Affinely Adjustable Robust Counterpart Research Articles

Adjustable Robust Optimization for Multi-Period Water Allocation in Droughts Under Uncertainty

Optimal water resource allocation can go some way to overcoming water deficiencies; however, its achievement is complex due to conflicting hierarchies and uncertainties, such as water availability (WA) and water demand (WD). This study develops a robust water withdrawal scheme for drought regions that can balance the trade-offs between the sub-areas and water use participants, ensure sustainable regional system development, and guarantee robust solutions for future uncertainties. A bi-level affinely adjustable robust counterpart (AARC) programming framework was developed in which the regional authority as the leader allocates water to the sub-areas to maximize the intra- and intergenerational equity, and the sub-areas as the followers allocate water to their respective water departments to maximize their economic benefits and minimize water shortages. A case study from Neijiang, China, is given to illustrate the applicability and feasibility of this framework. The novelty of this study is to propose a sustainable bi-level AARC regional water allocation framework which integrates intra- and inter-generational equity of regional water use and priority rules reflected by goal preference programming between water departments under uncertainties of WA and WD simultaneously in water deficient regions.

Affinely Adjustable Robust Volt/Var Control for Distribution Systems With High PV Penetration

We propose two novel two-stage Volt/Var control schemes based on the affinely adjustable robust counterpart (AARC) methodology, to mitigate the over-voltage issues caused by integration of photovoltaic panels into distribution systems. To cope with different grid code requirements, our first approach formulates the unused capacity of residential inverters to provide reactive power support based on the real power deviation, while the second approach formulates them based on voltage magnitude deviation. In the first stage of both schemes, we make central measurements throughout the network to determine a linear function, mapping the operating point deviations to the reactive power of inverters. In the second stage, the local controllers use the provided linear functions to determine the required reactive power to keep the voltages within the safe limits. Unlike similar approaches, voltage limit constraints are directly incorporated into the AARC problem, preventing the second stage controllers from unnecessarily use of reactive powers. We compare the performance of our schemes using a Monte-Carlo simulation with four other existing techniques on a real-world 27-bus and the IEEE 906-bus LV feeders. Our simulations show that our approaches decrease the real power loss, reactive power usage, and line congestion compared to the other Volt/Var control schemes.

Lagrangian Duality for Robust Problems with Decomposable Functions: The Case of a Robust Inventory Problem

We consider a class of min-max robust problems in which the functions that need to be “robustified” can be decomposed as the sum of arbitrary functions. This class of problems includes many practical problems, such as the lot-sizing problem under demand uncertainty. By considering a Lagrangian relaxation of the uncertainty set, we derive a tractable approximation, called the dual Lagrangian approach, that we relate with both the classical dualization approximation approach and an exact approach. Moreover, we show that the dual Lagrangian approach coincides with the affine decision rule approximation approach. The dual Lagrangian approach is applied to a lot-sizing problem, in which demands are assumed to be uncertain and to belong to the uncertainty set with a budget constraint for each time period. Using the insights provided by the interpretation of the Lagrangian multipliers as penalties in the proposed approach, two heuristic strategies, a new guided iterated local search heuristic, and a subgradient optimization method are designed to solve more complex lot-sizing problems in which additional practical aspects, such as setup costs, are considered. Computational results show the efficiency of the proposed heuristics that provide a good compromise between the quality of the robust solutions and the running time required in their computation. Summary of Contribution: The paper includes both theoretical and algorithmic contributions for a class of min-max robust optimization problems where the objective function includes the maximum of a sum of affine functions. From the theoretical point of view, a tractable Lagrangian dual model resulting from a relaxation of the well-known adversarial problem is proposed, providing a new perspective of well-known models, such as the affinely adjustable robust counterpart (AARC) and the dualization technique introduced by Bertsimas and Sim. These results are particularized to lot-sizing problems. From the algorithm point of view, efficient heuristic schemes—which exploit the information based on the interpretation of the Lagrangian multipliers to solve large size robust problems—are proposed, and their performance is evaluated through extensive computational results based on the lot-sizing problem. In particular, a guided iterated local search and a subgradient optimization method are proposed and compared against the dualization approach proposed by Bertsimas and Sim and with several heuristics based on the AARC approach, which include an iterated local search heuristic and a Benders decomposition approach. Computational results show the efficiency of the proposed heuristics, which provide a good compromise between the quality of the robust solutions and the running time.

