A hybrid combination of analytic hierarchy process and best-worst method: simulation and application

Sustainable development, especially in agriculture, is importantly linked to the UN Sustainable Development Goals (SDGs), particularly No Poverty (SDG1), Zero Hunger (SDG2), Responsible Consumption and Production (SDG12) and Life on Land (SDG15) (United Nations, 2018). Since agriculture systems involve multiple factors such as crops, land and water management (Nguyen et al., 2019; Cui and Xie, 2021; Cano et al., 2023), and because sustainable development is a multi-dimension issue itself (Ahmed et al., 2021; Boix-Fayos and de Vente, 2023; Nguyen-Anh et al., 2023), it is justified to evaluate sustainable development in agriculture from a multi-criteria perspective (Streimikis and Baležentis, 2020; Ngo et al., 2021; Dong and Mitchell, 2023).

It is acknowledged that multi-criteria decision making (MCDM), also known as multi-criteria decision analysis (MCDA), is a vital and popular branch of operations research as it helps decision-makers (DMs), in our case farmers, to make their (sustainable development) decisions (Hatami-Marbini and Tavana, 2011). Popular MCDM methods include the analytic hierarchy process (AHP), Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS), Elimination Et Choix Traduisant la REalité (ELECTRE), VlseKriterijumska Optimizacija I Kompromisno Resenje (VIKOR), Preference Ranking Organization METHod for Enrichment Evaluations (PROMETHEE) and so on (Brans et al., 1986; Saaty, 1986, 1990; Roy, 1991; Rao Tummala and Ling, 1998; Conejero et al., 2021). Consider a basic MCDM problem where a DM is facing m alternatives $(A1,A2,..,Am)$ each has n criteria $(C1,C2,..,Cn)$ whose score of alternative $Ai$ with respect to criterion $Cj$ is denoted as $pij$⁠, one can construct a decision matrix $D$ to rank each alternative $Ai$ as follows.

Although there are different approaches to constructing the decision matrix $D$⁠, a straightforward way is to use the pairwise comparison method originally proposed by Thurstone (1927) and then follow the AHP method to define the weight $wi$ for each criterion $Cj$⁠, given that the criteria usually have different importance (or weight) in each alternative (Saaty, 2004). The overall value of each alternative, $Vi$⁠, for which the alternatives can be ranked, can be consequently computed as in Equation (2).

As pointed out by Rezaei (2015) and Mi et al. (2019), although AHP is simple and popular, it is facing challenges in doing pairwise comparisons and the lack of consistency results, especially when the number of criteria and alternatives increases. Rezaei (2015) consequently proposed a best-worst method (BWM) as an improvement of AHP, where an alternative is compared with only the best and the worst alternatives instead of all alternatives as in AHP (details are presented in the following section). The BWM does not only help reduce the computation time (since there are only $2n−3$ pairwise comparisons in BWM compared to $n(n−1)/2$ pairwise comparisons in AHP for a set of $n$ criteria) but BWM also performs better than AHP in terms of consistency ratio, minimum violation, total deviation and conformity (Rezaei, 2015; Mi et al., 2019). As pointed out in Peykani et al. (2026) and Carpitella et al. (2024), among others, the trade-off between AHP and BWM regarding computation speed and information loss does exist, i.e. the former utilises more information (from all available pairwise comparisons) but is thus slower, while BWM is quicker but relies on less information (fewer comparisons as the best and worst criteria have been identified). However, given the rise of big data where studies increasingly using larger samples, the benefits of BWM (e.g. regarding consistency and violation) could outweigh this limitation of potential information loss. Since the introduction of BWM in 2015, it has therefore attracted many scholars' attention and has resulted in more than 120 studies applying the BWM method (Mi et al., 2019).

