Mixed-integer Linear Programming Reformulation Research Articles

Dynamic Modelling and Mutual Coordination of Electricity and Watershed Networks for Spatio-Temporal Operational Flexibility Enhancement Under Rainy Climates

This paper proposes a watershed-electricity nexus model to unlock the flexibility of watershed networks (WSNs) for supporting the operation of power distribution networks (PDNs) under rainy climates. The proposed model exploits the spatio-tem-poral flexibility of geographically dispersed pump clusters to pro-vide reserve services to PDNs, and a hyperbolic partial differential function derived from Saint-Venant hydrodynamic equations is formed to describe the dynamic processes of river stream flows. Be-sides, a flexibility evaluation method based on a composite sensitiv-ity matrix of water levels with respect to power injections is pre-sented to quantify the time-varying adjustable power domain of pump loads. Then, a multi-stage interactive coordinated scheduling strategy is developed for the mutual operation of WSNs and PDNs, where drainage pumps are jointly optimized to provide flexible power reserves, while an optimal PDN economic dispatch is per-formed to improve the power supply voltage of pump loads. Fur-thermore, an equivalent mixed-integer linear programming refor-mulation method is derived to cope with the original nonlinear par-tial differential optimization problem for computational tractabil-ity improvements. Comparative results have validated the effective-ness of the proposed strategy in eliminating voltage violations and shaving peak loads.

Two-stage stochastic programming with robust constraints for the logistics network post-disruption response strategy optimization

Logistics networks (LNs) are essential for the transportation and distribution of goods or services from suppliers to consumers. However, LNs with complex structures are more vulnerable to disruptions due to natural disasters and accidents. To address the LN post-disruption response strategy optimization problem, this study proposes a novel two-stage stochastic programming model with robust delivery time constraints. The proposed model jointly optimizes the new-line-opening and rerouting decisions in the face of uncertain transport demands and transportation times. To enhance the robustness of the response strategy obtained, the conditional value at risk (CVaR) criterion is utilized to reduce the operational risk, and robust constraints based on the scenario-based uncertainty sets are proposed to guarantee the delivery time requirement. An equivalent tractable mixed-integer linear programming reformulation is further derived by linearizing the CVaR objective function and dualizing the infinite number of robust constraints into finite ones. A case study based on the practical operations of the JD LN is conducted to validate the practical significance of the proposed model. A comparison with the rerouting strategy and two benchmark models demonstrates the superiority of the proposed model in terms of operational cost, delivery time, and loading rate.

An exact method for binary fortification games

A fortification game (FG) is a three-level, two-player hierarchical game, also known as defender-attacker-defender game, in which at the uppermost level, the defender selects some assets to be protected from potential malicious attacks. At the middle level, the attacker solves an interdiction game by depreciating unprotected assets, i.e., reducing the values of such assets for the defender, while at the innermost level the defender solves a recourse problem over the surviving or partially damaged assets. Fortification games have applications in various important areas, such as military operations, design of survivable networks, protection of facilities or power grid protection. In this work, we present an exact solution algorithm for FGs, in which the recourse problems correspond to (possibly NP-hard) combinatorial optimization problems. The algorithm is based on a new generic mixed-integer linear programming reformulation in the natural space of fortification variables. Our new model makes use of fortification cuts that measure the contribution of a given fortification strategy to the objective function value. These cuts are generated when needed by solving separation problems, which correspond to (modified) middle-level interdiction games. We design a branch-and-cut-based solution algorithm based on fortification cuts, their strengthened versions and other speed-up techniques. We present a computational study using the knapsack fortification game and the shortest path fortification game. For the latter one, we include a comparison with a state-of-the-art solution method from the literature. Our algorithm outperforms this method and allows us to solve previously unsolved instances with up to 330 386 nodes and 1 202 458 arcs to optimality.

Distributionally robust optimization for the closed‐loop supply chain design under uncertainty

AbstractThe closed‐loop supply chain network (CLSCN) contains reverse flows that collect products from customers and recycle or remanufacture usable parts. The CLSCN design problem is becoming more and more prominent under the context of Sustainable Development and Circular Economy. Parameters associated with a CLSCN including customer demands, transportation costs, or disposal rates are usually subject to uncertainty. Furthermore, natural or man‐made disruptions may cause part of the CLSCN to malfunction. We herein propose a hybrid stochastic and distributionally robust optimization (DRO) approach to hedge against discrete disruption scenarios and uncertain customer demands. We also tailor a Benders decomposition‐based algorithm to efficiently solve the resulting large‐scale mixed integer linear programming reformulations. Computational experiments demonstrate that the proposed algorithm can outperform commercial solvers such as CPLEX, and the DRO approach can produce solutions with low average costs and low variance in out‐of‐sample tests.

