Surgical scheduling under uncertainty by approximate dynamic programming

Surgical scheduling consists of selecting surgeries to be performed within a day, while jointly assigning operating rooms, starting times and the required resources. Patients can be elective or emergency/urgent. The scheduling of surgeries in an operating theatre with common resources to emergency or urgent and elective cases is highly subject to uncertainties not only on the duration of an intervention but mainly on the arrival of emergency or urgent cases. At the beginning of the day we are given a candidate set of elective surgeries with and an expected duration and a time window the surgery must start, but the expected duration and the time window of an emergency or urgent case become known when the surgery arrives. The day is divided into decision stages. Due to the dynamic nature of the problem, at the beginning of each stage the planner can make decisions taking into account the new information available. Decisions can be to schedule arriving surgeries, and to reschedule or cancel surgeries not started yet. The objective is to minimize the total expected cost composed of terms related to refusing arriving surgeries, to canceling scheduled surgeries, and to starting surgeries out of their time window. We address the problem with an approximate dynamic programming approach embedding an integer programming formulation to support decision making. We propose a dynamic model and an approximate policy iteration algorithm making use of basis functions to capture the impact of decisions to the future stages. Computational experiments have shown with statistical significance that the proposed algorithm outperforms a lookahead reoptimization approach.     

Design and Analysis of Hospital Admission Control for Operational Effectiveness

Variability in hospital occupancy negatively impacts the cost and quality of patient care delivery through increased emergency department (ED) congestion, emergency blockages and diversions, elective cancelations, backlogs in ancillary services, overstaffing, and understaffing. Controlling inpatient admissions can effectively reduce variability in hospital occupancy to mitigate these problems. Currently there are two major gateways for admission to a hospital: the ED and scheduled elective admission. Unfortunately, in highly utilized hospitals, excessive wait times make the scheduled gateway undesirable or infeasible for a subset of patients and doctors. As a result, this group often uses the ED gateway as a means to gain admission to the hospital. To better serve these patients and improve overall hospital functioning, we propose creating a third gateway: an expedited patient care queue. We first characterize an optimal admission threshold policy using controls on the scheduled and expedited gateways for a new Markov decision process model. We then present a practical policy based on insight from the analytical model that yields reduced emergency blockages, cancelations, and off‐unit census via simulation based on historical hospital data.     

Design of Flexible Multi‐Stage Processes

Faced with demand uncertainty across multiple product lines, many companies have recourse to flexible capacities which can process different products in order to better balance the trade‐off between capacity utilization and cost efficiency. Many studies demonstrated the potential benefit of using flexible capacity at the aggregate level by treating a whole plant or a whole process as a single stage. This paper extends these analyses by studying the benefits of flexible capacity while considering the multi‐stage structure of processes and consequently determining which stages should be flexible, which should be dedicated, and how much capacity to assign to each stage. We consider a two‐product firm which operates in a process‐to‐order environment and faces uncertain demand. Each stage of the process can be designed as dedicated or flexible. Dedicated resources are highly cost efficient but limited to the single product they are exclusively designed for, whereas flexible resources are versatile to handle several products but are more expensive. Using a general mathematical formulation our analysis shows that the optimal design may have some dedicated and some flexible stages along the process. Interestingly, this decision should be decoupled from the chronological order of the stages along the process.     

The Use of Flexible Manufacturing Capacity in Pharmaceutical Product Introductions*

This work considers the value of the flexibility offered by production facilities that can easily be configured to produce new products. We focus on technical uncertainty as the driver of this value, while prior works focused only on demand uncertainty. Specifically, we evaluate the use of process flexibility in the context of risky new product development in the pharmaceutical industry. Flexibility has value in this setting due to the time required to build dedicated capacity, the finite duration of patent protection, and the probability that the new product will not reach the market due to technical or regulatory reasons. Having flexible capacity generates real options, which enables firms to delay the decision about constructing product‐specific capacity until the technical uncertainty is resolved. In addition, initiating production in a flexible facility can enable the firm to optimize production processes in dedicated facilities. The stochastic dynamic optimization problem is formulated to analyze the optimal capacity and allocation decisions for a flexible facility, using data from existing literature. A solution to this problem is obtained using linear programming. The result of this analysis shows both the value of flexible capacity and the optimal capacity allocation. Due to the substantial costs involved with flexibility in this context, the optimal level of flexible capacity is relatively small, suggesting products be produced for only short periods before initiating construction of dedicated facilities.     

