This paper introduces a new bi-objective vehicle routing problem that integrates the Open Location Routing Problem (OLRP), recently presented in the literature, coupled with the growing need for fuel consumption minimization, named Green OLRP (G-OLRP). Open routing problems (ORP) are known to be NP-hard problems, in which vehicles start from the set of existing depots and are not required to return to the starting depot after completing their service. The OLRP is a strategic-level problem involving the selection of one or many depots from a set of candidate locations and the planning of delivery radial routes from the selected depots to a set of customers. The concept of radial paths allows us to use a set of constraints focused on maintaining the radiality condition of the paths, which significantly simplifies the set of constraints associated with the connectivity and capacity requirements and provides a suitable alternative when compared with the elimination problem of sub-tours traditionally addressed in the literature. The emphasis in the paper will be placed on modeling rather than solution methods. The model proposed is formulated as a bi-objective problem, considering the minimization of operational costs and the minimization of environmental effects, and it is solved by using the epsilon constraint technique. The results illustrate that the proposed model is able to generate a set of trade-off solutions leading to interesting conclusions about the relationship between operational costs and environmental impact.
The product-mix planning and the lot size decisions are some of the most fundamental research themes for the operations research community. The fact that markets have become more unpredictable has increaed the importance of these issues, rapidly. Currently, directors need to work with product-mix planning and lot size decision models by introducing stochastic variables related to the demands, lead times, etc. However, some real mathematical models involving stochastic variables are not capable of obtaining good solutions within short commuting times. Several heuristics and metaheuristics have been developed to deal with lot decisions problems, in order to obtain high quality results within short commuting times. Nevertheless, the search for an efficient model by considering product mix and deal size with stochastic demand is a prominent research area. This paper aims to develop a general model for the product-mix, and lot size decision within a stochastic demand environment, by introducing the Economic Value Added (EVA) as the objective function of a product portfolio selection. The proposed stochastic model has been solved by using a Sample Average Approximation (SAA) scheme. The proposed model obtains high quality results within acceptable computing times.