Efficient Stochastic Programming in Julia

We present StochasticPrograms.jl, a user-friendly and powerful open-source framework for stochastic programming written in the Julia language. The framework includes both modeling tools and structure-exploiting optimization algorithms. Stochastic programming models can be efficiently formulated using expressive syntax and models can be instantiated, inspected, and analyzed interactively. The framework scales seamlessly to distributed environments. Small instances of a model can be run locally to ensure correctness, while larger instances are automatically distributed in a memory-efficient way onto supercomputers or clouds and solved using parallel optimization algorithms. These structure-exploiting solvers are based on variations of the classical L-shaped and progressive-hedging algorithms. We provide a concise mathematical background for the various tools and constructs available in the framework, along with code listings exemplifying their usage. Both software innovations related to the implementation of the framework and algorithmic innovations related to the structured solvers are highlighted. We conclude by demonstrating strong scaling properties of the distributed algorithms on numerical benchmarks in a multi-node setup

Block-based Outpatient Clinic Appointments Scheduling Under Open-access Policy

Outpatient clinic appointment scheduling is an important topic in OR/IE studies. Open-access policy shows its strength in improving patient access and satisfaction, as well as reducing no-show rate. The traditional far-in-advance scheduling plays an important role in handling chronic and follow-up care. This dissertation discusses a hybrid policy under which a clinic deals with three types of patients. The first type of patients are those who request their appointments before the visit day. The second type of patients schedule their appointment on the visit day. The third type of patients are walk-in patients who go to the clinic without appointments and wait to see the physician in turn. In this dissertation, the online scheduling policy is addressed for the Type 2 and Type 3 patients, and the offline scheduling policy is used for the Type 1 patients. For the online scheduling policy, two stochastic integer programming (SIP) models are built under two different sets of assumptions. The first set of assumptions ignores the endogenous uncertainty in the problem. An aggregate assigning method is proposed with the deterministic equivalent problem (DEP) model. This method is demonstrated to be better than the traditional one-at-a-time assignment through both overestimation and underestimation numerical examples. The DEP formulations are solved using the proposed bound-based sampling method, which provides approximated solutions and reasonable sample size with the least gap between lower and upper bound of the original objective value. On the basis of the first set of assumptions and the SIP model, the second set of assumptions considers patient no-shows, preference, cancellations and lateness, which introduce endogenous uncertainty into the SIP model. A modified L-shaped method and aggregated multicut L-shaped method are designed to handle the model with decision dependent distribution parameter. Distinctive optimality cut generation schemes are proposed for three types of distribution for linked random variables. Computational experiments are conducted to compare performance and outputs of different methods. An alternative formulation of the problem with simple recourse function is provided, based on which, a mixed integer programming model is established as a convenient complementary method to evaluate results with expected value. The offline scheduling aims at assigning a certain number of Type 1 patients with deterministic service time and individual preferences into a limited number of blocks, where the sum of patients’ service time in a block does not exceed the block length. This problem is associated with bin packing problem with restrictions. Heuristic and metaheuristic methods are designed to adapt the added restrictions to the bin packing problem. Zigzag sorting is proposed for the algorithm and is shown to improve the performance significantly. A clique based construction method is designed for the Greedy Randomized Adaptive Search Procedure and Simulated Annealing. The proposed methods show higher efficiency than traditional ones. This dissertation offers a series of new and practical resolutions for the clinic scheduling problem. These methods can facilitate the clinic administrators who are practicing the open-access policy to handle different types of patients with deterministic or nondeterministic arrival pattern and system efficiency. The resolutions range from operations level to management level. From the operations aspect, the block-wise assignment and aggregated assignment with SIP model can be used for the same-day request scheduling. From the management level, better coordination of the assignment of the Type 1 patients and the same-day request patients will benefit the cost-saving control

