A 360-Degree Performance Appraisal Model Dealing with Heterogeneous Information and Dependent Criteria

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1 A 360-Degree Performance Appraisal Model Dealing with Heterogeneous Information and Dependent Criteria M. Espinilla, R. de Andrés, F.J. Martínez, and L. Martínez Department of Computer Sciences, University of Jaén Department of Economic and Economic History, University of Salamanca December 29, 2011 Abstract 360-degree performance appraisal process is used as a tool that provides an evaluation about employees performance. It is based on the opinion of different groups of reviewers who socialize with evaluated employees. Many criteria must be considered in performance appraisal that may have different nature and usually present uncertainty. Therefore, it seems necessary and appropriate to provide a heterogeneous framework adapted to the nature of the criteria. In this context, criteria can be asses by reviewers, according to their background and degree of knowledge about evaluated employees. Furthermore, the interaction among criteria, as well as reviewers weights should be taken into account to ensure an effective aggregation process of the information, providing interpretable and understandable evaluations. In this paper, we present an integrated model for 360-degree performance appraisal that can manage heterogenous information and computes a final linguistic evaluation for each employee, applying an effective aggregation that considers the interaction among criteria and reviewers weights. We also present a real case study to show the usefulness and effectiveness of the proposed model. 1 Introduction Performance appraisal is a key tool in companies that provides information about employees performance in order to make important decisions, such as salary adjustments, promotions, identification of training and development needs, documentation of performance levels or behaviors that may merit firing or sanctions. Furthermore, there is an evident link among performance appraisal and attitudes, efforts and behaviors of 1

2 the employees, that implies improvements in the financial results obtained by companies [12]. Currently, performance appraisal process is based on the opinion of different groups of reviewers who socialize with evaluated employees, since they can truly respond to how an employee develops his/her job. Moreover, the process includes the opinion of employee about her/himself (see Figure 1). This kind of performance appraisal process is so-called 360-degree appraisal or integral evaluation [10, 12, 24] and it overcomes some disadvantages from traditional evaluation such as lack of objectivity, prejudice or halo errors, etc. [1, 12]. The decision analysis [7] has been successfully applied to 360-degree performance appraisal process and others areas, such as sensory evaluation [25, 26] or, Figure 1: Points of view in 360-degree performance appraisal sustainable energy [19]. In this way, we can find in the literature, performance appraisal models proposed in [2, 12], where the set of criteria are evaluated in a quantitative way, providing numerical results. This fact implies a lack of accuracy in the performance appraisal process, due to the difficulty of expressing in a precise way, qualitative criteria and subjective judgments. Moreover, a numerical domain is not a very suitable expression domain for evaluation results, since these results should be interpreted in a easy way by different members of the company. Performance appraisal models presented in [8, 9] provide a flexible linguistic evaluation framework in which criteria are assessed in multiple linguistic scales, according to the degree of knowledge of the reviewers, providing results easy to interpret in a linguistic domain. However, the main issue of these models is that they do not consider quantitative criteria, which are very usual in performance evaluation. All the previously mentioned models fall into two classes that present either a strict evaluation framework or uninterpretable results. Additionally, these models only take into account a set of independent criteria, disregarding the possible interaction among them, which is very common in these processes. In order to obtain a successful 360- degree performance appraisal process to make right decisions in the company, three main aspects should be considered: 1. The evaluation framework should be flexible, such that the reviewers can provide their judgments within different domains (numerical, interval-valued and linguistic), according to the uncertainty and nature of criteria and the background of each reviewer, i.e., a heterogeneous evaluation framework. 2. The application of an adequate set of aggregation operators that take into account the interaction among criteria and reviewers weights. 3. The evaluation of the results should be close to human natural language in order to be understandable and interpretable by multiple members of the company. In this paper, we present an integrated model for 360-degree performance appraisal based on the decision analysis scheme [7] that can manage heterogenous information 2

