Mixed Integer Linear Programming Formulation Research Articles

Rebalancing in bike sharing systems: Application to the Lisbon case study

This work presents a mathematical programming formulation for the static Bike Rebalancing Problem to meet the practical constraints faced by the operators of Bike Sharing Systems. Adaptating a Mixed Integer Linear Programming formulation, it now includes different types of vehicles and bicycles, the autonomy of each vehicle, the duration of its route and the stations that require inspection actions. An intermediate station is introduced to temporarily store bicycles, relaxing the restrictions imposed by the vehicles’ capacity. After incorporating different acceleration procedures, a Branch-and-Cut scheme is run for 30 min to find optimized routes for the reposition vehicle on five real instances of the Lisbon Bike Sharing System. The computed routes resulted in a 30% to 75% decrease in total length, when compared to the real routes. This decrease is mainly attributed to the integration of all the mentioned factors in one global model, encouraging its future implementation.

Deployment of VNF service chains with grooming and resilience in elastic optical networks

Network function virtualization (NFV) uses virtual network functions (VNFs) and VNF service chains (VNF-SCs) as a flexible and cost-efficient way of providing network services. This new backbone plays a vital role in dealing with the outbreak of traffic patterns generated by diverse services. Elastic optical networks (EONs) can be used for maximum flexibility to interconnect the computing infrastructure and data centers (DCs) required to host these VNFs and effectively transport the heterogeneous traffic flowing through the VNF-SC. The aim of the paper is to provide a complete mixed integer linear programming (MILP) formulation for the problem of VNF-SC deployment with survivability to minimize the total number of functions, balance the load utilization through grooming, and avoid spectrum fragmentation in EONs. In addition, we propose an efficient heuristic solution in dealing with the problem of large networks. The performance of the proposed strategies is evaluated based on the availability of resources (function computing capability). Simulation results show that the proposed MILP and heuristic can reduce the total amount of allocated VNFs, thereby, allowing for a certain level of resilience.

A logic-based Benders decomposition solution approach for two covering problems that consider the underlying transportation

We investigate two maximal covering location problems with capacity restrictions, minimum workload, and transportation. The problems are inspired by a waste collection problem in which large waste containers are scattered throughout the municipality, and the residents bring their waste to these containers. We take the residents’ preferences into account when allocating them to locations. When a container is full, a vehicle transports an empty container from the disposal facility (depot) to that location and replaces it. We propose a mixed-integer linear programming formulation for the problems in which vehicles can carry one or two containers, and apply a logic-based Benders decomposition approach for the latter. Here, the sub problem is a multi-period minimum weight perfect matching problem. We show that our logic-based Benders decomposition approach outperforms the direct formulation in terms of solution quality and speed. We further show that transportation of two containers at a time reduces the distance to be driven by 29.5% on average, without compromising the covering level. Furthermore, we analyze the effect of imposing a minimum workload as well as the effect of changing the focus between transportation and covering.

A new approach to the joint order batching and picker routing problem with alternative locations

Abstract Accepted by: M. Zied Babai The clustered and generalized vehicle routing problem (CGVRP) extends the well-known vehicle routing problem by grouping the demand points into multiple distinct zones, and within each zone, further separation is made by forming clusters. The objective of the CGVRP is to determine the optimal routes for a fleet of vehicles dispatched from a depot, visiting all zones within each cluster. This requires making two simultaneous optimization decisions. Firstly, each zone must be visited by exactly one node, and secondly, all zones within a cluster must be visited by the same vehicle. In this paper, we introduce two mixed-integer linear programming formulations for the CGVRP, aimed at solving a joint order batching and picker routing problem with alternative locations in a warehouse environment featuring mixed-shelves configuration. Both formulations are tested on three scenarios of randomly generated small- and medium-sized instances. Additionally, we propose a general rule approach for calculating a cost matrix in a rectangular environment. The results demonstrate the effectiveness of the proposed mathematical formulations in efficiently solving problems with up to 180 nodes.

