A greedy algorithm for the two-level nested logit model

We consider the assortment optimization problem under the classical two-level nested logit model. We establish a necessary and sufficient condition for the optimal assortment and develop a simple and fast greedy algorithm that iteratively removes at most one product from each nest to compute an optimal solution.     

Price optimization under the extended nested logit model

We examine multi-product price optimization of the extended nested logit (ENL) model proposed by Kovach and Tserenjigmid, including the nested logit, increasing linear nested stochastic, and two-stage nested attraction models as special cases. The well-known constant adjusted markup and adjusted nest-level markup properties are extended to ENL. In addition, we present sufficient conditions under which the objective function is unimodal. We also provide the upper and lower bounds of the optimal adjusted nest-level markup when preconditions are not met.     

Approximations of choice probabilities in mixed logit models

This paper is concerned with the approximate computation of choice probabilities in mixed logit models. The relevant approximations are based on the Taylor expansion of the classical logit function and on the high order moments of the random coefficients. The approximate choice probabilities and their derivatives are used in conjunction with log likelihood maximization for parameter estimation. The resulting method avoids the assumption of an apriori distribution for the random tastes. Moreover experiments with simulation data show that it compares well with the simulation based methods in terms of computational cost.     

The nested consideration model: Investigating dynamic store consideration sets and store competition

The nested logit model has been widely used to study nested choice. A typical example of such nested choice is store patronage and brand choice. An important limitation of the nested logit model is that it assumes that all alternatives at both levels of the nest are available in the choice set of the consumer. While there is a wide literature on the incorporation of restricted choice sets into the logit model, the logical extension of this analysis to nested restricted choice sets has not been pursued in the literature. In this study we develop a nested consideration model that integrates store choice and brand choice incorporating the formation of dynamic restricted choice sets of both stores and brands. We term the model the nested consideration model and derive the related probabilities in a manner analogous to the well-known nested logit model. In an empirical illustration, we find that the nested consideration model shows better prediction than nested logit models (with the same explanatory variables). We find that it is important to account for dynamic store consideration sets rather than static sets or store loyalty measures. We also conduct simulations to demonstrate the importance of the nested consideration set model for correct pricing and store location decisions of business managers.     

A comparison of linear reformulations for multinomial logit choice probabilities in facility location models

In the last decade several papers appeared on facility location problems that incorporate customer demand by the multinomial logit model. Three linear reformulations of the original non-linear model have been proposed so far. In this paper, we discuss these models in terms of solvability. We present empirical findings based on synthetic data.     

A comparison of generalized multinomial logit and latent class approaches to studying consumer heterogeneity with some extensions of the generalized multinomial logit model

We calibrate and contrast the recent generalized multinomial logit model and the widely used latent class logit model approaches for studying heterogeneity in consumer purchases. We estimate the parameters of the models on panel data of household ketchup purchases, and find that the generalized multinomial logit model outperforms the best‐fitting latent class logit model in terms of the Bayesian information criterion. We compare the posterior estimates of coefficients for individual customers based on the two different models and discuss how the differences could affect marketing strategies (such as pricing), which could be affected by applying each of the models. We also describe extensions to the scale heterogeneity model that includes the effects of state dependence and purchase history. Copyright © 2011 John Wiley & Sons, Ltd.     

Evaluation of choice set generation algorithms for route choice models

This paper discusses choice set generation and route choice model estimation for large-scale urban networks. Evaluating the effectiveness of Advanced Traveler Information Systems (ATIS) requires accurate models of how drivers choose routes based on their awareness of the roadway network and their perceptions of travel time. Many of the route choice models presented in the literature pay little attention to empirical estimation and validation procedures. In this paper, a route choice data set collected in Boston is described and the ability of several different route generation algorithms to produce paths similar to those observed in the survey is analyzed. The paper also presents estimation results of some route choice models recently developed using the data set collected.     

Understanding and interpreting generalized ordered logit models

When outcome variables are ordinal rather than continuous, the ordered logit model, aka the proportional odds model (ologit/po), is a popular analytical method. However, generalized ordered logit/partial proportional odds models (gologit/ppo) are often a superior alternative. Gologit/ppo models can be less restrictive than proportional odds models and more parsimonious than methods that ignore the ordering of categories altogether. However, the use of gologit/ppo models has itself been problematic or at least sub-optimal. Researchers typically note that such models fit better but fail to explain why the ordered logit model was inadequate or the substantive insights gained by using the gologit alternative. This paper uses both hypothetical examples and data from the 2012 European Social Survey to address these shortcomings.     

Assortment optimization under the multinomial logit model with product synergies

In synergistic assortment optimization, a product’s attractiveness changes as a function of which other products are offered. We represent synergy structure graphically. Vertices denote products. An edge denotes synergy between two products, which increases their attractiveness when both are offered. Finding an assortment to maximize retailer’s expected profit is NP-hard in general. We present efficient algorithms when the graph is a path, a tree, or has low treewidth. We give a linear program to recover the optimal assortment for paths.     

