Uncertain supply chain network design considering carbon footprint and social factors using two-stage approach

Abstract

Sustainable development has become one of the leading global issues over the period of time. Currently, implementation of sustainability in supply chain has been continuously in center of attention due to introducing stringent legislations regarding environmental pollution by various governments and increasing stakeholders’ concerns toward social injustice. Unfortunately, literature is still scarce on studies considering all three dimensions (economical, environmental and social) of sustainability for the supply chain. An effective supply chain network design (SCND) is very important to implement sustainability in supply chain. This study proposes an uncertain SCND model that minimizes the total supply chain-oriented cost and determines the opening of plants, warehouses and flow of materials across the supply chain network by considering various carbon emissions and social factors. In this study, a new AHP and fuzzy TOPSIS-based methodology is proposed to transform qualitative social factors into quantitative social index, which is subsequently used in chance-constrained SCND model with an aim at reducing negative social impact. Further, the carbon emission of supply chain is estimated by considering a composite emission that consists of raw material, production, transportation and handling emissions. In the model, a carbon emission cap is imposed on total supply chain to reduce the carbon footprint of supply chain. To solve the proposed model, a code is developed in AMPL software using a nonlinear solver SNOPT. The applicability of the proposed model is illustrated with a numerical example. The sensitivity analysis examines the effects of reducing carbon footprint cap, negative social impacts and varying probability on the total cost of the supply chain. It is observed that a stricter carbon cap over supply chain network leads to opening of more plants across the supply chain. In addition, carbon footprint of supply chain is found to be decreased in certain extent with the reduction in negative social impacts from suppliers. The carbon footprint of the supply chain is found to be reduced with increasing certainty of material supply from the suppliers. The total supply chain cost is observed to be augmented with increasing probability.

Appendices

Appendix 1: Description of AHP

Suppose ${C = \{ C_{j} } |j = 1,2, \ldots ,n\}$ is the set of the criteria. Now the pairwise comparisons among the $n$ criteria are symbolized through a $(n \times n)$ matrix $A$. In matrix $A$, the elements $a_{ij} (i,j = 1,2, \ldots ,n)$ depict the weights assigned to various criteria (Albayrak and Erensal 2004; Dagdeviren et al. 2009).

$$A = \left[ {\begin{array}{*{20}c} {a_{11} } & {a_{12} } & \ldots & {a_{1n} } \\ {a_{21} } & {a_{22} } & \ldots & {a_{2n} } \\ \vdots & \vdots & \vdots & \vdots \\ {a_{n1} } & {a_{n2} } & \ldots & {a_{nn} } \\ \end{array} } \right],\quad a_{ii} = 1;\quad a_{ij} = \frac{1}{{a_{ji} }};\quad a_{ij} \ne 0$$

In AHP, pairwise comparisons are conducted in several phases (Albayrak and Erensal 2004). The number of pairwise comparisons relies on the structure of problem. For example, comparisons are made among the criteria with respect to the goal; similarly, comparisons are made among the alternatives with respect to individual criterion variable. The number of pairwise comparisons depends on structure of the problem. Subsequently, mathematical calculations are carried out to normalize and asses the relative weights of different factors for individual matrix. The relative weights are often calculated by the following formula $A_{w} = \lambda_{\hbox{max} } W$ where $(W)$ represents the right eigenvector corresponding to the largest eigenvalue $(\lambda_{\hbox{max} } )$. If pairwise assessments are found entirely consistent, then the matrix $A$ is having a rank of 1 and $\lambda_{\hbox{max} } = n$. Further, weights of the factors are computed through normalizing any of the rows or columns of $A$(Wang and Yang 2007). Validity of AHP entirely depends on the consistencies of pairwise comparisons. The consistency is based on the interaction among the entries of matrix $A:a_{ij} \times a_{jk} = a_{ik}$ (Dagdeviren et al. 2009). The consistency index (CI) is symbolized as follows: $CI = (\lambda_{\hbox{max} } - n)/(n - 1)$.

The next step in AHP is to calculate the consistency ratio (CR). It is calculated by taking ratio of CI and random index (RI). The acceptable number for CR is 0.1 or 10 percent. If the value of final consistency ratio goes beyond 0.1, then pairwise comparisons are need to be reiterated to bring the CR value below or equal to 0.1 (Wang and Yang 2007).

