Network Revenue Management Problem Research Articles

Network Revenue Management With Demand Learning and Fair Resource-Consumption Balancing

In addition to maximizing the total revenue, decision-makers in lots of industries would like to guarantee balanced consumption across different resources. For instance, in the retailing industry, ensuring a balanced consumption of resources from different suppliers enhances fairness and helps maintain a healthy channel relationship; in the cloud computing industry, resource-consumption balance helps increase customer satisfaction and reduce operational costs. Motivated by these practical needs, this paper studies the price-based network revenue management (NRM) problem with both demand learning and fair resource-consumption balancing. We introduce the regularized revenue, that is, the total revenue with a balancing regularization, as our objective to incorporate fair resource-consumption balancing into the revenue maximization goal. We propose a primal-dual-type online policy with the upper-confidence-bound demand learning method to maximize the regularized revenue. We adopt several innovative techniques to make our algorithm a unified and computationally efficient framework for the continuous price set and a wide class of balancing regularizers. Our algorithm achieves a worst-case regret of [Formula: see text], where [Formula: see text] denotes the number of products and [Formula: see text] denotes the number of time periods. Numerical experiments in a few NRM examples demonstrate the effectiveness of our algorithm in simultaneously achieving revenue maximization and fair resource-consumption balancing.

Performance Guarantees for Network Revenue Management with Flexible Products

Problem definition: We consider network revenue management problems with flexible products. We have a network of resources with limited capacities. To each customer arriving into the system, we offer an assortment of products. The customer chooses a product within the offered assortment or decides to leave without a purchase. The products are flexible in the sense that there are multiple possible combinations of resources that we can use to serve a customer with a purchase for a particular product. We refer to each such combination of resources as a route. The service provider chooses the route to serve a customer with a purchase for a particular product. Such flexible products occur, for example, when customers book at-home cleaning services but leave the timing of service to the company that provides the service. Our goal is to find a policy to decide which assortment of products to offer to each customer to maximize the total expected revenue, making sure that there are always feasible route assignments for the customers with purchased products. Methodology/results: We start by considering the case in which we make the route assignments at the end of the selling horizon. The dynamic programming formulation of the problem is significantly different from its analogue without flexible products as the state variable keeps track of the number of purchases for each product rather than the remaining capacity of each resource. Letting L be the maximum number of resources in a route, we give a policy that obtains at least [Formula: see text] fraction of the optimal total expected revenue. We extend our policy to the case in which we make the route assignments periodically over the selling horizon. Managerial implications: To our knowledge, the policy that we develop is the first with a performance guarantee under flexible products. Thus, our work constructs policies that can be implemented in practice under flexible products, also providing performance guarantees. Funding: The work of H. Topaloglu was partly funded by the National Science Foundation [Grant CMMI-1825406]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/msom.2022.0583 .

Network revenue management with online inverse batch gradient descent method

We consider a general class of price‐based network revenue management problems that a firm aims to maximize revenue from multiple products produced with multiple types of resources endowed with limited inventory over a finite selling season. A salient feature of our problem is that the firm does not know the underlying demand function that maps prices to demand rate, which must be learned from sales data. It is well known that for almost all classes of demand functions, such as linear, exponential, multinomial logit, and nested logit models, the revenue rate function is not concave in the products' prices but is concave in products' market shares (or price‐controlled demand rates). This creates challenges in adopting any stochastic gradient descent based methods in the price space. We propose a novel nonparametric learning algorithm termed the online inverse batch gradient descent (IGD) algorithm. This algorithm proceeds in batches. In each batch, the firm implements each product's perturbed prices and then uses the sales information to estimate the market shares. Leveraging these estimates, the firm carries out a stochastic gradient descent step in the market share space that takes into account the relative inventory scarcity for the entire horizon and then inversely maps the updated market shares back to the price space to obtain the prices for the next batch. Moreover, we also propose an inventory‐adjusted algorithm (IGD‐I) that the feasible market share set is dynamically adjusted to capture the real‐time relative inventory scarcity for the remaining season. For the large‐scale systems wherein all resources' inventories and the length of the horizon are proportionally scaled by a parameter k, we establish a dimension‐independent regret bound of [Formula: see text]. This result is independent of the number of products and resources and works for a continuum action‐set prices and the demand functions that are only once differentiable. Our theoretical result guarantees the efficacy of both algorithms in the high‐dimensional systems where the number of products or resources is large and the prices are continuous. Our algorithms also numerically outperform the existing algorithms in the literature.

