Generalized linear diophantine fuzzy Choquet integral with application to the project management and risk analysis

Abstract

A linear Diophantine fuzzy number (LDFN), by incorporating reference parameters (RPs), provides freedom to the decision makers in evaluating and analyzing the objects initially by membership and non-membership grades. The RPs provide an additional evaluation to these grades. LDFNs can express uncertain cognitive information in multi-criteria decision-making (MCDM) approach. To examine the uncertain cognition in MCDM and to improve interrelationship among criterion, we introduce aggregation operators named as linear Diophantine fuzzy Choquet integral aggregation (LDFCIA) operator and generalized linear Diophantine fuzzy Choquet integral aggregation (GLDFCIA) operators. Additionally, we associate the notion of $L^p$-spaces to GLDFCIA operators and extend them towards project management and risk analysis. A pre-chart based on the fuzzy interval [0, 1] to classify the types and stages of risk factors in project management is developed. To distinguish the most exquisite and successful project as well as to reduce the risk factors emerging in the project management, an innovative algorithm based on LDFNs, GLDFCIA operators, score functions LDFNs and max–min composition is constructed. Moreover, we examine the role of parameter “p” in GLDFCIA operators to facilitate decision-makers (DMs) to classify the symmetry of optimal decision, identification of risk factors, and reduction of risk factors in the project development schemes. The study reveals that by chosen a suitable AO as per the choice of the expert, it will provide a wide range of compromise solutions for the decision-maker.

Appendix

Proof of Theorem 4:

We prove this result by using Mathematical induction. For $q=1$ the the result obtained directly from Definition 3.3. Now we check it for $q=2$. The Eq. (3) for $q=2$ can be represented as

$$\begin{aligned}{} & {} GLDFCIA({\Im ^{\partial }}({\aleph ^{\hbar }}_1),{\Im ^{\partial }}({\aleph ^{\hbar }}_2))=\left( \int |{\Im ^{\partial }}|^p d\breve{\Upsilon }\right) ^{\frac{1}{p}}\\{} & {} \quad =\Big ({\sum \limits ^{2}_{k=1}}^{\oplus }[\breve{\Upsilon }({\vartheta ^{\kappa }}_{(k)}) -\breve{\Upsilon }({\vartheta ^{\kappa }}_{(k+1)})]({\Im ^{\partial }}({\aleph ^{\hbar }}_{(k)}))^p\Big ) ^{\frac{1}{p}} \end{aligned}$$

