Pricing and advertisement in a manufacturer–retailer supply chain

We use a game theoretical approach to study pricing and advertisement decisions in a manufacturer–retailer supply chain when price discounts are offered by both the manufacturer and retailer. When the manufacturer is the leader of the game, we obtained Stackelberg equilibrium with manufacturer’s local allowance, national brand name investment, manufacturer’s preferred price discount, retailer’s price discount, and local advertising expense. For the special case of two-stage equilibrium when the manufacturer’s price discount is exogenous, we found that the retailer is willing to increase local advertising expense if the manufacturer increases local advertising allowance and provides deeper price discount, or if the manufacturer decreases its brand name investment. When both the manufacturer and retailer have power, Nash equilibrium in a competition game is obtained. The comparison between the Nash equilibrium and Stackelberg equilibrium shows that the manufacturer always prefers Stackelberg equilibrium, but there is no definitive conclusion for the retailer. The bargaining power can be used to determine the profit sharing between the manufacturer and the retailer. Once the profit sharing is determined, we suggest a simple contract to help the manufacturer and retailer obtain their desired profit sharing.     

Optimal contracts and the manufacturer’s pricing strategies in a supply chain with an inequity-averse retailer

Studies in the supply chain literature have typically focused on profit or revenue maximization and assumed that agents within the supply chain are self-interested and only care about their own monetary payoffs. Research in these areas, however, rarely considers an important phenomenon called inequity aversion in which the object pursued by agents within the supply chain is not only their own profit maximization but also the equity of profit allocation. In fact, when agents within a supply chain collaborate with each other to serve a market, the scheme of profit allocation between them usually plays a determinate role in cooperation. Taking into account the impact of agents’ behavior of inequity aversion on the coordination of the supply chain, this paper investigates the optimal contracts and the manufacturer’s pricing strategies in a single-manufacturer and single-retailer supply chain. In this way, we obtain two interesting results: (1) the retailer’s equity aversion largely affects the manufacturer’s decision making, which is not always bad for the manufacturer; and (2) the retailer’s inequity aversion as well as the consumer’s price-sensitive coefficient plays a dominant role in the manufacturer’s decision making.     

Vertical cooperative advertising and pricing decisions in a manufacturer–retailer supply chain: A game-theoretic approach

Manufacturers can increase the advertising expenditures of their retailers by bearing a fraction of the occurring costs within the framework of a vertical cooperative advertising program. We expand the existing research which deals with advertising and pricing decisions in a manufacturer–retailer supply chain contemporaneously. By means of game theory, four different relationships between the channel members are considered: Firstly, three non-cooperative games with either symmetrical distribution of power or asymmetrical distribution with one player being the leader in each case, and one cooperative game where both players tend to maximize the total profit. The latter is complemented by a bargaining model, which proposes a fair split of profit on the basis of the players’ risk attitude and bargaining power. Our main findings are as follows: (a) In contrast to previous analyses, we do not limit the ratio between manufacturer’s and retailer’s margin, which provides more general insights into the effects of the underlying distribution of power within the channel. (b) The highest total profit is gained when both players cooperate. This behavior puts also the customers in a better position, as it produces the lowest retail price as well as the highest advertising expenditures compared to the other configurations.     

Coordination of a three-level supply chain (supplier–manufacturer–retailer) with permissible delay in payments and price discounts

Different coordination mechanisms are used to manage supply chains. This paper investigates a coordination of a three-level supply chain (supplier–manufacturer–retailer) by coupling two well-known trade credit mechanisms that are widely used in practice, permissible delay in payments and price discounts, where the length of the delay period and discounts offered along the supply chain are decision variables. The paper investigates nine different cases of delay in payments along with eight cases of price discounts among the three players in the supply chain. A numerical example was presented to compare between the cases considered. Also, extensive sensitivity analyses were performed to study the effect of changing the model parameters on the optimal decisions. In addition, we point out the limitations of each model developed in this paper. The numerical examples and the sensitivity analyses conclude that the coupling of delay in payments and price discounts increases the profit of the supply chain more than using only a single mechanism at a time.     

Principal–agent problem for returns handling in a reverse supply chain with one manufacturer and two competing dealers

Focusing on the principal–agent problem in handling new-products returned by customers in a reverse supply chain with one manufacturer and two competing dealers, we investigate how a manufacturer motivates her dealers to exert their efforts to handle and sell the returned new-products using mathematical modelling. By taking into consideration the dealers’ individual rationalities, we design optimal incentive contracts under both symmetric and asymmetric information and obtain the following insights. In the symmetric information situation, dealers’ effort levels are identical to those maximising the expected overall profit of the supply chain. In the asymmetric information situation, however, the dealers’ effort levels are lower than the supply chain optimum. Further, in the case of intensifying competition between the dealers, their effort levels increase, while each dealer’ revenue share received from the manufacturer decreases and the risk, incentive and total agency costs of the entire supply chain tend to diminish. The revenue share is inversely correlated with return uncertainties. When we introduce another random environmental factor affecting returns into the incentive contract, this revenue share increases with a rise in dealers’ effort levels, while the above-mentioned costs of the entire system decline and this improves the supply chain system coordination. The incorporation of fairness into the principal–agent model promotes a further increase in the dealers’ effort levels, which is proved using the fixed-point theorem. Finally, we provide an example to demonstrate the main results.     

