Adaptive policies for multi-mode project scheduling under uncertainty

We study stochastic variants of Packing Integer Programs (PIP) --- the problems of finding a maximum-value 0/1 vector x satisfying Ax ≤ b, with A and b nonnegative. Many combinatorial problems belong to this broad class, including the knapsack problem, maximum clique, stable set, matching, hypergraph matching (a.k.a. set packing), b-matching, and others. PIP can also be seen as a multidimensional knapsack problem where we wish to pack a maximum-value collection of items with vector-valued sizes. In our stochastic setting, the vector-valued sizes of each item is known to us apriori only as a probability distribution, and the size of an item is instantiated once we commit to including the item in our solution.Following the framework of [3], we consider both adaptive and non-adaptive policies for solving such problems, adaptive policies having the flexibility of being able to make decisions based on the instantiated sizes of items already included in the solution. We investigate the adaptivity gap for these problems: the maximum ratio between the expected values achieved by optimal adaptive and non-adaptive policies. We show tight bounds on the adaptivity gap for set packing and b-matching, and we also show how to find efficiently non-adaptive policies approximating the adaptive optimum. For instance, we can approximate the adaptive optimum for stochastic set packing to within O(d1/2), which is not only optimal with respect to the adaptivity gap, but it is also the best known approximation factor in the deterministic case. It is known that there is no polynomial-time d1/2-e approximation for set packing, unless NP = ZPP. Similarly, for b-matching, we obtain algorithmically a tight bound on the adaptivity gap of O(λ) where λ satisfies Σ λbj+1 = 1.For general Stochastic Packing, we prove that a simple greedy algorithm provides an O(d)-approximation to the adaptive optimum. For A ∈ [0, 1]dxn, we provide an O(λ) approximation where Σ 1/λbj = 1. (For b = (B, B,..., B), we get λ = d1/B.) We also improve the hardness results for deterministic PIP: in the general case, we prove that a polynomial-time d1-e-approximation algorithm would imply NP = ZPP. In the special case when A ∈ [0,1]dxn and b = (B,B,...,B), we show that a d1/B-∈-approximation would imply NP = ZPP. Finally, we prove that it is PSPACE-hard to find the optimal adaptive policy for Stochastic Packing in any fixed dimension d ≥ 2.

This paper presents scheduling algorithms for procrastinators, where the speed that a procrastinator executes a job increases as the due date approaches. We give optimal off-line scheduling policies for linearly increasing speed functions. We then explain the computational/numerical issues involved in implementing this policy. We next explore the online setting, showing that there exist adversaries that force any online scheduling policy to miss due dates. This impossibility result motivates the problem of minimizing the maximum interval stretch of any job; the interval stretch of a job is the job's flow time divided by the job's due date minus release time. We show that several common scheduling strategies, including the "hit-the-highest-nail" strategy beloved by procrastinators, have arbitrarily large maximum interval stretch. Then we give the "thrashing" scheduling policy and show that it is a \Theta(1) approximation algorithm for the maximum interval stretch.

We study the scheduling of multiple tasks under varying processing costs and derive a priority rule for optimal scheduling policies. Each task has a due date, and a non‐completion penalty cost is incurred if the task is not completely processed before its due date. We assume that the task arrival process is stochastic and the processing rate is capacitated. Our work is motivated by both traditional and emerging application domains, such as construction industry and freelance consulting industry. We establish the optimality of Shorter Slack time and Longer remaining Processing time (SSLP) principle that determines the priority among active tasks. Based on the derived structural properties, we also propose an effective cost‐balancing heuristic policy and demonstrate the efficacy of the proposed policy through extensive numerical experiments. We believe our results provide operators/managers valuable insights on how to devise effective service scheduling policies under varying costs.

We consider a stochastic variant of the NP-hard 0/1 knapsack problem, in which item values are deterministic and item sizes are independent random variables with known, arbitrary distributions. Items are placed in the knapsack sequentially, and the act of placing an item in the knapsack instantiates its size. Our goal is to compute a solution “policy” that maximizes the expected value of items successfully placed in the knapsack, where the final overflowing item contributes no value. We consider both nonadaptive policies (that designate a priori a fixed sequence of items to insert) and adaptive policies (that can make dynamic choices based on the instantiated sizes of items placed in the knapsack thus far). An important facet of our work lies in characterizing the benefit of adaptivity. For this purpose we advocate the use of a measure called the adaptivity gap: the ratio of the expected value obtained by an optimal adaptive policy to that obtained by an optimal nonadaptive policy. We bound the adaptivity gap of the stochastic knapsack problem by demonstrating a polynomial-time algorithm that computes a nonadaptive policy whose expected value approximates that of an optimal adaptive policy to within a factor of four. We also devise a polynomial-time adaptive policy that approximates the optimal adaptive policy to within a factor of 3 + ε for any constant ε > 0.