Data-driven distributionally robust surgery planning in flexible operating rooms over a Wasserstein ambiguity

We study elective surgery planning in flexible operating rooms (ORs) where emergency patients are accommodated in the existing elective surgery schedule. Specifically, elective surgeries can be scheduled weeks or months in advance. In contrast, an emergency surgery arrives randomly and must be performed on the day of arrival. Probability distributions of the actual durations of elective and emergency surgeries are unknown, and only a possibly small set of historical realizations may be available. To address distributional uncertainty, we first construct an ambiguity set that encompasses all possible distributions of surgery durations within a 1-Wasserstein distance from the empirical distribution. We then define a distributionally robust surgery assignment (DSA) problem to determine optimal elective surgery assignment decisions to available surgical blocks in multiple ORs, considering the capacity needed for emergency cases. The objective is to minimize the total cost consisting of the fixed cost related to scheduling or rejecting elective surgery plus the maximum expected cost associated with OR overtime and idle time over all distributions defined in the ambiguity set. Using the DSA model’s structural properties, we derive an equivalent mixed-integer linear programming (MILP) reformulation that can be implemented and solved efficiently using off-the-shelf optimization software. In addition, we extend the proposed model to determine the number of ORs needed to serve the two competing surgery classes and derive a MILP reformulation of this extension. We conduct extensive numerical experiments based on real-world surgery data, demonstrating our proposed model’s computational efficiency and superior out-of-sample operational performance over two state-of-the-art approaches. In addition, we derive insights into surgery scheduling in flexible ORs.

Building Load Control Using Distributionally Robust Chance-Constrained Programs with Right-Hand Side Uncertainty and the Risk-Adjustable Variants

Aggregation of heating, ventilation, and air conditioning (HVAC) loads can provide reserves to absorb volatile renewable energy, especially solar photo-voltaic (PV) generation. In this paper, we decide HVAC control schedules under uncertain PV generation, using a distributionally robust chance-constrained (DRCC) building load control model under two typical ambiguity sets: the moment-based and Wasserstein ambiguity sets. We derive mixed integer linear programming (MILP) reformulations for DRCC problems under both sets. Especially, for the Wasserstein ambiguity set, we use the right-hand side (RHS) uncertainty to derive a more compact MILP reformulation than the commonly known MILP reformulations with big-M constants. All the results also apply to general individual chance constraints with RHS uncertainty. Furthermore, we propose an adjustable chance-constrained variant to achieve tradeoff between the operational risk and costs. We derive MILP reformulations under the Wasserstein ambiguity set and second-order conic programming (SOCP) reformulations under the moment-based set. Using real-world data, we conduct computational studies to demonstrate the efficiency of the solution approaches and the effectiveness of the solutions. Summary of Contribution: The problem studied in this paper is motivated by a building load control problem that uses the aggregation of heating, ventilation, and air conditioning (HVAC) loads as flexible reserves to absorb uncertain solar photovoltaic (PV) generation. The problem is formulated as distributionally robust chance-constrained (DRCC) programs with right-hand side (RHS) uncertainty. In addition, we propose a risk-adjustable variant of the DRCC programs, where the risk level, instead of being predetermined, is treated as a decision variable. The paper aims to provide tractable reformulations and solution algorithms for both the (general) DRCC and the (general) adjustable DRCC models with RHS uncertainty.

Facility Location-Allocation Problem for Emergency Medical Service With Unmanned Aerial Vehicle

This paper models an operation problem of an unmanned aerial vehicle for the emergency medical service (UEMS) system. The model is set up as a location-allocation problem. The coverage distance and capacity of the UEMS facility are modeled as functions of UAVs assigned. The allocation of the demand point is constrained by the variable coverage distance of each facility. The robust optimization approach is used over the cardinality-constrained uncertain demand, which leads to a nonlinear optimization problem. The UEMS location-allocation problem (ULAP) is reformulated to a solvable problem. An extended formulation and corresponding branch-and-price (B&P) algorithm are also proposed, which strengthen the linear programming relaxation bound. The subproblem of the B&P algorithm is defined as a robust disjunctively constrained integer knapsack problem. Two solution approaches of mixed-integer linear programming reformulation and decomposed dynamic programming are designed for the subproblem. To provide time-efficient solutions for large-scale problems, a restricted master heuristic (RMH) is proposed based on the extended formulation. In computational experiments, the B&P algorithm provided a strong lower bound, and the RMH could find an effective feasible solution within an applicable computation time.

