Mixed Integer Programming Solver Research Articles

Optimal schedule recovery for the aircraft gate assignment with constrained resources

Efficient ground handling at airports contributes significantly to the performance of the entire air transport network. In this network, airports are coupled by flights that depend on passenger and crew connections, effective local airport operations, and efficient ground handling resource management. In addition, airport stakeholders must consider different time scales (look-ahead times), process estimates, and both bounded and multiple-dependent solution spaces for their decision-making processes. In this context, we aim to solve the NP-hard problem of combined aircraft turnaround time and airport gate allocation optimization. We developed a hybrid meta-heuristic algorithm based on the Shuffled Frog-Leaping Algorithm (SFLA) and the Grasshopper Optimization Algorithm (GOA). Our hybrid SFLA-GOA approach overcomes the drawback of each algorithm and achieves a preferred solution at a reduced computational cost. We test our approach with a scenario derived from Frankfurt Airport operations, including gate positions, terminal layout, and flight plan information. In all defined problem sizes, the hybrid SFLA-GOA algorithm outperforms an independent implementation of the SFLA algorithm. Also, the SFLA-GOA algorithm provides an improved solution by at least nearly 12% up to 40% over the standard mixed-integer programming solvers.

Siting renewable power generation assets with combinatorial optimisation

This paper studies the problem of siting renewable power generation assets using large amounts of climatological data while accounting for their spatiotemporal complementarity. The problem is cast as a combinatorial optimisation problem selecting a pre-specified number of sites so as to minimise the number of simultaneous low electricity production events that they experience relative to a pre-specified reference production level. It is shown that the resulting model is closely related to submodular optimisation and can be interpreted as generalising the well-known maximum coverage problem. Both deterministic and randomised algorithms are discussed, including greedy, local search and relaxation-based heuristics as well as combinations of these algorithms. The usefulness of the model and methods is illustrated by a realistic case study inspired by the problem of siting onshore wind power plants in Europe, resulting in instances featuring over ten thousand candidate locations and ten years of hourly-sampled meteorological data. The proposed solution methods are benchmarked against a state-of-the-art mixed-integer programming solver and several algorithms are found to consistently produce better solutions at a fraction of the computational cost. The physical nature of solutions provided by the model is also investigated, and all deployment patterns are found to be unable to supply a constant share of the electricity demand at all times. Finally, a cross-validation analysis shows that, except for an edge case, the model can successfully and reliably identify deployment patterns that perform well on previously unseen climatological data from historical data spanning a small number of weather years.

The transit time constrained fixed charge multi-commodity network design problem

This paper introduces the transit time constrained fixed charge multi-commodity network design problem. Transit times are origin-to-destination time limits for the commodities, which appear for example in transport systems with perishable goods. We discuss how to model the problem and present three different formulations of it. The first formulation is an exponential size path formulation, which we solve with a branch-and-price algorithm. Several speed up techniques from the literature on fixed charge multi-commodity network design problems are implemented, such as lifted cover inequalities and the recently proposed deep dual-optimal inequalities. In an extensive set of computational experiments, we show that these inequalities significantly improve the performance of the algorithm. The other two formulations are of polynomial size: one uses path indices and the other uses time indices. While the branch-and-price algorithm outperforms solving the compact formulations with a general-purpose mixed-integer programming solver, the study of compact models helps better understand the problem, and we can use them as benchmarks. A detailed sensitivity analysis of the branch-and-price algorithm shows that longer transit times and an increased ratio of fixed charge to flow cost increase the difficulty of solving the problem whereas the arc capacity has less impact. We further discuss in-depth implementational details.

