## Comparative Study AHP TOPSIS, and PROMETHEE-II of E-Commerce

A Comparative Study by AHP, TOPSIS, and PROMETHEE-II for the E-Commerce

*Abstract*– Now-a-Days the multi criteria decision making methods (MCDM) are gaining importance to solve decision making problem by choosing best alternatives from available alternatives. In this paper MCDM methods Analytic Hierarchy Process (AHP), Technique for Order Preference by Similarity to an Ideal Solution (TOPSIS) and Preference Ranking Organization METHod for Enrichment Evaluation (PROMETHEE) were applied to our decision problem. Multiple options have considered as alternatives and some criteria were considered.

Keywords-AHP, TOPSIS, PROMETHEE, MCDM

INTRODUCTION

Ever since in past from shopkeeper to Engineers to Scientist everyone facing a similar problem of finding the best decision alternatives to multi criteria problem. In 1980’s, T.L.Satty developed a hierarchical solution to these complex multi criteria problems named AHP (Analytical Hierarchical Process). AHP provides a hierarchical structure to criteria, sub-criteria, and alternatives and then perform a pair wise comparison. Since then AHP is widely used for multi criteria problems. AHP is quite easy and adaptable to apply on any decision-making problem. AHP uses a scale table for pair-wise comparison given by T.L.Satty. TOPSIS (Technique for Order Preference by the similarity to Ideal Solution) method was developed by Hwang and Yoon in 1981.TOPSIS is also MCDM method, used to solve multi criteria decision making problem. TOPSIS compares a set of alternative by identifying weight for each criterion and then aggregates them. PROMETHEE was developed by Professor J. Brans in 1982. PROMETHEE is another multi criteria decision making method which finds best alternatives. PROMEHTEE provides a comprehensive and relational framework for structuring decision problem. It highlights the main alternative by identifying and quantifying the conflicts. AHP, TOPSIS, and PROMETHEE methods are applied to the problem in many fields but not as a comparison to the problem for “SELECTION OF A CAR” in E-commerce. Hence, we took this problem into consideration to compare decisions obtained by AHP, TOPSIS and PROMEHTEE II.

METHODOLOGY

Analytical Hierarchy Process (AHP)

Early 1980’s T.L.Satty developed AHP(Analytical Hierarchy Process) which introduces a pair-wise and hierarchical model for decision making process. AHP is one of the most widely used MCDM (Multi Criteria Decision Making) methods. AHP decomposes the decision problem into criteria and sub-criteria forming a hierarchy which gives full understanding to the decision problem. For each criterion and sub-criteria a pairwise matrix taking Satty’s 9-point scale value and compare their relativeness to each other. Finding eigen vector and weight for each matrix which results in the overall ranking to the problem. AHP along with informative data it also takes human judgment into account to ensure the efficiency of the final result. Some of its application includes ranking, prioritization, resource allocation, benchmarking, quality management. The step for implementing AHP process are illustrated as follow:

*Step 1*: The problem is decomposed into decision making hierarchy in terms of goal, criteria, sub-criteria and alternatives. It is one of the important parts in AHP, to structure the decision problem as a hierarchy.

FIGURE 1. Decision-Making Hierarchy

FIGURE 2. SCALE FOR PAIR WISE COMPARISION

Hierarchy indicates a relationship between elements of one level with intermediate below level. This relationship prelocates down to lowest level of the hierarchy. At the root of the hierarchy is our goal followed by main criteria, sub-criteria and alternatives. When comparing elements of each level decision maker has to compare with respect to a level below and level above.

*Step 2*: Data are collected from decision makers and applied to the hierarchical structure, as a pairwise comparison using scale given below. Then decision maker can rate the comparison as linguistic scale values given in Table 1. Using specially designed format in figure 2 opinion can be collected.

The format indicates that “X” in the given format is “very strong” indicating that B is very strong compared to A. Similarly, each criterion is compared and converted into the quantitative number as shown below.

TABLE 1. SAATYS RATIO SCALE FOR PAIR WISE COMPARISION

*Option*

*Numerical value(s)*

Equal

1

Marginally Strong

3

Strong

5

Very Strong

7

Extremely Strong

9

Intermediate values to reflect fuzzy inputs

2,4,6,8

Reflecting dominance of second alternative Reciprocals compared with the first

Reciprocals

*Step 3*: The pairwise comparison for each and every criterion is generated using step 2 and organized into a square matrix where diagonal elements of the matrix are 1.

*Step 4*: For each comparison matrix principle eigen value and corresponding eigen vector is calculated. The elements of the normalized eigen vector are weights with respect to criteria and sub-criteria. To calculate eigen vector take matrix c[i][j], calculate sum of column sumc[j]. Then modify each matrix element c[i][j] by diving c[i][j] with sumc[j]. Now calculate the sum of row sumr[i] then get weight matrix W[i] by diving sumr[i] with no of row ‘n’.

