Heuristic solutions for nonlinear dynamic pricing in the presence of multiunit demand and two-dimensional customer heterogeneity

Schur, Rouven Article — Published Version Heuristic solutions for nonlinear dynamic pricing in the presence of multiunit demand and two-dimensional customer heterogeneity OR Spectrum Suggested Citation: Schur, Rouven (2025) : Heuristic solutions for nonlinear dynamic pricing in the presence of multiunit demand and two-dimensional customer heterogeneity, OR Spectrum, ISSN 1436-6304, Springer Berlin Heidelberg, Berlin/Heidelberg, Vol. 47, Iss. 4, pp. 1181-1215, https://doi.org/10.1007/s00291-025-00820-3 This Version is available at: https://hdl.handle.net/10419/333241 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by/4.0/ Vol.:(0123456789) OR Spectrum (2025) 47:1181–1215 https://doi.org/10.1007/s00291-025-00820-3 ORIGINAL ARTICLE Heuristic solutions fornonlinear dynamic pricing inthepresence ofmultiunit demand andtwo‑dimensional customer heterogeneity RouvenSchur1 Received: 27 February 2024 / Accepted: 30 April 2025 / Published online: 26 May 2025 © The Author(s) 2025 Abstract In this paper, we introduce a nonlinear dynamic pricing model in the presence of multiunit demand, enabling firms to quote separate prices for each batch size. This approach diverges from traditional models by accounting for two-dimensional customer heterogeneity in product attraction and batch size preference, each modeled by separate random variables in the calculation of customers’ willingness-to-pay. The underlying customer choice model results in a complex formulation of purchase probabilities, necessitating considerable effort for refinements to derive a manageable expression. We present optimality conditions for the state-wise optimization problem and introduce a modified formulation with reduced complexity that serves as an upper bound. We also prove that under specific conditions, the optimal solution to the modified model is optimal for the original problem. In our numerical study, these conditions were consistently met, offering a practical alternative for determining optimal prices. To address the computational challenges of solving the problem to optimality, we develop three efficient heuristics with significantly reduced runtimes. Benchmarking these heuristics against the optimal solution and other mechanisms demonstrates their near-optimal performance. We also evaluate the revenue potential of nonlinear, piecewise linear, and linear pricing schemes, providing firms with tools to weigh revenue maximization against pricing simplicity to inform strategic decisions. Notably, our analysis highlights the strong performance of piecewise linear pricing, offering a practical and easy-to-communicate alternative to full nonlinear pricing while achieving remarkably high revenues. Keywords Revenue management· Dynamic pricing· Nonlinear pricing· Multiunit demand· Customer choice * Rouven Schur rouv[email protected] 1 Chair ofProduction & Logistics Planning, Mercator School ofManagement, University ofDuisburg-Essen, Lotharstraße 65, 47057Duisburg, Germany 1182 R.Schur 1 Introduction Nonlinear pricing—such as volume discounts or special offers like "buy 3, pay 2"—were and still are a commonly applied pricing strategy in the ever-evolving field of retail. This approach is used both for individual products (e.g., cans of soda) and across product lines (e.g., shirts of different sizes, colors, or designs). Enabled by advancements in digital technologies (e.g., e-commerce and digital price tags), businesses can now adjust prices in real-time. This allows for swift responses to varying market demands and inventory levels. However, most traditional dynamic pricing models have a critical limitation: they usually assume that customers purchase only a single unit at a time. This assumption overlooks the complexities and opportunities presented by multiunit demand, which are prevalent in sectors ranging from groceries to clothing. Addressing this gap, our paper presents an approach that combines nonlinear and dynamic pricing to optimally quote prices for every possible batch size of a product (or variant within a product line). This framework is designed to maximize revenue in scenarios where the selling horizon is finite, the product inventory is scarce, and customers exhibit multiunit demand. In our context, multiunit demand does not necessarily imply that customers are predetermined on the number of units they will purchase. Instead, it refers to settings where customers consider various quantities based on pricing schemes, promotions, or the needs of multiple end-users (e.g., family members, colleagues, or friends). For instance, consider a supermarket dynamically adjusting batch prices for perishable products, such as bakery items or fresh produce. Depending on realized demand and remaining inventory, retailers dynamically raise batch prices to capitalize on strong demand or lower them to encourage bulk purchases as products approach expiration. Although this strategy naturally reduces waste and mitigates early stockouts, its primary goal remains revenue maximization given limited stock and a finite selling horizon. Similar dynamic nonlinear pricing schemes apply naturally in various other sectors, where the product loses its value at a specific point in time due to factors such as perishability (as in food or hotel stays) or the necessity to clear valuable shelf space for new inventory (as in fashion, seasonal goods, and consumer electronics). Central to our approach is our customer choice model that recognizes the twodimensional heterogeneity among customers, both in their attraction to the product and in their inclination to consume. We employ a utility maximization framework, where customer utility is modeled as a function of batch size and the two customer-specific attributes: product attraction and consumption. These attributes are customers’ private information and therefore unknown to the firm. However, we assume the firm has knowledge of the distribution of these attributes across the customer population. This allows us to treat these attributes as random variables, forming the basis for our random utility framework. With these assumptions, we align closely with a recent publication (see Schur 2024), where both studies aim to maximize expected revenues through dynamic nonlinear pricing and employ the same formulation of customers’ utility based 1183 Heuristic solutions fornonlinear dynamic pricing inthe… on two customer-specific attributes. Accordingly, both publications are rooted in the same problem and use identical mathematical formulations for the choice model and the general dynamic program. However, unlike this publication, we do not make the additional assumption that the firm can identify customers’ private information—specifically, their attraction to the product, their consumption indicator, or both. This assumption in Schur (2024) simplifies the optimization problem by utilizing customers’ known attributes. In contrast, our model does not rely on observing private information, resulting in a more complex optimization problem. This extends the framework to settings where customer identification is not possible. In this way, our work generalizes the framework of Schur (2024). While techniques similar to those used in Schur (2024), specifically action space reduction and upper bound investigation, can be adapted to reduce complexity in our setting, we must adopt a fundamentally different strategy that focuses on developing effective heuristics. By not relying on private information, our approach remains also applicable in scenarios where firms must quote prices before any interaction with customers occurs, as is typically the case in stores using digital price tags. This flexibility makes our model particularly valuable for scenarios where advance pricing decisions are necessary, or where the firm lacks access to customers’ private information. 