Methodology for productivity estimation of academic departments

3.3 Model Development

3.3.1 Methodology for productivity estimation of academic departments

Since most of the criteria to be evaluated are human intensive, the results are based on personal judgments and are therefore subjective (Shahroudi & Rouydel, 2012). One such example of subjectivity or personal judgment is: ‘rate this research publication in terms of poor, good or excellent’. In this study, all subjective evaluation criteria will be referred to as intangibles. An academic department will also have objective criteria that require evaluation. One such example is: ‘how many hours do you lecture per week?’ In this study, the objective criteria will be referred to as tangibles.

The methodology for the productivity estimation model is diagrammatically depicted in the flowchart (Figure 3-2). The steps on the methodology are discussed below the flowchart.

MCDM problem is identified, that is, the academic productivity estimation problem

Construct hierarchy structure

Set up reference scale according to management decisions

What kinds of sub-criteria?

Management sets up fuzzy sets for different levels

Attain fuzzy numbers by direct estimates, using historical data and the geometric mean method

Determine sub-score using triangular fuzzy number

Develop fuzzy judgment matrix

The fuzzy weight vector is determined using linguistic terms

Use fuzzy TOPSIS method Quantitative

(tangible) sub-criteria

Qualitative (Intangible) sub-criteria

The fuzzy performance matrix is determined by synthesizing the fuzzy judgment matrix with the fuzzy weight vector

Select optimal solution from the alternatives What kind of information is

required?

Selection and ranking Queries, reports,

comparisons, etc,

Defuzzify fuzzy performance matrix

Obtain crisp data

Output the information

Interpret the output

The fuzzy comparison matrix is attained for each decision-maker

The Consistency Ratio is calculated and revisions of fuzzy comparison matrix are done (if necessary)

The criteria are ranked according to importance intensity using BNP values

Figure 3-2: Methodology for productivity estimation for academic staff

Each step of the methodology (Figure 3-2) is discussed below:

1. Problem Identification:

This step of the development (problem identification) was completed in section 3.2 using design science research methodology.

2. The development of the hierarchical structure:

The general structure for the productivity estimation hierarchy is depicted in Figure 3-3.

The evaluation for the MCDM problem is bottom-up with the alternatives in the lowest level and the sub-criteria immediately above the alternatives. Each alternative is firstly measured using the sub-criteria. Each criterion (that is, each key performance attributes) combines with the sub- scores in order to attain the overall goal. An example of a criterion is 𝐶₁and an example of a sub-criterion is 𝐶₁₁as indicated in Figure 3-3.

3. Identify tangible and intangible operational indicators:

When estimating productivity of academic staff or an academic department as a whole, the attributes to be measured lend themselves more to a qualitative rather than a quantitative evaluation. As a result, most of the attributes to be measured are intangible. However, there are some tangible attributes (or quantitative attributes) that require evaluation. The artifact development will also show how tangible and intangible sub-criteria are handled in the AHP (sun, 2010).

Problem

𝐶₁ 𝐶₂ 𝐶_𝑘

𝐶₁₁ Goal

Criteria

Sub-criteria ^𝐶¹² ^… ^𝐶^1𝑝1 ^𝐶^𝑖1 ^𝐶^𝑖2 ^𝐶^𝑖𝑝𝑖 ^𝐶^𝑘1 ^𝐶^𝑘2 ^𝐶^𝑘𝑝𝑘

… .

Figure 3-3: The General AHP Hierarchy Structure

55 4. Evaluate the tangible sub-criteria

The ratings of alternatives are fuzzy numbers and are measured using the sub-criteria. Sub- criteria that are tangible (that is, with precise ratings) have to therefore be normalised into fuzzy numbers. Ding (2011) therefore two methods of converting the tangible sub-criteria into fuzzy numbers. The first method involves using the triangular fuzzy numbers directly if the sub- criteria can be appropriately estimated. For example, if the average teaching load of an academic is 10 lectures per week, with a minimum of 6 lectures and a maximum of 14 lectures, then this value (10) can be subjectively represented using triangular fuzzy numbers as (6, 10, 14). The second approach uses historical data. Let 𝑥₁, 𝑥₁,…, 𝑥_𝑘 represent H (that is, the average number of hours for community engagement) for k periods, then the geometric mean method can be used to express 𝐻as a fuzzy number (𝐿, 𝑀, 𝑈), where:

