Alternatives-Criteria Matrix (ACM)

• Alternatives-Criteria Matrix (ACM) is a structured data matrix that organizes and evaluates alternatives using numerical, ordinal, interval, or linguistic data.
• It integrates various normalization methods and weighting schemes to serve as the foundation for both reference-type and aggregation-type multi-criteria decision methods.
• Careful preprocessing, including normalization and weight selection, is crucial as these methodological choices significantly impact alternative rankings and overall decision outcomes.

An Alternatives–Criteria Matrix (ACM) is the canonical data structure for multi-criteria decision-making (MCDM), serving as the primary formalism for encoding, processing, and aggregating the performance of a finite set of alternatives under a finite set of evaluation criteria. The ACM underpins both classical and contemporary MCDM frameworks, enabling analysts to structure problems ranging from strategic planning and policy design to engineering selection and project prioritization. Its flexibility accommodates numerical, ordinal, interval, and linguistic data, as well as quantitative or qualitative importance preferences, thus allowing robust comparative analysis across disparate methodologies.

1. Formal Definition and Mathematical Structure

Let $A = \{a_1, ..., a_m\}$ denote the set of $m$ alternatives and $C = \{c_1, ..., c_n\}$ the set of $n$ criteria (also called attributes or properties). The ACM is the $m \times n$ matrix

$M = [a_{ij}], \quad i=1,\ldots,m; \; j=1,\ldots,n$

where $a_{ij}$ represents the evaluation or performance of alternative $a_i$ under criterion $c_j$.

Each row vector $M_{i*}$ encodes the full criterion profile for an alternative $a_i$, while each column $M_{*j}$ collects the set of performances on criterion $c_j$ across all alternatives. The type of entries $a_{ij}$ can span real numbers, intervals, ordinal categories (e.g., Low/Medium/High), or linguistic terms (e.g., “Very High,” “None”), contingent on the domain and methodological context (Agrawal, 2015, Yager, 2013, Wang et al., 22 Aug 2025, Wang et al., 8 Sep 2025).

In most formalizations, each criterion $c_j$ is endowed with an intra-criterion preference relation $\succ_j$ (total or partial order), such that $a_{ij} \succ_j a_{kj}$ means $a_i$ is strictly preferred to $a_k$ on $c_j$.

2. Construction, Data Types, and Preprocessing

Construction

The ACM is populated via measurement, elicitation from experts, simulation, or aggregation of prior evaluations. For numeric data, all $a_{ij} \in \mathbb{R}$. For fuzzy or qualitative settings, entries can be intervals $[L_{ij}, U_{ij}]$, symbolic adjectives, or elements of an ordered linguistic scale as in

$$L = \{\text{None}, \text{Very Low}, \text{Low}, \text{Medium}, \text{High}, \text{Very High}, \text{Perfect}\}$$

with a predefined order $S_i \succ S_j \iff i > j$ (Yager, 2013).

Preprocessing and Normalization

To permit coherent cross-criterion aggregation, column normalization is almost universally applied, using methods appropriate to the measurement scale and decision method:

Normalization	Formula	Used In
Max-normalization	$F_{ij} = x_{ij}/\max_k x_{kj}$ (benefit)	SAW, MEW, WASPAS (Wang et al., 8 Sep 2025)
Min–max normalization	$$F_{ij} = (x_{ij} - \min_k x_{kj})/\bigl(\max_k x_{kj} - \min_k x_{kj}\bigr)$$ (benefit)	GRA, MABAC (Wang et al., 22 Aug 2025)
Vector normalization	$F_{ij}=x_{ij} / \sqrt{\sum_k x_{kj}^2}$	TOPSIS, MOORA (Wang et al., 22 Aug 2025, Wang et al., 8 Sep 2025)
Sum-normalization	$F_{ij}=x_{ij}/\sum_k x_{kj}$	COPRAS (Wang et al., 8 Sep 2025)

All normalization aims to map diverse units to a common $[0,1]$ interval and enforce monotonicity (higher is better).

A plausible implication is that careful normalization selection can materially impact the resulting alternative rankings, especially when different criteria vary substantially in range or measurement modality.

3. Role in Multi-Criteria Decision Methodologies

The ACM forms the computational substrate for both reference-type and aggregation-type MCDM methods.

Reference-Type Methods

Reference-type methods such as TOPSIS, GRA, VIKOR, EDAS, MABAC, CODAS, PIV, MARCOS, and PROBID define explicit reference points (e.g., positive ideal solution, negative ideal solution, average) constructed from the ACM. Alternatives are ranked by evaluating their distance or relational position relative to these references (Wang et al., 22 Aug 2025). Reference points are calculated as:

• Positive ideal: $x_j^+ = \max_i F_{ij}$
• Negative ideal: $x_j^- = \min_i F_{ij}$
• Average: $\bar{x}_j = (1/m)\sum_{i=1}^m F_{ij}$

Aggregation-Type Methods

Aggregation-type methods (SAW, MEW, AHP, ANP, COPRAS, MOORA, FUCA, WASPAS) transform the ACM directly into a scalar performance score per alternative through additive, multiplicative, or rank-based manipulations, weighted by explicit or derived importance measures (Wang et al., 8 Sep 2025). For example:

