The vehicle routing and scheduling problem has been studied with much interest within the last four decades. In this paper, some of the existing literature dealing with routing and scheduling problems with environmental issues is reviewed, and a description is provided of the problems that have been investigated and how they are treated using combinatorial optimization tools.
In @cite_27 , the authors consider a multiobjective multimodal multicommodity flow problem with time windows and piecewise linear concave cost functions. Based on Lagrangian relaxation technique, the problem is broken into a set of smaller and easier subproblems and the subgradient optimization procedure is applied to solve the Lagrangian multipliers problem. Authors in @cite_84 proposed an origin-destination integer multi-commodity flow formulation with non-convex piecewise linear costs and use column generation based heuristic that provides both lower bounds and good quality feasible solutions. The author in @cite_1 deals with two objectives: the cost and the risk and develops a chance-constrained goal programming method to solve the problem.
@cite_27 @cite_1 @cite_84
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One central issue in practically deploying network coding is the adaptive and economic allocation of network resource. We cast this as an optimization, where the net-utility-the difference between a utility derived from the attainable multicast throughput and the total cost of resource provisioning-is maximized. By employing the MAX of flows characterization of the admissible rate region for multicasting, this paper gives a novel reformulation of the optimization problem, which has a separable structure. The Lagrangian relaxation method is applied to decompose the problem into subproblems involving one destination each. Our specific formulation of the primal problem results in two key properties. First, the resulting subproblem after decomposition amounts to the problem of finding a shortest path from the source to each destination. Second, assuming the net-utility function is strictly concave, our proposed method enables a near-optimal primal variable to be uniquely recovered from a near-optimal dual variable. A numerical robustness analysis of the primal recovery method is also conducted. For ill-conditioned problems that arise, for instance, when the cost functions are linear, we propose to use the proximal method, which solves a sequence of well-conditioned problems obtained from the original problem by adding quadratic regularization terms. Furthermore, the simulation results confirm the numerical robustness of the proposed algorithms. Finally, the proximal method and the dual subgradient method can be naturally extended to provide an effective solution for applications with multiple multicast sessions This paper considers the vehicle routing problem with time windows, where the service of each customer must start within a specified time interval. We consider the Lagrangian relaxation of the constraint set requiring that each customer must be served by exactly one vehicle yielding a constrained shortest path subproblem. We present a stabilized cutting-plane algorithm within the framework of linear programming for solving the associated Lagrangian dual problem. This algorithm creates easier constrained shortest path subproblems because less negative cycles are introduced and it leads to faster multiplier convergence due to a stabilization of the dual variables. We have embedded the stabilized cutting-plane algorithm in a branch-and-bound search and introduce strong valid inequalities at the master problem level by Lagrangian relaxation. The result is a Lagrangian branch-and-cut-and-price (LBCP) algorithm for the VRPTW. Making use of this acceleration strategy at the master problem level gives a significant speed-up compared to algorithms in the literature based on traditional column generation. We have solved two test problems introduced in 2001 by Gehring and Homberger with 400 and 1000 customers respectively, which to date are the largest problems ever solved to optimality. We have implemented the LBCP algorithm using the ABACUS open-source framework for solving mixed-integer linear-programs by branch, cut, and price. This paper presents fast algorithms that find approximate solutions for a general class of problems, which we call fractional packing and covering problems. The only previously known algorithms for solving these problems are based on general linear programming techniques. The techniques developed in this paper greatly outperform the general methods in many applications, and are extensions of a method previously applied to find approximate solutions to multicommodity flow problems. Our algorithm is a Lagrangian relaxation technique; an important aspect of our results is that we obtain a theoretical analysis of the running time of a Lagrangian relaxation-based algorithm.We give several applications of our algorithms. The new approach yields several orders of magnitude of improvement over the best previously known running times for algorithms for the scheduling of unrelated parallel machines in both the preemptive and the nonpreemptive models, for the job shop problem, for the Held and Karp bound for the traveling salesman problem, for the cutting-stock problem, for the network embedding problem, and for the minimum-cost multicommodity flow problem. This study focuses on one of the intermodal operational issues: how to select best routes for shipments through the international intermodal network. International intermodal routing is complicated by three important characteristics: (1) multiple objectives; (2) scheduled transportation modes and demanded delivery times; and (3) transportation economies of scale. In this paper, the international intermodal routing problem is formulated as a multiobjective multimodal multicommodity flow problem (MMMFP) with time windows and concave costs. The objectives of this paper are to develop a mathematical model encompassing all three essential characteristics, and to propose an algorithm that can effectively provide answers to the model. The problem is NP-hard. It follows that the proposed algorithm is a heuristic. Based on relaxation and decomposition techniques, the original problem is broken into a set of smaller and easier subproblems. The case studies show that it is important to incorporate the three characteristics into the international intermodal routing problem, and our proposed algorithm can effectively and efficiently solve the MMMFP with time windows and concave costs.