Chapter 10 Multicriteria Decision-Marking Models

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Chapter 10

Multicriteria Decision-Marking Models

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投票怎樣做會令教學的效果好一點？

用英文？

說話慢一點？

一般用中文，重點用英文？

中文說一片，重點用英文再說一片？

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投票方法

多輪舉手投票，每輪每人一票

每輪去掉票數最少的選項，直至剩下一選項為止

若兩個或以上的選項票數最少時，只考慮該等選項，仍以多

輪舉手投票，每輪每人一票去掉票數最少的選項

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投票結果選項第一輪第二輪第三輪第四輪第五輪

用英文？

說話慢一點？


先用中文，重點再用英文說一片？

選項第六輪第七輪第八輪第九輪第十輪

用英文？

說話慢一點？


先用中文，重點再用英文說一片？

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10.3 AHP (Analytical Hierarchy Process)

層級分析法

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AHP (Analytical Hierarchy Process)層級分析法

keys in the scoring method, identifying weights of factors (i.e., objectives) ratings of each alternative for each factor

weight microwave refers stoves

manuf. cap./cost 4 4 3 8market demand 5 8 4 2profit margin 3 6 9 5(long-term) prof./growth 5 3 6 7

transp. costs 2 9 2 4useful life 1 1 5 6

目標選擇 / 選項

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drawback of the scoring or weighting methods:

subjective

hard to simultaneously compare multiple items

questions: Is there any method

with more analytical basis?

easy to compare, e.g., each comparison is only between two options?

weight microwave refers stoves

manuf. cap./cost 4 4 3 8

market demand 5 8 4 2

profit margin 3 6 9 5

(long-term) prof./growth 5 3 6 7

transp. costs 2 9 2 4

useful life 1 1 5 6

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AHP each comparison between two factors or two altrenatives more analytical approach to get the weights

normalized weights of factors normalized priorities of each alternative for factors

Priorities

Alternative 3Alternative 2Alternative 1weightfactor

…………useful life

…………transp. costs

…………(long-term) prof./growth

…………profit margin

…………market demand

…………manuf. cap./cost

sum to 1

sum to 1

sum to 1

.

.

.

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AHP normalized weights of factors normalized priorities of each alternative for factors

Priorities

Alternative 3Alternative 2Alternative 1weightfactor

…………useful life

…………transp. costs

…………(long-term) prof./growth

…………profit margin

…………market demand

…………manuf. cap./cost

question: how to determine those

weights and priorities?

sum to 1

determining the relative importance (i.e., weights) of the factors

sum to 1

determining the relative

importance (i.e., the priorities) of the factors for the alternative

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ideas of AHP to determine the relative importance simple to compare for two alternatives combining the pairwise comparisons into

overall comparisons for weights of all factors, and for priorities of alternatives in each factor

每次都只比較兩樣東西，或是兩目標，或是兩選項，最終將這些兩兩相比轉化成所有選項的比較。

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Idea of AHP to Determine the Relative Importance

Table 10-1 Preference Scale for the Pairwise Comparisons

Extremely preferred

Very strongly to extremely preferred

Very strongly preferred

Strongly to very strongly preferred

Strongly preferred

Moderately to strongly preferred

Moderately preferred

Equally to moderately preferred

Equally preferred

Numerical ValueVerbal Statement of the preference

9

8

7

6

5

4

3

2

1

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Idea of AHP to Determine the Relative Importance

simple to compare for two items for relative importance of factors

1/3

1useful life

1Transp. costs

1(long-term) prof./growth

1profit margin

1market demand

1manuf. cap./cost

weightuseful lifetransp. costs(long-term) prof./growth

profit margin

market demand

manuf. cap./cost

3

market demandtransp. costs

row sum reflects the importance of a factor

see their relationship?

…………Options

…………Sound

…………Price

ClarityLuciditySharpweightsFactors

Alternatives

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Example 10-4

decision: stereo system to purchase brands (i.e., alternatives): Sharp, Lucidity, Clarity criteria (i.e., factors, objectives): sound, price, options

to find the relative importance of criteria to find the relative importance of brands in each criterion

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Example 10-4: To Find the Relative Importance of Criteria

TotalsOptionsSoundPrice

OptionsSoundPriceCriterion

Table 10-2: Pairwise Comparison Table for the Stereo System Selection Problem

84.3331.583311/31/4311/3431

TotalsOptionsSoundPrice

Average %OptionsSoundPriceCriterion

Table 10-3: Normalized Pairwise Comparison Table for the Stereo System Selection Problem

