Optimal advertising pulsation policies: a dynamic programming approach

This study formulates and solves an advertising pulsation problem for a monopolistic firm using dynamic programming (DP). The firm aims at maximising profit through an optimal allocation of the advertising budget in terms of rectangular pulses over a finite planning horizon. Aggregate sales response to the advertising effort is assumed to be governed by a modified version of the Vidale–Wolfe model in continuous time proposed by Little. Using a numerical example in which a planning horizon of one year is divided into one, two through ten equal time periods, computing routines are developed to solve 150 DP problems. Computational results show among other findings that the performance yielded by the DP policy dominates the uniform advertising policy (constant spending) for a concave advertising response function and the advertising pulsing policy (turning advertising on and off) for a linear or convex response function.     

On the superiority of pulsing under a concave advertising market potential function

The authors study the superiority of advertising pulsing policy (turning advertising on and off in a cyclic fashion) over its uniform (constant spending) counterpart that costs the same under the assumption that sales dynamics follow a modified Vidale–Wolfe aggregate advertising model. The authors show that pulsing can be superior if the product of the concave market potential function and the linear or concave advertising response function is convex in advertising. Similar to previous studies in the literature, the average undiscounted profit over the infinite planning horizon is considered as a performance measure according to which alternative advertising pulsation policies are compared.     

Flexible capacity strategy with multiple market periods under demand uncertainty and investment constraint

We establish a flexible capacity strategy model with multiple market periods under demand uncertainty and investment constraints. In the model, a firm makes its capacity decision under a financial budget constraint at the beginning of the planning horizon which embraces n market periods. In each market period, the firm goes through three decision-making stages: the safety production stage, the additional production stage and the optimal sales stage. We formulate the problem and obtain the optimal capacity, the optimal safety production, the optimal additional production and the optimal sales of each market period under different situations. We find that there are two thresholds for the unit capacity cost. When the capacity cost is very low, the optimal capacity is determined by its financial budget; when the capacity cost is very high, the firm keeps its optimal capacity at its safety production level; and when the cost is in between of the two thresholds, the optimal capacity is determined by the capacity cost, the number of market periods and the unit cost of additional production. Further, we explore the endogenous safety production level. We verify the conditions under which the firm has different optimal safety production levels. Finally, we prove that the firm can benefit from the investment only when the designed planning horizon is longer than a threshold. Moreover, we also derive the formulae for the above three thresholds.     

The impact of product recall on advertising decisions and firm profit while envisioning crisis or being hazard myopic

The occurrence of a product recall can have a disastrous effect on the firm responsible for the recall. Any major recall by a firm can negatively affect the goodwill of the firm. Consequently, the firm incurs a substantial indirect cost due to decline in sales and loss in profit. Moreover, a competitor’s opportunistic reaction can intensify the recalling firm’s damages. Strategic use of advertising recovers lost goodwill and mitigates the damages made by a product recall. In this paper, using a goodwill based model under a differential game framework, we analyze the equilibrium strategies of two competing manufacturers when either one firm or both can issue a product recall at a random time, and investigate (i) the firms’ equilibrium advertising strategies (ii) analyze the impact of the recall on a firm’s profit (iii) introduce and investigate the effect of “hazard myopia” (a firm’s inability to foresee the crisis likelihood) on a firm’s advertising decisions and profit. Our study finds that the equilibrium advertising strategies of competing firms depend on the impact and likelihood of the recall. Notably, we find that when both the firms are focal firms without the prior knowledge of who will recall first in a planning horizon, adjusting optimal advertising at an appropriate time is essential. Surprisingly, a product-recall with a minor impact can increase the focal firm’s long-term expected profit. On the other hand, hazard myopia can be profitable if the long-term effect of the recall is small. Our findings suggest that advertising levels of firms should differ in pre-recall and post-recall regimes depending on the impact and likelihood of the recall.     

Optimal Control of a Logarithmic Advertising Model

A logarithmic advertising model is posed and solved for an optimal dynamic advertising policy for both finite and infinite horizon cases by using optimal control theory. In the case of the infinite horizon, the optimal long-run stationary equilibrium rate of advertising is obtained. It is shown that the optimal advertising policy is independent of the initial level of sales for the logarithmic model.     

Multistage multiproduct advertising budgeting

We propose and analyze an effective model for the Multistage Multiproduct Advertising Budgeting problem. This model optimizes the advertising investment for several products, by considering cross elasticities, different sales drivers and the whole planning horizon. We derive a simple procedure to compute the optimal advertising budget and its optimal allocation. The model was tested to plan a realistic advertising campaign. We observed that the multistage approach may significantly increase the advertising profit, compared to the successive application of the single stage approach.     

Existence and uniqueness of optimal dynamic pricing and advertising controls without concavity

We consider a pricing and advertising dynamic-optimization problem where the goodwill dynamics evolve à la Nerlove–Arrow. The firm maximizes its profit over a finite-planning horizon corresponding to the product’s lifespan, and it turns out that the Hamiltonian is non-concave. We show the existence and uniqueness of an optimal solution under some mild conditions.     

