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Risk-Averse Bargaining in a stochastic Optimization Context

M&SOM-MANUFACTURING & SERVICE OPERATIONS MANAGEMENT 2022年第1期25卷 323-340页

作者： Gutjahr, Walter J. Kovacevic, Raimund M. Wozabal, David Univ Vienna Dept Stat & Operat Res A-1090 Vienna Austria Danube Univ Ctr Evidence Based Hlth Econ A-3500 Krems Austria Vienna Univ Technol Inst Stat & Math Methods Econ A-1040 Vienna Austria Tech Univ Munich Sch Management D-80333 Munich Germany

Problem definition: Bargaining situations are ubiquitous in economics and management. We consider the problem of bargaining for a fair ex ante distribution of random profits arising from a cooperative effort of a fixed set of risk-averse agents. Our approach integrates optimal managerial decision making into bargaining situations with random outcomes and explicitly models the impact of risk aversion. The proposed solution rests on a firm axiomatic foundation and yet allows to compute concrete bargaining solutions for a wide range of practically relevant problems. Methodology/results: We model risk preferences using coherent acceptability functionals and base our bargaining solution on a set of axioms that can be considered a natural extension of Nash bargaining to our setting. We show that the proposed axioms fully characterize a bargaining solution, which can be efficiently computed by solving a stochastic optimization problem. We characterize special cases where random payoffs of players are simple functions of overall project profit. In particular, we show that, for players with distortion risk functionals, the optimal bargaining solution can be represented by an exchange of standard options contracts with the project profit as the underlying asset. We illustrate the concepts in the paper with a detailed example of risk-averse households that jointly invest into a solar plant. Managerial implications: We demonstrate that there is no conflict of interest between players about management decisions and that risk aversion facilitates cooperation. Furthermore, our results on the structure of optimal contracts as a basket of option contracts provides valuable guidance for negotiators.

关键词： stochastic bargaining games coherent risk measures stochastic programming photovoltaics

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BIAS REDUCTION IN SAMPLE-BASED OPTIMIZATION

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SIAM JOURNAL ON OPTIMIZATION 2022年第1期32卷 130-151页

作者： Dentcheva, Darinka Lin, Yang Stevens Inst Technol Dept Math Sci Hoboken NJ 07030 USA

We consider the stochastic optimization problems which use observed data to estimate essential characteristics of the random quantities involved. Sample average approximation (SAA) or empirical (plug-in) estimation are very popular ways to use data in optimization. It is well known that SAA suffers from downward bias. Our proposal is to use smooth estimators rather than empirical ones in the optimization problems. We establish consistency results for the optimal value and the set of optimal solutions of the new problem formulation. The performance of the proposed approach is compared to SAA theoretically and numerically. We analyze the bias of the new problems and identify sufficient conditions for ensuring less biased estimation of the optimal value of the true problem. At the same time, the error of the new estimator remains controlled. Those conditions are satisfied for many popular statistical problems such as regression models, classification problems, and optimization problems with average (conditional) value at risk. We have proved that smoothing the least-squares objective in a regression problem by a normal kernel leads to a ridge regression. Our numerical experience shows that the new estimators also frequently exhibit smaller variance and smaller mean-square error than those of SAA.

关键词： kernel estimators stochastic programming sample average approximation strong law of large numbers smoothing regularized regression

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stochastic DECOMPOSITION METHOD FOR TWO-STAGE DISTRIBUTIONALLY ROBUST LINEAR OPTIMIZATION

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SIAM JOURNAL ON OPTIMIZATION 2022年第3期32卷 1901-1930页

作者： Gangammanavar, Harsha Bansal, Manish Southern Methodist Univ Dept Engn Management Informat & Syst Dallas TX 75275 USA Virginia Tech Dept Ind & Syst Engn Blacksburg VA 24061 USA