$$\begin{aligned}&\quad =\Big ([\breve{\Upsilon }({\vartheta ^{\kappa }}_{(1)})-\breve{\Upsilon }({\vartheta ^{\kappa }}_{(2)})] ({\Im ^{\partial }}({\aleph ^{\hbar }}_{(1)}))^p\oplus [\breve{\Upsilon }({\vartheta ^{\kappa }}_{(2)})-\breve{\Upsilon }({\vartheta ^{\kappa }}_{(3)})] ({\Im ^{\partial }}({\aleph ^{\hbar }}_{(2)}))^p\Big ) ^{\frac{1}{p}}\\&\quad =\Bigg (\Bigg \langle \Big (1-\Big (1-[{^{{\Im ^{\partial }}({\aleph ^{\hbar }}_{(1)})}{\lambda ^{\gimel }}}_{\beth }]^p\Big )^ {[\breve{\Upsilon }({\vartheta ^{\kappa }}_{(1)})-\breve{\Upsilon }({\vartheta ^{\kappa }}_{(2)})]} \Big (1-[{^{{\Im ^{\partial }}({\aleph ^{\hbar }}_{(2)})}{\lambda ^{\gimel }}}_{\beth }]^p\Big )^ {[\breve{\Upsilon }({\vartheta ^{\kappa }}_{(2)})-\breve{\Upsilon }({\vartheta ^{\kappa }}_{(3)})]}\Big )^{\frac{1}{p}},\\&\qquad 1-\Big (1-\Big (1-[1-{^{{\Im ^{\partial }}({\aleph ^{\hbar }}_{(1)})}{\xi ^{\varrho }}}_{\beth }]^p\Big )^ {[\breve{\Upsilon }({\vartheta ^{\kappa }}_{(1)})-\breve{\Upsilon }({\vartheta ^{\kappa }}_{(2)})]} \Big (1-[1-{^{{\Im ^{\partial }}({\aleph ^{\hbar }}_{(2)})}{\xi ^{\varrho }}}_{\beth }]^p\Big )^ {[\breve{\Upsilon }({\vartheta ^{\kappa }}_{(2)})-\breve{\Upsilon }({\vartheta ^{\kappa }}_{(3)})]} \Big )^{\frac{1}{p}}\Bigg \rangle ,\\&\qquad \times \Bigg \langle \Big (1-\Big (1-[{^{{\Im ^{\partial }}({\aleph ^{\hbar }}_{(1)})}\Psi }_{\beth }]^p\Big )^ {[\breve{\Upsilon }({\vartheta ^{\kappa }}_{(1)})-\breve{\Upsilon }({\vartheta ^{\kappa }}_{(2)})]} \Big (1-[{^{{\Im ^{\partial }}({\aleph ^{\hbar }}_{(2)})}\Psi }_{\beth }]^p\Big )^ {[\breve{\Upsilon }({\vartheta ^{\kappa }}_{(2)})-\breve{\Upsilon }({\vartheta ^{\kappa }}_{(3)})]}\Big )^{\frac{1}{p}},\\&\qquad 1-\Big (1-\Big (1-[1-{^{{\Im ^{\partial }}({\aleph ^{\hbar }}_{(1)})}\Re }_{\beth }]^p\Big )^ {[\breve{\Upsilon }({\vartheta ^{\kappa }}_{(1)})-\breve{\Upsilon }({\vartheta ^{\kappa }}_{(2)})]} \Big (1-[1-{^{{\Im ^{\partial }}({\aleph ^{\hbar }}_{(2)})}\Re }_{\beth }]^p\Big )^ {[\breve{\Upsilon }({\vartheta ^{\kappa }}_{(2)})-\breve{\Upsilon }({\vartheta ^{\kappa }}_{(3)})]} \Big )^{\frac{1}{p}}\Bigg \rangle \Bigg ) \\&\quad =\Bigg (\Bigg \langle \Big (1-\prod \limits ^{2}_{k=1}\Big (1-[{^{{\Im ^{\partial }} ({\aleph ^{\hbar }}_{(k)})}{\lambda ^{\gimel }}}_{\beth }]^p\Big )^ {[\breve{\Upsilon }({\vartheta ^{\kappa }}_{(k)})-\breve{\Upsilon } ({\vartheta ^{\kappa }}_{(k+1)})]}\Big )^{\frac{1}{p}},\\&\qquad 1-\Big (1-\prod \limits ^{2}_{k=1}\Big (1-[1-{^{{\Im ^{\partial }} ({\aleph ^{\hbar }}_{(k)})}{\xi ^{\varrho }}}_{\beth }]^p\Big )^ {[\breve{\Upsilon }({\vartheta ^{\kappa }}_{(k)})-\breve{\Upsilon } ({\vartheta ^{\kappa }}_{(k+1)})]}\Big )^{\frac{1}{p}}\Bigg \rangle ,\\&\qquad \times \Bigg \langle \Big (1-\prod \limits ^{2}_{k=1}\Big (1-[{^{{\Im ^{\partial }} ({\aleph ^{\hbar }}_{(k)})}\Psi }_{\beth }]^p\Big )^ {[\breve{\Upsilon }({\vartheta ^{\kappa }}_{(k)})-\breve{\Upsilon }({\vartheta ^{\kappa }}_{(k+1)})]}\Big )^{\frac{1}{p}},\\&\qquad 1-\Big (1-\prod \limits ^{2}_{k=1}\Big (1-[1-{^{{\Im ^{\partial }}({\aleph ^{\hbar }}_{(k)})}\Re }_{\beth }]^p\Big )^ {[\breve{\Upsilon }({\vartheta ^{\kappa }}_{(k)})-\breve{\Upsilon } ({\vartheta ^{\kappa }}_{(k+1)})]}\Big )^{\frac{1}{p}}\Bigg \rangle \Bigg ) \end{aligned}$$

