**3.1. WSM (Weighted Sum Model)**

**Bandwidth (BW):** This metric refers to the available bandwidth in WiMAX cell. It is simply

**Congestion delay:** This is the delay of packets due to queuing until they can be processed.

The cell selection problem is about selecting one BS for handover among limited number of candidate BSs with respect to a set of different criteria. This is a typical Multiple Criteria Decision Making (MCDM) problem. In the study of decision making, terms such as MCDM is the problem of choosing an alternative solution from a set of alternatives, which are charac‐ terized in terms of their attributes [14]. The most popular classical MCDM methods are:

**1.** *WSM (Weighted Sum Model):* the overall score of a candidate BS is determined by the

**2.** *TOPSIS (Technique for Order Preference by Similarity to Ideal Solution):* the chosen candidate BS is the one which is closest to ideal solution and the farthest from the worst case solution.

**3.** *AHP (Analytic Hierarchy Process):* decomposes the BS selection problem into several sub-

In this paper, we use TOPSIS as alternative score ranking based cell selection scheme and compare it with WSM, while AHP will be used for weighting the attributes or the criteria based

Cell selection could be considered as an MCDM problem. For instance, suppose a user is currently connected to BS A1 and has to make a decision among two alternative candidate BSs: A2 and A3. Handover criteria considered here are CINR, BW, and congestion delay, which are denoted as: X1, X2 and X3, respectively. The decision problem can be modelled in a decision

é ù ê ú <sup>=</sup> ê ú ê ú ë û

Where *x*11 is the CINR, *x*12 is the BW and *x*13 is the congestion delay of BS A1. In similar way, *x*21, *x*22and *x*23 are the CINR, BW and congestion delay values of BS A2. In addition, *x*31, *x*<sup>32</sup>

Assume the user is using two types of applications; real-time application such as VoIP and non-real-time application such as Media Content Download. The traffic or application

(1)

*xxx Ax x x Dm A x x x Ax x x*

matrix Dm as shown in (1), where the capabilities of each candidate are presented.

1 2 3

problems and assigns a weight value for each sub-problem.

on the importance of each criterion for the end users application.

and *x*33 are the CINR, BW and congestion delay of BS A3.

the difference between the total capacities and the aggregated used BW in Kbps.

**3. Multi-criteria decision making methods**

88 Selected Topics in WiMAX

weighted sum of all the attribute values.

WSM is the most popular multi criteria decision making (MCDM) method. It is the simplest way of evaluating the number of alternatives (*m*) in terms of a number of decision criteria (*n*) [17]. The overall score of an alternative is calculated as the weighted sum of all the attribute values as shown in equation (4).

$$A\_i^{\{\text{WSM}\}} = \sum\_{j}^{n} \pi\_j \mathbf{x}\_{ij}, i = 1, 2, 3, \dots \\ \text{m.} \tag{4}$$

Where *Ai* is the evaluated score of an alternative, *Wj* is the weight value for criteria *j*, *n* number of criteria. Because the decision matrix value could be in different scales such as BW could be 10 Mbps and the cell load could be 50% or 0.50 the decision matrix has to have a comparable scale (normalized) by using (5) for the benefit criteria (i.e. stronger CINR, larger BW) and (6) for cost criteria (i.e. more delay). In (5) and (6) *xij* is the performance score of alternative *Ai* with respect to criterion *xj* and *rij* is the normalization value of *xi <sup>j</sup>* .

$$r\_{ij} = \mathbf{x}\_{ij} \;/\; \mathbf{x}\_{j^{\text{MAX}}}, j = 1, \dots \\ m, j = 1, \dots \\ n \tag{5}$$

$$r\_{i\dot{j}} = \mathbf{x}\_{j\text{MIN}} \;/\; \mathbf{x}\_{i\dot{j}}, \mathbf{i} = \mathbf{1}, \dots \\ m, \dot{j} = \mathbf{1}, \dots \\ n \tag{6}$$