*Step 5*: The rating of each alternative is multiplied by the weights of the criteria and aggregated to get the final rating.

Technique for Order Preference by Similarity to Ideal Solution (TOPSIS)

The TOPSIS method was developed by Hwang and Yoon in 1981 and ranks the alternatives based on the distance from the positive and negative ideal solution, i.e. best alternatives should have the shortest distance from positive ideal solution and farthest distance from negative ideal solution.

The positive ideal solution is identified as an alternative that has the best value for all considered criteria whereas the negative ideal solution is identified as worst criteria values. The step for implementing TOPSIS process are illustrated as follow:

*Step 1*: Construct normalized decision matrix.

(1.1)

Where, Xij and rij are original and normalized score of decision matrix, respectively

*Step 2*: Determine the weighted decision matrix. The weighted decision matrix is constructed by multiplying the normalized decision matrix with random weights.

*V**ij** = W**j** x r**ij*(1.2)

Where, Wj is the weight of j criterion.

*Step 3*: Determine the positive ideal (S+) and negative ideal solutions (S–).

*S**+ **= {V**1**+** ,V**2**+**, A?a‚¬A¦, V**n**+** },*(1.3)

where:Vj+={(maxi(vij) if j A‘aˆ? J); (mini(vij) if j A‘aˆ? J’)}

*S**–** = {V**1**–** ,V**2**–** ,…, V**n**–** },*A‚A A‚A A‚A A‚A A‚A A‚A A‚A A‚A A‚A A‚A A‚A A‚A A‚A (1.4)

Where:Vj– ={(mini(vij) if j A‘aˆ? J); (maxi(vij) if j A‘aˆ? J’)}

*Step 4*: Calculate the separation measure for each alternative from the positive ideal and negative ideal solution.

(1.5)

(1.6)

*Step 5*: Calculate the relative closeness to the ideal solution Ci

(1.7)

Select the alternative with Ci closest to 1.

Preference Ranking Organization METHod for Enrichment Evaluation (PROMETHEE)

The PROMETHEE method was developed by Brans and Vincke in 1985. It is an outranking method and a special type of MCDM method which provides ranking order to the decision problems. PROMETHEE I method can only provide partial ordering to the decision problem, whereas PROMETHEE II methods can derive a full ranking of the alternatives. In this paper, PROMETHEE II method is applied to obtain a full ranking of the alternatives to our problem. The step for implementing TOPSIS process are illustrated as follow:

*Step 1*: Normalize the decision matrix using the following equation:

(i =1, 2…, n; j =1, 2…, m) (2.1)

where Xij is the performance measure of ith alternative with respect to the jth criterion.

*Step 2*: Calculate the evaluative differences of ith alternative with respect to other alternatives. This step involves the calculation of differences in criteria values between different alternatives pair-wise.

*Step 3*: Calculate the preference function, Pj(i, iA‚A?).

Brans and Mareschal proposed six types of preference functions but these preference functions require some parameters, such as preference and indifference thresholds. It is difficult for the decision maker to specify the preference function suitable for each criterion and determine the parameters. To avoid such problem, a simplified function is adopted here:

P j(i, I’) =0 if Rij <= Ri’j(2.2)

A‚A A‚A A‚A A‚A A‚A A‚A P j(i, I’) = (Rij – Ri’j) if Rij >Ri’jA‚A A‚A A‚A A‚A A‚A A‚A A‚A A‚A A‚A A‚A A‚A A‚A A‚A (2.3)

*Step 4*: Calculate the aggregated preference function taking into account the criteria weights.

Aggregated preference function,(2.4)

where wj is the relative importance (weight) of jth criterion.

*Step 5*: Determine the leaving and entering outranking flows as follows:

Leaving (or positive) flow for ith alternative,(2.5)

Entering (or negative) flow for ith alternative, (2.6)

where n is the number of alternatives.

*Step 6*: Calculate the net outranking flow for each alternative.

(2.7)

*Step 7*: Determine the ranking of the alternatives depending on the values of A?A?A?(i). The higher value of A?A?A?(i), the better is the alternative. Alternative with highest A?A?A?(i) value is the best alternative.

ILLUSTRATION

Here AHP, TOPSIS, and PROMETHEE II methods are applied for selection of a car by taking the subjective judgment of decision maker into consideration. Three criteria were identified for selecting a car and the alternatives are 3 car maker company.

AHP

TABLE 2.1. DECISION-MAKING PROBLEM

Car

Cost

Safety

Appearance

Honda

22000

28

Sporty

Mazda

28500

39

Slick

Volvo

33000

52

Dull

TABLE 2.2. PAIR WISE COMPARISION MATRICES FOR CRITERIA COST AND SAFETY

Cost

Safety

Honda

Mazda

Volvo

Priority Vector

Honda

Mazda

Volvo

Priority Vector

Honda

1

2

4

0.57

Honda

1

1/2

1/5

0.12

Mazda

A‚A?

