Aggregation of pragmatic operators to support probabilistic linguistic multi-criteria group decision-making problems

Abstract

In this paper, a multi-attribute group decision-making method based on aggregation operators is presented to solve the decision-making problems which the evaluation values take the form of probabilistic linguistic terms sets (PLTSs). Firstly, some properties of the PLTS are defined, such as the concept and the linguistic terms transformation function, the existing comparison methods and the proposed score function and distance. Secondly, some novel operators are proposed by combining the Heronian mean operator with the centered OWA operator and the power average operator under probabilistic linguistic environment, such as the probabilistic linguistic weighted centered order weighted generalized Heronian mean operator and the probabilistic linguistic weighted power generalized Heronian mean operator. Thirdly, the model of deriving the criteria weight is put forward based on the ideology of deviation maximizing and customized individual attitudinal. Furthermore, based on the proposed aggregation operators and EDAS method, a scientific group decision-making procedure is put forward under probabilistic linguistic environment. Finally, an illustrative example is also given to demonstrate the feasibility and practicality of the proposed method.

Appendices

Appendix A

Definition A1

(Pang et al. 2016) Let $S = \{ s_{ - \tau } , \ldots ,s_{0} , \ldots ,s_{\tau } \}$ be a LTS, a PLTS can be defined as:

$$ L(p) = \left\{ {L^{(k)} \left( {p^{(k)} } \right)|L^{(k)} \in S,\;p^{\left( k \right)} \ge 0,\;k = 1,2, \ldots ,\# L(p),\sum\nolimits_{k = 1}^{\# L(p)} {p^{(k)} } \le 1} \right\} , $$

(A1)

where $L^{(k)} (p^{(k)} )$ represents the LT $L^{(k)}$ associated with the probability $p^{(k)}$, and $\# L(p)$ be the number of LTs in $L(p)$.

Definition A2

(Gou et al. 2017) Let $S = \{ s_{ - \tau } , \ldots ,s_{0} , \ldots ,s_{\tau } \}$ and $S^{{\prime }} = \{ s_{{ - \tau^{{\prime }} }} , \ldots ,s_{0} , \ldots ,s_{{\tau^{{\prime }} }} \}$ be two LTS, $L_{S} (p)$ be a PLTS based on $S$, and $r^{\left( k \right)}$ be the subscript of LT $L^{(k)}$, then the transformation function $g$ can be defined to realize the reciprocal transformation between PLTSs with multi-granular linguistic information, which is shown as follows:

$$ g:{\kern 1pt} \,[ - \tau ,\tau ] \to [0,1]\quad g\left( {L_{S} (p)} \right) = \left\{ {\left( {\frac{{r^{(k)} }}{2\tau } + \frac{1}{2}} \right)(p^{(k)} )} \right\} = \left\{ {\eta^{(k)} (p^{\left( k \right)} )|\eta^{(k)} \in [0,1]} \right\} , $$

(A2)

$$ g^{ - 1} :\,{\kern 1pt} \left[ {0,1} \right] \to \left[ { - \tau^{{\prime }} ,\tau^{{\prime }} } \right]\quad g^{ - 1}_{{S^{{\prime }} }} \left( {g\left( {L_{S} (p)} \right)} \right) = \left\{ {\left( {s_{{\left( {2\eta^{(k)} - 1} \right) \cdot \tau^{{\prime }} }} (p^{k} )} \right)|\eta^{(k)} \in \left[ {0,1} \right]} \right\}_{{S^{{\prime }} }} , $$

(A3)

where $\eta \in [0,1]$ be a crisp value transformed from linguistic information. For the sake of the simplicity in calculation and application, we often normalize the PLTSs with multi-granularity linguistic information to the uniform maximal granularity, and the subscript $r^{\left( k \right)}$ will also be rounded. Further, the transformation function $g^{ - 1}$ is improved as:

$$ g^{ - 1} :\,{\kern 1pt} \left[ {0,1} \right] \to \left[ { - t,t} \right]\quad g^{ - 1}_{{\overline{S} }} \left( {g\left( {L_{S} (p)} \right)} \right) = \left\{ {\left( {s_{{[\left( {2\eta^{(k)} - 1} \right) \cdot t]}} (p^{k} )} \right)|\eta^{(k)} \in \left[ {0,1} \right]} \right\}_{{\overline{S} }} , $$

(A4)

where $\overline{S} = \{ s_{ - t} , \ldots ,s_{0} , \ldots ,s_{t} \left| {t = \hbox{max} (\tau ,\tau^{{\prime }} )} \right.\}$, and $[ \cdot ]$ represents the integration.

Example A1

Let $S = \{ s_{ - 3} = {\text{very}}\,{\text{low}},s_{ - 2} = {\text{low}},s_{ - 1} = {\text{slightly}}\,{\text{low}},s_{0} = {\text{fair}},s_{1} = {\text{slightly}}\,{\text{high}},s_{2} = {\text{high}},s_{3} = {\text{very}}\,{\text{high}}\}$, $S^{{\prime }} = \{ s_{ - 2} = {\text{very}}\,{\text{low}},s_{ - 1} = {\text{low}},s_{0} = {\text{fair}},s_{1} = {\text{high}},s_{2} = {\text{very}}\,{\text{high}}\}$ and $L_{S} (p) = \{ s_{ - 1} (0.2),s_{0} (0.3),s_{2} (0.4)\}$, $L{\prime }_{{S^{{\prime }} }} (p) = \{ s_{0} (0.6),s_{1} (0.4)\}$. After being transformed by Definition A2, we can obtain:

$$ \begin{aligned} & g\left( {L\prime_{S'} (p)} \right) = \{ 0.5(0.6),0.75(0.4)\} , \\ & g^{ - 1}_{{\overline{S} }} \left( {g\left( {L\prime_{{S^{\prime } }} (p)} \right)} \right) = \left\{ {s_{0} (0.6),s_{2} (0.4)} \right\}_{{\overline{S} }} . \\ \end{aligned} $$

