Transfer pricing in a dynamic marketing-operations interface

A transfer price mechanism is proposed to coordinate the strategies of the marketing and operations functional areas operating in a dynamic interface environment in a decentralized firm. Marketing and operations are strategic decision-makers in a differential game, in which marketing has price and advertising and operations has production as control variables, and advertising goodwill and production backlog are state variables. A constant transfer price is entered into the objective functionals for marketing and operations, and subgame perfect feedback strategies are derived for price, advertising, and production as functions of the state variables. The feedback strategies allow a solution for the dynamic system involving goodwill and backlog, and the total payoff to the firm, the sum of the payoffs to marketing and operations, is determined as a function of the transfer price. Finally, for certain parameter conditions an interior maximum of the payoff function is achieved, and the optimal transfer price is identified.     

Delayed effects of cooperative advertising in goodwill dynamics

This paper proposes a tool to control cooperative advertising which increases the goodwill of companies operating in a competitive market. We introduce the lag between advertising exposure and customer reaction in the goodwill dynamics evolved à la Nerlove–Arrow. As a result, we obtain a cooperative differential game with immediate and delayed effects of control variables for which we investigate the optimal solution. We examine the role the pre-coalition programmes and the length of delayed response in generating goodwill.     

考虑延迟现象的质量改进投入和营销努力动态协同策略

利用时滞微分方程刻画质量改进投入对品牌商誉提升的延迟现象,分别构建了制造商和零售商采取非合作博弈、合作博弈以及成本分担的部分合作博弈(制造商参与营销的单向部分合作博弈、零售商参与生产的单向部分合作博弈、制造商参与营销及零售商参与生产的双向部分合作博弈)五种决策模式下的微分博弈模型。借助哈密尔顿极大值原理,求解得到五种情形下的制造商最优质量改进投入策略和零售商的最优营销努力策略以及供应链利润。对比五种博弈模式下的结果发现:1)延时现象会降低制造商进行质量改进投入的积极性,但对零售商营销努力无影响;品牌商誉在延迟现象影响下出现先衰减后提升的演进规律;2)合作博弈对于供应链绩效总是最优的,三种成本分担的部分合作博弈契约虽不能实现供应链的完全协调,但可以对非合作博弈情形进行帕累托改进;3)对比两种单向部分合作博弈,在提高供应链利润方面,制造商参与营销的成本分担契约优于零售商参与生产的成本分担契约;4)三种成本分担契约中,双向合作的部分合作博弈是供应链的最优选择,但随着延迟时间增大,其帕累托改进效果将不再明显。     

Optimal dynamic advertising policies in the presence of continuously distributed time lags

The Nerlove-Arrow model of optimal dynamic advertising policies is generalized by incorporating a continuously distributed lag between advertising expenditures and increases in the stock of goodwill. This leads to a control problem where the equation of motion is given by an integro-differential equation. The transitory and steady-state properties of the optimal policies are examined, both for a general lag function and for a gamma distributed lag. The dependence of the steady-state solution on the parameters of the gamma distribution is also investigated. An example is given using specific demand and cost functions.     

Advertising a new product in a segmented market

We bring some market segmentation concepts into the statement of the “new product introduction” problem with Nerlove-Arrow’s linear goodwill dynamics. In fact, only a few papers on dynamic quantitative advertising models deal with market segmentation, although this is a fundamental topic of marketing theory and practice. In this way we obtain some new deterministic optimal control problems solutions and show how such marketing concepts as “targeting” and “segmenting” may find a mathematical representation. We consider two kinds of situations. In the first one, we assume that the advertising process can reach selectively each target group. In the second one, we assume that one advertising channel is available and that it has an effectiveness segment-spectrum, which is distributed over a non-trivial set of segments. We obtain the explicit optimal solutions of the relevant problems.     

Advertising and production of a seasonal good for a heterogeneous market

We bring some concepts from market segmentation, which is a fundamental topic of marketing theory and practice, into the statement of an advertising and production problem for a seasonal product with Nerlove–Arrow’s linear goodwill dynamics. We consider two kinds of situations. In the first one, the advertising process can reach selectively each segment. In the second one, one advertising medium is available which has a known effectiveness spectrum for a non-trivial set of segments. In both cases we solve, using the Pontryagin’s Maximum Principle conditions, the optimal control problems in which goodwill productivity of advertising is concave and good production cost is convex. Two special cases are discussed in detail.     

Optimal advertising strategies with age-structured goodwill

The problem of a firm willing to optimally promote and sell a single product on the market is here undertaken. The awareness of such product is modeled by means of a Nerlove–Arrow goodwill as a state variable, differentiated jointly by means of time and of age of the segments in which the consumers are clustered. The problem falls into the class of infinite horizon optimal control problems of PDEs with age structure that have been studied in various papers either in cases when explicit solutions can be found or using Maximum Principle techniques. Here, assuming an infinite time horizon, we use some dynamic programming techniques in infinite dimension to characterize both the optimal advertising effort and the optimal goodwill path in the long run. An interesting feature of the optimal advertising effort is an anticipation effect with respect to the segments considered in the target market, due to time evolution of the segmentation. We analyze this effect in two different scenarios: in the first, the decision-maker can choose the advertising flow directed to different age segments at different times, while in the second she/he can only decide the activation level of an advertising medium with a given age-spectrum.     

