Green supply chain design with emission sensitive demand: second order cone programming formulation and case study

Abstract

We propose a novel modeling framework for supply chain network design that models a prevailing trend in consumer choice in which demand is impacted by carbon footprint. To date, the literature lacks models that realistically account for and accurately calculate per unit emissions, i.e., carbon footprint. We develop a profit maximizing model that accounts for emissions at the different stages of the supply chain, locates facilities and selects their technology, and decides on the flow between echelons. To calculate the carbon footprint, fixed emissions are averaged over throughput, which results in a nonlinear optimization problem with fractional terms. To solve it, we provide a mixed integer second order cone programming reformulation. We perform extensive testing of the framework on a realistic case study and carry out detailed analysis. The proposed framework succeeds in capturing the trade-off between lost demand due to a high carbon footprint and investing in environmentally-friendly technology. The framework serves as a tool to induce organizations to invest in green technology and to allow regulating authorities to assess the impact of eco-labeling.

Appendix A. Trade-off between price and carbon emission elasticity

We investigate the relationship between changes in price to compensate for lost revenue due to increased demand elasticity to carbon footprint. Assume that sensitivity increases from $\gamma _k$ to $\gamma _k+\Delta ^\gamma _k$ and we would like to find the increase $\Delta ^\pi _k$ in price $\pi ^k$ that maintains the same revenue $\pi _k D_k$ for every customer zone k. Therefore,

$$\begin{aligned} \pi _k D_k= \pi _k \left( {\bar{D}}_k\sum _j y_{jk}-\gamma _k e_k\right) \end{aligned}$$

should be equal to

$$\begin{aligned} (\pi _k+ \Delta ^\pi _k)\left( \bar{D}_k\sum _j y_{jk}-(\gamma _k+\Delta ^\gamma _k) e_k\right) \end{aligned}$$

which implies that:

$$\begin{aligned} \Delta ^\pi _k\left( \bar{D}_k\sum _j y_{jk}-(\gamma _k+\Delta ^\gamma _k) e_k\right) =\pi _k\Delta ^\gamma _k e_k \end{aligned}$$

and therefore,

$$\begin{aligned} \Delta ^\pi _k =\frac{\pi _k\Delta ^\gamma _k e_k}{(\bar{D}_k\sum _j y_{jk}-\gamma _ke_k)-\Delta ^\gamma _k e_k} \end{aligned}$$

which is a function of the form $\frac{ax}{b-x}$ that is exponential for $0\le x<b$. This implies that as emission sensitivity increases linearly, the price has to increase exponentially to maintain the same revenue.