Construction of Equilibria in Strategic Stackelberg Games in Multi-Period Supply Chain Contracts

Almost every supplier faces uncertain and time-varying demand. E-commerce and online shopping have given suppliers unprecedented access to data on customers’ behavior, which sheds light on demand uncertainty. The main purpose of this research project is to provide an analytic tool for decentralized supply channel members to devise optimal long-term (multi-period) supply, pricing, and timing strategies while catering to stochastic demand in a diverse set of market scenarios. Despite its ubiquity in potential applications, the time-dependent channel optimization problem in its general form has received limited attention in the literature due to its complexity and the highly nested structure of its ensuing equilibrium problems. However, there are many scenarios where a single-period channel optimization solution may turn out to be myopic as it does not consider the after-effects of current pricing on future demand. To remedy this typical shortcoming, using general memory functions, we include the strategic customers’ cognitive bias toward pricing history in the supply channel equilibrium problem. In the form of two constructive theorems, we provide explicit solution algorithms for the ensuing Nash–Stackelberg equilibrium problems. In particular, we prove that our recursive solution algorithm can find equilibria in the multi-periodic variation of many standard supply channel contracts such as wholesale, buyback, and revenue-sharing contracts.publishedVersio

The newsvendor problem with advertising: an overview with extensions

The effect of advertising on sales has been the subject of recent studies as an important aspect in many demand-based problems. Herein, we deal with the newsvendor problem, due to its simple structure, as a suitable tool for illustrating how facets of marketing may affect decision-making concerning operational problems. In the setting presented, the newsvendor is faced with advertising-sensitive stochastic demand, where a demand-related random element comprises an advertising decision of the multiplicative or additive form. We assume that a suitable advertising strategy results in increased sales. Two advertising response functions are considered, these being concave downward and S-shaped. We review and extend the existing results relating to the newsvendor problem with marketing effects, which mostly pertain to the concave function. These are generalized by defining the S-shaped function, and some original insights into the effect of advertising are given. We establish that the optimal advertising expenditure for the multiplicative case is always less than or equal to the optimal amount in the equivalent deterministic model while it is always equal in the additive case. We finally illustrate the results that are obtained by providing numerical examples involving various advertising response functions, as well as management-related interpretations. © 2016, Springer-Verlag Berlin Heidelberg.European Regional Development Fund under the project CEBIA-Tech Instrumentation [CZ.1.05/2.1.00/19.0376]; specific research project entitled "Modern Methods of Applied Mathematics for the Use in Technical Sciences'' [FSI-S-14-2290, 25053]; Ministry of Education, Youth and Sports under National Sustainability Programme I [LO1202]; Norway Grants via the EEA Scholarship Programme: Bilateral Scholarship Programme; Institutional cooperation projects [NF-CZ07-ICP-4-345-2016

Competitive Multi-period Pricing with Fixed Inventories

This paper studies the problem of multi-period pricing for perishable products in a competitive (oligopolistic) market. We study non cooperative Nash equilibrium policies for sellers. At the beginning of the time horizon, the total inventories are given and additional production is not an available option. The analysis for periodic production-review models, where production decisions can be made at the end of each period at some production cost after incurring holding or backorder costs, does not extend to this model. Using results from game theory and variational inequalities we study the existence and uniqueness of equilibrium policies. We also study convergence results for an algorithm that computes the equilibrium policies. The model in this paper can be used in a number of application areas including the airline, service and retail industries. We illustrate our results through some numerical examples.Singapore-MIT Alliance (SMA

Inventory and pricing decisions in a single-period problem involving risky supply

Cataloged from PDF version of article.We explore an extension of the single-period (newsboy) inventory problem when supply is uncertain. We look into the negotiations between a newsvendor (retailer) and a manufacturer when there is competition from a second supplier. There is a chance that the second supplier will not be able to deliver the product. The retailer can maximize his expected profit by optimally allocating his order between the two suppliers. The retailer’s ordering problem is analyzed in conjunction with the manufacturer’s related pricing problem. The effects of demand and supply uncertainties on the optimal decisions of the parties are explored using numerical examples. We also explore extension of the retailer’s problem to the cases of order cancellation, price-dependent demand, and demand-dependent supply availability. & 2008 Elsevier B.V. All rights reserve

Dynamic Pricing through Sampling Based Optimization

In this paper we develop an approach to dynamic pricing that combines ideas from data-driven and robust optimization to address the uncertain and dynamic aspects of the problem. In our setting, a firm off ers multiple products to be sold over a fixed discrete time horizon. Each product sold consumes one or more resources, possibly sharing the same resources among di fferent products. The firm is given a fixed initial inventory of these resources and cannot replenish this inventory during the selling season. We assume there is uncertainty about the demand seen by the fi rm for each product and seek to determine a robust and dynamic pricing strategy that maximizes revenue over the time horizon. While the traditional robust optimization models are tractable, they give rise to static policies and are often too conservative. The main contribution of this paper is the exploration of closed-loop pricing policies for di fferent robust objectives, such as MaxMin, MinMax Regret and MaxMin Ratio. We introduce a sampling based optimization approach that can solve this problem in a tractable way, with a con fidence level and a robustness level based on the number of samples used. We will show how this methodology can be used for data-driven pricing or adapted for a random sampling optimization approach when limited information is known about the demand uncertainty. Finally, we compare the revenue performance of the di fferent models using numerical simulations, exploring the behavior of each model under diff erent sample sizes and sampling distributions.National Science Foundation (U.S.) (Grant 0556106-CMII)National Science Foundation (U.S.) (Grant 0824674-CMII)Singapore-MIT Allianc

Optimal Dynamic Procurement Policies for a Storable Commodity with L\'evy Prices and Convex Holding Costs

In this paper we study a continuous time stochastic inventory model for a commodity traded in the spot market and whose supply purchase is affected by price and demand uncertainty. A firm aims at meeting a random demand of the commodity at a random time by maximizing total expected profits. We model the firm's optimal procurement problem as a singular stochastic control problem in which controls are nondecreasing processes and represent the cumulative investment made by the firm in the spot market (a so-called stochastic "monotone follower problem"). We assume a general exponential L\'evy process for the commodity's spot price, rather than the commonly used geometric Brownian motion, and general convex holding costs. We obtain necessary and sufficient first order conditions for optimality and we provide the optimal procurement policy in terms of a "base inventory" process; that is, a minimal time-dependent desirable inventory level that the firm's manager must reach at any time. In particular, in the case of linear holding costs and exponentially distributed demand, we are also able to obtain the explicit analytic form of the optimal policy and a probabilistic representation of the optimal revenue. The paper is completed by some computer drawings of the optimal inventory when spot prices are given by a geometric Brownian motion and by an exponential jump-diffusion process. In the first case we also make a numerical comparison between the value function and the revenue associated to the classical static "newsvendor" strategy.Comment: 28 pages, 3 figures; improved presentation, added new results and section

オムニチャネル小売業におけるファッション商品の品切れに対する対応のモデル化分析

筑波大学 (University of Tsukuba)201