3 and computes a final linguistic evaluation for each employee, considering the interaction among criteria and reviewers weights. To do so, we propose the use of the linguistic 2-tuple representation model [15, 16] and its extended approach to deal with heterogeneous information [17], which is based on the unification of the heterogeneous information into linguistic information, thus facilitating understandable results. To obtain results close to human beings and taking into account the linguistic modeling of the unified information, we shall implement processes of computing with words (CWW) [28, 29], by using a multi-step aggregation process, considering the ordered weighted average operator, the weighted arithmetic average operator and the Choquet integral as aggregation operators, because they take into account reviewers weights and the interaction among criteria. Finally, the paper shows a real case study to illustrate the usefulness and effectiveness of the proposed model. The paper is organized as follows. Section 2 describes the relationship between decision analysis and evaluation problems. Section 3 reviews a CWW approach based on linguistic 2-tuple model to deal with heterogeneous information. Section 4 presents a novel heterogeneous model for 360-degree performance appraisal. Section 5 shows a real case study of the proposed model. Finally, in Section 6 some conclusions are pointed out. 2 Decision Analysis and Evaluation Process Decision analysis [7] is a discipline, belonging to decision theory, that has been successfully applied to solve evaluation problems in some different fields such as sustainable energy [18, 30, 31], sensory evaluation [25, 26], quality of service [5, 13] or digital libraries [3], engineering systems [27] and new product development [21, 22]. The goals of an evaluation process is to compute a set of overall assessments that summarizes the information gathered and to provide useful information about the set of evaluated elements. To do so, it is necessary to establish a set of evaluated elements in the evaluation framework, gather the information and finally, compute a final assessments for each element. It is therefore obvious that decision analysis is an excellent tool for evaluation, since it includes a wide variety of methods for the evaluation of a set of alternatives, considering all relevant criteria in a decision problem and involving experts [23]. Framework Gathering information Rating alternatives Make a decision Identify objetives Identify decision Identify alternatives Decision Analysis Rational way Decision Making Sometimes: Emotional way Figure 2: Decision making scheme 3

4 Thereby, decision analysis methods provide a rational analysis in a simple and quick way, since the highest final assessment would normally correspond to the best evaluated element [20]. A classical decision analysis scheme consists of the following phases (see Figure 2 [7]): Identify decision, objectives and alternatives of the problem. Framework: It defines the structure of the problem and the expression domains in which the preferences can be assessed. Gathering information: Decision makers provide their information. Rating alternatives: This phase obtains a collective assessment for each alternative. Make a decision. The application of the decision analysis scheme to an evaluation process does not necessarily involve all phases. The essential phases regarding an evaluation problem that will be used in our proposal are: establishing framework, gathering information and rating the alternatives. 3 Computing with words in a Heterogenous Framework CWW is a methodology in which objects of computation are words or sentences from a natural language. It emulates human cognitive processes to make reasoning processes and decisions in environments of uncertainty and imprecision [37]. Zadeh introduced the concept of linguistic variable [36] in order to deal with linguistic variables is necessary to choose the appropriate linguistic descriptors for the term set and their semantics. There are different possibilities to carry out these selections (see [34, 36]). The CWW methodology considers that inputs and output results should be expressed in a linguistic domain in order to be close to human natural language, providing interpretable and understandable results [28, 29]. This fact is key in performance appraisal where evaluation results are used to make hard decisions by the human resources department. The linguistic 2-tuple model [15, 16] and its extended approaches [11, 14, 17], providing methods to deal with complex decision making problems, following the CWW methodology. In our proposal, we consider a heterogenous framework, in which the reviewers could use numerical, linguistic and interval-valued information. In this section, we review the extended approach to deal with heterogeneous information presented in [17] based on the linguistic 2-tuple representation model [15, 16]. Figure 3 shows a basic scheme of this approach, whose steps are further described below. 3.1 Unification Process This step is based on the unification of heterogeneous information into linguistic information, thus facilitating comprehensive results for decision makers and it consists of the following states: 4

5 Heterogeneous Framework Numerical Values in [0,1] Interval Values in [0,1] T NS T IS Transformation into Fuzzy Sets s 0 Fuzzy Sets in the S s g Transformation into Linguistic 2-tuples S [0,g] (si, i) Methodology CWW Linguistic 2-tuple Computational Model Linguistic Values T SS (s k, k ),s k S UNIFICATION PROCESS AGGREGATION PROCESS APPROACH BASED ON FUSION OF THE INFORMATION Figure 3: Approach based on Information Fusion into Linguistic Values 1. Transformation of the information into Fuzzy Sets in S. The heterogeneous information will be unified into a specific linguistic domain, called Basic Linguistic Terms Set and noted as S = {s 0, s 1,..., s g }, that is selected with the aim of keeping as much knowledge as possible (see [17]). Each real number, interval value and linguistic value is then transformed into a fuzzy set on S, F(S), using an adequate transformation function for its expression domain [17]. (a) Numerical domain. When v [0, 1], a numerical transformation function, φ NS : [0, 1] F (S), is defined by: T NS (v) = g (s i /γ i ) (1) where γ i = µ si (v) [0, 1] is the membership degree of v to s i S. (b) Interval domain. When v P ([0, 1]), an interval transformation function, φ IS : P ([0, 1]) F (S), is defined by: T IS (v) = i=0 g (s i /γ i ) (2) i=0 where γ i = max min{µ I (y), µ si (y)}, i = 0,..., g, and y 0 if y < d, µ I (y) = 1 if d y e, 0 if y > e. (3) being y [0, 1]. 5