In this paper, we present a sequential sampling-based algorithm for the two-stage distributionally robust linear programming (2-DRLP) models. The 2-DRLP models are defined over a general class of ambiguity sets with discrete or continuous probability distributions. The algorithm is a distributionally robust version of the well-known stochastic decomposition algorithm of Higle and Sen [Math. Oper. Res., 16 (1991), pp. 650-669] for a two-stage stochastic linear program. We refer to the algorithm as the distributionally robust stochastic decomposition (DRSD) method. The key features of the algorithm include (1) it works with data-driven approximations of ambiguity sets that are constructed using samples of increasing size and (2) efficient construction of approximations of the worst-case expectation function that solves only two second-stage subproblems in every iteration. We identify conditions under which the ambiguity set approximations converge to the true ambiguity sets and show that the DRSD method asymptotically identifies an optimal solution, with probability one. We also computationally evaluate the performance of the DRSD method for solving distributionally robust versions of instances considered in stochastic programming literature. The numerical results corroborate the analytical behavior of the DRSD method and illustrate the computational advantage over an external sampling-based decomposition approach (distributionally robust L-shaped method).

关键词： distributionally robust optimization stochastic programming stochastic decomposition sequential sampling cutting plane method

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Predicting Tactical Solutions to Operational Planning Problems Under Imperfect Information

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INFORMS JOURNAL ON COMPUTING 2022年第1期34卷 227-242页

作者： Larsen, Eric Lachapelle, Sebastien Bengio, Yoshua Frejinger, Emma Lacoste-Julien, Simon Lodi, Andrea Univ Montreal Dept Comp Sci & Operat Res Montreal PQ H3T 1J4 Canada Univ Montreal CIRRELT Montreal PQ H3C 3J7 Canada Univ Montreal Mila Montreal PQ H2S 3H1 Canada Polytech Montreal CERC Montreal PQ H3T 1J4 Canada

This paper offers a methodological contribution at the intersection of machine learning and operations research. Namely, we propose a methodology to quickly predict expected tactical descriptions of operational solutions (TDOSs). The problem we address occurs in the context of two-stage stochastic programming, where the second stage is demanding computationally. We aim to predict at a high speed the expected TDOS associated with the second-stage problem, conditionally on the first-stage variables. This may be used in support of the solution to the overall two-stage problem by avoiding the online generation of multiple second-stage scenarios and solutions. We formulate the tactical prediction problem as a stochastic optimal prediction program, whose solution we approximate with supervised machine learning. The training data set consists of a large number of deterministic operational problems generated by controlled probabilistic sampling. The labels are computed based on solutions to these problems (solved independently and offline), employing appropriate aggregation and subselection methods to address uncertainty. Results on our motivating application on load planning for rail transportation show that deep learning models produce accurate predictions in very short computing time (milliseconds or less). The predictive accuracy is close to the lower bounds calculated based on sample average approximation of the stochastic prediction programs.

关键词： supervised learning deep learning integer linear programming stochastic programming

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On sample average approximation for two-stage stochastic programs without relatively complete recourse

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MATHEMATICAL programming 2022年第1-2期196卷 719-754页

作者： Chen, Rui Luedtke, James Univ Wisconsin Dept Ind & Syst Engn Madison WI 53706 USA

We investigate sample average approximation (SAA) for two-stage stochastic programs without relatively complete recourse, i.e., for problems in which there are first-stage feasible solutions that are not guaranteed to have a feasible recourse action. As a feasibility measure of the SAA solution, we consider the "recourse likelihood", which is the probability that the solution has a feasible recourse action. For epsilon is an element of (0, 1), we demonstrate that the probability that a SAA solution has recourse likelihood below 1 - epsilon converges to zero exponentially fast with the sample size. Next, we analyze the rate of convergence of optimal solutions of the SAA to optimal solutions of the true problem for problems with a finite feasible region, such as bounded integer programming problems. For problems with non-finite feasible region, we propose modified "padded" SAA problems and demonstrate in two cases that such problems can yield, with high confidence, solutions that are certain to have a feasible recourse decision. Finally, we conduct a numerical study on a two-stage resource planning problem that illustrates the results, and also suggests there may be room for improvement in some of the theoretical analysis.

