Two-stage Stochastic Integer Programming Research Articles

A disaggregated integer L-shaped method for stochastic vehicle routing problems with monotonic recourse

This paper proposes a new integer L-shaped method for solving two-stage stochastic integer programs whose first-stage solutions can decompose into disjoint components, each one having a monotonic recourse function. In a minimization problem, the monotonicity property stipulates that the recourse cost of a component must always be higher or equal to that of any of its subcomponents. The method exploits new types of optimality cuts and lower bounding functionals that are valid under this property. The stochastic vehicle routing problem is particularly well suited to be solved by this approach, as its solutions can be decomposed into a set of routes. We consider the variant with stochastic demands in which the recourse policy consists of performing a return trip to the depot whenever a vehicle does not have sufficient capacity to accommodate a newly realized customer demand. This work shows that this policy can lead to a non-monotonic recourse function, but that the monotonicity holds when the customer demands are modeled by several commonly used families of probability distributions. We also present new problem-specific lower bounds on the recourse that strengthen the lower bounding functionals and significantly speed up the solving process. Computational experiments on instances from the literature show that the new approach achieves state-of-the-art results.

Integrated stochastic optimisation of stope design and long-term production scheduling at an operating underground copper mine

ABSTRACT Conventional underground long-term mine planning is based on a deterministic stepwise framework, which is unable to effectively manage the synergies between the mine planning components or to manage the orebody risk in production schedules and forecasts. The present study enhances prior integrated optimisation approaches to a variation of the sublevel longhole open stoping mining method using backfilling, through a new two-stage stochastic integer programming formulation. A comparison to a sequential stochastic approach shows different stope layouts and extraction sequences for a copper mine with secondary elements being gold and uranium. The integrated approach shows lower horizontal development costs and 6% higher net present value.

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.

Distributed surgical scheduling across collaborating hospitals considering stochastic duration and emergency demand

In this paper, we address the need for distributed surgical scheduling across multiple hospitals and multiple days. We develop an effective ex-ante distributed schedule to determine elective service operation and elective patient assignment plan in a hospital alliance with the goal of minimizing the total expected cost considering possible realizations of treatment durations and emergency demands. The problem is formulated into a two-stage stochastic integer programming where uncertainty is characterized by a scenario set. Cancellations of scheduled elective patients and inter-hospital transfers of materialized emergency patients are explicitly modeled as second-stage decisions. To solve the problem, we propose an integer decomposition algorithm that incorporates a novel logic-based optimality cut along with a validity proof. Additionally, we introduce several enhancements to improve the algorithm. Through numerical experiments, we demonstrate that our proposed algorithm outperforms both the commercial solver and the existing integer decomposition algorithm for the proposed problem. It provides effective solutions that closely approach the best upper bounds achieved by the solver in less time, or even surpass the solver’s best upper bounds. Furthermore, we provide operational insights into other commonly-used scheduling approaches through numerical comparisons, including the strengths and limitations of the most common way of scheduling elective patients and two strategies for handling emergencies within a hospital alliance.

Efficient Algorithms for Stochastic Ride-Pooling Assignment with Mixed Fleets

Ride-pooling, which accommodates multiple passenger requests in a single trip, has the potential to substantially enhance the throughput of mobility-on-demand (MoD) systems. This paper investigates MoD systems that operate mixed fleets composed of “basic supply” and “augmented supply” vehicles. When the basic supply is insufficient to satisfy demand, augmented supply vehicles can be repositioned to serve rides at a higher operational cost. We formulate the joint vehicle repositioning and ride-pooling assignment problem as a two-stage stochastic integer program, where repositioning augmented supply vehicles precedes the realization of ride requests. Sequential ride-pooling assignments aim to maximize total utility or profit on a shareability graph: a hypergraph representing the matching compatibility between available vehicles and pending requests. Two approximation algorithms for midcapacity and high-capacity vehicles are proposed in this paper; the respective approximation ratios are [Formula: see text] and [Formula: see text], where p is the maximum vehicle capacity plus one. Our study evaluates the performance of these approximation algorithms using an MoD simulator, demonstrating that these algorithms can parallelize computations and achieve solutions with small optimality gaps (typically within 1%). These efficient algorithms pave the way for various multimodal and multiclass MoD applications. History: This paper has been accepted for the Transportation Science Special Issue on Emerging Topics in Transportation Science and Logistics. Funding: This work was supported by the National Science Foundation [Grants CCF-2006778 and FW-HTF-P 2222806], the Ford Motor Company, and the Division of Civil, Mechanical, and Manufacturing Innovation [Grants CMMI-1854684, CMMI-1904575, and CMMI-1940766]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/trsc.2021.0349 .

