Multistage Stochastic Linear Programming Research Articles

Two-stage linear decision rules for multi-stage stochastic programming

Multi-stage stochastic linear programs (MSLPs) are notoriously hard to solve in general. Linear decision rules (LDRs) yield an approximation of an MSLP by restricting the decisions at each stage to be an affine function of the observed uncertain parameters. Finding an optimal LDR is a static optimization problem that provides an upper bound on the optimal value of the MSLP, and, under certain assumptions, can be formulated as an explicit linear program. Similarly, as proposed by Kuhn et al. (Math Program 130(1):177–209, 2011) a lower bound for an MSLP can be obtained by restricting decisions in the dual of the MSLP to follow an LDR. We propose a new approximation approach for MSLPs, two-stage LDRs. The idea is to require only the state variables in an MSLP to follow an LDR, which is sufficient to obtain an approximation of an MSLP that is a two-stage stochastic linear program (2SLP). We similarly propose to apply LDR only to a subset of the variables in the dual of the MSLP, which yields a 2SLP approximation of the dual that provides a lower bound on the optimal value of the MSLP. Although solving the corresponding 2SLP approximations exactly is intractable in general, we investigate how approximate solution approaches that have been developed for solving 2SLP can be applied to solve these approximation problems, and derive statistical upper and lower bounds on the optimal value of the MSLP. In addition to potentially yielding better policies and bounds, this approach requires many fewer assumptions than are required to obtain an explicit reformulation when using the standard static LDR approach. A computational study on two example problems demonstrates that using a two-stage LDR can yield significantly better primal policies and modestly better dual policies than using policies based on a static LDR.

A Multi-Stage Stochastic Mixed-Integer Linear Programming to Design an Integrated Production-Distribution Network under Stochastic Demands

A Multi-Stage Stochastic Mixed-Integer Linear Programming to Design an Integrated Production-Distribution Network under Stochastic Demands

Two-Scale Stochastic Optimization for Controlling Distributed Storage Devices

This paper is motivated by a power system with storage devices at multiple locations which need to be controlled at a much finer timescale than that necessary for conventional generation units. We present a stochastic optimization model of the power system which captures interactions of decisions at these two timescales through a novel state-variable formulation. The model also includes transmission constraints approximated by a linearized dc network, fast response operating reserves, and renewable generation. To tackle this high-dimensional multistage stochastic optimization problem, we present a sequential sampling method which we refer to as the stochastic dynamic linear programming. This algorithm is a dynamic extension of regularized two-stage stochastic decomposition for stagewise independent multistage stochastic linear programs, and is targeted at the class of problems where decisions are made at two different timescales. We compare our algorithm with the stochastic dual dynamic programming (SDDP) which has been effectively applied in planning power systems operations. Our computational results show that our sequential Monte-Carlo approach provides prescriptive solutions and values which are statistically indistinguishable from those obtained from SDDP, while improving computational times significantly.

ON INTEGRATED CHANCE CONSTRAINTS IN ALM FOR PENSION FUNDS

AbstractWe discuss the role ofintegrated chance constraints(ICC) as quantitative risk constraints in asset and liability management (ALM) for pension funds. We define two types of ICC: theone periodintegrated chance constraint (OICC) and themultiperiodintegrated chance constraint (MICC). As their names suggest, the OICC covers only one period, whereas several periods are taken into account with the MICC. A multistage stochastic linear programming model is therefore developed for this purpose and a special mention is paid to the modeling of the MICC. Based on a numerical example, we first analyze the effects of the OICC and the MICC on the optimal decisions (asset allocation and contribution rate) of a pension fund. By definition, the MICC is more restrictive and safer compared to the OICC. Second, we quantify this MICC safety increase. The results show that although the optimal decisions from the OICC and the MICC differ, the total costs are very close, showing that the MICC is definitely a better approach since it is more prudent.

