1. ANALYTICAL HIERARCHY PROCESSEX
ANALYTICAL HIERARCHY
PROCESS
层次分析法
A Presentation By Professor Xie Kefan Business School of WUT 武汉理工大学管理学院 谢科范

2. Making a decision when there are multiple objectives or criteria to consider. For example:
Choosing which employment offer to accept.
Picking which computer (or car, etc.) to buy.
Deciding which new product to launch first.
Selecting a site for a new restaurant, hotel, etc.
Rating the best cities in which to live.
Choosing a new software package for your company.

3. A simple way to attack such a decision would be to assign weights to each of the criteria that were to be considered in making the decision.
Then, rank each decision alternative on a scale from 1 (worst) to 10 (best).
Finally, you would multiply the weights times the rankings for each criterion and sum them up.
The alternative with the highest score would be the most preferred.

4. For example, you are in charge of purchasing the next computer for the office. You have to choose between the following three computers:
1. Model A runs an AMD K6-II chip at 400 MHz
2. Model B runs a Celeron chip at 333 MHz
3. Model C runs a Pentium II chip at 450 MHz
The important criteria and their weights are:
Criteria Weight
Price 50%
Speed %
Hard-disk Size 20%
Warranty/Support %

5. Now, rank each of the three models on these four criteria
Now, rank each of the three models on these four criteria. Rank them on a scale from 1 to 10 as described earlier.
=SUM(C4:C7)
=SUMPRODUCT($C$4:$C$7,E4:E7)
Model B has the highest weighted score and thus would be the best computer to purchase.

6. This approach is quite simplistic and there are difficulties in setting the ranking scales on such different criteria.
加权平均
Analytic hierarchy process (AHP) also uses a weighted average approach idea, but it uses a method for assigning ratings (or rankings) and weights that is considered more reliable and consistent.
两两比较
(AHP) is based on pairwise comparisons between the decision alternatives on each of the criteria. 方案
Then, a similar set of comparisons are made to determine the relative importance of each criterion and thus produces the weights.

7. Analytic Hierarchy Process
Multiple criteria
quantitative
qualitative, “intangible”, subjective
provides measures of judgement consistency
derives priorities among criteria and alternatives
“user-friendly” pair-wise comparisons

8. Using AHP 1. Decompose the problem into a hierarchy
2. Make pairwise comparisons and establish priorities among the elements in the hierarchy
3. Synthesise the results (to obtain the overall ranking of alternatives to goal)
4. Evaluate the consistency of judgement
总排序

9. The basic procedure is as follows:
1. Develop the ratings for each decision alternative for each criterion by
developing a pairwise comparison matrix for each criterion
比较矩阵
normalizing the resulting matrix
正则化
averaging the values in each row to get the corresponding rating
calculating and checking the consistency ratio
一致性比率

10. 2. Develop the weights for the criteria by
developing a pairwise comparison matrix for each criterion
normalizing the resulting matrix
averaging the values in each row to get the corresponding rating
calculating and checking the consistency ratio
3. Calculate the weighted average rating for each decision alternative. Choose the one with the highest score.

11. Revenue Technology Corporation (RTC) PRAISE Strategic Solutions (PSS)
Consider the following example:
Sleepwell Hotels is looking for some help in selecting the “best” revenue management software package from among several vendors. The director of revenue management for this chain of hotels has been given this task.
供应商
Three vendors have been identified whose software meets the following basic needs:
Revenue Technology Corporation (RTC)
PRAISE Strategic Solutions (PSS)
El Cheapo (EC)

12. The important criteria are:
1. The total cost of the installed system
2. The follow-up service provided over the coming year
3. The sophistication of the underlying math engines
4. The amount of customization for Sleepwell

13. The first step in the AHP procedure is to make pairwise comparisons between the vendors for each criterion. Here is the standard scale for making these comparisons:
DESCRIPTION
1 Equally preferred
3 Moderately preferred
5 Strongly preferred
7 Very strongly preferred
9 Extremely strongly preferred
RATING
Values 2, 4, 6, or 8 may also be assigned and represent preferences halfway between the integers on either side.

14. Start with the total cost criterion and generate the following data in a spreadsheet:
The vendor in the row is being compared to the vendor in the column.
A value between 1 and 9 indicates that the vendor in the row is preferred to the vendor in the column.
If the vendor in the column is preferred to the vendor in the row, then the inverse of the rating is given.

