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The e-commerce industry has experienced significant growth in the past decade, particularly post-COVID. To accommodate such growth, the parcel delivery sector has also grown rapidly. However, there is a lack of study that properly evaluates its social and environmental impacts at a large scale. A model is proposed to analyze such impacts. A parcel generation process is presented to convert public data into parcel volumes and stops. A continuous approximation model is fitted to estimate the length of parcel service tours. A case study is conducted using New York City (NYC) data. The parcel generation is shown to be a valid fit. The continuous approximation model parameters have R2 values of 98% or higher. The model output is validated against UPS truck trips. Application of the model to 2021 suggests residential parcel deliveries contributed to 0.05% of total daily vehicle-kilometer-traveled (VKT) in NYC corresponding to 14.4 metric tons of carbon equivalent (MTCE) emissions per day. COVID-19 contributed to an increase in parcel deliveries that led to up to 1 064.3 MTCE of annual greenhouse gas (GHG) emissions in NYC (which could power 532 standard US households for a year). The existing bike lane infrastructure can support the substitution of 17% of parcel deliveries by cargo bikes, which would reduce VKT by 11%. Adding 3 km of bike lanes to connect Amazon facilities can expand their cargo bike substitution benefit from a VKT reduction of 5% up to 30%. If 28 km of additional bike lanes are made, parcel delivery substitution citywide could increase from 17% to 34% via cargo bike and save an additional 2.3 MTCE per day. Cargo bike priorities can be set to reduce GHG emissions for lower-income neighborhoods including Harlem, Sunset Park, and Bushwick.

This paper develops a continuous approximation model for analyzing the effect of road pricing on the spatial distribution of traffic flow. The traffic flow density, which describes traffic flow as a function of position, is derived for a rectangular city with a grid network. The analytical expression for the traffic flow density demonstrates how the toll level, the size and shape of the toll area, and the shape of the city affect the spatial distribution of traffic flow. As the size of the toll area increases, reducing the traffic flow density at the city center becomes difficult. As the aspect ratio of the toll area increases, the traffic flow density at the city center increases. The shape of the city has less impact on the traffic flow density than the shape of the toll area. [ABSTRACT FROM AUTHOR]

The upcoming wireless technology developments in the next generations are expected to substantially transform the vehicle-to-everything (V2X) communication network. The challenge of limited spectrum resources in V2X communication, caused by the need for high data rates, necessitates a thorough analysis of spectrum resource allocation and power control. This complex problem falls under the domain of mixed-integer nonlinear programming; a strategic approach is implemented to overcome these issues, which divides the main challenge into two sub-problems. The issue of resource allocation is addressed by implementing a multiaccess spectrum allocation method, which is deliberately designed to optimize the utilization of the spectrum resources that are currently accessible. Concurrently, the power control issue is resolved by employing a continuous convex approximation technique, which effectively converts non-convex power-allocation issues into convex equivalents. This approach helps to alleviate interference between users. Finally, the simulation results prove that the proposed approaches can improve vehicle performance. The algorithms proposed in this article significantly improve the system throughput and access rate of vehicular user equipment (VUEs) while ensuring the data rate of cellular user equipment (CUEs). [ABSTRACT FROM AUTHOR]

This work proposes a hybrid block numerical method of tenth order for the direct solution of fifth-order initial value problems. The formulas that constitute the block method are derived from a continuous approximation obtained through interpolation and collocation techniques. In order to obtain better accuracy, sixth-order derivatives are incorporated to develop the formulas. The main characteristics of the method are analyzed, namely, the order, local truncation errors, zero-stability, consistency and convergence. The proposed strategy performs well, as shown by some numerical examples and the corresponding efficiency curves. Compared to existing numerical methods in the literature, the proposed method is competitive and the numerical approximations it provides are significantly close to the precise solutions. [ABSTRACT FROM AUTHOR]

