Patient Customers Research Articles

Analyzing how inflation affects non-instantly decaying goods with demand linked to ads and selling price in a dual-warehouse setup

This study explores inventory management strategies specifically designed for non-instantaneous deteriorating goods under inflationary conditions, utilizing a dual-warehouse system—one owned and the other rented with limited storage capacity. We examine the effects of advertising frequency and product selling price on demand rates, taking into account the gradual decline in customer patience, which leads to partial backlogging of shortages. The primary objective is to determine optimal replenishment policies for retailers that effectively minimize total costs per unit time. In real life, managing oil inventory is crucial for industries where factors like gradual deterioration, demand fluctuations due to advertising and pricing, and sensitivity to inflation require sophisticated inventory models to optimize replenishment policies and minimize costs. To validate our proposed inventory model, we provide a numerical. A sensitivity study using MATLAB R2024a software highlights the impact of parameter changes, providing significant information for decision-makers across various industries.

Prioritization in the Presence of Self-ordering Opportunities in Omnichannel Services

Motivated by the popularity of mobile-order-and-pay applications, especially in fast-casual food restaurants and coffee shops, we study omnichannel service systems—where customers can employ mobile applications for self-ordering—with respect to sojourn times, throughput, and social welfare. Our models are two-stage queues with two customer classes: Walk-ins and mobiles. We identify Pareto efficient prioritization policies, highlighting trade-offs between each class’s mean sojourn times. We allow customers to make strategic joining decisions based on their anticipated delays under an information structure where walk-ins observe partial queue length information. We draw from a wide array of techniques, including steady-state, transient, busy period, hitting-time analyses, and matrix analytic methods. We showcase the significance of prioritization on the system throughput and social welfare. We demonstrate settings where a traditional service system’s (typically beneficial) transformation to an omnichannel reduces throughput. Our analysis highlights the importance of prioritization policy choice for an efficient transition to an omnichannel service system. The throughput-optimal policy choice highly depends on the operational parameters and customer patience levels; implementing a wrong policy can yield a significant loss in throughput and profitability.

Airline dynamic pricing with patient customers using deep exploration-based reinforcement learning

This paper addresses a crucial issue in the airline industry by tackling a dynamic pricing problem in the presence of patient customers, a scenario that has gained significance due to the revenue loss of airlines caused by customers’ non-myopic decision-making. To effectively capture this non-myopic characteristic, we propose a Markov decision process (MDP) including a history of offered prices as a state variable. In contrast to previous studies, distributions of customers’ properties are assumed to be unknown in advance for a more realistic representation of real-world scenarios. To deal with the new challenges of the problem, we propose utilizing a specific learning framework (i.e., deep exploration-based RL) that is unexplored in this domain. The numerical experiments demonstrate that its performance can be improved on the MDP we designed and show that it outperforms the benchmark algorithm. The structures of pricing policies generated from the bootstrapped deep Q-network algorithm imply that airlines should offer high and low prices alternately from the beginning of the sales period rather than increasing prices as time goes on. We also ascertain that more frequent consecutive high-priced periods can increase airlines’ revenue in environments with higher customer patience levels.

Co‐opetition strategies in supply chains with strategic customers

We study the co‐opetition strategy between a manufacturer of a proprietary component (MPC) and an original equipment manufacturer (OEM). The MPC can choose to supply components to the OEM and can also compete in the end market with its own final product. Customers can purchase from either firm and are strategic in that they choose whether to buy now or later. In a two‐period game theoretical model, we analyze the MPC's optimal co‐opetition strategy, which mainly concerns whether to sell to the OEM or to enter the end market. The optimal strategy can be characterized by the customer's patience to wait and the competitor's product quality. We then extend our model to investigate three key factors: new customers arriving late, cost differentiation between the two firms, and the existence of a third‐party supplier. While in the first two extensions the optimal strategies remain qualitatively the same as in the main model, the existence of a third‐party supplier impacts the MPC's choice of co‐opetition strategy crucially. We discuss how the MPC should balance cooperation and competition, and characterize the conditions when the MPC should change strategy.

