Heuristic Solution Methods Research Articles

Distributed solution of the day-ahead pump and valve scheduling problem for dynamically adaptive water distribution networks with storage

This paper investigates the computation of daily schedules of pumps and boundary valves for the minimization of energy costs in water distribution networks (WDN) with dynamically adaptive configurations. The considered problem combines integer (“on”/“off”) pump control variables, non-convex energy conservation constraints and time-coupling mass conservation constraints. For operational WDNs, the resulting non-convex mixed-integer non-linear program (MINLP) is too large to be solved using available methods. We propose a tailored heuristic solution method based on the Alternating Direction Method of Multipliers which distributes and coordinates the solution of smaller problems corresponding to individual time steps of the original MINLP. The proposed method is applied to a large-scale WDN from the UK. The daily schedule of pumps and boundary valves obtained for the dynamically adaptive network configuration, computed in 12 min, is shown to be at most 6% suboptimal and nearly 5% cheaper than the globally optimal schedule corresponding to the traditional (sectorized) network configuration. The proposed algorithm outperforms alternative off-the-shelf and tailored approaches, providing a scalable method to compute good solutions to the complex day-ahead pump and valve scheduling problem in operational dynamically adaptive WDNs.

Optimizing sequential decision-making under risk: Strategic allocation with switching penalties

This paper considers the multiarmed bandit (MAB) problem augmented with a critical real-world consideration: the cost implications of switching decisions. Our work distinguishes itself by addressing the largely unexplored domain of risk-averse MAB problems compounded by switching penalties. Such scenarios are not just theoretical constructs but are reflective of numerous practical applications. Our contribution is threefold: firstly, we explore how switching costs and risk aversion influence decision-making in MAB problems. Secondly, we present novel theoretical results, including the development of the Risk-Averse Switching Index (RASI), which addresses the dual challenges of risk aversion and switching costs, demonstrating its near-optimal efficacy. This heuristic solution method is grounded in dynamic coherent risk measures, enabling a time-consistent evaluation of risk and reward. Lastly, through rigorous numerical experiments, we validate our algorithm’s effectiveness and practical applicability, providing decision-makers with valuable insights and tools for navigating the multifaceted landscape of risk-averse environments with inherent switching costs.

Integrated crew organization and work zone scheduling for network-wide daily road pavement rehabilitation

This study develops a new integer-programming model to address the network-wide daily road pavement rehabilitation scheduling problem. In the model, the crew organization and work zone schedule are jointly optimized daily, with the objective of minimizing both the operational cost and user travel time. A day-to-day traffic dynamics model is applied to capture the non-equilibrium traffic evolution against network supply variation over the planning horizon, which leads to a simulation-based optimization problem. To solve this challenging problem, a two-stage hybrid heuristic solution method is proposed. In the first stage, hybrid tabu search (TS) meta-heuristics are comparatively developed to identify a group of active crew work routes without time slacks. The obtained crew routes are then fed to the second stage for work zone scheduling via a discrete compass search algorithm. Some important findings are obtained from numerical experiments. First, crew routing (or crew organization) is the dominant decision in the studied problem, and a desirable work zone schedule encourages a crew to execute the assigned tasks continually. The findings can be used to develop simplified and efficient solution algorithms. Second, the hybrid TS meta-heuristics developed for crew routing exhibit superior performance compared to other solution methods. Finally, a well-defined model for the current problem should consider both user travel time and operation cost. Our model enables decision-makers to make an effective trade-off between these two objectives. An effective measure is suggested to evaluate the cost-effectiveness of budget investment decisions when budgets are limited.