Last mile delivery routing problem using autonomous electric vehicles

This paper presents a study on the application of Autonomous Delivery Vehicles (ADVs) in last-mile delivery for urban logistics. Specifically, we focus on a routing problem using multi-stop ADVs, with the goal of minimizing route and vehicle usage costs while satisfying several constraints associated with load and battery capacities, maximum route duration for ADVs, and maximum walking distance for customers. We refer to this problem as the Autonomous Delivery Vehicle Routing Problem (ADVRP) and present its mixed-integer linear programming formulation. Due to the NP-hardness of the problem, we propose a two-phase metaheuristic approach that first clusters customers and determines stopping locations for the ADVs, followed by a phase that determines the optimal routes for the ADVs using hybrid variable neighborhood search and simulated annealing. To evaluate our proposed solution methodology, we conduct computational experiments on various related Vehicle Routing Problems (VRPs) from the literature and newly generated ADVRP instances. The results show that the proposed two-phase metaheuristic approach can produce high-quality solutions with minimal computational effort while outperforming an exact solver in 26 medium- and large-sized instances of ADVRP and reaching optimal solutions in most VRP instances and related problems. Furthermore, we conduct sensitivity analyses on selected problem parameters and present a case study in Istanbul, Turkey to provide managerial insights for implementing ADVs in urban logistics.

Modelling overflow using mixed integer programming in short-term hydropower scheduling

AbstractShort-term hydropower scheduling seeks to find a production schedule that maximizes profit, but must also consider the hydrological balance and risk of overflow. Overflow is by nature a non-linear and non-convex phenomenon. Common approximations and relaxations may cause non-physical results such as overflow from reservoirs that are not full. This paper presents a mixed-integer linear programming formulation that can be used to prevent non-physical overflow behaviour, but that comes at a cost of significant higher solving time. To achieve an acceptable solving time, we propose a heuristic to provide tight upper bounds on the overflow variables in each time step. When applied on the model of the Fossdal watercourse in Norway, the proposed method reduces the solving time with more than 90% compared to using a conservative fixed coefficient for all time steps.

Decentralized mixed-integer optimization for robust integrated electricity and heat scheduling

Electric power systems (EPSs) and district heating networks (DHNs) are always independently operated and dispatched but also coupled with each other at the interfaces of combined heat and power (CHP) generation, whereas the existing distributed scheduling methods for the integrated electricity and heat system (IEHS) under uncertainty are computationally expensive in practical applications. To handle this problem, this paper proposes a novel decentralized mixed-integer optimization method for robust coordination involving multiple stakeholders. Firstly, a centralized two-stage robust optimization (RO) scheduling model is installed for the IEHS considering the scheduling economy under the nominal scenario and the adjustment feasibility against uncertainty. Secondly, the Fourier-Motzkin elimination equivalently projects the second-stage feasible region of the two-stage RO scheduling model onto the first-stage optimization, thereby producing a concise centralized RO scheduling model in a mixed-integer linear programming (MILP) formulation. Finally, a dual decomposition algorithm derives the decentralized solution to the resulting MILP-type RO model with guaranteed convergence and optimality. This avoids setting up a coordination center for distributed scheduling. Case testing for two IEHSs validates that the computational efficiency of the proposed method is several tens of times speedup than the traditional distributed RO method with guaranteed solution optimality.

Strong cutting planes for the capacitated multi-pickup and delivery problem with time windows

In this paper, we propose an improved 2-index mixed integer linear programming formulation for capacitated multi-pickup and delivery problems with time windows. We develop new families of valid inequalities, present some simple separation heuristics and solve the benchmark problems using a cutting plane approach. Specifically, we propose and demonstrate the effectiveness of incompatible request sets based valid inequalities, exploiting thereby the time window and capacity constraints. Further, we introduce two sub-families of these inequalities: node-based and request-based incompatible request set cuts, along with their separation heuristics. The computational results show that the incompatible request set cuts are highly useful as cutting planes for the benchmark instances. Overall, there is a marked improvement in solution performance vis-à-vis the different formulations and solution approaches used in extant literature. The approach in this paper is able to solve 12%, 37% and 43% more instances to optimality, improve the root gap by 18%, 33% and 36%, and final gap by 36%, 45% and 43% vis-à-vis the asymmetric representative formulation, three-index and the two-index formulations respectively.