6 (c) Linguistic domain. Being v S, such that S = {s j, j = 1..., h} and h g, a linguistic transformation function, φ SS : S F (S), is defined by: g T SS (v) = (s i /γ i ) (4) i=0 where γ i = max min{µ sj (y), µ si (y)}, i = 0,..., g. y 2. Transformation of fuzzy sets, F (S), into Linguistic 2-tuples in S. The previous fuzzy sets are conducted into linguistic 2-tuples, which facilitate the processes of CWW and produce interpretable results [15, 16]. The linguistic 2-tuple representation model is based on the concept of symbolic translation [15] and represents the linguistic information through a 2-tuple (s, α), where s is a linguistic term and α is a numerical value representation of the symbolic translation [15]. Thereby, being β [0, g] the value generated by a symbolic aggregation operation, we can assign a 2-tuple (s, α) that expresses the equivalent information of that given by β. Definition 1 [15]. Let S = {s 0,..., s g } be a set of linguistic terms. The 2- tuple set associated with S is defined as S = S [ 0.5, 0.5). We define the function S : [0, g] S given by, S (β) = (s i, α), with { i = round (β), α = β i, (5) where round assigns to β the integer number i {0, 1,..., g} closest to β. We note that S is bijective [15, 16] and 1 S : S [0, g] is defined by 1 S (s i, α) = i + α. In this way, the 2-tuples of S will be identified with the numerical values in the interval [0, g]. As aforementioned, fuzzy sets are transform into linguistic 2-tuples in the S at this stage by using the function χ defined as: Definition 2 [17]. Given the linguistic term set S = {s 0, s 1,..., s g }, the function χ : F(S) S is defined by g j γ j j=0 χ ({γ 0, γ 1,..., γ g }) = S = (s g i, α) S. (6) j=0 γ j 6

7 3.2 Aggregation Process The linguistic 2-tuple representation model has a linguistic computing model associated [16], that accomplish processes of CWW in a precise way. Different aggregation operators have been proposed for linguistic 2-tuple [16, 35], in our proposal we will use the ordered weighted average operator, the weighted arithmetic average operator and the Choquet integral for linguistic 2-tuple that will be defined when they will be used Degree Performance Appraisal Model Our aim in this paper is to present a novel integrated model for 360-degree performance appraisal based on a decision analysis scheme with a flexible framework in which reviewers can express their judgments in different domains, providing linguistic results, considering the interaction of the criteria and reviewers weights. To do so, the 360-degree performance appraisal model consists of the phases shown in Figure 4 that cover the essential phases of decision analysis [7]. The following subsections present in further detail these phases of the performance appraisal model. Heterogeneous Evaluation Framework Numerical Interval Linguistic Gathering Information Supervisors Collaborators Customers Employee Rating Process Unification of the information Multi-step aggregation Figure 4: 360-degree performance appraisal model 4.1 Heterogeneous Evaluation Framework In this phase, the evaluation framework is determined to fix the structure of the 360- degree performance appraisal process. It is necessary to define the main features and terminology of this process in which personnel are evaluated from different points of view, using multiple expression domains. 7

8 Let us suppose there is a set of employees X = {x 1,..., x n } to be evaluated by three collectives: a set of supervisors (executive staff) A = {a 1,..., a r }, a set of collaborators (fellows) B = {b 1,..., b s } and a set of customers C = {c 1,..., c t }. Employees will be evaluated attending to a set of criteria Y = {Y 1,..., Y p }. The assessments provided by the members of the collectives a i A, b i B and C about the employee x j, according to the criterion Y k are denoted by a ik c i b ik j and c ik j, respectively. Moreover, xjkj is the assessment of x j about himself with respect to Y k. Therefore, there are (r + s + t + 1) p assessments, maximum, for each employee provided by different collectives and the employee himself. Thereby, we consider an heterogeneous evaluation framework in which each reviewer of the different collectives can use different domains to assess each criterion Y k, k = 1,..., p, depending on either the nature of the criteria or the reviewer s background, their degrees of knowledge, their perceptions and feelings about the evaluated employees. Our model includes in the heterogeneous evaluation framework: numerical values, v [0, 1], interval-values, v P ([0, 1]), and linguistic values with different granularity v S = {s 0,..., s h }. 4.2 Gathering Information Once the framework has been defined, the reviewers of the different collectives provide their judgment in a numerical, interval or linguistic domain, according to the evaluated employees, x j, j = {1,..., n} and the evaluated criteria Y k, k = {1,..., p}. The opinions of each collective are provided by means of assessment vectors: {â i1 j with i {1,..., r} for supervisors, {ˆb i1 j {ĉ i1 j,..., ĉip j ip,..., ˆb j } with i {1,..., t} for customers, and finally, {ˆxj1,..., âip j, j } } with i {1,..., s} for collaborators, j,..., ˆxjp j } for the evaluated employee, where ˆ represents an assessment in an expression domain (numerical, interval-valued or linguistic), according to the uncertainty and nature of criteria and the background of each reviewer. 4.3 Rating Process The aim of this process is to obtain a linguistic global assessment for each employee, easy to understand and interpretable to make right decisions by the company. To do so, we propose the application of the approach reviewed in Section 3. First the heterogenous information is unified into linguistic 2-tuple on the S, second a global assessment is computed for each employee in a multi-step aggregation process, and finally, a classification and a raking of employees are provided. The following subsections present in detail the rating process phases that were graphically described in Figure Unification of Heterogeneous Information According to the approach reviewed in Section 3, the heterogeneous information is conducted in a linguistic domain to carry out the CWW processes, providing linguistic results. 8