Robust planning of dynamic wireless charging infrastructure for battery electric buses

Battery electric buses with zero tailpipe emissions have great potential in improving environmental sustainability and livability of urban areas. However, the problems of high cost and limited range associated with on-board batteries have substantially limited the popularity of battery electric buses. The technology of dynamic wireless power transfer (DWPT), which provides bus operators with the ability to charge buses while in motion, may be able to effectively alleviate the drawbacks of electric buses. In this paper, we address the problem of simultaneously selecting the optimal location of the DWPT facilities and designing the optimal battery sizes of electric buses for a DWPT electric bus system. The problem is first constructed as a deterministic model in which the uncertainty of energy consumption and travel time of electric buses is ignored. The methodology of robust optimization (RO) is then adopted to address the uncertainty of energy consumption and travel time. The affinely adjustable robust counterpart (AARC) of the deterministic model is developed, and its equivalent tractable mathematical programming is derived. Both the deterministic model and the robust model are demonstrated with a real-world bus system. The results demonstrate that the proposed deterministic model can effectively determine the allocation of DWPT facilities and the battery sizes of electric buses for a DWPT electric bus system; and the robust model can further provide optimal designs that are robust against the uncertainty of energy consumption and travel time for electric buses.

Research on robust multi-period inventory inaccuracy based on RFID technology

Inventory inaccuracy has great influence on the supply chain performance and it has attracted much attention of large numbers of researchers. This paper mainly studies the robust multi-period inventory inaccuracy problem by attaching radio frequency identification (RFID) technology. In particular, the typical retailer-supplier flexible commitment (RSFC) problem is considered to cope with the uncertain environment. After modeling, robust optimization (RO) and the affinely adjustable robust counterpart (AARC) methodology are applied to solve the model. Finally, this paper uses a numerical example for the analysis of how RFID technology can be exploited in supply chain, and the effect of demand uncertainty on the systems. The results highlight the importance of inventory availability related rates and variable uncertainty in determining the profitability of RFID adoption, which can provide managerial guidelines to supply chain firms.

Robust Multi-Period OPF With Storage and Renewables

Renewable energy sources and energy storage systems present specific challenges to the traditional optimal power flow (OPF) paradigm. First, storage devices require the OPF to model charge/discharge dynamics and the supply of generated power at a later time. Second, renewable energy sources necessitate that the OPF solution accounts for the control of conventional power generators in response to errors of renewable power forecast, which are significantly larger than the traditional load forecast errors. This paper presents a sparse formulation and solution for the affinely adjustable robust counterpart (AARC) of the multi-period OPF problem. The AARC aims at operating a storage portfolio via receding horizon control; it computes the optimal base-point conventional generation and storage schedule for the forecasted load and renewable generation, together with the constrained participation factors that dictate how conventional generation and storage will adjust to maintain feasible operation whenever the renewables deviate from their forecast. The approach is demonstrated on standard IEEE networks dispatched over a 24-h horizon with interval forecasted wind power, and the feasibility of operation under interval uncertainty is validated via Monte Carlo analysis. The computational performance of the proposed approach is compared with a conventional implementation of the AARC that employs successive constraint enforcement.

Tractable approximations to a robust capacity assignment model in telecommunications under demand uncertainty

In a previous work, a min–max–min model has been proposed for robust capacity assignment in telecommunications where the demand is uncertain but belongs to a polyhedral set. As the problem appears hardly solvable, lower bounds and upper bounds computations have been proposed, but the latter were poor. It was then suggested that better upper bounds can be obtained using the so-called Affinely Adjustable Robust Counterpart (AARC) concept proposed by Ben-Tal et al. where the adjustable variables are restricted to depend affinely on the uncertain data. In this paper, we revisit this model from a dual perspective: Given an amount of traffic μ¯, we seek for an optimal link capacity assignment that, given an uncertainty set containing possible demand realizations, limits the loss of traffic to μ¯ in any realization of the demand. Our motivation is mainly computational, borrowing ingredients from earlier works and the AARC approach. We propose four tractable approximations to this problem and conduct some numerical experiments to compare them.

