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Optimization is a mathematical discipline or tool suitable for minimizing or maximizing a function. It has been largely used in every scientific field to solve problems where it is necessary to find a local or global optimum. In the engineering field of localization, optimization has been adopted too, and in the literature, there are several proposals and applications that have been presented. In the first part of this article, the optimization problem is presented by considering the subject from a purely theoretical point of view and both single objective (SO) optimization and multi-objective (MO) optimization problems are defined. Additionally, it is reported how local and global optimization problems can be tackled differently, and the main characteristics of the related algorithms are outlined. In the second part of the article, extensive research about local and global localization algorithms is reported and some optimization methods for local and global optimum algorithms, such as the Gauss–Newton method, Genetic Algorithm (GA), Particle Swarm Optimization (PSO), Differential Evolution (DE), and so on, are presented; for each of them, the main concept on which the algorithm is based, the mathematical model, and an example of the application proposed in the literature for localization purposes are reported. Among all investigated methods, the metaheuristic algorithms, which do not exploit gradient information, are the most suitable to solve localization problems due to their flexibility and capability in solving non-convex and non-linear optimization functions. [ABSTRACT FROM AUTHOR]

Background: Supply chains are the networks linking sources of supply with demand points and composed of so-called actors, i.e., producers, distributors/wholesalers, retailers, and customers/consumers. As in every network, supply chains contain vertices and arcs, the former represented by factories and warehouses (including distribution centers). Such facilities cause long-term and expensive investments. As a result, decisions on location and number of them belong to the strategic level of management and require quantitative analysis. To do this, mathematical models of the Facility Location Problem (FLP) are constructed to allow an application of optimization methods. Methods: Mathematical optimization or programming is the selection of the best solution, with regard to some criterion, from a set of feasible alternatives. The fundamental of mathematical optimization is the formulation of mathematical models of analyzed problems. Mathematical models are composed of objective function, decision variables, constraints, and parameters. These components are presented and compared in the paper concerning FLP from a supply chain perspective. Results: The ten mathematical models of the FLP are presented, including the two original ones. The models are classified according to such features as facility type they concern, including the desirable, neutral, and undesirable ones. The models and their components are characterized. In addition, their applicability and elasticity are analyzed. Finally, the models are compared and discussed from the supply chain point of view. Conclusions: However, the FLP mathematical models are relatively similar; the most important element of them for supply chain appropriate representation is an objective function. It strongly influences the possible applicability of FLP models and their solutions, as well. The objective functions having broader applicability turned out to be the maximized number of supply/demand points covered by facilities and the minimized number of facilities necessary to cover supply/demand points. However, not to locate all allowed facilities (use all the location sites) or as many as supply/demand points, but an appropriate number of them, it is necessary to take into account facility fixed costs. Thus, when locating logistics facilities, the minimized total cost of serving supply/demand points is the most appropriate objective function. [ABSTRACT FROM AUTHOR]

The berth allocation problem determining the berthing time and position for incoming vessels in port operations has garnered increased attention within the global transportation network. This study focuses on the berth allocation problem with a continuous quay and dynamic vessel arrivals. With the overarching goal of enhancing service quality and optimizing berth utilization rates, this article proposes a mathematical programming model that minimizes the total waiting time of vessels and the overall completion time of vessel service. The formulated model is a mixed-integer linear programming problem that deterministic optimization techniques can globally solve. For large-scale problems, this study develops a genetic algorithm optimization approach to improve computational efficiency in reaching a near-optimal solution. Several numerical experiments are conducted to demonstrate the effectiveness and efficiency of the proposed approach. [ABSTRACT FROM AUTHOR]

The geometric programming problem (GPP) is a beneficial mathematical programming problem for modeling and optimizing nonlinear optimization problems in various engineering fields. The structural configuration of the GPP is quite dynamic and flexible in modeling and fitting the reliability optimization problems efficiently. The work's motivation is to introduce a bounded solution approach for the GPP while considering the variation among the right-hand-side parameters. The bounded solution method uses the two-level mathematical programming problems and obtains the solution of the objective function in a specified interval. The benefit of the bounded solution approach can be realized in that there is no need for sensitivity analyses of the results output. The demonstration of the proposed approach is shown by applying it to the system reliability optimization problem. The specific interval is determined for the objective values and found to be lying in the optimal range. Based on the findings, the concluding remarks are presented. [ABSTRACT FROM AUTHOR]