Multistage Stochastic Linear Programming Research Articles

Sampling-Based Approximation Algorithms for Multistage Stochastic Optimization

Stochastic optimization problems provide a means to model uncertainty in the input data where the uncertainty is modeled by a probability distribution over the possible realizations of the actual data. We consider a broad class of these problems in which the realized input is revealed through a series of stages and hence are called multistage stochastic programming problems. Multistage stochastic programming and, in particular, multistage stochastic linear programs with full recourse, is a domain that has received a great deal of attention within the operations research community, mostly from the perspective of computational results in application settings. Our main result is to give the first fully polynomial approximation scheme for a broad class of multistage stochastic linear programming problems with any constant number of stages. The algorithm analyzed, known as the sample average approximation method, is quite simple and is the one most commonly used in practice. The algorithm accesses the input by...

A multi-stage stochastic supply network design problem with financial decisions and risk management

In this paper, a multi-period supply chain network design problem is addressed. Several aspects of practical relevance are considered such as those related with the financial decisions that must be accounted for by a company managing a supply chain. The decisions to be made comprise the location of the facilities, the flow of commodities and the investments to make in alternative activities to those directly related with the supply chain design. Uncertainty is assumed for demand and interest rates, which is described by a set of scenarios. Therefore, for the entire planning horizon, a tree of scenarios is built. A target is set for the return on investment and the risk of falling below it is measured and accounted for. The service level is also measured and included in the objective function. The problem is formulated as a multi-stage stochastic mixed-integer linear programming problem. The goal is to maximize the total financial benefit. An alternative formulation which is based upon the paths in the scenario tree is also proposed. A methodology for measuring the value of the stochastic solution in this problem is discussed. Computational tests using randomly generated data are presented showing that the stochastic approach is worth considering in these types of problems.

The value of rolling-horizon policies for risk-averse hydro-thermal planning

We consider the optimal management of a hydro-thermal power system in the mid and long terms. From the optimization point of view, this amounts to a large-scale multistage stochastic linear program, often solved by combining sampling with decomposition algorithms, like stochastic dual dynamic programming. Such methodologies, however, may entail prohibitive computational time, especially when applied to a risk-averse formulation of the problem. We propose instead a risk-averse rolling-horizon policy that is nonanticipative, feasible, and time consistent. The policy is obtained by solving a sequence of risk-averse problems with deterministic constraints for the current time step and future chance and CVaR constraints. The considered hydro-thermal model takes into account losses resulting from run-of-river plants efficiencies as well as uncertain demand and streamflows. Constraints aim at satisfying demand while keeping reservoir levels above minzones almost surely. We show that if the problem uncertainty is represented by a periodic autoregressive stochastic process with lag one, then the probabilistic constraints can be computed explicitly. As a result, each one of the aforementioned risk-averse problems is a medium-size linear program, easy to solve. For a real-life power system we compare our approach with three alternative policies. Namely, a robust nonrolling-horizon policy and two risk-neutral policies obtained by stochastic dual dynamic programming, implemented in nonrolling- and rolling-horizon modes, respectively. Our numerical assessment confirms the superiority of the risk-averse rolling-horizon policy that yields comparable average indicators, but with reduced volatility and with substantially less computational effort.

Staged Climate Change Adaptation Planning for Water Supply in Amman, Jordan

AbstractThis research develops a multistage stochastic linear programming (LP) model to assist in the process of water system planning and management under demographic and climate change in Amman, Jordan, over the next 75 years. Climate change is projected to have a gradual exacerbating effect on Amman’s water stress over the next century, and water resources management strategies and policies put in place now will likely influence water use patterns for generations to come. A multistage decision model allows the identification of both adaptation strategies that should be implemented now and actions likely to be needed later, depending on future climate and demographic conditions. For Amman, the model recommends that household water reuse be expanded immediately, large-scale wastewater reclamation begin within 25 years, and mega-scale water import projects be postponed for several decades. Although these recommendations for the future will almost certainly change as additional information is acquired, by ...