A Review of Robust Machine Scheduling

Robust optimization (RO) has been recognized as an effective means to deal with unanticipated events in highly uncertain and risky environments. This paper systematically reviews two types of emerging RO machine scheduling approaches—robust machine scheduling (R-MS) and distributionally R-MS (DR-MS) methods—which usually offer tractable formulations and analytical results for machine scheduling problems under uncertainty. First, after highlighting the advantages of RO methods over the stochastic approach in terms of tractability and robustness, we use the bibliometric method to analyze the literature related to R-MS/DR-MS problems and classify them from the following aspects: (1) uncertain factors, (2) uncertainty descriptions, (3) robustness criteria, (4) machine environments and (5) solution methods. Second, we discuss the uncertainty descriptions, and the robust feasibility and robust optimality criteria. We further provide a state-of-the-art review of R-MS/DR-MS models in different machine environments and discuss the performance of the R-MS/DR-MS models. Third, we review and discuss the existing exact, approximation, online, and heuristic solution methods for solving R-MS/DR-MS models. Finally, we present future research opportunities in two promising areas: green machine scheduling problems and machine learning-enabled algorithms. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —Machine scheduling plays an essential role in industrial and service systems, such as manufacturing, power generation, transportation and medical systems. However, in practice, scheduling systems usually operate in highly uncertain environments due to noisy measurements, prediction errors, and implementation deviations. To ensure robust feasibility and robust optimality, robust machine scheduling (R-MS) and distributionally R-MS (DR-MS) approaches have been recently proposed to hedge against the uncertainties related to processing time, release time, due date, machine breakdown, etc. This paper provides a comprehensive review of the R-MS/DR-MS models and algorithms in different machine environments from the aspects of uncertainty descriptions, robustness criteria and solution methods. This paper further highlights the challenges of R-MS problems and provides promising and valuable research opportunities in terms of problem formulations and algorithm designs.

A novel model for transfer synchronization in transit networks and a Lagrangian-based heuristic solution method

To realize the benefits of network connectivity in transfer-based transit networks, it is critical to minimize transfer disutility for passengers by synchronizing timetables of intersecting routes. We propose a mixed-integer linear programming timetable synchronization model that incorporates new features, such as dwell time determination and vehicle capacity consideration, which have been largely overlooked in the literature at the scheduling stage. We introduce a new concept of pre-planned holding time, called transfer buffer time, to reduce the transfer waiting time, particularly for transfers to low-frequency routes, while taking into account the penalty of extra in-vehicle time for onboard passengers and the possible consequences on headway regularity of a route. We develop a Lagrangian relaxation-based heuristic to obtain high-quality solutions efficiently for large instances. Our experiments on instances with up to 12 transfer nodes in the City of Toronto, with a mixture of low- and high-frequency routes, illustrate the potential benefits of the proposed model over the state of the art. The results indicate that incorporating transfer buffer time, dwell time determination, and vehicle capacity consideration improves model outcomes considerably, also demonstrating the computational efficiency of our Lagrangian-based solution method.

Optimal and heuristic solutions for placing multiple finite-size rectangular facilities in an existing layout

In a companion paper (Date, K., and R. Nagi. [(2023) Optimal Placement of Multiple Finite-size Rectangular Facilities in an Existing Layout. International Journal of Production Research, THIS ISSUE]) we investigated a new problem of optimal placement of multiple finite-size rectangular facilities with known dimensions in the presence of existing rectangular facilities. We introduced the requisite theory to arrive at a solution by dividing the feasible region into sub-regions whose boundaries provide the candidates for the optimal placement. In this paper, we focus on developing optimal and heuristic solution methods for solving problem instances with various facility counts. To solve small instances with fewer facilities, we develop two optimal methods: (1) efficient explicit enumeration based on a tree state-space representation and (2) an implicit enumeration scheme that uses a flow-decomposition-based lower bound to reduce the search space and time. To solve industrial-sized instances, we have to resort to a family of construction and improvement heuristics given that the problem is NP-hard. The heuristic procedures perform well for non-pathological cases with an acceptable optimality gap. The main contribution of this paper is a range of optimal to heuristic methods that can be applied by the practitioner depending on their problem characteristics and desired efficiency.

New features for customer classification in the Flying Sidekick Traveling Salesman Problem

The increasing interest in the deployment of truck and drone delivery systems leads to the definition of new and complex vehicle routing problems. In this context, the flying sidekick traveling salesman problem (FS-TSP) is the first truck-and-drone routing problem defined in the literature. Several variants appeared in the last years differing in the operating conditions and the structure of the hybrid truck-and-drone delivery system. Exact and heuristic solution methods have been proposed in the literature to solve these problems effectively. However, the exact solution methods can generally solve only small-size instances due to the complexity of these problems. On the other hand, heuristic solution methods are able to find feasible solutions with an acceptable computational burden but without any guarantee of the quality of the solution. This work aims to investigate the possibility of using data science and machine learning techniques to reduce the complexity of solving an FS-TSP instance. The idea is to determine a good/optimal customer-to-vehicle assignment apriori to reduce the number of decisions involved in the FS-TSP solution. The assignment is determined through the classification of customers based on a subset of features specifically defined for the FS-TSP. This information can be exploited by existing solution approaches for the FS-TSP to improve their performance. An extensive computational campaign on benchmark instances is carried out with a twofold objective. On the one hand, we aim to evaluate the impact of the features on customer classification. On the other hand, we intend to show the relationship between classification and combinatorial optimization results by observing the effect of the customer classification on the FS-TSP solution.

