Price differentiation is a common strategy in many markets. In this paper, we study a static multiproduct price optimization problem with demand given by a discrete mixed multinomial logit model. By considering a mixed logit model that includes individual-specific variables in the utility specification, our pricing problem reflects well the discrete choice models used in practice. To solve this pricing problem, we design an efficient iterative optimization algorithm that asymptotically converges to the optimal solution. To this end, a linear optimization (LO) problem is formulated, based on the trust-region approach, to find a "good" feasible solution and approximate the problem from below. Another LO problem is designed using piecewise linear relaxations to approximate the optimization problem from above. Then, we develop a new branching method to tighten the optimality gap. The effectiveness of our algorithm is demonstrated on a parking services pricing case, and a benchmark against solvers and existing contributions in the literature are performed.
In this paper we consider the problem of pricing multiple differentiated products. This is challenging as a price change in one product, not only changes the demand of that particular product, but also the demand for the other products. To address this problem, customer choice models have recently been introduced as these are capable of describing customer choice behavior across differentiated products. In the present paper the objective is to obtain the revenue-maximizing prices when the customer's decision making process is modelled according to a particular customer choice model, namely the…
In this paper, we investigate the capacitated assortment optimization problem with pricing under the paired combinatorial logit model, whose goal is to identify the revenue-maximizing subset of products as well as their selling prices subject to a known capacity limit. We model customers' purchase behavior using the paired combinatorial logit model, which allows for covariance among any pair of products. We formulate this problem as a non-linear mixed integer program. Then, we propose a two-step approach to obtain the optimal solution based on solving a mixed integer program and Lambert-W func…
Recently, there is growing interest and need for dynamic pricing algorithms, especially, in the field of online marketplaces by offering smart pricing options for big online stores. We present an approach to adjust prices based on the observed online market data. The key idea is to characterize optimal prices as minimizers of a total expected revenue function, which turns out to be convex. We assume that consumers face information processing costs, hence, follow a discrete choice demand model, and suppliers are equipped with quantity adjustment costs. We prove the strong smoothness of the tota…
Spatiotemporal Pricing and Fleet Management of Autonomous Mobility-on-Demand Networks: A Decomposition and Dynamic Programming Approach with Bounded Optimality Gap
In this paper, the problem of optimal dynamic pricing for retail electricity with an unknown demand model is considered. Under the day-ahead dynamic pricing (a.k.a. real time pricing) mechanism, a retailer obtains electricity in a twosettlement wholesale market and serves its customers in real time. Without knowledge on the aggregated demand function of its customers, the retailer aims to maximize its retail surplus by sequentially adjusting its price based on the behavior of its customers in the past. An online learning algorithm, referred to as piecewise linear stochastic approximation (PWLS…
With the rapidly increased penetration of renewable generations, incentive-based demand side management (DSM) shows great value on alleviating the uncertainty and providing flexibility for microgrid. However, how to price those demand resources becomes one of the most significant challenges for promoting incentive-based DSM under microgrid environments. In this paper, a flexible demand resource pricing scheme is proposed. Instead of using the utility function of end users like most existing literatures, the economic benefit of flexible demand resources is evaluated by the operation performance…
In this paper we investigate a dynamic pricing model for constant demand elasticity where customers have a probability distribution on the number of items they order. This is a generalization from standard models which restrict customers to buy only one item at a time. For the generalized model, we first obtain a closed form expression for the optimal expected revenue and optimal pricing strategy. This expression involves a recursively defined term for which we investigate the behavior. We call comparable models those which have the same demand, which is the customer arrival rate times the ave…
In this paper, we consider an infinite horizon, continuous-review, stochastic inventory system in which cumulative customers' demand is price-dependent and is modeled as a Brownian motion. Excess demand is backlogged. The revenue is earned by selling products and the costs are incurred by holding/shortage and ordering, the latter consists of a fixed cost and a proportional cost. Our objective is to simultaneously determine a pricing strategy and an inventory control strategy to maximize the expected long-run average profit. Specifically, the pricing strategy provides the price $p_t$ for any ti…
The problem of dynamic pricing of electricity in a retail market is considered. A Stackelberg game is used to model interactions between a retailer and its customers; the retailer sets the day-ahead hourly price of electricity and consumers adjust real-time consumptions to maximize individual consumer surplus. For thermostatic demands, the optimal aggregated demand is shown to be an affine function of the day-ahead hourly price. A complete characterization of the trade-offs between consumer surplus and retail profit is obtained. The Pareto front of achievable trade-offs is shown to be concave,…
This paper introduces a computationally efficient comparative approach to classical pricing rules for day-ahead electricity markets, namely Convex Hull Pricing, IP Pricing and European-like market rules, in a Power Exchange setting with non-convex demand bids. These demand bids can, for example, be useful to large industrial consumers, and extend demand block orders in use by European Power Exchanges. For this purpose, we show that Convex Hull Prices can be efficiently computed using continuous relaxations for bidding products involving start-up costs, minimum power output levels and ramp cons…
