The AHP is a systematic analysis technique developed for multi-criteria decisions [4]. Its operating mode lays on the decomposition and structuring of a complex issue into several levels, rigorous definition of manager priorities and computation of weights associated with the alternatives. The output of the AHP is a ranking that indicates the overall preference for each decision alternative.

The AHP technique involves quantitative and qualitative feature in a unique analysis structure that converts the natural thoughts of any human being into an explicit process. The latter is implemented in a decision-support tool that provides objective and reliable results, even under different scenarios.

Assuming that n decision factors are considered in the quantification process of the relative importance of each factor with respect to all the other factors, this problem can be set up as a hierarchy. Pairwise comparisons are then made between each pair of factors at a given level of the hierarchy, regarding their contribution toward the factor at the level immediately above. The comparisons are made on a scale of 1?9, as shown in Table 1. This scale was chosen to support comparisons within a limited range, but with sufficient sensitivity. These pairwise comparisons yield a reciprocal (n, n)-matrix A, where a_ii = 1 (diagonal elements) and a_ji = 1/a_ij.

The method supposes that only the first column of matrix A is required to state the relative importance of factors 2, 3, . . . , n with respect to factor 1. If the judgments were completely consistent, then the remaining columns in the matrix would be completely determined due to the transitivity of the relative importance of the factors. However, there was no consistency, with the exception of that obtained by setting a_ji = 1/a_ij. Therefore, the comparison needed to be repeated for each column of the matrix, i.e. independent judgments had to be made over each pair. If after all the comparisons are made, matrix A should include only exact relative weights.

Eq. (1) shows that multiplying the matrix by the vector of weights w = (w₁, w₂, . . . , w_n) yields

Therefore, to recover the overall scale from the matrix of ratios, the Eigenvector method was adopted [4]. According to the previous equation, the problem can be formulated as Aw = nw or (A-nI) = 0, which represents a system of homogenous linear equations (I is the identity matrix). This system has a nontrivial solution if; and only if, the determinant of (A-nI) vanishes, meaning that n has an Eigenvalue of A. Obviously, A has a unit rank, since every row is a constant multiple of the first row and; thus, all Eigenvalues except one will be equal to zero. The sum of the Eigenvalues of a matrix equals its trace, and in this case, the trace of A equals n. Therefore, n has an Eigenvalue of A and a nontrivial solution. The normalized vector is usually obtained by dividing all the entries, w_i, by their sum.

Thus, the scale can be recovered from the comparison matrix. In this exact case, the solution was any normalized column of A. Notably, matrix A in this case was consistent, indicating that its entries satisfied the condition a_jk = a_ji/a_ki (transitivity property).

However, in actual cases, precise values of w_i/w_j are not available, but their estimates, which in general differ from the ratios of the actual weights, are provided by the decision-maker. The matrix theory illustrates that a small perturbation of the coefficients implies a small perturbation of the Eigenvalues. Therefore, an Eigenvalue close to n, which is the largest Eigenvalue, λ_max, should be found, since the trace of the matrix (equal to n) remains equal to the sum of the Eigenvalues, while small errors of judgment are made and other Eigenvalues are non-zero.

The solution to the problem of the largest Eigenvalue, which is the weight Eigenvector, w, corresponding to λ_max when normalized, gives a unique estimate of the underlying ratio scale between the elements in the studied case. Furthermore, the matrix whose entries are w_i/w_j remains a consistent estimate of the "actual" matrix A, which may not be consistent. In fact, A is consistent if; and only if, λ_max= n. However, the inequality λ_max>; n always exists. Therefore, the average of the remaining Eigenvalues can be used as a "consistency index" (CI), which is the difference between λ_max and n divided by the normalizing factor (n-1).

The CI of the studied problem is compared with the average random index (RI) obtained from associated random matrices of order n to measure the error due to inconsistency [4]. As a rule of thumb, a consistency ratio (CR = CI/RI) value of 10% or less is considered acceptable, otherwise the pairwise comparisons should be revised.

