# Method Development Model Consensus on Analytic Hierarchy Process

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Published on International Journal of Biology, Physics & Mathematics
Publication Date: July 23, 2019

Maman Budiman, Dwi Gunarto & Randy Asmuni
Pancasakti University, Tegal
Widya Dharma University, Klaten
Indonesia

Journal Full Text PDF: Method Development Model Consensus on Analytic Hierarchy Process.

Abstract
This paper propose a model of consensus on the method of AHP (Analytic Hierarchy Process) to decision-making group. The consensus method can provide information about the level of agreement and disagreement, and disagreement range of individuals and groups, the structure of the cluster, the object identification on decision “problematic” and a marginal opinion. Formation of the consensus made by Delphi method approach, discussion (exchange of information) is performed to obtain a homogeneous opinion. Rules do when the termination discussion found more homogeneous the opinion of 66.6 percent, or based on time constraints.

Keyword: Analytic Hierarchy Process, consensus & homogeneous.

I. INTRODUCTION
Group decision-making process in general produces a complex decision. Each party involved in the decision making process, can have the values and principles are different. The difference in value or principle of life, can make a difference / conflict of goals / interests, which can cause differences in preferences between the parties involved, the situation is called a conflict. In conflict situations, ideally one party does not impose desires (strategy) to achieve his own wishes, but must be willing and able to work together (win-win) through negotiation.
The process of cooperation is based on the intention sincere and open, will achieve mutually beneficial results, even can give a synergy effect, so that the results will be obtained by each party will be better than the process that is competitive (win-loose), Putro and Tjakraatmadja [1].
The purpose of group decision making occurs when the owner of the decision (decision) subjectively included as participants in decision-making, which in some cases the participants did not know all the activities as a whole, Brugha [2]. In fact, to be able to describe the problem well in the evaluation, a decision, a decision maker needs to consider the opinions of others who understand the problem.
This paper propose a model of consensus on the method of AHP (Analytic Hierarchy Process) to decision-making group. The model developed is the development of Techniques for Analyzing Consensus Relevant Data / ACRD developed by, Ngwenyama [3].

II. LITERATURE REVIEW
Forman and Peniwati [4], said the process of synthesis by the AHP method can be done in one of two ways:
• Aggregation of individual assessment method (the aggregation of individual judgment / AIJ), or
• Method of aggregation of individual priorities (the aggregation of individual priorities / AIP).
For AIJ method it is assumed that each member of the group acts as a single entity and can no longer act on the individual’s identity. The assessment results of each individual in the group are aggregated with the geometric mean method. For this case, the Pareto principle is not relevant so it does not need to be considered.
In the AIP method, it is assumed that the group members act as individuals who are independent of the other group members. The priority sequence produced by each individual in the group are aggregated with the average method geometric or arithmetic average, and both of them do not violate the Pareto principle.
In producing the preferences of individual preference groups, to consider the fulfillment of the Pareto principle. According to the Pareto principle, if the two alternatives, the alternatives alternatives a1 and a2, compared and each individual group members preferred alternative compared to alternative a1 a2, then the group should prefer an alternative rather than an alternative a1 a2.
Zahir [5], stating that in a large group there are various possible patterns of thinking, namely:
a. All members of the group have the same thought
b. The views of members vary but they are a member of a coherent homogeneous group
c. There are several clusters or homogeneous sub-groups within a large group.
Homogeneous group does not demand identical preferences of each individual group members. In a homogeneous group, each individual opinion remains varied, but has a “similarity”.
The similarity of this argument can be seen by comparing the value of the cosine angle formed by a pair of individual preferences of group members with a limit of homogeneity (). If the value of the cosine angle formed by a pair of individual preference greater than or equal to the limit value homogeneity, Then a couple of individuals can be said to have a “similarity”.

