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This method solves the cutting-stock problem by starting with just a few patterns. It generates additional patterns when they are needed. For the one-dimensional case, the new patterns are introduced by solving an auxiliary optimization problem called the knapsack problem , using dual variable information from the linear program .
A number of software products are available for calculating cut and fill. A simple approach involves defining different earthworks features in a computer program and then adjusting elevations manually to calculate the optimal cut and fill. More sophisticated software is able to automatically balance cut and fill while also considering the ...
In that case, earthwork software is principally used to calculate cut and fill volumes which are then used for producing material and time estimates. Most products offer additional functionality such as the ability to takeoff terrain elevation from plans (using contour lines and spot heights ); produce shaded cut and fill maps; produce cross ...
This description assumes the ILP is a maximization problem.. The method solves the linear program without the integer constraint using the regular simplex algorithm.When an optimal solution is obtained, and this solution has a non-integer value for a variable that is supposed to be integer, a cutting plane algorithm may be used to find further linear constraints which are satisfied by all ...
More generally, if the objective function is not a quadratic function, then many optimization methods use other methods to ensure that some subsequence of iterations converges to an optimal solution. The first and still popular method for ensuring convergence relies on line searches , which optimize a function along one dimension.
In civil engineering, a cut or cutting is where soil or rock from a relative rise is removed. Cuts are typically used in road, rail, and canal construction to reduce a route's length and grade. Cut and fill construction uses the spoils from cuts to fill in defiles to create straight routes at steady grades cost-effectively.
Knapsack problems appear in real-world decision-making processes in a wide variety of fields, such as finding the least wasteful way to cut raw materials, [3] selection of investments and portfolios, [4] selection of assets for asset-backed securitization, [5] and generating keys for the Merkle–Hellman [6] and other knapsack cryptosystems.
Section through railway track and foundation showing the sub-grade. Grading in civil engineering and landscape architectural construction is the work of ensuring a level base, or one with a specified slope, [1] for a construction work such as a foundation, the base course for a road or a railway, or landscape and garden improvements, or surface drainage.