Objectives: To develop a new parametric measure of entropy for Intuitionistic Fuzzy Sets and to study its application in selecting the best project for a company. Methods: We have used fivepoint Likert scale to determine the feedback of different projects and devised a method based on parametric intuitionistic fuzzy entropy to select the best project for the company. Findings: In this study, a parametric measure of entropy is being proposed for intuitionistic fuzzy sets. It is a generalized version of entropy for intuitionistic fuzzy sets. Properties of the proposed entropy have also been discussed. The proposed entropy is compared with other existing measures of entropy for intuitionistic fuzzy sets using a numerical example. Finally, the application of proposed entropy in selecting the best project is studied. Novelty: The proposed entropy is stable with the structured linguistic variables which most of the entropies are not. We have used a standard method based on parametric intuitionistic fuzzy entropy measure to determine the weights of each criterion which makes it more generalized and flexible. Instead of depending just on their own knowledge, decisionmakers may use the suggested model to better comprehend the relative importance of each criteria for the selection of best project.
As a result of pandemics and geopolitical issues, the level of global business uncertainty has spiked in the past few years. These new conditions aim to employ various models in business that can deal with uncertainty, which is also pertinent to project management concerns. Typically, decisionmaker evaluations, including preference information, are expressed in linguistic terms. The values of a Linguistic variable are language elements that include words and phrases. Atanassov introduced the intuitionistic fuzzy sets theory, which permits vagueness to be represented quantitatively. When confronted with this decision environment, specialists cannot always express their opinions using
Numerous techniques such as TOPSIS, MULTIMOORA, TODIM, and VIKOR have been adopted for decisionmaking problem.
Managers in any organisation will only take on projects that they believe have a good chance of succeeding. Successful project selection is crucial for every business; hence this topic has received extensive attention. Finding the complicated tasks that need extra care and attention is essential for working on a successful project. A large amount of research has been devoted to the study of project selection because of its importance to managers. Hamdan et al.
Multicriterion project portfolio selection with an emphasis on environmental impact was the subject of research by Ma et al.
The objective of this paper is to introduce a new parametric measure of entropy for IFS and study its application in selecting the best project for an organisation. In Section 2, we discuss the methodology used in the paper. In section 3, the parametric measure of entropy for IFS is proposed with its proof of validity, properties of proposed entropy are studied, and an illustrative numerical instance is provided to compare the proposed entropy with existing measures of entropy and the application of proposed entropy in selecting the best project is studied.
A = {
where
A= {
where
(e1) e (A) =0 if A is a crisp set.
(e2) e (A) achieves its maximum at
(e3) e (A)
(e4) e (A)
(E1) E(A)=0 if A is a crisp set.
(E2) E(A) achieves its maximum at
(E3) E(A)
(E4) E(A)
Atanassov
A
A
A= B iff A
Now, we define some of the existing entropies on IFS.
Let A be an IFS in
Szmidt and Kacprzyk
E_{sk}(A)=
Burilli & Bustince
E_{BB} (A) =
Zeng and Li
E_{ZL}= 1(1/n)
Several writers, including Ma et al.
We first define a parametric measure of entropy on IFS and then illustrate with a numerical example to show that the behaviour of proposed intuitionistic fuzzy entropy
Step1: Criteria Selection: Firstly, the criteria are selected against which the performance of each project will be tested.
Step 2: Collecting the Project Performance Data: We use Likert five point scale to record the rating of each project against the criteria chosen in Step 1.
Step 3: Calculation of membership, non membership and hesitancy degree: We use the method given in to calculate Project Performance Data in terms of intuitionistic fuzzy values.
Step 4: Calculation of intuitionistic fuzzy entropy values for each project with respect to each criteria: We calculate the intuitionistic fuzzy entropy values for each project using the proposed intuitionistic fuzzy entropy given in equation (1)
Step 5: Determination of weight of each criteria: We determine the weight of each criteria by multiplying the normalized value of each row with the normalized quantity of corresponding column.
Step 6: Calculation of score of each project with respect to each criteria: We calculate the score of each project with respect to each criteria using equation
Step 7: Ranking of the projects: Calculating the sum of scores of each project, we give rank to the projects.
On the basis of fuzzy entropy
Let A be an IFS with universe of discourse
(P1)
(P2)
(P3)
Then,
Also, we know that
Then,
Conversely, Let
Then,
Where,
Differentiating (14) w.r.t
Again Differentiating, we get
At
and
Thus,
i.e
where
Taking partial derivative of
For finding critical point, we put
From (24) and (25)
Thus, from the monotonicity of function
Theorem 3.2: Let A and B be two IFS on a universe of discourse
Let A =
De et al.
Consider the AIFS A on
A = {<6, 0.1, 0.8>, <7, 0.3, 0.5>,<8, 0.5, 0.4>,<9, 0.9, 0.>,<10, 1.0, 0.0>}.
De et al.
Presently we consider these AIFSs to analyze the above entropy measures. From logical consideration, the entropies of these AIFSs are required to follow the following order pattern: E(


