Fuzzy-based approaches for agri-food supply chains: a mini-review

Abstract

A survey on the technologies employed in the modern agriculture and agri-food supply chains lately appeared, but only one paper using a fuzzy-based approach was cited. The aim of the present mini-review is to complement the above-mentioned survey and to show the application of different fuzzy-based approaches for agri-food supply chains. Agri-food supply chains represent linked events in the agricultural production of food, where all the stages of production, processing, trading, distribution and consumption are involved. These supply chains are expected to provide sustainable, affordable, safe and sufficient food and other derivatives to the consumers. Hence, it is critical to ensure that they operate properly and successfully in the volatile business environment. A first concern is to assess the service ability of the whole supply chain to preserve agri-food quality, eliminate deterioration and meet the demands of customers. Due to their complex structure, agri-food supply chains are susceptible to several vulnerabilities and risks, such as breakdowns, operational difficulties, and credit loss and economic losses due to various uncertain factors. A risk analysis can help to identify and categorize the risks. In this era characterized by the rapid industrialization of the agriculture and the increased global food demand, the sustainability and transparency of supply chains have become key factors. The new focus on sustainability emphasizes the issue of striking a balance between ecological and economic aspects in the agri-food business. In this context, problems such as the green supplier selection gained special attention.

Appendix: Fuzzy decision-making techniques

In order to offer a clear vision to the readers, in this section we recall the steps of the main fuzzy decision-making techniques mentioned in this survey.

1.1 Fuzzy AHP

Fuzzy analytic hierarchy process (fuzzy AHP) is a decision-making tool which has been used in several problems since it was introduced (Van Laarhoven and Pedrycz 1983). Let $C_i=\{c_1,...,c_i\}$ be the criteria set and $M=[{\tilde{M}}_{ij}]$ be the pairwise comparison matrix

$$\begin{aligned} {\tilde{M}}_{ij}= \left[ \begin{array}{ccc} {\tilde{M}}_{11} &{} \cdots &{} {\tilde{M}}_{1n}\\ \vdots &{} \ddots &{} \vdots \\ {\tilde{M}}_{n1} &{} \cdots &{} {\tilde{M}}_{nn} \\ \\ \end{array} \right] \end{aligned}$$

(1)

The method foresees the following steps

• Step 1 compute $S_k=(S_{k,l}, S_{k,m}, S_{k,u})$ values for each row as follows:

  $$\begin{aligned}&S_k=\sum _ {j=1}^n M_{kj} \times \left[ \sum _{i=1}^n \sum _ {j=1}^ n M_{ij}\right] ^{-1} \nonumber \\&\quad {for} \ \ k=1,...,n; \end{aligned}$$

  (2)

• Step 2 deduce the degree of possibility of $S_k\ge S_k'$ and $k\ne k'$ through the following equations; Let $S_1=(S_{1,l}, S_{1,m}, S_{1,u})$ and $S_2=(S_{2,l}, S_{2,m}, S_{2,u})$ then:

  $$\begin{aligned} \left\{ \begin{array}{ll} V(S_1 \ge S_2) \quad {if} \ \ S_{1,m} \ge S_{2,m} \\ V(S_1 \ge S_2)= \frac{S_{1,u}-S_{2,l}}{(S_{1,u}-s_{2,l})+(S_{2,m}-S_{1,m})} \end{array} \right. \end{aligned}$$

  (3)

• Step 3 calculate the criteria weight by

  $$\begin{aligned}&W'(c_i)=\min \{V(S_i\ge S_k)\} \ \ k=1,...,n \quad {and} \nonumber \\&\ \ k\ne i. \end{aligned}$$

  (4)

  and arranged into a vector

  $$\begin{aligned} W'=[W'(c_1),...,W'(c_n)] \end{aligned}$$

  (5)

• Step 4 compute the normalized weight

  $$\begin{aligned} W_i=\frac{W'(c_i)}{\sum _{i=1}^n W'(c_i)}. \end{aligned}$$

  (6)

1.2 Fuzzy DEMATEL

Decision-making trial and evaluation laboratory (DEMATEL) is considered as an efficient method for the identification of cause–effect of a complex system (Gabus and Fontela 1972). The categorization of the criteria helps to have a better understanding of the criteria. Moreover, it is used to assign importance weights to each of them. If the problem is consisting of n criteria, $C=\{C_1,C_2,...,C_n\}$, by the following steps we can compute importance weight of each of the criterion based on fuzzy DEMATEL.

