Wasserstein distance-based probabilistic linguistic TODIM method with application to the evaluation of sustainable rural tourism potential

Figures & data

Figure 1. Discrete (a) and semidiscrete (b) optimal transport in one dimension.

Source: The Authors.

Figure 2. The flowchart of the Wasserstein-based PL-TODIM method.

Source: The Authors.

Figure 3. Three rural settlements $\{a_1,a_2,a_3\}$ to be evaluated.

Source: The Authors.

Table 1. The LDM provided by the first expert.

	$c_1$	$c_2$	$c_3$	$c_4$
$x_1$	$s_0$	$s_1$	$s_1$	$s_2$
$x_2$	$s_0$	$s_0$	$s_{-1}$	$s_0$
$x_3$	$s_1$	$s_0$	–	$s_1$

Source: The Authors.

Table 2. The LDM provided by the second expert.

	$c_1$	$c_2$	$c_3$	$c_4$
$x_1$	$s_1$	$s_{-1}$	$s_1$	$s_2$
$x_2$	$s_0$	–	$s_{-2}$	$s_0$
$x_3$	$s_1$	$s_0$	$s_2$	$s_1$

Source: The Authors.

Table 3. The LDM provided by the third expert.

	$c_1$	$c_2$	$c_3$	$c_4$
$x_1$	$s_1$	$s_1$	$s_1$	$s_0$
$x_2$	$s_2$	$s_{-1}$	–	$s_1$
$x_3$	$s_0$	$s_0$	$s_1$	$s_3$

Source: The Authors.

Table 4. The LDM provided by the fourth expert.

	$c_1$	$c_2$	$c_3$	$c_4$
$x_1$	$s_1$	$s_1$	$s_1$	$s_0$
$x_2$	$s_0$	$s_1$	$s_0$	$s_0$
$x_3$	$s_0$	–	$s_0$	$s_1$

Source: The Authors.

Table 5. The LDM provided by the fifth expert.

	$c_1$	$c_2$	$c_3$	$c_4$
$x_1$	$s_0$	$s_1$	$s_0$	$s_2$
$x_2$	$s_0$	$s_0$	$s_{-1}$	$s_0$
$x_3$	$s_0$	$s_1$	–	$s_1$

Source: The Authors.

Table 6. The group PLDM.

	$c_1$	$c_2$	$c_3$	$c_4$
$x_1$	$\left\{s_0(0.4),s_1(0.6)\right\}$	$\left\{s_{-1}(0.2),s_1(0.8)\right\}$	$\left\{s_0(0.2),s_1(0.8)\right\}$	$\left\{s_0(0.4),s_2(0.6)\right\}$
$x_2$	$\left\{s_2(0.2),s_0(0.8)\right\}$	$\left\{s_{-1}(0.2),s_0(0.4),s_1(0.2)\right\}$	$\left\{s_{-2}(0.2),s_{-1}(0.4),s_0(0.2)\right\}$	$\left\{s_1(0.2),s_0(0.8)\right\}$
$x_3$	$\left\{s_0(0.6),s_1(0.4)\right\}$	$\left\{s_0(0.6),s_1(0.2)\right\}$	$\left\{s_0(0.2),s_1(0.2),s_2(0.2)\right\}$	$\left\{s_1(0.8),s_3(0.2)\right\}$

Source: The Authors.

Table 7. The normalised group PLDM.

	$c_1$	$c_2$	$c_3$	$c_4$
$x_1$	$\left\{s_0(0.40),s_1(0.60)\right\}$	$\left\{s_{-1}(0.20),s_1(0.80)\right\}$	$\left\{s_0(0.20),s_1(0.80)\right\}$	$\left\{s_0(0.40),s_2(0.60)\right\}$
$x_2$	$\left\{s_0(0.80),s_2(0.20)\right\}$	$\left\{s_{-1}(0.25),s_0(0.50),s_1(0.25)\right\}$	$\left\{s_{-2}(0.25),s_{-1}(0.5),s_0(0.25)\right\}$	$\left\{s_0(0.80),s_1(0.20)\right\}$
$x_3$	$\left\{s_0(0.60),s_1(0.40)\right\}$	$\left\{s_0(0.75),s_1(0.25)\right\}$	$\left\{s_0(0.33),s_1(0.33),s_2(0.34)\right\}$	$\left\{s_1(0.80),s_3(0.20)\right\}$

Source: The Authors.