AFFINELY ADJUSTABLE ROBUST OPTIMIZATION MODEL FOR MULTI-PERIOD PRODUCTION AND INVENTORY SYSTEM UNDER RISK PREFERENCE

A problem of optimizing multi-period centralized production and inventory system with waste disposal subjected to uncertain demands is investigated in this paper . Assuming limited information of distributions, that is, only the mean, support and some measures of deviations are available for the demand, a joint ellipsoid uncertainty set is construct ed to control the degree of conservatism of the production policies associated with the integrated manager’s risk preference. Using Affinely Adjustable Robust Counterpart method , we develop an uncertain optimization model in pursuit of maximizing the overall revenue t hrough adaptively control ling multi-period production policies, and equivalently convert it to one deterministic robust counterpart which is in fact a tractable second order cone problem .

Robust optimization for emergency logistics planning: Risk mitigation in humanitarian relief supply chains

This paper proposes a methodology to generate a robust logistics plan that can mitigate demand uncertainty in humanitarian relief supply chains. More specifically, we apply robust optimization (RO) for dynamically assigning emergency response and evacuation traffic flow problems with time dependent demand uncertainty. This paper studies a Cell Transmission Model (CTM) based system optimum dynamic traffic assignment model. We adopt a min–max criterion and apply an extension of the RO method adjusted to dynamic optimization problems, an affinely adjustable robust counterpart (AARC) approach. Simulation experiments show that the AARC solution provides excellent results when compared to deterministic solution and sampling based stochastic programming solution. General insights of RO and transportation that may have wider applicability in humanitarian relief supply chains are provided.

Robust multi-echelon multi-period inventory control

We consider the problem of minimizing the overall cost of a supply chain, over a possible long horizon, under demand uncertainly which is known only crudely. Under such circumstances, the method of choice is Robust Optimization, in particular the Affinely Adjustable Robust Counterpart (AARC) method which leads to tractable deterministic optimization problems. The latter is due to a recent re-parametrization technique for discrete time linear control systems. In this paper we model, analyze and test an extension of the AARC method known as the Globalized Robust Counterpart (GRC) in order to control inventories in serial supply chains. A simulation study demonstrates the merit of the methods employed here, in particular, it shows that a good control law that minimizes cost achieves simultaneously good control of the bullwhip effect.

Robust production management

The problem of production management can often be cast in the form of a linear program with uncertain parameters and risk constraints. Typically, such problems are treated in the framework of multi-stage Stochastic Programming. Recently, a Robust Counterpart (RC) approach has been proposed, in which the decisions are optimized for the worst realizations of problem parameters. However, an application of the RC technique often results in very conservative approximations of uncertain problems. To tackle this drawback, an Adjustable Robust Counterpart (ARC) approach has been proposed by Ben-Tal et al. In ARC, some decision variables are allowed to depend on past values of uncertain parameters. A restricted version of ARC, introduced by Ben-Tal et al. and which can be efficiently solved, is referred to as Affinely Adjustable Robust Counterpart (AARC).

Retailer-Supplier Flexible Commitments Contracts: A Robust Optimization Approach

We propose the use of robust optimization (RO) as a powerful methodology for multiperiod stochastic operations management problems. In particular, we study a two-echelon multiperiod supply chain problem, known as the retailer-supplier flexible commitment (RSFC) problem with uncertain demand that is only known to reside in some uncertainty set. We adopt a min-max criterion, whereby the cost function is minimized against the worst case demand occurrence. To solve the min-max RSFC problem we employ a recent extension of the RO method adapted to dynamic decision problems and known as the affinely adjustable robust counterpart (AARC) methodology. The AARC solution is tested by a large simulation study and found to provide excellent results.

Adjustable robust solutions of uncertain linear programs

We consider linear programs with uncertain parameters, lying in some prescribed uncertainty set, where part of the variables must be determined before the realization of the uncertain parameters (‘‘non-adjustable variables’’), while the other part are variables that can be chosen after the realization (‘‘adjustable variables’’). We extend the Robust Optimization methodology ([1, 3-6, 9, 13, 14]) to this situation by introducing the Adjustable Robust Counterpart (ARC) associated with an LP of the above structure. Often the ARC is significantly less conservative than the usual Robust Counterpart (RC), however, in most cases the ARC is computationally intractable (NP-hard). This difficulty is addressed by restricting the adjustable variables to be affine functions of the uncertain data. The ensuing Affinely Adjustable Robust Counterpart (AARC) problem is then shown to be, in certain important cases, equivalent to a tractable optimization problem (typically an LP or a Semidefinite problem), and in other cases, having a tight approximation which is tractable. The AARC approach is illustrated by applying it to a multi-stage inventory management problem.