Dynamic Analytic Hierarchy Process: AHP method adapted to a changing environment

Dynamic Analytic Hierarchy Process:

AHP method adapted to a changing environment

P. Viveros***, A. Crespo**. 

* General Dynamics – European Land Systems. Seville, Spain (e-mail: [email protected]). ** Department of Industrial Management, Escuela Superior de Ingenieros de Sevilla

(e-mail: [email protected], [email protected])

*** Department of Industries, Universidad Técnica Federico Santa María, Chile (e-mail: [email protected]) Abstract: This paper describes the Analytic Hierarchy Process as a tool that can facilitate decision-making related to some of the critical aspects in maintenance or after-sales area, permitting the alignment of actions with the business’ objectives. Actual situations evidence how adopted decisions can change the boundary conditions of the system where the AHP is being applied. Therefore, this research is intended to provide a dynamic view of the AHP method, considering the alternatives as temporary variables and obtaining finally not only one good choice for a specific moment but a range of decisions which present different trends with the time. With this purpose this paper starts with a short literature review and the general characteristics of the AHP method. Afterwards, the paper presents the problematic that appears frequently in actual situations which justify the development of this research. Once described the uncertainty appeared after the AHP implementation, the proposed methodology called DAHP (Dynamic Analytic Hierarchy Process) is presented. Finally, this paper concludes with the main points of the research suggesting applications and further extensions.

Keywords: After sales service, Analytic hierarchy process, Decision making, Dynamic environment, Maintenance, Warranty.

 1. INTRODUCTION

The Analytic Hierarchy Process (AHP) method establishes a series of scales from comparisons, where inputs can be measured as the price, weight, time, provisioning, etc., or even the subjective opinion on how the satisfaction and preference sentiments can be. AHP shows an approximate and realistic assessment of the best decision, including certain and small judgment incoherencies when subjective opinions are adopted, evidently due to the fact that human judgment is not always consistent. The AHP has been applied to multiple business situations, as well as political or personal issues, especially when it is necessary to synthesize the knowledge of different specialists to support decisions:

1. Personal situations, where everything is looking on organizing and reflecting internal preferences, for example: acquisition of resources, identification of tracks, etc.

2. Political situations in public administrations. Generally is orientated to consensus achievement or forecasting future, for example identification of public transport routes, public services allocation, etc.

3. Business situations in private companies. Orientated mainly by competitiveness and improvement, it is used in all situations to achieve objectives: organization, structuring projects, resources allocation, prediction, etc.

Therefore, AHP provides a picture of a specific situation where decisions are taken following alternatives and criteria which are in accordance to the boundary conditions of this specific moment. However, the adopted decisions may change the scenario under study and the effects of such implementations can change throughout the timeline the weight of the alternatives and, consequently, the AHP solutions will change with the time.

This paper intends to synthesize in a simple way how the Analysis Hierarchical Process at production and sale stage can help to make more appropriate decisions on strategic actions (for example, in regards to spare parts to properly assist customer claims or maintenance activities), including a dynamic view of the scenario where the boundary conditions change gradually with the adopted decisions. In other words, it is important to underline that the current development of identified criticalities are clearly static, i.e., it does not generate any feedback nor consider the type of production process. Therefore, one main goal of this document is to refer specifically to those important differences between static traditional processes, with highly dynamic ones during the process execution itself.

This last feature appears in processes such as mining, particularly during the extraction process, where there are many variables that make the system criticalities to vary over the time. E.g. in a production plan, variables can be the number of available suppliers, the copper prices, the mines expansion plans, or if the mineral to move is high-grade, low-grade or slag, among others.

2. SHORT LITERATURE REVIEW AND GENERAL CHARACTERISTICS

The AHP is a methodology developed by Thomas Saaty in 1970, based on facilitating the understanding of a complex problem through a breakdown in parts ordered hierarchically ranked (approaches and alternatives), quantifying and comparing variables through addition of views with geometrical average to synthesize a solution (Saaty, 1990), (Saaty, 1995), (Saaty and Vargas, 1982). The process has been used to assist numerous corporate and government decision makers. Some examples of decision problems are: choosing a telecommunication system, choosing a product marketing strategy, etc. In short, problems are decomposed into a hierarchy of criteria and alternatives (see Figure 1).

Fig. 1. Problems decomposition into criteria and alternatives In the decision matrix (Figure 2), it is synthesized decision maker information with resulted elements in pair compared criteria with a normalized and reciprocal scale of relative importance (see Table 1).

Fig. 2. AHP Decision matrix

The elements of decision matrix fulfill:

o Reciprocity: wij = 1/wji for all i, j = 1,..., n. o Consistency: wij = wik/wjk for all i, j, k = 1,..., n. o Σj = 1,…, n wj = 1.

The method has the following axioms:

Reciprocity. The importance of the relation Wij is the inverse

of Wji, so Wii=1.

Homogeneity. Compared elements in relation to the same property have the dame order of magnitude.

Dependence. To control the dependence among the elements from the same level and consecutive levels.

