Fuzzy Hamacher WASPAS decision-making model for advantage prioritization of sustainable supply chain of electric ferry implementation in public transportation

Abstract

Increasing the use of marine transportation has many benefits, ranging from the diversion of human traffic into different modes to the reduction in greenhouse gas (GHG) emissions because of reduced road traffic congestion. Since road traffic congestion has become a major issue for many cities, utilization of marine transportation is a point of interest for authorities. The supply chain of spare parts, batteries, etc., must also be carefully considered since the availability of materials is an important matter for such transitions. In this study, four alternatives, which are selected considering a sustainable transition to a more developed marine transportation system, are presented, namely optimizing the performance of ferry operations, converting current ferries into hybrid ferries, converting current ferries into electric ferries, and purchasing a new electric ferry fleet. Experts are consulted to receive their assessments of these alternatives according to various criteria, including sustainability measures and supply chain, defined under 4 main criteria topics, which are technical aspect, operational aspect, environmental aspect, and cost aspect. We aim to propose a novel weighted aggregated sum product assessment (WASPAS) approach based on the fuzzy Hamacher weighted averaging (FHWAA) function and weighted geometric averaging (FHWGA) function for advantage prioritization of the sustainable supply chain of the electric ferry. A case study is used to illustrate the formulation and solution of the problem. According to the expert views, it is seen that purchasing a new electric fleet is the most advantageous alternative. With a transition to more developed and sustainable marine transportation, this study provides the most advantageous alternative to authorities.

Appendix A

Proof of Theorem 1

Equation (8) is broken into segments to show the steps for performing Eq. (16).

From Eqs. (6) and (8), we obtain:

$$\tilde{w}_{j} \cdot \tilde{\chi }_{j} = \left( {w_{j}^{(l)} \cdot \chi_{j}^{(l)} ,\chi_{j}^{(m)} \cdot \chi_{j}^{(m)} ,\chi_{j}^{(u)} \cdot \chi_{j}^{(u)} } \right) = \left( \begin{gathered} \chi_{j}^{(l)} \frac{{\left( {1 + \left( {\gamma - 1} \right)f\left( {\chi_{j}^{(l)} } \right)} \right)^{{w_{j}^{(l)} }} - \left( {1 - f\left( {\chi_{j}^{(l)} } \right)} \right)^{{w_{j}^{(l)} }} }}{{\left( {1 + \left( {\gamma - 1} \right)f\left( {\chi_{j}^{(l)} } \right)} \right)^{{w_{j}^{(l)} }} + \left( {\gamma - 1} \right)\left( {1 - f\left( {\chi_{j}^{(l)} } \right)} \right)^{{w_{j}^{(l)} }} }}, \hfill \\ \chi_{j}^{(m)} \frac{{\left( {1 + \left( {\gamma - 1} \right)f\left( {\chi_{j}^{(m)} } \right)} \right)^{{w_{j}^{(m)} }} - \left( {1 - f\left( {\chi_{j}^{(m)} } \right)} \right)^{{w_{j}^{(m)} }} }}{{\left( {1 + \left( {\gamma - 1} \right)f\left( {\chi_{j}^{(m)} } \right)} \right)^{{w_{j}^{(m)} }} + \left( {\gamma - 1} \right)\left( {1 - f\left( {\chi_{j}^{(m)} } \right)} \right)^{{w_{j}^{(m)} }} }}, \hfill \\ \chi_{j}^{(u)} \frac{{\left( {1 + \left( {\gamma - 1} \right)f\left( {\chi_{j}^{(u)} } \right)} \right)^{{w_{j}^{(u)} }} - \left( {1 - f\left( {\chi_{j}^{(u)} } \right)} \right)^{{w_{j}^{(u)} }} }}{{\left( {1 + \left( {\gamma - 1} \right)f\left( {\chi_{j}^{(u)} } \right)} \right)^{{w_{j}^{(u)} }} + \left( {\gamma - 1} \right)\left( {1 - f\left( {\chi_{j}^{(u)} } \right)} \right)^{{w_{j}^{(u)} }} }} \hfill \\ \end{gathered} \right)$$

Then, applying expression (8) obtained a fuzzy Hamacher weighted averaging function (16).