We study the design of same-day delivery (SDD) systems under the assumption that service regions are allowed to vary over the course of the day; equivalently, that customers in different locations may have access to SDD for different lengths of time over the service day or may have no access at all. This contrasts with the bulk of the literature, in which a service region is defined in advance and all customers in the service region can place SDD orders during the same time window. Leveraging continuous approximation techniques to capture average-case system behavior, we derive optimal service region areas and corresponding SDD order cutoff times to maximize the expected number of orders served per day. We quantify the benefit of allowing the service regions to vary, both theoretically and empirically, and discuss related equity issues in SDD systems. We illustrate and validate our results with a case study set in the Phoenix, Arizona metropolitan area. • We study a same-day delivery (SDD) system with a time-varying service region. • We develop and rigorously analyze a continuous approximation model of the system. • We show that allowing service regions to vary increases the quantity of orders served. • We test and validate our model via simulations on a real road network. [ABSTRACT FROM AUTHOR]

The upcoming wireless technology developments in the next generations are expected to substantially transform the vehicle-to-everything (V2X) communication network. The challenge of limited spectrum resources in V2X communication, caused by the need for high data rates, necessitates a thorough analysis of spectrum resource allocation and power control. This complex problem falls under the domain of mixed-integer nonlinear programming; a strategic approach is implemented to overcome these issues, which divides the main challenge into two sub-problems. The issue of resource allocation is addressed by implementing a multiaccess spectrum allocation method, which is deliberately designed to optimize the utilization of the spectrum resources that are currently accessible. Concurrently, the power control issue is resolved by employing a continuous convex approximation technique, which effectively converts non-convex power-allocation issues into convex equivalents. This approach helps to alleviate interference between users. Finally, the simulation results prove that the proposed approaches can improve vehicle performance. The algorithms proposed in this article significantly improve the system throughput and access rate of vehicular user equipment (VUEs) while ensuring the data rate of cellular user equipment (CUEs).

This paper discusses the effect of road pricing on the spatial distribution of traffic flow. The traffic flow density is derived for a circular city with a radial-arc network. The traffic flow density describes the amount of traffic as a function of position and allows us to identify the location of potential congestion areas. The analytical expression for the traffic flow density demonstrates how the size of the toll area and the toll level affect the spatial distribution of traffic flow. As the size of the toll area increases, the decrease in traffic flow inside the toll area becomes smaller. As the toll level increases, the increase in traffic flow at the boundary of the toll area becomes greater. The effect of the travel cost on the spatial distribution of traffic flow is also examined. These findings can be used to determine the size of the toll area and the toll level required to achieve a certain level of traffic congestion. [ABSTRACT FROM AUTHOR]

The majority of research in the continuous approximation paradigm has emphasized routing problems, such as the travelling salesman problem and the vehicle routing problem. This article instead focuses on continuous approximation formulas for problems involving location, specifically the k$$ k $$‐medians, k$$ k $$‐dispersion, and generalized minimum spanning tree problems. We contribute bounds for constants that describe the growth rate of the cost of these problems as the number of demand points becomes large, and conduct computational experiments that verify that they provide a good approximation in practice. [ABSTRACT FROM AUTHOR]

Intercity Demand Responsive Transit (IDRT) systems, providing on-demand "door-to-door" services for intercity travelers, have been recently implemented in a few cities and gained popularity. To devise a theoretical tool for designing efficient IDRT systems, this study proposes a continuous approximation (CA) model to describe the operation metrics of the IDRT. In a two-city system, an optimization model is proposed to study the zoning strategies in the main city, the service area in the satellite city, and headways. The objective is to minimize the average system cost, including the costs of both the operators and the passengers. A two-stage partition algorithm is proposed to generate a near-optimal solution by combining the greedy algorithm and sensitivity-based coordination optimization method. Subsequently, we develop a direct solution algorithm to establish a lower bound for the optimal solution, facilitating a comparison with the near-optimal solution. Throughout all sensitivity analyses, the gaps between the near-optimal solution and the lower-bound solution remain below 3 %. The comparison results demonstrate that the computation time and solution quality of the two-stage partition algorithm are considered acceptable. Simulation is performed to test the validity of the CA model, and the errors in the average system costs are less than 1.16 %. Sensitivity analyses are carried out to evaluate the impacts of several key parameters (e.g., demand densities and line-haul distances) on the optimization results. By comparison with Intercity Fixed Route Transit (IFRT), it can be seen that the choice of IDRT depends on passengers' preference for door-to-door transportation. This study may serve as a valuable reference for the practical implementation of IDRT. [ABSTRACT FROM AUTHOR]

