## 7 Conclusions  
This work presented a location routing problem where the facility size is an additional variable, instead of a known parameter as the traditional LRP assumes. Moreover, we consider a hybrid fuzzy-stochastic setting in which some customers' demands are fuzzy  
Figure 7: Best-found solution by a nonflexible (**a**) and a flexible (**b**) fuzzy LRP for the instance _Cr40x5a-2_. and others are stochastic. Hence, a fuzzy simheuristic approach is proposed to solve this problem cost- and time- efficiently. Initially, our algorithm selects the best size for each open facility from a set of provided alternatives. We perform an iterative procedure in which a set of location-allocation-routing configurations are assessed in terms of opening and routing costs. Then a top list of complete LRP solutions is iteratively perturbed and simulated. The perturbation stage is performed by employing an iterated local search metaheuristic. The simulation stage is carried out by running a classic Monte Carlo simulation for the stochastic demands and a fuzzy simulation for the fuzzy demands. Failure costs are introduced as an additional performance indicator. Finally, a set of elite solutions is assessed through a refinement procedure where a larger number of simulation runs is executed. Our fuzzy simheuristic approach has been proved to be flexible enough not only to combine efficiently stochastic and fuzzy demands in a single execution but also to address less general scenarios in which demands of all customers are either deterministic or fuzzy. Our approach has also been proved to be a cost-efficient algorithm when considering uncertainty scenarios. It decreases route failure costs when compared with the best deterministic solution tested in a hybrid fuzzy-stochastic environment. The use of a safety stock policy as a protection against uncertainty has also contributed to this decrease. In order to design a time-efficient algorithm, our current approach employs stochastic and fuzzy simulation only to assess the designed routes. Hence, our algorithm results can be enhanced by introducing fuzzy-stochastic aspects from the construction stage. However, this approach might also increase computational times. To the best of our knowledge, this is the first time that a hybrid fuzzy-stochastic LRP with facility sizing decisions is addressed. Medium-sized benchmark instances considering three demand variability levels were used. Obtained results show that introducing such flexibility decreases total costs in two mutually nonextclusive ways: firstly, yielding savings in opening costs by locating facilities of smaller size; and secondly, yielding savings in routing costs by locating facilities of higher size, which drives a routes redesign that reduces the total traveled distance. We also have demonstrated that these savings are always incurred regardless of the demand variability level. Multiple challenges remain open for future research. Since we are considering that only routes fail when demands are higher than expected, future work can include the simulation of facility failures, which would prompt a revision of location-allocation decisions. In addition, failure costs are currently measured only by considering the distances traveled to perform round-trips and detours. Still, real-life customers might not allow a delivery delay, e.g., because a time windows constraint must be met. This delay may drive lost sales or a goodwill reduction. Hence, this type of costs can be included in the computation of failure costs. Finally, large-sized instances can be used to assess the influence of the number of nodes in our approach performance. Conceptualization, R.D.T. and A.A.J. ; methodology, R.D.T., P.J.C.-M. and J.P.; software, P.J.C.-M. and J.P.; formal analysis, R.D.T. ; investigation, R.D.T., P.J.C.-M., C.L.Q.-A. and J.R.M.-T.; writing--original draft preparation, all authors; writing--review and editing, R.D.T. and A.A.J. All authors have read and agreed to the published version of the manuscript.