**7. Develop the LP model**

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 to the ACBSE problem with data collected from the enterprise.

Definition of terms: i = Route number, where i ¼ 1, 2, 3, … *:*, n j = Shifts, j ¼ 1, 2, … , m *Dij* = Passenger demand of route i at shift j *Cx* and *Cy* = Capacity of bus type-I and bust type-II, respectively *Mj* = Minimum Number of trips required at a given shift j *Fx* and *FY* = Total available busses type-I and bus type-II, respectively *Tij*= Trip factor, maximum trips a bus can be made on route *i* per shift *j Pi* = Trip proportion for route *i*

X*m i*¼1

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

may affect real mathematical optimality.

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

**7.1 Minimum number of buses**

Therefore, *M <sup>j</sup>* is given by:

**7.2 Trip factor analysis**

**145**

four shifts are computed and presented in **Table 3**.

The objective function 6 will minimize the total number of busses required to serve the total demand. Constraint 7 represents the combined capacity of busses assigned to each route i at a given shift j. Eq. 8 and 9 represent the total number of busses that need to be assigned for each route. The total required number of busses must be equal or less than the available number of busses the enterprise has; Eq. 10 shows that the total number of busses to be allotted on each route per shift has to fulfill the minimum number of busses required per route per shift. Eq. 11 and 12 are the number of busses to be assigned to a given route. Moreover, Eq. 13 warrants that the total sum of a probability is always one; last Eq. 14 is nonnegativity constraints. This model guarantees at least one round trip for each route every 30 minutes to maintain a quality service level. It also contributes to the scientific body of knowledge by introducing different bus types in a single LP-model as a new constraint. In the model, the travel time of bus on a given route was considered as the total sum of passenger boarding and alighting time (dwell time), acceleration and deceleration at bus stops, traffic light, and transfer time between each stop. However, little attention was also given to consider the functionality of the model and its output. In particular, the model was run once for all of the four shifts. The number of busses will be checked to make sure, *M <sup>j</sup>* is met for all time slots. The results obtained may be fractional but rounded up later into upper integer values. All these

For clarity and understanding, moreover, some of the parameters were defined. Some of the parameters are, *Mj*, *Tij*, and *Pi*. They are explained and illustrated in detail below. The value of *Mj* is the minimum number of trips required at shift j to meet the maximum allowable waiting time. In this model, a single trip time for a given shift is 30 minutes in the length of the period which is 4 hours. For a given shift, then a minimum of 8 trips are required to limit the maximum waiting time of passengers to 30 minutes. This means, there should be at least one bus every 30 minutes or half an hour for each shift to provide quality service. The actual minimum number of busses required for this model is one bus because within a trip time of 30 minutes one bus can make eight trips during a 4 hour time interval.

*Mj* <sup>¼</sup> *Total Duration for* <sup>s</sup>*hift j minutes* ð Þ

Thus using Eq. 15 and the data reported in **Table 2** above, the value of *Mj* for the

The other parameter that needs to be defined and explained is the trip factor, *Tij*. It is computed using Eq. 16. This value is the minimum number of trips a bus can make on route *i* at a given shift *j*. Since the model computes the total trips that are

<sup>30</sup> *minutes* (15)

*Pi* ¼ 1 (13)

*xij*, *yij* ≥0 (14)

The objective is to minimize the total number of trips made by bus type-I and bust type-II. This objective is represented by Minimize : P*<sup>m</sup> j*¼1 P*<sup>n</sup> <sup>i</sup>*¼<sup>1</sup> *xij* <sup>þ</sup> *yij* h i but needs to fulfill other various constraints.

