Sequential Assignment Problem Research Articles

A greedy and distributable approach to the Lexicographic Bottleneck Assignment Problem with conditions on exactness

Solving the Lexicographic Bottleneck Assignment Problem (LexBAP) typically relies on centralised computation with order O(n4) complexity. We consider the Sequential Bottleneck Assignment Problem (SeqBAP), which yields a greedy solution to the LexBAP and discuss the relationship between the SeqBAP, the LexBAP, and the Bottleneck Assignment Problem (BAP). In particular, we reexamine tools used to analyse the structure of the BAP, and apply them to derive an O(n3) algorithm that solves the SeqBAP. We show that the set of solutions of the LexBAP is a subset of the solutions of the SeqBAP and analyse the conditions for which the solutions sets are identical. Furthermore, we provide a method to verify the satisfaction of these conditions. In cases where the conditions are satisfied, the proposed algorithm for solving the SeqBAP solves the LexBAP with computation that has lower complexity and can be distributed over a network of computing agents. The applicability of the approach is demonstrated with a case study where mobile robots are assigned to goal locations.

Asymptotics of the optimum in discrete sequential assignment

We consider the stochastic sequential assignment problem of Derman et al. (1972) corresponding to a discrete distribution supported on a finite set of points. We use large deviation estimates to compute the asymptotics of the optimal policy as the number of tasks N→∞.

Triply stochastic sequential assignment problem with the uncertainty in worker survival

Triply stochastic sequential assignment problem with the uncertainty in worker survival

Phase transition in a sequential assignment problem on graphs

We study the following sequential assignment problem on a finite graph $G=(V,E)$. Each edge $e\in E$ starts with an integer value $n_{e}\ge0$, and we write $n=\sum_{e\in E}n_{e}$. At time $t$, $1\le t\le n$, a uniformly random vertex $v\in V$ is generated, and one of the edges $f$ incident with $v$ must be selected. The value of $f$ is then decreased by $1$. There is a unit final reward if the configuration $(0,\ldots,0)$ is reached. Our main result is that there is a phase transition: as $n\to\infty$, the expected reward under the optimal policy approaches a constant $c_{G}>0$ when $(n_{e}/n:e\in E)$ converges to a point in the interior of a certain convex set $\mathcal{R}_{G}$, and goes to $0$ exponentially when $(n_{e}/n:e\in E)$ is bounded away from $\mathcal{R}_{G}$. We also obtain estimates in the near-critical region, that is when $(n_{e}/n:e\in E)$ lies close to $\partial\mathcal{R}_{G}$. We supply quantitative error bounds in our arguments.

Doubly Stochastic Sequential Assignment Problem

AbstractThis article introduces the Doubly Stochastic Sequential Assignment Problem (DSSAP), an extension of the Sequential Stochastic Assignment Problem (SSAP), where sequentially arriving tasks are assigned to workers with random success rates. A given number of tasks arrive sequentially, each with a random value coming from a known distribution. On a task arrival, it must be assigned to one of the available workers, each with a random success rate coming from a known distribution. Optimal assignment policies are proposed for DSSAP under various assumptions on the random success rates. The optimal assignment algorithm for the general case of DSSAP, where workers have distinct success rate distribution, has an exponential running time. An approximation algorithm that achieves a fraction of the maximum total expected reward in a polynomial time is proposed. The results are illustrated by several numerical experiments. © 2016 Wiley Periodicals, Inc. Naval Research Logistics 63: 124–137, 2016

Limiting Behavior of the Target-Dependent Stochastic Sequential Assignment Problem

The stochastic sequential assignment problem assigns distinct workers to sequentially arriving tasks with stochastic parameters. In this paper the assignments are performed so as to minimize the threshold probability, which is the probability of the long-run reward per task failing to achieve a target value (threshold). As the number of tasks approaches infinity, the problem is studied for independent and identically distributed (i.i.d.) tasks with a known distribution function and also for tasks that are derived from r distinct unobservable distributions (governed by a Markov chain). Stationary optimal policies are presented, which simultaneously minimize the threshold probability and achieve the optimal long-run expected reward per task.

