A genetic-algorithm-based approach to the two-echelon capacitated vehicle routing problem with stochastic demands in logistics service

This paper addresses the two-echelon capacitated vehicle routing problem (2E-CVRP) with stochastic demands (2E-CVRPSD) in city logistics. A stochastic program with recourse is used to describe the problem. This program aims to minimize the sum of the travel cost and the expected cost of recourse actions resulting from potential route failures. In a two-echelon distribution system, split deliveries are allowed at the first level but not at the second level, thereby increasing the difficulty of calculating the expected failure cost. Three types of routes with or without split deliveries are identified. Different methods are devised or adapted from the literature to compute the failure cost. A genetic-algorithm-based (GA) approach is proposed to solve the 2E-CVRPSD. A simple encoding and decoding scheme, a modified route copy crossover operator, and a satellite-selection-based mutation operator are devised in this approach. The numerical results show that for all instances, the expected cost of the best 2E-CVRPSD solution found by the proposed approach is not greater than that of the best-known 2E-CVRP solution with an average relative gap of 2.57%. Therefore, the GA-based approach can find high-quality solutions for the 2E-CVRPSD.     

A tabu search algorithm for the periodic vehicle routing problem with multiple vehicle trips and accessibility restrictions

In this paper, we consider a periodic vehicle routing problem that includes, in addition to the classical constraints, the possibility of a vehicle doing more than one route per day, as long as the maximum daily operation time for the vehicle is not exceeded. In addition, some constraints relating to accessibility of the vehicles to the customers, in the sense that not every vehicle can visit every customer, must be observed. We refer to the problem we consider here as the site-dependent multi-trip periodic vehicle routing problem. An algorithm based on tabu search is presented for the problem and computational results presented on randomly generated test problems that are made publicly available. Our algorithm is also tested on a number of routing problems from the literature that constitute particular cases of the proposed problem. Specifically we consider the periodic vehicle routing problem; the site-dependent vehicle routing problem; the multi-trip vehicle routing problem; and the classical vehicle routing problem. Computational results for our tabu search algorithm on test problems taken from the literature for all of these problems are presented.     

Topology optimization of plane frames in case of random external loadings

Using the first collapse–theorem, the necessary and sufficient survival conditions of an elasto–plastic structure consist of the yield condition and the equilibrium condition. In practical applications several random model parameters have to be taken into account. This leads to a stochastic optimization problem which cannot be solved using the traditional methods. Instead of that, appropriate (deterministic) substitute problems must be formulated. Here, the topology optimization of frames is considered, where the external load is supposed to be stochastic. The recourse problem will be formulated in general and in the standard form of stochastic linear programming (SLP). After the formulation of the stochastic optimization problem, the recourse problem with discretization and the expected value problem are introduced as representatives of substitute problems. Subsequently, numerical results using these methods are presented. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Solving multiobjective vehicle routing problem with stochastic demand via evolutionary computation

This paper considers the routing of vehicles with limited capacity from a central depot to a set of geographically dispersed customers where actual demand is revealed only when the vehicle arrives at the customer. The solution to this vehicle routing problem with stochastic demand (VRPSD) involves the optimization of complete routing schedules with minimum travel distance, driver remuneration, and number of vehicles, subject to a number of constraints such as time windows and vehicle capacity. To solve such a multiobjective and multi-modal combinatorial optimization problem, this paper presents a multiobjective evolutionary algorithm that incorporates two VRPSD-specific heuristics for local exploitation and a route simulation method to evaluate the fitness of solutions. A new way of assessing the quality of solutions to the VRPSD on top of comparing their expected costs is also proposed. It is shown that the algorithm is capable of finding useful tradeoff solutions for the VRPSD and the solutions are robust to the stochastic nature of the problem. The developed algorithm is further validated on a few VRPSD instances adapted from Solomon’s vehicle routing problem with time windows (VRPTW) benchmark problems.     

Some insights into the solution algorithms for SLP problems

We consider classes of stochastic linear programming problems which can be efficiently solved by deterministic algorithms. For two–stage recourse problems we identify two such classes. The first one consists of problems where the number of stochastically independent random variables is relatively low; the second class is the class of simple recourse problems. The proposed deterministic algorithm is successive discrete approximation. We also illustrate the impact of required accuracy on the efficiency of this algorithm. For jointly chance constrained problems with a random right–hand–side and multivariate normal distribution we demonstrate the increase in efficiency when lower accuracy is required, for a central cutting plane method. We support our argumentation and findings with computational results.     

