CC BY
9PHYSICS AND MATHEMATICS
A METHOD OF ENHANCING THE MOBILE VEHICLES DEPLOYMENT EFFECTIVENESS IN
BUILT-UP AREAS
Olesiuk I.
Master of the 2nd course of the Institute of Applied System Analysis National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute"
Mukhin V.
Doctor of engineering sciences, Professor National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute"
ABSTRACT
Nowadays bike-sharing systems are extremely popular all over the world. They have been installed in many cities. The main reason for installing such systems is to reduce vehicle pollution, difficulties with parking and frequent traffic congestion. The key to the success of such systems is the ability to meet the ever-changing demand for bicycles and vacant places at each station. This is achieved by rebalancing operation, which is done by a fleet of vehicles that are transferring bikes from station to station. In this paper, we address the Bike-sharing Rebalancing Problem (BRP), in which a fleet of capacitated vehicles is used with the objective of minimizing total cost of redistributing bicycles among the stations. An integer linear programming formulation to model bike-sharing static rebalancing problem is presented in this article. This formulation considers the number of damaged bicycles and the cost for maintenance of fleet vehicles that participate in rebalancing. The mathematical formulations of the BRP were first computationally tested using data obtained for the city of Arlington. Branch and cut algorithm is developed to solve this problem. The effectiveness of the presented approach was verified by a detailed numerical example.
Keywords: bike-sharing systems, static rebalancing, vehicle routing, pickup and delivery problems.
Introduction
Due to the global ongoing urbanization trends, local authorities face a host of problems like high traffic, energy consumption, noise pollution, air pollution, and so on. Bike-sharing systems play a prominent part in solving such problems. The first system for exchanging
mobile vehicles was a bicycle station installed in Amsterdam in 1965. During the last few years, the number of systems increased dramatically, reaching in 2011 more than 400 in Europe. In general, the number of bike-sharing systems in the world is growing rapidly (Fig. 1), with an overall growth rate of about 45% since 2007.
Fig. 1 - Number of European cities that have mobile vehicles sharing systems.
At the end of 2014, there were about 800 cities with bike-sharing systems. Detailed descriptions and examples can be found in [1, 2]. This is not surprising that bike-sharing systems have become extremely popular in recent years. Bikes are chosen by both young and elder generations to move through the city because everybody can easily learn to use them in just a couple of minutes. Moreover, taking into account the events in the modern world, society must pay attention to the problems of ecology and the environment.
Also, in research [3] it was stated that bike and scooter sharing systems help to save on road maintenance, transport infrastructure, etc. For instance, such systems in Washington, DC, allowed saving about
800$ per road, also each trip gave about $ 7 to the local economy.
In contrast, such rapid development of bike-sharing has resulted in many problems, such as damaged bicycles, bike parking needs, and bicycle maintenance. In many cities, streets are filled with a great number of bicycles parked everywhere. The parking problem is especially critical and has aroused people's concern. Currently, one of the main problems in the management of shared systems is to ensure the possibility of renting and parking bicycles at any station anytime. Rebalancing is used to solve this problem. This term refers to the repositioning of goods using vehicles with fixed capacity on the basis of the central depot. For example, trucks
or buses pick up bicycles from stations where demand is too low and deliver them to stations where the demand is high. A certain number of spare bikes are usually stored in the depot to allow wider distribution and replace some bikes in case of breakage.
Literature review
Even though bike-sharing is very popular worldwide, the literature on the BRP is rather sparse compared with those of the classical VRP and TSP. There are mainly two types of rebalancing - dynamic and static. The first type considers the case in which the number of bikes required by and presence in each station. This problem is always found in the daytime repositioning operation or the 24-h bike-sharing operation. In the static version, a snapshot of the level of occupation at the stations is taken and used to plan the redistribution. The static rebalancing is usually performed at the beginning of every night, based on the status of the system at definite time and the demand forecast for the next day.
There are a lot of papers where the bike-sharing problem is considered as static. One of them is
[4], where authors introduce the problem as one commodity TSP problem with pickup and delivery. The mathematical formulation is provided for both symmetric and asymmetric versions. Only one car is used to redistribute bicycles. The solution to the problem is carried out using branch and cut algorithm.
Later, the results of this work were improved in
[5], where the authors proposed two simple heuristics. Both works consider the case, where vehicles are allowed to leave depo with some load, but the resulting sum of requests should be zero.
