Estimating the overall efficiency of storage batteries in Fermatean picture fuzzy environment

. Electricity storage devices play a very vital role in e ﬃ cient energy management. Storage technologies vary depending on the speciﬁc needs and applications. Battery is a electric storage system which converts chemical energy into electrical energy and possess some speci ﬁc parameters that characterize them. Here, four types of storage batteries are evaluated based on their e ﬃ ciency, cost, life span and energy density properties using COPRAS and ARAS methods on Fermatean picture fuzzy sets. Firstly a decision matrix is constructed using Fermatean picture fuzzy sets and it is normalized. Entropy measure is deﬁned which helps in calculation of weight values. Next aggregated weighted normalized Fermatean picture fuzzy matrix is found and using this relative signiﬁcant and utility values are calculated in COPRAS method. In ARAS method optimal function values are determined and utility values are found. The alternatives are ranked based on the utility values. In both the methods it is found that the rank of the best alternative is the same whereas the ranking of other alternatives vary.


Introduction
Fuzzy Set introduced by Zadeh [1] was extended to IFS by Atanassov [2].Bathrinath et al. [3] expressed factors that influence sustainability in ship ports by F COPRAS.Bekar et al. [4] utilized F COPRAS method for evaluating the performance measure in TPM.Bokovic et al. [5] used CRITIC-ARAS method for selecting mobile network operators.Coung et al. [6] developed the concept of PF sets.Fouladgar et al. [7] applied AHP and F COPRAS for strategy selection.Garg et al. [8,9] applied SWARA-COPRAS for ERP selection and ranked e-learning websites using F COPRAS.Goswami et al. [10,11] developed COPRAS and ARAS model for optimizing EDM parameters and in the selection of best engineering materials.Further in [12,13] they implemented ARAS decision making methodology for Robot and mobile selection.Kumari et al. [14] solved MCDM problem with IF COPRAS method.San Martn et al. [15] used energy storage technology for electric applcations.Shaikh et al. [16] used integrated F COPRAS method for selection of optimum material to improve braking system in automobiles.Senapati et al. [17] proposed the concept of FFS and defined fundamental operations and Euclidean Distance on FFS.Motivated by these concepts we introduce Fermatean fuzzy sets and its application.Definition 1.1.Let X be the Universal set.A Fermatean picture fuzzy set (FPF) A in X is defined as follows :

Procedure of COPRAS Method
This section deals with the procedure for FPF COPRAS method and an illustrative example.
Step 1 : Construct the initial FPF decision matrix.
Step 2 : Compute the NFPFD matrix by Definition 1.3.
Step 3 : Compute the FPF entropy values Ej.
Step 4 : Calculate the FPF weight values Wj.
Step 6 : Compute AWNFPFD matrix and the weighted sum of beneficial and nonbeneficial attribute by Definitions 1.7 and 1.8.
Step 8 : Determine relative significance of alternatives by Definition 1.9.
Step 9 : Calculate the FPF utility values by Definition 1.10 and rank the alternatives.

Fig.1. Types of Batteries
Storage devices are integrated into electrical network to guarantee the authenticity and quality of power systems.Let FPF sets A1,A2,A3 and A4 represent four types of batteries such as Lithium Ion, Sodium sulphur, Nickel cadmium and Zinc bromine which are evaluated based on efficiency, cost, life span and operating temperature characteristics of the batteries.The best type of battery is found using FPF COPRAS and ARAS methods.
Step 1 : Construct the FPF decision matrix as given below.The highest utility value is the best choice among the alternatives.

Procedure for ARAS method:
Steps 1 to 6 are calculated as in COPRAS method.
Step 7: Compute optimality function values by Definition 3.1.
Step 8: Calculate the degree of utility by Definition 3.2 and rank the alternatives.

Illustrative Example:
FPF decision matrix of COPRAS method is taken.Steps 1 to 6 are repeated as in COPRAS method.

Conclusion
Here both COPRAS and ARAS methods on FPF sets are dealt with.FPF decision matrix is constructed.An entropy measure on FPF sets is defined and is used for calculating the weights.Next weighted normalized decision matrix is found and is aggregated.Using COPRAS method weighted sum of beneficial and non beneficial attributes are calculated and are used for finding the relative significant and utility values.In ARAS method optimal function values are used in calculating the utility values.In both the methods it is found that the rank of the best alternative is same.However, the ranking order of the other alternatives vary.
only condition that a FPF set must satisfy is 1

Definition 1 . 5 .Definition 1 . 6 .Definition 1 . 7 .Definition 1 . 8 .FQ
Weight values of FPF sets are determined using entropy measure as, Weighted normalized FPF decision matrix wnpf F is defined as, Weighted aggregate normalized FPF decision matrix afp F is defined as, and non-beneficial attributes of the matrix the aggregate values are added.the beneficial and non-beneficial attributes of the weighted aggregate normalized FPF decision matrix.Weighted sum of beneficial and non-beneficial attributes are defined as, The FPF significant relative value of each alternative is .is the FPF highest significance relative value.The utility values of the alternatives

Step 7 :Step 8 : 1 A
The optimality function values are The degree of the utility for the alternatives are, , which is Lithium Ion.