![]() ![]() But a solution to SMKP may actually be validated by solving a series of Subset-sum Problems, where we try to split the chosen items into the m knapsacks. The concept of bound-and-bound algorithms was intended for problems where feasibility of the upper bound solution is difficult to validate. In order to evade these problem, Martello and Toth proposed a bound-and-bound algorithm for MKP where at each New algorithm Some knowledge to guide the branching towards good feasible solutions is needed. The branch-and-bound algorithm needs good lower bounds for fathoming nodes in the enumeration. Generally it is difficult to verify feasibility of an upper bound obtained either by surrogate relaxation or Lagrangian relaxation. Martello and Toth focused on three aspects that make it difficult to solve MKP: ∑ j=1 n w j x ij ⩽c i, i=1,…,m, ∑ i=1 m x ij ⩽1, j=1,…,n, x ij ∈ for the relaxation, thus usually a subgradient optimization technique must be used, Bound-and-bound algorithm Thus we may formally define the 0–1 Multiple Knapsack Problem (MKP) as the ILP problem max ∑ i=1 m ∑ j=1 n p j x ij s.t. Each item j has an associated profit p j and weight w j, and the problem is to select m disjoint subsets of items, such that subset i fits into capacity c i and the total profit of the selected items is maximized. We consider the problem where n given items should be packed in m knapsacks of distinct capacities c i, i=1,…,m. A surprising result is that large instances with n=100 000 items may be solved in less than a second, and the algorithm has a stable performance even for instances with coefficients in a moderately large range. ![]() The developed algorithm is compared to the mtm algorithm by Martello and Toth, showing the benefits of the new approach. A new separable dynamic programming algorithm is presented for the solution of Subset-sum Problems, and we also use this algorithm for tightening the capacity constraints in order to obtain better upper bounds. The recursive branch-and-bound algorithm applies surrogate relaxation for deriving upper bounds, while lower bounds are obtained by splitting the surrogate solution into the m knapsacks by solving a series of Subset-sum Problems. A new exact algorithm for the MKP is presented, which is specially designed for solving large problem instances. The problem has several applications in naval as well as financial management. Finally, in the full version of this paper, we illustrate the proposed algorithm via trace-based experiments of EV charging.The Multiple Knapsack Problem (MKP) is the problem of assigning a subset of n items to m distinct knapsacks, such that the total profit sum of the selected items is maximized, without exceeding the capacity of each of the knapsacks. Moreover, our analysis provides a novel approach to online algorithm design based on an instance-dependent primal-dual analysis that connects the identification of worst-case instances to the design of algorithms. We introduce a new algorithm that achieves a competitive ratio within an additive factor of the best achievable competitive ratios for the general problem and matches or improves upon the best-known competitive ratio for special cases in the knapsack and one-way trading literatures. This problem generalizes variations of the knapsack problem and of the one-way trading problem that have previously been treated separately, and additionally finds application to the real-time control of electric vehicle (EV) charging. We introduce and study a general version of the fractional online knapsack problem with multiple knapsacks, heterogeneous constraints on which items can be assigned to which knapsack, and rate-limiting constraints on the assignment of items to knapsacks.
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