**7.5 Input parameter to the model**

needed per route per shift, the trip factor is used to calculate the available number

**Route No. Demand Demani (Dij) per Shift Trip factor**

*Concepts, Applications and Emerging Opportunities in Industrial Engineering*

 4126 0.014 1650 825 1444 206 7 12 8 3 3497 0.012 1399 699 1224 175 4 7 5 2 11,030 0.038 4412 2206 3860 551 4 7 5 2 3029 0.010 1212 606 1060 151 3 5 3 1 778 0.003 311 156 272 39 2 4 3 1

**Pi D1 (m1 = 7) D2 (m2 = 12) D3 (m3 = 8) D4 (m4 = 3) 1 2 3 4**

*Tij* <sup>¼</sup> *Total Duration for shift j*

*Fy*, it can help to find the maximum possible trips made by the total available

The other parameter that needs explanation is the trip proportion, *Pi*. It is required for route *i* during shift *j*. *Pi* is the trip proportion of a given route *i* from the overall routes also given by Eq. 17? *Pi* is used to determine trips to route *i* from the total available. The number of trips for each route is also computed based on the proportion of the total trips of all the routes. It is given by the following equation.

*Pi* <sup>¼</sup> *Daily Demand of Route i*

The last parameter the requires explanation is *Dij*. It is the average daily passengers of route *i* during shift *j* that requires transport services. It is collected from the secondary data sources of the enterprise. It is allocated per shifty by multiplying the average number of daily passengers of route *i* by the demand proportion of shift *j*

The next step is to run the model to obtain feasible solutions. The LP model is solved based on the data of the average daily passengers that have been transported for the last 19 months in four shifts. The daily passengers' that demand for transport for the last 19 months was collected and then the average daily passenger' demand

By multiplying the trip factor by the available number of busses, that is *Fx* and

*Single trip travel time for route i* (16)

**(Tij) per Shift**

*Total Daily Demand of all Routes* (17)

setting how many trips a single bus can make at a given period.

The trip factor is the maximum number of trips a bus can make on route *i* per shift *j*; this factor is used to get the available number of busses in terms of trips. This is because the model computes the total number of trips that are required per route per shift. The actual number of busses is, then, calculated from the trip factor by

of trips per route per shift.

*Input parameters for the LP model for some routes.*

busses.

**Table 3.**

**7.3 Trip proportion**

and is reported in **Table 2**.

**7.4 Solve the model**

**146**

In the process of running and solving the model, first, the input data has to be fitted. In this regard, to fit the LP-model with the input parameters that are involved in the model, first it the parameters needs to be determined. These parameters are either computed or collected from the enterprise. The sample input parametric values are shown in **Table 3**.

These inputs parameters are standard carrying capacity of busses, the operational number of busses, the passenger that demand transport services per route per shift (*Dij*), the trip factors (*Tij*), the minimum number of trips per shift (*Mj*) and the trip proportion per route (*Pi*). The sample input parametric values of a few routes are shown in **Table 3**.

There are four types of busses used by the enterprise (namely DAF, Mercedes, Single, and rigged Articulated busses) but they can be categorized in two based on their seat capacity. These are one bus with seat capacities of 30 passengers (DAF, Mercedes, Single, and rigged Articulated) and busses with seat capacity 50 passengers (the Articulated one).

While fitting to the LP-model busses with a seating capacity of 30 are classified as bus type-I (but can transport 60 passengers) and busses with a seating capacity of 50 are classified as bus type-II (but can transport 90 passengers). The maximum number of capacity, 60 passengers and 90 passengers are based on the standard capacity of public bus transportation [33]. The total capacity of each bus type is equal to the seating capacity plus the standing capacity. The enterprise has a total number of type-I and type-II is 600 and 90, respectively. Thus, the objective function of the research is used to compute the optimum trips and mixes of the two types of busses per route per shift.

