Introduction to stochastic programming

(2003). Risk Management: Value at Risk and Beyond. Journal of the American Statistical Association: Vol. 98, No. 462, pp. 494-494.

Risk Management: Value at Risk and Beyond

https://link.springer.com/content/pdf/10.1057%2Fjors.1994..pdf

Discrete Event Systems: Sensitivity Analysis and Stochastic Optimization by the Score Function Method

We study call routing policies for call centers with multiple call types and multiple agent groups. We introduce new weight-based routing policies where each pair (call type, agent group) is given a matching priority defined as an affine combination of the longest waiting time for that call type and the longest idle time or the number of idle agents in that agent group. The coefficients in this combination are parameters to be optimized. This type of policy is more flexible than traditional ones found in practice, and it performs better in many situations. We consider objective functions that account for the service levels, the abandonment ratios, and the fairness of occupancy across agent groups. We select the parameters of all considered policies via simulation-based optimization heuristics. This requires only the availability of a simulation model of the call center, which can be much more detailed and realistic than the models used elsewhere in the literature to study the optimality of certain types of routing rules. We offer a first numerical study of realistic routing rules that takes into account the complexity of real-life call centers.

Dynamic call center routing policies using call waiting and agent idle times

Research Guides: Prof. Anderson's Writing Guide: Bureau of Labor Statistics: Occupational Outlook Handbook

We consider a multistage stochastic discrete program in which constraints on any stage might involve expectations that cannot be computed easily and are approximated by simulation. We study a sample average approximation (SAA) approach that uses nested sampling, in which at each stage, a number of scenarios are examined and a number of simulation replications are performed for each scenario to estimate the next-stage constraints. This approach provides an approximate solution to the multistage problem. To establish the consistency of the SAA approach, we first consider a two-stage problem and show that in the second-stage problem, given a scenario, the optimal values and solutions of the SAA converge to those of the true problem with probability one when the sample sizes go to infinity. These convergence results do not hold uniformly over all possible scenarios for the second stage problem. We are nevertheless able to prove that the optimal values and solutions of the SAA converge to the true ones with probability one when the sample sizes at both stages increase to infinity. We also prove exponential convergence of the probability of a large deviation for the optimal value of the SAA, the true value of an optimal solution of the SAA, and the probability that any optimal solution to the SAA is an optimal solution of the true problem. All of these results can be extended to a multistage setting and we explain how to do it. Our framework and SAA results cover a large variety of resource allocation problems for which at each stage after the first one, new information becomes available and the allocation can be readjusted, under constraints that involve expectations estimated by Monte Carlo. As an illustration, we apply this SAA method to a staffing problem in a call center, in which the goal is to optimize the numbers of agents of each type under some constraints on the quality of service (QoS). The staffing allocation has to be decided under an uncertain arrival rate with a prior distribution in the first stage, and can be adjusted at some additional cost when better information on the arrival rate becomes available in later stages.

https://ink.library.smu.edu.sg/cgi/viewcontent.cgi?article=6284&context=sis_research

On a multistage discrete stochastic optimization problem with stochastic constraints and nested sampling

A simulation-based decomposition approach for two-stage staffing optimization in call centers under arrival rate uncertainty

We consider a staffing problem with probabilistic constraints in an emergency call center. The aim is to minimize the total cost of agents while satisfying chance constraints defined over the service level and the average waiting time, in a given set of time periods. We provide a mathematical formulation of the problem in terms of probabilities and expectations. We define a sample average approximation (SAA) version of this problem whose solution converges to that of the exact problem when the sample size increases. We also propose a quick and simple simulation-based (heuristic) algorithm to compute a good (nearly optimal) staffing solution for the SAA problem. We illustrate and validate our algorithm with a simulation model based on real data from the 911 emergency call center of Montreal, Canada.

https://hal.inria.fr/hal-01399507/document

Staffing optimization with chance constraints for emergency call centers

We consider a stochastic staffing problem with uncertain arrival rates. The objective is to minimize the total cost of agents under some chance constraints, defined over the randomness of the service level in a given time period. In the first stage, an initial staffing must be determined in advance based on imperfect forecast of the arrival rates. At a later time, when the forecast becomes more accurate, this staffing can be corrected with recourse actions, by adding or removing agents at the price of some penalty costs. We present a method that combines simulation, mixed integer programming, and cut generation to solve this problem.

Two-stage chance-constrained staffing with agent recourse for multi-skill call centers

In this thesis, we study the staffing optimization problem in multiskill call centers, in which we aim at minimizing the operating cost while delivering a high quality of service (QoS) to customers. We also introduce the use of chance constraints which require that the QoSs are met with a given probability. These constraints are adequate in the case when the performance is measured over a short time interval as QoS measures are random variables in a given time period. The proposed staffing problems are challenging in the sense that the stochastic constraints have no-closed forms and need to be approximated by simulation. In addition, the QoS functions are typically non-linear and non-convex. We consider staffing optimization problems in different settings and study the proposed models in both theoretical and practical aspects. The methodologies developed are general, in the sense that they can be adapted and applied to other staffing/scheduling problems in queuing-based systems. The thesis consists of three articles dealing with different challenges in modeling and solving staffing optimization problems in multiskill call centers. The first and second articles concern a two-stage staffing optimization problem under uncertainty. While in the first one, we study a general two-stage discrete stochastic programming model to provide a theoretical guarantee for the consistency of the sample average approximation (SAA) when the sample sizes go to infinity, the second one applies the SAA approach to solve the two-stage staffing optimization problem under arrival rate uncertainty. Both papers indicate the viability of the SAA approach in our context, in both theoretical and practical aspects. To be more precise, in the first article, we consider a general two-stage discrete stochastic problem with expected value constraints. We formulate its SAA with nested sampling. We show that under some assumptions that hold in call center examples, one can obtain the optimal solutions of the original problem by solving its SAA with large enough sample sizes. Moreover, we show that the probability that the optimal solution of the sample problem is an optimal solution of the original problem, approaches one exponentially fast as we increase the sample sizes. These theoretical findings are important, not only for call center applications, but also for other decision-making problems with discrete decision variables. The second article concerns solution methods to solve a two-stage staffing problem under arrival rate uncertainty. It is motivated by the fact that the SAA version of the two-stage staffing problem becomes expensive to solve with a large number of scenarios, as for each scenario, one needs to use simulation to approximate the QoS constraints. We develop an algorithm that combines simulation, cut generation, cut strengthening and Benders decomposition to solve the SAA problem. We show the efficiency of the approach, especially when the number of scenarios is large.

Thuy Anh Ta

Papers

On a multistage discrete stochastic optimization problem with stochastic constraints and nested sampling

A simulation-based decomposition approach for two-stage staffing optimization in call centers under arrival rate uncertainty

Staffing optimization with chance constraints for emergency call centers

Two-stage chance-constrained staffing with agent recourse for multi-skill call centers

Stochastic optimization of staffing for multiskill call centers