Balanced SPOTIS - Balanced Stable Preference Ordering Towards Ideal Solution

Overview

Lower P_i = better. α=0 → identical to standard SPOTIS (ISP only). α=1 → only ESP matters. α=0.5 balanced.

Data

Crisp

Weights

Needs a weight source

How it works

1. 1

   Determine ISP (Ideal Solution Point) from bounds based on criterion direction; load user-provided ESP.

   $$S_j^{\ast }=\{\begin{array}{ll}\max {\text{-bound}}_j& j\in J^{+}\\ \min {\text{-bound}}_j& j\in J^{-}\end{array}$$

   Shekhovtsov et al. 2025 §2

2. 2

   Normalised distance from ESP and ISP: d_ij = |x_ij − ref_j| / |bounds_min_j − bounds_max_j|.

   $$d_{ij}(S^{+})=\frac{|x_{ij}-S_j^{+}|}{|b_j^{\min }-b_j^{\max }|};d_{ij}(S^{\ast })=\frac{|x_{ij}-S_j^{\ast }|}{|b_j^{\min }-b_j^{\max }|}$$

   Shekhovtsov et al. 2025 §2

3. 3

   Weighted aggregated distances D_i(ESP) and D_i(ISP).

   $$D_i(S^{+})=\sum _j{w}_j{d}_{ij}(S^{+});D_i(S^{\ast })=\sum _j{w}_j{d}_{ij}(S^{\ast })$$

   Shekhovtsov et al. 2025 §2

4. 4

   Balanced score P_i = (1−α)·D_i(ISP) + α·D_i(ESP). Rank ascending (lower P = better).

   $$P_i=(1-\alpha )D_i(S^{\ast })+\alpha D_i(S^{+}),\text{rank ascending}$$

   Shekhovtsov et al. 2025 §2 Eq.(main)

Fits when

Edge cases and pitfalls

• ESP must be within bounds for each criterion; out-of-bounds ESP gives distances > 1. Paper §3.1 line 147-150: S+_j must satisfy Smin_j ≤ S+_j ≤ Smax_j for every j.

• α=0.5 indifference Theorem (Shekhovtsov et al. 2025 §4.1 Eqs.(7)-(8)): when all alternatives satisfy S*≤Si≤S+ (i.e., lie between ISP and ESP), P_i at α=0.5 reduces to (1/2)·Σ_j w_j·(S+_j − S*_j)/δ_j - INDEPENDENT of Si_j. All such alternatives are evaluated equally (total indifference). Recommendation: investigate multiple α values (e.g., 0.1, 0.5, 0.9 per §4.2) or ensure no alternative lies strictly between ESP and ISP.

• Special cases (paper §4.1 line 365-368): α=0.0 reproduces standard ISP-SPOTIS (Dezert 2020); α=1.0 reproduces ESP-SPOTIS. Use α ∈ (0,1) only for genuine ISP/ESP blending.

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