Appendix 2: Description of TOPSIS

Step 1 The first step of TOPSIS is to create a decision matrix for determining the ranking. The structure of the matrix is represented as follows:

$$\begin{aligned} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,F_{1} \,\,\,\,\,\,\,F_{2} \,\,\,\,\, \cdots \,\,\,\,\,\,\,F_{j} \,\,\,\,\,\, \cdots \,\,\,\,\,\,\,F_{n} \hfill \\ D = \begin{array}{*{20}c} {A_{1} } \\ {A_{2} } \\ \vdots \\ {A_{i} } \\ \vdots \\ {A_{J} } \\ \end{array} \left[ {\begin{array}{*{20}c} {f_{11} } & {f_{12} } & \cdots & {f_{1j} } & \cdots & {f_{1n} } \\ {f_{21} } & {f_{22} } & \cdots & {f_{2j} } & \cdots & {f_{2n} } \\ \vdots & \vdots & \cdots & \vdots & \cdots & \vdots \\ {f_{i1} } & {f_{i2} } & \cdots & {f_{ij} } & \cdots & {f_{{_{in} }} } \\ \vdots & \vdots & \cdots & \vdots & \cdots & \vdots \\ {f_{j1} } & {f_{j2} } & \cdots & {f_{jj} } & \cdots & {f_{jn} } \\ \end{array} } \right] \hfill \\ \end{aligned}$$

On the above matrix, $A_{j}$ connotes the available alternatives $j,\,\,j = 1,2, \ldots ,J;$ $F_{i}$ symbolizes the $i^{th}$ element or the criterion, pertaining to the $i^{th}$ alternative; and $f_{ij}$ is a crisp value representing the performance rating of individual alternative $A_{i}$ with respect to each criterion $F_{j}$ (Dagdeviren et al. 2009).

Step 2 The next step of TOPSIS is to estimate the normalized matrix $R( = [r_{ij} ]).$ The normalized values $r_{ij}$ are determined from the given equation.

$$r_{ij} = \frac{{f_{ij} }}{{\sqrt {\sum\nolimits_{j = 1}^{n} {f_{ij}^{2} } } }} = 1,2, \ldots ,J;\quad i = 1,2, \ldots ,n.$$

Step 3 Subsequently, normalized values are multiplied with its associated weights to estimate the weighted normalized values. The weighted normalized values ($v_{ij}$) are represented as follows: $v_{ij} = w_{i} \times r_{ij} ,j = 1,2, \ldots ,J;i = 1,2, \ldots ,n.$ In this equation, $w_{i}$ connotes the weight of the $i^{th}$ attribute or criterion.

Step 4 Next step of TOPSIS comprises of estimating the positive ideal and negative ideal solutions.

$$\begin{aligned} A^{*} = \{ v_{1}^{*} ,v_{2}^{*} , \ldots ,v_{i}^{*} \} \hfill \\ \,\,\,\,\,\,\, = \{ (\left. {\mathop {\hbox{max} }\limits_{j} \,v_{ij} } \right|i \in I^{'} ),(\left. {\mathop {\hbox{min} \,v_{ij} }\limits_{j} } \right|i \in I^{''} )\} \hfill \\ \end{aligned}$$

$$\begin{aligned} A^{ - } = \{ v_{1}^{ - } ,v_{2}^{ - } , \ldots ,v_{i}^{ - } \} \hfill \\ \,\,\,\,\,\,\, = \{ (\left. {\mathop {\hbox{min} }\limits_{j} \,v_{ij} } \right|i \in I^{'} ),(\left. {\mathop {\hbox{max} \,v_{ij} }\limits_{j} } \right|i \in I^{''} )\} \hfill \\ \end{aligned}$$

In the above formulation, $I^{'}$ denotes the criteria pertaining to benefits and $I^{''}$ represents the criteria related to costs.

Step 5 Next step is to calculate separation by using $n$ dimensional Euclidean distance. The distance between each alternative and positive ideal solution ($D_{j}^{*}$) is measured as follows:

$$D_{j}^{*} = \sqrt {\sum\limits_{i = 1}^{n} {(v_{ij} - v_{i}^{*} )^{2} } } \quad j = 1,2, \ldots ,J$$

Similarly, the distance between each alternative and negative ideal solution ($D_{j}^{ - }$) is estimated as follows:

$$D_{j}^{ - } = \sqrt {\sum\limits_{i = 1}^{n} {(v_{ij} - v_{i}^{ - } )^{2} } } \,\,\,\,\,\,\,\,\,\,\,\,\,j = 1,2, \ldots ,J$$

Step 6 Subsequently, relative closeness to the ideal solution for each alternative is calculated by using $D_{j}^{ - }$ and $D_{j}^{*}$ values. The relative closeness of alternative $A_{j}$ is represented as follows:

$$CC_{j}^{*} = \frac{{D_{j}^{*} }}{{D_{j}^{*} + D_{j}^{ - } }},\quad j = 1,2, \ldots ,J$$

The value of relative closeness ($CC_{j}^{*}$) falls in between 0 and 1. A larger value of the closeness index signifies the better alternative.