Definition A3

(Gou and Xu 2016) Let $S = \{ s_{ - \tau } , \ldots ,s_{0} , \ldots ,s_{\tau } \}$ and $S^{\prime } = \{ s_{{ - \tau^{\prime } }} , \ldots ,s_{0} , \ldots ,s_{{\tau^{\prime } }} \}$ be two different LTSs, and $\overline{S} = \{ s_{ - t} , \ldots ,s_{0} , \ldots ,s_{t} \left| {t = \hbox{max} (\tau ,\tau^{{\prime }} )} \right.\}$ be another LTS. $L_{S} (p)$ be an arbitrary PLTS based on LT $S$, and $L{\prime }_{{S^{{\prime }} }} (p)$ be another based on $S^{{\prime }}$. Suppose $\nu$ be a positive real value, and $\eta^{(i)} \in g(L_{S} )$, $\eta^{{{\prime }(j)}} \in g(L^{{\prime }}_{{S^{{\prime }} }} )$ and $i = 1,2, \ldots ,\# L_{S} (p)$, $j = 1,2, \ldots ,\# L^{{\prime }}_{{S^{{\prime }} }} (p)$. Then:

• (1) $L_{S} (p) \oplus L^{{\prime }}_{{S^{{\prime }} }} (p) = g^{ - 1}_{{\overline{S} }} \left( {\bigcup\limits_{{\eta^{(i)} \in g(L),\eta {\prime }^{(j)} \in g\left( {L^{{\prime }} } \right)}} {\left\{ {\left( {\eta^{(i)} + \eta^{{{\prime }(j)}} - \eta^{(i)} \eta^{{{\prime }(j)}} } \right)\left( {p^{(i)} p^{{{\prime }(j)}} } \right)} \right\}} } \right)$;

• (2) $L_{S} (p) \otimes L^{{\prime }}_{{S^{{\prime }} }} (p) = g^{ - 1}_{{\overline{S} }} \left( {\bigcup\limits_{{\eta^{(i)} \in g(L),\eta^{{{\prime }(j)}} \in g\left( {L^{{\prime }} } \right)}} {\left\{ {\left( {\eta^{(i)} \eta^{{{\prime }(j)}} } \right)\left( {p^{(i)} p^{{{\prime }(j)}} } \right)} \right\}} } \right)$;

• (3) $\nu L_{S} (p) = g^{ - 1}_{{\overline{S} }} \left( {\bigcup\limits_{{\eta^{(i)} \in g(L)}} {\left\{ {\left( {1 - \left( {1 - \eta^{(i)} } \right)^{\nu } } \right)\left( {p^{(i)} } \right)} \right\}} } \right)$;

• (4) $L_{S}^{\nu } (p) = g^{ - 1}_{{\overline{S} }} \left( {\bigcup\limits_{{\eta^{(i)} \in g(L)}} {\left\{ {\left( {\eta^{(i)} } \right)^{\nu } \left( {p^{(i)} } \right)} \right\}} } \right)$;

• (5) ${\text{neg}}(L_{S} (p)) = g^{ - 1}_{{\overline{S} }} \left( {\bigcup\limits_{{\eta^{(i)} \in g(L)}} {\left\{ {\left( {1 - \eta^{(i)} } \right)\left( {p^{(i)} } \right)} \right\}} } \right)$.

Example A2

By using Example A1, let $\nu = 2$, then:

• (1) $\begin{aligned} & L_{S} (p) \oplus L^{{\prime }}_{{S^{{\prime }} }} (p) = g^{ - 1}_{{\overline{S} }} \left( {\bigcup\limits_{{\eta^{(i)} \in g(L),\eta {\prime }^{(j)} \in g\left( {L^{{\prime }} } \right)}} {\left\{ {\left( {\eta^{(i)} + \eta^{{{\prime }(j)}} - \eta^{(i)} \eta^{{{\prime }(j)}} } \right)\left( {p^{(i)} p^{{{\prime }(j)}} } \right)} \right\}} } \right) \\ & \quad = g^{ - 1}_{{\overline{S} }} \left\{ {\frac{2}{3}(0.13),\frac{7}{9}(0.09),\frac{3}{4}(0.2),\frac{5}{6}(0.13),\frac{11}{12}(0.27),\frac{17}{18}(0.18)} \right\} \\ & \quad = \{ s_{1} (0.13),s_{2} (0.42),s_{3} (0.45)\}_{{\overline{S} }} ; \\ \end{aligned}$

• (2) $\begin{aligned} & L_{S} (p) \otimes L^{{\prime }}_{{S^{{\prime }} }} (p) = g^{ - 1}_{{\overline{S} }} \left( {\bigcup\limits_{{\eta^{(i)} \in g(L),\eta {\prime }^{(j)} \in g\left( {L^{{\prime }} } \right)}} {\left\{ {\left( {\eta^{(i)} \eta^{{{\prime }(j)}} } \right)\left( {p^{(i)} p^{{{\prime }(j)}} } \right)} \right\}} } \right) \\ & \quad = g^{ - 1}_{{\overline{S} }} \left\{ {\frac{1}{6}(0.13),\frac{2}{9}(0.09),\frac{1}{4}(0.2),\frac{1}{3}(0.13),\frac{5}{12}(0.27),\frac{5}{9}(0.18)} \right\} \\ & \quad = \{ s_{ - 2} (0.42),s_{ - 1} (0.4),s_{0} (0.18)\}_{{\overline{S} }} ; \\ \end{aligned}$