• Mixed car-sharing: station-based and free-floating layers working simultaneously. • Optimal design for each city context: vehicle fleet, parking, and repositioning. • Possibility of considering electrical vehicles not implying additional restrictions. • Part of the demand unserved if the system is to be economically sustainable. • Mixed systems will generally perform better. One-way car-sharing systems have become a popular urban transportation mode in many cities worldwide. First pilot implementations were originally station-based, so that trips needed to be station-to-station. This configuration created implementation difficulties in many cities and higher infrastructure costs for operating agencies. The appearance of free-floating on-street systems eased these limitations and implied an important growth of car-sharing implementations. In spite of these, free-floating systems are not free of planning and operative difficulties, which if not addressed carefully, might imply the economical unsustainability of the system. The idea of mixed systems, where a free-floating system and a station-based system complement each other, is new. The objective is to exploit the potentialities of both designs, while limiting their respective drawbacks. This paper presents a parsimonious model from which to derive the optimal strategical design variables for mixed car-sharing systems (i.e. the vehicle fleet size, the number of stations and the required intensity of rebalancing operations). This requires an integrated view of the system, allowing the optimization of the trade-off between the costs incurred by the operating agency and the level of service offered to users. The approach is based on the modelling technique of continuous approximations, which requires strong simplifications but allows obtaining very clear trade-offs and insights. The model has been applied to a case study taking the parameters from the city of Barcelona. Results prove the profitability of mixed car-sharing systems, which in particular contexts, is higher than that of pure free-floating or pure station-based systems on their own. Furthermore, if electrical cars are used, results show that battery recharging will not imply an active restriction to the system configuration. In conclusion, the proposed modeling approach represents a tool for the strategic design of car-sharing systems in the planning phase and provides guidelines for their adequate development, contributing to a more sustainable urban mobility. [ABSTRACT FROM AUTHOR]

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With recent advances in mobile technology, public transit agencies around the world have started actively experimenting with new transportation modes, many of which can be characterized as on-demand public transit. Design and efficient operation of such systems can be particularly challenging, because they often need to carefully balance demand volume with resource availability. We propose a family of models for on-demand public transit that combine a continuous approximation methodology with a Markov process. Our goal is to develop a tractable method to evaluate and predict system performance, specifically focusing on obtaining the probability distribution of performance metrics. This information can then be used in capital planning, such as fleet sizing, contracting, and driver scheduling, among other things. We present the analytical solution for a stylized single-vehicle model of first-mile operation. Then, we describe several extensions to the base model, including two approaches for the multivehicle case. We use computational experiments to illustrate the effects of the inputs on the performance metrics and to compare different modes of transit. Finally, we include a case study, using data collected from a real-world pilot on-demand public transit project in a major U.S. metropolitan area, to showcase how the proposed model can be used to predict system performance and support decision making. [ABSTRACT FROM AUTHOR]

This paper develops a continuous approximation model for analyzing the spatial distribution of traffic flow in the presence of a barrier to travel. The traffic flow density is derived for a rectangular city with a grid road network and a rectangular barrier. The traffic flow density describes traffic flow as a function of position and provides a fundamental understanding of the effect of the barrier. The analytical expression for the traffic flow density demonstrates how the location and shape of the barrier affect the spatial distribution of traffic flow. The effect of the shape of the city is also examined. The model is useful for transportation and logistics problems with barriers. [ABSTRACT FROM AUTHOR]

