SOMIT: Subjective-Objective Median Weighting

• SOMIT is a hybrid multi-criteria decision-making weighting technique that balances expert insights with objective data using median-based methods.
• It employs a three-stage process—subjective weighting, objective dispersion analysis, and multiplicative aggregation—to reduce bias and enhance robustness.
• Practical applications span renewable energy evaluation and policy analysis, offering improved cognitive efficiency and resistance to outlier effects compared to traditional methods.

The Subjective–Objective Median-Based Importance Technique (SOMIT) is a hybrid multi-criteria decision-making (MCDM) weighting methodology that synthesizes subjective expert judgments and objective data-driven insights for the robust determination of criteria weights. SOMIT emphasizes scalability, outlier-resistance, cognitive efficiency, and modularity, addressing the limitations of conventional subjective approaches (e.g., AHP) and purely objective weighting schemes (e.g., Entropy, CRITIC) in applications including renewable energy system evaluation, technology prioritization, and energy policy analysis (Ding et al., 6 Jan 2026).

1. Formal Problem Statement and Motivation

The MCDM context underlying SOMIT consists of $m$ alternatives $A = \{A_1, ..., A_m\}$ and $n$ criteria $C = \{C_1, ..., C_n\}$, where the alternatives–criteria matrix (ACM) $F = [f_{ij}]$ encodes the performance of $A_i$ against $C_j$. The core task is to determine a nonnegative, normalized weight vector $w = (w_1, ..., w_n)^\top$ such that $w_j \geq 0$, $\sum_j w_j = 1$, encoding the relative importance of each criterion.

Classic subjective methods (e.g., AHP) require $O(n^2)$ pairwise comparisons, demanding significant cognitive effort from experts and often resulting in bias or inconsistency. In contrast, purely objective methods derive weights solely from statistical properties of the ACM, disregarding expert knowledge and exhibiting sensitivity to noise and outliers. SOMIT was developed to mitigate both sets of limitations by integrating a median-based subjective weighting (minimizing expert burden), a median-based objective dispersion measure (ensuring outlier robustness), and a geometric combination (balancing both components) (Ding et al., 6 Jan 2026).

2. Mathematical Formulation

SOMIT operates in three distinct stages: subjective median-based weighting, objective median-based weighting, and multiplicative aggregation.

2.1. Subjective Median-Based Weights (Part I)

• The decision-maker selects a single “median” criterion $C_d$ as reference.
• For each $j \ne d$, an expert assigns a comparison $a_{j,d} \in [1/9, 9]$ reflecting the relative importance of $C_j$ to $C_d$.
• The most and least favored criteria $h$ and $l$ are identified: $h = \arg\max_j a_{j,d}$, $l = \arg\min_j a_{j,d}$, and $a_{h,l}$ is elicited.
• A set $A$ of all assessed comparisons is formed.
• The ideal relationship for weights: $d_{ij} = w^{(s)}_i / w^{(s)}_j$, which is approximated by solving the quadratic programming problem

$$\min_{w^{(s)}} \sum_{(i,j)\in A} \left(a_{i,j} w^{(s)}_i - w^{(s)}_j\right)^2$$

subject to $\sum_{j=1}^n w^{(s)}_j = 1$, $w^{(s)}_j \geq 0$.

2.2. Objective Median-Based Weights (Part II)

• Columns of $F$ are normalized to $[0,1]$ using max-min normalization:

$$F_{ij} = \frac{f_{ij} - \min_k f_{kj}}{\max_k f_{kj} - \min_k f_{kj}}$$

• For each criterion $j$, the column median $m_j$ is computed, followed by the average absolute deviation from the median (AADM):

$r_j = \frac{1}{m} \sum_{i=1}^m |F_{ij} - m_j|$

• Objective weights are the normalized $r_j$:

$w^{(o)}_j = \frac{r_j}{\sum_{k=1}^n r_k}$

2.3. Hybrid (Combined) Weights (Part III)

• Final criteria weights are obtained by geometric aggregation:

$$\tilde{w}_j = w^{(s)}_j \times w^{(o)}_j, \qquad w_j = \frac{\tilde{w}_j}{\sum_{k=1}^n \tilde{w}_k}$$

3. Algorithmic Structure and Computational Complexity

SOMIT’s algorithm is divided into three linear-complexity parts, with the primary computational costs as follows:

• Subjective Part: $O(n)$ queries, $O(n^3)$ quadratic programming for solving weights (practically $n \leq 20$).
• Objective Part: $O(mn + n \log m)$ for normalization, medians, and deviations.
• Hybrid Aggregation: $O(n)$ multiplicative and normalization steps.

