analityc hierarchy Process

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Chapter 15Chapter 15Multicriteria Decision ProblemsMulticriteria Decision Problems

Goal ProgrammingGoal Programming Goal Programming: Formulation Goal Programming: Formulation

and Graphical Solutionand Graphical Solution Scoring ModelsScoring Models Analytic Hierarchy Process (AHP)Analytic Hierarchy Process (AHP) Establishing Priorities Using AHPEstablishing Priorities Using AHP Using AHP to Develop an Overall Priority Using AHP to Develop an Overall Priority

RankingRanking


Analytic Hierarchy ProcessAnalytic Hierarchy Process

Analytic Hierarchy Process (AHP)Analytic Hierarchy Process (AHP), Adalah , Adalah prosedur yang dirancang untuk meng-angkakan prosedur yang dirancang untuk meng-angkakan pertimbangan manajerial kepentingan relatif dari pertimbangan manajerial kepentingan relatif dari setiap kriteria-kriteria yang memiliki arah dan setiap kriteria-kriteria yang memiliki arah dan kegunaan berbeda dalam pengambilan kegunaan berbeda dalam pengambilan keputusan. keputusan.



Step 1: Daftarkan semua tujuan, kriteria dan Step 1: Daftarkan semua tujuan, kriteria dan laternatif keputusan.laternatif keputusan.

Step 2: Buatlah matrix perbandingan pairwise Step 2: Buatlah matrix perbandingan pairwise (berpasangan) (berpasangan)

Tetapkan nilai tingkat kepentingan relatif Tetapkan nilai tingkat kepentingan relatif antara setiap pasangan dari alternatif keputusan. antara setiap pasangan dari alternatif keputusan. The matrix lists the alternatives horizontally and The matrix lists the alternatives horizontally and vertically and has the numerical ratings comparing vertically and has the numerical ratings comparing the horizontal (first) alternative with the vertical the horizontal (first) alternative with the vertical (second) alternative.(second) alternative.

nilai tingkat adalah sbb:nilai tingkat adalah sbb:

. . . continued. . . continued

------- For each criterion, perform steps 2 through 5 -------------- For each criterion, perform steps 2 through 5 -------------- For each criterion, perform steps 2 through 5 -------------- For each criterion, perform steps 2 through 5 -------



Step 2: Pairwise Comparison Matrix (continued)Step 2: Pairwise Comparison Matrix (continued)

Compared to the secondCompared to the secondalternative, the first alternative isalternative, the first alternative is: : Numerical Numerical ratingrating

Dipilih ekstrem Dipilih ekstrem 99

Dipilih secara sangat kuat 7Dipilih secara sangat kuat 7

dipilih kuat dipilih kuat 55

dipilih moderat dipilih moderat 33

dipilih secara sama 1dipilih secara sama 1



Step 2: Pairwise Comparison Matrix (continued)Step 2: Pairwise Comparison Matrix (continued)

Nilai tingkat angka antara yaitu 8, 6, 4, 2 Nilai tingkat angka antara yaitu 8, 6, 4, 2 dapat di ikutkan. A reciprocal rating (i.e. 1/9, dapat di ikutkan. A reciprocal rating (i.e. 1/9, 1/8, etc.) is assigned when the second 1/8, etc.) is assigned when the second alternative is preferred to the first. The value of alternative is preferred to the first. The value of 1 is always assigned when comparing an 1 is always assigned when comparing an alternative with itself. alternative with itself.



Step 3: Develop a Normalized MatrixStep 3: Develop a Normalized Matrix

bagilah setiap angka di dalam sebuah bagilah setiap angka di dalam sebuah kolom dari matrix perbandingan pairwise kolom dari matrix perbandingan pairwise (berpasangan) dengan jumlah kolomnya.(berpasangan) dengan jumlah kolomnya.

Step 4: Develop the Priority VectorStep 4: Develop the Priority Vector

Rata ratakan setiap baris dalam Rata ratakan setiap baris dalam normalized matrix. Hal ini membentuk vector normalized matrix. Hal ini membentuk vector prioritas dari pilihan alternatif dikaitkan prioritas dari pilihan alternatif dikaitkan dengan kriteria tertentu, jumlah vector adalah dengan kriteria tertentu, jumlah vector adalah 1. 1.



