An algorithm for solving a class of knapsack problems and its generalization

A class of knapsack problems which generalizes the classical ones is studied. Algorithms based on the dynamical programming and Branch and Bound methods are proposed. The correctness and time estimation of the algorithm are given. 1 Problem formulation We study a class of knapsack problems which generalizes problems from [1-2]. The main mathematical model we shall use is formulated as follows. Let a knapsack of size D and a set of items I = {1, 2, . . . , n} be given. For each item j ∈ I the cost cj , the size dj and the volume dj related to partition I are known. The set I is divided into p non-empty disjoint subsets I1, I2, . . . , Ip, I = I1 ∪ I2 ∪ · · · ∪ Ip; Ii ∩ Ij = ∅, i 6= j. For each subset Il, l = 1, p, the size Dl is known. According to this partition, from each of the subsets Il, l = 1, p, we include into the knapsack no more than kl items and the sum of volumes dj of these including items from Il does not exceed Dl. A sequence of items i1, i2, . . . , iq (q < n) from I we name the packing in the knapsack if the sum of items sizes dj of the sequence does not exceed D, each of the subsets Il, l = 1, p, contains at most kl items of the sequence and the sum of items volumes related to the partition dj