In [6] authors also consider the rebalancing operation as one commodity pickup and delivery problem with routing one single capacitated vehicle. A branch-and-cut algorithm is proposed for solving a relaxation of the problem. An upper bound of the optimal solution of the problem is obtained by a tabu search, which is based on some theoretical properties of the solution. The initial solution for a tabu search is provided by the greedy or relaxation method. Such approach allowed to solve large problems with more than 100 stations.
However, redistribution by a single vehicle is only effective for small stations with not too long distances. For large bike-sharing systems, several transport units are usually used, as one car is not enough to bypass all stations. Moreover, rebalancing usually takes a long time, and time resource is limited.
In [7] the authors consider the possibility to use several trucks and offer scalable approaches to find an accurate and approximate solution. Based on the modeling approach, time-indexed and arc-indexed formulations are offered. The chosen objective function is based on practical considerations and is quite unusual compared to typical routing problems as it focuses on minimizing the user dissatisfaction in face of stochastic demand. The authors provide examples of benchmarks for both formulations.
A dynamic version of the BRP where the fleet of vehicles is heterogeneous was considered by Contardo et al in [8]. They presented time-indexed formulations
that are solved via Dantzing-Wolfe and Benders' decomposition-based heuristics.
In this paper, a static approach is used. It allows for solving the rebalancing problem without considering traffic congestion and parking problems. This approach is useful for arranging the inventory of bicycles in the system for the next day and is widely used in small cities. It can also be combined with dynamic repositioning for reducing the amount of work required in the latter mode. The problem of rebalancing is considered as the vehicle routing problem with simultaneous pickup and delivery (VRPSPD). To solve it mixed-integer linear programming formulation, which extends the multicommodity VRPSPD is presented. In the current model, vehicles should deliver bikes to stations and also pick up broken bicycles. Also, the costs needed for the maintenance of a particular fleet vehicle are taken into account.
Problem description
Static BRP involves determining the best route for bypassing stations with the collection or supply of bikes with a fleet of vehicles that have limited capacity during the time of minimal user activity.
The problem statement is the following. A set of vertices V = {0,1,..., n} is given, where the vertex 0 is a depot and the set of vertices {1, ...,n} are stations. Each i station has qi, which can be positive or negative. qi stands for the number of bicycles, which are requested by i station, q0 = 0. When qi > 0, it means that i is a node, where bicycles should be removed (extra bikes are available at this station). In contrast, qt < 0 means that i is a node, where bicycles should be delivered (we have a shortage at this station). Bicycles, that have been removed from the stations can go to other delivery nodes or to the depot. Bicycles delivered to the station can be taken both from the depot and from the station with an excessive number of vehicles. There is a fleet of vehicles with capacity Q which are used for rebalancing operation. A matrix of distances d^ is given.
The main goal is to define such a route for m vehicles so that the costs are minimized, and the following restrictions are met:
1. The route of each vehicle begins and ends at the depot.
2. Each transport leaves the depot empty or with some load loadj < Q,j = (1, m)
3. The sum of stations requests visited together with the initial load is never negative or greater than Q on the route performed by the vehicle.
4. Each station is visited only once.
An important task for solving the problem of rebalancing is also the processing and analysis of data. Most of the existing methods of forecasting short-term demand at stations are based on the historical model of changes in the number of vehicles at stations. Over time, after the installation of a bicycle rental system in the city area, residents develop the habit of stationary travel. These habits are relatively stable on weekdays and a bit different on weekends.
The difference Bt between inflow and outflow Ot at the station i for timepoint t is used to determine the type of station:
Bk = ht
°it
Generally, stations can be classified into 3 types: supplier, receiver, supplier - receiver. The affiliation of the station to a specific type significantly depends on the type of area and proximity to the city center, as well as the availability of bus stations.
Demand prediction
The demand of bicycles can be either positive or negative and depends on station type. For the receiver station i the total demand for the next day can be deter-
mined using the minimum of Bit during the day. Consider Nd-1 be the number of bicycles at the end of the previous day. For receiver station bike request can be calculated as following:
Rid = Bt_min + Nd-1 where qid - demand for station i for the day d, Btmin - the minimum of Bt during the day d (Fig.2), Nd-1 - the number of bicycles at the end of previous day.
\ 100 200 300 400 500 600
B tjnin f—
Fig.2 Changing the value Bt for one of the stations during the working hours.