Step 5: Calculate a Consistency RatioStep 5: Calculate a Consistency RatioKonsistensi masukan subyektif dalam Konsistensi masukan subyektif dalam

pairwise matrix dapat diukur dengan pairwise matrix dapat diukur dengan menghitung ratio konsistensi. A consistency menghitung ratio konsistensi. A consistency ratio of less than .1 is good. Untuk rasio yang ratio of less than .1 is good. Untuk rasio yang lebih dari 0.1 maka harus di evaluasi kembali.lebih dari 0.1 maka harus di evaluasi kembali.

Step 6: Develop a Priority MatrixStep 6: Develop a Priority MatrixSetelah steps 2 through 5 telah Setelah steps 2 through 5 telah

dikerjakan untuk semua kriteria, hasil dari step dikerjakan untuk semua kriteria, hasil dari step 4 dijumlahkan dalam sebuah matrix priority 4 dijumlahkan dalam sebuah matrix priority dengan mendaftarkan pilihan keputusan dengan mendaftarkan pilihan keputusan (alternatif keputusan) secara horisontal dan (alternatif keputusan) secara horisontal dan kriteria secara vertikal. Masukan dalam kolom kriteria secara vertikal. Masukan dalam kolom adalah vektor vektor prioritas dari setiap adalah vektor vektor prioritas dari setiap kriteria. kriteria.



Step 7: Develop a Criteria Pairwise Development Step 7: Develop a Criteria Pairwise Development Matrix Matrix

Sama dengan bagian awal saja. This is Sama dengan bagian awal saja. This is done in the same manner as that used to done in the same manner as that used to construct alternative pairwise comparison construct alternative pairwise comparison matrices by using subjective ratings (step 2). matrices by using subjective ratings (step 2). Similarly, normalize the matrix (step 3) and Similarly, normalize the matrix (step 3) and develop a criteria priority vector (step 4). develop a criteria priority vector (step 4).

Step 8: Develop an Overall Priority VectorStep 8: Develop an Overall Priority Vector

Multiply the criteria priority vector (from Multiply the criteria priority vector (from step 7) by the priority matrix (from step 6).step 7) by the priority matrix (from step 6).


Determining the Consistency RatioDetermining the Consistency Ratio

Step 1:Step 1: For each row of the pairwise comparison For each row of the pairwise comparison

matrix, determine a weighted sum by summing matrix, determine a weighted sum by summing the multiples of the entries by the priority of its the multiples of the entries by the priority of its corresponding (column) alternative.corresponding (column) alternative.

Step 2:Step 2: For each row, divide its weighted sum by For each row, divide its weighted sum by

the priority of its corresponding (row) alternative.the priority of its corresponding (row) alternative. Step 3:Step 3:

Determine the average, Determine the average, maxmax, of the results , of the results of step 2.of step 2.


Determining the Consistency RatioDetermining the Consistency Ratio

Step 4:Step 4:

Compute the consistency index, CI, of the Compute the consistency index, CI, of the nn alternatives by: CI = (alternatives by: CI = (maxmax - - nn)/()/(nn - 1). - 1).

Step 5:Step 5:

Determine the random index, RI, as follows:Determine the random index, RI, as follows:

Number of RandomNumber of Random Number of RandomNumber of Random Alternative (Alternative (nn)) Index (RI)Index (RI) Alternative (Alternative (nn)) Index (RI)Index (RI)

3 0.583 0.58 6 6 1.24 1.24 4 0.904 0.90 7 7 1.32 1.32 5 1.125 1.12 8 8 1.41 1.41

Step 6:Step 6:

Compute the consistency ratio: CR = CR/RI.Compute the consistency ratio: CR = CR/RI.


Example: Gill GlassExample: Gill Glass

Designer Gill Glass must decide which of Designer Gill Glass must decide which of three manufacturers will develop his three manufacturers will develop his "signature" toothbrushes. Three factors seem "signature" toothbrushes. Three factors seem important to Gill: (1) his costs; (2) reliability of important to Gill: (1) his costs; (2) reliability of the product; and, (3) delivery time of the the product; and, (3) delivery time of the orders.orders.