Dynamic conformance and design quality in a supply chain: an assessment of contracts’ coordinating power

We consider a two-echelon supply chain involving one manufacturer and one supplier who collaborate on improving both design and conformance quality. Design quality is supposed to increase product desirability, and therefore market demand, while conformance quality should reduce the proportion of defective items, and therefore increase the manufacturer’s sales revenue. We investigate how the supply chain parties allocate effort between design and conformance quality under both cooperative and non-cooperative settings in an intertemporal framework. Furthermore, we evaluate wholesale price contracts and revenue-sharing contracts in terms of their performance and coordination power. We show that although a revenue-sharing contract enables the manufacturer to effectively involve the supplier in quality improvement, neither contract type allows for perfect coordination resulting in the quality that can be achieved by a cooperative supply chain. We thus suggest a reward-based extension to the revenue-sharing contract, to ensure system-wide optimal quality performance. Importantly, we find that the supplier would be better off adopting a reward-based revenue sharing contract and refusing a standard revenue-sharing contract, while the opposite would be true for the manufacturer.     

Solution of Hilbert’s problem for an annulus in a specific case and its application to an explosion problem

A method for solving Hilbert's boundary value problem with discontinuous coefficients is studied for a function single-valued and analytic in an annulus in the case in which the solution may have power-law singularities at finitely many points on the boundary of the annulus. To illustrate the results obtained, we consider an explosion problem for two pinching charges in a homogenous medium with a circular cylinder lying in the flow caused by the explosion. Translated fromMatematicheskie Zametki, Vol. 66, No. 1, pp. 135–144, July, 1999.     

A generalization of a conjecture due to Erd?s, Jacobson and Lehel

An r-graph is a loopless undirected graph in which no two vertices are joined by more than r edges. An r-complete graph on m+1 vertices, denoted by , is an r-graph on m+1 vertices in which each pair of vertices is joined by exactly r edges. A non-increasing sequence π=(d₁,d₂,…,d_n) of nonnegative integers is r-graphic if it is realizable by an r-graph on n vertices. Let be the smallest even integer such that each n-term r-graphic sequence with term sum of at least is realizable by an r-graph containing as a subgraph. In this paper, we determine the value of for sufficiently large n, which generalizes a conjecture due to Erd?s, Jacobson and Lehel.     

On a class of sofic affine invariant subsets of the 2-torus related to an Erd?s problem

Let 1 < β <?2 be a real number and G be the closed projection on the 2-torus of the (modified) Rademacher graph in base β. The smallest compact containing G and left invariant by the diagonal endomorphism ${(x,y)mapsto(2x,beta y)}$ (mod 1) is denoted by K. For β a simple Parry number of PV-type, K is proved to be a sofic affine invariant set with a fractal geometry closed to the one of G. When β is the golden number, we prove the uniqueness of the measure with full Hausdorff dimension on K.     

Local solvability and a priori estimates for classical solutions to an equation of Benjamin-Bona-Mahony-Bürgers type

We establish the local (in time) solvability in the classical sense for the Cauchy problem and first and second boundary-value problems on the half-line for a nonlinear equation similar to Benjamin-Bona-Mahony-Bürgers-type equation. We also derive an a priori estimate that implies sufficient blow-up conditions for the second boundary-value problem. We obtain analytically an upper bound of the blow-up time and refine it numerically using Richardson effective accuracy order technique.     

On a Sharper Lower Bound for a Percentile of a Student’s <Emphasis Type="Italic">t</Emphasis> Distribution with an Application

