Solving a green logistics bi-level bi-objective problem. - Maldonado Pinto, Carmen Sayuri (2017)
The situation here addressed is modelled as a bi-level programming problem with multiple objectives in the upper level and a single objective in the lower level. In this problem, a company (hereafter the leader) distribute a commodity over a selected subset of customers; while a manufacturer (hereafter the follower) will fabricate the commodities demanded by the selected customers. The leader has two objectives: the maximization of the profit gained by the distribution process and the minimization of CO2 emissions. The latter is important due to the regulations imposed by the government. It is clear that exists a compromise between both objectives, since the maximization of profit will attempt to include as much customers for being served as possible. Then, largest routes will be needed causing more CO2 emissions. For analyzing the problem, the single-commodity case is studied first. Under this assumption, the problem can be reduced into a single-level one. Hence, a tabu search algorithm for solving the aforementioned case is proposed. The tabu search is designed for solving two single-level simplifications of the problem: a monoobjective problem and the bi-objective one. After that, the multi-commodity bi-level case is studied and the respective adaptation of the tabu search is made. Then, a co-evolutionary algorithm is designed for obtaining good quality bi-level feasible solutions. The co-evolutionary approach is related with having two separated populations, one for each leader’s objective. Then, the solutions will evolve in each population and an interchange of information is made through the process. In other words, a swap between the best solutions from both populations in each generation is conducted. By doing this, the algorithm intends to find efficient solutions. The evolution performed in each population is done through a Biased Random Keys Genetic Algorithm( BRKGA). Furthermore, a path relinking algorithm is adapted in order to find the Pareto frontier for the bi-level bi-objective multi-commodity problem, in which the no dominated solutions of the tabu search and the co-evolutionary algorithms are used to initialize this procedure. Numerical experimentation showed the efficiency of the proposed methods for finding good quality solutions (for the mono-objective case) and for reaching a good approximation of the Pareto front (for the bi-objective cases) in reasonable computational time.