EDAS: Distance from Average Solution Method

• EDAS is a multi-criteria decision-making method that evaluates alternatives by comparing their performance to the arithmetic mean of each criterion.
• It computes positive and negative deviations using normalized differences, providing an intuitive alternative to ideal-based methods like TOPSIS and VIKOR.
• EDAS offers computational simplicity and robustness against outliers, though its rankings remain sensitive to the composition of the alternative set.

Searching arXiv for the cited EDAS-related papers to ground the article in current arXiv sources. arXiv Search Query: all:"Distance from Average Solution" OR all:EDAS Distance from Average Solution (EDAS) is a reference-type multi-criteria decision-making (MCDM) method introduced by Keshavarz Ghorabaee et al. (2015) that evaluates alternatives by calculating their positive and negative distances from the average solution. In EDAS, the reference point is the arithmetic mean of each criterion across the alternatives rather than an ideal extreme such as the best or worst observed value. Its basic ranking logic is criterion-wise: an alternative is assessed according to whether it performs better than the average or worse than the average, with better alternatives having higher Positive Distance from Average (PDA) and lower Negative Distance from Average (NDA). The final ranking is obtained by combining normalized forms of these two quantities into a single performance score (Wang et al., 22 Aug 2025). An arXiv application, KG-EDAS, retains this structure while repurposing EDAS as a meta-metric for knowledge graph completion (KGC) model evaluation across multiple datasets and metrics, yielding a single normalized score $M_i \in [0,1]$ (Gul et al., 21 Aug 2025).

1. Conceptual basis and reference-solution logic

EDAS is distinctive among reference-type methods because it uses the average solution rather than positive ideal solution (PIS) and negative ideal solution (NIS) type references. The intuition stated for this choice is direct: if an alternative is above average on a benefit criterion, that is favorable; if it is below average on a cost criterion, that is favorable; conversely, being below average on a benefit criterion or above average on a cost criterion is unfavorable. This makes the method easy to interpret because it asks whether each alternative is performing better or worse than the “typical” level represented by the arithmetic mean (Wang et al., 22 Aug 2025).

The chapter explicitly contrasts this logic with TOPSIS and VIKOR. TOPSIS relies on the positive ideal solution and negative ideal solution, preferring alternatives that are closer to the PIS and farther from the NIS. VIKOR also uses ideal-type references, focusing on closeness to the PIS while balancing group utility and individual regret. By contrast, EDAS does not compare alternatives to extreme best or worst values; it evaluates them relative to the average values across criteria. Because EDAS uses the average solution, it is less affected by extreme values than ideal-solution methods such as TOPSIS or VIKOR. However, it remains relative to the set of alternatives, so adding or removing alternatives can change the average and thus potentially alter rankings (Wang et al., 22 Aug 2025).

A frequent misconception is to treat “less affected by extreme values” as implying invariance to the choice set. The chapter does not support that interpretation. The method is explicitly relative rather than absolute: the benchmark is the set-average solution, not an external target. This suggests that EDAS trades some sensitivity to outliers for dependence on the composition of the comparison set.

2. Mathematical formulation and algorithmic structure

Using the notation stated in the chapter, $i \in \{1,2,\dots,m\}$ indexes alternatives, $j \in \{1,2,\dots,n\}$ indexes criteria, $f_{ij}$ is the value of the $j$-th criterion for the $i$-th alternative in the original alternatives-criteria matrix (ACM), and $w_j$ is the weight of criterion $j$, with

$\sum_{j=1}^{n} w_j = 1.$

The average solution is denoted by $\bar{A}$, and $i \in \{1,2,\dots,m\}$0 is the average value of criterion $i \in \{1,2,\dots,m\}$1 across all alternatives (Wang et al., 22 Aug 2025).

The first step is to determine the average solution: $i \in \{1,2,\dots,m\}$2 with

$i \in \{1,2,\dots,m\}$3

The second step is to compute PDA and NDA. For a maximization criterion,

$i \in \{1,2,\dots,m\}$4

$i \in \{1,2,\dots,m\}$5

and for a minimization criterion,

$i \in \{1,2,\dots,m\}$6

$i \in \{1,2,\dots,m\}$7

The third step is weighted aggregation: $i \in \{1,2,\dots,m\}$8

$i \in \{1,2,\dots,m\}$9

Here, $j \in \{1,2,\dots,n\}$0 measures the overall favorable deviation of alternative $j \in \{1,2,\dots,n\}$1 from the average, while $j \in \{1,2,\dots,n\}$2 measures the overall unfavorable deviation.

The fourth step is normalization: $j \in \{1,2,\dots,n\}$3

$j \in \{1,2,\dots,n\}$4

The normalization preserves the “larger is better” direction of $j \in \{1,2,\dots,n\}$5, while the subtraction from 1 reverses the direction of $j \in \{1,2,\dots,n\}$6, so that larger $j \in \{1,2,\dots,n\}$7 is better.

