Model for Microcirculation Transportation Network Design

The idea of microcirculation transportation was proposed to shunt heavy traffic on arterial roads through branch roads. The optimization model for designing micro-circulation transportation network was developed to pick out branch roads as traffic-shunting channels and determine their required capacity, trying to minimize the total reconstruction expense and land occupancy subject to saturation and reconstruction space constraints, while accounting for the route choice behaviour of network users. Since micro-circulation transportation network design problem includes both discrete and continuous variables, a discretization method was developed to convert two groups of variables discrete variables and continuous variables into one group of new discrete variables, transforming the mixed network design problem into a new kind of discrete network design problem with multiple values. The genetic algorithm was proposed to solve the new discrete network design problem. Finally a numerical example demonstrated the efficiency of the model and algorithm.


Introduction
Urban microcirculation transportation is an impersonate noun borrowed from human blood circulation system 1 . In the blood microcirculation system, blood flows from arterioles to microcirculation vessels and then flows from the microcirculation vessels back to venules. Similar to the blood microcirculation, traffic microcirculation can be defined as traffic flows from arterial roads to branch roads microcirculation roads and then flows from branch roads back to arterial roads. In general, most of vehicles run on the arterial roads, so arterial roads usually become very congested at peak hours. If microcirculation transportation network is designed around "jam points" of arterial roads, traffic on arterial roads can be shunted, and part of vehicles can go through the microcirculation roads branch roads .
In reality, owing to narrow road surface and complicated functions, some branch roads are primarily for in-area traffic e.g., pedestrian, bicycles, and few vehicles 2, 3 and to solve than the conventional MNDPs. This paper presented a discretization way to convert two groups of variables discrete variables and continuous variables into one group of new discrete variables, and then the MNDP is transformed into a new kind of DNDP. The new DNDP is different from the conventional 0-1 DNDP because the variable of the new DNDP can take multiple values. The genetic algorithm was proposed to solve the new DNDP.
Moreover, compared with the conventional NDPs, the microcirculation transportation network design problem has different objectives. Microcirculation transportation network is a little local network whose objective is to shunt traffic from arterial roads. Because the size of network is small, passing time of vehicles is very short if the network is not congested, so the factor of travel time may be ignored in the model which is usually taken into account by the conventional NDPs. The main objective of the microcirculation network design problem is to minimize the total reconstruction expense under a saturation constraint. Also, the objective of minimizing land occupancy is taken into account to minimize interference with in-area residents. In addition, microcirculation transportation network design problem considers some other constraints, such as reconstruction space constraint and restriction for the number of cross-points of microcirculation roads and arterial roads.
The remainder of the paper is organized as follows. Section 2 presents the optimization model for designing the microcirculation transportation network. Section 3 introduces a discretization way to solve the model. In Section 4, a numerical example is given to demonstrate the application of the model and algorithm. The final section concludes the paper.

Optimization Model for Designing Microcirculation Transportation Network
In Figure 1, road network N V, A∪B , V is the set of all nodes, n |V |. A is the set of arterial roads, and B is the set of candidate branch roads. q rs n×n is traffic distribution between origins and destinations. For branch road a a ∈ B , e a equals 1 if it is selected and 0 if not selected. All the selected branch roads construct the microcirculation transportation network. The existing capacity of each road is C a , a ∈ A ∪ B. For the selected branch road a a ∈ B , its required capacity after reconstruction is X a , a ∈ B. Apparently, X a ≥ C a , a ∈ B. e a , X a , a ∈ B are the optimization variables.
In general, there are two road links with opposite directions between two adjacent nodes, and their capacities are usually the same.
Before reconstruction the branch roads do not have the ability of shunting traffic from arterial roads. They are for in-area traffic and often crowded with pedestrians and bicycles and even occupied by some other temporary facilities, and so would not have their original designed capacity unless they are cleaned up or reconstructed.
The main optimization objective is to minimize the total reconstruction cost which lies on the length and capacity of the reconstructed roads. If the capacity is improved more, the reconstruction expense will become more.
In addition, the objective of minimizing land occupancy expressed as land use cost should be taken into account to reduce interference with the area. Although microcirculation transportation can shunt arterials' traffic, the shunted traffic will interfere with residents' life inside the area and may cause environmental pollution. Reducing land occupancy including road length and width of microcirculation transportation can reduce interference scope. So, for those unselected branch roads, some management measures need to be taken to bar the traversing traffic, making traffic shunting restricted within the selected roads.
From the previous analysis, the cost function of candidate branch road a can be expressed as P a e a l a p a e a l a h a , a ∈ B.

2.1
In 2.1 , item 1 of the right side is reconstruction expense and item 2 is land use cost of microcirculation transportation. The optimization goal is to minimize a∈B P a , namely: min Y a∈B P a a∈B e a l a p a h a .
2.2 l a is the length of candidate branch road a, p a is unit reconstruction expense, and h a is unit land use cost: In 2.3 , for branch road a, p a is an increasing function of X a ; namely, the required capacity is greater, the reconstruction expense is higher: Equation 2.4 implies that h a is an increasing function of X a , because land use of microcirculation roads depends on their length and width, while land use cost h a of unit length is decided by the road width which corresponds with the capacity after reconstruction X a . In general, for branch road a, if the capacity X a after reconstruction is greater, the road width should be greater, and so h a becomes greater.
The constraints are as follows.
of arterial roads should not be too small; otherwise, their capacity cannot be brought into full play. The key is to attain the goal of no more very congested: where u a , x a , a ∈ A are, respectively, the saturation and flow of arterial road a and U a is the allowed saturation of arterial road a.
2 Saturation constraint of branch roads: the saturation of microcirculation branch roads should also be under an allowed value to avoid traffic congestions on branch roads and ensure the safety of pedestrians and bicycles on the branch roads: where v a , x a , a ∈ B are, respectively, the saturation and flow of branch road a and V a is the allowed saturation of branch road a.
3 Capacity constraint of branch roads. Capacity enhancement of branch roads is affected by some actual conditions, such as land use restriction, building restriction and geological condition: where X 0 a is the available maximal capacity of branch road a after reconstruction. 2.8 x a , a ∈ A ∪ B is calculated via the user equilibrium UE traffic assignment model:

