A polynomial-time algorithm for optimizing over N-fold 4-block decomposable integer programs

In this paper we generalize N-fold integer programs and two-stage integer programs with N scenarios to N-fold 4-block decomposable integer programs. We show that for fixed blocks but variable N, these integer programs are polynomial-time solvable for any linear objective. Moreover, we present a polynomial-time computable optimality certificate for the case of fixed blocks, variable N and any convex separable objective function. We conclude with two sample applications, stochastic integer programs with second-order dominance constraints and stochastic integer multi-commodity flows, which (for fixed blocks) can be solved in polynomial time in the number of scenarios and commodities and in the binary encoding length of the input data. In the proof of our main theorem we combine several non-trivial constructions from the theory of Graver bases. We are confident that our approach paves the way for further extensions

On capacity expansion planning under strategic and operational uncertainties based on stochastic dominance risk averse management

A new scheme for dealing with uncertainty in scenario trees is presented for dynamic mixed 0–1 optimization problems with strategic and operational stochastic parameters. Let us generically name this type of problems as capacity expansion planning (CEP) in a given system, e.g., supply chain, production, rapid transit network, energy generation and transmission network, etc. The strategic scenario tree is usually a multistage one, and the replicas of the strategic nodes root structures in the form of either a special scenario graph or a two-stage scenario tree, depending on the type of operational activity in the system. Those operational scenario structures impact in the constraints of the model and, thus, in the decomposition methodology for solving usually large-scale problems. This work presents the modeling framework for some of the risk neutral and risk averse measures to consider for CEP problem solving. Two types of risk averse measures are considered. The first one is a time-inconsistent mixture of the chance-constrained and second-order stochastic dominance (SSD) functionals of the value of a given set of functions up to the strategic nodes in selected stages along the time horizon, The second type is a strategic node-based time-consistent SSD functional for the set of operational scenarios in the strategic nodes at selected stages. A specialization of the nested stochastic decomposition methodology for that problem solving is outlined. Its advantages and drawbacks as well as the framework for some schemes to, at least, partially avoid those drawbacks are also presentedThis research has been partially supported by the projects: MTM2015-63710 and MTM2016-79765 from the Spanish Ministry of Economy and Competitiveness. The authors like to thank the positive criticism of their colleagues Antonio Alonso-Ayuso, Luis Cadarso, F. Javier Martín-Campo and Angel Marín that helped to improve the presentation of the wor

An SDP approach for multiperiod mixed 0-1 linear programming models with stochastic dominance constraints for risk management *

Abstract In this paper we consider multiperiod mixed 0-1 linear programming models under uncertainty. We propose a risk averse strategy using stochastic dominance constraints (SDC) induced by mixed-integer linear recourse as the risk measure. The SDC strategy extends the existing literature to the multistage case and includes both first-order and second-order constraints. We propose a stochastic dynamic programming (SDP) solution approach, where one has to overcome the negative impact the cross-scenario constraints, due to SDC, have on the decomposability of the model. In our computational experience we compare our SDP against a commercial optimization package, in terms of solution accuracy and elapsed time. We use supply chain planning instances, where procurement, production, inventory, and distribution decisions need to be made under demand uncertainty. We confirm the hardness of the testbed, where the benchmark cannot find a feasible solution for half of the test instances while we always find one, and show the appealing tradeoff of SDP, in terms of solution accuracy and elapsed time, when solving medium-to-large instances

Models and algorithms for dominance-constrained stochastic programs with recourse

In der vorliegenden Dissertationsschrift befassen wir uns mit stochastischen Optimierungsproblemen unter Nebenbedingungen, die mithilfe stochastischer Ordnungen formuliert sind. Hierbei konzentrieren wir uns auf stochastische Dominanz erster Ordnung und die steigende konvexe Ordnung, wobei beide Ordnungen in unserem Fall auf Zufallsgrößen operieren, welche Optimalwerten zweistuger stochastischer Optimierungsprobleme mit Kompensation entsprechen. Wir stellen die theoretische Relevanz der vorliegenden Problemklasse heraus und tragen zur Entwicklung von effizienten Lösungsverfahren bei. Um Letzteres zu erreichen untersuchen und erweitern wir bestehende gemischt-ganzzahlige lineare Repräsentationen dieser Probleme und entwickeln maßgeschneiderte Dekompositionsverfahren. Der Schwerpunkt dieser Arbeit liegt dabei auf der Entwicklung und Implementierung besonders effizienter Lösungsansätze für den Fall mit linearer Kompensation.We consider optimization problems with stochastic order constraints of first and second order posed on random variables coming from two-stage stochastic programs with recourse. We clarify the theoretical relevance of these specific problems, and contribute to improving their computational tractability. For the latter, we review and enhance mixed-integer linear programming (MILP) equivalents. These exist for either mixed-integer or continuous variables in the second stage. Algorithmically, our focus is on developing tailored cutting-plane decomposition methods for these models