The cumulative school bus routing problem: Polynomial‐size formulations

AbstractThis article introduces the cumulative school bus routing problem, which concerns the transport of students from a school using a fleet of identical buses. The objective of the problem is to select a drop‐off point for each student among potential locations within a certain walking distance and to generate routes such that the sum of arrival times of all students from their school to their homes is minimized. The article describes six polynomial‐size mixed integer linear programming formulations based on original and auxiliary graphs, and the formulations are numerically compared on real instances. The article reports the results of computational experiments performed to evaluate the performance of the proposed models.

A Computationally Efficient Benders Decomposition for Energy Systems Planning Problems with Detailed Operations and Time-Coupling Constraints

Energy systems planning models identify least-cost strategies for expansion and operation of energy systems and provide decision support for investment, planning, regulation, and policy. Most are formulated as linear programming (LP) or mixed integer linear programming (MILP) problems. Despite the relative efficiency and maturity of LP and MILP solvers, large scale problems are often intractable without abstractions that impact quality of results and generalizability of findings. We consider a macro-energy systems planning problem with detailed operations and policy constraints and formulate a computationally efficient Benders decomposition separating investments from operations and decoupling operational timesteps using budgeting variables in the master model. This novel approach enables parallelization of operational subproblems and permits modeling of relevant constraints coupling decisions across time periods (e.g., policy constraints) within a decomposed framework. Runtime scales linearly with temporal resolution; tests demonstrate substantial runtime improvement for all MILP formulations and for some LP formulations depending on problem size relative to analogous monolithic models solved with state-of-the-art commercial solvers. Our algorithm is applicable to planning problems in other domains (e.g., water, transportation networks, production processes) and can solve large-scale problems otherwise intractable. We show that the increased resolution enabled by this algorithm mitigates structural uncertainty, improving recommendation accuracy. Funding: Funding for this work was provided by the Princeton Carbon Mitigation Initiative (funded by a gift from BP) and the Princeton Zero-carbon Technology Consortium (funded by gifts from GE, Google, ClearPath, and Breakthrough Energy). Supplemental Material: The e-companion is available at https://doi.org/10.1287/ijoo.2023.0005 .

Planning of three-level facilities network for collection and recycling of E-waste: A MILP-based modelling approach

The collection and recycling of E-waste are vital for the economy, the environment, and public health. From a financial perspective, planning the necessary E-waste facilities, including the collection centres and recycling centres, can be a challenging and laborious undertaking. The E-waste recycling facilities network planning problem involves three types of facilities related to different stages. Firstly, E-waste generated from households is collected at a location known as a collection point, from where it is transported to collection centres where it is sorted into several product groups, and lastly, it is sent to relevant recycling facilities. The recovered materials in recycling facilities are then forwarded to original manufacturers for reuse, while hazardous waste is dumped in landfills. The capacity and location decisions of recycling facilities are strategic and important as it involves huge capital investment, and they should be far from residential areas. Furthermore, to reduce transportation costs, these facilities should be located nearby collection centres and landfills. So, to reduce the overall E-waste recycling cost, the decision regarding the number and location of all three facilities plays an important role.In this study, the planning problem of determining the optimal number and location of collection points, collection centres, and recycling centres, as well as the flow of E-waste to these facilities, has been tackled by proposing a mixed-integer linear programming (MILP) formulation. The objective is to minimise the overall cost associated with these operations. The costs include the transportation cost and the cost of establishing all types of E-waste facilities. The application of the model has been demonstrated using a numerical example. The numerical case has been solved using GUROBI 8.0.1 optimisation solver. This study provides a framework to plan an effective and efficient network of E-waste facilities at minimum cost and lesser environmental impact.

Radar: Reliable Resource Scheduling for Composable/Disaggregated Data Centers

Hardware disaggregation decouples resources (e.g., processors and memory) from monolithic servers, potentially improving service reliability. However, from another perspective, directly exposing resource modules to a shared network may adversely affect service reliability. In this paper, we study a reliable resource allocation problem in disaggregated DCs (DDCs), considering network impact and different disaggregation scales. We provide a mixed-integer linear programming formulation and a resource allocation framework named Radar for this problem. Numerical results demonstrate that the benefits of hardware disaggregation may be adversely affected by an imperfect network. It also shows that both the hardware backup and a proposed migration-based restoration can be applied to overcome this potential adverse effect.