A 




0.761 
0.766 
0.742 
0.708 
0.674 

0.696 
0.704 
0.640 
0.567 
0.507 

0.646 
0.655 
0.559 
0.459 
0.397 

0.545 
0.528 
0.367 
0.262 
0 .208 

0.536 
0.516 
0.349 
0.247 
0.194 

0.523 
0.503 
0.332 
0.227 
0.176 

0.768 
0.771 
0.750 
0.721 
0.694 

0.707 
0.714 
0.655 
0.584 
0.524 

0.658 
0.662 
0.574 
0.484 
0.414 

0.538 
0.518 
0.353 
0.248 
0.196 

0.524 
0.503 
0.329 
0.224 
0.177 

0.505 
0.484 
0.300 
0.197 
0.154 

0.752 
0.756 
0.725 
0.686 
0.661 

0.683 
0.688 
0.613 
0.536 
0.474 

0.639 
0.640 
0.539 
0.446 
0.380 

0.558 
0.542 
0.389 
0.326 
0.227 

0.553 
0.535 
0.379 
0.275 
0.219 

0.549 
0.530 
0.371 
0.266 
0.212 
E_{s} 
0.404 
0.414 
0.348 
0.299 
0.264 
E_{sk} 
0.319 
0.307 
0.301 
0.212 
0.176 
E_{bb} 
0.462 
0.600 
0.660 
0.672 
0.680 
According to
One of the goals of a hightech software firm is to boost productivity by choosing an appropriate project. Before making the investment in developing new information systems, businesses should engage in strategic planning to ensure that they fully grasp the strategy and how the project will be used to achieve business goals. The process of choosing an information project is crucial to the success of many businesses. The goal of the selection procedure is to identify the best possible solution from among the available options.
Here, we illustrate the case of company’s selection of the difficult project. Often, the complexity of such projects is high, and the organization is having trouble keeping track of them all and allocating resources effectively. Optimal management and distribution of the company's scarce resources towards the most difficult tasks is of the utmost importance. Projects can be prioritised according to their level of complexity to assist the business stay organised and keep deliveries on schedule. After preliminary screening, four projects P1, P2, P3 and P4 are considered for further examination. To determine which project is best, we consider the four criterions for project selection: 1) profitability (C1), 2) organizational goals (C2) 3) competitive response (C3) and 4) availability of competent employees (C4). When traditional quantitative representations are inadequate or insufficient, language phrases or variables are frequently and effectively used to describe the circumstances. In our case, company’s managers including project managers give ratings to the projects P1, P2, P3 and P4 in terms of linguistic variables as “very poor”, “poor”, “fair”, “good” “very good”. Table 1 shows the rating of decisionmaking panel to the four projects P1, P2, P3 and P4.