The quantitative analysis of pulsed-chemical exchange saturation transfer (CEST) using a full model-based method is computationally challenging, as it involves dealing with varying RF values in pulsed saturation. A power equivalent continuous approximation of B 1 power was usually applied to accelerate the analysis. In line with recent consensus recommendations from the CEST community for pulsed-CEST at 3T, particularly recommending a high RF saturation power (B 1  = 2.0 µT) for the clinical application in brain tumors, this technical note investigated the feasibility of using average power (AP) as the continuous approximation. The simulated results revealed excellent performance of the AP continuous approximation in low saturation power scenarios, but discrepancies were observed in the z-spectra for the high saturation power cases. Cautions should be taken, or it may lead to inaccurate fitted parameters, and the difference can be more than 10% in the high saturation power cases.

• Formulated continuous approximation model for optimizing AB-type skip-stop service under heterogeneous demand. • Developed an efficient algorithm that solves the model to near-optimality. • Proposed a recipe for fitting the continuous approximation solution to real settings. • AB-type service outperforms all-stop service under various operating conditions. Transit vehicles operating under skip-stop service visit only a subset of the stops residing along a corridor. It is a strategy commonly used to increase vehicle speeds and reduce patron travel times. The present paper develops a continuous approximation model to optimally design a select form of skip-stop service, termed AB-type service. The model accounts for demand patterns that slowly vary over space. An efficient heuristic is developed to obtain solutions. These are shown to be near-optimal for a variety of numerical examples. Results also indicate that optimal AB-type designs outperform optimized all-stop service in a variety of cases. The AB-type service is found to be especially competitive when travel demands are high, trip origins are unevenly distributed along a corridor, and patrons have relatively high values of time. In these cases, AB-type service is found to reduce system costs by as much as 8%. [ABSTRACT FROM AUTHOR]

This paper seeks an analytical estimate of the expected distance for visiting an arbitrary subset of independently and uniformly distributed random points within a compact region. This problem has many real-world application contexts such as the emerging on-demand transportation and logistics services (e.g., ridesharing, customized buses). The lower bounds of the expected optimal tour length are analytically derived by considering a so-called "trapping effect", which explicitly addresses probabilistically the situation that some of the tour legs must connect points that are not neighbors. A parametric approach is developed to estimate the expected optimal tour length for both Euclidean and rectilinear metrics. Numerical experiments demonstrate the validity of these bounds, as well as the closeness of the proposed estimator to simulated results. • Parametric estimator of the expected minimum distance for visiting an arbitrary subset of random points in a compact region. • Closed-form formulas to gauge the expected extra distance due to a "trapping effect." • Theoretical and numerical analysis of the impacts of the region's shape and boundary. • High accuracy of the proposed estimator under the Euclidean and rectilinear metrics. [ABSTRACT FROM AUTHOR]

• Integrated optimization of electric scooter rebalancing and on-board charging operations. • Continuous approximation for local pickup, delivery, and charging. • Discrete-continuous hybrid model to solve large scale routing and inventory problems. • Non-linear transition of scooters' state of charge over time. • Numerical examples demonstrating effectiveness of the proposed approach. This paper presents a sequence of models for optimal overnight charging and rebalancing of shared electric scooters (e-scooters) by allowing e-scooters to be charged while being transported on rebalancing vehicles. This problem is first modeled as a mixed-integer program for the multi-commodity inventory routing problem, where commodities represent e-scooters with different states of charge. To avoid prohibitive computation burden, continuous approximation techniques are proposed to estimate costs associated with the pickup and drop-off operations in small local neighborhoods, and the formulation turns into a discrete-continuous hybrid model for the integrated operations at both local and line-haul levels. A series of numerical experiments are conducted to demonstrate that, as compared to direct application of the discrete formulation, the proposed hybrid approach can produce good quality solutions for large-scale instances in a much shorter computation time. [ABSTRACT FROM AUTHOR]

