A multidimensional intuitionistic fuzzy InterCriteria analysis in the restaurant

Abstract

The intuitionistic fuzzy InterCriteria analysis (ICrA) is a new method for correlation analysis, which is based on the concepts of index matrices (IMs) and intuitionistic fuzzy sets (IFSs), aiming at detecting of the dependencies between pairs of rating criteria in both clear and uncertain environments. In the present paper, which is an extension of [39], our aim is to extend ICrA to multidimensional ICrA (n-D ICrA) under intuitionistic fuzzy environment for situations where the evaluations of the objects against multidimensional criteria are completely unknown and to show its efficiency through an application in identifying correlations between pairs of criteria when referred to actual data gathered through estimates of a restaurant’s kitchen staff over a three-year period in Bulgaria. We also present a comparative analysis of the correlations between the evaluated criteria of the kitchen staff, on the basis the application of the correlation methods of ICrA, Pearson (PCA), Spearman (SCA) and Kendall (KCA). The four-correlation analysis yielded very similar correlation coefficients, but only the ICrA can be applied to intuitionistic fuzzy evaluations. It is observed that considerable divergence of the ICrA results from those obtained by the other classical correlation analyzes, is only found when the input data contains mistakes.

Keywords

Decision making index matrix InterCriteria analysis intuitionistic fuzzy logic rating criteria

1 Introduction

The correlation is an important statistical tool in desicion making. In real world data are often fuzzy [22] or intuitionistic fuzzy [10]. The measuring correlation coefficient between two variables involving fuzziness is a need and computational procedures are challenging. The concept of correlation coefficient of IFSs has been first studied by Gerstenkorn and Manko [29]. Using the statistical viewpoint, the IFS correlation coefficient was defined in [2 , 43]. In [25], a positive and negative type of a correlation was proposed, but the hesitation margin of IFSs does not participate in the calcullations. The proposed correlation coefficient in [6] takes into account the membership and non-membership values, and the hesitation margin of IFSs. The ICrA is a new statistical method [16, 20] for calculating pairwise dependencies (correlations) between each pair of criteria for evaluation of objects. The complexity of the ICrA algorithm is O (m²n²), which is polynomial in mn [30].

The advantages of the ICrA over the three classic correlation analyses PCA, SCA and KCA are:

It is only based on comparisons “<, > , =”, existing between the evaluations of the objects against the system of criteria, rather than on their numerical values, which makes computations faster than the other three correlation analysis methods.

It can be applied not only to clear data but also to incomplete and uncertain one.

The ICrA method [20] is based on the concepts of IFSs and IMs. IFSs were first defined by Atanassov [10, 14] as an extension of the concept of fuzzy sets defined by Zadeh [22]. The concept of IMs is introduced in [11] and described in [16, 37]. The intuitionistic based method of ICrA was developed in [20] to handle cases of multicriteria decision making, where measuring according to some of the criteria is slower, more expensive or unclear, which results in delaying or raising the cost of the process of decision making [27]. The ICrA allows finding statistically meaningful relations between a lot of criteria which characterize multiple objects. Later the ICrA was extended theoretically and was further applied to two-dimensional interval-valued and three-dimensional intuitionistic fuzzy data [21 , 40]. The approach has been discussed in a number of papers considering universities’ rankings [7, 32], electronics [23], neural networks [9, 27], metaheuristic algorithms [26], economic investigations [31, 39], medical and chemistry investigations [27, 28], etc.

The present paper, which is a continuation of [39], proposes for the first time a new form of ICrA (n-D ICrA) to be applied to n-dimensional intuitionistic fuzzy data in uncertain environments. Here, the ICrA approach is applied to the data over a three-year period with estimates of the kitchen staff at a fast food restaurant, part of a chain of restaurants in Burgas, Bulgaria. The staff assessment system in the restaurant is tested with four methods – ICrA, Pearson’s, Spearman’s and Kendall’s rank correlation analyzes to show the advantages of the proposed approach. The comparison of the results obtained shows that there is no comparative difference between those, obtained from the ICrA and the other three classic correlation analyzes. It is observed that considerable divergence of the ICrA results from those obtained by the other classical correlation analyzes, is only found when the input data contains mistakes. The originality of the paper comes from the proposed multidimensional ICrA over intuitionistic fuzzy evaluations of a set of objects by a set of criteria. The main contributions of the paper lie in its proposition for a new form of ICrA over n-dimensional intuitionistic fuzzy evaluations on the one hand, and its study of the effectiveness of the proposed hybrid method combining fuzzy logic with classic correlation analysis to optimize the kitchen staff rating system of the restaurant in Bulgaria on the other.

The rest of this paper is structured as follows: In Section 2, are discussed n-dimensional extended IMs (n-D EIM, see [17]) and the concepts of intuitionistic fuzzy pairs (IFPs, see [15]). Section 3 describes the n-D ICrA method, which is based on intuitionistic fuzzy logic and n-D EIM. In Section 4, we research the kitchen staff assessment system in a fast food restaurant and apply the 3-D ICrA to optimize the assessment system in the surveyed company. Finally, in Section 5, the obtained results are compared with those obtained from Pearson’s, Spearman’s and Kendall’s rank correlation analyzes. Section 6 offers the conclusion and outlines aspects for future research.

2 Short remarks on IMs and IFPs

2.1 Short notes on intuitionistic fuzzy pairs

Let us start with some remarks on intuitionistic fuzzy logic from [14, 15].

To each proposition (sentence) of classical logic (e.g., [3]), we juxtapose its truth value: truth – denoted by 1, or falsity – denoted by 0. In fuzzy logic [22], the truth value, called “truth degree”, is a real number in the interval [0, 1]. In the intuitionistic fuzzy case (see [13]), one more value – “falsity degree” – is added. It is again in the interval [0, 1]. Thus, to the proposition p, two real numbers, μ (p) and ν (p), are assigned with the following constraint: μ (p) + ν (p) ≤1 .

The IFP is an object with the form 〈a, b〉, where a, b ∈ [0, 1] and a + b ≤ 1, that is used as an evaluation of some object or process. Its components (a and b) are interpreted as degrees of membership and non-membership. Let us have two IFPs x = 〈a, b〉 and y = 〈c, d〉. In [15, 38] were defined the relations:

$\begin{matrix} x > y & iff & a > c and b < d \\ x \geq_{◊} y & iff b \leq d; & x \geq_{□} y iff a \geq c \\ x = y & iff & a = c and b = d \\ x \geq_{R} y & iff & R_{〈 a, b 〉} \leq R_{〈 c, d 〉}, \end{matrix}$ (1) where following [5] R_〈a,b〉 = 0, 5 (2 - a - b) 0, 5 (|1 - a| + |b| + |1 - a - b|) .

A set of the following operations # _p, (1 ≤ p ≤ 10) from [13] has been using for constructing a scale in [41], so that the index matrix elements are ordered: $\begin{matrix} x #_{1} y = 〈 ac, 1 - ac 〉, \\ x #_{2} y = 〈 ac, b + d - bd 〉, \\ x #_{3} y = 〈 min (a, c), max (b, d) 〉, \end{matrix}$

$\begin{matrix} x #_{4} y = 〈 min (a, 1 - d), max (b, d) 〉, \\ x #_{5} y = 〈 1 - max (b, d), max (b, d) 〉, \\ x #_{6} y = 〈 max (1 - b, c), 1 - max (1 - b, c) 〉, \\ x #_{7} y = 〈 max (1 - b, c), min (b, d) 〉, \\ x #_{8} y = 〈 1 - min (b, d), min (b, d) 〉, \\ x #_{9} y = 〈 1 - bd, bd 〉, \\ x #_{10} y = 〈 min (1, 2 - b - d), max (0, b + d - 1) 〉 . \end{matrix}$ (2)

Let for brevity $n #_{p}^{*} j = 1 x_{j} = #_{p}^{*} (x_{1}, x_{2}, . . ., x_{n})$ for p = 1, . . . , 10 .

