Valued Neutrosophic Score, Accuracy, and Certainty Functions We firstly present the most known and used in literature single-valued score, accuracy, and certainty functions. Let M be the set of single-valued neutrosophic triplet numbers, 𝑀 ={(𝑇,𝐼,𝐹),where 𝑇,𝐼,𝐹 ∈[0,1],0≤𝑇+𝐼+𝐹 ≤3}. (1) Let 𝑁 =(𝑇,𝐼,𝐹) ∊ M be a generic single-valued neutrosophic triplet number. Then: 𝑇 = truth (or membership) represents the positive quality of 𝑁; 𝐼 = indeterminacy represents a negative quality of 𝑁, hence 1−𝐼 represents a positive quality of 𝑁; 𝐹 = falsehood (or nonmembership) represents also a negative quality of 𝑁, hence 1−𝐹 represents a positive quality of 𝑁. We present the three most used and best functions in the literature: 2.1. The Single-Valued Neutrosophic Score Function 𝑠:𝑀 →[0,1] 𝑇+(1−𝐼)+(1−𝐹) 2+𝑇−𝐼−𝐹 𝑠(𝑇,𝐼,𝐹)= = (2) 3 3 that represents the average of positiveness of the single-valued neutrosophic components 𝑇, 𝐼, 𝐹. Florentin Smarandache, The Score, Accuracy, and Certainty Functions determine a Total Order on the Set of Neutrosophic Triplets (T, I, F) Neutrosophic Sets and Systems, Vol. 38, 2020 2 2.2. The Single-Valued Neutrosophic Accuracy Function 𝑎:𝑀 →[−1,1] 𝑎(𝑇,𝐼,𝐹)=𝑇−𝐹 (3) 2.3. The Single-Valued Neutrosophic Certainty Function 𝑐:𝑀 →[0,1] 𝑐(𝑇,𝐼,𝐹)=𝑇 (4) 3. Algorithm for Ranking the Single-Valued Neutrosophic Triplets Let (𝑇 ,𝐼 ,𝐹 ) and (𝑇 ,𝐼 ,𝐹 ) be two single-valued neutrosophic triplets from 𝑀, i.e. 1 1 1 2 2 2 𝑇 ,𝐼 ,𝐹 ,𝑇 ,𝐼 ,𝐹 ∈[0,1]. 1 1 1 2 2 2 Apply the Neutrosophic Score Function. 1. If 𝑠(𝑇 ,𝐼 ,𝐹 )>𝑠(𝑇 ,𝐼 ,𝐹 ), then (𝑇 ,𝐼 ,𝐹 )>(𝑇 ,𝐼 ,𝐹 ). 1 1 1 2 2 2 1 1 1 2 2 2 2. If 𝑠(𝑇 ,𝐼 ,𝐹 )<𝑠(𝑇 ,𝐼 ,𝐹 ), then (𝑇 ,𝐼 ,𝐹 )<(𝑇 ,𝐼 ,𝐹 ). 1 1 1 2 2 2 1 1 1 2 2 2 3. If 𝑠(𝑇 ,𝐼 ,𝐹 )=𝑠(𝑇 ,𝐼 ,𝐹 ), then apply the Neutrosophic Accuracy Function: 1 1 1 2 2 2 3.1 If 𝑎(𝑇 ,𝐼 ,𝐹 )>𝑎(𝑇 ,𝐼 ,𝐹 ), then (𝑇 ,𝐼 ,𝐹 )>(𝑇 ,𝐼 ,𝐹 ). 1 1 1 2 2 2 1 1 1 2 2 2 3.2 If 𝑎(𝑇 ,𝐼 ,𝐹 )<𝑎(𝑇 ,𝐼 ,𝐹 ), then (𝑇 ,𝐼 ,𝐹 )<(𝑇 ,𝐼 ,𝐹 ). 1 1 1 2 2 2 1 1 1 2 2 2 3.3 If 𝑎(𝑇 ,𝐼 ,𝐹 )=𝑎(𝑇 ,𝐼 ,𝐹 ), then apply the Neutrosophic Certainty Function. 1 1 1 2 2 2 3.3.1 If 𝑐(𝑇 ,𝐼 ,𝐹 )>𝑐(𝑇 ,𝐼 ,𝐹 ), then (𝑇 ,𝐼 ,𝐹 )>(𝑇 ,𝐼 ,𝐹 ). 1 1 1 2 2 2 1 1 1 2 2 2 3.3.2 If 𝑐(𝑇 ,𝐼 ,𝐹 )<𝑐(𝑇 ,𝐼 ,𝐹 ), then (𝑇 ,𝐼 ,𝐹 )<(𝑇 ,𝐼 ,𝐹 ). 1 1 1 2 2 2 1 1 1 2 2 2 3.3.1 If 𝑐(𝑇 ,𝐼 ,𝐹 )=𝑐(𝑇 ,𝐼 ,𝐹 ), then (𝑇 ,𝐼 ,𝐹 )≡(𝑇 ,𝐼 ,𝐹 ), i.e. 𝑇 =𝑇 , 𝐼 =𝐼 , 𝐹 =𝐹 . 