Use of Intuitionistic Fuzzy Numbers in Survey Sampling Analysis with Application in Electronic Data Interchange

Department of Mathematics and Statistics, International Islamic University Islamabad, Islamabad, Pakistan Department of Statistics, Quaid-i-Azam University, Islamabad, Pakistan Department of Mathematics, Faculty of Arts and Science, Yildiz Technical University, Esenler, 34210 Istanbul, Turkey Department of Logistics, University of Defence in Belgrade, Belgrade, Serbia Quantum Leap Africa (QLA), AIMS Rwanda Center, Remera Sector KN 3, Kigali, Rwanda Institut de Mathématiques et de Sciences Physiques (IMSP/UAC), Laboratoire de Topologie Fondamentale, Computationnelle et Leurs Applications (Lab-ToFoCApp), BP 613, Porto-Novo, Benin African Center for Advanced Studies, P.O. Box 4477, Yaounde, Cameroon


Introduction
e traditional way of conducting research in social sciences, marketing, business, economics, health services, and psychological research is survey sampling. In these fields, information is collected from a sample of individuals through their responses to certain questions. Survey sampling research is categorized into quantitative (using a questionnaire with numerically rated items), qualitative (using a questionnaire with binary, open-ended, closed-ended, and rating scale questions) and mixed research procedures (both numerically rated and rating scale questions). In qualitative research, binary and rating scale questions are coded and then statistically analyzed, which leads to bias results as human thinking is full of ambiguity. Such types of questions are fuzzy in nature and cannot be expressed by binary and ordered numbers. Survey questions can be of various types, for example, multiple choice questions, rating scale questions, Likert scale questions, and matrix questions. Likert scale questions are widely used in social sciences, marketing business, and psychological research as these are perfect in measuring respondent's attitudes and behavior. For example, "there were enough toppings on my pizza," and the ratings might strongly agree, agree, neutral, disagree, and strongly disagree. ese ratings are linguistic variables and the respondents are asked to pick just one answer; marked answer will get one numerical value while unmarked linguistic variables will be considered as zero. In multiple answer rating questions, respondents are asked to mark multiple answers. e representation of linguistic variables as a crisp variable does not represent the ambiguity of the respondent's mind. us, the linguistic variables are naturally vague. e problem of imprecise crisp response can be overcome by using an intuitionistic fuzzy (IF) response. In IF sense, respondents are requested to report responses in the form of the grade of membership and grade of nonmembership to linguistic variables at the given hesitancy level, where grade of membership and nonmembership are the values between zero and one, and their sum should be less than one. In 2020, Khani and Afkhami [1] conducted a sample survey using the interview method to analyze the issues which are related to the relationship between the language and the community. ey examined the phonological changes in the Turkish dialect of Qazvin, according to the effects of social variables, such as age and gender.
In 1965, Zadeh [2] defined the grade of membership of a component belonging to a universal set as a value between zero and one, but the grade of nonmembership is not a complement of the grade of membership because of the hesitancy grade. Later, the fuzzy number was introduced Zadeh [3,4] to represent the linguistic variables. Lin and Lee [5,6] discussed survey sampling in fuzzy environment. Li et al. [7] used fuzzy sampling survey to distinguish the endogenous and exogenous factors influencing regional tourism and developed theory to explain the basic countermeasures for promoting regional tourism cooperation in fuzzy sense. Tavares and Betti [8] used fuzzy set approach as complements to measure multidimensional poverty within the context of the coronavirus pandemic. ey used rank correlation analyzes and proposed indexes that can trace the trends in increasing infection and a higher mortality rate in vulnerable regions. Compared to headcount ratio results, the fuzzy measures have more precise outcomes and are better able to capture the evolution in mortality patterns.
In 1986, Atanassove [11] introduced the concept of intuitionistic fuzzy set(IFS), as an extension of fuzzy set theory to deal with imprecision. In 1994, Atanassove [9] proved its significance and importance in dealing with vagueness and described membership degree, nonmembership degree, and the degree of hesitation to each element of a set. Radhika and Parvathi [12] and angavelu et al. [13] described the stages of IFS as (i) Intuitionistic fuzzification (converting crisp into membership and nonmembership function) (ii) Application of operation rules of IFS (iii) Intuitionistic defuzzification (converting membership and nonmembership to crisp) In 2021, Isik and Kaya [14] combined the α-cut technique of intuitionistic fuzzy set theory in acceptance sampling with the linguistic approach to allow defining with multiple α values for different product segments with the help of numerical examples. In 2020, Yuhana et al. [15] applied proposed rough and fuzzy sets to investigate the important factors that affect the value of the minimum passing level (MPL) of competency achievement. ey predicted the category of the MPL using the combination of rough sets and fuzzy signatures method and evaluated its performance. ey collected data from fifteen headmasters and sixty teachers of elementary schools participated. Based on the experiment with 203 objects' data, their proposed method gave 97% accuracy in the identification of important factors and prediction of MPL. Precup et al. [16] evolved fuzzy models (FMs) to develop an incremental online identification algorithm that characterize the nonlinear finger dynamics of the human hand for the myoelectric (ME)-based control of a prosthetic hand. ey designed five simple Takagi-Sugeno proportional-integral (PI) controllers and showed that these fuzzy controllers are cost effective because of the simplicity of their structure. ey proved with the simulation study that the best evolved FMs of the process have good performance in the control system with fuzzy controllers than the control system with linear ones. e fuzzy set theory is rapidly applied in all fields of statistics to deal with imprecise data and response; however, to the best of our knowledge, IF is not applied in survey sampling. is paper shows the significant contribution of IF in survey sampling at different hesitancy levels to find the aggregative investment benefit rate for the selection of the best facility location of multinational enterprises and compares the results with the fuzzy assessment method. A standard electronic format that puts back paperbased documents such as purchase orders or invoices is known as electronic data interchange (EDI). By self-operating paperbased transactions, organizations can save time and remove costly errors caused by manual processing. A survey report, in 2015, contains short analysis of the EDI implementation trends, based on data between 2005 and 2016, as well as estimation for year 2017 in various industries. In this report, Pakistan was not included in the list of countries using EDI standards in transport, warehousing, logistics, and delivery. is study also discusses the intuitionistic fuzzy assessment with the hesitancy level showing ambiguity in experts' mind, on the implementation of EDI.