Exact algorithms for the vehicle routing problem,based on spanning tree and shortest path relaxations

We consider the problem of routing vehicles stationed at a central facility (depot) to supply customers with known demands, in such a way as to minimize the total distance travelled. The problem is referred to as the vehicle routing problem (VRP) and is a generalization of the multiple travelling salesman problem that has many practical applications. We present tree search algorithms for the exact solution of the VRP incorporating lower bounds computed from (i) shortest spanningk-degree centre tree (k-DCT), and (ii)q-routes. The final algorithms also include problem reduction and dominance tests. Computational results are presented for a number of problems derived from the literature. The results show that the bounds derived from theq-routes are superior to those fromk-DCT and that VRPs of up to about 25 customers can be solved exactly.     

Optimal design and sensitivity of large scale plane frames for the case of uncertainty

This paper deals with the optimal design and the robustness of large scale plane frames in dependence of their height. Using the first collapse theorem, the necessary and sufficient survival conditions of an elasto-plastic structure consist of the yield condition and the equilibrium condition. The basis for our consideration is provided by a plane n-storey frame which will be increased successively, and which is affected by applied random forces and moments. Taking into account these random applied loads, we get a stochastic structural optimization problem which cannot be solved using the traditional methods. Instead of that, an appropriate (deterministic) substitute problem is formulated. First, the recourse problem will be formulated in general and in the standard form of stochastic linear programming (SLP), and after the formulation of the stochastic optimization problem, the Recourse Problem based on Discretization (RPD) is introduced as a representative of substitute problems. The resulting (large scale) linear program (LP) can be solved efficiently by means of usual LP-solvers. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A priori policy evaluation and cyclic-order-based simulated annealing for the multi-compartment vehicle routing problem with stochastic demands

We develop methods to estimate and exactly calculate the expected cost of a priori policies for the multi-compartment vehicle routing problem with stochastic demands, an extension of the classical vehicle routing problem where customer demands are uncertain and products must be transported in separate partitions. We incorporate our estimation procedure into a cyclic-order-based simulated annealing algorithm, significantly improving the best-known solution values for a set of benchmark problems. We also extend the updating procedure for a cyclic order’s candidate route set to duration-constrained a priori policies.     

Exact algorithms for the chance-constrained vehicle routing problem

We study the chance-constrained vehicle routing problem (CCVRP), a version of the vehicle routing problem (VRP) with stochastic demands, where a limit is imposed on the probability that each vehicle’s capacity is exceeded. A distinguishing feature of our proposed methodologies is that they allow correlation between random demands, whereas nearly all existing exact methods for the VRP with stochastic demands require independent demands. We first study an edge-based formulation for the CCVRP, in particular addressing the challenge of how to determine a lower bound on the number of vehicles required to serve a subset of customers. We then investigate the use of a branch-and-cut-and-price (BCP) algorithm. While BCP algorithms have been considered the state of the art in solving the deterministic VRP, few attempts have been made to extend this framework to the VRP with stochastic demands. In contrast to the deterministic VRP, we find that the pricing problem for the CCVRP problem is strongly \(\mathcal {NP}\)-hard, even when the routes being priced are allowed to have cycles. We therefore propose a further relaxation of the routes that enables pricing via dynamic programming. We also demonstrate how our proposed methodologies can be adapted to solve a distributionally robust CCVRP problem. Numerical results indicate that the proposed methods can solve instances of CCVRP having up to 55 vertices.     

Dynamic Generalized Assignment Problems with Stochastic Demands and Multiple Agent--Task Relationships

The assignment problem is a well-known operations research model. Its various extensions have been applied to the design of distributed computer systems, job assignment in telecommunication networks, and solving diverse location, truck routing and job shop scheduling problems.This paper focuses on a dynamic generalization of the assignment problem where each task consists of a number of units to be performed by an agent or by a limited number of agents at a time. Demands for the task units are stochastic. Costs are incurred when an agent performs a task or a group of tasks and when there is a surplus or shortage of the task units with respect to their demands. We prove that this stochastic, continuous-time generalized assignment problem is strongly NP-hard, and reduce it to a number of classical, deterministic assignment problems stated at discrete time points. On this basis, a pseudo-polynomial time combinatorial algorithm is derived to approximate the solution, which converges to the global optimum as the distance between the consecutive time points decreases. Lower bound and complexity estimates for solving the problem and its special cases are found.     