It is noted that the questionnaire of BWM, however, differs from that of AHP in an important feature. In AHP, the criteria are freely compared against each other, whilst in BWM, the best and worst criteria were chosen in the first instance, and the rest of the criteria were then compared against those two. It means that data collected from AHP surveys could not be used in BWM studies and vice versa, and scholars who want to compare the results of the two methods will need to have two sets of questionnaires in place. Given that AHP is still dominating in MCDM studies, but BWM has several advantages, so far there is no solution for the re-production or combination of AHP studies using BWM, i.e. it is not possible to apply BWM analysis on the existing AHP data. Our study is therefore the first attempt to answer the question of “Can we conduct BWM analysis using AHP data?”, particularly in the field of sustainable agriculture development. It therefore contributes to the current debate on research reproducibility (Begley and Ioannidis, 2015; Baker, 2016), where AHP studies can be reproduced by the proposed AHP-BWM approach but with faster computation and better consistency ratio, minimum violation and total deviation.

The current study aims to answer the above question by proposing a new way to pre-process AHP data so that it can be used with BWM, namely the AHP-BWM method. Consequently, one important contribution of the AHP-BWM approach is that it combines the advantages of both AHP and BWM. Another important feature is that it allows researchers to reproduce BWM results based on AHP data. In this sense, the AHP-BWM approach is practically valuable for decision-making processes. The idea of AHP-BWM is simple: for a certain alternative $A$ in the AHP data, if we can determine the best and worst criteria, then we can use BWM instead of AHP. To do that, we use a Monte Carlo simulation with more than 5 million pairwise comparisons to illustrate the advantages of our AHP-BWM approach, compared to AHP; such results highlight the methodological contribution of this study. For our empirical application, we further utilise the AHP data of Ngo et al. (2021) to apply the AHP-BWM approach for assessing the important dimensions of sustainable agriculture development in Northern Vietnam. The empirical results also reveal that AHP-BWM could improve the BWM's limitation of information loss.

In the following section, basic principles of AHP and BWM are introduced, followed by the proposal of the new AHP-BWM method to select the best and worst criteria. Section 3 compares the simulation results between AHP and AHP-BWM. Section 4 provides an empirical application of the AHP-BWM method to examine the important factors of sustainable agriculture development in Bac Giang, a Northern province of Vietnam. The conclusions and suggestions for future research are presented in Section 5.

In AHP, for a certain alternative, its $n$ criteria $(C1,C2,..,Cn)$ are compared against each other via Saaty (1986, 1990)'s 1/9–9 scale as follows:

It is noted that $aij$ denotes the preference degree of criterion $Ci$ over criterion $Cj$ and thus $aij≠pij$⁠. Importantly, a value of $aij=1/9$ implies that criterion $Cj$ is extremely more important or absolutely preferred to criterion $Ci$⁠, whilst a value of $aij=9$ means that criterion $Ci$ is absolutely preferred to criterion $Cj$⁠. When $aij=1$⁠, the two criteria are equally important (i.e. $Ci=Cj$⁠), whereas other values of $aij$ represent the two-way preference between the two (i.e. $Ci>Cj$ or $Ci<Cj$⁠). Another note is that the matrix $A$ in Equation (3) is reciprocal, i.e. $aij=1/aji$ and $aii=1$⁠, for all $i$ and $j$⁠.

In the next step of AHP, each element in the matrix $A$ is normalised by the column sum, and then the weights (or principal eigenvectors) $wi$ can be calculated as the geometric mean of each row in the normalised matrix $A$ using the following equation:

Saaty (1986, 1990) argued that the original comparison problem in Equation (3) can now be transformed into $A′w′=λmaxw′$ where $λmax$ is the largest eigenvalue of $A′$⁠. It is further discussed that a consistency ratio (⁠$CR$⁠) could be calculated as in Equation (5) to determine if the pairwise comparison matrix $A$ is consistent and that the weights $wi$ are reliable; a small $CR$ of 0.1 or less is considered a positive evidence for informed judgment (Saaty, 2004; Ngo et al., 2021).