15. The next step is to normalize the matrix
The next step is to normalize the matrix. This is done by totaling the numbers in each column.
=SUM(B4:B6)
Each entry in the column is then divided by the column sum to yield its normalized score.
=B4/B8
=AVERAGE(B12:D12)
The average is calculated for the “Total Cost” criterion.
Highest average score

16. Now, calculate the consistency ratio and check its value
Now, calculate the consistency ratio and check its value. The purpose for doing this is to make sure that the original preference ratings were consistent.
There are 3 steps to arrive at the consistency ratio:
1. Calculate the consistency measure for each vendor.
2. Calculate the consistency index (CI).
3. Calculate the consistency ratio (CR=CI/RI where RI is a random index).
To calculate the consistency measure, we can take advantage of Excel’s matrix multiplication function =MMULT().

17. Multiply the average rating for each vendor times the scores in the first row one-at-a-time, sum these products up and divide this sum by the average rating for the first vendor.
=MMULT(B4:D4,$E$12:$E$14)/E12
=(AVERAGE(F12:F14)-3)/2
Provided by AHP (see next slide)
=F16/F18)

18. Approximation of the Consistency Index
1. Multiply each row of the pairwise comparison matrix by the corresponding weight.
2. Divide of sum of the row entries by the corresponding weight.
3. Compute the average of the values from step 2, denote it by Lmax.
4. The approximate CI is

19. RANDOM INDEX N
Random Index (RI)
the CI of a randomly-generated pairwise comparison matrix

20. If we are perfectly consistent, then the consistency measures will equal n and therefore, the CIs will be equal to zero and so will the consistency ratio.
If this ratio is very large (Saaty suggests > 0.10), then we are not consistent enough and the best thing to do is go back and revise the comparisons.
Now, continue for the other three criteria. You can easily do this by copying the “Total Cost” sheet into three other sheets (“Service,” “Sophistication,” and “Custom”) and then simply changing the pairwise comparisons.

21. Consistency ratio for “Service.”

22. Consistency ratio for “Sophistication.”

23. Consistency ratio for “Customization.”

24. In all three cases, the CR value ranges from 0. 0 to 0
In all three cases, the CR value ranges from 0.0 to which means that we are being consistent.
Note also that PSS is the winner on the Service criterion, RTC and PSS are tied for the best in terms of Sophistication, and PSS is considered the best on Customization.
All of this work concludes the first step in the procedure. The next step is to use similar pairwise comparisons to determine the appropriate weights for each of the criteria.
The process is the same in that we make comparisons, except that now we make the comparisons between the criteria not the vendors.

25. Consistency ratio for weights on criterion.
=1/D5
=1/E5
=1/E6
=1/D4
=1/E4
=AVERAGE(B12:E12)
=SUM(B4:B7)
=MMULT(B4:E4,$F$12:$F$15)/F12
=B4/B8
=(AVERAGE(G12:G15)-4)/3
=G16/G18)

26. The final step is to calculate the weighted average ratings of each decision alternative and use the results to decide from which vendor to purchase the software.
=WEIGHTS!F12
=TOTAL COST!E12
=TOTAL COST!E13
=TOTAL COST!E14
=SERVICE!E12
=SERVICE!E13
=SERVICE!E14
=SOPHISTICATION!E12
=SOPHISTICATION!E13
=SOPHISTICATION!E14
=CUSTOM!E12
=CUSTOM!E13
=CUSTOM!E14
=SUMPRODUCT($B$3:$B$6,C3:C6)
These results are pulled from all the other worksheets. From these results, we find that RTC barely edges out PSS for the software contract.

27. The mathematics of AHP
Suppose we already know the weights [w1, w2, w3, wn] of the n criteria and we form the following n x n pairwise-ratio matrix:

28. This pairwise-ratio matrix A and the vector of weights satisfy the following equation:

29. This equation is of the form: A w = l w
So w is an eigenvector of matrix A corresponding to eigenvalue l.
(In fact, l is the only non-zero eigenvalue, and w the unique eigenvector.)
Now, if we only know A, but not w, we can find what w is by solving for the eigenvalues and eigenvectors of A.
特征向量

30. Computing the weights for AHP
Eigenvector Method:
1. Find largest eigenvalue of the pairwise comparison matrix
2. Find corresponding eigenvector
Approximate Method:
1. Normalise each column (i.e. divide each entry by its column total)
2. The average values of row i in the normalised matrix is the estimate for weight i.

31. Consistency Index reflects the consistency of one’s judgement
CI = . lmax - n .
n - 1
Random Index (RI)
the CI of a randomly-generated pairwise comparison matrix
Tabulated by size of matrix:
. n RI .
2 0.0

32. Consistency Ratio CR = CI / RI
In practice, a CR of 0.1 or below is considered acceptable.
Any higher value at any level indicate that the judgements warrant re-examination.