1 1 1 2 2 2 1 1 1 2 2 2 1 2 1 2 1 2 3.1. Theorem We prove that the single-valued neutrosophic score, accuracy, and certainty functions all together form a total order relationship on 𝑀. Or: for any two single-valued neutrosophic triplets (𝑇 ,𝐼 ,𝐹 ) and (𝑇 ,𝐼 ,𝐹 ) we have: 1 1 1 2 2 2 a) Either (𝑇 ,𝐼 ,𝐹 )>(𝑇 ,𝐼 ,𝐹 ) 1 1 1 2 2 2 b) Or (𝑇 ,𝐼 ,𝐹 )<(𝑇 ,𝐼 ,𝐹 ) 1 1 1 2 2 2 c) Or (𝑇 ,𝐼 ,𝐹 )≡(𝑇 ,𝐼 ,𝐹 ), which means that 𝑇 =𝑇 , 𝐼 =𝐼 , 𝐹 =𝐹 . 1 1 1 2 2 2 1 2 1 2 1 2 Therefore, on the set of single-valued neutrsophic triplets 𝑀 ={(𝑇,𝐼,𝐹),with 𝑇,𝐼,𝐹 ∈ [0,1],0≤𝑇+𝐼+𝐹 ≤3}, the score, accuracy, and certainty functions altogether form a total order relationship. Proof. Firstly we apply the score function. The only problematic case is when we get equality: 𝑠(𝑇 ,𝐼 ,𝐹 )=𝑠(𝑇 ,𝐼 ,𝐹 ). (5) 1 1 1 2 2 2 That means: 2+𝑇1−𝐼1−𝐹1 =2+𝑇2−𝐼2−𝐹2 3 3 or 𝑇 −𝐼 −𝐹 =𝑇 −𝐼 −𝐹 . 1 1 1 2 2 2 Secondly we apply the accuracy function. Again the only problematic case is when we get equality: 𝑎(𝑇 ,𝐼 ,𝐹 )=𝑎(𝑇 ,𝐼 ,𝐹 ) or 𝑇 −𝐹 =𝑇 −𝐹 . 1 1 1 2 2 2 1 1 2 2 Florentin Smarandache, The Score, Accuracy, and Certainty Functions determine a Total Order on the Set of Neutrosophic Triplets (T, I, F) Neutrosophic Sets and Systems, Vol. 38, 2020 3 Thirdly, we apply the certainty function. Similarly, the only problematic case may be when we get equality: 𝑐(𝑇 ,𝐼 ,𝐹 )=𝑐(𝑇 ,𝐼 ,𝐹 ) or 𝑇 =𝑇 . 1 1 1 2 2 2 1 2 For the most problematic case, we got the following linear algebraic system of 3 equations of 6 variables: 𝑇 −𝐼 −𝐹 =𝑇 −𝐼 −𝐹 1 1 1 2 2 2 { 𝑇 −𝐹 =𝑇 −𝐹 1 1 2 2 𝑇 =𝑇 1 2 Let’s solve it. Since 𝑇 =𝑇 , replacing this into the second equation we get 𝐹 =𝐹 . 1 2 1 2 Now, replacing both 𝑇 =𝑇 and 𝐹 =𝐹 into the first equation, we get 𝐼 =𝐼 . 1 2 1 2 1 2 Therefore the two neutrosophic triplets are identical: (𝑇 ,𝐼 ,𝐹 )≡(𝑇 ,𝐼 ,𝐹 ), i.e. equivalent 1 1 1 2 2 2 (or equal), or 𝑇 =𝑇 , 𝐼 =𝐼 , and 𝐹 =𝐹 . 1 2 1 2 1 2 In conclusion, for any two single-valued neutrosophic triplets, either one is bigger than the other, or both are equal (identical). 4. Definition of Neutrosophic Negative Score Function We have introduce in 2017 for the first time [1] the Average Negative Quality Neutrosophic Function of a single-valued neutrosophic triplet, defined as: (1t)i f 1ti f s :[0,1]3 [0,1],s(t,i, f)  . (6) 3 3 4.1. Theorem The average positive quality (score) neutrosophic function and the average negative quality neutrosophic function are complementary to each other, or s(t,i, f)s(t,i, f) 1. (7) Proof. 