Introduction to Management Science 9 th Edition by Bernard W. Taylor III. Chapter 9 Multicriteria Decision Making

Introduction to Management Science 9 th Edition by Bernard W. Taylor III Chapter 9 Multicriteria Decision Making 2007 Pearson Education

Chapter Topics Goal Programming Interpretasi grafis dari Goal Programming Solusi komputer masalh Goal Programming dengan QM for Windows and Excel Analytical Hierarchy Process Scoring Models 2

Overview Pembelajaran permasalahan dengan beberapa kriteria, multiple criteria, bukan satu tujuan ketika membuat keputusan. Tiga teknik yang dibahas: goal programming, analytical hierarchy process dan scoring models. Goal programming adalah variasi dari program linier mempertimbangkan lebih dari satu tujuan (goals) dalam fungsi tujuan. Analytical hierarchy process (AHP) merupakan memberian skor untuk setiap alternatif keputusan berdasarkan perbandingan masingmasing di bawah kriteria yang berbeda yang mencerminkan preferensi pengambil keputusan.. Scoring models Model Scoring didasarkan pada teknik perkalian scoring yang relatif sederhana 3

Contoh Permasalahan Goal Programming ( of 2) Contoh Beaver Creek Pottery Company Maksimalkan Z = $40x 50x 2 Batasan: x 2x 2 40 jam kerja 4x 3x 2 20 pon tanah liat x, x 2 0 Dimana: x = Jumlah produksi mangkok x 2 = Jumlah produksi mug 4

Contoh Permasalahan Goal Programming (2 of 2) Menambahkan tujuan (goals) dalam urutan kepentingan, perusahaan: Tidak ingin menggunakan kurang dari 40 jam kerja per hari. Ingin mencapai tingkat laba yang memuaskan dari $.600 per hari. Memilih untuk tidak menyimpan lebih dari 20 pon tanah liat di tangan setiap hari. Ingin meminimalkan jumlah lembur. 5

Goal Programming Kendala Tujuan Semua kendala tujuan adalah kesetaraan yang meliputi variabel deviasi d dan d. Sebuah variabel deviasi positif (d ) adalah jumlah dimana tingkat tujuan terlampaui. Variabel deviasi negatif (d) adalah jumlah dimana tingkat tujuannya adalah di bawah tercapai. Setidaknya satu atau kedua variabel deviasi dalam kendala tujuan harus sama dengan nol. Fungsi tujuan dalam model goal programming untuk meminimalkan penyimpangan dari tujuan masingmasing dalam urutan prioritas tujuan. 6

Model Formulasi Goal Programming Kendala Tujuan ( of 3) Tujuan Jam Kerja: x 2x 2 d d = 40 (jam/hari) Tujuan Keuntungan: 40x 50 x 2 d 2 d 2 =,600 ($/hari) Tujuan Material: 4x 3x 2 d 3 d 3 = 20 (tanah liat/hari) 7

Model Formulasi Goal Programming Fungsi Objektif (2 of 3) Kendala Tujuan Jam Kerja (prioritas kurang dari 40 jam kerja, prioritas 4 minimum lembur ): Minimalkan P d, P 4 d Kendala Penambahan Tujuan keuntungan (prioritas 2 mencapai keuntungan sebesar $.600): Minimalkan P d, P 2 d 2, P 4 d Kendala Penambahan Tujuan Material (prioritas 3 menghindari menjaga lebih dari 20 pon tanah liat di tangan): Minimalkan P d, P 2 d 2, P 3 d 3, P 4 d 8

Model Formulasi Goal Programming Model Lengkap (3 of 3) Model Lengkap Goal Programming : Minimalkan P d, P 2 d 2, P 3 d 3, P 4 d batasan: x 2x 2 d d = 40 (jam kerja) 40x 50 x 2 d 2 d 2 =,600 (keuntungan) 4x 3x 2 d 3 d 3 = 20 (tanah liat) x, x 2, d, d, d 2, d 2, d 3, d 3 0 9

Goal Programming Bentuk alternatif Kendala Tujuan ( of 2) Mengubah keempat prioritas tujuan "batas lembur untuk 0 jam" bukannya meminimalkan lembur: d d 4 d 4 = 0 minimalkan P d, P 2 d 2, P 3 d 3, P 4 d 4 Penambahan kelima prioritas tujuan "penting untuk mencapai tujuan untuk mug": x d 5 = 30 mangkok x 2 d 6 = 20 mug minimalkan P d, P 2 d 2, P 3 d 3, P 4 d 4, 4P 5 d 5 5P 5 d 6 0