Modeling adaptation in multi-state resource systems

Previous chapter Next chapter Full AccessProceedings Proceedings of the 2011 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)Improved Approximation Results for Stochastic Knapsack ProblemsAnand Bhalgat, Ashish Goel, and Sanjeev KhannaAnand Bhalgat, Ashish Goel, and Sanjeev Khannapp.1647 - 1665Chapter DOI:https://doi.org/10.1137/1.9781611973082.127PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstract In the stochastic knapsack problem, we are given a set of items each associated with a probability distribution on sizes and a profit, and a knapsack of unit capacity. The size of an item is revealed as soon as it is inserted into the knapsack, and the goal is to design a policy that maximizes the expected profit of items that are successfully inserted into the knapsack. The stochastic knapsack problem is a natural generalization of the classical knapsack problem, and arises in many applications, including bandwidth allocation, budgeted learning, and scheduling. An adaptive policy for stochastic knapsack specifies the next item to be inserted based on observed sizes of the items inserted thus far. The adaptive policy can have an exponentially large explicit description and is known to be PSPACE-hard to compute. The best known approximation for this problem is a (3 + ∊)-approximation for any ∊ > 0. Our first main result is a relaxed PTAS (Polynomial Time Approximation Scheme) for the adaptive policy, that is, for any ∊ > 0, we present a poly-time computable (1 + ∊)-approximate adaptive policy when knapsack capacity is relaxed to 1+e. At a high-level, the proof is based on transforming an arbitrary collection of item size distributions to canonical item size distributions that admit a compact description. We then establish a coupling that shows a (1 + ∊)-approximation can be achieved for the original problem by a canonical policy that makes decisions at each step by observing events drawn from the sample space of canonical size distributions. Finally, we give a mechanism for approximating the optimal canonical policy. Our second main result is an (8/3 + ∊)-approximate adaptive policy for any ∊ > 0 without relaxing the knapsack capacity, improving the earlier (3 + ∊)-approximation result. Interestingly, we obtain this result by using the PTAS described above. We establish an existential result that the optimal policy for the knapsack with capacity 1 can be folded to get a policy with expected profit 3OPT/8 for a knapsack with capacity (1 − ∊), with capacity relaxed to 1 only for the first item inserted. We then use our PTAS result to compute the (1 + ∊)-approximation to such policy. Our techniques also yield a relaxed PTAS for non-adaptive policies. Finally, we also show that our ideas can be extended to yield improved approximation guarantees for multi-dimensional and fixed set variants of the stochastic knapsack problem. Previous chapter Next chapter RelatedDetails Published:2011ISBN:978-0-89871-993-2eISBN:978-1-61197-308-2 https://doi.org/10.1137/1.9781611973082Book Series Name:ProceedingsBook Code:PR138Book Pages:xviii-1788

This paper investigates the order-batching problem both as a theoretical problem, including its special variants, and as a practical problem, with possible heuristic solution procedures. It shows that certain special cases or variants of the order batching problem are solvable, while other cases of the problem are NP-hard. It also develops a new Mixed-Integer Programming (MIP) model to obtain near-exact solutions to the problem. The MIP provides good lower and upper bounds to the problem, which prove useful in the investigation of the heuristic solution procedures. The paper identifies and modifies slightly the order batching heuristic, which shows the strongest results in the numerical experiments. Although good solutions are obtained with this particular heuristic, the results indicate that it is computationally intensive. Furthermore, solutions obtained through the MIP model suggest there is still room to obtain even better solutions to the problem via heuristic procedures, although the challenge is to do so without further increasing the computational burden.

Adaptivity in the stochastic blackjack knapsack problem

Adaptive Lagrangian Policies for Inventory Control Consider the problem of managing inventory in a multiwarehouse, multistore setting where each store can be periodically replenished from potentially a set of nearby warehouses. This is a practically relevant problem, and its optimal policy is not easy to compute because of the curse of dimensionality. Existing works in the literature have proposed a policy based on Lagrangian relaxation of the original stochastic problem. This vanilla policy applies the control parameters derived from the Lagrangian model and has been shown to perform well in some cases. In “Adaptive Lagrangian Policies for a Multiwarehouse, Multistore Inventory System with Lost Sales,” we go one step farther and develop adaptive Lagrangian policies that update (at certain update times) the control parameters of the vanilla policy based on the historical demand realizations. We analytically show that our proposed adjustment significantly improves the performance of the vanilla policy.

To make scheduling problems more realistic, we develop a new scheduling model with penalty functions for earliness and tardiness. Instead of a fixed due date, the scheduling problem is approached by taking the due date as a time period, which can handle flexible due dates and flexible customer requirements with penalty parameter modifications. We solve this optimization problem by the simulated annealing method and demonstrate its effectiveness with three examples.

Based on several existing literatures that use mixed integer programming (MIP) to solve parallel machine scheduling problems, this study tests and compares three MIP models. Each order has its own ready date, due date and processing time. The completion time of the order cannot over the its due date. If not, penalties for tardiness will occur. Formulation 1 used in immediate-precedence variables [1]. Formulation 2 is an improved version of immediate-precedence variables original proposed by [2]. Formulation 3 used linear ordering variables[1]. The results reveal that Formulation 2 has better computational performance than other does.

Improved solutions for the traveling purchaser problem

This study is concerned with Markov Decision Processes with uncertain transition matrices. In the discounted case, the Bayesian analysis of this model is studied.We define an adaptive policy and a learning policy and show that there exists, for any ∈> 0 an ∈-optimal and learning policy. In the average case, the non-Bayesian analysis of this model is studied and an optimal adaptive policy is constructed.

In this paper, we present an analytical framework to jointly investigate MRC diversity and multiuser scheduling under Rayleigh fading channels with the effect of combining errors under a low SNR regime. We derive spectrum efficiency expressions for multiuser scheduling systems with two different cases of multiuser scheduling, (1) Full feedback user scheduling (2) Limited feedback user scheduling with effects of combining errors. A signal to noise ratio (SNR) based user selection scheme is considered. For combining errors, correlation is defined as the magnitude of the complex cross-correlation between the transmission coefficients of the medium associated with pilot and message frequencies. We derive the probability density function and cumulative density function for the above cases. Adaptation policies like (1) Optimal power and rate adaptation policy (OPRA), (2) Optimal rate adaptation policy, (3) Channel inversion with fixed rate policy, (4) Truncated channel inversion with fixed rate policy are employed to improve system performance. It is observed that limited feedback user scheduling along with employment of adaptation policies yields better system performance compared to full feedback user scheduling. Further, OPRA policy yields maximum capacity among the four policies.

The benefit of adaptivity in stochastic packing problems with probing