**2. Types of floors**

This section looks at the classification of floors. Any floor in a building can either be described as an entrance floor or an occupant floor (or both as presented in this paper). An entrance floor (referred to in short as *ent* floor) is a floor through which passengers can either enter or exit the building. An entrance floor can also be referred to an entrance/exit floor, as it becomes an exit floor when the traffic is out-going.

An occupant floor is a floor through which passengers cannot enter or exit the building, but where they will reside during their stay in the building (referred to in short as an *occ* floor). Based on the assumption above, all floors in the building would be denoted as either entrance floors or occupant floors (or both as is allowed in this paper). In some buildings, a floor could simultaneously be an entrance floor and an occupant floor (e.g., ground floor that has a population on it).

An example of a building that is represented in this form is shown below in **Table 1**. Each floor is either an entrance/exit floor or an occupant floor, although one floor is designated with both functions (entrance and occupant) which is the ground floor. The percentage of passengers entering the building via an entrance floor is denoted as the percentage arrival rate [*Prarr*(*i*)]. The population of a floor expressed as a percentage of the total building population is denoted as the percentage population [(*U(i)/U*)].

Where a floor is an entrance floor, it has a nonzero value for its percentage arrival rate and a zero-percentage population. Where a floor is an occupant floor, it has a nonzero percentage population but a zero-percentage arrival rate. Where it has mixed designation, it can have both.


#### **Table 1.** *Representation of traffic in a building.*

It is customary for the entrance floors to be contiguous and for the occupant floors to be contiguous, but this is not necessary. An example of a noncontiguous floor is where a restaurant is located on the top floor of a building and is classified as an entrance/exit floor (as it is functionally external to the building albeit not physically external). The design methodology presented later in this paper can cope with the general case where the entrance floors are noncontiguous and the occupant floors are noncontiguous.

**Table 1** shows a building with three entrances with unequal percentage arrival rates (0.7 ground floor denoted as G; 0.2 from the basement denoted as B1; 0.1 from the basement denoted as B2). There are six occupant floors (G and L1 to L5*)*. They have unequal population percentages.

The lowest floor in the building is denoted as floor 1 and the topmost floor as floor *N*. The following convention is followed in describing the values of arrival percentages and populations for the floors:

*Prarr(i)* is used to denote the arrival percentage of the *i th* floor, where *i* runs from 1 to *N.*

*U(i)/U* is used to denote the percentage population of the *i th* floor, where *i* runs from 1 to *N,* where *U(i)* is the population of the *i th* floor and *U* is the total building population.

The representation of traffic in a building as shown in **Table 1** is the default format and represents pure incoming traffic into the building. Under such a traffic condition, all passengers would be entering the building form the entrance floors and heading to the occupant floors.

A general format for representing the traffic in a building is shown in **Table 2** below. As a generalisation, the term arrival can be extended to arrival/departure to cover both passengers entering the building under incoming traffic conditions and passengers leaving the building under out-going traffic conditions. By setting a population percentage for a floor to zero, it is an indication that it is an entrance floor; by setting an arrival percentage for a floor to zero, it is an indication that it is an occupant floor. By setting a value to the population percentage and a value to the percentage arrival rate, it is an indication that it has dual function.

It is worth noting that the summation of the percentage arrival/departure rates is 1 and the summation of all the building percentage populations is 1 as shown in Eqs. (1) and (2) below.

$$\sum\_{i=1}^{N} \mathbf{Pr}\_{arr}(i) = \mathbf{1} \tag{1}$$


**Table 2.** *General representation format for a building.*


*A Universal Methodology for Generating Elevator Passenger Origin-Destination Pairs… DOI: http://dx.doi.org/10.5772/intechopen.93332*