P1 
G 
VG 
VG 
F 
P2 
VG 
F 
G 
VG 
P3 
P 
VG 
VG 
VP 
P4 
VG 
P 
P 
VG 
In





P1 
7 
9 
9 
5 
P2 
9 
5 
7 
9 
P3 
3 
9 
9 
1 
P4 
9 
3 
3 
9 
To convert, the above data into intuitionistic fuzzy values we use the method given by DengChan
Distances from each value to the lowest value were used to determine membership degrees, distances from each value to the highest value were used to determine nonmembership degrees, and distances from each value to the average of others were used to get the intuitionistic fuzzy index values. These estimated values are then each divided by the sum of their component parts. The intuitionistic fuzzy degrees, for instance, are computed here for P1:





P1 
(0.667,0.333,0.0) 
(0.706,0.0,0.294) 
(0.75,0.0,0.25) 
(0.444,0.444,0.111) 
P2 
(0.75,0.0,0.25) 
(0.267,0.533,0.20) 
(0.667,0.333,0.0) 
(0.727,0.0,0.273) 
P3 
(0.0,0.60,0.40) 
(0.706,0.0,0.294) 
(0.75,0.0,0.25) 
(0.0,0.615,0.385) 
P4 
(0.75,0.0,0.25) 
(0.0,0.632,0.368) 
(0.0,0.60,0.40) 
(0.727,0.0,0.273) 
Using equation (1), we calculate the intuitionistic fuzzy entropy values (for
(
Second Normalized value: (
Similarly, we can calculate the normalized values for horizontal group.







P1 
0.888 
0.502 
0.438 
1 
2.828 
1 
P2 
0.438 
0.929 
0.888 
0.471 
2.726 
1.102 
P3 
0.64 
0.502 
0.438 
0.622 
2.202 
1.626 
P4 
0.438 
0.600 
0.640 
0.471 
2.149 
1.679 
Sum 
2.404 
2.533 
2.404 
2.564 


Normalized value 
1.16 
1.031 
1.16 
1 


The coefficient of each criterion for each project is now represented by multiplying the normalised values of every row by the normalised values of every column. If we take P2 as an example, we find that C2's coefficient with regard to that P2 is = 1.136 (1.102





P1 
1.16 
1.031 
1.16 
1 
P2 
1.278 
1.136 
1.278 
1.102 
P3 
1.886 
1.676 
1.886 
1.626 
P4 
1.948 
1.731 
1.948 
1.679 
Sum 
6.272 
5.574 
6.272 
5.407 
Weights 
0.267 
0.237 
0.267 
0.230 
Lastly, the relative relevance of each criterion with regard to each project is calculated by multiplying the score of each intuitionistic fuzzy value with the weight of every criterion. Score value for each IFS value (







P1 
0.089 
0.167 
0.200 
0 
0.456 
1 
P2 
0.200 
0.063 
0.089 
0.167 
0.393 
2 
P3 
0.160 
0.167 
0.200 
0.141 
0.066 
3 
P4 
0.200 
0.150 
0.160 
0.167 
0.057 
4 
According to the data in
P1→ P2 →P3→ P4. Hence P1 is the best alternative. Also by Rahimi’s method
In this study, we have proposed a new parametric measure of entropy for IFS. The properties of proposed entropy have also been studied. A numerical model is demonstrated to check the reliability of proposed entropy at various values of α and β. The entropy is steady with the structured linguistic variables if value
The utilisation of enhanced intuitionistic fuzzy entropy enables a comprehensive and efficient depiction of fuzzy information in the assessment of innovation capability, encompassing uncertain and unknown factors. This approach enhances the precision and impartiality of evaluation outcomes to a certain degree, and presents a viable solution to the intuitionistic fuzzy multiattribute problem.
When selecting a project, every organization requires a costeffective selection procedure. Expertise alone cannot always decide. We utilized intuitionistic fuzzy entropy to evaluate projects. Expertise matters when selecting a service, particularly when determining a weight range. Our method reduces the influence of subject matter experts on decisionmaking by calculating weight differently.
The entropy measure presented in this paper incorporates the decision maker's outlook during the decisionmaking process by means of parameter selection. Proposed entropy is more generalised and provides flexibility in the approach.
Authors are thankful to the editor, referees of the journal and to all the authors whose names are mentioned in the reference list