$$HQ_{i}^{\gamma } = \left( {HQ_{i}^{\gamma (l)} ,HQ_{i}^{\gamma (m)} ,HQ_{i}^{\gamma (u)} } \right) = \sum\limits_{j = 1}^{n} {\tilde{w}_{j} \cdot \tilde{\chi }_{j} } = \left( \begin{gathered} \sum\limits_{j = 1}^{n} {\chi_{j}^{(l)} } \frac{{\prod\nolimits_{j = 1}^{n} {\left( {1 + \left( {\gamma - 1} \right)f\left( {\chi_{j}^{(l)} } \right)} \right)^{{w_{j}^{(l)} }} } - \prod\nolimits_{j = 1}^{n} {\left( {1 - f\left( {\chi_{j}^{(l)} } \right)} \right)^{{w_{j}^{(l)} }} } }}{{\prod\nolimits_{j = 1}^{n} {\left( {1 + \left( {\gamma - 1} \right)f\left( {\chi_{j}^{(l)} } \right)} \right)^{{w_{j}^{(l)} }} + \left( {\gamma - 1} \right)\prod\nolimits_{j = 1}^{n} {\left( {1 - f\left( {\chi_{j}^{(l)} } \right)} \right)^{{w_{j}^{(l)} }} } } }}, \hfill \\ \sum\limits_{j = 1}^{n} {\chi_{j}^{(m)} } \frac{{\prod\nolimits_{j = 1}^{n} {\left( {1 + \left( {\gamma - 1} \right)f\left( {\chi_{j}^{(m)} } \right)} \right)^{{w_{j}^{(m)} }} } - \prod\nolimits_{j = 1}^{n} {\left( {1 - f\left( {\chi_{j}^{(m)} } \right)} \right)^{{w_{j}^{(m)} }} } }}{{\prod\nolimits_{j = 1}^{n} {\left( {1 + \left( {\gamma - 1} \right)f\left( {\chi_{j}^{(m)} } \right)} \right)^{{w_{j}^{(m)} }} + \left( {\gamma - 1} \right)\prod\nolimits_{j = 1}^{n} {\left( {1 - f\left( {\chi_{j}^{(m)} } \right)} \right)^{{w_{j}^{(m)} }} } } }}, \hfill \\ \sum\limits_{j = 1}^{n} {\chi_{j}^{(u)} } \frac{{\prod\nolimits_{j = 1}^{n} {\left( {1 + \left( {\gamma - 1} \right)f\left( {\chi_{j}^{(u)} } \right)} \right)^{{w_{j}^{(u)} }} - \prod\nolimits_{j = 1}^{n} {\left( {1 - f\left( {\chi_{j}^{(u)} } \right)} \right)^{{w_{j}^{(u)} }} } } }}{{\prod\nolimits_{j = 1}^{n} {\left( {1 + \left( {\gamma - 1} \right)f\left( {\chi_{j}^{(u)} } \right)} \right)^{{w_{j}^{(u)} }} + \left( {\gamma - 1} \right)\prod\nolimits_{j = 1}^{n} {\left( {1 - f\left( {\chi_{j}^{(u)} } \right)} \right)^{{w_{j}^{(u)} }} } } }} \hfill \\ \end{gathered} \right)$$

where $\tilde{w}_{j} = (w_{j}^{(l)} ,w_{j}^{(m)} ,w_{j}^{(u)} )$ is the fuzzy vector of the weight coefficients of the criteria, while

$$f\left( {\tilde{\chi }_{j} } \right) = \left( {\frac{{\chi_{j}^{(l)} }}{{\sum\nolimits_{j = 1}^{n} {\chi_{j}^{(l)} } }},\frac{{\chi_{j}^{(m)} }}{{\sum\nolimits_{j = 1}^{n} {\chi_{j}^{(m)} } }},\frac{{\chi_{j}^{(u)} }}{{\sum\nolimits_{j = 1}^{n} {\chi_{j}^{(u)} } }}} \right)$$

Proof of Theorem 2

Equation (9) is broken into segments to show the steps for performing Eq. (16).

From Eqs. (7) and (9), we obtain:

$$\left( {\tilde{\chi }_{j} } \right)^{{\tilde{w}_{j} }} = \left( {\left( {\chi_{j}^{(l)} } \right)^{{w_{j}^{(l)} }} ,\left( {\chi_{j}^{(m)} } \right)^{{w_{j}^{(m)} }} ,\left( {\chi_{j}^{(u)} } \right)^{{w_{j}^{(u)} }} } \right) = \left( \begin{gathered} \chi_{j}^{(l)} \frac{{\gamma f\left( {\chi_{j}^{(l)} } \right)^{{w_{j}^{(l)} }} }}{{\left( {1 + \left( {\gamma - 1} \right)\left( {1 - f\left( {\chi_{j}^{(l)} } \right)} \right)} \right)^{{w_{j}^{(l)} }} + \left( {\gamma - 1} \right)f\left( {\chi_{j}^{(l)} } \right)^{{w_{j}^{(l)} }} }}, \hfill \\ \chi_{j}^{(m)} \frac{{\gamma f\left( {\chi_{j}^{(m)} } \right)^{{w_{j}^{(m)} }} }}{{\left( {1 + \left( {\gamma - 1} \right)\left( {1 - f\left( {\chi_{j}^{(m)} } \right)} \right)} \right)^{{w_{j}^{(m)} }} + \left( {\gamma - 1} \right)f\left( {\chi_{j}^{(m)} } \right)^{{w_{j}^{(m)} }} }}, \hfill \\ \chi_{j}^{(u)} \frac{{\gamma f\left( {\chi_{j}^{(u)} } \right)^{{w_{j}^{(u)} }} }}{{\left( {1 + \left( {\gamma - 1} \right)\left( {1 - f\left( {\chi_{j}^{(u)} } \right)} \right)} \right)^{{w_{j}^{(u)} }} + \left( {\gamma - 1} \right)f\left( {\chi_{j}^{(u)} } \right)^{{w_{j}^{(u)} }} }} \hfill \\ \end{gathered} \right)$$

Then, applying expression (9) obtained fuzzy Hamacher weighted geometric averaging function

$$HP_{i}^{\gamma } = \left( {HP_{i}^{\gamma (l)} ,HP_{i}^{\gamma (m)} ,HP_{i}^{\gamma (u)} } \right) = \prod\limits_{j = 1}^{n} {\left( {\tilde{\chi }_{j} } \right)^{{\tilde{w}_{j} }} } = \left( \begin{gathered} \sum\limits_{j = 1}^{n} {\chi_{j}^{(l)} } \frac{{\lambda \prod\nolimits_{j = 1}^{n} {f\left( {\chi_{j}^{(l)} } \right)^{{w_{j}^{(l)} }} } }}{{\prod\nolimits_{j = 1}^{n} {\left( {1 + \left( {\lambda - 1} \right)\left( {1 - f\left( {\chi_{j}^{(l)} } \right)} \right)} \right)^{{w_{j}^{(l)} }} + \left( {\lambda - 1} \right)\prod\nolimits_{j = 1}^{n} {f\left( {\chi_{j}^{(l)} } \right)^{{w_{j}^{(l)} }} } } }}, \hfill \\ \sum\limits_{j = 1}^{n} {\chi_{j}^{(m)} } \frac{{\lambda \prod\nolimits_{j = 1}^{n} {f\left( {\chi_{j}^{(m)} } \right)^{{w_{j}^{(m)} }} } }}{{\prod\nolimits_{j = 1}^{n} {\left( {1 + \left( {\lambda - 1} \right)\left( {1 - f\left( {\chi_{j}^{(m)} } \right)} \right)} \right)^{{w_{j}^{(m)} }} + \left( {\lambda - 1} \right)\prod\nolimits_{j = 1}^{n} {f\left( {\chi_{j}^{(m)} } \right)^{{w_{j}^{(m)} }} } } }}, \hfill \\ \sum\limits_{j = 1}^{n} {\chi_{j}^{(u)} } \frac{{\lambda \prod\nolimits_{j = 1}^{n} {f\left( {\chi_{j}^{(u)} } \right)^{{w_{j}^{(u)} }} } }}{{\prod\nolimits_{j = 1}^{n} {\left( {1 + \left( {\lambda - 1} \right)\left( {1 - f\left( {\chi_{j}^{(u)} } \right)} \right)} \right)^{{w_{j}^{(u)} }} + \left( {\lambda - 1} \right)\prod\nolimits_{j = 1}^{n} {f\left( {\chi_{j}^{(u)} } \right)^{{w_{j}^{(u)} }} } } }} \hfill \\ \end{gathered} \right)$$

where $\tilde{w}_{j} = (w_{j}^{(l)} ,w_{j}^{(m)} ,w_{j}^{(u)} )$ is the fuzzy vector of the weight coefficients of the criteria, while:

$$f\left( {\tilde{\chi }_{j} } \right) = \left( {\frac{{\chi_{j}^{(l)} }}{{\sum\nolimits_{j = 1}^{n} {\chi_{j}^{(l)} } }},\frac{{\chi_{j}^{(m)} }}{{\sum\nolimits_{j = 1}^{n} {\chi_{j}^{(m)} } }},\frac{{\chi_{j}^{(u)} }}{{\sum\nolimits_{j = 1}^{n} {\chi_{j}^{(u)} } }}} \right).$$