# Exploiting Counterfactuals for Scalable Stochastic Optimization

## Abstract

We propose a new framework for decision making under uncertainty to overcome the main drawbacks of current technology: modeling complexity, scenario generation, and scaling limitations. We consider three NP-hard optimization problems: the Stochastic Knapsack Problem (SKP), the Stochastic Shortest Path Problem (SSPP), and the Resource Constrained Project Scheduling Problem (RCPSP) with uncertain job durations, all with recourse. We illustrate how an integration of constraint optimization and machine learning technology can overcome the main practical shortcomings of the current state of the art.

The area of maritime transportation optimization has recently begun to achieve increasing success at solving large scale models, and industry is steadily adopting operations research-based models and algorithms. However, the parameters of models in the maritime domain, like many others, are beset with uncertainty. The travel times of ships, the handling times at port, the amounts of demand available at ports, fuel prices and more are all unknown and highly variable inputs to optimization methods. Recently, the maritime literature has started to address sources of uncertainty to provide higher quality decision making. We review this nascent area of the literature and provide a unifying view of different types of uncertainty across the main areas of maritime transport and varying problem types.

This paper aims at solving the stochastic shortest path problem in vehicle routing, the objective of which is to determine an optimal path that maximizes the probability of arriving at the destination before a given deadline. To solve this problem, we propose a data-driven approach, which directly explores the big data generated in traffic. Specifically, we first reformulate the original shortest path problem as a cardinality minimization problem directly based on samples of travel time on each road link, which can be obtained from the GPS trajectory of vehicles. Then, we apply an l1-norm minimization technique and its variants to solve the cardinality problem. Finally, we transform this problem into a mixed-integer linear programming problem, which can be solved using standard solvers. The proposed approach has three advantages over traditional methods. First, it can handle various or even unknown travel time probability distributions, while traditional stochastic routing methods can only work on specified probability distributions. Second, it does not rely on the assumption that travel time on different road segments is independent of each other, which is usually the case in traditional stochastic routing methods. Third, unlike other existing methods which require that deadlines must be larger than certain values, the proposed approach supports more flexible deadlines. We further analyze the influence of important parameters to the performances, i.e., accuracy and time complexity. Finally, we implement the proposed approach and evaluate its performance based on a real road network of Munich city. With real traffic data, the results show that it outperforms traditional methods.

A major issue in any application of multistage stochastic programming is the representation of the underlying random data process. We discuss the case when enough data paths can be generated according to an accepted parametric or nonparametric stochastic model. No assumptions on convexity with respect to the random parameters are required. We emphasize the notion of representative scenarios (or a representative scenario tree) relative to the problem being modeled.

We introduce an instance-weighting method to induce cost-sensitive trees. It is a generalization of the standard tree induction process where only the initial instance weights determine the type of tree to be induced-minimum error trees or minimum high cost error trees. We demonstrate that it can be easily adapted to an existing tree learning algorithm. Previous research provides insufficient evidence to support the idea that the greedy divide-and-conquer algorithm can effectively induce a truly cost-sensitive tree directly from the training data. We provide this empirical evidence in this paper. The algorithm incorporating the instance-weighting method is found to be better than the original algorithm in terms of total misclassification costs, the number of high cost errors, and tree size in two-class data sets. The instance-weighting method is simpler and more effective in implementation than a previous method based on altered priors.

The objectives of the airline crew-planning process are to allocate crews to flights and create work schedules for crew members. Most airlines solve their crew-planning problem in two steps. The first step, crew pairing, is to generate optimized, anonymous pairings that cover given flight schedules. In the second step, the resulting pairings are assigned to crew members. The general pairing problem is complex because flights may require an augmented crew for safety reasons. A flight’s crew-augmentation requirement varies, depending on the characteristics of the pairings that cover it. Furthermore, airlines often impose rules to govern the coverage of a flight by different pairings. Common approaches to the problem either fix the crew-augmentation requirement a priori, or add restrictions on how the augmentation requirement is satisfied. Crew augmentation is often overlooked from an optimization perspective because of the complexities involved. The Sabre® long-haul pairing optimizer explicitly models many types of crew-augmentation processes and simultaneously considers the relevant ranks of all members within the cockpit crew. It uses state-of-the-art large-scale optimization techniques, such as branch and price, to solve the problem. In this article, we introduce the long-haul pairing optimizer that Sabre developed in the mid-1990s, and share the evolution of the models and solution algorithms for the general crew-pairing problem with augmentation. We also compare our approach with four conventional approaches to show that we can effectively solve the general crew-augmentation problem and provide significant crew cost savings to airlines.

This paper presents an algorithm for finding all shortest routes from all nodes to a given destination in N N -node general networks (in which the distances of arcs can be negative). If no negative loop exists, the algorithm requires 1 2 M ( N − 1 ) ( N − 2 ) , 1 > M N − 1 \frac {1}{2}M\left ( {N - 1} \right ) \\ \left ( {N - 2} \right ),1 > MN - 1 , additions and comparisons. The existence of a negative loop, should one exist, is detected after 1 2 N ( N − 1 ) ( N − 2 ) \frac {1}{2}N\left ( {N - 1} \right )\left ( {N - 2} \right ) additions and comparisons.

We present a set of benchmark instances for the evaluation of solution procedures for single- and multi-mode resource-constrained project scheduling problems. The instances have been systematically generated by the standard project generator ProGen. They are characterized by the input-parameters of ProGen. The entire benchmark set including its detailed characterization and the best solutions known so-far are available on a public ftp-site. Hence, researchers can download the benchmark sets they need for the evaluation of their algorithms. Additionally, they can make available new results. Depending on the progress made in the field, the instance library will be continuously enlarged and new results will be made accessible. This should be a valuable and driving source for further improvements in the area of project type scheduling.

One of the challenges faced by liner operators today is to effectively operate empty containers in order to meet demand and to reduce inefficiency in an uncertain environment. To incorporate uncertainties in the operations model, we formulate a two-stage stochastic programming model with random demand, supply, ship weight capacity, and ship space capacity. The objective of this model is to minimize the expected operational cost for Empty Container Repositioning (ECR). To solve the stochastic programs with a prohibitively large number of scenarios, the Sample Average Approximation (SAA) method is applied to approximate the expected cost function. To solve the SAA problem, we consider applying the scenario aggregation by combining the approximate solution of the individual scenario problem. Two heuristic algorithms based on the progressive hedging strategy are applied to solve the SAA problem. Numerical experiments are provided to show the good performance of the scenario-based method for the ECR problem with uncertainties.

This paper gives an algorithm for L-shaped linear programs which arise naturally in optimal control problems with state constraints and stochastic linear programs (which can be represented in this form with an infinite number of linear constraints). The first section describes a cutting hyperplane algorithm which is shown to be equivalent to a partial decomposition algorithm of the dual program. The two last sections are devoted to applications of the cutting hyperplane algorithm to a linear optimal control problem and stochastic programming problems.