2.2 Short remarks on n-dimensional extended IMs

2.2.1 Definition of an n-dimensional extended IM (n-D EIM)

Let $I$ be again a fixed set of indices, $I^{n} = {〈 i_{1}, i_{2}, . . ., i_{n} 〉 | (\forall j : 1 \leq j \leq n) (i_{j} \in I)}$ and $I^{*} = \cup_{1 \leq n \leq \infty} I^{n} .$ Let $X$ be a fixed set of some objects. In the particular cases, they can be either natural, real or complex numbers; or only the numbers 0 or 1; or logical variables, propositions or predicates, IFPs, etc. An n-dimensional EIM (n-D EIM) with index sets K₁, K₂, . . . , K_n (K₁, K₂, . . . , K_n $\subseteq I^{*})$ and elements from set $X$ is called the object (see, [17]): $A = [K_{1}, K_{2}, . . ., K_{n}, {a_{k_{1, s_{1}}, k_{2, s_{2}}, . . ., k_{n, s_{n}}}}],$ where K_i = {k_i,1, k_i,2, . . . , k_{i,m_i}}, m_i ≥ 1 and $a_{k_{1, s_{1}}, k_{2, s_{2}}, . . ., k_{n, s_{n}}} \in X$ for 1 ≤ i ≤ n and 1 ≤ s_i ≤ m_i. The n-D EIM A has the following $\frac{n (n - 1)}{2}$ different representations as 2-D EIM: $A = \begin{matrix} 〈 k_{3, s_{3}}, . . ., k_{n, s_{n}} 〉 & k_{2, 1} & \dots \\ k_{1, 1} & a_{k_{1, 1}, k_{2, 1}, k_{3, s_{3}}, . . ., k_{n, s_{n}}} & \dots \\ ⋮ & ⋮ & ⋱ \\ k_{1, i} & a_{k_{1, i}, k_{2, 1}, k_{3, s_{3}}, . . ., k_{n, s_{n}}} & \dots \\ ⋮ & ⋮ & ⋱ \\ k_{1, m_{1}} & a_{k_{1, m_{1}}, k_{2, 1}, k_{3, s_{3}}, . . ., k_{n, s_{n}}} & \dots \end{matrix}$ $\begin{matrix} \dots & k_{2, m_{2}} \\ \dots & a_{k_{1, 1}, k_{2, m_{2}}, k_{3, s_{3}}, . . ., k_{n, s_{n}}} \\ ⋱ & ⋮ \\ \dots & a_{k_{1, i}, k_{2, m_{2}}, k_{3, s_{3}}, . . ., k_{n, s_{n}}} \\ ⋱ & ⋮ \\ \dots & a_{k_{1, m_{1}}, k_{2, m_{2}}, k_{3, s_{3}}, . . ., k_{n, s_{n}}} \end{matrix}$ $= \dots$ $\begin{matrix} = 〈 k_{1, s_{1}}, . . ., k_{n - 2, s_{n - 2}} 〉 & k_{n, 1} \\ k_{n - 1, 1} & \dots \\ ⋮ & ⋱ \\ k_{n - 1, i} & \dots \\ ⋮ & ⋱ \\ k_{n - 1, m_{n - 1}} & \dots \end{matrix}$ $\begin{matrix} \dots & k_{n, m_{n}} \\ \dots & a_{k_{1, s_{1}}, . . ., k_{n - 2, s_{n - 2}}, k_{n - 1, 1}, k_{n, m_{n}}} \\ ⋱ & ⋮ \\ \dots & a_{k_{1, s_{1}}, . . ., k_{n - 2, s_{n - 2}}, k_{n - 1, i}, k_{n, m_{n}}} \\ ⋱ & ⋮ \\ \dots & a_{k_{1, s_{1}}, . . ., k_{n - 2, s_{n - 2}}, k_{n - 1, m_{n - 1}}, k_{n, m_{n}}} \end{matrix} .$ In [11 , 34] a lot of operations are defined over the different types of IMs. When the elements of a given IM are real numbers and the operations are the standard ones, we obtain the partial cases of the standard matrices, but there are a lot of operations, relations and operators, defined over the different types of IMs, that do not have analogues in the classical matrix theory.

2.2.2 Scaled aggregation operations over n-D IFIM

This subsection introduces for the first time some scaled aggregation operations extensions, following the ideas from [35, 41] over n-D EIM with elements IFPs. Let us denote this type of IM as n-D IFIM.

Let an n-D IFIM A = [K₁, K₂, . . . , K_n, { a_{k_1,s₁,k_2,s₂,...,k_{n,s_n}} }] be given, where K_i = {k_i,1, k_i,2, . . . , k_{i,m_i}}, m_i ≥ 1 and a_{k_1,s₁,k_2,s₂,...,k_{n,s_n}} = 〈μ_{k_1,s₁,k_2,s₂,...,k_{n,s_n}}, ν_{k_1,s₁,k_2,s₂,...,k_{n,s_n}}〉

(for 1 ≤ i ≤ n and 1 ≤ s_i ≤ m_i), and let k_1,0 ∉ K₁, k_2,0 ∉ K₂, . . . , k_n,0 ∉ K_n .

Following [16, 36], let us start with the definition of the operation “transposition” of n-D IFIM.

For the standard 2-D IFIM, there are 2 (= 2!) IFIM, related to this operation: the standard IFIM and its transposed IFIM, while for n-D IFIM, there are (=n!) cases: the standard n-D IFIM and (n ! -1) different transposed n-D IFIM. Two of these analytical forms of the separate transposed n-D IFIMs are as follows: [K₁, K₂, . . . , K_n, { a_{k_1,s₁,k_2,s₂,...,k_{n,s_n}} }] ^[1,2,...n] = [K₁, K₂, . . . , K_n, { a_{k_1,s₁,k_2,s₂,...,k_{n,s_n}} }] ; $[K_{1}, K_{2}, . . ., K_{n}, {a_{k_{1, s_{1}}, k_{2, s_{2}}, . . ., k_{n, s_{n}}}}]^{[1, 2, . . ., n, n - 1]}$ = [K₁, K₂, . . . , K_n, K_n-1, {a_{k_1,s₁,k_2,s₂,...,k_{n,s_n},k_{n-1,s_n-1}} }] .

