Dynamic Pricing Problem Research Articles

Policy Optimization Using Semiparametric Models for Dynamic Pricing

In this paper, we study the contextual dynamic pricing problem where the market value of a product is linear in their observed features plus some market noise. Products are sold one at a time, and only a binary response indicating the success or failure of a sale is observed. Our model setting is similar to \cite{JN19} except that we expand the demand curve to a semiparametric model and need to learn dynamically both parametric and nonparametric components. We propose a dynamic statistical learning and decision-making policy that combines semiparametric estimation from a generalized linear model with an unknown link and online decision making to minimize regret (maximize revenue). Under mild conditions, we show that for a market noise c.d.f. $F(\cdot)$ with $m$-th order derivative, our policy achieves a regret upper bound of $\tilde{\cO}_{d}(T^{\frac{2m+1}{4m-1}})$ for $m\geq 2$, where $T$ is time horizon and $\tilde{\cO}_{d}$ is the order that hides logarithmic terms and the dimensionality of feature $d$. The upper bound is further reduced to $\tilde{\cO}_{d}(\sqrt{T})$ if $F$ is super smooth whose Fourier transform decays exponentially. In terms of dependence on the horizon $T$, these upper bounds are close to $\Omega(\sqrt{T})$, the lower bound where the market noise distribution belongs to a parametric class. We further generalize these results to the case when the product features are dynamically dependent, satisfying some strong mixing conditions.

Correlated Bandits for Dynamic Pricing Via the Arc Algorithm

The Asymptotic Randomised Control (ARC) algorithm provides a rigorous approximation to the optimal strategy for a wide class of Bayesian bandits, while retaining reasonable computational complexity. In particular, it allows a decision maker to observe signals in addition to their rewards, to incorporate correlations between the outcomes of different choices, and to have nontrivial dynamics for their estimates. The algorithm is guaranteed to asymptotically optimise the expected discounted payoff, with error depending on the initial uncertainty of the bandit. In this paper, we consider a batched bandit problem where observations arrive from a generalised linear model; we extend the ARC algorithm to this setting. We apply this to a classic dynamic pricing problem based on a Bayesian hierarchical model and demonstrate that the ARC algorithm outperforms alternative approaches.

Multi-Step Look-Ahead Optimization Methods for Dynamic Pricing With Demand Learning

Dynamic pricing is a beneficial strategy for firms seeking to achieve high revenues. It has been widely applied to various domains such as the airline industry, the hotel industry, and e-services. Dynamic pricing is basically the problem of setting time-varying prices for a certain product or service for the purpose of optimizing revenue. However, a major challenge encountered when applying dynamic pricing is the lack of knowledge of the demand-price curve. The demand-price curve defines the customer's response towards changing the price. In this work, we address the dynamic pricing problem in case of unknown demand-price relation. This work introduces a less myopic pricing approach based on looking ahead for one or several future steps in our quest to optimize revenue. Specifically, the proposed formulation maximizes the summation of the immediate revenue and the expected future revenues of one or multiple look-ahead steps. A key benefit of the proposed approach is that it automatically strikes a balance between the conflicting goals of revenue maximization and demand learning, by producing less myopic and profitable prices. We provide a formulation for the presented look-ahead pricing approach, and we implement two variants of it: one-step and two-step look-ahead methods. Experiments are conducted on synthetic and real datasets to compare the proposed pricing methods to other pricing strategies in literature. The experimental results indicate that the proposed look-ahead methods outperform their counterparts in terms of the achieved revenue gain.

Approximation of sell-out probability to estimate expected marginal value of capacity

In this study, the dynamic pricing problem is considered for single-leg airline revenue management. The dynamic programming formulation given for this problem is expressed in terms of the expected marginal revenue of capacity. In order to make the formulation applicable in practice, approximations are proposed in this study for estimating the expected marginal revenue term. Numerical tests based on simulating the sales process show that the proposed approximations work well as compared to the exact dynamic programming model.

