Standard Mixed-integer Programming Solver Research Articles

GAMMA: Graph Attention Model for Multiple Agents to Solve Team Orienteering Problem With Multiple Depots.

In this work, we present an attention-based encoder-decoder model to approximately solve the team orienteering problem with multiple depots (TOPMD). The TOPMD instance is an NP-hard combinatorial optimization problem that involves multiple agents (or autonomous vehicles) and not purely Euclidean (straight line distance) graph edge weights. In addition, to avoid tedious computations on dataset creation, we provide an approach to generate synthetic data on the fly for effectively training the model. Furthermore, to evaluate our proposed model, we conduct two experimental studies on the multi-agent reconnaissance mission planning problem formulated as TOPMD. First, we characterize the model based on the training configurations to understand the scalability of the proposed approach to unseen configurations. Second, we evaluate the solution quality of the model against several baselines-heuristics, competing machine learning (ML), and exact approaches, on several reconnaissance scenarios. The experimental results indicate that training the model with a maximum number of agents, a moderate number of targets (or nodes to visit), and moderate travel length, performs well across a variety of conditions. Furthermore, the results also reveal that the proposed approach offers a more tractable and higher quality (or competitive) solution in comparison with existing attention-based models, stochastic heuristic approach, and standard mixed-integer programming solver under the given experimental conditions. Finally, the different experimental evaluations reveal that the proposed data generation approach for training the model is highly effective.

New integer optimization models and an approximate dynamic programming algorithm for the lot-sizing and scheduling problem with sequence-dependent setups

In this paper, we propose new integer optimization models for the lot-sizing and scheduling problem with sequence-dependent setups, based on the general lot-sizing and scheduling problem. To incorporate setup crossover and carryover, we first propose a standard model that straightforwardly adapts a formulation technique from the literature. Then, as the main contribution, we propose a novel optimization model that incorporates the notion of time flow. We derive a family of valid inequalities with which to compare the tightness of the models’ linear programming relaxations. In addition, we provide an approximate dynamic programming algorithm that estimates the value of a state using its lower and upper bounds. Then, we conduct computational experiments to demonstrate the competitiveness of the proposed models and the solution algorithm. The test results show that the newly proposed time-flow model has considerable advantages compared with the standard model in terms of tightness and solvability. The proposed algorithm also shows computational benefits over the standard mixed integer programming solver.

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.

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.

Pilotage planning in seaports

Vessel pilotage is compulsory in most seaports around the world. When traveling into or out of the terminal of a seaport, vessels should be navigated by sea pilots in order to follow correct and safe waterways. This paper studies a pilotage planning problem that involves decisions of scheduling the vessel traffic in a seaport, assigning work shifts to pilots, and scheduling the pilots in each work shift for vessel navigation. We formulate the problem as a linear mixed-integer programming (MIP) model that aims at minimizing the cost incurred by the pilotage operations, and show that the problem is strongly NP-hard. For solving the problem, we develop a branch-and-price (B&P) algorithm in which the pricing problem is solved by a novel dynamic programming algorithm. We further propose several acceleration techniques to improve the efficiency of the B&P algorithm. The computational performance of the B&P algorithm is evaluated in extensive numerical experiments. Computational results demonstrate that the B&P algorithm is able to solve problem instances of practical sizes, and that the algorithm outperforms a standard MIP solver and a solution method commonly used in practice.

An application of the multi-depot heterogeneous fixed fleet open vehicle routing problem

This work describes an application of a multi-depot heterogeneous fixed fleet open vehicle routing problem. A contractor owns a fleet of vehicles with different capacities and running costs. The fleet is used to transport craftsmen from their homes to assigned project sites and back, with some of the craftsmen appointed as drivers while others are passengers. An optimisation model is described that enables the contractor to minimise the transportation costs, and a computational study shows that the model can be solved to optimality for realistically sized instances using a standard mixed-integer programming solver. A variant of the problem is also considered, where the assignment of craftsmen to projects is not fixed a priori. For this variant, several simple heuristic rules are investigated to generate project assignments, and computational results show that they are able to find improved assignments.