Heterogeneous Multi-resource Planning and Allocation Under Stochastic Demand

We study the capacity planning and allocation decisions for multiple heterogeneous resources, considering potential demand scenarios, where each demand requests a subset of the available resource types simultaneously at a specified time, location, and duration (smRmD). We model this problem as a two-stage stochastic integer program and consider two variants for the objective function: (a) maximize the expected reward of demands met over all scenarios, subject to a budget B for resources, and (b) maximize the expected reward of demands met over all scenarios minus the cost of resources. Contributions of this work include (i) a thorough complexity analysis of smRmD and its variants, (ii) analysis of structural properties, (iii) development of various approximation algorithms using the unique structural properties of smRmD and its variants, and (iv) an extensive computational study to explore the ease with which exact and approximate solutions may be found. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms–Discrete. Funding: This research has been supported in part by National Science Foundation (NSF) Graduate Research Fellowship [DGE-1650044], NSF [Grants CMMI-1538860, NSF-AF:1910423, and NSF-AF:1717947], and the following Georgia Tech benefactors: William W. George, Andrea Laliberte, Richard ”Rick” E. & Charlene Zalesky, and Claudia & Paul Raines.

Stochastic service network design for a platooning service provider

Automated driving is expected to support a wide range of transportation applications in the near future. However, especially in urban environments, not all streets may be feasible for automated driving which restricts the operation of autonomous vehicles (AVs) to certain zones. Outside of these zones, AVs may be safely navigated by a leading vehicle that is operated by a human driver using the concept of platooning. In this paper, we consider the service network design problem of a service provider that offers platooning services to AV operators in an urban environment in exchange for monetary compensation. Tactical planning decisions comprise determining the fleet size and scheduling platooning services for a planning horizon such as a season. Those decisions are supported by the quoting of rates to potential customers that reflect their willingness to pay depending on the received service level. A second stage of decisions, made on each day of the planning horizon, comprises the acceptance or rejection of individual platooning requests and the routing of AVs through the platooning services. Since the platooning demand is assumed to be stochastic, we model the problem as a two-stage stochastic integer program. On a real-world based network, we perform computational experiments that indicate potential profits for a platooning service provider and considerable savings for AV operators with different demand characteristics. We further assess the value of considering uncertainty.

Integrated stochastic optimization of stope design and long-term underground mine production scheduling

In the commonly used underground mine planning framework, mine design is first established and is the main input for the subsequent long-term mine production scheduling optimization. This sequential optimization approach cannot, therefore, capture the synergies between the involved planning steps, generating solutions that depart substantially from a global optimum. In addition, traditional underground mine planning methods for stope design and life-of-mine production scheduling are deterministic and are based on a single estimated orebody model. As a result, the uncertainty and variability in grades and material types are not incorporated into the optimization process, resulting in designs that misrepresent all high-, medium- and low-grade stoping volumes and production schedules with misleading forecasts. A two-stage stochastic integer program (SIP) for integrated optimization of stope and development network designs and an underground mine production scheduling are proposed for the sublevel open stoping mining method under grade uncertainty and variability, quantified by a set of geostatistical simulations of the mineral deposit considered. Assuming a mine is accessed through a shaft, the model defines a schedule of levels and stopes, which aims to maximize the discounted revenues, minimize development costs, and manage the risk of not meeting production targets, while satisfying geotechnical constraints. The practical aspects of the proposed method are presented through an application at an underground gold mine. A comparison with the stepwise framework, where the stope design is input to a subsequent optimization of the production schedule, shows that the proposed approach provides a physically different design and production schedule with an 11% higher net present value (NPV) and a life-of-mine that is two years shorter, affirming the advantages of the integrated optimization process.