33. AHP - Summary Easy to use Widely used Cost/Benefit Analysis
Vendor Selection
Strategic Planning

34. Case Study
Evaluating Banana and Mango Industries Location Option in Taiwan Using Analytic Hierarchy Process (AHP)

35. 1. INTRODUCTION
seg 7410
Mango and banana - source of revenue for the rural economy in Taiwan and Senegal
In the fruit industry the location can significantly improve the level of productivity
Agriculture industries location characterized by multiple and
conflicting factors
Use Analytic Hierarchy Process (AHP) multicriteria decision Model

36. 2.OBJECTIVES
To identify the most important factors (criteria) to consider in
selecting location for banana and mango in Taiwan;
To illustrate the use of Analytic Hierarchy Process to evaluate the
best alternative location for banana and mango industries in Taiwan.

37. 3.METHODOLOGY (banana & mango industries location)
1) To list the factors that can influence banana and mango industry location, obtained from a literature reviewing and interviewing experts on banana and mango.
Literature
Review
Experts
Interview
To list
factors

38. METHODOLOGY (banana and mango industries location)
2) Identify the respondents for the questionnaires: Academicians or Researchers and training officers of farmers association (TOFA) expert on banana and mango.
Identify in internet
Academician and
Researchers (50)
Identify the TOFA
Farmers associations
Name list (70)
Respondents

39. METHODOLOGY (banana & mango industries location)
3) Send by mail the 1st questionnaire with return envelop to all the 120 respondents, to identify the most important factors that should be considered for banana and mango industry location in Taiwan.
Table I. Questionnaire 1
Not
important
Very
Remarks
Accessibility to the market (town)
Transportation cost
Labor
Accessibility to airport or harbor
Water availability
Soil quality
Climate (temperature, wind, photoperiod, humidity, rain)
Topography of the land
Drainage
Material supply
Infrastructures
Law (regulation)
Traffic
Technology (research & extension services)
Other important factors

40. Table II. Results of the questionnaire 1a (banana industry location)
METHODOLOGY
Table II. Results of the questionnaire 1a (banana industry location)
Not important
Important
Very important
Total weight
Res. W.
Res
Soil infection 1 15 47 94
109**
Climate 2 17 44 88
105**
Drainage 26 35 70
96*
Soil quality 4 33 66
92*
Labor 31 30 60
91*
Technology 6 52 83
Water availability 24 48 81
Traffic 7 21 42 77
Topography of the land 37 20 40
Infrastructures 9 10 74
Transportation cost 8 39 16 32 71
Accessibility to airport or harbor 23 63
Accessibility to the market (town) 19 28
Material supply 13 41 18 59
Law 38 58
Res.= number of respondent, W. =weight
Weight (Not important= 0 , Important=1, Very important=2)
* one of the 5 highest values of importance

41. Table III. Results of Questionnaire 1b (mango industry location)
METHODOLOGY
Table III. Results of Questionnaire 1b (mango industry location)
Not important
Important
Very important
Total weight
Res. W.
Res
Climate 1 13 48 96
109*
Technology 6 18 38 76
94*
Soil quality 7 22 33 66
88*
Drainage 34 27 54
Labor 25 30 60
85*
Soil infection 14 26 52 74
Water availability 10 19 71
Traffic 5 36 70
Topography of the land
Transportation cost 15 31 16 32 63
Accessibility to the market (town) 28 20
Law 24 44
Infrastructure 11 42 9
Material supply 8
Accessibility to airport or harbor 23
Res.= number of respondent, W. =weight
Weight (Not important= 0 , Important=1, Very important=2)
* one of the 5 highest values

42. METHODOLOGY (banana industry location)
4a) 45 days after , another questionnaire with return envelope was sent to all the 63 respondents that had replied to the 1st questionnaire to establish:
- pair wise comparisons of the factors .
Climate
Soil infection
Soil quality
Labor
Drainage

43. METHODOLOGY (mango industry location)
4a) 45 days after, another questionnaire with return envelope was sent to all the 62 respondents that had replied to the 1st questionnaire to establish:
- pair wise comparisons of the factors .
Climate
Technology
Soil quality
Labor
Drainage

