Mean-based incomplete pairwise comparisons method with the reference values

Imagine you are the head chef at a busy restaurant, and you need to decide the prices for three new dishes you've created: a Mushroom Risotto, a Spicy Tortilla, and an Avocado Salad.

You don't have a price tag for these new dishes yet. However, you do have two "reference dishes" that have been on the menu for years: a Classic Soup (which you know sells for $6) and a Cheese Board (which sells for $4).

Your goal is to figure out the fair price for the three new dishes based on how they compare to each other and how they compare to your two known dishes.

The Problem: The "Missing Link"

In the old days, to price everything perfectly, you would have to ask your customers to compare every single dish against every other dish.

Soup vs. Tortilla
Tortilla vs. Salad
Risotto vs. Cheese Board
...and so on.

If you have 6 dishes, that's 15 comparisons. If you have 20 dishes, that's 190 comparisons! It's exhausting, time-consuming, and customers get tired of answering so many questions. Often, you end up with a "messy" list where some comparisons are missing (maybe the customer forgot to compare the Risotto to the Salad).

This is what the paper calls an Incomplete Pairwise Comparison Matrix. It's a puzzle with missing pieces.

The Old Way vs. The New Way

Previously, methods like AHP (Analytic Hierarchy Process) tried to fill in those missing pieces by guessing or by forcing the math to work, but they often struggled if the data was too sparse or inconsistent.

This paper introduces two new, smarter ways to solve this puzzle using Reference Values (your known $6 and $4 dishes). The authors call these methods HRE (Heuristic Rating Estimation). Think of HRE as a "smart calculator" that uses your known anchors to estimate the unknowns.

The paper proposes two flavors of this calculator:

1. The Arithmetic Method (The "Average" Approach)

Imagine you ask a customer: "How does the Tortilla compare to the Soup?" They say, "The Tortilla is half as good as the Soup."
Then they say, "The Tortilla is twice as good as the Cheese Board."

The Arithmetic Method takes these two opinions and simply averages them.

If the Soup is $6, half of that is $3.
If the Cheese Board is $4, twice that is $8.
The average of $3 and $8 is $5.50.

This method is intuitive. It treats every piece of feedback as equally important. If you have a mix of "optimistic" and "pessimistic" opinions, this method finds the middle ground.

2. The Geometric Method (The "Balanced" Approach)

The Geometric Method is a bit more sophisticated. Instead of adding the numbers and dividing by two, it multiplies them and takes a "root" (a mathematical way of finding a central balance).

Using the same example:

$3 \times 8 = 24$.
The square root of 24 is roughly $4.90.

Why use this? In the world of decision-making, extreme opinions (like saying something is "100 times better") can skew the average wildly. The Geometric method is more "resistant" to these outliers. It's like a wise judge who listens to all arguments but doesn't let one crazy exaggeration throw off the final verdict.

The Big Breakthroughs in the Paper

The authors didn't just invent these calculators; they proved they actually work mathematically.

The Geometric Method is "Optimal": They proved that the Geometric method is the best possible way to minimize errors. It's like finding the shortest path through a maze. No matter how messy your missing data is, this method finds the most logical, balanced set of prices.
It Always Works: They proved that as long as your "Reference Dishes" (the known prices) are connected to the new dishes in some way, the Geometric method will always give you an answer. It never gets stuck.
The Arithmetic Method is "Conditional": The Arithmetic method is great, but it only works if you have enough comparisons and your data isn't too crazy. The paper gives you a checklist to see if your specific situation is safe to use the Arithmetic method.

Why This Matters in Real Life

Think about a company trying to rank employees for a promotion, or a city council deciding where to build a new park.

Old Way: They need to compare every employee to every other employee (or every park location to every other). This takes forever.
New Way (This Paper): They pick a few "Star Employees" or "Ideal Park Locations" with known values. Then, they only ask for comparisons involving these stars and the new candidates.

The Benefits:

Less Work: You don't need to ask 100 questions; maybe 20 is enough.
Flexibility: You can add a new dish (or employee) later without recalculating the prices of the old ones. The "Reference Values" stay fixed, acting as a stable anchor.
No "Rank Reversal": In the old methods, adding a new, terrible option could accidentally make your best option look worse. This new method prevents that weird glitch.

The Final Verdict

The paper is essentially saying: "Stop trying to compare everything to everything. Use your known 'anchors' to estimate the unknowns."

If you want a result that feels intuitive and averages out opinions, use the Arithmetic method. If you want a mathematically perfect, error-minimizing result that is guaranteed to work, use the Geometric method.

It's like upgrading from a hand-drawn map with missing roads to a GPS that knows exactly where you are relative to a few known landmarks, and can calculate the rest of the route for you.

Here is a detailed technical summary of the paper "Mean-based incomplete pairwise comparisons method with the reference values" by Kułakowski et al.

1. Problem Statement

The paper addresses the challenge of deriving priority weights (a weight vector) from incomplete Pairwise Comparison (PC) matrices within the context of Multi-Criteria Decision Making (MCDM).

Context: Traditional methods like the Analytic Hierarchy Process (AHP) typically require a complete set of pairwise comparisons ( $n(n-1)/2$ for $n$ alternatives). However, in practice, obtaining all comparisons is often impossible or too burdensome for decision-makers (DMs).
Specific Gap: Existing methods for incomplete matrices (extensions of Eigenvalue Method or Geometric Mean Method) often lack the integration of reference values (alternatives with known, fixed priorities).
Goal: The authors propose extending the Heuristic Rating Estimation (HRE) method to handle incomplete data. HRE assumes that the priorities of a subset of alternatives (reference set $A_K$ ) are known, and the goal is to calculate the weights of the remaining unknown alternatives ( $A_U$ ) based on these references and the available comparisons.