The fifth step is the final performance score: $j \in \{1,2,\dots,n\}$8 The ranking rule is unambiguous: larger $j \in \{1,2,\dots,n\}$9 is better, and the alternative with the largest $f_{ij}$0 is top-ranked and recommended (Wang et al., 22 Aug 2025).

3. Criterion polarity, aggregation semantics, and implementation details

A central implementation feature of EDAS is its explicit treatment of mixed benefit and cost structures. For a benefit criterion, above-average performance contributes to PDA and below-average performance contributes to NDA. For a cost criterion, the sign logic is reversed: below-average cost is favorable and contributes to PDA, while above-average cost is unfavorable and contributes to NDA. The interpretation is therefore stable across criterion types: PDA always denotes favorable deviation from average, and NDA always denotes unfavorable deviation from average (Wang et al., 22 Aug 2025).

The chapter makes an additional structural point: for a given $f_{ij}$1, either PDA or NDA is zero. This follows directly from the $f_{ij}$2 construction. As a result, criterion-level interpretation is very transparent, because each criterion contributes either a favorable deviation, an unfavorable deviation, or neither when the alternative coincides with the average (Wang et al., 22 Aug 2025).

EDAS uses the original ACM directly; it does not require a preliminary normalization of the raw matrix before constructing the reference point. Instead, normalization occurs later at the aggregate-score stage via $f_{ij}$3 and $f_{ij}$4. Criterion weights $f_{ij}$5 must be assigned in advance, must sum to 1, and are applied only after computing PDA and NDA. The chapter identifies correct benefit/cost treatment as the most important implementation detail. From the algorithm as presented, EDAS assumes that a finite set of alternatives is available, criterion weights are known and sum to 1, average criterion values are meaningful as central references, and favorable and unfavorable deviations from average can be linearly aggregated (Wang et al., 22 Aug 2025).

4. Worked example and numerical behavior

The chapter’s common ACM example contains $f_{ij}$6 alternatives, $f_{ij}$7, and $f_{ij}$8 criteria, $f_{ij}$9. Criteria $j$0 and $j$1 are benefit criteria, while $j$2 is a cost criterion. The criterion weights are

$j$3

The reported average solution is

$j$4

Two calculations illustrate the mechanics. For $j$5 on $j$6, a benefit criterion,

$j$7

so $j$8 is favorably above average on $j$9. For $i$0 on $i$1, a cost criterion,

$i$2

so $i$3 receives an unfavorable deviation because its cost is above average (Wang et al., 22 Aug 2025).

The weighted sums reported in the chapter are $i$4, $i$5, $i$6, $i$7, $i$8, and $i$9, $w_j$0, $w_j$1, $w_j$2, $w_j$3. After normalization, the chapter reports $w_j$4, $w_j$5; $w_j$6, $w_j$7; $w_j$8, $w_j$9; $j$0, $j$1; and $j$2, $j$3 (Wang et al., 22 Aug 2025).

The final EDAS scores are as follows.

Alternative	$j$4
$j$5	0.5184
$j$6	0.5490
$j$7	0.0047
$j$8	0.5501
$j$9	0.5316

The resulting ranking is

$\sum_{j=1}^{n} w_j = 1.$0

Thus $\sum_{j=1}^{n} w_j = 1.$1 is the top-ranked alternative in the worked example. The example makes clear that EDAS rewards alternatives that simultaneously accumulate strong favorable deviations and avoid large unfavorable deviations, rather than merely excelling on isolated criteria (Wang et al., 22 Aug 2025).

5. Position within reference-type MCDM methods

The chapter situates EDAS among nine reference-type MCDM methods: TOPSIS, GRA, VIKOR, EDAS, MABAC, CODAS, PIV, MARCOS, and PROBID. Within that family, EDAS is characterized by a single average reference, in contrast to methods that rely on ideal extremes or multiple reference tiers (Wang et al., 22 Aug 2025).

Relative to TOPSIS, the difference is fundamental. EDAS uses the average solution and split deviations into positive and negative distances from average, whereas TOPSIS uses PIS and NIS together with Euclidean distances to those extreme reference points. The chapter states that EDAS is less affected by extreme values, while TOPSIS is more sensitive to outliers because PIS and NIS are extreme values. Relative to VIKOR, EDAS again uses the average solution, whereas VIKOR uses ideal best values and compromise relative to them, balancing overall utility and worst-case regret. The chapter further states that EDAS is simpler, while VIKOR is more involved because of $\sum_{j=1}^{n} w_j = 1.$2, $\sum_{j=1}^{n} w_j = 1.$3, $\sum_{j=1}^{n} w_j = 1.$4, and compromise conditions (Wang et al., 22 Aug 2025).