• Step 1 The pairwise influence matrix of the criteria is as following equation. Each of the influence matrix components describes the level of influence the criterion of that row has on the values of the criterion in that column.

  $$\begin{aligned} {\widetilde{IM}}_{kh}= \left[ \begin{array}{ccc} {\widetilde{IM}}_{11} &{} \cdots &{} {\widetilde{IM}}_{1n}\\ \vdots &{} \ddots &{} \vdots \\ {\widetilde{IM}}_{n1} &{} \cdots &{} {\widetilde{IM}}_{nn} \\ \\ \end{array} \right] \end{aligned}$$

  (7)

• Step 2 Normalizing the influence matrix IM by equation and obtaining the normalized influence matrix of NM:

  $$\begin{aligned} {\widetilde{NM}}_{kh}= \left[ \begin{array}{ccc} {\widetilde{NM}}_{11} &{} \cdots &{} {\widetilde{NM}}_{1n}\\ \vdots &{} \ddots &{} \vdots \\ {\widetilde{NM}}_{n1} &{} \cdots &{} {\widetilde{NM}}_{nn} \\ \\ \end{array} \right] \end{aligned}$$

  (8)

  where, ${\widetilde{NM}}_{kh}=\frac{{\widetilde{IM}}_{kh}}{{\tilde{R}}}=(\frac{{\widetilde{IM}}_{kh,l}}{{\tilde{R}}_l}, \frac{{\widetilde{IM}}_{kh,m}}{{\tilde{R}}_m}, \frac{{\widetilde{IM}}_{kh,u}}{{\widetilde{R}}_u})$ and

  $${\tilde{R}}=(\max ({\widetilde{IM}}_{kh,l}), \max ({\widetilde{IM}}_{kh,m}), \max ({\widetilde{IM}}_{kh,u})).$$

• Step 3 Obtaining the total-relation fuzzy matrix ${\tilde{T}}$ by:

  $$\begin{aligned} {\widetilde{T}}_{kh}= & {} \lim _{w \rightarrow \infty }({\widetilde{NM}}^1_{kh}+ {\widetilde{NM}}^2_{kh}+...+{\widetilde{NM}}^w_{kh}) \nonumber \\= & {} {\widetilde{NM}}_{kh}(1-{\widetilde{NM}}_{kh})^{-1} \end{aligned}$$

  (9)

  where ${\tilde{T}}_{kh}$ is a fuzzy number

  $$\begin{aligned} {\tilde{T}}_{kh}= \left[ \begin{array}{ccc} {\tilde{T}}_{11} &{} \cdots &{} {\tilde{T}}_{1n}\\ \vdots &{} \ddots &{} \vdots \\ {\tilde{T}}_{n1} &{} \cdots &{} {\tilde{T}}_{nn}. \\ \\ \end{array} \right] \end{aligned}$$

  (10)

• Step 4 Computing the sum of rows and columns of the total relation matrix and calling them $\tilde{D_i}$ and $\tilde{R_i}$.

• Step 5 Obtaining the weights ${\tilde{w}}_i=(w_{i,l}, w_{i,m}, w_{i,u})$ through

  $$\begin{aligned} w_{i,l}= & {} \sqrt{({\tilde{D}}_{i,l}+{\tilde{R}}_{i,l})^2+({\tilde{D}}_{i,l}-{\tilde{R}}_{i,l})^2} \end{aligned}$$

  (11)
  $$\begin{aligned} w_{i,m}= & {} \sqrt{({\tilde{D}}_{i,m}+{\tilde{R}}_{i,m})^2+({\tilde{D}}_{i,m}-{\tilde{R}}_{i,m})^2} \end{aligned}$$

  (12)
  $$\begin{aligned} w_{i,u}= & {} \sqrt{({\tilde{D}}_{i,u}+{\tilde{R}}_{i,u})^2+({\tilde{D}}_{i,u}-{\tilde{R}}_{i,u})^2} \end{aligned}$$

  (13)