(3.1) L = 𝑚𝑖𝑛

𝑖 {𝑥_𝑖} , M = [∏^𝑘_𝑖=1 𝑥_𝑖]^1/𝑘 and U = 𝑚𝑎𝑥

𝑖 {𝑥_𝑖}

For example if the 4 historical data available for H with respect to an alternative (such as A1) are 8, 12, 13, and 10, then after evaluation, the fuzzy triangular number is (8,⁴√8X10X12X13, 13)

= (8, 10.57,13).

5. Evaluate the intangible sub-criteria

Intangible sub-criteria lend themselves more to a qualitative rather than a quantitative evaluation.

These sub-criteria are difficult to evaluate because they are imprecise and vague. As a result, the evaluation is reliant on human judgment and decision-makers will have many diverse viewpoints for an alternative. A panel or a group with subjective judgments normally carries out the evaluation of an academic department. Therefore, in order to attain a more consistent outcome, a group decision method is proposed. Each decision-maker (𝐷_𝑝) grades each alternative (𝐴_𝑖) on the same sub-criteria (𝐶_𝑗𝑘). This approach enables an alternative (𝐴_𝑖) to attain many grades (𝐺̃_{𝑖𝑗𝑘𝑝}) from all the decision-makers (Tsaur et al., 2002). The matrix below shows the group decision method using intangible sub-criteria.

[

𝑫_𝟏 𝑫_𝟐 … 𝑫_𝒕

𝑨_𝟏 𝐺̃_1𝑗𝑘1 𝐺̃_1𝑗𝑘2 … 𝐺̃_{1𝑗𝑘𝑡} 𝑨_𝟐 𝐺̃_2𝑗𝑘1 𝐺̃_2𝑗𝑘2 … 𝐺̃_{2𝑗𝑘𝑡}

⋮ ⋮ ⋮ ⋱ ⋮

𝑨_𝒏 𝐺̃_{𝑛𝑗𝑘1} 𝐺̃_{𝑛𝑗𝑘2} … 𝐺̃_{𝑛𝑗𝑘𝑡}]

⋱⋱

A grade of alternate i with respect to decision-maker p on a sub-criterion jk is represented by(𝐺̃_{𝑖𝑗𝑘𝑝}). The fuzzy number (𝐺̃_𝑖𝑗𝑘) from (𝐺̃_{𝑖𝑗𝑘𝑝}) represent the sub-score of alternative i with respect to the sub-criterion 𝑗𝑘. The fuzzy number (𝐺̃_𝑖𝑗) represent a score of alternate i with respect to criterion 𝑗. Each element in the matrix has the subscript 𝑗𝑘 that indicates that the same sub-criterion is being evaluated by all decision-makers (𝐷_𝑡) for each alternative (𝐴_𝑛) under the criteria j (Tsaur et al., 2002). With(𝐺̃_{𝑖𝑗𝑘𝑝}), a fuzzy number for an intangible sub-criterion can be composed as follows: Let L, M, and U represent the lower middle and upper limits of a triangular fuzzy number. The lowest (L) and highest (U) score of all 𝑝 decision-makers are taken as the lower and upper limits respectively. The middle value (M) between the lowest and highest is calculated. The fuzzy number for an intangible sub-criterion can therefore be derived as follows (Tsaur et al., 2002):

(3.2) (𝐺̃_{𝑖𝑗𝑘𝑝}) = (𝐿_{𝑖𝑗𝑘𝑝}, 𝑀_{𝑖𝑗𝑘𝑝}, 𝑈_{𝑖𝑗𝑘𝑝} )

(3.3) 𝐿_𝑖𝑗𝑘 = min(𝐿_{𝑖𝑗𝑘𝑝}) with 𝑝 = 1, 2, … , 𝑡

(3.4) 𝑀_𝑖𝑗𝑘 = ^(∑ ^𝑀^{𝑖𝑗𝑘𝑝}

𝑡

𝑝=1 )