An Efficient Algorithm for Dynamic Pricing Using a Graphical Representation

We study a multi‐period, multi‐item dynamic pricing problem faced by a retailer. The objective is to maximize the total profit by choosing prices, while satisfying several business rules. The strength of our work lies in our graphical model reformulation, which allows us to use ideas from combinatorial optimization. We do not make any assumptions on the structure of the demand function. The complexity of our method depends linearly on the number of time periods but is exponential in the memory of the model (number of past prices that affect current demand) and in the number of items. We prove that the profit maximization problem is NP‐hard by showing an approximation preserving reduction from the weighted Max‐3‐SAT problem. We next introduce the discrete reference price model which is a discretized version of the reference price model, accounting for an exponentially smoothed contribution of all past prices. We show that our problem can be solved efficiently under this model. We then approximate common demand functions using the discrete reference price model. To handle cross‐item effects among multiple items, we propose to use a virtual reference price that assigns a reference price for each category of items (as opposed to a reference price for each item). To enhance the tractability of our approach, we cluster items into blocks and show how to adapt our method to include business constraints across blocks. Finally, we apply our solution approach using demand models calibrated with supermarket data and validate its practical performance.

The energy management strategies based on dynamic energy pricing for community integrated energy system considering the interactions between suppliers and users

The proliferation of cogeneration technology, information and communication technology has facilitated the integrations of different energy infrastructures and markets. In this paper, the energy management of an integrated energy system composed of multiple energy hub operators and numerous integrated energy providers is formulated as dynamic energy pricing problem. From the perspective of maximizing social welfare, a distributed gradient projection iterative algorithm is proposed to decide the dynamic energy prices and optimal operation strategies of all the participants. Furthermore, the dual decomposition method is employed to decouple the coupled constraints and the primal problem is decomposed into several independent subproblems which can be solved in a distributed manner without violating privacy. Simulation results indicate that the proposed energy pricing algorithm shows a good convergence and availability, which can guarantee the supply-demand balance in each time slot, as well as promote the interactions between the energy supply and consumption sides.

Robust Dynamic Pricing with Demand Learning in the Presence of Outlier Customers

This paper studies the dynamic pricing problem under model mis-specifi cation settings. To characterize the model mis-specification, we extend the eps-contamination model | the most fundamental model in robust statistics and machine learning, to the online setting. In particular, for a selling horizon of length T, the online eps-contamination model assumes that the demands are realized according to a typical unknown demand function only for (1-eps)T periods. For the rest of eps T periods, an outlier purchase can happen with arbitrary demand functions. Under this model, we develop new robust dynamic pricing policies to hedge against outlier purchase behavior. For the dynamic pricing problem, there are two critical prices, the revenue-maximizing price and inventory clearance price, and the optimal price is the larger price. The challenge is that the seller has no information about which price is larger, and the revenues near these two prices behave entirely differently. To address this challenge, we propose robust online policies for both cases when the optimal price is the revenue-maximizing price and when the optimal price is the clearance price, and then develop a meta algorithm that combines these two cases. Our algorithm is a fully adaptive policy that does not require any prior knowledge of the outlier proportion parameter . Our simulation study shows that our policy outperforms existing policies in the literature.

A Joint Dynamic Pricing, Advertising, and Production Model with Inventory-Level-Dependent Goodwill

Inventory level has a significant impact on the goodwill of products to customers, which seldom becomes the focus of previous studies. In this paper, joint dynamic pricing, advertising, and production decision-making problem is investigated, where the demand rate depends on sales price and goodwill. The inventory and backlog as well as advertisement are considered as goodwill-building factors. The optimal dynamic pricing, advertising, and production policies are derived by using Pontryagin’s maximum principle. Numerical examples are provided to demonstrate the obtained results, and sensitivity analysis of main system parameters is carried out to obtain some managerial insights. We find that when the initial goodwill is relatively high, the firm’s profit first decreases and then increases with respect to the impact intensity of inventory on goodwill; otherwise, the firm always benefits from a higher impact intensity of inventory on goodwill. Furthermore, the optimal production and advertising policies are complementary caused by the feature of inventory-dependent goodwill.