The total operational busses in bus type-I are 600 and that of bus type-II is 90 busses. The numbers of operational busses are not only 690, but the rest of the operational busses are kept for backups during failure and other services such as contract and employee service. Also, the 93 routes which are under analysis serve more than 90% of the demand during a day and thus the operational bus assignment is based on this proportion. After substituting the values of input parameters and constants into the LP model, the model can be re-written as:

$$\text{Minimize}: \sum\_{j=1}^{4} \sum\_{i=1}^{93} \left[ \mathbf{x}\_{ij} + \mathbf{y}\_{ij} \right] \tag{18}$$

Subject to

$$\mathbf{0}\mathbf{0}\mathbf{x}\_{\vec{\eta}} + \mathbf{9}\mathbf{0}\mathbf{y}\_{\vec{\eta}} \ge D\_{\vec{\eta}}\tag{19}$$

$$\sum\_{j=1}^{4} \sum\_{i=1}^{93} x\_{\vec{\eta}} \le 600 \ast \sum\_{j=1}^{4} \sum\_{i=1}^{93} T\_{\vec{\eta}} \tag{20}$$

$$\sum\_{j=1}^{4} \sum\_{i=1}^{93} \mathcal{y}\_{ij} \le \mathbf{90} \ast \sum\_{j=1}^{4} \sum\_{i=1}^{93} T\_{ij} \tag{21}$$

*Concepts, Applications and Emerging Opportunities in Industrial Engineering*

$$\sum\_{j=1}^{4} \sum\_{i=1}^{93} \left[ \varkappa\_{\vec{\imath}\vec{\jmath}} + \jmath\_{\vec{\imath}\vec{\jmath}} \right] \ge \Re 3 \ast \sum\_{j=1}^{4} M\_j \tag{22}$$

$$\mathbf{x}\_{\vec{\text{ij}}} \le \mathsf{600} \ast P\_i T\_{\vec{\text{ij}}} \tag{23}$$

from the model output by the trip factor (*Tij*). The sample of the number of busses per route per shift presented in **Table 4** and computed for all 93 routes in the same way.

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

Based on the results of the output of the LP-model, there are 4627 total trips required to serve the average daily passenger demand. The actual number of busses required for a given route i during a given shift *j* depends on the trip factor. That is the number of trips that can be made by each bus type during a given shift. Thus, the output needs to be transformed into the number of busses needed for each route per shift based on the feasible trips that a bus can make during a given shift *j* on each route *i*. From the output of the LP model shown in **Table 3**, the number of trips can be transformed into number busses using the following equation.

*Number of Bus* <sup>¼</sup> *Number of Trips*

For some routes, the output of the LP is small for some route, so adjusting the actual number of busses is required for such routes to have at least two busses on a given route per shift to allocate them on both ends of the route; that is one on the going trip and the other on the returning trip. This is because the demand during a given shift *j* on each route *i* lies in both directions of the route. Sample of the number of busses per route per shift reported in **Table 5** and computed for all 93

The results showed that the actual number of busses required during peak periods is higher than that of off-peak periods. Thus, some of the busses that operate during the morning peak period have to wait on bus stops until they are

> **Bus type-II**

1 31112111 2 51113111 3 16 3 2 3 10 4 0 3 4 61214120 5 41113111

� �������� � �������� � �������� 91 3 0 2 0 3 0 3 0 92 5 1 1 1 3 1 1 1 93 2 1 1 1 1 1 2 0 Total 396 94 114 88 269 94.8 121 79

**Shift 1 Shift 2 Shift 3 Shift 4**

**Bus type-I**

**Bus type-II**

**Bus type-I**

**Bus type-II**

*Tij*

(27)

**9. Assigning buses to routes**

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

routes in the same way.

**Route No.**

**Table 5.**

**149**

*Number of busses per route per shift.*

required for the evening peak.

**Bus type-I**

**Bus type-II**

**Bus type-I**

6 12 3 0 3 8 3

$$\mathcal{Y}\_{\vec{\text{ij}}} \le \mathbf{90} \ast P\_i T\_{\vec{\text{ij}}} \tag{24}$$

$$\sum\_{i=1}^{93} P\_i = \mathbf{1} \tag{25}$$

$$
\boldsymbol{x}\_{\vec{\boldsymbol{y}}}, \boldsymbol{y}\_{\vec{\boldsymbol{y}}} \ge \mathbf{0} \tag{26}
$$