𝑝 with 𝑝 = 1, 2, … , 𝑡 (3.5) 𝑈_𝑖𝑗𝑘 = max(𝑈_{𝑖𝑗𝑘𝑝}) with 𝑖 = 1, 2, … , 𝑡 Therefore the fuzzy number for an intangible sub-criterion is represented as:

(3.6) (𝐺̃_𝑖𝑗𝑘) = (𝐿_𝑖𝑗𝑘, 𝑀_𝑖𝑗𝑘, 𝑈_𝑖𝑗𝑘).

𝑤ℎ𝑒𝑟𝑒 𝑖 = 1, 2, … , 𝑛; 𝑗 = 1, 2, … , 𝑚; 𝑘 = 1, 2, … , 𝑞; 𝑝 = 1, 2, … , 𝑡

57 6. The fuzzy judgment matrix is obtained:

This step will judge the alternatives by examining the scores of the criteria. The sub-scores for all sub-criteria (𝐶_𝑗𝑘) belonging to the same criterion (𝐶_𝑗) of each alternative (𝐴_𝑗) are added and are represented as(𝐺̃_𝑖𝑗) (Tsaur et al., 2002). This process is carried out for every criterion that has sub-criteria. This calculation is derived as follows:

(3.7) 𝐺̃_𝑖𝑗 = ∑^𝑞 𝐺̃_𝑖𝑗𝑘

𝑘=1 , 𝑘 = 1, 2, … , 𝑞

where q is the number of sub criteria for each criteria (𝐶_𝑗). After applying this equation, the following decision matrix is derived (Sun, 2010):

[

𝑪_𝟏 𝑪_𝟐 … 𝑪_𝒎 𝑨_𝟏 𝐺̃₁₁ 𝐺̃₁₂ … 𝐺̃_1𝑚 𝑨_𝟐 𝐺̃₂₁ 𝐺̃₂₂ … 𝐺̃_2𝑚

⋮ ⋮ ⋮ ⋱ ⋮

𝑨_𝒏 𝐺̃_𝑛1 𝐺̃_𝑛2 … 𝐺̃_𝑛𝑚]

This matrix is then normalised so that the values can be matched with the weight vectors (the calculation of weights is discussed in step 7). The normalisation of each criterion (𝐶_𝑗) is done using the following equation (Chen & Hwang, 1992; Tsaur et al., 2002):

(3.8) 𝑎 ̃_𝑖𝑗 = ^𝐺̃^𝑖𝑗

√∑^𝑛 (𝐺̃_𝑖𝑗)² 𝑖=1

2 and the normalised matrix is now:

[

𝑪_𝟏 𝑪_𝟐 … 𝑪_𝒎 𝑨_𝟏 𝑎̃₁₁ 𝑎̃₁₂ … 𝑎̃_1𝑚 𝑨_𝟐 𝑎̃₂₁ 𝑎̃₂₂ … 𝑎̃_2𝑚

⋮ ⋮ ⋮ ⋱ ⋮

𝑨_𝒏 𝑎̃_𝑛1 𝑎̃_𝑛2 … 𝑎̃_𝑛𝑚]

The judgment score of alternate (𝐴_𝑖) with respect to (𝐶_𝑗) is denoted by 𝑎̃_𝑖𝑗. 7. The fuzzy performance vector is determined:

The overall fuzzy performance vector is attained when each alternative takes all the criteria into consideration. This is achieved when the fuzzy judgment matrix is multiplied by the fuzzy weight vector. The fuzzy judgment matrix was derived in step 6 above. The fuzzy weight vector

A =

has to be determined. There are many methods that can be used to attain the fuzzy weight vector. The two most popular ones are the use of absolute values from a Saaty scale that can be converted into fuzzy numbers. The second technique is the geometric mean method (Osman et al., 2013). The first method is discussed in sections 7.1 and 7.2 below. The geometric mean method uses linguistics values of the decision-makers to evaluate the criteria. This method is discussed in section 7.3. Although both methods have been successfully implemented, the nature of the problem should be taken into consideration before deciding on which technique to adopt (Osman et al., 2013). If it can be determined that decision-makers can get a more objective trade-off among all the criteria, then the first method should be used, otherwise the second method should be used (Tseun-Ho et al., 2012).

