**2. Bus scheduling techniques**

The concept of planning a minimum cost set of transporting routes to serve a group of customers is a fundamental constraint in the field of transport and logistics [5–7]. It is because, in the total cost of the product, transportation accounts for about 20% of the total costs of a product [8]. Therefore, the need for developing a better route plan that can reduce the cost of transportation is the concern of various industries in the field.

To address the above issues, the basic and well-studied routing model is the Traveling Salesman Problem (TSP), in which a salesman is to visit a set of cities and return to the city he started in [2, 9, 10]. The objective of the TSP is to minimize the total distance traveled by the salesman. Vehicle Routing Problem (VRP) is a generalization of the TSP in that the VRP consists of determining m vehicle, where a route is a tour that begins at the depot, visits a subset of the customers in a given order, and returns to the depot [9–11].

The activity of planning and designing a delivery or a pickup service to customers in the logistics and supply chain is known as a Vehicle Routing Problem [6]. The first time it was proposed by [12] under the title "Truck dispatching problem" to design the optimum routing of a fleet of gasoline delivery trucks between a bulk terminal and a large number of service stations supplied by the terminal. Often the context is that of delivering goods located at a central depot to customers who have placed orders for such goods, but the area of application of VRP is also so versatile and is used in many areas in real-world life.

"The VRP is defined by a depot, as a set of geographically scattered customers with known demands, and a set of vehicles with fixed capacity" [7, 13]. All depts must be visited just once and the total demand of a route must not exceed the total vehicle capacity. The VRP aims to minimize the overall distribution costs. In most real-life distribution contexts many side constraints complicate the VRP model. These side constraints can be time, that is the total route time and windows time within which the service must begin.

In the literature, VRP was also known as the "vehicle scheduling" (VSP) [6], or "Vehicle dispatching" or simply as the "delivery problem" [14]. It appears very frequently in real-world situations not directly related to the physical delivery of goods.

The VRP problem is a combination of the two well-known optimization problems: the Bin Packing Problem (BPP) and the Traveling Salesman Problem (TSP) [15–17]. The BPP is a problem given a finite set of numbers (the item sizes) and a

### *Linear Programming Optimization Techniques for Addis Ababa Public Bus Transport DOI: http://dx.doi.org/10.5772/intechopen.93629*

constant *K*, specifying the capacity of the bin, what is the minimum number of bins needed? [16].

Logically, all the items have to be inside exactly in one bin and the total capacity of items in each bin has to be within the capacity limits of the bin. This is known as the best packing version of BPP. The TSP [3] is about a traveling salesman who needs to visit several cities. The salesman has to visit each city exactly once, start and end location, commonly called depot in VRP. The issue is to search the shortest tour within all the cities [17]. Connecting this to the VRP, customers can be allotted to vehicles by solving BPP and the order in which they are visited can be found by solving TSP. A VRP with a single vehicle and infinite capacity is a TSP.

VRP is a common name given to a problem in which a set of routes for a fleet of vehicles based at one or several locations called a depot, must be determined for several dispersed cities or customers [18]. The motive is to service a set of customers with a minimum-cost [16, 19]. Vehicle routes originate and terminate at a depot. It is one of the most challenging combinatorial optimization problems in distribution, and logistics [7]. Customers may be in a dispersed location and a fleet of vehicles need to serve them from a depot and return to the depot [16]. The decision here is to determine the assignment of the vehicle (s) and route (s) that a vehicle will serve them best. The commonly used illustration of the input and output of VRP is given in **Figures 1** and **2**.

Since both BPP and TSP are the so-called NP-hard problems and since VRP is a combination of the two, it is also NP-hard [12, 16, 20, 21]. Since the last decades, VRP has got much interest from many scholars. Even in recent years, with the rapid advancements of globalization and supply chain systems, VRP is becoming one of the important research topics in the fields [4, 22, 23].

Moreover, the complexity and its application importance immense literature have devoted to the study and analysis of Bus Scheduling Problem (BSP) and many optimization models have been proposed [23]. The different models developed have tried to accomplish near-optimal solutions with an acceptable amount of computational effort and time [6]. There are many extensions for the Vehicle Schedule Problem (VSP) or VRP with several requirements in the literature over the last 50 years [16, 24]. Among many others, some of the examples are the existence of one depot [18] or more than one depots [4, 16], a heterogeneous fleet with multiple vehicle types [18] the permission of variable departure times of trips, VRP with deterministic demand which is commonly called classical VRP [13, 18].