While optimal forest harvesting regulations (regional forest regeneration schedules) have been established using the linear programming technique, its modeling restriction is still a challenge in practice. In this study, we developed an alternative framework for establishing an optimal forest harvesting regulation using the continuous approximation technique. The regulation problem was reformulated as a problem to find optimal smooth control of regeneration area, thereby overcoming the difficulties in finding the global optimum for the optimization models that include nonlinear or more complex models. A seven-dimensional optimization model was developed involving efficient assurance of the convergence to a stable forest state and prohibition of clearcutting in immature stands. Using this model, we established optimal forest harvesting regulations in Nagano Prefecture, Japan. Simulated annealing was utilized to explore optimal solutions. Analyses based on extreme value theory and comparison with solutions produced by grid search indicated that the best solutions may be sufficiently close to global optima. The best solutions suggested controlling timber supply to keep prices high in the early years, due to the supply–demand log price model and the use of net present value as the objective function. Because of the low profitability of the region, the solution suggested delaying the achievement of the target state as far as possible.

Using crowdsourced delivery capacity, that is, individuals offering their vehicle and their time to perform deliveries, can allow companies to provide faster delivery options and more easily accommodate fluctuations in demand. However, because of the uncertainty associated with crowdsourced delivery capacity, ensuring service quality is more challenging. To prevent or mitigate any negative effects of the uncertainty associated with crowdsourced delivery capacity, companies may choose to also have a scheduled delivery workforce that they can control more effectively. We investigate continuous approximation and value function approximation methods for scheduling this workforce, that is, deciding their shifts (start time and duration) to achieve a service level target at minimum cost. An extensive computational study demonstrates the efficacy of our methods and provides insights into the use of crowdsourced delivery capacity. [ABSTRACT FROM AUTHOR]

• The impact of route choice modeling on transit system performances is examined. • Optimal user costs are affected by route choice modeling, but agency costs are not. • The system design is often insensitive to route choice modeling. • Details of route choice are generally negligible, yet notable exceptions exist. Sketchy design models are used to examine the fundamental tradeoff in transit systems and to guide high-level decisions. Traditionally, passengers' route choice is greatly simplified in such models. This study aims to understand whether this simplification would compromise qualitatively the results expected from the sketchy models. To this end, three transit systems, which all offer competitive alternative routes, are analyzed using the continuous approximation approach. We test what the impact on transit system performance (e.g. optimal designs and system costs) would be if travelers somehow split between these routes, rather than concentrate on the "best" one. A random utility model is employed to enable a probabilistic assignment of passengers to different routes according to the "perceived" utility. Analytical methods are then developed to estimate the aggregate share of each route in each system, based on which the user cost is obtained. Numerical results show that, while stochastic route choice modestly increases the optimal user cost, it has a negligible effect on the agency cost. Furthermore, the actual system design is largely insensitive to route choice modeling. Thus, while the simplest deterministic route choice assumption may not be valid in all systems, transit planners can safely ignore route choice details in most cases, at least for the purpose of strategic planning. [ABSTRACT FROM AUTHOR]

Truck platooning involves a small convoy of freight vehicles using electronic coupling as an application in automated driving technology, and it is expected to represent a major solution for improving efficiency in truck transportation in the near future. Recently, there have been several trials regarding truck platooning with major truck manufacturers and logistics companies on public roads in the United States, European countries and Japan. There is a need to locate a facility for the formation of truck platooning to realize the unmanned operation of trucks following in a platoon. In this study, we introduce the current status of truck platooning in Japan and present the optimal location model for truck platooning using the continuous approximation model with a numerical experiment, considering the case in Japan. We derived the optimal locational strategy for the combination of the long-haul ratio and the cost factor of platooning. With parameters estimated for several scenarios for the deployment of truck platooning in Japan, the numerical results show that the optimal locational strategy for a platoon of manned vehicles and a platoon with unmanned following vehicles is the edge of the local region, and that for a platoon of fully automated vehicles is the center of the region.

