Approximating the Stochastic Knapsack Problem: The Benefit of Adaptivity

Dean, Brian C.; Goemans, Michel X.; Vondrák, Jan

doi:10.1287/moor.1080.0330

Cited by 196 publications

(181 citation statements)

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“…We note that the problems we consider in this work are by nature multi-stage stochastic problems, which are usually much harder (see [5] for a recent result on the stochastic knapsack problem).…”

Section: Introductionmentioning

confidence: 99%

Approximation Algorithms for Stochastic Inventory Control Models

et al. 2007

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We consider two classical stochastic inventory control models, the periodic-review stochastic inventory control problem and the stochastic lot-sizing problem. The goal is to coordinate a sequence of orders of a single commodity, aiming to supply stochastic demands over a discrete, finite horizon with minimum expected overall ordering, holding and backlogging costs. In this paper, we address the important problem of finding computationally efficient and provably good inventory control policies for these models in the presence of correlated and non-stationary (time-dependent) stochastic demands. This problem arises in many domains and has many practical applications in supply chain management.Our approach is based on a new marginal cost accounting scheme for stochastic inventory control models combined with novel cost-balancing techniques. Specifically, in each period, we balance the expected cost of over ordering (i.e, costs incurred by excess inventory) against the expected cost of under ordering (i.e., costs incurred by not satisfying demand on time). This leads to what we believe to be the first computationally efficient policies with constant worst-case performance guarantees for a general class of important stochastic inventory models. That is, there exists a constant C such that, for any instance of the problem, the expected cost of the policy is at most C times the expected cost of an optimal policy. In particular, we provide worst-case guarantee of 2 for the periodic-review stochastic inventory control problem and a worst-case guarantee of 3 for the stochastic lot-sizing problem. Our results are valid for all of the currently known approaches in the literature to model correlation and non-stationarity of demands over time.

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Section: Introductionmentioning

confidence: 99%

Approximation Algorithms for Stochastic Inventory Control Models

et al. 2007

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“…We consider the ordered adaptive model which is discussed in [9]. In this model we are given a sequence (i.e., ordered set) of n items.…”

Section: Theorem 51 (Fptas For Monotone Dp) Every Monotone Dynamic mentioning

confidence: 99%

“…In the ordered nonadaptive model all n decisions are made in advance. In [9] the authors give a polynomial time algorithm for the stochastic ordered adaptive knapsack problem. For every > 0, their algorithm gives a solution whose value is at least the optimal value, at the expense of a slight loss in terms of feasibility, i.e., the total volume of the items placed in the knapsack does not exceed (1 + )B.…”

Section: Theorem 51 (Fptas For Monotone Dp) Every Monotone Dynamic mentioning

confidence: 99%

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A Fully Polynomial-Time Approximation Scheme for Single-Item Stochastic Inventory Control with Discrete Demand

et al. 2009

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We develop a framework for obtaining (deterministic) Fully Polynomial Time Approximation Schemes (FPTASs) for stochastic univariate dynamic programs with either convex or monotone single-period cost functions. Using our framework, we give the first FPTASs for several NP-hard problems in various fields of research such as knapsack-related problems, logistics, operations management, economics, and mathematical finance. IntroductionDynamic Programming (DP). Dynamic Programming is an algorithmic technique used for solving sequential, or multi-stage, decision problems and is a fundamental tool in combinatorial optimization (e.g., [17], Section 2.5 in [3], and Chapter 8 in [30]). A discrete time finite time horizon dynamic program is to find an optimal policy over a finite time horizon that minimizes the average cost. At the beginning of a time period, the state of the system is observed and an action is taken. Based on exogenous stochastic information, the state, and the action, the system incurs a single-period cost and transitions into a new state. The goal is to find a policy that realizes the minimal total expected cost over the entire time horizon.We can formally model this by means of Bellman's optimality equation. Let z t (I t ) be the cost-to-go (also known as the value function). The value z t (I t ) is simply the cost of an optimal policy from time period t to the end of the time horizon, given that at the beginning of time period t the state is I t . The equation reads (1.1) z t (I t ) = min x t ∈A t (I t ) E Dt {g t (I t , x t , D t ) + z t+1 (f t (I t , x t , D t ))}.

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“…A full range of articles is concerned with criteria that guarantee the optimality of simple policies for special scheduling problems; see, e.g., Pinedo [26]. It is only recently that research also focuses on approximations for less restrictive problem settings (Möhring et al [24], Skutella and Uetz [37], Megow et al [19], Schulz [30], Dean et al [8]). All these results apply to nonpreemptive scheduling, and we are not aware of any approximation results when job preemption is allowed and processing times are stochastic.…”

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confidence: 99%

A Tight 2-Approximation for Preemptive Stochastic Scheduling

Megow

Vredeveld

2014

Mathematics of OR

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We consider dynamic stochastic scheduling of preemptive jobs with processing times that follow independent discrete probability distributions. We derive a policy with a guaranteed performance ratio of 2 for the problem of minimizing the sum of weighted completion times on identical parallel machines subject to release dates. The analysis is tight. Our policy as well as their analysis applies also to the more general model of stochastic online scheduling.In contrast to previous results for nonpreemptive stochastic scheduling, our preemptive policy yields an approximation guarantee that is independent of the processing time distributions. However, our policy extensively utilizes information on the distributions other than the first (and second) moments. We also introduce a new nontrivial lower bound on the expected value of an unknown optimal policy. It relies on a relaxation to the basic problem on a single machine without release dates, which is known to be solved optimally by the Gittins index priority rule. This dynamic priority index is crucial to the analysis and also inspires the design of our policy.Keywords: scheduling; stochastic; approximation algorithm; total weighted completion time MSC2000 subject classification: Primary: 90B36; secondary: 68W40 OR/MS subject classification: Primary: production/scheduling; secondary: approximation/heuristic History: Received August 15, 2011; revised March 9, 2014. Published online in Articles in Advance.1. Introduction. Stochastic scheduling problems have attracted researchers for about four decades. A full range of articles is concerned with criteria that guarantee the optimality of simple policies for special scheduling problems; see, e.g., Pinedo [26]. It is only recently that research also focuses on approximations for less restrictive problem settings (Möhring et al. All these results apply to nonpreemptive scheduling, and we are not aware of any approximation results when job preemption is allowed and processing times are stochastic.In this paper, we give the first constant factor approximation for the stochastic version of the classical problem of scheduling jobs preemptively, with or without release dates, on identical parallel machines. The objective is to minimize the expected sum of weighted completion times.We provide a very natural scheduling policy and show that it has an expected objective value of at most twice the expected optimal value. The analysis of our policy is tight. Both, the policy and its analysis are also valid in the model of stochastic online scheduling (Megow et al. [19], Chou et al. [5], Vredeveld [39]). They rely on the celebrated Gittins index priority rule (Gittins [9, 10]), which is an optimal policy for the setting when there is only one machine and there are no nontrivial release dates (Sevcik [33], Weiss [41]). Under deterministic input, our policy corresponds to the preemptive weighted shortest processing time (WSPT) rule, which is known to have a tight performance guarantee of 2 on identical parallel machines (Megow and Schulz...

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Approximating the Stochastic Knapsack Problem: The Benefit of Adaptivity

Cited by 196 publications

References 21 publications

Approximation Algorithms for Stochastic Inventory Control Models

Approximation Algorithms for Stochastic Inventory Control Models

A Fully Polynomial-Time Approximation Scheme for Single-Item Stochastic Inventory Control with Discrete Demand

A Tight 2-Approximation for Preemptive Stochastic Scheduling

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