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This paper proposes a methodology for the optimal design of series of sewer pipes including pumping stations. It employs a Shortest Path Algorithm to select the optimal combination of pipe diameters and invert elevations, as well as the optimal pumping features such as the number of pumps, location, and pumping head. The methodology is intended to be applied to the design of any sewer series. Although these are uncommon in real infrastructure, the study allows an analysis of the effect of pipe roughness, inflows, and pipe length on the total cost of the system and pumping features. The methodology was tested in theoretical series of 10 and 20 pipes as well as in a real series that is part of the sewer system of Bogotá, Colombia. The resulting designs suggested that using smooth pipes and reducing the pumping flow rate would decrease the cost of sewer systems in flat terrain. [ABSTRACT FROM AUTHOR]

The Shortest path problem is highly relevant in our daily lives, addressing uncertainties like traffic conditions and weather variations. To handle such uncertainties, we utilize Fuzzy Numbers. This paper focuses on Bipolar Neutrosophic Fuzzy Numbers, which have dual positive and negative aspects. They provide a robust framework for representing arc (node/edge) weights, signifying uncertain travel times between nodes. Importantly, these weights can change over time in bipolar neutrosophic fuzzy graphs. Our study introduces an extended Bellman-Ford Algorithm for identifying optimal paths and minimum times with time-dependent Bipolar Neutrosophic Fuzzy arc weights. We demonstrate its effectiveness through a step-by-step numerical example and conduct a comparative analysis to evaluate its efficiency. [ABSTRACT FROM AUTHOR]

The shortest path problem is a classic optimization problem in graph theory and computer technology. It involves identifying the shortest path between two nodes in a graph, where each edge has a numerical weight. In this paper, we put our effort into examining the use of the dynamic programming method to evaluate the shortest path (SP) between the two specified nodes in a multistage network where the parameter is a multi-value neutrosophic number (MVNN). Firstly, we propose an algorithm based on the forward and backward approach in an uncertain environment and also implement our approach in the Python-3 programming language. Furthermore, a numerical illustration has been provided to showcase the effectiveness and robustness of the novel model. [ABSTRACT FROM AUTHOR]