Compact Mixed-integer Linear Programming Research Articles

An optimization-based approach for an integrated forest fire monitoring system with multiple technologies and surveillance drones

Wildfires pose a significant threat to forests and the delicate natural balance. In this work, we present a novel optimization approach for designing forest fire monitoring systems that incorporate surveillance towers, monitoring balloons, and drone technology, which have gained popularity in recent years for monitoring, logistics, and civil applications. We develop a compact mixed-integer linear programming model that includes location decisions for multiple monitoring technologies and routing of drones among monitoring facilities to achieve the most extensive possible terrain coverage. Additionally, we propose a matheuristic algorithm that comprises six components: a solution procedure, perturbation procedures, local search procedures, a call to the general-purpose solver to solve a mixed-integer linear programming model, a global reset strategy, a local reset strategy, and an acceptance criterion. We test the proposed model and algorithm on a set of random instances and a real-life case study in Chile. While the proposed model can only solve small instances, the matheuristic can find good-quality solutions for all instances. This study can aid the government and private sector in designing an integrated fire monitoring system that leverages watchtowers, monitoring balloons, and drones.

Exact and heuristic solution approaches for energy-efficient identical parallel machine scheduling with time-of-use costs

Nowadays, energy-efficient scheduling has assumed a key role in ensuring the sustainability of manufacturing processes. In this context, we focus on the bi-objective problem of scheduling a set of jobs on identical parallel machines to simultaneously minimize the maximum completion time and the total energy consumption over a time horizon partitioned into a set of discrete slots. The energy costs are determined by a time-of-use pricing scheme, which plays a crucial role in regulating energy demand and flattening its peaks. First, we uncover a symmetry-breaking property that characterizes the structure of the solution space of the problem. As a consequence, we provide a novel, compact mixed-integer linear programming formulation at the core of an efficient exact solution algorithm. A thorough experimental campaign shows that the use of the novel mathematical programming formulation enables the solution of larger-scale instances and entails a reduction in the computational times as compared to the formulation already available in the literature. Furthermore, we propose a new heuristic that improves the state-of-the-art in terms of required computational effort and quality of solutions. Such a heuristic outperforms the existing heuristics for the problem and is also capable of speeding up the exact solution algorithm when used for its initialization. Finally, we introduce a novel dynamic programming algorithm that is able to compute the optimal timing of the jobs scheduled on each machine to further improve the performance of the new heuristic.

Optimization for integrated scheduling of intelligent handling equipment with bidirectional flows and limited buffers at automated container terminals

Although a large amount of intelligent handling equipment has been widely used in automated container terminals, few studies have explored the coordination optimization among them, which is critical to the overall operational efficiency of terminals. This paper investigates the integrated scheduling of intelligent handling equipment at automated container terminals, with challenging characteristics such as bidirectional flows and limited buffers. Besides, other characteristics such as blocking, machine eligibility, and separable sequence-dependent setup times also complicate the problem, making it difficult to find an optimal or near-optimal solution. This study formulates the problem as the blocking hybrid flow shop scheduling problem with bidirectional flows and limited buffers, and then develops a compact mixed-integer linear programming model that can obtain the optimal scheme on small-scale cases. To increase the size of solvable problems and improve the performance of the solution, an adaptive large neighborhood search is presented, which takes a high-quality coding scheme that specifies the equipment used by each job in various stages. Experiments have been conducted and computational results demonstrate the superiority of the proposed algorithm in terms of solution quality and convergence speed.

Exact methods for the traveling salesman problem with multiple drones

Drone delivery is drawing increasing attention in last-mile delivery. Effective solution methods to solve decision-making problems arising in drone delivery allow to run and assess drone delivery systems. In this paper, we focus on delivery systems with a single traditional vehicle and multiple drones working in tandem to fulfill customer requests. We address the Traveling Salesman Problem with Multiple Drones (TSP-MD) and investigate the modeling challenges posed by the presence of multiple drones, which have proven to be hard to handle in the literature. We propose a compact Mixed-Integer Linear Programming (MILP) model to formulate the TSP-MD and several families of valid inequalities. Moreover, we illustrate an exact decomposition approach based on the compact MILP and a branch-and-cut algorithm. We show that this exact approach can solve instances with up to 24 customers to proven optimality, improving upon existing exact methods that can solve similar problems with up to ten customers only.