Life-cycle asset allocation and consumption using stochastic linear programming

We consider optimal consumption and (strategic) asset allocation of an investor with uncertain lifetime. The problem is solved using a multi-stage stochastic linear programming (SLP) model to be able to generalize the closed-form solution obtained by Richard (1975). We account for aspects of the application of the SLP approach which arise in the context of life-cycle asset allocation, but are also relevant for other problems of similar structure. The objective is to maximize the expected utility of consumption over the lifetime and of bequest at the time of death of the investor. Since we maximize utility (rather than other objectives which can be implemented more easily) we provide a new approach to optimize the breakpoints required for the linearization of the utility function. The uncertainty of the problem is described by discrete scenario trees. The model solves for the rebalancing decisions in the first few years of the investor's lifetime, accounting for anticipated cash flows in the near future, and applies Richard's closed-form solution for the long, subsequent steady-state period. In our numerical examples we first show that available closed-form solutions can be accurately replicated with the SLP-based approach. Second, we add elements to the problem specification which are usually beyond the scope of closed-form solutions. We find that the SLP approach is well suited to account for these extensions of the classical Merton setting.

Cash management using multi-stage stochastic programming

We consider a cash management problem where a company with a given financial endowment and given future cash flows minimizes the Conditional Value at Risk of final wealth using a lower bound for the expected terminal wealth. We formulate the optimization problem as a multi-stage stochastic linear program (SLP). The company can choose between a riskless asset (cash), several default- and option-free bonds, and an equity investment, and rebalances the portfolio at every stage. The uncertainty faced by the company is reflected in the development of interest rates and equity returns. Our model has two new features compared to the existing literature, which uses no-arbitrage interest rate models for the scenario generation. First, we explicitly estimate a function for the market price of risk and change the underlying probability measure. Second, we simulate scenarios for equity returns with moment-matching by an extension of the interest rate scenario tree.

Back Testing Short-term Treasury Management Strategies Based on Multi-Stage Stochastic Programming

We show the practical viability of a short-term treasury management model which is formulated as a multi-stage stochastic linear program. A company minimises the Conditional Value at Risk of final wealth, subject to given future cash flows and the uncertain future development of interest rates and equity returns, choosing an asset allocation among cash, several bonds, and an equity investment. The scenario generation procedure includes an estimation of the market price of risk and a change of the underlying probability measure. We provide an out-of-sample back test for the proposed policy and compare the performance to alternative strategies. Our approach shows a better risk-return trade-off for different aggregated risk measures. Further, we perform several numerical studies based on a real market data set to test for the sensitivity to changes in the input parameters, e.g. shifts of the yield curve, changes in the equity spread or the cash flows. The resulting portfolios are well-diversified and the impact on the asset allocation follows economic intuition.

Stochastic optimization models for a single-sink transportation problem

In this paper, we study a single-sink transportation problem in which the production capacity of the suppliers and the demand of the single customer are stochastic. Shipments are performed by capacitated vehicles, which have to be booked in advance, before the realization of the production capacity and the demand. Once the production capacity and the demand are revealed, there is an option to cancel some of the booked vehicles against a cancellation fee; if the quantity shipped from the suppliers using the booked vehicles is not enough to satisfy the demand, the residual quantity is purchased from an external company. The problem is to determine the number of vehicles to book in order to minimize the total cost. We formulate a two-stage and a multistage stochastic mixed integer linear programming models to solve this problem and test them on a real case provided by Italcementi, the primary Italian cement producer and the fifth largest cement producer in the world. We test the influence of different scenario-tree structures on the solutions of the problem, as well as sensitivity of the results with respect to the cancellation fee.

On a Multi-stage Stochastic Programming Model for Inventory Planning

This paper considers a stochastic dynamic inventory problem involving a single item, linear cost structures, and finite distributions (but not necessarily independent) for the stochastic cost and demand parameters. We develop primal and dual algorithms for a multi-stage stochastic linear programming formulation for the problem. The complexity of the proposed algorithms is shown to be within O(N2), where N is the number of nodes in the scenario tree used to model the stochastic parameters.

