The Adaptive Knapsack Problem with Stochastic Rewards
نویسندگان
چکیده
Given a set of items with associated deterministic weights and random rewards, the Adaptive Stochastic Knapsack Problem (Adaptive SKP) maximizes the probability of reaching a predetermined target reward level when items are inserted sequentially into a capacitated knapsack before the reward of each item is realized. This model arises in resource allocation problems which permit or require sequential allocation decisions in a probabilistic setting. One particular application is in obsolescence inventory management. In this paper, the Adaptive SKP is formulated as a Dynamic Programming (DP) problem for discrete random rewards. The paper also presents a heuristic that mixes adaptive and static policies to overcome the “curse of dimensionality” in the DP. The proposed heuristic is extended to problems with Normally distributed random rewards. The heuristic can solve large problems quickly and its solution always outperforms a static policy. The numerical study indicates that a near-optimal solution can be obtained by using an algorithm with limited look-ahead capabilities.
منابع مشابه
Approximation Algorithms for Correlated Knaspacks and Non-Martingale Bandits
In the stochastic knapsack problem, we are given a knapsack of size B, and a set of jobs whose sizes and rewards are drawn from a known probability distribution. However, the only way to know the actual size and reward is to schedule the job—when it completes, we get to know these values. How should we schedule jobs to maximize the expected total reward? We know constant-factor approximations f...
متن کاملApproximation Algorithms for Stochastic k-TSP
This paper studies the stochastic variant of the classical k-TSP problem where rewards at the vertices are independent random variables which are instantiated upon the tour’s visit. The objective is to minimize the expected length of a tour that collects reward at least k. The solution is a policy describing the tour which may (adaptive) or may not (non-adaptive) depend on the observed rewards....
متن کاملStochastic Knapsack
Problem Definition This problem deals with packing a maximum reward set of items into a knapsack of given capacity, when the item-sizes are random. The input is a collection of n items, where each item i ∈ [n] := {1, · · · , n} has reward r i ≥ 0 and size S i ≥ 0, and a knapsack capacity B ≥ 0. In the stochastic knapsack problem, all rewards are deterministic but the sizes are random. The rando...
متن کاملOn a Stochastic Knapsack Problem
The deterministic knapsack problem is a well known and well studied NP-hard combinatorial optimization problem. It consists in filling a knapsack with items out of a given set such that the weight capacity of the knapsack is respected and the total reward maximized. For a review of references on the stochastic knapsack problem, stochastic gradient algorithms and branch-and-bound methods see [4]...
متن کاملThe Dynamic and Stochastic Knapsack Problem with Deadlines yJason
In this paper a dynamic and stochastic model of the well-known knapsack problem is developed and analyzed. The problem is motivated by a wide variety of real-world applications. Objects of random weight and reward arrive according to a stochastic process in time. The weights and rewards associated with the objects are distributed according to a known probability distribution. Each object can ei...
متن کامل