A heuristic algorithm to solve the one-warehouse multiretailer problem with an emission constraint

In this paper, we consider the one-warehouse multiretailer problem with a global carbon emission cap constraint (OWMR-EC). This constraint aims at limiting the carbon emissions related to the production, setup, and inventory-holding operations. We develop a penalized relaxation (PR) method to heuristically solve the considered problem, both with and without the possibility of having initial inventory. This heuristic uses in itself another heuristic that we propose to solve the standard one-warehouse multiretailer problem (OWMR). Our PR method is tested on numerous instances adapted from the literature. Our results indicate that the penalized method is able to find between 87.4% and 89.8% of feasible solutions for this NP-hard problem, with an average optimality gap of 2.1% and 2.2% depending on the algorithms we use to solve the different subproblems involved in the method. The results show that our method is highly effective in terms of run-time and solution quality, when a feasible solution is found. Furthermore, the results indicate that the heuristic for the standard OWMR is also very effective. We further perform a sensitivity analysis on the optimal solutions of the OWMR-EC to better understand the implications of the carbon emission cap constraint. The sensitivity analysis indicates that the marginal cost of reducing carbon emissions increases as the emission cap decreases. The analysis also shows that the correlation between the cost and emission parameters has an important impact on the potential to further lower the emissions, compared to the emission of the minimum cost solution.