Mixed Integer Linear Programming Formulation Research Articles

The Heterogeneous Flexible Periodic Vehicle Routing Problem: Mathematical formulations and solution algorithms

The aim of this paper is to introduce the Flexible Periodic Vehicle Routing Problem with Heterogeneous Fleet, a variant of the Periodic Vehicle Routing Problem. Flexibility is introduced in service schedules and delivered quantities, heterogeneity comes from different vehicles capacities and speeds. Three Mixed-Integer Linear Programming formulations and a matheuristic, based on Kernel Search, are proposed. Computational tests are made to evaluate the performance of the three formulations and to assess the quality of the solutions provided by the matheuristic.

Resilient scheduling of MESSs and RCs for distribution system restoration considering the forced cut-off of wind power

Mobile energy storage systems (MESSs), wind power and repair crews (RCs) are usually coordinated to restore distribution systems damaged by hurricanes. However, the forced cut-off of wind power occurs as wind speed exceeds the cut-off value, which can lead to sudden power shortage. To address this challenge, this paper proposes a restoration approach considering wind power cut-off to enhance distribution system resilience. The resilient coordination approach takes advantage of MESSs, RCs, network reconfiguration and distributed generators (DGs) to restore the outage distribution system. Considering the uncertainty of wind power cut-off, the proposed method is formulated as a two-stage model on two timescales. On the long timescale, MESSs, RCs, DGs and line switches are robustly optimized to reduce critical load loss considering wind power cut-off. On the short timescale, the output decisions of MESSs and DGs are re-dispatched to compensate for the first-stage strategies after uncertainties realization. In addition, the nonlinear output curve of wind power is linearized in mixed integer linear programming form to facilitate the solution of the robust model. Finally, the modified IEEE-69 distribution system is employed to verify the proposed method. Simulation results show that the proposed method can increase the amount of load restoration by 24.1%.

On Achilles Heel of some optical network designs and performance comparisons

This non-conventional paper represents the first attempt to uncover a possible vulnerability in some proposals for optical network designs and performance comparisons. While optical network designs and planning lie at the heart of achieving fiber capacity efficiency and/or operational efficiency, its combinatorial nature makes it computationally hard to reach optimal solutions for realistic scenarios. As a consequence, the well-established way that have been taken for granted by not-so-small number of research papers is that an optimization model based on mixed integer linear programming (MILP) formulation is first proposed and then due to the intractability of such combinatorial optimization model, a heuristic algorithm is offered as an approximation. The solution-quality comparison between the MILP and heuristic is then carried out on small-scale instances including topologies and traffic tests to verify the efficacy of the proposed heuristic. Next such allegedly verified heuristic are used for optical network designs of realistic scenarios. This approach may nevertheless leave a critical vulnerability as there is no guarantee that one performs well in small tests will generalize adequately for large-scale cases, a common pitfall widely referred as the peril of extrapolation and/or overfitting. Besides, it is not uncommon that in some research works, for benchmarking purpose, the comparison between a new design proposal whose performance is obtained from on one heuristic and a reference design based on another heuristic is carried out. As the consequence of lacking solution quality check, such performance comparison relied merely on heuristic solutions may be equally vulnerable, resulting to possibly unreliable conclusions. In this work, we pinpoint those issues and provide a realistic case study to highlight and demonstrate the impact of such vulnerabilities.

Scheduling of the inbound and outbound trucks at a Cross-Docking platform considering mutualization of docks

Paper aims This paper studies a new Cross-Docking (CD) scheduling problem inspired by the trend of resource mutualization at CD operations where a company owning a CD platform can lease its docks to other companies. Originality This work pioneers the studies of the resource mutualization at CD operations solving at scheduling problem, where there is a set of requirements for leasing the loading and unloading docks for a period, and the company owning the docks can accept or decline each requirement. Considering that the availability of the docks is linked to scheduling the trucks at the company’s loading and unloading docks, the decision to accept the requirements must be made along with the decisions for scheduling the trucks. Research method To solve this problem, this work proposes a Mixed-Integer Linear Programming (MILP) formulation. Main findings Using a commercial solver, the MILP formulation can optimally solve 65/128 instances based on a real operation. For the remaining instances, the MILP formulation obtains an average optimized gap of 8.95%. Implications for theory and practice The MILP formulation obtains acceptable results. Moreover, we found that all the instances accepted the requirements showing that it is economically interesting to lease docks.