The three manufacturers are Cornell The three manufacturers are Cornell Industries, Brush Pik, and Picobuy. Cornell Industries, Brush Pik, and Picobuy. Cornell Industries will sell toothbrushes to Gill Glass for Industries will sell toothbrushes to Gill Glass for $100 per gross, Brush Pik for $80 per gross, and $100 per gross, Brush Pik for $80 per gross, and Picobuy for $144 per gross. Gill has decided Picobuy for $144 per gross. Gill has decided that in terms of price, Brush Pik is moderately that in terms of price, Brush Pik is moderately preferred to Cornell and very strongly preferred preferred to Cornell and very strongly preferred to Picobuy. In turn Cornell is strongly to very to Picobuy. In turn Cornell is strongly to very strongly preferred to Picobuy.strongly preferred to Picobuy.



Hierarchy for the Manufacturer Selection Hierarchy for the Manufacturer Selection ProblemProblem

Select the Best Toothbrush ManufacturerSelect the Best Toothbrush ManufacturerSelect the Best Toothbrush ManufacturerSelect the Best Toothbrush Manufacturer

CostCost CostCost ReliabilityReliabilityReliabilityReliability Delivery TimeDelivery TimeDelivery TimeDelivery Time

CornellCornellBrush PikBrush PikPicobuyPicobuy






Overall GoalOverall Goal

CriteriaCriteria

DecisionDecisionAlternativesAlternatives



Forming the Pairwise Comparison Matrix For CostForming the Pairwise Comparison Matrix For Cost

•Since Brush Pik is moderately preferred to Since Brush Pik is moderately preferred to Cornell, Cornell's entry in the Brush Pik row is Cornell, Cornell's entry in the Brush Pik row is 3 and Brush Pik's entry in the Cornell row is 3 and Brush Pik's entry in the Cornell row is 1/3.1/3.

•Since Brush Pik is very strongly preferred to Since Brush Pik is very strongly preferred to Picobuy, Picobuy's entry in the Brush Pik row Picobuy, Picobuy's entry in the Brush Pik row is 7 and Brush Pik's entry in the Picobuy row is is 7 and Brush Pik's entry in the Picobuy row is 1/7.1/7.

•Since Cornell is strongly to very strongly Since Cornell is strongly to very strongly preferred to Picobuy, Picobuy's entry in the preferred to Picobuy, Picobuy's entry in the Cornell row is 6 and Cornell's entry in the Cornell row is 6 and Cornell's entry in the Picobuy row is 1/6.Picobuy row is 1/6.



Pairwise Comparison Matrix for CostPairwise Comparison Matrix for Cost

Cornell Brush Pik PicobuyCornell Brush Pik Picobuy

CornellCornell 1 1/3 1 1/3 6 6

Brush PikBrush Pik 3 3 1 1 7 7

PicobuyPicobuy 1/6 1/6 1/7 1/7 1 1



Normalized Matrix for CostNormalized Matrix for Cost

Bagi setiap masukan dalam pairwise Bagi setiap masukan dalam pairwise comparison matrix dengan jumlah kolom yang comparison matrix dengan jumlah kolom yang terkait. For example, for Cornell the column sum terkait. For example, for Cornell the column sum = 1 + 3 + 1/6 = 25/6. This gives:= 1 + 3 + 1/6 = 25/6. This gives:


CornellCornell 6/25 7/31 6/25 7/31 6/14 6/14

Brush PikBrush Pik 18/25 21/31 18/25 21/31 7/14 7/14

PicobuyPicobuy 1/25 3/31 1/25 3/31 1/14 1/14


Priority Vector For CostPriority Vector For Cost

The priority vector is determined by The priority vector is determined by averaging the row entries in the normalized averaging the row entries in the normalized matrix. Converting to decimals we get:matrix. Converting to decimals we get:

Cornell: ( 6/25 + 7/31 + 6/14)/3 Cornell: ( 6/25 + 7/31 + 6/14)/3 = .298 = .298

Brush Pik: (18/25 + 21/31 + 7/14)/3 Brush Pik: (18/25 + 21/31 + 7/14)/3 = .632 = .632

