Assessing the predictive effectiveness of the variety seeking and reinforcement models

Prediction of customer choice behaviour has been a big challenge for marketing researchers. They have adopted various models to represent customers purchase patterns. Some researchers considered simple zero–order models. Others proposed higher–order models to represent explicitly customers tendency to seek [variety] or [reinforcement] as they make repetitive choices. Nevertheless, the question [Which model has the highest probability of representing some future data?] still prevails. The objective of this paper is to address this question. We assess the predictive effectiveness of the well–known customer choice models. In particular, we compare the predictive ability of the [dynamic attribute satiation] (DAS) model due to McAlister (Journal of Consumer Research, 91, pp. 141–150, 1982) with that of the well–known stochastic variety seeking and reinforcement behaviour models. We found that the stochastic [beta binomial] model has the best predictive effectiveness on both simulated and real purchase data. Using simulations, we also assessed the effectiveness of the stochastic models in representing various complex choice processes generated by the DAS. The beta binomial model mimicked the DAS processes the best. In this research we also propose, for the first time, a stochastic choice rule for the DAS model.     

A model of consumer choice for national <Emphasis Type="Italic">vs</Emphasis> private label brands

Most consumer choice theory is built around the study of manufacturer brands but in recent years there has been a dramatic rise in the share of private label brands. This empirical study suggests that models of consumer choice need to recognise an asymmetry in competition when both manufacturer and retailer private label brands are available. A nested logit model is introduced which demonstrates asymmetric cross-brand substitutability in this type of market and leads to important implications for strategic brand management.     

Optimal ordering policies in inventory systems with random demand and random deal offerings

In this paper, we determine the optimal order policies for a firm facing random demand and random deal offerings. In a periodic review setting, a firm may first place an order at the regular price. Later in the period, if a price promotion is offered by the supplier (with a certain probability), the firm may decide to place another order. We consider two models in the paper. In the first model, the firm does not share the cost savings (due to the promotion offered by the supplier) with its own customers, i.e. its demand distribution remains fixed. In the second model, the cost savings are shared with the final customers. As a result, the demand distribution shifts to the right. For both the models, in a dynamic finite-horizon problem, the order policy structure is divided into three regions and is as follows. If the initial inventory level for the firm exceeds a certain threshold level, it is optimal not to order anything. If it is in the medium range, it is optimal to wait for the promotion and order only if it is offered. The order quantity when the promotion is offered has an ‘order up to’ policy structure. Finally, if the inventory level is below another threshold, it is optimal to place an order at the regular price, and to place a second order if the promotion is offered. The low initial inventory level makes it risky to just wait for the promotion to be offered. The sum of the order quantities in this case has an ‘order up to’ structure. Finally, we model the supplier's problem as a Stackelberg game and discuss the motivation for the supplier to offer a promotion for the case of uniform demand distribution for the firm. In the first model (when the firm does not share the cost savings with its customers), we show that it is rarely optimal for the supplier to offer a promotion. In the second model, the supplier may offer a promotion depending on the price elasticity of the product.     

The impact of loyalty and promotion effects on retention rate

Researching customer retention rates is made difficult because of the lack of suitable systems with which to measure it. From a first-order Markov brand-switching model, serial equations are derived to estimate the retention rate using market share and loyalty. To explore the impact of loyalty and promotion programme on retention rate, data from five leading brands for three product categories were obtained to fit the developed model. Findings suggest that: (1) when the loyalty effect remains unchanged over a period of time, the greater the effect of a promotion programme, the greater the retention rate; (2) when the promotion effects among brands are similar, the retention rate varies depending on level of loyalty; and (3) when the proportion of loyal customers is high, the proportion of potential switchers affected by promotion programmes is low. In this case, the impact of a promotion programme on the retention rate is not significant. This model successfully demonstrates how the Markov brand-switching model can be employed to estimate a measure of retention rate from consumer panel data as well as to better understand how promotion affects loyalty, switching behaviour, and retention rate.     