Multi-supply multi-capacitated p-median location optimization via a hybrid bi-level intelligent algorithm

The Capacitated P-Median Problem (CPMP) is a classic discrete facility location problem in which p capacitated facilities (medians) are selected from an economic point of view to serve a set of demand vertexes so that the total demand assigned to each of the facilities (medians) does not exceed its capacity. Since the capacity used by each facility is fixed and does not depend on the vertex demand, the traditional capacitated p-median problem’s total capacity is underutilized. To make full use of the total capacity, a more generalized model, namely, the Multi-supply Multi-Capacitated Location Problem (MMCLP), is considered in this paper, which assumes not only that each facility is allowed to open with different capacities while meeting the constraint of total capacity but also that each demand vertex can accept supplies from different facilities. To solve the MMCLP effectively, we propose a hybrid bi-level optimization framework based on decomposition. First, the original MMCLP is decomposed into numerous mixed-integer programming (MIP) sub-problems, i.e., inner optimization is performed, and for outer optimization, a Genetic Algorithm based on Co-Evolution of Population and Neighborhood (GACEPN) is proposed to find better sub-problems near the good sub-problems among all the candidate sub-problems. The inner optimization problem is solved by a commercial MIP solver. The results obtained are returned to the outer layer so that the neighborhood of the outer layer optimization can be adjusted appropriately to generate better sub-problems, thereby approaching the optimal solution of the original problem. The experiments show that for a large-scale MMCLP, compared to the Gurobi solver, the proposed bi-level algorithm (a hybrid of the GACEPN and Gurobi) can find a better solution in less time.

Learning Pseudo-Backdoors for Mixed Integer Programs

We propose a machine learning approach for quickly solving Mixed Integer Programs (MIP) by learning to prioritize a set of decision variables, which we call pseudo-backdoors, for branching that results in faster solution times. Learning-based approaches have seen success in the area of solving combinatorial optimization problems by being able to flexibly leverage common structures in a given distribution of problems. Our approach takes inspiration from the concept of strong backdoors, which corresponds to a small set of variables such that only branching on these variables yields an optimal integral solution and a proof of optimality. Our notion of pseudo-backdoors corresponds to a small set of variables such that only branching on them leads to faster solve time (which can be solver dependent). A key advantage of pseudo-backdoors over strong backdoors is that they are much amenable to data-driven identification or prediction. Our proposed method learns to estimate the solver performance of a proposed pseudo-backdoor, using a labeled dataset collected on a set of training MIP instances. This model can then be used to identify high-quality pseudo-backdoors on new MIP instances from the same distribution. We evaluate our method on the generalized independent set problems and find that our approach can efficiently identify high-quality pseudo-backdoors. In addition, we compare our learned approach against Gurobi, a state-of-the-art MIP solver, demonstrating that our method can be used to improve solver performance.

Mixed Integer Programming versus Evolutionary Computation for Optimizing a Hard Real-World Staff Assignment Problem

Assigning staff to engagements according to hard constraints while optimizing several objectives is a task encountered by many companies on a regular basis. Simplified versions of such assignment problems are NP-hard. Despite this, a typical approach to solving them consists of formulating them as mixed integer programming (MIP) problems and using a stateof-the-art solver to get solutions that closely approximate the optimum.In this paper, we consider a complex real-world staff assignment problem encountered by the professional service company KPMG, with the goal of finding an algorithm that solves it faster and with a better solution than a commercial MIP solver. We follow the evolutionary algorithm (EA) metaheuristic and design a search heuristic which iteratively improves a solution using domain-specific mutation operators. Furthermore, we use a flow algorithm to optimally solve a subproblem, which tremendously reduces the search space for the EA.For our real-world instance of the assignment problem, given the same total time budget of 100 hours, a parallel EA approach finds a solution that is only 1.7% away from an upper bound for the (unknown) optimum within under five hours, while the MIP solver Gurobi still has a gap of 10.5%.

An iterated local search procedure for the job sequencing and tool switching problem with non-identical parallel machines

In this paper, a new generalization of the uniform job sequencing and tool switching problem is presented. It considers non-identical parallel machines, which differ in tool magazine capacity and setup times. Applications of the problem can be found in the metal working or semiconductor manufacturing industries when the jobs require different tool sets for processing. The paper provides a Mixed Integer Programming formulation for the job sequencing and tool switching problem with machine-dependent processing and tool switching times under the consideration of three conflicting objectives (minimizing the total flowtime, minimizing the makespan and minimizing the total number of tool switches). Different efficient construction heuristics and Iterated Local Search methods are proposed and evaluated, using 640 new and publicly available instances. The results of the construction heuristics, the Iterated Local Search schemes and a Mixed Integer Programming Solver are discussed. The extensive computational experiments show the merit of the perturbation strategy in order to overcome local optima. It is shown that certain methods are more or less suitable depending on the objective function.