2.10
f rs k is the flow of path k between origin-destination OD pair r, s , L r, s is the number of paths between OD pair r, s , and q rs is traffic demand between OD pair r, s . x a is the flow of link a. δ rs a,k equals 1 if link a is on path k between OD pair r, s , otherwise 0. t a is travel time on link a. Here BPR bureau of public road link impedance function is applied: In 2.11 , M a is link capacity; for arterial roads, it is C a ; for branch roads, it is X a . α, β are parameters, and BPR suggested that α 0.15, β 4. t a0 is free-flow travel time of link a.

Solution Algorithms
There are two groups of variables e a , X a in the above model, so the solution is very hard. But if the two groups of variables can be converted into one, then the solution will become much easier.
For branch road a, a ∈ B the existing capacity C a , its capacity enhancement via reconstruction can be discretized if it is selected. Let capacity enhancements be 0, σ, 2σ, 3σ, 4σ, . . ., where σ denotes one added unit and 0 denotes that capacity enhancement is 0. This discretization way can accord with the real case. On the one hand, in reality, capacity enhancements via reconstruction are always discrete values instead of continuous; on the other hand, use of many discrete values is also able to reach the precision.
One group of new discrete variables λ a can be defined to convert two groups of variables e a , X a into one group of variables λ a : 0, select a, here e a 1, the added value is 0, X a C a ; 1, select a, here e a 1, the added value is σ, X a C a σ; 2, select a, here e a 1, the added value is 2σ, X a C a 2σ; 3, select a, here e a 1, the added value is 3σ, X a C a 3σ; . . . . . .

3.1
λ a , a ∈ B is the optimization variable. If λ a is calculated, e a and X a can be obtained.
The real coded genetic algorithm is applied to solve the optimization model. The chromosome is made up of λ 1 , λ 2 , λ 3 , . . ..  Steps of solving the model using genetic algorithm are as follows.
Step 1. Initialization: set population size E , chromosome length J , iteration number g max , probability of crossover P c , and probability of mutation P m .
Step 2. Construct a fitness function: F m C max − O m , where F m is the fitness of individual m, O m is the function value of individual m and C max is the estimated maximal function value. Randomly produce the initial population and set g 1.
Step 3. Calculate link flows via UE traffic assignment model, and then calculate the fitness and excess over constraints of each individual. If g g max , output the best individual; otherwise, turn to Step 4.
Step 4. Use roulette wheel selection operator based on ranking 23 to select the population of next generation. Feasible solutions rank from high to low by fitness, and then infeasible solutions rank from small to much by excess over constraints.
Step 5. According to probability of crossover P c , make multi-point crossover. Crossover points can be randomly selected without repeat. Variables between crossover points interchange alternately to produce two new individuals.
Step 6. According to probability of mutation P m , make single point mutation. Randomly produce an integer between -1 J J is the maximal value of λ a to replace the current value of the variable. Set g g 1, and return to Step 3.

A Numerical Example
In Figure 2, the thick lines around the area denote arterial roads and the thin lines inside the area denote candidate branch roads. Each line includes two links with opposite directions and equal capacity. Traffic distribution during peak hours is in Table 1. The capacity of arterial road is 3000 veh/h ; the existing capacity of candidate branch road is 500 veh/h . The length of each link is 1 km. Unit reconstruction cost function of branch roads is p a X a − 500 × 10 4 $/km ; unit land use cost function is h a 1/4 X a ×10 4 $/km . Road saturation should not exceed 1. The available maximal capacity of each branch road after reconstruction is 1000 veh/h . t a0 of arterial roads is 1 min and that of branch roads is 1.1 min.

Mathematical Problems in Engineering
Let σ 100; the solution set of λ a is {−1, 0, 1, 2, 3, 4, 5} since the available maximal capacity is 1000 and the existing capacity is 500. The selected branch roads are shown in Figure 3; the total cost is 8000 × 10 4 $. Saturations of arterial and branch roads are all less than 1. Flows and saturations of arterial roads are in Table 2; capacities, flows, and saturations of the selected branch roads for constructing the microcirculation network are in Table 3.
Comparatively, if only arterial roads exist without microcirculation road network , the saturation of arterial roads goes beyond 1 Table 4 .

Conclusions
This paper defined the concept of urban microcirculation transportation. Microcirculation transportation network is a little local network and can shunt traffic from arterial roads.   Through the microcirculation transportation network design model in this paper, the branch roads as traffic-shunting channels and their required capacity after reconstruction can be decided.

Mathematical Problems in Engineering
Since microcirculation transportation network design problem includes both discrete variables and continuous variables, this paper developed a discretization method to convert two groups of variables discrete variables and continuous variables into one group of new discrete variables, transforming the solution of MNDP into the solution of a new kind of DNDP with multiple values, and the genetic algorithm was proposed to solve the new DNDP.
A numerical example demonstrated the application of the model and algorithm and compared the results with or no microcirculation transportation network. The method and model proposed in this paper provided a new effective way for solving urban traffic congestions.