• (3) $\begin{aligned} & 2L_{S} (p) = g^{ - 1}_{{\overline{S} }} \left( {\bigcup\limits_{{\eta^{(i)} \in g(L)}} {\left\{ {\left( {1 - \left( {1 - \eta^{(i)} } \right)^{2} } \right)\left( {p^{(i)} } \right)} \right\}} } \right) \\ & \quad = g^{ - 1}_{{\overline{S} }} \{ 0.56(0.22),0.75(0.33),0.97(0.44)\} \\ & \quad = \{ s_{0} (0.22),s_{2} (0.33),s_{3} (0.44)\}_{{\overline{S} }} ; \\ \end{aligned}$

• (4) $\begin{aligned} & {\text{neg}}(L_{S} (p)) = g^{ - 1}_{{\overline{S} }} \left( {\bigcup\limits_{{\eta^{(i)} \in g(L)}} {\left\{ {\left( {1 - \eta^{(i)} } \right)\left( {p^{(i)} } \right)} \right\}} } \right) \\ & \quad = g^{ - 1}_{{\overline{S} }} \{ 0.67(0.22),0.5(0.33),0.17(0.44)\} \\ & \quad = \{ s_{ - 2} (0.44),s_{0} (0.33),s_{1} (0.22)\}_{{\overline{S} }} . \\ \end{aligned}$

Definition A4

(Bai et al. 2018) Let $S = \{ s_{t} \left| {t = - \tau , \ldots , - 1,0,1, \ldots ,\tau } \right.\}$ be an LTS, $L_{1} (p)$ and $L_{2} (p)$ be two PLTSs. We add any LT in $L_{i} (i = 1,2)$, so that the PLTSs have the same number of LTs. The transformed PLTSs are denoted as $L_{i}^{{\prime }} (i = 1,2)$. Let $P^{{\prime }}_{S(j,k)}$ be the probability degree of the common LTs in $L_{j}^{{\prime }}$ and $L_{k}^{{\prime }} \,(j = 1\,{\text{or}}\,{\kern 1pt} 2,k = 2\,{\text{or}}\,{\kern 1pt} 1)$, and $P^{{\prime }}_{S(j)}$ be the probability degree of all LTs in $L_{j}^{{\prime }}$ larger than the corresponding terms in $L_{k}^{{\prime }}$. The ratio

$$ p_{1} (L_{1} \ge L_{2} ) = \frac{{P^{{\prime }}_{S(1)} + 0.5P^{{\prime }}_{S(1,2)} }}{{P^{{\prime }}_{S(1)} + P^{{\prime }}_{S(2)} + 0.5(P^{{\prime }}_{S(1,2)} + P^{{\prime }}_{S(2,1)} )}} $$

(A5)

is defined as the possibility degree of $L_{1}$ being not less than $L_{2}$.

Definition A5

(Li et al. 2018a) Let $S = \{ s_{t} \left| {t = - \tau , \ldots , - 1,0,1, \ldots ,\tau } \right.\}$ be an LTS, $L_{1} (p)$ and $L_{2} (p)$ be two PLTSs, $p_{1} (L_{1} )$ and $p_{2} (L_{2} )$ be the probability distribution functions of $L_{1} (p)$ and $L_{2} (p)$, respectively, where $\sum\nolimits_{i = 1}^{{\# L_{1} (p)}} {p_{1} (L_{1}^{(i)} ) = 1}$ and $\sum\nolimits_{j = 1}^{{\# L_{2} (p)}} {p_{2} (L_{2}^{(j)} ) = 1}$. Then, the dominance degree of $L_{1} (p)$ being not less than $L_{2} (p)$ is determined as:

$$ p_{2} (L_{1} (p) \ge L_{2} (p)) = \sum\limits_{i = 1}^{{\# L_{1} (p)}} {\sum\limits_{{j = 1,{\kern 1pt} {\kern 1pt} r_{2}^{(j)} \le r_{1}^{(i)} }}^{{\# L_{2} (p)}} {p(L_{1}^{i} )p(L_{2}^{j} )} } - 0.5\sum\limits_{i = 1}^{{\# L_{1} (p)}} {\sum\limits_{{j = 1,{\kern 1pt} {\kern 1pt} r_{2}^{(j)} = r_{1}^{(i)} }}^{{\# L_{2} (p)}} {p(L_{1}^{i} )p(L_{2}^{j} )} } , $$

(A6)

where $r_{1}^{(i)}$ be the subscript of LT $L_{1}^{(i)}$, and $r_{2}^{(j)}$ be the subscript of LT $L_{2}^{(j)}$, $i = 1,2, \ldots ,\# L_{1} (p)$, $j = 1,2, \ldots ,\# L_{2} (p)$.

The comparison between $L_{1} (p)$ and $L_{2} (p)$ is as follows: if $p(L_{1} (p) \ge L_{2} (p)) > 0.5$, then $L_{1} (p)$ is superior to $L_{2} (p)$; if $p(L_{1} (p) \ge L_{2} (p)) < 0.5$, then $L_{1} (p)$ is inferior to $L_{2} (p)$; if $p(L_{1} (p) \ge L_{2} (p)) = 0.5$, then $L_{1} (p)$ is indifferent to $L_{2} (p)$.