Learning To Scale Mixed-Integer Programs

Many practical applications require the solution of numerically challenging linear programs (LPs) and mixed integer programs (MIPs). Scaling is a widely used preconditioning technique that aims at reducing the error propagation of the involved linear systems, thereby improving the numerical behavior of the dual simplex algorithm and, consequently, LP-based branch-and-bound. A reliable scaling method often makes the difference whether these problems can be solved correctly or not. In this paper, we investigate the use of machine learning to choose at the beginning of the solution process between two common scaling methods: Standard scaling and Curtis-Reid scaling. The latter often, but not always, leads to a more robust solution process, but may suffer from longer solution times. Rather than training for overall solution time, we propose to use the attention level of a MIP solution process as a learning label. We evaluate the predictive power of a random forest approach and a linear regressor that learns the (square-root of the) difference in attention level. It turns out that the resulting classification not only reduces various types of numerical errors by large margins, but it also improves the performance of the dual simplex algorithm. The learned model has been implemented within the FICO Xpress MIP solver and it is used by default since release 8.9, May 2020, to determine the scaling algorithm Xpress applies before solving an LP or a MIP.

Bounds on efficient outcomes for large-scale cardinality-constrained Markowitz problems

When solving large-scale cardinality-constrained Markowitz mean–variance portfolio investment problems, exact solvers may be unable to derive some efficient portfolios, even within a reasonable time limit. In such cases, information on the distance from the best feasible solution, found before the optimization process has stopped, to the true efficient solution is unavailable. In this article, I demonstrate how to provide such information to the decision maker. I aim to use the concept of lower bounds and upper bounds on objective function values of an efficient portfolio, developed in my earlier works. I illustrate the proposed approach on a large-scale data set based upon real data. I address cases where a top-class commercial mixed-integer quadratic programming solver fails to provide efficient portfolios attempted to be derived by Chebyshev scalarization of the bi-objective optimization problem within a given time limit. In this case, I propose to transform purely technical information provided by the solver into information which can be used in navigation over the efficient frontier of the cardinality-constrained Markowitz mean–variance portfolio investment problem.

Valid inequalities, preprocessing, and an effective heuristic for the uncapacitated three-level lot-sizing and replenishment problem with a distribution structure

We consider the uncapacitated three-level lot-sizing and replenishment problem with a distribution structure. In this problem, a single production plant sends the produced items to replenish warehouses from where they are dispatched to the retailers in order to satisfy their demands over a finite planning horizon. Transfers between warehouses or retailers are not permitted, each retailer has a single predefined warehouse from which it receives its items, and there is no restriction on the amount that can be produced or transported in a given period. The goal of the problem is to determine an integrated production and distribution plan minimizing the total costs, which comprehends fixed production and transportation setup as well as variable inventory holding costs. We describe new valid inequalities both in the space of a standard mixed integer programming (MIP) formulation and in that of a new alternative extended MIP formulation. We show that using such extended formulation, valid inequalities having similar structures to those in the standard one allow achieving tighter linear relaxation bounds. Furthermore, we propose a preprocessing approach to reduce the size of an extended multi-commodity MIP formulation available in the literature. Such preprocessing relies on the removal of variables based on the problem’s cost structure while preserving optimality guarantees. We also propose a multi-start randomized bottom-up dynamic programming-based heuristic. The heuristic employs greedy randomization via changes in certain costs and solves subproblems related to each level using dynamic programming. Computational experiments indicate that the use of the valid inequalities in a branch-and-cut approach significantly increase the ability of a MIP solver to solve instances to optimality. Additionally, the valid inequalities for the new alternative extended formulation outperform those for the standard one in terms of number of solved instances, running time and number of enumerated nodes. Moreover, the proposed heuristic is able to generate solutions with considerably low optimality gaps within very short computational times even for large instances. Combining the preprocessing approach with the heuristic, one can achieve an increase in the number of solutions solved to optimality within the time limit together with significant reductions on the average times for solving them.