• The basic trade-off in six distinctive transit systems is examined. • Ride-pooling is unlikely to change the economy of scale in transit design. • Ride-pooling promises modest improvements to the overall system efficiency. • Ride-pooling tilts the balance of trade-off in the user's favor. This paper examines the basic design trade-off in six distinctive transit systems, including two state-of-the-art fixed-route systems and four hybrid systems that use ride-pooling as an integrated feeder service. The systems are analyzed using a continuous approximation approach, and the resulting optimal design problem is formulated and solved as a mixed integer program. We find that ride-pooling changes little the fundamental laws inherent in transit design. Specifically, all six systems, despite their seemingly vastly different design features, display the following laws: (1) that the per capita agency cost correlates linearly with the per capita user cost, (2) that both costs are power functions of the demand density with an exponent close to − 1 3 and − 0.4 for agency and user costs, respectively, and (3) that the per capita agency cost is not significantly affected by city size but the user cost is. This finding suggests that ride-pooling may have a rather limited impact on the economy of scale in mass transit systems. However, mixing ride-pooling with fixed-route services does promise modest improvements to the overall system efficiency. It also tilts the balance of trade-off considerably in the user's favor, at the operator's expense. [ABSTRACT FROM AUTHOR]

Abstract The fabrication of biomedical devices close to hospitals using Additive Manufacturing (AM) technology has been garnering a lot of attention due to the many potential benefits it provides. Unlike biomedical devices fabricated via Traditional Manufacturing (TM), usually available from suppliers out of the immediate region, biomedical implants manufactured through AM enables surgeons to receive more customized, sophisticated, and patient-specific parts on-demand with faster response, lower inventory level, and reduced delivery costs. However, the AM deployment approach which determines how close the manufacturing point is to the hospitals has a huge impact on the supply chain, and influences the amount of these benefits reaped by the patient, hospitals, and also the AM providers. The choice of deployment approach (central, distributed, or hybrid) remains an open question requiring careful investigation due to the relatively high AM machine and personnel costs, as well as uncertainties in the future. This study proposes a Continuous Approximation (CA) model that quantifies the supply chain network cost associated with AM-produced biomedical implants. We present an optimization algorithm that calculates the locations of the AM machines and the hospitals that they serve (otherwise known as the AM facility's influence area), the quantity of raw materials to be kept in inventory at a central raw material warehouse (CRW), and the number of distributed AM facilities to minimize the total network cost and achieve a satisfactory patient satisfaction. We apply the cost model to a real-world case study that focuses on the use of biomedical implants in hospitals in 12 states of southeastern USA. Moreover, to demonstrate the effect of different parameters affecting the optimal decision, a number of different sensitivity analysis have been conducted. [ABSTRACT FROM AUTHOR]

This study presents a methodological framework for determining the most cost-effective design parameters for bike networks in cities with heterogenous travel demands. The suggested model utilizes continuous approximation techniques to calculate the optimal length and spacing between bike lanes, determine the number and location of bike-sharing stations, and estimate the appropriate fleet size (number of bikes). Throughout the process, both user and agency interests are considered. To validate the effectiveness of the proposed model, an example is provided using data from the City of Guayaquil, Ecuador. Additionally, sensitivity analysis is performed to gain further insights into the impact of significant parameters on the network design. The results indicate that agency costs constitute a relatively small proportion of the overall network costs compared to user costs. This finding suggests that the initial investment in facilities has been amortized over time, highlighting the network's long-term viability. Moreover, the study demonstrates that an increase in demand does not necessarily require a corresponding increase in investment. To strike a balance, the model tends to offer a reduced number of facilities, such as bikes, stations, and kilometers of lanes per person, while simultaneously densifying the network to reduce user access times, resulting in shorter total travel times for users. The proposed optimization model serves as a valuable tool for urban planners, aiding them in strategic design decisions during the planning stage of bike mobility systems. [ABSTRACT FROM AUTHOR]

