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1. Exploring the Effect of Clustering Algorithms on Sample Average Approximation

   Jacobson, Daniel ; Hassan, Menna ; Dong, Zhijie S. ( January 2021 , 2021 Institute of Industrial and Systems Engineers (IISE) Annual Conference & Expo)

   null (Ed.)

   Stochastic Programming (SP) is used in disaster management, supply chain design, and other complex problems. Many of the real-world problems that SP is applied to produce large-size models. It is important but challenging that they are optimized quickly and efficiently. Existing optimization algorithms are limited in capability of solving these larger problems. Sample Average Approximation (SAA) method is a common approach for solving large scale SP problems by using the Monte Carlo simulation. This paper focuses on applying clustering algorithms to the data before the random sample is selected for the SAA algorithm. Once clustered, a sample is randomly selected from each of the clusters instead of from the entire dataset. This project looks to analyze five clustering techniques compared to each other and compared to the original SAA algorithm in order to see if clustering improves both the speed and the optimal solution of the SAA method for solving stochastic optimization problems. 

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2. Optimizing Driver Menus Under Stochastic Selection Behavior for Ridesharing and Crowdsourced Delivery

   Horner, Hannah ; Pazour, Jennifer ; Mitchell, John E. ( January 2021 , Transportation research)

   null (Ed.)

   Peer-to-peer logistics platforms coordinate independent drivers to fulfill requests for last mile delivery and ridesharing. To balance demand-side performance with driver autonomy, a new stochastic methodology provides drivers with a small but personalized menu of requests to choose from. This creates a Stackelberg game, in which the platform leads by deciding what menu of requests to send to drivers, and the drivers follow by selecting which request(s) they are willing to fulfill from their received menus. Determining optimal menus, menu size, and request overlaps in menus is complex as the platform has limited knowledge of drivers' request preferences. Exploiting the problem structure when drivers signal willingness to participate, we reformulate our problem as an equivalent single-level Mixed Integer Linear Program (MILP) and apply the Sample Average Approximation (SAA) method. Computational tests recommend a training sample size for inputted SAA scenarios and a test sample size for completing performance analysis. Our stochastic optimization approach performs better than current approaches, as well as deterministic optimization alternatives. A simplified formulation ignoring `unhappy drivers' who accept requests but are not matched is shown to produce similar objective values with a fraction of the runtime. A ridesharing case study of the Chicago Regional Transportation network provides insights for a platform wanting to provide driver autonomy via menu creation. The proposed methods achieved high demand performance as long as the drivers are well compensated (e.g., even when drivers are allowed to reject requests, on average over 90% of requests are fulfilled when 80% of the fare goes to drivers; this drops to below 60% when only 40% of the fare goes to drivers). Thus, neither the platform nor the drivers benefit from low driver compensation due to its resulting low driver participation and thus low request fulfillment. Finally, for the cases tested, a maximum menu size of 5 is recommended as it produces good quality platform solutions without requiring much driver selection time. 

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3. A Mean-risk Mixed Integer Nonlinear Program for Network Protection

   https://doi.org/liojoin  

   Burton, Amy ( January 2020 , Thesis for Clemson University)