Consider Bday be the difference between bicycles at the end and the start of the day. For supplier stations Bday will be positive value. The request for this type of stations can be defined as following: Rid = Bday + Nd-i where qid - demand for station i for the day d, Bday is the difference between bicycles at the end and the start of the day d, Nd-1 - the number of bicycles at the end of previous day.
The stations with qid = 0 do not need to be included in rebalancing operation for the next day. Also, the total shortage of the system should be considered.
For computing the daily request of each station values Bday and Bt min should be predicted. There are a lot of methods that can be used for prediction of time series, such as Artificial neural networks, linear regression, ARIMA, etc.
In this article Gradient boosting method was used for prediction. It is a regression-based machine learning method that works by combining weak learners into a
single strong learner. The main idea is that many models are repeatedly trained and each new model progressively minimizes the loss function of the system. This is done by continually fitting a weak classifier, in our case it is decision tree, and join them to make the final prediction. The main advantages of this method are robustness, performance, reduced data cleaning and preparation times. Overfitting can be avoided by carefully selecting the regularization parameters.
For prediction, a small dataset obtained for the city of Arlington was used. The curve of request for one of the stations can be found on Fig.2. The main extracted features used for prediction are following: year - year extracted from the original date; month - month extracted from the original date; day - the day of week extracted from the original date; weather - temperature on that particular day; season - what time of season from 1 to 4 (being 1 winter and 4 autumn); peak - the time of maximum demand for the station; day type -weekday, weekend or holiday.
Fig.3 The curve of request for receiver station for 6 months (Aug 2017 - Jan 2018).
A feature importance function was used to measure the level of impact of each feature in the outcome of the predictions (Fig.4).
ÜavTvpe
Fig.4 Importance of each feature being trained in the dataset.
An example of prediction result for Btmin can be found in Table 1 and Fig.5.
Table 1
The result of Bt min prediction for one of the stations.
Date Predicted value Real value Absolute difference II*,;| - ^M Absolute percentage errors
01.02.2019 -15 -17 2 12%
02.02.2019 -20 -21 1 4%
03.02.2019 -2 -3 1 50%
04.02.2019 -3 -2 1 50%
05.02.2019 -27 -22 5 22%
06.02.2019 -26 -24 2 8%
07.02.2019 -25 -23 2 9%
o
-50
Fig. 5 Prediction results for Bt min value.
The average mean error for prediction was 2.8. Model formulation
Also, the absolute percent errors indicate a small abso- This article presents a formulation that considers
lute difference, so this is an acceptable prediction result the costs spent on vehicle fleet maintenance. The num-
for a bike-sharing system. ber of broken bikes is also taken into consideration.
In most cases, when performing the rebalancing operations, the expenditure needed for this operation is proportional to the distance to the stations. Bike-sharing companies usually do not pay attention to their own fleet vehicles for rebalancing operation, which, in fact, also play an important role in this process. Thus, when using a particular vehicle for rebalancing, it is necessary to take into account the technical condition of the vehicle and its drivability.
Consider Zaut0 as a vehicle amortization coefficient:
7 . = 1 - e(- (AT* Taut0 + AL* Lauto)) ¿•auto c
where AT- special coefficient depending on the life cycle of a vehicle for a particular brand; Taut0 - life cycle of a vehicle, in years; AL- coefficient calculated depending on the mileage, L - the value of vehicle mileage.
The price of using vehicle on kilometer can be calculated as following:
(Pricev)
Price! = (1-Zaut0)*---,
Lt
where Pricev is the market price of the vehicle; L1- vehicle mileage.
Consider Price2 as the price of vehicle service on kilometer:
. (Pricerec) Price2 =
U =
T
1 auto
where Pricerec- the average price of vehicle maintenance per year; L2- the average mileage of car per year.