Solution to dynamic economic dispatch with prohibited operating zones via MILP.

Dynamic economic dispatch (DED) problem considering prohibited operating zones (POZ), ramp rate constraints, transmission losses and spinning reserve constraints is a complicated non-linear problem which is difficult to solve efficiently. In this paper, a mixed integer linear programming (MILP) method is proposed to solve such a DED problem. Firstly, a novel MILP formulation for DED problem without considering the transmission losses, denoted by MILP-1, is presented by using perspective cut reformulation technique. When the transmission losses are considered, the quadratic terms in the transmission losses are replaced by their first order Taylor expansions, and then an MILP formulation for DED considering the transmission losses, denoted by MILP-2, is obtained. Based on MILP-1 and MILP-2, an MILP-iteration algorithm is proposed to solve the complicated DED problem. The effectiveness of the MILP formulation and MILP iteration algorithm are assessed by several cases and the simulation results show that both of them can solve to competitive solutions in a short time.

A MILP approach for the operational optimization of gas networks

Nowadays, natural gas is one of the primary energy resources employed in many sectors, and it can be regarded as a bridge fuel in the decarbonization process. The transport of natural gas relies on a complex infrastructure, which needs to be properly managed to minimize energy consumption and CO2 emissions. This study proposes a Mixed Integer Linear Programming formulation capable of optimizing the unit commitment (on/off) of the gas-turbine-driven compressors, their loads and the dynamic operation of the network. The gas network dynamic model has been discretized in space and time, and the friction term has been linearized. The performance maps of the gas-turbine-driven compressors have been linearized using the convex-hull approach. The model includes all the technical limitations of the gas-turbine-driven compressors and it can handle flow reversals on all pipes. Moreover it is suitable for cyclic networks. The MILP model is applied to a case study with 4 compressor stations, each containing multiple gas turbine driven compressors, and a cyclic network consisting of 29 pipes.

Mixed Integer Linear Programming formulations for the permutation flow shop scheduling problem with preventive maintenance activities

Mixed Integer Linear Programming formulations for the permutation flow shop scheduling problem with preventive maintenance activities

Continuous Covering on Networks: Strong Mixed Integer Programming Formulations

Covering problems are well-studied in the domain of Operations Research, and, more specifically, in Location Science. When the location space is a network, the most frequent assumption is to consider the candidate facility locations, the points to be covered, or both, to be discrete sets. In this work, we study the set-covering location problem when both candidate locations and demand points are continuous sets on a network. This variant has received little attention, and the scarce existing approaches have focused on particular cases, such as tree networks and integer covering radius. Here we study the general problem and present a Mixed Integer Linear Programming formulation (MILP) for networks with edges' lengths no greater than the covering radius. The model does not lose generality, as any edge not satisfying this condition can be partitioned into subedges of appropriate lengths without changing the problem. We propose a preprocessing algorithm to reduce the size of the MILP, and devise tight big-$M$ constants and valid inequalities to strengthen our formulations. Moreover, a second MILP is proposed, which admits edges' lengths greater than the covering radius. As opposed to existing formulations of the problem (including the first MILP proposed herein), the number of variables and constraints of this second model does not depend on the lengths of the network's edges. This second model represents a scalable approach that particularly suits real-world networks, whose edges are usually greater than the covering radius. Our computational experiments show the strengths and limitations of our exact approach on both real-world and random networks. Our formulations are also tested against an existing exact method.

A Mixed Integer Linear Formulation and a Grouping League Championship Algorithm for a Multiperiod‐Multitrip Order Picking System with Product Replenishment to Minimize Total Tardiness

Order picking, which is collecting a set of products from different locations in a warehouse, has repeatedly been described as one of the most laborious and time‐consuming internal logistic processes. Each order is issued to pick some products located at given locations in the warehouse. In this paper, we consider an order picking problem, in which a number of orders with different delivery due dates are going to be retrieved by a limited number of order pickers in multiperiods such that the total tardiness is minimized. The aim is to determine a retrieval plan in terms of order batching and order picker multitrip routing as decision variables. Besides, products are arrived and replenished at the predetermined locations at different periods. Therefore, products sitting in those locations should be delivered soon to provide empty rooms for replenishment. A mixed integer linear programming formulation is proposed for this new problem. The model is optimally solved for small‐size problems. For larger instances, grouping metaheuristic algorithms are proposed based on particle swarm optimization and the league championship algorithm that use group‐based operators to generate reasonable batches of orders. Improvement heuristics are designed as well. The performance of the MILP formulation and metaheuristic algorithms is analyzed for different problem instances whose designs are based on real data gathered from an auto parts warehouse. Results indicate that our algorithms can stably solve large instances of the problem in a reasonable time.

