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Abstract. In this paper, we model a research-and-development project as consisting of several modules, with each module containing one or more activities. We examine how to schedule the activities of such a project in order to maximize the expected profit when the activities have a probability of failure and when an activity's failure can cause its module and thereby the overall project to fail. A module succeeds when at least one of its constituent activities is successfully executed. All activities are scheduled on a scarce resource that is modeled as a single machine. We describe various policy classes, establish the relationship between the classes, develop exact algorithms to optimize over two different classes (one dynamic program and one branch-and-bound algorithm), and examine the computational performance of the algorithms on two randomly generated instance sets.Key words. scheduling, uncertainty, research and development, activity failures, modular precedence network AMS subject classifications. 90B35, 90B36, 90C27, 68M201. Introduction. Activities in a practical project are typically subject to many uncertainties; the most frequently studied types of uncertainty are resource breakdowns and duration variability. In research and development (R&D), activities may also fail altogether, for instance because the new technology under study does not perform as anticipated or because a toxicity test is not passed (in case of drug development). We model an R&D project as consisting of several modules, with each module containing one or more activities that pursue a homogeneous target, for instance representing repeated trials or technological alternatives. Each activity has a cost, a duration and a probability of success. A module is successful when at least one of its included activities succeeds. The successful completion of the whole project requires the successful completion of all the modules; project success equates with receiving a project payoff (cash inflow). This setup is subsequently referred to as 'modular project completion'. The objective is to schedule the activities in such a way that a maximum expected profit is attained. A solution to this scheduling problem is a policy, which is a dynamic decision rule that decides which activities are to be started at which time. We examine the scheduling of the project activities on a single machine, representing a scarce or bottleneck resource. Examples of such scarce resources are specialized equipment, or departments or individuals with specific areas of expertise (see Kavadias and Loch [14] for a similar motivation in a slightly different setting).The main contributions of this paper are fivefold: (1) we introduce and formulate a generic model for optimally scheduling R&D projects with modular completion; (2) we describe various scheduling policy classes and examine the relationship between the classes; (3) we provide an analysis of a number of properties; (4) we develop exact algorithms to optimize over two different classes (one dynamic program and o...

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In this paper, a multi depots capacitated electric vehicle routing problem where client demand is composed of two-dimensional weighted items (2L-MDEVRP) is addressed. This problem calls for the minimization of the transportation distance required for the delivery of the items which are demanded by the clients, carried out by a fleet of electric vehicles in several depots. Since the 2L-MDEVRP is an NP-hard problem, a heuristic algorithm combined variable neighborhood search algorithm (VNS) and space saving heuristic algorithm (SSH) is proposed. The VNS algorithm is used to solve the vehicle routing problem (VRP) sub-problem, and the SSH algorithm is used to solve the bin packing problem (BPP) subproblem. We propose the space saving heuristic to find the best matching solution between the next loading item and the feasible loading position. The SSH-VNS algorithm is tested by using benchmark instances available from the literature. The results show that the SSH-VNS algorithm has better performance compared with other published results for solving capacity vehicle routing problem (CVRP) and two-dimensional capacity vehicle routing problem (2L-CVRP). Some new best-known solutions of the benchmark problem are also found by SSH-VNS. Moreover, the effectiveness of the proposed algorithm on 2L-MDEVRP is demonstrated through numerical experiments and a practical logistic distribution case. In the last section, the managerial implications and suggestions for future research are also discussed.INDEX TERMS Vehicle routing problem, two-dimensional loading, electric vehicle, variable neighborhood search, space saving heuristic.

Tactical production-distribution planning models have attracted a great deal of attention in the past decades. In these models, production and distribution decisions are considered simultaneously such that the combined plans are more advantageous than the plans resolved in a hierarchical planning process. We consider a two-stage production process, where in the first stage raw materials are transformed into continuous resources that feed the discrete production of end products in the second stage. Moreover, the setup times and costs of resources depend on the sequence in which they are processed in the first stage. The minimum scheduling unit is the product family which consists of products sharing common resources and manufacturing processes. Based on different mathematical modelling approaches to the production in the first stage, we develop a sequence-oriented formulation and a product-oriented formulation, and propose decompositionbased heuristics to solve this problem efficiently. By considering these dependencies arising in practical production processes, our model can be applied to various industrial cases, such as the beverage industry or the steel industry. Computation tests on instances from an industrial application are provided at the end of the paper.