Appendix 3: Few relevant definitions on fuzzy set theory

Some preliminary definitions of the fuzzy set theory are discussed below.

Definition 1

A fuzzy set $\tilde{A}$ in a universe of discourse $X$ is characterized by a membership function $\mu_{{\tilde{A}}} (x)$ which associates with each element $x$ in $X$, a real number in the interval [0, 1]. The function value $\mu_{{\tilde{A}}} (x)$ is termed the grade of membership of $x$ in $\tilde{A}$.

Definition 2

A triangular fuzzy number $\tilde{a}$ is represented by triple points ($a_{1} ,a_{2} ,a_{3}$). The membership function is estimated as follows:

$$\mu_{{\tilde{a}}} (x) = \left\{ {\begin{array}{*{20}c} {0,\,\,\,\,\,\,\,\,\,\,\,\,\,x < a_{1} } \\ {\frac{{x - a_{1} }}{{a_{2} - a_{1} }},a_{1} \le x \le a_{2} } \\ {\frac{{x - a_{3} }}{{a_{2} - a_{3} }},\,a_{2} \le x \le a_{3} } \\ {0,\,\,\,\,\,\,\,\,\,\,\,\,\,x > a_{3} } \\ \end{array} } \right\}$$

Let $\tilde{a}$ and $\tilde{b}$ are the two triangular fuzzy numbers that are represented by three items ($a_{1} ,a_{2} ,a_{3}$) and ($b_{1} ,b_{2} ,b_{3}$), respectively. The operational laws of these two fuzzy numbers are as follows:

$$\tilde{a}( + )\tilde{b} = (a_{1} ,a_{2} ,a_{3} )( + )(b_{1} ,b_{2} ,b_{3} ) = (a_{1} + b_{1} ,a_{2} + b_{2} ,a_{3} + b_{3} )$$

$$\tilde{a}( - )\tilde{b} = (a_{1} ,a_{2} ,a_{3} )( - )(b_{1} ,b_{2} ,b_{3} ) = (a_{1} - b_{1} ,a_{2} - b_{2} ,a_{3} - b_{3} )$$

$$\tilde{a}( \times )\tilde{b} = (a_{1} ,a_{2} ,a_{3} )( \times )(b_{1} ,b_{2} ,b_{3} ) = (a_{1} .\,b_{1} ,a_{2} .\,b_{2} ,a_{3} .\,b_{3} )$$

$$\tilde{a}(/)\tilde{b} = (a_{1} ,a_{2} ,a_{3} )(/)(b_{1} ,b_{2} ,b_{3} ) = (a_{1} /\,b_{1} ,a_{2} /\,b_{2} ,a_{3} /b_{3} )$$

$$k( \times )\tilde{a} = (k.a_{1} ,k.a_{2} ,k.a_{3} )$$

Definition 3

Suppose $\tilde{a} = (a_{1} ,a_{2} ,a_{3} )$ and $\tilde{b} = (b_{1} ,b_{2} ,b_{3} )$ are the two triangular fuzzy numbers. The distance between these two fuzzy numbers can be computed as follows:

$$d(\tilde{a},\tilde{b}) = \sqrt {\frac{1}{3}[(a_{1} - b_{1} )^{2} + (a_{2} - b_{2} )^{2} + (a_{3} - b_{3} )^{2} }$$

Definition 4

Taking consideration of different importance values of each criterion, the weighted normalized fuzzy decision matrix can be constructed as follows:

• $\tilde{V} = [\tilde{v}_{ij} ]_{n \times J} \quad i = 1,2, \ldots ,n;\quad j = 1,2, \ldots ,J$ where $\tilde{v}_{ij} = \tilde{x}_{ij} \times w_{i}$

• A set of performance ratings of $A_{j} (j = 1,2, \ldots ,J)$ with respect to the criteria $C_{i} (i = 1,2, \ldots ,n)$ called $\tilde{X} = \{ \tilde{x}_{ij} ,\,i = 1,2, \ldots ,n;j = 1,2, \ldots ,J\} .$

• A set of importance weights of each criterion $w_{i} (i = 1,2, \ldots ,n).$

Appendix 4: Description of fuzzy TOPSIS

The steps of the fuzzy TOPSIS are discussed below (Önüt and Soner 2007).

Step 1 The first step of fuzzy TOPSIS is to assign linguistic values $\{ \tilde{x}_{ij} ,\,i = 1,2, \ldots ,n;j = 1,2, \ldots ,J\}$ to alternatives with respect to the criteria. The fuzzy linguistic rating $\tilde{x}_{ij}$ ensures that the ranges of the normalized triangular fuzzy numbers belong to [0, 1]; therefore, there is no requirement of normalization.

Step 2 The next step is to estimate the weighted normalized fuzzy decision matrix ($\tilde{v}_{ij}$).

