WASPAS: Hybrid Weighted Sum-Product Assessment

• WASPAS is a hybrid multi-criteria decision-making method that blends weighted sum and weighted product models using a tunable parameter (lambda) to balance compensability.
• It normalizes benefit and cost criteria via max/min scaling to transform raw data into a comparable, common benefit scale.
• The method efficiently ranks alternatives and supports sensitivity analysis by adjusting lambda to control trade-offs between additive transparency and multiplicative penalization.

Searching arXiv for recent and foundational material on WASPAS and related extensions. arxiv_search(query="WASPAS Zavadskas 2012 arXiv Weighted Aggregated Sum Product Assessment", max_results=10) arxiv_search(query="(Wang et al., 8 Sep 2025)", max_results=5) arxiv_search(query="(Bozyiğit et al., 21 Jan 2025) CASPAS WASPAS", max_results=5) arxiv_search(query="(Wan et al., 2022) CRITIC-WASPAS interval-valued q-rung orthogonal fuzzy", max_results=5) Weighted Aggregated Sum Product Assessment (WASPAS) is a hybrid multi-criteria decision-making (MCDM) method that converts an alternatives-criteria matrix into a single performance score per alternative by combining the additive logic of the weighted sum model with the multiplicative logic of the weighted product model. In the aggregation-type taxonomy, it belongs to methods that transform an alternatives-criteria matrix into one scalar score per alternative through additive, multiplicative, or hybrid manipulations for ranking purposes (Wang et al., 8 Sep 2025). WASPAS is designed to retain the transparency and proportional compensation of Simple Additive Weighting while also incorporating the “all-criteria-must-be-reasonable” property of Multiplicative Exponent Weighting, with the balance between these two components controlled by a parameter $\lambda \in [0,1]$ (Wang et al., 8 Sep 2025).

1. Methodological position within aggregation-type MCDM

WASPAS is presented alongside Simple Additive Weighting (SAW), Multiplicative Exponent Weighting (MEW), Analytic Hierarchy Process (AHP), Analytic Network Process (ANP), Complex Proportional Assessment (COPRAS), Multi-Objective Optimization on the basis of Ratio Analysis (MOORA), and Faire Un Choix Adequat (FUCA) as one of the selected aggregation-type MCDM methods that rank alternatives by reducing an alternatives-criteria matrix to a single performance score (Wang et al., 8 Sep 2025). Its defining characteristic is hybridity: it combines a weighted arithmetic aggregation and a weighted geometric aggregation rather than relying exclusively on one of them.

The method is motivated by the observation that additive and multiplicative aggregation encode different compensability regimes. The additive component preserves interpretability and proportional scaling, whereas the multiplicative component penalizes very low criterion performance more strongly (Wang et al., 8 Sep 2025). This suggests that WASPAS occupies an intermediate position between highly compensatory and less compensatory aggregation rules. A plausible implication is that it is especially attractive when neither pure linear trade-off behavior nor pure multiplicative strictness is normatively sufficient.

The 2025 solar-panel study situates WASPAS in the literature as a method proposed by Zavadskas et al. (2012) for MCDM problems with known importance weights, emphasizing that its core purpose is to strengthen ranking robustness by blending the Weighted Sum Model (WSM) and the Weighted Product Model (WPM) (Bozyiğit et al., 21 Jan 2025). In that account, the balancing parameter is denoted $\alpha$ rather than $\lambda$, but the functional role is the same: $\alpha=1$ yields the purely additive regime, $\alpha=0$ yields the purely multiplicative regime, and $\alpha=0.5$ is commonly used to assign equal importance to both components (Bozyiğit et al., 21 Jan 2025).

2. Formal structure, notation, and score construction

In the notation used for the method, alternatives are indexed by $i \in \{1,2,\dots,m\}$ and criteria by $j \in \{1,2,\dots,n\}$. The raw alternatives-criteria matrix entry is $f_{ij}$, the normalized criterion value is $F_{ij}$, and the criterion weight is $\alpha$0, with $\alpha$1 (Wang et al., 8 Sep 2025). Benefit criteria are to be maximized, whereas cost criteria are transformed so that larger normalized values are always preferable (Wang et al., 8 Sep 2025).

For the chapter formulation, WASPAS uses max normalization for benefit criteria and min/inverse normalization for cost criteria:

$\alpha$2

$\alpha$3

These transformations place all criteria on a common benefit scale in which larger $\alpha$4 indicates better performance (Wang et al., 8 Sep 2025). The solar-panel paper describes the same crisp WASPAS normalization in conventional $\alpha$5 notation:

$\alpha$6

It also notes that other normalization variants exist, including sum-based normalization (Bozyiğit et al., 21 Jan 2025).

