Recourse Problem Research Articles

An Interactive Fuzzy Satisficing Method for Multiobjective Stochastic Integer Programming with Simple Recourse

This paper considers multiobjective integer programming problems involving random variables in constraints. Using the concept of simple recourse, the formulated multiobjective stochastic simple recourse problems are transformed into deterministic ones. For solving transformed deterministic problems efficiently, we also introduce genetic algorithms with double strings for nonlinear integer programming problems. Taking into account vagueness of judgments of the decision maker, an interactive fuzzy satisficing method is presented. In the proposed interactive method, after determineing the fuzzy goals of the decision maker, a satisficing solution for the decision maker is derived efficiently by updating the reference membership levels of the decision maker. An illustrative numerical example is provided to demonstrate the feasibility and efficiency of the proposed method.

Robust location transportation problems under uncertain demands

In robust optimization, the multi-stage context (or dynamic decision-making) assumes that the information is revealed in stages. So, part of the decisions must be taken before knowing the real values of the uncertain parameters, and another part, called recourse decisions, is taken when the information is known. In this paper, we are interested in a robust version of the location transportation problem with an uncertain demand using a 2-stage formulation. The obtained robust formulation is a convex (not linear) program, and we apply a cutting plane algorithm to exactly solve the problem. At each iteration, we have to solve an NP-hard recourse problem in an exact way, which is time-consuming. Here, we go further in the analysis of the recourse problem of the location transportation problem. In particular, we propose a mixed integer program formulation to solve the quadratic recourse problem and we define a tight bound for this reformulation. We present an extensive computation analysis of the 2-stage robust location transportation problem. The proposed tight bound allows us to solve large size instances.

Optimal design and sensitivity of large scale plane frames for the case of uncertainty

AbstractThis paper deals with the optimal design and the robustness of large scale plane frames in dependence of their height. Using the first collapse theorem, the necessary and sufficient survival conditions of an elasto‐plastic structure consist of the yield condition and the equilibrium condition. The basis for our consideration is provided by a plane n‐storey frame which will be increased successively, and which is affected by applied random forces and moments. Taking into account these random applied loads, we get a stochastic structural optimization problem which cannot be solved using the traditional methods. Instead of that, an appropriate (deterministic) substitute problem is formulated. First, the recourse problem will be formulated in general and in the standard form of stochastic linear programming (SLP), and after the formulation of the stochastic optimization problem, the Recourse Problem based on Discretization (RPD) is introduced as a representative of substitute problems. The resulting (large scale) linear program (LP) can be solved efficiently by means of usual LP‐solvers. (© 2010 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Recourse problem of the 2-stage robust location transportation problem

Abstract In this paper, we are interested in the recourse problem of the 2-stage robust location transportation problem. We propose a solution process using a mixed-integer formulation with an appropriate tight bound.

Efficient, optimal stochastic-action selection when limited by an action budget

The problem that we consider here is a basic operations research problem, but it also a special case of the Stochastic Shortest Path with Recourse Problem and the Canadian Travellers Problem in the probabilistic path planning literature, and it is also a special case of maximizing a submodular set function subject to a matroid constraint. Specifically, suppose an agent has a task and suppose that there is a set of actions, any of which the agent might perform, with respective probabilities of the actions successfully accomplishing the task and respective rewards for the agent if the actions are successful; the agent is to select a sequence of some of these actions that will be performed sequentially, until the task is accomplished or the selected actions are exhausted, but there is a budget on the number of actions that can be performed. We provide an efficient algorithm that chooses a sequence of actions that, under the budget, maximize the agent’s expected reward. An example illustrates how, when conditioning on partial selection of actions, there can be changes to the order of the remaining actions’ adjusted utilities. However, we prove and exploit a nesting result involving solutions.