Reliable single allocation hub location problem under hub breakdowns

The design of hub-and-spoke transport networks is a strategic planning problem, as the choice of hub locations has to remain unchanged for long time periods. However, strikes, disasters or traffic breakdown can lead to the unavailability of a hub for a short period of time. Therefore it is important to consider such events already in the planning phase, so that a proper reaction is possible; once a hub breaks down, an emergency plan has to be applied to handle the flows that were scheduled to be served by this hub. In this paper, we develop a two-stage formulation for the single allocation hub location problem which includes the reallocation of sources to a backup hub in case the hub breaks down. In contrast to related problem formulations from the literature, we keep the non-linear structure of the problem in our model. A branch-and-cut framework based on Benders decomposition is designed to solve large scale instances to proven optimality. Thanks to our decomposition strategy, we keep the structure of the resulting formulation similar to the classical single allocation hub location problem, which in turn allows to use classical linearization techniques from the literature. Our computational experiments show that this approach leads to a significant improvement in the performance when embedded into a standard mixed-integer programming solver. We report optimal solutions for instances much bigger than those solved so far in the literature.

Shortest Path Based Decision Making Using Probabilistic Inference

We present a new perspective on the classical shortest path routing (SPR) problem in graphs. We show that the SPR problem can be recast to that of probabilistic inference in a mixture of simple Bayesian networks. Maximizing the likelihood in this mixture becomes equivalent to solving the SPR problem. We develop the well known Expectation-Maximization (EM) algorithm for the SPR problem that maximizes the likelihood, and show that it does not get stuck in a locally optimal solution. Using the same probabilistic framework, we then address an NP-Hard network design problem where the goal is to repair a network of roads post some disaster within a fixed budget such that the connectivity between a set of nodes is optimized. We show that our likelihood maximization approach using the EM algorithm scales well for this problem taking the form of message-passing among nodes of the graph, and provides significantly better quality solutions than a standard mixed-integer programming solver.

A computational comparison of formulations for the economic lot-sizing with remanufacturing

An important way to try reducing environmental damage in the manufacture of industrialized goods is through the use of production systems which deal with the reuse of returned materials such as reverse logistics. In this paper, we consider a production planning problem arising in the context of reverse logistics, namely the economic lot-sizing with remanufacturing (ELSR). In the ELSR, deterministic demand for a single item over a finite time horizon has to be satisfied, which can be performed from either newly produced or remanufactured items, and the goal consists in minimizing the total production costs. Our objective is to devise approaches to solve larger (more difficult) instances of the problem available in the literature to optimality using a standard mixed-integer programming (MIP) solver. We present a multicommodity extended formulation and a strengthened Wagner–Whitin based formulation, which makes use of a priori addition of newly described valid inequalities in the space of original variables. We also propose a novel dynamic heuristic measure based on the cost structure to automatically determine the size of a partial version of the Wagner–Whitin based formulation. Computational results show that the novel partial Wagner–Whitin based formulation with the size automatically determined in a heuristic way outperforms all the other tested approaches, including a best performing shortest path formulation available in the literature, when we consider the number of instances solved to proven optimality using a standard MIP solver. This new approach allowed to solve to optimality more than 96% of the tested instances for the ELSR with separate setups, including several instances that could not be solved otherwise.

An affine adjustable robust model for generation and transmission network planning

This work studies an electricity generation and transmission network planning problem where loads and cost parameters are uncertain. The problem is to first determine the generation and transmission capacities to install in the supply network. When the uncertainties are revealed, a flow plan is developed to minimize the total costs and to balance loads. A two-stage mixed-integer programming model is proposed to maximize the robustness of the plan in achieving a total cost budget target. The modeling approach in this study synthesizes recent developments in affine adjustable robust optimization technology and decision-making behavior under uncertainty. A novel solution approach is also proposed to achieve a safe and tractable approximation of the model. This involves the partitioning of the total cost budget target in order to transform the original problem into a small collection of mixed integer programming models that can be solved efficiently using standard mixed integer programming solvers. Numerical studies using a power generation network are performed, which demonstrate that the proposed robust planning model performs favorably compared to a stochastic programming model across different performance measures. The computational results strongly suggest the ability of the robust planning model to effectively mitigate the effect of uncertainties.

