Analysis of large scale linear programming problems with embedded network structures: Detection and solution algorithms

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.Linear programming (LP) models that contain a (substantial) network structure frequently arise in many real life applications. In this thesis, we investigate two main questions; i) how an embedded network structure can be detected, ii) how the network structure can be exploited to create improved sparse simplex solution algorithms. In order to extract an embedded pure network structure from a general LP problem we develop two new heuristics. The first heuristic is an alternative multi-stage generalised upper bounds (GUB) based approach which finds as many GUB subsets as possible. In order to identify a GUB subset two different approaches are introduced; the first is based on the notion of Markowitz merit count and the second exploits an independent set in the corresponding graph. The second heuristic is based on the generalised signed graph of the coefficient matrix. This heuristic determines whether the given LP problem is an entirely pure network; this is in contrast to all previously known heuristics. Using generalised signed graphs, we prove that the problem of detecting the maximum size embedded network structure within an LP problem is NP-hard. The two detection algorithms perform very well computationally and make positive contributions to the known body of results for the embedded network detection. For computational solution a decomposition based approach is presented which solves a network problem with side constraints. In this approach, the original coefficient matrix is partitioned into the network and the non-network parts. For the partitioned problem, we investigate two alternative decomposition techniques namely, Lagrangean relaxation and Benders decomposition. Active variables identified by these procedures are then used to create an advanced basis for the original problem. The computational results of applying these techniques to a selection of Netlib models are encouraging. The development and computational investigation of this solution algorithm constitute further contribution made by the research reported in this thesis.This study is funded by the Turkish Educational Council and Mugla University

Robust portfolio selection problem under temperature uncertainty

In this paper, we consider a portfolio selection problem under temperature uncertainty. Weather derivatives based on different temperature indices are used to protect against undesirable temperature events. We introduce stochastic and robust portfolio optimization models using weather derivatives. The investors’ different risk preferences are incorporated into the portfolio allocation problem. The robust investment decisions are derived in view of discrete and continuous sets that the underlying uncertain data in temperature model belong. We illustrate main features of the robust approach and performance of the portfolio optimization models using real market data. In particular, we analyze impact of various model parameters on different robust investment decisions

A robust asset–liability management framework for investment products with guarantees

This paper suggests a robust asset–liability management framework for investment products with guarantees, such as guaranteed investment contracts and equity-linked notes. Stochastic programming and robust optimization approaches are introduced to deal with data uncertainty in asset returns and interest rates. The statistical properties of the probability distributions of uncertain parameters are incorporated in the model through appropriately selected symmetric and asymmetric uncertainty sets. Practical data-driven approaches for implementation of the robust models are also discussed. Numerical results using generated and real market data are presented to illustrate the performance of the robust asset–liability management strategies. The robust investment strategies show better performance in unfavorable market regimes than traditional stochastic programming approaches. The effectiveness of robust investment strategies can be improved by calibrating carefully the shape and the size of the uncertainty sets for asset returns

Efficient solution selection for two-stage stochastic programs

Sampling-based stochastic programs are extensively applied in practice. However, the resulting models tend to be computationally challenging. A reasonable number of samples needs to be identified to represent the random data, and a group of approximate models can then be constructed using such a number of samples. These approximate models can produce a set of potential solutions for the original model. In this paper, we consider the problem of allocating a finite computational budget among numerous potential solutions of a two-stage linear stochastic program, which aims to identify the best solution among potential ones by conducting simulation under a given computational budget. We propose a two-stage heuristic approach to solve the computational resource allocation problem. First, we utilise a Wasserstein-based screening rule to remove potentially inferior solutions from the simulation. Next, we use a ranking and selection technique to efficiently collect performance information of the remaining solutions. The performance of our approach is demonstrated through well-known benchmark problems. Results show that our method provides good trade-offs between computational effort and solution performance

Dynamic pricing of flexible time slots for attended home delivery

In e-commerce, customers are usually offered a menu of home delivery time windows of which they need to select exactly one, even though at least some customers may be more flexible. To exploit the flexibility of such customers, we propose to introduce flexible delivery time slots, defined as any combination of such regular time windows (not necessarily adjacent). In selecting a flexible time slot (out of a set of windows that form the flexible product), the customer agrees to be informed only shortly prior to the dispatching of the delivery vehicle in which regular time window the goods will arrive. In return for providing this flexibility, the company may offer the customer a reduced delivery charge and/or highlight the environmental benefits. Our framework also can accommodate customized flexible slots where customers can self-select a set of regular slots in which a delivery may take place. The vehicle routing problem (VRP) in the presence of flexible time slots bookings corresponds to a VRP with multiple time windows. We build on literature on demand management and vehicle routing for attended home delivery, as well as on flexible products. These two concepts have not yet been combined, and indeed the results from the flexible products literature do not carry over directly because future expected vehicle routing implications need to be taken into account. The main methodological contribution is the development of a tractable linear programming formulation that links demand management decisions and routing cost implications, whilst accounting for customer choice behavior. The output of this linear program provides information on the (approximate) opportunity cost associated with specific orders and informs a tractable dynamic pricing policy for regular and flexible slots. Numerical experiments, based on realistically-sized scenarios, indicate that expected profit may increase significantly depending on demand intensity when adding flexible slots rather than using only regular slots