Goal Programming Alternative Forms of Goal Constraints (2 of 2) Model Lengkap dengan menambahkan Tujuan Baru: Minimalkan P d, P 2 d 2, P 3 d 3, P 4 d 4, 4P 5 d 5 5P 5 d 6 batasan: x 2x 2 d d = 40 40x 50x 2 d 2 d 2 =,600 4x 3x 2 d 3 d 3 = 20 d d 4 d 4 = 0 x d 5 = 30 x 2 d 6 = 20 x, x 2, d, d, d 2, d 2, d 3, d 3, d 4, d 4, d 5, d 6 0

Goal Programming Interpretasi Grafik ( of 6) Minimalkan P d, P 2 d 2, P 3 d 3, P 4 d batasan: x 2x 2 d d = 40 40x 50 x 2 d 2 d 2 =,600 4x 3x 2 d 3 d 3 = 20 x, x 2, d, d, d 2, d 2, d 3, d 3 0 Figure 9. Goal Constraints 2

Goal Programming Interpretasi Grafik (2 of 6) Minimalkan P d, P 2 d 2, P 3 d 3, P 4 d batasan : x 2x 2 d d = 40 40x 50 x 2 d 2 d 2 =,600 4x 3x 2 d 3 d 3 = 20 x, x 2, d, d, d 2, d 2, d 3, d 3 0 Figure 9.2 The FirstPriority Goal: Minimize 3

Goal Programming Interpretasi Grafik (3 of 6) Minimalkan P d, P 2 d 2, P 3 d 3, P 4 d batasan : x 2x 2 d d = 40 40x 50 x 2 d 2 d 2 =,600 4x 3x 2 d 3 d 3 = 20 x, x 2, d, d, d 2, d 2, d 3, d 3 0 Figure 9.3 The SecondPriority Goal: Minimize 4

Goal Programming Interpretasi Grafik (4 of 6) Minimalkan P d, P 2 d 2, P 3 d 3, P 4 d batasan : x 2x 2 d d = 40 40x 50 x 2 d 2 d 2 =,600 4x 3x 2 d 3 d 3 = 20 x, x 2, d, d, d 2, d 2, d 3, d 3 0 Figure 9.4 The ThirdPriority Goal: Minimize 5

Goal Programming Interpretasi Grafik (5 of 6) Minimalkan P d, P 2 d 2, P 3 d 3, P 4 d subject to: x 2x 2 d d = 40 40x 50 x 2 d 2 d 2 =,600 4x 3x 2 d 3 d 3 = 20 x, x 2, d, d, d 2, d 2, d 3, d 3 0 Figure 9.5 The FourthPriority Goal: Minimize 6

Goal Programming Interpretasi Grafik (6 of 6) Solusi goal programming tidak selalu mencapai semua tujuan dan tidak "optimal", mencapai yang terbaik atau yang paling memuaskan solusi yang mungkin. Minimalkan P d, P 2 d 2, P 3 d 3, P 4 d batasan: x 2x 2 d d = 40 40x 50 x 2 d 2 d 2 =,600 4x 3x 2 d 3 d 3 = 20 x, x 2, d, d, d 2, d 2, d 3, d 3 0 Solusi: x = 5 mangkok x 2 = 20 mug d = 5 jam 7

Goal Programming Computer Solution Using Excel ( of 3) Exhibit 9.4 8

Goal Programming Computer Solution Using Excel (2 of 3) Exhibit 9.5 9

Goal Programming Computer Solution Using Excel (3 of 3) Exhibit 9.6 20

Goal Programming Solution for Altered Problem Using Excel ( of 6) Minimize P d, P 2 d 2, P 3 d 3, P 4 d 4, 4P 5 d 5 5P 5 d 6 subject to: x 2x 2 d d = 40 40x 50x 2 d 2 d 2 =,600 4x 3x 2 d 3 d 3 = 20 d d 4 d 4 = 0 x d 5 = 30 x 2 d 6 = 20 x, x 2, d, d, d 2, d 2, d 3, d 3, d 4, d 4, d 5, d 6 0 2

