**Step 6:** End.

Before demonstrating the practical use of the proposed framework, Figure 1 is presented to obtain a clear view of the implementation process of the proposed framework. Initially, DMs' preference information is obtained as IVPHFEs for each object over a specific attribute. These matrices are aggregated using newly proposed IVPHFMM operator (refer Section 3.1) which extends the MM operator in the IVPHFS context. An evaluation matrix is obtained from the DMs for attribute weight calculation. Mathematical programming model (refer to Section 3.2) is proposed for calculating the weights of the attributes with the help of partially known information. Finally, a new systematic procedure (refer to Section 3.3) is developed for prioritizing objects which uses the aggregated matrix and weight vector as input for implementation.

**Figure 1.** Proposed decision framework in the IVPHFS context.

### **4. Numerical Example: Renewable Energy Source Selection from the Indian Perspective**

This section demonstrates the practicality of the proposed method by solving renewable energy source selection problem from an Indian perspective which is adapted from Reference [35]. India has a grea<sup>t</sup> appetite for energy owing to its high technological advancement and opportunities. Around 85,000 MW of energy demand is potentially satisfied with the help of biomass, small hydro, solar and wind energy. Recently, Mardani et al. [36] conducted a thorough survey of energy source selection using MADM methods and projected the key importance of MADM methods for energy selection. Luthra et al. [37] conducted a deep investigation of barriers to energy sources in India and suggested ideas for the mitigation of challenges from the Indian perspective. In a recent study performed by the Economic Times on energy demand and supply, they predicted that by 2040: (i) there will be a 30% increase in energy which crosses the conventional energy charts and forces an urgen<sup>t</sup> need for renewable energy sources; (ii) also by 2040, India will reach 49% renewable energy usage.

Motivated by the investigation, in this paper, we formulate the renewable source selection problem as an MADM problem and present a systematic procedure for the suitable selection of renewable energy sources.


Table 2 presents the preference information by di fferent DMs over renewable energy sources based on a set of attributes. IVPHFS based preference information is adopted.


By using Tables 4 and 5 the objective function is constructed from Model 1. The constraints are obtained from the DMs and the model is solved using optimization toolbox of MATLAB ®. From Model 1, we ge<sup>t</sup> the objective function as *MinZ* = 0.38 *w*1 − 0.008 *w*2 − 0.043 *w*3 + 0.112 *w*4 and the inequality constraints are given by *w*1 ≤ 0.3, *w*2 ≤ 0.3, *w*3 ≤ 0.3 and *w*4 ≤ 0.3. By solving we get, *w*1 = *w*2 = *w*3 = 0.3 and , *w*4 = 0.1.


**Table 2.** Interval-valued probabilistic hesitant fuzzy set (IVPHFS) based preference information for different DMs.

**Table 3.** Aggregation of preferences by using interval-valued probabilistic hesitant fuzzy Muirhead mean (IVPHFMM) operator.


**Table 4.** Evaluation matrix for attribute weight calculation.



**Table 5.** PIS & NIS for each attribute.

**Step 5:** Prioritize the energy sources by using the procedure given in Section 3.3. The cumulative ring sum value for each renewable energy source is given by 0.97, [0.86, 0.87], 0.93, [0.65, 0.87]; 0.93, [0.83, 0.89], 0.95, [0.87, 0.91]; 0.91,[0.7,0.93],0.9,[0.82,0.92];0.93,[0.84,0.93],0.92,[0.79,0.91]for unbiased attributes'

weights and 0.41, [0.33, 0.34], 0.34, [0.20, 0.33]; 0.37, [0.29, 0.34], 0.39, [0.31, 0.35]; 0.34, [0.23, 0.37], 0.34, [0.30, 0.37]; 0.37, [0.30, 0.36], 0.34, [0.27, 0.34] for biased attributes' weights. From Equation (12) we get, *a*1 = 1.54, *a*2 = 1.63, *a*3 = 1.52 and *a*4 = 1.6 for unbiased weight values and *a*1 = 0.23, *a*2 = 0.25, *a*3 = 0.23 and *a*4 = 0.23 for biased weight values. Thus, the ranking order is *a*2 *a*4 *a*1 *a*3 and the suitable renewable energy source fortheprocessissolarenergy*a*2.

**Step 6:** Perform sensitivity analysis on the risk appetite of each DM by varying the values within the predefined threshold value. Figures 2–4 depict 27 possible risk appetite values and the corresponding prioritized value for each renewable energy source. From the analysis, we can clearly observe that the prioritized order remains unchanged with the final order as *a*2 *a*4 *a*1 *a*3 with solar energy as a suitable renewable energy source for the process taken into consideration.

**Figure 2.** Sensitivity analysis of risk appetite of each DM ((1) 1,1,1 to (9) 1,3,3).

**Figure 3.** Sensitivity analysis of risk appetite of each DM ((10) 2,1,1 to (18) 2,3,3).

**Figure 4.** Sensitivity analysis of risk appetite of each DM ((19) 3,1,1 to (27) 3,3,3).
