Let G be a finite group. Then we denote ψ(G) = ∑ x∈G o(x) where o(x) is the order of the element x in G. In this paper we characterize some finite p-groups (p a prime) by ψ and their orders.Theorem 1.1. Suppose that P and Q are contained in CP 2 of the same order p n . Then the following statements are equivalent:(1) ψ(P ) = ψ(Q).(2(3) ψ(Ω i (P )) = ψ(Ω i (Q)) for all i ∈ N.
This paper presents a new Mixed-Integer Non-Linear Programming (MINLP) model for a Supply Vessel Planning (SVP) problem. The traditional SVP, which is a maritime transportation problem, is developed into a Maritime Fleet Sizing Mix Periodic Location-Routing Problem with Time Windows (MFSMPLRPTW) by considering suppliers, location of onshore base(s), and some real-life aspects. The objective of this model is to decide the composition of eets, optimal voyages, schedules, and the optimal location(s) for onshore base(s) in such a way that the total cost is minimized and the needs of operation regions are ful lled. The MFSMPLPRTW model is solved by an exact twophase solution approach for both small and medium cases. Moreover, two meta-heuristic algorithms are used to solve the large-sized instances. In order to justify and show how the model and solution can lead to signi cant economic improvements for real-life instances, a case study by the IOOC is considered, which is the only o shore oil and gas producer in Iran that has lots of installations and operation regions in the Persian Gulf and the Sea of Oman.
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