WASPAS then computes two component scores. The additive score is

$\alpha$7

and the multiplicative score is

$\alpha$8

The final aggregated score is

$\alpha$9

with $\lambda$0 (Wang et al., 8 Sep 2025). In the alternative notation of the solar-panel paper, these are written as

$\lambda$1

The notational difference does not alter the underlying mechanism (Bozyiğit et al., 21 Jan 2025).

The limiting cases are explicit. Setting $\lambda$2 recovers SAW, and setting $\lambda$3 recovers MEW (Wang et al., 8 Sep 2025). The same statement appears in the alternative notation with $\lambda$4 replacing $\lambda$5 (Bozyiğit et al., 21 Jan 2025). A common default is $\lambda$6, which weights the additive and multiplicative components equally (Wang et al., 8 Sep 2025).

3. Computational procedure and algorithmic interpretation

The chapter gives a stepwise algorithm for WASPAS. Given an alternatives-criteria matrix $\lambda$7, criterion weights $\lambda$8, and criterion types labeled as benefit or cost, the procedure is: normalize the matrix; compute the additive score $\lambda$9; compute the multiplicative score $\alpha=1$0; aggregate them into $\alpha=1$1 using $\alpha=1$2; and rank alternatives in descending order of $\alpha=1$3 (Wang et al., 8 Sep 2025). In schematic pseudocode form, the method computes column maxima and minima, populates the normalized matrix according to benefit/cost type, evaluates $\alpha=1$4, $\alpha=1$5, and $\alpha=1$6, and finally sorts by descending $\alpha=1$7 (Wang et al., 8 Sep 2025).

The method is computationally lightweight. Its operations are simple normalization, weighted sums, weighted products, and a linear blend (Wang et al., 8 Sep 2025). The chapter characterizes it as straightforward and computationally efficient and states that it scales well with the numbers of alternatives and criteria (Wang et al., 8 Sep 2025). The CASPAS paper makes a parallel point indirectly by contrasting WASPAS with its Choquet-generalized extension: when interaction modeling is introduced, per-alternative processing requires sorting criterion values and evaluating Choquet-based operators; the implication is that the classic WASPAS pipeline is materially simpler when interdependencies are not modeled (Bozyiğit et al., 21 Jan 2025).

Interpretively, the additive component $\alpha=1$8 is directly readable as a weighted sum, whereas the multiplicative component $\alpha=1$9 behaves as a weighted geometric-type aggregation in which very low normalized values can strongly depress the overall score (Wang et al., 8 Sep 2025). This difference is central to the role of $\alpha=0$0: it governs how much compensation across criteria is permitted. The chapter states that WASPAS allows compensation governed by the SAW-versus-MEW balance and implicitly assumes that criteria can be aggregated without interaction terms, that is, under an independence assumption (Wang et al., 8 Sep 2025).

4. Worked example and comparative ranking behavior

A common numerical example in the chapter uses four alternatives $\alpha=0$1 and three criteria $\alpha=0$2 (benefit), $\alpha=0$3 (benefit), and $\alpha=0$4 (cost), with weights $\alpha=0$5, $\alpha=0$6, and $\alpha=0$7 (Wang et al., 8 Sep 2025). The raw matrix entries are:

• $\alpha=0$8: $\alpha=0$9, $\alpha=0.5$0, $\alpha=0.5$1, $\alpha=0.5$2
• $\alpha=0.5$3: $\alpha=0.5$4, $\alpha=0.5$5, $\alpha=0.5$6, $\alpha=0.5$7
• $\alpha=0.5$8: $\alpha=0.5$9, $i \in \{1,2,\dots,m\}$0, $i \in \{1,2,\dots,m\}$1, $i \in \{1,2,\dots,m\}$2 (Wang et al., 8 Sep 2025)

After normalization, the chapter reports:

• $i \in \{1,2,\dots,m\}$3: $i \in \{1,2,\dots,m\}$4
• $i \in \{1,2,\dots,m\}$5: $i \in \{1,2,\dots,m\}$6
• $i \in \{1,2,\dots,m\}$7: $i \in \{1,2,\dots,m\}$8
• $i \in \{1,2,\dots,m\}$9: $j \in \{1,2,\dots,n\}$0 (Wang et al., 8 Sep 2025)

The corresponding SAW and MEW component scores are:

• $j \in \{1,2,\dots,n\}$1, $j \in \{1,2,\dots,n\}$2
• $j \in \{1,2,\dots,n\}$3, $j \in \{1,2,\dots,n\}$4
• $j \in \{1,2,\dots,n\}$5, $j \in \{1,2,\dots,n\}$6
• $j \in \{1,2,\dots,n\}$7, $j \in \{1,2,\dots,n\}$8 (Wang et al., 8 Sep 2025)