New Asset Liability Management Model with Decision Support System for Life Insurance Companies: Interface Design Issues for Database and Mathematical Models

We introduce a new asset liability management (ALM) model based on decision support system (DSS) for a life insurance firm. The DSS is based on multi-stage stochastic linear programming (SLP) with recourse for strategic planning. The model can be used with minimal knowledge of management sciences. The model maximises the expected value of total reserve (policy holders' reserve and shareholders' reserve) at the end of the time period of planning with constraints both on the asset and the liability side of the firm's balance sheet. We discuss issues related to DSS interface design, one to one correspondences between the SLP model and the database and the difficulty in multi stage DSS compared to two stage DSS. We also compare and contrast the similarities and differences with our earlier work on SLP based DSS for process industries.

Optimal operation of a CHP plant participating in the German electricity balancing and day-ahead spot market

During the last years, operators of combined heat and power (CHP) plants are challenged by changing circumstances in the electricity markets. While ensuring a stable heat and power supply for customers, CHP plant operators seek to maximise the profitability of the CHP plant operation. We propose a comprehensive concept and illustrate its potential for increasing the profitability when operating a CHP plant with heat storage by participating in multiple electricity markets. For this purpose, the structure of the decision-making process consisting of bid submission, market clearing and CHP plant operation planning is represented. A multistage stochastic mixed-integer linear programming (MILP) model is developed that simultaneously optimises the operation of the CHP plant with heat storage and bidding in sequential electricity markets. Price uncertainty is captured by means of stochastic processes. The proposed concept is applied to the real case of a municipal energy supply company that participates in the German day-ahead spot and balancing market. Results of the case study illustrate how trading in sequential electricity markets can result in an increased profitability of the CHP plant operation, where additional revenues from trading offset higher generation costs. Furthermore, we exemplify how the optimal operation of the CHP plant and heat storage device are adjusted according to revenue potential in multiple electricity markets.

Regularized Decomposition Methods for Deterministic and Stochastic Convex Optimization and Application to Portfolio Selection with Direct Transaction and Market Impact Costs

We define a regularized variant of the Dual Dynamic Programming algorithm called REDDP (REgularized Dual Dynamic Programming) to solve nonlinear dynamic programming equations. We extend the algorithm to solve nonlinear stochastic dynamic programming equations. The corresponding algorithm, called SDDP-REG, can be seen as an extension of a regularization of the Stochastic Dual Dynamic Programming (SDDP) algorithm recently introduced which was studied for linear problems only and with less general prox-centers. We show the convergence of REDDP and SDDP-REG. We assess the performance of REDDP and SDDP-REG on portfolio models with direct transaction and market impact costs. In particular, we propose a risk-neutral portfolio selection model which can be cast as a multistage stochastic second-order cone program. The formulation is motivated by the impact of market impact costs on large portfolio rebalancing operations. Numerical simulations show that REDDP is much quicker than DDP on all problem instances considered (up to 184 times quicker than DDP) and that SDDP-REG is quicker on the instances of portfolio selection problems with market impact costs tested and much faster on the instance of risk-neutral multistage stochastic linear program implemented (8.2 times faster).

Linear Decision Rules for Hydropower Scheduling Under Uncertainty

We investigate the hydropower scheduling problem, in which a price-taking producer determines a reservoir management strategy that maximises the present value of revenues from selling the produced electricity in a well-functioning market. Uncertainty is present both in market prices and in reservoir inflows. To solve the problem, we apply linear decision rules, which is an approximation method for solving multistage stochastic linear programming problems. Traditional methods for solving these types of problems suffer from computational efforts that grow exponentially with the number of stages and state variables. By restricting the decision variables to be affine functions of the realisations of the uncertain parameters, the original intractable problem is transformed into a problem with short computational time. The aim is to investigate feasibility of the framework. The approach is demonstrated on four Norwegian hydropower plants using recent inflow and price data over a ten year time horizon. We obtain flexible reservoir management strategies, providing feasible solutions where a deterministic approach fails, and otherwise improving expected profits by up to 4.5% compared to a deterministic approach. Solutions times are in the order of minutes.