Let us define for the first time the scaled aggregation operations over this type of IMs by one dimension K_i (for i = 1, . . . , n) and for $#_{p}^{*}$ defined in (2) (for p = 1, . . . , 10) as follows: $α_{K_{i}, #_{p}^{*}} (A, k_{i, 0})$ $= \begin{matrix} 〈 k_{3, s_{3}}, . . . k_{i, 0}, . . ., k_{n, s_{n}} 〉 & k_{2, 1} & \dots \\ k_{1, 1} & m_{i} #_{p}^{*} i = 1 a_{k_{1, 1}, . . ., k_{i, s_{i}}, . . ., k_{n, s_{n}}} & \dots \\ ⋮ & ⋮ & ⋱ \\ k_{1, i} & m_{i} #_{p}^{*} i = 1 a_{k_{1, 1}, . . ., k_{i, s_{i}}, . . ., k_{n, s_{n}}} & \dots \\ ⋮ & ⋮ & ⋱ \\ k_{1, m_{1}} & m_{i} #_{p}^{*} i = 1 a_{k_{1, 1}, . . ., k_{i, s_{i}}, . . ., k_{n, s_{n}}} & \dots \end{matrix}$ $\begin{matrix} \dots & k_{2, m_{2}} \\ \dots & m_{i} #_{p}^{*} i = 1 a_{k_{1, 1}, . . ., k_{i, s_{i}}, . . ., k_{n, s_{n}}} \\ ⋱ & ⋮ \\ \dots & m_{i} #_{p}^{*} i = 1 a_{k_{1, 1}, . . ., k_{i, s_{i}}, . . ., k_{n, s_{n}}} \\ ⋱ & ⋮ \\ \dots & m_{i} #_{p}^{*} i = 1 a_{k_{1, 1}, . . ., k_{i, s_{i}}, . . ., k_{n, s_{n}}} \end{matrix}$ $= (\begin{matrix} 〈 k_{2, s_{2}}, . . ., k_{i + 1, s_{i + 1}}, . . ., k_{n, s_{n}} 〉 & k_{1, 1} & \dots \\ k_{i, 0} & m_{i} #_{p}^{*} i = 1 a_{k_{1, 1}, . . ., k_{i, s_{i}}, . . ., k_{n, s_{n}}} \end{matrix}$ ${\begin{matrix} \dots & k_{1, m_{1}} \\ \dots & m_{i} #_{p}^{*} i = 1 a_{k_{1, 1}, . . ., k_{i, s_{i}}, . . ., k_{n, s_{n}}} \end{matrix})}^{[2, 3, . . ., i - 1, 1, i + 1, . . ., n]} .$ (3)

The operations $#_{p}^{*}$ in (2) generate a 10-element scale for the aggregation operations (3) for which we can denote, e.g., $#_{1}^{*}$ - super pessimistic aggregation operation, $#_{2}^{*}$ - very pessimistic aggregation operation, etc., $#_{10}^{*}$ - super optimistic aggregation operation.

Let be given index set K_* = {K_{d
₁}, . . . , K_{d
_x}, . . . , K_{d
_u}} and V_* = {k_d₁,0, …, k_{d_x,0}, …, k_{d_u,0}}, where k_{d_x,0} ∉ K_{d
_x} for x = 1, . . . , u. The definition of the aggregation operation by the dimensions, belong to K_* is: $\begin{matrix} α_{K_{*}, #_{p}^{*}} (A, V *) = α_{K_{d_{1}}, . . ., K_{d_{x}}, . . ., K_{d_{n - 1}, #_{p}^{*}}} \\ (α_{K_{d_{n}}, #_{p}^{*}} (A, k_{d_{n}, 0})) . \end{matrix}$

3 The n-dimensional intercriteria analysis method

In this section, a new extension of ICrA (n-D ICrA), based on n-dimensional intuitionistic fuzzy evaluations is introduced for the first time. Let us have an n-D EIM A [K₁, K₂, . . . , K_n, {a_{k_1,s₁,k_2,s₂,...,k_{n,s_n}}〉}] and $A = \begin{matrix} 〈 k_{3, s_{3}}, . . ., k_{n, s_{n}} 〉 & O_{2, 1} & \dots \\ C_{1, 1} & a_{C_{1, 1}, O_{2, 1}, k_{3, s_{3}}, . . ., k_{n, s_{n}}} & \dots \\ ⋮ & ⋮ & ⋱ \\ C_{1, i} & a_{C_{1, i}, O_{2, 1}, k_{3, s_{3}}, . . ., k_{n, s_{n}}} & \dots \\ ⋮ & ⋮ & ⋱ \\ C_{1, m_{1}} & a_{C_{1, m_{1}}, O_{2, 1}, k_{3, s_{3}}, . . ., k_{n, s_{n}}} & \dots \end{matrix}$ $\begin{matrix} \dots & O_{2, m_{x}} & \dots \\ \dots & a_{C_{1, 1}, O_{2, m_{x}}, k_{3, s_{3}}, . . ., k_{n, s_{n}}} & \dots \\ ⋱ & ⋮ & ⋱ \end{matrix}$ $\begin{matrix} \dots & a_{C_{1, i}, O_{2, m_{x}}, k_{3, s_{3}}, . . ., k_{n, s_{n}}} & \dots \\ ⋱ & ⋮ & ⋱ \\ \dots & a_{C_{1, m_{1}}, O_{2, m_{2}}, k_{3, s_{3}}, . . ., k_{n, s_{n}}} & \dots \end{matrix}$ $\begin{matrix} \dots & O_{2, m_{y}} & \dots \\ \dots & a_{C_{1, 1}, O_{2, m_{y}}, k_{3, s_{3}}, . . ., k_{n, s_{n}}} & \dots \\ ⋱ & ⋮ & ⋱ \\ \dots & a_{C_{1, i}, O_{2, m_{y}}, k_{3, s_{3}}, . . ., k_{n, s_{n}}} & \dots \\ ⋱ & ⋮ & ⋱ \\ \dots & a_{C_{1, m_{1}}, O_{2, m_{y}}, k_{3, s_{3}}, . . ., k_{n, s_{n}}} & \dots \end{matrix}$ $\begin{matrix} \dots & O_{2, m_{2}} \\ \dots & a_{C_{1, 1}, O_{2, m_{2}}, k_{3, s_{3}}, . . ., k_{n, s_{n}}} \\ ⋱ & ⋮ \\ \dots & a_{C_{1, i}, O_{2, m_{2}}, k_{3, s_{3}}, . . ., k_{n, s_{n}}} \\ ⋱ & ⋮ \\ \dots & a_{C_{1, m_{1}}, O_{2, m_{2}}, k_{3, s_{3}}, . . ., k_{n, s_{n}}} \end{matrix}$ (4) where for every p, q (p = 1, . . . , m₁, q = 1, . . . , m₂):

C_1,p is a criterion, O_2,q is an object,

a_{C
_{C_1,p,O_2,q,k_3,s₃,...,k_{n,s_n}}} is an evaluation of an object O_2,q (1 ≤ q ≤ m₂) against a criterion C_1,p (1 ≤ p ≤ m₁) in the conditions of other (n - 2) factors (time, location, etc.) and can be a real number, IFP or another object, including the empty place in the matrix, marked by ⊥, that is comparable about relation R with the other a-objects, so that for each x, y, i, k_3,s₃, . . . , k_{n,s_n}: R (a_{C_1,i,O_2,x,k_3,s₃,...,k_{n,s_n}}, a_{C_1,i,O_2,y,k_3,s₃,...,k_{n,s_n}}) is defined.

Let $\bar{R}$ be the dual relation of R in the sense that if R is satified, then $\bar{R}$ is not satisfied and vice versa. For example, if “R” is the relation “<”, then $\bar{R}$ is the relation “>” and vice versa. Let us suppose that some of the values of A are omitted (denoted by ⊥). Following [21], let the real numbers α^o, β^o, α^⊥, β^⊥ be fixed and let them satisfy the inequalities 0 ≤ α^o ≤ 1, 0 ≤ β^o ≤ 1, 0 ≤ α^⊥ ≤ 1, 0 ≤ β^⊥ ≤ 1, 0 ≤ α^o + β^o ≤ 1, 0 ≤ α^⊥ + β^⊥ ≤ 1 .

Let for each fixed ordered (n - 2)-index set 〈k_3,s₃, . . . , k_{d,s_d}, . . . , k_{n,s_n}〉(1 ≤ d ≤ m_d)

$S_{C_{1, k}, C_{1, l}, k_{3, s_{3}}, . . ., k_{d, s_{d}}, . . ., k_{n, s_{n}}}^{μ}$ be the number of cases in which R (a_{C_k,O_x,k_3,s₃,...,k_{d,s_d},...,k_{n,s_n}}, a_{C_k,O_y,k_3,s₃,k_{d,s_d},...,k_{n,s_n}}) and R (a_{C_l,O_x,k_3,s₃,...,k_{d,s_d},...,k_{n,s_n}}, a_{C_l,O_y,k_3,s₃,k_{d,s_d},...,k_{n,s_n}}) are simultaneously satisfied and the a-arguments are different from ⊥.

