Stochastic Integer Programming Model Research Articles

A stochastic optimization model for patient evacuation from health care facilities during hurricanes

We propose a rigorous modeling and methodological effort that integrates statistical implementation of hydrology models in predicting inland and coastal flood scenarios due to hurricanes and a scenario-based stochastic integer programming model which suggests resource and staging area decisions in the first stage and the evacuation decisions in the second stage. This novel study combines physics-based flood prediction models and stochastic optimization for large-scale multi-facility coordination of hospital and nursing home evacuations before impending hurricanes. The optimization model considers scenario-dependent evacuation demand, transport vehicles with varying capacities, and both critical and non-critical patients. Utilizing Hurricane Harvey of 2017 as a case study and actual healthcare facility locations in southeast Texas, we explore various evacuation policies, demonstrating the impact of routing strategies, staging area decisions, flood thresholds, and receiving facility capacities on evacuation outcomes. One of the findings is that choosing staging area(s) and deploying evacuation vehicles optimally considering the uncertainty of the hurricane’s path at the time of decision making could have significant effect on the total cost of the operation and evacuation time experienced by the evacuees. We also show the non-negligible value of the scenario-based staging and routing solution conservatively calculated in relation to a single scenario solution using the concept of value of stochastic solution.

Joint stochastic optimisation of stope layout, production scheduling and access network

The three main optimisation components of sublevel stoping methods are stope layout, production schedule (or stope sequencing) and access networks. The joint optimisation of these components could further add value to an underground mining project. This potential has not been considered in the literature due to computational difficulties, and the problem was solved sequentially. This paper proposes a new joint optimisation model to integrate these components. In addition, the proposed optimisation model incorporates stochastic simulations to capture uncertainty and variability associated with the grades of the related mineral deposits mined. The optimisation model is based on a two-stage stochastic integer programming (SIP) formulation that maximises the project's net present value (NPV) and minimises the planned dilution. Applying the proposed method at a small copper deposit shows that the SIP outperforms the results obtained from mixed integer programming. For a seven-year mine life, the SIP model generated ∼20% more NPV, demonstrating the importance of developing a joint optimisation formulation and accounting for grade uncertainty and variability.

New Exact Algorithm for the integrated train timetabling and rolling stock circulation planning problem with stochastic demand

This paper studies an integrated train timetabling and rolling stock circulation planning problem with stochastic demand and flexible train composition (TRSF). A novel stochastic integer programming model, which is formulated on a space-time underlying network to simultaneously optimize the train timetable and rolling stock circulation plan with flexible train composition, is proposed by explicitly considering the random feature of passenger distribution on an urban rail transit line. To solve this problem efficiently, the proposed model is decomposed into a master problem and a series of sub-problems regarding different stochastic scenarios. We further prove that each sub-problem model is equivalent to its linear programming relaxation problem, by proving that the coefficient matrix of each linear programming relaxation model is totally unimodular. Then, the classical Benders decomposition algorithm is applied to the studied problem. Based on the model characteristics, both single-cut and multi-cut methods with some speed-up techniques are developed to solve the proposed model in a novel and effective way. Numerical experiments are conducted on small-scale cases and large-scale cases derived from Shanghai Metro Line 17, and the results show that solving the stochastic problem can extract gains in efficiency and the value of stochastic solution tends to be high.

Designing variable-sized block appointment system under time-varying no-shows

We consider the tactical problem of optimizing the capacity of a block appointment system with time-varying no-show and random service duration. The problem characteristics are motivated by a real-life case study of two outpatient specialty clinics in the US, where time-of-day variation in appointment attendance was observed. We aim to determine the block size (patients to be assigned in each time period) such that the weighted sum of block-wise waiting- and idle-times (or total cost) are minimized. First, we consider optimizing the capacity of a single block appointment system (SBAS) as a special case of the inverse newsvendor problem, and develop an analytical closed-form solution under the assumption of normally distributed service time. Also, a stochastic integer programming (SIP) model is developed to solve SBAS for any service time distribution. Subsequently, the SIP model is extended to determine the block size of the variable-sized multi-block appointment system (VSMBAS) by treating it as a sequential inverse newsvendor problem. Owing to the computational complexity of the SIP, we employ sample average approximation to estimate the expected total cost. Numerical studies considered several realistic clinic settings, and the results demonstrated that integrating time-varying no-shows for block size determination will considerably improve schedule efficiency as opposed to ignoring it. We also found the cost ratios (waiting-time to idle-time penalty), service time variation, and no-show pattern have a substantial influence on the block size of VSMBAS. Finally, we provide several practical implications based on our analysis.

