PSI - Preference Selection Index

Overview

Higher I means better. PSI derives criterion weights from Preference Variation Values rather than consuming user-specified weights. Because φ_j=1−V_j, larger squared variation V_j produces a smaller weight basis φ_j.

Output

utility, higher is better

Data

Crisp, complete numeric matrix

Weights

Derived internally, no weight source needed

Size

2+ alternatives, 3-12 criteria works best

Used for

Alternative selection, Supplier evaluation

How it works

1. 1

   Linear normalisation per criterion direction.

   $$r_{ij}=\{\begin{array}{ll}{x}_{ij}/{\max }_i{x}_{ij}& j\in J^{+}\\ {\min }_i{x}_{ij}/x_{ij}& j\in J^{-}\end{array}$$

   Maniya-Bhatt 2010, p.1786 Eqs.(2)-(3)

2. 2

   Column means r̄_j and squared deviations Σ (r_ij − r̄_j)².

   $${\bar{r}}_j=\frac{1}{m}\sum _i{r}_{ij},V_j=\sum _{i=1}^m(r_{ij}-{\bar{r}}_j{)}^2$$

   Maniya-Bhatt 2010, p.1786 Eqs.(4)-(5)

3. 3

   Preference variation φ_j = 1 − V_j and self-derived weights w_j = φ_j / Σ φ_k.

   $${\varphi }_j=1-V_j,w_j=\frac{{\varphi }_j}{{\sum }_{k=1}^n{\varphi }_k}$$

   Maniya-Bhatt 2010, p.1786 Eqs.(6)-(7)

4. 4

   Preference Selection Index I_i = Σ w_j r_ij and descending ranking.

   $$I_i=\sum _{j=1}^n{w}_j{r}_{ij}$$

   Maniya-Bhatt 2010, p.1786 Eq.(8)

Look elsewhere when

Assumptions to verify

• Criteria preferences are independent (no synergistic interactions)
• Compensation is acceptable: high score on one criterion can offset low on another
• Decision matrix is complete (no missing values)

Edge cases and pitfalls

• Zero is valid only on benefit criteria whose column maximum is positive; cost criteria must be strictly positive. Constant positive columns remain mathematically defined. If φ_j=1−V_j becomes negative/non-finite or the φ sum is non-positive, the kernel fails closed instead of emitting invalid weights.

Works with

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