9 Linguistic 2-tuple To do so, we fix the unification domain, i.e., the basic linguistic term set, S. The heterogeneous information is then conducted by fuzzy sets on the S according to the expression domain, by using the respective transformation functions (see equations (1), (2) and (4)). In order to facilitate the understandability of the results, the fuzzy sets are transformed in S into linguistic 2-tuples, S, by using the equations (5) and (6) (see Figure 3). In this way, the heterogeneous information provided by the different collectives has been already unified into linguistic 2-tuples in S: {a i1 j {b i1 j {c i1 j {x j1 j,..., aip j }, i {1,..., r} with ai1 j = (s i, α i ) i1 j S.,..., bip j }, i {1,..., s} with bi1 j = (s i, α i ) i1 j S.,..., cip j }, i {1,..., t} with ci1 j = (s i, α i ) i1 j S.,..., xjpj } with xj1j = (s i, α i ) j1 j S Multi-step Aggregation Process The aim of the performance appraisal process is to obtain a global assessment for each evaluated employee that summarizes her/his weaknesses and strengths, considering the interaction among criteria and reviewers weights. These assessments will be expressed in a linguistic domain that facilitates their interpretation and understandability. Multi-Step Aggregation Process Step 1 OWA operator Collective criteria values Supervisors Collaborators Customers Employees Step 2 Weighted average operator Global criteria values Step 3 Choquet Integral Global values Figure 5: Steps of multi-step aggregation process To do so, our integrated model proposes the aggregation of the unified information into linguistic 2-tuples by means of a multi-step aggregation process [4], in which an adequate set of aggregation operators is used, according to the needs of the aggregation phase (see Figure 5). A point worthy of mention is that usually the opinions of employees about themselves are not include in the aggregation process because can bias the result of the process. These opinions are used by companies in the rating phase with the purpose of improving their performance and synchronizing companies goals with employees goals (see Figure 4). This part of the performance appraisal procedure is so-called feedback and coaching process [12]. Following, we present in further detail each stage of the multi-step aggregation process. 9

10 Computing collective criteria values, v k (x j ). For each reviewers collective, their assessments about a given criterion Y k are aggregated by means of the 2-tuple OWA operator, G w. We suggest this operator because it does not distinguish the origin of the assessments, since it is anonymous. Therefore, we use it in the first step of the multi-step aggregation process. Definition 3 [15]. Let ((s 1, α 1 ),..., (s m, α m )) S m be a vector of linguistic 2- tuples, and w = (w 1,..., w m ) [0, 1] m be a weighting vector such that m i=1 w i = 1. The 2-tuple OWA operator associated with w is the function G w : S m S defined by G w( (s 1, α 1 ),..., (s m, α m ) ) ( m ) = S w i βi = (s l, α l ) S, where β i { is the i-th largest element of 1 S i=1 } (s 1, α 1 ),..., 1 (s S m, α m ). Under these assumptions, the supervisors assessment about criterion k, regarding the employee x j, v k A (x j) is computed by the function G w A,k : S r S defined by v k A(x j ) = G w A,k(a 1k j,..., a rk j ) S where a ik j S for i = 1... r are linguistic 2-tuples belong to S. In the same way, it is possible to obtain the collective criteria assessments about criterion Y k for the others reviewers collectives: v k B (x j) (collaborators), v k C (x j) (customers), and v k X (x j) (evaluated employees themselves) Computing global criteria values, v k (x j ). In this step of the multi-step aggregation process, the previous collective assessments for each collective and each criterion are aggregated by means of a 2-tuple weighted average operator to obtain a global value for each criterion and for each employee. We propose this operator to aggregate the information because it allows companies to establish different weights for each reviewers collective, taking into account their knowledge about the evaluated criteria and their significance in performance appraisal process. Definition 4 [15]. Let ((s 1, α 1 ),..., (s m, α m )) S m be a vector of linguistic 2- tuples, and w = (w 1,..., w m ) [0, 1] m be a weighting vector such that m i=1 w i = 1. The 2-tuple weighted average operator associated with w is the function F w : S m S defined by F w( (s 1, α 1 ),..., (s m, α m ) ) ( m ) = S w i 1 S (s i, α i ) = (s l, α l ) S. i=1 The collective assessments then obtained in the previous step, v k A (x j), v k B (x j) and v k C (x j) for every k {1,..., p}, are aggregated by F w k : S 3 S, where w = (w A, w B, w C ) and w = 1, obtaining a global value for each criterion Y k modeled by a linguistic 2-tuple: v k (x j ) = Fk w ( v k A (x j ), vb(x k j ), vc(x k j ) ) S. 10

