Public Transport

, Volume 11, Issue 2, pp 321–340

# Locating key stations of a metro network using bi-objective programming: discrete and continuous demand mode

Original Paper

## Abstract

This study proposes two bi-objective optimization problems for locating key stations of a metro network in both discrete and continuous demand modes. Traditionally, designing a metro network based on optimization techniques consists of two approaches. The first approach locates a number of alignments and their stations simultaneously, while the second approach involves locating key stations, designing a core network, and locating secondary stations. In locating key stations processed by a single objective model, the number of produced and attracted trips to the key stations is maximized. This paper considers a second objective for this stage to maximize the coverage of key stations on origin/destination (OD) trips. A fuzzy goal programming model is established to solve the bi-objective model and provide some Pareto-optimal solutions. The previous single objective model and the proposed model with continuous demand mode are applied to a real network. Results show that the proposed model significantly increases the coverage of key stations on OD trips with only a slight reduction in the number of produced and attracted trips.

## Keywords

Rail rapid transit Metro network design Fuzzy goal programming

## Sets/Indices

$$S$$

The set of proposed stations

$$D$$

The set of demand points

$$C$$

The set of demand areas

$$L$$

The set of catchment levels

$$s,s^{\prime}$$

The index for proposed stations

$$i,j$$

The index for demand points/areas

$$l,l^{\prime}$$

The index for catchment levels

## Input parameters

$$w_{s}$$

The coverage of station s of network demand points

$$w_{ij}^{{ss^{\prime}}}$$

The coverage of stations s and $$s^{\prime}$$ of demand flow i to j where s and $$s^{\prime}$$ are situated around the demand points/areas i and j, respectively

$$w^{{ss^{\prime}}}$$

The coverage of stations s and $$s^{\prime}$$ of total demand flows

$$P$$

The number of key stations in the network

$$\theta_{l}$$

The cover intensity at the catchment level l

$$v_{i}$$

The total number of trip production and attraction in the demand point i

$$v_{ij}$$

The demand flow from point/area i to point/area j

$$\delta_{isl}$$

A binary input parameter, which is equal to one, if station i belongs to catchment level l of station s, and zero, otherwise

$$d_{{ss^{\prime}}}$$

The Euclidian distance between stations s and $$s^{\prime}$$

$$d_{min}$$

The minimum permitted Euclidian distance between two key stations

$$\beta$$

A constant value based on the gravity model

$$a_{i}$$

The total area of demand area i

$$a_{isl}$$

The common area of demand area i and catchment area l of station s

## Decision variables

$$Z_{s}$$

A binary decision variable; it is equal to one if station s is selected as key station; otherwise it is equal to zero

$$Y^{{ss^{\prime}}}$$

A binary decision variable; it is equal to one if both stations s and $$s^{\prime}$$ are selected as key stations; otherwise it is equal to zero

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