Team Orienteering with Time-Varying Profit

This paper studies the team orienteering problem, where the arrival time and service time affect the collection of profits. Such interactions result in a nonconcave profit function. This problem integrates the aspect of time scheduling into the routing decision, which can be applied in humanitarian search and rescue operations where the survival rate declines rapidly. Rescue teams are needed to help trapped people in multiple affected sites, whereas the number of people who could be saved depends as well on how long a rescue team spends at each site. Efficient allocation and scheduling of rescue teams is critical to ensure a high survival rate. To solve the problem, we formulate a mixed-integer nonconcave programming model and propose a Benders branch-and-cut algorithm, along with valid inequalities for tightening the upper bound. To solve it more effectively, we introduce a hybrid heuristic that integrates a modified coordinate search (MCS) into an iterated local search. Computational results show that valid inequalities significantly reduce the optimality gap, and the proposed exact method is capable of solving instances where the mixed-integer nonlinear programming solver SCIP fails in finding an optimal solution. In addition, the proposed MCS algorithm is highly efficient compared with other benchmark approaches, whereas the hybrid heuristic is proven to be effective in finding high-quality solutions within short computing times. We also demonstrate the performance of the heuristic with the MCS using instances with up to 100 customers. Summary of Contribution: Motivated by search and rescue (SAR) operations, we consider a generalization of the well-known team orienteering problem (TOP) to incorporate a nonlinear time-varying profit function in conjunction with routing and scheduling decisions. This paper expands the envelope of operations research and computing in several ways. To address the scalability issue of this highly complex combinatorial problem in an exact manner, we propose a Benders branch-and-cut (BBC) algorithm, which allows us to efficiently deal with the nonconcave component. This BBC algorithm is computationally enhanced through valid inequalities used to strengthen the bounds of the BBC. In addition, we propose a highly efficient hybrid heuristic that integrates a modified coordinate search into an iterated local search. It can quickly produce high-quality solutions to this complex problem. The performance of our solution algorithms is demonstrated through a series of computational experiments.

Computational aspects of infeasibility analysis in mixed integer programming

The analysis of infeasible subproblems plays an important role in solving mixed integer programs (MIPs) and is implemented in most major MIP solvers. There are two fundamentally different concepts to generate valid global constraints from infeasible subproblems: conflict graph analysis and dual proof analysis. While conflict graph analysis detects sets of contradicting variable bounds in an implication graph, dual proof analysis derives valid linear constraints from the proof of the dual LP’s unboundedness. The main contribution of this paper is twofold. Firstly, we present three enhancements of dual proof analysis: presolving via variable cancellation, strengthening by applying mixed integer rounding functions, and a filtering mechanism. Further, we provide a comprehensive computational study evaluating the impact of every presented component regarding dual proof analysis. Secondly, this paper presents the first combined approach that uses both conflict graph and dual proof analysis simultaneously within a single MIP solution process. All experiments are carried out on general MIP instances from the standard public test set Miplib 2017; the presented algorithms have been implemented within the non-commercial MIP solver SCIP and the commercial MIP solver FICO Xpress.

Enhancing large neighbourhood search heuristics for Benders\u2019 decomposition

A general enhancement of the Benders’ decomposition (BD) algorithm can be achieved through the improved use of large neighbourhood search heuristics within mixed-integer programming solvers. While mixed-integer programming solvers are endowed with an array of large neighbourhood search heuristics, few, if any, have been designed for BD. Further, typically the use of large neighbourhood search heuristics is limited to finding solutions to the BD master problem. Given the lack of general frameworks for BD, only ad hoc approaches have been developed to enhance the ability of BD to find high quality primal feasible solutions through the use of large neighbourhood search heuristics. The general BD framework of SCIP has been extended with a trust region based heuristic and a general enhancement for large neighbourhood search heuristics. The general enhancement employs BD to solve the auxiliary problems of all large neighbourhood search heuristics to improve the quality of the identified solutions. The computational results demonstrate that the trust region heuristic and a general large neighbourhood search enhancement technique accelerate the improvement in the primal bound when applying BD.