Performance of expectations. The structure of criteria and alternatives has to represent the expectations in a ranking.

Table 1. Saaty scale to compare criteria in pairs

SAATY SCALE RECIPROCAL SAATY SCALE

1 Equal importance of _{both elements} --- --- 3 Weak importance of one _{element over another} 1/3 Slightly less importance _{of one over another} 5

Essential or strong importance of one element over another

1/5 Less importance of one _{over another}

7

Demonstrated and very strong importance of one element over another

1/7 Far less importance of _{one over another}

9

Absolute importance of one element over another

1/9

Absolute less importance of one over another

2, 4, 6,

8 Intermediate values

½, ¼,

1/6, 1/8 Intermediate values

As it has previously stated, in the application of this method, subjective values can be used. This subjectivity is presented, implying a degree of uncertainty or lack of reliability. That makes it necessary to measure the sensitivity in changes of parameters. To measure here the reliability (CR), we use the ratio between consistency rate CI of a comparisons array into pairs, and the value of the same index of a comparisons array into pairs randomly generated:

(1) Reliability is sufficient if CR is less than or equal to 0.10, otherwise, it must be reviewed to improve its consistency. As commented, AHP has been applied to multiple situations. In our case with a dynamic environment, AHP method is orientated to a continuous improvement, taking also into account the following points (Harker and Vargas 1990), (Saaty 1994a), (Saaty 1994b):

o If the number of alternatives grows, comparisons grow exponentially, and the use of method can be made cumbersome (Moffett et al. 2005).

o It does not consider variation of criteria ranges in a specific t.

o It is more a comparing tool for management than a statistical method (Dyer 1990).

o Valuations of comparisons can be interpreted differently by different subjects.

o Individually comparison may lead to conflicts, because if A> B and B> C, it may not occur A < C.

o Inclusion of a new irrelevant approach may affect management of two relevant criteria (Dyer 2005). This would contradict axiom of MAVT (Multi Attribute Value Theory) on irrelevant alternatives (Arrow and Raynaud 1986).

Alternative 1

PROBLEM

CRITERION 1.1

Alternative 2 Alternative … Alternative n CRITERION 1 CRITERION 2 CRITERION … CRITERION n

Alternative 1

PROBLEM

CRITERION 1.1

Alternative 2 Alternative … Alternative n CRITERION 1 CRITERION 2 CRITERION … CRITERION n

CI CR = ≤ 0,1 w CIrandom 11 w 12 … w 1n w 21 w 22 … w 2n W = … … … … w n1 w n2 … w nn

o Asymmetrical inconsistency in eigenvectors by Saaty scales (Donegan et al. 1992), (Donegan et al. 1995).

To correct this lack of consistency, the Modified AHP Method was developed (Donegan et al. 1992). (Donegan et al. 1995), but also it has its criticisms (Zanakis et al. 1998), indicating that improvements in real cases are not as crucial as Saaty AHP original method. Other published articles about AHP method on decision-making are authored by Belton (1986), Bevilacqua and Braglia, (2000) or González-Prida et al. (2011).

3. UNCERTAINTY AFTER THE AHP IMPLEMENTATION

The Analytic Hierarchy Process provides a logical framework to determine the benefits of each alternative. Other deductions are possible to obtain from the numerical results. With the conventional AHP method, a maintenance or after sales manager can check the influence, for instance, of:

o Purchasing additional spare part, which would change the weight of the stock level criterion but also the supply term and the supply cost.

o Repairing a failed piece although it would have a longer repair term, instead of purchasing a new one.

o Performing an opportunistic maintenance while any corrective repair, that increases the reliability criterion.

o Etc.

In other words, this tool allow us to use it as a sensitivity analysis, showing how projected choices can change with qualitatively or quantitatively variations in the input of key assumptions on which the decision-making is based, and showing in some way a criticality analysis of the issue.

Fig. 3. Relationship between AHP and DAHP

The uncertainty appears when the AHP is implemented and the decision adopted from it can change itself the initial conditions under which, the method were applied. Consequently, once the decision taken is adopted, the boundary conditions of the system under study can change, and the original hypotheses considered at the beginning are no more valid. In addition to this, sometimes the adopted decision requires a time to be properly implemented (for example, in terms of provisioning times). Therefore, any possible delay adopting the decision can cause that, once the action is implemented, the proper decision could be another one completely different. These situations appear in actual cases so frequently that requires a specific adaptation of the AHP to this changing environment. This adaptation is what it is here called as Dynamic Analytic Hierarchy Process (DAHP). In short, the DAHP applies the same AHP

methodology but considers the influence of the decisions in the boundary conditions. Figure 3 tries to illustrate this relationship.

In other words, while the AHP provides a fixed picture of a system in a specific moment with its best local decision, the DAHP provides a motion picture of the system where the best decision can be different to the ones calculated in determined moments. It is important here to add that, although costs have been included in AHP and DAHP, in many complex decisions, costs should be set aside until the benefits of the alternatives are evaluated. Otherwise it could happen that the general costs of the maintenance or warranty assistance were too high, taking not care about its benefits. In other words, discussing costs together with benefits can sometimes bring forth many political and emotional results.