$S_{C_{1, k}, C_{1, l}, k_{3, s_{3}}, . . ., k_{d, s_{d}}, . . ., k_{n, s_{n}}}^{ν}$ be the number of cases in which R (a_{C_k,O_x,k_3,s₃,...,k_{d,s_d},...,k_{n,s_n}}, a_{C_k,O_y,k_3,s₃,k_{d,s_d},...,k_{n,s_n}}) and $\bar{R} (a_{C_{l}, O_{x}, k_{3, s_{3}}, . . ., k_{d, s_{d}}, . . ., k_{n, s_{n}}},$ a_{C_l,O_y,k_3,s₃,k_{d,s_d},...,k_{n,s_n}}) are simultaneously satisfied and the a-arguments are different from ⊥.

$S_{C_{1, k}, C_{1, l}, k_{3, s_{3}}, . . ., k_{d, s_{d}}, . . ., k_{n, s_{n}}}^{⊥}$ be the number of cases in which R (a_{C_k,O_x,k_3,s₃,...,k_{d,s_d},...,k_{n,s_n}}, a_{C_k,O_y,k_3,s₃,k_{d,s_d},...,k_{n,s_n}}) and $\bar{R} (a_{C_{l}, O_{x}, k_{3, s_{3}}, . . ., k_{d, s_{d}}, . . ., k_{n, s_{n}}},$ a_{C_l,O_y,k_3,s₃,k_{d,s_d},...,k_{n,s_n}}) is satisfied, because at least one of the elements a_{C_k,O_x,k_3,s₃,...,k_{d,s_d},...,k_{n,s_n}} and a_{C_k,O_y,k_3,s₃,k_{d,s_d},...,k_{n,s_n}} or a_{C_l,O_x,k_3,s₃,...,k_{d,s_d},...,k_{n,s_n}} and a_{C_l,O_y,k_3,s₃,k_{d,s_d},...,k_{n,s_n}} is equal to ⊥ (is empty).

$S_{C_{1, k}, C_{1, l}, k_{3, s_{3}}, . . ., k_{d, s_{d}}, . . ., k_{n, s_{n}}}^{o}$ be the number of cases in which a_{C_k,O_x,k_3,s₃,...,k_{d,s_d},...,k_{n,s_n}} = = a_{C_k,O_y,k_3,s₃,k_{d,s_d},...,k_{n,s_n}}, or a_{C_l,O_x,k_3,s₃,...,k_{d,s_d},...,k_{n,s_n}} = = a_{C_l,O_y,k_3,s₃,k_{d,s_d},...,k_{n,s_n}} and all a-arguments are different from ⊥.

Let $\begin{matrix} S_{C_{1, k}, C_{1, l}, k_{3, s_{3}}, . . ., k_{d, s_{d}}, . . ., k_{n, s_{n}}}^{π} \\ = S_{C_{1, k}, C_{1, l}, k_{3, s_{3}}, . . ., k_{d, s_{d}}, . . ., k_{n, s_{n}}}^{⊥} \\ + S_{C_{1, k}, C_{1, l}, k_{3, s_{3}}, . . ., k_{d, s_{d}}, . . ., k_{n, s_{n}}}^{o} . \end{matrix}$

Let $N_{p} = \frac{m_{2} (m_{2} - 1)}{2} .$ Obviously, $\begin{matrix} S_{C_{1, k}, C_{1, l}, k_{3, s_{3}}, . . ., k_{d, s_{d}}, . . ., k_{n, s_{n}}}^{μ} \\ + S_{C_{1, k}, C_{1, l}, k_{3, s_{3}}, . . ., k_{d, s_{d}}, . . ., k_{n, s_{n}}}^{ν} \\ + S_{C_{1, k}, C_{1, l}, k_{3, s_{3}}, . . ., k_{d, s_{d}}, . . ., k_{n, s_{n}}}^{π} = N_{p} . \end{matrix}$

Now, for every k, l, such that 1 ≤ k < l ≤ m₁ and for m₂ ≥ 2, we define $\begin{matrix} μ_{C_{1, k}, C_{1, l}, k_{3, s_{3}}, . . ., k_{d, s_{d}}, . . ., k_{n, s_{n}}} \\ = \frac{S_{C_{1, k}, C_{1, l}, k_{3, s_{3}}, . . ., k_{d, s_{d}}, . . ., k_{n, s_{n}}}^{μ}}{N_{p}} \\ + \frac{α^{o} S_{C_{1, k}, C_{1, l}, k_{3, s_{3}}, . . ., k_{d, s_{d}}, . . ., k_{n, s_{n}}}^{o}}{N_{p}} \\ + \frac{α^{⊥} S_{C_{1, k}, C_{1, l}, k_{3, s_{3}}, . . ., k_{d, s_{d}}, . . ., k_{n, s_{n}}}^{⊥}}{N_{p}}, \\ ν_{C_{1, k}, C_{1, l}, k_{3, s_{3}}, . . ., k_{d, s_{d}}, . . ., k_{n, s_{n}}} \\ = \frac{S_{C_{1, k}, C_{1, l}, k_{3, s_{3}}, . . ., k_{d, s_{d}}, . . ., k_{n, s_{n}}}^{ν}}{N_{p}} \\ + \frac{β^{o} S_{C_{1, k}, C_{1, l}, k_{3, s_{3}}, . . ., k_{d, s_{d}}, . . ., k_{n, s_{n}}}^{o}}{N_{p}} \\ + \frac{β^{⊥} S_{C_{1, k}, C_{1, l}, k_{3, s_{3}}, . . ., k_{d, s_{d}}, . . ., k_{n, s_{n}}}^{⊥}}{N_{p}} . \end{matrix}$ (5)

Hence, $\begin{matrix} μ_{C_{1, k}, C_{1, l}, k_{3, s_{3}}, . . ., k_{d, s_{d}}, . . ., k_{n, s_{n}}} \\ + ν_{C_{1, k}, C_{1, l}, k_{3, s_{3}}, . . ., k_{d, s_{d}}, . . ., k_{n, s_{n}}} \\ \leq \frac{S_{C_{1, k}, C_{1, l}, k_{3, s_{3}}, . . ., k_{d, s_{d}}, . . ., k_{n, s_{n}}}^{μ}}{N_{p}} \\ + \frac{S_{C_{1, k}, C_{1, l}, k_{3, s_{3}}, . . ., k_{d, s_{d}}, . . ., k_{n, s_{n}}}^{ν}}{N_{p}} \\ + \frac{S_{C_{1, k}, C_{1, l}, k_{3, s_{3}}, . . ., k_{d, s_{d}}, . . ., k_{n, s_{n}}}^{o}}{N_{p}} \\ + \frac{S_{C_{1, k}, C_{1, l}, k_{3, s_{3}}, . . ., k_{d, s_{d}}, . . ., k_{n, s_{n}}}^{⊥}}{N_{p}} \leq 1 . \end{matrix}$

In the simple form of the optimistic approach the α- and β-constants are α^o = 1, α^⊥ = 1, β^o = 0, β^⊥ = 0 .