Decomposition techniques for large scale stochastic linear programs

Stochastic linear programming is an effective and often used technique for incorporating uncertainties about future events into decision making processes. Stochastic linear programs tend to be significantly larger than other types of linear programs and generally require sophisticated decomposition solution procedures. Detailed algorithms based uponDantzig-Wolfe and L-Shaped decomposition are developed and implemented. These algorithms allow for solutions to within an arbitrary tolerance on the gap between the lower and upper bounds on a problem\u27s objective function value. Special procedures and implementation strategies are presented that enable many multi-period stochastic linear programs to be solved with two-stage, instead of nested, decomposition techniques. Consequently, abroad class of large scale problems, with tens of millions of constraints and variables, can be solved on a personal computer. Myopic decomposition algorithms based upon a shortsighted view of the future are also developed. Although unable to guarantee an arbitrary solution tolerance, myopic decomposition algorithms may yield very good solutions in a fraction of the time required by Dantzig-Wolfe/L-Shaped decomposition based algorithms.In addition, derivations are given for statistics, based upon Mahalanobis squared distances,that can be used to provide measures for a random sample\u27s effectiveness in approximating a parent distribution. Results and analyses are provided for the applications of the decomposition procedures and sample effectiveness measures to a multi-period market investment model

Dynamic sequencing and cut consolidation for the parallel hybrid-cut nested L-shaped method

Abstract The Nested L-shaped method is used to solve two-and multi-stage linear stochastic programs with recourse, which can have integer variables on the first stage. In this paper we present and evaluate a cut consolidation technique and a dynamic sequencing protocol to accelerate the solution process. Furthermore, we present a parallelized implementation of the algorithm, which is developed within the COIN-OR framework. We show on a test set of 48 two-stage and 42 multi-stage problems, that both of the developed techniques lead to significant speed ups in computation time

Parallel algorithms for two-stage stochastic optimization

We develop scalable algorithms for two-stage stochastic program optimizations. We propose performance optimizations such as cut-window mechanism in Stage 1 and scenario clustering in Stage 2 of benders method for solving two-stage stochastic programs. A naive implementation of benders method has slow convergence rate and does not scale well to large number of processors especially when the problem size is large and/or there are integer variables in Stage 1. Parallelization of stochastic integer programs pose very unique characteristics that make them very challenging to parallelize. We develop a Parallel Stochastic Integer Program Solver (PSIPS) that exploits nested parallelism by exploring the branch-and-bound tree vertices in parallel along with scenario parallelization. PSIPS has been shown to have high parallel efficiency of greater than 40% at 120 cores which is significantly greater than the parallel efficiency of state-of-the-art mixed-integer program solvers. A significant portion of the time in this branch-and-bound solver is spent in optimizing the stochastic linear program at the root vertex. Stochastic linear programs at other vertices of the branch-and-bound tree take very less iterations to converge because they can inherit benders cut from their parent vertices and/or the root. Therefore, it is important to reduce the optimization time of the stochastic linear program at the root vertex. We propose two decomposition schemes namely the Split-and-Merge (SAM) method and the Lagrangian Decomposition and Merge (LDAM) method that significantly increase the convergence rate of benders decomposition. SAM method gives up to 64% reduction in solution time while also giving significantly higher parallel speedups as compared to the naive benders method. LDAM method, on the other hand, has made it possible to solve otherwise intractable stochastic programs. We further provide a computational engine for many real-time and dynamic problems faced by US Air Mobility Command. We first propose a stochastic programming solution to the military aircraft allocation problem with consideration for disaster management. Then, we study US AMC's dynamic mission re-planning problem and propose a mathematical formulation that is computationally feasible and leads to significant savings in cost as compared to myopic and deterministic optimization. It is expected that this work will provide the springboard for more robust problem solving with HPC in many logistics and planning problems

A Stochastic Benders Decomposition Scheme for Large-Scale Data-Driven Network Design

Network design problems involve constructing edges in a transportation or supply chain network to minimize construction and daily operational costs. We study a data-driven version of network design where operational costs are uncertain and estimated using historical data. This problem is notoriously computationally challenging, and instances with as few as fifty nodes cannot be solved to optimality by current decomposition techniques. Accordingly, we propose a stochastic variant of Benders decomposition that mitigates the high computational cost of generating each cut by sampling a subset of the data at each iteration and nonetheless generates deterministically valid cuts (as opposed to the probabilistically valid cuts frequently proposed in the stochastic optimization literature) via a dual averaging technique. We implement both single-cut and multi-cut variants of this Benders decomposition algorithm, as well as a k-cut variant that uses clustering of the historical scenarios. On instances with 100-200 nodes, our algorithm achieves 4-5% optimality gaps, compared with 13-16% for deterministic Benders schemes, and scales to instances with 700 nodes and 50 commodities within hours. Beyond network design, our strategy could be adapted to generic two-stage stochastic mixed-integer optimization problems where second-stage costs are estimated via a sample average