Integrated Optimization of Planning and Operations for Shared Autonomous Electric Vehicle Systems

Shared autonomous electric vehicle systems will be a promising alternative for sustainable urban mobility. This study investigates an integrated optimization problem for shared autonomous electric vehicle systems that determines the long-term charging facility deployment at the planning level (e.g., the sizing and configurations of charging facilities), while the vehicle assignment, relocation, and charging decisions in the short term are also optimized at the operational level. A two-stage stochastic integer program is formulated to capture the demand uncertainty, in which an event-activity space-time-battery network is proposed for tracking the charging choices and battery states of vehicles and determining the optimal operational decisions. For dealing with a large number of scenarios in the stochastic program, the sample average approximation scheme is applied as the sampling strategy. An accelerated two-phase Benders decomposition-based algorithm is proposed for solving the two-stage program. The modeling and algorithm approach is tested on a large-scale case in Shanghai City in China. Extensive experiments show that the proposed approach can always find high-quality solutions in an efficient way. Numerical results indicate that deployment of both normal- and fast-charging infrastructure can increase the system profit and improve its operational performance. Funding: This work was supported by the Singapore Ministry of Education Academic Research Fund Tier 1 [Grant R-266-000-135-114] and by the National Natural Science Foundation of China [Grants 71971021 and 72101019]. Supplemental Material: The e-companion is available at https://doi.org/10.1287/trsc.2022.1156 .

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.

On Generating Lagrangian Cuts for Two-Stage Stochastic Integer Programs

We investigate new methods for generating Lagrangian cuts to solve two-stage stochastic integer programs. Lagrangian cuts can be added to a Benders reformulation and are derived from solving single scenario integer programming subproblems identical to those used in the nonanticipative Lagrangian dual of a stochastic integer program. Although Lagrangian cuts have the potential to significantly strengthen the Benders relaxation, generating Lagrangian cuts can be computationally demanding. We investigate new techniques for generating Lagrangian cuts with the goal of obtaining methods that provide significant improvements to the Benders relaxation quickly. Computational results demonstrate that our proposed method improves the Benders relaxation significantly faster than previous methods for generating Lagrangian cuts and, when used within a branch-and-cut algorithm, significantly reduces the size of the search tree for three classes of test problems.

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.

Stochastic Last-Mile Delivery with Crowd-Shipping and Mobile Depots

This paper proposes a two-tier last-mile delivery model that optimally selects mobile depot locations in advance of full information about the availability of crowd-shippers and then transfers packages to crowd-shippers for the final shipment to the customers. Uncertainty in crowd-shipper availability is incorporated by modeling the problem as a two-stage stochastic integer program. Enhanced decomposition solution algorithms including branch-and-cut and cut-and-project frameworks are developed. A risk-averse approach is compared against a risk-neutral approach by assessing conditional-value-at-risk. A detailed computational study based on the City of Toronto is conducted. The deterministic version of the model outperforms a capacitated vehicle routing problem on average by 20%. For the stochastic model, decomposition algorithms usually discover near-optimal solutions within two hours for instances up to a size of 30 mobile depot locations, 40 customers, and 120 crowd-shippers. The cut-and-project approach outperforms the branch-and-cut approach by up to 85% in the risk-averse setting in certain instances. The stochastic model provides solutions that are 3.35%–6.08% better than the deterministic model, and the improvements are magnified with increased uncertainty in crowd-shipper availability. A risk-averse approach leads the operator to send more mobile depots or postpone customer deliveries to reduce the risk of high penalties for nondelivery.

