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Scoring Model

Overview

Decision OS uses a Weighted Sum Model (WSM) for multi-criteria decision analysis. This document specifies the exact formulas, rules, and examples.

Definitions

Symbol Definition
$O$ Set of options ${o_1, o_2, \ldots, o_m}$
$C$ Set of criteria ${c_1, c_2, \ldots, c_n}$
$w_i$ Raw weight for criterion $c_i$ (0–100)
$\hat{w}_i$ Normalized weight for criterion $c_i$
$s_{j,i}$ Raw score for option $o_j$ on criterion $c_i$ (0–10)
$e_{j,i}$ Effective score (adjusted for criterion type)
$T_j$ Total weighted score for option $o_j$

Step 1: Normalize Weights

Raw weights can be any value 0–100. They are normalized to sum to 1.0:

$$\hat{w}_i = \frac{\max(0, w_i)}{\sum_{k=1}^{n} \max(0, w_k)}$$

Edge case: If all raw weights are 0, each normalized weight = $\frac{1}{n}$ (equal weighting).

Edge case: Negative weights are treated as 0.

Step 2: Compute Effective Scores

Each criterion has a type: benefit (higher is better) or cost (lower is better).

$$e_{j,i} = \begin{cases} s_{j,i} & \text{if } c_i \text{ is benefit} \ 10 - s_{j,i} & \text{if } c_i \text{ is cost} \end{cases}$$

Scores are clamped to [0, 10] before the effective score is computed.

Step 3: Compute Total Score

$$T_j = \sum_{i=1}^{n} \hat{w}_i \times e_{j,i}$$

The maximum possible total score is 10.00 (if all effective scores are 10).

Step 4: Rank Options

Options are sorted by $T_j$ descending. Ties are broken by insertion order (first added wins).

Display Rounding

All displayed scores are rounded to 2 decimal places using JavaScript's Number.toFixed(2).

Worked Example

Setup

Decision: Which laptop to buy?

Option Price (cost, w=40) Performance (benefit, w=35) Portability (benefit, w=25)
MacBook Pro 8 9 7
ThinkPad X1 5 7 8

Step 1: Normalize Weights

Total raw weight = 40 + 35 + 25 = 100

Criterion Raw Weight Normalized
Price 40 0.40
Performance 35 0.35
Portability 25 0.25

Step 2: Effective Scores

Option Price (cost: 10-s) Performance Portability
MacBook Pro 10 - 8 = 2 9 7
ThinkPad X1 10 - 5 = 5 7 8

Step 3: Total Scores

MacBook Pro: (0.40 × 2) + (0.35 × 9) + (0.25 × 7) = 0.80 + 3.15 + 1.75 = 5.70

ThinkPad X1: (0.40 × 5) + (0.35 × 7) + (0.25 × 8) = 2.00 + 2.45 + 2.00 = 6.45

Step 4: Rankings

Rank Option Score
1 ThinkPad X1 6.45
2 MacBook Pro 5.70

Winner: ThinkPad X1 — it scores higher primarily because of better value on Price (the highest-weighted criterion).

Sensitivity Analysis

The weight-swing analysis tests robustness by adjusting each criterion's weight by ±N% and recomputing:

  1. For each criterion $c_i$:

    • Compute $w_i^{+} = w_i \times (1 + \text{swing}/100)$
    • Compute $w_i^{-} = \max(0, w_i \times (1 - \text{swing}/100))$
    • Recompute results with modified weight
    • Check if winner changes

  2. Report: "X out of Y swings change the winner"

A robust result means no reasonable weight adjustment flips the outcome.

Determinism Guarantee

The scoring engine is deterministic: identical inputs always produce identical outputs. There is no randomness, no floating-point-dependent branching, and all edge cases produce defined results. This is verified by unit tests.