Cormac Lucas

Computational aspects of alternative portfolio selection models in the presence of discrete asset choice constraints

We consider the mean-variance (M-V) model of Markowitz and the construction of the risk-return efficient frontier. We examine the effects of applying buy-in thresholds, cardinality constraints and transaction roundlot restrictions to the portfolio selection problem. Such discrete constraints are of practical importance but make the efficient frontier discontinuous. The resulting quadratic mixed-integer (QMIP) problems are NP-hard and therefore computing the entire efficient frontier is computationally challenging. We propose alternative approaches for computing this frontier and provide insight into its discontinuous structure. Computational results are reported for a set of benchmark test problems.

Computational solution of capacity planning models under uncertainty

Abstract The traditional supply chain network planning problem is stated as a multi-period resource allocation model involving 0–1 discrete strategic decision variables. The MIP structure of this problem makes it fairly intractable for practical applications, which involve multiple products, factories, warehouses and distribution centres (DCs). The same problem formulated and studied under uncertainty makes it even more intractable. In this paper we consider two related modelling approaches and solution techniques addressing this issue. The first involves scenario analysis of solutions to “wait and see” models and the second involves a two-stage integer stochastic programming (ISP) representation and solution of the same problem. We show how the results from the former can be used in the solution of the latter model. We also give some computational results based on serial and parallel implementations of the algorithms.

Heuristic algorithms for the cardinality constrained efficient frontier

This paper examines the application of genetic algorithm, tabu search and simulated annealing metaheuristic approaches to finding the cardinality constrained efficient frontier that arises in financial portfolio optimisation. We consider the mean-variance model of Markowitz as extended to include the discrete restrictions of buy-in thresholds and cardinality constraints. Computational results are reported for publicly available data sets drawn from seven major market indices involving up to 1318 assets. Our results are compared with previous results given in the literature illustrating the effectiveness of the proposed metaheuristics in terms of solution quality and computation time.

Tools for reformulating logical forms into zero-one mixed integer programs

Abstract A systematic procedure for transforming a set of logical statements or logical conditions imposed on a model into an Integer Linear Programming (ILP) formulation or a Mixed Integer Programming (MIP) formulation is presented. A reformulation procedure which uses the extended reverse Polish notation to represent a compound logical form is then described. The syntax of an LP modelling language is extended to incorporate statements in propositional logic forms with linear algebraic forms whereby 0–1 MIP models can be automatically formulated. A prototype user interface by which logical forms can be reformulated and the corresponding MIP constructed and analysed within an existing Mathematical Programming modelling system is illustrated. Finally, the steps to formulate a discrete optimisation model in this way are illustrated by means of an example.

A review of portfolio planning: Models and systems

Publisher Summary This chapter provides an overview of a number of portfolio planning models which have been proposed and investigated over the past 50 years. The mean-variance (MV) model of Markowitz is a single period static portfolio planning model, and, in recent times, it has become the core decision engine of many portfolio analytics and planning systems in the construction of the risk-return efficient frontier. The estimation of the underlying parameters which are required as the input to MV analysis is an important modeling step. Small changes in the input can have a large impact on the optimal asset weights. Diagonal models are of interest as the corresponding quadratic forms can then be expressed as variable separable functions, which in turn are approximated as piecewise linear functions. The use of factor models to describe asset returns can also lead to a diagonal form provided the composition of the covariance matrix is appropriately exploited.

A prototype decision support system for strategic planning under uncertainty

Strategic planning of the supply chain is an important decision problem determining the long‐term survival and prosperity of companies in the manufacturing, retail, and other industrial sectors. In general such companies rely on their information systems to acquire the essential data that are used in their planning models. The interaction of information systems and decision modelling, and the progressive transformation of data, into information, and knowledge is a key process underlying any decision support system (DSS) for strategic, tactical or operational planning. In this paper we consider a DSS for supply chain planning (SCP) decisions. The SCP system has an embedded decision engine that uses a two‐stage stochastic program as a paradigm for optimisation under uncertainty. The system has been used for decision making in diverse domains, including automotive manufacturing and consumer products.

Robust solutions and risk measures for a supply chain planning problem under uncertainty

We consider a strategic supply chain planning problem formulated as a two-stage stochastic integer programming (SIP) model. The strategic decisions include site locations, choices of production, packing and distribution lines, and the capacity increment or decrement policies. The SIP model provides a practical representation of real-world discrete resource allocation problems in the presence of future uncertainties which arise due to changes in the business and economic environment. Such models that consider the future scenarios (along with their respective probabilities) not only identify optimal plans for each scenario, but also determine a hedged strategy for all the scenarios. We 1)exploit the natural decomposable structure of the SIP problem through Benders’ decomposition,2)approximate the probability distribution of the random variables using the generalized lambda distribution, and3)through simulations, calculate the performance statistics and the risk measures for the two models, namely the expected-value and the here-and-now.

An application of Lagrangian relaxation to a capacity planning problem under uncertainty

A supply chain network-planning problem is presented as a two-stage resource allocation model with 0-1 discrete variables. In contrast to the deterministic mathematical programming approach, we use scenarios, to represent the uncertainties in demand. This formulation leads to a very large scale mixed integer-programming problem which is intractable. We apply Lagrangian relaxation and its corresponding decomposition of the initial problem in a novel way, whereby the Lagrangian relaxation is reinterpreted as a column generator and the integer feasible solutions are used to approximate the given problem. This approach addresses two closely related problems of scenario analysis and two-stage stochastic programs. Computational solutions for large data instances of these problems are carried out successfully and their solutions analysed and reported. The model and the solution system have been applied to study supply chain capacity investment and planning.

Computer-Assisted Mathematical Programming (Modelling) System: CAMPS

A Computer-Assisted Mathematical Programming (Modelling) System (CAMPS) is described in this paper. The system uses program-generator techniques for model creation and contrasts with earlier approaches, which use a special-purpose language to construct models. Thus no programming skill is required to formulate a model. In designing the system we have first analysed the salient components of the mathematical programming modelling activity. A mathematical programming model is usually constructed by progressive definition of dimensions, data tables, model variables, model constraints and the matrix coefficients which connect the last two entities. Computer assistance is provided to structure the data and the resulting model in the above sequence. In addition to this novel feature and the automatic documentation facility, the system is in line with recent developments, and incorporates a friendly and flexible user interface.

Treasury Management Model with Foreign Exchange Exposure

In this paper we formulate a model for foreign exchange exposure management and (international) cash management taking into consideration random fluctuations of exchange rates. A vector error correction model (VECM) is used to predict the random behaviour of the forward as well as spot rates connecting dollar and sterling. A two-stage stochastic programming (TWOSP) decision model is formulated using these random parameter values. This model computes currency hedging strategies, which provide rolling decisions of how much forward contracts should be bought and how much should be liquidated.The model decisions are investigated through ex post simulation and backtesting in which value at risk (VaR) for alternative decisions are computed. The investigation (a) shows that there is a considerable improvement to “spot only” strategy, (b) provides insight into how these decisions are made and (c) also validates the performance of this model.