Assign-to-Seat: Dynamic Capacity Control for Selling High-Speed Train Tickets

Problem definition: We consider a revenue management problem that arises from the selling of high-speed train tickets in China. Compared with traditional network revenue management problems, the new feature of our problem is the assign-to-seat restriction. That is, each request, if accepted, must be assigned instantly to a single seat throughout the whole journey, and later adjustment is not allowed. When making decisions, the seller needs to track not only the total seat capacity available, but also the status of each seat. Methodology/results: We build a modified network revenue management model for this problem. First, we study a static problem in which all requests are given. Although the problem is NP-hard in general, we identify conditions for solvability in polynomial time and propose efficient approximation algorithms for general cases. We then introduce a bid-price control policy based on a novel maximal sequence principle. This policy accommodates nonlinearity in bid prices and, as a result, yields a more accurate approximation of the value function than a traditional bid-price control policy does. Finally, we combine a dynamic view of the maximal sequence with the static solution of a primal problem to propose a “re-solving a dynamic primal” policy that can achieve uniformly bounded revenue loss under mild assumptions. Numerical experiments using both synthetic and real data document the advantage of our proposed policies on resource-allocation efficiency. Managerial implications: The results of this study reveal connections between our problem and traditional network revenue management problems. Particularly, we demonstrate that by adaptively using our proposed methods, the impact of the assign-to-seat restriction becomes limited both in theory and practice. Funding: S. Liu’s research is partly supported by the National Natural Science Foundation of China (NSFC) [Grant NSFC-72072117]. Z. Wang’s research is partly supported by the NSFC [Grant NSFC-72150002]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/msom.2023.1188 .

Reductions of non-separable approximate linear programs for network revenue management

We suggest a novel choice of non-separable basis functions for an approximate linear programming approach to the well-known network revenue management problem. Considering non-separability is particularly important when interdependencies between resources are large. Such a situation can be illustrated for example by a bus line, where different origin-destination pairs have many overlapping segments. Traditional separable approximation approaches tend to ignore the resulting interactions.We suggest to group resources into non-separable subnetworks. For each chosen subnetwork, basis functions either span the whole function space or consist of linear functions. Given this more general choice of basis functions, we extend existing reductions of approximate linear programs. If there is only one subnetwork, for which the basis functions span the whole function space, we prove the equivalence to a compact linear program of polynomial size. For the general case, we suggest an approximate reduction. Numerical examples illustrate our novel upper bounds for the maximum expected revenue and the corresponding competitive policies. In particular, we find that the added benefit of non-separability heavily depends on the network structure and the capacity.Our work helps to better understand the impact of assuming separability in network revenue management. The polynomial sized reductions make it possible to estimate the added average revenue resulting from incorporating interactions between resources. The theory we develop demonstrates how the interpretation of dual variables as state-action probabilities can be applied to reduce exponentially large approximate linear programs via variable aggregation.

Bid price controls for car rental network revenue management

We consider a car rental network revenue management (RM) problem, accounting for the key operational characteristics of car rental services such as the varying length of rentals and mobility of inventories, which imply the intertemporal and spatial correlations of rental demands for inventories across different locations and days. The problem is formulated as an infinite‐horizon cyclic stochastic dynamic program to account for the time‐varying and cyclic nature of car rental businesses. To tackle the curse of dimensionality, we propose a Lagrangian relaxation (LR) approach with product‐ and time‐dependent Lagrangian multipliers to decomposing the dynamic network problem into multiple single‐station single‐day subproblems. We show that the Lagrangian dual problem is a convex program and then develop a subgradient‐based algorithm to solve the dual problem and derive an LR‐based bid price policy. To improve the scalability of the LR approach, we further propose three simpler LR‐based bid price policy variants with either location‐dependent or leadtime‐dependent Lagrangian multipliers, or both. Our numerical study indicates that the LR‐based bid price policies can outperform some commonly used heuristics. Using a set of real‐world booking data, we provide a case study in which we empirically demonstrate the operational characteristics of car rental services, calibrate the arrival process of booking requests using a Poisson regression model, and demonstrate that the LR‐based bid price policies indeed outperform other heuristics consistently in both in‐sample and out‐of‐sample horizons.