   d. Many of the infrastructure sectors that are considered to be crucial by the Department of Homeland Security include networked systems (physical and temporal) that function to move some commodity like electricity, people, or even communication from one location of importance to another. The costs associated with these flows make up the price of the network’s normal functionality. These networks have limited capacities, which cause the marginal cost of a unit of flow across an edge to increase as congestion builds. In order to limit the expense of a network’s normal demand we aim to increase the resilience of the system and specifically the resilience of the arc capacities. Divisions of critical infrastructure have faced difficulties in recent years as inadequate resources have been available for needed upgrades and repairs. Without being able to determine future factors that cause damage both minor and extreme to the networks, officials must decide how to best allocate the limited funds now so that these essential systems can withstand the heavy weight of society’s reliance. We model these resource allocation decisions using a two-stage stochastic program (SP) for the purpose of network protection. Starting with a general form for a basic two-stage SP, we enforce assumptions that specify characteristics key to this type of decision model. The second stage objective—which represents the price of the network’s routine functionality—is nonlinear, as it reflects the increasing marginal cost per unit of additional flow across an arc. After the model has been designed properly to reflect the network protection problem, we are left with a nonconvex, nonlinear, nonseparable risk-neutral program. This research focuses on key reformulation techniques that transform the problematic model into one that is convex, separable, and much more solvable. Our approach focuses on using perspective functions to convexify the feasibility set of the second stage and second order conic constraints to represent nonlinear constraints in a form that better allows the use of computational solvers. Once these methods have been applied to the risk-neutral model we introduce a risk measure into the first stage that allows us to control the balance between an efficient, solvable model and the need to hedge against extreme events. Using Benders cuts that exploit linear separability, we give a decomposition and solution algorithm for the general network model. The innovations included in this formulation are then implemented on a transportation network with given flow demand 

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4. A Decision Rule Approach for Two-Stage Data-Driven Distributionally Robust Optimization Problems with Random Recourse

   https://doi.org/10.1287/ijoc.2021.0306  

   Fan, Xiangyi ; Hanasusanto, Grani A. ( November 2023 , INFORMS Journal on Computing)

   We study two-stage stochastic optimization problems with random recourse, where the coefficients of the adaptive decisions involve uncertain parameters. To deal with the infinite-dimensional recourse decisions, we propose a scalable approximation scheme via piecewise linear and piecewise quadratic decision rules. We develop a data-driven distributionally robust framework with two layers of robustness to address distributional uncertainty. We also establish out-of-sample performance guarantees for the proposed scheme. Applying known ideas, the resulting optimization problem can be reformulated as an exact copositive program that admits semidefinite programming approximations. We design an iterative decomposition algorithm, which converges under some regularity conditions, to reduce the runtime needed to solve this program. Through numerical examples for various known operations management applications, we demonstrate that our method produces significantly better solutions than the traditional sample-average approximation scheme especially when the data are limited. For the problem instances for which only the recourse cost coefficients are random, our method exhibits slightly inferior out-of-sample performance but shorter runtimes compared with a competing approach. 

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5. Data Pooling in Stochastic Optimization

   https://doi.org/10.1287/mnsc.2020.3933  

   Gupta, Vishal ; Kallus, Nathan ( January 2021 , Management Science)

   null (Ed.)

   Managing large-scale systems often involves simultaneously solving thousands of unrelated stochastic optimization problems, each with limited data. Intuition suggests that one can decouple these unrelated problems and solve them separately without loss of generality. We propose a novel data-pooling algorithm called Shrunken-SAA that disproves this intuition. In particular, we prove that combining data across problems can outperform decoupling, even when there is no a priori structure linking the problems and data are drawn independently. Our approach does not require strong distributional assumptions and applies to constrained, possibly nonconvex, nonsmooth optimization problems such as vehicle-routing, economic lot-sizing, or facility location. We compare and contrast our results to a similar phenomenon in statistics (Stein’s phenomenon), highlighting unique features that arise in the optimization setting that are not present in estimation. We further prove that, as the number of problems grows large, Shrunken-SAA learns if pooling can improve upon decoupling and the optimal amount to pool, even if the average amount of data per problem is fixed and bounded. Importantly, we highlight a simple intuition based on stability that highlights when and why data pooling offers a benefit, elucidating this perhaps surprising phenomenon. This intuition further suggests that data pooling offers the most benefits when there are many problems, each of which has a small amount of relevant data. Finally, we demonstrate the practical benefits of data pooling using real data from a chain of retail drug stores in the context of inventory management. This paper was accepted by Chung Piaw Teo, Special Issue on Data-Driven Prescriptive Analytics. 

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