44. METHODOLOGY (banana industry location)
Central Area
South Central Area
Eastern
Area
Southern Area

45. METHODOLOGY (mango industry location)

46. 4b) pair wise comparisons of the sites according to each factor
METHODOLOGY (banana industry location)
4b) pair wise comparisons of the sites according to each factor
Selection of the best
site for banana industry
Climate
Soil infection
Soil quality
Drainage
Labor
Centrale Area
(Taichung, Chuanghwa
And Nanto)
Central South
Area
( Chiai and Tainan)
South Area
(Kaohsiung and
Pingtung)
East Area
(Taitung)

47. 4b) pair wise comparisons of the sites according to each factor
METHODOLOGY (mango industry location)
4b) pair wise comparisons of the sites according to each factor
Selection of the best
site for mango industry
Climate
Technology
Soil quality
Drainage
Labor
Tainan
Pingtung
Kaohsiung
Taitung

48. 4.RESULTS OF THE SURVEY- Banana Industry
Table V . Comparison Weights of Importance of the Criteria By the
TOFA and the Experts for Banana Industry Location in Taiwan
Climate
Soil infection
Soil quality
Drainage
Labor
 TOFA
(CR = 0.02)
Experts
(CR = 0.01)
Overall
 0.194
0.178
0.185
 0.235
0.391
0.347
 0.246
0.150
0.175
 0.208
0.176
0.184
 0.116
0.105
0.109

49. RESULTS OF THE SURVEY
Table VI. Comparison Weights of Importance of Four Sites According to
the Factor Climate by the TOFA and the Experts for Banana Industry
Location in Taiwan
Central area
South central area
Southern area
Eastern area
 TOFA
(CR = 0.03)
Experts
Overall
 0.180
0.187
0.186
0.246
0.260
0.254
 0.410
0.454
0.433
 0.164
0.099
0.127

50. RESULTS OF THE SURVEY
Table VII. Comparison Weights of Importance of Four Sites According
to the Factor Soil Infection by the TOFA and the Experts for Banana
Industry Location in Taiwan
Central area
South central area
Southern area
Eastern area
 TOFA
(CR = 0.01)
Experts
(CR = 0.00)
Overall
 0.284
0.211
0.232
 0.217
0.208
0.212
 0.259
0.233
0.242
0.239
0.347
0.314

51. RESULTS OF THE SURVEY
Table VIII. Comparison Weights of Importance of Four Sites According to the
Factor Soil Quality by the TOFA and the Experts for Banana Industry in Taiwan
Central area
South central area
Southern area
Eastern area
 TOFA
(CR = 0.00)
Experts
(CR = 0.02)
Overall
(CR = 0.01)
 0.237
0.203
0.212
 0.246
0.276
0.267
 0.356
0.368
0.366
 0.161
0.153
0.155

52. Table IX. Comparison Weights of Importance of Four Sites According
RESULTS OF THE SURVEY
Table IX. Comparison Weights of Importance of Four Sites According
to the Factor Drainage by the TOFA and the Experts for Banana
Industry in Taiwan
Central area
South central area
Southern area
Eastern area
 TOFA
(CR = 0.00)
Experts
(CR = 0.01)
Overall
 0.164
0.292
0.258
 0.277
0.205
0.226
 0.354
0.263
0.290
 0.204
0.241

53. Table X. Comparison Weights of Importance of Four Sites According
RESULTS OF THE SURVEY
Table X. Comparison Weights of Importance of Four Sites According
to the Factor Labor by the TOFA and the Experts for Banana Industry
in Taiwan
Central area
South central area
Southern area
Eastern area
 TOFA
(CR = 0.04)
Experts
(CR = 0.01)
Overall
 0.160
0.211
0.196
 0.182
0.235
 0.394
0.312
0.335
 0.264
0.241
0.258

54. RESULTS OF THE SURVEY (summary)
Comparison of Criteria
Climate
Soil infection
Soil quality
Drainage
Labor
TOFA 
Experts
Overall
Comparison of Sites according to each criteria (Summary)
TOFA
Experts
overall
In term of
Climate
Central area
South C. area
Southern area 
Eastern area
  In term of
Soil infection
 In term of
Soil quality   
 In term of Drainage
Labor

55. Obtaining Composite Weights
Overall priority of site: j =  wi (qij)
i= 1
wi weight of criteria, qij weight of the site j with respect to criteria i
Table XI . Evaluation of the Composite Priority Weights for the Four Alternative Sites for Banana Industry in Taiwan
Central area
south central area
Southern area
Eastern area
TOFA
Experts
Overall
0.213
0.220
0.221
0.238
0.230
0.232
0.348
0.306
0.318
0.201
0.244
0.229
The preferable site is the one with the highest overall priority weight, in this case is the South area.
The statistics of banana production in Taiwan show that area giving the highest volume.

