The Sequential Stochastic Assignment Problem With Postponement Options

Abstract

The sequential and stochastic assignment problem (SSAP) has wide applications in logistics, finance, and health care management, and has been well studied in the literature. It assumes that jobs with unknown values arrive according to a stochastic process. Upon arrival, a job's value is made known and the decision-maker must immediately decide whether to accept or reject the job and, if accepted, to assign it to a resource for a reward. The objective is to maximize the expected reward from the available resources. The optimal assignment policy has a threshold structure and can be computed in polynomial time. In reality, there exist situations in which the decision-maker may postpone the accept/reject decision. In this research, we study the value of postponing decisions by allowing a decision-maker to hold a number of jobs which may be accepted or rejected later. While maintaining this queue of arrivals significantly complicates the analysis, optimal threshold policies exist under mild assumptions when the resources are homogeneous. We illustrate the benefits of delaying decisions through higher profits and lower risk in both cases of homogeneous and heterogeneous resources. 27Sakaguchi [24] extends the work of Albright [2] to the case in which jobs are finite in number and arrive according to a non-homogeneous Poisson process. Sakaguchi [25] permits the number of available resources to be unknown. Albright [3] studies the "secretary problem" where the best M secretaries are to be selected from N available secretaries. The model in this work assumes the value of the secretaries come from different distributions, with two successive secretaries' values governed by a Markov Chain. Nakai [15,16] studies the case with the distribution of resources varying according to a partially observable Markov process. Albright [4] considers the case when the parameters of the distribution of the job value are not fully known and allows the parameters to be updated through a Bayesian model. Kennedy [12] permits job values to be dependent. Righter [20] studies the case where each person has independent deadlines and compares this model with the discount model. Righter [21] permits the arrival rate, the job's value, and the variability of job values to change according to independent Markov processes. Additionally, Albright and Derman [1] and Saario [23] studied the asymptotic results of the SSAP.The dynamic and stochastic knapsack problem (DSKP) with homogeneous sized items studied by Kleywegt and Papastavrou [13] is closely related to SSAP. By setting the costs appropriately, the threshold policy developed there holds for the SSAP with homogeneous resources. Nikolaev and Jacobson [17] extended the results of the SSAP and the DSKP to the case with a random number of arrivals.A variety of applications utilize the SSAP. For example, the resource can be replaced by houses to sell and jobs can be replaced by purchase offers with random values arriving at random times. Elfving [7] uses this interpretation for the case with M = 1. McLa...

The dynamic and stochastic knapsack Problem with homogeneous‐sized items and postponement options

“…To avoid technical inconveniences, we use the sign of the right first-order derivative to argue monotonicity of functions in this article. This, however, requires a property of real functions, which is rarely mentioned in standard mathematical analysis texts but introduced in Feng and Hartman [11]. We reiterate this property in the following lemma, and henceforth, do not distinguish right derivatives and derivatives.…”

“…Other extensions include Albright , Kennedy , and Righter . Feng and Hartman studied the model with postponement options.…”

“…Note that may not exist at , leading to technical inconvenience, when arguing the monotonicity of . In this article, we instead use the sign of the right first‐order derivative to argue the monotonicity of functions, based on a property of real functions introduced in Feng and Hartman (see Appendix A). For convenience, we henceforth do not differentiate the right derivatives and the derivatives.…”

“…Spivey and Powell point out the potential benefit of postponing assignments. Feng and Hartman study the stochastic sequential assignment problem (SSAP) with the ability to delay. Real options analysis generally evaluates the delay option in investment analysis , but only when considering a single investment decision.…”

Doubly Stochastic Sequential Assignment Problem

“…Baharian and Jacobson perform the task assignment under a threshold criterion, which minimizes the probability that the total reward fails to achieve a given value. Feng and Hartman allow the decision maker to hold a number of jobs that may be rejected or accepted later to study the value of postponing decisions. Khatibi et al study the SSAP with random success rates, which are assumed to take on the same value during the entire assignment process.…”