Scatter search for capacitated minimum spanning tree

We develop a scatter search algorithm for the classical capacitated minimum spanning tree (CMST) problem. We address both the homogeneous and the more difficult heterogeneous variant of the problem. The CMST is central in a wide range of network design applications in industrial engineering, routing and logistics, and communication networks. Since the problem is NP-Complete, heuristic solution methods are often required to find high-quality solutions within practical time limits. Research efforts have been focused on complex algorithm designs combining advanced neighborhood search techniques with exact solution methods or hybrids of genetic algorithms with various alternative search techniques and likewise supplemented with optimization procedures. In this study we go back to the roots and offer an alternative algorithm that competes with the best algorithms in the literature and both avoids complicated artifacts and is simpler to implement. Key features of the proposed algorithm include an oscillation neighborhood search method characterized by two classes of neighborhoods and an evolutionary search procedure based on a scatter search framework.

A machine learning enhanced multi-start heuristic to efficiently solve a serial-batch scheduling problem

Serial-batch scheduling problems are widespread in several industries (e.g., the metal processing industry or industrial 3D printing) and consist of two subproblems that must be solved simultaneously: the grouping of jobs into batches and the sequencing of the created batches. This problem’s NP-hard nature prevents optimally solving large-scale problems; therefore, heuristic solution methods are a common choice to effectively tackle the problem. One of the best-performing heuristics in the literature is the ATCS–BATCS(β) heuristic which has three control parameters. To achieve a good solution quality, most appropriate parameters must be determined a priori or within a multi-start approach. As multi-start approaches performing (full) grid searches on the parameters lack efficiency, we propose a machine learning enhanced grid search. To that, Artificial Neural Networks are used to predict the performance of the heuristic given a specific problem instance and specific heuristic parameters. Based on these predictions, we perform a grid search on a smaller set of most promising heuristic parameters. The comparison to the ATCS–BATCS(β) heuristics shows that our approach reaches a very competitive mean solution quality that is only 2.5% lower and that it is computationally much more efficient: computation times can be reduced by 89.2% on average.

Energy consumption optimization for sustainable flexible robotic cells: Proposing exact and metaheuristic methods

Many manufacturing companies are always looking for a way to reduce energy consumption by utilizing energy-efficient production methods. These methods can be different depending on the type of products and production technology. For instance, one of the ways to increase energy efficiency and keep the precision of production is to use robots for the transportation of the parts among the machines and loading/unloading the machines. This technology is affordable compared to the technologies used in manufacturing companies. Manufacturing companies that rely on robotics technology must have a strategy to reduce energy costs and at the same time increase production by adjusting the intensity of processing or controlling the production rate. This study presents an exact solution method for flexible robotic cells to control the production rate and minimize energy consumption, which aims to both reduce electricity prices and minimize greenhouse gas (GHG) emissions under a lead time of production. Then, considering the NP-hardens nature of the problem, a heuristic solution method based on the genetic algorithm (GA) is proposed. Using the proposed approach, manufacturing companies will be able to make more accurate decisions about processing intensity and process scheduling while ensuring sustainability.

