Two-stage quadratic integer programs with stochastic right-hand sides

We consider two-stage quadratic integer programs with stochastic right-hand sides, and present an equivalent reformulation using value functions. We propose a two-phase solution approach. The first phase constructs value functions of quadratic integer programs in both stages. The second phase solves the reformulation using a global branch-and-bound algorithm or a level-set approach. We derive some basic properties of value functions of quadratic integer programs and utilize them in our algorithms. We show that our approach can solve instances whose extensive forms are hundreds of orders of magnitude larger than the largest quadratic integer programming instances solved in the literature.     

Multistage stochastic convex programs: Duality and its implications

In this paper, we study alternative primal and dual formulations of multistage stochastic convex programs (SP). The alternative dual problems which can be traced to the alternative primal representations, lead to stochastic analogs of standard deterministic constructs such as conjugate functions and Lagrangians. One of the by-products of this approach is that the development does not depend on dynamic programming (DP) type recursive arguments, and is therefore applicable to problems in which the objective function is non-separable (in the DP sense). Moreover, the treatment allows us to handle both continuous and discrete random variables with equal ease. We also investigate properties of the expected value of perfect information (EVPI) within the context of SP, and the connection between EVPI and nonanticipativity of optimal multipliers. Our study reveals that there exist optimal multipliers that are nonanticipative if, and only if, the EVPI is zero. Finally, we provide interpretations of the retroactive nature of the dual multipliers. This work was supported by NSF grant DMII-9414680.     

L-shaped decomposition of two-stage stochastic programs with integer recourse

We consider two-stage stochastic programming problems with integer recourse. The L-shaped method of stochastic linear programming is generalized to these problems by using generalized Benders decomposition. Nonlinear feasibility and optimality cuts are determined via general duality theory and can be generated when the second stage problem is solved by standard techniques. Finite convergence of the method is established when Gomory’s fractional cutting plane algorithm or a branch-and-bound algorithm is applied.     

BBPH: Using progressive hedging within branch and bound to solve multi-stage stochastic mixed integer programs

Progressive hedging, though an effective heuristic for solving stochastic mixed integer programs (SMIPs), is not guaranteed to converge in this case. Here, we describe BBPH, a branch and bound algorithm that uses PH at each node in the search tree such that, given sufficient time, it will always converge to a globally optimal solution. In addition to providing a theoretically convergent “wrapper” for PH applied to SMIPs, computational results demonstrate that for some difficult problem instances branch and bound can find improved solutions after exploring only a few nodes.     

A generalized knapsack problem with variable coefficients

The ordinary knapsack problem is to find the optimal combination of items to be packed in a knapsack under a single constraint on the total allowable resources, where all coefficients in the objective function and in the constraint are constant.In this paper, a generalized knapsack problem with coefficients depending on variable parameters is proposed and discussed. Developing an effective branch and bound algorithm for this problem, the concept of relaxation and the efficiency function introduced here will play important roles. Furthermore, a relation between the algorithm and the dynamic programming approach is discussed, and subsequently it will be shown that the ordinary 0–1 knapsack problem, the linear programming knapsack problem and the single constrained linear programming problem with upper-bounded variables are special cases of the interested problem. Finally, practical applications of the problem and its computational experiences will be shown.     

Computations with disjunctive cuts for two-stage stochastic mixed 0-1 integer programs

Two-stage stochastic mixed-integer programming (SMIP) problems with recourse are generally difficult to solve. This paper presents a first computational study of a disjunctive cutting plane method for stochastic mixed 0-1 programs that uses lift-and-project cuts based on the extensive form of the two-stage SMIP problem. An extension of the method based on where the data uncertainty appears in the problem is made, and it is shown how a valid inequality derived for one scenario can be made valid for other scenarios, potentially reducing solution time. Computational results amply demonstrate the effectiveness of disjunctive cuts in solving several large-scale problem instances from the literature. The results are compared to the computational results of disjunctive cuts based on the subproblem space of the formulation and it is shown that the two methods are equivalently effective on the test instances.     

Solving stochastic programs with integer recourse by enumeration: A framework using Gröbner basis

In this paper we present a framework for solving stochastic programs with complete integer recourse and discretely distributed right-hand side vector, using Gröbner basis methods from computational algebra to solve the numerous second-stage integer programs. Using structural properties of the expected integer recourse function, we prove that under mild conditions an optimal solution is contained in a finite set. Furthermore, we present a basic scheme to enumerate this set and suggest improvements to reduce the number of function evaluations needed.     

Integer programming duality: Price functions and sensitivity analysis

Recently a duality theory for integer programming has been developed. Here we examine some of the economic implications of this theory, in particular the necessity of using price functions in place of prices, and the possibility of carrying out sensitivity analysis of optimal solutions. In addition we consider the form of price functions that are generated by known algorithms for integer programming.This research was supported in part by a Senior Visiting Research Fellowship from the Science Research Council at the London School of Economics while the author was on leave from CORE, Université Catholique de Louvain, at Louvain-la-Neuve.     