A mixed integer formulation and an efficient metaheuristic for the unrelated parallel machine scheduling problem: Total tardiness minimization

In this paper, the unrelated parallel machine scheduling problem with the objective of minimizing the total tardiness is addressed. For such a problem, a mixed-integer linear programming (MILP) formulation, that considers assignment and positional variables, is presented. In addition, an iterated local search (ILS) algorithm that produces high-quality solutions in reasonable times is proposed for higher size instances. The ILS robustness was determined by comparing its performance with the results provided by the MILP. The instances used in this paper were constructed under a new approach which results in tighter due dates than the previous generation method for this problem. The proposed MILP formulation was able to solve instances of up to 150 jobs and 20 machines. Regarding the ILS, it yielded high-quality solutions in a reasonable time, solving instances of a size up to 400 jobs and 20 machines. Experimental results confirm that both approaches are efficient and promising. • A new mixed integer programming formulation for the Unrelated Parallel Machine Scheduling Problem. • The formulation uses positional variables, is linear and does not use the Big-M technique. • The formulation solves instances up to 150 jobs and 20 machines to optimality. • An efficient iterated local search metaheuristic is presented. • A potential issue in the commonly used instance-generation procedure is discussed and a new set of instances is proposed.

A MATHEMATICAL MODEL AND SOLUTION METHOD FOR THE BERTH ALLOCATION PROBLEM WITH VARIABLE HANDLING TIME AND CONTINUOUS TIME HORIZON

In this paper, we present the integration of two problems related to the operations in a port terminal: the Berth Allocation Problem (BAP) integrated with the Machine Assignment Problem. We present a mixed-integer linear programming (MILP) formulation, capable of assigning and scheduling incoming vessels to berthing positions and the assignment of machines for handling the vessels. The machines can be quay cranes, mobile cranes, straddle carriers, forklifts, trucks, and any other machine. The problem aims to minimize the waiting time plus the handling time of the vessels. To solve the problem, we developed a heuristic algorithm, capable of solving a problem instance in seconds. To compare the results, we generate several instance problems based on real data and solve them with our MILP formulation implemented in a solver, our heuristic, and a First In First Out (FIFO) algorithm. The solver was able to find solutions only in small-scale instances, and the heuristic was able to find good solutions for all instances.

Deep Space Network Scheduling Using Quantum Annealing

NASA's Deep Space Network (DSN) is responsible for communication and navigation for several NASA and international missions. The DSN comprises three complexes located in Goldstone (California, USA), Cambera (Australia) and Madrid (Spain). This distribution in longitude guarantees a full sky coverage. Each complex has one 70-meter and several 34-meter antennas. The network routinely serves a few dozens missions. The scheduling of the DSN is complex and involves humans interventions as well as automated solutions. In order to increase the level of automation, different computing paradigms have been explored. In this paper, we report on the quadratic unconstrained binary optimization (QUBO) formulation of the DSN scheduling, as well as a custom classical solver designed around some of the unique features of this scheduling problem. Thanks to a hybrid framework that extends the size of the problems that can be solved with a quantum annealer, we are able to generate a schedule from the QUBO formulation of the problem for one week's worth of user antenna requests, which represents the time period scheduled during operation. In other words, this work describes a real- world application of quantum annealing using real-world, operational data. We compare the resulting schedules' quality to solutions obtained using a mixed-integer linear programming formulation on a commercial solver. Our custom solver, based on a quantum-inspired optimization technique called Substochatic Monte-Carlo, while much faster in generating schedules, could only treat a subset of requests and hence we report its results independently.