1.1 Results Our research achieves three interconnected goals: deriving the optimal solution, developing effective heuristics, and evaluating the trade-offs between nonlinear pricing and simpler pricing schemes. First, we establish optimality conditions that enable the computation of the optimal solution. While the optimal solution provides an important benchmark for evaluating the effectiveness of our heuristics, its computation remains impractical for real-world applications due to its substantial runtimes, requiring nearly five days in our largest setting. Second, to overcome these computational challenges, we propose three heuristics that result in well-performing solutions while significantly reducing runtimes. For example, one of these heuristics computes a policy in just 2.5min with an optimality gap below 0.4% in our largest setting, making it nearly 3000 times faster than computing the optimal solution. Similarly, the other heuristics also perform well, offering near-optimal solutions with runtimes suitable for practical application. Finally, using the optimal and heuristic solutions as benchmarks, we evaluate the revenue potential of different pricing schemes—nonlinear, piecewise linear, and linear pricing. Our numerical study demonstrates that nonlinear pricing consistently achieves the highest revenue. However, simpler pricing schemes, particularly piecewise linear pricing, can perform very strongly. This observation is supported by an analysis of the strategies employed by our best-performing heuristics, which reveal two key patterns. First, the pricing schemes exhibit an almost piecewise linear structure. This structure explains the strong performance of piecewise linear pricing in our study and suggests that firms could adopt this simpler pricing scheme to 1184 R.Schur reduce communication complexity. By quoting just two prices—one for the first unit and another for additional units—firms can achieve high revenues while maintaining ease of implementation. Second, in scenarios with high initial inventory ( C T≥2 ) prices for smaller batches decrease gradually over time. This gradual adjustment implies that less frequent price changes can still deliver strong performance, making such strategies viable for firms constrained by technical limitations, customer preferences, or strategic considerations. Conversely, in low-stock scenarios ( T C≥2 ), the revenue advantage of nonlinear pricing decreases, and standard dynamic pricing approaches become more effective. 1.2 Contribution andoutline We contribute to the sparse literature on multiunit dynamic pricing by addressing the challenging optimization problem of dynamically quoting batch prices while considering two-dimensional customer heterogeneity, represented by two random variables in the customer choice model. Our main contributions are as follows: C1. Optimality conditions: We derive optimality conditions by reducing the complexity of the customer choice model. Notably, we prove that the optimality condition holds whether the optimal solution is located in the interior or on the boundary of the action space, allowing us to focus on this condition without needing to separately investigate boundary cases. C2. Problem modification and upper bound: We introduce a modified version of the original problem that can be solved more efficiently and serves as an upper bound. We further prove that under specific conditions, the solution of the modified problem is optimal for the original problem. C3. Development of heuristics: We develop three efficient heuristics that produce well-performing pricing policies across various problem sizes. These heuristics achieve substantially shorter runtimes compared to finding the optimal solution, enabling their use in practical applications. C4. Numerical study: Our numerical study demonstrates the effectiveness and efficiency of all three heuristics. The study also highlights the revenue potential of linear, piecewise linear, and nonlinear pricing schemes, enabling firms to balance revenue maximization with pricing simplicity. C5. Insights into pricing schemes: Our study provides insights into the strategy of well-performing pricing schemes. Notably, fewer price changes and simpler pricing schemes, such as piecewise linear pricing, may still perform very well. In Sect.2, we review the relevant literature, positioning our research within the context of existing studies and identifying gaps that our work addresses. Section3 introduces the customer choice model, emphasizing the two-dimensional approach, and presents the optimization model. In Sect.4, we reduce the complexity of the probability function, derive optimality conditions, and introduce the modified problem that serves as an upper bound. This section addresses the first two contributions, C1 and C2. It also achieves our first 1185 Heuristic solutions fornonlinear dynamic pricing inthe… goal (outlined in Sect.1.1) by enabling the computation of the optimal solution and builds the foundation for evaluating different pricing schemes, contributing to our third goal. Section5 discusses the heuristics we developed, addressing the third contribution, C3. By providing efficient and practical alternatives to computing the optimal solution, this section achieves our second goal from Sect.1.1. Finally, Sect. 6 presents the numerical study and its findings. This section addresses the last two contributions, C4 and C5, by demonstrating the practical applicability of the proposed heuristics and gaining insights into their strategies. It also fulfills our third goal by evaluating the revenue potential of nonlinear pricing and simpler pricing schemes. 2 Literature review In this study, we bridge the domains of dynamic pricing and nonlinear pricing, integrating the strengths of both. In Sect. 2.1, we provide a review of works within these two distinct yet interconnected domains. This foundational overview sets the stage for a deeper exploration into the specialized segments of multiunit and multiproduct dynamic pricing, as well as bundle pricing, in Sect. 2.2. Multiunit dynamic pricing, which also covers nonlinear pricing, represents a relatively new field of research with sparse literature. Our research contributes to this niche, acknowledging the critical role of nonlinear pricing in catering to multiunit demands. Additionally, we examine multiproduct dynamic pricing and bundle pricing, domains aligned with our research due to their consideration of customer choice, where customers select from multiple options. However, all these studies differ from our approach in a key aspect: they do not account for two-dimensional customer heterogeneity. The assumption of one-dimensional heterogeneity implies that all customers share the same relative utility when comparing two options, which is an unrealistic assumption in multiunit demand scenarios. To underline this gap, we incorporate several studies that, regardless of their specific field of application, address similar choice behaviors. We compare these approaches with our own in Sect. 2.3. 2.1 Foundations ofnonlinear pricing anddynamic pricing Nonlinear pricing is a widespread strategy across various sectors, including telecommunications, transportation, energy, supply chains, and retail. Consequently, there is a rich and diverse literature on the subject. Wilson (1993) (Part I) provides a comprehensive overview of the application fields, economic principles, and marketing insights related to nonlinear pricing. While much of the existing literature focuses on static pricing models, a small subset of researchers has turned their attention to dynamic environments, which align more closely with the thematic focus of our study (e.g., Dhebar and Oren 1986, and Braden and Oren 1994). 