11 Computing a global value, v(x j ). Eventually, companies work out a global value v(x j ), for each employee regarding all criteria, taking into account the interaction among them. This value is obtained by aggregating the global criteria values related to each employee x j X, by means of a Choquet integral [6] that considers the mutual interaction among criteria. The use of the Choquet integral requires the computation of the coefficients that represent criteria relations, i.e, fuzzy measures [32]. In our model the relations among criteria are fixed by company s Human Resource Department. Definition 5 [32]. Let N = {1,..., m} be a set of m criteria. A fuzzy measure is a set function ϕ : N [0, 1] that satisfies the following conditions: ϕ( ) = 0, ϕ(n) = 1 ϕ(s) ϕ(t ) whenever S T (ϕ is monotonic). Definition 6 [35]. Let ((s 1, α 1 ),..., (s m, α m )) S m be a vector of linguistic 2- tuples, N be the set of attributes and ϕ be a fuzzy measure on [0, 1], the linguistic 2-tuple Choquet integral is given by M ϕ : S m S M ϕ ((s 1, α 1 ),..., (s m, α m )) = S ( n i=1 [ (ϕ(h)σ(i) ) (ϕ(h) σ(i 1) ) ] 1 S (s i, α i ) where (σ(1), σ(2),..., σ(n)) is a permutation of (1,2,...,n) such that ((s σ(1), α σ(1) ) (s σ(2), α σ(2) )... (s σ(m), α σ(m) )), being x σ(i) the criterion corresponding to the (s σ(1), α σ(1) ) and (H) σ(i) = {x σ(k) k i} for i 1 with (H) σ(0) =. Considering the previous definitions, the multi-step aggregation process is carried out as follows: v(x j ) = M ϕ ( v 1 (x j ), v 2 (x j ),..., v p (x j ) ) S. The global results obtained in each step of the multi-step aggregation process, v k A (x j), v k B (x j), v k C (x j), v k (x j ), for k = 1,..., p, and v(x j ), are used either for sorting and ranking the employees or to establish the companies policy in the ranking phase. In our integrated model, linguistic results are provided at all the stages. These results are easy to understand and interpret as required by the human resources department Ranking Phase Usually, the global assessments are used by the company for sort and rank the employees for applying some specific policy. For this fact, we propose this phase in our integrated model in which we can sort employees in each step of the multi-step aggregation process. Furthermore in this phase, the opinions of employees about themselves are used in order to improve their performance and synchronizing companies goals with employees goals (feedback and coaching process) [12]. ), 11

12 Thus, the employees ranking is obtained by the comparison of the corresponding linguistic 2-tuple in S, according to the ordinary lexicographic order presented in [15]. It provides a complete ranking of the set of evaluated employees from the best to the worst according to their computed assessments in each step. 5 Case Study In this section, we show a real case study to show the usefulness and effectiveness of the proposed integrated model that can manage heterogenous evaluation framework and provides linguistic assessments in order to be close to human natural language, while considering the interaction among criteria and reviewers weights. 5.1 Heterogeneous Evaluation Framework We have developed our study on a well-known multinational clothing company, which each month carries out an integral evaluation process over their employees in each store. Our case study involves opinions from supervisors, collaborators/co-workers, customers and employees themselves. Specifically, the reviewers collectives are the following: A set of supervisors that, in the case of this company, is made up of three subset: 1. The area manager subset: A A = {a A 1 }. 2. The direct supervisors subset: A D = {a D 1, a D 2, a D 3 }. 3. The non-direct supervisors subset: A ND = {a ND 1, a ND 2, a ND 3 }. Therefore, the collective of supervisors reviewers is defined as A = A A A D A ND where A A A D A ND =. A set of collaborators or/and co-workers consists of: 1. A subset of selling employees: B S = {b S 1, b S 2, b S 3, b S 4, b S 5, b S 6, b S 7 }. 2. A subset of non-selling employees: B NS = {b NS 1, b NS 2, b NS 3, b NS 4 }. Consequently, the set of collaborators/co-workers reviewers is defined as B = B S B NS, where B S B NS =. And finally, a set of customers: C = {c 1,..., c 20 }. Obviously, such amount is not the total of customers that visited the shop for a month. We have selected twenty-representative of them with the purpose of simplifying the case study. In this company, the employee s performance is characterized by eleven criteria Y = {Y 1,..., Y 11 }, which are shown below: Y 1 : Productivity. This criterion is the amount of employee s sales divides among the employee s hour worked. Y 2 : Level of sales. This criterion is the employee s sales amount. 12