Assessing policy quality in a multistage stochastic program for long-term hydrothermal scheduling

We consider a multistage stochastic linear program in which we aim to assess the quality of an operational policy computed by means of a stochastic dual dynamic programming algorithm. We perform policy assessment by considering two strategies to compute a confidence interval on the optimality gap: (i) using multiple scenario trees and (ii) using a single scenario tree. The first approach has already been considered in several applications, while the second approach has been discussed previously only in a two-stage framework. The second approach is useful in practical applications in order to more quickly assess the quality of a policy. We present these ideas in the context of a multistage stochastic program for Brazilian long-term hydrothermal scheduling, and use numerical instances to compare the confidence intervals on the optimality gap computed via both strategies. We further consider the relative merits of using naive Monte Carlo sampling, randomized quasi Monte Carlo sampling, and Latin hypercube sampling within our framework for assessing the quality of a policy.

Convergence Analysis of Sampling-Based Decomposition Methods for Risk-Averse Multistage Stochastic Convex Programs

We consider a class of sampling-based decomposition methods to solve risk-averse multistage stochastic convex programs. We prove a formula for the computation of the cuts necessary to build the outer linearizations of the recourse functions. This formula can be used to obtain an efficient implementation of Stochastic Dual Dynamic Programming applied to convex nonlinear problems. We prove the almost sure convergence of these decomposition methods when the relatively complete recourse assumption holds. We also prove the almost sure convergence of these algorithms when applied to risk-averse multistage stochastic linear programs that do not satisfy the relatively complete recourse assumption. The analysis is first done assuming the underlying stochastic process is interstage independent and discrete, with a finite set of possible realizations at each stage. We then indicate two ways of extending the methods and convergence analysis to the case when the process is interstage dependent.

Combining sampling-based and scenario-based nested Benders decomposition methods: application to stochastic dual dynamic programming

Nested Benders decomposition is a widely used and accepted solution methodology for multi-stage stochastic linear programming problems. Motivated by large-scale applications in the context of hydro-thermal scheduling, in 1991, Pereira and Pinto introduced a sampling-based variant of the Benders decomposition method, known as stochastic dual dynamic programming (SDDP). In this paper, we embed the SDDP algorithm into the scenario tree framework, essentially combining the nested Benders decomposition method on trees with the sampling procedure of SDDP. This allows for the incorporation of different types of uncertainties in multi-stage stochastic optimization while still maintaining an efficient solution algorithm. We provide an illustration of the applicability of our method towards a least-cost hydro-thermal scheduling problem by examining an illustrative example combining both fuel cost with inflow uncertainty and by studying the Panama power system incorporating both electricity demand and inflow uncertainties.

Optimizing Strategic Allocation of Vehicles for One-Way Car-sharing Systems Under Demand Uncertainty

Car-sharing offers an environmentally sustainable, socially responsible and economically feasible mobility form in which a fleet of shared-use vehicles in a number of locations can be accessed and used by many people on as-needed basis at an hourly or mileage rate. To ensure its sustainability, car-sharing operators must be able to effectively manage dynamic and uncertain demands, and make the best decisions on strategic vehicle allocation and operational vehicle reallocation both in time and space to improve their profits while keeping costs under control. This paper develops a stochastic optimization method to optimize strategic allocation of vehicles for one-way car-sharing systems under demand uncertainty. A multi-stage stochastic linear programming model is developed and solved for use in the context of car-sharing. A seven-stage experimental network study is conducted. Numerical results and computational insights are discussed.

Database Structure for a Multi Stage Stochastic Optimization Based Decision Support System for Asset Liability Management of a Life Insurance Company

We introduce a stochastic optimization based decisi on support system (DSS) for asset-liability management of a life insurance firm using a multi-s tage, stochastic optimization model. The DSS is based on a multi-stage stochastic linear pro gram (SLP) with recourse for strategic planning. The model can be used with little or no k nowledge of management sciences. The model maximizes the expected value of total reserve (policy holders’ reserve and shareholders’ reserve) at the end of the time period of planning. We discuss the issues related to database design structure, DSS interface design, database up dating procedure, and solution reporting.