1186 R.Schur The conceptual foundations of dynamic pricing trace back to seminal studies on intertemporal price discrimination conducted 30–40years ago, with notable contributions by Stokey (1979), Landsberger and Meilijson (1985), and Wilson (1988). A significant milestone was achieved by Gallego and van Ryzin (1994), who were the first to explore optimal dynamic pricing for a single product under stochastic demand over a finite selling horizon. This pioneering work led to a vast amount of follow-up research, which was reviewed and summarized by many authors, including Bitran and Caldentey (2003), Chiang etal. (2007), and, with a special focus, Gönsch etal. (2013) and den Boer (2015), as well as in textbooks such as Talluri and van Ryzin (2004) (Chapter5). 2.2 Multiunit dynamic pricing, multiproduct dynamic pricing, andbundle pricing While dynamic pricing has been extensively explored, the specific area of multiunit dynamic pricing is still emerging. Elmaghraby et al. (2008) introduce markdown pricing mechanisms within a multiunit demand framework, assuming complete information about customers and their willingness-to-pay. Levin etal. (2014) expanded the discussion to a dynamic pricing model characterized by stochastic customer demand for batches. In their framework, customers request a specific batch size and the seller’s subsequent pricing response influences whether the requested batch size is purchased. However, unlike their approach, our research introduces a more flexible decision-making process in which customers can review all available prices prior to requesting the purchase quantity. This advancement not only gives customers greater flexibility but also provides firms with a mechanism to strategically steer customers’ buying decisions and purchase volumes. Gallego et al. (2020) investigate three dynamic pricing strategies: nonlinear, linear, and block pricing. In their proposed choice model, customers seek to maximize their utility and are characterized by a single random variable, representing one-dimensional customer heterogeneity. The authors develop optimality conditions and show structural properties. However, our research diverges fundamentally by modeling customer behavior with two independent variables, thereby offering a more granular representation of customer decision-making processes and allowing for two-dimensional heterogeneity. This two-dimensional approach represents a significant departure from the conventional one-dimensional models (refer to Sect.2.3 for a broader overview of choice models applied in similar demand settings). Schur (2024) explores a scenario similar to ours but diverges in assuming firms’ access to some or all private information about arriving customers. This assumption paves the way for personalized pricing strategies and the evaluation of the strategic value of customer information. Contrary to this, our research operates under the premise of private information, enabling the universal applicability of our pricing model without relying on the availability of detailed customer insights. Multiunit dynamic pricing can be compared to the better-explored field of multiproduct dynamic pricing by defining batches of a single product as several different “products”. Several articles (see, e.g., Zhang and Cooper 2009, Dong etal. 2009, and Akçay et al. 2010, or, for a review, Chen and Chen 2015) have investigated 1187 Heuristic solutions fornonlinear dynamic pricing inthe… dynamic pricing of substitutes. However, studies focusing on horizontally differentiated products are less relevant to our setting, as they lack a common order in product valuations, which violates a core assumption in multiunit demand setting: customers do not pay more for fewer units. In this context, general studies like Maglaras and Meissner (2006) and research on vertically differentiated products, such as Akçay etal. (2010) and Liu and Zhang (2013), are more closely aligned with our setting. While Maglaras and Meissner (2006) do not provide a specific willingness-to-pay function, they demonstrate the asymptotic optimality of a solution of a deterministic fluid approximation. Akçay etal. (2010) and Liu and Zhang (2013) rely on onedimensional customer heterogeneity regarding quality, which, in a multiunit setting, would imply that marginal willingness-to-pay for additional units decreases at a fixed rate across all customers. A special case of multiproduct pricing is found in the bundle pricing literature, where multiple items are combined into a bundle. Bundle-size pricing involves the firm quoting a price for each bundle-size, allowing customers to choose which items to include in their bundle. In this context, items can be viewed as single units of the same product, and bundles as batches. Bundle pricing aligns well with multiunit demand, as customer utility typically increases with bundle-size, reflecting one of the core assumptions of multiunit demand. However, related literature, such as Honhon and Pan (2017) and Song and Xue (2021), often assumes one-dimensional customer heterogeneity. Other studies, such as Abdallah etal. (2021), Ettl etal. (2020), and Chen etal. (2024), do not ensure decreasing marginal utilities with increasing bundle size, another key assumption in multiunit demand. 2.3 Literature withsimilar customer choice considerations Literature generally agrees on the fundamental concept that customers evaluate all available options by assigning a monetary value, often referred to as willingness-topay, to each option and comparing these values to quoted prices. The option with the highest utility, i.e. the difference between willingness-to-pay and price, is chosen. Since willingness-to-pay is private information known only to the customer, various models have been introduced, typically accounting for customer heterogeneity using one of two prominent approaches. The first approach models willingness-to-pay using various customer segmentspecific nominal parameters, where heterogeneity within a segment is captured through independently and identically distributed error terms. Hanemann (1984) formulates a willingness-to-pay function that is linear in quantity, leading to constant marginal utilities. Subsequent studies, such as Allenby and Rossi (1991), Chiang (1991), Bell etal. (1999), and Nair et al. (2005), propose non-constant marginal utilities to better reflect decreasing customer valuations for additional quantity. Quadratic formulations for those non-constant marginal utilities can be found in Lambrecht etal. (2007) and Iyengar etal. (2008). Iyengar and Jedidi (2012) introduce a generalization that encompasses both quadratic and power utility functions as special cases. These studies often employ multinomial logit models, assuming Gumbel-distributed error terms. While this model has certain advantages, it presents 1188 R.Schur a significant limitation: it cannot enforce a common order of valuations, leading to instances where customers may be willing to pay more for fewer units. Similarly, in multiproduct pricing, this model is less suited for vertically differentiated products (refer to Song and Xue 2021). The second approach often relies on a single random variable to represent customers’ preferences, as opposed to the multiple parameters combined with random error terms used in the first approach. This one-dimensional heterogeneity is applied by Mussa and Rosen (1978), where willingness-to-pay increases linearly in quantity. Further applications of this approach, with non-constant marginal utilities, can be seen in Braden and Oren (1994), Sundararajan (2004), Banciu etal. (2010), and Biyalogorsky and Koenigsberg (2014). Rochet and Stole (2002) incorporate twodimensional heterogeneity, where willingness-to-pay is modeled as linear in quantity, weighted by a random variable representing customer type, along with an additive random variable that accounts for brand preferences. We follow the second approach to model customer choice, addressing twodimensional heterogeneity through two random variables, but not in the linear form proposed by Rochet and Stole (2002). Instead, our willingness-to-pay function is based on the generalized formulation of Iyengar and Jedidi (2012), incorporating non-linearity in quantity and resulting in non-constant marginal utilities. 