Step 3 Subsequently, positive ideal ($A^{*}$) and negative ideal ($A^{ - }$) solutions are identified. The fuzzy positive ideal solution ($FPIS,A^{*}$) and fuzzy negative ideal solution $(FNIS,A^{ - } )$ are depicted in the following equations:

$$A^{*} = \{ \tilde{v}_{1}^{*} ,\tilde{v}_{2}^{*} , \ldots ,\tilde{v}_{i}^{*} \} = \{ ( {\mathop {max}\limits_{j} v_{ij} } |i \in I^{'} ) \quad \times ( {\mathop {min}\limits_{j} v_{ij} } |i \in I^{''} )\} \quad i = 1,2, \ldots ,n;\quad j = 1,2, \ldots ,J$$

$$A^{ - } = \{ \tilde{v}_{1}^{ - } ,\tilde{v}_{2}^{ - } , \ldots ,\tilde{v}_{i}^{ - } \} = \{ ( {\mathop {\hbox{min} }\limits_{j} v_{ij} } |i \in I^{'} ) \quad \times ( {\mathop {\hbox{max} }\limits_{j} v_{ij} } |i \in I^{''} )\} \quad i = 1,2, \ldots ,n;\quad j = 1,2, \ldots ,J$$

Here, $I^{'}$ depicts the benefit criteria and $I^{''}$ denotes the cost criteria.

Step 4 The next step is to calculate distance of each alternative from $A^{*}$ and $A^{ - }$ using the following equations:

$$D_{j}^{*} = \sum\limits_{j = 1}^{n} {d(\tilde{v}_{ij} } ,\tilde{v}_{i}^{*} )\quad j = 1,2, \ldots ,J$$

$$D_{j}^{ - } = \sum\limits_{j = 1}^{n} {d(\tilde{v}_{ij} } ,\tilde{v}_{i}^{ - } )\quad j = 1,2, \ldots ,J$$

Step 5 The next step is to calculate the similarity to ideal solution.

$$CC_{j} = \frac{{D_{j}^{ - } }}{{D_{j}^{*} + D_{j}^{ - } }}\quad j = 1,2, \ldots ,J$$

Step 6 Rank the alternatives as per the values of $CC_{j}$ and choose the largest value.

Appendix 5: Variations of chance-constrained programming

When a _ij are assumed as random variables

Suppose variables $\overline{{a_{ij} }}$ and $\sigma_{{a_{ij} }}$ are means and standard deviations of normally distributed random variables $a_{ij}$. The stochastic constraints of the problem can be converted to the deterministic equivalent equations. The constraints in deterministic form are shown below (Nazemi and Tahmasbi 2013).

$$\sum\limits_{j = 1}^{n} {\overline{{a_{ij} }} } *x_{j} - {\Phi}^{ - 1} (1 - p_{i} )\sqrt {\sum\limits_{j = 1}^{n} {(\sigma_{{a_{ij} }} )^{2} } *(x_{j} )^{2} } \, \le b_{i}$$

Variables in ${\Phi}^{ - 1} (1 - p_{i} )$ denote the inverses of cumulative standard normal distributions.

When $b_{i}$ are assumed as random variables

Suppose variables $\overline{{b_{i} }}$ and $\sigma_{{b_{i} }}$ connote the means and standard deviations of normally distributed random variables $b_{i}$. The stochastic constraints of the problem can be transformed to equivalent deterministic equations. The modified constraints are shown in the following equation (Nazemi and Tahmasbi 2013).

$$\sum\limits_{j = 1}^{n} {a_{ij} x_{j} \le \overline{{b_{i} }} } + {\Phi}^{ - 1} (1 - p_{i} )*\sigma_{{b_{i} }}$$

Appendix 6: Calculations related to AHP and fuzzy TOPSIS

See Tables 2, 3, 4, 5, 6 and Fig. 2.

Table 4 Fuzzy evaluation matrix for suppliers

Table 5 Weighted evaluation for the alternative suppliers

Table 6 Fuzzy TOPSIS results

Appendix 7: Data and results of mathematical modeling

See Tables 7, 8, 9, 10, 11, 12, 13, 14.

Table 7 Data related to suppliers’ capacities, variable costs and variable emissions

Table 8 Variable costs at different echelons of supply chain

Table 9 Variable emissions at different echelons of supply chain

Table 10 Data associated with capacity, fixed cost, demand and fixed emission

Table 11 Data related to social sustainability and probabilities

Table 12 Standard deviations of the emissions from suppliers to plants

Table 13 Standard deviations of the emissions at different echelons of supply chain

Table 14 Results of the problem

Appendix 8: Sensitivity plots

See Figs. 3, 4, 5, 6, 7, 8 and 9.