**Figure 1.** *VRP inputs.*

high variability during each period which requires fluctuating the number of assigned busses in each route. But the enterprise uses mainly a fixed number of busses scheduled per route in its operation throughout the day. This resulted in some busses moving empty while others are being overcrowded, which subsequently results in poor performance and service quality. Moreover, the transportation service in ACBSE has many challenges such as low bus utilization, unsatisfied

*Concepts, Applications and Emerging Opportunities in Industrial Engineering*

To address the challenges of bus assignment and scheduling problems in ACBSE, this paper first focuses to develop a demand-oriented Linear Programming (LP). Linear Programming is a well-accepted technique within the field of Operations Research, a specialty area within the broader field of Industrial Engineering. Then the LP-model is used to solve and optimally satisfy the existing passengers' demand in four operating periods in a day using 93 selected routes. For simplicity purpose,

The concept of planning a minimum cost set of transporting routes to serve a group of customers is a fundamental constraint in the field of transport and logistics [5–7]. It is because, in the total cost of the product, transportation accounts for about 20% of the total costs of a product [8]. Therefore, the need for developing a better route plan that can reduce the cost of transportation is the concern of various

To address the above issues, the basic and well-studied routing model is the Traveling Salesman Problem (TSP), in which a salesman is to visit a set of cities and return to the city he started in [2, 9, 10]. The objective of the TSP is to minimize the total distance traveled by the salesman. Vehicle Routing Problem (VRP) is a generalization of the TSP in that the VRP consists of determining m vehicle, where a route is a tour that begins at the depot, visits a subset of the customers in a given

The activity of planning and designing a delivery or a pickup service to customers in the logistics and supply chain is known as a Vehicle Routing Problem [6]. The first time it was proposed by [12] under the title "Truck dispatching problem" to design the optimum routing of a fleet of gasoline delivery trucks between a bulk terminal and a large number of service stations supplied by the terminal. Often the context is that of delivering goods located at a central depot to customers who have placed orders for such goods, but the area of application of VRP is also so versatile

"The VRP is defined by a depot, as a set of geographically scattered customers with known demands, and a set of vehicles with fixed capacity" [7, 13]. All depts must be visited just once and the total demand of a route must not exceed the total vehicle capacity. The VRP aims to minimize the overall distribution costs. In most real-life distribution contexts many side constraints complicate the VRP model. These side constraints can be time, that is the total route time and windows time

In the literature, VRP was also known as the "vehicle scheduling" (VSP) [6], or "Vehicle dispatching" or simply as the "delivery problem" [14]. It appears very frequently in real-world situations not directly related to the physical delivery of

The VRP problem is a combination of the two well-known optimization problems: the Bin Packing Problem (BPP) and the Traveling Salesman Problem (TSP) [15–17]. The BPP is a problem given a finite set of numbers (the item sizes) and a

passengers' demand, and higher operating costs.

**2. Bus scheduling techniques**

order, and returns to the depot [9–11].

and is used in many areas in real-world life.

within which the service must begin.

goods.

**138**

industries in the field.

in this paper, the four operating periods are named as shifts.

service time at each customer are also known, it can be defined and formulated in

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

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*

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

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

Minimize <sup>¼</sup> <sup>X</sup>*<sup>n</sup>*

X*n i*, *j qi xk*

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

*m* = The number of vehicles available

*Q* = The vehicle capacity.

Time Windows (VRPTW) [18, 21].

inequality such that Eq. (3):

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

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

*k*¼1

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

In many realistic cases, the cost or the distance matrix satisfies the triangular

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

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

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.

*cijx<sup>k</sup>*

*ij* (1)

*ij* ≤ *Q*, *k* ¼ 1, 2, *::n* (2)

*cik* þ *ckj* ≥*cij*; ∀*<sup>i</sup>*,*j*,*<sup>k</sup>* ∈*V:* (3)

*A* ¼ f g ð Þj *i*, *j i*∈*V*, *j*∈*V*, *i* ¼ *j* (4)

*A* ¼ f g ð Þj *i*, *j i* ∈*V*, *j*∈*V*, *i* <*j* (5)

this section as a general VRP model [28].

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

to represent asymmetric or undirected graph.

presented as follow:

Subject to

Where:

two cases [30].

**141**

*xk*

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