Solving a class of stochastic mixed-integer programs with branch and price

We begin this paper by identifying a class of stochastic mixed-integer programs that have column-oriented formulations suitable for solution by a branch-and-price algorithm (B&P). We then survey a number of examples, and use a stochastic facility-location problem (SFLP) for a detailed demonstration of the relevant modeling and solution techniques. Computational results with a scenario representation of uncertain costs, demands and capacities show that B&P can be orders of magnitude faster than solving the standard formulation by branch and bound. We also demonstrate how B&P can solve SFLP exactly – as exactly as a deterministic mixed-integer program – when demands and other parameters can be represented as certain types of independent, random variables, e.g., independent, normal random variables with integer means and variances. Kevin Wood thanks the Office of Naval Research, Air Force Office of Scientific Research, the Naval Postgraduate School (NPS) and the University of Auckland for their support. Eduardo Silva thanks NPS and the Brazilian Navy for their support. Both authors are grateful to the COIN-OR team for assistance with computational issues, as well as to two anonymous referees for highly useful, constructive criticism.     

Solving integer programs with a few important binary gub constraints

During a branch and bound search of an integer program, decisions have to be taken about which subproblem to solve next and which variable or special ordered set to branch on. Both these decisions are usually based on some sort of estimated change in the objective caused by different branching. When the next subproblem is chosen, the estimated change in the objective is often found by summing the change caused by changing all integer variables with non-integer values, as if they were independent. For special ordered sets the estimation is done for each set as a whole. The purpose of this paper is to report some results from trying to do a simultaneous estimation for all the variables in a binary gub constraint. By this, the analysed problems contain one or a few constraints saying that the sum ofn binary variables should be equal tom (<n).I am grateful to Scicon Ltd. for giving me access to the SCICONIC source code.     

Modeling methods and a branch and cut algorithm for pharmaceutical clinical trial planning using stochastic programming

We discuss methods for the solution of a multi-stage stochastic programming formulation for the resource-constrained scheduling of clinical trials in the pharmaceutical research and development pipeline. First, we present a number of theoretical properties to reduce the size and improve the tightness of the formulation, focusing primarily on non-anticipativity constraints. Second, we develop a novel branch and cut algorithm where necessary non-anticipativity constraints that are unlikely to be active are removed from the initial formulation and only added if they are violated within the search tree. We improve the performance of our algorithm by combining different node selection strategies and exploring different approaches to constraint violation checking.     

Parking buses in a depot with stochastic arrival times

Given buses of different types arriving at a depot during the evening, the bus parking problem consists of assigning these buses to parking slots in such a way that they can be dispatched adequately to the next morning routes without moving them between their arrivals and departures. In practice, the bus arrival times deviate stochastically from the planned schedule. In this paper, we introduce for this problem two solution approaches that produce solutions which are robust to variations in the arrival times. The first approach considers that each arrival can deviate from its planned arrival order (sooner or later) by at most k positions, where k is a predefined parameter. In the second approach, the objective aims at minimizing the expectation of a function positively correlated with the number of buses that make the planned solution infeasible because they arrive too late or too early. In both approaches, the problem is modeled as an integer linear program that can be solved by a commercial mip solver. Computational results obtained on instances derived from a real-world dataset are reported.     

Balancing stochastic two-sided assembly lines: A chance-constrained,piecewise-linear,mixed integer program and a simulated annealing algorithm

Two-sided assembly lines are often designed to produce large-sized products, such as automobiles, trucks and buses. In this type of a production line, both left-side and right-side of the line are used in parallel. In all studies on two-sided assembly lines, the task times are assumed to be deterministic. However, in real life applications, especially in manual assembly lines, the tasks may have varying execution times defined as a probability distribution. The task time variation may result from machine breakdowns, loss of motivation, lack of training, non-qualified operators, complex tasks, environment, etc. In this paper, the problem of balancing two-sided assembly lines with stochastic task times (STALBP) is considered. A chance-constrained, piecewise-linear, mixed integer program (CPMIP) is proposed to model and solve the problem. As a solution approach a simulated annealing (SA) algorithm is proposed. To assess the effectiveness of CPMIP and SA algorithm, a set of test problems are solved. Finally, computational results indicating the effectiveness of CPMIP and SA algorithm are reported.     

Solving a class of low rank d.c. programs via a branch and bound approach: A computational experience

Various classes of d.c. programs have been studied in the recent literature due to their importance in applicative problems. In this paper we consider a branch and bound approach for solving a class of d.c. problems. Both stack policies and partitioning rules are analyzed, pointing out their performance effectiveness by means of the results of a computational experience.     

Two-stage vehicle routing problem with arc time windows: A mixed integer programming formulation and a heuristic approach

In this paper, we introduce a new variant of the Vehicle Routing Problem (VRP), namely the Two-Stage Vehicle Routing Problem with Arc Time Windows (TS_VRP_ATWs) which generally emerges from both military and civilian transportation. The TS_VRP_ATW is defined as finding the vehicle routes in such a way that each arc of the routes is available only during a predefined time interval with the objective of overall cost minimization. We propose a Mixed Integer Programming (MIP) formulation and a heuristic approach based on Memetic Algorithm (MA) to solve the TS_VRP_ATW. The qualities of both solution approaches are measured by using the test problems in the literature. Experimental results show that the proposed MIP formulation provides the optimal solutions for the test problems with 25 and 50 nodes, and some test problems with 100 nodes. Results also show that the proposed MA is promising quality solutions in a short computation time.