IMA Journal of Management Mathematics - Advance Access

A survey of effective heuristics and their application to a variety of knapsack problems

Wilbaut, C., Hanafi, S., Salhi, S. — 2008-04-15

We present a family of knapsack problems (KPs) while highlighting their particular applications. Though most of the problems are derived from the classical KP, the differences arise in the addition or modification of the constraints or in the way the objective function is defined. Appropriate techniques that were found to be successful in solving these problems are briefly reviewed. Hybrid methods that combine the strengths of different methods such as exact and heuristics are also briefly discussed. Some research avenues that we believe to be useful and challenging are also pointed out.

Forecasting aggregate time series with intermittent subaggregate components: top-down versus bottom-up forecasting

Viswanathan, S, Widiarta, H., Piplani, R. — 2008-02-27

In this paper, we evaluate the performance of top-down (TD) and bottom-up (BU) forecasting strategies in estimating the aggregate data series when the subaggregate time series components are intermittent. The findings of our simulation-based study are as follows. When the variability of the inter-order intervals of the subaggregate time series is low, BU forecasting that is carried out using Croston's method outperformed TD for estimating the aggregate data series. However, when the inter-order intervals and the demand sizes of the subaggregate components are highly variable and the aggregate data series is composed of many subaggregate components, TD forecasting outperformed BU. Finally, for forecasting the aggregate demand using TD forecasting, the simple exponential smoothing technique outperformed Croston's method in a majority of the cases.

Advertising in a segmented market: comparison of media choices

Grosset, L., Viscolani, B. — 2008-01-27

Segmentation is a core strategy in modern marketing but, to the best of our knowledge, it is not considered in most dynamic advertising models. In this paper, we aim to fill such a gap by presenting a dynamic advertising model which includes market segmentation. First, we model goodwill evolution in a segmented market under the assumption that the decision maker may independently choose the advertising intensity directed at each different segment. Then, we assume that the decision maker must use a single medium, which reaches several segments with different effectiveness. We obtain the explicit solutions of the relevant optimal control problems. These results permit us to compare the two different contexts and to obtain a preference index for advertising media.

Hidden Markov models for scenario generation

Messina, E., Toscani, D. — 2007-10-08

We consider the problem of modelling processes sequentially changing behaviour and unexpected changes that can hinder finding the best approximation function. These dynamics cannot be observed directly either because they are masked by observational noise or because the process generating them is too complex and involves too many variables. In this paper, the problem of modelling financial time series has been approached using hidden Markov models (HMMs), which have been shown to be suitable for sequential data analysis and in particular for financial time series modelling and forecasting. HMMs are essentially data-driven models that allow us to focus attention on the observation generation process, which is indeed final objective. The goal of our time series analysis model is the generation of scenarios to be included in decision models. Therefore, our focus will not be on determining the best forecast but in capturing the generation process behaviour in order to characterize its possible evolutions.

Scenario generation for stochastic programming and simulation: a modelling perspective

Di Domenica, N., Lucas, C., Mitra, G., Valente, P. — 2007-08-22

Stochastic programming (SP) brings together models of optimum resource allocation and models of randomness and thereby creates a robust decision-making framework. The models of randomness with their finite, discrete realizations are known as scenario generators. In this report, we consider alternative approaches to scenario generation in a generic form which can be used to formulate (a) two-stage (static) and (b) multi-stage dynamic SP models. We also investigate the modelling structure and software issues of integrating a scenario generator with an optimization model to construct SP recourse problems. We consider how the expected value and SP decision model results can be evaluated within a descriptive modelling framework of simulation. Illustrative examples and computational results are given in support of our investigation.