So Eq. (4) is true for $q=2$. Now we suppose that it is true for $q=z$ and prove that it holds for $q=z+1$. For $q=z$ we can write it as

$$\begin{aligned}&\quad =\Bigg (\Bigg \langle \Big (1-\prod \limits ^{z}_{k=1}\Big (1-[{^{{\Im ^{\partial }}({\aleph ^{\hbar }}_{(k)})} {\lambda ^{\gimel }}}_{\beth }]^p\Big )^ {[\breve{\Upsilon }({\vartheta ^{\kappa }}_{(k)})-\breve{\Upsilon }({\vartheta ^{\kappa }}_{(k+1)})]}\Big )^{ \frac{1}{p}},\\&\qquad 1-\Big (1-\prod \limits ^{z}_{k=1}\Big (1-[1-{^{{\Im ^{\partial }}({\aleph ^{\hbar }}_{(k)})}{\xi ^{\varrho }}}_ {\beth }]^p\Big )^ {[\breve{\Upsilon }({\vartheta ^{\kappa }}_{(k)})-\breve{\Upsilon }({\vartheta ^{\kappa }}_{(k+1)})]}\Big )^ {\frac{1}{p}}\Bigg \rangle ,\\&\qquad \times \Bigg \langle \Big (1-\prod \limits ^{z}_{k=1}\Big (1-[{^{{\Im ^{\partial }}({\aleph ^{\hbar }}_{(k)})}\Psi }_{\beth }]^p\Big )^ {[\breve{\Upsilon }({\vartheta ^{\kappa }}_{(k)})-\breve{\Upsilon }({\vartheta ^{\kappa }}_{(k+1)})]}\Big )^{\frac{1}{p}},\\&\qquad 1-\Big (1-\prod \limits ^{z}_{k=1}\Big (1-[1-{^{{\Im ^{\partial }}({\aleph ^{\hbar }}_{(k)})}\Re }_{\beth }]^p \Big )^ {[\breve{\Upsilon }({\vartheta ^{\kappa }}_{(k)})-\breve{\Upsilon }({\vartheta ^{\kappa }}_{(k+1)})]}\Big ) ^{\frac{1}{p}}\Bigg \rangle \Bigg ) \end{aligned}$$

