{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,28]],"date-time":"2026-02-28T18:39:17Z","timestamp":1772303957190,"version":"3.50.1"},"reference-count":40,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2019,7,29]],"date-time":"2019-07-29T00:00:00Z","timestamp":1564358400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["IJGI"],"abstract":"<jats:p>In order to improve the dynamic optimization of fleet size and standardized management of dockless bike-sharing, this paper focuses on using the Markov stochastic process and linear programming method to solve the problem of bike-sharing fleet size and rebalancing. Based on the analysis of characters of bike-sharing, which are irreducible, aperiodic and positive-recurrence, we prove that the probability limits the state (steady-state) of bike-sharing Markov chain only exists and is independent of the initial probability distribution. Then a new \u201cMarkov chain dockless bike-sharing fleet size solution\u201d algorithm is proposed. The process includes three parts. Firstly, the irreducibility of the bike-sharing transition probability matrix is analyzed. Secondly, the rank-one updating method is used to construct the transition probability random prime matrix. Finally, an iterative method for solving the steady-state probability vector is therefore given and the convergence speed of the method is analyzed. Furthermore, we discuss the dynamic solution of the bike-sharing steady-state fleet size according to the time period, so as improving the practicality of the algorithm. To verify the efficiency of this algorithm, we adopt the linear programming method for bicycle rebalancing analysis. Experiment results show that the algorithm could be used to solve the disordered deployment of dockless bike-sharing.<\/jats:p>","DOI":"10.3390\/ijgi8080334","type":"journal-article","created":{"date-parts":[[2019,7,29]],"date-time":"2019-07-29T11:20:18Z","timestamp":1564399218000},"page":"334","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":23,"title":["Fleet Size and Rebalancing Analysis of Dockless Bike-Sharing Stations Based on Markov Chain"],"prefix":"10.3390","volume":"8","author":[{"given":"Yong","family":"Zhai","sequence":"first","affiliation":[{"name":"National Geomatics Center of China, Beijing 100830, China"}]},{"given":"Jin","family":"Liu","sequence":"additional","affiliation":[{"name":"National Geomatics Center of China, Beijing 100830, China"}]},{"given":"Juan","family":"Du","sequence":"additional","affiliation":[{"name":"National Geomatics Center of China, Beijing 100830, China"}]},{"given":"Hao","family":"Wu","sequence":"additional","affiliation":[{"name":"National Geomatics Center of China, Beijing 100830, China"}]}],"member":"1968","published-online":{"date-parts":[[2019,7,29]]},"reference":[{"key":"ref_1","unstructured":"(2019, April 26). The Bike-Sharing Blog. Available online: http:\/\/bike-sharing.blogspot.com\/."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"159","DOI":"10.3141\/2143-20","article-title":"Bikesharing in Europe, the Americas, and Asia: Past, present, and future","volume":"2143","author":"Shaheen","year":"2010","journal-title":"Transp. Res. Rec."},{"key":"ref_3","first-page":"92","article-title":"Bikeshare: A Review of Recent Literature","volume":"36","author":"Fishman","year":"2016","journal-title":"Urban. Transport. China"},{"key":"ref_4","unstructured":"Shaheen, S.A., Martin, E.W., Cohen, A.P., and Finson, R.S. (2012). Public Bikesharing in North America: Early Operator and User Understanding, Mineta Transportation Institute. Quadrige."},{"key":"ref_5","unstructured":"(2019, April 26). Social Bicycles. Available online: http:\/\/socialbicycles.com\/."},{"key":"ref_6","unstructured":"(2019, April 26). The Past and Present of Bike-Sharing\u2014Incomplete Competing Analysis. Available online: https:\/\/www.jianshu.com\/p\/68c0d1a3208e."},{"key":"ref_7","first-page":"36","article-title":"The transportation system of Hangzhou\u2019s public bicycle","volume":"27","author":"Zhou","year":"2011","journal-title":"Traffic Transp."