7.1 The fuzzy weight vector is determined using a Saaty scale with absolute values The relative importance among the criteria is determined using a weight vector. This is achieved in the AHP with pair-wise comparisons of elements (based on human judgment) and is represented using precise values attained from a nominal scale such as the one indicated in Table 2-1 (Saaty, 2008). These precise values will be used to determine priorities or weights that will be used to select the best alternative. However, using precise values of individual decision- makers does not provide a credible evaluation of an intangible attribute. In order to address this deficiency, the scores of all decision-makers (a group decision method) based on triangular fuzzy numbers is proposed (Sun, 2010).

In order to achieve such triangular fuzzy numbers, a scale (see Table 2-1) with precise ratings to carry out pair-wise comparisons of the criteria is used. This is indicated as follows (Saaty, 2008):

[

𝑪_𝟏 𝑪_𝟐 … 𝑪_𝒎

𝑪_𝟏 𝑏_11𝑝 𝑏_12𝑝 … 𝑏_1𝑚𝑝 𝑪_𝟐 𝑏_21𝑝 𝑏_22𝑝 … 𝑏_2𝑚𝑝

⋮ ⋮ ⋮ ⋱ ⋮

𝑪_𝒎 𝑏_𝑚1𝑝 𝑏_𝑚2𝑝 … 𝑏_𝑚𝑚𝑝]

If C₁represents “teaching” and C2 represents “administration” and using a value of 9 in a Saaty scale means that teaching is 9 times more important than administration. The converse means that administration is 9 times less important than teaching, that is, the value for (administration,

𝑫_𝒑 =

teaching) is ¹₉ (the reciprocal value). The “equally preferred” values are 𝑏_11𝑝, 𝑏_22𝑝,..., 𝑏_𝑚𝑚𝑝 and will yield a value of 1. The above situation can be represented as follows (Sun, 2010): 𝑏_𝑗𝑒𝑝

= 𝑏_𝑒𝑗𝑝⁻¹ if j ≠ e and 𝑏_𝑗𝑒𝑝= 1 if j = e (j = 1, 2...m and e = 1, 2….m) where a decision-maker 𝐷_𝑝 measures the relative importance between criteria e and j. In order to attain triangular fuzzy numbers from all the decision makers, the following equations are used:

(3.9) 𝐿_𝑗𝑒 = min(𝑏_𝑗𝑒𝑝) with 𝑝 = 1, 2, … , 𝑡; 𝑗 = 1, 2, … , 𝑚 𝑎𝑛𝑑 𝑒 = 1, 2, … , 𝑚

(3.10) 𝑀_𝑗𝑒 = ^(∑ ^𝑏^𝑗𝑒𝑝

𝑡

𝑝=1 )

𝑝 with 𝑝 = 1, 2, … , 𝑡; 𝑗 = 1, 2, … , 𝑚 𝑎𝑛𝑑 𝑒 = 1, 2, … , 𝑚 (3.11) 𝑈_𝑗𝑒 = max(𝑏_𝑗𝑒𝑝) with 𝑝 = 1, 2, … , 𝑡; 𝑗 = 1, 2, … , 𝑚 𝑎𝑛𝑑 𝑒 = 1, 2, … , 𝑚 The fuzzy numbers are now represented as:

(3.12) (𝑏̃_𝑗𝑒) = (𝐿_𝑗𝑒, 𝑀_𝑗𝑒, 𝑈_𝑗𝑒) , 𝑗 = 1, 2, … , 𝑚; 𝑒 = 1, 2, … , 𝑚.