1

2

0.29

Mazda

2

1

1/4

0.20

Volvo

A‚A?

1/2

1

0.14

Volvo

5

4

1

0.68

TABLE 2.3. PAIR WISE COMPARISION MATRICES FOR CRITERIA APPEARANCE

Appearance

Honda

Mazda

Volvo

Priority Vector

Honda

1

5

9

0.76

Mazda

1/5

1

2

0.16

Volvo

1/9

1/2

1

0.08

TABLE 2.4. PAIR WISE COMPARISION MATRICES FOR ALTERNATIVES

All Criteria

Cost

Safety

Appe.

Priority Vector

Cost

1

1/3

1/2

0.16

Safety

3

1

2

0.54

Appea.

2

1/2

1

0.29

TABLE 2.3. PRIORITY VECTOR AND RANKING OF ALTERNATIVES

Car

Cost

Safety

Appearance

Final Weight

Rank

Honda

0.558

0.117

0.761

0.380

2

Mazda

0.320

0.200

0.158

0.207

3

Volvo

0.122

0.683

0.082

0.412

1

TOPSIS

TABLE 3.1. NORMALIZED DECISION MATRIX

Cost

Safety

Appearance

Wi

0.1

0.3

0.2

Honda

0.4504

0.3956

0.8452

Mazda

0.5835

0.5510

0.5071

Volvo

0.6757

0.7347

0.1690

TABLE 3.2. WEIGHTED DECISION MATRIX

Cost

Safety

Appearance

Wi

0.1

0.3

0.2

Honda

0.4504

0.1187

0.1690

Mazda

0.5835

0.1653

0.1014

Volvo

0.6757

0.2204

0.0338

TABLE 3.3. POSITIVE AND NEGATIVE IDEAL SOLUTION

Maximum

Minimum

0.6757

0.4504

0.2204

0.1187

0.1690

0.0338

TABLE 3.4. SEPARATION DISTANCE FROM IDEAL SOLUTION

Alternatives

S+

S-

Honda

0.2472

0.1352

Mazda

0.1269

0.1564

Volvo

0.1352

0.2472

TABLE 3.5. RELATIVE CLOSENESS AND RANKING OF ALTERNATIVES

Alternatives

Relative Closeness

Rank

Honda

0.3536

3

Mazda

0.5521

2

Volvo

0.6468

1

PROMETHEE II

TABLE 4.1. OBJECTIVE DATA AND NORMALIZED DECISION MATRIX

Evaluation Matrix

Normalized Matrix

Cost

Safety

Appearance

Cost

Safety

Appearance

Honda

20000

28

Sporty/5

Honda

0

0

1

Mazda

28500

39

Slick/3

Mazda

0.591

0.4583

0.5

Volvo

33000

52

Dull/1

Volvo

1

1

0

TABLE 4.2. PREFERENCE FUNCTIONS FOR ALL THE PAIRS OF ALTERNATIVES

Pair

Cost

Safety

Appearance

Honda, Mazda

0

0

0.5

Honda, Volvo

0

0

1

Mazda, Honda

0.591

0.4583

0

Mazda, Volvo

0

0

0.5

Volvo, Honda

1

1

0

Volvo, Mazda

0.409

0.5417

0

TABLE 4.3. AGGREGATED PREFERENCE FUNCTION

Car

Honda

Mazda

Volvo

Honda

–

0.1667

0.3333

Mazda

0.328

–

0.1667

Volvo

0.6667

0.3390

–

TABLE 4.4. LEAVING AND ENTERING FLOW FOR DIFFERENT ALTERNATIVES

Car

Entering Flow

Leaving Flow

Honda

0.25

0.4973

Mazda

0.2473

0.2528

Volvo

0.5028

0.25

TABLE 4.5. NET OUTRANKING FLOW VALUES FOR DIFFERENT ALTERNATIVES

Car

Net Outranking

Rank

Honda

-0.2473

3

Mazda

-0.006

2

Volvo

0.2528

1

TABLE 5. FINAL ALTERNATIVE RANKING OF ALL THREE METHODS

Alternatives

AHP

TOPSIS

PROMETHEE II

Honda

2

3

3

Mazda

3

2

2

Volvo

1

1

1

CONCLUSION

In this comparative study, it is clear that the result obtained from AHP is different than TOPSIS and PROMETHEE II. AHP uses hierarchy structure to define the problem and creates a comparison matrix for all its criteria and sub-criteria, which results a more efficient and better ranking. AHP uses complex method to solve the problem and time taking, whereas TOPSIS and PROMETHEE II are faster and simpler. So, it is quite useful to understand the decision problem first and then decide which MCDM method to apply.

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