MEASURING AND COMPARING VOLUME FLEXIBILITY IN THE CAPITAL GOODS INDUSTRY

This paper presents a theoretical framework for measuring volume flexibility and relating these measures to firm performance. We develop four metrics using the principle that a volume flexible firm can handle similar levels of uncertainty (as measured by sales variability) with smaller fluctuations in inputs (as measured by variability in cost of goods sold and variability in inventory levels). Then, using 20 years of Compustat data on 550 firms in the capital goods industry, we find that on three of four process‐based measures, small firms are more volume flexible. However, when we incorporate financial performance into our fourth metric, we find that large firms are more volume flexible. We conclude that, to be volume flexible is one thing, but to benefit from this flexibility, firms need to focus on the cost of being flexible.     

Beyond Patient Classification: Using Individual Patient Characteristics in Appointment Scheduling

Scheduling patients involves a trade‐off between the productivity of the service provider and customer service. This study considers how outpatient medical facilities can improve their appointment scheduling by incorporating individual patient information in the scheduling process. Specifically, we obtain data on patient characteristics and examination durations from a health clinic, describe how that data can be used to predict patient examination durations in the clinic's appointment scheduling system, and evaluate the benefit of using individual patient characteristics over a conventional classification method. Computational results illustrate this method of patient scheduling reduces an overall cost function comprised of patient wait time, physician idle time, and over time by up to 24.2%, particularly when patients are sequenced with short duration patients being scheduled first. Several environmental characteristics are found to play critical roles in determining the magnitude of the benefit, including patient punctuality, no‐show probability, the clinic duration, the appointment rule used for scheduling, and the ratio of the physician's idle time cost to the patient wait cost. We also detail and evaluate a practical procedure for using heterogeneous scheduling under a fixed schedule.     

A dynamic Bayesian network-based emergency decision-making framework highlighting emergency propagations: Illustrated using the Fukushima nuclear accidents and the Covid-19 pandemic

When facing public emergencies, human societies need to make decisions rapidly in order to mitigate the problems. However, this process can be difficult due to complexity of the emergency scenarios and lack of systematic methods for analyzing them. In the work reported here, we develop a framework based upon dynamic Bayesian networks in order to simulate emergency scenarios and support corresponding decisions. In this framework, we highlight the importance of emergency propagation, which is a critical factor often ignored by decisionmakers. We illustrate that failure of considering emergency propagation can lead to suboptimal mitigation strategies. By incorporating this critical factor, our framework enables decisionmakers to identify optimal response strategies minimizing emergency impacts. Scenarios developed from two public emergencies: the 2011 Fukushima nuclear power plant accidents and the Covid-19 pandemic, are utilized to illustrate the framework in this paper. Capabilities of the framework in supporting decision making in both events illustrate its generality and adaptability when dealing with complex real-world situations. Our analysis results reveal many similarities between these two seemingly distinct events. This indicates that seemingly unrelated emergencies can share many common features beyond their idiosyncratic characteristics. Valuable mitigation insights can be obtained by analyzing a broad range of past emergencies systematically.     

Reducing Surgical Ward Congestion Through Improved Surgical Scheduling and Uncapacitated Simulation

High surgical bed occupancy levels often result in heightened staff stress, frequent surgical cancellations, and long surgical wait times. This congestion is in part attributable to surgical scheduling practices, which often focus on the efficient use of operating rooms but ignore resulting downstream bed utilization. This paper describes a transparent and portable approach to improve scheduling practices, which combines a Monte Carlo simulation model and a mixed integer programming (MIP) model. For a specified surgical schedule, the simulation samples from historical case records and predicts bed requirements assuming no resource constraints. The MIP model complements the simulation model by scheduling both surgeon blocks and patient types to reduce peak bed occupancies. Scheduling guidelines were developed from the optimized schedules to provide surgical planners with a simple and implementable alternative to the MIP model. This approach has been tested and delivered to planners in a health authority in British Columbia, Canada. The models have been used to propose new surgical schedules and to evaluate the impact of proposed system changes on ward congestion.     