A granular iterated local search for the asymmetric single truck and trailer routing problem with satellite depots at DHL Group

AbstractTo plan the postal deliveries of our industry partner DHL Group (DHL), the single truck and trailer routing problem with satellite depots (STTRPSD) is solved to optimize mail carriers routes. In this application context, instances feature a high number of customers and satellites, and they are based on real street networks. This motivates the study of the asymmetric STTRPSD (ASTTRPSD). The heuristic solution methods proposed in the literature for the STTRPSD can either solve only the symmetric problem variant, or it is unclear whether they can also be used to solve the ASTTRPSD. We introduce an iterated local search, called ILS‐ASTTRPSD, which generates different first‐level tours in the perturbation phase, and improves the second‐level tours in the local search phase. To speed up the search, granular neighborhoods are used. The computational results on instances from the literature prove the capability of ILS‐ASTTRPSD to return high‐quality solutions. On DHL instances, ILS‐ASTTRPSD significantly decreases total travel times of the mail carriers and returns solutions with a different structure compared to the ones provided by DHL. Based on these differences, we give recommendations on how DHL could design more efficient mail carrier practices. Dedicated computational experiments reveal that considering parking and loading times when solving the ASTTRPSD leads to lower travel times, and that ignoring parking times is more counterproductive than ignoring loading times. Moreover, we assess the robustness of our solutions under parking time fluctuations. Finally, we derive properties of instances for which optimal solutions contain multiple second‐level tours rooted at the same parking spot and for which the optimal solutions of the ASTTRPSD correspond to the ones of a pure traveling salesman problem.

Order Acceptance and Scheduling Model Based on Improved Genetic Algorithm

As the manufacturing industry’s industrial structure continues to evolve, many enterprises have shifted from traditional make-to-store modes (MTS) to make-to-order modes (MTO). This paper focuses on MTO manufacturing enterprises’ order acceptance and scheduling problem, namely, how to exclude low-margin orders under capacity constraints and schedule them. We establish a hypothetical model with the goal of maximizing corporate profits, taking into account production capacity constraints and delivery time constraints. To solve the model, we propose a heuristic solution method based on genetic algorithms. The feasibility of the method is verified through randomly generated examples. The research results demonstrate that our proposed solution method can quickly solve large-scale order decision-making problems and has practical significance as a guiding tool.

Worst-case analysis of heuristic approaches for the temporal bin packing problem with fire-ups

We consider the temporal bin packing problem with fire-ups (TBPP-FU), a branch of operations research recently introduced in multi-objective cloud computing. In this scenario, any item is equipped with a resource demand and a lifespan meaning that it requires the bin capacity only during that time interval. We then aim at finding a schedule minimizing a weighted sum of the total number of bins required and the number of switch-on processes (so-called fire-ups) caused during operation. So far, research on the TBPP-FU has mainly focused on exact approaches and their improvement by valid cuts or variable reduction techniques. Although these studies have revealed the problem considered here to be very difficult to cope with, theoretical contributions to heuristic solution methods have not yet been presented in the available literature. Hence, in this article we investigate the worst-case behavior of some approximation algorithms, ranging from classic online algorithms to a more sophisticated look-ahead heuristic specifically designed for the TBPP-FU. In addition, we theoretically study three heuristics the ideas of which are inspired by solution methods for generalized bin packing problems in the field of logistics. As a main contribution, we constructively show that the feasible solutions obtained by all these approaches can be arbitrarily bad. By doing so, we (i) identify a new open problem in cutting and packing, and (ii) establish another previously unknown difference between the classical TBPP and the extended problem with fire-ups, rendering the latter the more difficult problem even from a heuristic point of view.

An energy-aware combinatorial auction-based virtual machine scheduling model and heuristics for green cloud computing

Considering the increasing demand for cloud computing, and the financial and environmental impact of the increasing energy consumption trend of data centers, improving energy efficiency is vital for cloud service providers. In this study, an energy-aware virtual machine scheduling model is proposed which is based on the multi-unit nondiscriminatory combinatorial auction. The model includes a powerful bidding language that allows users to declare their complicated virtual machine requests using logical AND and OR relations along with the time constraints. The study also presents the formal definition of the model and the associated optimization problem for determining the optimum schedule and energy-efficient placement of VMs on physical servers. The optimization problem is formulated using integer linear programming and several heuristic solution methods including the Genetic Algorithm are proposed for this problem. The performances of the model and the proposed heuristics are assessed on a comprehensive test suite. The proposed model is estimated to provide approximately a 37% improvement in revenues, and the solution methods are estimated to provide high-quality solutions within only 5% of the optimum which enable the model to be deployed in large-scale clouds.

