Abstract
Performance measurement is a key issue when a company is designing new strategies to improve resource allocation. This paper offers a new methodology inspired by classic importance–performance analysis (IPA) that provides a global index of importance versus performance for firms. This index compares two rankings of the same set of features regarding importance and performance, taking into account underperforming features. The marginal contribution of each feature to the proposed global index defines a set of isocurves that represents an improvement in the IPA diagram. The defined index, together with the new version of the diagram, will enable the assessment of a firm’s overall performance and, therefore, enhance decision making in the allocation of resources. The proposed methodology has been applied to a Taiwanese multiformat retailer and managerial perceptions of performance and importance are compared to assess the firm’s overall performance.
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Acknowledgments
This work was partially supported by the SENSORIAL Research Project (TIN201020966C0201, 02), funded by the Spanish Ministry of Science and Information Technology.
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Communicated by V. Loia.
Appendices
Appendix A: The qualitative absolute orderofmagnitude model
Qualitative reasoning techniques, specifically orderofmagnitude models, are considered an appropriate mathematical framework to represent expert opinions or preferences through a hierarchical model with linguistic labels (Andrés et al. 2010; Soto 2011; Herrera et al. 2008).
The onedimensional absolute orderofmagnitude qualitative model (Agell et al. 2012; TravéMassuyès and Dague 2003) works with a finite number of qualitative labels corresponding to an ordinal scale of measurement. The number of labels chosen to describe a real problem is not fixed, but depends on the characteristics of each represented variable.
Let us consider an ordered finite set of basic labels \(S_m^*=\{B_1,\ldots ,B_m\}\), which is totally ordered as a chain: \(B_1<\cdots < B_m\), each basic label corresponding to a linguistic term, for instance, “very bad” \(<\) “bad” \(<\) “acceptable” \(<\) “good” \(<\) “very good”. The complete universe of description for the orderofmagnitude space OM(\(m\)), with granularity \(m\), is the set \({\mathbb {S}}_m={\mathbb {S}}^*_m \cup \{[B_i,B_j] \; B_i,B_j \in S^*_m, i<j\},\) where the labels \([B_i,B_j]\) with \(i < j\) are defined \([B_i,B_j]=\{B_i,B_{i+1},\ldots ,B_j\}\) and named nonbasic labels (see Fig. 9).
The order considered in the set of basic labels \(S_m^*\) induces a partial order \(\le \) in \({\mathbb {S}}_m\) defined as:
considering the convention \([B_i,B_i]=B_i\).
This relation is trivially an order relation in \({\mathbb {S}}_m\), but a partial order, since there are pairs of noncomparable labels. Moreover, as Fig. 9 shows, there is another partial order relation in \({\mathbb {S}}_m\) “to be more precise than”; given two qualitative labels \(X_1\) and \( X_2\) in \({\mathbb {S}}_m\), we say that \(X_1\) is more precise than \(X_2\) if \(X_1\varsubsetneq X_2\). The least precise label (most abstract description) is \(?=[B_1,B_m]\) and basic labels are the most precise labels.
Appendix B: A ranking method using qualitative linguistic descriptions
In the proposed ranking method, each feature is characterized by the judgments of \(k\) evaluators, and each evaluator makes his/her judgements by means of qualitative labels belonging to an orderofmagnitude space \({\mathbb {S}}_{m_h}\) with granularity \(m_h\) for \(h=1,\ldots ,k\). The evaluations are then synthesized by means of the distance to a reference \(k\)dimensional vector of labels. This reference \(k\)dimensional label is given by the supreme of the sets of evaluations of each feature. The distances between evaluations and their supreme give the ranking of features directly. In this way, the process considered for ranking features assessed by \(k\) expert evaluators can be split in the following four steps:

1.
Representing features as \(k\)dimensional vectors of labels.

2.
Defining a distance \(d\) between \(k\)dimensional vectors of labels.

3.
Building a reference \(k\)dimensional vector of labels \(\mathbf X ^\mathrm{sup}\).

4.
Obtaining the ranking of the features from the values \(d(\mathbf X , \mathbf X ^{\mathrm{sup}} )\).
The subsections below describe each of the above steps.
B.1. Feature representation as \(k\)dimensional vectors of labels
Features are represented by a \(k\)dimensional vectors of labels belonging to the set \({\mathbb {X}}\), which is defined as:
For every component, monotonicity is assumed, i.e., \(X_{h}\le X'_{h}\) indicates that the evaluation made by the evaluator \(h\) corresponding to the feature \(X'\) is better or equal to the one corresponding to \(X\). The order relation defined in each \({\mathbb {S}}_{m_h}\) is extended to the Cartesian product \({\mathbb {X}} \):
This order relation in \({\mathbb {X}} \) is partial, since there are pairs of noncomparable \(k\)dimensional vectors of labels. And \(\mathbf X < \mathbf X '\), that is to say, \(\mathbf X \le \mathbf X '\) and \(\mathbf X \ne \mathbf X '\), means that feature \(\mathbf X \) is preferred to feature \(\mathbf X '\) by all the evaluators.
B.2. A distance between \(k\)dimensional vectors of labels
A method for computing distances between \(k\)dimensional vectors of labels is presented in Agell et al. (2012) via a codification of the labels in each \({\mathbb {S}}_{m_h}\) given by a location function. The location function codifies each element \(X_h =[B_i,B_j]\) in \({\mathbb {S}}_{m_h}\) by a pair of integers \((l_1(X_h), l_2(X_h))\), where \(l_1(X_h)\) is the opposite of the number of basic elements in \({\mathbb {S}}_{m_h}\) that are “between” \(B_1\) and \(B_i\), that is, \(l_1(X_h) = (i1)\), and \(l_2(X_h)\) is the number of basic elements in \({\mathbb {S}}_{m_h}\) that are “between” \(B_j\) and \(B_{m_h}\), i.e., \(l_2(X_h) = m_hj\).
The extension of the location function to the set \({\mathbb {X}}\) of \(k\)dimensional vectors of labels is defined in the following way:
A distance \(d\) between labels \(\mathbf X , \mathbf X '\) in \({\mathbb {X}}\) is then defined via a weighted Euclidian distance in \({\mathbb {R}}^{2k}\) between their codifications:
where \(w_i\) are considered to be the weights assigned to the \(k\) evaluators and \(\sum ^k_{h=1}w_h=1\). This function inherits all the properties of the weighted Euclidian distance in \({\mathbb {R}}^{2k}\).
B.3. Building a reference \(k\)dimensional vector of labels
The reference \(k\)dimensional vector of labels considered in this ranking method is the supreme with respect to the order relation \(\le \) of the set of feature representations.
Let \(\{\mathbf{X }^1,\ldots ,\mathbf{X }^n\}\subset {\mathbb {X}}\) be the set of \(n\) features representations to be ranked, then the supreme of the set \(\mathbf X ^\mathrm{sup}\), i.e., the minimum label in \({\mathbb {X}}\) which satisfies \(\mathbf{X }^r \le \mathbf X ^\mathrm{sup}, r = 1, \ldots , n,\) is computed as follows:
Given \(\mathbf X ^r=(X_1^r,\ldots ,X_k^r)\), with \(X_h^r=[B_{i_h}^r,B_{j_h}^r]\) for all \(h=1,\ldots ,k\), and for all \(r=1,\ldots ,n\), then,
where:
B.4. Obtaining the features ranking from the values \(d(\mathbf X , \mathbf X _{\mathrm{sup}} )\)
Let \(d\) be the distance defined in \({\mathbb {X}}\) in Formula (5) and \(\mathbf X ^\mathrm{sup}\) the reference label defined in Formula (6). Then, the following binary relation in \({\mathbb {X}}\):
is a preorder, i.e., it is reflexive and transitive. This preorder relation induces an equivalence relation \(\equiv \) in \({\mathbb {X}}\) by means of:
In the quotient set \({\mathbb {X}}/\!\!\equiv \), the following relation between equivalence classes is:
is an order relation. It is trivially a total order.
In this way, a set of features \(\mathbf X ^1,\ldots ,\mathbf X ^n\) can be ordered as a chain with respect to their proximity to the supreme: \(\text {class}\,(\mathbf X ^{i_1})\unlhd \cdots \unlhd \,\text {class}\, (\mathbf X ^{i_n})\).
If each \(\text {class}\, (\mathbf X ^{i_j}), j=1,\ldots n\), contains only a feature representation \(\mathbf X ^{i_j}\), the process is finished and we obtain the ranking \(\mathbf X ^{i_1}\unlhd \cdots \unlhd \,\mathbf X ^{i_n}\). If there is some \(\text {class}\, (\mathbf X ^{i_j})\) with more than one feature representation, then the same ranking process is applied to the set of the feature representations belonging to \(\text {class}\, (\mathbf X ^{i_j})\), and continued until an iteration of the process gives the same ranking as the previous iteration. The final ranking \(\mathbf X ^{m_1}\unlhd \cdots \unlhd \,\mathbf X ^{m_n}\) is then obtained.
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Sayeras, J.M., Agell, N., Rovira, X. et al. A measure of perceived performance to assess resource allocation. Soft Comput 20, 3201–3214 (2016). https://doi.org/10.1007/s0050001516963
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Keywords
 Performance evaluation
 Reasoning under uncertainty
 Fuzzy operator
 Similarity index