2.4 Summary Our literature review highlights a general gap in research on nonlinear dynamic pricing, with only a few exceptions, which differ from our work in their specific assumptions. Broadening the scope, it becomes apparent that even in related fields such as vertically differentiated multiproduct pricing and bundle pricing, two-dimensional customer heterogeneity—allowing for customer-specific, non-fixed rates of marginal utility decrease—is not considered. In this regard, our approach could be of interest to these fields as well. 3 Problem definition We first present the random utility framework that defines our customer choice model in Sect.2.3. Then, we introduce the dynamic programming formulation that describes our nonlinear dynamic pricing problem. 3.1 Customer choice model In our setting, customers exhibit multiunit demand but are not predetermined in the number of units they consider purchasing. Instead, they evaluate all available options, i.e., possible batch sizes, ranging from a batch size of zero to the entire remaining stock c . This evaluation results in a personal monetary valuation for each option, which is known as willingness-to-pay and denoted by Xj for j units. It is 1195 Heuristic solutions fornonlinear dynamic pricing inthe… identifying these specific intervals, we know precisely which curve defines the upper bound for realizations w at any given realization l . Lemma1 For every i≠k with i,k>j and ri−rj≠0≠rk−rj , there is at most one l∈(0,1) where r i −r j ∑ i−1 m=j lm = r k −r j ∑ k−1 m=j l m . For every i≠k with i,k<j and ri−rj≠0≠rk−rj , there is at most one l∈[0,1] where r j −r k ∑ j−1 m=k lm = r j −r i ∑ j−1 m=i l m . Proof: See Supplement S.1. Lemma 1, in conjunction with the observation that the equation r k −r j ∑ k−1 m = j lm = 1 is satisfied by at most one l∈[0,1] , implies that each curve can only serve as the minimum or maximum exclusively within a specific interval. At some point, it will intersect with another curve and will consistently remain either above or below the intersecting curve. Consequently, for the various sets of interest – namely Λj( r ) for instances where 1≥ min j+1≤k≤c � rk−rj ∑ k−1 m=jlm � , Λmin ij (r ) for cases where i>j , and Λmax ji ( r ) for cases where j>i —we define their corresponding intervals as [ lj(r),lj(r) ] , [ lmin ij (r),l min ij (r) ] , and [ lmax ji (r),l max ji (r) ] , respectively. It is important to note that some of these intervals may be empty, and thus, boundary values have to be selected accordingly. Remark 1 By definition, 𝑚𝑎𝑥 i > j{ l min ij (r) } =𝑚𝑎𝑥 i < j{ l max ji (r) } . We denote this largest boundary as lend j(r)=𝑚𝑎𝑥 i > j{ l min ij (r) }. Moreover, it holds that: ⋃ i>j � lmin ij (r),l min ij (r) � = � lj(r),l end j(r) � ⋃ i<j � lmax ji (r),l max ji (r) � = � lj(r),l end j(r) � Currently, our model requires extensive notation to accurately represent the probability function. Nevertheless, the introduction of the subsequent lemma will streamline the notation required, thereby enhancing the brevity and clarity of our presentation. Lemma2 For every i>j it holds that [ lmin ij (r),l min ij (r) ] = [ lmax ij (r),l max ij (r) ] . Proof: See Supplement S.3. This lemma not only simplifies our notation but also carries another implication: for i>j , the interval [ lmin ij (r),l min ij (r) ] defines the region for realization l where the curve r i −r j ∑ i−1 m = j lm marks the upper bound for all realizations w that specify, in combina- 1196 R.Schur tion with l , all customers who choose to purchase j units. From Lemma 2, we know that [ lmin ij (r),l min ij (r) ] = [ lmax ij (r),l max ij (r) ] , which implies that Λmin ij (r)=Λ max ij (r) . By definition of Λmax ij ( r ) , the same curve ( r i −r j ∑ i−1 m=j l m ) also serves as the lower bound for all realizations w that, in combination with l , represent all customers who prefer purchasing i units. Consequently, this curve marks the boundary between two decision regions (refer to Fig.1): one associated with customers opting to purchase j units and the other with those preferring i units. We can now shorten our notation to lij (r)=l min ij (r)=l max ij (r ) and l ij(r)=l min ij (r)=l max ij (r ) . The probability function can be written as: With this refinement of the probability function, we can better explore how prices and their variations impact selling probability. This is crucial for determining optimal prices in the state-wise optimization (7). However, some complexity remains due to the implicit definition of the bounds of the integrals. To address this, it is reasonable to gain deeper insights into these bounds. One way to achieve this is by reducing the action space, which will result in a closed-form expression of some of the bounds. 4.2 Action space reduction Upon closer examination of our choice model, it becomes evident that we only need to consider a specific subset of prices to maximize expected revenue. Consequently, we aim to refine the definition of the action space Rc by excluding prices that do not lead to a unique outcome in terms of selling probabilities and expected revenues. By doing so, we simplify our optimization problem without the risk of excluding a potentially unique optimal solution. The argumentation for deeming certain prices as irrelevant is as follows: Maintaining multiple different values for price rj that effectively nullify demand for j units (i.e., pj( r )=0 ) is unnecessary. It suffices to have at least one value for rj (depending on r 1 ,…,rj− 1 ,rj+ 1 ,…,rc ) to preserve the option of pricing out j units. Irrelevant prices can be identified by any of the following four criteria. 1. Exclusion of higher prices for smaller batches: We exclude any price rj with rj>rj+1 because customers almost surely have a higher willingness-to-pay for (8) pj(r)= lj(r) ∫ lj(r) f𝜆(l)dl + c � i=j+1 lij(r) ∫ lij(r) F𝜔�ri−rj ∑i−1 m=jlm�f𝜆(l)dl − j−1 � i=0 lji(r) ∫ l ji (r) F𝜔�rj−ri ∑ j−1 m=ilm�f𝜆(l)dl for j=1, …,c . 1197 Heuristic solutions fornonlinear dynamic pricing inthe… j+1 units than for j units. Therefore, setting rj=rj+1 is sufficient to eliminate demand for j units. 2. Exclusion of prices exceeding batch size threshold: Any price rj exceeding j is irrelevant, as the maximal willingness-to-pay for j units is j . 3. Exclusion of prices with excessive margins: We exclude any price rj for which rj −rj− 1 >1 , since the maximal marginal utility for the j -th unit is one. 4. Exclusion of prices with excessive comparative margins: We omit prices rj that satisfy the condition ( r j −r j− 1 ) 1 j−1> ( r j+ 1−r j) 1 j . This inequality implies that a customer with a positive marginal utility for purchasing the j -th unit has almost surely also a positive marginal utility for purchasing the (j+1) -th unit. The following lemma confirms that reducing the action space based on these four criteria only eliminates prices that do not result in unique outcomes in terms of selling probabilities and revenues. Lemma3 The search for the optimal price vector can be restricted to the set . Proof: See Supplement S.4. In Sect.3.1, we have seen that lk and lk play a crucial role in calculating selling probabilities. With the action space reduction, we are now able to shed more light on the definition of these parameters. Lemma4 It holds that l1( r )=0 , lk (r)= ( r k −r k−1) 1 k−1=l k−1 (r ) , 2≤k≤c , lc (r)= 1 for all r∈Rc . Proof: See Supplement S.5. With Lemma 4, we obtain a closed-form expression for lk( r ) and lk( r ) ( 1≤k≤c ), which simplifies our probability function. Additionally, the proof of this lemma shows that lk , k− 1 ( r )=lk( r ) , extending the closed-form expression to these specific bounds. While the implicit definitions of other bounds, such as lkj( r ) and lkj (r ) , generally cannot be replaced by explicit definitions, we can still explore how these bounds change with small variations in prices, laying the groundwork for calculating partial derivatives and finding optimal solutions. These bounds correspond to the intervals Λmin kj (r ) and Λmax kj ( r ) for k>j and are defined as the intersection points of two curves, provided these intervals are nonempty (for an illustrative reference, see Fig. 2). By identifying these bounds as R c= { r∈ℝc∶0≤r1≤⋯≤rc≤c,rj≤j∀j,rj−rj−1≤1 for j≥2, and ( rj−rj−1 ) 1 j−1≤ ( rj+1−rj ) 1 jfor 2 ≤j≤c−1 }. 