We consider a knapsackproblem with precedence constraints imposed on pairs of items, known as the precedence constrained knapsack problem (PCKP). This problem has applications in manufacturing and mining, and also ap...
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We consider a knapsackproblem with precedence constraints imposed on pairs of items, known as the precedence constrained knapsack problem (PCKP). This problem has applications in manufacturing and mining, and also appears as a subproblem in decomposition techniques for network design and related problems. We present a new approach for determining facets of the PCKP polyhedron based on clique inequalities. A comparison with existing techniques, that lift knapsack cover inequalities for the PCKP, is also presented. It is shown that the clique-based approach generates facets that cannot be found through the existing cover-based approaches, and that the addition of clique-based inequalities for the PCKP can be computationally beneficial, for both PCKP instances arising in real applications, and applications in which PCKP appears as an embedded structure.
We study an extension of the precedence constrained knapsack problem where the knapsack can be filled in multiple periods. This problem is known in the mining industry as the open-pit mine production scheduling proble...
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The open pit mine production scheduling problem (OPMPSP) consists of scheduling the extraction of a mineral deposit that is broken into a number of smaller segments, or blocks, such that the net present value (NPV) of...
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The open pit mine production scheduling problem (OPMPSP) consists of scheduling the extraction of a mineral deposit that is broken into a number of smaller segments, or blocks, such that the net present value (NPV) of the operation is maximised. This problem has been formulated as an integer programming (IP) model, involving both knapsack and precedence constraints. However, due to the large number of blocks and precedence constraints, this model has remained impractical in real planning applications. In this paper, we propose a new method to quickly generate near optimum feasible (integer) solutions by using the fractional solutions from the linear programming (LP) relaxation of the IP model. To be applicable to real sized problems, a new heuristic that quickly computes a feasible LP solution is also proposed. Our methodology is tested on a set of both academically designed and real-world mine deposits, and shows better performance than the heuristic used to tackle the same deposits in the literature. Interestingly, the proposed methodology improves the best known solutions for the majority of the instances. (C) 2017 Elsevier Ltd. All rights reserved.
This paper considers the minimization version of a class of nonconvex knapsackproblems with piecewise linear cost structure. The items to be included in the knapsack have a divisible quantity and a cost function. An ...
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This paper considers the minimization version of a class of nonconvex knapsackproblems with piecewise linear cost structure. The items to be included in the knapsack have a divisible quantity and a cost function. An item can be included partially ill the given quantity range and the cost is a nonconvex piecewise linear function of quantity. Given a demand, the optimization problem is to choose an optimal quantity for each item such that the demand is satisfied and the total cost is minimized. This problem and its close variants are encountered in manufacturing planning, supply chain design, volume discount procurement auctions, and many other contemporary applications. Two separate mixed integer linear programming formulations of this problem are proposed and are compared with existing formulations. Motivated by different scenarios in which the problem is useful, the following algorithms are developed: (1) a fast polynomial time, near-optimal heuristic using convex envelopes;(2) exact pseudo-polynomial time dynamic programming algorithms;(3) a 2-approximation algorithm;and (4) a fully polynomial time approximation scheme. A comprehensive test suite is developed to generate representative problem instances with different characteristics. Extensive computational experiments show that the proposed formulations and algorithms are faster than the existing techniques. (C) 2007 Elsevier B.V. All rights reserved.
In this paper, a new Specific Breakpoint Algorithm (SBA), which can efficiently search appropriate breakpoints of parametric maximum-flow-related problems, is presented. The algorithm is used to solve Lagrangian Relax...
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In this paper, a new Specific Breakpoint Algorithm (SBA), which can efficiently search appropriate breakpoints of parametric maximum-flow-related problems, is presented. The algorithm is used to solve Lagrangian Relaxed precedence constrained knapsack problem (LRPCKP) and Linear Programming Relaxed precedence constrained knapsack problem (LPRPCKP) in mine pushback design. The relaxed solutions are then processed through Rounded Topo-Sort (RoTS) heuristic to produce feasible solutions. The study results on seven bench mark datasets on Minelib for two approaches, referred here as LRPCKP-SBA and LPRPCKP-SBA, indicate that LRPCKP-SBA in spite of being faster, produces inferior quality solutions than well known BZ and CPLEX solutions. However, LPRPCKP-SBA produces a comparable quality of solutions as BZ in a computationally more efficient manner. Furthermore, the RoTS heuristics operated on relaxed solutions produce a better quality of feasible solutions than an existing technique, Expected Topo-Sort heuristic (ExTS).
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