Picobuy: ( 1/25 + 3/31 + 1/14)/3 Picobuy: ( 1/25 + 3/31 + 1/14)/3 = .069 = .069



Checking ConsistencyChecking Consistency

•Multiply each column of the pairwise Multiply each column of the pairwise comparison matrix by its priority:comparison matrix by its priority:

1 1/3 1 1/3 6 .923 6 .923

.298 3 + .632 1 + .069 7 = .298 3 + .632 1 + .069 7 = 2.009 2.009

1/6 1/7 1/6 1/7 1 .209 1 .209

•Divide these number by their priorities to get:Divide these number by their priorities to get:

.923/.298 = 3.097.923/.298 = 3.097

2.009/.632 = 3.1792.009/.632 = 3.179

.209/.069 = 3.029.209/.069 = 3.029





•Average the above results to get Average the above results to get maxmax..

maxmax = (3.097 + 3.179 + 3.029)/3 = 3.102 = (3.097 + 3.179 + 3.029)/3 = 3.102

•Compute the consistence index, CI, for two Compute the consistence index, CI, for two terms.terms.

CI = (CI = (maxmax - - nn)/()/(nn - 1) = (3.102 - 3)/2 - 1) = (3.102 - 3)/2 = .051= .051

•Compute the consistency ratio, CR, by CI/RI, Compute the consistency ratio, CR, by CI/RI, where RI = .58 for 3 factors:where RI = .58 for 3 factors:

CR = CI/RI = .051/.58 = .088CR = CI/RI = .051/.58 = .088

CR nya kurang dari 0.1 maka konsisten. CR nya kurang dari 0.1 maka konsisten.



Gill Glass has determined that for Gill Glass has determined that for reliabilityreliability, Cornell is very strongly preferable to , Cornell is very strongly preferable to Brush Pik and equally preferable to Picobuy. Brush Pik and equally preferable to Picobuy. Also, Picobuy is strongly preferable to Brush Pik.Also, Picobuy is strongly preferable to Brush Pik.



Pairwise Comparison Matrix for ReliabilityPairwise Comparison Matrix for Reliability


CornellCornell 1 7 1 7 2 2

Brush PikBrush Pik 1/7 1/7 1 1 5 5

PicobuyPicobuy 1/2 1/2 1/5 1/5 1 1



Normalized Matrix for ReliabilityNormalized Matrix for Reliability

Divide each entry in the pairwise Divide each entry in the pairwise comparison matrix by its corresponding column comparison matrix by its corresponding column sum. For example, for Cornell the column sum = sum. For example, for Cornell the column sum = 1 + 1/7 + 1/2 = 23/14. This gives:1 + 1/7 + 1/2 = 23/14. This gives:


CornellCornell 14/23 35/41 14/23 35/41 2/8 2/8


PicobuyPicobuy 7/23 1/41 7/23 1/41 1/8 1/8



Priority Vector For ReliabilityPriority Vector For Reliability

The priority vector is determined by The priority vector is determined by averaging the row entries in the normalized matrix. averaging the row entries in the normalized matrix. Converting to decimals we get: Converting to decimals we get:

Cornell: (14/23 + 35/41 + 2/8)/3 = .571 Cornell: (14/23 + 35/41 + 2/8)/3 = .571

Brush Pik: ( 2/23 + 5/41 + 5/8)/3 = .278 Brush Pik: ( 2/23 + 5/41 + 5/8)/3 = .278

Picobuy: ( 7/23 + 1/41 + 1/8)/3 = .151 Picobuy: ( 7/23 + 1/41 + 1/8)/3 = .151


Gill Glass’ responses to reliability could be Gill Glass’ responses to reliability could be checked for consistency in the same manner as checked for consistency in the same manner as was cost.was cost.



Gill Glass has determined that for Gill Glass has determined that for delivery delivery timetime, Cornell is equally preferable to Picobuy. , Cornell is equally preferable to Picobuy. Both Cornell and Picobuy are very strongly to Both Cornell and Picobuy are very strongly to extremely preferable to Brush Pik.extremely preferable to Brush Pik.