2. Methodology

The authors propose two quantitative extensions of the HRE method: Arithmetic Incomplete HRE (aiHRE) and Geometric Incomplete HRE (giHRE).

Core Assumptions

The set of alternatives $A$ is divided into $A_U$ (unknown weights) and $A_K$ (known reference weights).
Missing comparisons (denoted as ?) are treated as consistent with the final solution. Specifically, if $c_{ij}$ is missing, it is assumed that $c_{ij} = w(a_i)/w(a_j)$ .

A. Arithmetic Incomplete HRE (aiHRE)

Principle: The weight of an alternative is the arithmetic mean of the weighted priorities of all other alternatives it is compared with.
Formulation:
- The method constructs a system of linear equations $Cw = b$ .
- For an alternative $a_i$ , the equation is derived by summing known comparisons and substituting missing ones with the ratio $w(a_i)/w(a_j)$ .
- This results in a linear system where the unknown weights $w(a_i)$ are isolated.
Solution: The system is solved algebraically. If the matrix is singular or conditions are not met, the authors suggest formulating it as an optimization problem (minimizing least squares) to find an approximate solution.

B. Geometric Incomplete HRE (giHRE)

Principle: The weight of an alternative is the geometric mean of the weighted priorities of other alternatives.
Formulation:
- The method starts with a nonlinear system: $w(a_i) = (\prod c_{ij}w(a_j))^{1/(n-1)}$ .
- By applying a logarithmic transformation ( $\bar{w} = \ln w$ ), the nonlinear system is converted into a linear system: $\bar{C}\bar{w} = \bar{b}$ .
- The matrix $\bar{C}$ is constructed based on the number of missing comparisons and the known reference values.
Solution: The linear system is solved to find $\bar{w}$ , which is then exponentiated to retrieve the final weights.

3. Key Contributions

1. Theoretical Proofs of Existence and Optimality

Geometric Approach (giHRE):
- Existence: The authors prove that a unique, real, and positive solution always exists for giHRE provided the comparison graph is irreducible (all alternatives are connected directly or indirectly) and at least one non-reference alternative is compared to a reference alternative.
- Optimality: They prove that the giHRE solution is optimal. It minimizes the sum of squared logarithmic errors (Least Squares) between the estimated weights and the provided pairwise comparisons, inheriting the optimality of the Geometric Mean Method (GMM).
Arithmetic Approach (aiHRE):
- Existence: The authors provide sufficient conditions for the existence of a unique, positive solution. These conditions involve the matrix being irreducibly diagonally dominant. They note that while these conditions are sufficient, solutions may still exist even if they are not strictly met (as shown in numerical examples).

2. Flexibility in Data Collection

The proposed methods allow decision-makers to select a flexible number of reference alternatives and perform a flexible number of comparisons (from a minimum of $n-1$ to a full set).
This reduces the cognitive load on the DM compared to requiring a full matrix.

3. Integration with AHP

The methods are designed to integrate seamlessly with the AHP hierarchical model. Reference values can be assigned at any node in the hierarchy, allowing for "local" reference points rather than global ones.
This integration helps prevent the rank reversal phenomenon, as adding new alternatives does not alter the weights of existing reference alternatives.

4. Results and Numerical Examples

The paper validates the methods through several numerical examples:

Refrigerator Selection: A complete matrix example comparing standard EVM/GMM with HRE to show consistency.
Advertising Campaign: A complete HRE example demonstrating how to rank new designs against known historical performance data.
Party Appetizers (Incomplete Data):
- The authors apply both aiHRE and giHRE to a scenario where guests cannot compare all appetizers directly (missing ? values).
- Result: Both methods produced consistent rankings (Mushrooms > Tortilla > Soup > Gyros > Avocado), though the absolute weight values differed slightly.
- Discussion: The arithmetic method yielded higher weights for items with mixed high/low comparisons (averaging effect), while the geometric method yielded lower weights (geometric mean dampens extreme values).
Disjoint Graphs: The paper demonstrates that even if the unknown alternatives form disjoint groups (no direct comparisons between groups), as long as each group connects to the reference set, a solution exists (particularly for giHRE).

5. Significance and Implications

Practical Utility: The methods significantly reduce the effort required for decision-making by allowing incomplete data and leveraging known benchmarks (reference values).
Theoretical Robustness: The proof of optimality for giHRE establishes it as a mathematically superior choice for minimizing error in inconsistent data, while aiHRE offers an intuitive arithmetic interpretation.
Decision Support: The paper provides a framework for DMs to choose between methods based on their specific needs:
- Use giHRE if mathematical optimality, guaranteed solution existence, and ranking consistency are the primary goals.
- Use aiHRE if the decision-maker prefers the intuitive interpretation of the arithmetic mean and the data is consistent enough to satisfy the sufficient conditions for a solution.
Rank Reversal Protection: By anchoring the model with fixed reference values, the methods offer a safeguard against rank reversal when new alternatives are introduced, a common criticism of standard AHP.

In conclusion, this paper successfully generalizes the Heuristic Rating Estimation method to incomplete datasets, providing rigorous mathematical proofs for the geometric variant and practical sufficient conditions for the arithmetic variant, thereby enhancing the applicability of pairwise comparison methods in complex, real-world decision-making scenarios.