Designing corridor systems with modular autonomous vehicles enabling station-wise docking: Discrete modeling method

Jointly designing vehicle dispatch headway and capacity is a relatively new solution to the demand–supply asymmetry in urban mass transportation studies. This paper studies a new problem of this kind where a transportation corridor operates with modular autonomous vehicles (MAV) enabling station-wise docking; i.e., vehicles can change their formations (or capacity) at any station along the corridor. We formulate the problem into a compact mixed integer linear programming model where the passenger boarding order is explicitly modeled. Due to the multiple station system structure and the station-wise docking operation, the solution space of the model increases rapidly with the instance size, making it very challenging to solve the model with existing commercial solvers. To improve the solution efficiency, we design a customized branch and bound (B&B) algorithm with theoretical properties of the investigated problem. These properties offer upper and lower bounds to the optimal vehicle formation, reveal the relationship between the passenger queue and vehicle dispatch headway, and identify a dominance rule between any two feasible solutions to the investigated problem. They greatly reduce the number of nodes in the B&B tree that would grow dramatically without these properties. Further, the lower and upper bounds to the objective value at each node of the B&B tree are computed analytically, allowing us to search through the solution space very quickly. Consequently, the computation speed of the B&B algorithm is greatly improved. With numerical experiments, we show that the customized B&B algorithm outperforms a state-of-the-art commercial solver, Gurobi, and solves relatively large instances in real-world applications efficiently. The station-wise docking operation is shown to reduce system costs compared with existing fixed capacity operation. Further, its performance is affected by system parameters related to the vehicle operational cost and passenger waiting cost. Overall, this study contributes to the literature by extending the urban mass transportation design methodology from traditional fixed capacity design to the MAV-based station-wise docking design under various operational factors (e.g., minimum dispatch headway). The algorithm proposed can be used as a benchmark to verify the solution accuracy and computation performance for research efforts that aim to develop other solution algorithms for the investigated problem.

Routing Optimisation of Urban Medical Waste Recycling Network considering Differentiated Collection Strategy and Time Windows

The increasing gap between medical waste production and disposal stresses the urgency of further development of urban medical waste recycling. This paper investigates an integrated optimisation problem in urban medical waste recycling network. It combines the vehicle routing problem of medical facilities with different requirements and the collection problem of clinics’ medical waste to the affiliated hospital. To solve this problem, a compact mixed-integer linear programming model is proposed, which takes account of the differentiated collection strategy and time windows. Since the medical waste recycling operates according to a two-day pattern, the periodic collection plan is also embedded in the model. Moreover, we develop a particle swarm optimisation (PSO) solution approach for problem-solving. Numerical experiments are also conducted to access the solution efficiency of the proposed algorithm, which can obtain a good solution in solving large-scale problem instances within a reasonable computation time. Based on the results, some managerial implications can be recommended for the third-party recycling company.

Benders decomposition for a node-capacitated Virtual Network Function placement and routing problem

In this paper we study a problem faced by network service providers in which a set of Virtual Network Functions (VNFs) has to be installed in a telecommunication network at minimum cost. For each given origin-destination pair of nodes (commodities), a latency-constrained routing path has to be found that visits the required VNFs in a pre-defined order. A limited number of functions can be installed at each node. We first prove that the problem is NP-hard in a strong sense, even for a single commodity and without node-capacity, latency and precedence constraints. We then provide a compact Mixed Integer Linear Programming (MILP) formulation, along with several families of valid inequalities. To tackle the problem from a computational perspective, we provide theoretical results that allow us to derive Benders reformulation of the problem. We also exploit an alternative path-based MILP formulation to derive heuristic solutions. All these ingredients are combined in a Branch-and-Benders-Cut framework and computationally tested on a wide range of realistic instances. Our results are also compared with the Automatic Benders decomposition provided by Cplex. Computational results indicate that our decomposition approach is more efficient compared to the two methods provided by the off-the-shelf solver, both in terms of the CPU time and the overall solution quality. The results also indicate that our MILP-heuristic provides high-quality solutions.