An Optimal Pricing Strategy with Cannibalization

Cannibalization in pricing happens when the market share of some product categories decreases as a result of changes in the prices of other product categories. Despite the conventional wisdom of revenue management in avoiding cannibalization, we show that under certain conditions, it can be a part of an optimal pricing strategy in revenue maximization. In this paper, we focus on a capacity- constrained dynamic pricing problem faced by a firm in selling differentiated classes of products with price ranges. In case there are price overlaps across product classes, the higher class will cannibalize the market share of the lower class. We present real- world examples of such situations. Using a Markovian decision problem (MDP) approach, we determine the optimal price ranges for product classes. The result shows that having overlaps in price ranges is part of the optimal pricing strategy. Next, using data collected from a European storage rental company, we use simulations to better understand how firms can benefit from our recommendations. Our results show that when the capacity constraints are tight in the lower class and less stringent in the higher class, in other words when firms profit from scattering the demand over different product classes, cannibalization is part of the optimal pricing strategy. Firms can use these insights to nudge their customers to act in a way that makes both customers and the firm better off.

A Control Theoretic Approach to Simultaneously Estimate Average Value of Time and Determine Dynamic Price for High-Occupancy Toll Lanes

The dynamic pricing problem of a freeway corridor with high-occupancy toll (HOT) lanes was formulated and solved based on a point queue abstraction of the traffic system <xref ref-type="bibr" rid="ref1" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[1]</xref> . However, existing pricing strategies cannot guarantee that the closed-loop system converges to the optimal state, in which the HOT lanes’ capacity is fully utilized but there is no queue on the HOT lanes, and a well-behaved estimation and control method is quite challenging and still elusive. This paper attempts to fill the gap by making three fundamental contributions: (i) to present a simpler formulation of the point queue model based on the new concept of residual capacity, (ii) to propose a simple feedback control theoretic approach to estimate the average value of time and calculate the dynamic price, and (iii) to analytically and numerically prove that the closed-loop system is stable and guaranteed to converge to the optimal state, in either Gaussian or exponential manners.

Dynamic pricing in electricity and natural gas distribution networks: An EPEC model

Abstract Independent clearing of coupled retail electricity and natural gas markets generally results in a loss of market efficiency. This study investigates the problem of dynamic pricing in retail electricity and natural gas markets in the presence of network constraints by developing a two-leader multiple-follower bi-level model. Here, upper-level electricity and gas utility companies serve as leaders in the model, and determine their optimal dynamic retail prices in anticipation of the demand response (DR) provided by multiple lower-level integrated third-party DR providers or aggregators. Moreover, the model considers the dynamic gas prices provided by gas utility companies to electric utility companies for distributed gas-fired power units, which coordinates the operations of the separate utilities. The resulting model is recast as an equilibrium problem with equilibrium constraints (EPEC). Numerical results for a test system composed of an electric grid with distributed gas-fired power units, a gas distribution network, and multiple integrated DR aggregators indicate that the economic value of the proposed dynamic pricing scheme is up to 5.5% compared with existing regular pricing schemes.

Approximate dynamic programming for stochastic resource allocation problems

A stochastic resource allocation model, based on the principles of Markov decision processes &#x0028 MDPs &#x0029 , is proposed in this paper. In particular, a general-purpose framework is developed, which takes into account resource requests for both instant and future needs. The considered framework can handle two types of reservations &#x0028 i.e., specified and unspecified time interval reservation requests &#x0029 , and implement an overbooking business strategy to further increase business revenues. The resulting dynamic pricing problems can be regarded as sequential decision-making problems under uncertainty, which is solved by means of stochastic dynamic programming &#x0028 DP &#x0029 based algorithms. In this regard, Bellman’s backward principle of optimality is exploited in order to provide all the implementation mechanisms for the proposed reservation pricing algorithm. The curse of dimensionality, as the inevitable issue of the DP both for instant resource requests and future resource reservations, occurs. In particular, an approximate dynamic programming &#x0028 ADP &#x0029 technique based on linear function approximations is applied to solve such scalability issues. Several examples are provided to show the effectiveness of the proposed approach.