On upper bounds of sequential stochastic production planning problems

In this paper we consider a class of problems that determine production, inventory and work force levels for a firm in order to meet fluctuating demand requirements. A production planning problem arises because of the need to match, at the firm level, supply and demand efficiently. In practice, the two common approaches to counter demand uncertainties are (i) carrying a constant safety stock from period to period, and (ii) planning with a rolling horizon. Under the rolling horizon (or sequential) strategy the planning model is repeatedly solved, usually at the end of every time period, as new information becomes available and is used to update the model parameters. The costs associated with a rolling horizon strategy are hard to compute a priori because the solution of the model in any intermediate time period depends on the actual demands of the previous periods.In this paper we derive two a priori upper bounds on the costs for a class of production planning problems under the rolling horizon strategy. These upper bounds are derived by establishing correspondences between the rolling horizon problems and related deterministic programs. One of the upper bounds is obtained through Lagrangian relaxation of the service level constraint. We propose refinements to the non-Lagrangian bounds and present limited computational results. Extensions of the main results to the multiple item problems are also discussed. The results of this paper are intended to support production managers in estimating the production costs and value of demand information under a rolling horizon strategy.     

An Application of Multiple Regression Analysis to the Greek Beer Market

The object of this paper is to present a quantitative analysis of the Greek beer industry, through the use of various single regressions. The first part presents the relationship between advertising and sales through disaggregating advertising outlays, media and firm advertised, while a number of indices referring to advertising effectiveness are also estimated. The second part extends the discussion by presenting an illustrative example with regard to the implied duration interval and its importance in advertising.     

Allocation of advertising budget between multiple channels to support sales in multiple markets

We consider the allocation of a limited marketing budget between multiple channels in order to promote sales at multiple markets. The channels differ in their type, level of targetability (or reach), and costliness. We incorporate the “threshold effect” from each market in our resource allocation which requires some positive advertising investment in each market before much sales impact can be observed from it. The increased number of channels in recent years with the advent of digital advertising, along with the added complexity created by the threshold effect, necessitates the development of new allocation approaches. In this paper, we formulate the firm’s resource allocation decision as a nonlinear and nonseparable knapsack problem. We develop a branch and cut solution method which is enhanced by a heuristic approach. A set of numerical experiments illustrate the performance of our methods and evaluate the usefulness of two rule-of-thumb strategies commonly used in practice.     

Optimal policy for a dynamic,non-stationary,stochastic inventory problem with capacity commitment

This paper studies a single-product, dynamic, non-stationary, stochastic inventory problem with capacity commitment, in which a buyer purchases a fixed capacity from a supplier at the beginning of a planning horizon and the buyer’s total cumulative order quantity over the planning horizon is constrained with the capacity. The objective of the buyer is to choose the capacity at the beginning of the planning horizon and the order quantity in each period to minimize the expected total cost over the planning horizon. We characterize the structure of the minimum sum of the expected ordering, storage and shortage costs in a period and thereafter and the optimal ordering policy for a given capacity. Based on the structure, we identify conditions under which a myopic ordering policy is optimal and derive an equation for the optimal capacity commitment. We then use the optimal capacity and the myopic ordering policy to evaluate the effect of the various parameters on the minimum expected total cost over the planning horizon.     

Advertising Strategies in a Differential Game with Negative Competitor’s Interference

We consider a duopolistic industry where the current sales of each firm is proportional to its goodwill stock. The evolution of the latter depends positively on own advertising effort and negatively on competitor’s advertising. A standard assumption in the literature in differential games of advertising is that the players remain active throughout the whole (infinite) duration of the game. We relax this assumption and characterize the circumstances under which a firm finds it optimal to remain or exit the industry. Among other things, it is shown that, if both players are “strong”, then the unique Nash equilibrium is the same that one would obtain in the absence of interference from competitor’s advertising. Research supported by Ministry of University and Research of Italy, University of Padua and NSERC, Canada.     

The Application of Preprocessing and Cutting Plane Techniques for a Class of Production Planning Problems

This paper investigates properties of integer programming models for a class of production planning problems. The models are developed within a decision support system to advise a sales team of the products on which to focus their efforts in gaining new orders in the short term. The products generally require processing on several manufacturing cells and involve precedence relationships. The cells are already (partially) committed with products for stock and to satisfy existing orders and therefore only the residual capacities of each cell in each time period of the planning horizon are considered. The determination of production recommendations to the sales team that make use of residual capacities is a nontrivial optimization problem. Solving such models is computationally demanding and techniques for speeding up solution times are highly desirable. An integer programming model is developed and various preprocessing techniques are investigated and evaluated. In addition, a number of cutting plane approaches have been applied. The performance of these approaches which are both general and application specific is examined.     

On optimal bidding and inventory control in sequential procurement auctions: the multi period case

We consider the problem of a firm that in each cycle of a planning horizon builds inventory of identical items that it acquires by participating in auctions in order to satisfy its own market demand. The firm’s objective is to have a procurement strategy that maximizes the expected present value of the profit for an infinite planning horizon of identical cycles. We formulate this problem as a Markov decision process. We establish monotonicity properties of the value function and of the optimal bidding rule.     