On Controlled Linear Diffusions with Delay in a Model of Optimal Advertising under Uncertainty with Memory Effects

We consider a class of dynamic advertising problems under uncertainty in the presence of carryover and distributed forgetting effects, generalizing the classical model of Nerlove and Arrow (Economica 29:129–142, 1962). In particular, we allow the dynamics of the product goodwill to depend on its past values, as well as previous advertising levels. Building on previous work (Gozzi and Marinelli in Lect. Notes Pure Appl. Math., vol. 245, pp. 133–148, 2006), the optimal advertising model is formulated as an infinite-dimensional stochastic control problem. We obtain (partial) regularity as well as approximation results for the corresponding value function. Under specific structural assumptions, we study the effects of delays on the value function and optimal strategy. In the absence of carryover effects, since the value function and the optimal advertising policy can be characterized in terms of the solution of the associated HJB equation, we obtain sharper characterizations of the optimal policy.     

Optimal Control of the Fokker–Planck Equation with Space-Dependent Controls

This paper is devoted to the analysis of a bilinear optimal control problem subject to the Fokker–Planck equation. The control function depends on time and space and acts as a coefficient of the advection term. For this reason, suitable integrability properties of the control function are required to ensure well posedness of the state equation. Under these low regularity assumptions and for a general class of objective functionals, we prove the existence of optimal controls. Moreover, for common quadratic cost functionals of tracking and terminal type, we derive the system of first-order necessary optimality conditions.     

Error analysis of discontinuous Galerkin finite element method for optimal control problem governed by the transport equation

This article discusses a priori and a posteriori error estimates of discontinuous Galerkin finite element method for optimal control problem governed by the transport equation. We use variational discretization concept to discretize the control variable and discontinuous piecewise linear finite elements to approximate the state and costate variable. Based on the error estimates of discontinuous Galerkin finite element method for the transport equation, we get a priori and a posteriori error estimates for the transport equation optimal control problem. Finally, two numerical experiments are carried out to confirm the theoretical analysis.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1493–1512, 2017     

Cooperative Advertising in a Marketing Channel

This paper examines dynamic advertising and promotion strategies in a marketing channel where the retailer promotes the manufacturer product and the manufacturer spends on advertising to build a stock of goodwill. We assume that sales depend on goodwill and promotion activities and that there are decreasing marginal returns to goodwill. Two scenarios are studied. First, the manufacturer and retailer determine noncooperatively their respective strategies. Second, the game is played à la Stackelberg with the manufacturer as the leader who supports partially the cost of the promotion activities of the retailer. In both cases, stationary Markovian strategies are characterized. These scenarios are examined also in the absence of decreasing marginal effect of goodwill on sales. The results show that, whether or not the goodwill stock has a decreasing marginal effect on sales, the cooperative advertising program is a coordinating mechanism in the marketing channel, i.e., both players receive higher payoffs.     

Impulsive optimal control model for the trajectory of horizontal wells

This paper presents an impulsive optimal control model for solving the optimal designing problem of the trajectory of horizontal wells. We take fully into account the effect of unknown disturbances in drilling. The optimal control problem can be converted into a nonlinear parametric optimization by integrating the state equation. We discuss here that the locally optimal solution depends in a continuous way on the parameters (disturbances) and utilize this property to propose a revised Hooke–Jeeves algorithm. The uniform design technique is incorporated into the revised Hooke–Jeeves algorithm to handle the multimodal objective function. The numerical simulation is in accordance with theoretical results. The numerical results illustrate the validity of the model and efficiency of the algorithm.     

Optimal control and synchronization of Lotka–Volterra model

In this paper we discuss the problem of optimal control for the steady state of Lotka–Volterra model. The conditions of the asymptotic stability of the steady state of this model are used to obtain the optimal control functions. In such study, the optimal Lyapunov function is used. The general solution of the equations of the perturbed state is obtained as a function of time. In addition, the optimal control is also applied to achieve the state synchronization of two identical Lotka–Volterra systems. Graphical and numerical simulation studies of the obtained results are presented.     

Necessary Optimality Conditions for Some Control Problems of Elliptic Equations with Venttsel Boundary Conditions

In this paper we derive a necessary optimality condition for a local optimal solution of some control problems. These optimal control problems are governed by a semi-linear Vettsel boundary value problem of a linear elliptic equation. The control is applied to the state equation via the boundary and a functional of the control together with the solution of the state equation under such a control will be minimized. A constraint on the solution of the state equation is also considered.     

Optimal control of nonlinear one-dimensional periodic wave equation with x-dependent coefficients

This paper is concerned with an optimal control problem governed by the nonlinear one dimensional periodic wave equation with x-dependent coefficients. The control of the system is realized via the outer function of the state. Such a model arises from the propagation of seismic waves in a nonisotropic medium. By investigating some important properties of the linear operator associated with the state equation, we obtain the existence and regularity of the weak solution to the state equation. Furthermore, the existence of the optimal control is proved by means of the Arzelà-Ascoli lemma and Sobolev compact imbedding theorem.     