While ever-growing bio-ethanol production poses considerable challenges to the bioenergy supply chain, the risk of refinery operation disruptions further compromises the efficiency and reliability of the energy supply system. This paper applies discrete and continuous reliable facility location models to the design of reliable bio-ethanol supply chains so that the system can hedge against potential operational disruptions. The discrete model is shown to be suitable for obtaining the exact optimality for small or moderate instances, while the continuous model has superior computational tractability for large-scale applications. The impacts of both site-independent and dependent disruptions (i.e., due to flooding) are analyzed in empirical case study for the State of Illinois (one of the main biomass supply states in the U.S.). The reliable solution is compared with a deterministic solution under the same setting. It is found that refinery disruptions, especially those site-dependent ones, affect both optimal refinery deployment and the supply chain cost. Sensitivity analysis is also conducted to show how refinery failure probability and fixed cost (for building biorefineries) affect optimal supply chain configuration and the total expected system cost.

Dry port can speed the flow of cargo between ships and major land transportation networks while reducing carbon emissions and relieving seaport congestion. This study introduces a continuous approximation model to facilitate the design of a seaport-dry-port network that takes carbon emissions into account. Multimodal transport with two transport modes: rail and road are considered. Containers are delivered quickly from shippers to dry ports through road transport; then transferred to seaport through rail transport to save money and reduce carbon emission. For decision-making scheme, a game theory is used to help the seaport, dry ports, and shippers optimize their own benefits and guide them to determine their decision variables. A nonlinear optimization technique is used to solve this seaport-dry-port network design problem. The closed-form of all decision variables is found directly. A numerical study is conducted to derive managerial insights. The results show that the development of dry port concepts and multimodal transports could reduce the carbon cost from road transport. [ABSTRACT FROM AUTHOR]

In this paper, the dynamics of local finite-time Lyapunov exponents of a 4D hyperchaotic system of integer or fractional order with a discontinuous right-hand side and as an initial value problem, are investigated graphically. It is shown that a discontinuous system of integer or fractional order cannot be numerically integrated using methods for continuous differential equations. A possible approach for discontinuous systems is presented. To integrate the initial value problem of fractional order or integer order, the discontinuous system is continuously approximated via Filippov's regularization and Cellina's Theorem. The Lyapunov exponents of the approximated system of integer or fractional order are represented as a function of two variables: as a function of two parameters, or as a function of the fractional order and one parameter, respectively. The obtained three-dimensional representation leads to comprehensive conclusions regarding the nature, differences and sign of the Lyapunov exponents in both integer order and fractional order cases. [ABSTRACT FROM AUTHOR]

In this paper, it is shown numerically that a class of fractional-order piece-wise continuous systems, which depend on a single real bifurcation parameter, have no zero Lyapunov exponents but can be chaotic or hyperchaotic with hidden attractors. Although not analytically proved, this conjecture is verified on several systems including a fractional-order piece-wise continuous hyperchaotic system, a piece-wise continuous chaotic Chen system, a piece-wise continuous variant of the chaotic Shimizu-Morioka system and a piece-wise continuous chaotic Sprott system. These systems are continuously approximated based on results of differential inclusions and selection theory, and numerically integrated with the Adams-Bashforth-Moulton method for fractional-order differential equations. It is believed that the obtained results are valid for many, if not most, fractional-order PWC systems. [ABSTRACT FROM AUTHOR]