2.1 Techniques for Analyzing Consensus Relevant Data/ ACRD (Ngwenyama et. Al., 2006)
The decision making process in computer-supported group allows anonymous process takes place with the help of a facilitator as a steering discussions. To generate information for the facilitator, [3] proposed a number of techniques and approaches for analyzing data group preferences in the decision process.
Data analysis techniques relevant to the consensus (the techniques for analyzing the consensus of relevant data / ACRD) proposed by [3], is utilizing the preference data individually produced by each decision maker (value and sequence) in a group against a set of alternative decisions using AHP.
The analysis was performed based on the similarity of individual partner preferences of group members, which is indicated by the cosine of the angle is greater than or equal to the value limits of agreement.
The disagreement between a pair of individuals reached if the cosine angle value less than or equal to the limit value of disagreement.
Value, which probably is 0985 (cosine angle 10o) and value which probably is 0.966 (cosine angle 15o). Rationalization of making these angles is that the largest possible angle between two vectors weight is 90o. So 0o 10o-90o on a scale equivalent to 1 on a scale of 1-9, and 15o equivalent to 1.5 on a scale of 1-9.
Conceptually, this approach to support this ACRD techniques can be divided into three stages, namely:
a. The pre-evaluation stage
b. Stage generate preferences
c. Phase data analysis and reporting
Pre-evaluation stage includes the selection of alternatives for evaluation and determination of the evaluation criteria, as well as the delimitation agreement to define the rules of the termination of the decision-making process, such as the achievement of an allocated time interval or a certain level of agreement on the issue of (partial or full).
While on stage do the sorting preference produce alternative and presentation of the data comparison using AHP.
At the stage of data analysis and reporting analyzing the preference data performed by the decision makers to identify their position in group decision-making process. In this phase also the identification of possible coalitions, identify problematic decision alternatives, and identification of key individuals who have a preference position that enables the negotiation of consensus.
This approach allows the facilitator in the group to assess the level of group consensus at every stage of the group, so that the resulting information can help the facilitator to negotiate the formation of a consensus within the group. Consensus can be achieved when the consensus map has been identified that describe the preferences that can be accepted by the group.
In an ideal situation, should be reached complete agreement within the group. But in general, taking into account the differences of opinion, this is not possible, so it takes the rules of termination.
Individual indicators expressed by the Consensus Individual Vector (ICVt), which is used to identify individuals who have a good level of agreement with other group members and do not have barriers that make it difficult. The key individuals have the greatest ICVt value (ICVtmaks). Individual keys are used to facilitate the formation of a group consensus.

2.2 Cluster algorithm Zahir (2009)
For medium-sized groups (intermidiate-sized group) or a group of large-sized (large group), group homogeneity can not be guaranteed or achieved, so [5] proposed a clustering algorithm based on the method VAHP (the Vector Space Formulation of the Analytic Hierarchy Process).
By using this algorithm, in a group consisting of N members can be formed  each cluster homogeneous, where 1 N,
Clusters are naturally determined by the value of cluster membership boundary (). The value of cluster membership boundary () Varies depending on the type of problem. This cluster membership limit values set by agreement of members of the group.
To determine the membership of a decision-maker to a cluster, cosine of the angle between the weight vector of the decision makers and the resultant weight vector of all decision makers in the cluster than the cluster membership boundary value ().
If the cosine of the angle between the weight vector of the decision makers and the resultant weight vector is greater than or equal to the limit value of cluster membership (), Then the decision maker was elected to be a member of the cluster.
Vice versa, if the cosine of the angle formed between the decision makers of the weight vector and the resultant weight vector is smaller than the limit value of cluster membership, the decision was delayed to have become a member of the cluster.
This delayed decision makers should wait for a re-elected into the cluster until there is another group of decision makers who are elected cluster. If not selected, the pending decision makers have to wait to be placed in another cluster.
The formation of clusters in the algorithm, [5] using a Monte Carlo simulation.

III. METHOD DEVELOPMENT MODEL
Systematics design modeling cycle as a model to follow;
As an initial step in the development of the model is done the problem definition. Furthermore, based on the definition of the problem formulated a conceptual model that shows the relationship between the variables that determine the behavior of the model. This model includes verbal model which only outlines the relationship issue, a system, and the purpose of the study.
Objective studies provide indications of performance to be achieved and provides a framework of conceptual models that form the expected performance. To operationalize the conceptual model of symbolization and determination of quantitative rules. Idealization and simplification linkage model variables referred to as the characterization phase models. Model formulation conducted as early development of formal models that show the size of the model performance as a function of the variables of the model.
In the formulation of the model used teleologik principle (review the modeling purposes) for memfungsionalkan attributes by looking at the destination (Teleos) of the system. Through a systems approach, the existence of the system and its environment can be understood by knowing the elements of the system, the relationship between elements and attributes of each element.
Environment system is a collection of objects outside the limits (boundaries) system that affects (affected) systems.
After the initial formulation of the model is complete, then the model’s ability to reproduce the properties and behavior of the real system testing. In this case the testing is based on three criteria to evaluate the model, namely:
a. tested the suitability of the model behavior with the behavior of the real system represents
b. testing the structure between the model variables.
c. estimates for variables, testing the availability of estimated values for key variables.

