Robust Mixed-integer Programming Research Articles

A study of distributionally robust mixed-integer programming with Wasserstein metric: on the value of incomplete data

This study addresses a class of linear mixed-integer programming (MILP) problems that involve uncertainty in the objective function parameters. The parameters are assumed to form a random vector, whose probability distribution can only be observed through a finite training data set. Unlike most of the related studies in the literature, we also consider uncertainty in the underlying data set. The data uncertainty is described by a set of linear constraints for each random sample, and the uncertainty in the distribution (for a fixed realization of data) is defined using a type-1 Wasserstein ball centered at the empirical distribution of the data. The overall problem is formulated as a three-level distributionally robust optimization (DRO) problem. First, we prove that the three-level problem admits a single-level MILP reformulation, if the class of loss functions is restricted to biaffine functions. Secondly, it turns out that for several particular forms of data uncertainty, the outlined problem can be solved reasonably fast by leveraging the nominal MILP problem. Finally, we conduct a computational study, where the out-of-sample performance of our model and computational complexity of the proposed MILP reformulation are explored numerically for several application domains.

A novel and efficient exact technique for integrated staffing, assignment, routing, and scheduling of home care services under uncertainty

We model and solve integrated multi-period staffing, assignment, routing, and scheduling of caregivers for home care services and obtain insights for the case under uncertainty. The goal is to construct a weekly schedule that adheres to related operational considerations and determines optimal staffing of caregivers by minimizing caregivers’ fixed- and overtime costs. For tractability, we incorporate a priori-generated visit pattern—an existing practical approach that deals effectively with hard assignment decisions. First, we propose a novel mixed-integer program (MIP) for the nominal (deterministic) problem. We then incorporate uncertainty in service and travel times and develop a robust counterpart by hybridizing interval and polyhedral uncertainty sets. Second, we show that there is a special mathematical structure within the model that allows us to develop a novel logic-based Benders branching-decomposition algorithm that systematically delays the resolution of difficult routing/ scheduling problems and efficiently solves both the nominal and robust MIP models, i.e., our solution approach for the deterministic case allowed us to solve the robust model. Using a benchmark and newly generated instances from the literature, we show that CPLEX can solve our nominal model with average optimality gap of 56%. On the other hand, our new exact technique can solve our nominal model to an average optimality gap of 5%. Third, we provide practical insights into (i) the price of robustness and (ii) the impacts of nurse flexibility and overtime. The average total cost does not increase beyond 18% than the nominal solution and the cost-savings of nurse flexibility is about 19% on average and higher than overtime’s. We also demonstrate that nurse flexibility impact is high and comparable in both cases, deterministic and stochastic.

Semantically-Aware Pedestrian Intent Prediction With Barrier Functions and Mixed-Integer Quadratic Programming

We develop algorithms for inferring long-term intentions and parameters of local collision-avoidance behavior of agents in a multiagent system from their trajectories. This problem is challenging because an agent's observed trajectory only partially manifests its long-term task; it also contains adjustments made by the agent to ensure collision avoidance with other agents and obstacles in the environment. Since an observer would have no means to determine the magnitude of these adjustments, it is difficult to isolate the task-oriented component from the observed motion. To circumvent this problem, we model the agent's dynamics using a reactive optimization whose objective function captures the long-term task while its constraints capture collision-avoidance behavior. We develop two robust mixed-integer programming algorithms that infer the task and safety related parameters of this optimization problem from the positions and velocities of the agents. These algorithms are validated on synthetic datasets using parameter estimation errors, displacement errors and computation time as metrics. We further test these algorithms on a dataset of real human trajectories. We show that the learned parameters capture the true underlying pedestrian dynamics by rolling out the learned model and showing similarity between the ground truth trajectories and the reconstructed trajectories.

An exact robust approach for the integrated berth allocation and quay crane scheduling problem under uncertain arrival times

We consider an integrated berth allocation and quay crane assignment and scheduling problem where the arrival times of the vessels may be affected by uncertainty. The problem is modelled as a two-stage robust mixed integer program where the berth allocation decisions are taken before the exact arrival times are known, and the crane assignment and scheduling operations are adjusted to the arrival times. To solve the robust two-stage model, we follow a decomposition algorithm that decomposes the problem into a master problem and a separation problem. A new scenario reduction procedure for solving the separation problem is proposed as well as a warm start technique for reducing the number of iterations performed by the decomposition algorithm. To scale the proposed decomposition algorithm for large size instances, it is combined with a rolling horizon heuristic.The efficiency and effectiveness of the proposed algorithms are demonstrated through extensive computational experiments carried out on randomly generated instances with both homogeneous and heterogeneous cranes as well as on instances from the literature.

