Combinatorial Optimization

Combinatorial optimization is an area in discrete optimization where an optimal solution must be identified from a finite set of solutions. Combinatorial optimization covers a wide range of real problems, such as routing, network design, the travel- ing salesman, and facility location problems. Most of combinatorial optimization problems can be formulated as graph problems, that can normally be well defined as integer programs (IP) and be solved by an IP solver.

An example of combinatorial optimization problem is the shortest path (SP) prob- lem, that is to find a possible path with the minimum length on a given graph G(V, E) from a given source node s to a given destination node t. The SP problem

2.6. TRAFFIC ASSIGNMENT MODELING 19

can be formulated as an IP as follows

Minimize X (i,j)∈E wijxij (2.12) Subject to X j xij − X k xki =        1 if i = s −1 if i = t 0 otherwise ∀i ∈ V (2.13)

xij ∈ {0, 1} for all (i, j) ∈ E, (2.14)

where wij is the length of the link (i, j) and the each variables xij receive either the

value 1, if (i, j) is on the determined shortest path from s to t, or 0 in other cases. Constraints (2.13) are so-called the flow constraints. The optimal solution of the SP problem can be found by solving its IP formulation using an IP solver, however, there are also dedicated efficient algorithms for the problem, such as Dijkstra’s al- gorithm and A-Star algorithm. The details of these algorithms will be discussed in Chapter 3.

2.6 Traffic Assignment Modeling

Traffic assignment modeling (TAM) is the problem of forecasting traffic flows on the links (roads) of a given transportation network with given travel demands between each pair of population zones. It plays a key role in urban traffic plan- ning. In this section we introduce shortly basic knowledge of TAM and it will be investigated deeply in Chapter 5. In Subsection 2.6.1 a number of definitions and terminologies for traffic systems are presented including a review of traveling time functions. Subsection 2.6.2 gives an overview of popular traffic assignment models with an emphasis on the user equilibrium model, that is the most well known model being used in traffic planning.

2.6.1

Introduction

The traffic flow on a link corresponds to the average number of vehicles passing a certain point on the link within a specified time interval, e.g. 12 cars pass the entrance point of the link within every 10 seconds. A free flow on a link is a possible flow when vehicles do not interact with each other, such that they can run as freely as without other vehicles. Traffic flows are measured in a flow unit, that usually corresponds to one vehicle of the most common kind of vehicles in the traffic system. For instance, in a traffic system containing mostly cars, the traffic flow unit should be the passenger car unit (PCU). Similarly, the motorcycle unit (MCU) should be the flow unit for traffic systems containing mostly motorcycles. In the

case of mixed traffic systems (MTS), i.e. traffic systems consist of various kinds of vehicles traveling together without dedicated lanes for each kind, it is necessary to estimate the equivalent value of each kind of vehicle in the flow unit corresponding to a given kind of vehicle. For instance, in an MTS consisting of cars and motorcycles where MCU is chosen to be the flow unit, in order to estimate the flows on the links, the equivalent value of a car in MCU must be investigated, e.g. one car equals 3.67 MCU. The equivalent value of a vehicle is normally depended on the physical characteristics of the vehicle, such as the size, the speed, the acceleration, and the safety distance to others.

The link capacity of a link is the maximum traffic flow on the link, such that some conditions are satisfied. For instance, the steady capacity of a link is the maximum steady-state flow on the link, i.e., the capacity of the point on the link with the minimum capacity. This point is usually the end point (at an intersection) or a bottle-neck point on the link. The practical capacity of a link is defined as the maximum traffic flow, that can go through the link, such that there is no dense traffic or congestion.

The traveling time on a link depends on many factors, e.g. the capacity of the link, the traffic flow, and geographical shape of the link, etc. Given a link, the relationship between traveling time and the traffic flow on the link is called under different names, e.g. traveling time function, travel cost function, and speed- flow equation. Branston [8] gave a good review of traveling time functions by 1976 summarized shortly in Table 2.3 with an addition of the conical volume-delay function proposed by Spiess [62]. Each of proposed functions is based on some certain observations and data. Thus, they may be accurate for estimating traveling time on some specific kinds of links where the data was collected and the observations are made, but may not really accurate on links with different characteristics.

Table 2.3: Overview of a number of popular traveling time functions.

Type Authors Comment

N -line Irwin, Dodd, Cube [41] Simple, but hard to identify with-

out data

Curvilinear Smock [61], Soltman Expensive in computation

Overgaard [57] Logarithmic

Mosher [54] Not suitable for iterative assign-

ments exponential

BPR BPR (USA) [9] Simple, easily and quickly inte-

grable. Suitable for UE models Steenbrink [63]

Conical Spiess [62] Simple, easily and quickly inte-

2.6. TRAFFIC ASSIGNMENT MODELING 21

The function, that is most accepted in general, is the BPR function, which has been developed by the Bureau of Public Roads (USA) [9]—one of the former sections of the Federal Highway Administration. The BPR function computes the traveling time on a link as t(x) = T0  1 + ρ x Cp
β , (2.15)

where ρ, β are the parameters suggested by BPR engineers to be 0.15 and 4 without explanation, T0 is the traveling time at free flow, and Cp is the practical capacity of

the link.

In TAM, each of traffic assignment (TA) model has its own assumptions about rout- ing behaviors of drivers or about transportation networks. The principles proposed by Wardrop [67] are widely used in TAM as basic traffic assumptions.

Principle 2.6.1. (Wandrop’s first principle) The journey times in all routes actually used are equal and less than those which would be experienced by a single vehicle on any unused route. Each user non-cooperatively seeks to minimize his cost of transportation.

Principle 2.6.2. (Wandrop’s second principle) At equilibrium the average journey time is minimum. This implies that each user behaves cooperatively in choosing his own route to ensure the most efficient use of the whole system.