Optimal Design of Plane Frames under Uncertainty: A Comparison of Two Approaches

AbstractUsing the first collapse theorem, the necessary and sufficient survival conditions of an elasto‐plastic structure consist of the yield condition and the equilibrium condition. In practical applications several random model parameters have to be taken into account. This leads to a stochastic optimization problem which cannot be solved using the traditional methods. Instead of that, appropriate (deterministic) substitute problems must be formulated. Here, the design of plane frames is considered, where the applied load is supposed to be stochastic. In the first approach the recourse problem will be formulated in the standard form of stochastic linear programming (SLP). In order to apply efficient numerical solution procedures (LP‐solvers), approximate recourse problems based on discretization (RPD) and the expected value problem (EVP) are introduced. In the second approach – based on the yield condition – a quadratic cost function will be introduced. After the formulation of the stochastic optimization problem, the expected cost based optimization problem (ECBOP) and the minimum expected cost problem (MECP) are formulated as representatives of appropriate substitute problems. Subsequently, comparative numerical results using these methods are presented. (© 2009 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Optimal Allocation of Surgery Blocks to Operating Rooms Under Uncertainty

The allocation of surgeries to operating rooms (ORs) is a challenging combinatorial optimization problem. There is also significant uncertainty in the duration of surgical procedures, which further complicates assignment decisions. In this paper, we present stochastic optimization models for the assignment of surgeries to ORs on a given day of surgery. The objective includes a fixed cost of opening ORs and a variable cost of overtime relative to a fixed length-of-day. We describe two types of models. The first is a two-stage stochastic linear program with binary decisions in the first stage and simple recourse in the second stage. The second is its robust counterpart, in which the objective is to minimize the maximum cost associated with an uncertainty set for surgery durations. We describe the mathematical models, bounds on the optimal solution, and solution methodologies, including an easy-to-implement heuristic. Numerical experiments based on real data from a large health-care provider are used to contrast the results for the two models and illustrate the potential for impact in practice. Based on our numerical experimentation, we find that a fast and easy-to-implement heuristic works fairly well, on average, across many instances. We also find that the robust method performs approximately as well as the heuristic, is much faster than solving the stochastic recourse model, and has the benefit of limiting the worst-case outcome of the recourse problem.

Regular Analog/RF Integrated Circuits Design Using Optimization With Recourse Including Ellipsoidal Uncertainty

Long design cycles due to the inability to predict silicon realities are a well-known problem that plagues analog/RF integrated circuit product development. As this problem worsens for nanoscale IC technologies, the high cost of design and multiple manufacturing spins causes fewer products to have the volume required to support full-custom implementation. Design reuse and analog synthesis make analog/RF design more affordable; however, the increasing process variability and lack of modeling accuracy remain extremely challenging for nanoscale analog/RF design. We propose a regular analog/RF IC using metal-mask configurability design methodology Optimization with Recourse of Analog Circuits including Layout Extraction (ORACLE), which is a combination of reuse and shared-use by formulating the synthesis problem as an optimization with recourse problem. Using a two-stage geometric programming with recourse approach, ORACLE solves for both the globally optimal shared and application-specific variables. Furthermore, robust optimization is proposed to treat the design with variability problem, further enhancing the ORACLE methodology by providing yield bound for each configuration of regular designs. The statistical variations of the process parameters are captured by a confidence ellipsoid. We demonstrate ORACLE for regular Low Noise Amplifier designs using metal-mask configurability, where a range of applications share common underlying structure and application-specific customization is performed using the metal-mask layers. Two RF oscillator design examples are shown to achieve robust designs with guaranteed yield bound.

Finding and identifying optimal inventory levels for systems with common components

In this article, we consider the problem of finding the optimal inventory level for components in an assembly system where multiple products share common components in the presence of random demand. Previously, solution procedures that identify the optimal inventory levels for components in a component commonality problem have been considered for two product or one common component systems. We will here extend this to a three products system considering any number of common components. The inventory problem considered is modeled as a two stage stochastic recourse problem where the first stage is to set the inventory levels to maximize expected profit while the second stage is to allocate components to products after observing demand. Our main contribution, and the main focus of this paper, is the outline of a procedure that finds the gradient for the stochastic problem, such that an optimal solution can be identified and a gradient based search method can be used to find the optimal solution.