The chapter also provides shorter contrasts with MABAC and PROBID. EDAS uses an arithmetic mean reference, while MABAC uses a geometric mean-based border approximation area. In that sense, EDAS is generally more direct and easier to interpret. PROBID uses multiple ideal tiers plus an average solution, whereas EDAS uses one average reference. Accordingly, PROBID is described as richer but more computationally intensive, while EDAS is simpler (Wang et al., 22 Aug 2025).

These comparisons clarify the design space of reference-type methods. A plausible implication is that EDAS is especially suitable when interpretability in terms of above-average and below-average performance is preferred over explicit distance-to-ideal or compromise formulations.

6. Advantages, limitations, assumptions, and recurrent issues

The chapter lists three advantages of EDAS. First, it uses a simple and intuitive reference solution: the average of each criterion, making “better-than-average” and “worse-than-average” straightforward to interpret. Second, because it relies on the average solution rather than ideal extremes, it is less sensitive to outliers than methods such as TOPSIS or VIKOR. Third, it has computational simplicity, since the calculations are based mainly on deviations from average values and weighted summation (Wang et al., 22 Aug 2025).

The same source identifies three limitations. First, possible rank reversal remains: adding or removing alternatives changes the average solution, so rankings may still change. Second, EDAS provides relative rather than absolute benchmarking, because it compares alternatives to the average rather than to an absolute best or worst benchmark. Third, the final score

$\sum_{j=1}^{n} w_j = 1.$5

gives equal weight by default to the positive and negative components, and the chapter notes that this may not always reflect the true preferences of decision-makers (Wang et al., 22 Aug 2025).

These points delimit the method’s scope. EDAS is not an absolute-target method, and it does not eliminate dependence on the candidate set. It also embeds a specific aggregation convention at the last stage. This helps resolve another common misunderstanding: the interpretability of the average reference does not remove the need for careful weight selection or sensitivity analysis.

7. Adaptation as a benchmarking meta-metric: KG-EDAS

The paper “KG-EDAS: A Meta-Metric Framework for Evaluating Knowledge Graph Completion Models” adapts EDAS to produce a single meta-metric for KGC model evaluation across multiple metrics and datasets. In this setting, the alternatives are KGC models, denoted

$\sum_{j=1}^{n} w_j = 1.$6

and the criteria are standard KGC evaluation metrics, explicitly MR, MRR, Hits@1, and Hits@10. Across the experiments, the methodology effectively treats dataset-metric pairs as separate criteria, so if there are $\sum_{j=1}^{n} w_j = 1.$7 datasets and $\sum_{j=1}^{n} w_j = 1.$8 metrics, the total number of criteria is effectively $\sum_{j=1}^{n} w_j = 1.$9. The decision matrix is

$\bar{A}$0

with $\bar{A}$1 the score of model $\bar{A}$2 on criterion $\bar{A}$3 (Gul et al., 21 Aug 2025).

The average solution is computed as

$\bar{A}$4

and the paper retains standard EDAS-style PDA and NDA formulas, with MRR and Hits@k treated as benefit criteria and MR treated as a cost or non-beneficial criterion. Equal weights are used in the experiments, so effectively $\bar{A}$5 for all $\bar{A}$6. The weighted sums are

$\bar{A}$7

followed by normalization and the final score

$\bar{A}$8

Models are ranked by sorting $\bar{A}$9 in descending order (Gul et al., 21 Aug 2025).

The paper presents EDAS in this setting as a robust and interpretable meta-metric that synthesizes model performance across multiple datasets and diverse evaluation criteria into a single normalized score. For relation prediction, example rankings include RotatE with $i \in \{1,2,\dots,m\}$00 and rank 1, TuckER with $i \in \{1,2,\dots,m\}$01 and rank 2, and TorusE with $i \in \{1,2,\dots,m\}$02 and rank 10. For tail prediction, TransR is reported with $i \in \{1,2,\dots,m\}$03 and rank 1, and AMIE with $i \in \{1,2,\dots,m\}$04 and rank 7. The paper also reports strong correlations between EDAS and conventional metrics, including EDAS vs Mean MRR with Pearson $i \in \{1,2,\dots,m\}$05, $i \in \{1,2,\dots,m\}$06, and Kendall $i \in \{1,2,\dots,m\}$07, $i \in \{1,2,\dots,m\}$08, as well as EDAS vs Mean Hit@1 with Pearson $i \in \{1,2,\dots,m\}$09, $i \in \{1,2,\dots,m\}$10, and Kendall $i \in \{1,2,\dots,m\}$11, $i \in \{1,2,\dots,m\}$12. At the same time, the paper records several caveats: no explicit epsilon formula is given for zero-average cases, equal weighting may not suit all downstream tasks, benchmark dependence remains because the average solution depends on the participating models, and direct empirical comparison against multiple alternative aggregation schemes is not provided (Gul et al., 21 Aug 2025).