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Assignment Problem Ca Final Nov

Can anyone please solve this Assignment Problem of ICWAI Final. Actually I know how to solve assignment problem but the blank space in the matrix is a bit puzzling to me. this is from Operational Research Paper. Thanks

Q: A travelling salesman has to visit 5 cities. He wishes to start from a particular city, visit each city once and then return to his starting point. The travelling cost for each city from a particular city is given below:

                                        to city 

from city     A         B          C        D       E         

A                 -         4           7        3         4

B                4         -           6          3        4

C               7          6         -           7         5

D               3         3          7          -          7

E              4          4           5         7         -

What is the sequence of the visit of the salesman, so that the cost is optimal?

1 Introduction

The multi-class traffic assignment problem is an important extension of the classical traffic equilibrium problem (i.e., the Beckmann-type mathematical programming formulation) that aims to improve the realism of traffic assignment models by explicitly modeling multiple-user classes in the same transportation network with individual cost functions contributing to its own and possibly other classes's cost functions in an individual way [1]. Applications of multi-class traffic assignment problem include modeling market segmentation of travelers' characteristics such as users with different valuations of travel times [2, 3], different risk-taking behaviors [4, 5], and different knowledge levels of network travel times [6] and modeling different vehicle types (e.g., trucks and passenger cars) sharing the same highway network [7-12]. Particularly, modeling the asymmetric interactions among different vehicle types (e.g., the impact of trucks on the travel times of cars is typically higher than the impact of cars on the travel times of trucks) has the potential to improve the realism of traffic assignment models (i.e., better capture the travel times required by different vehicle types to traverse a link). On the contrary, inadequate considerations of vehicle interactions in the traffic assignment models could result in inaccurate travel time estimates, which could affect flow allocations to routes and network equilibrium process. In addition, there is an increasing interest in addressing the impacts of truck traffic on congestion, infrastructure deterioration, safety, and environmental concerns in urban cities as truck traffic is expected to grow because of the increasing freight shipments transported by trucks.

In the literature, only a few studies explored the algorithmic development and computation efforts required for solving the multi-class traffic assignment problem with asymmetric interactions among different vehicle types sharing the same highway network. Mahmassani and Mouskos [12] investigated the application of the diagonalization algorithm (or Jacobi method) involving asymmetric interactions between cars and trucks (i.e., the impacts of cars and trucks are not reciprocal) in real transportation networks. Wu et al. [10] provided an algorithm based on the method of successive averages (MSA) for solving a multi-class network equilibrium problem in which trucks are converted to Passenger Car Equivalent (PCE) units based on the composition of roadway conditions (i.e., traffic volume, proportion of trucks, and road grade). Noriega and Florian [11] compared three solution algorithms (i.e., MSA, the Jacobi method (or diagonalization method), and the Gauss–Seidel method) for solving the asymmetric multi-class network equilibrium problems with different class delay relationships.

However, none of the aforementioned studies considered the route overlapping and/or vehicle restriction problems in a stochastic multi-class user equilibrium (UE) framework. The purpose of this paper is to develop a customized path-based algorithm for solving the stochastic multi-class traffic assignment problem with different sizes of trucks, different operating characteristics, and different travel patterns with considerations of random perceptions of network conditions, route overlapping, and vehicle restrictions. Similar to the single-user class, the inputs to the algorithm are the transportation network and origin–destination (O–D) trip table, except that the multiple-user classes require multiple copies to characterize the specific characteristics of each user class. The multi-class traffic assignment problem considered in this paper includes asymmetric interactions among different vehicle types through the link travel time functions, route overlapping using the path-size logit (PSL) model for accounting random perceptions of network conditions in a stochastic UE (SUE) framework, and various vehicle restrictions (e.g., lane restriction, height restriction, and weight restriction for trucks) in a transportation network. The outputs generally include algorithmic performance measures, class-specific and total link-flow and path-flow pattern results, as well as network-wide performance measures.

The remainder of this paper is organized as follows. In Section 2, the stochastic multi-class traffic assignment problem with asymmetric interactions, route overlapping, and vehicle restrictions is discussed. Section 3 presents a customized path-based algorithm specifically designed for solving asymmetric interactions among different vehicle types, overlapping routes using the PSL model, and various vehicle restrictions considered in the stochastic multi-class traffic assignment problem. Section 4 presents two sets of experiments to demonstrate the computational performance of the path-based algorithm and its effectiveness with respect to different model parameters. Finally, some concluding remarks are given in Section 5.

2 Stochastic Multi-class Traffic Assignment Problem with Asymmetric Interactions and Vehicle Restrictions

This section presents the stochastic multi-class traffic assignment problem, which includes asymmetric interactions among different vehicle types in the link travel time functions, route overlapping in the PSL model, and various vehicle restrictions in a transportation network.

2.1 Multiple-user classes and link travel time functions

A typical method used to combine multiple vehicle classes (or modes) to assess traffic flow rate on a highway is the concept of PCE or also known as passenger car unit. A PCE assesses the impact that a mode of transport (e.g., truck) has on traffic variables (e.g., speed, density, and flow) compared with a single car. In the traffic assignment procedure, PCE factors are used in the link travel time function to implement the stochastic multi-class traffic assignment. Truck volumes are converted to passenger car volumes to account for the fact that larger trucks take up more capacity on the roads than passenger cars. The Bureau of Public Road's (BPR) function is perhaps the most widely used link travel time function in transportation-planning applications. In this paper, the following multi-class BPR function is adopted to model the interactions among the vehicle classes:

(1)

where is the travel time on link a of class m; M is the set of vehicle classes; A is the set of links; pm is the travel time weight parameter of class m; is the free-flow travel time of link a; PCEm is the PCE factor of class m; is the flow on link a of class m; is the total flow on link a; Ca is the capacity of link a; and αa and βa are the parameters of travel time function on link a.