Notations
V - set of vertices, where V0 QV \ {0}, S & 0
n - number of stations
K - vehicle fleet
Q- vehicle capacity
m - number of vehicles
k - index of vehicle, k = 1,2,..., m
A - set of arcs
qi - request of station i
Pj - number of damaged bicycles at the station j dtj - distance between station i and j Load0k - initial load of vehicle k Loadk0 - final load of vehicle k Loadkj - load of vehicle k after visiting station j, j e7o
Qtot - total request Xijk - binary variable Price3 - fuel cost per km The formulation for bike-sharing static rebalancing by considering the vehicle fleet details and collection of bicycles in need of repair is given as follows:
riceik + Price2k + Price3k)dij
xijk
X;
12
min'i e vZj e v'k e K(P
'i e v'k e Kxijk = 1,V j e Vo ' evxihk = 'j e vxhjk,V h eVo; V k e K
'j e v0 'k e Kxojk < m, = ' e v0xiok ,V k e K Qtot = 'j ev0 Qj,
' j e v0 xojk
Loadok = max{0, - ' e^' ev0 Rjxijk },Vk e K ' eK Loadok > max{ 0,-Qtot },
Loadko = ' ev'j ev0 Pj * xijk, V k e K 'fc exLoadko = 'j ev0 Pj,
*0k + Ij +Pj-M(t — xojk),vk e K,vj e Vo
Loadjk > Loadn
Loadkj > Loadki + qj + pj — M (l — '
fc eK xijk
o,
vi e v0,vj ev0,i ± j
Load0k <Q,Vk eK Loadkj < Q,j eV0,k eK
ji E Vq ¿j E VQ Zk E K Xijk
_ (1, if arc (i,j) is used by vehicle k,
ljk { 0, in other case
Objective function (1) minimizes the travelling repair. Constraints (11) costs. It makes possible considering the price and feasibility of using a particular car for rebalancing operations. (2) - (3) impose that every node but the depot is visited exactly once. Constraints (4) and (5) ensure the fact that at most m vehicles leave the depot and all vehicles return to the depot at the end of their route. (6) shows the calculation of Qtot. The initial load and fact that total load leaving the depot initially should be nonnegative and lower than Qtot in case it takes a negative value is imposed by (7) - (8).
Constraints (9) - (10) impose the final load of the vehicle k and ensure that the final load of all vehicles is the number of all collected bicycles in need of
<N— [
-max{0,-Qtot}+ ' ev0 Pi
Q
] ■
(1) (2)
(3)
(4)
(5)
(6)
(7)
(8) (9)
(10) (11) (12)
(13)
(14)
(15)
(16)
(12) impose the vehicle load after the first and j station. (13) - (14) specify upper bounds. Constraint (15) is the classical subtour elimination constraint that impose the connectivity of the solution. (16) describes the binary variable that stands for usage of arc(i,j) by a vehicle.
Computational result and analysis
For finding the solution branch-and-cut algorithm with some inequalities simplifications and separation procedures from [9] was used. The result of finding a rebalancing route for a small dataset for the city of Arlingon can be found on Fig.6.
L
Fig. 6 Rebalancing route for 10 bike-sharing stations.
Computation using branch-and-cut algorithm produces four routes for rebalancing:
route1 = depot — 7 — 6 — 5 — 4 — depot, route2 = depot — 1 — 2 — depot, route3 = depot — 3 — depot, route4 = depot — 8 — 9 — 11 — depot . The total cost value for each route is 103.34 $, 17.82$, 24.2$, 76.25$. The total cost is 241.61$. The interesting fact is among the 5 vehicles available for rebalancing the most expensive car wasn't selected for rebalancing operation and two cheaper cars were selected for longer routes.
Due to not having access to real-time values from the past, the data of past trips was used to simulate how many bikes would arrive and depart from each station every hour and how many bikes were in each station. Also, the mean percentage of empty and full stations was calculated to determine the effectiveness of rebalancing operation. Using static rebalancing proposed in this article the percentage of empty and full stations dropped significantly (Fig.7). The mean percentage of full or empty stations can be decreased by combining 2 types of rebalancing - static and dynamic.
Fig. 7 Mean percentage of empty or full stations per hour
Conclusions and future work
In this paper we considered the Bike Rebalancing Problem, which calls for the repositioning of bikes on the stations for a bike sharing system at minimum cost, using a fleet of capacitated vehicles. The model formulation is provided in such a way, that it takes into account the special details of vehicle fleet, total costs needed for the maintenance of vehicles and also broken bicycles, which need to be replaced to fully satisfy customers' request. The demand for each station is predicted using Gradient boosting method. Provided computational results and visualizations prove the effectiveness of rebalancing operation.
BSSs are highly dynamic and riders' behavior is difficult and complex to predict so there are always existing opportunities to improve certain features. An interesting future research direction is to try to solve these instances using heuristic and metaheuristic algorithms and improve the quality of demand predictions and performance.