• SAW: $P_i = \sum_{j=1}^n w_j F_{ij}$
• MEW: $P_i = \prod_{j=1}^n (F_{ij})^{w_j}$
• FUCA: $R_i = \sum_{j=1}^n w_j r_{ij}$ (where $r_{ij}$ is the rank of $a_i$ in $c_j$)

Qualitative and Non-Numeric Methods

Some frameworks, such as those of Agrawal or Yager, leverage the ACM for decision making without requiring cardinal weights or numeric scoring. Agrawal’s qualitative dominance test utilizes the ACM, intra-criterion orders, and an interval-order partial preference over criteria to construct a strict partial order on alternatives (Agrawal, 2015). Yager’s fuzzy/linguistic aggregation operates directly on a matrix of categorical evaluations via ordinal logic and fuzzy-implication operators (Yager, 2013).

4. Integration of Criteria Importance

Importance (weighting) of criteria is incorporated in numerous ways:

• Explicit weights: In most reference- and aggregation-type methods, a normalized weight vector $W = (w_1, ..., w_n)$, often derived via AHP, pairwise judgments, or direct assignment, scales the columns of the normalized ACM.
• Partial or qualitative orders: Agrawal’s framework employs only a partial order $\triangleleft$ over criteria, i.e., $c_p \triangleleft c_q$ implies $c_p$ is strictly more important than $c_q$, but transitivity may be incomplete (interval order). No numeric weights are instantiated (Agrawal, 2015).
• Linguistic or fuzzy importance: Yager’s approach allows direct linguistic weighting, where criteria weights $w_j\in L$ are processed through order-preserving ordinal negation and fuzzy-implication, obviating numeric scaling (Yager, 2013).

A plausible implication is that the ACM supports both complete and incomplete, quantitative and qualitative statements of criterion importance, permitting robust modeling under information uncertainty or DM preference ambiguity.

5. Algorithmic Implementation and Complexity

The ACM facilitates algorithmic ranking through well-defined procedures:

• Pairwise Dominance (Qualitative): $O(m^2 n^2)$: For each pair $(a_i, a_k)$, scan criteria for witness columns and verify “no-worse” conditions on relevant criteria based on the partial importance order; outputs a strict partial order via graph traversal/topological sort (Agrawal, 2015).
• Reference-type MCDM (e.g., TOPSIS, VIKOR, GRA): $O(mn)$ to compute weighted, normalized matrices, reference vector(s), and distances/grades; $O(m\log m)$ for sorting.
• Aggregation-type MCDM (e.g., SAW, MEW, COPRAS, MOORA): $O(mn)$ for scoring all alternatives.
• Consensus in multi-expert, non-numeric settings: Per Yager, involves matrix construction per expert, per-alternative minimum aggregation, then OWA-based fusion across experts (Yager, 2013).

The ACM thus presents no super-polynomial bottlenecks, and its tabular structure enables straightforward translation into computational workflows.

6. Illustrative Examples and Comparative Method Outcomes

Worked examples using a common ACM across nine reference-type methods (Wang et al., 22 Aug 2025) and eight aggregation-type methods (Wang et al., 8 Sep 2025) reveal pronounced variability in the top-ranking alternative depending on method and normalization. For instance, in a five-alternative, three-criterion problem, A$_2$ dominates in TOPSIS, MARCOS, and PROBID, whereas A$_4$ is the leader under GRA, EDAS, MABAC, PIV, etc. This suggests that method selection and preprocessing decisions, all of which are applied at the ACM level, can be as consequential as the raw evaluation data.

Example Method	Top-Ranked Alternative (5×3 Example (Wang et al., 22 Aug 2025))
TOPSIS	A$_2$
GRA	A$_4$
VIKOR	A$_1$
MARCOS	A$_2$

Such patterns underscore the ACM's centrality to MCDM reproducibility and sensitivity analysis.

7. Extensions, Robustness, and Limitations

Robustness of the ACM derives from its generality: it accepts data of arbitrary scale, missingness, and imprecision, provided only that intra-criterion orderings are specified. When expert disagreement or multi-source data drive the evaluations, consensus and fusion approaches (e.g., Yager’s OWA over expert ACMs) can be implemented directly at the matrix level (Yager, 2013). Limitations include susceptibility to rank reversal (the addition or removal of alternatives may reorder rankings), dependence on normalization scheme, and the potentially strong impact of subjective weight and scale choices (Wang et al., 22 Aug 2025, Wang et al., 8 Sep 2025).

A plausible implication is that cross-method ACM-based analyses, sensitivity checks, and explicit transparency in preprocessing protocols are essential for defensible MCDM applications.

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References:

• (Agrawal, 2015) Agrawal, M. et al., "Qualitative Decision Methods for Multi-Attribute Decision Making"
• (Yager, 2013) Yager, R. R., "A Non-Numeric Approach to Multi-Criteria/Multi-Expert Aggregation Based on Approximate Reasoning"
• (Wang et al., 22 Aug 2025) Wang et al., "Chapter 7 Multi-Criteria Decision-Making: Reference-Type Methods"
• (Wang et al., 8 Sep 2025) Wang et al., "Chapter 8 Multi-Criteria Decision-Making: Aggregation-Type Methods"