Optimization model for remanufacturing in a real sawmill

Sawmills are an important part of the forest supply chain, and as at any company, their production planning is highly complex. Planning in the remanufacturing area, in terms of its economic contribution to the sawmill and the supply chain, has not been studied in the scientific literature. The goal of this study was to develop and solve a mixed-integer linear programming model by employing an efficient allocation of cutting patterns on in-stock logs to maximize profits. To quantify the impact of an appropriate use of raw materials in the remanufacturing area in a sawmill, real and generated data were used. The model considers fixed and variable production costs, the availability of raw material, the capacity of the processes, the sale price of the products and the demand, for a process period of one month. The proposed compact mixed-integer linear programming model was solved using the commercial solver IBM ILGO CPLEX12.8. It was determined that the additional margin in USD earned in the remanufacturing area for the considered scenarios amounted to an average of 21.6%. The proposed method facilitates evaluating the economic contribution of remanufacturing while identifying bottlenecks and assessing proposed scenarios.

The Robust Bulk Ship Routing Problem with Batched Cargo Selection

Maritime transportation forms the backbone of the world merchandise trade. In this paper, we consider a problem that combines three interconnected subproblems in tramp shipping: the fleet adjustment problem, the cargo selection problem, and the ship routing problem. For cargo selection, we consider the decision behaviors under the setting of Contract of Affreightment (COA), in which cargoes should be rejected or accepted as a batch. In view of the uncertainties observed in maritime transportation, we formulate the problem in a robust way so that the solutions can protect the profitability of shipping companies against variations in voyage costs. We first provide compact mixed integer linear programming formulations for the problem and then convert them into a strengthened set covering model. A tailored branch-and-price-and-cut algorithm is developed to solve the set covering model. The algorithm is enhanced by a multi-cut generation technique aimed at tightening the lower bounds and a primal heuristic aimed at finding high-quality upper bounds. Extensive computational results show that our algorithm yields optimal or near-optimal solutions to realistic instances within short computing times and that the enhancement techniques significantly improve the efficiency of the algorithm.

A branch‐and‐cut approach and alternative formulations for the traveling salesman problem with drone

AbstractIn this paper, we are interested in studying the traveling salesman problem with drone (TSP‐D). Given a set of customers and a truck that is equipped with a single drone, the TSP‐D asks that all customers are served exactly once and minimal delivery time is achieved. We provide two compact mixed integer linear programming formulations that can be used to address instances with up to 10 customer within a few seconds. Notably, we introduce a third formulation for the TSP‐D with an exponential number of constraints. The latter formulation is suitable to be solved by a branch‐and‐cut algorithm. Indeed, this approach can be used to find optimal solutions for several instances with up to 20 customers within 1 hour, thus challenging the current state‐of‐the‐art in solving the TSP‐D. A detailed numerical study provides an in‐depth comparison on the effectiveness of the proposed formulations. Moreover, we reveal further details on the operational characteristics of a drone‐assisted delivery system. By using three different sets of benchmark instances, consideration is given to various assumptions that affect, for example, technological drone parameters and the impact of distance metrics.

An enhanced lower bound for the Time-Dependent Travelling Salesman Problem

Given a graph whose arc traversal times vary over time, the Time-Dependent Travelling Salesman Problem amounts to find a Hamiltonian tour of least total duration. In this paper we exploit a new degree of freedom in the Cordeau et al. (2014) speed decomposition. This approach results in a parameterized family of lower bounds. The parameters are chosen by fitting the traffic data. The first model is nonlinear and difficult to solve. Hence, we devise a linearization which gives rise to a compact Mixed Integer Linear Programming model. Then, we develop an optimality condition which allows to further reduce the size of the model. Computational results show that, when embedded into a branch-and-bound procedure, this lower bounding mechanism allows to solve to optimality a larger number of instances than state-of-the-art algorithms.