2ti f 1ti f s(t,i, f)s(t,i, f)   1. (8) 3 3 The Neutrosophic Accuracy Function has been defined by: h: [0, 1]3  [-1, 1], h(t, i, f) = t - f. (9) We have also introduce [1] for the first time the Extended Accuracy Neutrosophic Function, defined as follows: he: [0, 1]3  [-2, 1], he(t, i, f) = t – i – f, (10) which varies on a range: from the worst negative quality (-2) [or minimum value], to the best positive quality (+1) [or maximum value]. 4.2. Theorem If s(T1, I1, F1) = s(T2, I2, F2), a(T1, I1, F1) = a(T2, I2, F2), and c(T1, I1, F1) = c(T2, I2, F2), then T1 = T2, I1 = I2, F1 = F2, or the two neutrosophic triplets are identical: (T1, I1, F1) ≡ (T2, I2, F2). Florentin Smarandache, The Score, Accuracy, and Certainty Functions determine a Total Order on the Set of Neutrosophic Triplets (T, I, F) Neutrosophic Sets and Systems, Vol. 38, 2020 4 Proof: It results from the proof of Theorem 3.1. 5. Xu and Da’s Possibility Degree Xu and Da [3] have defined in 2002 the possibility degree 𝑃(.) that an interval is greater than another interval: [𝑎 ,𝑎 ]≥[𝑏 ,𝑏 ] 1 2 1 2 for 𝑎 ,𝑎 ,𝑏 ,𝑏 ∈[0,1] and 𝑎 ≤𝑎 ,𝑏 ≤𝑏 , in the following way: 1 2 1 2 1 2 1 2 𝑃([𝑎 ,𝑎 ]≥[𝑏 ,𝑏 ])=𝑚𝑎𝑥{1−𝑚𝑎𝑥( 𝑏2−𝑎1 ,0),0}, 1 2 1 2 𝑎2−𝑎1+𝑏2−𝑏1 where 𝑎 −𝑎 +𝑏 −𝑏 ≠0 (i.e. 𝑎 ≠𝑎 𝑜𝑟 𝑏 ≠𝑏 . 2 1 2 1 2 1 2 1 They proved the following: 5.1. Properties 1) 𝑃([𝑎 ,𝑎 ]≥[𝑏 ,𝑏 ])∈[0,1]; 1 2 1 2 2) 𝑃([𝑎 ,𝑎 ]≈[𝑏 ,𝑏 ])=0.5; 1 2 1 2 3) 𝑃([𝑎 ,𝑎 ]≥[𝑏 ,𝑏 ])+𝑃([𝑏 ,𝑏 ]≥[𝑎 ,𝑎 ])=1. 1 2 1 2 1 2 1 2 5.2. Example Let [0.4,0.7] and [0.3,0.6] be two intervals. Then, 0.6−0.4 0.2 𝑃([0.4,0.7]≥[0.3,0.6])=𝑚𝑎𝑥{1−𝑚𝑎𝑥( ,0),0}=𝑚𝑎𝑥{1−𝑚𝑎𝑥( ),0} 0.7−0.4+0.6−0.3 0.6 0.2 0.4 =𝑚𝑎𝑥{1− ,0}= ≈0.66>0.50, 0.6 0.6 therefore [0.4,0.7]≥[0.3,0.6]. The opposite: 0.7−0.3 0.4 𝑃((0.3,0.6)≥([0.4,0.7]))=𝑚𝑎𝑥{1−𝑚𝑎𝑥( ,0),0}=𝑚𝑎𝑥{1−𝑚𝑎𝑥( ,0),0} 0.6−0.3+0.7−0.4 0.6 0.4 0.2 =𝑚𝑎𝑥{1− ,0}= ≈0.33<0.50, 0.6 0.6 therefore [0.3,0.6]≤[0.4,0.7]. We see that 0.4 0.2 𝑃([0.4,0.7]≥[0.3,0.6])+𝑃([0.3,0.6]≥[0.4,0.7])= + =1. 0.6 0.6 Another method of ranking two intervals is the midpoint one. 6. Midpoint Method Let A = [a1, a2] and B = [b1, b2] be two intervals included in or equal to [0, 1], with 𝑚𝐴 = (a1 + a2)/2 and 𝑚𝐵 = (b1 + b2)/2 the midpoints of A and respectively B. Then: 1) If 𝑚 <𝑚 then 𝐴<𝐵. 𝐴 𝐵 2) If 𝑚 >𝑚 then 𝐴>𝐵. 