Where $RIAHP$ is extracted from Table 11 of Saaty (1990).

Equations (3) and (4) suggest that there are $n(n−1)/2$ pairwise comparisons in AHP, and when $n$ increases, the number of comparisons as well as the probability of matrix $A$ being inconsistent also increase. To overcome those defects, the BWM proposed that it is justified to compare the criterion $Ci$ with only two other criteria, the best (⁠$CB$⁠) and the worst (⁠$CW$⁠) ones – given that those two are identified beforehand (Rezaei, 2015, 2016; Mi et al., 2019). However, one can also argue that this identification as well as the relevant BWM's weights/rankings are objectively affected by the participants, whilst AHP results are more subjective and data-driven (as long as CR ≤ 0.1). The preference degree $aij$ will then get the values from $1$ to $9$ as it is now only reflecting the one-way direction (i.e. $Ci<CB$ and $Ci>CW$⁠). In this sense, only two vectors of comparisons are involved in BWM (and thus there are only $2n−3$ comparisons in total) as follows:

Where $BO$ denotes the best-to-others vector comparing the best criterion $CB$ and the rest, whilst $OW$ denotes the others-to-worst vector comparing other criteria to the worst criterion $CW$ (note that $aBB=aWW=1$⁠).

The weights of the criteria are the set $(w1,w2,..,wn)$ that satisfies the following linear programming model (Rezaei, 2016; Mi et al., 2019):

Where $wB$ denotes the weight corresponding to the best criterion and $wW$ denotes the weight corresponding to the worst criterion.

Similar to AHP, BWM also uses the consistency ratio CR to determine if those weights are reliable or not using the same threshold value of 0.1, whereas the corresponding CRBWM is an extension of Equation (5) using the optimal value $ξ*$ derived from solving Equation (8).

Where $RIBWM$ is extracted from Table 1 of Rezaei (2015).

More importantly, Rezaei (2015) statistically proved that BWM outperforms AHP with respect to the consistency ratio (⁠$CR$⁠), minimum violation (⁠$MV$⁠), total deviation (⁠$TD$⁠) and conformity (⁠$CF$⁠). The formulas for the last three measures are presented in Equations (10)-(12) as follows.

Where

Where $Rankj$ represents the rank of criterion $Cj$ computed by BWM and $Rankj0$ represents the intuitive rank of criterion $Cj$ defined beforehand by the participants.

AHP-BWM is an extension of BWM to utilise data from AHP; previous BWM studies need to have their own data (Mi et al., 2019). The basic idea of AHP-BWM is, therefore, to determine the best and worst criteria using AHP data. While AHP determines the weights using the principal eigenvector of the comparison matrix, in BWM, however, the identification of the best and worst criteria does not require such calculations. For instance, in Step 2 of the BWM process, Rezaei (2015) mentioned that the two were arbitrarily selected by the participating DMs without any computation or comparison requirement. This step is further illustrated in Appendix A of Rezaei (2015), where the BWM questionnaire indicates that the participants were to simply select the most and least important criteria from the list of six criteria provided. In other words, BWM does not rely on the eigenvector to determine the best and worst criteria, nor does AHP-BWM in the current study. As shown below, one can select the best and worst criteria using the preference degrees of those criteria; a simple numerical illustration in Table 1 and intensive simulations in Section 3 empirically verify this approach.

It is noted that in AHP, the relationship between the criteria of matrix A is $aij$ (see Equation (3) above). Regardless of the scale of $aij$ to be in the range of [1,7], [1/9] or [1/9–9], what it represents is the (in)equality between criterion $Ci$ and $Cj$ such that the higher value of $aij$ the higher gap or difference between the two criteria. For instance, Rezaei (2015) argued that if criterion A is preferred over criterion B (i.e. $aAB>1$⁠) and criterion B outweighs criterion C (i.e. $aBC>1$⁠), then it also implies that A is preferred against C. Consequently, the summation of the preference degrees (⁠$aAB+aBC$⁠) should also reflect the importance of criterion A over C. We, therefore, argue that the criterion accompanied by the highest value of total preference degree $∑aij$ (⁠$i≠j$⁠) is the best criterion $CB$ while the criterion accompanied by the lowest value of $∑aij$ will be the worst criterion $CW$⁠. We illustrate this idea using data from Table 5 of Saaty (1990) for the case of three criteria as follows.