关键词： stochastic programming Mixed-integer programming Sample average approximation Chance constraints

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Transitivity and approximate consistency threshold determination for reciprocal preference relations in group decision making

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JOURNAL OF THE OPERATIONAL RESEARCH SOCIETY 2022年第7期73卷 1649-1666页

作者： Mi, Xiaomei Liao, Huchang Zeng, Xiao-Jun Sichuan Univ Chengdu Peoples R China Univ Manchester Manchester Lancs England

In a group decision-making process, contradictory pairwise comparisons may exist in individual preferences or the group preferences even though the consensus level is reached. To avoid such a contradictory phenomenon, this study presents an approximate transitivity-based consistency threshold for reciprocal preference relations to ensure the reliability of the ranking of alternatives with reciprocal preference relations in group decision making. The natural inconsistency or intransitivity of reciprocal preference relations is analyzed and verified by numerical experiments. Then, using the results of numerical experiments, approximate transitivity-based consistency thresholds are introduced based on the objectives of minimising the Type I error, Type II error and total error in statistics. Moreover, a transitivity checking process regarding individual reciprocal preference relations and group reciprocal preference relations is incorporated in the group decision-making process. A transitivity-checking integrated group decision-making model is given for application. An example about the station selection for high-speed railway line is provided to show the necessity of the approximate transitivity-based consistency threshold in checking the transitivity of reciprocal preference relations for group decision making.

关键词： Decision analysis group decision making multi-criteria stochastic programming data mining

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Optimal follow-up policies for monitoring chronic diseases based on virtual age

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INTERNATIONAL JOURNAL OF PRODUCTION RESEARCH 2022年第15期60卷 4712-4726页

作者： Li, Mei Liu, Zixian Liu, Yiliu Li, Xiaopeng Lv, Ling Tianjin Univ Coll Management & Econ Tianjin Peoples R China Norwegian Univ Sci & Technol Dept Mech & Ind Engn Trondheim Norway Nanjing Univ Finance & Econ Sch Management & Engn Nanjing Peoples R China Tianjin Childrens Hosp Dept Endocrinol 238 Longyan St Tianjin 300134 Peoples R China

Follow-up policies following treatment are indispensable and effective in reducing the number of complications of chronic diseases, and hence the cost of treating complications, but bring additional follow-up cost inevitably. This paper introduces the virtual age method to measure the effect of follow-up on the patient's risk of developing a complication, and further proposes a mixed integer nonlinear programming model to develop the optimal periodic follow-up policies from a cost perspective. By means of the proposed model, the optimal timing and type of follow-up checkups for heterogeneous patients can be derived, achieving a tradeoff between costs of treating complications and follow-up. A case study of pediatric type 1 diabetes mellitus patients is presented to illustrate the applicability of the proposed method and analyse the impacts of significant input parameters on the optimal model solutions. The findings form the basis to design flexible and effective follow-up policies for monitoring patients with chronic diseases.

关键词： Follow-up policy disease monitoring stochastic programming virtual age patient heterogeneity

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A stochastic optimization approach for the supply vessel planning problem under uncertain demand

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TRANSPORTATION RESEARCH PART B-METHODOLOGICAL 2022年 162卷 209-228页

作者： Santos, A. M. P. Fagerholt, Kjetil Laporte, Gilbert Soares, C. Guedes Univ Lisbon Ctr Marine Technol & Ocean Engn CENTEC Inst Super Tecn Ave Rovisco Pais P-10490 Lisbon Portugal NTNU Dept Ind Econ & Technol Management Alfred Getz veg 3 N-7491 Trondheim Norway HEC Montreal 3000 Chemin Cote St Catherine Montreal PQ H3T 2A7 Canada