Increases or discounts: Price strategies based on customers’ patience times

This study analyzes dynamic pricing in a non-preemptive M/M/1 service system in which customers exhibit patience in regard to waiting time. We use a stylized, single-server queue to incorporate customer patience into the queuing problem, with the assumption that customers will not enter the queue when the waiting time exceeds their inherent patience time, reflecting the decision whether to wait. We focus on queue design, structure, and revenue and consider two homogeneous and heterogeneous customer scenarios. Our findings show that, for homogeneous customers, price-raising and price-discounting strategies can mitigate revenue loss. We examined which policy is better for service providers in different scenarios. For heterogeneous customers, we draw two conclusions. First, a revenue-maximizing service provider should increase her priority fee if customers have patience in regard to waiting time, regardless of whether they are regular customers, priority customers, or a mix of both. Second, contrary to conventional wisdom, we show that the service provider should abandon a portion of his or her regular customers to attract more customers who are willing to pay a higher priority fee for better service when the value of regular customers is small. Under this situation, customers’ patience times can bring more revenue to the service provider. If the value of regular customers is high, however, then an increased priority fee cannot compensate for the revenue loss from abandoning these customers and losing their revenue. Overall, our findings confirm that customers’ patience times result in a high net monetary cost for the time-sensitive customer.

Queuing Models for Analyzing the Steady-State Distribution of Stochastic Inventory Systems with Random Lead Time and Impatient Customers

In material management, the inventory systems may have good management aspects in terms of materials; however, this negatively affects the relationship between the facility and customers. In classical inventory models, arriving demands are satisfied immediately if there is enough on-hand inventory. Traditional inventory models consider optimization problems and find the optimal policy of decision variables without computing the stationary distribution of the inventory states for random demand. Hence, a detailed analysis of inventory management systems requires a joint distribution of system stock levels and the number of requests to be investigated thoroughly. This research provides a new stochastic mathematical model for inventory systems with lead times and impatient customers under deterministic and uniform order sizes. The proposed model identifies the performance measures in a stochastic environment, analyzing the properties of the inventory system with stochastic and probabilistic parameters, and finally, validating the model’s accuracy. To analyze the system, balance equations were derived from a mathematical characterization of the underlying queuing model dependent on the Markov chain formalism. The precise performance was achieved by examining the graphical representation of the service process in a steady-state as a function of both arrival distribution and the customer patience coefficient, while it was challenging to derive an optimal curve fit in a three-dimensional space that features two input variables and a single output variable.

Revenue Management of Group Service Considering Different Quality Levels and Stochastic Customer Patience

We consider a service that provides different quality levels to customers who are served in groups (referred to as “group service” (GS)). We consider stochastic customer patience and build a multitype service model to maximize service profit. First, we consider a fixed number of service personnel and decompose the model into several subproblems. We then transform the model into a service personnel allocation problem that can be efficiently solved. Furthermore, we use the water-filling theory, together with a dichotomy approach, to design the numerical algorithm to conduct a numerical test based on actual data. The main contributions of this study are as follows. First, we provide the service provider with a service capacity allocation mechanism that can determine the optimal choice of service quality levels, number of service groups of different quality levels, time length of the service registration period, and service prices. Second, we find that under scarce service capacity, the single-type service can be more profitable than the multitype service, and low-quality services have higher marginal profits than high-quality services. However, when service capacity increases to a certain level, the multitype service can be more profitable than the single-type service. Third, we construct a service capacity decision mechanism and demonstrate the influence of service quality on the total number of service personnel and service groups. The conclusions are potentially useful to GS service providers to effectively make better managerial decisions and improve service profits.

Performance Approximation for Time-Dependent Queues with Generally Distributed Abandonments

Many stochastic systems face a time-dependent demand. Especially in stochastic service systems, for example, in call centers, customers may leave the queue if their waiting time exceeds their personal patience. As discussed in the extant literature, it can be useful to use general distributions to model such customer patience. This paper analyzes the time-dependent performance of a multiserver queue with a nonhomogeneous Poisson arrival process with a time-dependent arrival rate, exponentially distributed processing times, and generally distributed time to abandon. Fast and accurate performance approximations are essential for decision support in such queueing systems, but the extant literature lacks appropriate methods for the setting we consider. To approximate time-dependent performance measures for small- and medium-sized systems, we develop a new stationary backlog-carryover (SBC) approach that allows for the analysis of underloaded and overloaded systems. Abandonments are considered in two steps of the algorithm: (i) in the approximation of the utilization as a reduced arrival stream and (ii) in the approximation of waiting-based performance measures with a stationary model for general abandonments. To improve the approximation quality, we discuss an adjustment to the interval lengths. We present a limit result that indicates convergence of the method for stationary parameters. The numerical study compares the approximation quality of different adjustments to the interval length. The new SBC approach is effective for instances with small numbers of time-dependent servers and gamma-distributed abandonment times with different coefficients of variation and for an empirical distribution of the abandonment times from real-world data obtained from a call center. A discrete-event simulation benchmark confirms that the SBC algorithm approximates the performance of the queueing system with abandonments very well for different parameter configurations. Summary of Contribution: The paper presents a fast and accurate numerical method to approximate the performance measures of a time‐dependent queueing system with generally distributed abandonments. The presented stationary backlog carryover approach with abandonment combines algorithmic ideas with stationary queueing models for generally distributed abandonment times. The reliability of the method is analyzed for transient systems and numerically studied with real‐world data.