Masking primal and dual models for data privacy in network revenue management

We study a collaborative revenue management problem where multiple decentralized parties agree to share some of their capacities. This collaboration is performed by constructing a large mathematical programming model that is available to all parties. The parties then use the solution of this model in their own capacity control systems. In this setting, however, the major concern for the parties is the privacy of their input data, along with their individual optimal solutions. We first reformulate a general linear programming model that can be used for a wide range of network revenue management problems. Then we address the data privacy concern of the reformulated model and propose an approach based on solving an equivalent data-private model constructed with input masking via random transformations. Our main result shows that, after solving the data-private model, each party can safely access only its own optimal capacity allocation decisions. We also discuss the security of the transformed problem in the considered multi-party setting. Simulation experiments are conducted to support our results and evaluate the computational efficiency of the proposed data-private model. Our work provides an analytical approach and insights on how to manage shared resources in a network problem while ensuring data privacy. Constructing and solving a collaborative network problem requires information exchange between parties that may not be possible in practice. Including data privacy in decentralized collaborative network revenue management problems with capacity sharing is new to the literature and relevant to practice.

Technical Note—Product-Based Approximate Linear Programs for Network Revenue Management

A Novel and Promising Approximation for Network Revenue Management In “Product-Based Approximate Linear Programs for Network Revenue Management,” Zhang, Samiedaluie, and Zhang propose a novel separable piecewise linear (SPL) approximation for the network revenue management problem. The coefficients of the proposed SPL approximation can be interpreted as each product’s revenue contribution to the value of each resource in a given period, which provides more granular information compared with the existing resource-based SPL approximation in the literature. The new approximation provides more flexibility for policy construction. Furthermore, the new approximation opens the opportunity to derive a set of valid inequalities to further improve the computational performance and achieve additional gains in the expected revenue. Computational experiments with instances of various network structures and parameters demonstrate its efficacy: the new approximation leads to bid-price policies generating higher expected revenues and demonstrates better performance in terms of both computational efficiency and numerical stability.

Constant Regret Resolving Heuristics for Price-Based Revenue Management

Title: Constant Regret Resolving Heuristics for Price-Based Revenue Management Network revenue management (NRM) and its corresponding pricing question is one of the most fundamental problems in operations management. To alleviate the curse of dimensionality and the prohibitive cost of computing an exact solution using dynamic programming, computationally efficient resolving algorithms are proposed. The state-of-the-art analysis of the resolving heuristic establishes a logarithmic additive regret for price-based NRM problems. In “Constant Regret Resolving Heuristics for Price-Based Revenue Management,” Y. Wang and H. Wang from the University of Florida and Georgia Institute of Technology, respectively, significantly advance the state-of-the-art analysis by showing a constant regret for resolving heuristics. Their theoretical advance is made possible by a novel, direct analysis of the exact DP solution.

Efficient Algorithms for Minimizing Compositions of Convex Functions and Random Functions and Its Applications in Network Revenue Management

