Parametric Evaluation and Mean-Standard Deviation Analysis in Stochastic Programming Models.

Stochastic programming models have been proposed for capacity planning problems in different environments, including energy, telecommunication networks, distribution networks, and manufacturing systems. In this chapter we give an introductory tutorial to stochastic linear programming models, with emphasis on modeling techniques, rather than specialized solution methods. We consider two-stage and multi-stage stochastic programming models with recourse for manufacturing related applications, such as production planning and capacity planning with uncertainty on demand. We stress the importance of proper model formulation from two points of view: the first one is building strong mixed-integer formulations; the second one is generating scenario trees in order to suitably represent uncertainty while keeping them to a manageable size. We also compare the stochastic programming approach to traditional dynamic programming and to robust optimization.

Multiperiod stochastic programming for biomass supply chain design under spatiotemporal variability of feedstock supply

Stochastic optimization models have been extensively applied to financial portfolios and have proven their effectiveness in asset and asset-liability management; hardly however they have been applied to decision universe including not only financial but also derivatives written on the underlying, derivatives such as options or futures with their dedicated risk profiles and associated modeling complexities. Including options in the portfolio gives us the opportunity to hedge the underlying and to speculate on them to increase the profit potential given a certain risk level. The modeling of options in multistage stochastic programming framework would have many advantages, for instance, a put option can be used for insuring a portfolio against any downside movement in the market, high volatility in options prices is risky but rewarding if captured accurately, in-the-money call/put options at maturity can be used to buy/sell the underlying security at a price lower/higher than the market price, optimally increasing and reducing inventory. We present here multistage models to include cash and physical settled call and put options in a portfolio along with other asset classes.

Many planning problems involve choosing a set of optimal decisions for a system in the face of uncertainty of elements that may play a central role in the way the system is analyzed and operated. During the past decade, there has been a renewed interest in the modelling, analysis, and solution of such problems due to a remarkable development of both new theoretical results and novel computational techniques in stochastic optimization. A prominent approach is to develop upper and lower bounding approximations to the problem along with procedures to sharpen bounds until an acceptable tolerance is satisfied. The contributions of this dissertation are concerned with the latter approach. The thesis first studies the stochastic linear programming problem with randomness in both the objective coefficients and the constraints. A convex concave saddle property of the value function is utilized to derive new bounding techniques which generalize previously known results. These approximations require discretizing bounded domains of the random variables in such a way that tight upper and lower bounds result. Such techniques will prove attractive with the recent advances in large scale linear programming. The above results are also extended to obtain new upper and lower bounds when the domains of random variables are unbounded. While these bounds are tight, the approximating models are large-scale deterministic linear programs. In particular, with a proposed order-cone decomposition for the domains, these linear programs are well-structured, thus enabling one to use efficient techniques for solution, such as parallel computation. The thesis next considers convex stochastic programs. Using aggregation concepts from the deterministic literature, new bounds are developed for the problem which are computable using standard convex programming algorithms. Finally, the discussion is focused on a stochastic convex program arising in a certain resource allocation problem. Exploiting the problem structure, bounds are developed via the Karush-Kuhn-Tucker conditions. Rather than discretizing domains, these approximations advocate replacing difficult multidimensional integrals by a series of simple univariate integrals. Such practice allows one to preserve differentiability properties so that smooth convex programming methods can be applied for solution.

A stochastic program approach for path reconstruction oriented sensor location model

ABSTRACTThis paper introduces a two-stage stochastic integer linear programming model to improve phlebotomist scheduling in laboratory facilities of healthcare delivery systems. The model developed enables laboratory management to determine optimal scheduling policies that minimize work overload. The stochastic programming model considers the uncertainty associated with the blood collection demand in laboratory environments when optimizing phlebotomist scheduling. The paper presents an application of the model to a hospital laboratory in urban North Carolina as a case study discussing the implications for hospital laboratory management.

Although there have been numerous studies on the economics of weed control, relatively few of these studies have focused on natural ecosystems. The aims of this papers are: A) to identify the combination of control options that is optimal for blackberry (Rubus anglocandicans) in Australian natural ecosystems, B) to assess whether an integrated control strategy is superior to chemical-only strategies, C) to evaluate the economic net benefits of specific biologically oriented control methods (a rust and grazing by goats), D) to determine how changes in model parameters affect the optimal control strategy. To address these aims, a stochastic dynamic simulation model and a stochastic dynamic programming model are developed. The results indicate that, while an integrated strategy combining chemical and non-chemical control methods may be optimal in certain circumstances, it is not necessarily superior to a chemical-dominant strategy in all cases. Results show that in most cases a combination of herbicides and one physical control method (mowing) is optimal in the base-case scenario. However, if chemical efficacy is greater than in the base-case, the optimal strategy involves chemicals only. The results indicate that grazing goats for control of blackberry can be optimal despite its stochastic effectiveness and relatively low efficacy.