The score (𝑏̃_𝑗𝑒) indicates the comprehensive judgment scores of all decision-makers regarding all the criteria and is represented in the matrix as:

[

𝑪_𝟏 𝑪_𝟐 … 𝑪_𝒎 𝑪_𝟏 𝑏̃₁₁ _𝑏̃₁₂ … 𝑏̃_1𝑚 𝑪_𝟐 𝑏̃₂₁ _𝑏̃₂₂ … 𝑏̃_2𝑚

⋮ ⋮ ⋮ ⋱ ⋮

𝑪_𝒎 𝑏̃_𝑚1 _𝑏̃_𝑚2 … 𝑏̃_𝑚𝑚]

The purpose of using a Saaty scale (Table 2-1) is to show that the criteria have varying degrees of importance. It is therefore important to acquire weights or priorities (𝑤̃_𝑗) that correspond to a criterion (𝐶_𝑗). The following equation is used to calculate these weights (Rana et al., 2012):

(3.13) 𝑤̃j =

∑^𝑚 𝑏̃_𝑗𝑒 𝑒=1

∑ ∑^𝑚 𝑏̃_𝑗𝑒

𝑒=1 𝑚

𝑗=1

, 𝑗 = 1, 2, … , 𝑚 𝑎𝑛𝑑 𝑒 = 1, 2, … , 𝑚

The criteria weights (𝑤̃_𝑗) collectively will make up the fuzzy weight vector 𝑊 = (𝑤̃₁,𝑤̃₂, … ,𝑤̃_𝑚).

𝑫 =

7.2 Synthesize the fuzzy judgment matrix with the fuzzy weight vector:

The next step involves calculating the fuzzy performance matrix. However, before the fuzzy performance matrix is calculated, the pair-wise comparison matrix will have to be checked for inconsistencies in the decision-makers choices (Lee, 2010). Consistency checking is discussed in number 8 below. After the pair-wise comparison matrix has been checked for any consistencies (and revisions implemented), the fuzzy performance matrix is calculated by multiplying the fuzzy judgment matrix by the fuzzy weight vector. The fuzzy-judgment matrix provides an overall score of alternate 𝐴_𝑖 with respect to criteria 𝐶_𝑗. However, the relative weights between each criterion are not taken into consideration. The fuzzy judgment matrix and the fuzzy weight vector have to therefore be synthesized and this is done by multiplying each criterion weight 𝑤̃_𝑗 to its corresponding criterion of the fuzzy judgment matrix as indicated

below (Lee, 2010).

[

𝑪_𝟏 𝑪_𝟐 … 𝑪_𝒎

𝑨_𝟏 𝑤̃₁𝑎̃₁₁ 𝑤̃₂𝑎̃₁₂ … 𝑤̃_𝑚𝑎̃_1𝑚 𝑨_𝟐 𝑤̃₁𝑎̃₂₁ 𝑤̃₂𝑎̃₂₂ … 𝑤̃_𝑚𝑎̃_2𝑚

⋮ ⋮ ⋮ ⋱ ⋮

𝑨_𝒏 𝑤̃₁𝑎̃_𝑛1 𝑤̃₂𝑎̃_𝑛2 … 𝑤̃_𝑚𝑎̃_𝑛𝑚] =

[

𝑪_𝟏 𝑪_𝟐 … 𝑪_𝒎 𝑨_𝟏 ℎ̃₁₁ ℎ̃₁₂ … ℎ̃_1𝑚 𝑨_𝟐 ℎ̃₂₁ ℎ̃₂₂ … ℎ̃_2𝑚

⋮ ⋮ ⋮ ⋱ ⋮

𝑨_𝒏 ℎ̃_𝑛1 ℎ̃_𝑛2 … ℎ̃_𝑛𝑚]

The fuzzy performance score with alternate (𝐴_𝑖) corresponding to (𝐶_𝑗) with fuzzy numbers (𝐿_𝑖𝑗, 𝑀_𝑖𝑗, 𝑈_𝑖𝑗) is denoted by ℎ̃_𝑖𝑗_.The overall fuzzy performance scores of each alternative with all criteria are represented by H.