• Step 6 Defuzzification of fuzzy weights through equation:

  $$\begin{aligned} w_i=\frac{w_{i,l}+2w_{i,m}+w_{i,u}}{4}. \end{aligned}$$

  (14)

1.3 Fuzzy TOPSIS

The technique for order of preference by similarity to ideal solution (TOPSIS) is a multi-criteria decision method based on the concept that the chosen alternative should have the shortest distance from the positive ideal solution and the longest distance from the negative ideal solution (Hwang and Yoon 1981). The Euclidean distance measure is used to this end. Let $A=(l_1, p_1, u_1)$ and $B=(l_2, p_2, u_2)$ two triangular fuzzy numbers (TFNs). We recall that the distance between two TFNs is calculated by:

$$\begin{aligned} d(A,B)= \sqrt{\frac{1}{3}[(l_1-l_2)^2+ (p_1-p_2)^2+(u_1-u_2)^2]}. \nonumber \\ \end{aligned}$$

(15)

Fuzzy TOPSIS works as follows. Potential alternatives (suppliers in the above-mentioned supplier selection problem) and evaluation factors (criteria) are created based on expert preferences (decision makers). Let m and n be the number of alternatives and criteria through which the performance of criteria is going to be evaluated, respectively. Let ${\tilde{X}}\in {\mathbb {R}}^{m\times n}$ be the fuzzy decision matrix. It is first normalized to form the normalized decision matrix ${\tilde{R}}$ as follows:

$$\begin{aligned}&{\tilde{R}}= [{\tilde{r}}_ {ij}]_{m \times n} \end{aligned}$$

(16)

$$\begin{aligned}&{\tilde{r}}_{ij}=\left( \frac{l_{ij}}{\sqrt{\sum _i {u_{ij}}^2}}, \frac{p_{ij}}{\sqrt{\sum _i {u_{ij}}^2}}, \frac{u_{ij}}{\sqrt{\sum _i {u_{ij}}^2}}\right) \end{aligned}$$

(17)

Weighted normalized decision matrix ${\tilde{U}}$ is computed by multiplying the weights of criteria ${\tilde{W}}=[{\tilde{w}}_1,\ldots ,{\tilde{w}}_n]$ by the corresponding elements of the normalized decision matrix ${\tilde{R}}$.

$$\begin{aligned} {\tilde{U}}= [{\tilde{u}}_ {ij}]_{m\times n} \end{aligned}$$

(18)

where ${\tilde{u}}_ {ij}$ is formulated as ${\tilde{u}}_{ij }= {\tilde{r}}_ {ij} \times {\tilde{w}}_j$. The positive ideal solution (PIS) is determined by the largest normalized and weighted score for each criterion. Similarly, the negative ideal solution (NIS) is determined by selecting the least normalized and weighted score of each criterion as follows:

$$\begin{aligned} PIS= & {} \{ \tilde{u_1}^+, \tilde{u_2}^+, ..., \tilde{u_n}^+\}, \end{aligned}$$

(19)

$$\begin{aligned} NIS= & {} \{ \tilde{u_1}^-, \tilde{u_2}^-, ..., \tilde{u_n}^-\}. \end{aligned}$$

(20)

The distances of each alternative to the positive ideal solution $d^+$ and negative ideal solution $d^-$ for alternative $i = 1, 2,..., m$ are calculated according to (15) as

$$\begin{aligned} d_i^+= & {} \sum _{j=1}^n d_u(u_{ij}, u_j^+) \end{aligned}$$

(21)

$$\begin{aligned} d_i^-= & {} \sum _{j=1}^n d_u(u_{ij}, u_j^-) \end{aligned}$$

(22)

Using these distance values, closeness index CI for each alternative is:

$$\begin{aligned} CI=\frac{d_i^-}{d_i^+-d_i^-} \end{aligned}$$

(23)

The values of CI range from 0 to 1, and an alternative with the highest CI is selected as the best alternative.

