Two-stage Stochastic Integer Programming Research Articles

A two‐stage stochastic integer programming model for a thermal power system expansion

AbstractIn this paper the obtaining of an optimum policy in the capacity expansion planning of a particular thermal‐electric power system is proposed. Therefore, a two‐stage stochastic integer programming is formulated. The model includes, through a finite group of scenarios, the existent uncertainty related to the future availability of the thermal plants currently under operation. The resultant model is solved numerically by the application of the L‐shaped method, whose implementation and development were executed using the software AMPL, with CPLEX as a solver. The results reached are shown, which validate the use of the methodology adopted in this work.

Joint Inventory Replenishment and Component Allocation Optimization in an Assemble-to-Order System

This paper considers a multicomponent, multiproduct periodic-review assemble-to-order (ATO) system that uses an independent base-stock policy for inventory replenishment. Product demands in each period are integer-valued correlated random variables, with each product being assembled from multiple units of a subset of components. The system quotes a prespecified response time window for each product and receives a reward if the demand for that product is filled within its response time window. We formulate a two-stage stochastic integer program with recourse to determine the optimal base-stock policy and the optimal component allocation policy for the ATO system. We show that the component allocation problem is a general multidimensional knapsack problem (MDKP) and is NP-hard. We propose a simple, order-based component allocation rule and show that it can be solved in either polynomial or pseudopolynomial time. We also use the sample average approximation method to determine the optimal base-stock levels and compare it with two variations of the equal fractile heuristic. Intensive testing indicates that our solution method for each stage of the stochastic program is robust, effective, and that it significantly outperforms existing methods. Finally, we discuss several managerial implications of our findings.

Joint Inventory Replenishment and Component Allocation Optimization in an Assemble-to-Order System

This paper considers a multicomponent, multiproduct periodic-review assemble-to-order (ATO) system that uses an independent base-stock policy for inventory replenishment. Product demands in each period are integer-valued correlated random variables, with each product being assembled from multiple units of a subset of components. The system quotes a prespecified response time window for each product and receives a reward if the demand for that product is filled within its response time window. We formulate a two-stage stochastic integer program with recourse to determine the optimal base-stock policy and the optimal component allocation policy for the ATO system. We show that the component allocation problem is a general multidimensional knapsack problem (MDKP) and is NP-hard. We propose a simple, order-based component allocation rule and show that it can be solved in either polynomial or pseudopolynomial time. We also use the sample average approximation method to determine the optimal base-stock levels and compare it with two variations of the equal fractile heuristic. Intensive testing indicates that our solution method for each stage of the stochastic program is robust, effective, and that it significantly outperforms existing methods. Finally, we discuss several managerial implications of our findings.

Modeling and solving real-time scheduling problems by stochastic integer programming

This contribution deals with scheduling problems of flexible chemical batch processes with a special emphasis on their real-time character. This implies not only the need for sufficiently short response times, but in particular the burden of in-complete knowledge about the future. To solve such problems, the application of two-stage stochastic integer programming techniques on moving horizons is proposed. They reflect the need for immediately applicable decisions and the potential of later recourse actions to cope with realized uncertainties. In addition to the classical expected value objective, simple measures of risk can be included. Motivated by an example process, some essential modeling prerequisites are discussed. As an important first step, the master scheduling problem is studied and a number of master scheduling models are presented. Large mixed-integer linear problems arise, which are well-suited for a dual decomposition approach. Numerical experiments with a problem-specific solution algorithm demonstrate the applicability of the method to real-world problems.

A finite branch-and-bound algorithm for two-stage stochastic integer programs

This paper addresses a general class of two-stage stochastic programs with integer recourse and discrete distributions. We exploit the structure of the value function of the second-stage integer problem to develop a novel global optimization algorithm. The proposed scheme departs from those in the current literature in that it avoids explicit enumeration of the search space while guaranteeing finite termination. Computational experiments on standard test problems indicate superior performance of the proposed algorithm in comparison to those in the existing literature.

Two-stage simulation optimization for agile manufacturing capacity planning

Capacity planning involves the selection of manufacturing technologies and the allocation of budget to specific equipment acquisitions. In today's highly volatile manufacturing world, an agile capacity-planning tool is required. This tool must provide the mechanism for a company to thrive in an environment of uncertainty. Uncertain future demands make capacity planning and technology selection difficult tasks, whether they are caused by variations in forecasts of direct demand or by upstream variability in a supply chain. In this paper, a practical modelling technique for minimizing the required investment in capacity planning for discrete manufacturing sites under an uncertain demand stream is presented. The method consists of a two-stage stochastic integer program. The first stage characterizes the optimal response of the system under uncertainty. The second stage selects a tool set based on the characterization from the first stage, with the addition of budget constraints. The model is scalable, allowing for multiple products, multiple operations, multiple flow paths including re-entrant flow, and multiple tool types. A simple example is introduced to explain the methodology, followed by the results of a large-scale real-world application in the semiconductor industry.

Decomposition of test sets in stochastic integer programming

We study Graver test sets for linear two-stage stochastic integer programs and show that test sets can be decomposed into finitely many building blocks whose number is independent on the number of scenarios of the stochastic program. We present a finite algorithm to compute the building blocks directly, without prior knowledge of test set vectors. Once computed, building blocks can be employed to solve the stochastic program by a simple augmentation scheme, again without explicit knowledge of test set vectors. Finally, we report preliminary computational experience.

Program Does Not Equal Program: Constraint Programming and Its Relationship to Mathematical Programming

Arising from research in the computer science community, constraint programming is a fairly new technique for solving optimization problems. For those familiar with mathematical programming, a number of language barriers make it difficult to understand the concepts of constraint programming. In this short tutorial on constraint programming, we explain how it relates to familiar mathematical programming concepts and how constraint programming and mathematical programming technologies are complementary. We assume a minimal background in linear and integer programming.

Stochastic Network Interdiction

Using limited assets, an interdictor attempts to destroy parts of a capacitated network through which an adversary will subsequently maximize flow. We formulate and solve a stochastic version of the interdictor's problem: Minimize the expected maximum flow through the network when interdiction successes are binary random variables. Extensions are made to handle uncertain arc capacities and other realistic variations. These two-stage stochastic integer programs have applications to interdicting illegal drugs and to reducing the effectiveness of a military force moving materiel, troops, information, etc., through a network in wartime. Two equivalent model formulations allow Jensen's inequality to be used to compute both lower and upper bounds on the objective, and these bounds are improved within a sequential approximation algorithm. Successful computational results are reported on networks with over 100 nodes, 80 interdictable arcs, and 180 total arcs.