Randomized Constraints Consensus for Distributed Robust Mixed-Integer Programming

In this article, we consider a network of processors aiming at cooperatively solving mixed-integer convex programs subject to uncertainty. Each node only knows a common cost function and its local uncertain constraint set. We propose a randomized, distributed algorithm working under asynchronous, unreliable, and directed communication. The algorithm is based on a local computation and communication paradigm. At each communication round, nodes perform two updates: 1) A verification in which they check—in a randomized fashion—the robust feasibility of a candidate optimal point, and 2) an optimization step in which they exchange their candidate basis (the minimal set of constraints defining a solution) with neighbors and locally solve an optimization problem. As a main result, we show that processors can stop the algorithm after a finite number of communication rounds (either because verification has been successful for a sufficient number of rounds or because a given threshold has been reached) so that candidate optimal solutions are consensual. The common solution has proven to be—with high confidence—feasible and, hence, optimal for the entire set of uncertainty except a subset having an arbitrarily small probability measure. We show the effectiveness of the proposed distributed algorithm using two examples: a random, uncertain mixed-integer linear program and a distributed localization in wireless sensor networks. The distributed algorithm is implemented on a multicore platform in which the nodes communicate asynchronously.

Robust Mixed 0-1 Programming and Submodularity

The paper studies a robust mixed integer program with a single unrestricted continuous variable. The purpose of the paper is the polyhedral study of the robust solution set using submodularity. A submodular function is a set function with a diminishing returns property, and little work has been studied on the utilization of submodularity in the study of optimization problems considering data uncertainty. In this paper, we propose valid inequalities using submodularity. Valid inequalities for the robust mixed integer program are defined. A polynomial separation algorithm is proposed, and we show that the convex hull of the problem can be completely described using the proposed inequalities. In computational tests, we showed the proposed cuts are effective when they are applied to general robust discrete optimization problems with one or multiple constraints.

A Novel and Efficient Exact Technique for Integrated Staffing, Assignment, Routing, and Scheduling of Home Care Services Under Uncertainty

We model and solve an integrated multi-period staffing, assignment, routing, and scheduling for home care services under uncertainty. The goal is to construct a weekly schedule that adheres to related operational considerations and determines optimal staffing of caregivers by minimizing caregivers’ fixed- and overtime costs. For tractability, we incorporate a priori generated visit patterns—an existing practical approach that deals effectively with hard assignment decisions in. First, we propose a novel mixed-integer program (MIP) for the nominal problem. We then incorporate uncertainty in service and travel times and develop a robust counterpart by hybridizing interval and polyhedral uncertainty sets. Second, we show that there is a special mathematical structure within the model that allows us to develop a novel logic-based Benders branching-decomposition algorithm that systematically delays the resolution of difficult routing/ scheduling problems and efficiently solves both the nominal and robust MIP models. Using a dataset from the literature, we show that CPLEX can solve our nominal and robust models with an average optimality gaps of 44.56% and 45.53%, respectively. Using the same dataset, we demonstrate that our new exact technique can solve our nominal and robust mixed-integer models to an average optimality gap of 2.8% and 4.5%, respectively. Third, we provide practical insights into (i) the price of robustness and (ii) the impacts of nurse flexibility and overtime. The average total cost does not increase beyond 12.7% than the nominal solution and the cost-savings of nurse flexibility is approximately five times higher than that of overtime.

Multistage distributionally robust mixed-integer programming with decision-dependent moment-based ambiguity sets

We study multistage distributionally robust mixed-integer programs under endogenous uncertainty, where the probability distribution of stage-wise uncertainty depends on the decisions made in previous stages. We first consider two ambiguity sets defined by decision-dependent bounds on the first and second moments of uncertain parameters and by mean and covariance matrix that exactly match decision-dependent empirical ones, respectively. For both sets, we show that the subproblem in each stage can be recast as a mixed-integer linear program (MILP). Moreover, we extend the general moment-based ambiguity set in Delage and Ye (Oper Res 58(3):595–612, 2010) to the multistage decision-dependent setting, and derive mixed-integer semidefinite programming (MISDP) reformulations of stage-wise subproblems. We develop methods for attaining lower and upper bounds of the optimal objective value of the multistage MISDPs, and approximate them using a series of MILPs. We deploy the Stochastic Dual Dynamic integer Programming (SDDiP) method for solving the problem under the three ambiguity sets with risk-neutral or risk-averse objective functions, and conduct numerical studies on multistage facility-location instances having diverse sizes under different parameter and uncertainty settings. Our results show that the SDDiP quickly finds optimal solutions for moderate-sized instances under the first two ambiguity sets, and also finds good approximate bounds for the multistage MISDPs derived under the third ambiguity set. We also demonstrate the efficacy of incorporating decision-dependent distributional ambiguity in multistage decision-making processes.