Obviously, the preference degrees associated with Model I are 5 (Model I vs Model II) and 3 (Model I vs Model III), so the total degree for Model I is $5+3=8$⁠. Similarly, the preference degrees associated with Model II are 1/5 (Model II vs Model I) and 1/3 (Model II vs Model III), making a total degree of $1/5+1/3=0.53$⁠. For Model III, the total preference degree is thus $1/3+3=3.33$⁠. As such, the best criterion is Model I and the worst criterion is Model II.

Generally, consider a basic AHP problem where $n$ criteria $(C1,C2,..,Cn)$ are compared against each other via their preference degree $aij$⁠, similar to the matrix A in Equation (3). The sum of the preference degrees for a certain criterion $Ci$ can be computed using Equation (13) below and then the best (⁠$CB$⁠) and worst (⁠$CW$⁠) criteria can be consequently determined as in Equations (14) and (15), respectively. After the two criteria $CB$ and $CW$ are determined, BWM can be applied to examine the dataset, similar to what has been discussed in Section 2.2 above.

Simulation is an effective way to generate reliable data for numerical analysis for MCDM studies (Besharati et al., 2006; Banker and Natarajan, 2007; Hammami et al., 2022) and particularly in AHP (Hauser and Tadikamalla, 1996; Rao Tummala and Ling, 1998; Aull-Hyde et al., 2006; Yaraghi et al., 2015; Bose, 2022), because it allows researchers to generate their own data set and compare the “true” values (underlined by the data generating process) with the simulated ones. The basic idea of AHP simulation studies is to randomly generate the AHP matrix $A$ of Equation (3), then randomly adjust its elements based on several indicators such as the range of adjustments, the number of simulations and so on (Hauser and Tadikamalla, 1996; Carmone et al., 1997). The simulation results will then tell how sensitive or robust a certain measure (e.g. the weight of each criterion $Ci$ or the rank of a certain alternative $Aj$⁠) is. To the best of our knowledge, however, there is no proper data-generating process for AHP simulation in the literature; we just randomly generated the AHP matrix, then checked for its consistency before including it in the simulation. Figure 1 below presents the basic algorithm of an AHP simulation with five steps: (1) Randomly generate a consistent AHP matrix A (i.e. $CR≤0.1$⁠) so that the relevant results are reliable; (2) Compute the original results and store them in $AHP0$⁠; (3) Use the simulation technique to adjust the matrix $A$ for a total of $K$ times, making sure that each time the new matrix is also consistent; (4) Re-compute the results and store them in $AHPk$ and (5) Finally compare the results $AHP0$ and $AHPk$ (⁠$k=1,2,..,K$⁠).

The Monte-Carlo simulation technique is also used in this study to illustrate the robustness of our AHP-BWM approach. Different from AHP simulations where all the results $AHP0$ and $AHPk$ are AHP bases (see Figure 1); however, our AHP-BWM simulation aims to compare AHP and AHP-BWM results. It is noted that if one has BWM data, the results of the traditional BWM and AHP-BWM are the same because Equations (14) and (15) coincide with Equations (6) and (7), respectively; thus, there is no need to compare between BWM and AHP-BWM. If one has AHP data, however, BWM is traditionally not applicable, but only the proposed AHP-BWM approach is appropriate. Therefore, this study only compares the results between AHP and AHP-BWM. In this sense, if the two results are consistent, we can conclude that AHP-BWM is a reliable MCDM method such that it can be an alternative to AHP (or even superior to AHP since it has the advantages of BWM).