This paper focuses on solving two-stage stochastic mixed integer programs (SMIPs) with general mixed integer decision variables in both stages. We develop a decomposition algorithm in which the first stage approximation is solved using a branch-and-bound tree with nodes inheriting Benders' cuts that are valid for their ancestor nodes. In addition, we develop two closely related convexification schemes which use multi-term disjunctive cuts to obtain approximations of the second stage mixed-integer programs. We prove that the proposed methods are finitely convergent. One of the main advantages of our decomposition scheme is that we use a Benders-based branch-and-cut approach in which linear programming approximations are strengthened sequentially. Moreover as in many decomposition schemes, these subproblems can be solved in parallel. We also illustrate these algorithms using several variants of an SMIP example from the literature, and present preliminary evidence of the scalability of these algorithms as the number of scenarios increases.

In this dissertation we develop numerical algorithms and software tools to facilitate parallel solutions of nonlinear programming (NLP) problems. In particular, we address large-scale, block-structured problems with an intrinsic decomposable configuration. These problems arise in a great number of engineering applications, including parameter estimation, optimal control, network optimization, and stochastic programming. The structure of these problems can be leveraged by optimization solvers to accelerate solutions and overcome memory limitations, and we propose variants to two classes of optimization algorithms: augmented Lagrangian (AL) schemes and Schur-complement interior-point methods. The convergence properties of augmented Lagrangian decomposition schemes like the alternating direction method of multipliers (ADMM) and progressive hedging (PH) are well established for convex optimization but convergence guarantees in non-convex settings are still poorly understood. In practice, however, ADMM and PH often perform satisfactorily in complex non-convex NLPs. In this work, we study connections between the method of multipliers (MM), ADMM, and PH to derive benchmarking metrics that explain why PH and ADMM work in practice. We illustrate the concepts using challenging dynamic optimization problems. Our exposition seeks to establish more formalism in benchmarking ADMM, PH, and AL schemes and to motivate algorithmic improvements. The effectiveness of nonlinear interior-point solvers for solving large-scale problems relies quite heavily on the solution of the underlying linear algebra systems. The schur-complement decomposition is very effective for parallelizing the solution of linear systems with modest coupling. However, for systems with large number of coupling variables the schur-complement method does not scale favorably. We implement an approach that uses a Krylov solver (GMRES) preconditioned with ADMM to solve block-structured linear systems that arise in the interior-point method. We show that this ADMM-GMRES approach overcomes the well-known scalability issues of Schur decomposition. One important drawback of using decomposition approaches like ADMM and PH is their convergence rate. Unlike Schur-complement interior-point algorithms that have super-linear convergence, augmented Lagrangian approaches typically exhibit linear and sublinear rates. We exploit connections between ADMM and the Schur-complement decomposition to derive an accelerated version of ADMM. Specifically, we study the effectiveness of performing a Newton-Raphson algorithm to compute multiplier estimates for augmented Lagrangian methods. We demonstrate using two-stage stochastic programming problems that our multiplier update achieves convergence in fewer iterations for MM on general nonlinear problems. In the case of ADMM, the newton update significantly reduces the number of subproblem solves for convex quadratic programs (QPs). Moreover, we show that using newton multiplier updates makes the method robust to the selection of the penalty parameter. Traditionally, state-of-the-art optimization solvers are implemented in low-level programming languages. In our experience, the development of decomposition algorithms in these frameworks is challenging. They present a steep learning curve and can slow the development and testing of new numerical algorithms. To mitigate these challenges, we developed PyNumero, a new open source framework implemented in Python and C++. The package seeks to facilitate development of optimization algorithms for large-scale optimization within a high-level programming environment while at the same time minimizing the computational burden of using Python. The efficiency of PyNumero is illustrated by implementing algorithms for problems arising in stochastic programming and optimal control. Timing results are presented for both serial and parallel implementations. Our computational studies demonstrate that with the appropriate balance between compiled code and Python, efficient implementations of optimization algorithms are achievable in these high-level languages.

This paper proposes a stochastic dependent-chance programming model for vendor selection problem under the condition that the capacity, quality level, service level and lead time of each vendor are considered to be stochastic. Since stochastic programming is hard to solve by traditional methods, a hybrid adaptive genetic algorithm, which embeds the neutral network and stochastic simulation, is presented. To improve the performance of the algorithm, the probability of crossover and mutation will be adjusted according to the stage of evolution and fitness of the population. The solution procedure is tested on several randomly generated problems with varying parameters. The experimental results demonstrate that the hybrid adaptive genetic algorithm has strong adaptability.

