**3. Modeling and vehicle routing problem**

Vehicle Routing Problem has been used in different applications and practices but with many constraints. The major constraints according to [16] are the network [25], demand and customers, depot locations, the type of vehicle used, and sometimes drivers. it is not possible to satisfy all the constraints in one model. In this case, some of the constraints can be reduced without loss of generality. When some customers left unserved due to this reason, in VRP, it called route failure [7]. In some cases, by introducing different penalties or priorities, the situations can be handled [6, 16].

The routing operations are performed to serve customers to start and end at one or more locations, located at the road [5]. Each location or depot is identified by the number and types of vehicles associated with it and by the number of goods it can deal with [23]. Transportation of goods is carried out by using a fleet of vehicles with fixed composition and size according to the necessities of the customers [25, 26].

The vehicle used in VRP is also one constraint in the model assumptions. According to [16], the vehicles may be characterized by the capacity of the vehicle, expressed as the maximum weight, or volume, or the number of pallets, the vehicle can carry. In most application areas, the least practice constraint is the drivers [16].

VRP is one of the most studied combinatorial optimization problems and is concerned with the optimal design of routes to be used by a fleet of vehicles to serve a set of customers [26]. VRP is not only focusing on the delivery and collection of goods but involves also in different application areas arising from the transportation and logistics system [16].

Different VRP models have been developed for different applications to study the Routing Problem [1, 22]. The most recent is the one that has been devoted to more complex variants of the VRP occasionally called "rich" VRPs. These are closer to the VRP models [23] and are used to design an optimum allocation system that can improve the level of their services [27]. Toth Paolo and Vigo Daniele [16] have reported that the use of computerized methods in the VRP problems has significantly saved the computational effort ranging from 5 to 20%.

VRP can be designed as a directed or undirected graph subject to the problem environment [21, 28]. In the case of classical VRP where customers or customer's demands are known in advance and the driving time between the customer and the *Linear Programming Optimization Techniques for Addis Ababa Public Bus Transport DOI: http://dx.doi.org/10.5772/intechopen.93629*

service time at each customer are also known, it can be defined and formulated in this section as a general VRP model [28].

It is designated in graph theory. To define the general model of VRP, let *G* ¼ ð Þ *v*, *A* be an asymmetric graph where *V* ¼ f0, 1, … , *n*Þ is a set of vertices representing cities or depot situated at the vertex 0, and *A* is the set of arcs. In every arcð Þ *<sup>i</sup>*, *<sup>j</sup>* ; *<sup>i</sup>* <sup>¼</sup> *<sup>j</sup>* is related to a nonnegative distance matrix *<sup>C</sup>* <sup>¼</sup> *Cij* � ). In some cases, *Cij* can be understood as a travel cost or as a travel time [21, 29, 30], it is often appropriate to substitute *A* by *E*. Where *E* is a set of undirected edges in the graph to represent asymmetric or undirected graph.

The common VRP comprises a set of at most *K* delivery or collection routes plan such that each route starts and ends at the depot, each customer is visited exactly once by exactly one vehicle, the total demand of each route does not exceed the vehicle capacity and the total routing cost is minimized. With all these assumptions, according to Stewart and Golden [31], a compact representation of VRP can be presented as follow:

$$\text{Minimize} = \sum\_{k=1}^{n} c\_{\vec{\eta}} \mathbf{x}\_{\vec{\eta}}^{k} \tag{1}$$

Subject to

**3. Modeling and vehicle routing problem**

*Concepts, Applications and Emerging Opportunities in Industrial Engineering*

handled [6, 16].

**Figure 2.** *VRP outputs.*

drivers [16].

**140**

and logistics system [16].

Vehicle Routing Problem has been used in different applications and practices but with many constraints. The major constraints according to [16] are the network [25], demand and customers, depot locations, the type of vehicle used, and sometimes drivers. it is not possible to satisfy all the constraints in one model. In this case, some of the constraints can be reduced without loss of generality. When some customers left unserved due to this reason, in VRP, it called route failure [7]. In some cases, by introducing different penalties or priorities, the situations can be

The routing operations are performed to serve customers to start and end at one or more locations, located at the road [5]. Each location or depot is identified by the number and types of vehicles associated with it and by the number of goods it can deal with [23]. Transportation of goods is carried out by using a fleet of vehicles with fixed composition and size according to the necessities of the customers [25, 26]. The vehicle used in VRP is also one constraint in the model assumptions. According to [16], the vehicles may be characterized by the capacity of the vehicle, expressed as the maximum weight, or volume, or the number of pallets, the vehicle can carry. In most application areas, the least practice constraint is the

