1 Introduction

Signal processing (SP) methods have a wide range of applications in various fields of engineering and natural sciences, such as acoustics [12], and biomedical data analysis [23]. SP methods handle diverse signal characteristics such as non-stationarity and noise, making them suitable for tasks like denoising and adaptive prediction [21].

Given their advantages, SP techniques can effectively address decision problems across different domains that require analysis of time-series data. These domains include economics, healthcare, environmental science, and other scientific fields. For instance, [22] showed the potential of Kalman filters in decision problems found in economy. In [19], the authors show that Independent Component Analysis (ICA) can be used as a pre-processing step in a class of decision problems comprising multiple criteria. Moreover, the least-mean-square algorithm was employed in decision making related to the COVID-19 pandemic in Italy [16]. Besides, [18] applied an adaptive Kalman filter technique for diabetic recommendation systems.

This study aims to investigate the potential of a comprehensive approach that combines signal processing methods with Multi-Criteria Decision Analysis (MCDA) [13] in a tensorial-based data structure.

MCDA methods provide a basis for informed decision-making in the process of ranking alternatives according to multiple criteria. The data structure used in MCDA methods is usually a decision matrix, where each row corresponds to an alternative, and each column corresponds to a criterion. The entries in the matrix represent the evaluation of each alternative according to each criterion. An example of the application of MCDA methods is to rank government projects based on criteria such as environmental impact, economic feasibility, and social benefit.

The entries (evaluations) in the decision matrix usually considers static information comprising evaluations acquired in the instant decision is taken. This approach is referred to as current or single period analysis. Although the current period analysis may be suitable for certain types of decision-making, many real-world decisions require analyzing the evolution of the criteria over time (their time-series or signals). In the example of ranking government projects, the current analysis corresponds to considering the values of environmental impact, economic feasibility, and social benefit at the moment the project is implemented. However, the long-term implications of these criteria values in the years following project implementation can provide valuable information for the final decision [10].

Few studies in MCDA have been considering time-series-based approaches, as showed in [4]. Examples of time-based approaches in MCDA include the studies by [10, 17], which proposed a methodology for dealing with future time series in sustainable contexts, and [26], which propose a long-term planning horizons in a case study from the German energy sector. Other recent studies deal with historical time series analysis in MCDA, such as [4, 24, 25], and [7].

Fig. 1.
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Tensorial proposal in which predictive signals are obtained using an adaptive prediction method.

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The works mentioned above highlight a gap in the field of MCDA when it comes to handling time series data for criteria. To address this gap, reference [5] proposed an extension of the classical decision matrix to a tensor (often employed in signal processing when dealing with multiple signals [3, 15]). The tensor in [5] has three dimensions, with the first dimension representing the alternatives, the second dimension representing the criteria, and the third dimension representing the historical time series of the criteria. For each time series, an adaptive prediction method was applied to predict the values of the criteria for specific future periods. Subsequently, a decision matrix was constructed using these predicted values. Finally, a MCDA method, called Preference Ranking Organization Method for Enrichment Evaluation II (PROMETHEE II) [1], was applied to obtain a predicted ranking for a specific future period.

Although the approach in [5] is useful in certain contexts, it may not provide an effective decision-making when the decision-makers must consider the long-term consequences of their choices. We address this case in the present work by obtaining future prediction signals for each criterion, instead of a single prediction value as done in [5]. The proposed methodology is illustrated in Fig. 1. Once the data are structured as a tensor, an adaptive prediction method is applied to each criterion to obtain q (\(q > 1\)) predicted values and form a q-dimensional prediction signal for each criterion. We then apply the PROMETHEE II method to the predicted signals dimension, then to the criteria dimension, to obtain the ranking of the alternatives. Moreover, we investigate how the prediction error influences the final ranking.

The paper is organized as follows. Section 2 provides an overview of the tensor notation adapted for the MCDA context. Section 3 presents a brief background on the MCDA problem and the PROMETHEE II method. The proposed algorithm is described in Sect. 4. The results and discussion are presented in Sect. 5. Finally, Sect. 6 concludes the study.

2 Tensorial Structure for Support Multi-criteria Decision Analysis

Tensors are widely used in SP [8], and can be defined as functions of three or more indices. We consider in this study a third-order tensor denoted by \(\mathcal {H} \in \mathbb {R}^{m \times n \times T}\). The first dimension of this tensor is related to the alternatives, the second to the criteria, and the third to the time-series. There are three types of slices in this tensor, represented as frontal \({\textbf {H}}(:,:,t)\), horizontal \({\textbf {H}}(:,j,:)\) and vertical \({\textbf {H}}(i,:,:)\).