A joint dynamic pricing and advertising model of perishable products

Advertising and dynamic pricing play key roles in maximizing profit of a firm. In this paper a joint dynamic pricing and advertising problem for perishable products is investigated, where the time-varying demand rate is decreasing in sales price and increasing in goodwill. A dynamic optimization model is proposed to maximize total profit by setting a joint pricing and advertising policy under the constraint of a limited advertising capacity. By solving the dynamic optimization problem on the basis of Pontryagin’s maximum principle, the analytical solutions of the optimal joint dynamic pricing and advertising policy are obtained. Additionally, to highlight the advantage of the joint dynamic strategy, the case of the optimal advertising with static pricing policy is considered. Numerical examples are presented to illustrate the validness of the theoretical results, and some managerial implications for the pricing and advertising of the perishable products are provided.     

服务水平约束下动态定价与库存管理

本文研究服务水平约束下的动态定价与库存管理问题。企业在有限期内销售某种产品,产品的需求为随机需求,且期望需求依赖于产品价格。在每一期期初,企业需要在满足服务水平约束的条件下同时决定订货量和产品价格。本文首先构建了动态定价和订购联合决策的随机动态规划模型,并证明了最优解的存在性。进一步,通过对最优解的结构进行刻画,将原问题的求解转化为若干子问题的求解,降低了问题求解的难度。通过对最优解的分析发现,当期初库存增大时,产品最优价格降低。通过分析目标服务水平对利润的影响,证明了服务水平与利润之间存在权衡,实现高的服务水平需要承受利润损失。数值模拟表明,相对于传统的静态定价策略,采用动态定价策略可以降低追求服务水平所带来的利润损失,验证了动态定价策略的有效性。     

Optimal pricing and inventory policies with reliable and random-yield suppliers: characterization and comparison

In this paper, we study the joint pricing and inventory replenishment problem for a periodic-review inventory system with random demand and dual suppliers, one of the suppliers is reliable but more expensive, the other supplier is less expensive but is unreliable with random yield. We characterize the firm’s optimal policies that simultaneously determine the optimal ordering and pricing decisions in each period over a finite planning horizon, and investigate the impacts of supply source diversification and supplier reliability on the firm and on its customers. We show that having source diversification or higher reliability of suppliers not only increases the firm’s expected profit, but also results in a lower optimal selling price, thus they benefit both the firm and its customers.     

Assessing the effect of advertising expenditures upon sales: A Bayesian structural time series model

We propose a robust implementation of the Nerlove‐Arrow model using a Bayesian structural time series model to explain the relationship between advertising expenditures of a countrywide fast‐food franchise network with its weekly sales. Due to the flexibility and modularity of the model, it is well suited to generalization to other markets or situations. Its Bayesian nature facilitates incorporating a priori information reflecting the manager's views, which can be updated with relevant data. This aspect of the model will be used to support the decision of the manager on the budget scheduling of the advertising firm across time and channels.     

Performance evaluation of distribution strategies for the inventory routing problem

Direct shipping strategy is an easy-to-implement distribution strategy frequently used in industrial distribution systems. In this paper, an analytic method is developed for performance evaluation of the strategy for the infinite horizon inventory routing problem with delivery frequency constraint. With the method, the effectiveness of direct shipping strategy can be represented as a function of some system parameters. We demonstrate that the effectiveness of direct shipping is at least the square root of the smallest utilization ratio of vehicle capacity. This implies that the effectiveness of the strategy can reach 100% (respectively, 94.86%) whenever the demand rate of each retailer is 100% (respectively, 90%) of the vehicle capacity multiplied by the upper bound of the delivery frequency. This insight can help a firm answer questions such as: under what conditions direct shipping strategy is effective and why, and how effective the strategy is under a specific condition? In case direct shipping strategy is proven ineffective, a more general Fixed Partition Policy (FPP) that combines direct shipping strategy and multiple-stop shipping strategy must be used. An analytic method is also developed for performance evaluation of general FPPs. We demonstrate that the effectiveness of an FPP depends on the total demand rate of the retailers in each partition (each retailer set) and their closeness level. This insight provides a useful guideline to the design of effective FPPs. The analytic methods make the performance improvement of a distribution system possible through adjusting its system parameters.     

Multiple level production planning in rolling horizon assembly environments

In this paper, nine multiple level planning heuristics are evaluated to characterize how rolling horizon results relate to fixed horizon results in a deterministic demand environment. The weighted order cycle (WOC) is introduced as a single expression of cost structure within a multiple item bill of materials. When planning horizons are restated in terms of WOC (versus time buckets), it becomes apparent that the cost performance of the majority of the heuristics is essentially the same for fixed horizon and rolling horizon conditions when the planning horizon is at least two WOC in length. The horizon sensitive logic of the best two heuristics in cost performance also exhibited less nervousness then several horizon myopic rules, a counter intuitive result. An established multiple level cost modification technique was found to reduce the nervousness of single item rules, in addition to its original goal of schedule cost reduction. To gauge cost performance, Lagrangian relaxation of a binary formulation of the problem was used to find bounds within an average of 1% of the optimal solution cost of each simulation.