## 4 Conclusions and Suggestions for Future Work  
Modeling and optimizing the supply chain in order to locate an optimum material transferring system has attracted many researchers' attention. In this paper, the supply chain has three levels including suppliers, cross-docks, and factories. Raw materials are transferred to several cross-docks via several suppliers. Each channel is inclusively for a special product in a cross-dock. Goods entrance rate from supplier to cross-docks is stochastic and follows a general probability distribution. On the other hand, we assume that startup and delivery time in cross-docks are stochastic and follow a general probability distribution. Service time in each cross-dock depends on the number of existing doors; therefore, we can consider each cross-dock as the G/G/m queue system in which m is the number of doors. The mathematical model is expanded by considering variables and their relations. Our main aim is to present an analytical model using the cross-docks and queue theory technique. Thus, the minimization of two main objectives is evaluated; the former is to minimize materials transferring whereas the latter is to minimize customer waiting time.  
Regarding the fact that the proposed model is NP-hard, three metaheuristics including multi-objective water flow, non-dominated sorting genetic, and multi-objective simulated annealing algorithm are used to optimize the proposed model. After tuning the algorithm parameters via the Taguchi method, the obtained results from the algorithms are analyzed  
Figure 10: Confidence intervals of 95% for algorithms RPD in different criteria.  
by statistical test and the algorithm efficiencies are evaluated. The results vividly demonstrate that in terms of SNS, MID, and Diversity criteria, NSGA-II has higher efficiency than the other two algorithms. Furthermore, the MOSA algorithm performs better than MOWFA in the NOEF criterion. Regarding the resulted tables obtained from algorithms solutions, we can surely conclude that NSGA-II performs highly better than MOSA and MOWFA and reaches better solutions. Examining and analyzing the obtained results from analysis and variance tests prove this claim.  
In this paper, we evaluated fundamental costs which are related to transportation and cross-docks. As a result, the most important impact of this research is to identify factors that have high costs. This can help organizations reduce their costs. The special suggestions that this research has for future researchers are derived from the current needs in today's world. If transportation paths are of high significance, routing problems can be added as a scope for future research. Some other assumptions can be added to the model such as operational storage expenses for products and considering time limitations and distance for each route. Designing another model which is able to plan all of the events in a cross-dock can be considered another practical scope for future research. Replacing metaheuristic algorithms with simulation methods and neural networks and also evaluating other metaheuristic algorithms is another way to enlarge our knowledge in this scope.  
All authors contributed to the study concept and methodology. Material preparation, data collection and analysis were performed by [PERSON] The supervision and leading the work was conducted by [PERSON]. The first draft of the manuscript was written by [PERSON] All corrections and editing are done by [PERSON] All authors have read and agreed to the published version of the manuscript.  
The authors declare that no funds, grants, or other support were received during the preparation of this manuscript.  
Not applicable.  
The authors confirm that the data supporting the findings of this study are available within the article.  
The authors have no competing interest to declare that are relevant to the content of this article.  
The authors have no relevant ethical approval to disclose for this article.