Computational Testing of a Separation Procedure for the Knapsack Set with a Single Continuous Variable

We study an exact separation procedure—SEP-MK—for the knapsack set with a single continuous variable XMK. Then, we address the question of whether SEP-MK can be of practical use in tightening mixed-integer programming (MIP) formulations when using standard (floating-point) MIP solvers. To this purpose, we present a separation procedure for MIP problems—SEP-MIPMK—where we derive knapsack sets of the form XMK by aggregating the continuous variables in the mixed knapsack inequalities of the formulation. Then, we use SEP-MK to generate cutting planes. Before the continuous variables are aggregated, the mixed knapsack inequalities are modified through the use of a bound substitution procedure to take into account fixed and variable bounds on the continuous variables. Bound substitution is made according to some heuristic rules, so even if its basic component SEP-MK is “exact,” the overall separation procedure for MIP problems, SEP-MIPMK, is heuristic. We perform a computational study on a wide set of mixed-integer programming instances from the MIPLIB 2003 [Achterberg, T., T. Koch, A. Martin. 2006. Mixed Integer Problem Library (MIPLIB) 2003. Konrad-Zuse-Zentrum für Informationstechnik Berlin, Berlin. http://miplib.zib.de] and Mittelmann [Mittelmann, H. 2010. MILP testcases. http://plato.asu.edu/ftp/milp] benchmark sets. Computational experiments confirm that lifted cover and mixed-integer rounding (MIR) inequalities are effective from a computational viewpoint. Nevertheless, there are several instances where SEP-MIPMK is able to significantly raise the lower bounds given by lifted cover and MIR inequalities.

Fix-and-optimize heuristics for capacitated lot-sizing with sequence-dependent setups and substitutions

In this paper, we consider a capacitated single-level dynamic lot-sizing problem with sequence-dependent setup costs and times that includes product substitution options. The model is motivated from a real-world production planning problem of a manufacturer of plastic sheets used as an interlayer in car windshields. We develop a mixed-integer programming (MIP) formulation of the problem and devise MIP-based Relax&Fix and Fix&Optimize heuristics. Unlike existing literature, we combine Fix&Optimize with a time decomposition. Also, we develop a specialized substitute decomposition and devise a computation budget allocation scheme for ensuring a uniform, efficient usage of computation time by decompositions and their subproblems. Computational experiments were performed on generated instances whose structure follows that of the considered practical application and which have rather tight production capacities. We found that a Fix&Optimize algorithm with an overlapping time decomposition yielded the best solutions. It outperformed the state-of-the-art approach Relax&Fix and all other tested algorithm variants on the considered class of instances, and returned feasible solutions with neither overtime nor backlogging for all instances. It returned solutions that were on average only 5% worse than those returned by a standard MIP solver after 4 hours and 19% better than those of Relax&Fix.

A MIP‐based local search method for the railway rescheduling problem

AbstractFor operational and unpredictable reasons, many small incidents occur day after day in rail transportation systems. Most of them have a local impact, but, in some cases, mainly in dense networks, minimal disruptions can spread out through the whole network and affect significantly the train schedules. In this article, we present the railway rescheduling problem as the problem of finding a new schedule of trains after one or several incidents by minimizing some measure of the effect. We investigate the solution of this problem through a mixed‐integer programming (MIP) formulation. Because of the impossibility for solving it exactly just using a standard MIP solver, we propose to limit the search space around the original nondisrupted schedule by hard and soft fixing of integer variables with local‐branching‐type cuts. Different variations of the method are compared to a right‐shift rescheduling policy in two different networks located in France and Chile. The experimental results are also used to study the impact of different objectives on the total delay. © 2010 Wiley Periodicals, Inc. NETWORKS, 2011