Bi‐Criteria Scheduling of Surgical Services for an Outpatient Procedure Center

Uncertainty in the duration of surgical procedures can cause long patient wait times, poor utilization of resources, and high overtime costs. We compare several heuristics for scheduling an Outpatient Procedure Center. First, a discrete event simulation model is used to evaluate how 12 different sequencing and patient appointment time‐setting heuristics perform with respect to the competing criteria of expected patient waiting time and expected surgical suite overtime for a single day compared with current practice. Second, a bi‐criteria genetic algorithm (GA) is used to determine if better solutions can be obtained for this single day scheduling problem. Third, we investigate the efficacy of the bi‐criteria GA when surgeries are allowed to be moved to other days. We present numerical experiments based on real data from a large health care provider. Our analysis provides insight into the best scheduling heuristics, and the trade‐off between patient and health care provider‐based criteria. Finally, we summarize several important managerial insights based on our findings.     

Measuring the Impact of Increased Product Substitution on Pricing and Capacity Decisions Under Linear Demand Models

We consider two substitutable products and compare two alternative measures of product substitutability for linear demand functions that are commonly used in the literature. While one leads to unrealistically high prices and profits as products become more substitutable, the results obtained using the other measure are in line with intuition. Using the more appropriate measure of product substitutability, we study the optimal investment mix in flexible and dedicated capacities in both monopoly and oligopoly settings. We find that the optimal investment in manufacturing flexibility tends to decrease as the products become closer substitutes; this is because (1) pricing can be used more effectively to balance supply and demand, and (2) the gains obtained by shifting production to the more profitable product are reduced due to increased correlation between the price potentials of the substitutable products. The value of flexibility always increases with demand variability. We also show that, as long as the optimal investments in dedicated capacity for both products are positive, the optimal expected prices and production quantities do not depend on the cost of the flexible capacity. Manufacturing flexibility simply allows the firm to achieve those expected values with lower capacity, while leading to higher expected profits.     

Worst-case analysis for on-line service policies

This paper considers an on-line scheduling problem occurred in a service system with two classes (ordinary and special) of customers and two types (dedicated and flexible) of servers. The customers line up with a list according to the order of their arrivals. All customers are assumed to be in the system when service starts and no preemption is allowed. The objective is to complete all services in list as early as possible. We study three classes of service policies that are commonly used in practice and analyze their worst-case performance in terms of makespan. Tight worst-case ratios have been derived for these service policies.     

Commitment vs. Flexibility

We study the optimal trade‐off between commitment and flexibility in a consump‐ tion–savings model. Individuals expect to receive relevant information regarding tastes and thus they value the flexibility provided by larger choice sets. On the other hand, they also expect to suffer from temptation, with or without self‐control, and thus they value the commitment afforded by smaller choice sets. The optimal commitment problem we study is to find the best subset of the individual's budget set. This problem leads to a principal–agent formulation. We find that imposing a minimum level of savings is always a feature of the solution. Necessary and sufficient conditions are derived for minimum‐savings policies to completely characterize the solution. We also discuss other applications, such as the design of fiscal constitutions, the problem faced by a paternalist, and externalities.     

