Two-stage Stochastic Programming Model Research Articles

A stochastic programming approach to surgery scheduling under parallel processing principle

Parallel processing is a principle which enables simultaneous implementation of anesthesia induction and operating room (OR) turnover with the aim of improving OR utilization. In this article, we study the problem of scheduling surgeries for multiple ORs and induction rooms (IR) that function based on the parallel processing principle under uncertainty. We propose a two-stage stochastic mixed-integer programming model considering the uncertainty in induction, surgery and turnover durations. We sequence patients and set appointment times for surgeries in the first stage and assign patients to IRs at the second stage of the model. We show that an optimal myopic policy can be used for IR assignment decisions due to the special structure of the model. We minimize the expected total cost of patient waiting time, OR idle time and IR idle time in the objective function. We enhance the model formulation using bounds on variables and symmetry-breaking constraints. We implement a novel progressive hedging algorithm by proposing a penalty update method and a variable fixing mechanism. Based on real data of a large academic hospital, we compare our solution approach with several scheduling heuristics from the literature. We assess the additional benefits and costs associated with the implementation of parallel processing using near-optimal schedules. We examine how the benefits are inflated by increasing the number of IRs. Finally, we estimate the value of stochastic solution to underline the importance of considering uncertainty in durations.

Optimizing inland container logistics through dry ports: A two-stage stochastic mixed-integer programming approach considering volume discounts and consolidation in rail transport

It is essential for container shipping companies to plan for the optimum number and locations of dry ports as well as efficient inland container logistics operations to reduce costs and environmental impacts while improving customer service. The plan for optimum inland container transportation network design must account for demand uncertainty to eliminate financial difficulty due to redundant investment. In this regard, this study proposed a two-stage stochastic mixed-integer programming model for optimizing inland container logistics through dry ports. The model contributes to the state of the art in current research by including a piecewise-linear cost function for railway transportation to account for volume discounts, integrating full container scheduling with transport mode selection and empty container relocation for consolidation, and reflecting the fact that the amount of import and export full containers transported between customers (consignees and consignors) and seaports are exogenously decided. The solution results demonstrated the definite performance superiority of the progressive hedging algorithm over the extensive form solution. Additionally, the value of stochastic solution calculation showed that the application of stochastic solution might result in significant cost savings compared to the application of mean value deterministic solution. The proposed model can be applied for practical necessities to design robust optimum inland container logistics operations using intermodal rail transport with the progressive hedging algorithm as an efficient solution approach.

EMS location-allocation problem under uncertainties

Emergencies, especially those considered routine (i.e., occurring on a daily basis), pose great threats to health, life, and property. Immediate response and treatment can greatly mitigate these threats. This research is conducted to optimize the locations of ambulance stations, deployment of ambulances, and dispatch of vehicles under demand and traffic uncertainty, which are the main factors that influence emergency response time. The research problem is formulated as a dynamic scenario-based two-stage stochastic programming model, aiming to minimize the total cost while responding to as much demand as possible. The Sample Average Approximation is proposed to approximate the original problem using a limited number of scenarios, and a two-phase Benders Decomposition solution scheme is proposed to accelerate computation, especially when solving a large-sized problem. Numerical experiments using real-world emergency data are conducted to validate the performance of the solution method. The results demonstrate the effectiveness and efficiency of the proposed algorithm. We additionally conduct a sensitivity analysis to evaluate the influences of crucial parameters, including the response time standard, facility capacity, service capacity, and facility heterogeneity. The managerial insights derived from sensitivity analysis will provide valuable guidance for the design of an emergency response system in practice.

Çoklu terk ediş noktası bulunan pistler için stokastik sıralama planlama modeli

The runway exit points (REPs) of the airport are constructed considering the operational performance of different types of aircraft based on historical flight data. In sequence planning, it is assumed that aircraft will vacate the runway from an expected exit point. However, real performance can be uncertain, and the same type of aircraft may vacate the runway from different exit points rather than the expected point. In addition, the runway occupancy times (ROTs) of aircraft that vacate the runway from the same exit point may not be equal. This situation brings two types of uncertainty when making traffic plans in an airport with several REPs. The first uncertainty is the REP of the aircraft, and the second is the ROT uncertainty considering the exit points. In this study, a two-stage stochastic programming model was developed for aircraft sequencing in an airport that has multiple runway exit points. In the model, both runway exit and ROT uncertainties are considered. A runway with multiple exit points at an airport in Turkey was selected and flight track data of 154 arrival flights to this runway was examined. Various expected time of arrival and departure (ETAD) scenarios were generated based on real data and integrated into the mathematical models. The proposed model was then compared with deterministic and first come first serve (FCFS) approaches in terms of total delay. As a result of the comparison and analyses, the presented stochastic programming model provided robust solutions and delay savings compared to the other approaches.