Scalable relaxation techniques to solve stochastic dynamic multi-product pricing problems with substitution effects

In many businesses, firms are selling different types of products, which share mutual substitution effects in demand. To compute effective pricing strategies is challenging as the sales probabilities of each of a firm’s products can also be affected by the prices of potential substitutes. In this paper, we analyze stochastic dynamic multi-product pricing models for the sale of perishable goods. To circumvent the limitations of time-consuming optimal solutions for highly complex models, we propose different relaxation techniques, which allow to reduce the size of critical model components, such as the state space, the action space, or the set of potential sales events. Our heuristics are able to decrease the size of those components by forming corresponding clusters and using subsets of representative elements. Using numerical examples, we verify that our heuristics make it possible to dramatically reduce the computation time while still obtaining close-to-optimal expected profits. Further, we show that our heuristics are (i) flexible, (ii) scalable, and (iii) can be arbitrarily combined in a mutually supportive way.

An Intelligent Computation Demand Response Framework for IIoT-MEC Interactive Networks

The joint optimization problem of a 5G-inspired IIoT-MEC interactive network aims to maximize revenue of MNOs and minimize IIoT operators’ economic cost is formulated, which is challenging to be solved. This letter proposes a dynamic pricing strategy for IIoT-MEC network to maximize MNOs’ revenue while providing acceptable service prices for IIoT mobile devices (MDs). The dynamic pricing problem is first modeled as a discrete finite Markov decision process (MDP). Then Q-learning algorithm is utilized to solve this problem. The results show that the proposed dynamic pricing strategy can significantly enhance MNOs’ revenue and decrease IIoT operators’ economic cost.

Coordinating Inventory and Pricing Decisions Under Total Minimum Commitment Contracts

A total minimum commitment contract is a supply contract under which a firm commits to buying a minimum quantity of a product from its supplier during the contract duration (e.g., one year). Such contracts are widely used in industries, because they provide the buyer with flexibility in terms of the timing and size of each individual order and the supplier with a guaranteed total order volume. Previous studies on such contracts are primarily concerned with the firm's inventory decisions, but none of them has considered the coordination of inventory and pricing decisions. In this paper, we fill this gap by studying dynamic inventory and pricing problems under a commitment contract. Under a general demand model with backlogging and zero lead time, we prove that the optimal policy is a committed-inventory-position-dependent base-stock list-price policy and characterize its structural properties. We also conduct an extensive numerical study to derive further managerial insights. In particular, we find that dynamic pricing can substantially improve the firm's profitability and enables it to select contracts with large committed quantities and price discount rates, and ignoring the commitment in joint pricing and ordering decisions can result in substantial profit loss. Finally, we partly extend our results to a lost-sales model and a backlogging model with positive lead times.

Dynamic Pricing and Optimal Control for a Stochastic Inventory System with Non-Instantaneous Deteriorating Items and Partial Backlogging

In this paper, we consider a problem of the dynamic pricing and inventory control for non-instantaneous deteriorating items with uncertain demand, in which the demand is price-sensitive and governed by a diffusion process. Shortages and remains are permitted, and the backlogging rate is variable and dependent on the waiting time for the next replenishment. In order to maximize the expected total profit, the problem of dynamic pricing and inventory control is described as a stochastic optimal control problem. Based on the dynamic programming principle, the stochastic control model is transformed into a Hamilton-Jacobi-Bellman (HJB) equation. Then, an exact expression for the optimal dynamic pricing strategy is obtained via solving the HJB equation. Moreover, the optimal initial inventory level, the optimal selling pricing, the optimal replenishment cycle and the optimal expected total profit are achieved when the replenishment cycle starts at time 0. Finally, some numerical simulations are presented to demonstrate the analytical results, and the sensitivities analysis on system parameters are carried out to provide some suggestions for managers.