Furthermore, the possibility degree or dominance degree formulae previously defined satisfy the following properties:

• (1) $0 \le p(L_{1} \ge L_{2} ) \le 1$

• (2) $p(L_{1} \ge L_{2} ) + p(L_{2} \ge L_{1} ) = 1$

• (3) $p(L_{1} \ge L_{2} ) = 1$, only if $\hbox{min} (r_{1}^{(i)} ) \ge \hbox{max} (r_{2}^{(j)} )\;\left( {i = 1,2, \ldots ,\# L_{1} ;j = 1,2, \ldots ,\# L_{2} } \right)$

• (4) $p(L_{1} \ge L_{2} ) = 0{\kern 1pt}$, only if $\hbox{max} (r_{1}^{(i)} ) \le \hbox{min} (r_{2}^{(j)} )\;\left( {i = 1,2, \ldots ,\# L_{1} ;j = 1,2, \ldots ,\# L_{2} } \right)$

Appendix B

Proof of Theorem 1

For convenience, let $\chi_{i} = w_{i}^{{\prime }} K\int_{{\frac{i - 1}{n}}}^{{\frac{i}{n}}} {\varphi \left( y \right){\text{d}}y}$, then Theorem 1 can be proven via the arithmetical operations of PLTSs presented in Sect. 2.

• (1) When n = 2, we have

  $$ \begin{aligned} & {\text{PLCOWA}}(L_{1} (p),L_{2} (p)) = \chi_{1} L_{1}^{{\prime }} (p) \oplus \chi_{2} L_{2}^{{\prime }} (p) \\ & \quad = g^{ - 1} \left( {\bigcup\limits_{{\eta_{1}^{{{\prime }(k)}} \in g\left( {L_{1}^{{\prime }} (p)} \right)}} {\left\{ {\left( {1 - \left( {1 - \eta_{1}^{{{\prime }(k)}} } \right)^{{\chi_{1} }} } \right)\left( {p^{{{\prime }(k)}} } \right)} \right\}} } \right) \oplus g^{ - 1} \left( {\bigcup\limits_{{\eta_{2}^{{{\prime }(t)}} \in g\left( {L_{2}^{{\prime }} (p)} \right)}} {\left\{ {\left( {1 - \left( {1 - \eta_{2}^{{{\prime }(t)}} } \right)^{{\chi_{2} }} } \right)\left( {p^{{{\prime }(t)}} } \right)} \right\}} } \right) \\ & \quad = g^{ - 1} \left( {\bigcup\limits_{ \eta_{1}^{{{\prime }(k)}} \in g\left( {L_{1}^{{\prime }} (p)} \right), \eta_{2}^{{{\prime }(t)}} \in g\left( {L_{2}^{{\prime }} (p)} \right)} {\left\{ {\left( {1 - \left( {1 - \eta_{1}^{{{\prime }(k)}} } \right)^{{\chi_{1} }} + 1 - \left( {1 - \eta_{2}^{{{\prime }(t)}} } \right)^{{\chi_{2} }} - \left( {1 - \left( {1 - \eta_{1}^{{{\prime }(k)}} } \right)^{{\chi_{1} }} } \right) \cdot \left( {1 - \left( {1 - \eta_{2}^{{{\prime }(t)}} } \right)^{{\chi_{2} }} } \right)} \right) \cdot \left( {p_{1}^{{{\prime }(k)}} p_{2}^{{{\prime }(t)}} } \right)} \right\}} } \right) \\ & \quad = g^{ - 1} \left( {\bigcup\limits_{{\eta_{1}^{{{\prime }(k)}} \in g\left( {L_{1}^{{\prime }} (p)} \right),\eta_{2}^{{{\prime }(t)}} \in g\left( {L_{2}^{{\prime }} (p)} \right)}} {\left\{ {\left( {1 - \left( {1 - \eta_{1}^{{{\prime }(k)}} } \right)^{{\chi_{1} }} \cdot \left( {1 - \eta_{2}^{{{\prime }(k)}} } \right)^{{\chi_{2} }} } \right) \cdot \left( {p_{1}^{{{\prime }(k)}} p_{2}^{{{\prime }(t)}} } \right)} \right\}} } \right). \\ \end{aligned} $$

  Therefore, when n = 2, Theorem 1 is true.

• (2) Assume that when n = k, Theorem 1 is true. Thus,

  $$ {\text{PLCOWA}}(L_{1} (p),L_{2} (p), \ldots ,L_{k} (p)) = g^{ - 1} \left( {\bigcup\limits_{{\eta_{1}^{{\prime }} \in g(L_{1}^{{\prime }} (p)),\eta_{2}^{{\prime }} \in g(L_{2}^{{\prime }} (p)), \ldots ,\eta_{k}^{{\prime }} \in g(L_{k}^{{\prime }} (p))}} {\left\{ {\left( {1 - \prod\limits_{i = 1}^{k} {(1 - \eta_{i}^{{\prime }} )^{{\chi_{i} }} } } \right) \cdot \left( {p_{1}^{{\prime }} p_{2}^{{\prime }} \ldots p_{k}^{{\prime }} } \right)} \right\}} } \right) . $$

