# Applying the ANP Model for Selecting the Optimal Full-service Advertising Agency

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3 50 relationship between two nodes (components) in a system (Meade and Sarkis, 1999). Let the components of a decision system be Ck; k =1,, n, where each component k has mk elements, denoted by ek1, ek2,, ekmk. The local priority vectors derived in Step 2 are grouped and located in appropriate positions in a supermatrix based on the flow of influence from one component to another, or from a component to itself, as in the loop. The standard form of a supermatrix resembles that in (5) (Saaty, 2001). C1 Ck Cn e11e1 e 1 1e e 1 e k e11 C W W W 1 e 1m1 ek1 W C k Wk1 Wkk Wkn ekmk en1 C n Wn1 Wnk Wnn e nmn m k km n nm n 11 1k 1n If the criteria are interrelated, the (2, 2) entry of Wn given by W22 would indicate the interdependency, and the supermatrix would be (Saaty, 1996) Wn w21 w22 0 (6) 0 w32 I Notably, any zero in the supermatrix can be replaced by a matrix if there is an interrelationship among the elements in a component or between two components. Since interdependence generally exists among clusters in a network, the columns of a supermatrix usually total more than one. The supermatrix must be transformed first to make it stochastic; that is, each column of the matrix sums to unity. Saaty (2001) recommended determining the relative importance of the clusters in the supermatrix with the column cluster (block) as the controlling component (Meade and Sarkis, 1999). That is, the row components with nonzero entries for their blocks in that column block are compared according to their impact on the component of that column block (Saaty, 1996). Through a pair-wise comparison matrix of the row components with respect to the column component, an eigenvector can be obtained. This process obtains an eigenvector for each column block. For each column block, the first entry of the respective eigenvector is multiplied by all the elements in the first block of that column, the second entry is multiplied by all the elements in the second block of that column and so on. The block in each column of the supermatrix is thus weighted, and the result is termed the weighted supermatrix, which is stochastic. Raising a matrix to powers gives the long-term relative influences of elements on each other. To achieve convergence of the importance weights, the weighted supermatrix is raised to the power of 2k + 1; where k is an arbitrarily large number, and this new matrix is called the limit supermatrix (Saaty, 1996). The limit supermatrix has the same form as the weighted supermatrix, but all the columns of the limit supermatrix are the same. Normalizing each block of this supermatrix can obtain the final priorities of all the elements in the matrix. Step 4: Selection the best alternatives. If the supermatrix formed in Step 3 covers the whole network, the priority weights of alternatives can be found in the column of alternatives in the normalized supermatrix. On the other hand, if a supermatrix is only comprised of interrelated components, additional calculations must be performed to obtain the overall priorities of the alternatives. The alternative with the largest overall priority should be the one selected. This study applies the first method, and a supermatrix that covers the whole network, as shown by the bracket in Figure 1, is then formed. (5)

5 52 The problem of Ad agency selection is broken into four levels: first that of achieving the ultimate goal of selecting the optimal Ad agency, followed by five criteria, 10 sub-criteria and finally the alternatives, as shown in Figure 2. According to the suggestions of nine experts, Figure 3 and 4 show the interdependence among criteria based on the hierarchical structure. Level 1: Goal Level 2: Criteria Level 3: Sub-criteria Level 4: Alternatives Marketing research (S 1 ) Strategic planning ability (C 1 ) Whole planning (S 2 ) Business understand(s 3 ) Agency A Media ability (C 2 ) Media planning (S 4 ) Agency B Select the optimal advertising agency Creativity (C 3 ) Media buying (S 5 ) Creative work that sells (S 6 ) Advertising awards (S 7 ) Agency C Service level (C 4 ) Service range (S 8 ) Personnel quality (S 9 ) Agency D Compatibility/timing (S 10 ) Agency E Cost consciousness (C 5 ) Figure 2 Hierarchical structure required to selecting the optimal Ad agency Strategic planning ability Media ability Creativity Service level Cost consciousness Figure 3 Interdependence criteria of ANP model Marketing research Whole planning Business understand Media planning Media buying Creative work that sells Personnel quality Figure 4 Interdependence Sub-criteria of ANP model Step 3. Establish the pair-wise comparisons matrices and determine eigenvectors

6 53 Nine experts make a pair-wise comparison of the decision criteria and assign relative scores, and the geometry mean value is used to calculate comprehensive decision-making community scores, and the eigenvectors are then calculated using Eg. (2). Egs. (3) and (4) are used to calculate the criteria comparison matrix of consistency for each hierarchy. The results of the consistency test, the CR of the comparison matrix from each of the nine experts, are all smaller than 0.1, indicating consistency. Furthermore, the CR of the aggregate matrix is also below 0.1, again indicating consistency. Table 1 lists the aggregate pair-wise comparison matrix of criteria and consistency test. Table 2 presents the aggregate pair-wise comparison matrix for sub-criteria and consistency test. Table 1: Pair-wise comparison matrix for criteria of Level 2 Level 2 Criteria C 1 C 2 C 3 C 4 C 5 e Vecter (W 21 ) Strategic planning (C 1 ) Media ability (C 2 ) Creativity (C 3 ) Service level (C 4 ) Cost consciousness (C 5 ) λ max =5.007; CI=0.002; CR= consistency Table2: Pairwise comparison matrix for sub-criteria of Level 3 Strategic planning ability (C 1 ) Service level (C 4 ) S 1 S 2 S 3 e Vecter (W 32 ) S 8 S 9 S 10 e Vecter (W 32 ) S S S S S S λ max =3.000; CI=0; CR= λ max =3.002; CI=0.001; CR= Media ability (C 2 ) Creativity (C 3 ) S 4 S 5 e Vecter (W 32 ) S 6 S 7 e Vecter (W 32 ) S S S S λ max =2.000; CI=0; CR= λ max =2.000; CI=0; CR=0 0.1 Step 4. Establish pair-wise comparisons matrices of interdependencies In order to consider interdependencies between second level criteria, and between third level sub-criteria, this study also invited the above nine experts to assign interdependent weights, and then integrate the geometric mean for these interdependent weights. Table 3 lists the aggregate pair-wise comparison matrix of interdependence between criteria and consistency test. Table 4 lists the interdependence matrix of sub-criteria. Table 3: Pair-wise comparison matrix for interdependent weights between criteria Strategic planning ability (C1) C2 C3 C4 C5 e Vecter (W22) Media ability (C2) Creativity (C3) Service level (C4) Cost consciousness (C5) CI= 0.029; CR= consistency Media ability (C2) C1 C3 C4 C5 e Vecter (W22) Strategic planning (C1) Creativity (C3) Service level (C4) Cost consciousness (C5) CI= 0.021; CR= consistency Creativity (C3) C1 C2 C4 C5 e Vecter (W22) Strategic planning (C1) Media ability (C2) Service level (C4) Cost consciousness (C5)

9 56 Table 5 The supermatix Goal C1 C2 C3 C4 C5 S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 Goal C C C C C S S S S S S S S S S A B C D E

10 57 Table 6 The weighted supermatrix Goal C1 C2 C3 C4 C5 S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 Goal C C C C C S S S S S S S S S S A B C D E

11 58 Table 7 The limit supermatrix Goal C1 C2 C3 C4 C5 S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 Goal C C C C C S S S S S S S S S S A B C D E

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