## **3.2. Technique for Order Preference by Similarity to Ideal Solution (TOPSIS)**

TOPSIS is one of MCDM methods based on the concept that the chosen alternative should have the shortest distance from the positive ideal solution (PIS) and the farthest from the negative ideal solution (NIS) for solving a multiple criteria decision making problem. Briefly, PIS is made up of all best values attainable criteria, whereas NIS is composed of all worst values incurred from criteria [14]. The calculation processes of this method are as follows. Normalize the decision matrix.

$$r\_{ij} = \mathbf{x}\_{ij} \land \sqrt{\sum\_{i=1}^{m} \mathbf{x}\_{ij}}, i = 1, \dots, m, j = 1, \dots, n \tag{7}$$

The AHP method main steps are as follows:

shown in Equation (12). Noticeably, the *aij*

the hierarchy.

**1.** Define the decision criteria and decompose the decision problem into different levels of

**2.** Compare each factor to all the other factors within the same level through pairwise

11 12 13 1 21 22 23 2

é ù ê ú

*aaa a aaa a*

*iii ij*

**1.** After creating matrix *A* of comparison, the next step is to determine the weights of the criteria, in which *wi* is the weight of objective *i* in the weight vector *w* = [*w*1, *w*<sup>2</sup> , …, *wn* ]

ë û

1

**1.** Then we divide each element of the matrix with the sum of its column, we have normalized

'

*i*

**2.** Check the Consistency Index (CI). The CI can be computed as the difference between the

*ij*

*a*

*n i ij j a a* =

'

''

*a*

*ij*

relative weight. The sum of each column is 1 as shown in (14).

**3.** Previous steps are repeated until we get a very small value of CI.

current and previous computed eigenvectors.

**4.** Determine the weights of the criteria

*aaa a*

123

tance of criterion *i* to that of criterion *j*. The criteria are compared pair-wise with respect to the goal. *A* is a matrix indicating the importance of criterion *i* relative to criterion *j* as

> ... ...

> ...

= 1 when *i*= *j*, while *a ji*

Hybrid AHP and TOPSIS Methods Based Cell Selection (HATCS) Scheme for Mobile WiMAX

*j j* , relating the impor‐

http://dx.doi.org/10.5772/55894

, which reflects the

(12)

91

= 1/*aij*

<sup>=</sup> å (13)

*<sup>a</sup>* <sup>=</sup> (14)

comparison matrix. The judgments in the AHP are made in pairs *aij*

reciprocal importance of criterion *j* relative to criterion *i*.

*A*

for *n* criteria. To get this, the eigenvector is used.

**1.** Raise the *A* matrix to powers square.

**2.** Sum each column of the squared matrix.

The eigenvector can be calculated using the following steps:

=

.


$$A^\* = \left(\max\_i v\_{i\\_j} \mid j \in I\right) \cdot \left(\min\_i v\_{i\\_j} \mid j \in I\right) \tag{8}$$

$$A^{-} = \left(\min\_{i} v\_{ij} \mid j \in J\right) / \left(\max\_{i} v\_{ij} \mid j \in J^{\prime}\right) \tag{9}$$

Where *vi <sup>j</sup>* is the weighted and normalized of the *xij* , while *J* is associated with the benefit criteria and *J* ' is associated with the cost criteria.

**•** Calculate the separation of each alternative from the ideal solution, and the negative ideal solution.

$$\begin{aligned} \mathcal{S}\_{i+} &= \sqrt{\sum\_{j=1}^{m} (a\_{ij} - a\_{j+})^2}, i = 1, \dots, n\\ \mathcal{S}\_{i-} &= \sqrt{\sum\_{j=1}^{m} (a\_{ij} - a\_{j-})^2}, i = 1, \dots, n \end{aligned} \tag{10}$$

**•** Relative closeness to the ideal solution is calculated.

$$\mathbf{C}\_{i+} = \mathbf{S}\_{i-} / (\mathbf{S}\_{i-} + \mathbf{S}\_{i+}) / i = \mathbf{1}, \ldots, n \tag{11}$$

#### **3.3. Analytic Hierarchy Process (AHP)**

The Analytic Hierarchy Process (AHP) is a structured technique for organizing and analyzing complex decision problems. It has a variety of applications used around the world in different fields of decision situations such as business, industry, healthcare, engineering and education.

