Convex Prophet Inequalities

We introduce a new class of prophet inequalities-convex prophet inequalities-where a gambler observes a sequence of convex cost functions ci (xi ) and is required to assign some fraction 0 ≤ x_i ≤ 1 to each, such that the sum of assigned values is exactly 1. The goal of the gambler is to minimize the sum of the costs. We provide an optimal algorithm for this problem, a dynamic program, and show that it can be implemented in polynomial time when the cost functions are polynomial. We also precisely characterize the competitive ratio of the optimal algorithm in the case where the gambler has an outside option and there are polynomial costs, showing that it grows as θ(n^(p-1)/ℓ), where n is the number of stages, p is the degree of the polynomial costs and the coefficients of the cost functions are bounded by [ℓ,u]

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

Convex Prophet Inequalities

We introduce a new class of prophet inequalities-convex prophet inequalities-where a gambler observes a sequence of convex cost functions ci (xi ) and is required to assign some fraction 0 ≤ x_i ≤ 1 to each, such that the sum of assigned values is exactly 1. The goal of the gambler is to minimize the sum of the costs. We provide an optimal algorithm for this problem, a dynamic program, and show that it can be implemented in polynomial time when the cost functions are polynomial. We also precisely characterize the competitive ratio of the optimal algorithm in the case where the gambler has an outside option and there are polynomial costs, showing that it grows as θ(n^(p-1)/ℓ), where n is the number of stages, p is the degree of the polynomial costs and the coefficients of the cost functions are bounded by [ℓ,u]

Stochastic Stackelberg equilibria with applications to time dependent newsvendor models

In this paper we prove a sufficient maximum principle for general stochastic differential Stackelberg games, and apply the theory to continuous time newsvendor problems. In the newsvendor problem a manufacturer sells goods to a retailer, and the objective of both parties is to maximize expected profits under a random demand rate. Our demand rate is an Ito-Levy process, and to increase realism information is delayed, e.g., due to production time. We provide complete existence and uniqueness proofs for a series of special cases, including geometric Brownian motion and the Ornstein-Uhlenbeck process, both with time variable coefficients. Moreover, these results are operational because we are able to offer explicit solution formulas. An interesting finding is that more precise information may be a considerable disadvantage for the retailer.Stochastic differential games; newsvendor model; delayed information; Ito-Levy processes

Multiproduct Multiperiod Newsvendor Problem with Dynamic Market Efforts

We study a multiperiod multiproduct production planning problem where the production capacity and the marketing effort on demand are both considered. The accumulative impact of marketing effort on demand is captured by the Nerlove and Arrow (N-A) advertising model. The problem is formulated as a discrete-time, finite-horizon dynamic optimization problem, which can be viewed as an extension to the classic newsvendor problem by integrating with the N-A model. A Lagrangian relaxation based solution approach is developed to solve the problem, in which the subgradient algorithm is used to find an upper bound of the solution and a feasibility heuristic algorithm is proposed to search for a feasible lower bound. Twelve kinds of instances with different problem size involving up to 50 products and 15 planning periods are randomly generated and used to test the Lagrangian heuristic algorithm. Computational results show that the proposed approach can obtain near optimal solutions for all the instances in very short CPU time, which is less than 90 seconds even for the largest instance