Table 8. The PIS and NIS of each attribute and the distance between them.

	$c_1$	$c_2$	$c_3$	$c_4$
PIS	$\left\{s_0(0.40),s_1(0.60)\right\}$	$\left\{s_{-1}(0.20),s_1(0.80)\right\}$	$\left\{s_0(0.33),s_1(0.33),s_2(0.34)\right\}$	$\left\{s_1(0.80),s_3(0.20)\right\}$
NIS	$\left\{s_0(0.80),s_2(0.20)\right\}$	$\left\{s_{-1}(0.25),s_0(0.50),s_1(0.25)\right\}$	$\left\{s_{-2}(0.25),s_{-1}(0.50),s_0(0.25)\right\}$	$\left\{s_0(0.80),s_1(0.20)\right\}$

Source: The Authors.

Table 9. The relative weight ${\tilde{w}}_j(j=1,2,3,4).$

	$c_1$	$c_2$	$c_3$	$c_4$
${\tilde{w}}_j=w_j/\widehat{w}$	0.2823	1	0.3814	0.7376

Source: The Authors.

Table 10. Effect of the parameter $\theta$ on the ranking results.

	Global prospect value vector $\delta =({\delta }_1,{\delta }_2,{\delta }_3)$	Ranking order of alternatives
$\theta =1$	$(1,0,0.9851)$	$a_1\succ a_3\succ a_2$
$\theta =1.25$	$(1,0,0.9838)$	$a_1\succ a_3\succ a_2$
$\theta =1.5$	$(1,0,0.9827)$	$a_1\succ a_3\succ a_2$
$\theta =1.75$	$(1,0,0.9816)$	$a_1\succ a_3\succ a_2$
$\theta =2$	$(1,0,0.9806)$	$a_1\succ a_3\succ a_2$
$\theta =2.25$	$(1,0,0.9797)$	$a_1\succ a_3\succ a_2$
$\theta =2.5$	$(1,0,0.9788)$	$a_1\succ a_3\succ a_2$

Source: The Authors.

Table 11. Comparison results of the existence distances and the PL-Wasserstein distance.

Counterexamples	Existing distances	Defects	Wasserstein distance
$L_1(p)=\left\{s_0(0.4),s_1(0.6)\right\}$ $L_2(p)=\left\{s_0(0.8),s_3(0.2)\right\}$	$d_1(L_1,L_2)=0$ (Pang et al., 2016)	Contrary to our intuition	$d_W(L_1,L_2)=0.134$
$L_3(p)=\left\{s_{-3}(0.5),s_3(0.5)\right\}$ $L_4(p)=\left\{s_{-3}(0.4),s_2(0.6)\right\}$	$d_2(L_3,L_4)=0$ (Mao et al., 2019)	Contrary to our intuition	$d_6(L_3,L_4)=0.168$
$L_1(p)=\left\{s_0(0.4),s_1(0.6)\right\}$ $L_2(p)=\left\{s_0(0.8),s_3(0.2)\right\}$	$d_3(L_1,L_2)=0$ (Wang, Wang, et al., 2019)	Contrary to our intuition	$d_W(L_1,L_2)=0.134$
$L_5(p)=\left\{s_0(0.2),s_1(0.8)\right\}$ $L_6(p)=\left\{s_0(0.5),s_1(0.5)\right\}$	$d_4(L_5,L_6)=0$ (Zhang, Xu, et al., 2016)	Contrary to our intuition	$d_W(L_5,L_6)=0.051$
$L_7(p)=\left\{s_0(0.2),s_1(0.8)\right\}$ $L_8(p)=\left\{s_0(0.2),s_1(0.8)\right\}$	$d_5(L_7,L_8)=0.05$ (Zhang, Xu, et al., 2017)	Not reasonable in some cases	$d_W(L_7,L_8)=0$
$L_9(p)=\left\{s_0(1)\right\}$ $L_{10}(p)=\left\{s_{-3}(1)\right\}$ $L_{11}(p)=\left\{s_{-1}(1)\right\}$	$d_6(L_9,L_{10})=0.5$ $d_6(L_9,L_{11})=0.6$ (Lin et al., 2019)	Not reasonable in some cases	$d_W(L_9,L_{10})=0.5$ $d_W(L_9,L_{11})=0.34$
$L_5(p)=\left\{s_0(0.2),s_1(0.8)\right\}$ $L_6(p)=\left\{s_0(0.5),s_1(0.5)\right\}$	$d_7(L_5,L_6)=0$ (Zhang et al., 2019)	Contrary to our intuition	$d_W(L_5,L_6)=0.051$