A heuristic for a special case of the generalized assignment problem with additional conditions

We consider the following variant of the generalized assignment problem (GAP). There are a set of agents and a set of jobs for which a single resource use. Each job is to assign to one and only one agent subject to the constraints on the capacity and loading of agents. The resource expense for executing any job is independent of the agents choice unlike the profit from the job. Each job has a certain type (or color). For every agent, the maximum possible number of the job types given. It is necessary to find a feasible assignment of agents to jobs so that all jobs were completed and total profit was maximized. Finding a feasible solution to this problem is NP-hard. We present a heuristic algorithm based on the ideas of random search and local improvement of solutions. Used the mixed-integer programming (MIP) relaxation and variables fixing, we construct a set of integer linear programming (ILP) subproblems similar to the original problem and solve them by the general MIP solver. The results of a computational experiment for tasks with random initial data are presented.

Weighted proximity search

Proximity search is an iterative method to solve complex mathematical programming problems. At each iteration, the objective function of the problem at hand is replaced by the Hamming distance function to a given solution, and a cutoff constraint is added to impose that any new obtained solution improves the objective function value. A mixed integer programming solver is used to find a feasible solution to this modified problem, yielding an improved solution to the original problem. This paper introduces the concept of weighted Hamming distance that allows to design a new method called weighted proximity search. In this new distance function, low weights are associated with the variables whose value in the current solution is promising to change in order to find an improved solution, while high weights are assigned to variables that are expected to remain unchanged. The weights help to distinguish between alternative solutions in the neighborhood of the current solution, and provide guidance to the solver when trying to locate an improved solution. Several strategies to determine weights are presented, including both static and dynamic strategies. The proposed weighted proximity search is compared with the classic proximity search on instances from three optimization problems: the p-median problem, the set covering problem, and the stochastic lot-sizing problem. The obtained results show that a suitable choice of weights allows the weighted proximity search to obtain better solutions, for 75% of the cases, than the ones obtained by using proximity search and for 96% of the cases the solutions are better than the ones obtained by running a commercial solver with a time limit.

Solving the maximum edge disjoint path problem using a modified Lagrangian particle swarm optimisation hybrid

This paper presents a new method to solve the Maximum Edge Disjoint Paths (MEDP) problem. Given a set of node pairs within a network, the MEDP problem is the task of finding the largest number of pairs that can be connected by paths, using each edge within the network at most once. We present a heuristic algorithm that builds a hybridisation of Lagrangian Relaxation and Particle Swarm Optimisation, referred to as LaPSO. This hybridisation is combined with a new repair heuristic, called Largest Violation Perturbation (LVP). We show that our LaPSO method produces better heuristic solutions than both current state-of-the-art heuristic methods, as well as the primal solution found by a standard Mixed Integer Programming (MIP) solver within a limited runtime. Significantly, when run with a limited runtime, our LaPSO method also produces strong bounds which are superior to a standard MIP solver for the larger instances tested, whilst being competitive for the remainder. This allows our LaPSO method to prove optimality for many instances and provide optimality gaps for the remainder, making it a “quasi-exact” method. In this way our LaPSO algorithm, which draws on ideas from both mathematical programming and evolutionary algorithms, is able to outperform both MIP and metaheuristic solvers that only use ideas from one of these areas.

Collision Avoidance Modelling in Airline Traffic Based on the Change of Airplane Movements and Dynamic Clouds

An Air Traffic Controller (ATC) system aims to manage airline traffic to prevent collision of the airplane, called the Collision Avoidance (CA). The study on CA, called Conflict Detection and Resolution (CDR), becomes more critical as the airline traffic has grown each year significantly. Previous studies used optimization algorithms for CDR and did not involve the presence of cumulonimbus clouds. Many such clouds can be found in tropical regions like in Indonesia. Therefore, involving such clouds in the CDR optimization algorithms will be significant in Indonesia. We developed a CDR-based CA modelling that involves the Cumulonimbus (CB) clouds by considering three airplane maneuvers, i.e., Velocity, angle Turn and Altitude level Change (VTAC). Our optimization algorithm is developed based on a Mixed-Integer Programming (MIP) solver due to its efficiency. This proposed algorithm requires two input data, namely the initial airplane and cloud states input and the flight parameter such as velocity, angle and altitude levels. The outputs of our VTAC optimization algorithm are the optimum speed, altitude and angle turn of an airplane that is determined based on the currently calculated variables. Extensive experiments have been conducted to validate the proposed approach and the experiment results show that collisions between airplanes and clouds can be avoided with minimum change of the initial airplane velocity, angle and altitude levels. The VTAC algorithm produced longer distance to avoid collision between airplanes by at least 1 Nautical Mile (NM) compared to the VAC algorithm. The addition of angle in the VTAC algorithm has improved the result significantly.