1198 R.Schur intersection points, we establish that for every upper bound lkj (r ) (with a single exception as noted in Remark 1), there exists a corresponding lower bound lij( r ) such that l kj(r)=l ij (r ) and r k −r j ∑ k−1 m=j � lkj(r) � m= r i −r j ∑ i−1 m=j � lkj(r) � m. We now want to discuss the impact of a small price variation, specifically changing rm while keeping other batch prices constant. For m∈{j,k,i} , this would obviously shift the location where both curves intersect. However, as long as the price variation is sufficiently small, it would not affect which intervals are adjacent and connected at the intersection point, i.e. the equation l kj(r)=l ij (r ) would still hold, albeit with a different value. For m∉{j,k,i} , a sufficiently small price variation would neither change the matching of l kj(r ) and lij( r ) nor affect the value of these two bounds. However, there are rare instances of ambiguity that we have not explicitly addressed. Specifically, when the price vector results in three curves intersecting at a single point, any two of these three curves can be used to determine the intersection point. In this scenario, one of the corresponding intervals consists of exactly one element, which is the intersection point. This ambiguity resolves with even arbitrarily small changes in one of the prices rm associated with these curves, as the three curves no longer intersect at the same spot, causing the one-element interval to either become empty or expand to cover an infinite number of elements. In the first case, we would use the other two curves to determine the intersection point. In the second case, we would determine two intersection points using the combinations of the curve belonging to the one-element interval with both other curves. While these cases of ambiguity are not explicitly covered in the remainder of this section, please note that they can be easily resolved. The following remark summarizes the observations above and will come in handy in the development of optimality conditions. Remark 2 It holds that: • For every l kj(r) ≠ l end j (r ) there is l ij ( r ) such that l kj(r)=l ij (r ) and r k −r j ∑ k−1 m=j � lkj(r) � m= r i −r j ∑ i−1 m=j � lkj(r) �m . Moreover, 𝜕 𝜕 r m lkj(r)=𝜕 𝜕r m lij(r ) for all m and 𝜕 𝜕 r m lkj(r)= 𝜕 𝜕r m lij(r)= 0 for m∉{j,k,i} . • lj+ 1 ( r )=lj+ 1, j( r ) with r j+1 −r j ( lj+1(r) ) j= 1 . Moreover, 𝜕 𝜕 r m lj+1(r)= 𝜕 𝜕r m lj+1,j(r ) for all m and 𝜕 𝜕rm lj+1(r)= 𝜕 𝜕 rm lj+1,j(r)= 0 for m∉{j,j+1} . • lj( r )=lj,j−1( r ) with r j −r j−1 ( lj(r) ) j−1= 1 . Moreover, 𝜕 𝜕rm lj(r)= 𝜕 𝜕 rm lj,j−1(r ) for all m and 𝜕 𝜕 r m lj(r)= 𝜕 𝜕r m lj,j−1(r)= 0 for m∉{j−1, j} . 1199 Heuristic solutions fornonlinear dynamic pricing inthe… 4.3 Optimality conditions With the previous section, we gathered enough information regarding the probability function to advance to one of our main goal: finding the optimal solution to (7). Before we engage in the partial differentiation of the objective function, we first want to elaborate more on the partial differentiation of probability function (8). The calculation of 𝜕 𝜕 r i pj(r ) varies a little depending on the following three cases: i>j , i<j , and i=j . Lemma5 It holds: (1) For i>j , 𝜕 𝜕 ri pj(r)= lij(r) ∫ l ij (r) 1 ∑ i−1 m=jlmf𝜔 � ri−rj ∑ i−1 m=jlm � f𝜆(l) dl . (2) For i<j , 𝜕 𝜕 ri pj(r)= lji(r) ∫ l ji (r) 1 ∑ j−1 m=ilmf𝜔 � rj−ri ∑ j−1 m=ilm � f𝜆(l) dl . (3) For i=j , 𝜕 𝜕ri pj(r)=− ∑ c k=i+1 lki(r) ∫ l ki (r) 1 ∑ k−1 m=i lmf𝜔 � rk−ri ∑ k−1 m=i lm � f𝜆(l)dl − ∑ i−1 k=0 lik(r) ∫ l ik (r) 1 ∑ i−1 m=k lmf𝜔 � ri−rk ∑ i−1 m=k lm � f𝜆(l) dl . Proof: See Supplement S.6. With additional knowledge about the probability function, we now can turn our focus on the first-order condition. Therefore, we calculate the partial derivatives of the objective function of (7): 𝜕 𝜕 ri �c � j=1 pj(r)⋅�rj−Δ jVt−1(c)� � =pi(r)+ c � j=1�𝜕 𝜕ri pj(r)�⋅�rj−Δ jVt−1(c)� =pi(r)+ i−1 � j=1  lij(r) ∫ l −ij (r) 1 ∑i−1 m=jlmf𝜔�ri−rj ∑i−1 m=jlm�f𝜆(l)dl ⋅�rj−Δ jVt−1(c) � − c � k=i+1  lki(r) ∫ l −ki (r) 1 ∑k−1 m=ilmf𝜔�rk−ri ∑k−1 m=ilm�f𝜆(l)dl ⋅�ri−Δ iVt−1(c)� − i−1 � k=0  lik(r) ∫ l −ik (r) 1 ∑i−1 m=klmf𝜔�ri−rk ∑i−1 m=klm�f𝜆(l)dl ⋅�ri−Δ iVt−1(c)� + c � j=i+1  lji(r) ∫ l − ji (r) 1 ∑ j−1 m=i lmf𝜔�rj−ri ∑ j−1 m=i lm�f𝜆(l)dl ⋅ � rj−Δ jVt−1(c) � . 1200 R.Schur While the first-order condition is a necessary condition for local maxima, the global maximum over a closed and bounded space does not generally need to satisfy this condition. However, the following propositions states that, regardless of whether the optimal solution lies on the boundary of the action space Rc or within its interior, it always satisfies the first-order condition. Proposition1 The optimal solution of (7) meets for every batch size i the first-order condition: Proof: See Supplement S.7. Remark 3 For 𝜔∼U[0, 1] the first-order condition simplifies to 𝜕 𝜕r i � ∑ c j=1pj(r)⋅ � rj−Δ jVt−1(c) � �=2pi(r)− � F𝜆 � li+1(r) � −F𝜆 � li(r) �� −∑c k=i+1 lki(r) ∫ l ki (r) ΔkVt−1(c)−Δ iVt−1(c) ∑k−1 m=i lmf𝜆(l)dl +∑i−1 k=0 lik(r) ∫ l ik (r) ΔiVt−1(c)−Δ kVt−1(c) ∑i−1 m=k lmf𝜆(l) dl . Moreover, it holds that ∑ c i=1pi(r∗)=1 2− ∑ c k=1 lk0(r∗) ∫ l k0 (r∗) ΔkVt−1(c) ∑ k−1 m=0lmf𝜆(l) dl . Thus, the overall selling probability is less than or equal to 0.5 and decreasing with opportunity costs. In the final period ( t=1 ), there are no opportunity costs, so Proposition 1 applies with ΔjVt−1(c)=0 for all j in these states. To conclude this section, we focus on a scenario without opportunity costs, where both random variables 𝜆 and 𝜔 are uniformly distributed. Under these conditions, finding the optimal solution in the final states further simplifies. Moreover, the optimal solution does not lie on the boundary of our action space, implying that none of the corresponding selling probabilities are reduced to zero. Proposition2 Let 𝜔,𝜆∼U[0, 1] . If ΔkVt−1(c)=0 for every k≤c, the optimal solution for (7) is an interior point of Rc and fulfills pi(r)= 1 2( l i+ 1(r)−l i (r) ) for every i≤c . Proof: See Supplement S.8. Although we have formulated the optimality conditions, finding the solution that fulfills these equations within each state is a difficult task. We are facing a system with c nonlinear equations that are additionally plagued by integrals and implicitly defined variables. Many of these challenges arise from the analytical intractability of the choice model. To address this, we explore a modified optimization problem in the following section. This alternative formulation is simpler to solve and provides 𝜕 𝜕 ri ( c ∑ j= 1 pj(r)⋅ ( rj−Δ jVt−1(c) )) = 0. 1201 Heuristic solutions fornonlinear dynamic pricing inthe… an upper bound for the original problem. Notably, under specific conditions, the optimal solution of the modified formulation also constitutes the optimal solution to our original problem. 