On the convergence of stochastic dual dynamic programming and related methods

We discuss the almost-sure convergence of a broad class of sampling algorithms for multistage stochastic linear programs. We provide a convergence proof based on the finiteness of the set of distinct cut coefficients. This differs from existing published proofs in that it does not require a restrictive assumption.

Multistage stochastic linear programming model for daily coordinated multi-reservoir operation

Operation planning for a coordinated multi-reservoir is a complex and challenging task due to the inherent uncertainty in inflow. In this study, we suggest the use of a new, multi-stage and scenario-based stochastic linear program with a recourse model incorporating the meteorological weather prediction information for daily, coordinated, multi-reservoir operation planning. Stages are defined as prediction lead-time spans of the weather prediction system. The multi-stage scenarios of the stochastic model are formed considering the reliability of rainfall prediction for each lead-time span. Future inflow scenarios are generated by a rainfall–runoff model based on the rainfall forecast. For short-term stage (2 days) scenarios, the regional data assimilation and prediction system (RDAPS) information is employed, and for mid-term stage (more than 2 days) scenarios, precipitation from the global data assimilation and prediction system (GDAPS) is used as an input for the rainfall–runoff model. After the 10th day (third stage), the daily historical rainfall data are used following the ensemble streamflow prediction (ESP) procedure. The model is applied to simulate the daily reservoir operation of the Nakdong River basin in Korea in a real-time operational environment. The expected benefit of the stochastic model is markedly superior to that of the deterministic model with average rainfall information. Our study results confirm the effectiveness of the stochastic model in real-time operation with meteorological forecasts and the presence of inflow uncertainty.

Capturing dynamics in integrated supply chain management

A major challenge for an enterprise to stay competitive in today’s highly competitive market environment is to be able of capturing and handling the dynamics of its entire supply chain (SC). This work incorporates uncertainty and process dynamics into enterprise wide models which also contemplate cross-functional decisions. The SC integrated solution developed includes a design–planning and a financial formulations. A model predictive control (MPC) methodology is proposed that comprises a stochastic optimization approach. A scenario based multi-stage stochastic mixed integer linear programming (MILP) model is employed to address the problem. The novel control framework introduced constitutes a step-forward in closing the loop for the dynamic supply chain management (SCM) and a supporting platform for the supervisory module handling the incidences that may arise in the SC. The potential of the presented approach is highlighted through a case study, where the results of the deterministic MPC and the joint control framework are compared. It is emphasized the significance of merging uncertainty treatment and control strategies to improve the SC performance.

Life-Cycle Asset Allocation and Optimal Consumption Using Stochastic Linear Programming

We consider optimal consumption and (strategic) asset allocation of an investor with uncertain lifetime. The problem is solved using a multi-stage stochastic linear programming (SLP) model to be able to generalize the closed-form solution obtained by Richard (1975). We account for aspects of the application of the SLP approach which arise in the context of life-cycle asset allocation, but are also relevant for other problems of similar structure. The objective is to maximize the expected utility of consumption over the lifetime and of bequest at the time of death of the investor. Since we maximize utility (rather than other objectives which can be implemented more easily) we provide a new approach to optimize the breakpoints required for the linearization of the utility function. The uncertainty of the problem is described by discrete scenario trees. The model solves for the rebalancing decisions in the first few years of the investor's lifetime, accounting for anticipated cash flows in the near future, and applies Richard's closed-form solution for the long, subsequent steady-state period. In our numerical examples we first show that available closed-form solutions can be accurately replicated with the SLP-based approach. Second, we add elements to the problem specification which are usually beyond the scope of closed-form solutions. We find that the SLP approach is well suited to account for these extensions of the classical Merton setting.

Scenario Formulation of Stochastic Linear Programs and the Homogeneous Self-Dual Interior-Point Method

We consider a homogeneous self-dual interior-point algorithm for solving multistage stochastic linear programs. The algorithm is particularly suitable for the so-called “scenario formulation” of the problem, whose constraint system consists of a large block-diagonal matrix together with a set of sparse nonanticipativity constraints. Due to this structure, the major computational work required by the homogeneous self-dual interior-point method can be split into three steps, each of which is highly decomposable. Numerical results on some randomly generated problems and a multistage production-planning problem are reported.