In this paper, we study a class of nonconvex stochastic optimization in the form of $\min_{x\in\mathcal{X}} F(x):=\mathbb{E}_\xi [f(\phi(x,\xi))]$, where the objective function $F$ is a composition of a convex function $f$ and a random function $\phi$. Leveraging an (implicit) convex reformulation via a variable transformation $u=\mathbb{E}[\phi(x,\xi)]$, we develop stochastic gradient-based algorithms and establish their sample and gradient complexities for achieving an $\epsilon$-global optimal solution. Interestingly, our proposed Mirror Stochastic Gradient (MSG) method operates only in the original $x$-space using gradient estimators of the original nonconvex objective $F$ and achieves $\tilde{\mathcal{O}}(\epsilon^{-2})$ sample and gradient complexities, which matches the lower bounds for solving stochastic convex optimization problems. Under booking limits control, we formulate the air-cargo network revenue management (NRM) problem with random two-dimensional capacity, random consumption, and routing flexibility as a special case of the stochastic nonconvex optimization, where the random function $\phi(x,\xi)=x\wedge\xi$, i.e., the random demand $\xi$ truncates the booking limit decision $x$. Extensive numerical experiments demonstrate the superior performance of our proposed MSG algorithm for booking limit control with higher revenue and lower computation cost than state-of-the-art bid-price-based control policies, especially when the variance of random capacity is large. KEYWORDS: stochastic nonconvex optimization, hidden convexity, air-cargo network revenue management, gradient-based algorithms

On the Strength of Relaxations of Weakly Coupled Stochastic Dynamic Programs

Many stochastic dynamic programs (DPs) have a weakly coupled structure in that a set of linking constraints in each period couple an otherwise independent collection of subproblems. Two widely studied approximations of such problems are approximate linear programs (ALP), which involve optimizing value function approximations that additively separate across subproblems, and Lagrangian relaxations, which involve relaxing the linking constraints. It is well-known that both of these approximations provide upper bounds on the optimal value function in all states and that the ALP provides a tighter upper bound in the initial state. The purpose of this short paper is to provide theoretical justification for the fact that these upper bounds are often close if not identical. We show: (i) For any weakly coupled DP, the difference between these two upper bounds --- the relaxation gap --- is bounded from above in terms of the integrality gap of the constraint separation problems within the ALP; (ii) If subproblem rewards are uniformly bounded and some broadly applicable conditions on the linking constraints hold, the relaxation gap is bounded from above by a constant that is independent of the number of subproblems; and (iii) When subproblem actions are binary and the linking constraints have a unimodular structure, the relaxation gap equals zero. The conditions for (iii) hold in several widely studied problems: restless bandit problems, online stochastic matching problems, network revenue management problems, and price-directed control of relocating resources. These findings generalize and unify existing results.

Network Revenue Management with Time and Precedence Constraints

This paper explores the network revenue management problem where a firm sells products composed of multiple resources. Unlike the traditional network revenue management problem, the products have time and precedence constraints. This means that some products may be infeasible to deliver. To filter out those infeasible products, this paper proposes a simple linear programming. The firm starts the sales horizon only with the feasible products. The dynamic programming can be formulated to have the optimal policy, but it may suffer from the curse of dimensionality. To overcome this difficulty, this paper proposes the linear programming based heuristic. The numerical study shows that the heuristic can save the computation time greatly while providing a reasonably good solution.

A Linear Programming approach for robust network revenue management in the airline industry

The classical revenue management problem consists of allocating a fixed network capacity to different customer classes, so as to maximize revenue. This area has been widely applied in service industries that are characterized by a fixed perishable capacity, such as airlines, cruises, hotels, etc.It is traditionally assumed that demand is uncertain, but can be characterized as a stochastic process (See Talluri and van Ryzin (2005) for a review of the revenue management models). In practice, however, airlines have limited demand information and are unable to fully characterize demand stochastic processes. Robust optimization methods have been proposed to overcome this modeling challenge. Under robust optimization framework, demand is only assumed to lie within a polyhedral uncertainty set (Lan et al. (2008); Perakis and Roels (2010)).In this paper, we consider the multi-fare, network revenue management problem for the case demand information is limited (i.e. the only information available is lower/upper bounds on demand). Under this interval uncertainty, we characterize the robust optimal booking limit policy by use of minimax regret criterion. We present an LP (Linear Programming) solvable mathematical program for the maximum regret so our model is able to solve large-scale problems for practical use. A genetic algorithm is proposed to find the booking limit control to minimize the maximum regret. We provide computational experiments and compare our methods to existing ones. The results demonstrate the effectiveness of our robust approach.

