Sustainability assessment of biomass-based energy supply chain using multi-objective optimization model

Abstract

In recent years, population growth and lifestyle changes have led to an increase in energy consumption worldwide. Providing energy from fossil fuels has negative consequences, such as energy supply constraints and overall greenhouse gas emissions. As the world continues to evolve, reducing dependence on fossil fuels and finding alternative energy sources becomes increasingly urgent. Renewable energy sources are the best way for all countries to reduce reliance on fossil fuels while reducing pollution. Biomass as a renewable energy source is an alternative energy source that can meet energy needs and contribute to global warming and climate change reduction. Among the many renewable energy options, biomass energy has found a wide range of application areas due to its resource diversity and easy availability from various sources all year round. The supply assurance of such energy sources is based on a sustainable and effective supply chain. Simultaneous improvement of the biomass-based supply chain's economic, environmental and social performance is a key factor for optimum network design. This study has suggested a multi-objective goal programming (MOGP) model to optimize a multi-stage biomass-based sustainable renewable energy supply chain network design. The proposed MOGP model represents decisions regarding the optimal number, locations, size of processing facilities and warehouses, and amounts of biomass and final products transported between the locations. The proposed model has been applied to a real-world case study in Istanbul. In addition, sensitivity analysis has been conducted to analyze the effects of biomass availability, processing capacity, storage capacity, electricity generation capacity, and the weight of the goals on the solutions. To realize sensitivity analysis related to the importance of goals, for the first time in the literature, this study employed a spherical fuzzy set-based analytic hierarchy method to determine the weights of goals.

Appendices

Appendix 1

Spherical fuzzy sets: preliminaries

Spherical fuzzy sets (SFs) as a generalization of Pythagorean fuzzy sets and neutrosophic sets were presented by Kutlu and Kahraman in 2018. In spherical fuzzy sets, while the squared sum of membership, nonmembership, and hesitancy parameters can be between 0 and 1, each of them can be defined between 0 and 1 independently (Büyüközkan & Güler, 2020; Kutlu Gündoğdu & Kahraman, 2019, 2020). Thus, SFS provides a larger preference domain for decision-makers through the novel concept. For instance, a decision-maker may assign his/her preference for an alternative with respect to a criterion as (0.5, 0.4, 0.6). In this case, the sum of the parameters is larger than one, whereas the squared sum is 0.77. In SFS, the decision-maker should define a hesitancy degree just like other dimensions, with membership and nonmembership degrees.

The basic definitions and notations of the linguistic variable SFS and its operations as follows:

Definition 1

In SFS, ${\widetilde{A}}_{s}$ of the universe of discourse U is defined by the following expression;

$$u_{{\widetilde{{A_{s} }}}} :U \to \left[ {0,1} \right],v_{{\tilde{A}_{s} }} :U \to \left[ {0,1} \right],\pi_{{\tilde{A}_{s} }} :U \to \left[ {0,1} \right]$$

and

$$0 \le u_{{\tilde{A}_{s} }}^{2} \left( u \right) + v_{{\tilde{A}_{s} }}^{2} \left( u \right) + \pi_{{\tilde{A}_{s} }}^{2} \left( u \right) \le 1 \quad (u \in U)$$

$$\tilde{A}_{s} = \left\{ {u,\left( {u_{{\tilde{A}_{s} }} \left( u \right),v_{{\tilde{A}_{s} }} \left( u \right),\pi_{{\tilde{A}_{s} }} \left( u \right)} \right)|u \in U} \right\}$$

For each $u$, the value ${u}_{{\widetilde{A}}_{s}}\left(u\right),{v}_{{\widetilde{A}}_{s}}\left(u\right),\mathrm{and}\; {\pi }_{{\widetilde{A}}_{s}}\left(u\right)$ are the degree of membership, nonmembership, and hesitancy of u to ${\widetilde{A}}_{s}$, respectively (Kutlu Gündoğdu & Kahraman, 2020).