Stochastic RWA and Lightpath Rerouting in WDM Networks

In a telecommunication network, routing and wavelength assignment (RWA) is the problem of finding lightpaths for incoming connection requests. When facing a dynamic traffic, greedy assignment of lightpaths to incoming requests based on predefined deterministic policies leads to a fragmented network that cannot make use of its full capacity because of stranded bandwidth. At this point, service providers try to recover the capacity via a defragmentation process. We study this setting from two perspectives: (i) while granting the connection requests via the RWA problem and (ii) during the defragmentation process by lightpath rerouting. For both problems, we present the first two-stage stochastic integer programming model incorporating incoming request uncertainty to maximize the expected grade of service. We develop a decomposition-based solution approach, which uses various relaxations of the problem and a newly developed problem-specific cut family. Simulation of two-stage policies for a variety of instances in a rolling-horizon framework of 52 stages shows that our stochastic models provide high-quality solutions when compared with traditionally used deterministic ones. Specifically, the proposed provisioning policies yield improvements of up to 19% in overall grade of service and 20% in spectrum saving, while the stochastic lightpath rerouting policies grant up to 36% more requests, using up to just 4% more bandwidth spectrum.Summary of Contribution: For handling the intrinsic uncertainty of demand in the telecommunications industry, this paper proposes novel stochastic models and solution methodology for two fundamental problems in telecommunications at operational level: (i) routing and wavelength assignment (RWA) and (ii) lightpath rerouting problem. Despite the vast literature on the RWA problem, stochastic optimization has not been considered as a viable solution for resource allocation in optical networks. We propose two-stage stochastic programming models for both problems and design efficient decomposition-based solution methods that use various relaxations of the models and a new family of cutting planes. Our extensive and rigorous numerical experiments show the significant merit of incorporating uncertainty into decision making, as well as the effectiveness of the decomposition framework and our newly designed family of cuts in enhancing the solvability of both models. This work opens new avenues to explore where the powerful stochastic programming literature can be leveraged to make operational decisions in telecommunications problems, a field that currently relies mostly on deterministic and heuristic solution methods.

Integrated berth and yard space allocation under uncertainty

Improving container ports’ operational efficiency to reduce the delays at ports concerns global port operators significantly. Optimizing the tactical level allocation of quay side and yard side resources to vessel calls is a typical lever to mitigate the delays in operations. Another lever is by strategical level planning that re-adjust the preferred visiting time windows of vessel calls. It aims to absorb the delays at the scheduling stage rather than in the operations stage. This paper integrates the planning and operations at container ports to jointly optimize strategical level planning and tactical level berth and yard space allocation under uncertain vessel arrival times and uncertain numbers of loading/unloading containers. The problem is formulated as a two-stage stochastic integer programming model. To solve it, we develop an original decomposition algorithm that passes columns of second-stage problems to the first-stage problem to approximate the second-stage decision making. Numerical experiments are conducted to validate the effectiveness and efficiency of our proposed algorithm. Some managerial implications for port operators are also obtained.

Short-Term nurse schedule adjustments under dynamic patient demand

We study two-stage short-term staffing adjustments for the upcoming nursing shift. Our proposed adjustments are first used at the beginning of each 4-hour nursing shift, shift t, for the upcoming shift, shift t + 1. Then, after observing actual patient demand for nursing at the start of shift t + 1, we make our final staffing adjustments to meet the patient demand. We model six different adjustment options for the two-stage stochastic programming model, five options available as first-stage decisions and one option available as the second-stage decision. We develop a two-stage stochastic integer programming model, which minimizes total nurse staffing costs and the cost of adjustments to the original schedules, while ensuring the coverage of nursing demand. Our experimental results, using the data from an urban Children’s Hospital, indicate that the developed stochastic nurse schedule adjustment model can deliver cost savings up to 18% for the medical units, compared to alternative no short-term adjustment scheduling models. The proposed stochastic adjustments model successfully keeps average understaffing percentages under 2% throughout the staffing horizon.