7.3 The fuzzy weight vector is determined using linguistic values:

The first step involves constructing pair-wise comparison matrices among all the criteria as indicated below. These matrices indicate the judgment scores of all decision-makers regarding all the criteria. The decision-makers will use a linguistic scale (like the one depicted in Table 3- 1) to rate all the criteria.

H =

[

𝑪_𝟏 𝑪_𝟐 … 𝑪_𝒎 𝑪_𝟏 1 𝑎̃₁₂ … 𝑎̃_1𝑚 𝑪_𝟐 𝑎̃₂₁ 1 … 𝑎̃_2𝑚

⋮ ⋮ ⋮ ⋱ ⋮

𝑪_𝒎 𝑎̃_𝑚1 𝑎̃_𝑚2 … 1 ] =

[

𝑪_𝟏 𝑪_𝟐 … 𝑪_𝒎 𝑪_𝟏 1 𝑎̃₁₂ … 𝑎̃_1𝑚 𝑪_𝟐 _𝑎̃¹

12 1 … 𝑎̃_2𝑚

⋮ ⋮ ⋮ ⋱ ⋮

𝑪_𝒎 _𝑎̃¹

1𝑚 1

𝑎̃_2𝑚 … 1 ]

where

𝑎̃_𝑖𝑗 = {9̃⁻¹, 8̃⁻¹, 7̃⁻¹, 6̃⁻¹, 5̃⁻¹, 4̃⁻¹, 3̃⁻¹, 2̃⁻¹, 1̃⁻¹, 1̃, 2̃, 3̃, 4̃, 5̃, 6̃, 7̃, 8̃, 9̃ 𝑖 ≠ 𝑗 1 𝑖 = 𝑗

The next step involves using the geometric mean technique to define the fuzzy geometric mean and fuzzy weights of each criterion as follows (Sun, 2010):

(3.14) 𝑟̃_𝑖 = (𝑎̃_𝑖1⊗ … ⊗ 𝑎̃_𝑖𝑗 ⊗ … ⊗ 𝑎̃_𝑖𝑛)^1/𝑛 (3.15) 𝑤̃_𝑖 = 𝑟̃_𝑖⊗ [𝑟̃₁⊕ … ⊕𝑟̃_𝑖 ⊕ … ⊕ 𝑟̃_𝑛]⁻¹

where 𝑎̃_𝑖𝑗 is a fuzzy comparison value of dimension i to criterion j. Thus 𝑟̃_𝑖 is a geometric mean of fuzzy comparison value of the criterion i to each criterion. The fuzzy weight of the 𝑖^𝑡ℎ criterion is 𝑤̃_𝑖 and can be indicated as a triangular fuzzy number as 𝑤̃_𝑖 = (𝑙𝑤_𝑖, 𝑚𝑤_𝑖, 𝑢𝑤_𝑖) where 𝑙𝑤_𝑖, 𝑚𝑤_𝑖 and 𝑢𝑤_𝑖 stands for the lower, middle and upper values of the fuzzy number.

D =

The fuzzy judgment matrix has to be synthesized with the fuzzy weight vector during the multiplication process. Synthesizing is done by multiplying each criterion weight 𝑤̃_𝑖 to its corresponding criterion of the fuzzy judgment matrix. This process is discussed in section 7.2.

8. Consistency test of the fuzzy pair-wise comparison matrix

Since the criteria to be measured are vague and imprecise, the evaluations are reliant on human judgment. As a result of the varying viewpoints from different decision-makers, the AHP does not demand perfect consistency but allows for a small measure of inconsistency (Ramik &

Korviny, 2013). A Consistency Ratio (CR) ≤ 10% (or CR ≤ 0.1) is considered acceptable (Osman et al., 2013). If the CR is > 10% (or CR > 0.1) then the pair-wise judgments will have to be revised by the decision-makers. The consistency of the pair-wise comparison matrix has to therefore be measured, checked and revised (if necessary) before the fuzzy performance matrix is calculated (Rostamy et al., 2012).

In order to measure consistency, a triangular fuzzy positive reciprocal (TFPR) matrix is used.