Limiting Behavior of the Target-Dependent Stochastic Sequential Assignment Problem

The stochastic sequential assignment problem assigns distinct workers to sequentially arriving tasks with stochastic parameters. In this paper the assignments are performed so as to minimize the threshold probability, which is the probability of the long-run reward per task failing to achieve a target value (threshold). As the number of tasks approaches infinity, the problem is studied for independent and identically distributed (i.i.d.) tasks with a known distribution function and also for tasks that are derived from r distinct unobservable distributions (governed by a Markov chain). Stationary optimal policies are presented, which simultaneously minimize the threshold probability and achieve the optimal long-run expected reward per task.

On the complexity of short-term production planning and the near-optimality of a sequential assignment problem heuristic approach

In semiconductor manufacturing, the process of short-term production planning requires setting clear and yet challenging and doable goals to each operation and toolset in the process flow per each product type. We demonstrate the complexity of this problem using an experimental study performed with proficient workforce, and then show how the problem can be decomposed, aggregated, and solved using sequential recurrent linear programming assignment problems. We also refer to the improvements that the proposed algorithm has achieved in practice when applied to multiple semiconductor production facilities, and discuss its efficiency and uniqueness as a fast heuristic relative to other proposed methods.

Limiting behavior of the stochastic sequential assignment problem

AbstractThe stochastic sequential assignment problem (SSAP) considers how to allocate available distinct workers to sequentially arriving tasks with stochastic parameters such that the expected total reward obtained from the sequential assignments is maximized. Implementing the optimal assignment policy for the SSAP involves calculating a new set of breakpoints upon the arrival of each task (i.e., for every time period), which is impractical for large‐scale problems. This article studies two problems that are concerned with obtaining stationary policies, which achieve the optimal expected reward per task as the number of tasks approaches infinity. The first problem considers independent and identically distributed (IID) tasks with a known distribution function, whereas in the second problem tasks are derived from r different unobservable distributions governed by an ergodic Markov chain. The convergence rate of the expected reward per task to the optimal value is also obtained for both problems. © 2013 Wiley Periodicals, Inc. Naval Research Logistics, 2013

STOCHASTIC SEQUENTIAL ASSIGNMENT PROBLEM WITH THRESHOLD CRITERIA

The stochastic sequential assignment problem (SSAP) allocates distinct workers to sequentially arriving tasks with stochastic parameters to maximize the expected total reward. In this paper, the assignment of tasks is performed under the threshold criterion, which seeks a policy that minimizes the probability of the total reward failing to achieve a target value. A Markov-decision-process approach is employed to model the problem, and sufficient conditions for the existence of a deterministic Markov optimal policy are derived, along with fundamental properties of the optimal value function. An algorithm to approximate the optimal value function is presented, and convergence results are established.

Evaluating the effectiveness of sequential aviation security screening policies

Passenger and baggage screening is an essential component of aviation security systems designed to detect and remove threats endangering the safety of air transportation. Upon entering the security checkpoint, passengers are assigned to a multilevel security class system defined by sets of screening devices. The sequential passenger assignment problem has been formulated as a stochastic process, with the objective of maximizing the overall true alarm rate, subject to device capacity constraints. This article introduces three metrics for evaluating the performance of sequential passenger assignment policies with respect to the retrospective optimal solution. The concepts of under-screening and over-screening passengers are defined, from which expressions for the expectation and variance of the number of under-screened passengers are obtained in part through a Markov chain. A conditional probability inequality is used to develop an upper bound on attaining the set of optimal assignments for a given realization of passenger risk. Estimators for the performance metrics are presented for the efficient simulation of cases involving a large number of passenger assignments. The key result is that for populations containing a majority of low-risk passengers, an initial underestimation of the overall risk level produces fewer under-screened passengers in comparison to that which results if the true risk level lies below what was anticipated. Furthermore, fewer passengers are incorrectly assigned to undergo the appropriate screening intensity if security class capacities are biased toward screening the majority of low-risk passengers using a combination of low-intensity detection devices, while reserving the high-intensity, time-consuming devices for the limited number of high-risk passengers.

STOCHASTIC SEQUENTIAL ASSIGNMENT PROBLEM WITH ARRIVALS

We extend the classic sequential stochastic assignment problem to include arrivals of workers. When workers are all of the same type, we show that the socially optimal policy is the same as the individually optimal policy for which workers are given priority according to last come–first served. This result also holds under several variants in the model assumptions. When workers have different types, we show that the socially optimal policy is determined by thresholds such that more valuable jobs are given to more valuable workers, but now the individually optimal policy is no longer socially optimal. We also show that the overall value increases when worker or job values become more variable.