It provides a comprehensive and rational framework for structuring a decision problem, for weighting the decision criteria, and for evaluating alternative solutions [14, 18]. In this paper, we use the AHP method for weighting the decision criteria. The calculation processes of this method are as follows and shown in Figure 2.

The AHP method main steps are as follows:

PIS is made up of all best values attainable criteria, whereas NIS is composed of all worst values incurred from criteria [14]. The calculation processes of this method are as follows. Normalize

/ , 1,..., , 1,....

= == å (7)

.

, while *J* is associated with the benefit

(10)

) (8)

and the negative-ideal solutions *A*<sup>−</sup>

*vi <sup>j</sup>* <sup>|</sup> *<sup>j</sup>* <sup>∈</sup> *<sup>J</sup>* '

*A v jJ v jJ* - =Î Î (9)

/ ( ), 1,..., *C S S Si n ii ii* +- -+ = += (11)

*r x x i mj n*

*vi <sup>j</sup>* <sup>|</sup> *<sup>j</sup>* <sup>∈</sup> *<sup>J</sup>*), (min*<sup>i</sup>*

(min | , max | ' *i j* ) ( *i j* ) *<sup>i</sup> <sup>i</sup>*

**•** Calculate the separation of each alternative from the ideal solution, and the negative ideal

2

( ) , 1,...,

2

( ) , 1,...,

The Analytic Hierarchy Process (AHP) is a structured technique for organizing and analyzing complex decision problems. It has a variety of applications used around the world in different fields of decision situations such as business, industry, healthcare, engineering and education.

It provides a comprehensive and rational framework for structuring a decision problem, for weighting the decision criteria, and for evaluating alternative solutions [14, 18]. In this paper, we use the AHP method for weighting the decision criteria. The calculation processes of this

1

+ + =

*S aa i n*

= -=

= -=

*S aa i n*

*m i ij j j m i ij j j*

å

å

1


1

=

**•** Decision matrix is weighted using the weighting factor.

*<sup>A</sup>*<sup>+</sup> =(max*<sup>i</sup>*

is the weighted and normalized of the *xij*

criteria and *J* ' is associated with the cost criteria.

**•** Relative closeness to the ideal solution is calculated.

**3.3. Analytic Hierarchy Process (AHP)**

method are as follows and shown in Figure 2.

**•** Determine the ideal solutions *A*<sup>+</sup>

*m ij ij ij i*

the decision matrix.

90 Selected Topics in WiMAX

Where *vi <sup>j</sup>*

solution.


$$A = \begin{bmatrix} a\_{11} & a\_{12} & a\_{13} & \dots & a\_{1j} \\ a\_{21} & a\_{22} & a\_{23} & \dots & a\_{2j} \\ \vdots \\ a\_{i1} & a\_{i2} & a\_{i3} & \dots & a\_{ij} \end{bmatrix} \tag{12}$$

**1.** After creating matrix *A* of comparison, the next step is to determine the weights of the criteria, in which *wi* is the weight of objective *i* in the weight vector *w* = [*w*1, *w*<sup>2</sup> , …, *wn* ] for *n* criteria. To get this, the eigenvector is used.

The eigenvector can be calculated using the following steps:


$$a^\*\_{\ i} = \sum\_{j=1}^n a\_{ij} \tag{13}$$

**1.** Then we divide each element of the matrix with the sum of its column, we have normalized relative weight. The sum of each column is 1 as shown in (14).

$$a^{""}\_{\
u} = \frac{a\_{ij}}{a^{"}\_{\
u}}\tag{14}$$


**Figure 2.** AHP calculation processes