With $j \in \{1,2,\dots,n\}$9, the final WASPAS scores are:

• $f_{ij}$0
• $f_{ij}$1
• $f_{ij}$2
• $f_{ij}$3 (Wang et al., 8 Sep 2025)

The resulting ranking is

$f_{ij}$4

For the same alternatives-criteria matrix, the chapter reports the following rankings for other methods (Wang et al., 8 Sep 2025):

Method	Ranking
SAW	$f_{ij}$5
MEW	$f_{ij}$6
COPRAS	$f_{ij}$7
MOORA	$f_{ij}$8
FUCA	$f_{ij}$9
WASPAS ($F_{ij}$0)	$F_{ij}$1

This comparison is significant because it shows that WASPAS may align with either its additive or multiplicative component on some positions while differing on others. In the common example, WASPAS agrees with SAW on the full ordering, whereas MEW and MOORA swap the relative positions of $F_{ij}$2 and $F_{ij}$3 (Wang et al., 8 Sep 2025). The chapter uses this consolidated outcome to reinforce a broader recommendation: multiple MCDM methods may identify the same top alternative while disagreeing on lower ranks, so sensitivity analyses and method triangulation are often appropriate (Wang et al., 8 Sep 2025).

5. Strengths, weaknesses, and recurrent implementation issues

The chapter lists several practical strengths. WASPAS is straightforward and computationally efficient; its hybrid structure combines interpretability and proportional scaling from the additive component with anti-compensatory pressure from the multiplicative component (Wang et al., 8 Sep 2025). The solar-panel paper reiterates that it is used when decision makers want a simple, transparent, and computationally efficient method that balances sum-type and product-type aggregation (Bozyiğit et al., 21 Jan 2025).

Its principal weaknesses are also explicit. First, the method is sensitive to the choice of $F_{ij}$4, and selecting $F_{ij}$5 is itself an additional decision problem (Wang et al., 8 Sep 2025). Second, zero or extremely small normalized values can cause the multiplicative component $F_{ij}$6 to collapse or become overly punitive (Wang et al., 8 Sep 2025). Third, like most normalization-based aggregation methods, WASPAS is vulnerable to rank reversal when alternatives are added or removed, because the normalization denominators depend on observed maxima and minima (Wang et al., 8 Sep 2025).

Several implementation nuances follow directly from these properties. The normalization choice matters: the chapter uses max normalization, but it notes that different schemes such as vector or sum normalization can change $F_{ij}$7 and therefore $F_{ij}$8, $F_{ij}$9, and $\alpha$00 (Wang et al., 8 Sep 2025). Benefit-versus-cost handling also matters because the method assumes monotonicity after normalization: once transformed, larger normalized values must always be better (Wang et al., 8 Sep 2025). The chapter further notes that ties are not explicitly addressed; a reasonable practice is to report ties or apply secondary criteria (Wang et al., 8 Sep 2025).

Other issues are discussed as practical rather than formal components of the method. Missing values and zero-variance criteria are not handled in the chapter; it states that, in practice, missing data require imputation or exclusion, while zero-variance criteria contribute no discrimination and can be dropped (Wang et al., 8 Sep 2025). For very small normalized values, the multiplicative score may underflow numerically, and log-domain computations can mitigate this even though such stabilization is not discussed in the chapter itself (Wang et al., 8 Sep 2025). These points help clarify a common misconception: WASPAS is not merely a matter of substituting a formula for a ranking. Its behavior depends materially on preprocessing, normalization, and the numerical properties of the data matrix.

6. Parameter sensitivity, interpretation of compensability, and method selection

The balancing parameter is the central control variable in WASPAS. The chapter states that varying $\alpha$01 between $\alpha$02 and $\alpha$03 smoothly transitions the ranking toward MEW or SAW, respectively, and recommends $\alpha$04 as a common default (Wang et al., 8 Sep 2025). No detailed $\alpha$05-sensitivity results are reported for the common alternatives-criteria matrix, but an exercise in the chapter suggests performing such an analysis (Wang et al., 8 Sep 2025). The solar-panel paper makes the same recommendation in the classical setting: if there is no strong reason to privilege sum-type or product-type aggregation, $\alpha$06 is a natural starting point (Bozyiğit et al., 21 Jan 2025).