Then μ_{C_1,k,C_1,l,k_3,s₃,...,k_{d,s_d},...,k_{n,s_n}} $\begin{matrix} = \frac{S_{C_{1, k}, C_{1, l}, k_{3, s_{3}}, . . ., k_{d, s_{d}}, . . ., k_{n, s_{n}}}^{μ}}{N_{p}} \\ + \frac{S_{C_{1, k}, C_{1, l}, k_{3, s_{3}}, . . ., k_{d, s_{d}}, . . ., k_{n, s_{n}}}^{o}}{N_{p}} \\ + \frac{S_{C_{1, k}, C_{1, l}, k_{3, s_{3}}, . . ., k_{d, s_{d}}, . . ., k_{n, s_{n}}}^{⊥}}{N_{p}}, \\ ν_{C_{1, k}, C_{1, l}, k_{3, s_{3}}, . . ., k_{d, s_{d}}, . . ., k_{n, s_{n}}} \\ = \frac{S_{C_{1, k}, C_{1, l}, k_{3, s_{3}}, . . ., k_{d, s_{d}}, . . ., k_{n, s_{n}}}^{ν}}{N_{p}} . \end{matrix}$

In the simple form of the pessimistic approach the α- and β-constants are α^o = 0, α^⊥ = 0, β^o = 1, β^⊥ = 1 .

Then μ_{C_1,k,C_1,l,k_3,s₃,...,k_{d,s_d},...,k_{n,s_n}} $\begin{matrix} = \frac{S_{C_{1, k}, C_{1, l}, k_{3, s_{3}}, . . ., k_{d, s_{d}}, . . ., k_{n, s_{n}}}^{μ}}{N_{p}}, \\ ν_{C_{1, k}, C_{1, l}, k_{3, s_{3}}, . . ., k_{d, s_{d}}, . . ., k_{n, s_{n}}} \\ = \frac{S_{C_{1, k}, C_{1, l}, k_{3, s_{3}}, . . ., k_{d, s_{d}}, . . ., k_{n, s_{n}}}^{ν}}{N_{p}} \\ + \frac{S_{C_{1, k}, C_{1, l}, k_{3, s_{3}}, . . ., k_{d, s_{d}}, . . ., k_{n, s_{n}}}^{o}}{N_{p}} \\ + \frac{S_{C_{1, k}, C_{1, l}, k_{3, s_{3}}, . . ., k_{d, s_{d}}, . . ., k_{n, s_{n}}}^{⊥}}{N_{p}} . \end{matrix}$

Follow the ideas from [21], let us propose the uniform (α - β)-approach as follows: let $α, β \in [0, \frac{1}{4}]$ be two fixed numbers, so that $α + β \leq \frac{1}{2}$ and let α^o = α^⊥ = α, β^o = β^⊥ = β .

Then $\begin{matrix} μ_{C_{1, k}, C_{1, l}, k_{3, s_{3}}, . . ., k_{d, s_{d}}, . . ., k_{n, s_{n}}} \\ = \frac{S_{C_{1, k}, C_{1, l}, k_{3, s_{3}}, . . ., k_{d, s_{d}}, . . ., k_{n, s_{n}}}^{μ}}{N_{p}} \\ + \frac{α S_{C_{1, k}, C_{1, l}, k_{3, s_{3}}, . . ., k_{d, s_{d}}, . . ., k_{n, s_{n}}}^{o}}{N_{p}} \\ + \frac{α S_{C_{1, k}, C_{1, l}, k_{3, s_{3}}, . . ., k_{d, s_{d}}, . . ., k_{n, s_{n}}}^{⊥}}{N_{p}}, \end{matrix}$ $\begin{matrix} ν_{C_{1, k}, C_{1, l}, k_{3, s_{3}}, . . ., k_{d, s_{d}}, . . ., k_{n, s_{n}}} \\ = \frac{S_{C_{1, k}, C_{1, l}, k_{3, s_{3}}, . . ., k_{d, s_{d}}, . . ., k_{n, s_{n}}}^{ν}}{N_{p}} \\ + \frac{β S_{C_{1, k}, C_{1, l}, k_{3, s_{3}}, . . ., k_{d, s_{d}}, . . ., k_{n, s_{n}}}^{o}}{N_{p}} \\ + \frac{β S_{C_{1, k}, C_{1, l}, k_{3, s_{3}}, . . ., k_{d, s_{d}}, . . ., k_{n, s_{n}}}^{⊥}}{N_{p}} . \end{matrix}$

We can construct the IM R that determines the degrees of correspondence between criteria C_1,1, . . . , C_1,m₁ using the values for pairs (5)

〈μ_{C_1,k,C_1,l,k_3,s₃,...,k_{d,s_d},...,k_{n,s_n}}, ν_{C_1,k,C_1,l,k_3,s₃,...,k_{d,s_d},...,k_{n,s_n}}〉:

$R = \begin{matrix} 〈 k_{3, s_{3}}, . . ., k_{n, s_{n}} 〉 & C_{1, 1} & \dots \\ C_{1, 1} & a_{C_{1, 1}, C_{1, 1}, k_{3, s_{3}}, . . ., k_{n, s_{n}}} & \dots \\ ⋮ & ⋮ & ⋱ \\ C_{1, i} & a_{C_{1, i}, C_{1, 1}, k_{3, s_{3}}, . . ., k_{n, s_{n}}} & \dots \\ ⋮ & ⋮ & ⋱ \\ C_{1, m_{1}} & a_{C_{1, m_{1}}, C_{1, 1}, k_{3, s_{3}}, . . ., k_{n, s_{n}}} & \dots \end{matrix}$ } $\begin{matrix} \dots & C_{1, m_{1}} \\ \dots & a_{C_{1, 1}, C_{1, m_{1}}, k_{3, s_{3}}, . . ., k_{n, s_{n}}} \\ ⋱ & ⋮ \\ \dots & a_{C_{1, i}, C_{1, m_{1}}, k_{3, s_{3}}, . . ., k_{n, s_{n}}} \\ ⋱ & ⋮ \\ \dots & a_{C_{1, m_{1}}, C_{1, m_{1}}, k_{3, s_{3}}, . . ., k_{n, s_{n}}} \end{matrix},$ where a_{C_1,k,C_1,l,k_3,s₃,...,k_{d,s_d},...,k_{n,s_n}} = 〈μ_{C_1,k,C_1,l,k_3,s₃,...,k_{d,s_d},...,k_{n,s_n}}, ν_{C_1,k,C_1,l,k_3,s₃,...,k_{d,s_d},...,k_{n,s_n}}〉 .

Let be given index set V_* = {k_3,0, …, k_{d_x,0}, …, k_n,0} , where k_{d_x,0} ∉ K_{d
_x} for 1 ≤ x ≤ u . Let us apply the following (scaled) aggregation operation (3) to the n-D IM R by the dimensions k_3,s₃, . . . , k_{n,s_n} $R_{agg} = α_{k_{3, s_{3}}, . . ., k_{n, s_{n}}, #_{p}^{*}} (R, V *),$ $= \begin{matrix} 〈 k_{3, 0}, . . ., k_{n, 0} 〉 & C_{1, 1} & \dots \\ C_{1, 1} & m_{n} #_{p}^{*} i = 1 . . . m_{3} #_{p}^{*} i = 1 a_{C_{1, 1}, C_{1, 1}, k_{3, s_{3}}, . . ., k_{n, s_{i}}} & \dots \\ ⋮ & ⋮ & ⋱ \\ C_{1, i} & m_{n} #_{p}^{*} i = 1 . . . m_{3} #_{p}^{*} i = 1 a_{C_{1, i}, C_{1, 1}, k_{3, s_{3}}, . . ., k_{n, s_{i}}} & \dots \\ ⋮ & ⋮ & ⋱ \\ C_{1, m_{1}} & m_{n} #_{p}^{*} i = 1 . . . m_{3} #_{p}^{*} i = 1 a_{C_{1, m_{1}}, C_{1, 1}, k_{3, s_{3}}, . . ., k_{n, s_{n}}} & \dots \end{matrix}$ } $\begin{matrix} \dots & c_{1, m_{1}} \\ \dots & m_{n} #_{p}^{*} i = 1 . . . m_{3} #_{p}^{*} i = 1 a_{C_{1, 1}, C_{1, m_{1}}, k_{3, s_{3}}, . . ., k_{n, s_{i}}} \\ ⋱ & ⋮ \\ \dots & m_{n} #_{p}^{*} i = 1 . . . m_{3} #_{p}^{*} i = 1 a_{C_{1, i}, C_{1, m_{1}}, k_{3, s_{3}}, . . ., k_{n, s_{i}}}, \\ ⋱ & ⋮ \\ \dots & m_{n} #_{p}^{*} i = 1 . . . m_{3} #_{p}^{*} i = 1 a_{C_{1, m_{1}}, C_{1, m_{1}}, k_{3, s_{3}}, . . ., k_{n, s_{n}}} \end{matrix}$