A Novel Tractable Methodology to Stochastic Multi-Period AC OPF in Active Distribution Systems Using Sequential Linearization Algorithm

The ongoing transition of passive distribution networks to active distribution systems (ADSs) calls for the development of sophisticated and tractable tools to address the needs and challenges of future ADSs. In this regard, this work proposes a novel tractable methodology for flexibility procurement via stochastic multi-period AC optimal power flow (S-MP-OPF). The first novelty of the proposed methodology is the formulation of a new mixed-integer linear programming (MILP) model, which develops novel linear approximations for all nonlinear constraints (i.e., active and reactive power flows and branch currents) that ensure tractability. The proposed linearization approach relies on the second-order Taylor series expansion of trigonometric terms, does not hinge on the flat-voltage, near-voltage and small angle assumptions, and formulates the linear approximations using square of voltage magnitude and voltage angle difference variables. The second novelty is a sophisticated heuristic sequential linearization algorithm (SLA) which incorporates and improves the accuracy of the MILP model in an iterative manner. The bench-marking of proposed SLA is done on a 34-bus ADS for a comprehensive set of flexible options, whereas its versatility and scalability is demonstrated on real-world 31-bus weakly-meshed and 191-bus radial ADSs. Extensive numerical analyses show that the proposed SLA leads to a highly accurate and feasible solution with small optimality gap within few iterations, under normal and stressed operating conditions, outperforms alternative methods in terms of solution accuracy and computational efficiency, and is scalable to large S-MP-OPF problems.

A Path Relinking Method for the Joint Online Scheduling and Capacity Allocation of DL Training Workloads in GPU as a Service Systems

The Deep Learning (DL) paradigm gained remarkable popularity in recent years. DL models are used to tackle increasingly complex problems, making the training process require considerable computational power. The parallel computing capabilities offered by modern GPUs partially fulfill this need, but the high costs related to GPU as a Service solutions in the cloud call for efficient capacity planning and job scheduling algorithms to reduce operational costs via resource sharing. In this work, we jointly address the online capacity planning and job scheduling problems from the perspective of cloud end-users. We present a Mixed Integer Linear Programming (MILP) formulation, and a path relinking-based method aiming at optimizing operational costs by (i) rightsizing Virtual Machine (VM) capacity at each node, (ii) partitioning the set of GPUs among multiple concurrent jobs on the same VM, and (iii) determining a due-date-aware job schedule. An extensive experimental campaign attests the effectiveness of the proposed approach in practical scenarios: costs savings up to 97% are attained compared with first-principle methods based on, e.g., Earliest Deadline First, cost reductions up to 20% are obtained with respect to a previously proposed Hierarchical Method and up to 95% against a dynamic programming-based method from the literature. Scalability analyses show that systems with up to 100 nodes and 450 concurrent jobs can be managed in less than 7 seconds. The validation in a prototype cloud environment shows a deviation below 5% between real and predicted costs.

Sizing and Siting of Battery Energy Storage Systems: A Colombian Case

This paper presents a mixed-integer linear programming (MILP) formulation for sizing and siting of battery energy storage systems (BESSs). The problem formulation seeks to minimize both operation costs and BESS investment. The proposed model includes restrictions of the conventional security-constrained unit commitment problem, a piece-wise linear approximation to model power losses, and a linear model of hydro generation units. The proposed model is tested in a 6-bus test system and a 15-bus system representing the Colombian power system. For the two studied systems, simulation results show that the reduction of operation costs due to the installation of BESSs compensates the investments, under some of the considered technical cost cases. Additionally, results show that adequate sizing and siting of BESSs reduce renewable energy curtailment in the Colombian power system with high penetration of fluctuating renewable generation.

Integrated Passive-Active Model Identification With Tunable Model Discrimination for Affine Discrete-Time Systems

This letter proposes a passive-active model identification algorithm for affine discrete-time systems that integrates active model discrimination (AMD) and model invalidation (MI). A look-up tree consisting of control inputs is constructed offline for this integrated model identification (IMI) technique to discriminate among models in a time-varying model set, which is only known at run time when repeatedly applying MI online. Furthermore, a novel tunable AMD (TAMD), with its mixed-integer linear programming (MILP) formulation, is proposed and combined with the IMI algorithm, which can improve model discrimination performance. The effectiveness of the proposed IMI algorithm is demonstrated through simulations for identifying intention models of human-driven vehicles in a lane changing scenario.

