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# Application of a mathematical programming model for development of Tehran metro network

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## Abstract

Designing or Development of a transit network is one of the most important challenge of transportation engineers and urban planners. Among various types of transit, metro is so noticeable because it is green and massive transit. Metro lines or network design problem is so complicated because a metro network has numerous stakeholders with different purposes and it is very costly. In this paper, two binary non-linear mathematical programming models are proposed for designing circular and non-circular lines. The objective is to maximize the ratio of population coverage per construction cost. A heuristic method is introduced for solving these problems. Four non-circular corridors are considered to improve population coverage and accessibility of the Tehran metro network, and two circular corridors are defined to enhance connectivity and robustness. Results indicate that the total length of new lines and the total number of stations are equal to 140.2 km and 102, respectively. The total construction cost is 47.2 thousand billion Tomans. Consequently, for one billion Tomans investment to construct these new lines, the total population coverage of the Tehran metro network is increased equal to 36 persons.
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armamdoohi@modares.ac.ir
Grove School of EngineeringCity College of New Yorkmallahviranloo@ccny.cuny.edu
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ij
x
ij
i
C
ij
c
iijij
ij
d

min
l
max
l

i
piTS
N

max Z= piyi(Ciyi+ Cij𝑑𝑖𝑗xij)
s.t.
xij = yi-1
xiji,j Syi
i S / {k} kSN, |S|≥2
lmin ≤ dij ≤ lmax
yi=1 iTN
yi=0 , 1 iN
xij=0 , 1 (i,j)E




max Z= piyi(Ciyi+ Cij𝑑𝑖𝑗xij)



5
s.t.
𝑥𝑖𝑗𝑖 ∈𝑁/{𝑖} = yi
𝑥𝑖𝑗𝑗 ∈𝑁/{𝑗} = y𝑗
xiji,j δS ≥y𝑘 kSN















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7
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8
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
[1]
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Bank Group Press, 2018.
[2]
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[3]
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[4]
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[5]
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exact algorithm.”, Networks: An International Journal, Vol.43, No. 3, pp.177-189, 2004.
[6]
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simulated annealing.”, 12th World conference of transportation research, Lisbon, Portugal, 2010.
[7]
Gutiérrez-Jarpa, G., Obreque, C., Laporte, G. and Marianov, V., “Rapid transit network design for optimal
cost and origin–destination demand capture.”, Computers & Operations Research, Vol. 40, No, 12,
pp.3000-3009, 2013.
[8]
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Operations Research, Vol. 62, pp.78-94, 2015.
[9]
Canca, D., De-Los-Santos, A., Laporte, G. and Mesa, J.A., “A general rapid network design, line planning
and fleet investment integrated model.”, Annals of Operations Research, Vol. 246, No. 1, pp.127-144, 2016.
[10]
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Wei, Y., Jin, J.G., Yang, J. and Lu, L., “Strategic network expansion of urban rapid transit systems: A bi‐
objective programming model.”, Computer‐Aided Civil and Infrastructure Engineering, pp.1-13, 2018.
[12]
Costa, A., Cordeau, J.F. and Laporte, G., Steiner tree problems with profits.”, INFOR: information systems
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[13]
Feillet, D., Dejax, P. and Gendreau, M., Traveling salesman problems with profits.”, Transportation
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