Source: The Authors.

Table 12. Ranking results of different methods.

Methods	Comprehensive values			Ranking
	$a_1$	$a_2$	$a_3$
The proposed method ($\theta =1$)	1	0	0.9851	$a_1\succ a_3\succ a_2$
PL-TODIM method ($\theta =1$)	1	0	0.7952	$a_1\succ a_3\succ a_2$
Extended PL-TOPSIS method	0	–1.8	–0.6	$a_1\succ a_3\succ a_2$
PL-GLDS method	0.3094	–0.3862	0.0585	$a_1\succ a_3\succ a_2$
Traditional HFLTS-based TOPSIS method	–12.43	–20.4	0	$a_3\succ a_1\succ a_2$

Source: The Authors.

Pang, Q., Wang, H., & Xu, Z. (2016). Probabilistic linguistic term sets in multi-attribute group decision making. Information Sciences, 369, 128–143. https://doi.org/10.1016/j.ins.2016.06.021

 Web of Science ®Google Scholar

Mao, X. B., Wu, M., Dong, J. Y., Wan, S. P., & Jin, Z. (2019). A new method for probabilistic linguistic multi-attribute group decision making: Application to the selection of financial technologies. Applied Soft Computing, 77, 155–175. https://doi.org/10.1016/j.asoc.2019.01.009

 Web of Science ®Google Scholar

Wang, X., Wang, J., & Zhang, H. (2019). Distance-based multicriteria group decision-making approach with probabilistic linguistic term sets. Expert Systems, 36(2), e12352. https://doi.org/10.1111/exsy.12352

 Web of Science ®Google Scholar

Zhang, Y., Xu, Z., Wang, H., & Liao, H. (2016). Consistency-based risk assessment with probabilistic linguistic preference relation. Applied Soft Computing, 49, 817–833.

 Web of Science ®Google Scholar

Zhang, Y., Xu, Z., & Liao, H. (2017). A consensus process for group decision making with probabilistic linguistic preference relations. Information Sciences, 414, 260–275. https://doi.org/10.1016/j.ins.2017.06.006

 Web of Science ®Google Scholar

Lin, M., Chen, Z., Liao, H., & Xu, Z. (2019). ELECTRE II method to deal with probabilistic linguistic term sets and its application to edge computing. Nonlinear Dynamics, 96(3), 2125–2143. https://doi.org/10.1007/s11071-019-04910-0

 Web of Science ®Google Scholar

Zhang, Y., Xu, Z., & Liao, H. (2019). Water security evaluation based on the TODIM method with probabilistic linguistic term sets. Soft Computing, 23(15), 6215–6230. https://doi.org/10.1007/s00500-018-3276-9

 Web of Science ®Google Scholar