Optimally Scheduling Heterogeneous Impatient Customers

We study the question of scheduling impatient customers in parallel server queuing systems. At the time of arrival, customers can be identified as one of many classes, where the class represents the service time and patience time distributions, and cost characteristics. From the system's perspective, customers that are of the same type at time of arrival get further differentiated on their residual patience time as they wait in the system. This is due to the fact that at time of arrival, the system only knows the overall patience distribution from which the customer's patience value is drawn, and as time elapses, this estimate can be further updated for customers who are still in the system using the information that the customer has not yet abandoned. For non-exponential patience distributions, such an update indeed reveals additional information. In this paper, we use a fluid approach to characterize the cost-minimizing policy that schedules customers on two dimensions of heterogeneity: class and time-in-queue information. We propose a multi-class time-in-queue policy that prioritizes both across customer classes, and within each customer class using a simple rule, and further we show that most of the gains of such a policy can be achieved by deviating from within-class First Come First Serve (FCFS) for at most one customer class. Finally, for systems with exponential abandonment times, our policy reduces to a simple priority-based policy, which we prove to be asymptotically optimal with an optimality gap that does not grow with system scale.

An Invitation Control Policy for Proactive Service Systems: Balancing Efficiency, Value, and Service Level

Problem definition : We study the problem of designing a dynamic invitation policy for proactive service systems with finite customer patience under scarce capacity. In such systems, prior knowledge regarding customer value or importance is used to decide whether the company should offer service or not. Academic/practical relevance : Proactive service systems are becoming more popular, as data availability and machine-learning techniques are developed to forecast customer needs. However, very little is known about the efficient use of such tools to promote and manage service systems. Methodology : We use fluid approximation and the Filippov convex method to analyze system dynamics and develop approximations for important performance measures. Results : We show that whereas prioritizing customers in descending order of their r-μ ranking (as long as there are idle servers in the system) is optimal on the fluid level, refinements are necessary in the presence of abandonment on the stochastic level. We propose an r-μ-N policy to account for customer patience. Managerial implications : Our policy can be used to promote service effectivenes and allow decision makers the means to trade off service level against costs in such systems explicitly. Using a case study of a transportation service provider, we show that such a policy can triple revenue compared with random arrivals (nonproactive policy).

Hazard rate estimation for call center customer patience time

Estimating the hazard function of customer patience time has become a necessary component of effective operational planning such as workforce staffing and scheduling in call centers. When customers get served, their patience times are right-censored. In addition, the exact event times in call centers are sometimes unobserved and naturally binned into time intervals, due to the design of data collection systems. We develop a TunT (Transform-unTransform) estimator that turns the difficult problem of nonparametric hazard function estimation into a regression problem on binned and right-censored data. Our approach starts with binning event times and transforming event count data with a mean-matching transformation, which enables a simpler characterization of the heteroscedastic variance function. A nonparametric regression technique is then applied to the transformed data. Finally, the estimated regression function is back-transformed to yield an estimator for the original hazard function. The proposed estimation procedure is illustrated using call center data to reveal interesting customer patience behavior, and health insurance plan trial data to compare the effect between treatment and control groups. The numerical study shows that our approach yields more accurate estimates and better staffing decisions than existing methods.