Goal Programming Solution for Altered Problem Using Excel (2 of 6) Exhibit 9.7 22

Goal Programming Solution for Altered Problem Using Excel (6 of 6) Exhibit 9. 26

Analytical Hierarchy Process Overview AHP merupakan metode untuk merangking beberapa alternatif keputusan dan menyeleksi yang terbaik pengambil keputusan mempunyai banyak tujuan dan kriteria yang mendasari keputusan. Pengambil keputusan membuat keputusan berdasarkan bagaimana alternatif dibandingkan menurut beberapa kriteria. Pembuat keputusan akan alternatif yang paling memenuhi kriteria keputusan nya. AHP adalah proses untuk mengembangkan nilai numerik untuk menentukan peringkat masingmasing alternatif keputusan berdasarkan seberapa baik alternatif memenuhi kriteria pembuat keputusan. 27

Analytical Hierarchy Process Contoh Southcorp Development Company shopping mall site selection. Three potential sites: Atlanta Birmingham Charlotte. Perbandingan kriteria untuk tempat : Customer market base. Income level Infrastructure Transportation 28

Analytical Hierarchy Process Hierarchy Structure Top of the hierarchy: the objective (select the best site). Second level: how the four criteria contribute to the objective. Third level: how each of the three alternatives contributes to each of the four criteria. 29

Analytical Hierarchy Process General Mathematical Process Mathematically determine preferences for sites with respect to each criterion. Mathematically determine preferences for criteria (rank order of importance). Combine these two sets of preferences to mathematically derive a composite score for each site. Select the site with the highest score. 30

Analytical Hierarchy Process Pairwise Comparisons ( of 2) In a pairwise comparison, two alternatives are compared according to a criterion and one is preferred. A preference scale assigns numerical values to different levels of performance. 3

Analytical Hierarchy Process Pairwise Comparisons (2 of 2) Table 9. Preference Scale for Pairwise Comparisons 32

Analytical Hierarchy Process Pairwise Comparison Matrix A pairwise comparison matrix summarizes the pairwise comparisons for a criteria. Customer Market Site A B C A B C /3 /2 3 5 2 /5 Income Level Infrastructure Transportation A B C /6 3 6 9 /3 /9 3 /3 /7 7 3 2 /3 /4 /2 4 33

Analytical Hierarchy Process Developing Preferences Within Criteria ( of 3) In synthesization, decision alternatives are prioritized with each criterion and then normalized: Customer Market Site A B C A B C /3 /2 /6 3 5 9 2 /5 6/5 Customer Market Site A B C A 6/ 3/9 5/8 B C 2/ 3/ /9 5/9 /6 5/6 34

Analytical Hierarchy Process Developing Preferences Within Criteria (2 of 3) The row average values represent the preference vector Table 9.2 The Normalized Matrix with Row Averages 35

Analytical Hierarchy Process Developing Preferences Within Criteria (3 of 3) Preference vectors for other criteria are computed similarly, resulting in the preference matrix Table 9.3 Criteria Preference Matrix 36

Analytical Hierarchy Process Ranking the Criteria ( of 2) Pairwise Comparison Matrix: Criteria Market Income Infrastructure Transportation Market Income Infrastructure Transportation 5 /3 /4 /5 /9 /7 3 9 /2 4 7 2 Table 9.4 Normalized Matrix for Criteria with Row Averages 37

Analytical Hierarchy Process Ranking the Criteria (2 of 2) Preference Vector for Criteria: Market Income Infrastructure Transportation 0.993 0.6535 0.0860 0.062 38

Analytical Hierarchy Process Developing an Overall Ranking Overall Score: Site A score =.993(.502).6535(.289).0860(.790).062(.56) =.309 Site B score =.993(.85).6535(.0598).0860(.6850).062(.696) =.595 Site C score =.993(.3803).6535(.6583).0860(.360).062(.2243) =.534 Overall Ranking: Site Charlotte Atlanta Birmingham Score 0.534 0.309 0.595.0000 39

Analytical Hierarchy Process Summary of Mathematical Steps Develop a pairwise comparison matrix for each decision alternative for each criteria. Synthesization Sum the values of each column of the pairwise comparison matrices. Divide each value in each column by the corresponding column sum. Average the values in each row of the normalized matrices. Combine the vectors of preferences for each criterion. Develop a pairwise comparison matrix for the criteria. Compute the normalized matrix. Develop the preference vector. Compute an overall score for each decision alternative Rank the decision alternatives. 40