Optimal Design of Plane Frames under Uncertainty: A Comparison of Two Approaches

Using the first collapse theorem, the necessary and sufficient survival conditions of an elasto-plastic structure consist of the yield condition and the equilibrium condition. In practical applications several random model parameters have to be taken into account. This leads to a stochastic optimization problem which cannot be solved using the traditional methods. Instead of that, appropriate (deterministic) substitute problems must be formulated. Here, the design of plane frames is considered, where the applied load is supposed to be stochastic. In the first approach the recourse problem will be formulated in the standard form of stochastic linear programming (SLP). In order to apply efficient numerical solution procedures (LP-solvers), approximate recourse problems based on discretization (RPD) and the expected value problem (EVP) are introduced. In the second approach – based on the yield condition – a quadratic cost function will be introduced. After the formulation of the stochastic optimization problem, the expected cost based optimization problem (ECBOP) and the minimum expected cost problem (MECP) are formulated as representatives of appropriate substitute problems. Subsequently, comparative numerical results using these methods are presented. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stochastic Material and Topology Design

In order to find a robust optimal topology or material design with respect to stochastic variations of the model parameters of a mechanical structure, the basic optimization problem under stochastic uncertainty must be replaced by an appropriate deterministic substitute problem. Starting from the equilibrium equation and the yield/strength conditions, the problem can be formulated as a stochastic (linear) program “with recourse”. Hence, by discretization the design space by finite elements, linearizing the yield conditions, in case of discrete probability distributions the resulting deterministic substitute problems are linear programs with a dual decomposition data structure.     

The Sample Average Approximation Method Applied to Stochastic Routing Problems: A Computational Study

The sample average approximation (SAA) method is an approach for solving stochastic optimization problems by using Monte Carlo simulation. In this technique the expected objective function of the stochastic problem is approximated by a sample average estimate derived from a random sample. The resulting sample average approximating problem is then solved by deterministic optimization techniques. The process is repeated with different samples to obtain candidate solutions along with statistical estimates of their optimality gaps.We present a detailed computational study of the application of the SAA method to solve three classes of stochastic routing problems. These stochastic problems involve an extremely large number of scenarios and first-stage integer variables. For each of the three problem classes, we use decomposition and branch-and-cut to solve the approximating problem within the SAA scheme. Our computational results indicate that the proposed method is successful in solving problems with up to 2¹⁶⁹⁴ scenarios to within an estimated 1.0% of optimality. Furthermore, a surprising observation is that the number of optimality cuts required to solve the approximating problem to optimality does not significantly increase with the size of the sample. Therefore, the observed computation times needed to find optimal solutions to the approximating problems grow only linearly with the sample size. As a result, we are able to find provably near-optimal solutions to these difficult stochastic programs using only a moderate amount of computation time.     

The complexity of branch-and-price algorithms for the capacitated vehicle routing problem with stochastic demands

The capacitated vehicle routing problem with stochastic demands (CVRPSD) is a variant of the deterministic capacitated vehicle routing problem where customer demands are random variables. While the most successful formulations for several deterministic vehicle-routing problem variants are based on a set-partitioning formulation, adapting such formulations for the CVRPSD under mild assumptions on the demands remains challenging. In this work we provide an explanation to such challenge, by proving that when demands are given as a finite set of scenarios, solving the LP relaxation of such formulation is strongly NP-Hard. We also prove a hardness result for the case of independent normal demands.     

Deterministic Routing With Bounded Buffers: Turning Offline Into Online Protocols

In this paper we present a deterministic protocol for routing arbitrary permutations in arbitrary networks. The protocol is analyzed in terms of the size of the network and the routing number of the network. Given a network H of n nodes, the routing number of H is defined as the maximum over all permutations on {1, ..., n} of the minimal number of steps to route offline in H. We show that for any network H of size n with routing number R our protocol needs time to route any permutation in H using only constant size edge buffers. This significantly improves all previously known results on deterministic routing. In particular, our result yields optimal deterministic routing protocols for arbitrary networks with diameter or bisection width , constant. Furthermore we can extend our result to deterministic compact routing. This yields, e.g., a deterministic routing protocol with runtime O(R logn) for arbitrary bounded degree networks if only O(logn) bits are available at each node for storing routing information. Our protocol is a combination of a generalized ``routing via simulation' technique with an new deterministic protocol for routing h-relations in an extended version of a multibutterfly network. This protocol improves upon all previous routing protocols known for variants of the multibutterfly network. The ``routing via simulation' technique used here extends a method previously introduced by the authors for designing compact routing protocols. Received July 18, 1997     

A hybrid metaheuristic for the vehicle routing problem with stochastic demand and duration constraints