Also, there are three types of fibers, \({\textbf {h}}(i, j,:)\), \({\textbf {h}}(i, :, t)\) and \({\textbf {h}}(:, j, t)\). The fiber \({\textbf {h}}(i, j,:)\) is illustrated in Fig. 1, representing the time-series related to the criteria. Finally, each element of this tensor is denoted by \(h_{ijt}\), and corresponds to the evaluation of alternative i in criterion j at instant t.

It is worth mentioning that the employ of tensor structures in MCDA is rather recent. The decision tensor was introduced as an alternative to the traditional decision matrix in [5, 6]. Additionally, an unbalanced three-dimensional decision tensor was proposed in [20], in such a way as to allow the third dimension of the tensor to have varying sizes.

3 Multi-criteria Decision Analysis Background

MCDA methods have become increasingly popular for ranking alternatives based on multiple criteria. The data structure in MCDA is the decision matrix denoted by \({\textbf {H}} \in \mathbb {R}^{m \times n}\). Here, the rows of \({\textbf {H}}\) correspond to a set of m alternatives denoted as \(A = \{a_1, a_2, \cdots , a_m\}\), while the columns represent n criteria denoted by \(C = \{c_1, c_2, \cdots , c_n\}\). Each element in the matrix, represented by \(h_{ij}\), represents the evaluation of alternative i with respect to criterion j.

Some MCDA techniques utilize the concept of matrix aggregation, which involves transforming the decision matrix \({\textbf {H}} \in \mathbb {R}^{m \times n}\) into a scoring vector \({\textbf {f}} \in \mathbb {R}^{m}\). This process maps each row i of matrix \({\textbf {H}}\) (which corresponds to alternative i) to a score \(f_i\), which is then used to rank the alternatives. There are numerous aggregation methods available [27], including both linear and nonlinear approaches [11]. In addition, the method parameters can be trained in a supervised, semi-supervised, or unsupervised [9] way.

A well-established MCDA aggregation technique is the PROMETHEE family of methods [2]. This approach is based on pairwise comparisons, which involves comparing the evaluations of two alternatives on a particular criterion for all pairs of alternatives in all criteria [1]. The method also uses a weight vector \(\Gamma = \{\gamma _1, \gamma _2, \cdots , \gamma _n\}\) which model relative importance of each criterion; assuming that \(\gamma _n \ge 0\) and \(\sum _{j=1}^{n} \gamma _j = 1\). PROMETHEE II [1] is a specific variant of the PROMETHEE method that takes the decision matrix H as input and outputs a score vector \({\textbf {f}} \in \mathbb {R}^m\).

The initial step in PROMETHEE II is to compute the difference between the performance of each pair of alternatives \(a_i\) and \(a_k\) in each criterion j using the following equation:

$$\begin{aligned} d_j(a_i, a_k) = h_{ij} - h_{kj} \ \ \forall j = 1,\cdots , n. \end{aligned}$$
(1)

By applying Eq. (1), we can generate a tensor \(\mathcal {D} \in \mathbb {R}^{m \times m \times n}\). Each matrix slice of \(\mathcal {D}\), denoted as \({\textbf {D}}(:,:,j) \in \mathbb {R}^{m \times m}\), represents a pairwise comparison matrix.

In the second step, a preference function P is applied to each pairwise comparison matrix \({\textbf {D}}(:,:,j)\) using the following formula:

$$\begin{aligned} P_j(d_j(a_i, a_k)) = P_j({\textbf {D}}(:,:,j)) \ \ \forall j = 1,\cdots , n. \end{aligned}$$
(2)

where \(0 \le P_j({\textbf {D}}(:,:,j)) \le 1\). There are various types of preference functions available, as described in [1]. For our study, we use the most common preference function known as Usual: \(P_j(d_j(a_i, a_k)) = \dfrac{\text{ sgn }(d_j(a_i, a_k))+1}{2}\), where sgn(\( d_j(a_i, a_k)) \) represents the sign function.

The next step is to consider a mapping function \(\mathbb {R}^{m \times m \times n} \rightarrow \mathbb {R}^{m \times m}\) that provides a global preference index of \(a_i\) over \(a_k\) \(\forall \) i, k:

$$\begin{aligned} \pi (a_i, a_k) = \sum _{j=1}^{n}\gamma _j P_j({\textbf {D}}(:,:,j)). \end{aligned}$$
(3)

Because \(P_j({\textbf {D}}(:,:,j)) \ge 0\) and \(\gamma _j \ge 0\), then \(\pi (a_i, a_k) \ge 0\).