},{"key":"ref_8","first-page":"9","article-title":"Problems and suggestions for bike-sharing development","volume":"9","author":"Li","year":"2017","journal-title":"Manag. Manag."},{"key":"ref_9","unstructured":"(2019, April 26). China Bike-Sharing Industry Research Report. 2017\u20132021. Available online: http:\/\/www.pday.com.cn\/Htmls\/Report\/201704\/24516157.html."},{"key":"ref_10","unstructured":"(2019, April 26). What is Dockless Bike Share. Available online: https:\/\/altaplanning.com\/dockless-bike-share\/."},{"key":"ref_11","unstructured":"Banerjee, S., Freund, D., and Lykouris, T. (2016). Multi-objective pricing for shared vehicle systems. arXiv."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"41","DOI":"10.5038\/2375-0901.12.4.3","article-title":"Bike-sharing: History, impacts, models of provision, and future","volume":"12","author":"DeMaio","year":"2009","journal-title":"J. Public Transp."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"120","DOI":"10.1016\/j.disopt.2012.11.005","article-title":"Bike sharing systems: Solving the static rebalancing problem","volume":"10","author":"Chemla","year":"2013","journal-title":"Discret. Optim."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1016\/j.tra.2016.12.007","article-title":"An empirical analysis of bike sharing usage and rebalancing: Evidence from barcelona and seville","volume":"97","author":"Hampshire","year":"2017","journal-title":"Transp. Res. Part. A Policy Pract."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"7","DOI":"10.1016\/j.omega.2013.12.001","article-title":"The bike sharing rebalancing problem: Mathematical formulations and benchmark instances","volume":"45","author":"Hadjicostantinou","year":"2014","journal-title":"Omega"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"235","DOI":"10.1016\/j.apgeog.2012.07.002","article-title":"Optimizing the location of stations in bike-sharing programs: A GIS approach","volume":"35","author":"Latorre","year":"2012","journal-title":"Appl. Geogr."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"513","DOI":"10.1016\/j.sbspro.2012.09.769","article-title":"An optimisation algorithm to establish the location of stations of a mixed fleet biking system: An application to the city of Lisbon","volume":"54","author":"Martinez","year":"2012","journal-title":"Proc. Soc. Behav. Sci."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1007\/s13676-012-0017-6","article-title":"Static repositioning in a bike-sharing system: Models and solution approaches","volume":"2","author":"Raviv","year":"2013","journal-title":"EURO J. Transp. Logist."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Rainer-Harbach, M., Papazek, P., Hu, B., and Raidl, G.R. (2013). Balancing Bicycle Sharing Systems: A Variable Neighborhood Search Approach, Springer. Lecture Notes in Computer Science.","DOI":"10.1007\/978-3-642-37198-1_11"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Kloim\u00fcllner, C., Papazek, P., Hu, B., and Raidl, G.R. (2014). Balancing Bicycle Sharing Systems: An Approach for the Dynamic Case. European Conference on Evolutionary Computation in Combinatorial Optimization, Springer.","DOI":"10.1007\/978-3-662-44320-0_7"},{"key":"ref_21","unstructured":"McKenzie, G. (2018, January 28\u201331). Docked vs. Dockless Bike-sharing: Contrasting Spatiotemporal Patterns (Short Paper). Proceedings of the 10th International Conference on Geographic Information Science (GIScience 2018), Melbourne, VIC, Australia."},{"key":"ref_22","unstructured":"(2019, April 26). Mobike About. Available online: https:\/\/mobike.com\/global\/about."},{"key":"ref_23","unstructured":"Xiong, S. (2017). An Internet-Based Short-Distance Travel Solution in China. [Bachelor\u2019s Thesis in Business, Lahti University of Applied Sciences]."