FlowSort-H: Multi-Criteria Sorting with Choquet Integral

• FlowSort-H is a multi-criteria sorting methodology that extends the original FlowSort by using the Choquet integral to aggregate preference degrees on heterogeneous scales.
• It transforms diverse criteria into a common scale using PROMETHEE-type preference functions to avoid the need for prior normalization.
• The method applies a 2-additive capacity model to explicitly capture criterion interactions, enabling robust assignment of alternatives to ordered categories.

FlowSort-H is a methodology for multi-criteria sorting that extends the FlowSort method to accommodate interacting criteria evaluated on heterogeneous scales. Utilizing the Choquet integral as an aggregation operator within the outranking framework, FlowSort-H addresses two significant limitations of classical approaches: (i) the necessity of a common scale for all criteria, and (ii) the inability to explicitly model synergy or redundancy (interactions) between criteria. By constructing a common, commensurate scale via PROMETHEE-type preference functions and applying a 2-additive Choquet integral for aggregation, FlowSort-H enables robust assignment of alternatives to ordered categories in complex decision-aiding scenarios (Pelissari et al., 2019).

1. Theoretical Foundation: Choquet Integral on Heterogeneous Scales

FlowSort-H employs a capacity (fuzzy measure) $\mu: 2^G \to [0,1]$ defined on the set of criteria $G = \{1, \dots, n\}$, satisfying $\mu(\emptyset) = 0$, $\mu(G) = 1$ and monotonicity constraints. The Möbius representation $m: 2^G \to \mathbb{R}$ expresses $\mu(S)$ as the sum of the Möbius measures $m(T)$ over $T \subseteq S$, with a 2-additive restriction ($m(T)=0$ if $|T|>2$):

$$\mu(S) = \sum_{j\in S} m(\{j\}) + \sum_{\{j,s\}\subseteq S} m(\{j,s\}).$$

Instead of applying the Choquet integral to direct criterion scores $g_j(\cdot)$ (often on incompatible scales), FlowSort-H computes all aggregations on the basis of preference degrees $P_j(x,y)\in[0,1]$, constructed for each pair of entities $x, y$ (alternative or profile). This ensures commensurability required for valid Choquet integration.

2. Construction of Commensurate Scales: Preference Functions

FlowSort-H systematically transforms heterogeneous criteria into a common unit interval by employing PROMETHEE-type preference functions:

$$P_j(x, y) = P\left[ g_j(x) - g_j(y) \right] \in [0,1]$$

DMs choose suitable preference function families (e.g., Type I: $P_j(x, y)=1$ iff $g_j(x)>g_j(y)$, else $0$), tailored to each criterion. The mapping

$\phi_j:\;g_j(x)\mapsto P_j(x,y)$

is used to construct all subsequent aggregations. This approach circumvents the need to normalize $g_j$ a priori and directly models intensity-of-preference in a scale-agnostic manner.

3. Choquet-Based Outranking Degree

The core of FlowSort-H is the replacement of the classical weighted-sum outranking degree with a Choquet-based measure that incorporates both importance (via Shapley indices) and interaction (via pairwise interaction indices):

$$CI_{\pi}(x, y) = \sum_{j \in G} m(\{j\}) P_j(x, y) + \sum_{\{j, s\} \subseteq G} m(\{j, s\}) \min\{P_j(x, y), P_s(x, y)\}$$

Alternatively, using Shapley values $I_j$ and interaction indices $I_{js}$:

$$CI_{\pi}(x,y) = \sum_{I_{js}>0} \min\{P_j, P_s\} I_{js} + \sum_{I_{js}<0} \max\{P_j, P_s\} |I_{js}| + \sum_{j=1}^n P_j(x, y) \left(I_j - \tfrac12 \sum_{s\neq j}|I_{js}|\right)$$

where $$I_j = m(\{j\}) + \frac12 \sum_{s \ne j} m(\{j, s\})$$, $I_{js} = m(\{j, s\})$. For $I_{js}=0$, the expression reduces to classic additive aggregation.