Risk-Based Policies for Airport Security Checkpoint Screening

Passenger screening is an important component of aviation security that incorporates real-time passenger screening strategies designed to maximize effectiveness in identifying potential terrorist attacks. This paper identifies a methodology that can be used to sequentially and optimally assign passengers to aviation security resources. An automated prescreening system determines passengers' perceived risk levels, which become known as passengers check in. The levels are available for determining security class assignments sequentially as passengers enter security screening. A passenger is then assigned to one of several available security classes, each of which corresponds to a particular set of screening devices. The objective is to use the passengers' perceived risk levels to determine the optimal policy for passenger screening assignments that maximize the expected total security, subject to capacity and assignment constraints. The sequential passenger assignment problem is formulated as a Markov decision process, and an optimal policy is found using dynamic programming. The general result from the sequential stochastic assignment problem is adapted to provide a heuristic for assigning passengers to security classes in real time. A condition is provided under which this heuristic yields the optimal policy. The model is illustrated with an example that incorporates data extracted from the Official Airline Guide (Official Airline Guide. 1998. OAG Business Travel Planner: North American Edition. Official Airline Guides, Bedfordshire, UK).

Rotorcraft Dispatching in Large-Scale Emergencies Using Sequential Linear Assignment Techniques

The paper presents a methodology for the dispatching of helicopters in a large scale emergency, based on an iterative application of simulation and solution of linear assignment problems. Two examples of different complexity are shown, including sensitivity analyses. The sequential linear assignment problem methodology is especially suitable because it does not rely on the existence of a pre-existing network of routes, and can cope with highly dynamic situations. Because of the efficiency of the solution algorithms, the computation times are small and not a factor in the dispatching procedure. The computer times required for the solution of each LAP depend on the number of helicopters involved in the rescue, and the number of rescue locations, but not on the number of victims to be rescued (which, however, affects the time required for the entire solution). Through sensitivity analyses it is possible to study the effects of one or more design parameters, of one or more helicopters involved in the rescue, on the time required to complete the rescue. The problem proves nonlinear, at least in the design parameters considered in the study. The characteristics of the rescue operation, and the assumptions made in modeling the helicopters and the rescue operations themselves, play a major role.

A rank-based approach to the sequential selection and assignment problem

In the classical sequential assignment problem, “machines” are to be allocated sequentially to “jobs” so as to maximize the expected total return, where the return from an allocation of job j to machine k is the product of the value x j of the job and the weight p k of the machine. The set of m machines and their weights are given ahead of time, but n jobs arrive in sequential order and their values are usually treated as independent, identically distributed random variables from a known univariate probability distribution with known parameter values. In the paper, we consider a rank-based version of the sequential selection and assignment problem that minimizes the sum of weighted ranks of jobs and machines. The so-called “secretary problem” is shown to be a special case of our sequential assignment problem (i.e., m = 1). Due to its distribution-free property, our rank-based assignment strategy can be successfully applied to various managerial decision problems such as machine scheduling, job interview, kidney allocations for transplant, and emergency evacuation plan of patients in a mass-casualty situation.

Protein sequential resonance assignments by combinatorial enumeration using 13C alpha chemical shifts and their (i, i-1) sequential connectivities.

The need for the structural characterization of proteins on a genomic scale has brought with it demands for new technology to speed the structure determination process. In NMR, one bottleneck is the sequential assignment of backbone resonances. In this paper, we explore the computational complexity of the sequential assignment problem using only 13C alpha chemical shift data and C alpha (i, i - 1) sequential connectivity information, all of which can potentially be obtained from a single three-dimensional NMR spectrum. Although it is generally believed that there is too much ambiguity in such data to provide sufficient information for sequential assignment, we show that a straightforward combinatorial search algorithm can be used to find correct and unambiguous sequential assignments in a reasonable amount of CPU time for small proteins (approximately 80 residues or smaller) when there is little missing data. The deleterious effect of missing or spurious peaks and the dependence on match tolerances is also explored. This simple algorithm could be used as part of a semi-automated, interactive assignment procedure, e.g., to test partial manually determined solutions for uniqueness and to extend these solutions.