Time window optimization for attended home service delivery under multiple sources of uncertainties

We consider a vehicle routing problem variant to optimize time window assignments together with vehicle routing and scheduling decisions, under the uncertainties of trip time, service time and possible customers’ cancellations. We minimize the expected cost of vehicles’ overtime, idleness, and customer waiting, when also allowing to add new customers to existing schedules. We formulate a two-stage stochastic mixed-integer programming model using finite samples of the uncertain parameters, where in the first stage, we optimize vehicle routes and assign service time windows to customers, and in the second stage, we construct a linear program to compute the resultant undesirable cost given routes and time windows. We also propose a re-optimization method and an insertion-based linear program for accommodating real-time requests in a rolling horizon way for dynamic operations. To speed up computation, we further decompose the problem into three phases and propose Assignment–Routing–Scheduling heuristics. We first design three clustering algorithms based on spatial similarities to assign customers to vehicles, and then combine the nearest-neighbor and smallest-variance rules to decide the route for each vehicle. Finally, we cast the scheduling part as a Newsvendor problem variant and apply inventory approximations to derive closed-form solutions for determining time windows. We conduct numerical studies on diverse instances generated using both well-established benchmark data sets and Ford’s mobile service data, to compare different approaches and demonstrate the benefits of allowing flexible time-window assignments.

Analytics with stochastic optimisation: experimental results of demand uncertainty in process industries

This study reports the test results of a two-stage stochastic linear programming (SLP) model with recourse using a user-friendly generic decision support system (DSS) in a North American steel company. This model has the flexibility to configure multiple material facilities, activities and storage areas in a multi-period and multi-scenario environment. The value of stochastic solution (VSS) with a real-world example has a potential benefit of US$ 24.61 million. Experiments were designed according to the potential joint probability distribution scenarios and the magnitude of demand variability. Overall, 144 SLP optimisation model instances were solved across four industries, namely, steel, aluminium, polymer and pharmaceuticals. The academic contribution of this research is two-fold: first, the potential contribution to profit in a steel company using an SLP model; and second, the optimisation empirical experiments confirm a pattern that the VSS and expected value of perfect information (EVPI) increase with the increase in demand variability. This study has implications for practicing managers seeking business solutions with prescriptive analytics using stochastic optimisation-based DSS. This study will attract more industry attention to business solutions, and the prescriptive analytics discipline will garner more scholarly and industry attention.

A two-stage stochastic programming model for seaport berth and channel planning with uncertainties in ship arrival and handling times

The limited availability of berths and channels is generally the bottleneck restricting the capacity of a seaport and thus resulting in traffic congestion. Optimizing the operations of the berths and channels has been recognized as a more economic avenue for mitigating seaport traffic congestion compared with channel dredging and berth expanding that needs significant capital and time costs. This paper presents a two-stage stochastic mixed integer linear programming model for the seaport berth and channel planning, aiming to minimize the expected total weighted completion times of ships under uncertain ship arrival times and ship handling durations. The first stage decides the berth allocation of ships under uncertainty. In the second stage, the channel planning, including the selection of lanes, assignment of tugboats, and sequencing of ships, is determined after the uncertainty has been realized. To effectively solve the model, we propose two tailored decomposition methods, that is, the stage decomposition method and the decomposition-based heuristic algorithm (DHA). Then, a lower bound of the problem is derived to evaluate the quality of the solution. Numerical experiments on Tianjin Port of China show the satisfactory performance of these two proposed methods. Especially, the DHA is able to obtain near-optimal solutions with the average optimality gap less than 3% within four-hour computational time for the instances up to 500 scenarios and 190 ship movements. Some managerial insights are obtained to guide the operations of the port.

E-commerce supply chain network design using on-demand warehousing system under uncertainty

During the COVID-19 pandemic, e-commerce retailers have had trouble satisfying the growing demand because of limited warehouse capacity constraints. Fortunately, an on-demand warehousing system has emerged as a new alternative to mitigate warehouse capacity issues. In recent years, several studies have focused on the supply chain problem considering on-demand warehousing. However, there is no study that deals simultaneously with inherent uncertainties and the property of commitment, which is the main advantage of on-demand warehousing. To fill these research gaps, this paper presents an e-commerce supply chain network design problem considering an on-demand warehousing and decisions for commitment periods. We propose the two-stage stochastic programming model that captures the inherent uncertainties to formulate the presented problem. We solve the proposed model utilizing sample average approximation combined with the Benders decomposition algorithm. Of particular note, we develop a method to generate effective initial cuts for improving the convergence speed of the Benders decomposition algorithm. Computational results show that the developed method could find an effective feasible solution within a reasonable computational time for problems of practical size. Furthermore, we show the significant cost-saving effects, based on experiment results, that occur when an on-demand warehousing system is used for designing supply chain networks.

