Shannon Entropy

In this section we explain the methodology to use the entropy method to determine the weights of attributes.

Entropy is an important concept in information theory proposed by Shannon. It is degree of uncertainty formulated in terms of discrete probability distribution. Shannon entropy became significant in quantifying the randomness in various scientific fields such as data compression , financial analysis , information theory , statistics and engineering science . The entropy theory can be effectively employed in decision making process since it measures existent contrasts among sets of data. However, most of the existing approaches like AHP, ANP, SD, are used to obtain the attribute weights. Among those, entropy which is a scientific tool is used to determine the weight for each attribute since it considers important knowledge of different services performance in different QoS aspects and avoids equivalent (similar) preferences among decision makers towards the attributes. In particular, the degree of diversification of information provided by values of each attribute can be used to determine the attribute weights.

The following procedure should be adopted to determine weight for each attribute through Shannon entropy procedure involved in entropy is as follows:

Step-1: Normalize the given performance matrix  by transforming every element in  into a corresponding element in the normalized matrix through linear normalization method.

(1)

Step-2: Calculate the entropy value  for each attribute  based on Eqn. (1)

(2)

Where the parameter  is Boltzmann’s constant or normalization factor equal to . Suppose,  then  ensure the validity of Eqn. (2).

Step-3: Obtain the attribute weights using Eqn. (3)

(3)

TOPSIS

The Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) is a unique approach used to obtain the optimal alternative or rank the alternatives should have the shortest distance from the reliable ideal (best) solution and the farthest distance from non-reliable ideal (worst) solution. A solution is said to be reliable ideal solution if it maximizes the benefit criteria or minimizes the cost criteria. On the other hand, the solution which maximizes the cost criteria or minimizes the benefit criteria is a non-reliable ideal solution.

Suppose an MADM problem consists of  alternatives represented as  and  decision attributes denoted as . The performance ratings assigned to the alternatives with respect to each attributes constitutes a performance matrix or evaluation matrix or decision matrix represented by . Let  be the weight vector of each attribute satisfying . Then, the TOPSIS approach can be summarized as in definition 1.

Definition 1. Establish a performance matrix  with a set of alternatives and criteria. Normalize the performance matrix as defined in Eqn. (1). Compute the weighted normalization matrix using Eqn. (4)

(4)

Obtain the reliable ideal and non-reliable ideal solutions represented as  and  respectively as mentioned in Eqn. (5) and (6)

(5)

(6)

Where  denotes benefit criteria and  represents cost criteria.

Compute the separation measures of each alternative from the reliable ideal solution and non-reliable ideal solution using Euclidean distance given in Eqn. (7) and (8).

(7)

(8)

Calculate the relative closeness of each alternative to the ideal solution using Eqn. (9)

(9)

Where . The larger the , the Shannon Entropy

In this section we explain the methodology to use the entropy method to determine the weights of attributes.

Entropy is an important concept in information theory proposed by Shannon. It is degree of uncertainty formulated in terms of discrete probability distribution. Shannon entropy became significant in quantifying the randomness in various scientific fields such as data compression , financial analysis , information theory , statistics and engineering science . The entropy theory can be effectively employed in decision making process since it measures existent contrasts among sets of data. However, most of the existing approaches like AHP, ANP, SD, are used to obtain the attribute weights. Among those, entropy which is a scientific tool is used to determine the weight for each attribute since it considers important knowledge of different services performance in different QoS aspects and avoids equivalent (similar) preferences among decision makers towards the attributes. In particular, the degree of diversification of information provided by values of each attribute can be used to determine the attribute weights.

The following procedure should be adopted to determine weight for each attribute through Shannon entropy procedure involved in entropy is as follows:

Step-1: Normalize the given performance matrix  by transforming every element in  into a corresponding element in the normalized matrix through linear normalization method.

(1)

Step-2: Calculate the entropy value  for each attribute  based on Eqn. (1)

(2)

Where the parameter  is Boltzmann’s constant or normalization factor equal to . Suppose,  then  ensure the validity of Eqn. (2).

Step-3: Obtain the attribute weights using Eqn. (3)

(3)

TOPSIS

The Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) is a unique approach used to obtain the optimal alternative or rank the alternatives should have the shortest distance from the reliable ideal (best) solution and the farthest distance from non-reliable ideal (worst) solution. A solution is said to be reliable ideal solution if it maximizes the benefit criteria or minimizes the cost criteria. On the other hand, the solution which maximizes the cost criteria or minimizes the benefit criteria is a non-reliable ideal solution.

Suppose an MADM problem consists of  alternatives represented as  and  decision attributes denoted as . The performance ratings assigned to the alternatives with respect to each attributes constitutes a performance matrix or evaluation matrix or decision matrix represented by . Let  be the weight vector of each attribute satisfying . Then, the TOPSIS approach can be summarized as in definition 1.

Definition 1. Establish a performance matrix  with a set of alternatives and criteria. Normalize the performance matrix as defined in Eqn. (1). Compute the weighted normalization matrix using Eqn. (4)

(4)

Obtain the reliable ideal and non-reliable ideal solutions represented as  and  respectively as mentioned in Eqn. (5) and (6)

(5)

(6)

Where  denotes benefit criteria and  represents cost criteria.

Compute the separation measures of each alternative from the reliable ideal solution and non-reliable ideal solution using Euclidean distance given in Eqn. (7) and (8).

(7)

(8)

Calculate the relative closeness of each alternative to the ideal solution using Eqn. (9)

(9)

Where . The larger the , the better the alternative  will be.

Finally a set of alternatives can be ranked by the descending order of the value of .

(Finally, a set of alternatives can be ranked according to the relative closeness to the ideal solution)better the alternative  will be.

Finally a set of alternatives can be ranked by the descending order of the value of .

(Finally, a set of alternatives can be ranked according to the relative closeness to the ideal solution)

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