Two-stage stochastic program for disassembly lot-sizing under random ordering lead time

This study addresses disassembly lot-sizing that determines the ordering and disassembly schedules of end-of-life (EOL) products and subassemblies to satisfy items demands. A stochastic version with uncertain ordering lead time (OLT) is considered for the first time. Here, OLT represents the time elapsed between placing an order and receiving it (only an EOL product can be ordered). Stochastic integer programming model is developed for the objective of minimizing the expected sum of setup, purchase, inventory and backlog costs. To test the performance of the proposed model, computational experiments were done on various test instances and are reported. The results show that the performance of the proposed model depends on the size of the ordering lead times, the number of periods and components.

Capacity expansion strategies for electric vehicle charging networks: Model, algorithms, and case study

AbstractGovernments in many jurisdictions are taking measures to promote the use of electric vehicles. As part of this goal, it is crucial to provide a sufficient number of charging stations to alleviate drivers' anxieties associated with the range of the vehicle. The goal of this research is to help governments develop vehicle charging networks for public use via the application of multistage stochastic integer programming model that determines both the locations and capacities of charging facilities over finite planning horizons. The logit choice model is used to estimate drivers' choices of nearby charging stations. Moreover, we characterize the charging demand as a function of the charging station quantity to reflect the range anxiety of consumers. The objective of the model is to minimize the expected total cost of installing and operating the charging facilities. An approximation algorithm, a heuristic algorithm, and a branch‐and‐price algorithm are designed to solve the model. We conduct numerical experiments to test the efficiency of these algorithms. Importantly, each algorithm has advantages over the CPLEX MIP solver. Finally, the City of Oakville in Ontario, Canada, is used to demonstrate the effectiveness of this model.

Stochastic stope design optimisation under grade uncertainty and dynamic development costs

ABSTRACT Stope design optimisation defines three-dimensional extraction volumes aiming to maximise cashflows, subject to geotechnical and operational constraints. Available stope layout methods are deterministic, failing to account for grade uncertainty and variability that affect stope locations and sizes, as well as value. A two-stage stochastic integer programming model for stope design optimisation is proposed, integrating grade uncertainty quantified through geostatistical simulations, level allocation, variable stope and pillar sizes for different geotechnical zones, and development costs. An application at an underground gold mine employing sublevel open stoping highlights the integration and management of grade uncertainty to define risk-resilient stope design.

State-Variable Modeling for a Class of Two-Stage Stochastic Optimization Problems

This paper considers a class of two-stage stochastic mixed-integer optimization problems where, for a given first-stage solution, we can determine the optimal values of recourse variables sequentially. This class of problems arises in a wide variety of applications. In the case of multivariate discrete distributions for uncertain parameters, a standard stochastic programming formulation of these problems involves an exponential number of scenarios, therefore an exponential number of variables and constraints. We propose a new mixed-integer programming modeling approach where the number of variables and constraints is independent of the number of scenarios and scales at most pseudopolynomially with the problem size. The proposed modeling approach relies on state variables that track the system’s state as the uncertainty realizes sequentially. We demonstrate the advantages of the proposed approach in two applications arising in project scheduling and operating room allocation. Summary of Contribution: This paper proposes a new modeling approach for a class of two-stage stochastic optimization problems that is computationally more efficient than the traditional scenario-based stochastic integer programming models. The proposed modeling approach relies on state variables that track the system's state as the uncertainty realizes sequentially. We demonstrated the efficiency of the proposed approach by computational results on two applications in project scheduling and operating room allocation.

Stochastic Programming for Liner Ship Routing and Scheduling under Uncertain Sea Ice Conditions

It is anticipated that in the foreseeable future the Northern Sea Route (NSR) will be able to serve commercial shipping as an alternative transportation shortcut between East Asia and Europe, especially in the summer season. The sailing time, however, is heavily subject to the variation of sea ice conditions along this route. Any participating shipping company must consider how to mitigate the ill effects on itinerary planning caused by sailing time and cost uncertainty. Finding a good trade-off between the benefit from a tight schedule and the risk caused by an unexpected delay is a key element in relevant routing and scheduling decisions, and may be beyond the reach of traditional deterministic planning models. With the aim of maximizing profit over all possible shipping environment scenarios, this article proposes a two-stage stochastic nonlinear integer programming model for liner ship routing and scheduling with uncertain shipping time and cost, the nonlinearity of which arises from the coexistence of schedule-sensitive shipping demand and uncertain arrival time variables in the objective function. The model is converted into an equivalent linear integer programming counterpart by introducing a set of nominal delay variables, and Benders decomposition is applied to solve the linearized problem. Numerical experiments and sensitivity analyses are conducted to validate the efficacy and effectiveness of the model, the results of which suggest several managerial insights that can be used to guide liner ship route and schedule planning under uncertain shipping conditions.