1.4 Fuzzy VIKOR

VIseKriterijumska Optimizacija I Kompromisno Resenje (VIKOR) is the Serbian translation of multi-criteria optimization and compromise solution (Opricovic and Tzeng 2004). The VIKOR method first establishes a compromise ranking-list, then a compromise solution, and finally the weight stability intervals for the compromise solution. Fuzzy VIKOR works as follows. The fuzzy best value ${\tilde{f}}_j^* = (l_j^*, m_j^*,u_j^*)$ and fuzzy worst value ${\tilde{f}}_j^\circ = (l_j^\circ , m_j^\circ ,u_j^\circ )$ are determined, respectively, as

$$ {\tilde{f}}_j^*= \max _i {{\tilde{x}}_{ij}},\quad {\tilde{f}}_j^\circ = \min _i {{\tilde{x}}_{ij}} $$

The fuzzy difference ${\tilde{d}}_{ij}$ between ${\tilde{x}}_{ij}$ and fuzzy best value ${\tilde{x}}_{ij}$ and ${\tilde{f}}_j^*$ (respectively, worst value ${\tilde{f}}_j^\circ )$ is obtained by

$$\begin{aligned} {\tilde{d}}_{ij}=\frac{{\tilde{f}}_j^*-{\tilde{x}}_{ij}}{u_j^*-l_j^\circ } \end{aligned}$$

(24)

The separation ${\tilde{S}}_ i$ of supplier $A_i$ from the fuzzy best value ${\tilde{f}}_j^*$, and the separation of $R_i$ of supplier $A_i$ from the fuzzy worst value ${\tilde{f}}_j^\circ$ can be obtained from

$$\begin{aligned} {\tilde{S}}_i= & {} (S_ i^l , S_i^m, S_i^u)=\sum _{j=1}^m ({\tilde{w}}_j \otimes {\tilde{d}}_{ij}) \end{aligned}$$

(25)

$$\begin{aligned} {\tilde{R}}_i= & {} (R_ i^l , R_i^m, R_i^u)=\max _j ({\tilde{w}}_j \otimes {\tilde{d}}_{ij}), \end{aligned}$$

(26)

where ${\tilde{S}}_i$ is a fuzzy weighted sum that is the separation measure of $A_i$ from the fuzzy best value, ${\tilde{R}}_j$ is a fuzzy operator max denoting the separation measure of $A_i$ from the fuzzy worst value, and $w_i$ is the average importance weight of criterion $C_j$. The relation ${\tilde{Q}}_i = (l_i , m_i , u_ i )$ is given by

$$\begin{aligned} {\tilde{Q}}_i= & {} K\left[ \frac{({\tilde{S}}_i- \min _ i {{\tilde{S}}_i}}{\max _i{S}_i^u- \min _i S_{i}^l})\right] \nonumber \\&\oplus (1-K) \left[ \frac{({\tilde{R}}_i- \min _ i {{\tilde{R}}_i}}{\max _i{R}_i^u- \min _i R_{i}^l}) \right] , \end{aligned}$$

(27)

where $K= \frac{n+1}{2n}$ is the criteria weight and $\oplus$ is the fuzzy sum. The value of ${\tilde{S}}_i, {\tilde{R}}_i$ and ${\tilde{Q}}_i$ can be defuzzified and converted into crisp numbers $S_i, R_i$ and $Q_ i$. Consequently, alternatives are ranked in ascending order of S, R and Q. If the following two conditions are satisfied, the compromise solution $(A^{(1)})$ is proposed as the best ranked by measure Q:

• I. sustainable advantage, $Q(A^{ (2)} ) -Q(A^{ (1)} ) \ge \frac{1}{m-1}$, where $A^{(2)}$ is the alternative with second position;

• II. sustainable stability in decision making, alternative $A^{(1)}$ must also be the best ranked by S and/or R; this may imply “voting by majority rule” (when $K> 0.5$ is needed), or “by consensus $K\approx 0.5$ , or “with veto” ($K< 0.5$), where K denotes the weight of decision-making strategy of the maximum group utility.

In case that one condition is not satisfied, then a set of compromise solutions is developed, such as:

• III. alternatives $A^{(1)}$ and $A^{(2)}$, if condition II is not satisfied;

• IV. alternatives $A^{ (1) }, A^{ (2)},..., A^{ (M)}$, if condition I is not satisfied. $A^{(M)}$ is determined by the relation $Q(A^{(M)}) -Q(A^{(1)}) < \frac{1}{m-1}$ for maximum M.