Dynamic Pricing under Cost Reduction in the Presence of Myopic and Strategic Consumers

After release, a product usually suffers cost reductions during its whole lifespan. Compared to the myopic, strategic consumers may have stronger incentive to delay the purchase once they perceive that a significant cost reduction will result in a markdown. The strategic (compared to the myopic) properties influence the seller both quantitatively in terms of proportion of strategic consumers and qualitatively in terms of customer patience. To forecast the reaction of the whole market under cost reduction, it is necessary to acquire the strategic properties. In this paper, we study the impacts of proportion of strategic consumers, customer patience, and cost reduction on dynamic pricing strategy when cost reduction comes from technology advancement. The seller makes pricing strategies when facing unknown future cost, and the buyer makes purchase decisions when facing unknown future price. Our study shows that generally both higher strategic consumer proportion and customer patience contribute to a delay in sales. Further, profit diversion happens under great combination of strategic properties. In addition, with the increase of customer patience, not only strategic but also myopic consumers will buy less. Finally, the strategic properties moderate the pricing strategy in latter stage when there is a cost reduction. This indicates a threshold as combination of strategic properties, upon which seller tends to offer a smaller markdown to discourage strategic waiting, and under which seller tends to offer a greater markdown to divert strategic consumers to the latter period.

Design heuristic for parallel many server systems

We study a parallel queueing system with multiple types of servers and customers. A bipartite graph describes which pairs of customer-server types are compatible. We consider the service policy that always assigns servers to the first, longest waiting compatible customer, and that always assigns customers to the longest idle compatible server if on arrival multiple compatible servers are available. For a general renewal stream of arriving customers, general service time distributions that depend both on customer and on server types, and general customer patience distributions, the behavior of such systems is very complicated. Key quantities for their performance are the matching rates, the fraction of services for each pair of compatible customer-server. Calculation of these matching rates in general is intractable, it depends on the entire shape of service time distributions. We suggest through a heuristic argument that if the number of servers becomes large, the matching rates are well approximated by matching rates calculated from the tractable bipartite infinite matching model. We present simulation evidence to support this heuristic argument, and show how this can be used to design systems with desired performance requirements.

To Wait or Not to Wait: Two-Way Functional Hazards Model for Understanding Waiting in Call Centers

ABSTRACTTelephone call centers offer a convenient communication channel between businesses and their customers. Efficient management of call centers needs accurate modeling of customer waiting behavior, which contains important information about customer patience (how long a customer is willing to wait) and service quality (how long a customer needs to wait to get served). Hazard functions offer dynamic characterization of customer waiting behavior, and provide critical inputs for agent scheduling. Motivated by this application, we develop a two-way functional hazards (tF-Hazards) model to study customer waiting behavior as a function of two timescales, waiting duration and the time of day that a customer calls in. The model stems from a two-way piecewise constant hazard function, and imposes low-rank structure and smoothness on the hazard rates to enhance interpretability. We exploit an alternating direction method of multipliers algorithm to optimize a penalized likelihood function of the model. We carefully analyze the data from a U.S. Bank call center, and provide informative insights about customer patience and service quality patterns along waiting time and across different times of a day. The findings provide primitive inputs for call center agent staffing and scheduling, as well as for call center practitioners to understand the effect of system protocols on customer waiting behavior. Supplementary materials for this article are available online.

Optimal allocation of spares to maximize the window fill rate in a two-echelon exchangeable-item repair system

We solve the spares allocation problem in a two-echelon, exchangeable-item repair system in which the lower echelon comprises multiple locations and the higher echelon is a single depot. We assume that customers tolerate a certain wait and therefore the optimization criterion is the window fill rate, i.e., the expected portion of customers who are served within the tolerable wait. We develop two algorithms to solve this problem. The first algorithm (FTEA) is formula-based and is suboptimal. The second algorithm (HTEA) combines simulations into the first algorithm and obtains a higher degree of accuracy at the cost of extra running time. We characterize the near-optimal solution by its degree of pooling and concentration. Pooling happens when spares are allocated to the depot and are therefore shared by all the lower-echelon locations. Concentration takes place when spares are allocated to only a few lower-echelon locations whereas the other lower-echelon locations receive no spares. We use numerical examples to compare the algorithms and to illustrate how the budget, shipment time, local repair and customer patience affect the optimal solution and degree of pooling in varying ways. Using the numerical results, we propose a third algorithm (ETEA) that obtains HTEA’s output in 30% of the time.