where 1 ≤ p ≤ 10. If we use the operation $#_{1}^{*}$ , then we will get super pessimistic evalution of the corellation dependencies between criteria; if we use $#_{10}^{*}$ , then we will get super optimistic forecast of correlation dependencies between criteria, etc.

The last step of the algorithm is to determine the degrees of correlation between groups of indicators depending of the chosen thresholds for μ and ν from the user. We call that criteria C_1,k and C_1,l are in

(γ, δ)-positive consonance, if μ_{C_1,k,C_1,l,k_3,s₃,...,k_{d,s_d},...,k_{n,s_n}} > γ and ν_{C_1,k,C_1,l,k_3,s₃,...,k_{d,s_d},...,k_{n,s_n}}< δ ;

(γ, δ)-negative consonance, if μ_{C_1,k,C_1,l,k_3,s₃,...,k_{d,s_d},...,k_{n,s_n}} < γ and μ_{C_1,k,C_1,l,k_3,s₃,...,k_{d,s_d},...,k_{n,s_n}}> δ ;

(γ, δ)-dissonance, otherwise.

The ICrA pairs are calculated using the software described in [24]. The n-D ICrA works with two IMs $R_{agg}^{μ}$ and $R_{agg}^{ν}$ , rather than with the IM R_agg. The IM $R_{agg}^{μ}$ contains as elements the first components of the IFPs of R_agg, while $R_{agg}^{ν}$ contains the second components of the IFPs of R_agg. Finally let us select the best ICrA correlations according to (1), which simultaneously takes into account both the membership and the non-membership components of the IFPs. The number of criteria can be reduced by determining the degrees of correlations between the criteria using the scale [19] with values of γ = 0, 85 and δ = 0, 15 . If some (slow or expensive) criteria exhibit positive consonance with some of the rest of the criteria (that are faster or cheaper) the decision maker may decide to omit them in the further decision-making process [27].

4 Application of the ICrA to the kitchen staff’s evaluation criteria at a fast food restaurant

In this section, the ICrA method is applied to data containing values of the kitchen staff’s evaluation criteria at a fast food restaurant over a three year period. The evaluation system contains the following criteria: – Criterion C₁ – Knows all the dishes; – Criterion C₂ – Complies with product weights; – Criterion C₃ – Respects ways to prepare; – Criterion C₄ – Follows the process; – Criterion C₅ – Rotational principle; – Criterion C₆ – Performs activities outside his duties; – Criterion C₇ – Hygiene, cleaning; – Criterion C₈ – Works with machines and equipment; – Criterion C₉ – Filing sheets; – Criterion C₁₀ – Adheres to the restaurant rules; – Criterion C₁₁ – Working spirit; – Criterion C₁₂ – Skills for correct presentation of information; – Criterion C₁₃ – Positivism towards the working environment.

The input data for the ICrA are presented in a 3-D IFIM A [C, O, H] with a structure such as (4), where its elements are assessments of the employees O = {O₁, O₂, …, O_m} by criteria from the set C = {C₁, C₂, …, C₁₃} in a time-moment h_g ∈ H for g = 1, 2, 3. The initial data deviation is equal to 0,70. The results, obtained from the application of the software that implements ICrA [24], are in the form of two IMs (see Figs. 1, 2), containing, respectively, the membership and the nonmembership parts of the intuitionistic fuzzy correlations detected between each pair of criteria. The all cells of the matrix are coloured in the greyscale, with the highest values coloured in the darkest shade of grey, while the lowest ones are coloured in white. Of course, every criteria perfectly correlates with itself, so for any i (1 ≤ i ≤ 13) the value μ_{C_i,C_i} = 1, and ν_{C_i,C_i} = π_{C_i,C_i} = 0 . The two IMs are obviously symmetrical according to the main diagonals.

Fig. 1

Membership parts of the IFPs, giving the InterCriteria correlations.

Fig. 2

Non-membership parts of the IFPs, giving the InterCriteria correlations.

Taking the results produced by the 3-D ICrA, we plot them onto the intuitionistic fuzzy (IF) interpretational triangle in Fig. 3.

Fig. 3

The results of ICrA, plotted onto the IF interpretational triangle.

Table 1 describes the type of correlations, obtained between the pairs of criteria follow the scale from [19]:

Table 1

Correlations between the pairs of criteria

Type of correlation	Pairs of criteria
Strong positive consonance [0, 95 ; 1]	–
Positive consonance[0, 85 ; 0, 95)	–
Weak positive consonance [0, 75 ; 0, 85)	–
Weak dissonance [0, 67 ; 0, 75)	12-6, 12-10
Dissonance [0, 57 ; 0, 67)	12-4
Strong dissonance [0, 43 ; 0, 57)	1-2, 1-4, 1-9, 1-11, 2-8, 3-4, 3-12, 5-7, 6-10, 8-9, 8-13, 10-13, 11-13
Dissonance [0, 33 ; 0, 43)	1-6, 1-7, 1-12, 2-4, 2-5, 2-7, 2-10, 2-13, 3-8, 3-9, 3-11, 3-13, 4-6, 4-8, 4-9, 4-10, 4-11, 5-6, 5-9, 5-10, 5-12, 6-11, 7-8, 7-9, 7-11, 8-10, 8-12, 9-11, 9-13, 12-13
Weak dissonance [0, 25 ; 0, 33)	1-3, 1-5, 1-8, 1-13, 2-9, 2-11, 2-12, 3-10, 4-5, 4-7, 6-9, 7-10, 7-12, 7-13, 8-11, 9-12, 10-11, 11-12
Weak negative consonance [0, 15 ; 0, 25)	1-10, 2-3, 2-6, 3-5, 3-7, 4-13, 5-8, 5-11, 5-13, 6-8, 9-10
Negative consonance [0, 05 ; 0, 15)	–
Strong negative consonance [0 ;0, 05]	–

Calculating the distances [5] of the intercriteria IFPs to the pair 〈1, 0〉 using $d_{〈 a, b 〉} = \sqrt{(1 - a)^{2} + b^{2}},$ (6) we obtain in Table 2 the ordering of the pairs.