The interpretation of $\alpha$07 is not purely technical; it encodes a choice about compensability. Increasing $\alpha$08 emphasizes interpretability and proportionality associated with the additive component, whereas decreasing it emphasizes the stronger penalization of low-performing criteria associated with the multiplicative component (Wang et al., 8 Sep 2025). The CASPAS paper phrases the same distinction as a choice between less punitive “sum-like” behavior and stronger multiplicative/geometric effects (Bozyiğit et al., 21 Jan 2025). This suggests that the parameter can be read as a policy variable governing tolerance for trade-offs across criteria.

The chapter also situates WASPAS relative to alternative MCDM families. It states that WASPAS is preferable to AHP or ANP when the problem is represented by an objective alternatives-criteria matrix and the aim is fast, transparent aggregation with tunable compensability, whereas AHP or ANP are appropriate when subjective pairwise comparisons, hierarchies, or network interdependencies are central (Wang et al., 8 Sep 2025). Relative to MOORA and COPRAS, the chapter states that WASPAS is often preferable when both additive and multiplicative perspectives are relevant and a single parameter is desired to control that balance (Wang et al., 8 Sep 2025).

A second misconception concerns robustness. The 2022 interval-valued q-rung orthogonal fuzzy study reports that, in its case study, rankings were invariant to changes in $\alpha$09, with WASPAS scores remaining between the weighted-sum and weighted-product components (Wan et al., 2022). That result should not be generalized indiscriminately: the chapter explicitly identifies sensitivity to $\alpha$10 as a weakness of the method in general (Wang et al., 8 Sep 2025). The combined reading implies that robustness to $\alpha$11 is empirical and dataset-dependent rather than guaranteed by the method’s structure.

7. Extensions beyond the classical independent-criteria formulation

A major line of extension generalizes WASPAS to environments where classical weighted aggregation is inadequate. The CASPAS methodology, introduced for solar panel selection, is a Choquet Aggregated Sum Product Assessment method that generalizes WASPAS by replacing WSM and WPM with Choquet-based sum and product aggregations in order to model interdependencies among criteria via fuzzy measures (Bozyiğit et al., 21 Jan 2025). In this framework, the final significance degree is

$\alpha$12

with $\alpha$13 playing a role analogous to $\alpha$14 in classical WASPAS (Bozyiğit et al., 21 Jan 2025). The extension is built in a disc intuitionistic fuzzy set (D-IFS) environment, where each evaluation carries membership, non-membership, and a radius component, and ranking is performed through D-IFS score and accuracy functions (Bozyiğit et al., 21 Jan 2025).

The conceptual importance of CASPAS lies in a limitation of standard WASPAS that is explicit in the chapter: classical WASPAS assumes criteria can be aggregated without interaction terms (Wang et al., 8 Sep 2025). CASPAS addresses precisely this issue by using capacities and discrete Choquet integrals to model synergy and redundancy among criteria (Bozyiğit et al., 21 Jan 2025). The paper proves a reduction property: when the capacity is additive and there are no interactions, the Choquet arithmetic operator reduces to weighted arithmetic aggregation and the Choquet geometric operator reduces to weighted geometric aggregation, so CASPAS becomes consistent with classical WASPAS under criterion independence (Bozyiğit et al., 21 Jan 2025).

Another extension integrates WASPAS with CRITIC weighting, Yager operators, and interval-valued q-rung orthogonal fuzzy numbers for group decision-making with unknown weights (Wan et al., 2022). In this formulation, decision-maker weights are derived through similarity-based weighting, attribute weights are obtained objectively through CRITIC and then converted to real-valued weights, and WASPAS is implemented by replacing the classical weighted-sum and weighted-product steps with interval-valued q-rung orthogonal fuzzy Yager weighted average and Yager weighted geometric operators (Wan et al., 2022). The final mixture retains the familiar form

$\alpha$15

followed by score-based ranking (Wan et al., 2022). The hypertension risk-management case yields the ranking

$\alpha$16

with $\alpha$17, and the result is reported as consistent with the experts’ judgments (Wan et al., 2022).

These developments clarify the boundary of the original method. Classical WASPAS is a hybrid scalar aggregation procedure for an objective alternatives-criteria matrix under independent criteria and explicit weights (Wang et al., 8 Sep 2025). Its later variants preserve the same sum-product fusion idea while modifying the representational layer: Choquet capacities are introduced when criterion interactions matter (Bozyiğit et al., 21 Jan 2025), and interval-valued q-rung orthogonal fuzzy structures are introduced when uncertainty, group heterogeneity, and unknown weights must be handled jointly (Wan et al., 2022). A plausible implication is that the enduring core of WASPAS is not a particular normalization formula alone, but the controlled interpolation between additive and multiplicative evaluation regimes.