Formulating multi-class user equilibrium using mixed-integer linear programming

We introduce an approach to formulate and solve the multi-class user equilibrium traffic assignment as a mixed-integer linear programming (MILP) problem. Compared to simulation approaches, the analytical MILP formulation makes the solution of network assignment problems more tractable. When applied in a multi-class context, it obviates the need to assume a symmetrical influence between classes and thereby allows richer traffic behavior to be taken into account. Also, it integrates naturally in optimization problems such as maintenance planning and traffic management. We develop the model and apply it for the Sioux Falls network, showing that it outperforms the traditional Beckmann-based and MSA approaches in smaller-scale problems. Further research opportunities lie in developing extensions of MILP-based assignment, with different variants of user equilibrium or dynamic assignment, and in improving the model and solution algorithms to allow large-scale application. • We formulate multi-class static traffic assignment using MILP. • The new formulation is analytical and therefore prescriptive. • We compare the performance of MILP with other existing methods in the Sioux Falls network. • This formulation opens a new axis for future works.

A hybrid genetic algorithm for the hybrid flow shop scheduling problem with machine blocking and sequence-dependent setup times

This study contributes to the hybrid flow shop due to a lack of consideration of characteristics existing in real-world problems. Prior studies are neglecting identical machines, explicit and sequence-dependent setup times, and machine blocking. We propose a hybrid genetic algorithm to solve the problem. Furthermore, we also propose a mixed-integer linear programming formulation. We note a predominance of the mathematical model for small instances, with five jobs and three machines because of how fast there is convergence. The objective function adopted is to minimize the makespan, and relative deviation is used as a performance criterion. Our proposal incorporates two metaheuristics in this process: a genetic algorithm to generate sequences (the flow shop subproblem) and a GRASP to allocate the jobs in the machines (the parallel machines subproblem). The extensive computational experience carried out shows that the proposed hybrid genetic algorithm is a promising procedure to solve large-sized instances.

Coordinated Vehicle Platooning with Fixed Routes: Adaptive Time Discretization, Strengthened Formulations and Approximation Algorithms

We consider the coordinated vehicle platooning problem over a road network with time constraints and the routes of vehicles are given. The problem is to coordinate the departure time of each vehicle to enable platoon formation hence maximizing the total fuel saving. We first focus on the case that the routes form a tree. In this case, the flexibility time window of each vehicle admits a unified representation, under which the synchronization of departure time is equivalent to the management of time-window overlapping. Based on this observation, a time-window-overlapping induced mixed-integer linear program formulation has been established which is equivalent to the continuous-time formulation established in the previous work. By imposing an adaptive time-discretization procedure, we can further reformulate the time coordination problem using a vehicle-to-time-bucket assignment induced mixed-integer linear program. Compared to the continuous-time formulation, the assignment formulation is free of big-M coefficients, and it is demonstrated by numerical experiments that the assignment formulation is indeed more effective in computational performance. As an independent interest, three approximation algorithms have been derived with provable competitive ratios under a certain regularity assumptions, which is for the first time on this discrete optimization problem. The solution approach has been extended to general cases in which the graph formed by vehicle routes can have loops, by adding a systematic loop breaking scheme. An insightful discovery is that whether there exists an exact equivalent assignment formulation for a general problem instance has a close connection to the pattern of lattice groups.

Decomposition of the vehicle routing problem with time windows on the time dimension

The Vehicle Routing Problem with Time Windows (VRPTW) asks for the optimal set of routes to be performed by a fleet of vehicles to serve a set of customers within their assigned time windows. In this work, we propose a matheuristic for the VRPTW which utilizes the sub-problem constituted by optimizing only a selected time window of the VRPTW whereas the tours outside of this time window are regarded as fixed. We call this problem the Single Time Window Vehicle Routing Problem (STWVRP). For applying the STWVRP, we must assume that several customers are assigned to the same time window, i.e., the number of time windows is much smaller than the number of customers. An exact problem description of the STWVRP is given in the form of a Mixed-Integer Linear Programming formulation. We apply this exact formulation within a matheuristic for the VRPTW. The paper concludes with extensive computational experiments.