Global solutions of nonconvex standard quadratic programs via mixed integer linear programming reformulations

A standard quadratic program is an optimization problem that consists of minimizing a (nonconvex) quadratic form over the unit simplex. We focus on reformulating a standard quadratic program as a mixed integer linear programming problem. We propose two alternative formulations. Our first formulation is based on casting a standard quadratic program as a linear program with complementarity constraints. We then employ binary variables to linearize the complementarity constraints. For the second formulation, we first derive an overestimating function of the objective function and establish its tightness at any global minimizer. We then linearize the overestimating function using binary variables and obtain our second formulation. For both formulations, we propose a set of valid inequalities. Our extensive computational results illustrate that the proposed mixed integer linear programming reformulations significantly outperform other global solution approaches. On larger instances, we usually observe improvements of several orders of magnitude.

Data-driven distributionally robust chance-constrained optimization with Wasserstein metric

We study distributionally robust chance-constrained programming (DRCCP) optimization problems with data-driven Wasserstein ambiguity sets. The proposed algorithmic and reformulation framework applies to all types of distributionally robust chance-constrained optimization problems subjected to individual as well as joint chance constraints, with random right-hand side and technology vector, and under two types of uncertainties, called uncertain probabilities and continuum of realizations. For the uncertain probabilities (UP) case, we provide new mixed-integer linear programming reformulations for DRCCP problems. For the continuum of realizations case with random right-hand side, we propose an exact mixed-integer second-order cone programming (MISOCP) reformulation and a linear programming (LP) outer approximation. For the continuum of realizations (CR) case with random technology vector, we propose two MISOCP and LP outer approximations. We show that all proposed relaxations become exact reformulations when the decision variables are binary or bounded general integers. For DRCCP with individual chance constraint and random right-hand side under both the UP and CR cases, we also propose linear programming reformulations which need the ex-ante derivation of the worst-case value-at-risk via the solution of a finite series of linear programs determined via a bisection-type procedure. We evaluate the scalability and tightness of the proposed MISOCP and (MI)LP formulations on a distributionally robust chance-constrained knapsack problem.

Distributionally robust facility location problem under decision-dependent stochastic demand

While the traditional facility location problem considers exogenous demand, in some applications, locations of facilities could affect the willingness of customers to use certain types of services, e.g., carsharing, and therefore they also affect realizations of random demand. Moreover, a decision maker may not know the exact distribution of such endogenous demand and how it is affected by location choices. In this paper, we consider a distributionally robust facility location problem, in which we interpret the moments of stochastic demand as functions of facility-location decisions. We reformulate a two-stage decision-dependent distributionally robust optimization model as a monolithic formulation, and then derive exact mixed-integer linear programming reformulation as well as valid inequalities when the means and variances of demand are piecewise linear functions of location solutions. We conduct extensive computational studies, in which we compare our model with a decision-dependent deterministic model, as well as stochastic programming and distributionally robust models without the decision-dependent assumption. The results show superior performance of our approach with remarkable improvement in profit and quality of service under various settings, in addition to computational speed-ups given by formulation enhancements. These results draw attention to the need of considering the impact of location decisions on customer demand within this strategic-level planning problem.

Scenario Grouping and Decomposition Algorithms for Chance-Constrained Programs

A lower bound for a finite-scenario-based chance-constrained program is the quantile value corresponding to the sorted optimal objective values of scenario subproblems. This quantile bound can be improved by grouping subsets of scenarios at the expense of solving larger subproblems. The quality of the bound depends on how the scenarios are grouped. In this paper, we formulate a mixed-integer bilevel program that optimally groups scenarios to tighten the quantile bounds. For general chance-constrained programs, we propose a branch-and-cut algorithm to optimize the bilevel program, and for chance-constrained linear programs, a mixed-integer linear-programming reformulation is derived. We also propose several heuristics for grouping similar or dissimilar scenarios. Our computational results demonstrate that optimal grouping bounds are much tighter than heuristic bounds, resulting in smaller root-node gaps and better performance of scenario decomposition for solving chance-constrained 0-1 programs. Also, the optimal grouping bounds can be greatly strengthened using larger group size. Summary of Contribution: Chance-constrained programs are in general NP-hard but widely used in practice for lowering the risk of undesirable outcomes during decision making under uncertainty. Assuming finite scenarios of uncertain parameter, chance-constrained programs can be reformulated as mixed-integer linear programs with binary variables representing whether or not the constraints are satisfied in corresponding scenarios. A useful quantile bound for solving chance-constrained programs can be improved by grouping subsets of scenarios at the expense of solving larger subproblems. In this paper, we develop algorithms for optimally and heuristically grouping scenarios to tighten the quantile bounds. We aim to improve both the computation and solution quality of a variety of chance-constrained programs formulated for different Operations Research problems.