10
Application of a mathematical programming model for development of Tehran metro
network
Amir R. Mahdavi1, A.R. Mamdoohi2 and M. Allahviranloo3
1- MSc Graduate of Transportation Planning, Tarbiat Modares University, Tehran, Iran. Email:
amirrezamahdavi@modares.ac.ir
2- Associate Professor, Civil and Environment Faculty, Tarbiat Modares University, Tehran, Iran. Email:
armamdoohi@modares.ac.ir
3- Assistant Professor, Grove School of Engineering, City College of New York, New York, USA. Email:
mallahviranloo@ccny.cuny.edu
Abstract
Designing or Development of a transit network is one of the most important challenge of transportation
engineers and urban planners. Among various types of transit, metro is so noticeable because it is green and
massive transit. Metro lines or network design problem is so complicated because a metro network has
numerous stakeholders with different purposes and it is very costly. In this paper, two binary non-linear
mathematical programming models are proposed for designing circular and non-circular lines. The
objective is to maximize the ratio of population coverage per construction cost. A heuristic method is
introduced for solving these problems. Four non-circular corridors are considered to improve population
coverage and accessibility of the Tehran metro network, and two circular corridors are defined to enhance
connectivity and robustness. Results indicate that the total length of new lines and the total number of
stations are equal to 140.2 km and 102, respectively. The total construction cost is 47.2 thousand billion
Tomans. Consequently, for one billion Tomans investment to construct these new lines, the total population
coverage of the Tehran metro network is increased equal to 36 persons.
Key words: Metro line design model, Metro network development model, Development heuristic
algorithm, Binary non-linear mathematical programming
ResearchGate has not been able to resolve any citations for this publication.
Article
Full-text available
Traveling Salesman Problems with Profits (TSPs with Profits) are a generalization of the Traveling Salesman Problem (TSP) where it is not necessary to visit all vertices. With each vertex is associated a profit. The objective is to find a route with a satisfying collected profit (maximized) and travel cost (minimized). Applications of these problems arise in contexts such as traveling salesman problems, job scheduling or carrier transportation. In this paper, the existing literature about TSPs with Profits is surveyed.
Article
Full-text available
This is a survey of the Steiner tree problem with profits, a variation of the classical Steiner problem where, besides the costs associated with edges, there are also revenues associated with vertices. The relationships between these costs and revenues are taken into consideration when deciding which vertices should be spanned by the solution tree. The survey contains a classification of the problems falling within this category and an overview of the methods developed to solve them. It also lists several graph preprocessing procedures and analyzes their validity for the different variants of the problem. Finally, a brief comparison is made between the profit versions of the Steiner tree problem and of the travelling salesman problem.
Article
With the development of urbanization and the extension of city boundaries, the expansion of rapid transit systems based on the existing lines becomes an essential issue in urban transportation systems. In this study, the network expansion problem is formulated as a bi‐objective programming model to minimize the construction cost and maximize the total travel demand covered by the newly introduced transit lines. To solve the bi‐objective mixed‐integer linear program, an approach called minimum distance to the utopia point is applied. Thus, the specific trade‐off is suggested to the decision makers instead of a series of optimal solutions. A real‐world case study based on the metro network in Wuxi, China, is conducted, and the results demonstrate the effectiveness and efficiency of the proposed model and solution method. It is found that the utopia method can not only provide a reasonable connecting pattern of the network expansion problem but also identify the corridors with high priority under the limited budget condition.
Rapid transit network design is highly dependent on the future system usage. These spatially distributed systems are vulnerable to disruptions: during daily operations different incidents may occur. Despite the unpredictable nature of them, effective mitigation methods from an engineering perspective should be designed. In this paper, we present two new approaches to the rapid transit network design problem. The first one aims at minimizing the impacts of the worst scenario in the network operation. The second one takes into account different risk profiles and also minimizes the impacts of the worst scenario across all the risk profiles.
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Traditionally, network design and line planning have been studied as two different phases in the planning process of public transportation. At the strategic level approaches dealing with the network design problem minimize travel time or maximize trip coverage, whereas at the tactical level, in the case of line planning, most models minimize cost or the number of transfers. The main novelty of this paper is the integration of the strategic and tactical phases of the rapid transit planning process. Specifically, a mathematical programming model that simultaneously determines the infrastructure network, line planning, train capacity of each line, fleet investment and personnel planning is defined. Moreover, the demand is assumed to be elastic and, therefore it is split into the rapid transit network and a competing mode according to a generalized cost. A rigorous analysis for the calibration of the different concepts that appear as consequence of the integration of phases is presented. Our approach maximizes the total profit of the network by achieving a balance between the maximum trip coverage and the minimum total cost associated to the network. Numerical results taking into account data based on real-world instances are presented.
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This paper proposes a practical methodology for the problem of designing a metro configuration under two criteria: population coverage and construction cost. It is assumed that a set of corridors defining a rough a priori geometric configuration is provided by the planners. The proposed algorithm consists of fine tuning the location of single alignments within each corridor. This is achieved by means of a bicriteria methodology that generates sets of non-dominated paths. These alignments are then combined to form a metro network by solving a bicriteria integer linear program. Extensive computational experiments confirm the efficiency of the proposed methodology.
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This article addresses the problem of locating a metro or rapid transit line. An alignment comprising n stations must be located in a territory, subject to minimum and maximum station interspacings. The objective is to maximize the total population covered by the alignment. A tabu search heuristic is developed and computational tests on randomly generated instances are presented. Sensitivity analyses are performed on a number of parameters used in the algorithm.
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Rapid transit construction projects are major endeavours that require long-term planning by several players, including politicians, urban planners, engineers, management consultants, and citizen groups. Traditionally, operations research methods have not played a major role at the planning level but several tools developed in recent years can assist the decision process and help produce tentative network designs that can be submitted to the planners for further evaluation. This article reviews some indices for the quality of a rapid transit network, as well as mathematical models and heuristics that can be used to design networks.
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This article presents a mathematical model and a two-phase heuristic for the location of a rapid transit alignment in an urban setting. This method can be viewed as a building block for the multi-line network design problem. Computational results on randomly generated instances and on some Milan real data confirm the efficiency of the proposed approach. When designing rapid transit lines, a common objective is the maximization of the total population covered by the alignment, subject to to interstation spacing constraints. The article describes an efficient and robust heuristic for generating good alignments. These can then be assessed according to other criteria before a final decision is made.