Table 2

Ordering of the correlating pairs, with respect to the distance from 〈1, 0〉

No	d _{C_i,C_j}	Criteria	μ _{C_i,C_j}	ν _{C_i,C_j}
1.	0.318	C₆ - C₁₂	0.68	0.02
2.	0.359	C₁₀ - C₁₂	0.68	0.17
3.	0.409	C₄ - C₁₂	0.59	0
4.	0.459	C₅ - C₇	0.55	0.06
5.	0.515	C₃ - C₁₂	0.49	0
6.	0.515	C₆ - C₁₃	0.68	0
7.	0.527	C₂ - C₈	0.5	0.17
8.	0.537	C₁ - C₁₁	0.49	0.15
9.	0.538	C₁ - C₉	0.47	0.09

The following conclusions can be obtained after the application of the ICrA:

There is no pair of criteria that are in a strong positive, a positive and a weak positive consonance, a negative and a strong negative consonance. The lowest detected correlations are between the pairs of criteria C₁ - C₁₀, C₂ - C₃, C₃ - C₅, C₂ - C₆, C₃ - C₇, C₅ - C₁₁, C₅ - C₁₃ and C₂ - C₆ .

The highest correlation has been observed between pairs C₁₂ “skills for correct presentation of information” and C₆ “performs activities outside his duties”, and C₁₀ “adheres to the restaurant rules”. The other observed dissonance is between the criteria pairs C₁₂ and C₄ “follows the process”. The open dependencies between the criteria help the manager in the efficient management of kitchen staff in the restaurant. The positivism towards the working environment is related to the willingness of the kitchen workers to perform activities outside its duties, to follow the restaurant rules and to present correct the information.

The indicators C₁ “know all the dishes” is in strong dissonance with C₁₁ “working spirit”, C₃ “respects preparation rules” is in a strong dissonance with C₁₂ “skills for correct presentation of information”. The criteria C₁₀ “adheres to the restaurant rules” and C₁₁ “working spirit” are related with C₁₃ “positivism towards the working environment”.

No correlation dependencies of a strong positive and a positive consonance between the pairs of criteria after the application of the ICrA, which means that the assessment system in the fast food restaurant is optimized.

5 A comparison of the results, obtained after application of ICrA, PCA, SCA and KCA

In this section is shown that the ICrA, PCA, SCA and KCA can be helpful for the in-depth analysis of data sets with kitchen staff evaluations in a restaurant in Bulgaria. These four approaches complement each other, but only ICrA can be applied to vague and incomplete data. The section has compared the results obtained by the classical Pearson’s, Spearman’s and Kendall’s rank correlation [1] analyzes with these from ICrA after their applying to the real dataset, containing the kitchen staff evalutions against criteria in a fast food restaurant. Let the confdence interval for correlation is 95% and p-value indicates the risk of concluding that a correlation exists - when actually, no correlation exists - is 5%.

The Table 3 describes the strongest correlations between the criteria obtained after the application of ICrA, PCA, SCA and KCA.

Table 3
The strongest correlations between the pairs of criteria according ICrA, PCA, SCA and KCA

Criterion ICrA PCA SCA KCA

C ₁ C ₉ C ₉ C ₉ C ₉

C ₂ C ₈ C ₈ C₄, C₈ C₈, C₄

C ₃ C ₁₂ C ₁₂ C ₁₂ C ₁₂

C ₄ C ₁₂ C ₁₂ C ₁₂ C ₁₂

C ₅ C ₇ C ₇ C ₇ C ₇

C ₆ C ₁₂ C ₁₁ C ₁₁ C ₁₁

C ₇ C ₅ C ₅ C ₅ C ₅

C ₈ C ₂ C ₂ C ₂ C₁₂, C₂

C ₉ C ₁ C ₁ C ₁ C ₁

C ₁₀ C ₁₂ C ₁₂ C ₁₂ C ₁₂

C ₁₁ C ₁ C₆, C₁₃, C₁ C₆, C₁ C₁₃, C₆, C₁

C ₁₂ C ₁₀ C ₁₀ C ₁₀ C ₁₀

C ₁₃ C ₆ C₁₁, C₃, C₆ C₃, C₆ C ₆

Criterion	ICrA	PCA	SCA	KCA
C ₁	C ₉	C ₉	C ₉	C ₉
C ₂	C ₈	C ₈	C₄, C₈	C₈, C₄
C ₃	C ₁₂	C ₁₂	C ₁₂	C ₁₂
C ₄	C ₁₂	C ₁₂	C ₁₂	C ₁₂
C ₅	C ₇	C ₇	C ₇	C ₇
C ₆	C ₁₂	C ₁₁	C ₁₁	C ₁₁
C ₇	C ₅	C ₅	C ₅	C ₅
C ₈	C ₂	C ₂	C ₂	C₁₂, C₂
C ₉	C ₁	C ₁	C ₁	C ₁
C ₁₀	C ₁₂	C ₁₂	C ₁₂	C ₁₂
C ₁₁	C ₁	C₆, C₁₃, C₁	C₆, C₁	C₁₃, C₆, C₁
C ₁₂	C ₁₀	C ₁₀	C ₁₀	C ₁₀
C ₁₃	C ₆	C₁₁, C₃, C₆	C₃, C₆	C ₆

The following minor differences are observed between these four methods of correlation analysis:

according to the ICrA, the criterion C₆ “performs activities outside his duties” is in the strongest correlation with the criterion C₁₂ “skills for correct presentation of information” (〈μ (C₆, C₁₂) , ν (C₆, C₁₂) 〉 = 〈0, 68 ; 0, 02〉); according to the PCA, SCA and KCA, C₆ is in the strongest correlation relation 0,43 with the criterion C₁₁ “working spirit”, and the correlation coefficient between C₆ and C₁₂ is equal to -0,09; -0,09 and -0,09 respectively according to the method PCA, SCA and KCA;

according to the ICrA, the criterion C₁₁ “working spirit” is in the strongest correlation relation with the criterion C₁ “know all the dishes” (〈μ (C₁₁, C₁) , ν (C₁₁, C₁) 〉 = 〈0, 49 ; 0, 15〉; according to the PCA, SCA and KCA, the criterion C₁₁ is in the strongest correlation relation 0,43 with the criterion C₆ “performs activities outside his duties”; the correlation coefficient between C₁₁ and C₁ is equal respectively to 0,27; 0,34 and 0,31 according to the method PCA, SCA and KCA.

The last two differences between the criteria correlations are due to high hesitancy degree π of the IFPs 〈μ (C₆, C₁₂) , ν (C₆, C₁₂) 〉 (π (C₆, C₁₂) =0, 30) and 〈μ (C₁₁, C₁) , ν (C₁₁, C₁) 〉 (π (C₁₁, C₁) =0, 36).

The Table 4 describes the strongest correlations between the criteria obtained after the application of ICrA, PCA, SCA and KCA above the original data and the data with mistakes. The columns with labels “ICrA1”, “PCA1”, “SCA1” and “KCA1” contain the criteria with the strongest correlations after applying these analyzes above the original data. The comparison of the results obtained shows that there is no comparative difference between those, obtained from the ICrA and the other three classic correlation analyzes.