For $q=z+1$

$$\begin{aligned}&\left( \int |{\Im ^{\partial }}|^p d\breve{\Upsilon }\right) ^{\frac{1}{p}}\\&\quad =\Bigg (\Bigg \langle \Big (1-\prod \limits ^{z}_{k=1}\Big (1-[{^{{\Im ^{\partial }}({\aleph ^{\hbar }}_{(k)})} {\lambda ^{\gimel }}}_{\beth }]^p\Big )^ {[\breve{\Upsilon }({\vartheta ^{\kappa }}_{(k)})-\breve{\Upsilon }({\vartheta ^{\kappa }}_{(k+1)})]} \Big )^{\frac{1}{p}},\\&\qquad 1-\Big (1-\prod \limits ^{z}_{k=1}\Big (1-[1-{^{{\Im ^{\partial }}({\aleph ^{\hbar }}_{(k)})}{\xi ^{\varrho }}}_{\beth }] ^p\Big )^ {[\breve{\Upsilon }({\vartheta ^{\kappa }}_{(k)})-\breve{\Upsilon }({\vartheta ^{\kappa }}_{(k+1)})]}\Big )^{\frac{1}{p}}\Bigg \rangle ,\\&\qquad \Bigg \langle \Big (1-\prod \limits ^{z}_{k=1}\Big (1-[{^{{\Im ^{\partial }}({\aleph ^{\hbar }}_{(k)})}\Psi }_{\beth }]^p\Big )^ {[\breve{\Upsilon }({\vartheta ^{\kappa }}_{(k)})-\breve{\Upsilon }({\vartheta ^{\kappa }}_{(k+1)})]}\Big )^{\frac{1}{p}},\\&\qquad 1-\Big (1-\prod \limits ^{z}_{k=1}\Big (1-[1-{^{{\Im ^{\partial }}({\aleph ^{\hbar }}_{(k)})}\Re }_{\beth }]^p\Big )^ {[\breve{\Upsilon }({\vartheta ^{\kappa }}_{(k)})-\breve{\Upsilon }({\vartheta ^{\kappa }}_{(k+1)})]}\Big ) ^{\frac{1}{p}}\Bigg \rangle \Bigg )\\&\qquad \oplus \Bigg (\Bigg \langle \Big (1-\Big (1-[{^{{\Im ^{\partial }}({\aleph ^{\hbar }}_{(z+1)})} {\lambda ^{\gimel }}}_{\beth }]^p\Big )^ {[\breve{\Upsilon }({\vartheta ^{\kappa }}_{(z+1)})-\breve{\Upsilon }({\vartheta ^{\kappa }}_{(z+2)})]}\Big ) ^{\frac{1}{p}},\\&\qquad 1-\Big (1-\Big (1-[1-{^{{\Im ^{\partial }}({\aleph ^{\hbar }}_{(z+1)})}{\xi ^{\varrho }}}_{\beth }]^p\Big )^ {[\breve{\Upsilon }({\vartheta ^{\kappa }}_{(z+1)})-\breve{\Upsilon }({\vartheta ^{\kappa }}_{(z+2)})]}\Big ) ^{\frac{1}{p}}\Bigg \rangle ,\\&\qquad \Bigg \langle \Big (1-\Big (1-[{^{{\Im ^{\partial }}({\aleph ^{\hbar }}_{(z+1)})}\Psi }_{\beth }]^p\Big )^ {[\breve{\Upsilon }({\vartheta ^{\kappa }}_{(z+1)})-\breve{\Upsilon }({\vartheta ^{\kappa }}_{(z+2)})]}\Big ) ^{\frac{1}{p}},\\&\qquad 1-\Big (1-\Big (1-[1-{^{{\Im ^{\partial }}({\aleph ^{\hbar }}_{(z+1)})}\Re }_{\beth }]^p\Big )^ {[\breve{\Upsilon }({\vartheta ^{\kappa }}_{(z+1)})-\breve{\Upsilon }({\vartheta ^{\kappa }}_{(z+2)})]}\Big ) ^{\frac{1}{p}}\Bigg \rangle \Bigg )\\&\qquad =\Bigg (\Bigg \langle \Big (1-\prod \limits ^{z+1}_{k=1}\Big (1-[{^{{\Im ^{\partial }}({\aleph ^{\hbar }}_{(k)})} {\lambda ^{\gimel }}}_{\beth }]^p\Big )^ {[\breve{\Upsilon }({\vartheta ^{\kappa }}_{(k)})-\breve{\Upsilon }({\vartheta ^{\kappa }}_{(k+1)})]}\Big )^{\frac{1}{p}},\\&\qquad 1-\Big (1-\prod \limits ^{z+1}_{k=1}\Big (1-[1-{^{{\Im ^{\partial }}({\aleph ^{\hbar }}_{(k)})}{\xi ^{\varrho }}}_{\beth }] ^p\Big )^ {[\breve{\Upsilon }({\vartheta ^{\kappa }}_{(k)})-\breve{\Upsilon }({\vartheta ^{\kappa }}_{(k+1)})]}\Big )^ {\frac{1}{p}}\Bigg \rangle ,\\&\qquad \Bigg \langle \Big (1-\prod \limits ^{z+1}_{k=1}\Big (1-[{^{{\Im ^{\partial }}({\aleph ^{\hbar }}_{(k)})}\Psi }_{\beth }] ^p\Big )^ {[\breve{\Upsilon }({\vartheta ^{\kappa }}_{(k)})-\breve{\Upsilon }({\vartheta ^{\kappa }}_{(k+1)})]}\Big )^{\frac{1}{p}},\\&\qquad 1-\Big (1-\prod \limits ^{z+1}_{k=1}\Big (1-[1-{^{{\Im ^{\partial }}({\aleph ^{\hbar }}_{(k)})}\Re }_{\beth }]^p\Big )^ {[\breve{\Upsilon }({\vartheta ^{\kappa }}_{(k)})-\breve{\Upsilon }({\vartheta ^{\kappa }}_{(k+1)})]}\Big ) ^{\frac{1}{p}}\Bigg \rangle \Bigg ) \end{aligned}$$

This implies that Eq. (4) is true for $q=z+1$. $\square$