**3.2 Modeling uncertainty**

A total of 16 experts from the network for Biodigesters in Latin America and the Caribbean (RedBioLAC, 2020 edition) participated in a survey to define the importance of each criterion using the MIMDU procedure. As an example, the assessments


*Data and Decision Sciences – Recent Advances and Applications*

> **Table 6.** *Criteriaand*

 *subcriteria*

 *defined and evaluation*

 *of their importance.*

#### *Perspective Chapter: A Novel Method for Integrated Multicriteria… DOI: http://dx.doi.org/10.5772/intechopen.106589*

of three experts are shown in **Table 6**, reflecting their differences according to the technical or academic background of each expert (E1 and E2 have industry technical background and E16 has academic training). When looking at the socioeconomic criteria, it is observed that experts E1 and E2 evaluated their importance with less confidence level than E16, who is either sure or very sure (S and VS) of their high importance. Oppositely, E16 has different opinions on the importance of the technical criteria (for example, he/she assigns a 5 to the digestate content of heavy metals and pathogens, and a 2 to the dry, organic matter and nutrients contents), but is in all cases indecisive (I) and unsure (U) about the evaluations. Hence, the use of MIMDU allows to capture that uncertainty and modeling the responses consequently.

Regarding the evaluations of the alternatives, the uncertainty modeling is tackled differently according to the quantitative and qualitative nature of each (sub)criterion, and they are given below.

Quantitative (sub)criteria are T1, T3, T4, E2.2, E2.3, and S1. For these (sub) criteria, a reference value of the evaluations is obtained with real data collected in situ. Such real data embraced parameters of the construction and operation of the full-scale digesters in Colombia (e.g., biogas production and quality characteristics of the digestate, including heavy metals, pathogens, organic matter, nutrients, etc.). On the other hand, the impact of the alternatives on those digestate parameters, such as reduction rates for the metal or pathogens content, and increase rates for the dry, organic matter and nutrients, are taken from the literature [11, 20, 21]. The alternatives are also sized (determining the surface and volume required to process the digestate coming from the biogas digester, and the materials needed) to obtain an initially estimated of the initial investment and maintenance cost from the amount of materials needed in each alternative. Finally, to define a TFN, a 10% deviation from the reference value is considered to account for uncertainty on the inherent data obtained. This 10% was agreed among the experts involved in the decision-making as an appropriate estimation of the deviation of measures of biogas digester's and digestate's parameters. The specific detail of the evaluation of the alternatives according to the quantitative (sub)criteria for the specific case study can be found in [19].

Qualitative (sub)criteria are T2, E1, E2.1, S2, S3, and S4. For these (sub)criteria, a similar procedure explained in Section 2 is used to evaluate the alternatives. An assessment of 0–5 is assigned to each pair alternative-criterion according to how much the alternative fulfills the criterion, and a 1 deviation is considered to account for the potential uncertainty in the human judgment due to hesitance. Similarly, specific details of the evaluation of the alternatives according to the qualitative (sub)criteria can be found in [19].

#### **3.3 Alternatives ranking and discussion**

From the experts' opinions on the criteria weights and the evaluations of the alternatives, the F-CRM is applied using Eqs. (1) and (2), and the corresponding MPMI is calculated for each alternative. **Table 7** shows the results of the crisp and the fuzzy-based ranking of all the alternatives considered for digestate post-treatment in small-scale farms located in rural. The vermifilter (A3) appears to be the best posttreatment alternative for both rankings, followed by recirculation (A4) and sand filtration (A2) alternatives. The similarity between the crisp and fuzzy rankings confirms the robustness of the results, since it means that the uncertainty included in both weights and evaluations does not modify the result achieved due to the crisp opinions


**Table 7.**

*Crisp and fuzzy rankings of the alternatives for digestate post-treatment [19].*

**Figure 5.** *FN for the distance of A1 – A1 + A5 to the ideal solution [19].*

of experts. **Figure 5** offers a representation of the FN and is in accordance with the conclusions provided. The results show that the distance of A3 to the ideal solution is clearly lower than the others alternatives, i.e., the corresponding FN is placed more to the left. These results should ease and increase the confidence of decision-makers.

The overall predominance of the vermifilter alternative (A3) relies on its capacity of generating a final product (vermicompost), which is easier to implement, manage, and transport, and at the same time, it is a high-quality biofertilizer that can increase the agriculture production and has itself market potential for being sold [22]. In consequence, it accounts for the best evaluation in some of the environmental and socioeconomic criteria, such as the sustainability of materials needed for its implementation (basically wood, E2.1) and its capacity of generating income for the beneficiary population (S2.1). Alternatively, coupling a degassing tank and a vermifilter in series (i.e., combined A1 and A3, a.k.a. A1 + A3) enhances even more the quality of the digestate, since diluted methane is highly recovered (T1.6), but represents significantly greater economic investments for implementation and day-to-day operation.

Other well-ranked alternatives are recirculating the digestate alternative (A4) and implementing a sand filter alternative (A2). A4 is very easy to implement, does not require skilled labor (T2.1, T2.2) nor surface area (T3), and reduces the amount of water that feeds the digester (E2.2). Meanwhile, A2 drastically reduces the heavy metals content (T1.1) and has a long life span (T4).

## **4. Conclusions**

MIMDU is a novel Methodology for Integrated Multicriteria Decision-Making with Uncertainty that focuses on integrating the experts' level of confidence into their responses. The method is divided into three phases, namely modeling opinions (P1), ranking alternatives (P2), and interpreting the results (P3). Compared with other multicriteria methods available in the literature (such as VIKOR and TOPSIS [1]), MIMDU offers two key features, including (1) generate better estimation of the opinions collected from experts incorporating their various levels of confidence through predefined TFN, and (2) provide complimentary information for a robust decision-making, including a crisp ranking without uncertainty consideration and a fuzzy-based ranking incorporated uncertainty considerations. These MIMDU's features enable a robust decision-making process.

To ease comprehension, MIMDU was demonstrated for a generic example case with reduced size. An example using three criteria and three alternatives was provided. Results obtained from this example showed that the proposed MIMDU procedure helps decision-makers to choose the most reliable alternative, as significant differences in the ranking "without" and "with uncertainty" can be quantified and compared. Specifically, for the example use case, the crisp ranking showed that alternative A1 is 6.58% better than alternative A3; but when the level of confidence is considered, A3 turns out to be 10.64% better; and hence A3 is selected as the best alternative as compared with A1 and A2. Also, the effect of lower or higher confidence in the response is tackled within a sensitivity analysis. Results show that increasing the confidence when evaluating an alternative can significantly improve its performance in the final ranking.

Finally, the proposed MIMDU was demonstrated for digestate posttreatment in small-scale farms with low-cost biodigesters. Both the crisp and fuzzy-based ranking results pointed out that the vermifilter alternative is the best option, followed by recirculating the digestate and the sand filter alternatives. In particular, the vermifilter is confirmed as an environmental-friendly technology that is allowed to create a high-quality product (vermicompost) to increase agricultural productivity and also generate incomes to the families due to sales. The consideration of uncertainty in both the experts' opinions and the alternatives evaluation demonstrated that MIMDU is a robust decision-making method for agriculture applications. The proposed MIMDU procedure described in this chapter can be extended to other applications.

## **Acknowledgements**

This research was possible thanks to the grant FPU18/05389 and the research project RTI2018-097962-B-I00, funded by the Spanish Ministry of Science, Innovation and Universities MCIN/AEI/10.13039/501100011033 and FEDER. The research was cofunded by the UPC Centre for Development Cooperation (CCD2021-G006).