The algorithm of AHP-BWM is presented in Figure 2 as follows: (1) Use the simulation technique to randomly generate an AHP matrix B for a total of $K$ times; (2) Each time calculate the AHP and AHP-BWM results and store them in $AHP1$ and $AHP‐BWM1$⁠, respectively; and (3) Finally conduct the pairwise comparison between $AHPk$ and $AHP‐BWMk$ (⁠$k=1,2,..,K$⁠). It is noted that in our simulations, we do not require the matrix B to be consistent as in the case of matrix A in the AHP simulation because we would like to see if AHP-BWM can outperform the traditional AHP method in terms of providing better consistency ratio (⁠$CR$⁠), beside the other performance indicators of (lower) minimum violation (⁠$MV$⁠) and total deviation (⁠$TD$⁠) (see previous discussions in Section 2.2 above).

The simulation described above was conducted using different parameters on the number of criteria or size of the matrix B (⁠$Criteria$⁠), the number of alternatives or observations in each simulation (⁠$Observations$⁠) and the number of simulations (⁠$Simulations$⁠). In particular, we follow Aull-Hyde et al. (2006) to set $Criteria$ in the range from 3 to 6 (Saaty (1980) recommended that a dimension of less than 6 x 6 criteria is manageable) while $Observations$ ranges from 5 to 100 and $Simulations$ ranges from 100 to 1000 - different from Aull-Hyde et al. (2006), here we allow $Simulations$ to have different values to see if the number of simulations can affect our results or not, as normally observed from Monte Carlo simulation studies such as Cross and Färe (2009) and Azadeh et al. (2016). We ended up using 5,130,000 pairwise comparison matrices for our simulation analysis as reported in Table 2, and some key simulation results are presented in the subsequent paragraphs. For space-saving purpose, we do not report all comparison results between AHP and BWM-AHP in this study – that information is available upon reasonable request. Note also that the results indicated that AHP-BWM outperforms AHP when the parameters increase; it is not necessary to extend the simulation, e.g. for more observations or larger dimensions.

Table 3 presents the results of comparing the consistency ratio (⁠$CR$⁠) between AHP and AHP-BWM for the cases that $CRAHP‐BWM$ are smaller than $CRAHP$⁠, i.e. AHP-BWM produces more reliable results than AHP. It is noted that for the cases where AHP resulted with a $CRAHP>0.1$ (i.e. the result is inconsistent and the observation therefore should be excluded from AHP analysis), AHP-BWM also ended up with inconsistent results but with smaller CR values (i.e. $0.1<CRAHP‐BWM≤CRAHP$⁠). Note also that the differences, in this case is $CRAHP‐BWM<CRAHP$⁠, or any other cases in this section such as $MVAHP‐BWM=MVAHP$ or $TDAHP<TDAHP‐BWM$⁠, are all statistically tested using the paired t-test at a 5% level of significance. Accordingly, it is observed that when the number of observations (⁠$Observations$⁠), number of simulations (⁠$K$⁠) and especially the number of criteria to be compared (⁠$Criteria$⁠), increase, AHP-BWM could produce more reliable results (i.e. lower $CR$⁠) than the traditional AHP approach. For example, for the case of only three $Criteria$⁠, 100 simulations using five observations resulted in 4% cases in which AHP-BWM can outperform AHP, whilst 1,000 simulations on 100 observations ended up with 87.3% of cases in which AHP can produce lower $CR$ values than AHP. For the case of $Criteria=4$⁠, AHP-BWM dominates AHP when the number of observations is greater than or equal to 15. When $Criteria$ increases to five or six, AHP-BWM absolutely performs better than AHP with 100% cases of lower $CR$⁠. We therefore conclude that AHP-BWM, in comparison to AHP, leads to more consistent comparisons and hence can produce more reliable results.