Flexible Backup Supply and the Management of Lead‐Time Uncertainty

We consider replenishment decisions for a constant rate demand environment from a supplier with uncertain lead times. We study the potential use of a flexible backup supplier as an emergency response to accurate lead‐time information arriving at (or close after) the beginning of the demand interval and well after an original order with the stochastic lead‐time supplier has been placed. The emergency response decisions involve whether to order and how much from the flexible backup supplier, with the objective of minimizing the cost of meeting demand. We derive the optimal emergency‐response policy and clearly outline its implications on the optimized safety lead time of the original order placement and on the cost of meeting demand. We examine the impact on the use of the flexible backup supplier of factors like the arrival time of accurate lead‐time information and the response lead time of the backup supplier. We further study the potential benefits of the use of the flexible backup supplier in a dual role: as one of the two suppliers in a redundant supply system assigned to originally meet the demand and as an emergency response to later‐arriving lead‐time information. Our numerical studies illustrate the benefits from the use of the flexible backup supplier as an emergency response, but for reasonable purchase premiums and short lead times of flexible backup supply options, their use in a dual (regular and emergency response) role often leads to improved performance over safety lead‐time single and uncertain lead‐time supplier‐replenishment strategies. The benefits of the backup supply options are accentuated the higher the lead‐time uncertainty of the stochastic lead‐time supplier is.     

Flexible home care scheduling

Home care services are in high demand given how they are steadily becoming the primary source of care for the elderly. Powerful decision support tools are indispensable for effectively managing available staff in the context of ever-increasing demand for care and limited caregiver availability. This paper advances home care literature by introducing flexible task durations, thereby enabling tasks to be completed faster and ultimately more care to be scheduled. This new concept, which originates from practice, introduces an additional decision to be made when creating a schedule, thereby greatly increasing the scheduling complexity. Consequently, this paper introduces a new optimization-based decision support model which allows for scheduling with flexible task duration, as well as other types of flexibility. A computational study quantifies the impact of: (i) scheduling with a finer task granularity thereby enabling accurate prioritization of high and low priority care, (ii) flexibility in task duration enabling tasks to be completed faster and more care to be scheduled, and (iii) increasing the number of different locations visited by a caregiver thereby enabling a trade-off between the number of serviced clients and caregiver workload. A new publicly available real-world data set is used, obtained directly from home care organizations operating in Flanders. Analysis of the computational results demonstrates that significant improvements in operational efficiency may be realized with minimal effort required by organizations. Furthermore, the proposed algorithm’s performance is confirmed by comparison against the bounds obtained by solving an integer programming formulation of the problem. Finally, a management policy scheme is proposed which, when gradually implemented in a home care organization, results in a more efficient and therefore cost-effective deployment of its workforce.     

Optimal Sequencing of Unpunctual Patients in High‐Service‐Level Clinics

Even though patients often arrive early and out of turn for scheduled appointments in outpatient clinics, no research has been undertaken to establish whether an available provider should see an early patient right away (preempt) or wait for the patient scheduled next. This problem, which we call the “Wait‐Preempt Dilemma,” is particularly relevant for “high‐service‐level” clinics (such as psychotherapy, chiropractic, acupuncture), where preempting may cause the missing patient to wait for an excessively long time, should she show up soon. Typically, the dilemma is resolved by preemption, where the provider starts serving the patient who has already arrived to avoid staying idle. By contrast, we analytically determine the time intervals where it is optimal to preempt and those where it is optimal to wait, and find that in some cases the provider should in fact stay idle, even in the presence of waiting patients. Our results suggest that the proposed analytical method outperforms the always‐preempt policy in clinics that do not overbook and have service times longer than 30 minutes. In these cases, the analytical method dramatically reduces patient waiting times at the cost of a modest increase in overtime. By contrast, in clinics that overbook or have short service times, the two policies perform similarly, and hence the always‐preempt policy is preferable due to its simplicity. A software application is provided that clinics can readily use to solve the wait‐preempt dilemma.     