**Table 3.** *Types of traffic.*

$$\sum\_{i=1}^{N} \left( \frac{U(i)}{U} \right) = \mathbf{1} \tag{2}$$

Two row vectors (arrays) can be developed based on the arrival percentages and the floor population percentages, as shown below. The first row vector is the percentage arrivals, denoted as *Prarr*:

$$\text{Pr}\_{\text{arrr}} = \begin{bmatrix} \text{Pr}\_{\text{arrr}}(\mathbf{1}) \text{ } \text{Pr}\_{\text{arrr}}(\mathbf{2}) & \dots & \dots & \dots & \text{Pr}\_{\text{arrr}}(\mathbf{N}-\mathbf{1}) \text{ } \text{Pr}\_{\text{arrr}}(\mathbf{N}) \text{ } \end{bmatrix} \tag{3}$$

The convention that will be used in this paper is that the indexing will start from the lowest floor in the building and increase upwards. The percentage population can be also be organised into a row vector:

*Unor* ¼ ½ � *U*ð Þ1 *=U U*ð Þ2 *=U* ……… *U N*ð Þ � 1 *=U UN*ð Þ*=U* (4)

The "nor" subscript stands for "normalised".

### **3. Types of traffic**

This section classifies the possible types of journeys and thus the possible types of traffic. Every passenger journey must logically have an origin and a destination. Considering that any floor can either be an entrance/exit floor or an occupant floor, there can exist in theory four types of journeys depending on the classification of the origin and destination floors for each journey.

A journey that starts from an entrance/exit floor and terminates at an occupant floor is denoted as an incoming traffic journey. A journey that starts from an occupant floor and terminates at an entrance/exit floor is denoted as an outgoing traffic journey. A journey that starts from an occupant floor and terminates at an occupant floor is denoted as an inter-floor journey. A journey that starts from an entrance/exit floor and terminates at an entrance/exit floor is denoted as an inter-entrance journey. These four types of traffic are listed in **Table 3** below.

### **4. Description of the traffic in a building**

It has become customary to describe the prevailing traffic in a building at any one point in time as a mixture of the four types of traffic described in the previous section.

The percentage of the traffic that is incoming at any one point in time is denoted as *ic*; the percentage of the traffic that is outgoing at any one point in time is denoted as *og*; and the traffic that is inter-floor at any one point in time is denoted as *if*; and the traffic that is inter-entrance is denoted as *ie*. The combination of these four numbers can be used to describe the traffic mix as shown below (where any of these parameters can vary between 0 and 1):

$${ic: og: \#\, : ie}$$

As expected, the sum of all four numbers should add up to 1 as shown in Eq. (5) below. Thus, assigning values for three of these numbers automatically sets the value of the fourth parameter.

$$
\dot{a}\mathbf{\dot{c}} + \mathbf{og} + \dot{\mathbf{f}}\mathbf{\dot{f}} + \dot{\mathbf{e}}\mathbf{\dot{e}} = \mathbf{1} \tag{5}
$$

As an example, one suggested composition of the lunchtime traffic conditions can be described by the following representation [11]:

$${\{i:\text{og}:\text{if}\,:\text{je}\,\text{as}\,\text{0.45}:0.45:0.10:0.0\text{ respectively}}}.$$

#### **5. The origin-destination (***OD***) matrix**

At the end of Section 2, the percentage floor populations were compiled in a concise normalised format in the shape of a row vector denoted as *Unor***.** In addition, the percentage arrivals were compiled in a concise format in the shape of a row vector denoted as *Prarr*.

In order to compile the overall final origin destination, it is first necessary to develop the four origin-destination matrices that contain the contribution from the four types of traffic (incoming, outgoing, inter-floor and inter-entrance). Once these four matrices have been fully developed, the final overall origin-destination matrix is found by adding these four matrices.