Table 4

The strongest correlations (CORs) between the pairs of criteria according ICrA, PCA, SCA and KCA

Criterion	ICrA1	ICRA2	ICrA3	PCA1	PCA2	PCA3	SCA1	SCA2	SCA3	KCA1	KCA2	KCA3
C ₁	C ₉	C ₉	C ₉	C ₉	C ₉	C ₉	C ₉	C ₉	C ₉	C ₉	C ₉	C ₉
COR	〈0, 47 ; 0, 09〉	〈0, 44 ; 0, 06〉	〈0, 56 ; 0, 17〉	0, 42	0, 42	0, 42	0, 47	0, 47	0, 52	0, 41	0, 41	0, 41
C ₂	C ₈	C ₄	C ₄	C ₈	C ₈	C ₈	C₄, C₈	C ₄	C ₇	C ₈	C ₄	C ₄
COR	〈0, 5 ; 0, 17〉	〈0, 5 ; 0, 11〉	〈0, 56 ; 0, 17〉	0, 48	0, 48	0, 48	0, 40	0, 40	0, 41	0, 33	0, 33	0, 30
C ₃	C ₁₂	C ₈	C ₈	C ₁₂	C ₁₃	C ₁₂	C ₁₂	C ₁₁	C ₁₃	C ₁₂	C ₁₃	C ₈
COR	〈0, 49 ; 0〉	〈0, 44 ; 0, 06〉	〈0, 56 ; 0, 11〉	0, 43	0, 38	0, 42	0, 44	0, 44	0, 27	0, 42	0, 40	0, 23
C ₄	C ₁₂	C ₁₂	C ₁₂	C ₁₂	C ₁₂	C ₁₂	C ₁₂	C ₁₂	C ₁₂	C ₁₂	C ₁₂	C ₁₂
COR	〈0, 59 ; 0〉	〈0, 72 ; 0〉	〈0, 72 ; 0〉	0, 52	0, 52	0, 51	0, 52	0, 52	0, 53	0, 53	0, 50	0, 52
C ₅	C ₇	C ₇	C ₇	C ₇	C ₇	C ₇	C ₇	C ₇	C ₇	C ₇	C ₇	C ₇
COR	〈0, 47 ; 0, 09〉	〈0.61 ; 0〉	〈0, 56 ; 0, 06〉	0, 46	0, 46	0, 46	0, 45	0, 45	0, 66	0, 43	0, 43	0, 58
C ₆	C ₁₂	C ₁₂	C ₁₀	C ₁₁	C ₁₁	C ₁₁	C ₁₁	C ₁₀	C ₁₁	C ₁₁	C ₁₃	C ₁₁
COR	〈0, 68 ; 0, 02〉	〈0, 72 ; 0, 06〉	〈0, 78 ; 0〉	0, 43	0, 43	0, 44	0, 49	0, 42	0, 80	0, 40	0, 40	0, 43
C ₇	C ₅	C ₅	C ₅	C ₅	C ₅	C ₅	C ₅	C ₅	C ₅	C ₅	C ₅	C ₅
COR	〈0, 55 ; 0, 06〉	〈0, 56 ; 0, 06〉	〈0, 61 ; 0〉	0, 46	0, 46	0, 44	0, 45	0, 45	0, 66	0, 43	0, 43	0, 58
C ₈	C ₂	C ₁₃	C ₁₃	C ₂	C ₂	C ₂	C ₂	C ₂	C ₁₂	C ₁₂	C ₁₂	C ₁₂
COR	〈0, 50 ; 0, 17〉	〈0, 83 ; 0, 11〉	〈0, 72 ; 0.11〉	0, 48	0, 48	0, 48	0, 38	0, 38	0, 69	0, 35	0, 33	0, 33
C ₉	C ₁	C ₁₂	C ₆	C ₁	C ₁	C ₁	C ₁	C ₁	C ₁	C ₁	C ₁	C ₁
COR	〈0, 47 ; 0, 09〉	〈0, 56 ; 0, 01〉	〈0, 56 ; 0, 11〉	0, 42	0, 42	0, 42	0, 47	0, 47	0, 42	0, 41	0, 41	0, 42
C ₁₀	C ₁₂	C ₄	C ₆	C ₂	C ₂	C ₂	C ₁₃	C ₁₃	C ₆	C ₁₃	C ₁₃	C ₆
COR	〈0, 68 ; 0, 17〉	〈0, 67 ; 0, 17〉	〈0, 78 ; 0〉	0, 42	0, 43	0, 43	0, 37	0, 37	0, 80	0, 35	0, 35	0, 75
C ₁₁	C ₁	C ₁₃	C ₁	C ₆	C ₆	C ₆	C ₆	C ₆	C ₁₂	C ₁₃	C ₁₃	C ₆
COR	〈0, 49 ; 0, 15〉	〈0, 50 ; 0, 17〉	〈0, 56 ; 0, 17〉	0, 43	0, 43	0, 44	0, 42	0, 42	0, 49	0, 37	0, 40	0, 43
C ₁₂	C ₁₀	C ₄	C₄, C₆	C ₄	C ₄	C ₄	C ₄	C ₈	C ₄	C ₄	C ₈	C ₄
COR	〈0, 68 ; 0, 17〉	〈0, 72 ; 0〉	〈0, 72 ; 0〉	0, 52	0, 52	0, 51	0, 42	0, 52	0, 69	0, 50	0, 51	0, 59
C ₁₃	C ₆	C ₈	C ₈	C ₁₁	C ₁₁	C ₁₁	C₃, C₆	C ₃	C ₆	C ₆	C₆, C₁₁	C ₁₁
COR	〈0, 68 ; 0〉	〈0, 72 ; 0, 11〉	〈0, 83 ; 0, 11〉	0, 42	0, 42	0, 42	0, 41	0, 42	0, 50	0, 39	0, 39	0, 43

With a view to compare the results of ICrA with those obtained by the other classical correlation analyzes on the input data containing the errors (for example, shifting the decimal separator) let us simulate mistakes in the initial data as a result of misplacing the decimal separator with one position to the right in the estimates by the first two criteria to all the kitchen staff of the restaurant. The data deviation is equal to 17,20. We apply the four statistical analyzes to the data to determine that the differences in results between ICrA and the other statistical analyzes are due to data mistakes. The columns in the Table 4 with labels “ICrA2”, “PCA2”, “SCA2” and “KCA2” contain the criteria with the strongest correlations after applying these analyzes above the data with mistakes.

With a view to compare the results of ICrA with those obtained by other classical correlation analyzes on the input data, let us change the accuracy of the estimates of the first three kitchen workers against the evaluation criteria to a second digit after the decimal point. The data deviation is 0,50. The columns with labels “ICrA3”, “PCA3”, “SCA3” and “KCA3” in the Table 4 contain the criteria with the strongest correlations after applying these analyzes above the data with diffrent accuracy.

Comparing the results of ICrA, PCA, SCA and KCA on the initial data and these with mistakes and different accuracy, it is observed that considerable divergence of the ICrA results from those obtained by the other classical correlation analyzes, defined in [1, 4], is only found when the input data contain mistakes. So the use of them together can be taken as a way of detecting errors in the input data (e.g., shift of the decimal separator) or the information noise. Although the volume of testing data is small and its numerical interval is small, differences between the four statistical analyses were still obtained. In the future, we plan to perform similar tests on a larger volume of data belonging to a wider numerical range.

6 Conclusion

In this paper it is proposed for the first time to extend the ICrA to multidimensional over intuitionistic fuzzy evaluations against n - dimensional criteria in uncertain environments. The n-D ICrA approach is applied to establish relations and dependencies between staff’s evaluation criteria over a three-year period at a fast food chain of restaurants in Bulgaria and optimize its rating system. No correlation dependencies of positive consonance between the pairs of criteria were found after the application of 3-D ICrA, which means that the kitchen staff assessment system in the fast food restaurant is optimized. The comparison of the results obtained after the application of the classic statistical rank correlation analyses to the real data, containing the evaluation of the kitchen staff against the criteria of the evaluation system, shows that there is no comparative difference between those, obtained from the ICrA and the other three classical correlation analyses, but only the ICrA can be applied to retrieve information of other types of multidimensional fuzzy data. It is observed that considerable divergence of the ICrA results from those obtained by the other classical correlation analyzes, is only found when the input data contains mistakes. So the use of them together can be taken as a way of detecting errors in the input data (e.g., shift of the decimal separator) or the information noise.

In our next piece of research, we will describe ICrA with n-dimensional interval-valued intuitionistic fuzzy evaluations [12 , 18]. In the future, new types of ICrA method will be applied in different areas.

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