A Two-Stage Stochastic Programming Model for Emergency Supplies Pre-Position under the Background of Civil-Military Integration

The pre-positioning of emergency supplies is of great significance for a timely and effective rescue after a disaster. Using the background of civil-military integration in China, this paper puts the military storage facilities into the layout scheme of emergency supplies reserve and builds a two-stage stochastic programming model of emergency supplies location–allocation optimization, the aim of which is to effectively use the reserve resources of both military and civilian sides to reduce the reserve cost. Then, an improved whale optimization algorithm (IWOA) with more strategies is designed to solve the model. The applicability of the model is proved via a real-world case study in Tangshan, China. The case study shows that when the unit storage cost of military storage facilities is less than 1.5 times of that of civilian emergency storage, the military and civilian joint reserve mode can reduce the reserve cost-effectively; decision-makers can set different maximum rescue times according to different preferences to adapt to different emergency decision-making needs.

Integrated consultation and chemotherapy scheduling with stochastic treatment times

This paper studies the integrated scheduling of consultation and treatment appointments for chemotherapy patients, while taking into account the stochastic duration of injection. Patients may require one or both types of consultation and treatment appointments. The objective is to minimize the clinic’s overtime and the waiting time of patients in the clinic. To formulate the problem, we develop two two-stage stochastic programming models. We also propose a sample average approximation algorithm as the solution method. To improve the efficiency of our solution approach, we devise a specialized algorithm that quickly evaluates a given first-stage solution for a large set of scenarios, without solving the second-stage models. Several computational experiments are carried out to evaluate the performance of proposed models and algorithms.

Stochastic optimization and scenario generation for peak load shaving in Smart District microgrid: sizing and operation

Microgrids play an essential role in the integration of multiple distributed energy resources in buildings. They can meet critical loads in buildings while reducing peak loads and congestion and providing other types of ancillary services to the main electrical grid. Sizing the components of microgrids and scheduling their optimal operation while jointly integrating uncertain renewable energy generation and loads can significantly affect its ability to meet these objectives. In fact, microgrids face great challenges due to renewable energies and load uncertainties. In this paper, a two-stage stochastic programming model for optimal sizing and operation of various distributed energy resources for peak load shaving in district buildings is proposed. The first stage is related to the planning of photovoltaic panels and batteries, while the second stage aims to find the optimal operation of the system in grid-connected mode. The uncertainties are related to photovoltaic power generation and loads. The Generative Adversarial Network (GAN) is implemented to generate the scenarios of uncertain parameters, and the k-medoids classical method for scenario reduction is applied to decrease the number of scenarios. The proposed two-stage stochastic optimization is tested for a real case study of a university campus in Canada.

A two-stage stochastic programming model for collaborative asset protection routing problem enhanced with machine learning: a learning-based matheuristic algorithm

ABSTRACT In this paper, a two-stage stochastic mathematical model is developed for an asset protection routing problem under a wildfire. The main aim of this study is to reduce the negative impact of a wildfire. Some parameters, such as travel and service times, obtaining profit by protecting an asset, and upper bounds of time windows, are considered as stochastic parameters. Generating proper scenarios for uncertain parameters has a large impact on the accuracy of the obtained solutions. Therefore, artificial neural networks are employed to extract possible scenarios according to previous actual wildfire events. The problem cannot be solved by exact solvers for large instances, so two matheuristic algorithms are proposed in this study to solve the problem in a reasonable time. In the first algorithm, a set of feasible routes is generated based on a heuristic approach, then a route-based mathematical model is used to obtain the final solution. Also, another matheuristic algorithm based on adaptive large neighbourhood search (ALNS) is proposed. In this algorithm, routing decisions are determined using the ALNS algorithm while other decisions are achieved by solving an intermediate mathematical model. The numerical analysis confirms the efficiency of both proposed algorithms; however, the first algorithm performs more efficiently.