On the Effect of Product Variety in Production–Inventory Systems

In this paper, we examine the effect of product variety on inventory costs in a production–inventory system with finite capacity where products are made to stock and share the same manufacturing facility. The facility incurs a setup time whenever it switches from producing one product type to another. The production facility has a finite production rate and stochastic production times. In order to mitigate the effect of setups, products are produced in batches. In contrast to inventory systems with exogenous lead times, we show that inventory costs increase almost linearly in the number of products. More importantly, we show that the rate of increase is sensitive to system parameters including demand and process variability, demand and capacity levels, and setup times. The effect of these parameters can be counterintuitive. For example, we show that the relative increase in cost due to higher product variety is decreasing in demand and process variability. We also show that it is decreasing in expected production time. On the other hand, we find that the relative cost is increasing in expected setup time, setup time variability and aggregate demand rate. Furthermore, we show that the effect of product variety on optimal base stock levels is not monotonic. We use the model to draw several managerial insights regarding the value of variety-reducing strategies such as product consolidation and delayed differentiation.     

An Optimal Congestion and Cost-sharing Pricing Scheme for Multiclass Services

We study in this paper a social welfare optimal congestion-pricing scheme for multiclass queuing services which can be applied to telecommunication networks. Most of the literature has focused on the marginal price. Unfortunately, it does not share the total cost among the different classes. We investigate here an optimal Aumann–Shapley congestion-price which verifies this property. We extend the work on the Aumann–Shapley price for priority services, based on the results on the marginal price: instead of just determining the cost repartition among classes for given rates, we obtain the rates and charges that optimize the social welfare.     

Brand positioning under lexicographic choice rules

This paper discusses a brand positioning model in which two brands of a product are to be positioned in a price-quality space under a new behavioral assumption. This assumption asserts that customers determine the highest-quality product within their reservation price and purchase it, provided its quality does not fall short of a minimum standard. The model also includes producers' costs that are incurred for delivering a certain quality. We first delineate reaction functions for the optimal location of one brand, give a location of its competitor. We then show that Nash equilibria do not exist as long as price and quality are both variable. Finally, we consider a two phase model: in the first phase, the duopolists sequentially choose their quality levels under the assumption that both competitors know that in the second phase, a Nash equilibrium in prices follows. Single-variable mathematical programming formulations are presented to solve the problem. A numerical example is also given to illustrate the working of the model.     

Stochastic modelling of household behaviour before,during and after a consumer price promotion

A stochastic model is developed to study household behaviour with regard to purchase quantity, brand choice and purchase timing before, during and after a price change and a price promotion such as price-offs and price-cuts. The basic assumption of the model is that price promotion and levels of consumer inventory influence a household's purchase-timing and brand-switching decisions. The model incorporates market segments and brand switching on aggregated demand for the brands by the use of multivariate Markov processes. A transient stochastic model is employed to analyse the dynamic process of household behaviour before, during, and after a price promotion. The interpurchase time that is derived from the model does not require any assumptions and is not independently, identically distributed. An empirical analysis using the Information Resources Incorporated cracker data indicated that (1) price promotion does affect household purchase of the brand and (2) households with larger family size tend to purchase promoted items. We conjecture that households with larger family size take advantage of the lower price of the promoted brands while smaller households tend to remain loyal to one brand.     

Positioning of new brands in an experiment

This paper deals with an experimental investigation of positioning new brands. For this purpose, a management game was carried out with students. The brands introduced in the cause of the game were analysed in respect to their positions in a two-dimensional feature space. We try to find out which of the two strategies, niche policy and imitation, is more frequently used in complex decision situations and if there is a difference in profits. Furthermore, we want to find out, whether differences of prices and advertising exist in our experiment depending on the positioning strategy used.     

Using store level scanner data to improve category management decisions: Developing positioning maps

This paper provides evidence of the usefulness of aggregated point-of-sale scanner data to infer the positioning of competing brands, providing valuable information for category management and hence facilitating decision making. Specifically, the authors propose a methodology to study the internal market structure based on market share models with latent heterogeneity when only macro-level time series data (not individual choices) are available. The proposed approach assumes a multidimensional decomposition, latent in the preference structure that is implicit to these types of models. By empirically applying this approach, the authors (1) simultaneously identify both latent dimensions of competing brands and latent segments with different brand preferences, (2) explain the competitive positioning of brands without using disaggregated consumer panel data, and (3) achieve greater predictive performance. The findings offer insights to academics and practitioners interested in improving the practice of category management.     