An economic-environmental asset planning in electric distribution networks considering carbon emission trading and demand response

Initiatives such as government programs and investment in low-carbon technologies have been adopted to mitigate the carbon emissions in the electricity sector. These initiatives have resulted in new challenges in the power sector, and to address them adequately, innovative frameworks are required in the electric distribution network (EDN) expansion planning and operation problems. Therefore, this work proposes an environmentally committed asset planning approach to remedy the existing issues to some extent. The proposed strategy investigates the benefits of the simultaneous allocation of several assets such as capacitor banks (CBs), distributed generation (DG) units based on renewable energy sources, and energy storage systems (ESSs). Moreover, an innovative carbon emission trading scheme is formulated in the planning stage to mitigate the CO2 emissions while a demand response program is applied to modify the consumption behavior. The proposed approach is formulated as a two-stage robust mixed-integer programming model, which considers uncertainties associated with the electricity demand and renewable-based DG. To cope with the difficulties of this complex model, utilizing an efficient decomposition algorithm, such as the C&CG decomposition algorithm, is essential. The potential of the proposed approach is studied under different operating conditions and via several test cases on a 137-node EDN. In addition, to validate the performance of the proposed carbon emission scheme, a multi-region 54-node distribution network is adequately evaluated. Results show that by considering simultaneously multiple planning alternatives, carbon emission trading scheme, and the demand response program, the total CO2 emissions are reduced by up to 15%.

Two-stage robust mixed integer programming problem with objective uncertainty

Many real problems with uncertain parameters can be modeled as two-stage robust mixed integer programming problems (RMIPs). Due to the complex nature of this kind of problems, this paper focuses on the two-stage RMIPs with objective uncertainty. Based on the results that the augmented Lagrangian is a strong dual for integer programming Boland and Eberhard (Math Program 150(2):491–509, 2015), we present the upper and lower bounds. In a special case, we show that the two-stage RMIPs can be equivalently reformulated as a solvable minimax problem.

Electric power system planning with renewable energy accommodation for supporting the sustainable development of Tangshan City, China

In this study, considering break the original energy structure dominated by coal and make the most use of renewable energy in the future, an inexact stochastic robust mixed-integer programming (ITSRMP) method was developed for supporting regional electric power system management in Tangshan City, China. The developed method incorporated interval-parameter programming (IPP), stochastic robust optimization (SRO), two-stage stochastic programming (TSP), and mixed integer programming (MIP) within a general optimization framework to reflect uncertainties expressed as interval values and probability distributions in the regional electric power system. Three scenarios corresponding to different subsidy price levels and three cases associated with different pollutants emission reduction levels were designed. The electricity generation schemes, facility-expansion, pollutant emission, and system cost considering the subsidy policy and air pollution mitigation control had been obtained. The results indicated that subsidy policy would exert an important influence on the development of Tangshan's electric power system, which can reduce the cost advantage of conventional power generation and enhance the development enthusiasm of renewable power generation to power enterprises. In detail, the electricity generation amount of renewable energy would increase with the improvement of subsidy price level. Moreover, decision makers could identify the possible policy implementations and enforcements under considering the trade-off among system economy, security and environmental objectives. The modeling results were valuable for promoting new energy accommodation and supporting the sustainable development of social economy.

An integrated approach to solve a robust forward/reverse supply chain for short lifetime products

Supply chain management plays a key role in optimizing manufacturing costs in competitive markets. In this paper, a multi period, multi product, multi echelon and capacitated supply chain problem for short lifetime products was solved using both exact method and metaheuristic algorithms. The considered echelons in a forward/reverse supply chain network design include suppliers, hybrid production–inspection centers, hybrid warehouse-collection centers, retailers, disposal centers and recovery centers. To tackle both strategic and tactical nature of the problem and due to its complexity, the location problem is first solved using an exact method in which strategic decisions are made aiming to minimize the location costs. Then the allocation problem is solved using both exact and metaheuristic algorithms, namely Tabu Search and Differential Evolution, in which based on the results obtained from the location problem, the profit is maximized and tactical decisions are made. A cyclic feedback procedure is proposed in between the location and allocation models to obtain the best solution. The allocation model is linearized using a new method and to handle the uncertainty, a robust mixed integer programming model is developed which can retain the linearity of the model. The computational results confirm the effectiveness of the exact method for the location problem while testifies the proposed algorithms can obtain effective solutions compared with that of the exact solutions for the allocation problem in both deterministic and uncertain environment. In addition, the results show that in terms of preciseness and convergence, the proposed Differential Evolution algorithm outperforms the other algorithm.

