Abstract
The knapsack feasibility problems have been intensively studied both because of their immediate applications in industry and financial management, but more pronounced for theoretical reasons, as knapsack problems frequently occur by relaxation of various integer programming problems. In this work, the large-scale knapsack feasibility problem is divided into two subproblems. The first subproblem is transforming of the knapsack feasibility problem into a polytope judgement problem which is based on lattice basis reduction. In the next subproblem, a distributed implementation of Dang and Ye’s fixed-point iterative algorithm is introduced to solve the polytope judgement problem generated in the former subproblem. Compared with the branch and bound method, numerical results show that this distributed fixed-point method is effective.
Highlights
The knapsack problems, which have been intensively studied since the pioneering work of Dantzig (1957), have played an important role in industry, financial management, and so on
Various integer programming problems can be relaxed to the knapsack problems
Inspired by region partition of items, an effective hybrid algorithm based on greedy degree and expectation efficiency is constructed in the paper (Lv et al 2016). Combining advanced features both from the path relinking method and the responsive threshold search algorithm, the first evolutionary path relinking approach is introduced in paper (Chen et al 2016) for solving the quadratic multiple knapsack problem approximately
Summary
The knapsack problems, which have been intensively studied since the pioneering work of Dantzig (1957), have played an important role in industry, financial management, and so on. Based on the harmony search method, a new binary-coded version of harmony search (Kong et al 2015) is presented to solve large-scale multidimensional knapsack problem. In this proposed algorithm, attention is paid to the probability distribution rather than the exact value of each decision variable, and the concept of mean harmony is developed in the memory consideration. The paper (Wu et al 2017b) uses the Dang and Ye’s iterative fixed-point method for integer programming to generate feasible flight routes which are used to construct an aircraft reassignment in response to the grounding of one aircraft. No jk s.t. x j P ( y q , k ), j k 1, k 2 , , n
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