IV. DEVELOPMENT MODEL CONSENSUS ON AHP METHOD
For group decision making, decision-making is done by a group of individuals who are considered worthy to determine a decision. The decision group is considered better than individual opinions. Assessment conducted by many participants will be possible to produce a different opinion from one another, Anonymous [6].
Preferences group with AHP method is generated by synthesizing the preference of each individual group members. However, to carry out the synthesis of individual preferences, there are prerequisites that its decision-making should be ensured homogeneous [5].
Analysis of the achievement of group consensus can be done with attention to the “sameness” of individual partner’s opinion partisan group members / respondents, the opinions of those individuals can be incorporated into a homogeneous clusters,
Formation of the cluster structure can be made by utilizing the information on individual agreements, the level of individual disagreements and disagreements range of individuals who will be included in a cluster. The resulting cluster structure can provide information to identify the possible presence of a marginal opinion of individual members of the group, which can be used as a reference for the group members to obtain the consensus of the group.
The structure of the cluster can be used to identify objects either problematic decision criteria and alternatives, taking advantage of the aggregate value of the cosine on that object. Based on this information, for objects that have a low weight, do a study on the possibility of the removal of that element from the process of decision analysis.

4.2 Formulation Model
4.2.1 Analysis of Consensus in Group Decision Making
Development of consensus methods made to the development of group decision based on concepts that have been previously known. Development of the consensus method is divided in four main stages, namely:
a. Pre-evaluation
At this stage an agreement to establish the boundaries that form clusters. Objects decisions made for each criteria and alternatives. While the agreement stipulated limit is the value of the deal retang individuals in the group, The value of disagreement, And the limit value of cluster membership, And the rules of termination, either a time limit or level of consensus reached.
b. Generate preferences
At this stage, the individual preferences of each member of the group of objects decisions made based on the method of AHP, including testing for consistency.
c. Data analysis
At this stage, the determination of the value of the aggregation of individual preferences into preference groups, the analysis of the level of agreement and disagreement individuals, as well as the formation of cluster structures.
d. Formation of consensus
Formation of the consensus made Delphi method.

4.2.2 Preferences group
Preferences group is the value of “average” individual preferences. preferences group AIP is calculated based method [4], it is assumed that the group members act as individuals who are independent of the other group members. The priority sequence produced by each individual in the group are aggregated with the average method arithmetic average.
… (1)
Where:
= Weighting preferences to-element group i
= Weight of individual preferences to-element i
n = Number of individuals in the group
To see the relationship between the value of individual preferences with the preferences of the group is determined based on the amount of the angle, To facilitate the analysis is then performed the conversion value the value cos, Under the condition:
– cos approaches a value of 1, indicating strong agreement between the preferences of individuals with a preference group
– cos approaching a value of 0, indicating a weak agreement between individual preferences with the preferences of the group

Figure 1. Individual preferences and the preferences of the group in Vector Spaces
For the purposes of data analysis, carried out the determination of the level of agreement and disagreement individuals, as well as the formation of cluster structures.

4.2.3 Range of agreement
The range of agreements taken in this paper is the value ranges of the agreement Of 1.0 on a scale of AHP, or the difference 10oin a vector space. The basis used to determine the range of the deal is the rationalization of making these angles is that the largest possible angle between two vectors weight is 90o. So 0o 10o-90o on a scale equivalent to 1 on a scale of 1-9, and 15o equivalent to 1.5 on a scale of 1-9, [3].