A robust bi-objective model for concurrent planning of project scheduling and material procurement

Simultaneous planning of project scheduling and material procurement can lead to the project execution costs improvement. Hence, the issue has been addressed in this paper by a robust mixed-integer programming mathematical model, which aims to minimize the corresponding costs and maximize the schedule robustness. The given approach is able to control the degree of solution conservatism, in regard to probabilistic bounds on constraint violation. The proposed model takes the uncertainty issue into account from both viewpoints of activities duration time and execution costs. The NSGA-II and a modified version of multi-objective differential evolution algorithm have been applied as the solution methodologies. Moreover, the principal factors are calibrated by the Taguchi method to provide robustness to the obtained results. Finally, the performance of the solution methods is compared according to a varied set of instances to test their applicability and efficiency.

Multi-Stage Robust Unit Commitment Considering Wind and Demand Response Uncertainties

With the increasing penetration of wind power into the power grid, maintaining system reliability has been a challenging issue for ISOs/RTOs, due to the intermittent nature of wind power. In addition to the traditional reserves provided by thermal, hydro, and gas generators, demand response (DR) programs have gained much attention recently as another reserve resource to mitigate wind power output uncertainty. However, the price-elastic demand curve is not exactly known in advance, which provides another dimension of uncertainty. To accommodate the combined uncertainties from wind power and DR, we allow the wind power output to vary within a given interval with the price-elastic demand curve also varying in this paper. We develop a robust optimization approach to derive an optimal unit commitment decision for the reliability unit commitment runs by ISOs/RTOs, with the objective of maximizing total social welfare under the joint worst-case wind power output and demand response scenario. The problem is formulated as a multi-stage robust mixed-integer programming problem. An exact solution approach leveraging Benders' decomposition is developed to obtain the optimal robust unit commitment schedule for the problem. Additional variables are introduced to parameterize the conservatism of our model and avoid over-protection. Finally, we test the performance of the proposed approach using a case study based on the IEEE 118-bus system. The results verify that our proposed approach can accommodate both wind power and demand response uncertainties, and demand response can help accommodate wind power output uncertainty by lowering the unit load cost.

Robust Inventory Routing Under Demand Uncertainty

This paper introduces a robust inventory routing problem where a supplier distributes a single product to multiple customers facing dynamic uncertain demands over a finite discrete time horizon. The probability distribution of the uncertain demand at each customer is not fully specified. The only available information is that these demands are independent and symmetric random variables that can take some value from their support interval. The supplier is responsible for the inventory management of its customers, has sufficient inventory to replenish the customers, and distributes the product using a capacitated vehicle. Backlogging of the demand at customers is allowed. The problem is to determine the delivery quantities as well as the times and routes to the customers, while ensuring feasibility regardless of the realized demands, and minimizing the total cost composed of transportation, inventory holding, and shortage costs. Using a robust optimization approach, we propose two robust mixed integer programming (MIP) formulations for the problem. We also propose a new MIP formulation for the deterministic (nominal) case of the problem. We implement these formulations within a branch-and-cut algorithm and report results on a set of instances adapted from the literature.

Gomory cuts revisited

We investigate the use of Gonory's mixed integer cuts within a branch-and-cut framework. It has been argued in the literature that “a marriage of classical cutting planes and tree search is out of the question as far as the solution of large-scale combinatorial optimization problems is concerned” [16] because the cuts generated at one node of the search tree need not be valid at other nodes. We show that it is possible, by using a simple lifting procedure, to make Gomory cuts generated at a node of the enumeration tree globally valid in the case of mixed 0–1 programs. The procedure essentially amounts to treating the variables fixed at 0 or 1 as if they were free. We also show why this lifting procedure is not valid for general (other than 0–1) mixed integer programs. Other issues addressed in the paper are of a computational nature, such as strategies for generating the cutting planes, deciding between branching and cutting, etc. The result is a robust mixed integer program solver.