Mature offshore oil field development: Solving a real options problem using stochastic dual dynamic integer programming

Oil and gas companies are facing low output prices and are forced to focus on the development of mature fields. Relevant investment decisions for operators include lifetime-enhancing activities, such as drilling new wells or permanent shutdown. We study the problem of optimal timing of investments in mature oil and gas fields in the presence of price uncertainty, which is an example of a complex real options problem consisting of a portfolio of interdependent options. We formulate a multistage stochastic integer programming model that incorporates a detailed representation of the uncertain oil price, and demonstrate how such a complex real options problem can be efficiently solved using the Stochastic Dual Dynamic Integer Programming algorithm. The paper presents a numerical example based on realistic data and discusses our computational results. We find that only a small number of Markov states are required to represent the uncertain price process, while obtaining convergence of the lower and upper bounds of the objective function. The value of stochastic solution of 11% is considerable in this example. It is concluded that the shutdown decision tends to be postponed as a result of high decommissioning costs, high discount rates, high price uncertainty and low operational expenditures, while it generally is accelerated if the decommissioning costs increase over time.

Mathematical Model for Cargo Allocation Problem in Synchromodal Transportation

Synchromodality is a freight transport process in which information is exchanged expeditiously in order to maximize the benefits of different modes of transport and transport nodes in terms of efficiency and environmental impact. The aim of the study is to analyze the problems of synchronized intermodal traffic management between the main port and inland transport nodes in European transport corridors and to find reliable solutions to these problems. Therefore, the main purpose of this article is to investigate the problem of the distribution of containers transported by rail between two transport terminals in a synchronous transport network. A specific optimization model is presented in this article. This optimization task is formulated as a stochastic integer programming model between the terminals located in Vilnius and Klaipeda Seaport, the essence of which is as follows: (a) to minimize the waiting time for container cargo at the location—terminal No. 1; (b) to minimize the total journey time of the train; (c) to minimize the waiting time for containerized cargo at the point of arrival—terminal No. 2.

A multi-stage stochastic programming approach for supply chain risk mitigation via product substitution

Trends like globalization, shorter product life-cycles, and cost reduction strategies in the global business environment have exposed many supply chains to various risks. Disruptions are one of the supply chain risks that can interrupt product flow, delay customer deliveries, and reduce supply chain revenues considerably. Prior planning for disruptions could greatly alleviate these consequences. A method to cope with disruptions is to use product substitution in the case of a product shortage. In this research, the supply chain of a livestock-drug distribution company in Iran, facing demand disruptions, has been chosen as a case study. For this purpose, a multi-stage stochastic integer programming model is proposed and solved using a customized progressive hedging algorithm. Moreover, the effect of uncertainty on the supply chain performance is measured using the value of the stochastic solution (VSS) and the expected value of perfect information (EVPI) metrics. Based on the different instances of the problem solved, the VSS metric shows that modeling and solving the proposed stochastic model could enhance the company profit by about 3.27 percent on average. In addition, the EVPI metric demonstrates that planning and investing in proactive demand management could enhance the profit up to 9.42 percent. Finally, analyses indicate that when dealing with increased demand uncertainty levels, the importance of using the proposed method increases as the profitability of the company decreases.