A Re-Solving Heuristic with Uniformly Bounded Loss for Network Revenue Management

We consider a canonical quantity-based network revenue management problem where a firm accepts or rejects incoming customer requests irrevocably in order to maximize expected revenue given limited resources. Because of the curse of dimensionality, the exact solution to this problem by dynamic programming is intractable when the number of resources is large. We study a family of re-solving heuristics that periodically re-optimize an approximation to the original problem known as the deterministic linear program (DLP), where random customer arrivals are replaced by their expectations. We find that, in general, frequently re-solving the DLP produces the same order of revenue loss as one would get without re-solving, which scales as the square root of the time horizon length and resource capacities. By re-solving the DLP at a few selected points in time and applying thresholds to the customer acceptance probabilities, we design a new re-solving heuristic with revenue loss that is uniformly bounded by a constant that is independent of the time horizon and resource capacities. This paper was accepted by Kalyan Talluri, revenue management and market analytics.

Managing Flexible Products on a Network

A flexible product is a menu of two or more alternatives products serving the same market. Purchasers of flexible products are assigned to one of the alternatives at a later date. Gallego and Phillips show that capacitated suppliers, such as airlines and hotels, can potentially improve revenue by offering flexible products in addition to traditional specific products. In this paper, we extend the concept of flexible products to networks. We study the network revenue management problem with flexible products in two different settings: one where the demand for each product is independently and exogenously generated; and the other where the demand is driven by a consumer choice model. We show that in both these settings the optimal value of the stochastic optimization problem can be closely approximated by the optimal value of a deterministic linear program. In the independent demand case the corresponding linear program is of modest size. When the demand is driven by a customer choice model, the linear program has exponentially many variables; however, we show that for an important class of consumer choice models the linear program can be efficiently solved using column-generation. We report the findings of numerical experiments with a real airline subnetwork and show how the results vary as a function of the key inputs.

Near-Optimal Primal-Dual Algorithms for Quantity-Based Network Revenue Management

We study the canonical quantity-based network revenue management (NRM) problem where the decision-maker must irrevocably accept or reject each arriving customer request with the goal of maximizing the total revenue given limited resources. The exact solution to the problem by dynamic programming is computationally intractable due to the well-known curse of dimensionality. Existing works in the literature make use of the solution to the deterministic linear program (DLP) to design asymptotically optimal algorithms. Those algorithms rely on repeatedly solving DLPs to achieve near-optimal regret bounds. It is, however, time-consuming to repeatedly compute the DLP solutions in real time, especially in large-scale problems that may involve hundreds of millions of demand units. In this paper, we propose innovative algorithms for the NRM problem that are easy to implement and do not require solving any DLPs. Our algorithm achieves a regret bound of $O(\log k)$, where $k$ is the system size. To the best of our knowledge, this is the first NRM algorithm that (i) has an $o(\sqrt{k})$ asymptotic regret bound, and (ii) does not require solving any DLPs.

Bifurcating Constraints to Improve Approximation Ratios for Network Revenue Management with Reusable Resources

In the network revenue management (NRM) problem, itineraries consisting of capacity-constrained are sold to a stream of online demand. Approximate dynamic programming (ADP) and prophet inequalities represent two broad methods for deriving approximation algorithms for NRM, and in this paper we dissect their pros and cons. If every itinerary uses at most L legs, then ADP guarantees a 1/(1+L)-approximation. If the legs correspond to the availabilities of reusable resources over time, then a 1/2-approximation is known, but only for L=1. Meanwhile, prophet inequalities guarantee a 1/2-approximation when the capacity constraints follow a matroid structure, but they cannot handle reusable resources. Accordingly, we propose a policy which bifurcates the capacity constraints, and uses each method to handle its specialties. Our guarantee is 1/(2(1+L')) under both matroid constraints and other capacity constraints from which each itinerary uses at most L' capacities; we note that L' could be much smaller than L. Our guarantee generalizes to reusable resources, choice models, and improves to 1/(1+L') with the matroid constraint removed. We conduct simulations on NRM instances containing legs corresponding to a sequence of which satisfies the matroid structure. We find that bifurcating the add-on constraints always improves the empirical performance, and moreover, the improvement from bifurcation is linear in the number of add-ons, which is consistent with our improvement in theoretical guarantee. Whereas existing work has proposed one-size-fits-all methods that handle all the constraints generically, our paper demonstrates the benefit of analyzing network structure and using different methods to handle different constraints.