Universal linear prediction by model order weighting

A common problem that arises in adaptive filtering, autoregressive modeling, or linear prediction is the selection of an appropriate order for the underlying linear parametric model. We address this problem for linear prediction, but instead of fixing a specific model order, we develop a sequential prediction algorithm whose sequentially accumulated average squared prediction error for any bounded individual sequence is as good as the performance attainable by the best sequential linear predictor of order less than some M. This predictor is found by transforming linear prediction into a problem analogous to the sequential probability assignment problem from universal coding theory. The resulting universal predictor uses essentially a performance-weighted average of all predictors for model orders less than M. Efficient lattice filters are used to generate the predictions of all the models recursively, resulting in a complexity of the universal algorithm that is no larger than that of the largest model order. Examples of prediction performance are provided for autoregressive and speech data as well as an example of adaptive data equalization.

Automated analysis of protein NMR assignments using methods from artificial intelligence

An expert system for determining resonance assignments from NMR spectra of proteins is described. Given the amino acid sequence, a two-dimensional 15N- 1H heteronuclear correlation spectrum and seven to eight three-dimensional triple-resonance NMR spectra for seven proteins, AUTOASSIGN obtained an average of 98% of sequence-specific spin-system assignments with an error rate of less than 0.5%. Execution times on a Sparc 10 workstation varied from 16 seconds for smaller proteins with simple spectra to one to nine minutes for medium size proteins exhibiting numerous extra spin systems attributed to conformational isomerization. AUTOASSIGN combines symbolic constraint satisfaction methods with a domain-specific knowledge base to exploit the logical structure of the sequential assignment problem, the specific features of the various NMR experiments, and the expected chemical shift frequencies of different amino acids. The current implementation specializes in the analysis of data derived from the most sensitive of the currently available triple-resonance experiments. Potential extensions of the system for analysis of additional types of protein NMR data are also discussed.

Stochastically maximizing the number of successes in a sequential assignment problem

In the classical sequential assignment problem as introduced by Derman et al. (1972) there are n workers who are to be assigned a finite number of sequentially arriving jobs. If a worker of value p is assigned a job of value x the return is px, where we interpret the return as the probability that the given worker correctly completes the given job. The job value is a random value that is observed upon arrival, and jobs must be assigned or rejected when they arrive. Each worker can only do one job. Derman et al. showed that when the objective is to maximize the expected return, i.e., the expected number of correctly completed jobs, the optimal policy is a simple threshold policy, which does not depend on the worker values. Their result was extended by Albright (1974) to allow job arrivals according to a Poisson process and a single random deadline for job completion (which is equivalent to discounting). Righter (1987) further extended the result to permit workers to have independent random deadlines for job completions. Here we show that when there are independent deadlines a simple threshold policy that is independent of the worker values stochastically maximizes the number of correctly completed jobs, and therefore maximizes the expected number of correctly completed jobs. We also show that there is no policy that stochastically maximizes the number of correctly completed jobs when there is a single deadline. However, when there is single deadline and the objective is to maximize the probability that n jobs are done correctly by n workers, then the optimal policy is determined by a single threshold that is independent of n and of the worker values.

Stochastically maximizing the number of successes in a sequential assignment problem

In the classical sequential assignment problem as introduced by Derman et al. (1972) there are n workers who are to be assigned a finite number of sequentially arriving jobs. If a worker of value p is assigned a job of value x the return is px, where we interpret the return as the probability that the given worker correctly completes the given job. The job value is a random value that is observed upon arrival, and jobs must be assigned or rejected when they arrive. Each worker can only do one job. Derman et al. showed that when the objective is to maximize the expected return, i.e., the expected number of correctly completed jobs, the optimal policy is a simple threshold policy, which does not depend on the worker values. Their result was extended by Albright (1974) to allow job arrivals according to a Poisson process and a single random deadline for job completion (which is equivalent to discounting). Righter (1987) further extended the result to permit workers to have independent random deadlines for job completions. Here we show that when there are independent deadlines a simple threshold policy that is independent of the worker values stochastically maximizes the number of correctly completed jobs, and therefore maximizes the expected number of correctly completed jobs. We also show that there is no policy that stochastically maximizes the number of correctly completed jobs when there is a single deadline. However, when there is single deadline and the objective is to maximize the probability that n jobs are done correctly by n workers, then the optimal policy is determined by a single threshold that is independent of n and of the worker values.