Pairwise Comparison Matrix for Delivery TimePairwise Comparison Matrix for Delivery Time


CornellCornell 1 8 1 8 1 1

Brush PikBrush Pik 1/8 1/8 1 1 1/8 1/8

PicobuyPicobuy 1 1 8 8 1 1



Normalized Matrix for Delivery TimeNormalized Matrix for Delivery Time

Divide each entry in the pairwise Divide each entry in the pairwise comparison matrix by its corresponding column comparison matrix by its corresponding column sum. sum.


CornellCornell 8/17 8/17 8/17 8/17 8/17 8/17


PicobuyPicobuy 8/17 8/17 8/17 8/17 8/17 8/17



Priority Vector For Delivery TimePriority Vector For Delivery Time


Cornell: (8/17 + 8/17 + 8/17)/3 = .471 Cornell: (8/17 + 8/17 + 8/17)/3 = .471

Brush Pik: (1/17 + 1/17 + 1/17)/3 = .059 Brush Pik: (1/17 + 1/17 + 1/17)/3 = .059

Picobuy: (8/17 + 8/17 + 8/17)/3 = .471 Picobuy: (8/17 + 8/17 + 8/17)/3 = .471


Gill Glass’ responses to delivery time could Gill Glass’ responses to delivery time could be checked for consistency in the same manner be checked for consistency in the same manner as was cost.as was cost.



The accounting department has The accounting department has determined that in terms of determined that in terms of criteriacriteria, cost is , cost is extremely preferable to delivery time and very extremely preferable to delivery time and very strongly preferable to reliability, and that strongly preferable to reliability, and that reliability is very strongly preferable to delivery reliability is very strongly preferable to delivery time.time.



Pairwise Comparison Matrix for CriteriaPairwise Comparison Matrix for Criteria

Cost Reliability DeliveryCost Reliability Delivery

CostCost 1 7 1 7 9 9

ReliabilityReliability 1/7 1/7 1 1 7 7

DeliveryDelivery 1/9 1/9 1/7 1/7 1 1



Normalized Matrix for CriteriaNormalized Matrix for Criteria

Divide each entry in the pairwise Divide each entry in the pairwise comparison matrix by its corresponding column comparison matrix by its corresponding column sum.sum.

Cost Reliability Cost Reliability DeliveryDelivery

CostCost 63/79 49/57 63/79 49/57 9/179/17

ReliabilityReliability 9/79 7/57 9/79 7/57 7/177/17

DeliveryDelivery 7/79 1/57 7/79 1/57 1/17 1/17



Priority Vector For CriteriaPriority Vector For Criteria


Cost: Cost: (63/79 + 49/57 + 9/17)/3 (63/79 + 49/57 + 9/17)/3 = .729 = .729

Reliability: Reliability: ( 9/79 + 7/57 + 7/17)/3 ( 9/79 + 7/57 + 7/17)/3 = .216 = .216

Delivery: Delivery: ( 7/79 + 1/57 + 1/17)/3 ( 7/79 + 1/57 + 1/17)/3 = .055= .055


Overall Priority VectorOverall Priority Vector

The overall priorities are determined by The overall priorities are determined by multiplying the priority vector of the criteria by multiplying the priority vector of the criteria by the priorities for each decision alternative for the priorities for each decision alternative for each objective.each objective.

Priority VectorPriority Vector

for Criteriafor Criteria [ .729 .216 [ .729 .216 .055 ] .055 ]

Cost Reliability DeliveryCost Reliability Delivery

Cornell Cornell .298 .571 .298 .571 .471 .471

Brush PikBrush Pik .632 .278 .632 .278 .059 .059

PicobuyPicobuy .069 .151 .069 .151 .471 .471




Overall Priority Vector (continued)Overall Priority Vector (continued)

Thus, the overall priority vector is:Thus, the overall priority vector is:

Cornell:Cornell: (.729)(.298) + (.216)(.571) + (.055)(.729)(.298) + (.216)(.571) + (.055)(.471) = .366(.471) = .366

Brush Pik: (.729)(.632) + (.216)(.278) + (.055)Brush Pik: (.729)(.632) + (.216)(.278) + (.055)(.059) = .524(.059) = .524

Picobuy: (.729)(.069) + (.216)(.151) + (.055)Picobuy: (.729)(.069) + (.216)(.151) + (.055)(.471) = .109(.471) = .109

Brush Pik appears to be the overall Brush Pik appears to be the overall recommendation.recommendation.

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