The vehicle routing problem with stochastic demands (VRPSD) consists in designing optimal routes to serve a set of customers with random demands following known probability distributions. Because of demand uncertainty, a vehicle may arrive at a customer without enough capacity to satisfy its demand and may need to apply a recourse to recover the route’s feasibility. Although travel times are assumed to be deterministic, because of eventual recourses the total duration of a route is a random variable. We present two strategies to deal with route-duration constraints in the VRPSD. In the first, the duration constraints are handled as chance constraints, meaning that for each route, the probability of exceeding the maximum duration must be lower than a given threshold. In the second, violations to the duration constraint are penalized in the objective function. To solve the resulting problem, we propose a greedy randomized adaptive search procedure (GRASP) enhanced with heuristic concentration (HC). The GRASP component uses a set of randomized route-first, cluster-second heuristics to generate starting solutions and a variable-neighborhood descent procedure for the local search phase. The HC component assembles the final solution from the set of all routes found in the local optima reached by the GRASP. For each strategy, we discuss extensive computational experiments that analyze the impact of route-duration constraints on the VRPSD. In addition, we report state-of-the-art solutions for a established set of benchmarks for the classical VRPSD.     

p -Sparse BEM for weakly singular integral equation with random data

We consider the weakly singular boundary integral equation 𝒱u = g on a randomly perturbed smooth closed surface Γ(ω) with deterministic g or on a deterministic closed surface Γ with stochastic g (ω). The aim is the computation of the centered moments ℳ︁^k u, k ⩾ 1, if the corresponding moments of the perturbation are known. The problem on the stochastic surface is reduced to a problem on the nominal deterministic surface Γ with the random perturbation parameter κ (ω). Resulting formulation for the k th moment is posed in the tensor product Sobolev spaces and involve the k -fold tensor product operators. The standard full tensor product Galerkin BEM requires 𝒪(N^k) unknowns for the k th moment problem, where N is the number of unknowns needed to discretize the nominal surface Γ. Based on [1], we develop the p -sparse grid Galerkin BEM to reduce the number of unknowns to 𝒪(N (log N)^{k –1}) (cf. [2], [3] for the wavelet approach). (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Queueing Framework for Routing Problems with Time-dependent Travel Times

Assigning and scheduling vehicle routes in a dynamic environment is a crucial management problem. Despite numerous publications dealing with efficient scheduling methods for vehicle routing, very few addressed the inherent stochastic and dynamic nature of travel times. In this paper, a vehicle routing problem with time-dependent travel times due to potential traffic congestion is considered. The approach developed introduces the traffic congestion component based on queueing theory. This is an innovative modelling scheme to capture the stochastic behavior of travel times as it generates an analytical expression for the expected travel times as well as for the variance of the travel times. Routing solutions that perform well in the face of the extra complications due to congestion are developed. These more realistic solutions have the potential to reduce real operating costs for a broad range of industries which daily face routing problems. A number of datasets are used to illustrate the appropriateness of the novel approach. Moreover it is shown that static (or time-independent) solutions are often infeasible within a congested traffic environment which is generally the case on European road networks. Finally, the effect of travel time variability (obtained via the queueing approach) is quantified for the different datasets.      

随机需求的车辆路线问题的新模型

本主要研究随机需求的VRP问题,其中服务需求量满足二项式分布,根据期望值的大小我们提出了在一条路线上理想最大服务点数的新概念,并在此基础上建立了三种VRP问题的新模型,由于允许服务失败两次和部分服务使得模糊能适应多种实际问题,以模拟退火思想为基础的两阶段方法经修正后用于解新模型并取得较好的数值结果,理论分析和数值结果表明,新模型较好地描述随机需求的VRP问题,并且容易求解。     

Multi-level Monte Carlo Finite Element method for elliptic PDEs with stochastic coefficients

In Monte Carlo methods quadrupling the sample size halves the error. In simulations of stochastic partial differential equations (SPDEs), the total work is the sample size times the solution cost of an instance of the partial differential equation. A Multi-level Monte Carlo method is introduced which allows, in certain cases, to reduce the overall work to that of the discretization of one instance of the deterministic PDE. The model problem is an elliptic equation with stochastic coefficients. Multi-level Monte Carlo errors and work estimates are given both for the mean of the solutions and for higher moments. The overall complexity of computing mean fields as well as k-point correlations of the random solution is proved to be of log-linear complexity in the number of unknowns of a single Multi-level solve of the deterministic elliptic problem. Numerical examples complete the theoretical analysis.