Extending Algebraic Modelling Languages for Stochastic Programming

Algebraic modelling languages have gained wide acceptance and use by researchers and practitioners in mathematical programming. At a basic level, these languages can define stochastic programming (SP) models by constructing their deterministic equivalents. Unfortunately, this leads to very large model data instances. We propose instead a direct approach in which the random values of the model coefficients and the stage structure of the decision variables and constraints are “overlaid” on the underlying deterministic (core) model of the SP problems. This leads to a natural definition of the SP model, while the resulting generated instance is also a compact representation of the otherwise large problem data. As an example of the workability of our approach, we describe the design of a stochastic extension for the AMPL language that enables the formulation of two-stage and multistage scenario-based recourse problems. This extended language, which we call SAMPL, is in turn embedded in a stochastic programming integrated environment that facilitates modelling and investigation of SP problems.

Topology optimization of plane frames in case of random external loadings

AbstractUsing the first collapse–theorem, the necessary and sufficient survival conditions of an elasto–plastic structure consist of the yield condition and the equilibrium condition. In practical applications several random model parameters have to be taken into account. This leads to a stochastic optimization problem which cannot be solved using the traditional methods. Instead of that, appropriate (deterministic) substitute problems must be formulated. Here, the topology optimization of frames is considered, where the external load is supposed to be stochastic. The recourse problem will be formulated in general and in the standard form of stochastic linear programming (SLP). After the formulation of the stochastic optimization problem, the recourse problem with discretization and the expected value problem are introduced as representatives of substitute problems. Subsequently, numerical results using these methods are presented. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Optimal design of frames under stochastic uncertainties

AbstractIn the following the design of frames is treated where the load is supposed to be stochastic. The recourse problem will be formulated in the standard form of stochastic linear programming (SLP). After this formulation the recourse problem with discretization and the expected value problem are introduced as representatives of substitute problems. Subsequently, optimization results using these methods for the robust optimal design of a two‐storey frame are presented. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

The Approximation Method for Two-Stage Fuzzy Random Programming With Recourse

In this paper, a new class of fuzzy random optimization problem called two-stage fuzzy random programming or fuzzy random programming with recourse (FRPR) problem is first presented; then its deterministic equivalent programming problem is characterized. Because the FRPR problems include fuzzy random variable parameters with an infinite support, they are inherently infinite-dimensional optimization problems that can rarely be solved directly. Therefore, an approximation approach to the fuzzy random variables with infinite supports by finitely supported ones is proposed, which results in finite-dimensional FRPR problems. After that, this paper is devoted to establishing the conditions under which the objective value (optimal objective value, and minimizers) of such finite-dimensional FRPR problem can be shown to converge to the objective value (respectively, optimal objective value and minimizers) of the original infinite-dimensional FRPR problem.

A complementarity model for solving stochastic natural gas market equilibria

This paper presents a stochastic equilibrium model for deregulated natural gas markets. Each market participant (pipeline operators, producers, etc.) solves a stochastic optimization problem whose optimality conditions, when combined with market-clearing conditions give rise to a certain mixed complementarity problem (MiCP). The stochastic aspects are depicted by a recourse problem for each player in which the first-stage decisions relate to long-term contracts and the second-stage decisions relate to spot market activities for three seasons. Besides showing that such a market model is an instance of a MiCP, we provide theoretical results concerning long-term and spot market prices and solve the resulting MiCP for a small yet representative market. We also note an interesting observation for the value of the stochastic solution for non-optimization problems.

Solving two-stage stochastic programming problems with level decomposition

We propose a new variant of the two-stage recourse model. It can be used e.g., in managing resources in whose supply random interruptions may occur. Oil and natural gas are examples for such resources. Constraints in the resulting stochastic programming problems can be regarded as generalizations of integrated chance constraints. For the solution of such problems, we propose a new decomposition method that integrates a bundle-type convex programming method with the classic distribution approximation schemes. Feasibility and optimality issues are taken into consideration simultaneously, since we use a convex programming method suited for constrained optimization. This approach can also be applied to traditional two-stage problems whose recourse functions can be extended to the whole space in a computationally efficient way. Network recourse problems are an example for such problems. We report encouraging test results with the new method.