  Then, when n = k + 1, we have

  $$ \begin{aligned} & {\text{PLCOWA}}(L_{1} (p),L_{2} (p), \ldots ,L_{k} (p),L_{k + 1} (p)) \\ & \quad = {\text{PLPWA}}(L_{1} (p),L_{2} (p), \ldots ,L_{k} (p)) \oplus \chi_{i + 1} L_{k + 1} (p) \\ {\kern 1pt} & \quad = g^{ - 1} \left( {\bigcup\limits_{{\eta_{1}^{{\prime }} \in g(L_{1}^{{\prime }} (p)),\eta_{2}^{{\prime }} \in g(L_{2}^{{\prime }} (p)), \ldots ,\eta_{k}^{{\prime }} \in g(L_{k}^{{\prime }} (p))}} {\left\{ {\left( {1 - \prod\limits_{i = 1}^{k} (1 - \eta_{i}^{{\prime }} )^{{\chi_{i} }} } \right) \cdot \left( {p_{1}^{{\prime }} p_{2}^{{\prime }} \ldots p_{k}^{{\prime }} } \right)} \right\}} } \right) \oplus \\ & \quad \quad g^{ - 1} \left( {\bigcup\limits_{{\eta_{k + 1}^{{\prime }} \in g\left( {L_{k + 1}^{{\prime }} (p)} \right)}} {\left\{ {\left( {1 - \left( {1 - \eta_{k + 1}^{{\prime }} } \right)^{{\chi_{i + 1} }} } \right)\left( {p_{k + 1}^{{\prime }} } \right)} \right\}} } \right) \\ & \quad = g^{ - 1} \left( {\bigcup\limits_{{\eta_{1}^{{\prime }} \in g(L_{1}^{{\prime }} (p)),\eta_{2}^{{\prime }} \in g(L_{2}^{{\prime }} (p)), \ldots ,\eta_{k}^{{\prime }} \in g(L_{k}^{{\prime }} (p)),\eta_{k + 1}^{{\prime }} \in g\left( {L_{k + 1}^{{\prime }} (p)} \right)}} {\left\{ {\left( {1 - \prod\limits_{i = 1}^{k} (1 - \eta_{i}^{{\prime }} )^{{\chi_{i} }} \cdot \left( {1 - \eta_{k + 1}^{{\prime }} } \right)^{{\chi_{i + 1} }} } \right) \cdot \left( {p_{1}^{{\prime }} p_{2}^{{\prime }} \ldots p_{k}^{{\prime }} p_{k + 1}^{{\prime }} } \right)} \right\}} } \right) \\ & \quad = g^{ - 1} \left( {\bigcup\limits_{{\eta_{1}^{{\prime }} \in g(L_{1}^{{\prime }} (p)),\eta_{2}^{{\prime }} \in g(L_{2}^{{\prime }} (p)), \ldots ,\eta_{k}^{{\prime }} \in g(L_{k}^{{\prime }} (p)),\eta_{k + 1}^{{\prime }} \in g\left( {L_{k + 1}^{{\prime }} (p)} \right)}} {\left\{ {\left( {1 - \prod\limits_{i = 1}^{k + 1} (1 - \eta_{i}^{{\prime }} )^{{\chi_{i} }} } \right) \cdot \left( {p_{1}^{{\prime }} p_{2}^{{\prime }} \ldots p_{k}^{{\prime }} p_{k + 1}^{{\prime }} } \right)} \right\}} } \right). \\ \end{aligned} $$

  Thus, when n = k + 1, Theorem 1 is true.

Proof of Theorem 2

For convenience, let $\varphi_{i} = \frac{{(1 + T(L_{i} (p)))w_{i} }}{{\sum\nolimits_{i = 1}^{n} {(1 + T(L_{i} (p)))w_{i} } }}$, then Theorem 2 can be proven via the arithmetical operations of PLTSs presented in Sect. 2.

• (1) When n = 2, we have

  $$ \begin{aligned} & {\text{PLPWA}}(L_{1} (p),L_{2} (p)) = \varphi_{1} L_{1} (p) \oplus \varphi_{2} L_{2} (p) \\ {\kern 1pt} & \quad = g^{ - 1} \left( {\bigcup\limits_{{\eta_{1}^{(k)} \in g\left( {L_{1} (p)} \right)}} {\left\{ {\left( {1 - \left( {1 - \eta_{1}^{\left( k \right)} } \right)^{{\varphi_{1} }} } \right)\left( {p^{\left( k \right)} } \right)} \right\}} } \right){\kern 1pt} {\kern 1pt} {\kern 1pt} \oplus g^{ - 1} \left( {\bigcup\limits_{{\eta_{2}^{\left( t \right)} \in g\left( {L_{2} (p)} \right)}} {\left\{ {\left( {1 - \left( {1 - \eta_{2}^{\left( t \right)} } \right)^{{\varphi_{2} }} } \right)\left( {p^{\left( t \right)} } \right)} \right\}} } \right){\kern 1pt} {\kern 1pt} \\ & \quad = g^{ - 1} \left( {\bigcup\limits_{ \eta_{1}^{\left( k \right)} \in g\left( {L_{1} (p)} \right), \eta_{2}^{\left( t \right)} \in g\left( {L_{2} (p)} \right) } {\left\{ {\left( {1 - \left( {1 - \eta_{1}^{\left( k \right)} } \right)^{{\varphi_{1} }} + 1 - \left( {1 - \eta_{2}^{\left( t \right)} } \right)^{{\varphi_{2} }} - \left( {1 - \left( {1 - \eta_{1}^{\left( k \right)} } \right)^{{\varphi_{1} }} } \right) \cdot \left( {1 - \left( {1 - \eta_{2}^{\left( t \right)} } \right)^{{\varphi_{2} }} } \right)} \right) \cdot \left( {p_{1}^{\left( k \right)} p_{2}^{\left( t \right)} } \right)} \right\}} } \right){\kern 1pt} \\ & \quad = g^{ - 1} \left( {\bigcup\limits_{{\eta_{1}^{\left( k \right)} \in g\left( {L_{1} (p)} \right),\eta_{2}^{\left( t \right)} \in g\left( {L_{2} (p)} \right)}} {\left\{ {\left( {1 - \left( {1 - \eta_{1}^{\left( k \right)} } \right)^{{\varphi_{1} }} \cdot \left( {1 - \eta_{2}^{\left( t \right)} } \right)^{{\varphi_{2} }} } \right) \cdot \left( {p_{1}^{\left( k \right)} p_{2}^{\left( t \right)} } \right)} \right\}} } \right). \\ \end{aligned} $$

  Therefore, when n = 2, Theorem 2 is true.