13 Y 3 : Average closing time. After the closing time, each employee has to arrange a specific shop zone. This criterion measures how it takes employee to arrange her/his zone. Y 4 : Identification with the company. Identification with the product that company sales. Think company s values. Trust and defend the company. Like fashion. Y 5 : Training interest. Positive attitude to learn and improve. Y 6 : Work tasks. Open truck and alarmed. Organization of stock. Replacement of products. Coordinate and shape. Neat and clean work. Serve and sell in changing room. Y 7 : Customers service. Available for posting. Always say hello. Thank the purchase. Y 8 : Responsibility. Achieve daily work correctly. Not leave the shop before finishing work. Do tasks that are not her/his responsibility. Assume the consequences of her/his acts. Be always on time. Y 9 : Initiative. Get in before work needs. Propose ideas and solutions. Make decisions within their responsibilities. Accept supervision. Help collaborators and co-workers. Y 10 : To serve as an example. Fulfil all the shop rules and methods. Serve as a model for co-workers and collaborators. Be positive. Look forward solutions. Constructive attitude. Y 11 : Personal image. Dresses stylishly. Be polite. Have pleasant manners. Dress with taste. We can note that each group of reviewers express their opinions on selling employees about each criterion according to the uncertainty and nature of such criteria and the background of each reviewer, through a specific domain: Numerical (N), Interval (I) or Linguistic (L). Specifically in this case study, reviewers can express their linguistic assessments in four linguistic domains with different number of terms {S 3, S 5, S 7, S 9 }. Each linguistic term set is symmetrically and uniformly distributed and its syntax is as follows: S 3 ={Null (N), Medium (M), Perfect (P)} S 5 ={Null (N), Low (L), Medium (M), High (H), Perfect (P)} S 7 ={Null (N), VeryLow (VL), Low (L), Medium (M), High (H), VeryHigh (VH), Perfect (P)} S 9 ={Null (N), AlmostNull (AN), VeryLow (VL), Low (L), Medium (M) High (H) SlightlyHigh (VH), VeryHigh (VH), Perfect (P)} Obviously, all reviewers do not evaluate all the criteria, as there are collectives that do not have sufficient knowledge to evaluate certain criteria. The collective of reviewers for each criterion and the expression domain used for evaluating are shown in Table 1. 13

14 A A A D A ND B S B NS C Cost Y 1 N Y 2 N Y N Benefit Y 4 S 7 S 7 S 7 S 5 S 5 - Y 5 S 5 S 5 S Y 6 S 9 S 9 S Y 7 S 3 S 3 S 3 S 3 S 3 S 5 Y 8 S 7 S 7 S Y 9 S 7 S 7 S 7 S 3 S 3 - Y 10 S 7 S 7 S 7 S 3 S 3 - Y 11 S 5 S 5 S 5 S 5 S 5 S 3 Table 1: Reviewers collective for each criterion 5.2 Gathering Information Once the evaluation framework has been fixed, the reviewers express their opinions about evaluated employees. In this case study, we manage a great amount of information: seven evaluated employees, eleven criteria and 38 reviewers, being seven supervisors, eleven collaborators and 20 customers. Due to this fact, we omit the assessments provided by the reviewers about selling employees for the set of criteria in this paper, but it can be consulted in the additional material Rating Process According to the integrated proposed model in Section 4, we carry out the three steps of the rating process Unification the Heterogeneous Information The heterogenous information is unified into a basic linguistic terms set, S. In our case study the S corresponds to the linguistic terms set with nine labels, S 9, in order to keep as much knowledge as possible. This domain is key in our case study because the global assessments will be expressed in it in order to offer results easy to interpret, following the CWW methodology. An important aspect to consider in our case study is that criteria Y 1, Y 2 and Y 3 are cost criteria. Therefore, preferences about these criteria must be normalized according to their expression domains. Once this is done, we unify the gathered information, according to the revised approach in the Section Currently, the additional material is a document to the reviewers of the paper, but if the paper is accepted, this material will be available to interested readers in order that they can find additional information and include their own results. 2 In the same way that the gathered information, the unified information into linguistic 2-tuple in the S can be consulted in the additional material for the reviewers of the paper. 14