Preliminaries
In 1965, Zadeh defined the fuzzy set as follows.
is called a fuzzy set; the function μ A is called the membership function of the set A. More precisely, it is defined as A fuzzy set is called normalized if μ A (X) � 1, for some x ϵ X. For α ϵ [0, 1], the set, is called α − level set of A. For example, if X is the set of real number and we define μ A : X ⟶ [0, 1] as μ A (x) � |sin x|∀x ϵ X, then A � (x, μ A ) is a fuzzy set, and for α � 0.5, A α � x ϵ X: |Sin x| ≥ 0.5 { } is an α − level set of A. e fuzzy number is a normalized fuzzy set A of the set of real numbers if, for any Definition 2. triangular fuzzy number was introduced with arithmetic properties to get better conclusion in decisionmaking and real-life uncertain situations than fuzzy logic. A � (a 1 , a 2 , a 3 ) is defined to be a triangular fuzzy number if its membership of μ A : R ⟶ [0, 1] is equal to Atanassov [10] modified the concept of standard fuzzy set by introducing the notion of nonmembership function ] A : X ⟶ [0, 1]. He termed his finding as intuitionistic fuzzy set(IFS). So, by an IFS, we mean the triplet (x, μ A , ] A ) and the relation between the elements of x, and their images under μ A and ] A can be expressed as (4) e idea of nonmembership function is not identical to the standard concept of probability of occurrence P and nonoccurrence (1 − P) of a certain event in classical probability theory. at is, in IFS, μ A (x) + ] A (x) ≤ 1 which means that there must be some degree of hesitancy in giving response to a particular event. For instance, X � N, Atanassov [9,10] defined the algebraic operations on IFS as follows: Definition 4. Zhang and Liu [17] described the operations on the intuitionistic fuzzy number as follows.