The first constraint is to set the overall bus capacity of the Enterprise. This is the capacity of the two types of busses assigned in route *i* during shift *j*. This capacity should be greater than or equals to the total demand required by passengers that require ta rip in route *i* at shift *j*. It is Mathematically expressed as *Cxxij* þ *Cyyij* ≥ *Dij*. The second and third constraints are used to check the total number of busses for each type of bus. To this effect, the total trip required by bus type-I and bus type-II should be less than or equal to the total trips available by bus type-I and bus type-II respectively. These are Mathematically expressed as: P<sup>m</sup> *j*¼1 P*<sup>n</sup> <sup>i</sup>*¼<sup>1</sup>*xij* <sup>≤</sup>*Fx* P<sup>m</sup> *j*¼1 P*<sup>n</sup> <sup>i</sup>*¼<sup>1</sup>*Tij*; and P<sup>m</sup> *j*¼1 P*<sup>n</sup> <sup>i</sup>*¼<sup>1</sup>*yij* <sup>≤</sup>*FY* P<sup>m</sup> *j*¼1 P*<sup>n</sup> <sup>i</sup>*¼<sup>1</sup>*Tij*.

The fourth constraint will determine the total minimum number of trips required in every 30 minutes for each of the 93 routes. This constraint is given b, P<sup>m</sup> *j*¼1 P*<sup>n</sup> <sup>i</sup>*¼<sup>1</sup> *xij* <sup>þ</sup> *yij* h i <sup>≥</sup>93P<sup>4</sup> *<sup>j</sup>*¼<sup>1</sup>*Mj* . The trip made by bus type-I and bus type-II for each route i at each shift j should also be less than the total proportion of the trip available. This constrain is mathematically stated as: *xij* ≤ *FxPiTij*; and *yij* ≤*FyPiTij*. The last constraint that shows sum of proportion. It must be summed and become one. This is written as P*<sup>m</sup> <sup>i</sup>*¼<sup>1</sup>*Pi* <sup>¼</sup> <sup>1</sup>*:*

After compiling all the above constraints, the overall LP-model that determines the optimum number of trips i required per route per shift j is formulated as follows.

$$\text{Minimize}: \sum\_{j=1}^{m} \sum\_{i=1}^{n} \left[ \mathbf{x}\_{ij} + \mathbf{y}\_{ij} \right] \tag{6}$$

Subject to

$$\mathbf{C}\_{\mathbf{x}}\mathbf{x}\_{\circ j} + \mathbf{C}\_{\mathbf{y}}\mathbf{y}\_{\circ j} \ge D\_{\circ j} \tag{7}$$

$$\sum\_{j=1}^{\text{m}} \sum\_{i=1}^{n} x\_{ij} \le F\_x \sum\_{j=1}^{\text{m}} \sum\_{i=1}^{n} T\_{ij} \tag{8}$$

$$\sum\_{j=1}^{\text{m}} \sum\_{i=1}^{n} \mathbf{y}\_{ij} \le F\_{\text{y}} \sum\_{j=1}^{\text{m}} \sum\_{i=1}^{n} T\_{ij} \tag{9}$$

$$\sum\_{j=1}^{\mathbf{m}} \sum\_{i=1}^{n} \left[ \mathbf{x}\_{ij} + \mathbf{y}\_{ij} \right] \ge \mathbf{m} \sum\_{j=1}^{\mathbf{m}} \mathbf{M}\_j \tag{10}$$

$$\mathcal{X}\_{\vec{\eta}} \le F\_{\mathfrak{x}} P\_i T\_{\vec{\eta}} \tag{11}$$

$$\mathcal{Y}\_{\vec{\eta}} \le F\_{\mathcal{Y}} P\_i T\_{\vec{\eta}} \tag{12}$$

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

$$\sum\_{i=1}^{m} P\_i = \mathbf{1} \tag{13}$$

$$
\boldsymbol{x}\_{\vec{\eta}}, \boldsymbol{y}\_{\vec{\eta}} \ge \mathbf{0} \tag{14}
$$

The objective function 6 will minimize the total number of busses required to serve the total demand. Constraint 7 represents the combined capacity of busses assigned to each route i at a given shift j. Eq. 8 and 9 represent the total number of busses that need to be assigned for each route. The total required number of busses must be equal or less than the available number of busses the enterprise has; Eq. 10 shows that the total number of busses to be allotted on each route per shift has to fulfill the minimum number of busses required per route per shift. Eq. 11 and 12 are the number of busses to be assigned to a given route. Moreover, Eq. 13 warrants that the total sum of a probability is always one; last Eq. 14 is nonnegativity constraints.