Definition 2

Let ${U}_{1}$ and ${U}_{2}$ be two universes. Let ${\widetilde{A}}_{s}$ and ${\widetilde{B}}_{s}$ be two SFSs of the universe of discourse ${U}_{1}$ and ${U}_{2}$. Geometrical representation of SFS and distances between ${\widetilde{A}}_{s}$ and ${\widetilde{B}}_{s}$ is given in Fig. 12 (Yang & Chiclana, 2009).

Fig. 12

3D geometrical representation of SFs

$$D \left( {\tilde{A}_{s} , \tilde{B}_{s} } \right) = \frac{2}{\pi }\mathop \sum \limits_{i:1}^{n} \arccos \left( {1 - 0.5 \times \left[ {\left( {u_{{\tilde{A}_{s} }} - u_{{\tilde{B}_{s} }} } \right)^{2} + \left( {v_{{\tilde{A}_{s} }} - v_{{\tilde{B}_{s} }} } \right)^{2} + \left( {\pi_{{\tilde{A}_{s} }} - \pi_{{\tilde{B}_{s} }} } \right)^{2} } \right]} \right)$$

$$0 \le {D}(\tilde{A}_{s} ,\,\tilde{B}_{s} ) \le {n}$$

by utilizing $u_{{\tilde{A}}}^{2} + v_{{\tilde{A}}}^{2} + \pi_{{\tilde{A}}}^{2} = 1$, we can find the normalized distances between $\widetilde{{A_{s} }}$ and $\tilde{B}_{s}$ as follows:

$$D_{n } \left( {\tilde{A}_{s} , \tilde{B}_{s} } \right) = \frac{2}{n\pi }\mathop \sum \limits_{i:1}^{n} \arccos \left( {u_{{\tilde{A}_{s} }} \left( {u_{i} } \right) \times u_{{\tilde{B}_{s} }} \left( {u_{i} } \right) + v_{{\tilde{A}_{s} }} \left( {u_{i} } \right) \times v_{{\tilde{B}_{s} }} \left( {u_{i} } \right) + \pi_{{\tilde{A}_{s} }} \left( {u_{i} } \right) \times \pi_{{\tilde{B}_{s} }} \left( {u_{i} } \right)} \right)$$

$$0 \le D_{n} (\tilde{A}_{s} ,\tilde{B}_{s} ) \le 1$$

Definition 3

The algebraic operations are defined as follows (Kutlu Gündoğdu & Kahraman, 2019).

Addition:

$$\tilde{A}_{s} \oplus \tilde{B}_{s} = \left\{ {\sqrt {u_{{\tilde{A}_{s} }}^{2} + u_{{\tilde{B}_{s} }}^{2} - u_{{\tilde{A}_{s} }}^{2} \cdot u_{{\tilde{B}_{s} }}^{2} } ,v_{{\tilde{A}_{s} }}^{2} \cdot v_{{\tilde{B}_{s} }}^{2} ,\sqrt {\left( {\left( {1 - u_{{\tilde{B}_{s} }}^{2} } \right)\pi_{{\tilde{A}_{s} }}^{2} + \left( {1 - u_{{\tilde{A}_{s} }}^{2} } \right)\pi_{{\tilde{B}_{s} }}^{2} - \pi_{{\tilde{A}_{s} }}^{2} .\pi_{{\tilde{B}_{s} }}^{2} } \right)} } \right\}$$

Multiplication;

$$\tilde{A}_{s} \otimes \tilde{B}_{s} = \left\{ {u_{{\tilde{A}_{s} }}^{2} \cdot u_{{\tilde{B}_{s} }}^{2} ,\sqrt {v_{{\tilde{A}_{s} }}^{2} + v_{{\tilde{B}_{s} }}^{2} - v_{{\tilde{A}_{s} }}^{2} \cdot v_{{\tilde{B}_{s} }}^{2} } ,\sqrt {\left( {\left( {1 - v_{{\tilde{B}_{s} }}^{2} } \right)\pi_{{\tilde{A}_{s} }}^{2} + \left( {1 - v_{{\tilde{A}_{s} }}^{2} } \right)\pi_{{\tilde{B}_{s} }}^{2} - \pi_{{\tilde{A}_{s} }}^{2} .\pi_{{\tilde{B}_{s} }}^{2} } \right)} } \right\}$$