𝐴 𝐵 3) If 𝑚 =𝑚 then 𝐴= 𝐵, i.e. A is neutrosophically equal to B. 𝐴 𝐵 𝑁 Florentin Smarandache, The Score, Accuracy, and Certainty Functions determine a Total Order on the Set of Neutrosophic Triplets (T, I, F) Neutrosophic Sets and Systems, Vol. 38, 2020 5 6.1. Example 1) We take the previous example, 0.4+0.7 where 𝐴=[0.4,0.7], and 𝑚 = =0.55; 𝐴 2 0.3+0.6 and B=[0.3,0.6], and 𝑚 = =0.45. 𝐵 2 Since 𝑚 =0.55>0.45=𝑚 , we have 𝐴>𝐵. 𝐴 𝐵 Let 𝐶 =[0.1,0.7] and 𝐷 =[0.3,0.5]. 0.1+0.7 0.3+0.5 Then 𝑚 = =0.4, and 𝑚 = =0.4. 𝐶 2 𝐷 2 Since 𝑚 =𝑚 =0.4, we get 𝐶 = 𝐷. 𝐶 𝐷 𝑁 Let’s verify the ranking relationship between C and D using Xu and Da’s possibility degree method. 0.5−0.1 0.4 𝑃([0.1,0.7]≥[0.3,0.5])=𝑚𝑎𝑥{1−𝑚𝑎𝑥( ,0),0}=𝑚𝑎𝑥{1−𝑚𝑎𝑥( ,0),0} 0.7−0.1+0.5−0.3 0.8 0.4 0.4 =𝑚𝑎𝑥{1− ,0}=𝑚𝑎𝑥{ ,0}=0.5; 0.8 0.8 0.7−0.3 and 𝑃([0.3,0.5]≥[0.1,0.7])=𝑚𝑎𝑥{1−𝑚𝑎𝑥( ,0),0}=𝑚𝑎𝑥{1− 0.5−0.3+0.7−0.1 0.4 0.4 0.4 𝑚𝑎𝑥( ,0),0}=𝑚𝑎𝑥{1− ,0}=𝑚𝑎𝑥{ ,0}=0.5; 0.8 0.8 0.8 thus, [0.1,0.7]= [0.3,0.5]. 𝑁 6.2. Corollary The possibility method for two intervals having the same midpoint gives always 0.5. For example: p([0.3, 0.5] ≥ [0.2, 0.6]) = max{1 - max( ((0.6-0.3) / (0.5-0.3 + 0.6-0.2)), 0 ), 0} = = max{1 - max( ((0.3) / (0.6)), 0 ), 0} = max{1 - max( 0.5, 0 ), 0} = 0.5. Similarly, p([0.2, 0.6] ≥ [0.3, 0.5]) = max{1 - max( ((0.5-0.2) / (0.6-0.2 + 0.5-0.3)), 0 ), 0} = 0.5. Hence, none of the intervals [0.3, 0.5] and [0.2, 0.6]) is bigger than the other. Therefore, we may consider that the intervals [0.3, 0.5] = [0.2, 0.6] are neutrosophically equal 𝑁 (or neutrosophically equivalent). 7. Normalized Hamming Distance between Two Intervals Let’s consider the Normalized Hamming Distance between two intervals [a1, a2] and [b1, b2] h : int([0, 1])⨯ int([0, 1])  [0, 1] defined as follows: h([a1, b1], [a2, b2])= ½(|a1 - b1| + |a2 - b2|). 7.1. Theorem 7.1.1. The Normalized Hamming Distance between two intervals having the same midpoint and the negative-ideal interval [0, 0] is the same. Florentin Smarandache, The Score, Accuracy, and Certainty Functions determine a Total Order on the Set of Neutrosophic Triplets (T, I, F) Neutrosophic Sets and Systems, Vol. 38, 2020 6 7.1.2. The Normalized Hamming Distance between two intervals having the same midpoint and the positive-ideal interval [1, 1] is also the same (Jun Ye [4, 5]). Proof. Let