Parallel machine scheduling with nested processing set restrictions

We consider the problem of scheduling a set of n independent jobs on m parallel machines, where each job can only be scheduled on a subset of machines called its processing set. The machines are linearly ordered, and the processing set of job j   is given by two machine indexes _aj$a_j$ and _bj$b_j$; i.e., job j   can only be scheduled on machines _aj,_aj+1,…,_bj$a_j\text{,}{a}_j+1\text{,}\dots \text{,}{b}_j$. Two distinct processing sets are either nested or disjoint. Preemption is not allowed. Our goal is to minimize the makespan. It is known that the problem is strongly NP-hard and that there is a list-type algorithm with a worst-case bound of 2-1/m$2-1/m$. In this paper we give an improved algorithm with a worst-case bound of 7/4. For two and three machines, the algorithm gives a better worst-case bound of 5/4 and 3/2, respectively.     

An adaptive Monte Carlo algorithm for computing mixed logit estimators

Researchers and analysts are increasingly using mixed logit models for estimating responses to forecast demand and to determine the factors that affect individual choices. However the numerical cost associated to their evaluation can be prohibitive, the inherent probability choices being represented by multidimensional integrals. This cost remains high even if Monte Carlo or quasi-Monte Carlo techniques are used to estimate those integrals. This paper describes a new algorithm that uses Monte Carlo approximations in the context of modern trust-region techniques, but also exploits accuracy and bias estimators to considerably increase its computational efficiency. Numerical experiments underline the importance of the choice of an appropriate optimisation technique and indicate that the proposed algorithm allows substantial gains in time while delivering more information to the practitioner. Fabian Bastin: Research Fellow of the National Fund for Scientific Research (FNRS)     

Optimal worst-case pricing for a logit demand model with network effects

We consider optimal pricing problems for a product that experiences network effects. Given a price, the sales quantity of the product arises as an equilibrium, which may not be unique. In contrast to previous studies that take a best-case view when there are multiple equilibrium sales quantities, we maximize the seller’s revenue assuming that the worst-case equilibrium quantity will arise in response to a chosen price. We compare the best- and worst-case solutions, and provide asymptotic analysis of revenues.     

Estimation of failure probability using semi-definite logit model

We will propose a new and practical method for estimating the failure probability of a large number of small to medium scale companies using their balance sheet data. We will use the maximum likelihood method to estimate the best parameters of the logit function, where the failure intensity function in its exponent is represented as a convex quadratic function instead of a commonly used linear function. The reasons for using this type of function are : (i) it can better represent the observed nonlinear dependence of failure probability on financial attributes, (ii) the resulting likelihood function can be maximized using a cutting plane algorithm developed for nonlinear semi-definite programming problems.We will show that we can achieve better prediction performance than the standard logit model, using thousands of sample companies.Revised: December 2002,     

A two step algorithm for solving a large scale semi-definite logit model

This paper proposes a two step algorithm for solving a large scale semi-definite logit model, which is appreciated as a powerful model in failure discriminant analysis. This problem has been successfully solved by a cutting plane (outer approximation) algorithm. However, it requires much more computation time than the corresponding linear logit model. A two step algorithm to be proposed in this paper is intended to reduce the amount of computation time by eliminating a certain portion of the data based on the information obtained by solving an associated linear logit model. It will be shown that this algorithm can generate a solution with almost the same quality as the solution obtained by solving the original large scale semi-definite model within a fraction of computation time.     

Robust assortment optimization using worst-case CVaR under the multinomial logit model

We consider robust assortment optimization problems with partial distributional information of parameters in the multinomial logit choice model. The objective is to find an assortment that maximizes a revenue target using a distributionally robust chance constraint, which can be approximated by the worst-case Conditional Value-at-Risk. We show that our problems are equivalent to robust assortment optimization problems over special uncertainty sets of parameters, implying the optimality of revenue-ordered assortments under certain conditions.     

GeneralizedF-tests for unbalanced nested designs under heteroscedasticity

Two-factor fixed-effect unbalanced nested design model without the assumption of equal error variance is considered. Using the generalized definition ofp-values, exact tests under heteroscedasticity are derived for testing main effects of both factors. These generalizedF-tests can be utilized in significance testing or in fixed level testing under the Neyman-Pearson theory. Two examples are given to illustrate the proposed test and to demonstrate its advantages over the classicalF-test. Extensions of the procedure for three-factor nested designs are briefly discussed.     

Analysing head coach dismissals in the German “Bundesliga” with a mixed logit approach

This article analyzes the dismissal of head coaches in the top tier of German professional football, the “Bundesliga”. A random parameter logit model is used to analyse the characteristics that are associated with the probability of a coach to be dismissed, taking into account uncontrolled heterogeneity of the data. Policy implications are presented.     

A method for portfolio choice

This paper shows how one can use the theory of hidden Markov models for portfolio optimization. We illustrate our method by a ball and urn experiment. An application to historical data is examined. Copyright © 2003 John Wiley & Sons, Ltd.