Pseudocode (Python-like style):

w_s = somit_i() # expert-completed w_o = somit_ii(ACM) # input ACM as m x n array w = somit_iii(w_s, w_o)

4. Robustness, Modularity, and Comparative Advantages

SOMIT demonstrates several technical advantages over competing methodologies:

• Cognitive Efficiency: Reduces expert inputs from $O(n^2)$ (AHP) to $O(n)$ through single-reference (median) comparison.
• Outlier Robustness: Utilization of the column median and AADM ensures minimal weight shift under severe outlier contamination. For example, shifting a solar emissions input from $48$ to $480$ gCO₂/kWh altered SOMIT weights by only $1.9\%$ (AAFD$_w$), compared to CRITIC’s $2.5\%$ change.
• Bias Mitigation: The geometric aggregation curtails dominance by either the subjective or objective component, enforcing balance.
• Modularity: Each SOMIT part is functional as a standalone weighting scheme and is compatible with any weighted MCDM framework.

5. Empirical Performance in Renewable Energy Evaluation

SOMIT’s efficacy is illustrated by two renewable energy case studies:

India: 4 Alternatives × 10 Criteria

• Alternatives: Solar, Wind, Hydro, Biomass.
• Criteria grouped as financial, technical, environmental, and social.
• Resulting SOMIT weights yield the following performance scores (TOPSIS):

| Alternative | $P$ (Score) | |-------------|------------| | Solar | 0.5725 | | Biomass | 0.5293 | | Wind | 0.4511 | | Hydro | 0.4279 |

Saudi Arabia: 5 Alternatives × 8 Criteria

• Alternatives: Solar PV, Solar Thermal, Wind, Geothermal, Biomass.
• SOMIT weights and TOPSIS performance:

| Alternative | $P$ (Score) | |----------------|------------| | Solar PV | 0.7562 | | Solar Thermal | 0.7556 | | Wind | 0.4997 | | Geothermal | 0.3312 | | Biomass | 0.2281 |

In both cases, solar-related alternatives achieved highest performance rankings (Ding et al., 6 Jan 2026).

6. Implementation and Integration

A dedicated Python package, pysomit, provides practical access to the SOMIT workflow:

• Installation: pip install pysomit
• Interfaces:
  • somit_i() (interactive subjective input, returns $w^{(s)}$)
  • somit_ii(ACM) (objective weights from ACM, returns $w^{(o)}$)
  • somit_iii(w_s, w_o) (hybrid combination)

Sample workflow:

from pysomit import somit_i, somit_ii, somit_iii import numpy as np w_s = somit_i() # subjective ACM = np.array([...]) # m × n matrix w_o = somit_ii(ACM) # objective w = somit_iii(w_s, w_o) # combined print("Final weights:", np.round(w,4))

These weights plug directly into pyDecision’s TOPSIS implementation or other weighted MCDM packages:

from pyDecision.algorithm import topsis_method scores = topsis_method(ACM, weights=w, types=types)

7. Practical Considerations and Generalizations

SOMIT is recommended for:

• Scenarios with moderate to large $n$ ($n \geq 6$–10) to maintain manageable expert burden.
• Datasets with potential outliers or skewness.
• MCDM problems where neither fully subjective nor fully objective weighting is sufficient.
• Applications beyond renewable energy, including site selection, technology portfolios, energy storage planning, and multi-criteria policy analysis.

Its modular structure supports selective application of subjective, objective, or hybrid weighting components, and enables straightforward coupling with popular MCDM aggregation methods such as TOPSIS, MARCOS, and PROBID (Ding et al., 6 Jan 2026).