Stochastically maximizing the number of successes in a sequential assignment problem

1990 ◽  
Vol 27 (2) ◽  
pp. 351-364 ◽  
Author(s):  
Rhonda Righter
Keyword(s):  
Finite Number ◽  
Poisson Process ◽  
Optimal Policy ◽  
Assignment Problem ◽  
Sequential Assignment ◽  
Expected Number ◽  
Expected Return ◽  
Threshold Policy ◽  
The Given

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

1990 ◽  
Vol 27 (02) ◽  
pp. 351-364 ◽  
Author(s):  
Rhonda Righter
Keyword(s):  
Finite Number ◽  
Poisson Process ◽  
Optimal Policy ◽  
Assignment Problem ◽  
Sequential Assignment ◽  
Expected Number ◽  
Expected Return ◽  
Threshold Policy ◽  
The Given

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.

The Stochastic Sequential Assignment Problem With Random Deadlines

1987 ◽  
Vol 1 (2) ◽  
pp. 189-202 ◽  
Author(s):  
Rhonda Righter
Keyword(s):  
Simple Form ◽  
Optimal Policy ◽  
Assignment Problem ◽  
Independent Random Variables ◽  
Sequential Assignment ◽  
Activity System ◽  
Combined System ◽  
Expected Return ◽  
Batch Arrivals ◽  
Activity Models

Resources are to be allocated sequentially to activities to maximize the total expected return, where the return from an allocation is the product of the value of the resource and the value of the activity. The set of activities and their values are given ahead of time, but the resources arrive according to a Poisson process and their values are independent random variables that are observed upon arrival. It is assumed that either there is a single random deadline for all activities, which is the same as discounting the returns, or the activities have independent random deadlines. The model has applications machine scheduling, packet switching, and kidney allocation for transplant. It is known that the optimal policy in the discounted case has a very simple form that does not depend on the activity values. We show that this is also true when the deadlines are independent and in this case the solution can expressed in terms of solutions to single activity models. These results also hold when there are batch arrivals of resources. The effects of pooling separate identical systems with a single activity into a combined system is investigated for both models. When activities have independent deadlines it is optimal to reject a resource in the combined system if and only if it is optimal to reject it in the single activity system. However, when returns are discounted, it is sometimes optimal to accept a resource in the combined system that would be rejected in the single activity system.

STOCHASTIC SEQUENTIAL ASSIGNMENT PROBLEM WITH ARRIVALS

2011 ◽  
Vol 25 (4) ◽  
pp. 477-485 ◽  
Author(s):  
Rhonda Righter
Keyword(s):  
Optimal Policy ◽  
Assignment Problem ◽  
Sequential Assignment ◽  
Different Types ◽  
Job Values ◽  
Stochastic Assignment ◽  
Sequential Stochastic Assignment

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.

Triply stochastic sequential assignment problem with the uncertainty in worker survival

2021 ◽  
Author(s):  
Siddhartha Nambiar ◽  
Alexander Nikolaev ◽  
Eduardo Pasiliao
Keyword(s):  
Assignment Problem ◽  
Sequential Assignment

scholarly journals Hermite–Hadamard-Type Inequalities for Product of Functions by Using Convex Functions

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Tariq Nawaz ◽  
M. Asif Memon ◽  
Kavikumar Jacob
Keyword(s):  
Convex Function ◽  
Finite Number ◽  
Convex Functions ◽  
The Many ◽  
The Given

One of the many techniques to obtain a new convex function from the given functions is to calculate the product of these functions by imposing certain conditions on the functions. In general, the product of two or finite number of convex function needs not to be convex and, therefore, leads us to the study of product of these functions. In this paper, we reframe the idea of product of functions in the setting of generalized convex function to establish Hermite–Hadamard-type inequalities for these functions. We have analyzed different cases of double and triple integrals to derive some new results. The presented results can be viewed as the refinement and improvement of previously known results.