VRP is one of the most studied combinatorial optimization problems and is concerned with the optimal design of routes to be used by a fleet of vehicles to serve a set of customers [26]. VRP is not only focusing on the delivery and collection of goods but involves also in different application areas arising from the transportation

Different VRP models have been developed for different applications to study the Routing Problem [1, 22]. The most recent is the one that has been devoted to more complex variants of the VRP occasionally called "rich" VRPs. These are closer to the VRP models [23] and are used to design an optimum allocation system that can improve the level of their services [27]. Toth Paolo and Vigo Daniele [16] have reported that the use of computerized methods in the VRP problems has signifi-

VRP can be designed as a directed or undirected graph subject to the problem environment [21, 28]. In the case of classical VRP where customers or customer's demands are known in advance and the driving time between the customer and the

cantly saved the computational effort ranging from 5 to 20%.

$$\sum\_{i,j}^{n} q\_i \mathbf{x}\_{ij}^k \le \mathbf{Q}, k = 1, 2, \dots \\ n \tag{2}$$

Where:

*cij* = The cost/distance of traveling, from *i* to *j:*

*xk ij* = 1 if vehicle k travels from *i* to *j*; 0 otherwise

*m* = The number of vehicles available

*Sm* = The set of all feasible solutions in the m-traveling salesman problem (m-TSP)

*qi* = The amount demanded at location *i*

*Q* = The vehicle capacity.

In many realistic cases, the cost or the distance matrix satisfies the triangular inequality such that Eq. (3):

$$c\_{ik} + c\_{kj} \ge c\_{ij}; \forall\_{i,j,k} \in V. \tag{3}$$

In the VRP models, a differentiation has to be made between symmetric Eq. (4) and asymmetric Eq. (5). Solution approaches can vary significantly between these two cases [30].

$$A = \{(i, j) \mid i \in V, j \in V, i = j\} \tag{4}$$

$$A = \{(i, j) \mid i \in V, j \in V, i < j\} \tag{5}$$

In the real world, however, the general VRP model is enhanced by various constraints or side-constraints, [5]. The constraints can be such as vehicle capacity or time interval in which each customer has to be served [16], revealing the Capacitated Vehicle Routing Problem (CVRP) [20] and the Vehicle Routing Problem with Time Windows (VRPTW) [18, 21].

VRP models, whether they are used for public transport or transit, as well as distribution and logistics, they share certain mutual features. That is, they focus on the optimization of cost (working cost), distance covered, waiting time, etc.


occurs after the morning peak and the evening peak shift in which 20% and 5% for

**Shift Time interval Duration (minute) Demand proportion (%)**

Morning peak hours 6:15–9:30 195 40% First off-peak hours 9:30–15:30 360 20% Evening peak hours 15:30–19:30 240 35% Second off-peak hours 19:30–21:00 90 5% **Total** 870 100

*Linear Programming Optimization Techniques for Addis Ababa Public Bus Transport*

The researchers have investigated the existing operating systems of the ACBSE, and a linear programming (LP) model was developed. The model helps to achieve a solution for the scheduling problems of the enterprise. The LP model developed in this study is a new approach in the literature of VRP that considers the trips made by two different types of busses to address the demand distribution of passengers in

The LP model was fitted with data. To achieve its object, the model was coded and run using the General Algebraic Modeling System (GAMS) optimization software. The GAMS code was running within 0.15 seconds on 3.10 GHz, Window7 Home Premium, 4GB RAM, and core (i5) Dell Personal computer (Optiplex 790-Model). The resulting solution of the LP-model was first the number of trips per route per shift for each type of bus. After obtaining the solution, then, the number of trips was

translated into the number of busses per route per shift for each type of bus.

total distance traveled, and the different operating costs.

to the ACBSE problem with data collected from the enterprise.

The LP model was investigated and validated to identify potential areas of improvement in the scheduling and assignment problem of ACBSE. It is also used to determine the number of busses to be assigned in a given route at a given shift that can address the demand distribution of passengers each shift for the 93 routes. The results of the LP-model were validated by comparing the current schedule and performances of the enterprise. The validation was made using four performance measuring parameters namely bus utilization, the total number of trips made, the

In order to develop the LP mode, leti be the number of trips per route per shiftj made by bus type-I is represented by "*xij*" and by that of bus type-II is marked by "*yij*". The model is developed as a general model, at this section, then later is fitted

the demand proportion is allocated respectively.

*Demand proportion and duration of each shift.*

*DOI: http://dx.doi.org/10.5772/intechopen.93629*

**6. Model development**

**Table 2.**

93 routes in four working shifts.

**6.1 Running the model**

**6.2 Model validation**

**7. Develop the LP model**

**143**

**Table 1.**

*The fixed number of busses assigned on selected routes (as of 2011).*