Beyond kemeny rank aggregation: A parameterizable-penalty framework for robust ranking aggregation with ties

Rank Aggregation has ubiquitous applications in operations research, artificial intelligence, computational social choice, and various other fields. Interest in this problem has increased due in part to the need to consolidate lists of rankings and scores output by different decision-making processes and algorithms. Although most attention has focused on the variant of this problem induced by the Kemeny-Snell distance, other robust rank aggregation problems have been proposed. This work delves into the rank aggregation problem under the generalized Kendall-tau distance —a parameterizable-penalty distance measure for comparing rankings with ties— which contains Kemeny aggregation as a special case. First, it derives exact and heuristic solution methods. Second, it introduces a social choice property (GXCC) that encloses existing variations of the Condorcet criterion as special cases, thereby expanding this seminal social choice concept beyond Kemeny aggregation for the first time. GXCC offers both computational and theoretical advantages. In particular, GXCC may help to divide the original problem into smaller subproblems, while still ensuring that solving them independently yields the optimal solution to the original problem. Experiments on two benchmark datasets conducted herein show that the featured exact and heuristic solution methods can benefit from GXCC. Finally, this work derives new theoretical insights into the effects of the generalized Kendall-tau distance penalty parameter on the optimal ranking and on the proposed social choice property.

From School Buses to Carpooling: A Multifaceted Approach to Solving Vehicle Routing Problems in People Transportation

This paper delves into the complex realm of vehicle routing problems (VRPs) with a specific focus on the transportation of people, highlighting the nuanced challenges and optimization opportunities distinct from those associated with goods transportation. Among these, the Dial-a-Ride Problem (DARP) is emphasized as a pivotal area of study. DARP caters to providing personalized, accessible transportation services, particularly for individuals unable to use standard transit options, such as the elderly and handicapped. This research meticulously explores the mathematical formulation of DARP, and operational challenges and extends the discourse to encompass related VRPs in people transportation, thereby broadening the scope of investigation. Operational intricacies of DARP include managing time windows for pickups and deliveries, adhering to vehicle capacity limits, and ensuring tailored transportation solutions that optimally satisfy both operational costs and customer convenience. The paper reviews a wide spectrum of heuristic and metaheuristic solution methods developed over the years, tracing the evolution of these algorithms from their inception to the sophisticated techniques currently employed to address DARP's dynamic, static, and stochastic variants. Additionally, the research extends to related problems such as the School Bus Routing Problem (SBRP), which emphasizes the efficient and timely transportation of students, and the Carpooling Problem, focusing on organizing shared rides to reduce vehicle kilometers and enhance mobility. Demand Responsive Transportation (DRT) problems are also discussed, highlighting their importance in areas where fixed-route services are inefficient. Conclusively, the paper discusses the future scope of research in this domain, underscoring the necessity for novel solution concepts to address the growing complexity and scale of transportation demands. The integration of intermodal aspects, combining public transportation with demand-responsive services, and the advancement of hybrid algorithms are identified as key areas for future exploration. This comprehensive study not only contributes to the academic discourse on optimizing vehicle routing for people transportation but also aims to inform the development of more efficient, accessible, and customer-oriented transportation systems.

Mixed‐model assembly lines with variable takt and open stations

Industry trends such as product customization, radical innovation, and local production accelerate the adoption of mixed‐model assembly lines (MMALs) that can cope with a widening gap between model processing times and true build to order capabilitiy. The existing high work content deviations on such assembly lines stress production planning, especially the assembly line sequencing. Most manufacturers set the launching rate for all assembly line products to a fixed launching rate resulting in rising utility work and idle time when system load increases. We present an “ideal” variable rate launching (VRL) case resulting in minimal computation and achieving 100% productivity (full elimination of idle time and utility work) for balanced assembly times and homogeneous station lengths. Managers should foster the ideal circumstances where operators need not wait for a preceding task to be completed and product sequence restrictions are eliminated, thus enabling unmatched production flexibility. Furthermore, we present a mixed‐integer model to analyze both closed and open workstations on an MMAL for fixed rate launching and VRL. This model incorporates costs not only for labor inefficiencies but also for extending the line length. We present a heuristic solution method when process times and station lengths are heterogeneous and demonstrate that the variable takt dominates the fixed takt. In a numerical, industrial benchmark study, we illustrate that a VRL strategy with open stations has significantly lower labor costs as well as a substantially reduced total line length and thus lower throughput time.