Block-based state-expanded network models for multi-activity shift scheduling

This paper presents new mixed-integer linear programming formulations for multi-activity shift scheduling problems (MASSP). In these formulations, the rules governing shift feasibility are encoded in block-based state-expanded networks in which nodes are associated with states and arcs represent assignments of blocks of work or break periods inducing state transitions. A key advantage of these formulations is that for the anonymous MASSP in which all employees are considered as equal only a single network with integer flow variables is needed as long as the network encodes all shift composition rules. A challenging aspect is that the networks can become very large, yielding huge models that are hard to solve for large problem instances. To address this challenge, this paper proposes two exact modeling techniques that substantially reduce the size of the model instances: First, it introduces a set of aggregate side constraints enforcing that an integer flow solution can be decomposed into paths representing feasible shifts. Second, it proposes to decouple the shift composition from the assignment of concrete activities to blocks of work periods, thereby removing a large amount of symmetry from the original model. In a computational study with two MASSP instance sets from the literature dealing with shift scheduling problems, we demonstrate the effectiveness of these techniques for reducing the both size of the model instances and the solution time: We are able to solve all instances, including more than 70 previously open instances, to optimality–the vast majority of them in less than 30 min on a notebook computer.

An Adaptive Large Neighbourhood Search algorithm for a real-world Home Care Scheduling Problem with time windows and dynamic breaks

This paper presents a Home Care Scheduling Problem (from now on HCSP) based on a real case of a care company for elderly and dependent people located in the North of Spain. The problem incorporates many of the common features addressed in the HCSP literature, such as soft and hard time windows for the services, available working time of the caregivers or affinity levels between users and caregivers. However, it also includes other novel characteristics that increase the difficulty of the problem significantly, since the breaks between services will play a key role in the quality of the solutions. To evaluate the solutions, the users welfare will be prioritized over the cost associated with the schedule.The problem has been formulated as a Mixed Integer Linear Programming (MILP) one but, due to the complexity of the model, it is not possible to solve it for real size instances. Therefore, we have designed a method that combines the Adaptive Large Neighbourhood Search (ALNS) methodology with a heuristic approach necessary to evaluate the objective functions. To analyse the behaviour of the algorithm, a set of computational experiments are carried out under different configurations. First, the MILP formulation and the algorithm have been compared over some standard instances from the literature. Finally, the performance of the algorithm is evaluated over a real case study based on the timetables of the company during some weeks from 2016 to 2017.

Linear model decision trees as surrogates in optimization of engineering applications

Machine learning models are promising as surrogates in optimization when replacing difficult to solve equations or black-box type models. This work demonstrates the viability of linear model decision trees as piecewise-linear surrogates in decision-making problems. Linear model decision trees can be represented exactly in mixed-integer linear programming (MILP) and mixed-integer quadratic constrained programming (MIQCP) formulations. Furthermore, they can represent discontinuous functions, bringing advantages over neural networks in some cases. We present several formulations using transformations from Generalized Disjunctive Programming (GDP) formulations and modifications of MILP formulations for gradient boosted decision trees (GBDT). We then compare the computational performance of these different MILP and MIQCP representations in an optimization problem and illustrate their use on engineering applications. We observe faster solution times for optimization problems with linear model decision tree surrogates when compared with GBDT surrogates using the Optimization and Machine Learning Toolkit (OMLT).

Optimizing bilinear multi-source water supply systems using mixed-integer linear programming approximations: An analysis of the Israeli seawater desalination array

Decision support systems (DSSs) in water resources management often use optimization models to make decisions. Therefore, the quality of these models is crucial for the success of DSSs. However, optimization models can struggle with reaching global optima, resulting in inconsistent and untrustworthy results for decision-makers. Additionally, current solution methods can be slow and inefficient, hindering real-time interaction with the DSS. This study addresses the optimal regional Multisource Water Supply Systems (MWSS) management using efficient global optimization techniques. The MWSS problem is formulated as a nonlinear optimization program due to using bilinear terms, which can result in multiple local optima. To overcome this challenge, four Mixed Integer Linear Program (MILP) formulations are proposed, which can be globally solved in a computationally effective manner using readily available MILP solvers. Our proposed formulations are tested on the Israeli Water Supply System (IWSS), which presents a unique challenge due to the inclusion of the seawater desalination array. In addition, this system must balance conventional and non-conventional water sources with varying qualities and characteristics. The results show that the proposed formulations provide a computationally efficient solution for the IWSS problem, allowing for the optimal utilization of different water sources in the IWSS.