In terms of the minimum violation (⁠$MV$⁠), we report the aggregated results for all $Criteria$ (i.e. full simulation of 5,130,000 comparison matrices) between AHP and AHP-BWM in Table 4. Our findings are similar to those of the CR above, where AHP-BWM tends to violate the ordinal preferences less than AHP when the simulation's parameters (e.g. $Observations$ and $K$⁠) increase. Specifically, when these parameters increase, the number of similar $MV$ results between the two approaches decreases (see Table 4A), the number of cases that AHP can outperform AHP-BWM also decreases (see Table 4C), but the cases that AHP-BWM performs better than AHP increase (see Table 4B). It is therefore suggested that AHP-BWM can better preserve the ordinal preferences or rankings among the criteria, which in turn assures that AHP-BWM results are more reliable.

In terms of the total deviation (⁠$TD$⁠), we have also found that AHP-BWM can produce more results that are closer to the original pairwise comparison ratios, although its calculation does not involve those ratios (see Equations (5)-(7)), whilst AHP does (see Equation (4)). We argue that since AHP is less reliable (regarding the $CR$ and $MV$ above), it could not produce reliable weights $wi$ such that the deviations of AHP weight ratios $wi/wj$ to the original pairwise comparison ratio $aij$ are still larger than those of AHP-BWM. We report the findings for the case that $Criteria=3$ in Table 5 where AHP-BWM performs better than AHP, especially when the simulation parameters increase. One can easily find that when $Observations≥20$ and/or $K≥300$⁠, AHP-BWM starts dominating AHP to produce 100% cases with lower $TD$ values. For the cases that $Criteria>3$⁠, AHP-BWM continues to produce 100% cases with lower $TD$ than AHP and thus are not reported here.

Overall, we could summarize our simulation results by arguing that AHP-BWM performs better than AHP in terms of $CR$⁠, $MV$ and $TD$⁠, so that AHP-BWM can provide more reliable results than AHP. This argument is stronger when the parameters of simulation (i.e. $Criteria$⁠, $Observations$ and $Simulations$⁠) increase, although AHP can be equally good (compared to AHP-BWM) when the $Criteria$ is small (e.g. 3 or 4) and for fewer alternatives or participants (e.g. $Observations≤30$⁠).

As discussed earlier, the main purpose of the AHP-BWM method is to utilise BWM analysis on AHP data. This section applies the proposed method to the survey AHP data for sustainable agriculture development in Northern Vietnam (Ngo et al., 2021) to illustrate the empirical use of AHP-BWM. Unlike Ngo et al. (2021), who examined a wide set of variables to provide a detailed picture of the agriculture practices towards sustainability of Vietnamese farmers, we only analyse the top three dimensions of sustainable agriculture (Zahm et al., 2008, 2018), including Agroecology (AGRO), Socio-territorial (SOTE) and Economic (ECON). In other words, we would like to use the AHP-BWM method to evaluate the contributions of the three dimensions above to the overall sustainable agriculture development in Vietnam. It is noted that, however, our method can be easily extended to examine other factors or dimensions of the dataset at different levels of hierarchies – we leave this task for future research.

Proposed in Zahm et al. (2008) and then expanded in Zahm et al. (2018), the sustainability of agriculture farm can be assessed using 53 multi-dimensional indicators covering a wide range of factors such as diversity of crops, managing use of water, animal well-being and economic capacity. Those indicators belong to the three main dimensions of AGRO, SOTE and ECON, which reflect a “top-down” view of experts in the field that those dimensions are the most important factors defining a sustainable farm (Krank et al., 2013; Candido et al., 2015). Ngo et al. (2021) proposed a “bottom-up” approach to examine whether farmers are really concerned about those indicators and dimensions in their day-to-day activities, and how those factors contribute to the overall sustainability of their farms. Empirically, they surveyed 673 farmers from Bac Giang province in the Northern region of Vietnam; however, 29 farmers were excluded from the analysis due to having their consistency ratio (CR) greater than 0.1 (see Section 2.1 above). Accordingly, Ngo et al. (2021) found that ECON is the most important dimension (accounts for 54% of the overall sustainability development), followed by SOTE (29%) and then AGRO (17%) (Ngo et al., 2021). Other studies on sustainable agriculture development in Vietnam focused on different aspects such as land sustainability (D'haeze et al., 2005), the Green Revolution (Tran and Kajisa, 2006), knowledge and human capital (Nguyen et al., 2019), agricultural extension services (Le et al., 2020), among others.