Non-profit resource allocation and service scheduling with cross-subsidization and uncertain resource consumptions

We consider a mixture of for-profit and non-profit requests that share multiple resources at random consumption rates. The revenue from fulfilling for-profit requests is used to cross-subsidize the cost of non-profit operations, and we aim to maximize the number of completed non-profit service requests. We consider two problems that respectively optimize resource allocation and service schedules, and employ chance constraints to restrict the probability of undesired outcomes such as resource over-utilization and service delay. For the allocation model, we propose three approximations of the chance-constrained program, and derive their variants to allow variable resource capacities. For the scheduling model, we derive a mixed-integer linear programming reformulation and develop a two-phase algorithm that separately decides allocation decisions and the start time of each service request. We conduct numerical studies on randomly generated instances of non-profit surgery planning to demonstrate the computational results of different models, and the impact of varying parameters and cross-subsidization on non-profit operations.

On Robust Fractional 0-1 Programming

We study single- and multiple-ratio robust fractional 0-1 programming problems (RFPs). In particular, this work considers RFPs under a wide range of disjoint and joint uncertainty sets, where the former implies separate uncertainty sets for each numerator and denominator and the latter accounts for different forms of interrelatedness between them. We first demonstrate that unlike the deterministic case, a single-ratio RFP is nondeterministic polynomial-time hard under general polyhedral uncertainty sets. However, if the uncertainty sets are imbued with a certain structure, variants of the well-known budgeted uncertainty, the disjoint and joint single-ratio RFPs are polynomially solvable when the deterministic counterpart is. We also propose mixed-integer linear programming (MILP) formulations for multiple-ratio RFPs. We conduct extensive computational experiments using test instances based on real and synthetic data sets to evaluate the performance of our MILP reformulations as well as to compare the disjoint and joint uncertainty sets. Finally, we demonstrate the value of the robust approach by examining the robust solution in a deterministic setting and vice versa.

On Enhancing Resilience to Cascading Failures via Post-Disturbance Tweaking of Line Reactances

The primary goal of this paper is to develop an optimization framework for studying the efficacy of transmission line reactance tweaking as a mechanism for post-disturbance control in a transmission network. We start by developing a mixed-integer linear programming (MILP) formulation for tracking the redistribution of direct current (DC) flows and the graph-theoretic evolution of a network topology over the course of cascading failures. Next, we propose a min–max setup for studying the impact of post-disturbance reactance tweaking on the resilience of the system to a worst-case disturbance. We devise a MILP reformulation scheme for the underlying bilevel non-convex mixed-integer nonlinear program to facilitate the computation of its globally optimal solution. We then develop a MILP framework for computing a tight upper bound (best-case scenario) on the efficacy of post-disturbance reactance tweaking for a given set of bus loads. Our numerical case study suggests that post-disturbance reactance tweaking, even on only a small number of lines, can be effective in reducing the amount of load shed in some scenarios in the tested system.

Chance-Constrained Surgery Planning Under Conditions of Limited and Ambiguous Data

Surgery planning decisions include which operating rooms (ORs) to open, allocation of surgeries to ORs, sequence, and time to start each surgery. They are often made under uncertain surgery durations with limited data that lead to unknown distributional information. Moreover, cost parameters for criteria such as overtime and surgery delays are often difficult or impossible to estimate in practice. In this paper, we formulate distributionally robust (DR) chance constraints on surgery waiting and OR overtime, which recognize practical limitations on data availability and cost parameter accuracy. We use [Formula: see text]-divergence measures to build an ambiguity set of possible distributions of random surgery durations, and derive a branch-and-cut algorithm for optimizing a mixed-integer linear programming reformulation based on finite samples of the random surgery durations. We test instances generated from real hospital-based surgery data. The results show computational efficacy of our approaches, and provide insights for DR surgery planning.