Slot allocation and exchange for container shipping alliance under profit-sharing agreement and uncertain demand

Shipping alliance has become a prevalent cooperation form among major container shipping companies in recent decades. By forming an alliance, shipping companies can exchange their container slots to better satisfy the shipping demand and improve capacity utilization. This paper considers the slot allocation and exchange for a shipping alliance with consideration of profit-sharing agreement and uncertain demand. The profit-sharing agreement indicates the profit segmentation that should be kept by each alliance member among the total alliance profit. To tackle this problem, a two-stage stochastic programming model is subsequently developed to maximize the total profit of the alliance and minimize the deviation of the profit percentage of each member from the prescribed value indicated in the profit-sharing agreement. In the first stage, the number of slots allocated and exchanged within the alliance is determined. In the second stage, after the uncertain shipping demand is realized, each alliance member separately determines their container shipping plan to obtain profit. A tailored pattern-search-based solution algorithm is developed to solve the model. A series of numerical experiments demonstrate the applicability of the programming model and deliver several interesting managerial insights.

Stochastic tramp ship routing with speed optimization: analyzing the impact of the Northern Sea Route on $$\mathrm {CO_2}$$ emissions

AbstractTo address the decarbonization challenge, we consider a tramp ship routing problem with speed optimization where the availability of future cargoes is uncertain. We propose a two-stage stochastic programming model to solve it. The first stage of the model decides on the deterministic cargoes with the known availability, while the second stage decides on the detailed routing and scheduling plans with the available spot cargoes revealed. Since sailing speed heavily influences on the fuel consumption, and hence costs and emissions, we include the ships’ speeds along the different sailing legs as decision variables. The opening of the Northern Sea Route (NSR) provides a shorter alternative in connecting Asia and Europe, which may benefit the shipping industry in reducing $$\mathrm {CO_2}$$ CO 2 emissions. We use our model to evaluate the impact of NSR on $$\mathrm {CO_2}$$ CO 2 emissions in tramp shipping considering speed optimization and possible future carbon tax schemes, such as fixed and progressive carbon taxes. The computational results indicate that using the NSR improves the gross margins and reduces the $$\mathrm {CO_2}$$ CO 2 emissions for tramp operators.

Air traffic management in parallel-point merge systems under wind uncertainties

The point merge system (PMS) was developed by EUROCONTROL in 2006 to enable controllers to implement systematic sequences and replace the traditional vector technique in Terminal Maneuver Area (TMA) even during peak hours. A conventional PMS includes two arc-shaped route segments referred to as sequencing legs and a single merge point to merge arrivals. When necessary, arrivals can be delayed along these legs using path stretching before they are directed to the merge point. After merging, aircraft join the final approach. The parallel-point merge system (P-PMS), however, has a more complex route structure which consists of two oppositely located PMS and a set of common points located between the merge points and final paths of runways. This system increases the capacity of the airspace and provides the advantages of the single PMS. However, especially in simulation studies where P-PMS has been tested, the emphasis on wind sensitivity came to the fore as one of the main issues affecting the use of the system. The use of P-PMS was stated as feasible but more difficult due to the loss of symmetry, especially when the wind is perpendicular to the sequencing legs. Controllers may need help in providing safe and efficient sequences in this type of structure. In this study, a multi-objective two-stage stochastic programming model is developed for P-PMS to obtain robust aircraft sequences, schedules, and runway assignments considering the uncertainties of both wind direction and speed. The model was implemented on the existing layout of Istanbul Airport having a P-PMS serving five parallel runways using the real traffic and wind data. A scenario-based approach was adopted to represent the uncertainties of the model. Also, to meet the demands of the various stakeholders in the air traffic system, the minimization of total fuel consumption, total flight time, and total delays were considered as single and multi-objectives. As a result, it was found that the stochastic approach provides better solutions in terms of objective values and emission outputs compared to deterministic and FCFS approaches under wind uncertainties. The differences between these solutions are quantified through numerical experiments.

Routing in offshore wind farms: A multi-period location and maintenance problem with joint use of a service operation vessel and a safe transfer boat

This paper studies an interesting logistical problem related to maintaining a set of selected turbines in offshore wind farms. A service operation vessel (SOV), used to support maintenance activities, also carries a safe transfer boat (STB), which is used to transport technicians, parts, and equipment to the turbines. The maintenance activities are optimised over a planning horizon by simultaneously coordinating the SOV and the STB. A mixed-integer linear programming model is designed, where the objective function is to minimise the total maintenance cost. As the proposed mathematical model is hard to solve with a commercial solver, a decomposition approach is developed. A two-stage stochastic programming model is also designed to tackle uncertain conditions, including the duration of maintenance and the STB’s travel time. The performance of the solution method is evaluated using the Thanet Offshore Wind Farm, in the southeast of the UK. The results of our experiments show that joint use of an SOV and an STB can yield a lower total maintenance cost than only using an SOV or crew travel vessels (CTVs) that are currently the common practice in maintaining offshore turbines.