This paper presents a two-stage stochastic programming with recourse methodology to solve the Supply Vessel Planning Problem with stochastic Demands (SVPPSD), a problem arising in offshore logistics and which generalizes the Periodic Vehicle Routing Problem with stochastic Demands and Time Windows. In the SVPPSD, a fleet of vessels is used to deliver a regular supply of commodities to a set of offshore installations to ensure continuous production, with each installation requiring one or more visits per week and having stochastic demands. Both the onshore depot where the product to be distributed is kept and the offshore installations have time windows, and voyages are allowed to span more than one day. A solution to the SVPPSD consists in the identification of an optimal fleet of vessels and the corresponding weekly schedule. As a solution methodology, we embed a discrete-event simulation engine within a genetic search procedure to approximate the cost of recourse and arrive at the minimized expected cost solution. We make comparisons with two alternative approaches: an expected value problem with upscaled demand, and a chance-constrained algorithm. While alternative methodologies yield robust schedules, robustness is achieved mainly through an increase in fleet size. In contrast, a two-stage stochastic programming with recourse algorithm, by accounting for the cost of recourse in the search phase, and exploring a wider solution space, allows arriving at robust schedules with a smaller fleet size, thereby yielding significant cost savings. For the tested problem instances, the proposed algorithm leads to savings of approximately 10 to 15 million USD per year.

关键词： Supply vessel planning problem stochastic programming stochastic demands

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Robust Traffic Control Using a First Order Macroscopic Traffic Flow Model

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IEEE TRANSACTIONS ON INTELLIGENT TRANSPORTATION SYSTEMS 2022年第7期23卷 8048-8062页

作者： Liu, Hao Claudel, Christian Machemehl, Randy Univ Texas Austin Ctr Transportat Res Austin TX 78712 USA Univ Texas Austin Dept Civil Architectural & Environm Engn Austin TX 78712 USA

Traffic control is at the core of research in transportation engineering because it is one of the hest practices for reducing traffic congestion. It has been shown in recent years that the traffic control problem involving Lighthill-Whitham-Richards (LWR) model can be formulated as a Linear programming (LP) problem given that the corresponding initial conditions and the model parameters in the fundamental diagram are fixed. However, the initial conditions can be uncertain when studying actual control problems. This paper presents a stochastic programming formulation of the boundary control problem involving chance constraints, to capture the uncertainty in the initial conditions. Different objective functions are explored using this framework, and the proposed model is validated by conducting case studies for both a single highway link and a highway network. In addition, the accuracy of relaxed optimal results is proved using Monte Carlo simulation.

关键词： Traffic control linear programming stochastic programming chance constraints optimal control

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Demand response potential of industrial processes considering uncertain short-term electricity prices

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AICHE JOURNAL 2022年第11期68卷 e17828-e17828页

作者： Germscheid, Sonja H. M. Mitsos, Alexander Dahmen, Manuel Forschungszentrum Julich Inst Energy & Climate Res Energy Syst Engn IEK 10 D-52425 Julich Germany Rhein Westfal TH Aachen Aachen Germany JARA ENERGY Julich Germany Rhein Westfal TH Aachen Proc Syst Engn AVT SVT Aachen Germany

Time-varying electricity prices on the day-ahead and intraday market incentivize demand response of industrial processes. In prior work (Schafer et al. AIChE J. 2020;66:1-14), we studied the demand response potential with a generalized process model, but neglected the intraday market. Extending our prior investigation, we account for uncertain intraday prices in a mixed-integer linear stochastic programming-based scheduling, that is, we minimize expected cost and conditional value-at-risk in a bi-objective optimization. We find that for very broad variations of the generalized process parameters, the conditional value-at-risk can be reduced significantly without drastically increasing the expected cost. Furthermore, simultaneously improving multiple process parameter leads to synergetic benefits. Moreover, the savings of three electrolysis processes can be more than doubled by marketing flexibility on the intraday market in addition to the day-ahead market. Overall, our model allows for a rapid early assessment of the demand response potential considering the two markets.

关键词： conditional value-at-risk demand response electricity price uncertainty production scheduling stochastic programming

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