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The deflection of a simply supported beam that carries a central point load can be analyzed using structural mechanics principles, particularly by applying Castigliano's theorem. This theorem relates the displacement at the load application point to the partial derivatives of the strain energy in the structure. The simply supported beam with a point load at its center has symmetric reaction forces at the supports, each bearing half of the load. The bending moment at any point along the beam...
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Consider a man with a mass of 70 kg seated in a chair connected to a pin support through a member BC. If the man maintains an upright position, the task is to determine the horizontal and vertical reactions of the chair on the man when the member makes a 45° angle with the horizontal. At this moment, the man has a speed of 5 m/s, increasing at a rate of 1 m/s².
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Statically indeterminate problems are those where statics alone can not determine the internal forces or reactions. Consider a structure comprising two cylindrical rods made of steel and brass. These rods are joined at point B and restrained by rigid supports at points A and C. Now, the reactions at points A and C and the deflection at point B are to be determined. This rod structure is classified as statically indeterminate as the structure has more supports than are necessary for maintaining...
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Pappus and Guldinus's theorems are powerful mathematical principles that are used for finding the surface area and volume of composite shapes. For example, consider a cylindrical storage tank with a conical top. Finding the surface area or volume can be challenging for such complex shapes. These theorems are particularly useful in calculating the volume and surface area of such systems. Here, the cylindrical storage tank with a conical top can be broken down into two simple shapes: a...
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The dot product is a powerful tool in problem-solving involving vectors, given that the dot product of two vectors is the product of their magnitudes and the cosine of the angle between them measured anti-clockwise. Solving problems involving the dot product requires understanding its properties and developing a step-by-step process to solve them. Here are the main steps to follow when solving any general problem involving the dot product:
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Evaluating the Effect of Roadside Parking on a Dual-Direction Urban Street
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Cumulative Capacitated Colored Traveling Salesman Problem.

Xiangping Xu, Jinde Cao, Xinli Shi

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    |December 20, 2023
    PubMed
    Summary

    A new Cumulative Capacitated Colored Traveling Salesman Problem (C2-CTSP) model is introduced for efficient routing. A General Variable Neighborhood Search (GVNS) metaheuristic significantly outperforms existing algorithms in solving this complex problem.

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    Area of Science:

    • Operations Research
    • Combinatorial Optimization
    • Logistics

    Background:

    • The Colored Traveling Salesman Problem (CTSP) extends the classic TSP by incorporating city accessibility constraints based on colors.
    • Real-world routing problems often require fast response times and consider customer capacities, necessitating specialized models.

    Purpose of the Study:

    • To introduce a novel city/customer-centric model, the Cumulative Capacitated CTSP (C2-CTSP), addressing practical routing challenges.
    • To develop the first hypergraph and mathematical programming formulations for the C2-CTSP.
    • To design and evaluate a General Variable Neighborhood Search (GVNS) metaheuristic for solving the C2-CTSP.

    Main Methods:

    • Developed hypergraph and mathematical programming formulations for the C2-CTSP.
    • Designed a GVNS metaheuristic incorporating greedy backtracking for initialization, random perturbation operations (2-swap, reinsertion, double-bridge), and variable neighborhood descent for local search (neighborhood-list-2-opt, relocation, generalized partition crossover).
    • Conducted extensive experiments comparing GVNS against genetic algorithms, ant colony systems, and other VNS methods on 20 test cases.

    Main Results:

    • The proposed GVNS demonstrated superior search ability and convergence rate compared to all competing algorithms, including tuned genetic algorithms and ant colony systems.
    • Statistical analysis confirmed the significant performance advantage of GVNS.
    • Analysis of GVNS variants validated the crucial contribution of each operator to its overall effectiveness.

    Conclusions:

    • The C2-CTSP model effectively addresses practical routing problems with fast response and capacity constraints.
    • The developed GVNS metaheuristic is a highly effective and efficient method for solving the C2-CTSP.
    • Each component of the GVNS plays a vital role in achieving its superior performance in complex routing scenarios.