Decision-making models in marketing: dynamic optimization and differential games

ISBN 978-5-7525-1858-

Trade Discount Policies in the Differential Games Framework

We consider a vertical control distribution channel in which a manufacturer sells a single kind of good to a retailer. We assume that a wholesale price discount increases the retailer's sale motivation thus improving sales. The optimal control of manufacturer's profit via trade discounts is embedded in a differential game framework; in the special case of constant controls we compare the Stackelberg equilibria obtained considering manufacturer and retailer respectively as leaders of the game with Nash equilibrium points

Minimization of communication expenditure for seasonal products

MathSciNet: MR 195798

A model for communication in a multi-segment market

We consider a linear optimal control model for the marketing of seasonal products which are produced by the same firm and sold by retailers in different market segments The horizon is divided in two consecutive non-intersecting intervals, called production and selling periods, respectively. The production period state variables are the inventory levels and two kinds of goodwills (consumers' and retailers' goodwill, respectively) while the selling period state variables are the sales levels and the two kinds of goodwills. In the production interval there are three kinds of controls: on production, quality and advertising, while in the selling one the controls are on communication via advertising, promotion addressed to consumers and incentives given to retailers. We consider the case of several kinds of communications. The optimal control problem is transformed into an equivalent nonlinear programming problem

Why are losses from trade unlikely?

© 2015. Examining a standard monopolistic competition model with unspecified utility/cost functions, we find necessary and sufficient conditions on their elasticities for welfare losses to arise from trade or market expansion. Two numerical examples explain the losses (under unrealistic elasticities)

A fractional optimal control problem for maximizing advertising efficiency

ISBN: 978-5-7525-1858-

A global optimization approach to fractional optimal control

In this paper, we consider a fractional optimal control problem governed by system of linear differential equations, where its cost function is expressed as the ratio of convex and concave functions. The problem is a hard nonconvex optimal control problem and application of Pontriyagin's principle does not always guarantee finding a global optimal control. Even this type of problems in a finite dimensional space is known as NP hard. This optimal control problem can, in principle, be solved by Dinkhelbach algorithm [10]. However, it leads to solving a sequence of hard D.C programming problems in its finite dimensional analogy. To overcome this difficulty, we introduce a reachable set for the linear system. In this way, the problem is reduced to a quasiconvex maximization problem in a finite dimensional space. Based on a global optimality condition, we propose an algorithm for solving this fractional optimal control problem and we show that the algorithm generates a sequence of local optimal controls with improved cost values. The proposed algorithm is then applied to several test problems, where the global optimal cost value is obtained for each case

Dinkelbach Approach to Solving a Class of Fractional Optimal Control Problems

We consider optimal control problems with functional given by the ratio of two integrals (fractional optimal control problems). In particular, we focus on a special case with affine integrands and linear dynamics with respect to state and control. Since the standard optimal control theory cannot be used directly to solve a problem of this kind, we apply Dinkelbach’s approach to linearize it. Indeed, the fractional optimal control problem can be transformed into an equivalent monoparametric family { P q } of linear optimal control problems. The special structure of the class of problems considered allows solving the fractional problem either explicitly or requiring straightforward classical numerical techniques to solve a single equation. An application to advertising efficiency maximization is presented

Исследование влияния фазового состава на термическое расширение и механические свойства сплавов Al–Сu–Li

The study employed high-temperature X-ray diffraction, quantitative phase analysis, and tensile mechanical property measurements to investigate the relationship between coefficient of thermal expansion (CTE) and phase composition, along with the average yield strengths and Young's moduli of Al–Cu–Li alloys in three different sheet orientations: 1441, V-1461, V-1469, V-1480, and V-1481. The copper content within the solid solution and the mass fractions of the T1(Al2CuLi) and δ′(Al3Li) phases were determined using an innovative technique based on measuring the lattice distance of the α solid solution, Vegard's law, and balance equations for the elemental and phase compositions of the alloys. It was observed that as the lithium-to-copper ratio in the alloys increased from 0.32 to 1.12, the proportion of the δ′(Al3Li) phase increases from 6.3–8.4 wt.% in V-1481, V-1480 and V-1469 alloys to 16.0–17.3 wt.% in 1441 and V-1461 alloys, accompanied by a decrease in the T1(Al2CuLi) phase from 5 to 1 wt.%. This led to an increase in the Young's modulus from 75 to 77 GPa due to higher overall proportion of intermetallic compounds and a reduction in yield strength from 509 to 367 MPa due to the decrease in the T1 phase. This decrease in yield strength resulted from the fact that the hardening effect of the T1 phase was 3–4 times greater than that of the δ′ phase, and this couldn't be offset by an increase in the total intermetallic compound proportion. The observed increase in Young's modulus indicated that the elastic properties of the intermetallic phases were similar, and the rise in the total fraction of intermetallic compounds compensated for the decrease in the T1 phase. Furthermore, it was demonstrated that СTE, as measured based on the thermal expansion of the solid solution, also depended on the characteristics of the intermetallic phases present in the alloy. This expanded the potential interpretations of СTE measurement results. Методами высокотемпературной рентгенографии, количественного фазового анализа и измерения механических свойств при растяжении определяли корреляционные соотношения характеристик термического расширения (ТКЛР) и фазового состава с усредненными значениями по 3-м направлениям в листах пределов текучести и модулей Юнга сплавов системы Al–Cu–Li: 1441, В-1461, В-1469, В-1480 и В-1481. Содержание меди в твердом растворе и массовые доли фаз T1(Al2CuLi) и δ′(Al3Li) оценивали с помощью оригинальной методики, основанной на измерении периода решетки α-твердого раствора, законе Вегарда и уравнениях баланса элементного и фазового составов сплавов. Показано, что с увеличением отношения лития к меди в сплавах от 0,32 до 1,12 повышается доля δ′(Al3Li)-фазы от 6,3–8,4 мас.% в сплавах В-1481, В-1480 и В-1469 до 16,0–17,3 мас.% в сплавах 1441 и В-1461 за счет снижения количества T1(Al2CuLi)-фазы от 5 до 1 мас.%. Это приводит к увеличению модуля Юнга от 75 до 77 ГПа из-за возрастания суммарной доли интерметаллидов и к снижению предела текучести от 509 до 367 МПа из-за уменьшения количества Т1-фазы, поскольку эффект упрочнения T1-фазы в 3–4 раза превосходит упрочнение от выделения δ′-фазы, что не может быть скомпенсировано повышением суммарной доли интерметаллидов. Тот факт, что модуль Юнга при этом увеличивается, свидетельствует о том, что упругие свойства интерметаллидных фаз близки и возрастание суммарной доли интерметаллидов компенсирует снижение количества T1-фазы. Показано, что величина ТКЛР, измеренная на основании термического расширения твердого раствора, зависит также от характеристик присутствующих в сплаве интерметаллидных фаз, что расширяет возможности интерпретации результатов измерения ТКЛР.