An effective heuristic for multistage linear programming with a stochastic right-hand side

The multistage Stochastic Linear Programming (SLP) problem may become numerically intractable for huge instances, in which case one can solve an approximation for example the well known multistage Expected Value (EV) problem. We introduce a new approximation to the SLP problem that we call the multistage Event Linear Programming (ELP) problem. To obtain this approximation the SLP constraints are aggregated by means of the conditional expectation operator. Based on this new problem we derive the ELP heuristic that produces a lower and an upper bound for the SLP problem. We have assessed the validity of the ELP heuristic by solving large scale instances of the network revenue management problem, where the new approach has clearly outperformed the EV approach. One limitation of this paper is that it only considers randomness on the right-hand side, which is assumed to be discrete and stagewise independent.

In This Issue

In This Issue

On Solving Multistage Stochastic Programs with Coherent Risk Measures

We consider a class of multistage stochastic linear programs in which at each stage a coherent risk measure of future costs is to be minimized. A general computational approach based on dynamic programming is derived that can be shown to converge to an optimal policy. By computing an inner approximation to future cost functions, we can evaluate an upper bound on the cost of an optimal policy, and an outer approximation delivers a lower bound. The approach we describe is particularly useful in sampling-based algorithms, and a numerical example is provided to show the efficacy of the methodology when used in conjunction with stochastic dual dynamic programming.

Dynamic Appointment Scheduling of a Stochastic Server with Uncertain Demand

We formulate and solve two new stochastic linear programming formulations of appointment scheduling problems that are motivated by the management of health services. We assume that service durations and the number of customers to be served on a particular day are uncertain. In the first model, customers may fail to show up for their appointments (“no-show”). This model is formulated as a two-stage stochastic linear program. In the second model, customers are scheduled dynamically, one at a time, as they request appointments. This model is formulated as a multistage stochastic linear program with stages defined by customer appointment requests. We analyze the structure of the models and adapt decomposition-based algorithms to solve the problems efficiently. We present numerical results that illustrate the impact of uncertainty on dynamic appointment scheduling, and we identify useful insights that can be applied in practice. We also present a case study based on real data for an outpatient procedure center.

Management of Dynamic Vehicle Allocation for Carsharing Systems

Carsharing offers innovative mobility solutions and has been gaining popularity around the world as an environmentally sustainable, socially responsible, and economically feasible form of mobility. Carsharing allows members to benefit from private vehicle use without the costs and responsibilities of ownership and provides individuals with access to a fleet of shared-use vehicles in a network of locations on a short-term, as-needed basis. This paper seeks to develop a stochastic optimization framework to address the dynamic vehicle allocation problem for carsharing systems, in which the service operator needs to manage and determine the optimal vehicle allocation in both time and space to maximize profits. A multistage stochastic linear programming model with recourse, which can account for system uncertainties such as car-sharing demand variation, is formulated and solved. Numerical results are discussed and computational insights are presented on the basis of a seven-stage experimental network pilot study.

Exploiting the structure of autoregressive processes in chance-constrained multistage stochastic linear programs

We consider an interstage dependent stochastic process whose components follow an autoregressive model with time varying order. At a given time, we give some recursive formulæ linking future values of the process with past values and noises. We then consider multistage stochastic linear programs with uncertain sets depending affinely on such processes. At each stage, dealing with uncertainty using probabilistic constraints, the recursive relations can be used to obtain explicit expressions for the feasible set.

SDDP for multistage stochastic linear programs based on spectral risk measures

We consider risk-averse formulations of multistage stochastic linear programs. For these formulations, based on convex combinations of spectral risk measures, risk-averse dynamic programming equations can be written. As a result, the Stochastic Dual Dynamic Programming (SDDP) algorithm can be used to obtain approximations of the corresponding risk-averse recourse functions. This allows us to define a risk-averse nonanticipative feasible policy for the stochastic linear program. Formulas for the cuts that approximate the recourse functions are given. In particular, we show that some cut coefficients have analytic formulas.