• (2) Assume that when n = k, Theorem 2 is true. Thus,

  $$ {\text{PLPWA}}(L_{1} (p),L_{2} (p), \ldots ,L_{k} (p)) = g^{ - 1} \left( {\bigcup\limits_{{\eta_{1} \in g(L_{1} (p)),\eta_{2} \in g(L_{2} (p)), \ldots ,\eta_{k} \in g(L_{k} (p))}} {\left\{ {\left( {1 - \prod\limits_{i = 1}^{k} (1 - \eta_{i} )^{{\varphi_{i} }} } \right) \cdot \left( {p_{1} p_{2} \ldots p_{k} } \right)} \right\}} } \right) . $$

  Then, when n = k + 1, we have

  $$ \begin{aligned} & {\text{PLPWA}}(L_{1} (p),L_{2} (p), \ldots ,L_{k} (p),L_{k + 1} (p)) = {\text{PLPWA}}(L_{1} (p),L_{2} (p), \ldots ,L_{k} (p)) \oplus \varphi_{i + 1} L_{k + 1} (p) \\ {\kern 1pt} & \quad = g^{ - 1} \left( {\bigcup\limits_{\eta_{1} \in g(L_{1} (p)),\eta_{2} \in g(L_{2} (p)), \ldots ,\eta_{k} \in g(L_{k} (p)) } {\left\{ {\left( {1 - \prod\limits_{i = 1}^{k} (1 - \eta_{i} )^{{\varphi_{i} }} } \right) \cdot \left( {p_{1} p_{2} \ldots p_{k} } \right)} \right\}} } \right) \oplus g^{ - 1} \left( {\bigcup\limits_{{\eta_{k + 1} \in g\left( {L_{k + 1} (p)} \right)}} {\left\{ {\left( {1 - \left( {1 - \eta_{k + 1} } \right)^{{\varphi_{i + 1} }} } \right)\left( {p_{k + 1} } \right)} \right\}} } \right) \\ & \quad = g^{ - 1} \left( {\bigcup\limits_{\eta_{1} \in g(L_{1} (p)),\eta_{2} \in g(L_{2} (p)), \ldots ,\eta_{k} \in g(L_{k} (p)),\eta_{k + 1} \in g\left( {L_{k + 1} (p)} \right) } {\left\{ {\left( {1 - \prod\limits_{i = 1}^{k} (1 - \eta_{i} )^{{\varphi_{i} }} \cdot \left( {1 - \eta_{k + 1} } \right)^{{\varphi_{i + 1} }} } \right) \cdot \left( {p_{1} p_{2} \ldots p_{k} p_{k + 1} } \right)} \right\}} } \right) \\ & \quad = g^{ - 1} \left( {\bigcup\limits_{\eta_{1} \in g(L_{1} (p)),\eta_{2} \in g(L_{2} (p)), \ldots ,\eta_{k} \in g(L_{k} (p)),\eta_{k + 1} \in g\left( {L_{k + 1} (p)} \right) } {\left\{ {\left( {1 - \prod\limits_{i = 1}^{k + 1} (1 - \eta_{i} )^{{\varphi_{i} }} } \right) \cdot \left( {p_{1} p_{2} \ldots p_{k} p_{k + 1} } \right)} \right\}} } \right). \\ \end{aligned} $$

  Thus, when n = k + 1, Theorem 2 is true.

Proof of Theorem 3

Theorem 3 can be proven based on the arithmetical operations of PLTSs presented in Sect. 2.