4. FlowSort-H Algorithmic Process

Given a set of alternatives $A = \{a_1, ..., a_m\}$, ordered categories $K_1 \succ \cdots \succ K_k$ operationalized via profiles $R = \{r_1, ..., r_{k+1}\}$, the FlowSort-H assignment process proceeds as:

1. For each $a_i$, set $R_i := R \cup \{ a_i \}$.
2. For all pairs $x, y \in R_i$, and for each criterion $j$, compute $P_j(x, y)$ using the selected preference function.
3. Compute $CI_\pi(x, y)$ for all $x, y \in R_i$ per the Choquet-based outranking degree.
4. For all $x \in R_i$, calculate the positive and negative flows:

$$\phi^+_{R_i}(x) = \frac{1}{|R_i| - 1} \sum_{y \neq x} CI_\pi(x, y), \quad \phi^-_{R_i}(x) = \frac{1}{|R_i| - 1} \sum_{y \neq x} CI_\pi(y, x)$$

Net flow: $\phi_{R_i}(x) = \phi^+_{R_i}(x) - \phi^-_{R_i}(x)$.

1. Assign $a_i$ to category $K_h$ using a chosen rule (positive, negative, or net-flow) by comparing $\phi^+_{R_i}(a_i)$ (resp. $\phi^-_{R_i}(a_i)$, $\phi_{R_i}(a_i)$) to profiles' flows.

Capacity parameters $(m(\{j\}), m(\{j, s\}))$ or equivalently $(I_j, I_{js})$ are elicited from DM judgments or via indirect approaches such as Robust Ordinal Regression or SMAA.

5. Modeling and Interpreting Criterion Interactions

In FlowSort-H, the 2-additive Möbius coefficients $m(\{j, s\})=I_{js}$ represent the pairwise interactions:

• $I_{js}>0$ (synergic): the criteria are superadditive, boosting the $\min\{P_j, P_s\}$ term.
• $I_{js}<0$ (redundant): the criteria are subadditive, so the $\max\{P_j, P_s\}$ term is emphasized.
• The Shapley value $I_j$ ensures $\sum_j I_j=1$ (normalization).

Monotonicity is guaranteed under the constraint:

$$I_j - \frac12 \sum_{s \ne j} |I_{js}| \geq 0, \quad \forall j.$$

This structure enables modeling the influence of both individual criteria and their mutual interactions within the assignment process.

6. Numerical Example

Consider three alternatives and two criteria with heterogeneous measurement units, assigned to two categories via three profiles:

$\begin{array}{c|cc} & g_1(\cdot) \text{ (km/h)} & g_2(\cdot) \text{ (km/l)} \ \hline a_1 & 210 & 10 \ a_2 & 202 & 12 \ a_3 & 200 & 12 \ \hline r_1 & 215 & 12.5 \ r_2 & 195 & 11.0 \ r_3 & 180 & 10.0 \end{array}$

With Type I preference functions ($P_j(x, y) = 1$ iff $g_j(x) > g_j(y)$, $0$ otherwise) and 2-additive capacity parameters $I_1=0.5$, $I_2=0.5$, $I_{12}=0.2$ (indicating synergy), the Möbius components are $m(\{1\})=0.4$, $m(\{2\})=0.4$, $m(\{1,2\})=0.2$.

For $a_1$ vs $r_2$:

$$P_1(a_1, r_2) = 1,\quad P_2(a_1, r_2) = 0\ CI_\pi(a_1, r_2) = 0.4 \times 1 + 0.4 \times 0 + 0.2 \times \min\{1,0\} = 0.4$$

Positive flows are computed analogously; if $\phi^+(a_1) = 0.7$, $\phi^+(r_1) = 1$, $\phi^+(r_2)=0.5$, the positive-flow rule yields $a_1\in K_1$ since $0.7$ lies between $1$ and $0.5$.

7. Implementation and Practical Considerations

FlowSort-H provides a detailed operational procedure suitable for direct implementation, enabling researchers to build preference-function transformations, apply the Choquet aggregation directly on pairwise preference degrees, and execute the full sorting workflow without preliminary normalization of heterogeneous scales. Capacities can be assessed either through direct DM interaction or inferred by indirect means. The approach is conceptually simple to implement and designed to facilitate the inclusion of criterion interactions in multi-criteria sorting, supporting robust and interpretable classification in the presence of conflicting, non-commensurate data (Pelissari et al., 2019).