References
1. DeMaio, Paul. Bike-sharing: History, Impacts, Models of Provision, and Future. // Journal of Public Transportation, 12 (4), 2009.
2. OBIS (Optimizing bike sharing in European cities) Project Handbook., [Electronic resource]. Access mode: URL: https://ec.europa.eu/energy/intelli-gent/projects/en/projects/obis (last access date: 20.11.2020).
3. LDA Consulting, Capital Bikeshare Member Survey Report 2011, [Electronic resource]. Access mode: URL: https://www.capitalbikeshare.com/sys-tem-data (last access date: 18.11.2020).
4. Hernández-Pérez H, Salazar-González J-J. A branch-and-cut algorithm for a traveling salesman problem with pickup and delivery. Discrete Applied Mathematics 2004;145:126-139.
5. Hernández-Pérez H, Salazar-González J-J. The one-commodity pickup and delivery traveling salesman
problem: inequalities and algorithms. Networks 2007;50:258-272.
6. Chemla, D., Meunier, F., Wolfler Calvo, R., Bike sharing systems: Solving the static rebalancing problem, Discrete Optimization, 2013. 120-146 p.
7. Raviv T., Tzur M., Forma I.A, Static repositioning in a bike-sharing system: models and solution approaches, EURO Journal on Transportation and Logistics, 2013.187-229 p.
8. C. Contardo, C. Morency, and L.-M. Rousseau, Balancing a dynamic public bike-sharing system. Cirrelt Montreal, 2012.
9. M. Dell'Amico, E. Hadjicostantinou, M. Iori, S. Novellani, The bike sharing rebalancing problem: Mathematical formulations and benchmark instances, Omega, Volume 45, 2014, 7-19 p.
ФИЗИЧЕСКИЕ ЗАКОНЫ ИЗЛУЧЕНИЯ КАК КЛЮЧ К ВЫЯВЛЕНИЮ КОСМОЛОГИЧЕСКИХ ТАЙН ВСЕЛЕННОЙ
Кошман В.С.
канд. техн. наук, доцент, Пермский государственный аграрно-технологический университет,
Пермь, Россия
PHYSICAL LAWS OF RADIATION AS A KEY TO REVEALING THE COSMOLOGICAL SECRETS
OF THE UNIVERSE
Koshman V.
Cand. Tech. Sci., Associate Professor, Perm State Agrarian and Technological University,
Perm, Russia
АННОТАЦИЯ
Вселенная рассматривается как газовая смесь фотонов, барионов и скрытых материальных частиц. Приняты во внимание присутствие во Вселенной реликтового излучения, его спектр абсолютно черного тела, а также физические законы фотонного излучения. Получена формула для количественной оценки изменчивости температуры Вселенной в функции от времени и переменной величины фотон - барионного отношения. В согласие с формулой Планка, выраженной через планковские величины энергии, объема и частоты, приведены аргументы в пользу состояния Вселенной на момент окончания эпохи Планка. Обозначены контуры физического явления взрывной ядерной природы у истока космологического расширения Вселенной.
ABSTRACT
The universe is viewed as a gas mixture of photons, baryons, and hidden material particles. The presence of relic radiation in the Universe, its blackbody spectrum, and the physical laws of photon radiation are taken into account. A formula is obtained for quantifying the variability of the temperature of the Universe as a function of time and the variable photon-baryon ratio. In agreement with the Planck formula, expressed in terms of Planck values of energy, volume, and frequency, arguments are given in favor of the state of the Universe at the end of the Planck epoch. The contours of the physical phenomenon of explosive nuclear nature at the source of the cos-mological expansion of the Universe are outlined.
Ключевые слова: модель Вселенной, реликтовое излучение, физические законы излучения, планков-ские величины, эпоха Планка, взрыв, рождение реликтовых фотонов.
Keywords: model of the Universe, relic radiation, physical laws of radiation, Planck values, Planck epoch, explosion, birth of relic photons.
«Когда мы рассуждаем о характере физических законов,
мы можем по крайней мере предполагать, что говорим о самой природе» Ричард Фейнман
Если у кого - либо из вас возникнет необходимость построить свою личную модель космологической эволюции Вселенной, то это, в лучшем случае, сопряжено с необходимость ответить на массу всевозможных вопросов. Так, у вас могут спросить: «Честно говоря, пока для меня не ясно, какую задачу Вы решаете. То есть: в чем состоит проблема. Ведь в каждой из областей познания проблем