A = [m – a, m + a] and B = [m – b, m + b] be two intervals from [0, 1], where m-a, m+a, m-b, m+b, a, b, m ∊ [0, 1]. A and B have the same midpoint m. 7.1.1. h([m - a, m + a], [0, 0])= ½(|m – a - 0| + |m + a - 0|) = ½(m – a + m + a) = m, and h([m - b, m + b], [0, 0])= ½(|m – b - 0| + |m + b - 0|) = ½(m – b + m + b) = m, 7.1.2. h([m - a, m + a], [1, 1])= ½(|m – a - 1| + |m + a - 1|) = ½(1 - m + a + 1 - m - a) = 1 - m, and h([m - b, m + b], [1, 1])= ½(|m – b - 1| + |m + b - 1|) = ½(1 - m + b + 1 - m - b) = 1 – m. 8. Xu and Da’s Possibility Degree Method is equivalent to the Midpoint Method We prove the following: 8.1. Theorem The Xu and Da’s Possibility Degree Method is equivalent to the Midpoint Method in ranking two intervals included in [0, 1]. Proof. Let A and B be two intervals included in [0,1]. Without loss of generality, we write each interval in terms of each midpoint: 𝐴=[𝑚 −𝑎,𝑚 +𝑎] and 𝐵 =[𝑚 −𝑏,𝑚 +𝑏], 1 1 2 2 where 𝑚 ,𝑚 ∈[0,1] are the midpoints of A and respectively B, and 𝑎,𝑏 ∈[0,1], 𝐴,𝐵 ⊆[0,1]. 1 2 0.4+0.7 (For example, if 𝐴=[0.4,0.7], 𝑚 = =0.55, 0.55-0.4=0.15, then 𝐴=[0.55−0.15,0.55+ 𝐴 2 0.15]). 1) First case: 𝑚 < 𝑚 . According to the Midpoint Method, we get 𝐴<𝐵. Let’s prove the same 1 2 inequality results with the second method. Let’s apply Xu and Da’s Possibility Degree Method: 𝑃(𝐴≥𝐵)=𝑃([𝑚 −𝑎,𝑚 +𝑎]≥[𝑚 −𝑏,𝑚 +𝑏]) 1 1 2 2 (𝑚 +𝑏)−(𝑚 −𝑎) 2 1 =𝑚𝑎𝑥{1−𝑚𝑎𝑥( ,0),0} (𝑚 +𝑎)−(𝑚 −𝑎)+(𝑚 +𝑏)−(𝑚 −𝑏) 1 1 2 2 𝑚 −𝑚 +𝑎+𝑏 2 1 =𝑚𝑎𝑥{1−𝑚𝑎𝑥( ,0),0} 2𝑎+2𝑏 𝑚 −𝑚 +𝑎+𝑏 2 1 =𝑚𝑎𝑥{1− ,0},because 𝑚 <𝑚 , 2𝑎+2𝑏 1 2 2𝑎+2𝑏−𝑚 +𝑚 −𝑎−𝑏 𝑎+𝑏+𝑚 −𝑚 2 1 1 2 =𝑚𝑎𝑥{ ,0}=𝑚𝑎𝑥{ ,0} 2𝑎+2𝑏 2𝑎+2𝑏 i) If 𝑎+𝑏+𝑚 −𝑚 ≤0, then 𝑝(𝐴 ≥𝐵)=𝑚𝑎𝑥{𝑎+𝑏+𝑚1−𝑚2,0}=0, hence 𝐴<𝐵. 1 2 2𝑎+2𝑏 ii) If 𝑎+𝑏+𝑚 −𝑚 >0, 1 2 then 𝑝(𝐴≥𝐵)=𝑚𝑎𝑥{𝑎+𝑏+𝑚1−𝑚2,0}=𝑎+𝑏+𝑚1−𝑚2 >0. 2𝑎+2𝑏 2𝑎+2𝑏 Florentin Smarandache, The Score, Accuracy, and Certainty Functions determine a Total Order on the Set of Neutrosophic Triplets (T, I, F) Neutrosophic Sets and Systems, Vol. 38, 2020 7 We need to prove that 𝑎+𝑏+𝑚1−𝑚2 <0.5, 2𝑎+2𝑏 or 𝑎+𝑏+𝑚 −𝑚 <0.5(2𝑎+2𝑏), 1 2 or 𝑎+𝑏+𝑚 −𝑚 <𝑎+𝑏, 1 2 or 𝑚 −𝑚 <0, 1 2 or 𝑚 <𝑚 , which is true according to the first case assumption. 1 2 2) Second case: 𝑚 =𝑚 . According to the Midpoint Method, A is neutrosophically equal to B 1 2 (we write 𝐴= 𝐵). 𝑁 Let’s prove that we get the same result with Xu and Da’s Method. Then 𝐴=[𝑚 −𝑎,𝑚 +𝑎], and 𝐵 =[𝑚 −𝑏,𝑚 +𝑏]. 