SINGLE-MACHINE MULTIPLE-RECIPE PREDICTIVE MAINTENANCE

2013 ◽  
Vol 27 (2) ◽  
pp. 209-235 ◽  
Author(s):  
Yiwei Cai ◽  
John J. Hasenbein ◽  
Erhan Kutanoglu ◽  
Melody Liao
Keyword(s):  
Optimal Policy ◽  
Predictive Maintenance ◽  
Production Costs ◽  
Threshold Policy ◽  
Discrete Time Markov Chain ◽  
System A ◽  
Long Run ◽  
Monotonicity Result ◽  
Maintenance Policies ◽  
Discounted Costs

This paper studies a multiple-recipe predictive maintenance problem with M/G/1 queueing effects. The server degrades according to a discrete-time Markov chain and we assume that the controller knows both the machine status and the current number of jobs in the system. The controller's objective is to minimize total discounted costs or long-run average costs which include preventative and corrective maintenance costs, holdings costs, and possibly production costs. An optimal policy determines both when to perform maintenance and which type of job to process. Since the policy takes into account the machine's degradation status, such control decisions are known as predictive maintenance policies. In the single-recipe case, we prove that the optimal policy is monotone in the machine status, but not in the number of jobs in the system. A similar monotonicity result holds in the two-recipe case. Finally, we provide computational results indicating that significant savings can be realized when implementing a predictive maintenance policies instead of a traditional job-based threshold policy for preventive maintenances.

scholarly journals Limiting Behavior of the Target-Dependent Stochastic Sequential Assignment Problem

2014 ◽  
Vol 51 (4) ◽  
pp. 943-953 ◽  
Author(s):  
Golshid Baharian ◽  
Sheldon H. Jacobson
Keyword(s):  
Distribution Function ◽  
Markov Chain ◽  
Assignment Problem ◽  
Sequential Assignment ◽  
Threshold Probability ◽  
Limiting Behavior ◽  
Target Value ◽  
Long Run ◽  
Stochastic Parameters ◽  
Optimal Policies

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.

Waiting times for multiple occurrences of several events

1988 ◽  
Vol 25 (03) ◽  
pp. 624-629
Author(s):  
Stephen Scheinberg
Keyword(s):  
Asymptotic Behavior ◽  
Finite Number ◽  
Waiting Times ◽  
Expected Number

Consider an ‘experiment' which can be repeated indefinitely often resulting in independent random outcomes. Fix attention on a finite number of possible (sets of) outcomes E 1, E 2, … and define W = W(N 1, N 2, …) to be the expected number of repetitions needed to ensure that E 1 has occurred (at least) N 1 times, E 2 has occurred (at least) N 2 times, etc. This article examines the asymptotic behavior of W as a function of the sum Σ j N j, as the latter grows without bound.

scholarly journals An On-Path Caching Scheme Based on the Expected Number of Copies in Information-Centric Networks

Electronics ◽  
2020 ◽  
Vol 9 (10) ◽  
pp. 1705
Author(s):  
Yuanhang Li ◽  
Jinlin Wang ◽  
Rui Han
Keyword(s):  
Network Architectures ◽  
Expected Number ◽  
Placement Problem ◽  
Mobility Support ◽  
Allocation Method ◽  
Caching Scheme ◽  
Content Popularity ◽  
Content Placement ◽  
The Given ◽  
Cache Allocation

The Information-Centric Network (ICN) is one of the most influential future network architectures and in-network caching in ICN brings some helpful features, such as low latency and mobility support. How to allocate cache capacity and place content properly will greatly influence the performance of ICN. This paper focuses on the cache allocation problem and content placement problem under the given cache space budget. Firstly, a lightweight allocation method utilizing information of both topology and content popularity is proposed, to allocate cache space and get the expected number of copies of popular content. The expected number of copies represents the number of content copies placed in the topology. Then, an on-path caching scheme based on the expected number of copies is proposed to handle the content placement problem. In the cache allocation scenario, the lightweight allocation method performs better than other baseline methods. In the content placement scenario, Leave Copy Down (LCD) based on the expected number of copies performs the second-best and is very close to Optimal Content Placement (OCP).

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