The last step involves mapping from \(\mathbb {R}^{m \times m} \rightarrow \mathbb {R}^{m}\). This is achieved by comparing each \(a_i\) with the remaining \((m-1)\) alternatives, and obtaining its final score, denoted by \(\phi (a_i)\). The score is calculated as the difference between the mean preference of \(a_i\) over the other alternatives and the mean preference of all alternatives over \(a_i\):

$$\begin{aligned} \phi (a_i) = \dfrac{1}{m-1}\sum _{a \in A}\pi (a, a_i) - \dfrac{1}{m-1}\sum _{a \in A}\pi (a_i, a) \ \ \forall i = 1,\cdots , m. \end{aligned}$$
(4)

The vector f is formed by \(\phi (a_i) \in [-1,1]\), and it is sorted in descending order to obtain a vector with the ranking r of the alternatives.

4 The Proposed MCDA Method Using Tensor-Based Structure and Adaptive Prediction

The proposed algorithm is composed by three main steps: (1) MCDA data tensor structuring; (2) Prediction; (3) Aggregation. Steps (1) and (2) are illustrated in Fig. 1.

In step (2), we use the Recursive Least Square (RLS) [21] due to its fast convergence and its ability to track eventual dynamic behaviors of the data. The required computational complexity of the RLS is thoroughly compatible with our problem. Next, we provide a detailed description of each step and formalize it in Algorithm 1.

  • (1) MCDA data tensorial structuration: The input for this step is the data related to the evaluation of the alternatives over the years. These data are structured in a tensor denoted by \(\mathcal {H} \in \mathbb {R}^{m \times n \times T}\), with m alternatives, n criteria, and T samples. This tensor is the output of this step;

  • (2) Prediction: For this step, the input is the tensor \(\mathcal {H}\). For each fiber \({\textbf {h}}(i, j,:)\), the RLS algorithm is applied with \(\lambda \) samples to be predicted ahead, where \(\lambda \) ranges from 1 to q. As a result, prediction signals are obtained, which are represented by \(\hat{{\textbf {h}}}(i, j,:)\). A prediction tensor \(\mathcal {\hat{H}} \in \mathbb {R}^{m \times n \times q}\) is then constructed using these prediction signals. This tensor is the output of this step;

  • (3) Aggregation: For this step, the input is the tensor \(\mathcal {\hat{H}}\). For each slice \(\hat{{\textbf {H}}}(:,j,:)\), where \(j = 1,\ldots , n\), represented in Fig. 2, we apply the PROMETHEE II method. As a result, n vectors f are outputted, which compose a matrix \(\hat{{\textbf {F}}} \in \mathbb {R}^{m \times n}\). Next, PROMETHE II is applied in matrix \(\hat{{\textbf {F}}}\), to obtain a final vector f. By sorting f, the ranking r of the alternatives is obtained, which is the output of this step.

Fig. 2.
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Horizontal slices of the tensor \(\mathcal {\hat{H}}\).

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figure a

5 Numerical Experiments

This section presents experimental results obtained from synthetic data to demonstrate the advantages of the proposed approach for decision-making, which incorporates ranking based on future prediction signals compared to ranking based on single-period data. The code and data utilized in these experiments are publicly available on the following Github repository: https://github.com/BSCCampello/Tensor-analytics-prediction.

In order to assess the dissimilarities between two rankings in our experiments, we employ the Kendall tau rank correlation coefficient [14]: \( \tau _{{\textbf {r}}\times \hat{{\textbf {r}}}} = \left\| {\textbf {r}} - \hat{ {\textbf {r}}} \right\| _\kappa \). This coefficient measures the distance between two rankings, \( {\textbf {r}}\) and \(\hat{ {\textbf {r}}}\), and is defined as follows:

$$\begin{aligned} \tau _{{\textbf {r}}\times \hat{{\textbf {r}}}} = \dfrac{2}{n(n-1)} | \{(p, u): p<u, (\tau _{\textbf {r}}(p) < \tau _{\textbf {r}}(u)\ \wedge \tau _{\hat{{\textbf {r}}}}(p) > \tau _{\hat{{\textbf {r}}}}(u)) \\ \vee (\tau _{\textbf {r}}(p) > \tau _{\textbf {r}}(u)\ \wedge \tau _{\hat{{\textbf {r}}}}(p) < \tau _{\hat{{\textbf {r}}}}(u))\} |, \end{aligned}$$

where \( \tau _{\textbf {r}}(p) \) and \(\tau _{\hat{{\textbf {r}}}}(p)\) means the positions of the p-th element in vectors r and \(\hat{{\textbf {r}}}\). The \(\tau _{{\textbf {r}}\times \hat{{\textbf {r}}}}\) value is bounded on [0, 1]; being zero when the rankings are the same, and one if the rankings are opposite.