Definition 3. A triangular intuitionistic fuzzy number (TIFN) is
} be two triangular intuitionistic fuzzy numbers. e arithmetic operation on A * and B * are given below: Addition: Definition 5. angavelu et al. [13] defined average function to defuzzify a triangular intuitionistic fuzzy number.

Questionnaire in Sample Surveys
At the beginning of twenty first century, the decisionmaking problem about the selection of manufacturing facility locations is a demanding task. e conventional factors used in facility site selection were manufacturing site wages, infrastructure, education, workforce development, proximity to market, etc. Many new factors such as political stability, social harmony, trade regulations, and nature of governments' environmental consideration are added as pivotal factors in the decision-making problem of facility site selection. e achievement of a site selection scheme can be directly imputed to projects that begin with a checklist of matters and factors, appropriately weighted and estimated for the client's particular demands. Lin and Lee [6] proposed a new fuzzy assessment model to assist in finding the best facility site of multinational enterprises (MNEs) based on various investment environments. ey proposed the fuzzy assessment model which is not only easier but also closer to evaluator real thinking and more useful than the ones presented before. For such type of surveys, questionnaire may include main survey items and subitems may exist under each main item. We define them as Complexity 3 Main item: C 1 , C 2 , . . . , C n with weights: c 1 , c 2 , . . . , c n , respectively, subject to: 0 ≤ c i ≤ 1, i � 1, 2, . . . , n, and n i�1 a i � 1, Let A u , for u � 1, 2, . . . , k, be the k different linguistic variables such as very low, low, medium, high, and very high.
ese linguistic variables are rating scale question which requires a person to rate it along a well-defined, evenly spaced continuum. Rating scales are often used to measure the direction and intensity of attitudes of interviewee or respondents. e above questionnaire can be described as in Table 1. e rating scale questions can be answered in two ways, single rating choice answer and multiple-rating choice answer. In single rating choice answer, the respondent/interviewee is requested to say yes to only one option and zero to other options. And, in multiple-rating answer questions, respondents can be requested to say yes for more than one option.

e Crisp Mode.
Suppose we have a sample of t respondents drawn from the population; each respondent rate more than 1 option among A u , for u � 1, 2, . . . , k, respectively, denotes the selected option by 1, otherwise denoted by zero in the crisp case. e sample data are shown in Table 2.
Multiple-rating choice answer:

e Intuitionistic Fuzzy
Mode. e linguistic variables express the attitude of respondents, but the answer in yes coded as 1 and not coded as 0 cannot express the ambiguity in the mind of interviewee. If we request to interviewee to use intuitionistic fuzzy sense to give his response, the sample data can be obtained in the form of Table 3. In Table 3, μ ijlu ϵ [0, 1] and ] ijlu ϵ [0, 1] such that μ ijlu + ] ijlu + ϵ � 1, where 0 ≤ ϵ < 1 and satisfies the following conditions.
Single rating choice answer: Multiple rating choice answer: e average of the membership function and nonmembership function of t samples according to the above definitions, for each A u , is denoted by M ijlu and N ijlu , respectively:

Weights' Selection.
Every respondent/interviewee has to assign weights to each main item C i and subitem C ij according to relative importance. e traditional way is to assign ranks to each item. However, the crisp weight cannot express the hesitancy of the interviewee. So, we use the intuitionistic fuzzy number to express the grading of items. e range of grades is 0 to 10, and we express the grades into triangular intuitionistic fuzzy numbers, as listed in Table 4. e membership and nonmembership of grades are Suppose there are t interviewees which assign weights w ijl ϵ [W 0 , W 1 , . . . W 10 ], l � 1, 2, . . . , t, to each subitem C ijl . Let for the l th expert w ijl � (a 1l , a 2l , a 3l ), (a * 1l , a * 2l , a * 3l ) . e average weight of t experts is 4 Complexity and represented by w ij � (a 1 , a 2 , a 3 ), (a * 1 , a * 2 , a * 3 ) . en, defuzzified by equation (7), we have D ij � D w ij � a 1 + 4a 2 + a 3 + a * 1 + 4a * 2 + a * 3 12 .
. C nm n c nm n Table 2: e sample data in the crisp case with multiple-rating choice answer. Table 3: e sample data in intuitionistic fuzzy case with multiple-rating choice answer.