Motivated by demand-side management in smart grids, a decentralized controlled Markov chain formulation is proposed to model a homogeneous population of users with binary demands (i.e., off or on). The binary demands often arise in scheduling applications such as plug-in hybrid vehicles. Normally, an independent service operator (ISO) has a finite number of options when it comes to providing the users with electricity. The options represent various incentive means, generation resources, and price profiles. The objective of the ISO is to find optimal options in order to keep the distribution of…
This paper is concerned with the determination of pricing strategies for a firm that in each period of a finite horizon receives replenishment quantities of a single product which it sells in two markets, e.g., a long-distance market and an on-site market. The key difference between the two markets is that the long-distance market provides for a one period delay in demand fulfillment. In contrast, on-site orders must be filled immediately as the customer is at the physical on-site location. We model the demands in consecutive periods as independent random variables and their distributions depe…
To guarantee the well-functioning of electricity distribution networks, it is crucial to constantly ensure the demand-supply balance. To do this, one can control the means of production, but also influence the demand: demand-side management becomes more and more popular as the demand keeps increasing and getting more chaotic. In this work, we propose a bilevel model involving an energy supplier and a smart grid operator (SGO): the supplier induces shifts of the load controlled by the SGO by offering time-dependent prices. We assume that the SGO has contracts with consumers and decides their co…
This paper provides a framework to quantify the sensitivity associated with behavioral models based on Cumulative Prospect Theory (CPT). These are used to design dynamic pricing strategies aimed at maximizing performance metrics of the Shared Mobility On-Demand Service (SMoDS), as solutions to a constrained nonlinear optimization problem. We analyze the sensitivity of both the optimal tariff as well as the optimal objective function with respect to CPT model parameters. In addition to deriving analytical solutions under certain assumptions, more general numerical results are obtained via compu…
We consider a profit maximization problem in an urban mobility on-demand service, of which the operator owns a fleet, provides both exclusive and shared trip services, and dynamically determines prices of offers. With knowledge of the traveler preference and the distribution of future trip requests, the operator wants to find the pricing strategy that optimizes the total operating profit of multiple trips during a specific period, namely, a day in this paper. This problem is first formulated and analyzed within the dynamic programming framework, where a general approach combining parametric ro…
We study robust versions of pricing problems where customers choose products according to a generalized extreme value (GEV) choice model, and the choice parameters are not known exactly but lie in an uncertainty set. We show that, when the robust problem is unconstrained and the price sensitivity parameters are homogeneous, the robust optimal prices have a constant markup over products, and we provide formulas that allow to compute this constant markup by bisection. We further show that, in the case that the price sensitivity parameters are only homogeneous in each partition of the products, u…
When randomness in demand affects the sales of a product, retailers use dynamic pricing strategies to maximize their profits. In this article, we formulate the pricing problem as a continuous-time stochastic optimal control problem and find the optimal policy by solving the associated Hamilton-Jacobi-Bellman (HJB) equation. We propose a new approach to modelling the randomness in the dynamics of sales based on diffusion processes. The model assumes a continuum approximation to the stock levels of the retailer which should scale much better to large-inventory problems than the existing Poisson …
In a Shared Mobility on Demand Service (SMoDS), dynamic pricing plays an important role in the form of an incentive for the decision of the empowered passenger on the ride offer. Strategies for determining the dynamic tariff should be suitably designed so that the incurred demand and supply are balanced and therefore economic efficiency is achieved. In this manuscript, we formulate a discrete time Markov Decision Process (MDP) to determine the probability desired by the SMoDS platform corresponding to the acceptance rate of each empowered passenger at each state of the system. We use Estimated…
Price based demand response schemes may significantly improve power system efficiency. Additionally, it is desired that such schemes yield improved power operation, by reducing the peak consumption. This paper proposes the Intraday Block Pricing (IBP) scheme, aiming to promote effective demand response among consumers by charging their electricity usage based on intraday time-slots. To design the prices associated with the proposed scheme, we formulate a bilevel optimization problem that aims to minimize the Peak-to-Average Ratio (PAR) and simultaneously benefit the consumers and the utility c…
We consider the setting in which an electric power utility seeks to curtail its peak electricity demand by offering a fixed group of customers a uniform price for reductions in consumption relative to their predetermined baselines. The underlying demand curve, which describes the aggregate reduction in consumption in response to the offered price, is assumed to be affine and subject to unobservable random shocks. Assuming that both the parameters of the demand curve and the distribution of the random shocks are initially unknown to the utility, we investigate the extent to which the utility mi…