1 1 1 1 Let’s apply Xu and Da’s Method: (𝑚 +𝑏)−(𝑚 −𝑎) 𝑃(𝐴 ≥𝐵)=𝑚𝑎𝑥{1−𝑚𝑎𝑥( 1 1 ,0),0} (𝑚 +𝑎)−(𝑚 −𝑎)+(𝑚 +𝑏)−(𝑚 −𝑏) 1 1 1 1 𝑎+𝑏 1 =𝑚𝑎𝑥{1−𝑚𝑎𝑥( ,0),0}=𝑚𝑎𝑥{1− ,0}=0.5 2𝑎+2𝑏 2 Similarly: (𝑚 +𝑎)−(𝑚 −𝑏) 𝑃(𝐵 ≥𝐴)=𝑚𝑎𝑥{1−𝑚𝑎𝑥( 1 1 ,0),0} (𝑚 +𝑏)−(𝑚 −𝑏)+(𝑚 +𝑎)−(𝑚 −𝑎) 1 1 1 1 𝑎+𝑏 =𝑚𝑎𝑥{1−𝑚𝑎𝑥( ,0),0}=0.5 2𝑎+2𝑏 Therefore, again 𝐴= 𝐵. 𝑁 3) If 𝑚 >𝑚 , according to the Midpoint Method, we get 𝐴>𝐵. 1 2 Let’s prove the same inequality using Xu and Da’s Method. 𝑃(𝐴≥𝐵)=𝑃([𝑚 −𝑎,𝑚 +𝑎]≥[𝑚 −𝑏,𝑚 −𝑏])= 1 1 2 2 (𝑚 +𝑏)−(𝑚 −𝑎) 2 1 𝑚𝑎𝑥{1−𝑚𝑎𝑥( ,0),0} (𝑚 +𝑎)−(𝑚 −𝑎)+(𝑚 +𝑏)−(𝑚 −𝑏) 1 1 2 2 𝑚 −𝑚 +𝑎+𝑏 2 1 =𝑚𝑎𝑥{1−𝑚𝑎𝑥( ,0),0} 2𝑎+2𝑏 i) If 𝑚 −𝑚 +𝑎+𝑏 ≤0, then 𝑃(𝐴≥𝐵)=𝑚𝑎𝑥{1−0,0}=1, therefore 𝐴>𝐵. 2 1 ii) If 𝑚 −𝑚 +𝑎+𝑏 >0, then 2 1 𝑚 −𝑚 +𝑎+𝑏 2𝑎+2𝑏−𝑚 +𝑚 −𝑎−𝑏 𝑎+𝑏+𝑚 −𝑚 𝑃(𝐴 ≥𝐵)=𝑚𝑎𝑥{1− 2 1 ,0}= 2 1 = 1 2 2𝑎+2𝑏 2𝑎+2𝑏 2𝑎+2𝑏 We need to prove that 𝑎+𝑏+𝑚1−𝑚2 >0.5, 2𝑎+2𝑏 or 𝑎+𝑏+𝑚 −𝑚 >0.5(2𝑎+2𝑏) 1 2 or 𝑎+𝑏+𝑚 −𝑚 >𝑎+𝑏 1 2 or 𝑚 −𝑚 >0 1 2 or 𝑚 >𝑚 , which is true according to the third case. Thus 𝐴>𝐵. 1 2 8.2. Consequence All intervals, included in [0, 1], with the same midpoint are considered neutrosophically equal. 𝐶(𝑚)={[𝑚−𝑎,𝑚+𝑎],where all 𝑚,𝑎,𝑚−𝑎,𝑚+𝑎 ∈[0,1]} represents the class of all neutrosophically equal intervals included in [0, 1] whose midpoint is 𝑚. Florentin Smarandache, The Score, Accuracy, and Certainty Functions determine a Total Order on the Set of Neutrosophic Triplets (T, I, F) Neutrosophic Sets and Systems, Vol. 38, 2020 8 i) If 𝑚=0 or 𝑚 =1, there is only one interval centered in 0, i.e. [0, 0], and only one interval centered in 1, i.e. [1, 1]. ii) If 𝑚∉{0,1}, there are infinitely many intervals from [0, 1], centered in 𝑚. 8.3. Consequence Remarkably we can rank an interval [𝑎,𝑏]⊆[0,1] with respect to a number 𝑛 ∈[0,1] since the number may be transformed into an interval [𝑛,𝑛] as well. For example [0.2,0.8]>0.4 since the midpoint of [0.2, 0.8] is 0.5, and the midpoint of [0.4, 0.4]= 0.4, hence 0.5>0.4. Similarly, 0.7>(0.5,0.8). 9. Interval (-Valued) Neutrosophic Score, Accuracy, and Certainty Functions Let 𝑇,𝐼,𝐹 ⊆[0,1] be three open, semi-open / semi-closed, or closed intervals. Let 𝑇𝐿 =𝑖𝑛𝑓𝑇 and 𝑇𝑈 =𝑠𝑢𝑝𝑇; 𝐼𝐿 =𝑖𝑛𝑓𝐼 and 𝐼𝑈 =𝑠𝑢𝑝𝐼; 𝐹𝐿 =𝑖𝑛𝑓𝐹 and 𝐹𝑈 =𝑠𝑢𝑝𝐹. Let 𝑇𝐿,𝑇𝑈,𝐼𝐿,𝐼𝑈,𝐹𝐿,𝐹𝑈 ∈[0,1], with 𝑇𝐿 ≤𝑇𝑈,𝐼𝐿 ≤𝐼𝑈,𝐹𝐿 ≤𝐹𝑈. We consider all possible types of intervals: open (a, b), semi-open / semi-closed (a, b] and [a, b), and closed [a, b]. For simplicity of notations, we are using only [a, b], but we understand all types. Then 𝐴=([𝑇𝐿,𝑇𝑈],[𝐼𝐿,𝐼𝑈],[𝐹𝐿,𝐹𝑈]) is an Interval Neutrosophic Triplet. 