Multiplication by a scalar;

$$\tilde{A}_{s} \otimes x = \left\{ {\sqrt {1 - \left( {1 - u_{{\tilde{A}_{s} }}^{2} } \right)^{x} } ,v_{{\tilde{A}_{s} }}^{x} ,\sqrt {\left( {1 - u_{{\tilde{A}_{s} }}^{2} } \right)^{x} - \left( {1 - u_{{\widetilde{{A_{s} }}}}^{2} - \pi_{{\tilde{A}_{s} }}^{2} } \right)^{x} } } \right\}$$

x. Power of $\tilde{A}_{s}$:

$$\tilde{A}_{s}^{x} = \left\{ {u_{{\tilde{A}_{s} }}^{x} ,\sqrt {1 - \left( {1 - v_{{\tilde{A}_{s} }}^{2} } \right)^{x} } ,\sqrt {\left( {1 - v_{{\tilde{A}_{s} }}^{2} } \right)^{x} - \left( {1 - v_{{\tilde{A}_{s} }}^{2} - \pi_{{\tilde{A}_{s} }}^{2} } \right)^{x} } } \right\}$$

Union;

$$\tilde{A}_{s} \cup \tilde{B}_{s} = \left\{ {{\text{max}}\left( {u_{{\tilde{A}_{s} }}^{2} , u_{{\tilde{B}_{s} }}^{2} } \right),{\text{min}}\left( {v_{{\tilde{A}_{s} }}^{2} \cdot v_{{\tilde{B}_{s} }}^{2} } \right), {\text{min}}\left( {1 - \left( {\left( {\max \left( {u_{{\tilde{A}_{s} }}^{2} , u_{{\tilde{B}_{s} }}^{2} } \right)} \right)^{2} + \left( {\min \left( {v_{{\tilde{A}_{s} }}^{2} , v_{{\tilde{B}_{s} }}^{2} } \right)} \right)^{2} } \right),{\text{max}}\left( {\pi_{{\tilde{A}_{s} }}^{2} ,\pi_{{\tilde{B}_{s} }}^{2} } \right)} \right)} \right\}$$

Intersection;

$$\tilde{A}_{s} \cap \tilde{B}_{s} = \left\{ {{\text{min}}\left( {u_{{\tilde{A}_{s} }}^{2} , u_{{\tilde{B}_{s} }}^{2} } \right),{\text{max}}\left( {v_{{\tilde{A}_{s} }}^{2} \cdot v_{{\tilde{B}_{s} }}^{2} } \right), {\text{min}}\left( {1 - \left( {\left( {\min \left( {u_{{\tilde{A}_{s} }}^{2} , u_{{\tilde{B}_{s} }}^{2} } \right)} \right)^{2} + \left( {\max \left( {v_{{\tilde{A}_{s} }}^{2} , v_{{\tilde{B}_{s} }}^{2} } \right)} \right)^{2} } \right),{\text{min}}\left( {\pi_{{\tilde{A}_{s} }}^{2} ,\pi_{{\tilde{B}_{s} }}^{2} } \right)} \right)} \right\}$$

Definition 4

The basic operators in SFSs are defined as follows (Kutlu Gündoğdu & Kahraman, 2019).