We performed the simulations by carrying out the two stages described below \(\ell \) = 1,000 times, , and we averaged the results of these \(\ell \) simulations: , assuming a set of \(m=5\) alternatives, \(n=4\) criteria, \(T=25\) samples, and \(q = 5\) samples to be predicted.

Stage 1 – generate the input data: The data used to compose the tensor \(\mathcal {H} \in \mathbb {R}^{m \times n \times T}\) is generated. Given a continuous uniform distribution U and a set \(Z = [1,\)..., T], with \(t \in Z\), each fiber \({\textbf {h}}(i, j,:)\) of the tensor \(\mathcal {H}\) is obtained as follows: Criterion 1, \({\textbf {h}}(i, 1,:)\) \(\rightarrow \) \(h_{i1t} = a_i + b_it \), \(a_i \sim U[2, 20]\), \(b_i \sim U[0.5,0.8]\), \(\forall t, i\); Criterion 2, \({\textbf {h}}(i, 2,:)\) \(\rightarrow \)] \(h_{i2t} = a_i + sin(f_it)\), \(a_i \sim U[2, 20]\), \(f_i\sim U[0.7\pi , 0.75\pi ]\), \(\forall t, i\); Criterion 3, \({\textbf {h}}(i, 3,:)\) \(\rightarrow \) \(h_{i3t} = a_i + (-1)^t\), \(a_i \sim U[2, 20]\), \(\forall t, i\); Criterion 4, \({\textbf {h}}(i, 4,:)\) \(\rightarrow \)] \(h_{i4t} = a_i + t^{0.075}\), \(a_i \sim U[2, 20]\), \(\forall t, i\). Since there is a white noise component in actual signals, we add random noise \(\alpha \boldsymbol{\mu }\), \(\boldsymbol{\mu }\sim N(0,1)\), to the signals \({\textbf {h}}(i, j,:)\), being \(\alpha \) calculated as:

\( \alpha = 10^{(\frac{10log\sigma _{{\textbf {h}}(i, j,:)}^2 -SNR}{20}) }\),

where \(\sigma _{{\textbf {h}}(i, j,:)}^2\) is the variance of \( {\textbf {h}}(i, j,:)\), and the SNR is the signal-to-noise ratio, which assumes the values from 0 to 50 dB.

Stage 2 – Computing the results: To compute the results, let us assume that the current period is at the instant of time T, as shown in Fig. 3. The known signal is assumed to be from instant 1 to \(T = 25\), and the years to be predicted are for the next 5 years, which means that the algorithm considers \(q = 5\) samples. Considering four rankings to compute the Tau distances \(\tau ^c_{{\textbf {r}}^* \times {\textbf {r}}^c}\), \(\hat{\tau }_{{\textbf {r}}^* \times \hat{{\textbf {r}}}}\), and \(\hat{\tau }_{{\textbf {g}}_\lambda \times \hat{{\textbf {r}}}}\), obtained as:

Fig. 3.
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Signals and the current period.

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– Current ranking (\({\textbf {r}}^c\)): We construct a matrix \({\textbf {H}} \in \mathbb {R}^{m \times n}\) with the data at time instant \(T = 25\) (current values) and apply the PROMETHEE II method. This ranking is commonly used in MCDA literature as the classical ranking.

– Prediction ranking (\(\hat{{\textbf {r}}}\)): This ranking is calculated by applying Algorithm 1 using the data generated in Stage 1 as input.

– Reference ranking or benchmark (\({\textbf {r}}^*\)): We construct a tensor \(\mathcal {H}^* \in \mathbb {R}^{m \times n \times q}\) containing the actual data from period \(T+1\) to \(T + 5\), and apply the PROMETHEE II method to both time and criteria dimensions. This ranking serves as the benchmark for the prediction one. Indeed, \(\hat{{\textbf {r}}}\) and \({\textbf {r}}^*\) should be the same if the prediction errors do not influence the final ranking.