𝑇𝐿 is the lower limit of the interval 𝑇, 𝑇𝑈 is the upper limit of the interval 𝑇, and similarly for 𝐼𝐿,𝐼𝑈, and 𝐹𝐿,𝐹𝑈 for the intervals 𝐼, and respectively 𝐹. Hong-yu Zhang, Jian-qiang Wang, and Xiao-hong Chen [2] in 2014 defined the Interval Neutrosophic Score, Accuracy, and Certainty Functions as follows. Let’s consider int([0, 1]) the set of all (open, semi-open/semi-closed, or closed) intervals included in or equal to [0, 1], where the abbreviation and index int stand for interval, and Zhang stands for Hong-yu Zhang, Jian-qiang Wang, and Xiao-hong Chen. 9.1. Zhang Interval Neutrosophic Score Function sZhang :{int([0,1])}3 int([0,1]) int 𝑆𝑍ℎ𝑎𝑛𝑔(𝐴)=[𝑇𝐿+1−𝐼𝑈+1−𝐹𝑈, 𝑇𝑈+1−𝐼𝐿+1−𝐹𝐿] (11) 𝑖𝑛𝑡 9.2. Zhang Interval Neutrosophic Accuracy Function aZhang :{int([0,1])}3 int([0,1]) int aZhang(𝐴)=[𝑚𝑖𝑛{𝑇𝐿−𝐹𝐿,𝑇𝑈−𝐹𝑈},𝑚𝑎𝑥{𝑇𝐿−𝐹𝐿,𝑇𝑈−𝐹𝑈}] (12) int 9.3. Zhang Interval Neutrosophic Certainty Function cZhang :{int([0,1])}3 int([0,1]) int 𝑐𝑍ℎ𝑎𝑛𝑔(𝐴)=[𝑇𝐿, 𝑇𝑈] (13) 𝑖𝑛𝑡 Florentin Smarandache, The Score, Accuracy, and Certainty Functions determine a Total Order on the Set of Neutrosophic Triplets (T, I, F) Neutrosophic Sets and Systems, Vol. 38, 2020 9 9. New Interval Neutrosophic Score, Accuracy, and Certainty Functions Since comparing/ranking two intervals is equivalent to comparing/ranking two members (i.e. the intervals’ midpoints), we simplify Zhang Interval Neutrosophic Score (𝑆𝑍ℎ𝑎𝑛𝑔), Accuracy 𝑖𝑛𝑡 (𝑎𝑍ℎ𝑎𝑛𝑔), Certainty (𝑐𝑍ℎ𝑎𝑛𝑔) functions, as follows: 𝑖𝑛𝑡 𝑖𝑛𝑡 sFS :{int([0,1])}3 [0,1] int aFS :{int([0,1])}3 [1,1] int cFS :{int([0,1])}3 [0,1] int where the upper index FS stands for our name’s initials, in order to distinguish these new functions from the previous ones: 10.1. New Interval Neutrosophic Score Function sFS(([𝑇𝐿,𝑇𝑈],[𝐼𝐿,𝐼𝑈],[𝐹𝐿,𝐹𝑈]))=𝑇𝐿+𝑇𝑈+(1−𝐼𝐿)+(1−𝐼𝑈)+(1−𝐹𝐿)+(1−𝐹𝑈)=4+𝑇𝐿+𝑇𝑈−𝐼𝐿−𝐼𝑈−𝐹𝐿−𝐹𝑈, int 6 6 which means the average of six positivenesses; 10.2. New Interval Neutrosophic Accuracy Function aFS(([𝑇𝐿,𝑇𝑈],[𝐼𝐿,𝐼𝑈],[𝐹𝐿,𝐹𝑈]))=𝑇𝐿+𝑇𝑈−𝐹𝐿−𝐹𝑈 , which means the average of differences int 2 between positiveness and negativeness; 10.3. New Interval Neutrosophic Certainty Function cFS(([𝑇𝐿,𝑇𝑈],[𝐼𝐿,𝐼𝑈],[𝐹𝐿,𝐹𝑈]))=𝑇𝐿+𝑇𝑈, int 2 which means the average of two positivenesses. 