2.2 Route overlapping

In the SUE problem, route overlapping is one of the major concerns in modeling route choice decisions (see Prashker and Bekhor [13], Chen et al. [14], and Zhou et al. [15] for a detailed description of the different approaches for handling the route-overlapping problem). In this paper, the path-size (PS) factor is adopted to handle the route-overlapping problem because of its simplicity and relatively well performance compared with other closed-form models. The PS factor accounting for different path sizes is determined by the length of links within a path and the relative lengths of paths that share a link as follows:

(2)

where is the PS factor of path k between O–D pair rs of class m; a on k of the first summation in Equation (2) means link a is on path k; RS is the set of O–D pairs; is the set of paths between O–D pair rs for class m; la is the length of link a; is the length on path k between O–D pair rs of class m; and is the path-link indicator, 1 if link a is on path k between O–D pair rs for class m and 0 otherwise. Paths with a heavy overlapping with other routes have a smaller PS value, while a larger PS value indicates the paths are more distinct. For other functional forms of the PS factor, see Bovy et al. [16] and Prato [17]. With the derived PS value in Equation (2), the PS logit (PSL) probability for the stochastic multi-class traffic assignment problem can be expressed as

(3)

where is the probability of selecting path k between O–D pair rs of class m; θm is the dispersion parameter for class m; and is the cost of path k between O–D pair rs of class m.

2.3 Vehicle restrictions

Trucks are often perceived to restrict the flows or safety on certain road segments. In general, there are three types of restriction: lane restriction, height restriction, and weight restriction. For lane restriction on highways, truck flows are typically restricted to the right or outside lane (e.g., curb lane) to improve the efficiency of passenger car travel. For modeling lane restriction, network modification is typically needed to restrict the specific vehicle type on a particular lane. Figure 1 provides a simple lane modification to account for slow-moving vehicles (e.g., trucks) traversing on the right or outside lane.

For height restriction, trucks are restricted on tunnels or underpasses if the vehicles exceed certain height requirements. For weight restriction, heavy trucks exceeding certain weight limit are typically banned on bridges or elevated roadways. These vehicle restrictions that are used to improve safety hazards can be posed as constraints as follows:

(4)

where xm is a link flow vector of class m and Âm is a subset of links of class m in the network, which involves imposing vehicle restriction constraint on vehicle class m. In Equation (4), vehicles of class m are restricted from using link a by setting . As an example of weight restriction constraints, heavy trucks over 17 tons (i.e., vehicle class m) are not allowed to use the designated bridges (i.e., link a denoted as a bridge). These vehicle restrictions, particularly for trucks, are often used to improve highway operations and safety hazards including pavement and structural considerations, work-zone construction restrictions, and crashes.

2.4 Variational inequality formulation

Let denote the equivalent cost of the PSL model on path k between O–D pair rs of class m as follows:

(5)

Denote η and f as the vectors of equivalent path costs and path flows , respectively. Then, the stochastic multi-class traffic assignment problem with asymmetric interactions and vehicle restrictions can be formulated as a variational inequality (VI) problem as follows:

Find a path flow solution f* ∈ Ω, such that

(6)

where Ω represents the feasible set defined by Equations (7)-(11):

(7)
(8)
(9)
(10)
(11)

Equation (7) is the travel demand conservation constraint between O–D pair rs of class m; Equation (8) is a definitional constraint that sums up all path flows of class m that pass through a given link a; Equation (9) is the total flow on link a using the PCE factors to convert the different vehicle classes to equivalent passenger car units; Equation (10) is a non-negativity constraint on the flow of path k between O–D pair of class m; and Equation (11) is the vehicle restriction constraint describing the traffic restraint constraint on link a of class m. Compared with Equation (4), the vehicle restriction constraint is written as an inequality for generality because any equality constraint can be rewritten as two inequality constraints (i.e., x = 0 is equivalent to x ≥ 0 and x ≤ 0). Alternatively, it can be restricted to a sufficiently small number that is equivalent to zero. Note that the side constraint in Equation (11) can also be used to model physical link capacity constraint that restricts the total flow of all vehicle classes to be less than or equal to the capacity on link a (i.e., xa ≤ Ca, where xa is the total flow on link a and Ca is the capacity on link a). For other physical and environmental constraints, see Chen et al. [18] and Xu et al. [19]. This paper only considers different vehicle restriction constraints, particularly for trucks, to enhance the realism of multi-class traffic assignment models.

It should be mentioned that the previous VI formulation is different from the traditional VI formulation for the asymmetric traffic assignment problem [20, 21] and the mathematical programming formulation for the standard traffic assignment model [22, 23] by including an array of important modeling features that are required for the real-world applications of multi-class traffic assignment with multiple vehicle types in urban networks. These features include the following: (1) asymmetric interactions among different vehicle types, (2) a PSL model for modeling different random perceptions and accounting for overlapping routes, and (3) various vehicle restriction constraints (e.g., lane restriction, height restriction, and weight restriction for different types of trucks). These modeling considerations can enhance the realism of traffic assignment models for real-world applications.

3 Solution Procedure

This section presents a customized path-based algorithm specifically designed for solving the stochastic multi-class traffic assignment problem with considerations of asymmetric vehicle interactions through the link travel time functions, route overlapping with a PSL model, and vehicle restriction constraints on specific vehicle classes. This customized path-based algorithm has three main steps: direction finding, line search, and column generation. These three modules work together to solve the stochastic multi-class traffic assignment problem.

3.1 Direction finding

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