The classic Dial-A-Ride Problem (DARP) aims at designing the minimum-cost routing that accommodates a set of user requests under constraints at an operations planning level, where users' preferences and revenue management are often overlooked. In this paper, we present a mechanism for accepting/rejecting user requests in a Demand Responsive Transportation (DRT) context based on the representative utilities of alternative transportation modes. We consider utility-maximizing users and propose a mixed-integer programming formulation for a Chance Constrained DARP (CC-DARP), that captures users' pr…
In an electric power system, demand fluctuations may result in significant ancillary cost to suppliers. Furthermore, in the near future, deep penetration of volatile renewable electricity generation is expected to exacerbate the variability of demand on conventional thermal generating units. We address this issue by explicitly modeling the ancillary cost associated with demand variability. We argue that a time-varying price equal to the suppliers' instantaneous marginal cost may not achieve social optimality, and that consumer demand fluctuations should be properly priced. We propose a dynamic…
University of Essex（埃塞克斯大学）；University of Manchester（曼彻斯特大学）
In this paper, we propose a realistic multiple dynamic pricing approach to demand response in the retail market. First, an adaptive clustering-based customer segmentation framework is proposed to categorize customers into different groups to enable the effective identification of usage patterns. Second, customized demand models with important market constraints which capture the price-demand relationship explicitly, are developed for each group of customers to improve the model accuracy and enable meaningful pricing. Third, the multiple pricing based demand response is formulated as a profit m…
This paper studies optimal pricing and rebalancing policies for Autonomous Mobility-on-Demand (AMoD) systems. We take a macroscopic planning perspective to tackle a profit maximization problem while ensuring that the system is load-balanced. We begin by describing the system using a dynamic fluid model to show the existence and stability of an equilibrium (i.e., load balance) through pricing policies. We then develop an optimization framework that allows us to find optimal policies in terms of pricing and rebalancing. We first maximize profit by only using pricing policies, then incorporate re…
Park-and-ride facilities are car parks where users can transfer to public transportation. Commuters can use P&R facilities or choose to travel by car to their destinations, and individual choice behavior is assumed to follow a logit model. The P&R facility location problem identifies locations for a fixed number of P&R facilities from among potential locations such that the number of users of the P&R facilities is maximized. This problem has previously been formalized under a multinomial logit demand function. However, as it imposes the strong condition of the independence of irrelevant altern…
Renewable sources are taking center stage in electricity generation. However, matching supply with demand in a renewable-rich system is a difficult task due to the intermittent nature of renewable resources (wind, solar, etc.). As a result, Demand Response (DR) programs are an essential part of the modern grid. An efficient DR technique is to devise different pricing schemes that encourage customers to reduce or shift the electric load. In this paper, we consider a market model for DR using Block Rate Pricing (BRP) of two blocks. We use a utility maximization approach in a competitive market. …
Dynamic Pricing with Adversarially-Censored Demands
Carnegie Mellon University（卡内基梅隆大学）；University of Texas at Dallas（德克萨斯大学达拉斯分校）；New York University（纽约大学）；University of California San Diego（加州大学圣地亚哥分校）
The problem of robust dynamic pricing of an abstract commodity, whose inventory is specified at an initial time but never subsequently replenished, originally studied by Perakis and Sood (2006) in discrete time, is considered from the perspective of continuous time. We use a multiplicative demand function to model the uncertain demand, and develop a robust counterpart to replace the uncertain demand constraint. The sellers' robust best response problem yields a generalized Nash equilibrium problem, which can be formulated as an equivalent, continuous-time quasi-variational inequality. We demon…
Pricing decisions are often made when market information is still poor. In turn, existing theoretical models often reason about the response of optimal prices to changing market characteristics without exploiting all available information about the demand distribution. Our aim is to develop a theory for the optimization and systematic comparison of prices between different instances of the same market under various forms of knowledge about the corresponding demand distributions. We revisit the classic problem of monopoly pricing under demand uncertainty in a vertical market with an upstream su…
Dynamic Pricing in Smart Grids under Thresholding Policies: Algorithms and Heuristics
Masdar Institute of Science and Technology（马斯达尔理工学院）；Brigham Young University（杨百翰大学）；New York University, Abu Dhabi（纽约大学阿布扎比分校）；Massachusetts Institute of Technology（麻省理工学院）
We study a problem of an online retailer who observes the unit sales of a product, and dynamically changes the retail price, in order to maximize the expected revenue. Assuming the demand of the product is price sensitive, we are interested in the optimal pricing policy when future demand is uncertain. We build a system to investigate the relationship between retail price and demand and estimate the demand function. The system predicts demand and revenue at a given retail price. We formulate a revenue maximization problem over a discrete finite time horizon with discrete retail price. The opti…
A large fraction of the total electric load is comprised of end-use devices whose demand for energy is inherently deferrable in time. Of interest is the potential to leverage on such latent flexibility in demand to absorb variability in power supplied from intermittent renewable generation. The challenge, however, lies in designing incentives to reliably induce the desired response in demand. With an eye to electric vehicle charging, we propose a novel forward market for differentiated electric power services, where consumers consent to deferred service of pre-specified loads in exchange for a…