An interval-parameter multi-stage stochastic programming model for water resources management under uncertainty

In this study, an interval-parameter multi-stage stochastic linear programming (IMSLP) method has been developed for water resources decision making under uncertainty. The IMSLP is a hybrid methodology of inexact optimization and multi-stage stochastic programming. It has three major advantages in comparison to the other optimization techniques. Firstly, it extends upon the existing multi-stage stochastic programming method by allowing uncertainties expressed as probability density functions and discrete intervals to be effectively incorporated within the optimization framework. Secondly, penalties are exercised with recourse against any infeasibility, which permits in-depth analyses of various policy scenarios that are associated with different levels of economic consequences when the promised water-allocation targets are violated. Thirdly, it cannot only handle uncertainties through constructing a set of scenarios that is representative for the universe of possible outcomes, but also reflect dynamic features of the system conditions through transactions at discrete points in time over the planning horizon. The developed IMSLP method is applied to a hypothetical case study of water resources management. The results are helpful for water resources managers in not only making decisions of water allocation but also gaining insight into the tradeoffs between environmental and economic objectives.

The Scenario Generation Algorithm for Multistage Stochastic Linear Programming

A multistage stochastic linear program (MSLP) is a model of sequential stochastic optimization where the objective and constraints are linear. When any of the random variables used in the MSLP are continuous, the problem is infinite dimensional. To numerically tackle such a problem, we usually replace it with a finite-dimensional approximation. Even when all the random variables have finite support, the problem is often computationally intractable and must be approximated by a problem of smaller dimension. One of the primary challenges in the field of stochastic programming deals with discovering effective ways to evaluate the importance of scenarios and to use that information to trim the scenario tree in such a way that the solution to the smaller optimization problem is not much different from the problem stated with the original tree. The scenario generation (SG) algorithm proposed in this paper is a finite-element method that addresses this problem for the class of MSLP with random right-hand sides.

Modeling a Hydrothermal System with Hydro- and Snow Reservoirs

This paper presents a stochastic model for analyzing the effects of variable precipitation on the Nordic power system. The model is formulated as a multistage stochastic linear program and solved by the ReSa sampling based Benders decomposition method. The stochastic parameter representing the inflow to the hydro reservoirs has been split into two in the model to describe separately the inflow from rain and snowmelt. This captures the fact that information about the snow reservoirs reduces the decision maker’s level of uncertainty about future inflow levels. Computational analysis of the Nordic power market undertaken using the model confirms the importance of modeling multiple inflow sources, with the addition of the snow reservoirs reducing the span between the highest and lowest expected future spot price.

On the Convergence of Sampling-Based Decomposition Algorithms for Multistage Stochastic Programs

The paper presents a convergence proof for a broad class of sampling algorithms for multistage stochastic linear programs in which the uncertain parameters occur only in the constraint right-hand sides. This class includes SDDP, AND, ReSa, and CUPPS. We show that, under some independence assumptions on the sampling procedure, the algorithms converge with probability 1.

Approximation to Multistage Stochastic Optimization in Multiperiod Batch Plant Scheduling under Demand Uncertainty

We consider the problem of scheduling under demand uncertainty a multiproduct batch plant represented through a state−task network. Given a scheduling horizon consisting of several time periods in which product demands are placed, the objective is to select a schedule that maximizes the expected profit. We present a multistage stochastic mixed integer linear programming (MILP) model, wherein certain decisions are made irrespective of the realization of the uncertain parameters and some decisions are made upon realization of the uncertainty. To overcome the computational expense associated with the solution of the large-scale stochastic multistage MILP for large problems, we examine an approximation strategy based on the solution of a series of a two-stage models within a shrinking-horizon approach. Computational results indicate that the proposed approximation strategy provides an expected profit within a few percent of the multistage stochastic MILP result in a fraction of the computation time and provid...

A New Decomposition Technique in Solving Multistage Stochastic Linear Programs by Infeasible Interior Point Methods

Multistage stochastic linear programming (MSLP) is a powerful tool for making decisions under uncertainty. A deterministic equivalent problem of MSLP is a large-scale linear program with nonanticip...