Design Principles for Flexible Systems

A fundamental aspect of designing systems with dedicated servers is identifying and improving the system bottlenecks. We extend the concept of a bottleneck to networks with heterogeneous, flexible servers. In contrast with a network with dedicated servers, the bottlenecks are not a priori obvious, but can be determined by solving a number of linear programming problems. Unlike the dedicated server case, we find that a bottleneck may span several nodes in the network. We then identify some characteristics of desirable flexibility structures. In particular, the chosen flexibility structure should not only achieve the maximal possible capacity (corresponding to full server flexibility), but should also have the feature that the entire network is the (unique) system bottleneck. The reason is that it is then possible to shift capacity between arbitrary nodes in the network, allowing the network to cope with demand fluctuations. Finally, we specify when certain flexibility structures (in particular chaining, targeted flexibility, and the “N” and “W” structures from the call center literature) possess these desirable characteristics.     

Mitigating overtime risk in tactical surgical scheduling

Overtime is a common phenomenon in surgery departments, causing stress to physicians, dissatisfaction to patients, and financial loss to hospitals. We help risk-averse managers of operating rooms (ORs) to mitigate overtime in tactical surgery scheduling, which determines the assignment of elective patients to available ORs in upcoming time periods. We model the uncertain surgical durations via partial, full, or empirical distributions. To mitigate overtime, our model maximizes the risk aversion level of the OR manager (and thus the risk-hedging ability of the solution) while ensuring that the certainty equivalent of surgery duration in each OR at each time period does not exceed the stipulated working hours. The corresponding decision criterion, termed the maximized risk aversion level, is demonstrated in theory and in numerical experiments to be able to mitigate both the overtime probability and the expected overtime duration. To solve the problem, we develop an exact hill-climbing algorithm and demonstrate its convergence and correctness. Numerical experiments based on real-life surgery data show that our method outperforms the existing methods in several indicators that of concern to OR managers. In particular, this method is computationally amiable and hence is applicable to larger-scale instances.     

Intelligent Procedures for Intra‐Day Updating of Call Center Agent Schedules

For nearly all call centers, agent schedules are typically created several days or weeks before the time that agents report to work. After schedules are created, call center resource managers receive additional information that can affect forecasted workload and resource availability. In particular, there is significant evidence, both among practitioners and in the research literature, suggesting that actual call arrival volumes early in a scheduling period (typically an individual day or week) can provide valuable information about the call arrival pattern later in the same scheduling period. In this paper, we develop a flexible and powerful heuristic framework for managers to make intra‐day resource adjustment decisions that take into account updated call forecasts, updated agent requirements, existing agent schedules, agents' schedule flexibility, and associated incremental labor costs. We demonstrate the value of this methodology in managing the trade‐off between labor costs and service levels to best meet variable rates of demand for service, using data from an actual call center.     

突发性产出下的供应链协调应对策略

针对突发事件影响制造商产出的情形，研究了收益共享契约协调应对突发产出事件的有效性问题。首先论证了无突发产出事件时收益共享契约协调供应链的有效性，进而分析了突发产出事件对供应链最优生产计划、定价决策、绩效和协调性的影响。然后，建立了突发产出事件下的收益共享契约协调应对模型，并对原收益共享契约和修正后的收益共享契约进行了对比分析。研究表明：当突发产出事件导致产出扰动范围较小时，供应链的最优生产计划、批发价格和零售价格几乎保持不变，仅当产出扰动范围较大时，三者才需要同时调整，此时原供应链的协调性也将被打破，而修正后的收益共享契约具有良好的抗突发性。最后，通过算例进行了验证。     

Impact of Price Postponement on Capacity and Flexibility Investment Decisions

Investments in dedicated and flexible capacity have traditionally been based on demand forecasts obtained under the assumption of a predetermined product price. However, the impact on revenue of poor capacity and flexibility decisions can be mitigated by appropriately changing prices. While investment decisions need to be made years before demand is realized, pricing decisions can easily be postponed until product launch, when more accurate demand information is available. We study the effect of this price decision delay on the optimal investments on dedicated and flexible capacity. Computational experiments show that considering price postponement at the planning stage leads to a large reduction in capacity investments, especially in the more expensive flexible capacity, and a significant increase in profits. Its impact depends on demand correlation, elasticity and diversion, ratio of fixed to variable capacity costs, and uncertainty remaining at the times the pricing and production decisions are made.