Uniqueness and zeros of <Emphasis Type="Italic">q</Emphasis>-shift difference polynomials

In this paper, we consider the zero distributions of q-shift difference polynomials of meromorphic functions with zero order, and obtain two theorems that extend the classical Hayman results on the zeros of differential polynomials to q-shift difference polynomials. We also investigate the uniqueness problem of q-shift difference polynomials that share a common value.     

Operationalizing technology improvements in product development decision-making

Achieving competitive advantage and price premiums in many technology-based markets requires the incorporation of current technology in new products. To do so, firms in hyper-competitive environments increasingly plan and design their products concurrent with the independent development and validation of underlying technologies. Simultaneous validation of a core technology has important implications for a company’s product positioning and launch sequence decisions making these traditional marketing decisions relevant to operations managers. Prior research has shown that to minimize cannibalization in the absence of such improvements in technology, a firm should not launch low-end products before high-end products. However, concurrent evolution of technology can make it desirable and even necessary to introduce low-end products before high-end products. This is because in technology-based industries, improvements in technology delay the introduction of a high-end product, and a firm must trade-off the benefit of launching the low-end product earlier (greater discounted profits) against the cost of cannibalization of high-end product sales. High-end product cannibalization can be further reduced by offering the customer an option to upgrade from the low-end to high-end product, with important implications for the firm’s product positioning and introduction sequence decisions. Based on our study in the high technology industry, we model the product positioning and introduction sequence decisions under the simultaneous evolution of technology. Our analysis indicates that it may be optimal in a variety of circumstances for a firm to launch products in an increasing order of performance, even in the absence of network externalities. Besides presenting analytical results for product positioning and profit from different introduction sequences, the paper also makes a contribution to managerial practice by providing insights in the form of a conceptual framework.     

Inventory and shelf-space optimization for fresh produce with expiration date under freshness-and-stock-dependent demand rate

It is well documented that the demand for fresh produce, to a great extent, depends on how fresh it is and an increase in shelf space for displayed stocks may induce more purchase of the produce. However, relatively little attention has been paid to the effect of expiration date despite the fact that produce deteriorates over time and expiration dates are often an important factor in consumers’ purchase decision. In this paper, we propose an economic order quantity model in which we explicitly specify the demand for fresh produce to be a function of its freshness-expiration date and displayed volume. With the demand being freshness-and-stock dependent, it may be profitable to maintain high stock level at the end of the replenishment cycle. Hence, we relax the traditional assumption of zero ending inventory to non-zero ending inventory. Consequently, the objective here is to determine the optimal level of shelf space size, replenishment cycle time, and/or ending inventory level in an effort of maximizing the total annual profit. We found that the total annual profit is strictly pseudo-concave with regard to the three decision variables, which simplifies the search for the global solution to a local optimal. Numerical examples are then presented to highlight the theoretical implications and managerial insights.     

$\boldsymbol{\mathcal{G}-}$Inhomogeneous Markov Systems of High Order

In the present, we introduce and study the G-\mathcal{G-}inhomogeneous Markov system of high order, which is a more general in many respects stochastic process than the known inhomogeneous Markov system. We define the inhomogeneous superficial razor cut mixture transition distribution model extending for the homogeneous case the idea of the mixture transition model. With the introduction of the appropriate vector stochastic process and the establishment of relationships among them, we study the asymptotic behaviour of the G-\mathcal{G-}inhomogeneous Markov system of high order. In the form of two theorems, the asymptotic behaviour of the inherent G-\mathcal{G-}inhomogeneous Markov chain and the expected and relative expected population structure of the G-\mathcal{G-}inhomogeneous Markov system of high order, are provided under assumptions easily met in practice. Finally, we provide an illustration of the present results in a manpower system.     

On Nonexistence of any λ0-invariand positive harmonic function,a counter example to stroock' conjecture

Consider a sub-Markovian semigroup such that λ₀, the border number between recrrence and transience, equals zero. In 1982, D. W. Stroock conjectured that under general hypotheses on the semi-group the corresponding process always admits an invariant measure.