$$\tilde{A}_{s} \oplus \tilde{B}_{s} = \tilde{B}_{s} \oplus \tilde{A}_{s}$$

$$\tilde{A}_{s} \otimes \tilde{B}_{s} = \tilde{B}_{s} \otimes \tilde{A}_{s}$$

$$x\left( {\tilde{A}_{s} \oplus \tilde{B}_{s} } \right) = x \cdot \tilde{A}_{s} \oplus x \cdot \tilde{B}_{s}$$

$$x_{1} \cdot \tilde{A}_{s} \oplus x_{2} \cdot \tilde{A}_{s} = \left( {x_{1} + x_{2} } \right)\tilde{A}_{s}$$

$$\left( {\tilde{A}_{s} \otimes \tilde{B}_{s} } \right)^{x} = \tilde{A}_{s}^{x} .\tilde{B}_{s}^{x}$$

$$\tilde{A}_{s}^{ - x} \otimes \tilde{A}_{s}^{ - y} = \tilde{A}_{s}^{ - x - y}$$

Definition 5

Spherical weighted arithmetic mean (SWAM) with respect to $w = (w_{1} ,w_{2} , \ldots , w_{n} );\mathop \sum \limits_{i:1}^{n} w_{i} = 1,$ is defined as follows (Kutlu Gündoğdu & Kahraman, 2019).

$${\text{SWAM}}_{w} \left( {\tilde{A}_{s1} ,\tilde{A}_{s2} , \ldots ,\tilde{A}_{sn} } \right) = w_{1} \tilde{A}_{s1} + w_{2} \tilde{A}_{s2} + \ldots + w_{n} \tilde{A}_{sn}$$

$$= \left\{ {\sqrt {1 - \mathop \prod \limits_{i:1}^{n} \left( {1 - u_{{\widetilde{{A_{si} }}}}^{2} } \right)^{{w_{i} }} } ,\mathop \prod \limits_{i:1}^{n} v_{{\widetilde{{A_{si} }}}}^{{w_{i} }} ,\sqrt {\mathop \prod \limits_{i:1}^{n} \left( {1 - u_{{\widetilde{{A_{si} }}}}^{2} } \right)^{{w_{i} }} - \mathop \prod \limits_{i:1}^{n} \left( {1 - u_{{\widetilde{{A_{si} }}}}^{2} - \pi_{{\widetilde{{A_{si} }}}}^{2} } \right)^{{w_{i} }} } } \right\}$$

Definition 6

Spherical weighted geometric mean (SWGM) with respect to $w = (w_{1} ,w_{2} , \ldots , w_{n} );\mathop \sum \limits_{i:1}^{n} w_{i} = 1$ is defined as follows [13]:

$${\text{SWGM}}_{w} \left( {\tilde{A}_{s1} ,\tilde{A}_{s2} , \ldots ,\tilde{A}_{sn} } \right) = \tilde{A}_{s1}^{{w_{1} }} + \tilde{A}_{s2}^{{w_{2} }} + \ldots + \tilde{A}_{sn}^{{w_{n} }}$$

$$= \left\{ {\mathop \prod \limits_{i:1}^{n} u_{{\widetilde{{A_{si} }}}}^{{w_{i} }} ,\sqrt {1 - \mathop \prod \limits_{i:1}^{n} \left( {1 - v_{{\widetilde{{A_{si} }}}}^{2} } \right)^{{w_{i} }} } ,\sqrt {\mathop \prod \limits_{i:1}^{n} \left( {1 - v_{{\widetilde{{A_{si} }}}}^{2} } \right)^{{w_{i} }} - \mathop \prod \limits_{i:1}^{n} \left( {1 - v_{{\widetilde{{A_{si} }}}}^{2} - \pi_{{\widetilde{{A_{si} }}}}^{2} } \right)^{{w_{i} }} } } \right\}$$

Definition 7

Score functions and accuracy function of sorting SFS are defined with [13];

Score $\left( {\tilde{A}_{s} } \right) = \left( {u_{{\widetilde{{A_{s} }}}} - \pi_{{\widetilde{{A_{s} }}}} } \right)^{2} - \left( {v_{{\widetilde{{A_{s} }}}} - \pi_{{\widetilde{{A_{s} }}}} } \right)^{2}$.