15 5.3.2 Multi-step Aggregation Process In this stage, we compute a global assessment for each selling employee in a multi-step aggregation process that considers an adequate set of aggregation operators in order to meet the needs of performance assessment, according to the interaction among criteria and reviewers weights Computing collective criteria values. In the first step of this process, we apply the 2-tuple OWA operator, which requires a OWA weighting vector. This vector can be determined by means of different ways, in our case study, the Human Resource Department uses a non-decreasing linguistic quantifiers [33], specifically, the linguistic quantifier Most. Table 2: Collectives assessments for each selling employee about each criterion x 1 x 2 x 3 x 4 x 5 x 6 x 7 Y 1 A (V L,.3) (L,.5) (N, 0) (AN,.2) (M,.4) (L, 0) (P, 0) Y 2 A (V L,.5) (L,.5) (N, 0) (AN, 0) (H, 0) (M,.5) (P, 0) Y 3 A (P, 0) (L,.5) (N, 0) (AN, 0) (H, 0) (M,.5) (P, 0) Y 4 A (SH,.3) (M, 0) (H,.2) (V L,.3) (M, 0) (SH,.5) (V H,.1) Y 4 B (SH, 0) (H,.3) (H,.3) (AN,.4) (L,.4) (H, 0) (V H,.4) Y 5 A (H,.4) (SH,.5) (M,.2) (L,.1) (V L,.2) (L,.1) (V H,.1) Y 6 A (SH,.3) (V H, 0) (L, 0) (L,.3) (SH, 0) (M, 0) (V H,.3) Y 7 A (V H, 0) (M, 0) (M, 0) (L, 0) (M, 0) (M, 0) (P, 0 Y 7 B (SH, 0) (M, 0) (L,.2) (AN,.3) (M, 0) (H,.3) (P, 0) Y 7 C (M,.4) (V L, 0) (N, 0) (V L, 0) (V L, 0) (H,.2) (SH,.2) Y 8 A (SH,.2) (SH,.5) (L,.3) (L, 0) (L,.3) (V L,.5) (P, 0) Y 9 A (L, 0) (L, 0) (AN,.3) (L,.1) (SH,.1) (H,.2) (V H,.4) Y 9 B (M, 0) (H,.3) (V L, 0) (N, 0) (L,.2) (N, 0) (V H,.2) Y 10 A (M,.2) (L,.3) (L,.2) (AN, 0) (AN,.3) (AN,.3) (L,.3) Y 10 B (N,.4) (M, 0) (N, 0) (N, 0) (V L,.4) (V L,.4) (SH,.4) Y 11 A (SH, 0) (M,.5) (M, 0) (V H,.1) (M, 0) (SH,.5) (P, 0) Y 11 B (H, 0) (L,.3) (M,.3) (H,.3) (L,.4) (M,.3) (V H,.3) Y 11 C (SH,.4) (SH,.4) (M,.4) (M,.4) (M, 0) (H,.2) (P,.4) In this way, collectives assessments for each selling employee about each criterion are computed and these are shown in the Table Computing global criteria values. In the second step of this process, we apply the linguistic 2-tuple weighted average operator, which also requires a weighting vector. This vector is defined by the Human Resource Department, attending to the evaluated criterion and the reviewer collective. Therefore, global criteria values obtained with the weights fixed by Human Resource Department are shown in Table 3. Y 4 : w A = 0.5 w B = 0.5 Y 7 : w A = 0.25 w B = 0.25 w C = 0.5 Y 9 : w A = 0.75 w B = 0.25 Y 10 : w A = 0.6 w B = 0.4 Y 11 : w A = 0.33 w B = 0.33 w C =