A novel modeling approach for the fleet deployment problem within a short-term planning horizon

This paper is concerned with model development for a short-term fleet deployment problem of liner shipping operations. We first present a mixed integer nonlinear programming model in which the optimal vessel speeds for different vessel types on different routes are interpreted as their realistic optimal travel times. We then linearize the proposed nonlinear model and obtain a mixed integer linear programming (MILP) model that can be efficiently solved by a standard mixed integer programming solver such as CPLEX. The MILP model determines the optimal route service frequency pattern and take into account the time window constraints of shipping services. Finally, we report our numerical results and performance of CPLEX on randomly generated instances.

On the Selection of the Globally Optimal Prototype Subset for Nearest-Neighbor Classification

The nearest-neighbor classifier has been shown to be a powerful tool for multiclass classification. We explore both theoretical properties and empirical behavior of a variant method, in which the nearest-neighbor rule is applied to a reduced set of prototypes. This set is selected a priori by fixing its cardinality and minimizing the empirical misclassification cost. In this way we alleviate the two serious drawbacks of the nearest-neighbor method: high storage requirements and time-consuming queries. Finding this reduced set is shown to be NP-hard. We provide mixed integer programming (MIP) formulations, which are theoretically compared and solved by a standard MIP solver for small problem instances. We show that the classifiers derived from these formulations are comparable to benchmark procedures. We solve large problem instances by a metaheuristic that yields good classification rules in reasonable time. Additional experiments indicate that prototype-based nearest-neighbor classifiers remain quite stable in the presence of missing values.

A general quantity discount and supplier selection mixed integer programming model

Recently, new models and heuristics for exploiting quantity discounts have been proposed that are applicable in classical purchasing as well as in an e-business environment and can be implemented as part of an advanced planning system. These models can now handle both the single- and multi-item case with fixed cost to be shared among several products ordered at the same point in time from a single supplier. Furthermore, the supplier selection problem is addressed, i.e., how to best exploit quantity discounts over time offered by several suppliers. Last but not least, additional constraints on the buyer’s or on the supplier’s side may be included. While so far only purpose-built heuristics have been proposed for this generalized problem, we present a linear mixed integer programming (MIP) model, which not only represents the all-units discount but also the incremental discount case. Furthermore, the objective function chosen resolves (former) conflicts among proponents of a purely cost oriented and a cash flow oriented modeling approach. Computational tests show that our model yields near optimal solutions within a given CPU time limit by making use of a standard MIP solver.

A mixed integer programming formulation for the 1-maximin problem

In this paper, I present a mixed integer programming (MIP) formulation for the 1-maximin problem with rectilinear distance. The problem mainly appears in facility location while trying to locate an undesirable facility. The rectilinear distance is quite commonly used in the location literature. Our numerical experiments show that one can solve reasonably large location problems using a standard MIP solver. We also provide a linear programming formulation that helps find an upper bound on the objective function value of the 1-maximin problem with any norm when extreme points of the feasible region are known. We discuss various extension alternatives for the MIP formulation.

Optimization-based planning heuristic for material flow congestion avoidance in conveyor network design

A model is developed to plan material flow congestion avoidance on a network of conveyors for handling discrete batches of items, e.g. scanned packages. Minimizing maximum material flow congestion is used as a guide to design the conveyor network by evenly distributing the flow of such batches by considering capacities of the conveyors and existing flow. Problems of this type are typically NP-hard which makes them unsolvable by standard mixed integer programming solvers. A method based on results from probability theory is used to solve the formulated problem. Experimental results are presented to demonstrate convergence within 10-100 iterations of the described method for most problems. The increase in CPU time is observed to be approximately linear with respect to the number of constraints and with respect to the number of binary variablesfor a constant number ofiterationsofthe algorithm. A procedure for estimation of bounds is derived to show the proximity of the obtained solution to an optimal solution.