The elements of a TFPR matrix are positive triangular positive numbers 𝑎̃_𝑖𝑗= (𝑎_𝑖𝑗^𝐿, 𝑎_𝑖𝑗^𝑀, 𝑎_𝑖𝑗^𝑈) where

Linguistic scale for importance

Triangular fuzzy scale Triangular fuzzy reciprocal scale

Just equal (1, 1, 1) (1, 1, 1)

Equally important (EI) (1/2, 1, 3/2) (2/3, 1, 2)

Weakly more important (WMI)

(1, 3/2, 2) (1/2, 2/3, 1)

Strongly more important (SMI)

(3/2, 2, 5/2) (2/5, 1/2, 2/3)

Very strongly more important (VSMI)

(2, 5/2, 3) (1/3, 2/5, 1/2)

Absolutely more important (AMI)

(5/2, 3, 7/2) (2/7, 1/3, 2/5)

Table 3-1: Linguistic terms for criteria ratings

𝑎_𝑖𝑗^𝐿 > 0 and by using the formula for the reciprocal of fuzzy numbers, that is, (_𝑎̃¹^̃ = (¹

𝑎^𝑈,¹

𝑎^𝑀 ,¹

𝑎^𝐿)), the following matrix is constructed:

[

(1, 1, 1) (𝑎₁₂^𝐿 , 𝑎₁₂^𝑀, 𝑎₁₂^𝑈 ) … (𝑎_1𝑛^𝐿 , 𝑎_1𝑛^𝑀, 𝑎_1𝑛^𝑈)

( ¹

𝑎₁₂^𝑈 , ¹

𝑎₁₂^𝑀, ¹

𝑎₁₂^𝐿 ) (1, 1, 1) … (𝑎_2𝑛^𝐿 , 𝑎_2𝑛^𝑀, 𝑎_2𝑛^𝑈)

⋮ ⋮ ⋱ ⋮

( ¹

𝑎_1𝑛^𝑈 , ¹

𝑎_1𝑛^𝑀 , ¹

𝑎_1𝑛^𝐿 ) (¹

𝑎_2𝑛^𝑈 , ¹

𝑎_2𝑛^𝑀 , ¹

𝑎_2𝑛^𝐿 ) … (1, 1, 1) ]

Using triangular fuzzy numbers are considered to be the most appropriate technique in group decision making. This is so because 𝑎^𝐿 is interpreted as the minimum possible values, 𝑎^𝑈 is interpreted as the maximum possible value and 𝑎^𝑀 is the geometric mean, that is, the mean value or the most possible value of all decision makers. Hence triangular fuzzy numbers can therefore be effectively used to measure the consistency of the pair-wise comparison matrix.

When calculating the Consistency Ratio (CR), Saaty’s absolute value method is adapted in order to take fuzzy requirements into consideration. Saaty’s method is chosen as a reference because it can handle multiple criteria with ease as well as calculate the weights and Consistency Index (CI) without cumbersome mathematics. His method uses the eigenvector method to determine the weights of the various criteria by implementing a pair-wise reciprocal comparison matrix (Odeyale et al., 2014). The largest eigenvector is defined as λ_𝑚𝑎𝑥 of matrix A and the weight 𝑤_𝑖 as a component of the normalised vector corresponding to λ_𝑚𝑎𝑥 where 𝑤_𝑖 = _(r ^r^𝑖

1+ r2 +⋯+ r𝑛) and 𝑟_𝑖 is the geometric mean of each row with 𝑟_𝑖 = [∏^𝑛_𝑗=1 𝑎_𝑖𝑗]^1/𝑛 (Aly & Vrana, 2008). The consistency of the pair-wise comparisons can be assessed through calculating the consistency ratio (CR) from the consistency index and the random (RI) as follows:

(3.16) CI = ^λ^{max − 𝑛}_𝑛−1

(3.17) CR = ^𝐶𝐼_𝑅𝐼

The random index values provide different values for n (that is, n = the number of factors or criteria) as indicated in Table 3-2.

𝐴̃ =

N 1 2 3 4 5 6 7 8 9 10 11 12

RI 0 0 0.58 0.9 1.12 1.24 1.32 1.41 1.45 1.49 1.51 1.58

The value in this table is simply a look-up table for the values of n from 1 to 12 (Rostamy et al., 2012). These values have been determined experimentally and have been universally accepted.