Simple integer recourse models: convexity and convex approximations

We consider the objective function of a simple recourse problem with fixed technology matrix and integer second-stage variables. Separability due to the simple recourse structure allows to study a one-dimensional version instead. Based on an explicit formula for the objective function, we derive a complete description of the class of probability density functions such that the objective function is convex. This result is also stated in terms of random variables. Next, we present a class of convex approximations of the objective function, which are obtained by perturbing the distributions of the right-hand side parameters. We derive a uniform bound on the absolute error of the approximation. Finally, we give a representation of convex simple integer recourse problems as continuous simple recourse problems, so that they can be solved by existing special purpose algorithms.

Some insights into the solution algorithms for SLP problems

We consider classes of stochastic linear programming problems which can be efficiently solved by deterministic algorithms. For two–stage recourse problems we identify two such classes. The first one consists of problems where the number of stochastically independent random variables is relatively low; the second class is the class of simple recourse problems. The proposed deterministic algorithm is successive discrete approximation. We also illustrate the impact of required accuracy on the efficiency of this algorithm. For jointly chance constrained problems with a random right–hand–side and multivariate normal distribution we demonstrate the increase in efficiency when lower accuracy is required, for a central cutting plane method. We support our argumentation and findings with computational results.

Computational aspects of minimizing conditional value-at-risk

We consider optimization problems for minimizing conditional value-at-risk (CVaR) from a computational point of view, with an emphasis on financial applications. As a general solution approach, we suggest to reformulate these CVaR optimization problems as two-stage recourse problems of stochastic programming. Specializing the L-shaped method leads to a new algorithm for minimizing conditional value-at-risk. We implemented the algorithm as the solver CVaRMin. For illustrating the performance of this algorithm, we present some comparative computational results with two kinds of test problems. Firstly, we consider portfolio optimization problems with 5 random variables. Such problems involving conditional value at risk play an important role in financial risk management. Therefore, besides testing the performance of the proposed algorithm, we also present computational results of interest in finance. Secondly, with the explicit aim of testing algorithm performance, we also present comparative computational results with randomly generated test problems involving 50 random variables. In all our tests, the experimental solver, based on the new approach, outperformed by at least one order of magnitude all general-purpose solvers, with an accuracy of solution being in the same range as that with the LP solvers.

Scenario-Based Management of Air Traffic Flow: Developing and Using Capacity Scenario Trees

Recent studies of the single-airport ground-holding problem use static or dynamic optimization to manage uncertainty about future airport capacities. Scenario trees of airport capacity profiles provide the basis for formulating multistage recourse problems. In this paper, methodologies are presented for generating scenario trees from empirical data, and the performance of scenario-based models is examined with the scenario trees. Most U.S. airports have capacity profiles that can be classified into some nominal scenarios, and for many airports, these scenarios can be naturally combined into scenario trees. The delay costs yielded from using dynamic optimization are consistently and considerably lower than from static optimization. The results illustrate the benefit of the wait-and-see strategy in a real-world setting and suggest the need for further research on implementing scenario-based dynamic strategies.

Scenario-Based Management of Air Traffic Flow

Recent studies of the single-airport ground-holding problem use static or dynamic optimization to manage uncertainty about future airport capacities. Scenario trees of airport capacity profiles provide the basis for formulating multistage recourse problems. In this paper, methodologies are presented for generating scenario trees from empirical data, and the performance of scenario-based models is examined with the scenario trees. Most U.S. airports have capacity profiles that can be classified into some nominal scenarios, and for many airports, these scenarios can be naturally combined into scenario trees. The delay costs yielded from using dynamic optimization are consistently and considerably lower than from static optimization. The results illustrate the benefit of the wait-and-see strategy in a real-world setting and suggest the need for further research on implementing scenario-based dynamic strategies.