This model guarantees at least one round trip for each route every 30 minutes to maintain a quality service level. It also contributes to the scientific body of knowledge by introducing different bus types in a single LP-model as a new constraint.

In the model, the travel time of bus on a given route was considered as the total sum of passenger boarding and alighting time (dwell time), acceleration and deceleration at bus stops, traffic light, and transfer time between each stop. However, little attention was also given to consider the functionality of the model and its output. In particular, the model was run once for all of the four shifts. The number of busses will be checked to make sure, *M <sup>j</sup>* is met for all time slots. The results obtained may be fractional but rounded up later into upper integer values. All these may affect real mathematical optimality.

#### **7.1 Minimum number of buses**

Definition of terms:

and P<sup>m</sup>

P<sup>m</sup> *j*¼1 P*<sup>n</sup>*

*j*¼1 P*<sup>n</sup>*

Subject to

**144**

j = Shifts, j ¼ 1, 2, … , m

*Pi* = Trip proportion for route *i*

needs to fulfill other various constraints.

*<sup>i</sup>*¼<sup>1</sup>*yij* <sup>≤</sup>*FY*

*<sup>i</sup>*¼<sup>1</sup> *xij* <sup>þ</sup> *yij* h i

one. This is written as P*<sup>m</sup>*

i = Route number, where i ¼ 1, 2, 3, … *:*, n

*Dij* = Passenger demand of route i at shift j

bust type-II. This objective is represented by Minimize :

respectively. These are Mathematically expressed as: P<sup>m</sup>

*<sup>i</sup>*¼<sup>1</sup>*Pi* <sup>¼</sup> <sup>1</sup>*:*

Minimize :

Xm *j*¼1

Xm *j*¼1

Xm *j*¼1 X*n i*¼1

X*n i*¼1

X*n i*¼1

P<sup>m</sup> *j*¼1 P*<sup>n</sup> <sup>i</sup>*¼<sup>1</sup>*Tij*.

≥93P<sup>4</sup>

*Cx* and *Cy* = Capacity of bus type-I and bust type-II, respectively *Mj* = Minimum Number of trips required at a given shift j

*Concepts, Applications and Emerging Opportunities in Industrial Engineering*

*Fx* and *FY* = Total available busses type-I and bus type-II, respectively *Tij*= Trip factor, maximum trips a bus can be made on route *i* per shift *j*

The objective is to minimize the total number of trips made by bus type-I and

The first constraint is to set the overall bus capacity of the Enterprise. This is the capacity of the two types of busses assigned in route *i* during shift *j*. This capacity should be greater than or equals to the total demand required by passengers that require ta rip in route *i* at shift *j*. It is Mathematically expressed as *Cxxij* þ *Cyyij* ≥ *Dij*. The second and third constraints are used to check the total number of busses for each type of bus. To this effect, the total trip required by bus type-I and bus type-II should be less than or equal to the total trips available by bus type-I and bus type-II

The fourth constraint will determine the total minimum number of trips required in every 30 minutes for each of the 93 routes. This constraint is given b,

each route i at each shift j should also be less than the total proportion of the trip available. This constrain is mathematically stated as: *xij* ≤ *FxPiTij*; and *yij* ≤*FyPiTij*. The last constraint that shows sum of proportion. It must be summed and become

After compiling all the above constraints, the overall LP-model that determines the optimum number of trips i required per route per shift j is formulated as follows.