Supply chain network design under uncertainty and risk

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.We consider the research problem of quantitative support for decision making in supply chain network design (SCND). We first identify the requirements for a comprehensive SCND as (i) a methodology to select uncertainties, (ii) a stochastic optimisation model, and (iii) an appropriate solution algorithm. We propose a process to select a manageable number of uncertainties to be included in a stochastic program for SCND. We develop a comprehensive two-stage stochastic program for SCND that includes uncertainty in demand, currency exchange rates, labour costs, productivity, supplier costs, and transport costs. Also, we consider conditional value at risk (CV@R) to explore the trade-off between risk and return. We use a scenario generator based on moment matching to represent the multivariate uncertainty. The resulting stochastic integer program is computationally challenging and we propose a novel iterative solution algorithm called adaptive scenario refinement (ASR) to process the problem. We describe the rationale underlying ASR, validate it for a set of benchmark problems, and discuss the benefits of the algorithm applied to our SCND problem. Finally, we demonstrate the benefits of the proposed model in a case study and show that multiple sources of uncertainty and risk are important to consider in the SCND. Whereas in the literature most research is on demand uncertainty, our study suggests that exchange rate uncertainty is more important for the choice of optimal supply chain strategies in international production networks. The SCND model and the use of the coherent downside risk measure in the stochastic program are innovative and novel; these and the ASR solution algorithm taken together make contributions to knowledge

Multi-stage stochastic optimization and reinforcement learning for forestry epidemic and covid-19 control planning

This dissertation focuses on developing new modeling and solution approaches based on multi-stage stochastic programming and reinforcement learning for tackling biological invasions in forests and human populations. Emerald Ash Borer (EAB) is the nemesis of ash trees. This research introduces a multi-stage stochastic mixed-integer programming model to assist forest agencies in managing emerald ash borer insects throughout the U.S. and maximize the public benets of preserving healthy ash trees. This work is then extended to present the first risk-averse multi-stage stochastic mixed-integer program in the invasive species management literature to account for extreme events. Significant computational achievements are obtained using a scenario dominance decomposition and cutting plane algorithm.The results of this work provide crucial insights and decision strategies for optimal resource allocation among surveillance, treatment, and removal of ash trees, leading to a better and healthier environment for future generations. This dissertation also addresses the computational difficulty of solving one of the most difficult classes of combinatorial optimization problems, the Multi-Dimensional Knapsack Problem (MKP). A novel 2-Dimensional (2D) deep reinforcement learning (DRL) framework is developed to represent and solve combinatorial optimization problems focusing on MKP. The DRL framework trains different agents for making sequential decisions and finding the optimal solution while still satisfying the resource constraints of the problem. To our knowledge, this is the first DRL model of its kind where a 2D environment is formulated, and an element of the DRL solution matrix represents an item of the MKP. Our DRL framework shows that it can solve medium-sized and large-sized instances at least 45 and 10 times faster in CPU solution time, respectively, with a maximum solution gap of 0.28% compared to the solution performance of CPLEX. Applying this methodology, yet another recent epidemic problem is tackled, that of COVID-19. This research investigates a reinforcement learning approach tailored with an agent-based simulation model to simulate the disease growth and optimize decision-making during an epidemic. This framework is validated using the COVID-19 data from the Center for Disease Control and Prevention (CDC). Research results provide important insights into government response to COVID-19 and vaccination strategies

Supply chain network design under uncertainty and risk

We consider the research problem of quantitative support for decision making in supply chain network design (SCND). We first identify the requirements for a comprehensive SCND as (i) a methodology to select uncertainties, (ii) a stochastic optimisation model, and (iii) an appropriate solution algorithm. We propose a process to select a manageable number of uncertainties to be included in a stochastic program for SCND. We develop a comprehensive two-stage stochastic program for SCND that includes uncertainty in demand, currency exchange rates, labour costs, productivity, supplier costs, and transport costs. Also, we consider conditional value at risk (CV@R) to explore the trade-off between risk and return. We use a scenario generator based on moment matching to represent the multivariate uncertainty. The resulting stochastic integer program is computationally challenging and we propose a novel iterative solution algorithm called adaptive scenario refinement (ASR) to process the problem. We describe the rationale underlying ASR, validate it for a set of benchmark problems, and discuss the benefits of the algorithm applied to our SCND problem. Finally, we demonstrate the benefits of the proposed model in a case study and show that multiple sources of uncertainty and risk are important to consider in the SCND. Whereas in the literature most research is on demand uncertainty, our study suggests that exchange rate uncertainty is more important for the choice of optimal supply chain strategies in international production networks. The SCND model and the use of the coherent downside risk measure in the stochastic program are innovative and novel; these and the ASR solution algorithm taken together make contributions to knowledge.EThOS - Electronic Theses Online ServiceGBUnited Kingdo