The dynamic and stochastic knapsack Problem with homogeneous‐sized items and postponement options

Abstract

This article generalizes the dynamic and stochastic knapsack problem by allowing the decision-maker to postpone the accept/reject decision for an item and maintain a queue of waiting items to be considered later. Postponed decisions are penalized with delay costs, while idle capacity incurs a holding cost. This generalization addresses applications where requests of scarce resources can be delayed, for example, dispatching in logistics and allocation of funding to investments. We model the problem as a Markov decision process and analyze it through dynamic programming. We show that the optimal policy with homogeneoussized items possesses a bithreshold structure, despite the high dimensionality of the decision space. Finally, the value (or price) of postponement is illustrated through numerical examples.

Multidimensional Knapsack Problem Based on Uncertain Measure

“…Based on Theorem 4, the expected value model for UMKP is equivalent to the following model: x j 2 f0; 1g; y j 2 f0; 1g; for j = 1; 2; ; 16. (9) With the help of mathematical software LINGO, we can solve this 0-1 programming problem (9). The result shows that the decision-maker should carry goods 2, 3, 5, 7, 9, 12, 13, and 14 into the bag, and the expected maximum total value of the items that can be carried in the bag is 296.…”

“…For instance, Kosuch [8] studied a two-stage stochastic knapsack problem with random weights. Feng and Hartman [9] investigated a stochastic knapsack problem with homogeneous-sized items and postponement options. Han et al [10] considered a certain class of chance-constrained knapsack problems where each item has an independent normally-distributed random weight.…”