• (1) When n = 2, we have

  $$ \begin{aligned} & {\text{PLHWM}}^{p,q} (L_{1} (p),L_{2} (p)) = \left( {\frac{1}{3}\left( {\sum\limits_{i = 1}^{n} {\sum\limits_{j = i}^{n} {\left( {nw_{i} L_{i} (p)} \right)^{p} \left( {nw_{j} L_{j} (p)} \right)^{q} } } } \right)} \right)^{{\frac{1}{p + q}}} \\ & \quad = \left( {\frac{1}{3}\left( {\sum\limits_{i = 1}^{2} {\sum\limits_{j = i}^{2} {\left( {g^{ - 1} \left( {\bigcup\limits_{{\eta_{i} \in g(L_{i} (p))}} {\left\{ {\left( {1 - (1 - \eta_{i} )^{{2w_{i} }} } \right) \cdot \left( {p_{i} } \right)} \right\}} } \right)^{p} \otimes g^{ - 1} \left( {\bigcup\limits_{{\eta_{j} \in g(L_{j} (p))}} {\left\{ {\left( {1 - (1 - \eta_{j} )^{{2w_{j} }} } \right) \cdot \left( {p_{j} } \right)} \right\}} } \right)^{q} } \right)} } } \right)} \right)^{{\frac{1}{p + q}}} \\ & \quad = \left( {\frac{1}{3}\left( \begin{aligned} g^{ - 1} \left( {\bigcup\limits_{{\eta_{1} \in g(L_{1} (p)),\eta_{2} \in g(L_{2} (p))}} {\left\{ {\left( {\left( {1 - (1 - \eta_{1} )^{{2w_{1} }} } \right)^{p} \left( {1 - (1 - \eta_{2} )^{{2w_{2} }} } \right)^{q} } \right) \cdot \left( {p_{1} p_{2} } \right)} \right\}} } \right) \hfill \\ \oplus g^{ - 1} \left( {\bigcup\limits_{{\eta_{1} \in g(L_{1} (p))}} {\left\{ {\left( {\left( {1 - (1 - \eta_{1} )^{{2w_{1} }} } \right)^{p + q} } \right) \cdot \left( {p_{1} } \right)} \right\}} } \right) \oplus g^{ - 1} \left( {\mathop U\limits_{{\eta_{2} \in g(L_{2} (p))}} \left\{ {\left( {\left( {1 - (1 - \eta_{2} )^{{2w_{2} }} } \right)^{p + q} } \right) \cdot \left( {p_{2} } \right)} \right\}} \right) \hfill \\ \end{aligned} \right)} \right)^{{\frac{1}{p + q}}} \\ & \quad = \left( {\frac{1}{3}\left( \begin{aligned} g^{ - 1} \left( {\bigcup\limits_{{\eta_{1} \in g(L_{1} (p)),\eta_{2} \in g(L_{2} (p))}} {\left\{ {\left( {\left( {1 - (1 - \eta_{1} )^{{2w_{1} }} } \right)^{p} \left( {1 - (1 - \eta_{2} )^{{2w_{2} }} } \right)^{q} } \right) \cdot \left( {p_{1} p_{2} } \right)} \right\}} } \right) \oplus \hfill \\ g^{ - 1} \left( {\bigcup\limits_{{\eta_{1} \in g(L_{1} (p))}} {\left\{ {\left( {\left( {1 - (1 - \eta_{1} )^{{2w_{1} }} } \right)^{p + q} + \left( {1 - (1 - \eta_{2} )^{{2w_{2} }} } \right)^{p + q} - \left( {1 - (1 - \eta_{1} )^{{2w_{1} }} } \right)^{p + q} \left( {1 - (1 - \eta_{2} )^{{2w_{2} }} } \right)^{p + q} } \right) \cdot \left( {p_{1} p_{2} } \right)} \right\}} } \right) \hfill \\ \end{aligned} \right)} \right)^{{\frac{1}{p + q}}} \\ & \quad = \left( {\frac{1}{3}\left( {g^{ - 1} \left( {\bigcup\limits_{ \eta_{1} \in g(L_{1} (p)), \eta_{2} \in g(L_{2} (p))} {\left\{ {\left( \begin{aligned} \left( {1 - (1 - \eta_{1} )^{{2w_{1} }} } \right)^{p} \left( {1 - (1 - \eta_{2} )^{{2w_{2} }} } \right)^{q} + \left( {1 - (1 - \eta_{1} )^{{2w_{1} }} } \right)^{p + q} + \left( {1 - (1 - \eta_{2} )^{{2w_{2} }} } \right)^{p + q} \hfill \\ - \left( {1 - (1 - \eta_{1} )^{{2w_{1} }} } \right)^{p + q} \left( {1 - (1 - \eta_{2} )^{{2w_{2} }} } \right)^{p + q} - \left( {1 - (1 - \eta_{1} )^{{2w_{1} }} } \right)^{p} \left( {1 - (1 - \eta_{2} )^{{2w_{2} }} } \right)^{q} \hfill \\ \left( \begin{aligned} \left( {1 - (1 - \eta_{1} )^{{2w_{1} }} } \right)^{p + q} + \left( {1 - (1 - \eta_{2} )^{{2w_{2} }} } \right)^{p + q} \hfill \\ - \left( {1 - (1 - \eta_{1} )^{{2w_{1} }} } \right)^{p + q} \left( {1 - (1 - \eta_{2} )^{{2w_{2} }} } \right)^{p + q} \hfill \\ \end{aligned} \right) \hfill \\ \end{aligned} \right) \cdot \left( {p_{1} p_{2} } \right)} \right\}} } \right)} \right)} \right)^{{\frac{1}{p + q}}} \\ & \quad = g^{ - 1} \left( {\bigcup\limits_{ \eta_{1} \in g(L_{1} (p)), \eta_{2} \in g(L_{2} (p)) } {\left\{ {\left( {1 - \left( {\prod\limits_{i = 1}^{2} \prod\limits_{j = i}^{2} \left( {1 - \left( {1 - (1 - \eta_{i} )^{{2w_{i} }} } \right)^{p} \left( {1 - (1 - \eta_{j} )^{{2w_{j} }} } \right)^{q} } \right)} \right)^{{\frac{1}{3}}} } \right)^{{\frac{1}{p + q}}} \cdot \left( {p_{1} p_{2} } \right)} \right\}} } \right). \\ \end{aligned} $$

  Therefore, when n = 2, Theorem 3 is true.

