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The purpose of this paper is to present a polynomial time algorithm which determines the lot sizes for purchase component in Material Requirement Planning (MRP) environments with deterministic time-phased demand with zero lead time. In this model, backlog is not permitted, the unit purchasing price is based on the all-units discount system and resale of the excess units is possible at the ordering time. The properties of an optimal order policy are argued and on the basis of them, a branch and bound algorithm is presented to construct an optimal sequence of order policies. In the proposed B&B algorithm, some useful fathoming rules have been proven to make the algorithm very efficient. By defining a rooted tree graph, it has been shown that the worst-case time complexity function of the presented algorithm is polynomial. Finally, some test problems which are randomly generated in various environments are solved to show the efficiency of the algorithm.

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The prevalent strategy in the topology optimization phase is to select a subset of members existing in an excessively connected truss, called Ground Structure, such that the overall weight or cost is minimized. Although finding a good topology significantly reduces the overall cost, excessive growth of the size of topology space combined with existence of varied types of design variables challenges applicability of evolutionary algorithms tailored for simultaneous optimization of topology, shape and size (TSS) in more complicated cases which are of great practical interest. In practice, large-scale truss structures are often modular, formed by joining periodically repeated units. This article organizes a novel simulation approach for this class of truss structures where the main drawbacks of the ground structure-based simulation approach are greatly moderated. The two approaches are independently employed for simultaneous TSS optimization of a modular truss example and the size of topology space as well as the required computation budget to generate an acceptable candidate design is compared. Result comparison reveals by employing the novel approach, problem complexity grows linearly with respect to the number of modules which allows for expanding application of TSS optimizers to complex modular trusses. Use of relative coordinates is also warranted for shape optimization which concludes to a more efficient optimization process.

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Existing project selection models do not consider the complexity of projects as a selection criterion, while their complexity may prolong the project duration and even result in its failure. In addition, existing models cannot formulate the aggregate complexity of the selected projects. The aggregated complexity is not always equal to summation of complexity of projects because of possible synergies or conflicts between them may increase or decrease the total complexity. In this paper, a model is proposed for measuring the aggregate complexity in the selection of project portfolios. A case study is presented to show the usefulness of the model and its applicability in practice. Moreover, several large-sized numerical examples have been tested showing the capability of the model to solve such problems in logical computational time.