A computational comparison of compact MILP formulations for the zero forcing number

Consider a graph where some of its vertices are colored. A colored vertex with a single uncolored neighbor forces that neighbor to become colored. A zero forcing set is a set of colored vertices that forces all vertices to become colored. The zero forcing number is the size of a minimum forcing set. Finding the minimum forcing set of a graph is NP-hard. We give a new compact mixed integer linear programming formulation (MILP) for this problem, and analyze this formulation and establish relation to an existing compact formulation and to two variants. In order to solve large size instances we propose a sequential search algorithm which can also be used as a heuristic to derive upper bounds for the zero forcing number. A computational study using Xpress (a MILP solver) is conducted to test the performances of the discussed compact formulations and the sequential search algorithm. We report results on cubic, Watts–Strogatz and randomly generated graphs with 10, 20 and 30 vertices.

A Kernel Search for a Patient Satisfaction-oriented Nurse Routing Problem with Time-Windows

We study a variant of the Nurse Routing Problem where each patient may require more than one service at possible different times during the day. Each executed service yields a profit, and an additional reward is gained if all services associated with a patient are fulfilled. The problem looks for nurse routes, each one not exceeding a predefined working time limit, that maximize the global collected service profits plus the patient rewards, while respecting the time windows associated with services. We first provide a compact Mixed-Integer Linear Programming formulation for the problem. Then, we develop an Iterative Kernel Search to solve the problem heuristically. Finally, we compare the heuristic performance on several instances with that of the plain model solved through a state-of-the-art exact solver and strengthened by the separation of valid inequalities. The obtained results clearly show that our heuristic algorithm finds fairly good solutions in terms of quality, despite the use of shorter computational times.

Min-ordering and max-ordering scalarization methods for multi-objective robust optimization

Several robustness concepts for multi-objective uncertain optimization have been developed during the last years, but not many solution methods. In this paper we introduce two methods to find min–max robust efficient solutions based on scalarizations: the min-ordering and the max-ordering method. We show that all point-based min–max robust weakly efficient solutions can be found with the max-ordering method and that the min-ordering method finds set-based min–max robust weakly efficient solutions, some of which cannot be found with formerly developed scalarization based methods. We then show how the scalarized problems may be approached for multi-objective uncertain combinatorial optimization problems with special uncertainty sets. We develop compact mixed-integer linear programming formulations for multi-objective extensions of bounded uncertainty (also known as budgeted or Γ-uncertainty). For interval uncertainty, we show that the resulting problems reduce to well-known single-objective problems.

Solving a selective dial-a-ride problem with logic-based Benders decomposition

Today’s society is facing an ever-growing demand for mobility. To a high degree these needs can be fulfilled by individual and public transport. People that do not have access to the former and cannot use the latter require additional means of transportation. This is where dial-a-ride services come into play. The dial-a-ride problem considers transportation requests of people from pick-up to drop-off locations. Users specify time windows with respect to these points. Requests are served by a given vehicle fleet with limited capacity and tour duration per vehicle. Moreover, user inconvenience considerations are taken into account by limiting the travel time between origin and destination for each request.Previous research on the dial-a-ride problem primarily focused on serving a given set of requests with a fixed-size vehicle fleet at minimal traveling costs. It is assumed that the request set is sufficiently small to be served by the available vehicles. We consider a different scenario in which a maximal number of requests shall be served under the given constraints, i.e., it is no longer guaranteed that all requests can be accepted. For this new problem variant we propose a compact mixed integer linear programming model as well as algorithms based on Benders decomposition. In particular, we employ logic-based Benders decomposition and branch-and-check using mixed integer linear programming and constraint programming algorithms. We consider different variants on how to generate Benders cuts as well as heuristic boosting techniques and different types of valid inequalities. Computational experiments illustrate the effectiveness of the suggested algorithms.

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.

