*1.3. Contributions*

The main contribution of this paper is the representation of performance metrics as measured through KPIs as discrete probability distributions. Embedding these distributions in the "Wasserstein space" enables the comparison and ranking of different stores. In addition, through the definition of Wasserstein barycenters, it is possible to perform clustering in the Wasserstein space with the aim of finding groups of similar stores. Moreover, since some KPIs are correlated with each other, in this paper, a subset of the most "informative" ones are chosen using feature selection and information gain. To further motivate the usage of the Wasserstein distance, a barycenter-based measure of how KPI data are not Euclidean is proposed; the computational results show that the discrepancy between the analysis in the Euclidean space and the WST space grows with the size of the subset.

#### **2. Key Performance Indicators and the Formulation of the Problem**

The focus of this paper is on a multinational retailer company which operates through a network of stores. The performance of each store is characterized in terms of service to the customer and is evaluated by the customers themselves through a number of Key Performance Indicators. Each store receives its evaluation through a survey composed of a number of questions. For each question, a customer can answer with a number on a scale from −100 to 100, which represents the satisfaction of a specific service. Each *KPIi*, with *i* = 1, ... , *K*, is computed as the average of a set of questions and captures one feature of the customer experience. Figure 1 shows an example considering the experience of a customer inside a store. This aspect of the CX can be evaluated through seven different KPIs, each of which is obtained from the answers to a set of different questions.

**Figure 1.** An example of the KPI tree related to the experience inside a store.

The objective of this study is to propose a system to assess stores' performances while simultaneously considering different KPIs. As a case study, a network of 50 stores owned

and operated by a multinational retailer is considered. In this paper, the seven KPIs related to the customer experience inside the store are considered. The following list of KPIs provides an idea of the scope of this study:


Usually, the mean of each KPI for each store is analyzed to build a ranking or to evaluate different aspects of the stores. A very effective way to visualize these means is by using the parallel coordinates plot, as shown in Figure 2. This chart enables the easy and clear visualization of a set of points (stores) in a multi-dimensional space (KPIs).

**Figure 2.** Parallel coordinates plot showing the seven KPIs of five stores (each line represents a store).

#### **3. Space of Data and Distributional Representation**
