Cross-docking is considered as a method to manage and control the inventory flow, which is essential in the context of supply chain management. This paper studies the integration of the vehicle routing problem with cross-docking, namely VRPCD which has been extensively studied due to its ability to reduce the overall costs occurring in a supply chain network. Given a fleet of homogeneous vehicles for delivering a single type of product from suppliers to customers through a cross-dock facility, the objective of VRPCD is to determine the number of vehicles used and the corresponding vehicle routes, such that the vehicle operational and transportation costs are minimized. An adaptive large neighborhood search (ALNS) algorithm is proposed to solve the available benchmark VRPCD instances. The experimental results show that ALNS is able to improve 80 (out of 90) best known solutions and obtain the same solution for the remaining 10 instances within short computational time. We also explicitly analyze the added value of using an adaptive scheme and the implementation of the acceptance criteria of Simulated Annealing (SA) into the ALNS, and also present the contribution of each ALNS operator towards the solution quality.

This paper considers a multi-objective fixed-charge transportation problem (MOFCTP) in which the parameters of the objective functions are random rough variables, while the supply and the demand parameters are rough variables. In real-life situations, the parameters of a multi-objective fixed-charge transportation problem may not be defined precisely, because of globalization of the market, uncontrollable factors, etc. As such, the multi-objective fixed-charge transportation problem is proposed under rough and random rough environments. To tackle uncertain (rough and random rough) parameters, the proposed model employs an expected value operator. Furthermore, a procedure is developed for converting the uncertain multi-objective fixed-charge transportation problem into a deterministic form and then solving the deterministic model. Three different methods, namely, the fuzzy programming, global criterion, and ϵ-constrained methods, are used to derive the optimal compromise solutions of the suggested model. To provide the preferable optimal solution of the formulated problem, a comparison is drawn among the optimal solutions that are extracted from different methods. Herein, the ϵ-constrained method derives a set of optimal solutions and generates an exact Paretofront. Finally, in order to show the applicability and feasibility of the proposed model, the paper includes a real-life example of a multi-objective fixed-charge transportation problem. The main contribution of the paper is that it deals with MOFCTP using two types of uncertainties, thus making the decision making process more flexible.