Social equity-based distribution networks design for the COVID-19 vaccine

This study aims to investigate the role of social equity in vaccine distribution network design problems. Inspired by the current COVID-19 vaccine allocation in-country context, we capture social equity-based distribution by modeling three theories: Rawls’ theory, Sadr’s theory, and utilitarianism. We consider various social groups based on degree of urbanization, including inhabitants of cities, towns and suburbs, and rural areas. The distribution problem is subject to, on the one hand, demand-side uncertainty characterized by the daily contamination rate and its space–time propagation that anticipate the in-need population. On the other hand, supply-side uncertainty characterized by the stochastic arrival of vaccine doses for the supply period. To tackle this problem, we propose a novel bi-objective two-stage stochastic programming model using the sample average approximation (SAA) method. We also develop a lexicographic goal programming approach where the social equity objective is prioritized, thereafter reaching an efficiency level. Using publicly available data on COVID-19 in-country propagation and the case of two major provinces in Iran as example of middle-income country, we provide evidence of the benefits of considering social equity in a model-based decision-making approach. The findings suggest that the design solution produced by each social equity theory matches its essence in social science, differing considerably from the cost-based design solution. According to the general results, we can infer that each social equity theory has its own merits. Implementing Rawls’ theory brings about a greater coverage percentage in rural areas, while utilitarianism results in a higher allocation of vaccine doses to social groups compared to the Sadr and Rawls theories. Finally, Sadr’s theory outperforms Rawls’ in terms of both the allocation and cost perspective. These insights would help decision-makers leverage the right equity approach in the COVID-19 vaccine context, and be better prepared for any pandemic crisis that the future may unfold.

Two-stage stochastic programming models for the extended aircraft arrival management problem with multiple pre-scheduling points

Extended aircraft arrival management under uncertainty has been previously studied in the literature using two-stage stochastic optimization in the case of a single initial approach fix (IAF) and a single runway. In this paper, we propose an extension taking into account: (i) multiple IAFs feeding the landing runway, (ii) aircraft having different initial flight status (at-departure-gate or airborne) when making first-stage decisions, and (iii) a time-deviation cost function to minimize that is based on reference values depending on aircraft type and flight phase. Two problem variants are modeled according to the degree of freedom on IAF assignment to aircraft. In the first variant, IAFs are to be assigned to aircraft, as a first-stage decision. In the second variant, IAF assignment is fixed and considered as a problem input. Numerical results on realistic instances from Paris Charles-de-Gaulle airport confirm the benefit of taking into account uncertainty through two-stage stochastic programming, and through re-assignment of IAFs.

Optimisation of containerised air cargo forwarding plans considering a hub consolidation process with cargo loading

Air cargo plays an important role in supporting global supply chains; this becomes more vital when facing uncertainties in a crisis such as the COVID-19 pandemic. This motivates our study on air cargo forwarding plans, considering demand uncertainties and economic conditions. Cargos are placed into air containers based on weights and volumes, and then flown from regional collection points into a hub, for consolidation before transporting to onward destinations. Decisions are made in advance by cargo forwarders as to the containers to book, both in regions and in the hub, since airlines offer discounts on containers booked in advance; however, cargo quantities are uncertain when advance bookings are made. We develop a two-stage stochastic programming model, where the first stage determines both the quantities and types of air containers to book; the second stage deals with ordering any extra containers, at higher cost, or returning unused containers, as well as making loading and consolidation plans. The objective is to minimise the total expected costs. We then extend it into a multistage case and design a genetic algorithm as the solution method. Experimental results demonstrate that the proposed approaches provide a cost-efficient plan and responsive to demand as it arises.

The development of input-monitoring system on biofuel economics and social welfare analysis.

Biofuel production relies on stable supply of biomass which would be significantly influenced by climate-induced impacts. Since the actual agricultural outputs are relatively unpredictable in the face of uncertain environmental conditions and can only be realized in the harvest season, providing useful information regarding the stability of biomass supply to the downstream biofuel industry is crucial. This study firstly illustrates a theoretical framework to explore the resultant market equilibrium and optimal conditions of agricultural and bioenergy production in the face of highly uncertain environmental risks and then employs a two-stage stochastic programming model to investigate the optimal biofuel development and associated economic and environmental effects. The results show that total welfare may not always increase because the loss of other agricultural commodities induced by climate impacts may be greater than the gains received by biofuel production and emission reduction. This study provides insights into the area where artificial intelligence monitoring system can be implemented to analyze the input data associated with agricultural activities and help the biofuel industry to improve its production possibilities.