10.4. Theorem Let ℳ ={(𝑇,𝐼,𝐹),where 𝑇,𝐼,𝐹 ⊆[0,1],and 𝑇,𝐼,𝐹 are intervals} , be the set of interval 𝑖𝑛𝑡 neutrosophic triplets. The New Interval Neutrosophic Score, Accuracy, and Certainty Functions determine a total order relationship on the set ℳ of Interval Neutrosophic Triplets. 𝑖𝑛𝑡 Proof. Let’s assume we have two interval neutrosophic triplets: 𝑃 =([𝑇𝐿,𝑇𝑈],[𝐼𝐿,𝐼𝑈],[𝐹𝐿,𝐹𝑈]), 1 1 1 1 1 1 1 and 𝑃1 =([𝑇2𝐿,𝑇2𝑈],[𝐼2𝐿,𝐼2𝑈],[𝐹2𝐿,𝐹2𝑈]), both from Mint. We have to prove that: either 𝑃 >𝑃 , or 𝑃 <𝑃 , or 𝑃 =𝑃 . 1 2 1 2 1 2 Apply the new interval neutrosophic score function (sFS) to both of them: int sFS(𝑃 )=4+𝑇1𝐿+𝑇1𝑈−𝐼1𝐿−𝐼1𝑈−𝐹1𝐿−𝐹1𝑈 int 1 6 sFS(𝑃 )=4+𝑇2𝐿+𝑇2𝑈−𝐼1𝐿−𝐼1𝑈−𝐹1𝐿−𝐹1𝑈 int 2 6 Florentin Smarandache, The Score, Accuracy, and Certainty Functions determine a Total Order on the Set of Neutrosophic Triplets (T, I, F) Neutrosophic Sets and Systems, Vol. 38, 2020 10 If sFS(𝑃 )> sFS(𝑃 ), then 𝑃1 >𝑃2. int 1 int 2 If sFS(𝑃 )< sFS(𝑃 ), then 𝑃1 <𝑃2. int 1 int 2 If sFS(𝑃 )= sFS(𝑃 ), then we get from equating the above two equalities that: int 1 int 2 𝑇𝐿+𝑇𝑈−𝐼𝐿−𝐼𝑈−𝐹𝐿−𝐹𝑈 =𝑇𝐿+𝑇𝑈−𝐼𝐿−𝐼𝑈−𝐹𝐿−𝐹𝑈 1 1 1 1 1 1 2 2 2 2 2 2 In this problematic case, we apply the new interval neutrosophic accuracy function (aFS) to both 𝑃 and 𝑃 , and we get: int 1 2 TL TU FL FU aFS(P) 1 1 1 1 int 1 2 TL TU FL FU aFS(P) 2 2 2 2 int 2 2 If aFS(𝑃 )> aFS (𝑃 ), then 𝑃1 >𝑃2. int 1 int 2 If aFS(𝑃 )< aFS (𝑃 ), then 𝑃1 <𝑃2. int 1 int 2 If aFS(𝑃 )= aFS (𝑃 ), then we get from equating the two above equalities that: int 1 int 2 𝑇𝐿+𝑇𝑈−𝐹𝐿−𝐹𝑈 =𝑇𝐿+𝑇𝑈−𝐹𝐿−𝐹𝑈 1 1 1 1 2 2 2 2 Again, a problematic case, so we apply the new interval neutrosophic certainty function (cFS) int to both 𝑃 and 𝑃 , and we get: 1 2 cFS(𝑃 )=𝑇𝐿+𝑇𝑈 int 1 1 1 cFS(𝑃 )=𝑇𝐿+𝑇𝑈 int 2 2 2 If cFS(𝑃 )> cFS(𝑃 ), then 𝑃1 >𝑃2. int 1 int 2 If cFS(𝑃 )< cFS(𝑃 ), then 𝑃1 <𝑃2. int 1 int 2 If cFS(𝑃 )= cFS(𝑃 ), then we get: int 1 int 2 𝑇𝐿+𝑇𝑈 =𝑇𝐿+𝑇𝑈 1 1 2 2 We prove that in the last case we get: 𝑃 = 𝑃 (or 𝑃 is neutrosophically equal to 𝑃 ). 1 𝑁 2 1 2 We get the following linear algebraic system of 3 equations and 12 variables: 𝑇𝐿+𝑇𝑈−𝐼𝐿−𝐼𝑈−𝐹𝐿−𝐹𝑈 =𝑇𝐿+𝑇𝑈−𝐼𝐿−𝐼𝑈−𝐹𝐿−𝐹𝑈 1 1 1 1 1 1 2 2 2 2 2 2 { 𝑇𝐿+𝑇𝑈−𝐹𝐿−𝐹𝑈 =𝑇𝐿+𝑇𝑈−𝐹𝐿−𝐹𝑈 1 1 1 1 2 2 2 2 𝑇𝐿+𝑇𝑈 =𝑇𝐿+𝑇𝑈 1 1 2 2 Florentin Smarandache, The Score, Accuracy, and Certainty Functions determine a Total Order on the Set of Neutrosophic Triplets (T, I, F)