In this paper we present an example of a second order elliptic operator P with a generalized principal eigenvalue λ₀ which equals zero such that the parabolic equation does not admit any positive invariant P—harmonic function and also any invariant measure. This gives a counter example to Stroock's conjecture for diffusion processes. We also present an example of a complete Riemannian manifold M which does not admit any positive invariant harmonic function while λ₀, the bottom of the spectrum of M, is zero. This gives a partial answer to a question of Stroock and Sullivan.     

Asymptotic modelling of conductive thin sheets

We derive and analyse models which reduce conducting sheets of a small thickness ε in two dimensions to an interface and approximate their shielding behaviour by conditions on this interface. For this we consider a model problem with a conductivity scaled reciprocal to the thickness ε, which leads to a nontrivial limit solution for ε → 0. The functions of the expansion are defined hierarchically, i.e. order by order. Our analysis shows that for smooth sheets the models are well defined for any order and have optimal convergence meaning that the H ¹-modelling error for an expansion with N terms is bounded by O(ε ^N+1) in the exterior of the sheet and by O(ε ^N+1/2) in its interior. We explicitly specify the models of order zero, one and two. Numerical experiments for sheets with varying curvature validate the theoretical results.     

Efficient visit frequencies for polling tables: minimization of waiting cost

Polling systems have been used as a central model for the modeling and analysis of many communication systems. Examples include the Token Ring network and a communications switch. The common property of these systems is the need to efficiently share a single resource (server) amongN entities (stations). In spite of the massive research effort in this area, very little work has been devoted to the issue of how toefficiently operate these systems.In the present paper we deal with this problem, namely with how to efficiently allocate the server's attention among theN stations. We consider a framework in which a predetermined fixed visit order (polling table) is used to establish the order by which the server visits the stations, and we address the problem of how to construct an efficient (optimal) polling table. In selecting a polling table the objective is to minimize the mean waiting cost of the system, a weighted sum of the mean delays with arbitrary cost parameters. Since the optimization problem involved is very hard, we use an approximate approach. Using two independent analyses, based on a lower bound and on mean delay approximations, we derive very simple rules for the determination of efficient polling tables. The two rules are very similar and even coincide in most cases. Extensive numerical examination shows that the rules perform well and that in most cases the system operates very close to its optimal operation point.     

A local limit theorem for the probability of ruin

In this paper, we give a result on the local asymptotic behaviour of the probability of ruin in a continuous-time risk model in which the inter-claim times have an Erlang distribution and the individual claim sizes have a distribution that belongs to S(v) with v≥ 0, but where the Lundberg exponent of the underlying risk process does not exist.     

On convergence of solutions to difference equations with additive perturbations

Various types of stabilizing controls lead to a deterministic difference equation with the following property: once the initial value is positive, the solution tends to the unique positive equilibrium. Introducing additive perturbations can change this picture: we give examples of difference equations experiencing additive perturbations which have solutions staying around zero rather than tending to the unique positive equilibrium. When perturbations are stochastic with a bounded support, we give an upper estimate for the probability that the solution can stay around zero. Applying extra conditions on the behaviour of the map function f at zero or on the amplitudes of stochastic perturbations, we prove that the solution tends to the unique positive equilibrium almost surely. In particular, this holds either for all amplitudes when the right derivative of the map f at zero exceeds one or, independently of the behaviour of f at zero, when the amplitudes are not square summable.     

Projective varieties admitting an embedding with Gauss map of rank zero

We introduce an intrinsic property for a projective variety as follows: there exists an embedding into some projective space such that the Gauss map is of rank zero, which we call (GMRZ) for short. It turns out that (GMRZ) imposes strong restrictions on rational curves on projective varieties: In fact, using (GMRZ), we show that, contrary to the characteristic zero case, the existence of free rational curves does not imply that of minimal free rational curves in positive characteristic case. We also focus attention on Segre varieties, Grassmann varieties, and hypersurfaces of low degree. In particular, we give a characterisation of Fermat cubic hypersurfaces in terms of (GMRZ), and show that a general hypersurface of low degree does not satisfy (GMRZ).