56. 5.RESULTS OF THE SURVEY- Mango Industry
Table XII. Comparison Weights of Importance of Criteria by the TOFA and the
Experts for Mango Industry Location in Taiwan
Climate
Technology
Soil quality
Drainage
Labor
 TOFA
(CR = 0.05)
Experts
(CR = 0.01)
Overall
(CR = 0.02)
 0.308
0.302
0.310
 0.257
0.199
0.224
 0.229
0.135
0.164
 0.119
0.154
0.142
 0.080
0.210
0.159

57. RESULTS OF THE SURVEY
Table XIII. Comparison Weighs of Importance of Sites According
to the Factor Climate by the TOFA and the Experts for Mango
Industry Location in Taiwan
Tainan
Pingtung
Kaohsiung
Taitung
 TOFA
(CR = 0.00)
Experts
(CR = 0.04)
Overall
(CR = 0.02)
 0.450
0.195
0.277
 0.261
0.518
0.427
 0.185
0.194
 0.103
0.093
0.101

58. RESULTS OF THE SURVEY
Table XIV. Comparison Weighs of Importance of Four Sites According to the
Factor Technology by the TOFA and the Experts for Mango Industry Location
in Taiwan
Tainan
Pingtung
Kaohsiung
Taitung
 TOFA
(CR = 0.04)
Experts
(CR = 0.01)
Overall
 0.440
0.316
0.357
 0.268
0.378
0.339
 0.167
0.192
0.185
 0.124
0.114
0.119

59. RESULTS OF THE SURVEY
Table XV. Comparison Weighs of Importance of Four Sites According to
the Factor Soil Quality by the TOFA and the Experts for Mango Industry
Location in Taiwan
Tainan
Pingtung
Kaohsiung
Taitung
 TOFA
(CR = 0.05)
Experts
(CR = 0.01)
Overall
(CR = 0.02)
 0.441
0.335
0.370
 0.233
0.318
0.288
 0.217
0.197
0.206
 0.110
0.150
0.136

60. RESULTS OF THE SURVEY
Table XVI. Comparison Weighs of Importance of Four Sites According
to the Factor Drainage by the TOFA and the Experts for Mango Industry
Location in Taiwan
Tainan
Pingtung
Kaohsiung
Taitung
 TOFA
(CR = 0.02)
Experts
(CR = 0.00)
Overall
 0.324
0.325
0.327
 0.295
0.302
 0.264
0.185
0.211
 0.117
0.188
0.161

61. RESULTS OF THE SURVEY
Table XVII. Comparison Weighs of Importance of Sites four According
to the Factor Labor by the TOFA and the Experts for Mango Industry
Location in Taiwan
Tainan
Pingtung
Kaohsiung
Taitung
 TOFA
(CR = 0.01)
Experts
Overall
(CR = 0.00)
 0.346
0.313
0.328
 0.285
0.291
0.292
 0.252
0.179
0.203
 0.117
0.217
0.177

62. RESULTS OF THE SURVEY (summary)
Comparison of Criteria
Climate
Technology
Soil quality
Drainage
Labor
TOFA 
Experts
Overall
Comparison of Sites according to each criteria (Summary)
TOFA
Experts
overall
In term of
Climate
Tainan 
Pingtung
Kaohsiung
Taitung   
Technology
 In term of
Soil quality
Drainage
Labor

63. Obtaining Composite Weights
Overall priority of site: j =  wi (qij)
i= 1
wi weight of criteria, qij weight of the site j with respect to criteria i
Table XIII. Evaluation of the Composite Priority Weights for the Four
Alternative Sites for Mango Industry location in Taiwan
Tainan
Pingtung
Kaohsiung
Taitung
TOFA
Experts
Overall
0.422
0.283
0.325
0.263
0.382
0.345
0.203
0.189
0.198
0.113
0.146
0.131
The preferable site is the one with the highest overall priority weight, in this case is Pingtung and
Tainan with very close weights .
The statistics of mango production in Taiwan show those 2 areas giving the highest volumes

64. 6.CONCLUSION & SUGGESTIONS
This study demonstrated AHP as a decision tool in agribusiness,
particularly in locating banana and mango industries.
The results confirm tropical origin of mango tree and banana plant by eliciting Pingtung (for mango industry) and the southern area (for banana industry) as the best locations.
The factors soil infection, soil quality and labor were also reported by Hwang (1992) as important factor for banana industry in Taiwan.