Crowdsourced on-demand food delivery: An order batching and assignment algorithm

Since the early 2010s, the meal delivery business went through a veritable revolution due to online food delivery platforms. By allowing customers to quickly order from a wide range of restaurants and outsourcing currently available couriers using their vehicles (crowdsourcing), this typology of service dynamically bridges demand and supply. The main goal of online food delivery platforms consists of matching couriers to meal orders within short time intervals to provide an efficient, reliable, and sustainable service. A way to increase efficiency consists of consolidating orders into batches, such that the same courier can serve several orders in multiple pickup and drop-off routes. Since such an assignment-batching problem becomes computationally prohibitively costly in real-world scenarios characterized by a large number of customer orders as well as uncertain demand and supply, heuristic solution methods come into play. This study proposes an order batching and assignment algorithm that leverages a graph-based approach after decomposing the original problem into more tractable sub-problems employing clustering. The solution is improved by local search moves and re-optimization procedures and integrated with advanced policies to improve solutions over time. An ‘Insertion policy’ aimed at increasing batch size, and a ‘Swap policy’ aimed at identifying more efficient assignments, are implemented and compared to a ‘Myopic policy’ that does not involve any re-optimization over time. An agent-based simulation framework is developed to implement dynamic policies where couriers’ operations and movements are realistically reproduced. The performance of the developed solution approach is tested through experiments based on a real-world case study. Results show that the algorithm allows for high-quality solutions in several configurations characterized by different demand and supply patterns (e.g., density levels, couriers availability) and problem sizes.In particular, the two advanced policies investigated considerably improve the solutions.

Sustainable cooling cycles by algorithmically supported design of decentral pump systems

Accounting for up to 25% of electricity consumption in industry, pump systems are significant electricity consumers. Designing pump systems is a complex process because of the multitude of parameters and their combinations (pump types, interconnection variants, various load scenarios). Moreover, decentral pumps can be integrated to reduce the required hydraulic power, which further increases the complexity. This complexity can no longer be mastered with conventional planning methods. The aim of this paper is to tackle the complexity of pump systems design using algorithm-based design methods. The design problem is formulated as a mixed-integer nonlinear optimisation program with the objective of minimising the life cycle costs. The solution involves three steps: (i) the placement of decentral pump stations to reduce the hydraulic power, (ii) the preselection of suitable pump types, and (iii) the optimisation of the selection and operation of pumps. To solve the optimisation problem, an exact off-the-shelf solver and a problem-specific heuristic solution method are used. These are compared with conventional design methods. Both the algorithm-based and the conventional design methods are applied to an industrial cooling cycle operated by BASF SE. The results show that due to decentralisation, up to 38% (248 kW) of hydraulic power can be saved. With the help of the algorithm-based design methods, savings can also be achieved with regard to the efficiency and costs: up to 21% of the life-cycle costs can be saved while increasing the net efficiency from 47.2% to 57.8%. Furthermore, the algorithm-based design methods are superior, leading to 15% (approx. 770 000 €) lower life-cycle costs compared to conventional methods. Conventional methods based on the best efficiency point or mean efficiency show the lowest level of performance. The advantages are particularly evident in system variants with a high complexity. The trade-off between investment costs and energy demand is presented transparently using a Pareto front.

Kinematic Orienteering Problem With Time-Optimal Trajectories for Multirotor UAVs

In many unmanned aerial vehicle (UAV) applications for surveillance and data collection, it is not possible to reach all requested locations due to the given maximum flight time. Hence, the requested locations must be prioritized and the problem of selecting the most important locations is modeled as an Orienteering Problem (OP). To fully exploit the kinematic properties of the UAV in such scenarios, we combine the OP with the generation of time-optimal trajectories with bounds on velocity and acceleration. We define the resulting problem as the Kinematic Orienteering Problem (KOP) and propose an exact mixed-integer formulation together with a Large Neighborhood Search (LNS) as a heuristic solution method. We demonstrate the effectiveness of our approach based on Orienteering instances from the literature and benchmark against optimal solutions of the Dubins Orienteering Problem (DOP) as the state-of-the-art. Additionally, we show by simulation \color{black} that the resulting solutions can be tracked precisely by a modern MPC-based flight controller. Since we demonstrate that the state-of-the-art in generating time-optimal trajectories in multiple dimensions is not generally correct, we further present an improved analytical method for time-optimal trajectory generation.