4.4 Modified formulation In this section, we propose a modification to our choice model, which leads to a simplified problem that acts as an upper bound to our original problem. The main aspect of this formulation, however, is the potential for its optimal solution to also be the optimal solution to the original problem. Specifically, we develop easyto-verify conditions for this solution to ensure its optimality with respect to the original problem. The modification of the customer choice model involves reducing the competition between different options. In particular, instead of calculating the probability that U k=max j= 0, … , c{ Uj } , we now calculate the probability that U k=max j= 0, k− 1, k , k+ 1 { Uj } . Thereby, we reduce competition between available options and, thus, only compare four instead of c+1 options. We denote demand by this modified choice model as pm j( r ) and write pm j(r)=ℙ ( Uj=max k=0,j−1,j,j+1 { Uk })≥ pj(r ) . A state-wise optimization with pm j( r ) therefore serves as an upper bound to (7). Technically, through the modification, the choice model is no longer a choice model. By summing up all probabilities, we get a value greater than or equal to 1 , i.e. ∑C j=0 pm j (r) ≥∑C j=0 pj(r)= 1 , which does not satisfy one of the main properties of choice models. Similar considerations as in Sect.4.3 lead to the following remark. Remark 4 When pm j( r ) is used instead of pj( r ) , Propositions 1 and 2 still hold. This modification significantly reduces complexity in our choice model (see Supplement S.9 for an exemplary calculation of pm j( r ) for a scenario where 𝜔,𝜆∼U[0, 1] ), thus improving the determination of the optimal solution rm for the modified formulation compared to finding the optimal solution of the original problem. Utilizing the fact that the modified model constitutes an upper bound to the original model, it suffices to verify that its optimal solution rm is feasible for the original model and results in the same objective value. To facilitate this verification, we propose an approach that avoids the exhaustive checking whether pm j( r m)=p j ( r m) for every j . Instead, we introduce a set of straightforward and verifiable conditions, designed to simplify the process of proving rm ’s optimality for the original model. Proposition 3 If r fulfills lend j (r) ≤ l end j+ 1(r ) for every j , with 1≤j<c , then pm j( r )=p j ( r ) for every j . Proof: See Supplement S.10. 1202 R.Schur Remark 5 The conditions of Proposition3 are satisfied by linear pricing schemes. Additionally, piecewise linear pricing schemes, where a distinct price is set for the first unit and a constant marginal price applies to any additional unit, also satisfy the conditions of Proposition3. In our numerical study, we calculated the optimal solution for the modified model for every combination of T≤40 and C≤120 , with 𝜔,𝜆∼U[0, 1] . The conditions of Proposition 3 were always satisfied, showing that, in these instances, the optimal solution obtained from the modified model is also optimal for the original problem. However, our numerical study reveals that solving each state of the modified model leads to long computation times for medium-sized problems (see Sect.6.2). Therefore, we aim to address these drawbacks by developing efficient and effective heuristics. 5 Heuristic design Dynamic pricing decisions often need to be made online and in real-time. Given the long computation times required to find the optimal solution, we developed three heuristics to tackle the dynamic optimization problem (5). These heuristics aim to compute well-performing policies efficiently, with short runtimes, ensuring practical applicability in dynamic nonlinear pricing scenarios. In this section, we introduce three heuristics to address the dynamic optimization problem (5). Two of these approaches, presented in Sect.5.1, build on the results of Schur (2024) and use the optimal solution in a setting where the firm has access to customers’ private information, i.e., their base willingness-to-pay and their consumption indicator, respectively. The third approach, presented in Sect.5.2, can be described as a decomposition in units. Thereby, we allow customers to buy the j -th unit of the product without buying the units 1, 2, …,j−1 . Even though this does not reflect reality, it constitutes an easy to solve optimization problem, enabling us to devise batch pricing strategies for the original problem based on the solutions obtained. 5.1 Approaches 1 and2: expected optimal batch prices We can efficiently compute realization-dependent optimal batch prices rjt(c|w) and rjt(c|l) for realization w and l , respectively. Technically, these realization-dependent batch prices are themselves random variables, raising the following idea: By calculating the expected value of these realization-dependent optimal batch prices, we construct a policy for optimization problem (5). Both approaches follow the same idea, differing only in the determination of the realization-dependent optimal batch prices: rjt(c|w) for Approach 1 and rjt(c|l) for Approach 2. Beyond this distinction, both approaches follow the same subsequent steps. Consequently, we will explain the remaining steps without distinguishing 1203 Heuristic solutions fornonlinear dynamic pricing inthe… between both approaches and write rjt(c|x) instead of rjt(c|w) and rjt(c|l) to denote the realization-dependent optimal batch prices. This framework is applicable across a wide spectrum of distribution functions, including, but not limited to, uniform, triangular, normal, exponential, Weibull, Gumbel, and gamma distributions, along with their truncated versions, albeit with some constraints on parameter selections. Notably, for scenarios where 𝜔∼U[0, 1] , Approach 2 provides a closed-form expression for the realization-dependent optimal batch prices: r jt(c � l)=1 2 �∑ j−1 k=0lk+Δ jVE t−1(c) � , with j ≤ Nt(c | l)=max j= 1, … , c{ j∶Δ 1V E t−1(c−j+1)<l j−1} . Building on these realizations-dependent optimal batch prices, we compute expected optimal batch prices: More precisely, this formulation results in conditional expected optimal batch prices, where we only take realizations of 𝜆 and 𝜔 into account that lead to possible economic sales, i.e. rj+ 1, t (c | ⋅)−r jt (c | ⋅) ≥ Δ1V E t−1 (c+1−j ) . Other realizations are economically irrelevant and can distort results, given the lack of a clear pricing strategy in these cases. Thus, these events where we refrain from selling are not used to compute our policy. Except for the scenario where we have a closed-form expression of rjt(c|l) , we resort to numerically calculating rjt(c|x) for several realizations x to accurately derive rE jt (c ) (in Sect.6, we use a sample size of 100 ; for an analysis of the tradeoff between accuracy and runtimes, refer to Supplement S.11). Finally, we calculate the expected revenue-to-go derived by expected optimal batch prices rE jt (c ) : with the same boundary conditions as the original problem (5). Remark 6 By using suboptimal batch prices, we get a lower bound to optimization problem (5). Thus, it holds that Vt (c) ≥ V E t (c ) for every (t,c) . We sum up the first two heuristics that are based upon the idea of expected optimal batch prices by the following pseudo code: (9) r E jt (c)= 1 ∫ 0 rjt(c|x)⋅1{j<Nt(c|x)}f(x)dx 1 ∫ 0 1{j<Nt(c | x)}f(x)dx for j=1, …,c . (10) V E t(c)= c ∑ j=1 pj ( rE t(c) ) ⋅ ( rE jt (c)+VE t−1(c−j) ) + ( 1− c ∑ j=1 pj ( rE t(c) )) ⋅VE t−1(c) , 1204 R.Schur 5.2 Approach 3: decomposition inunits Our next algorithm employs a decomposition strategy. The basic idea is that customers have the flexibility to buy the j -th unit of the product even though they might not buy units 1 to j−1 . As every unit of the product is the same, there is no distinction between the 1 st, 2 nd or j -th unit other than the number customers have already in their basket. Thus, this decomposition is merely theoretical without having immediate practical applicability. However, it results in a greatly simplified optimization problem. A hypothetical customer now faces c distinct binary decisions instead of one decision with c+1 options. This, in turn, enables us to solve c distinct and rather simple independent optimization problems instead of one complex problem. With this simplification, we can derive batch prices that can form a policy to address