4.2.4 Value limits of agreement and disagreement
Agreements limit value set at (For a strong agreement) are set to the value of 0985 (cosine angle 10o) and  (For strong disagreement) is determined by the value of 0.966 (cosine angle 15o), the members of the group said to have a strong agreement if the cosine smaller than the value and a strong disagreement if cosine greater than,

4.2.5 Formation of the cluster structure
The structure of the cluster using cluster membership delimitation. Ordinance on the development of consensus analysis in this paper uses membership limits as big as 10o. Cluster structure was formulated on the development of the model are presented in Table 1.

Table 1 Cluster structure
clusters restriction limitation cosine
Cluster-1 0 ° – 5 ° 1000-0996
Cluster-2 5o – 15o 0996-0966
Cluster 2 ‘
Cluster 3rd 15o – 25o 0966-0906
Cluster to-3 ‘
Cluster 4th 25o – 35o 0906-0819
4th cluster ‘
Cluster 5th 35o – 45o 0819-0707
Cluster to-5 ‘
Cluster 6th 45o – 55o 0707-0574
Cluster 6th ‘
Cluster 7th 55o – 65o 0574-0423
Cluster 7th ‘
Cluster 8th 65o – 75o 0423-0259
Cluster 8th ‘
Cluster 9th 75o – 85o 0259-0087
Cluster 9th ‘
Cluster 10th 85o – 90o 0087-0000
Cluster 10th ‘

Information:
(A) Cluster to-n indicates the direction of the horizontal axis
(B) Cluster to-n ‘indicates the direction of the vertical axis

As an illustration, if the preference groups showed a weight of 0.5 for the ith element and 0.5 for element j, then the cluster structure for this problem is shown in Figure 2.

Figure 2 Individual preferences in space Cluster Group

4.2.6 Coherence individual preferences
To identify individual coherence, the analysis is done by calculating the coherence of individual preferences ( ) based on the average value of cos for each element based on individual preferences to-i
… (2)
Where:
= to-individual coherence i
= Angle formed between the preferences of individuals and groups
n = Number of comparisons related preferences to-element i
Elements that have values  0.966 () Indicate that the individual is “problematic”.

4.2.7 Coherence elements
To identify objects problematic decisions, the analysis done by calculating coherence elements ( ) based on the average value of cos to each individual’s preferences with regard to the i-th element.

Where:
= coherence to-element i
= Angle formed between the preferences of individuals and groups related to the i-th element
n = Number of comparisons related preferences to-element i
m = The number of individual members of the group
Elements that have values 0.966. Indicate that the element (object decision) are “problematic”. Objects problematic decision is a decision if the object is removed from the preference vector will increase the aggregate value of coherence elements.

4.2.8 Termination Rule
Rules set before the termination discussion proper assessment of the decision object. This termination rule can be in the form of a certain level of agreement or a certain time limit which if achieved by the group, then the group should do the synthesis of individual decisions even though the level of agreement has not been reached.
The level of agreement of 0.66 or more indicates that there is a majority in the group (more than two-thirds of the members of the group have the same level of agreement). This means that members of the group had an order of preference are very close. However, if the level of agreement (individuals who are in cluster 1) is less than 0.66, then the group repeats the stages of generating preference.

4.2.9 Formation of Consensus
Formation of the consensus on this model using the rules of the Delphi method. Discussion (exchange of information) was first performed by distributing a marginal opinion on group preferences. The second iteration is done by distributing the opinion that had the most powerful closeness value to the preference group formed in the previous stage (after iteration 1).
If consensus is not formed after two iterations, the facilitator identifies the object decision “problematic”, then held discussions with the members of the group about the possibility of the removal of the element. The next iteration is done and stop until a homogeneous common preference or based on time constraints.
The method developed consensus on this method can provide information about the level of agreement and disagreement, and disagreement range of individuals and groups, the structure of vector preference clusters and each cluster, objects mengidentifikansi decision “problematic” and a marginal opinion.

V. CONCLUSION
Consensus method developed in this paper can provide information about the level of agreement and disagreement, and disagreement range of individuals and groups, cluster structure, coherence individual preferences, and the coherence of the object elements of the decision. This analysis provides decision object information “problematic” and that the marginal individual opinion. Formation of the consensus made by Delphi method approach.