Accuracy $\left( {\tilde{A}_{s} } \right) = u_{{\tilde{A}s}}^{2} + v_{{\tilde{A}s}}^{2} + \pi_{{\tilde{A}s}}^{2}$.

Note that: $\tilde{A}_{s} < \tilde{B}_{s}$ if and only if ${\text{Score }}\left( {{ }\tilde{A}_{s} } \right){ } < {\text{Score }}\left( {\tilde{B}_{s} } \right)$ or ${\text{Score }}\left( {{ }\tilde{A}_{s} } \right) = {\text{Score }}\left( {\tilde{B}_{s} } \right)$ and ${\text{Accuracy }}\left( {{ }\tilde{A}_{s} } \right){ } < {\text{Accuracy }}\left( {\tilde{B}_{s} } \right)$.

2.1 SF-AHP steps

SF-AHP includes four steps, as given below.

• Step 1 Determine criteria weights using SF-AHP.

• Step 2 Establish the hierarchical structure of the DMM.

• Step 3 Construct a pairwise comparison matrix with spherical fuzzy judgment matrices based on the linguistic terms given in Table 2. Equations (27) and (28) are used to obtain the score indices (SI) in Table 9.

Table 9 Linguistic scale and corresponding SFs (Kutlu Gündoğdu & Kahraman, 2020)

For AMI, VHI, HI, SMI, and EI

$${\text{SI}} = \sqrt {\left| {100 \times \left( {\left( {u_{{\tilde{A}_{s} }} - \pi_{{\tilde{A}_{s} }} } \right)^{2} - \left( {v_{{\tilde{A}_{s} }} - \pi_{{\tilde{A}_{s} }} } \right)^{2} } \right)} \right|}$$

(27)

For EI; SLI; LI; VLI; and ALI;

$${\text{SI}}^{ - 1} = {\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 {\sqrt {\left| {100 \times \left( {\left( {u_{{\tilde{A}_{s} }} - \pi_{{\tilde{A}_{s} }} } \right)^{2} - \left( {v_{{\tilde{A}_{s} }} - \pi_{{\tilde{A}_{s} }} } \right)^{2} } \right)} \right|} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\sqrt {\left| {100 \times \left( {\left( {u_{{\tilde{A}_{s} }} - \pi_{{\tilde{A}_{s} }} } \right)^{2} - \left( {v_{{\tilde{A}_{s} }} - \pi_{{\tilde{A}_{s} }} } \right)^{2} } \right)} \right|} }$}}$$

(28)

Step 4. Estimate the spherical fuzzy weights of criteria using SWAM operator given in Definition (v). The weighted arithmetic mean is used to compute the spherical fuzzy weights.

Determining the weight of goals by employing SF-AHP

In this stage, SF-AHP determines the relative importance of economic, environmental, and social goal weights.

• Step 1 Establish the Hierarchical structure

  The hierarchical structure of determining the weights of goals consists of three main criteria, as depicted in Fig. 13. 

  Fig. 13

  

  Hierarchical structure for determining the weights of goals

• Step 2 Construct pairwise comparisons matrix

  The pairwise comparison matrices for the main criteria are determined by three experts using the linguistic scale in Table 9 (Kutlu Gündoğdu & Kahraman, 2020). The data were collected from three experts through a structured survey. Table 10 gives the experts’ opinions on the pairwise matrices of the main criteria.

  Table 10 Pairwise matrices for each expert’s opinion

  Aggregated fuzzy pairwise comparison matrix for the main criteria is constructed as per Table 11.

  Table 11 Aggregated evaluations of three experts on the main criteria

• Step 3 Estimate the spherical fuzzy global and local weights of criteria:

  Table 12 The spherical fuzzy weights of the criteria

  We use the SWAM operator given in Definition (v). The weighted arithmetic mean is used to compute the spherical fuzzy weights, as given in Table 12.