Assign-to-Seat: Dynamic Capacity Control for Selling Train Tickets

We consider a revenue management problem that arises from the selling of high-speed train tickets in China. Compared with traditional network revenue management problems, the new feature of our problem is that each request, if accepted, must be assigned instantly to a single seat throughout the whole journey, and later adjustment is not allowed. When making decisions, the seller needs to track not only the total seat capacity available but also the status of each seat. We build a modified network revenue management model for this problem. First, we study a static problem in which all requests are given. Although the problem is NP-hard in general, we identify conditions for solvability in polynomial time. We then introduce a bid-price control policy based on a novel maximal sequence principle. This policy accommodates nonlinearity in bid prices and, as a result, yields a more accurate approximation of the value function than a traditional bid-price control policy does. Finally, we combine a dynamic view of the maximal sequence with the static solution of a primal problem to propose a re-solving a dynamic primal'' policy that can achieve uniformly bounded revenue loss under mild assumptions. The results of this study also reveal connections between our problem and traditional network revenue management problems. Numerical experiments using both synthetic and real data document the advantage of the policies we propose for improving the efficiency of capacity allocation.

Choice Network Revenue Management Based on New Tractable Approximations

The choice network revenue management model incorporates customer purchase behavior as probability of purchase as a function of the offered products and is appropriate for airline and hotel network revenue management, dynamic sales of bundles, and dynamic assortment optimization. The optimization problem is a stochastic dynamic program and is intractable. Consequently, a linear programming approximation called choice deterministic linear program (CDLP) is usually used to generate controls. Tighter approximations, such as affine and piecewise-linear relaxations, have been proposed, but it was not known if they can be solved efficiently even for simple models, such as the multinomial logit (MNL) model with a single segment. We first show that the affine relaxation (and, hence, the piecewise-linear relaxation) is NP-hard even for a single-segment MNL choice model. By analyzing the affine relaxation, we derive a new linear programming approximation that admits a compact representation, implying tractability, and prove that its value falls between the CDLP value and the affine relaxation value. This is the first tractable relaxation for the choice network revenue management problem that is provably tighter than CDLP. This approximation, in turn, leads to new policies that, in our numerical experiments, show very good promise: a 2% increase in revenue on average over CDLP and the values typically coming very close to the affine relaxation. We extend our analysis to obtain other tractable approximations that yield even tighter bounds. We also give extensions to the case with multiple customer segments with overlapping consideration sets in which choice by each segment is according to the MNL model.

Network Revenue Management with Limited Switches: Known and Unknown Demand Distributions

Our work is motivated by a common business constraint in online markets. While firms respect the advantages of dynamic pricing and price experimentation, they must limit the number of price changes (i.e., switches) to be within some budget due to various practical reasons. We study both the classical price-based network revenue management problem in the distributionally-unknown setup, and the bandits with knapsacks problem. In these problems, a decision-maker (without prior knowledge of the environment) has finite initial inventory of multiple resources to allocate over a finite time horizon. Beyond the classical resource constraints, we introduce an additional switching constraint to these problems, which restricts the total number of times that the decision-maker makes switches between actions to be within a fixed switching budget. For such problems, we show matching upper and lower bounds on the optimal regret, and propose computationally-efficient limited-switch algorithms that achieve the optimal regret. Our work reveals a surprising result: the optimal regret rate is completely characterized by a piecewise-constant function of the switching budget, which further depends on the number of resource constraints --- to the best of our knowledge, this is the first time the number of resources constraints is shown to play a fundamental role in determining the statistical complexity of online learning problems. We conduct computational experiments to examine the performance of our algorithms on a numerical setup that is widely used in the literature. Compared with benchmark algorithms from the literature, our proposed algorithms achieve promising performance with clear advantages on the number of incurred switches. Practically, firms can benefit from our study and improve their learning and decision-making performance when they simultaneously face resource and switching constraints.