Extension Vikor for Priority Orders Based on Three Parameters Interval Fuzzy Number

In this study, an improved VIKOR method was presented to deal with multi-attribute decision-making based on three parameters interval fuzzy number. The attribute weights were unknown but alternative priority of object preference was given. A new non-linear rewards and punishment method in positive interval was proposed to make the attributes normal, information covered reliability and relative superiority degree two methods were used to compare and sort the Three Parameters Interval Fuzzy Number (TPIFN) and a quadratic programming based on contribution was constructed to get attribute weights, then defined the information entropy distance between TPIFN and the optimum object orders was obtained by VIKOR. The numerical example was provided to demonstrate the feasibility and validity.


INTRODUCTION
Fuzzy analysis leads an important role in Multi-Attribute Decision Making (MADM) and it has made a great progress.Interval fuzzy number is usually be used to describe the fuzzy value, when it is difficult to describe the attribute value assuredly, but is easy to get the scale.The assumption of average but not fuzzy sets in interval is implicated in common two parameters interval fuzzy number and its lacking of fuzzy degree controlling lead to distortion and departure easily.Therefore some scholars define Three Parameters Interval Fuzzy Number (TPIFN) to overcome the shortcoming above (Bu and Zhang, 2001).The research of for TPIFN has just begun.The information aggregation of TPIFN is investigated (Wang, 2008).The method of three parameters interval grey correlation degree is presented (Luo, 2009).Lan and Fan (2009) define the operation and distance formula of TPIFN and advance the relevant TOPSIS method.But professor Opricovic (1998) indicated that there are bugs in TOPSIS and VIKOR can overcome the shortage and make the decision more in reason as a compromise sort method (Opricovic and Tzeng, 2004).The research of VIKOR based on TPIFN has not been reported for the moment.Besides, the relation research insufficient include: firstly, the existing research all most focus on specific values or relation to describe preference (Zeng, 2005;Xu and Zhao, 2004), but it is less in form of priority orders.Secondly, the normalizing methods are chiefly based on the principles of only award but no punishment, leading to low resolution.Moreover, it brings limitation for data processing because of negative data by rewards and punishment.Thirdly, the distance formulas are most use of Euclidean and other geometrical distance, their discrimination is low.In view of above, improved non-linear rewards and punishment method in [0, 1] is proposed to normalize the TPIFN.With alternative priority of known in advance, information covered reliability and relative superiority degree methods are given and a quadratic programming is constructed to get attribute weights, then TPIFN distance formula based on information entropy is defined and is introduced to VIKOR to make decision.
In this study, an improved VIKOR method was presented to deal with multi-attribute decision-making based on three parameters interval fuzzy number.The attribute weights were unknown but alternative priority of object preference was given.A new non-linear rewards and punishment method in positive interval was proposed to make the attributes normal, information covered reliability and relative superiority degree two methods were used to compare and sort the Three Parameters Interval Fuzzy Number (TPIFN) and a quadratic programming based on contribution was constructed to get attribute weights, then defined the information entropy distance between TPIFN and the optimum object orders was obtained by VIKOR.The numerical example was provided to demonstrate the feasibility and validity.

DESCRIPTION OF MODEL
TPIFN is the interval fuzzy number with three parameters, marked as . The known preference priority order is A is in the k-th position.1 2 , , , n i i i  is the sequence numbers of all object.Decision-maker make determination of object orders based on the information above.

MADM BASED ON EXTENSION VIKOR
Rewards and punishment standardization for TPIFN: The original matrix need to be transformed to eliminate dimension difference and repeated information.The existing "rewarding good and punishing bad" method make the data in [-1, 1], it ensure the justice, but the negative lead to programming data limit.Such as the distance of information entropy after, it demands evaluate data is positive.So improve the non-linear rewarding and punishing transform according to method of "rewarding good and punishing bad", but the normalizing value are still in [0,1].Based on translation and scaling independence (Peng and Xiao, 2011), the basic thought is: translating and scale [-1, 1] interval, if the objects are better than average, assignment scores for higher than 0.5 and lower contrarily.Suppose: And 1 I is fixation attribute sets, 4 I is deviation attribute sets: The improved methods enlarge the differences between values and it increases the resolving degree.
Attribute weights based on alternative priority: By principle of Linear assignment method, quadratic programming model is constructed based on contribution and aim to make the difference between subjective and objective factors minimum, then compute the weight by Lagrange method.
First of all, get the single ranking alternative of all object in every attribute and construct the single ranking There are two methods to get the orders alternative for TPIFN.
First is based on information covered reliability of TPIFN.There are n information in information sets of TPIFN a and i E is the cover of information i , is the length of cover E .
Definition 1: is the reliability of 1 2 a a ≥ .0 1 α ≤ ≤ as a constant reflect the recognition degree of decision-maker for taking center of gravity as compared criterion.is the reliability rate of center of gravity compared with TPIFN: Second is based on relative superiority degree.Define the relative superiority degree about TPIFN.
Base on the compared relationship of θ , p′ with 0.5, the two methods above can be used to make the order in every attribute to construct single order matrix D .The total difference between objective evaluation and subjective preference can be described by σ and when it is the minimum, the result achieve optimum.Quadratic programming model is constructed based above:

If the object priority orders is 1 2
ω can be achieved by solve the model。

VIKOR based on distance of information entropy:
The basic thoughts of VIKOR are definition of positive and negative ideal solution and give the orders of object according to the close degree between objects and ideal ones.The distance between TPIFN is the foundation of MADM.The existing researches adopt geometrical distances mostly, but they are difficult to rank the point lie in perpendicular bisector between two objects accurately, it is low in discrimination.Based on information entropy theory, measurement the difference between X and Y by K-L distance, give the definition of the relative entropy distance.
Definition 3: The relative entropy of TPIFN between . When, ) log 0 1 0.5( ) • The thought of object priority orders is adopted to construct quadratic programming model based on covered reliability or relative superiority degree two kinds of methods and weights * j ω are got • Get positive and negative ideal solution ) ) show decision aim to make the most of i S ; 0.5 λ = show decision according to make the balanced of i S and i R .
0.5 λ < show decision aim to make the minimum of i R • Get the order by i Q .The value of i Q is smaller, the corresponding object is more excellent.

CONCLUSION
An improved VIKOR method is given to research MADM with preference alternative priority based on TFIFN.The common normalizing method and distance formula are improved to advance discrimination.The result of example satisfies the fact, it is feasible and effective.And the principle is simple and easy to operate.The method can be applied to many varieties of decision-making.
i A is at k-th position in priority orders, otherwise 0 ik c = .[ ] ik n n π × is the matrix of total contribution for all objects, Improve traditional VIKOR based above and the process of extending method is:• Transform TPIFN matrix to eliminate dimension difference based on rewards and punishment rules,