Approximate Collaborative Fleet Routing with a Pointer Generation Neural Network Approach

With rapid emergence of high-performance computing platforms, the availability of client big data and new machine learning techniques, the application domain of platform-based mobility services supports the research of new optimization techniques for discrete combinatorial optimization problems. Within this research field, particularly large-scale transportation domain specific problems, e.g. multi-vehicle-request matching and collaborative vehicle fleet routing problems are of high interest. In this contribution we present our novel combinatorial Deep Reinforcement Learning for solving symmetric and asymmetric multi-vehicle-request weighted assignment or matching problems by learning and predicting an efficient solution heuristic automatically. The solved assignment problem is characterized by defining different node classes for vehicles, requests and service stations. Our results contain algorithm benchmarks based on reproducible artificial data and statistical evaluations towards solution accuracy and efficiency with respect to different problem complexities. We contribute additional comparisons between different algorithms and heuristics such as naive Greedy, k-Regret and exact (globally optimal) solutions with the simplex method by the Mixed-Integer Programming (MIP) solver Cplex. Further, we compare the results with our model with respect to solution accuracy and efficiency We conclude that our proposed model solves the presented problem setting globally optimal up to an upper graph complexity bound defined via node degree. Furthermore, our results show that our proposed method outperforms all other algorithms with respect to the solution time required.

Data-Driven Optimization of Reward-Risk Ratio Measures

We investigate a class of fractional distributionally robust optimization problems with uncertain probabilities. They consist in the maximization of ambiguous fractional functions representing reward-risk ratios and have a semi-infinite programming epigraphic formulation. We derive a new fully parameterized closed-form to compute a new bound on the size of the Wasserstein ambiguity ball. We design a data-driven reformulation and solution framework. The reformulation phase involves the derivation of the support function of the ambiguity set and the concave conjugate of the ratio function. We design modular bisection algorithms which enjoy the finite convergence property. This class of problems has wide applicability in finance, and we specify new ambiguous portfolio optimization models for the Sharpe and Omega ratios. The computational study shows the applicability and scalability of the framework to solve quickly large, industry-relevant-size problems, which cannot be solved in one day with state-of-the-art mixed-integer nonlinear programming (MINLP) solvers.

Dual Probability Learning Based Local Search for the Task Assignment Problem

The task assignment problem (TAP) is concerned with assigning a set of tasks to a set of agents subject to the limited processing and memory capacities of each agent. The objective to be minimized is the total assignment cost and total communication cost. TAP is a relevant model for many practical applications, yet solving the problem is computationally challenging. Most of the current metaheuristic algorithms for TAP adopt population-based search frameworks, whose search behaviors are usually difficult to analyze and understand due to their complex features. In this work, unlike previous population-based solution methods, we concentrate on a single trajectory stochastic local search model to solve TAP. Especially, we consider TAP from the perspective of a grouping problem and introduce the first probability learning-based local search algorithm for the problem. The proposed algorithm relies on a dual probability learning procedure to discover promising search regions and a gain-based neighborhood search procedure to intensively exploit a given search region. We perform extensive computational experiments on a set of 180 benchmark instances with the proposed algorithm and the general mixed integer programming solver CPLEX. We assess the composing ingredients of the proposed algorithm to shed light on their impacts on the performance of the algorithm. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —This work is motivated by the problem of program modules designing and task allocation in parallel and distribution systems. It can also be applied to deal with job (task) grouping problems in practical industrial applications. This article presents a novel and effective learning-based local search algorithm to obtain high-quality solutions for the considered problem. The results of numerical experiments and comparisons show that our algorithm can achieve good search performances on problem instances of different scales and difficulties. Afterward, we use the proposed solution method to solve a real-life open-order slab assignment problem, which is derived from the production planning of silicon steel for an iron and steel company. The learning techniques of the proposed algorithm are of general interest and can be used in search algorithms for solving other real-life optimization problems with grouping features. For future research, we will design solution methods based on these learning techniques to address other practical optimization problems.