16 Table 3: Global criteria and collectives weights fixed by human resource department x 1 x 2 x 3 x 4 x 5 x 6 x 7 Y 1 (V L,.3) (L,.5) (N, 0) (AN,.2) (M,.4) (L, 0) (P, 0) Y 2 (V L,.5) (L,.5) (N, 0) (AN, 0) (H, 0) (M,.5) (P, 0) Y 3 (P, 0) (L,.5) (N, 0) (AN, 0) (H, 0) (M,.5) (P, 0) Y 4 (SH,.2) (M,.3) (H,.1) (V L,.1) (M,.3) (H,.3) (V H,.2) Y 5 (H,.1) (SH,.5) (M,.2) (L,.1) (V L,.2) (L,.1) (V H,.1) Y 6 (SH,.3) (V H, 0) (L, 0) (L,.3) (SH, 0) (M, 0) (V H,.3) Y 7 (H,.3) (L, 0) (V L,.3) (V L,.1) (L, 0) (H,.3) (V H,.1) Y 8 (SH,.2) (SH,.5) (L,.3) (L, 0) (L,.3) (V L,.5) (P, 0) Y 9 (L,.3) (M,.5) (V L,.5) (V L,.3) (H,.3) (M,.1) (V H,.4) Y 10 (L,.3) (L,.2) (V L,.4) (AN,.4) (V L,.3) (V L,.3) (M,.2) Y 11 (SH,.5) (M,.1) (M,.2) (H,.3) (M,.2) (H,.3) (V H,.4) Computing a global value. This last step integrates a Choquet integral for linguistic 2-tuple, which is able to cope with interaction among criteria. In our case study, fuzzy measures have been established after several meetings by company s Human Resource Department as follows: ϕ(y 1 ) = 0.1 ϕ(y 2 ) = 0.1 ϕ(y 1, Y 2) = 0.43 ϕ(y 3 ) = 0.02 ϕ(y 4 ) = 0.02 ϕ(y 5 ) = 0.02 ϕ(y 6 ) = 0.1 ϕ(y 7 ) = 0.1 ϕ(y 8 ) = 0.02 ϕ(y 11 ) = 0.02 ϕ(y 6, Y 7, Y 8 ) = 0.45 ϕ(y 9 ) = 0.02 ϕ(y 10 ) = 0.02 In other case ϕ(s T )= ϕ(s) + ϕ(t ), S, T Y, S T = We can see that criteria Y 1 and Y 2 are positively correlated, since ϕ(y 1 Y2 ) > ϕ(y 1 ) + ϕ(y 2 ). Due to the fact that these criteria present some degree of complementarity. In the same way to the criteria Y 6, Y 7 and Y 8. Finally, global values for each selling employees are computed, using the Choquet integral for linguistic 2-tuple, which are shown in Table 4. Table 4: Assessments for each selling employee x 1 x 2 x 3 x 4 x 5 x 6 x 7 (M,.2) (M,.5) (AN,.3) (V L,.1) (M, 0) (L, 0) (V H,.2) In this phase of our case study, we have obtained linguistic results for each employee in each step of the multi-step aggregation process, following the CWW methodology. These results are close to human natural language and providing interpretability and understandability Ranking Phase Finally, we put in order all computed values and we establish a ranking among evaluated employees with the purpose of identifying the best ones. In the case study, (V H,.2) = (V eryhigh,.2) S is the highest global value and it corresponds to the employee x 7. Therefore, x 7 is the best selling employee for this evaluation and the total ranking is: 16

17 x 7 x 5 x 1 x 2 x 6 x 4 x 3 6 Concluding remarks Performance appraisal is a relevant process used for companies in order to make important decisions, such as promotions, salaries, needs, etc. In this paper, we have proposed an integrated 360-degree performance appraisal model that provides a flexible evaluation framework in which reviewers can provide their assessments within different domains, according to the uncertainty and nature of such criteria and the background of each reviewer. Moreover, with the aim of ensure an effective aggregation of the information, the proposed model applies an adequate set of aggregation operators to cope with the interaction among criteria and reviewers weights, providing results close to human natural language by using the computing with words methodology in order to be understandable and interpretable by different members of the company. Finally, we have present a real case study to show the usefulness and effectiveness of the integrated proposed model. Acknowledgements This paper has been partially supported by the Consejería de Innovación, Ciencia y Empresa (Junta de Andalucía) with the Research Project AGR-6487 and by the Ministerio de Ciencia e Innovación with the Research Project TIN and the Research Project ECO References [1] C.G. Banks and L. Roberson. Performance appraisers as test developers. Academy of Management Review, 10: , [2] J. N. Baron and D. M. Kreps. Strategic Human Resources, Frameworks for General Managers. Wiley & Sons, Nueva York, [3] F.J. Cabrerizo, J. López-Gijón, A.A. Ruíz, and E. Herrera-Viedma. A model based on fuzzy linguistic information to evaluate the quality of digital libraries. International Journal of Information Technology and Decision Making, 9(3): , [4] T. Calvo, R. Mesiar, and R.R. Yager. Quantative weight and aggregation. IEEE Transactions on Fuzzy Systems, 12:62 69, [5] C.T. Chen. Applying linguistic decision-making method to deal with service quality evaluation problems. International Journal of Uncertainty, Fuzziness and Knowlege-Based Systems, 9(SUPPL.): ,

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