The profit-oriented hub line location problem with elastic demand

This paper deals with an extension of the hub line location problem considering demand elasticity with respect to travel times. The proposed model aims to capture the impact the hub network topology has on demand. The objective is to maximize the total revenue generated by each unit of demand using the hub line. We propose mixed-integer nonlinear formulations to model this problem. We study some properties of the nonlinear objective function associated with these formulations. Due to the inherent complexity involved in solving these nonlinear formulations with state-of-the-art solvers, we also present alternative mixed-integer linear programming formulations. Computational results compare the proposed formulations and the benefits of the presented model using benchmark instances commonly used in hub location. Moreover, a sensitivity analysis study is carried out with real data from the city of Montreal, Canada, to demonstrate the added value of incorporating demand elasticity when using the proposed model for public transportation planning.

Secure expansion of energy storage and transmission lines considering bundling option under renewable penetration

This paper presents a multi-stage expansion model for the co-planning of transmission lines, battery energy storage (ES), and wind power plants (WPP). High penetration of renewable energy sources (RES) is integrated into the proposed model concerning renewable portfolio standard (RPS) policy goals. The possibility of bundling existing transmission lines to uprate power flow capacity is considered. Renewable energy curtailment and load shedding are included in the model to assess the system operation more precisely. Battery ES devices are co-planned to defer transmission expansion and renewable management. To make the time complexity of the problem tractable and capture the uncertainties of load and RES in an hourly resolution, a chronological time-period clustering algorithm is used to extract the representative hours of each planning stage. Additionally, the flexible ramp reserve is utilized to handle the uncertainty of RES. An accelerated Benders dual decomposition (BDD) algorithm is developed to solve the proposed model mixed-integer linear programming (MILP) formulation. The N-1 security criterion is evaluated by considering a designed contingency screening (CS) algorithm to identify higher risk contingencies. The effectiveness of the proposed co-planning model is evaluated using IEEE RTS 24-bus and IEEE 118-bus test systems.

Forward and Line-Based Cycle Bases for Periodic Timetabling

The optimization of periodic timetables is an indispensable planning task in public transport. Although the periodic event scheduling problem (PESP) provides an elegant mathematical formulation of the periodic timetabling problem that led to many insights for primal heuristics, it is notoriously hard to solve to optimality. One reason is that for the standard mixed-integer linear programming formulations, linear programming relaxations are weak, and the integer variables are of pure technical nature and in general do not correlate with the objective value. While the first problem has been addressed by developing several families of cutting planes, we focus on the second aspect. We discuss integral forward cycle bases as a concept to compute improved dual bounds for PESP instances. To this end, we develop the theory of forward cycle bases on general digraphs. Specifically for the application of timetabling, we devise a generic procedure to construct line-based event-activity networks and give a simple recipe for an integral forward cycle basis on such networks. Finally, we analyze the 16 railway instances of the benchmark library PESPlib, match them to the line-based structure, and use forward cycle bases to compute better dual bounds for 14 out of the 16 instances.

Stackelberg Security Games with Contagious Attacks on a Network: Reallocation to the Rescue

In the classic network security games, the defender distributes defending resources to the nodes of the network, and the attacker attacks a node, with the objective of maximizing the damage caused. In this paper, we consider the network defending problem against contagious attacks, e.g., the attack at a node u spreads to the neighbors of u and can cause damage at multiple nodes. Existing works that study shared resources assume that the resource allocated to a node can be shared or duplicated between neighboring nodes. However, in the real world, sharing resource naturally leads to a decrease in defending power of the source node, especially when defending against contagious attacks. Therefore, we study the model in which resources allocated to a node can only be transferred to its neighboring nodes, which we refer to as a reallocation process. We show that the problem of computing optimal defending strategy is NP-hard even for some very special cases. For positive results, we give a mixed integer linear program formulation for the problem and a bi-criteria approximation algorithm. Our experimental results demonstrate that the allocation and reallocation strategies our algorithm computes perform well in terms of minimizing the damage due to contagious attacks.