Software Implementation of the Optimal Temporal Intuitionistic Fuzzy Algorithm for Franchisee Selection

The selection of the most suitable franchisee applicant in an uncertain environment in a particular moment of time is a key decision for a franchisor and the success of a franchising business. In this work, for the first time, we describe a problem for choosing the optimal candidate for the franchise chain and algorithm for a solution in terms of temporal intuitionistic fuzzy pairs and index matrices as a means for data analysis in uncertain conditions over time. We also use our software utility to demonstrate the proposed algorithm and to apply the decision support approach to a franchisee selection for the largest fast food restaurant chain in Bulgaria.


I. INTRODUCTION
Franchising is an effective business strategy for entering new markets. The franchisor grants the right to its franchisees to use the brand, the business concept, and the products or services within a specific time frame [1]. The concept of fuzzy [10] and intuitionistic fuzzy logic [6], provides such a tool for creating an optimal algorithm for choosing a franchisee in conditions of ambiguity. The studies [12], [13] present fuzzy franchisee selection models using an Analytic Hierarchy Process (AHP) and neural networks.
In [16], we presented an optimal interval-valued intuitionistic fuzzy multicriteria decision-making problem in outsourcing and a software utility for its solution. We have also introduced in [19] an intuitionistic fuzzy approach (IFIMFr) to select the most suitable candidates for franchising in a patisserie using the theory of index matrices (IMs, [5]). The aim of the paper is to expand the IFIMFr approach so that it can be applied to temporal intuitionistic fuzzy data [3]. The work uses our custom programs to implement the proposed algorithm and to apply it to the largest fast food restaurant chain in Bulgaria. The remainder of our study includes 4 sections: Section II describes some definitions and properties of temporal intuitionistic fuzzy IMs and pairs. Section III describes a problem for choosing the optimal franchise candidate and algorithm Work is supported by the Asen Zlatarov University through project Ref.
No. NIX-449/2021 "Index matrices as a tool for knowledge extraction". for solution in terms of temporal IFPs an IMs as a means for uncertain data analysis over time and the basic characteristics of our software utility. Section IV sets out the conclusions and aspects for future research.

II. DEFINITION AND PROPERTIES OF TEMPORAL
INTUITIONISTIC FUZZY IMS AND PAIRS Let us briefly give the definitions of temporal intuitionistic fuzzy IMs and TIFPs and some of their properties [3].

Three-Dimensional Temporal Intuitionistic Fuzzy Index Matrices (3-D TIFIM)
where I be a fixed set of indices, (K, L, T ¢ I ), and its elements ïµ k i ,l j ,t g , ν k i ,l j ,t g ð are TIFPs. T is a fixed temporal scale and its elements t g (g = 1, . . . , g, . . . f ) are time moments.
, where for each k i * W, l j * V and t g * U, ïR p r ,q s ,e d , S p r ,q s ,e d ð = ïµ k i ,l j ,t g , ν k i ,l j ,t g ð. A Level Operator for Decreasing the Number of Elements of TIFIM: Let ïα(t), β (t)ð is an TIFP, then according to [9] where

III. AN OPTIMAL TEMPORAL INTUITIONISTIC FUZZY ALGORITHM FOR SELECTION OF THE MOST ELIGIBLE FRANCHISEE (OTIFAFR)
This section proposes an OTIFAFr, used the concepts of IMs and TIFPs. Let us formulate the optimal problem as follows: A large franchise has decided to turn to experts to select the best franchisee candidate for expanding its brand. The franchise candidates for studied brand v e need to be evaluated by the experts. The experts assess the IF priorities pk c j ,v e ,t f of the criteria c j in the evaluation system of the franchise chain v e at a particular moment t f . The IF ratings of the experts are defined on the basis of their participation in previous franchise evaluation procedures and given to the experts at a time t f . All candidates for a franchisee have been evaluated by the experts at a particular moment t f and their evaluations ev k i ,c j ,d s ,t f at a current moment t f are temporal intuitionistic fuzzy data. The estimations of the same applicants from previous evaluation procedures are given as elements of TIFIM at time points t 1 , ...,t g , ...,t f 21 . The aim of the problem is to select the most eligible franchisee for this brand.