Uncertainty representation and risk management for direct segmented marketing

Mining for truly responsive customers has become an integral part of customer portfolio management, and, combined with operational tactics to reach these customers, requires an integrated approach to meeting customer needs that often involves the application of concepts from traditionally distinct fields: marketing, statistics, and operations research. This article brings such concepts together to address customer value and revenue maximization as well as risk minimization for direct marketing decision making problems under uncertainty. We focus on customer lift optimization given the uncertainty associated with lift estimation models, and develop risk management and operational tools for the multiple treatment (recommendation) problem using stochastic and robust optimization techniques. Results from numerical experiments are presented to illustrate the effect of incorporating uncertainty on the performance of recommendation models

Worst-case robust decisions for multi-period mean-variance portfolio optimization

In this paper, we extend the multi-period mean-variance optimization framework to worst-case design with multiple rival return and risk scenarios. Our approach involves a min-max algorithm and a multi-period mean-variance optimization framework for the stochastic aspects of the scenario tree. Multi-period portfolio optimization entails the construction of a scenario tree representing a discretised estimate of uncertainties and associated probabilities in future stages. The expected value of the portfolio return is maximized simultaneously with the minimization of its variance. There are two sources of further uncertainty that might require a strengthening of the robustness of the decision. The first is that some rival uncertainty scenarios may be too critical to consider in terms of probabilities. The second is that the return variance estimate is usually inaccurate and there are different rival estimates, or scenarios. In either case, the best decision has the additional property that, in terms of risk and return, performance is guaranteed in view of all the rival scenarios. The ex-ante performance of min-max models is tested using historical data and backtesting results are presented

Robust optimal decisions with imprecise forecasts

A robust minimax approach for optimal investment decisions with imprecise return forecasts and risk estimations in financial portfolio management is considered. Single-period and multi-period mean-variance optimization models are extended to worst-case design with multiple rival risk estimations and return forecasts. In multi-period stochastic formulation of classical mean-variance portfolio optimization problem, an investor makes an investment decision based on expectations and/or scenarios up to some intermediate times prior to the horizon and, consequently, rebalances or restructures the portfolio. Multi-period portfolio optimization entails the construction of a scenario tree representing a discretized estimate of uncertainties and associated probabilities in future stages. It is well known that return forecasts and risk estimations are inherently inaccurate and there are different rival estimates, or scenario trees. Robust optimization models are presented and imprecise nature of moment forecasts to reduce the risk of making a decision based on the wrong scenario is addressed. The worst-case performance is guaranteed in view of all rival risk and return scenarios and will only improve when any scenario other than the worst-case is realized. The ex-ante performance of minimax models is tested using historical data and backtesting results are presented. (c) 2006 Elsevier B.V. All rights reserved

Modelling oil and gas supply disruption risks using extreme-value theory and copula

In this paper, we are concerned with modelling oil and gas supply disruption risks using extreme-value theory and copula. We analyse financial and volumetric losses due to both oil and gas supply disruptions and investigate their dependence structure using real data. In order to illustrate the impact of crude oil and natural gas supply disruptions on an energy-dependent economy, Nigeria is considered as a case study. Computational studies illustrate that the generalized extreme-value distribution anticipates higher financial losses and extreme-value copulas produce the best fit for financial and volumetric losses compared with normal copulas. Moreover, multivariate financial losses exhibit stronger positive dependence than volumetric losses

Robust optimization approaches to single period portfolio allocation problem

Portfolio management is one of the fundamental problems in financial decision making. In a typical portfolio management problem, an investor is concerned with an optimal allocation of the capital among a number of available financial assets to maximize the return on the investment while minimizing the risk. This problem was formulated in the mean-variance portfolio management framework proposed by Markowitz in 1952. Since then, it has been widely studied by researchers and the practitioners. However, the solution is sensitive to model parameters due to data uncertainty. In this chapter, we review robust approaches to deal with data uncertainty for a single-period portfolio allocation problem. We first introduce the main ideas of robust optimization using symmetric and asymmetric uncertainty sets where the uncertain asset returns belong to. We then focus on data driven and distributionally robust optimization approaches