Data-Driven Distributionally Robust Chance-Constrained Optimization With Wasserstein Metric

We study distributionally robust chance-constrained programming (DRCCP) optimization problems with data-driven Wasserstein ambiguity sets. The proposed algorithmic and reformulation framework applies to distributionally robust optimization problems subjected to individual as well as joint chance constraints, with random right-hand side and technology vector, and under two types of uncertainties, called uncertain probabilities and continuum of realizations. For the uncertain probabilities case, we provide new mixed-integer linear programming reformulations for DRCCP problems and derive a set of precedence optimality cuts to strengthen the formulations. For the continuum of realizations case with random right-hand side, we propose an exact mixed-integer second-order cone programming (MISOCP) reformulation and a linear programming (LP) outer approximation. For the continuum of realizations case with random technology vector, we propose two MISOCP and LP outer approximations. We show that all proposed relaxations become exact reformulations when the decision variables are binary or bounded general integers. We evaluate the scalability and tightness of the proposed MISOCP and (MI)LP formulations on a distributionally robust chance constrained knapsack problem.

Minimizing the weighted sum of completion times under processing time uncertainty

We address the robust counterpart of a classical single machine scheduling problem by considering a budgeted uncertainty and an ellipsoidal uncertainty. We prove that the problem is NP-hard for arbitrary ellipsoidal uncertainty sets. Then, a mixed-integer linear programming reformulations and a second order cone programming reformulations are provided. We assess the reformulations on randomly generated instances, comparing them with branch-and-cut algorithms.

An Enhanced Network-Constrained UC Model for Leveraging System Operation Cost and Financial Profitability of Incentive-Based DR Loads

Demand response (DR) programs provide consumers opportunities to play an active role in electricity market operation by encouraging them to reduce and/or shift their electricity usage during peak periods and/or when system security is jeopardized. There are primarily two types of DR programs, including incentive-based DR and price-based DR. This paper focuses on the incentive-based DR. For this type of DR program, although shifting peak loads to off-peak periods could reduce system operation cost, it may raise off-peak electricity prices and the incentives may not be enough to cover the increased payments of DR loads. This would diminish consumers' enthusiasm to the continued participation in DR programs. Federal Energy Regulatory Commission Order 745 states that when dispatching a DR resource as an alternative to a generation resource is cost-effective, which is determined by the Independent System Operator's (ISO) net benefits test, the DR must be compensated at full locational marginal price (LMP) value. However, the order focuses more on load curtailment, but not load shifting. In addition, the net benefits test mainly focuses on consumers' savings, while ignoring operation costs and the value lost by consumers, which may cause system inefficiency. This paper proposes a systematic model for leveraging system operation cost and consumer payment concerns in electricity market operation. The proposed model minimizes the system operation cost, while incorporating explicit LMP formulations and consumer payment requirements into the network-constrained unit commitment formulation. Thus, it determines the proper amount of DR loads to be shifted from peak hours to off-peaks under the ISO direct load control for reducing the operation cost and ensuring that DR load payments will not deteriorate significantly after load shifting. The proposed model is solved in its original mixed-integer nonlinear programming formulation and the mixed-integer linear programming reformulation via piece-wise linear approximation. Numerical case studies illustrate the effectiveness of the proposed model and compare the computational performance of the two formulations.

A dynamic reformulation heuristic for Generalized Interdiction Problems

We consider a subfamily of mixed-integer linear bilevel problems that we call Generalized Interdiction Problems. This class of problems includes, among others, the widely-studied interdiction problems, i.e., zero-sum Stackelberg games where two players (called the leader and the follower) share a set of items, and the leader can interdict the usage of certain items by the follower. Problems of this type can be modeled as Mixed-Integer Nonlinear Programming problems, whose exact solution can be very hard. In this paper we propose a new heuristic scheme based on a single-level and compact mixed-integer linear programming reformulation of the problem obtained by relaxing the integrality of the follower variables. A distinguished feature of our method is that general-purpose mixed-integer cutting planes for the follower problem are exploited, on the fly, to dynamically improve the reformulation. The resulting heuristic algorithm proved very effective on a large number of test instances, often providing an (almost) optimal solution within very short computing times.

Mixed-integer linear programming approach for global discrete sizing optimization of frame structures

This paper focuses on discrete sizing optimization of frame structures using commercial profile catalogs. The optimization problem is formulated as a mixed-integer linear programming (MILP) problem by including the equations of structural analysis as constraints. The internal forces of the members are taken as continuous state variables. Binary variables are used for choosing the member profiles from a catalog. Both the displacement and stress constraints are formulated such that for each member limit values can be imposed at predefined locations along the member. A valuable feature of the formulation, lacking in most contemporary approaches, is that global optimality of the solution is guaranteed by solving the MILP using branch-and-bound techniques. The method is applied to three design problems: a portal frame, a two-story frame with three load cases and a multiple-bay multiple-story frame. Performance profiles are determined to compare the MILP reformulation method with a genetic algorithm.