– Rankings obtained using the approach proposed in [5] (\({\textbf {g}}_1\), \({\textbf {g}}_2\), \({\textbf {g}}_3\), \({\textbf {g}}_4\), \({\textbf {g}}_5\)): To generate these rankings, we first construct a matrix of predicted values for a single future period, \(T+1\), using the RLS technique. We then apply the PROMETHEE II method to this matrix to obtain a ranking, denoted by \({\textbf {g}}_1\), at period \(T+1\). We repeat this process for periods \(T+2\) through \(T+5\), generating a total of 5 rankings, one for each period.

At the end of Stages 1 and 2, we compute the average of \(\ell \) simulations for each SNR value of \(\tau ^c_{{\textbf {r}}^* \times {\textbf {r}}^c}\), \(\hat{\tau }_{{\textbf {r}}^* \times \hat{{\textbf {r}}}}\), and \(\hat{\tau }_{{\textbf {g}}_\lambda \times \hat{{\textbf {r}}}}\), and these results are shown in Fig. 4 and Fig. 5.

Fig. 4.
figure 4

\(\tau ^c_{{\textbf {r}}^* \times {\textbf {r}}^c}\), and \(\hat{\tau }_{{\textbf {r}}^* \times \hat{{\textbf {r}}}}\).

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A preliminary analysis of Fig. 4 concerns the comparison between the ranking typically computed in MCDA literature, referred to as the current ranking, and the ranking obtained by considering future values of the criteria, known as the benchmark ranking (\(\tau ^c_{{\textbf {r}}^* \times {\textbf {r}}^c}\)). It is evident that these rankings differ with Tau values higher than 0.11. This implies that the classical approach used in MCDA may not yield appropriate solutions when it is crucial to analyze the future impact of a decision.

In real-world decision-making, future criterion values are often unknown, and predictive methods should be employed to estimate them. A second analysis of Fig. 4 is therefore necessary to evaluate the influence of prediction error on the ranking. Specifically, this analysis compares the benchmark ranking and the predicted ranking through the Tau value \(\hat{\tau }_{{\textbf {r}}^* \times \hat{{\textbf {r}}}}\), which quantifies the impact of the prediction error in the ranking outcomes. We may observe that, for SNR higher than 15, Tau values are less than 0.08, which can be acceptable if future rankings are relevant for the decision-making process.

The results in Fig. 4 show that the difference between the benchmark and prediction rankings (\(\hat{\tau }_{{\textbf {r}}^* \times \hat{{\textbf {r}}}}\)) is smaller than the difference between the benchmark and current rankings (\(\tau ^c_{{\textbf {r}}^* \times {\textbf {r}}^c}\)). This suggests that, for decisions with future consequences, a prediction-based approach may be more suitable than one relying on the current data values, even if there is an error in the prediction and, in consequence, in the ranking.

Fig. 5.
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Comparison between the approach proposed in [5] and this proposal.

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To compare the approach proposed in [5] with the one presented in this work, we obtained additional results, as illustrated in Fig. 5. We calculated \(\hat{\tau }_{{\textbf {g}}_\lambda \times \hat{{\textbf {r}}}}\) for \(\lambda = 1, \cdots , q\), where \({\textbf {g}}_\lambda \) denotes the ranking obtained by the adaptive approach proposed in [5]. For instance, the metric \(\hat{\tau }_{{\textbf {g}}_1 \times \hat{{\textbf {r}}}}\) quantifies the difference between the rankings \({\textbf {g}}_1\) (obtained using the single-period prediction for \(T+1\)) and \(\hat{{\textbf {r}}}\). Notably, the ranking \(\hat{{\textbf {r}}}\) is distinct from all the rankings obtained using the single-period approach. These results suggest that the proposed approach is capable of providing different rankings than those obtained by the single-period approach. Therefore, the proposed approach can be a valuable tool in decision-making processes that consider long-term consequences.

6 Conclusion

The present paper proposes a new decision-making method based on tensor data structure and adaptive prediction, thus extending previous work that provide cross-fertilization between the fields of signal processing and MCDA. The proposed approach showed promising results in generating rankings suitable for decision-making that are relevant to the analysis of long-term future consequences.

The simulation results revealed that the proposed approach can provide rankings close to the benchmark, indicating that prediction error does not significantly affect ranking when SNR is higher than 15 dB. Furthermore, the results suggest that this approach can be more suitable for making decisions with future consequences, even if there is an error in the ranking, than relying solely on current data values. Additionally, the proposed approach is also more suitable than the approach proposed in [5], when future decision should consider long-term analysis.

In summary, the proposed approach can be a valuable tool for decision-making in diverse domains, including finance, healthcare, and sustainable development. Further research can be conducted to evaluate the applicability of the proposed approach in real-life decision-making scenarios, and to investigate the influence of different prediction methods on its performance.