A. An Optimal Temporal Intuitionistic Fuzzy Algorithm for Assignment of a Franchisee
The procedure of OTIFAFr includes the following steps: Step 1. The team of experts needs to evaluate the candidates for the brand v e according to the approved criteria in the company at a particular moment t f . The estimations of the candidate at a particular moment t f . Expert assessments are uncertain due to galloping inflation and the existing pandemic. The data values are transformed into TIFPs as demonstrated in [3], [20]. The TIFP {ev k i ,c j ,d s ,t f } presents the degrees of perception (the positive evaluation of the d s -th expert for the k i -th candidate by the c j -th criterion divided by the (maximum-minimum) evaluation) and non-perception (the negative evaluation of the d s -th expert for the k i -th candidate by the c j -th criterion divided by the (maximum-minimum) evaluation) of the d s -th expert for the k i -th candidate by the c j -th criterion at a particular moment t f . The hesitation degree corresponds to the uncertain evaluation of the d s -th expert for the k i -th candidate by the c j -th criterion at a particular moment t f .
The experts have the opportunity to include assessments for the same candidates from the previous evaluation procedures at time points t 1 , ...,t g , ..
. . ,t f and the element {ev k i ,c j ,d s ,t g } is the estimate of the d s -th expert for the k i -th candidate by the c j -th criterion at a particular moment t g . Let us apply the α T -th aggregation operation (4) to find the aggregated evaluation of the d s -th expert (s = 1, ..., D) for the k i -th candidate for the period T.
The result IM α EV s (T ),# q has the form Then we go to Step 2.
Step 2. Let the present score coefficient r s (t f ) of each expert (s * E) is defined by an TIFP ïδ s (t f ), ε s (t f )ð, which elements can be interpreted respectively as his degree of competence and of incompetence at a particular moment t f . Then is created contains the final score of each franchise candidate at a particular moment t f .
The total assessment of the k i -th candidate on the c j -th criterion at a particular moment t f / * E is calculated by an application of the α E -th aggregation operation as follows where (1 f q f 3). If we use # 7 1 = ïmin, maxð, then we accept super pessimistic aggregation operation, with # 7 2 = ïaverage, averageð we assume averaging aggregation operation and with # 7 3 = ïmax, minð we accept super optimistic aggregation operation for the assessment of the applicant.
If the franchise chain has a requirement for the candidates, so that their total score is not less than a predetermined TIFP ïα(t f ), β (t f )ð, then in this case it is necessary to apply the level operator (5) to TIFIM B to remove from the ranking candidates who do not meet this requirement. Go to Step 3.
Step 3. This step creates a TIFIM PK(h 0 ) with the coefficients determining the importance of the evaluation criteria for the franchisor v e at a particular moment t f by : c n pk c n ,v e ,h 0 , Then we calculate the evaluation TIFIM containing the total estimates of the k i -th candidate (for 1 f i f m) at a particular moment t f for the brand v e , where ïç, 7ð is an operation from (4). Go to Step 4.
Step 4. At this step we choose the most optimal franchisee for v e by using the aggregation operation by K -α K,# q (B(h 0 ), k 0 ) using pessimistic, average or optimistic scenarios where k 0 / * K, 1 f q f 3. Go to Step 5.
Let ïα, β ð is an TIFP. The criteria C k and C l are in: ICrA is applied over the matrix R(h 0 ) to find the criteria, which are in a consonance. More complex criteria are reduced from the evaluation franchise system using the IM reduction operation over R(h 0 ). Go to Step 6.
Step 6. This step obtains the new rank coefficients of the experts. Let the expert d s (s = 1, ..., D) has participated in γ s evaluation procedures for the selection of a franchisee, on the basis of which his score r s (t f ) = ïδ s (t f ), ε s (t f )ð is determined, then after his participation in the next procedure, his new score ïδ ð will be changed by [6]: if the expert had not given any estimation ï if the expert has assessed incorrectly (9) The compexity of OTIFAFr algorithm is O(Dm 2 n 2 ) [15]). For the application of the OTIFAFr algorithm, we will use an updated version of the C++ utility we previously developed for the IFIMOA and IVIFIMOA algorithms. As we outlined before [2], [16], it is based on a template class which allows us to replace its type with any C++ type or class that implements basic comparison and arithmetic operators. This has allowed us to use the same code for both IFPs (as we will do here) and IVIFPs (as we have done in [16], [21]). The program is command-line based. It expects the following input arguments: a 3-D TIFIM of the experts' evaluations, a matrix of the experts' rating coefficients and a matrix of the weight coefficients of each criterion for each service. The expert evaluations can be given either directly as an index matrix of IFPs, or as a matrix of mark intervals. For the latter case, the first argument of the program must be "-interval" followed by the lowest and highest possible mark that a expert can give [21].