In this communication, we first compare z _α and t _ν,α , the upper 100α% points of a standard normal and a Student’s t _ν distributions respectively. We begin with a proof of a well-known result, namely, for every fixed 0 < a < \frac120<\alpha <\frac{1}{2} and the degree of freedom ν, one has t _ν,α  > z _α . Next, Theorem 3.1 provides a new and explicit expression b _ν ( > 1) such that for every fixed 0 < a < \frac120<\alpha < \frac{1}{2} and ν, we can conclude t _ν,α  > b _ν z _α . This is clearly a significant improvement over the result that is customarily quoted in nearly every textbook and elsewhere. A proof of Theorem 3.1 is surprisingly simple and pretty. We also extend Theorem 3.1 in the case of a non-central Student’s t distribution (Section 3.3). In the context of Stein’s (Ann Math Stat 16:243–258, 1945; Econometrica 17:77–78, 1949) 100(1 − α)% fixed-width confidence intervals for the mean of a normal distribution having an unknown variance, we have examined the oversampling rate on an average for a variety of choices of m, the pilot sample size. We ran simulations to investigate this issue. We have found that the oversampling rates are approximated well by t_n,a/2²z_a/2^-2t_{\nu ,\alpha /2}^{2}z_{\alpha /2}^{-2} for small and moderate values of m( ≤ 50) all across Table 2 where ν = m − 1. However, when m is chosen large (≥ 100), we find from Table 3 that the oversampling rates are not approximated by t_n,a/2²z_a/2^-2t_{\nu ,\alpha /2}^{2}z_{\alpha /2}^{-2} very well anymore in some cases, and in those cases the oversampling rates either exceed the new lower bound of t_n,a/2²z_a/2^-2,t_{\nu ,\alpha /2}^{2}z_{\alpha /2}^{-2}, namely b_n²,b_{\nu }^{2}, or comes incredibly close to b_n²b_{\nu }^{2} where ν = m − 1. That is, the new lower bound for a percentile of a Student’s t distribution may play an important role in order to come up with diagnostics in our understanding of simulated output under Stein’s fixed-width confidence interval method.     

The Poisson integral and Green’s function for one strongly elliptic system of equations in a circle and an ellipse

For a strongly elliptic system of second-order equations of a special form, formulas for the Poisson integral and Green’s function in a circle and an ellipse are obtained. The operator under consideration is represented by the sum of the Laplacian and a residual part with a small parameter, and the solution to the Dirichlet problem is found in the form of a series in powers of this parameter. The Poisson formula is obtained by the summation of this series.     

Iteration complexity of an inexact Douglas-Rachford method and of a Douglas-Rachford-Tseng’s F-B four-operator splitting method for solving monotone inclusions

Numerical Algorithms - In this paper, we propose and study the iteration complexity of an inexact Douglas-Rachford splitting (DRS) method and a Douglas-Rachford-Tseng’s forward-backward (F-B)...     

Combining Binary Search and Newton′s Method to Compute Real Roots for a Class of Real Functions

We generalize a hybrid algorithm of binary search and Newton′s method to compute real roots for a class of real functions. We show that the algorithm computes a root inside (0, R] with error ϵ in O(log log(R/ϵ)) time, where one function evaluation or one arithmetic operation counts for one unit of time. This work is based on Smale′s criterion for using Newton′s method and Renegar′s result of approximating roots of polynomials.     

Mathematical Analysis and Pattern Formation for a Partial Immune System Modeling the Spread of an Epidemic Disease

Our motivation is a mathematical model describing the spatial propagation of an epidemic disease through a population. In this model, the pathogen diversity is structured into two clusters and then the population is divided into eight classes which permits to distinguish between the infected/uninfected population with respect to clusters. In this paper, we prove the weak and the global existence results of the solutions for the considered reaction-diffusion system with Neumann boundary. Next, mathematical Turing formulation and numerical simulations are introduced to show the pattern formation for such systems.     

Erratum to “A genetic algorithm approach for solving a closed loop supply chain model: A case of battery recycling” [Appl. Math. Modell. 34 (2010) 655–670]

This study analyzes Mixed Integer Linear Program (MILP) proposed by G. Kannan, P. Sasikumar M. Devika, (2010) in their paper titled ‘A genetic algorithm approach for solving a closed loop supply chain model: A case of battery recycling’, Applied Mathematical Modelling, (34, 655–670). The model in Kannan et al. (2010) is found to be inadequate for the problem described. It is erroneous/infeasible in terms of constraints, objective and variables. In this work, we list down the flaws in the published work and propose modifications to rectify the flaws. The revised model is presented and illustrated using hypothetical problems.     

Uniform Estimates for the Fourier Transform of Surface Carried Measures in ?3 and an Application to Fourier Restriction

Let S be a hypersurface in \BbbR³{\Bbb{R}}^{3} which is the graph of a smooth, finite type function φ, and let μ=ρ  dσ be a surface carried measure on S, where dσ denotes the surface element on S and ρ a smooth density with sufficiently small support. We derive uniform estimates for the Fourier transform [^(m)]\hat{\mu} of μ, which are sharp except for the case where the principal face of the Newton polyhedron of φ, when expressed in adapted coordinates, is unbounded. As an application, we prove a sharp L ^p -L ² Fourier restriction theorem for S in the case where the original coordinates are adapted to φ. This improves on earlier joint work with M. Kempe.