> X*m j*¼1

*xij* ≤*Fx*

*yij* ≤*Fy*

*xij* þ *yij* h i

X*n i*¼1

> Xm *j*¼1

Xm *j*¼1 X*n i*¼1

X*n i*¼1

<sup>≥</sup> <sup>m</sup>X<sup>m</sup>

*j*¼1

*xij* þ *yij* h i

*Cxxij* þ *Cyyij* ≥ *Dij* (7)

*xij* ≤*FxPiTij* (11) *yij* ≤*FyPiTij* (12)

*Tij* (8)

*Tij* (9)

*Mj* (10)

P*<sup>m</sup> j*¼1 P*<sup>n</sup>*

*j*¼1 P*<sup>n</sup>*

*<sup>j</sup>*¼<sup>1</sup>*Mj* . The trip made by bus type-I and bus type-II for

*<sup>i</sup>*¼<sup>1</sup>*xij* <sup>≤</sup>*Fx*

P<sup>m</sup> *j*¼1 P*<sup>n</sup> <sup>i</sup>*¼<sup>1</sup>*Tij*;

*<sup>i</sup>*¼<sup>1</sup> *xij* <sup>þ</sup> *yij* h i

but

(6)

For clarity and understanding, moreover, some of the parameters were defined. Some of the parameters are, *Mj*, *Tij*, and *Pi*. They are explained and illustrated in detail below. The value of *Mj* is the minimum number of trips required at shift j to meet the maximum allowable waiting time. In this model, a single trip time for a given shift is 30 minutes in the length of the period which is 4 hours. For a given shift, then a minimum of 8 trips are required to limit the maximum waiting time of passengers to 30 minutes. This means, there should be at least one bus every 30 minutes or half an hour for each shift to provide quality service. The actual minimum number of busses required for this model is one bus because within a trip time of 30 minutes one bus can make eight trips during a 4 hour time interval. Therefore, *M <sup>j</sup>* is given by:

$$M\_j = \frac{\text{Total Duration for shift } j \text{ (minutes)}}{\text{30 minutes}} \tag{15}$$

Thus using Eq. 15 and the data reported in **Table 2** above, the value of *Mj* for the four shifts are computed and presented in **Table 3**.

#### **7.2 Trip factor analysis**

The other parameter that needs to be defined and explained is the trip factor, *Tij*. It is computed using Eq. 16. This value is the minimum number of trips a bus can make on route *i* at a given shift *j*. Since the model computes the total trips that are


**Table 3.**

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

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

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

$$T\_{\vec{\eta}} = \frac{\text{Total Duration for shift } j}{\text{Single trip travel time for route i}} \tag{16}$$

of each month was computed per route per shift based on the trip proportion (*Pi*) of

In the process of running and solving the model, first, the input data has to be

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

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 passen-

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

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

> X 4

X 93

*xij* <sup>þ</sup> *yij* h i (18)

*Tij* (20)

*Tij* (21)

60*xij* þ 90*yij* ≥ *Dij* (19)

*i*¼1

X 4

X 93

*i*¼1

X 93

*i*¼1

*j*¼1

X 4

*j*¼1

*j*¼1

*xij* ≤ 600 ∗

*yij* ≤90 ∗

and constants into the LP model, the model can be re-written as:

Minimize :

X 4

X 93

*i*¼1

X 93

*i*¼1

*j*¼1

X 4

*j*¼1

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

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

route *i*, and reported in **Table 3**.

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

**7.5 Input parameter to the model**

parametric values are shown in **Table 3**.

routes are shown in **Table 3**.

gers (the Articulated one).

types of busses per route per shift.

Subject to

**147**

By multiplying the trip factor by the available number of busses, that is *Fx* and *Fy*, it can help to find the maximum possible trips made by the total available busses.

#### **7.3 Trip proportion**

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.

$$P\_i = \frac{\text{Daily Demand of Rute i}}{\text{Total Daily Demand of all Rouse}} \tag{17}$$

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* and is reported in **Table 2**.

#### **7.4 Solve the model**

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

of each month was computed per route per shift based on the trip proportion (*Pi*) of route *i*, and reported in **Table 3**.