B. An Application of OTIFAFr to the Largest Fast Food Restaurant Chain in Bulgaria
In this section, the proposed OTIFAFr model from Sect. III-A is demonstrated with a real case study for choosing a franchisee for the largest fast food restaurant chain in Bulgaria. The optimal problem is defined below: The largest fast food restaurant chain in Bulgaria has given a decision to expand its business through the selection of a franchisee. The franchisor decides to invite a team of 3 experts to evaluate the 3 candidates at a particular moment t f . The evaluation system consists of 4 groups of criteria: C 1 -owner profitability and business experience level; C 2brand marketing and franchise brand development concept; C 3 -opportunities to quickly start a franchise and actively participate in the management of the restaurant and C 4restaurant traffic management and parking options, strategic location of the restaurant and successful traffic management around it with provided parking opportunities. Each criteria has priority coefficient as TIFPs pk c j ,v e ,t f according to their importance from the franchisor's point of view at a current moment t f . The experts' ratings are defined by TIFP {r 1 (t f ), r 2 (t f ), r 3 (t f )} at a particular moment t f . In the final ranking we admit only candidates with an overall score higher than ï0.6, 0.01ð Now we need to optimally rank the candidates and select the most eligible one. Solution of the problem: Step 1. At this step, we create the expert evaluation TIFIM Step 2. The rating coefficients of the experts at t f are: , which contains the final score of each franchisee at a current moment t f , is constructed by Then we apply the operation α E, is equal to 1, 2 or 3 depending on whether the pessimistic, averaging or optimistic scenarios are chosen. The franchise chain has a requirement for the candidates, so that their total score is not less than a predetermined TIFP ï0.6, 0.01ð. The level operator (5) is applied to the TIVIFIM B and it is established that all candidates meet this requirement.
Step 3. At this step, a TIFIM PK(h 0 ) of the weight coefficients of the assessment criterion according to its antecedence is created from the franchisor v e : .656, 0.0148ð k 2 ï0.661, 0.0136ð k 3 ï0.669, 0.0142ð Step 4. The optimistic aggregation operation α K,# 3 (B, k 0 ) finds that k 3 is the optimal franchisee for the franchise chain of fast food restaurants in Bulgaria v e with the maximum degree of acceptance (d.a.) 0.669 and the minimum degree of rejection (d.r.) 0.0142 in an optimistic scenario, in an pessimistic scenariok 1 with the minimum d.a. 0.656 and the maximum d.r. 0.0148. The closest to the average scenario is k 2 with the d.a. 0.661 and the d.r. 0.0136.
Step 5. At this step, we apply the ICrA with α = 0.8 and β = 0.10 over R(h 0 ). The conclusion is that the evaluation system in the chain is optimized.
The results, obtained from the ICrA application [11], are in the form of IM in µν view result matrix: Step 6. At last step, the experts' assessments are correct from the point of view of IF logic [6] and their new rating coefficients are equal to {ï0.82, 0.05ð, ï0.64, 0.05ð, ï0.81, 0.05ð}.
IV. CONCLUSION In the study, we have defined the OTIFAFr procedure for selection of the most suitable franchisee over temporal IF evaluations. A software implementation of the proposed algorithm was presented and the decision making procedure applied for a franchisee selection. In the future, the study will continue with the development of OTIFAFr approach, so that it can be applied over the data, saved in extended TIFIMs [5] and also with software for its implementation.