Recourse in Kidney Exchange Programs

We introduce the problem of selecting patient-donor pairs in a kidney exchange program to undergo a crossmatch test, and we model this selection problem as a two-stage stochastic integer programming problem. The optimal solutions of this new formulation yield a larger expected number of realized transplants than previous approaches based on internal recourse or subset recourse. We settle the computational complexity of the selection problem by showing that it remains NP-hard even for maximum cycle length equal to two. Furthermore, we investigate to what extent different algorithmic approaches, including one based on Benders decomposition, are able to solve instances of the model. We empirically investigate the computational efficiency of this approach by solving randomly generated instances and study the corresponding running times as a function of maximum cycle length, and of the presence of nondirected donors. Summary of Contribution: This paper deals with an important and very complex issue linked to the optimization of transplant matchings in kidney exchange programs, namely, the inherent uncertainty in the assessment of compatibility between donors and recipients of transplants. Although this issue has previously received some attention in the optimization literature, most attempts to date have focused on applying recourse to solutions selected within restricted spaces. The present paper explicitly formulates the maximization of the expected number of transplants as a two-stage stochastic integer programming problem. The formulation turns out to be computationally difficulty, both from a theoretical and from a numerical perspective. Different algorithmic approaches are proposed and tested experimentally for its solution. The quality of the kidney exchanges produced by these algorithms compares favorably with that of earlier models.

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.

About the complexity of two-stage stochastic IPs

We consider so called 2-stage stochastic integer programs (IPs) and their generalized form, so called multi-stage stochastic IPs. A 2-stage stochastic IP is an integer program of the form max { c^T x mid {mathcal {A}} x = b, ,l le x le u,, x in {mathbb {Z}}^{s + nt} } where the constraint matrix {mathcal {A}} in {mathbb {Z}}^{r n times s +nt} consists roughly of n repetitions of a matrix A in {mathbb {Z}}^{r times s} on the vertical line and n repetitions of a matrix B in {mathbb {Z}}^{r times t} on the diagonal. In this paper we improve upon an algorithmic result by Hemmecke and Schultz from 2003 [Hemmecke and Schultz, Math. Prog. 2003] to solve 2-stage stochastic IPs. The algorithm is based on the Graver augmentation framework where our main contribution is to give an explicit doubly exponential bound on the size of the augmenting steps. The previous bound for the size of the augmenting steps relied on non-constructive finiteness arguments from commutative algebra and therefore only an implicit bound was known that depends on parameters r, s, t and Delta , where Delta is the largest entry of the constraint matrix. Our new improved bound however is obtained by a novel theorem which argues about intersections of paths in a vector space. As a result of our new bound we obtain an algorithm to solve 2-stage stochastic IPs in time f(r,s,Delta ) cdot mathrm {poly}(n,t), where f is a doubly exponential function. To complement our result, we also prove a doubly exponential lower bound for the size of the augmenting steps.

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.

Design of a biofuel supply network under stochastic and price-dependent biomass availability

This article presents a framework for profit-maximizing strategic bio-energy supply chain design by taking into account variability in biomass as a response to price set as well as uncertainty in biomass yield. We present our model as a two-stage stochastic integer program for a multi-period integrated design of a network in which the here-and-now strategic decisions include biorefinery locations and size as well as base biomass price. To efficiently solve our model, we suggest an L-shaped-based algorithm along with a Sample Average Approximation approach. Finally, we demonstrate our results in a case study in Texas using realistic data. Within our framework, we present the relationship between biomass and biofuel price, as well as the optimal network design for the biofuel producer.

Multiproduct Newsvendor Problem with Customer-Driven Demand Substitution: A Stochastic Integer Program Perspective

This paper studies a multiproduct newsvendor problem with customer-driven demand substitution, where each product, once run out of stock, can be proportionally substituted by the others. This problem has been widely studied in the literature; however, because of nonconvexity and intractability, only limited analytical properties have been reported and no efficient approaches have been proposed. This paper first completely characterizes the optimal order policy when the demand is known and reformulates this nonconvex problem as a binary quadratic program. When the demand is random, we formulate the problem as a two-stage stochastic integer program, derive several necessary optimality conditions, prove the submodularity of the profit function, and also develop polynomial-time approximation algorithms and show their performance guarantees. We further propose a tight upper bound via nonanticipativity dual, which is proven to be very close to the optimal value and can yield a good-quality feasible solution under a mild condition. Our numerical investigation demonstrates effectiveness of the proposed algorithms. Moreover, several useful findings and managerial insights are revealed from a series of sensitivity analyses.