Complexity 5
Let where weights of the subitems C i1 , C i2 , . . . , C im i are c i1 , c i2 , . . . , c im i . Similarly, for t interviewees, we assign weights en, it is defuzzified as Let where weights of the main items C 1 , C 2 , . . . , C n are c i , c i , . . . , c n .

The Intuitionistic Aggregative Assessment for Survey Sampling
Let A � A 1 , A 2 , . . . , A k be the set of linguistic variables for each subitem. Using intuitionistic triangular fuzzy number and getting the rating (both single-and multiple-rating choice answer) in terms of the intuitionistic fuzzy mode, the results of survey sampling can be analyzed as follows: Step 1: averaging assessment results of all respondents, an IF (intuitionistic fuzzy) evaluation matrix is as follows: where g ij (R u ) � R u : μ iju , ] iju 〉|i � 1, 2, 3, . . . , n; j � 1, 2, . . . , m i }.
Step 2: the normalize weighted vector for subitems is c ij � (c i1 , c i2 , . . . , c im i ), and for the main item C i , the first stage aggregative assessment is as where is is called the first stage intuitionistic fuzzy assessment matrix R * .
Step 3: the normalized weight vector for main items is c i � (c 1 , c 2 , . . . , c n ).
e second stage intuitionistic fuzzy assessment is as follows: where is second stage aggregative assessment with respect to the set of intuitionistic rating A 1 , A 2 , . . . A k is

Numerical Example
In this section, we use example of Lin and Lee [6] to determine the aggregative benefit of investment for global facility site selection. In identifying the best facility site of multinational enterprises based on various environments and based on identifying the critical factors that will affect the ongoing business, the linguistic variables used by Lin and Lee [6] to rate factors are � very high, and A 7 � extrahigh to get aggregative benefit rate of facility site. ese variables represented in triangular intuitionistic fuzzy numbers are shown in Table 5.
Assume two evaluators give weights and responses of factors effecting investment benefit with hesitancy error 0.05 in Tables 6 and 7, respectively. en, by equations (7), (17), and (20), the aggregative of responses of two evaluators is shown in Table 8.
Step 1: the intuitionistic fuzzy average response matrix R is Step 2: the first stage intuitionistic fuzzy aggregative assessment for each main item C i is as   And, this IF aggregative benefit for facility site selection is represented as Defuzzifing the above, the IF aggregative investment membership and nonmembership rate, we have the IF investment benefit rate with 0.05 hesitancy rate is (0.452331, 0.021309). e fuzzy investment benefit rate calculated by Lin and Lee was 0.2039. us, the IF proposed methodology gives better results than the fuzzy approach. With the hesitancy level, the decision-making become more reliable in uncertain situations. e intuitionistic fuzzy aggregative investment benefit with different hesitancy levels are shown in Table 9.

Application.
Pakistan can boost its imports, exports, industry, and business by using electronic data interchange. e major cause of the stagnation of Pakistan's economy is the less use of EDI. To determine the intuitionistic fuzzy EDI implementation rate, the opinion is collected from experts rather than conducting a large opinion survey on the implementation of EDI. e hierarchical model on the implementation of EDI Pakistan is shown in Figure 1. e linguistic variables on the use of implementation of EDI in different fields are extra low, very low, low, middle, high, very high, and extra high.   [7,8,9]) F 41 ( [5,6,7], [8,9,10]