optimization problem (5). To consider this decomposition, we must change the customer choice model. Customers still strive to maximize their utility. But, instead of purchasing j units if and only if U j=max j=0,…,c{ Uj } with U0=0 denoting the no-purchase option, they decide for every single unit whether they want to purchase it or not. This decision is based upon whether the additional willingness-to-pay for the j -th unit is at least as high as the additional price the customer has to pay, i.e. Xj −X j− 1=𝜔⋅(𝜆) j−1≥ r j −r j−1 . If the customers decide to purchase the j -th unit, they must pay rj −rj−1 . For example, for given batch prices, a customer might only be willing to purchase the second and fourth unit due to the willingness-to-pay curve. In this case, the customer pays ( r2−r1 ) + ( r4−r3 ) to get 2 units of the product in total. The decomposition approach reduces the complexity of the choice model. The model itself becomes easier as the decision between several options is broken down to several binary independent decisions. This method avoids the need to determine a single price vector that encompasses all batch prices and to predict the customer’s 1211 Heuristic solutions fornonlinear dynamic pricing inthe… To summarize, mechanism D closely aligns with the optimal mechanism, mirroring mean revenues and mean purchases over time. In contrast, mechanism E(𝜔) adopts a more aggressive pricing strategy, selling more units than D and the optimal policy over the entire selling horizon, seemingly by offering lower prices. This pattern is also evident in Fig.6, where we have depicted the evolution of batch prices for the scenario where no customer makes a purchase. Specifically, the figure shows batch prices in the states (c,t) for c=20 and t=10, 9, …,1 . In the figure, the lowest curve corresponds to the price for a batch of size one, the second-lowest line represents the price for a batch of size two, and so on. Over the entire selling horizon, mechanism D quotes higher prices than mechanism E(𝜔) . Additionally, prices from both mechanisms are decreasing over time. Despite the distinct approaches used to calculate rD t (c ) and r E(𝜔) t (c ) , the resulting curves share a similar structure, indicating an underlying structure well-performing policies have in common. Batch prices for small numbers of units are virtually linear in batch size (apart from the first unit, the second, third, etc. units cost nearly the same). For large numbers of units, batch prices are convexly increasing in batch size. With a concavely increasing willingness-to-pay curve, these pricing schemes automatically prevent selling a large batch or even the whole stock ( C=20 ) to only one customer. Finally, it is notable that prices for small batches merely change over time whereas prices for big batches noticeably decrease. In Fig.5, we have seen that both heuristics result in selling processes where a customer purchases approximately 1.5 units on average. Therefore, we also want to examine the evolution of batch prices in a scenario where alternately two units and one unit are sold, starting with a purchase of 2 units in t=10 . As a result, we analyze the states (c,t)=(20,10) , (18,9) , (17,8) ,…, (6,1) . As the firm cannot offer batches that are not covered by capacity any longer, most of the curves are terminated during the selling horizon. The pattern the curves draw looks nearly the same for both heuristics. Batch prices obtained by E(𝜔) are lower than those obtained by D. The gaps between batch prices are nearly same-sized for smaller batches (again, starting with the two-unit Fig. 5 Mean revenues (left) and mean purchases (right) at every point in time, t=10, 9, …,1 with C=20 , and 𝜔,𝜆∼U[0, 1] 1212 R.Schur batch) and are increasing for bigger batches. After selling, batch prices for bigger batches increase. This effect is more pronounced after selling two units in comparison to selling one unit. It is a well-observed pricing behavior in (standard) dynamic pricing that prices increase after a sale took place. However, this only holds partially in our multiunit setting as prices for small batches usually decrease slowly and steady over the selling horizon. In conclusion, we have observed two notable effects in Figs.6 and 7. First, prices for small batches (relative to the remaining stock) decrease over time, regardless of whether a sale takes place or not. This holds for the discussed settings with a reasonably large stock, i.e., C T = 2 . In scenarios with a small stock, i.e., T C = 2 , pr ices for small batches do not always decrease from one period to the next; to maintain brevity, we have excluded the corresponding figures. Second, prices for small and medium batches increase approximately linearly with batch size, starting from the two-unit batch. This linearity explains the strong performance of a piecewise linear pricing scheme observed in our numerical study. Such a pricing scheme is easy to communicate, as the firm can quote two prices—one for the first unit and one for each additional unit—instead of a long list of prices for every possible batch size. Fig. 6 Evolution of batch prices without a purchase for D (left) and E( 𝜔 ) (right) over t=10,9, …,1 with C=20 , and 𝜔,𝜆∼U[0, 1] Fig. 7 Evolution of batch prices with purchases at every period for D (left) and E(𝜔) (right) over t=10,9, …,1 with C=20 , and 𝜔,𝜆∼U[0, 1] 1213 Heuristic solutions fornonlinear dynamic pricing inthe… 7 Conclusion In this study, we introduced a nonlinear dynamic pricing model, based on a customer choice model that captures two-dimensional customer heterogeneity in terms of product attraction and consumption inclination. Despite the complexity of the resulting probability function, we were able to simplify it by leveraging structural properties and removing irrelevant batch prices from the action space. We then presented optimality conditions for the state-wise optimization model. Additionally, we introduced a modified model that, under certain conditions, yields an optimal solution that is also optimal for the original problem. However, finding the optimal solution for the state-wise optimization model remains computationally challenging. To address these difficulties, we developed three novel heuristics: one that uses a decomposition approach and two that calculate expected optimal prices. In our numerical study, two heuristics performed exceptionally well, with an optimality gap of less than 0.4% and 0.6% , while the third heuristic was only slightly behind. Moreover, they significantly outperformed other mechanisms. The results also highlight the revenue potential of nonlinear pricing, allowing firms to compare straightforward pricing schemes, such as linear and piecewise linear pricing, with nonlinear pricing. We further analyzed both best performing heuristics and found several interesting characteristics of well-performing pricing policies: In cases with reasonably large stocks (in our setting with, e.g., C T≥2 ), batch prices for small batches are slowly decreasing over time. This is particularly interesting as it is an indicator that changing prices at a lower rate (not after every customer) might still perform well in the presence of multiunit demand. This makes the obtained policies also applicable in settings where the firm cannot sustain frequent changes in batch prices due to, e.g., technical reasons, customers’ reluctance, or strategic considerations. On the other hand, in scenarios with a limited stock (in our setting with, T C≥2 ), the importance of nonlinear pricing is declining, whereas a typical (standard) dynamic pricing structure becomes more and more relevant. Another finding is that batch prices are nearly linear for lowand medium-sized batches starting with the two-unit batch. This explains the strong performance of a piecewise linear pricing scheme in our numerical study, which has the benefit of being easy-to-communicate. Instead of displaying a long list containing prices for every possible batch size, the firm could quote two prices—one for the first unit and one for additional units. Computing the optimal piecewise linear pricing scheme efficiently remains a challenge at this stage. Supplementary Information The online version contains supplementary material available at https:// doi. org/ 10. 