The demonstration in Chapter 5 will therefore make use of this table.

The λ_𝑚𝑎𝑥 value for absolute values is a single precise value. The λ_𝑚𝑎𝑥 value for fuzzy inputs is computed as the modal value of the resulting fuzzy number and fuzzy operations are used in all the calculations. Annexure C shows detailed calculations involving Saaty’s method to calculate the weights and the CR for precise values. The purpose of Annexure C is to demonstrate the steps involved when precise values are involved so that this technique can be adapted to take fuzzy data into consideration.

9. Apply fuzzy TOPSIS method for ranking and selection:

The fuzzy performance matrix H in step 7 above was derived after applying fuzzy logic and fuzzy set theory to the AHP. Since each element of the matrix is a triangular fuzzy number, it will be used as the input for the fuzzy TOPSIS method (for selection and ranking). The fuzzy TOPSIS method was discussed in detail in section 2.7.2.

10. Selecting the optimal solution:

This step in the model development selects the optimal solution from all the alternatives with respect to all the criteria. The ranking and selection process is incorporated in the discussion on fuzzy TOPSIS in section 2.7.2.

11. Defuzzify the fuzzy performance matrix:

Defuzzification is a technique that converts fuzzy numbers into crisp real values in order to determine the best non-fuzzy performance (BNP) value (Tsaur et al., 2002). Although there are many methods of accomplishing this, the most popular method is the Centre-of-Area method (also called the centroid method). This method is popular because it is simple to implement.

Table 3-2: The random index RI for number of factors/criteria n

Also, the analyst’s personal judgment is not required when this equation is used (Zhao &

Govind, 1991). The equation for defuzzification is:

(3.18) 𝐵𝑁𝑃_𝑖𝑗 = ^[(𝑈^𝑖𝑗^{− 𝐿}^𝑖𝑗^)+(𝑀₃ ^𝑖𝑗^{− 𝐿}^𝑖𝑗^)]+ 𝐿_𝑖𝑗 (Tsaur et al., 2002) In this study, the centroid method will be used for defuzzification to:

 Rank each performance criteria according to its importance intensity. This is done in section 5.5.5 (e)

 Evaluate the performance of an academic based on each criterion. A BNP value will indicate how an academic has performed when a certain criteria is considered. This is discussed in section 5.6. For example: “Show the evaluation of alternative 𝐴₁on

“Administration”. This evaluation can be attained from the fuzzy performance matrix if alternative 𝐴₁represents an academic and criteria 𝐶₁represent “Administration”. The fuzzy performance score for the academic with respect to Administration is therefore ℎ_𝑖𝑗.

Defuzzification of the fuzzy data ℎ̃_𝑖𝑗will be done using the BNP equation.

3.4 Conclusion

In Chapter 2, a detailed discussion was provided as to why a fuzzy-based multi-criteria decision making method is most suitable for estimating productivity of academic staff. This chapter (chapter 3) discussed in detail the methodology for developing a productivity estimation model of academic staff and academic departments using a fuzzy-based approach. A design science research methodology (DSRM) with six activities was proposed. The design was accomplished using a Multi-Criteria Decision Making (MCDM) method called Fuzzy Analytic Hierarchy Process (FAHP) and a ranking and selection method called Fuzzy TOPSIS (Technique for Order Preference by Similarity to Ideal Solution). The first three activities of the design science research (DSR) approach were adopted in the model development phase. The demonstration activity (4^th activity) will show how the model can be used to solve the productivity estimation problem of academic staff and an academic department. This activity is discussed in Chapter 5.

However, before the demonstration activity, it is imperative to show how imprecise and fuzzy data can be modeled using fuzzy objects in an object-oriented programming environment. A fuzzy object-oriented approach to programming is therefore discussed in chapter 4.

Dalam dokumen Computer-based productivity estimation of academic staff using the fuzzy analytic hierarchy process and fuzzy topsis method. (Halaman 68-82)