Fuzzy criteria programming approach for optimising the TBL performance of closed loop supply chain network design problem

Abstract

Immense concern for sustainability and increasing stakeholders’ involvement has sparked tremendous interest towards designing optimal supply chain networks with significant economic, environmental, and social influence. Central to the idea, this study aims to design a closed loop supply chain (CLSC) network for an Indian laptop manufacturer. The network configuration, which involves a manufacturer, suppliers, third party logistics providers (forward and reverse), retailers, customers and a non-government organisation (NGO), is modelled as a mixed integer linear programming problem with fuzzy goals of minimising environmental impact and maximising net profit and social impact, subject to fuzzy demand and capacity constraints. Profit is generated from the sale of primary and secondary laptops, earned tax credits, and revenue sharing with reverse logistics providers. The environmental implications are investigated by measuring the carbon emitted due to activities of manufacturing, assembling, dismantling, fabrication, and transportation. The social dimension is quantified in terms of the number of jobs created, training hours, community service hours, and donations to NGO. The novelty of the model rests on its quantification of the three triple bottom line (TBL) indicators and on its use of AHP–TOPSIS for modelling the multi-criteria perspectives of the stakeholders. Numerical weights for the triple lines of sustainability are utilized. Further, a fuzzy multi-objective programming approach that integrates fuzzy set theory with goal programming techniques is utilised to yield properly efficient solutions to the multi-objective problem and to provide a trade-off set for conflicting objectives. The significance of the CLSC model is empirically established as a decision support tool for improving the TBL performance of a particular Indian laptop manufacturer. Practical and theoretical implications are derived from the result analysis, and a generalised quantitative closed-loop model can be effectively adapted by other electronic manufacturers to increase their competitiveness, profitability, and to improve their TBL.

Appendix A

1.1 AHP–TOPSIS methodology

The integrated AHP–TOPSIS technique is utilised for generating the weights of importance for the three TBL objectives (Perçin 2009). AHP is used for deriving weights $v_{i }$ (normalized) for the 11 criteria (alternatives) with respect to the TBL goals. Further, AHP is used again for finding the normalized priority vectors $u_{i }$ representing the importance of the three TBL objectives (alternatives) with respect to each of the 11 criteria (goals). The AHP method is explained in Steps 1–4 given below. Utilising these normalized priorities in the initial decision matrix of TOPSIS, the final weights of the TBL objectives (alternatives) are then calculated as per the criteria $\hbox {C}_{1}$-$\hbox {C}_{11}$, following the steps 5–10.

The steps of AHP are as follows:

• 1. Suppose there are ‘n’ alternatives whose importance weights need to be calculated with respect to a given goal ‘g.’ The alternatives are compared pair-wise on a scale of 1–9 representing subjective judgments: ‘1—equally important’, ‘3— moderately important’, ‘5—important’, ‘7—very important’, ‘9— extremely important’, with the intermediate subjectivities represented by 2, 4, 6, and 8. This leads to construction of nxn matrix A whose (i, j)th element ($a_{{ ij}}$) is the quantified value of the pair-wise comparison of the ith and jth criteria.

• 2. Normalize the matrix A by dividing each (i, j)th element ($a_{{ ij}}$) by the jth column sum $a_{j}$ to derive the matrix A* as shown below:

  

• 3. Determine the consistency of the matrix A* by calculating the maximal eigen value, consistency index (CI), and the consistency ratio (CR) utilising the equations given below:

  $$\begin{aligned} { Aw}= & {} \lambda w, { CI}=\frac{\lambda _{\max } -n}{n-1}, { CR}=\frac{CI}{RI} {\textit{where n is order of A}} \\ { and}\,\lambda _{\max }= & {} \max \left\{ {\lambda _i ,\;i=1,\ldots n} \right\} \end{aligned}$$

• 4. Calculate the averages $a_{i}*$ of rows of the matrix A*. The normalised priority vector is obtained by normalising the vector ($a_{1}*, a_{2}*, {\ldots }.a_{n}*)$

Using Steps 1–4, the normalised priority vector $\{\hbox {v}_{1}, {\ldots }.\hbox {v}_{11}\}$ is determined. Based on each criterion, the normalised priority vector $\{\hbox {u}_{1j}$, $\hbox {u}_{2j}$, $\hbox {u}_{3j},\}$ is determined. Table 19 provides the value of RI.

Further, to determine the importance rankings of the TBL objectives $(k=3)$ with respect to the criteria $\hbox {C}_{1}$–$\hbox {C}_{11}$ using TOPSIS, the following steps are utilised:

• 5. Construct the IDM for TOPSIS as shown in Table 20.

Table 19 Random index (RI)

Table 20 Initial decision matrix

• 6. Construct the nxn weighted normalised matrix $\bar{{U}}=(\bar{{u}}_{ij} )=wU=(w_i u_{ij} )$

• 7. Calculate the ideal solution and the negative-ideal solution as follows:

  $$\begin{aligned} \bar{{U}}^{+}= & {} \left\{ {(\mathop {\max \;\bar{{u}}_{kj} }\limits _k |\;j\in J^{{\prime }}),(\mathop {\min \;\bar{{u}}_{kj} }\limits _k |\;j\in J^{{\prime }{\prime }})|k=1,2,3} \right\} =\left\{ {\bar{{u}}_1^+ ,\bar{{u}}_2^+ ,\bar{{u}}_3^+ } \right\} ,\\ \bar{{U}}^{-}= & {} \left\{ {(\mathop {\min \;\bar{{u}}_{kj} }\limits _k |\;j\in J^{{\prime }}),(\mathop {\max \;\bar{{u}}_{kj} }\limits _k |\;j\in J^{{\prime }{\prime }})|k=1,2,3} \right\} =\left\{ {\bar{{u}}_1^- ,\bar{{u}}_2^- ,\bar{{u}}_3^- } \right\} \end{aligned}$$

  where $J^{{\prime }} \,{ and}\, J^{{\prime }{\prime }}$ are associated with benefit and cost criteria respectively.

• 8. Calculate the separation measure of each alternative as follows:

  $$\begin{aligned} d_k^+= & {} \sqrt{\sum _{k=1}^3 {\left( {\bar{{u}}_{kj} -\bar{{u}}_k^+ } \right) ^{2}} },\;d_k^- =\sqrt{\sum _{k=1}^3 {\left( {\bar{{u}}_{kj} -\bar{{u}}_k^- } \right) ^{2}} }, k=1,2,3 \end{aligned}$$

• 9. Calculate the relative closeness of each goal as:

$$\begin{aligned} CC_k^*=\frac{d_k^- }{d_k^+ +d_k^- }\;k=1, 2, 3 \end{aligned}$$

Clearly, $0\le CC_i^*\le \,1 i=1,2,,3$, and higher value implies closeness to the ideal solution.

• 10. The weighting vector $w=\{\hbox {w}_{1}, \hbox {w}_{2}, \hbox {w}_{3}\}$ of the goals is obtained by normalising the closeness vector CC*.