HEPrpms Strategy in Multi-Criteria Decision Analysis

• HEPrpms Strategy is a hybrid method that combines empirical ratings with expert AHP evaluations to integrate objective data and subjective judgments.
• The approach employs both additive and geometric formulations, solving linear and logarithmic systems to efficiently estimate unknown priorities.
• Its hierarchical embedding allows for criterion-specific reference sets, reducing the need for extensive pairwise comparisons and enhancing decision credibility.

The HEPrpms strategy refers to the structured integration of Heuristic Rating Estimation (HRE) within the multi-criteria Analytic Hierarchy Process (AHP) framework. This approach addresses scenarios in multi-criteria decision analysis where the priority (ranking) of some alternatives is externally known (measured or established through other means), and leverages these known values directly, reducing the reliance on subjective pairwise comparisons and enhancing both the accuracy and efficiency of the decision process (Kędzior et al., 2022).

1. Conceptual Foundation and Motivation

The HEPrpms strategy emerges from the limitation of classical AHP, where all alternatives are assigned priorities solely through subjective pairwise comparisons—even when empirically measured values for some alternatives are available. HRE is designed to accommodate partial knowledge by anchoring the solution to known reference values while estimating unknown priorities based on expert comparisons. In the context of HEPrpms, this duality allows the integration of objective quantitative data and subjective expert input within a single, coherent multi-criteria ranking.

The methodology supports two primary approaches: the additive HRE (structurally similar to the Eigenvector Method) and the geometric HRE (parallel to the Geometric Mean Method).

2. Mathematical Framework

Additive HRE Formulation

Given a set of alternatives $A$ split into known references $A_k$ and unknowns $A_u$, with $n = |A_k| + |A_u| = m + k$, HRE postulates that the weight $w(a_i)$ of an unknown $a_i \in A_u$ is the average of the weighted ratings of all other alternatives:

$$w(a_i) = \frac{1}{n-1} \sum_{j \neq i} c_{ij} \cdot w(a_j)$$

This yields a linear system:

$M \cdot w = b$

where $M$ is a $k \times k$ matrix with $M_{ii} = 1$ and $M_{ij} = -\frac{1}{n-1} c_{ij}$ for $i \neq j$, and $b_i = \frac{1}{n-1} \sum_{j \in A_k} c_{ij} w(a_j)$. After solving for the unknown weights, the complete priority vector is formed by appending the reference values.

Geometric HRE Formulation

Alternatively, HRE may use the geometric mean, positing:

$$w(a_i) = \Bigg[ \prod_{j \neq i} c_{ij} w(a_j) \Bigg]^{\frac{1}{n-1}}$$

Taking logarithms transforms the nonlinear system into:

$N \cdot \mu = d$

where $\mu_i = \log w(a_i)$, $N_{ii} = n - 1$, $N_{ij} = -1$ for $i \neq j$, and $d$ is constructed from known log-weights and comparisons. Recovering $w(a_i)$ is done via exponentiation.

3. Integration within AHP Hierarchy

HEPrpms generalizes HRE to multi-criteria environments by embedding it within each subproblem of the hierarchical AHP structure. For each criterion $q_t$, alternatives are split into $A_k^{(q_t)}$ and $A_u^{(q_t)}$, and the appropriate HRE system is solved (either linear or geometric). Each criterion yields a local priority vector $w^{(q_t)}$. The global aggregation follows the weighted sum:

$w = \sum_{t=1}^{s} \hat{w}(q_t) \cdot w^{(q_t)}$

where $\hat{w}(q_t)$ are criterion weights derived from standard AHP pairwise comparisons of the criteria.

Distinct criteria can employ different partitions of known and unknown alternatives, allowing high flexibility in engineering and decision analysis contexts.

4. Strategic and Computational Advantages

Key strengths of the HEPrpms framework include:

• Retention of Objective Values: Where alternatives have empirically measured or externally validated priority values, these are directly preserved in the final ranking, ensuring that the resulting prioritization aligns with ground truth metrics.
• Reduction of Judgments: Known alternatives do not require further comparisons, reducing the elicitation workload and the exposure to subjective bias in large-scale decision scenarios.
• Improved Accuracy and Credibility: Blending measured data with expert comparisons increases the trustworthiness of rankings, which is especially relevant in rigorous technical or regulatory settings.
• Criterion-Specific Reference Sets: Different criteria within the hierarchy may employ different sets of reference alternatives, maximizing data utilization.

5. Illustrative Examples

The HEPrpms approach is demonstrated in the multi-criteria selection of sports facility investments and ranking collectible porcelain cups.

• In sports facility planning, criteria such as profitability, durability, and increase in popularity are considered. Profitability uses known monthly revenues as anchor values, whereas subjective assessment is used for new facilities. After solving the HRE system and aggregating across criteria, the facility with the highest strategic fit is objectively and efficiently identified.
• In collectibles ranking, reference values such as manufacturer reputation or product quality (if known) are used directly, while secondary or subjective criteria (like uniqueness) are evaluated via standard AHP.

These examples substantiate the method’s flexibility and its ability to produce rankings that are both interpretable and robust.

6. Broader Implications and Practical Use

HEPrpms represents a systematic response to the challenge of integrating heterogeneous information into multi-criteria decision-making frameworks. The underlying mathematical rigor guarantees consistency, while the hierarchical embedding provides the scalability necessary for high-dimensional evaluation problems. A direct implication is its potential application in sectors where empirical performance data routinely coexists with subjective expert judgment, including infrastructure assessment, technology selection, and resource allocation.

By preserving measured values, HEPrpms also enhances auditability, a crucial requirement in regulated industries and public-sector decision-making.

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In summary, the HEPrpms strategy encapsulates the analytical integration of Heuristic Rating Estimation with Analytic Hierarchy Process, yielding a hybrid methodology that fortifies classical multi-criteria analysis with measured benchmarks, reduces elicitation burden, and increases the interpretability and validation of decision outcomes (Kędzior et al., 2022).