4. PROPOSED METHODOLOGY

The methodology is here described in a very generic way and shall be adapted to each scenario under study. Therefore, further research on this area will be the application of this methodology to a case study where the proposed method can be more easily illustrated through a practical example. The Dynamic Analytic Hierarchy Process (DAHP) application will be performed following these next steps:

1. State the objective (it can be for instance to obtain spares part ranking for maintenance or warranty assistances).

2. Define the criteria (in the a.m. example could be the stock level, supply term, repair term, reliability, costs etc.).

3. Selection of alternatives or variables in function of time (f(t)).

4. Conventional AHP solving for each value of t. 5. Analysis of the variables trends.

When defining the criteria, both qualitative and quantitative criteria can be compared using informed judgments to derive weights and priorities. In order to define the relative importance of the criteria it will be used judgments, determining by this way the ranking of the mentioned criteria. Using pairwise comparisons (see Table 1) the relative importance of one criterion over another can be expressed in cij according to Table 2.

Table 2. Matrix of relative importance between criteria

C1 C2 Cj Cn

C1 --- c12 c1j c1n C2 c21 --- c2j c2n Ci ci1 ci1 --- Cin Cn cn1 cn2 cnj ---

In order to turn this matrix (n x n) into a ranking of criteria, the eigenvector solution is the best approach (Saaty, 1990). In order to review its consistency, it is necessary to apply the already stated formulas CI and CR to λmax, which is obtained from the previously indicated eigenvalues and the matrix of DAHP METHOD

ALTERNATIVES

CRITERIA

AHP METHOD DECISION

DECISIONS TREND

pairsiwe comparisons (Table 2), once it is normalized (Donegan et al. 1995). The value CIRando depends on the number of criteria being compared where Dr. Saaty proposed the use of an appropriate consistency index (also called RCI “Random Consistency Index”).

In terms of the alternatives, a pairwise comparison determines the preference of each alternative over another, according to each criterion. Therefore, it is possible to constitute a 3D-matrix (m x m x n), where each slides (see Figure 4) refers to pairwise comparison according to a specific criteria. The novelty here is that values for each matrix cell are not a fixed value. They on the contrary depend on those decisions taken previously. This dependency on historical data yields finally in a dependency on time. With this point of view, a conventional AHP can be applied for each t, but getting together all these pictures for each t, we finally obtain a dynamical representation of the system behaviour where the system feeds back itself like in an iterative process.

Fig. 4. Variables matrices function of time

With this proposal, the new methodology not only combines both qualitative and quantitative information, it also introduces a dynamic view of the solving process due to the fact that the adopted decisions themselves make influence in the later system behaviour. With the mentioned context, and computing the eigenvector in each moment, the method determines the relative ranking of alternatives under each criterion and time (see Table 3).

Table 3. Ranking values for each variable and time

t0 t1 … tf

V1 v10 v11 … v1f V2 v20 v21 … v2f Vs vs0 vs1 … vsf

Vm vm0 vm1 … vmf

With the above chart related to the ranking position of each variable in comparison to the rest of them, it is possible to illustrate graphically these evolutions in order to obtain trends (see Figure 5).

Therefore, the best decision under a dynamic context is not always what the AHP offers at a first glance, but that variable which remain in prior positions in the ranking of alternatives with the development of the time. The ranking obtained with

the conventional AHP will refer to good decisions from a local point of view, but not necessary under a temporary context. Therefore de DAHP considers the time factor as a crucial aspect for taking decisions properly in a long term.

Fig. 5. Example of variables evolution with time 5. CONCLUSIONS

As it has been hinted at, the application of the AHP method in the decision making process involves the acquisition of the following objectives:

o Improve organization, structuring and

documentation (feedback) of the process.

o Use a rational and logical analysis, which minimizes emotional burden of trials.

o Classify and compare alternatives.

o Employ quantitative and qualitative criteria, accurate or measurable, and inaccurate or estimated.

o Consensus and satisfaction in group decision-making.

o Predict at times likely results.

That means the traditional AHP method tries to answer questions (in our area) as: What information should be relevant to make decisions regarding the maintenance or the after-sales service? How to select a policy for maintenance or warranty assistances in order to improve the profit and the image of the company? What are the future investigations that arise from this line? From the last question, this paper is intended to go one step beyond trying to track the performance of each adopted decisions taking them in function of the time variable. In few words, this research develops the problem of a changing environment where the AHP method is applied, appearing in consequence a new modified version of the methodology called Dynamic Analytic Hierarchy Process (DAHP). The fact of boundary conditions that change gradually with each adopted decision is a phenomenon that occurs in highly dynamic systems, so the criticalities are affected by the production plan as, for example, in mining fleets. Therefore, further extensions to this research will be the development of a case study on this area that can illustrate the described methodology.

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