Vehicle Routing Problems with Synchronized Visits and Stochastic Travel and Service Times: Applications in Healthcare

This paper, for the first time, studies vehicle routing problems with synchronized visits (VRPS) and stochastic travel and service times. In addition to considering a home healthcare scheduling problem, we introduce an operating room scheduling problem with stochastic durations as a novel application of VRPS. We formulate VRPS with stochastic times as a two-stage stochastic integer programming model that, unlike the deterministic models in the VRPS literature, does not have any big-M constraints. This advantage comes at the cost of a large number of second-stage integer variables. We prove that the integrality constraints on second-stage variables can be relaxed, and therefore, we can apply the L-shaped algorithm and its branch-and-cut implementation to solve the problem. We enhance the model by developing valid inequalities and a lower bounding functional. We analyze the subproblems of the L-shaped algorithm and devise a specialized algorithm for them that is significantly faster than standard linear programming algorithms. Computational results show that the branch-and-cut algorithm optimally solves stochastic home healthcare scheduling instances with 15 patients and 10%–30% of synchronized visits. It also finds solutions with an average optimality gap of 3.57% for instances with 20 patients. Furthermore, the branch-and-cut algorithm optimally solves stochastic operating room scheduling problems with 20 surgeries.

Equity and Strength in Stochastic Integer Programming Models for the Dynamic Single Airport Ground-Holding Problem

We study stochastic integer programming models for assigning delays to flights that are destined for an airport whose capacity has been impacted by poor weather or some other exogenous factor. In the existing literature, empirical evidence seemed to suggest that a proposed integer programming model had a strong formulation, but no existing theoretical results explained the observation. We apply recent results concerning the polyhedra of stochastic network flow problems to explain the strength of the existing model, and we propose a model whose size scales better with the number of flights in the problem and that preserves the strength of the existing model. Computational results are provided that demonstrate the benefits of the proposed model. Finally, we define a type of equity property that is satisfied by both models.

A stochastic particle swarm based model for long term production planning of open pit mines considering the geological uncertainty

Long term production planning (LTPP) plays a critical role to achieve success of a mining operation. LTPP, as an optimization problem, aims to maximize the net present value (NPV) of a mine subject to a set of constraints. One of the main reasons for not achieving production targets is the uncertainty of the LTPP's inputs. Geological uncertainty as the main sources of uncertainty is considered in this research. In this regard, a set of equiprobable scenarios of orebody and derived two new block model called “risk block model” and “EType” were used as inputs. Then, a stochastic integer programming (SIP) model was developed to integrate the geological uncertainty. Finally, a PSO-based algorithm was developed to solve the SIP model. Four different strategies were developed, according to the population topology and how to use the risk block model. Population topology defines the subset of particles that effect on each particle. Implementation the proposed approach on a large scale mine demonstrate its performance to develop a unique schedule considering geological uncertainties with maximum NPV and minimum risk of not achieving production targets. Investigations show that Gbest based PSO is more susceptible to trap in local optima. Multiple risk based strategies are able to generate better solutions, however, single risk based strategies are good practices when companies are looking for flexible or specific risk based designs.

Two-stage bond portfolio optimization and its application to Saudi Sukuk Market

We consider a two-stage stochastic bond portfolio optimization problem, where an investor aims to optimize the cost of bond portfolio under different scenarios while ensuring predefined liabilities during a given planning horizon. The investor needs to optimally decide whether to buy, hold, or sell bonds based upon present market conditions under different scenarios and varying assumptions, where the scenarios are determined based on interest rates and buying prices of the bonds. Three stochastic integer programming models are proposed and tested using real data from Saudi Sukuk (Bond) Market. The obtained results demonstrate the varying optimal decisions made to manage bond portfolio over the two stages. In addition, the three stochastic programming models for bond portfolio optimization are tested on a large set of randomly generated instances similar to the Saudi Sukuk (Bond) Market. The results of computational experiments attest the efficiency of the proposed models.

A HYBRID MODEL FOR MIX-BANK BUFFER CONTENT DETERMINATION IN AUTOMOBILE INDUSTRY

In mixed-model automobile assembly lines, paint defective vehicles are the main reason for unintentional sequence alteration that stirs up the production sequence so that resequencing is required. This study aims to decide optimal spare vehicles that are held in the mix-bank buffer to be replaced instead of defective vehicles in order to regain the production sequence. We develop a hybrid solution methodology in which optimal spare vehicle content is determined by genetic algorithm (GA) and the releasing order of the vehicles to final assembly (FA) are decided by stochastic mixed integer programming (MIP) model. In addition to discussing the efficiency of the hybrid model, the following insights were gained: 1) not only FA but also paint shop constraints should be considered in production sequence determination when mix-bank buffer is efficiently used; 2) the decrease in defect rate improves sequence restoration linearly; 3) effect of an additional lane on SSAR increase is diminishing. [Received: 22 December 2018; Accepted: 28 October 2019]