As shown in Fig. 1, a hierarchical scheme for the AHP [4] was designed to achieve successful utilization of the IMS for sewerage facilities in the upper reaches of multi-dams using an AHP. Three alternatives were selected to decide who should control the IMS for sewerage facilities:

Alternative 1: commission to a specialized institute, i.e. public corporations, such as the EMC. The association of related local governments still plays a role as a regulator. This alternative secures specialty, and derives independent and responsible management.

Alternative 2: commission to private corporations. This alternative promotes cost saving, application of advanced technology, and service quality control, etc.

Alternative 3: role-sharing partnership, which suggests that KMOE, local government, and a specialized institute, such as the EMC, share the role of managing, operating and maintaining the IMS for sewerage facilities. Table 2 shows the pros and cons when each alternative is implemented.

The seven factors influencing the decision as to who should control the IMS, as well as achieving the goal of successful utilization of IMS, were selected via a brainstorming conference. Table 3 shows the descriptions of the seven factors.

A questionnaire, composed of pairwise comparisons, was prepared based on the predetermined hierarchical scheme and sent to experts (engineering consultants, operators of STPs, and university professors) engaged in the field of sewerage design and construction. In this study, 31 different individuals provided independent comparison values. The comparison framework, as outlined above, was carefully explained to each evaluator, who was then asked to quantify accordingly the comparison values for all factors and alternatives.

Tables 2 and 3 were used to provide a common basis for the comparisons, although each evaluator was free to make their own decision regarding the consequences. A typical comparison matrix is shown in Table 4.

After receiving 31 answers, the analysis was performed using MS-Excel, with the rates of inconsistency for each factor evaluated. The CI was calculated as 0.01, whereby the ratio CI/RI = 0.014/1.320 = 0.011<0.1. In all cases, the evaluators stayed within this constraint; thus, the consistency level regarding the weights was satisfactory.

Table 5 shows the calculated weights and ranks of each factor. Public interest received the highest weight; whereas, receptiveness received the lowest weight for successful IMS of sewerage system.

The evaluated results for each factor and each alternative are shown in Table 6. Commissioning to a special institute (Alternative 1) received the highest score for sustainability, but the lowest for grievance mediation. Commissioning to a private corporation (Alternative 2) received the highest score for economics, but the lowest for grievance mediation. The role-sharing partnership (Alternative 3) received the highest score for grievance mediation, but the lowest for economics.

Table 7 shows the final scores and ranks for the three alternatives. As a final result, the role-sharing partnership (Alternative 3) received the highest score (conversed score of 100). Although commissioning to a special institute (Alternative 1) received the medium score, the difference in the scores between alternatives 1 and 3 was very slight. Thus, alternative 1 cannot be overlooked as a possibility without careful consideration. It was notice that the complete commissioning to a private corporation showed a significantly lower preference than the other alternatives.

In this study, an attempt was been made to compare different entity alternatives for the successful implementation of the management, operation and maintenance of the IMS for sewerage facilities in the upper reaches of multi-purpose dams in Korea. Three alternatives were considered in deciding who should control the IMS for sewerage facilities: commissioning to public corporations, commissioning to private corporations, or a role-sharing partnership. In using the AHP technique, the emphasis was on comparing public interests, economics, efficiency, sustainability, specialty, grievance mediation and receptiveness.

The role-sharing partnership (Alternative 3) received the highest score (conversed score of 100). Although commissioning to a special institute (Alternative 1) received the second highest score of the three alternatives, the difference in the scores between alternatives 1 and 3 was not significant. Thus, alternative 1 cannot be easily dismissed as a method without careful consideration. These results lead us to the conclusion that decision-makers can choose either alternative 1 or 3.

This research was supported by a Chung-Ang University ResearchGrants in 2005-2006.