Extending algebraic modelling languages to support algorithm development for solving stochastic programming models

Karabuk, S. — 2007-04-26

An algebraic modelling language (AML) is a domain-specific computer programming language for describing and solving mathematical programming models. We propose extending AMLs so that solution algorithms that are based on iteratively manipulating, modifying and solving a model are supported at a high abstraction level. We specifically focus on stochastic programming models with random parameters formulated as discrete scenarios and mathematical decomposition algorithms, which are commonly applied to solve such models. We identify the necessary language constructs and develop a design based on the open-source modelling software APLEpy. The proposed design, although specifically addressing decomposition algorithms, proves useful for implementing heuristic solution algorithms as well. The object-oriented nature of the design enables the algorithms that are coded with the proposed extensions to work with any other model that satisfies the assumptions of the initial model. This flexible and robust design helps inexperienced modellers to easily apply an advanced solution algorithm, and experienced modellers to build sophisticated algorithms quickly within the same development environment that is used to describe the model under consideration.

Modelling health-based restrictions on pesticides in groundwater under uncertainty: a policy application

Willett, K., Willett, D. — 2007-03-14

This paper has two objectives. First, the opportunity cost of health-based restrictions on pesticides in groundwater under uncertainty is examined under varying conditions. The determination of opportunity cost is based on a protocol which combines a stochastic simulation model with mathematical programming techniques that incorporate health-based risk considerations as well as elements of uncertainty. The model structure used is a safety-rule framework and begins with the establishment of a target for human health-related exposure to pesticides in groundwater. The changes in key parameters that impact the opportunity cost for a specific health-based level of pesticides in groundwater are identified and discussed. An empirical implementation of the model along with an analysis of the results is presented. The second objective of this paper is to develop an optimal tax based on the standards and charges approach. The numerical results from the empirical model are then used to calculate the various tax rates.

Robust stochastic programming with uncertain probabilities

Thiele, A. — 2007-03-06

Stochastic programming has traditionally assumed the exact knowledge of the underlying scenario probabilities. In practice, however, such probabilities are difficult to estimate accurately and the optimal decision variables may be quite sensitive to the assumed distributions. This motivates the use of minimax stochastic models, where the decision maker minimizes the maximum expected cost over the set of possible probability distributions. We use ideas from the field of robust optimization to reformulate the minimax stochastic programming problem when probabilities belong to a polyhedral uncertainty set as a single convex problem, and show that it can be solved efficiently using the traditional techniques developed to address sequential decision making under uncertainty. In the two-stage setting, we describe how the Benders decomposition algorithm can be modified to solve the robust formulation. In the case of multiple stages, we build upon the recursive equations of dynamic programming to formulate an approach as tractable as the multi-stage stochastic problem where the probabilities are known exactly. Key contributions of this work are the following: (i) we show that the minimax approach is equivalent to the nominal stochastic programming problem with a penalty term, which measures the cost volatility due to the ambiguity on the probability estimates, and (ii) we provide deeper insights into the connection between the value of the recourse function in a given scenario and the worst-case probability associated with that outcome. The robust approach also allows the decision maker to adjust the parameters defining the uncertainty set in order to better capture his own trade-off between ambiguity and performance.

The SMPS format explained

Gassmann, H. I., Kristjansson, B. — 2007-03-06

Recent extensions to the SMPS format have vastly increased the range of stochastic linear programs that can be expressed within the format. This paper illustrates some of the features of SMPS using sample problems from the literature. For each problem, we give the general mathematical formulation, a small illustrative instance and the SMPS core, time and stoch files.

A stochastic optimization model for a gas sale company

Allevi, E, Bertocchi, M., Innorta, M, Vespucci, M. — 2007-02-27

In this paper, the authors develop a stochastic optimization model, named Optimization Modelling for Gas Seller (OMoGas), to assist companies dealing with gas retail commercialization. Stochasticity is due to the dependence of consumption on temperature uncertainty. Nonlinearities are present in both the objective function and the constraints. The model can be classified as a non-linear programming (NLP) mixed integer model, with the profit function depending on the number of contracts with the final consumers, the characteristics of such consumers and the cost supported to meet the final demand. Constraints related to a maximum daily gas consumption, yearly maximum and minimum consumption in order to avoid penalties and consumption profiles are included. The model is implemented in the General Algebraic Modeling System (GAMS) environment and the results obtained by the stochastic version, based on consumption scenarios, are compared with the deterministic solution.