1.00.15790.2105

1.00.11990.27210.6079

1.00.1250.375

0.5

1.00.07690.23080.69230.6316

…………Options

…………Sound

…………Price


Alternatives

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Example 10-4: To Find the Relative Importance of Brands in Price

3.33381.75Totals131/2Clarity

1/311/4Lucidity241Sharp

ClarityLuciditySharpCriterion

Table 10-6: Pairwise Comparison Matrix Price

1.01.01.01.0Totals0.32920.300.3750.2857Options0.12260.100.1250.1429Sound0.55710.60.50.5714Price


Table 10-7: Proportion Percentage Matrix for Price

…………Options

…………Sound

…………Price


Alternatives 這步驟是說，以 Price 作

目標，你較喜歡那個選項。留意，喜歡與否，只與喜好有關，沒特別說是較便宜還

是較貴。

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Example 10-4: To Find the Relative Importance of Brands in Sound

1.58334.57Totals

134Clarity

1/312Lucidity

1/41/21Sharp


Pairwise Comparison Matrix Sound



Proportion Percentage Matrix for Sound

…………Options

…………Sound

…………Price


Alternatives

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Example 10-4: To Find the Relative Importance of Brands in Options

461.75Totals111/2Clarity111/4Lucidity241Sharp


Pairwise Comparison Matrix Options



Proportion Percentage Matrix for Options

…………Options

…………Sound

…………Price


Alternatives

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Example 10-4: To Find the Overall Importance of the Brands

overall importance: by weighted score of brands

0.39840.16210.4456Weighted score

0.23410.18650.57940.1199Options

0.62320.23950.13730.2721Sound

0.32020.12260.55710.6079Price

ClarityLuciditySharpweight (%)Criterion

Weights of Factors, Priorities of Brands in Factors, and Weighted Score of Brands

0.1621 = (0.6079)(0.1226)+(0.2721)(0.2395)+(0.1199)(0.1865)

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Inconsistency in Pairwise Comparison

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Possibility of Inconsistency in a Pairwise Comparison Matrix

the pairwise comparisons may not be consistent any method to check whether a pairwise comparison

matrix is consistent or not?

A B C

A 1 2 1

B 1/2 1 3

C 1 1/3 1

these pairwise comparisons are not consistent

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Random Index to Check the Consistency of an AHP

general idea (with detail given later)

consistency index, CI: an index calculated from a pairwise comparison matrix

random index, RI: an index calculated from randomly generated pairwise comparison matrices

The pairwise comparison matrix is inconsistent if CI/RI > some critical value

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Consistency Index for a Pairwise Comparison Matrix

procedure (for an n-dimensional comparison matrix)

1 Construct a pairwise comparison matrix P

2 Find the normalized weights or priorities for P

3 Calculate = P

4 Calculate ratios for each element of and of , i.e., calculate i/i for i = 1, …, n

5 Calculate average ratio, A = (1/1 + … + n/n)/n

6 Calculate CI = (An)/(n1)

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Example on Consistency Index

84.3331.5833Totals11/31/4c311/3b431acbaCriterion

1.01.01.01.0Totals0.11990.1250.07690.1579c0.27210.3750.23080.2105b0.60790.50.69230.6316a

Average %cbaCriterion

1 3 4

1/ 3 1 3

1/ 4 1/ 3 1

P

1

0.6079

0.2721

0.1199

2

1 3 4 0.6079 1.904

1/ 3 1 3 0.2721 0.8345

1/ 4 1/ 3 1 0.1199 0.3626

= P 3

1

1

1.9043.13136

0.608=

2

2

0.83453.0669

0.2721=

3

3

0.36263.0242

0.1199=

4

53.13136 3.0669 3.0242

3

average ratio,

3.0742

A

=

6 1

3.0742 33 1

CI=

0.0371

A nn

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Random Index and the Criterion of Consistency

RI, the consistency index for a pairwise matrix where each pairwise comparison is randomly generated

RI as a function of n in Table 10-5Table 10-5 Random Index Values for the Comparison of n items

consistent if CI/RI < 0.1 consistent because CI = 0.0371, RI = 0.58 for n = 3

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Final Remarks

Ours is a simplified version of AHP. For example:

AHP is more for a decision problem with hierarchical decisions;

The theory of AHP is related to the maximum eigenvalue and the corresponding eigenvector of a pairwise comparison matrix, something that we have skipped.

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Chapter 10: Homework for AHP

Problem 16

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Chapter 10 Multicriteria Decision-Marking Models