Stochastic Knapsack Revisited: The Service Level Perspective

A key challenge in the resource allocation problem is to find near-optimal policies to serve different customers with random demands/revenues, using a fixed pool of capacity (properly configured). Three classes of allocation policies, responsive (with perfect hindsight), adaptive (with information updates), and anticipative (with forecast information) policies, are widely used in practice. We analyze and compare the performances of these policies for both capacity minimization and revenue maximization models. In both models, the performance gaps between optimal anticipative policies and adaptive policies are shown to be bounded when the demand and revenue of each item are independently generated. In contrast, the gaps between the optimal adaptive policies and responsive policies can be arbitrarily large. More importantly, we show that the techniques developed, and the persistency values obtained from the optimal responsive policies can be used to design good adaptive and anticipative policies for the other two variants of resource allocation problems.

A Risk-Aversion Approach for the Multiobjective Stochastic Programming Problem

Multiobjective stochastic programming is a field that is well suited to tackling problems that arise in many fields: energy, financial, emergencies, among others; given that uncertainty and multiple objectives are usually present in such problems. A new concept of solution is proposed in this work, which is especially designed for risk-averse solutions. The proposed concept combines the notions of conditional value-at-risk and ordered weighted averaging operator to find solutions protected against risks due to uncertainty and under-achievement of criteria. A small example is presented in order to illustrate the concept in small discrete feasible spaces. A linear programming model is also introduced to obtain the solution in continuous spaces. Finally, computational experiments are performed by applying the obtained linear programming model to the multiobjective stochastic knapsack problem, gaining insight into the behaviour of the new solution concept.

The Risk-Averse Static Stochastic Knapsack Problem

This research proposes and analyzes new models for a stochastic resource allocation problem that arises in a variety of operations contexts. One of the primary contributions of the paper lies in providing a succinct, robust, and general model that can address a range of different risk-based objectives and cost assumptions under uncertainty. Although the model expression is relatively simple, it embeds a reasonably high degree of underlying complexity, as the analysis shows. In addition, in-depth analysis of the model, both in its general form and under various specific risk measures, uncovers some interesting and powerful insights regarding the problem trade-offs. Furthermore, this analysis leads to a highly efficient class of heuristic algorithms for solving the problem, which we demonstrate via numerical experimentation to provide close-to-optimal solutions. This computational benefit is a critical element for solving a class of broadly applicable larger problems for which our problem arises as a subproblem that requires repeated solution.

New Bounds for the Stochastic Knapsack Problem

The knapsack problem is a problem in combinatorial optimization that seeks to maximize the objective function \(\sum_{i = 1}^{n} v_ix_i\) subject to the constraints \(\sum_{i = 1}^{n} w_ix_i \leq W\) and \(x_i \in \{0, 1\}\), where \(\mathbf{x}, \mathbf{v} \in \mathbb{R}^{n}\) and \(W\) are provided. We consider the stochastic variant of this problem in which \(\mathbf{v}\) remains deterministic, but \(\mathbf{x}\)is an \(n\)-dimensional vector drawn uniformly at random from \([0, 1]^{n}\). We establish a sufficient condition under which the summation-bound condition is almost surely satisfied. Furthermore, we discuss the implications of this result on the deterministic problem.

New Bounds for the Stochastic Knapsack Problem

The knapsack problem is a problem in combinatorial optimization that seeks to maximize the objective function \(\sum_{i = 1}^{n} v_ix_i\) subject to the constraints \(\sum_{i = 1}^{n} w_ix_i \leq W\) and \(x_i \in \{0, 1\}\), where \(\mathbf{x}, \mathbf{v} \in \mathbb{R}^{n}\) and \(W\) are provided. We consider the stochastic variant of this problem in which \(\mathbf{v}\) remains deterministic, but \(\mathbf{x}\)is an \(n\)-dimensional vector drawn uniformly at random from \([0, 1]^{n}\). We establish a sufficient condition under which the summation-bound condition is almost surely satisfied. Furthermore, we discuss the implications of this result on the deterministic problem.

Logarithmic Regret in the Dynamic and Stochastic Knapsack Problem with Equal Rewards

We study a dynamic and stochastic knapsack problem in which a decision maker is sequentially presented with items arriving according to a Bernoulli process over n discrete time periods. Items have equal rewards and independent weights that are drawn from a known nonnegative continuous distribution F. The decision maker seeks to maximize the expected total reward of the items that the decision maker includes in the knapsack while satisfying a capacity constraint and while making terminal decisions as soon as each item weight is revealed. Under mild regularity conditions on the weight distribution F, we prove that the regret—the expected difference between the performance of the best sequential algorithm and that of a prophet who sees all of the weights before making any decision—is, at most, logarithmic in n. Our proof is constructive. We devise a reoptimized heuristic that achieves this regret bound.