A data-driven neural network approach to optimal asset allocation for target based defined contribution pension plans

A data-driven Neural Network (NN) optimization framework is proposed to determine optimal asset allocation during the accumulation phase of a defined contribution pension scheme. In contrast to parametric model based solutions computed by a partial differential equation approach, the proposed computational framework can scale to high dimensional multi-asset problems. More importantly, the proposed approach can determine the optimal NN control directly from market returns, without assuming a particular parametric model for the return process. We validate the proposed NN learning solution by comparing the NN control to the optimal control determined by solution of the Hamilton–Jacobi–Bellman (HJB) equation. The HJB equation solution is based on a double exponential jump model calibrated to the historical market data. The NN control achieves nearly optimal performance. An alternative data-driven approach (without the need of a parametric model) is based on using the historic bootstrap resampling data sets. Robustness is checked by training with a blocksize different from the test data. In both two and three asset cases, we compare performance of the NN controls directly learned from the market return sample paths and demonstrate that they always significantly outperform constant proportion strategies.     

Taylor expansions of the value function associated with a bilinear optimal control problem

A general bilinear optimal control problem subject to an infinite-dimensional state equation is considered. Polynomial approximations of the associated value function are derived around the steady state by repeated formal differentiation of the Hamilton–Jacobi–Bellman equation. The terms of the approximations are described by multilinear forms, which can be obtained as solutions to generalized Lyapunov equations with recursively defined right-hand sides. They form the basis for defining a suboptimal feedback law. The approximation properties of this feedback law are investigated. An application to the optimal control of a Fokker–Planck equation is also provided.     

Optimal fishery with coastal catch

In many spatial resource models, it is assumed that an agent is able to harvest the resource over the complete spatial domain. However, agents frequently only have access to a resource at particular locations at which a moving biomass, such as fish or game, may be caught or hunted. Here, we analyze an infinite time‐horizon optimal control problem with boundary harvesting and (systems of) parabolic partial differential equations as state dynamics. We formally derive the associated canonical system, consisting of a forward–backward diffusion system with boundary controls, and numerically compute the canonical steady states and the optimal time‐dependent paths, and their dependence on parameters. We start with some one‐species fishing models, and then extend the analysis to a predator–prey model of the Lotka–Volterra type. The models are rather generic, and our methods are quite general, and thus should be applicable to large classes of structurally similar bioeconomic problems with boundary controls. Recommedations for Resource Managers

• Just like ordinary differential equation‐constrained (optimal) control problems and distributed partial differential equation (PDE) constrained control problems, boundary control problems with PDE state dynamics may be formally treated by the Pontryagin's maximum principle or canonical system formalism (state and adjoint PDEs).
• These problems may have multiple (locally) optimal solutions; a first overview of suitable choices can be obtained by identifying canonical steady states.
• The computation of canonical paths toward some optimal steady state yields temporal information about the optimal harvesting, possibly including waiting time behavior for the stock to recover from a low‐stock initial state, and nonmonotonic (in time) harvesting efforts.
• Multispecies fishery models may lead to asymmetric effects; for instance, it may be optimal to capture a predator species to protect the prey, even for high costs and low market values of the predators.

     

Mitigating the curse of dimensionality: sparse grid characteristics method for optimal feedback control and HJB equations

We address finding the semi-global solutions to optimal feedback control and the Hamilton–Jacobi–Bellman (HJB) equation. Using the solution of an HJB equation, a feedback optimal control law can be implemented in real-time with minimum computational load. However, except for systems with two or three state variables, using traditional techniques for numerically finding a semi-global solution to an HJB equation for general nonlinear systems is infeasible due to the curse of dimensionality. Here we present a new computational method for finding feedback optimal control and solving HJB equations which is able to mitigate the curse of dimensionality. We do not discretize the HJB equation directly, instead we introduce a sparse grid in the state space and use the Pontryagin’s maximum principle to derive a set of necessary conditions in the form of a boundary value problem, also known as the characteristic equations, for each grid point. Using this approach, the method is spatially causality free, which enjoys the advantage of perfect parallelism on a sparse grid. Compared with dense grids, a sparse grid has a significantly reduced size which is feasible for systems with relatively high dimensions, such as the 6-D system shown in the examples. Once the solution obtained at each grid point, high-order accurate polynomial interpolation is used to approximate the feedback control at arbitrary points. We prove an upper bound for the approximation error and approximate it numerically. This sparse grid characteristics method is demonstrated with three examples of rigid body attitude control using momentum wheels.     

Optimal control problem of a generalized Ginzburg–Landau model equation in population problems

In this paper, we consider the problem for distributed optimal control of the generalized Ginzburg–Landu model equation in population. The optimal control under boundary condition is given, the existence of optimal solution to the equation is proved, and the optimality system is established. Copyright © 2013 John Wiley & Sons, Ltd.