Refined Models for Efficiency-Driven Queues with Applications to Delay Announcements and Staffing

Data has revealed a noticeable impact of delay-time-related information on phone-customers; for example and somewhat surprisingly, delay announcements can abruptly increase the likelihood to abandon (hang up). Our starting point is that the latter phenomena can be used to support the control of queue lengths and delays. We do so by timing the announcements appropriately and determining the staffing levels accordingly. To this end, we model a service system as an overloaded GI/M/s + GI queue, in which we seek to minimize the number of servers, s, subject to quality-of-service constraints (e.g., fraction abandoning), while accounting for the instantaneous (hence discontinuous) impact of an announcement on the distribution (hazard rate) of customer patience. For tractability, our analysis is asymptotic as s increases indefinitely, and it is naturally efficiency-driven (namely the servers are highly busy, and hence essentially all customers are delayed in queue prior to service). This requires one to go beyond existing theory, which turns out to be too crude for our needs (e.g., it requires a continuous hazard rate of impatience and hence cannot be applied). We thus develop a refined process and steady-state models, and use them to solve our minimization problem and more. The value and accuracy of our models are demonstrated via extensive numerical experiments. The online appendix is available at https://doi.org/10.1287/opre.2017.1619 .

Linking Delay Announcements, Abandonment, and Staffing: A Behavioral Perspective

Using field data from a bank's call center, we study the behavioral determinants of customers' queue abandonment decisions in the presence of delay announcements. We relax traditional behavioral assumptions on how customers should make abandonment decisions and empirically explore how customers actually behave. Our flexible model can account for multiple delay announcements, reference-dependent behavior, sunk costs in time, and heterogeneity in customer patience. We show that customers exhibit loss aversion with respect to time losses in queue, becoming much more likely to abandon if forced to wait longer than the announced waiting time. We also find that customers may fall for the sunk cost effect, becoming less likely to abandon with more time sunk waiting in queue. Additionally, we find that longer announced wait times increase customer abandonment behavior and that customers are often strategic in their arrival decisions, with impatient customers avoiding the busy time of day. In a counterfactual study, we show that, by accounting for our behavioral insights, firms may significantly reduce their staffing levels without increasing customers' overall waiting time.

Optimal spares allocation to an exchangeable-item repair system with tolerable wait

In a multi-location, exchangeable-item repair system with stochastic demand, the expected waiting time and the fill rate measures are oftentimes used as the optimization criteria for the spares allocation problem. These measures, however, do not take into account that customers will tolerate a reasonable delay and therefore, a firm does not incur reputation costs if customers wait less than their tolerable wait. Accordingly, we generalize the expected waiting time and fill rate measures to reflect customer patience. These generalized measures are termed the truncated waiting time and the window fill rate, respectively. We develop efficient algorithms to solve the problem for each of the criteria and demonstrate how incorporating customer patience provides considerable savings and profoundly affects the optimal spares allocation.

The timing of green product introduction in relation to technological evolution

As a result of the rising awareness of environmental issues and increasingly stringent regulations, the ability of a business to manage environmentally friendly products represents an important competitive edge. This paper presents a study of the impacts of two types of technological solutions, namely, Zero Sum and Synergy, on strategies for timing the introduction of green products. We have developed mathematical models to determine the optimal price, traditional quality, and environmental quality required to maximize profit. In addition, we discuss how differentiation in environmental quality and customer patience impact the choice of product introduction strategies under the commitment and no-commitment scenarios. We also investigate how Zero Sum and Synergy interact with choice. The results show that a business tends to adopt a simultaneous introduction strategy when the differentiation in market condition for environmental quality is high, customer is patient, and/or the business employs the Synergy technology. In addition, we determine that this strategy change is less likely in the commitment scenario, when the technology changes from Zero Sum to Synergy.

A Unified Approach to Diffusion Analysis of Queues with General Patience-Time Distributions

We propose a unified approach to establishing diffusion approximations for queues with impatient customers within a general framework of scaling customer patience time. The approach consists of two steps. The first step is to show that the diffusion-scaled abandonment process is asymptotically close to a function of the diffusion-scaled queue length process under appropriate conditions. The second step is to construct a continuous mapping not only to characterize the system dynamics using the system primitives, but also to help verify the conditions needed in the first step. The diffusion approximations can then be obtained by applying the continuous mapping theorem. The approach has two advantages: (i) it provides a unified procedure to establish the diffusion approximations regardless of the structure of the queueing model or the type of patience-time scaling; and (ii) it makes the diffusion analysis of queues with customer abandonment essentially the same as the diffusion analysis of queues without customer abandonment. We demonstrate the application of this approach via the single-server system with Markov-modulated service speeds in the traditional heavy-traffic regime and the many-server system in the Halfin-Whitt regime and the nondegenerate slowdown regime.