Matrix factorization is among the most popular approaches for matrix completion, with recent advances including gradient-based and deep-learning-based methods. Even though many applications involve matrices with discrete values, most of the existing matrix factorization models focus on the continuous domain. Discretization is applied as an additional step, often using a heuristic mapping that results in sub-optimal solutions, which either do not take into account the structure of the matrix or introduce significant quantization errors. In this letter, we propose a novel method that allows gradient-based and deep-learning-based methods to jointly learn both the matrix factorization model and a discretization operator. By introducing a loss function that accounts for the reconstruction error with respect to the discrete predictions, we obtain a discrete matrix completion algorithm with high reconstruction accuracy. Experiments using well-known datasets show the improvement obtained by the proposed algorithm over the state of the art. [ABSTRACT FROM AUTHOR]

In this paper, a continuous approximation to studying a class of PWC systems of fractional-order is presented. Some known results of set-valued analysis and differential inclusions are utilized. The example of a hyperchaotic PWC system of fractional order is analyzed. It is found that without equilibria, the system has hidden attractors. [ABSTRACT FROM AUTHOR]

Growing awareness of environmental issues is prompting the development of sustainable supply chain management. Closed-loop supply chains in which used products can be returned for remanufacture are becoming increasingly popular. This paper introduces a two-phase approach to the design of supply chain networks taking into account carbon emission and remanufacturing. In the first phase, a continuous approximation model is used to design the forward supply chain network. The objective is to minimize the total forward network cost by simultaneously determining the number and the service areas of distribution centers (DCs) and the replenishment cycle time for DCs. A nonlinear optimization technique is used to solve the forward supply chain network design problem. In the second phase, a reverse supply chain network is formulated based on the results of the first phase to determine the optimal number and service areas of remanufacturing centers (RCs) and the replenishment cycle time for RCs. Finally, numerical analyses are conducted to show the solution approach and provide some managerial insights. [ABSTRACT FROM AUTHOR]

This paper studies the design problem of a demand adaptive paired-line hybrid transit system with a radial network structure. Depending on whether the demand adaptive service is operated along circular or radial transit lines, two variants of such systems are considered: a Circular Model (or C-Model) and a Radial Model (or R-Model). A continuous approximation approach is employed to develop the optimal design problem, which is formulated as a mixed integer program. A comprehensive numerical experiment is performed to compare various cost components corresponding to the optimal design of the two systems, and a discrete-event simulation is developed to validate the analysis. Numerical and simulation results suggest that the radial route network outperforms the grid route network, which was analyzed previously in the literature, with a significant margin, while the C-Model and R-Model offer similar performance. [ABSTRACT FROM AUTHOR]

Continuous approximation (CA) is an efficient and parsimonious technique for modeling complex logistics problems. In this paper, we review recent studies that develop CA models for transportation, distribution and logistics problems with the aim of synthesizing recent advancements and identifying current research gaps. This survey focuses on important principles and key results from CA models. In particular, we consider how these studies fill the gaps identified by the most recent literature reviews in this field. We observe that CA models are used in a wider range of applications, especially in the areas of facility location and integrated supply chain management. Most studies use CA as an alternative and a complement to discrete solution approaches; however, CA can also be used in combination with discrete approaches. We conclude with promising areas of future work. [ABSTRACT FROM AUTHOR]

The acceleration of global e-commerce brings an increasing environmental burden to urban last-mile logistics. Autonomous delivery robots (ADRs) have often been considered as an attractive solution to this challenge but, to date, their environmental impact had not been fully assessed. To fill this gap, a life-cycle analysis of two-echelon and business-as-usual distribution strategies is proposed in this paper. To model ADR production, primary data from an actual prototype is used. The mathematical formulation of the use stage is done using the continuous approximation methodology. Finally, some managerial insights are obtained. Two-echelon operations would generate between 60 and 130 gCO 2 -eq per parcel delivery depending on the considered operation scenario. The ADR fleet production and renewal are the biggest contributors to this total global warming potential (GWP). As a consequence, the three main leverages to decrease the GWP of an ADR-based two-echelon delivery scheme are an improvement of the ADR production processes, the maximization of the robot lifespan (both for mechanical parts and battery), and the optimization of delivery operations to minimize the robot fleet size. [ABSTRACT FROM AUTHOR]