A green transportation location-inventory-routing problem by dynamic regional pricing

Non-uniform distribution of customers in a region and variation of their maximum willingness to pay at distinct areas make regional pricing a practical method to maximize the profit of the distribution system. By subtracting the classic objective function, which minimizes operational costs from revenue function, profit maximization is aimed. A distribution network is designed by determining the number of trucks to each established distribution center, allocating customers in routes, and inventory levels of customers. Also, environmental impacts, including fuel consumption and CO2 emission, aimed to be minimized. So, a new quadratic mixed-integer programming model is presented for the Green Transportation Location-Inventory-Routing Problem integrated with dynamic regional pricing problem (GTLIRP+DRP). The model is applied to the real case study, to show its competent application. To tackle this problem, a Hybrid Bees Algorithm (HBA) is developed and verified by the genetic algorithm. Finally, managers suggested using HBA that achieves better solutions in the less computational time.

Machine learning applications in nonlife insurance

AbstractThe literature on analytical applications in insurance tends to be either very general or rather technical, which may hold back the adoption of new important tools by industrial practitioners. Our goal is to stress that machine learning (ML) algorithms will play a significant role in the insurance industry in the near future and thus to encourage practitioners to learn and apply these techniques. After discussing the increasing relevance of data for nonlife insurance and briefly reviewing the major impact of digital technology on this business, we restrict our discussion to technical analytical applications and indicate where ML algorithms can add most value. We present two real examples: first a comparison of retention models for household insurance and then a dynamic pricing problem for online motor insurance. Both applications illustrate the advantages but also some of the difficulties of applying ML tools in practice. Finally, we mention some challenges posed by the use of ML in the industry and formulate a few recommendations for successful applications in insurance. This article is neither a tutorial nor an exhaustive review of technical ML applications in nonlife insurance. However, references for additional learning materials are provided.

Technical note: Finite‐time regret analysis of Kiefer‐Wolfowitz stochastic approximation algorithm and nonparametric multi‐product dynamic pricing with unknown demand

AbstractWe consider the problem of nonparametric multi‐product dynamic pricing with unknown demand and show that the problem may be formulated as an online model‐free stochastic program, which can be solved by the classical Kiefer‐Wolfowitz stochastic approximation (KWSA) algorithm. We prove that the expected cumulative regret of the KWSA algorithm is bounded above by where κ1, κ2 are positive constants and T is the number of periods for any T = 1, 2, …. Therefore, the regret of the KWSA algorithm grows in the order of , which achieves the lower bounds known for parametric dynamic pricing problems and shows that the nonparametric problems are not necessarily more difficult to solve than the parametric ones. Numerical experiments further demonstrate the effectiveness and efficiency of our proposed KW pricing policy by comparing with some pricing policies in the literature.

Dynamic Pricing for Electric Vehicle Extreme Fast Charging

Significant developments and advancement pertaining to electric vehicle (EV) technologies, such as extreme fast charging (XFC), have been witnessed in the last decade. However, there are still many challenges to the wider deployment of EVs. One of the major barriers is its availability of fast charging stations. A possible solution is to build a fast charging sharing system, by encouraging small business owners or even householders to install and share their fast charging devices, by reselling electricity energy sourced from traditional utility companies or their own solar grid. To incentivize such a system, a smart dynamic pricing scheme is needed to facilitate those growing markets with fast charging stations. The pricing scheme is expected to take into account the dynamics intertwined with pricing, demand, and environment factors, in an effort to maximize the long-term profit with the optimal price. To this end, this paper formulates the problem of dynamic pricing for fast charging as a Markov decision process and accordingly proposes several algorithmic schemes for different applications. Experimental study is conducted with useful and interesting insights.