#### **5.1 Finding the initial values of the four OD matrices**

The first matrix (the incoming traffic matrix) is obtained by multiplying the transpose of the percentage arrival vector (*Prarr*) by the normalised population vector (*Unor*). As the dimensions of the percentage arrival row vector is 1 by N, when it is transposed, it dimensions become N by 1. The dimensions of the population normalised row vector is 1 by N. When an N by 1 vector is multiplied by a 1 by N vector, this results in an N by N matrix.

$$OD\_{ic\\_ini} = Pr\_{arr}{}^T \bullet U\_{nor} \tag{6}$$

The second matrix (the outgoing traffic matrix) is obtained by multiplying the transpose of the normalised population vector (*Unor*) by the percentage arrival vector (*Prarr*) as shown in Eq. (7) below.

$$OD\_{\text{og\\_}ini} = U\_{nor} \, ^T \bullet \, Pr\_{arr} \tag{7}$$

The third matrix (the inter-floor traffic matrix) is obtained by multiplying the transpose of the normalised population vector (*Unor*) by the normalised population vector (*Unor*) as shown in Eq. (8) below.

*A Universal Methodology for Generating Elevator Passenger Origin-Destination Pairs… DOI: http://dx.doi.org/10.5772/intechopen.93332*

$$OD\_{\text{if\\_ini}} = U\_{nor} \, ^T \bullet U\_{nor} \tag{8}$$

The fourth matrix (the inter-entrance traffic matrix) is obtained by multiplying the transpose of the percentage arrival vector (*Prarr*) by the percentage arrival vector (*Prarr*) as shown in Eq. (7) below.

$$OD\_{ie\\_ini} = \Pr\_{arr} \, ^T \bullet \Pr\_{arr} \tag{9}$$

where the subscript T denotes the transpose of a matrix.

It is worth noting that each of these matrices that result from the multiplication have a total sum of all the elements that is equal to 1.

#### **5.2 Adjusting the four OD matrices**

Once the four initial Origin-Destination matrices have been produced by multiplying the relevant vectors, the next step is to remove the irrational traffic (travelling from a floor back to the same floor). This is done in three steps as discussed below:

a. Finding the value of the adjusting factor, M, for each matrix. The value of M is necessary in order to re-adjust the sum of the matrix back to 1, later on. M is calculated as shown below:

$$M = 1 - \left(\sum\_{i=1}^{N} (pr\_{ii})\right) \tag{10}$$

b. The second step is to zero all the diagonal elements of the four matrices. This removes the irrational behaviour of a passenger going from a floor back to the same floor.

$$p\_{ii} = 0 \quad for \quad i = j \tag{11}$$

c. But as the diagonal items have been zeroed, the sum of all the elements in the matrix no longer adds up to 1. Thus, the third step is to re-adjust the remaining nonzero elements of the matrices in order to restore the sum of all elements in the matrix back to 1. This is done by dividing each element in the matrix by the value of M, found in Eq. (10) above.

Once these three steps have been carried out, the resulting four OD matrices become the final matrices.

$$OD\_{ic\\_fin}, OD\_{og\\_fin}, OD\_{if\\_fin}, OD\_{ic\\_fin}$$

#### **5.3 Finding the final OD matrix from the four OD matrices**

The final origin–destination matrix can now be calculated by adding the weighted sum of the four final matrices. Each one is multiplied by the percentage of the traffic that is represents. This is shown in Eq. (12) below.

$$\text{OD}\_{\text{fin}} = \text{ic} \bullet \text{OD}\_{\text{ic\\_fin}} + \text{og} \bullet \text{OD}\_{\text{og\\_fin}} + \text{if} \bullet \text{OD}\_{\text{if\\_fin}} + \text{ic} \bullet \text{OD}\_{\text{ig\\_in}} \tag{12}$$

This matrix can also be referred to as the "*normalised* origin-destination matrix". It is referred to as *normalised* as it only depends on the arrival percentages of the floors, the population percentages of the floors and the mix of traffic. It does not depend on the passenger arrival intensity. The matrix in this form can now be used in order to generate random passenger origin–destination pairs for evaluating the round-trip time using the Monte Carlo Simulation (*MCS*) method. It can also be used to evaluate the round-trip time using formula by calculation. The sum of all the elements of the matrix is equal to 1. The diagonal element of the matrix must be equal to zero.