• (2) Assume that when n = k, Theorem 3 is true. Thus,

  $$ \begin{aligned} & {\text{PLHWM}}^{p,q} (L_{1} (p),L_{2} (p), \ldots ,L_{k} (p)) \\ & \quad = g^{ - 1} \left( {\bigcup\limits_{\eta_{1} \in g(L_{1} (p)),\eta_{2} \in g(L_{2} (p)), \ldots ,\eta_{k} \in g(L_{k} (p))} {\left\{ {\left( {1 - \left( {\prod\limits_{i = 1}^{k} \prod\limits_{j = i}^{k} \left( {1 - \left( {1 - (1 - \eta_{i} )^{{nw_{i} }} } \right)^{p} \left( {1 - (1 - \eta_{j} )^{{nw_{j} }} } \right)^{q} } \right)} \right)^{{\frac{2}{k(k + 1)}}} } \right)^{{\frac{1}{p + q}}} \cdot \left( {p_{1} p_{2} \ldots p_{k} } \right)} \right\}} } \right) \\ \end{aligned} . $$

  Then, when n = k + 1, we have

  $$ \begin{aligned} & {\text{PLHWM}}^{p,q} (L_{1} (p),L_{2} (p), \ldots ,L_{k} (p),L_{k + 1} (p)) \\ & \quad = \left( {\frac{2}{(k + 1)(k + 2)}\left( {\sum\limits_{i = 1}^{k} {\sum\limits_{j = i}^{k} {\left( {nw_{i} L_{i} (p)} \right)^{p} \left( {nw_{j} L_{j} (p)} \right)^{q} } } \oplus \sum\limits_{t = 1}^{k + 1} {\left( {nw_{t} L_{t} (p)} \right)^{p} \left( {nw_{k + 1} L_{k + 1} (p)} \right)^{q} } } \right)} \right)^{{\frac{1}{p + q}}} \\ & \quad = \left( {\frac{2}{(k + 1)(k + 2)}\left( \begin{aligned} g^{ - 1} \left( {\bigcup\limits_{\eta_{1} \in g(L_{1} (p)),\eta_{2} \in g(L_{2} (p)), \ldots ,\eta_{k} \in g(L_{k} (p))} {\left\{ {\left( {1 - \prod\limits_{i = 1}^{k} \prod\limits_{j = i}^{k} \left( {1 - \left( {1 - (1 - \eta_{i} )^{{nw_{i} }} } \right)^{p} \left( {1 - (1 - \eta_{j} )^{{nw_{j} }} } \right)^{q} } \right)} \right) \cdot \left( {p_{1} p_{2} \ldots p_{k} } \right)} \right\}} } \right) \hfill \\ \oplus g^{ - 1} \left( {\bigcup\limits_{ \eta_{t} \in g(L_{t} (p)), \eta_{k + 1} \in g(L_{k + 1} (p)) } {\left\{ {\left( {1 - \prod\limits_{t = 1}^{k + 1} \left( {1 - \left( {1 - (1 - \eta_{t} )^{{nw_{t} }} } \right)^{p} \left( {1 - (1 - \eta_{k + 1} )^{{nw_{k + 1} }} } \right)^{q} } \right)} \right) \cdot \left( {p_{1} p_{2} \ldots p_{k} p_{k + 1} } \right)} \right\}} } \right) \hfill \\ \end{aligned} \right)} \right)^{{\frac{1}{p + q}}} \\ & \quad = \left( {\frac{2}{(k + 1)(k + 2)}g^{ - 1} \left( {\bigcup\limits_{ \eta_{1} \in g(L_{1} (p)),\eta_{2} \in g(L_{2} (p)), \ldots , \eta_{k} \in g(L_{k} (p)),\eta_{k + 1} \in g\left( {L_{k + 1} (p)} \right)} {\left\{ {\left( {1 - \prod\limits_{i = 1}^{k + 1} \prod\limits_{j = i}^{k + 1} \left( {1 - \left( {1 - (1 - \eta_{i} )^{{nw_{i} }} } \right)^{p} \left( {1 - (1 - \eta_{j} )^{{nw_{j} }} } \right)^{q} } \right)} \right) \cdot \left( {p_{1} p_{2} \ldots p_{k + 1} } \right)} \right\}} } \right)} \right)^{{\frac{1}{p + q}}} \\ & \quad = g^{ - 1} \left( {\bigcup\limits_{ \eta_{1} \in g(L_{1} (p)),\eta_{2} \in g(L_{2} (p)), \ldots , \eta_{k} \in g(L_{k} (p)),\eta_{k + 1} \in g\left( {L_{k + 1} (p)} \right) } {\left\{ {\left( {1 - \left( {1 - \left( {1 - \prod\limits_{i = 1}^{k + 1} \prod\limits_{j = i}^{k + 1} \left( {1 - \left( {1 - (1 - \eta_{i} )^{{nw_{i} }} } \right)^{p} \left( {1 - (1 - \eta_{j} )^{{nw_{j} }} } \right)^{q} } \right)} \right)} \right)^{{\frac{2}{(k + 1)(k + 2)}}} } \right)^{{\frac{1}{p + q}}} \cdot \left( {p_{1} p_{2} \ldots p_{k + 1} } \right)} \right\}} } \right) \\ & \quad = g^{ - 1} \left( {\bigcup\limits_{ \eta_{1} \in g(L_{1} (p)),\eta_{2} \in g(L_{2} (p)), \ldots , \eta_{k} \in g(L_{k} (p)),\eta_{k + 1} \in g\left( {L_{k + 1} (p)} \right)} {\left\{ {\left( {1 - \left( {\prod\limits_{i = 1}^{k + 1} \prod\limits_{j = i}^{k + 1} \left( {1 - \left( {1 - (1 - \eta_{i} )^{{nw_{i} }} } \right)^{p} \left( {1 - (1 - \eta_{j} )^{{nw_{j} }} } \right)^{q} } \right)} \right)^{{\frac{2}{(k + 1)(k + 2)}}} } \right)^{{\frac{1}{p + q}}} \cdot \left( {p_{1} p_{2} \ldots p_{k + 1} } \right)} \right\}} } \right) \\ \end{aligned} . $$

  Thus, when n = k + 1, Theorem 3 is true.