1007/ s0029102500820-3. Acknowledgements The author would like to thank the anonymous referees for their valuable suggestions and feedback which contributed to an improved quality of the results of the paper. Funding Open Access funding enabled and organized by Projekt DEAL. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative 1214 R.Schur Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/ licenses/by/4.0/. References Abdallah T, Asadpour A, Reed J (2021) Large-scale bundle-size pricing: a theoretical analysis. Oper Res 69(4):1158–1185. https:// doi. org/ 10. 1287/ opre. 2020. 2097 Akçay Y, Natarajan HP, Xu SH (2010) Joint dynamic pricing of multiple perishable products under consumer choice. Manag Sci 56(8):1345–1361. https:// doi. org/ 10. 1287/ mnsc. 1100. 1178 Allenby GM, Rossi PE (1991) Quality perceptions and asymmetric switching between brands. Market Sci 10(3):185–204. https:// doi. org/ 10. 1287/ mksc. 10.3. 185 Banciu M, Gal-Or E, Mirchandani P (2010) Bundling strategies when products are vertically differentiated and capacities are limited. Manag Sci 56(12):2207–2223. https:// doi. org/ 10. 1287/ mnsc. 1100. 1242 Baucells M, Sarin RK (2007) Satiation in discounted utility. Oper Res 55(1):170–181. https:// doi. org/ 10. 1287/ opre. 1060. 0322 Bell DR, Chiang J, Padmanabhan V (1999) The decomposition of promotional response: an empirical generalization. Market Sci 18(4):504–526. https:// doi. org/ 10. 1287/ mksc. 18.4. 504 Bitran G, Caldentey R (2003) An overview of pricing models for revenue management. M&SOM 5(3):203–229. https:// doi. org/ 10. 1287/ msom.5. 3. 203. 16031 Biyalogorsky E, Koenigsberg O (2014) The design and introduction of product lines when consumer valuations are uncertain. Prod Oper Manag 23(9):1539–1548. https:// doi. org/ 10. 1111/ poms. 12167 Braden DJ, Oren SS (1994) Nonlinear pricing to produce information. Market Sci 13(3):310–326. https:// doi. org/ 10. 1287/ mksc. 13.3. 310 Chen M, Chen Z-L (2015) Recent developments in dynamic pricing research: multiple products, competition, and limited demand information. Prod Oper Manag 24(5):704–731. https:// doi. org/ 10. 1111/ poms. 12295 Chen N, Li X, Li Z, Wang C (2024) Component pricing with bundle size discount. SSRN J. https:// doi. org/ 10. 2139/ ssrn. 40322 47 Chiang J (1991) A simultaneous approach to the whether, what and how much to buy questions. Market Sci 10(4):297–315. https:// doi. org/ 10. 1287/ mksc. 10.4. 297 Chiang WC, Chen JC, Xu X (2007) An overview of research on revenue management: current issues and future research. IJRM 1(1):97. https:// doi. org/ 10. 1504/ IJRM. 2007. 011196 den Boer AV (2015) Dynamic pricing and learning: Historical origins, current research, and new directions. Surv Oper Res Manag Sci 20(1):1–18. https:// doi. org/ 10. 1016/j. sorms. 2015. 03. 001 Dhebar A, Oren SS (1986) Dynamic nonlinear pricing in networks with interdependent demand. Oper Res 34(3):384–394. https:// doi. org/ 10. 1287/ opre. 34.3. 384 Dong L, Kouvelis P, Tian Z (2009) Dynamic pricing and inventory control of substitute products. M&SOM 11(2):317–339. https:// doi. org/ 10. 1287/ msom. 1080. 0221 Elmaghraby W, Gülcü A, Keskinocak P (2008) Designing optimal preannounced markdowns in the presence of rational customers with multiunit demands. M&SOM 10(1):126–148. https:// doi. org/ 10. 1287/ msom. 1070. 0157 Ettl M, Harsha P, Papush A, Perakis G (2020) A data-driven approach to personalized bundle pricing and recommendation. Manuf Serv Oper Manag 22(3):461–480. https:// doi. org/ 10. 1287/ msom. 2018. 0756 Gallego G, van Ryzin G (1994) Optimal dynamic pricing of inventories with stochastic demand over finite horizons. Manag Sci 40(8):999–1020. https:// doi. org/ 10. 1287/ mnsc. 40.8. 999 Gallego G, Li MZF, Liu Y (2020) Dynamic nonlinear pricing of inventories over finite sales horizons. Oper Res 68(3):655–670. https:// doi. org/ 10. 1287/ opre. 2019. 1891 1215 Heuristic solutions fornonlinear dynamic pricing inthe… Goldman MB, Leland HE, Sibley DS (1984) Optimal nonuniform prices. Rev Econ Stud 51(2):305. https:// doi. org/ 10. 2307/ 22976 94 Gönsch J, Klein R, Neugebauer M, Steinhardt C (2013) Dynamic pricing with strategic customers. J Bus Econ 83(5):505–549. https:// doi. org/ 10. 1007/ s115730130663-7 Hanemann WM (1984) Discrete/continuous models of consumer demand. Econometrica 52(3):541. https:// doi. org/ 10. 2307/ 19134 64 Honhon D, Pan XA (2017) Improving profits by bundling vertically differentiated products. Prod Oper Manag 26(8):1481–1497. https:// doi. org/ 10. 1111/ poms. 12686 Iyengar R, Jedidi K (2012) A conjoint model of quantity discounts. Market Sci 31(2):334–350. https:// doi. org/ 10. 1287/ mksc. 1110. 0702 Iyengar R, Jedidi K, Kohli R (2008) A conjoint approach to multipart pricing. J Market Res 45(2):195– 210. https:// doi. org/ 10. 1509/ jmkr. 45.2. 195 Lambrecht A, Seim K, Skiera B (2007) Does uncertainty matter? Consumer behavior under three-part tariffs. Market Sci 26(5):698–710. https:// doi. org/ 10. 1287/ mksc. 1070. 0283 Landsberger M, Meilijson I (1985) Intertemporal price discrimination and sales strategy under incomplete information. RAND J Econ 16(3):424. https:// doi. org/ 10. 2307/ 25555 68 Levin Y, Nediak M, Bazhanov A (2014) Quantity premiums and discounts in dynamic pricing. Oper Res 62(4):846–863. https:// doi. org/ 10. 1287/ opre. 2014. 1285 Liu Q, Zhang D (2013) Dynamic pricing competition with strategic customers under vertical product differentiation. Manag Sci 59(1):84–101. https:// doi. org/ 10. 1287/ mnsc. 1120. 1564 Maglaras C, Meissner J (2006) Dynamic pricing strategies for multiproduct revenue management problems. M&SOM 8(2):136–148. https:// doi. org/ 10. 1287/ msom. 1060. 0105 Mussa M, Rosen S (1978) Monopoly and product quality. In Journal of Economic Theory 18(2):301–317. https:// doi. org/ 10. 1016/ 00220531(78) 90085-6 Nair H, Dubé J-P, Chintagunta P (2005) Accounting for primary and secondary demand effects with aggregate data. Market Sci 24(3):444–460. https:// doi. org/ 10. 1287/ mksc. 1040. 0101 Powell WB (2011) Approximate dynamic programming. Wiley, Hoboken Rochet J-C, Stole LA (2002) Nonlinear pricing with random participation. Rev Econ Stud 69(1):277– 311. https:// doi. org/ 10. 1111/ 1467937X. 00206 Schur R (2024) Multiunit dynamic pricing with different types of observable customer information. Or Spectrum 46(2):589–636. https:// doi. org/ 10. 1007/ s0029102400759-x Song J-S, Xue Z (2021) Demand shaping through bundling and product configuration: a dynamic multiproduct inventory-pricing model. Oper Res 69(2):525–544. https:// doi. org/ 10. 1287/ opre. 2020. 2062 Stokey NL (1979) Intertemporal price discrimination. Quarterly J Econ 93(3):355. https:// doi. org/ 10. 2307/ 18831 63 Sundararajan A (2004) Nonlinear pricing of information goods. Manag Sci 50(12):1660–1673. https:// doi. org/ 10. 1287/ mnsc. 1040. 0291 Talluri, KT, van Ryzin GJ (2004) The theory and practice of revenue management (International series in operations research & management science) Wilson CA (1988) On the optimal pricing policy of a monopolist. J Polit Econ 96(1):164–176. https:// doi. org/ 10. 1086/ 261529 Wilson R (1993): Nonlinear pricing. 1. publ. New York, NY: Oxford Univ. Press. Available online at http:// www. loc. gov/ catdir/ enhan cemen ts/ fy0639/ 91032 603-d. html Zhang D, Cooper WL (2009) Pricing substitutable flights in airline revenue management. Eur J Oper Res 197(3):848–861. https:// doi. org/ 10. 1016/j. ejor. 2006. 10. 067 Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.