Finance: Longevity Risk Management for Individual Investors Optimal SP-Based Personal Financial Planning with Intermediate and Long Term Goals Intertemporal Surplus Management with Jump Risks Jump-Diffusion Risk-Sensitive Benchmarked Asset Management Dynamic Portfolio Optimization Under Regime-Based Firm Strength Options Portfolio Management as a Chance Constrained Problem Stochastic Models for Optimizing Immunization Strategies in Fixed-Income Security Portfolios Under Some Sources of Uncertainty Stochastic Programming and Optimization in Horserace Betting Production Planning and Logistics: Multi-Stage Stochastic Programming for Natural Gas Infrastructure Design with a Production Perspective Stochastic Programming Model for Optimizing the Production of Farmed Atlantic Salmon Prioritizing Network Interdiction of Nuclear Smuggling Sawmill Production Planning Under Uncertainty: Modelling and Solution Approaches Energy: An Electricity Procurement Model with Energy and Peak Charges Value of Flexibility in Hydroelectric Reservoir Operations Multi-Lag Benders Decomposition for Power Generation Planning with Nonanticipativity Constraints on the Dispatch of LNG Thermal Plants Theory Papers: Stochastic Second-Order Cone Programming in Mobile Ad-Hoc Networks Stochastic Frequency Assignment Problem.

With the complications of modern agricultural production, how to formulate scientific and reasonable planting strategies in the limited land resources and highly uncertain market environment has become a key research topic in the agricultural field. The diversity of land types (such as terraced fields, sloping fields, and irrigated land) and the variety of crops in North China make the problem of planting optimization more complicated and practical. In this paper, a bi-level optimization model combining mixed integer programming (MIP) and stochastic programming (SP) is proposed to solve the problems of resource allocation and income maximization in crop planting. Under certain conditions, the planting scheme is constructed by a mixed integer programming model, and the effects of two over-processing strategies, namely “slow sales” and “lower price sales”, are analyzed. Under the condition of uncertainty, various random scenarios are generated by Monte Carlo simulation, and the influence of market demand and yield fluctuation per mu on planting strategy is explored by combining with a stochastic programming model.The experimental results show that the stochastic programming model is superior to the deterministic model in key indicators such as average income, land utilization rate, and income stability. Specifically, compared with the deterministic model, the land utilization rate of the stochastic programming model is increased by about 15% on average, the average income is increased by about 8%, and the fluctuation range of income is reduced by more than 50%. Sensitivity analysis further reveals the significant influence of market demand and the fluctuation range of yield per mu on income, and the stochastic programming model shows strong flexibility and robustness in adapting to uncertainty. At the same time, this study provides multi-dimensional decision support for agricultural operators by introducing multi-scenario weight optimization, which not only performs well in improving economic benefits but also provides an effective reference in resource allocation efficiency and risk management.This paper provides a new theoretical framework for agricultural planting optimization and has important practical value. The stochastic programming method can effectively deal with extreme market conditions and environmental uncertainties, and provide a decision-making basis for agricultural policy makers in subsidy distribution and market regulation. In addition, the research results provide technical support for the digital transformation and intelligent management of agricultural production and promote the development of precision agriculture. The optimization model proposed in this study also provides a reference for resource allocation and optimization in other fields.

How should water resources be allocated for shale gas development? An exploratory study in China

This paper develops a mathematical model for post-disaster planning with human casualties, which can be considered as operational guidance for the proper use of emergency resources. For this purpose, a stochastic mixed-integer programming model is provided to formulate the problem. The objective functions of the model are (1) maximizing the survival probability of patients, (2) minimizing the maximum of completion time of treatment of all patients, and (3) minimizing the total cost of operations. The model is solved with the ϵ-constraint method. Due to the NP-hardness of the problem which is a significant challenge in the literature, two innovative meta-heuristic algorithms are proposed, i.e. a non-dominated sorting genetic algorithm (NSGA-II) and a multi-objective simulated annealing (MOSA). Finally, a comprehensive computational analysis is performed for evaluation purposes. Also, a case study is made on the earthquake in Iran, which illustrates the real-world application of the model. Highlights A stochastic multi-objective mathematical programming model to allocate patients to hospitals and treat them. NSGA-II and MOSA are proposed, in addition to ϵ-constraint method as solution methods. Performance of two meta-heuristic algorithms is measured with five evaluation metrics. The study showed that NSGA-II is more effective than MOSA. The model was implemented on a real-world case, Tabriz earthquake in Iran.

For the companies adopting assemble-to-order (ATO) production strategy, providing accurate and reliable order promising is an important issue especially subject to the uncertainty. The ATO production system has the following characteristics. There are no finished goods in the strict sense. Inventory is holding at component level. The material and capacity planning is driven by the sales forecasting. Assembly scheduling is driven by the customer orders. We reviewed the existing order-promising researches which are mostly determined by the environment. In the base of analysis the influence mechanism of uncertainty and available-to-promise/capable-to-promise (ATP/CTP) allocation approach, we establish a stochastic dependent-chance programming model considering the available resources which are fluctuant. The objective is to maximise the chance of acceptance of the orders as much as possible. This means higher utilisation of assembly capacity. The available amount of resources obeyed some kind of common distribution. We develop a hybrid genetic algorithm (HGA) for solving the stochastic programming model; the main steps are chance function construction, Monte Carlo simulation, neural network approach and genetic algorithm (GA). We implement the model and algorithm in an automobile manufacturer, the uncertainty variables are fitted using historical data. The experiment result verified that the model and solver are valid.