Such a large sample is suitable for our AHP-BWM application, as our simulations in the previous section have shown the efficacy of this approach over the traditional AHP approach when the number of observations increases. In our empirical application, several improvements are observed as follows. Firstly, the AHP-BWM (our model) outperforms the AHP (the model of Ngo et al., 2021) in all CR, MV and TD statistics, in both full and sub-samples (see Table 5). Importantly, the statistics for the full sample in Table 6 show that the AHP-BWM can provide more consistent (via a lower average CR value) and less biased (via lower MV and TD values) results, allowing fewer participants to be excluded from the analysis. Particularly, our AHP-BWM analysis can utilise information from 662 participants, compared to the case of 644 participants of Ngo et al. (2021). Such improvement is important for any empirical analysis as the more participants remain, the more information can be extracted and analysed. In this sense, AHP-BWM empirically helps improve both AHP and BWM in a hybrid fashion: it is quicker than AHP but also more consistent and utilise more information; it allows BWM to utilise AHP data and also improve BWM's limitation of information loss, as introduced earlier, as more participants could be examined.

Secondly, we also looked at the cases when the AHP and AHP-BWM produce similar results (i.e. the statistics of the two models are equal), when AHP outperforms AHP-BWM (i.e. AHP’ statistics are lower than those of AHB-BWM), and when AHP-BWM outperforms AHP (i.e. AHP’ statistics are higher than those of AHP-BWM). Table 7 shows that while the equal cases consist of about 10–20% of the full sample (for all CR, MV and TD statistics), the majority of cases show that the AHP-BWM can produce better statistics than the AHP (around 51% of the cases for MV and more than 80% for both CR and TD). This finding, therefore, strengthens our argument in Section 3 that the AHP-BWM approach can provide more reliable results than the AHP.

Thirdly, when we further examine the estimated results regarding the contributions of the three dimensions AGRO, SOTE and ECON towards sustainable agriculture development, we found that (1) the AHP-BWM produces consistent results for both the full and sub-sample and (2) the ranking orders of the three dimensions identified by AHP and AHP-BWM are similar, although their values are slightly different. Particularly, the second and third rows of Table 8 show that the AHP method identified conflicting results regarding the contribution of AGRO: this dimension accounts for 34% of sustainable agriculture development in Northern Vietnam if one relies on the information from a full sample of 673 participants but dropped to only 17% when utilising a sub-sample of 644 participants. Of course, one may argue that the estimations based on the full sample are not as reliable as the ones based on the sub-sample, given the consistency ratio issue. However, Table 8 also shows that this problem does not occur in the AHP-BWM approach, where the values and the ranks of the three dimensions are similar, even when we ignore the consistency issue and use the full sample of 673 participants. Once again, we argue that the AHP-BWM can utilise more information from the data, compared to the AHP, to produce better results. We thus conclude that while our findings are still in line with Ngo et al. (2021) in confirming that ECON is the most important dimension of sustainable agriculture development in Northern Vietnam (accounts for 52%), followed by SOTE (32%) and then AGRO (16%); however, our AHP-BWM results are more reliable because they are based on a larger (sub)sample (i.e. 662 or even 673 versus 644 participants).