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Luo, Q., Dou, X., Di, X., and Hampshire, R.C. (2018, January 4\u20137). Multimodal Connections between Dockless Bikesharing and Ride-Hailing: An Empirical Study in New York City. Proceedings of the 2018 21st International Conference on Intelligent Transportation Systems (ITSC), Maui, HI, USA.","DOI":"10.1109\/ITSC.2018.8569896"},{"key":"ref_25","unstructured":"Cui, W. (2018). The Effects of Urban Density on the Efficiency of Dockless Bike Sharing System\u2014A Case Study of Beijing, China. [Master\u2019s Theses, Arizona State University]."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Pan, L., Cai, Q., Fang, Z., Tang, P., and Huang, L. (2018). A Deep Reinforcement Learning Framework for Rebalancing Dockless Bike Sharing Systems. arXiv.","DOI":"10.1609\/aaai.v33i01.33011393"},{"key":"ref_27","first-page":"72","article-title":"The traffic value and development path of bike-sharing","volume":"205","author":"Wang","year":"2017","journal-title":"City"},{"key":"ref_28","first-page":"37","article-title":"A brief analysis of the sustainable profit model of the Bike-sharing","volume":"39","author":"Jiao","year":"2017","journal-title":"Consum. Guide"},{"key":"ref_29","unstructured":"Hu, J. (2018). User Behaviour Analysis and Optimization in Bike-Sharing Systems. [China Master\u2019s Theses, Zhejiang University]."},{"key":"ref_30","first-page":"67","article-title":"Design and research on Crime Prevention Through Environmental about Bike-sharing","volume":"9","author":"Yang","year":"2017","journal-title":"Chin. Criminol. Rev."},{"key":"ref_31","unstructured":"Zhai, Y., Liu, J., Du, J., and Chen, J. (2018, January 1\u20135). Solution to fleet size of dockless bike-sharing station based on matrix analysis. Proceedings of the 2018 ISPRS TC IV Mid-term Symposium \u201c3D Spatial Information Science\u2014The Engine of Change\u201d, Delft, The Netherlands."},{"key":"ref_32","unstructured":"Liu, C. (2008). Stochastic Process, Huazhong University of Science and Technology Press. [4th ed.]."},{"key":"ref_33","unstructured":"Ching, W.K., and Ng, M.K. (2006). Markov Chains. Models, Algorithms and Applications, Springer."},{"key":"ref_34","unstructured":"Langville, A.N., and Meyer, C.D. (2011). Google\u2019s PageRank and Beyond: The Science of Search Engine Rankings, Princeton University Press."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"968","DOI":"10.1137\/040619028","article-title":"Updating Markov chains with an eye on GooglE\u2019s PageRank","volume":"27","author":"Langville","year":"2006","journal-title":"Siam J. Matrix Anal. Appl."},{"key":"ref_36","unstructured":"(2019, April 26). The Secret of Codoon\u2019s Bike-Sharing Data. Available online: https:\/\/www.codoon.com\/iec\/article\/PwdL_gm_O5E=."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"146","DOI":"10.1137\/0201010","article-title":"Depth-first search and linear graph algorithms","volume":"1","author":"Tarjan","year":"1972","journal-title":"SIAM J. Comput."},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Stewart, W.J. (1994). Introduction to the Numerical Solution of Markov Chains, Princeton University Press.","DOI":"10.1515\/9780691223384"},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Mao, D., Hao, Z., Wang, Y., and Fu, S. (2019). A Novel Dynamic Dispatching Method for Bicycle-Sharing System. ISPRS Int. J. GeoInf., 8.","DOI":"10.3390\/ijgi8030117"},{"key":"ref_40","unstructured":"Dantzig, G. (2016). Linear Programming and Extensions, Princeton University Press."}],"container-title":["ISPRS International Journal of Geo-Information"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2220-9964\/8\/8\/334\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T13:10:45Z","timestamp":1760188245000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2220-9964\/8\/8\/334"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,7,29]]},"references-count":40,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2019,8]]}},"alternative-id":["ijgi8080334"],"URL":"https:\/\/doi.org\/10.3390\/ijgi8080334","relation":{},"ISSN":["2220-9964"],"issn-type":[{"value":"2220-9964","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,7,29]]}}}