Minimum Linear Arrangements

Let G = (V, E) be a simple undirected graph. Given distinct labels in {1,⋯,|V|} to the vertices of G, we define the weight of an edge uv∈E as the absolute difference between the labels assigned to u and v. The minimum linear arrangement problem (MinLA) consists in finding a labeling of the vertices of G such that the sum of the weights of its edges is minimized. It is an NP-Hard problem whose corresponding polyhedron has a factorial number of extreme points. We propose a quadratic model for MinLA and use it to obtain a novel compact mixed integer linear programming (MILP) formulation for the problem, featuring O(|V|2) variables and O(|V|2) constraints. We show the correctness of the new model and discuss valid inequalities for the problem. Our findings open new insights on the study of effective exact approaches to the problem. Computational experiments show that the new quadratic and mixed linear models performed better than existing ones in the literature for new and benchmark instances of this problem.

A Unit Commitment Algorithm and a Compact MILP Model for Short-Term Hydro-Power Generation Scheduling

This paper presents a unit commitment algorithm that defines each unit discharge given the water head, the total plant downstream flow, the variable discharge upper limit, the unit efficiency curves, and the restricted operating zones in order to maximize power efficiency. This algorithm is part of the preprocessing phase that is intended to approximate a hydro-power production function that represents individualized unit decisions. A compact mixed-integer linear programming formulation, with fewer integer variables, based on an equivalent unit model and a piecewise linear generation function, is proposed. The unit commitment is integrated without increasing the model size and complexity due to the preprocessing phase. Moreover, the optimal aggregate decision is automatically converted to unit decisions by the proposed algorithm. The coordination with mid/long-term planning is performed by taking into account the power demand allocated to the hydro-power plants. Numerical tests on Brazilian hydro-power plants demonstrate that the proposed formulation has lower computational cost than unit individualized models considering a given accuracy level for the generation function approximation.

Improved handling of uncertainty and robustness in set covering problems

Set Covering Problems are one of the classical combinatorial optimization problems and part of many different practical applications, e. g., the location of emergency service facilities. In many real-world problems the required statistical parameters are not precisely known and the obtained solutions may reveal a non-adequate performance. We introduce a robust formulation of the uncertain/probabilistic Set Covering Problem which combines the concepts of robust and probabilistic optimization by introducing ’Γ-robust α-covering’ constraints. This Robust Uncertain Set Covering Problem can be stated as a compact mixed-integer linear programming model. Additionally, two non-compact integer linear model formulations are developed. As the strength of these formulations is not known a priori, we analyze the performance of these formulations in an extensive computational study. A case study for the location of emergency service facilities highlights the benefits of our approach in comparison to a formulation neglecting these uncertainties.

Full‐load route planning for balancing bike sharing systems by logic‐based benders decomposition

Public bike sharing systems require some kind of rebalancing to avoid too many rental stations of running empty or entirely full, which would make the system ineffective and annoy customers. Most frequently, a fleet of vehicles with trailers is used for this purpose, moving bikes among the stations. Previous works considered different objectives and modeled the underlying routing problem in different ways, but they all allow an arbitrary number of bikes to be picked up at some stations and delivered to other stations, just limited by the vehicles’ capacities. Observations in practice, however, indicate that in larger well‐working bike sharing systems drivers almost never pickup or deliver only few bikes, but essentially always approximately full vehicle loads. Many stations even require several visits with full loads. Due to budgetary reasons, typically only just enough drivers and vehicles are employed to achieve a reasonable balance most of the time, but basically never an ideal one where single bikes play a substantial role. Consequently, we investigate here a simplified problem model, in which only full vehicle loads are considered for movement among the rental stations. This restriction appears to have only a minor impact on the achieved quality of the rebalancing in practice but eases the modeling substantially. More specifically, we formulate the rebalancing problem as a selective unit‐capacity pickup and delivery problem with time budgets on a bipartite graph and present a compact mixed integer linear programming model, a logic‐based Benders decomposition and a variant thereof, namely branch‐and‐check for it. For the general case, instances with up to 70 stations, and for the single‐vehicle case instances with up to 120 stations are solved to proven optimality. A comparison to leading metaheuristic approaches considering flexible vehicle loads indicates that indeed the restriction to full loads has only a very small impact on the finally achieved balance in typical scenarios of Citybike Wien. © 2017 Wiley Periodicals, Inc. NETWORKS, Vol. 69(3), 270–289 2017