All in all, our AHP-BWM results suggest that economic considerations play a significant role in sustainable agriculture development. Therefore, factors such as economic/financial viability and overall efficiency (which belong to the ECON dimension) are crucial for (Northern) Vietnamese farmers in determining the sustainability of agricultural practices. On the other hand, factors such as local development and circular economy or employment and quality of work (belong to the SOTE dimension) are also important to the farmers. The AGRO dimension involves factors such as restraint in the use of resources or reduced impact on human health and ecosystems, attracting the least concern from Vietnamese farmers – we argue that it can be improved alongside the development of the country's agriculture system. Overall, these weightings indicate that achieving sustainable agriculture requires a balanced approach that considers economic viability, social and territorial aspects, and agroecological principles. Policy and DMs in Vietnam, therefore, should strive for an inclusive and holistic approach that integrates economic, social and environmental dimensions to promote sustainable agricultural practices.

AHP is a popular MCDM method to evaluate the weights or preference degrees of different criteria that help DMs in their (rational) decision-making processes. When the number of criteria (within a certain alternative or project) and especially when the number of alternatives increases, AHP tends to suffer from the computation burden and consequently the consistency problem (Rezaei, 2015; Mi et al., 2019). The BWM proposed by Rezaei (2015) can overcome those issues; however, it requires different data and thus cannot be used on existing AHP data but as a new independent research. In this article, we proposed a new AHP-BWM approach to apply the BWM to AHP data. AHP-BWM is based on a simple argument that is if we can determine the best and worst criteria from AHP data, then we can use BWM on those data. The AHP-BWM therefore can (1) combine the advantages of both AHP and BWM and (2) allow researchers to reproduce BWM analysis utilising existing AHP data. In this sense, the AHP-BWM approach is practically valuable for decision-making processes.

To test for the reliability of the proposed AHP-BWM approach, we used a Monte Carlo simulation to compare results derived from AHP and AHP-BWM in terms of the consistency ratio (⁠$CR$⁠), minimum violation (⁠$MV$⁠) and total deviation (⁠$TD$⁠). This simulation involved 5,130,000 pairwise comparison matrices where the number of criteria ranges from 3 to 6, the number of alternatives ranges from 5 to 100, and the number of simulations ranges from 100 to 1,000. The results of this simulation suggested that AHP-BWM performs better and hence, its results are more reliable than AHP. For small samples, i.e. when the number of criteria equals 3 or 4 only and/or the number of alternatives is below 30, AHP can still be equally good as AHP-BWM.

For empirical illustration, the AHP-BWM was applied to surveyed data from 673 farmers in Bac Giang province in the Northern region of Vietnam regarding their practices and experiences towards sustainable agriculture development (Ngo et al., 2021). By evaluating the contributions of the three dimensions of Agroecology (AGRO), Socio-territorial (SOTE) and Economic (ECON), our AHP-BWM results confirmed the findings of Ngo et al. (2021) that ECON is the most important dimension of sustainable agriculture development in Northern Vietnam, followed by SOTE and then AGRO. However, our results are more reliable than those of Ngo et al. (2021) by having better CR, MV and TD statistics, which helped utilise more information from a larger (sub)sample, i.e. 662 participants in our AHP-BWM analysis compared to 644 participants in Ngo et al. (2021).

There are several potential ways to improve and extend this study. One important aspect is to define a proper data-generating process to derive the simulation data, whereas researcher could identify the “true” weights of the criteria and then compare the AHP and AHP-BWM results to those true values, instead of comparing AHP to AHP-BWM as in the current study. Others could examine different ways of defining the best and worst criteria from AHP data using data envelopment analysis (DEA) (Lin et al., 2011; Rashidi, 2020) or using the common set of weights DEA (Hammami et al., 2022). We are also interested in experiments involving recent developments of AHP such as fuzzy AHP (Rezaei and Ortt, 2013) or Bayesian AHP (Altuzarra et al., 2007). In terms of empirical applications, we look forward to the use of larger datasets in wider settings, e.g. manufacturing firms, financial institutions or even big data context (Ngo et al., 2019; Le et al., 2022; Zhu, 2022). We leave those tasks for future research.