Analytical Hierarchy Process: Consistency ( of 3) Consistency Index (CI): Check for consistency and validity of multiple pairwise comparisons Example: Southcorp s consistency in the pairwise comparisons of the 4 site selection criteria Step : Multiply the pairwise comparison matrix of the 4 criteria by its preference vector Market Income Infrastruc. Transp. Criteria Market /5 3 4 0.993 Income 5 9 7 X 0.6535 Infrastructure /3 /9 2 0.0860 Transportation /4 /7 /2 0.062 ()(.993)(/5)(.6535)(3)(.0860)(4)(.062) = 0.8328 (5)(.993)()(.6535)(9)(.0860)(7)(.062) = 2.8524 (/3)(.993)(/9)(.6535)()(.0860)(2)(.062) = 0.3474 (/4)(.993)(/7)(.6535)(/2)(.0860)()(.062) = 0.2473 4

Analytical Hierarchy Process: Consistency (2 of 3) Step 2: Divide each value by the corresponding weight from the preference vector and compute the average 0.8328/0.993 = 4.786 2.8524/0.6535 = 4.3648 0.3474/0.0860 = 4.040 0.2473/0.062 = 4.0422 6.257 Average = 6.257/4 = 4.564 Step 3: Calculate the Consistency Index (CI) CI = (Average n)/(n), where n is no. of items compared CI = (4.5644)/(4) = 0.052 (CI = 0 indicates perfect consistency) 42

Analytical Hierarchy Process: Consistency (3 of 3) Step 4: Compute the Ratio CI/RI where RI is a random index value obtained from Table 9.5 Table 9.5 Random Index Values for n Items Being Compared CI/RI = 0.052/0.90 = 0.0580 Note: Degree of consistency is satisfactory if CI/RI < 0.0 43

Analytical Hierarchy Process Excel Spreadsheets ( of 4) Exhibit 9.2 44

Analytical Hierarchy Process Excel Spreadsheets (2 of 4) Exhibit 9.3 45

Scoring Model Overview Each decision alternative graded in terms of how well it satisfies the criterion according to following formula: where: S i = g ij w j w j = a weight between 0 and.00 assigned to criterion j;.00 important, 0 unimportant; sum of total weights equals one. g ij = a grade between 0 and 00 indicating how well alternative i satisfies criteria j; 00 indicates high satisfaction, 0 low satisfaction. 48

Scoring Model Example Problem Mall selection with four alternatives and five criteria: Grades for Alternative (0 to 00) Weight Decision Criteria (0 to.00) Mall Mall 2 Mall 3 Mall 4 School proximity 0.30 40 60 90 60 Median income 0.25 75 80 65 90 Vehicular traffic 0.25 60 90 79 85 Mall quality, size 0.0 90 00 80 90 Other shopping 0.0 80 30 50 70 S = (.30)(40) (.25)(75) (.25)(60) (.0)(90) (.0)(80) = 62.75 S 2 = (.30)(60) (.25)(80) (.25)(90) (.0)(00) (.0)(30) = 73.50 S 3 = (.30)(90) (.25)(65) (.25)(79) (.0)(80) (.0)(50) = 76.00 S 4 = (.30)(60) (.25)(90) (.25)(85) (.0)(90) (.0)(70) = 77.75 Mall 4 preferred because of highest score, followed by malls 3, 2,. 49

Scoring Model Excel Solution Exhibit 9.6 50

Goal Programming Example Problem Problem Statement Public relations firm survey interviewer staffing requirements determination. One person can conduct 80 telephone interviews or 40 personal interviews per day. $50/ day for telephone interviewer; $70 for personal interviewer. Goals (in priority order): At least 3,000 total interviews. Interviewer conducts only one type of interview each day. Maintain daily budget of $2,500. At least,000 interviews should be by telephone. Formulate a goal programming model to determine number of interviewers to hire in order to satisfy the goals, and then solve the problem. 5

Analytical Hierarchy Process Example Problem Problem Statement Purchasing decision, three model alternatives, three decision criteria. Pairwise comparison matrices: Price Bike X Y Z X Y Z /3 /6 3 /2 6 2 Gear Action Bike X Y Z X Y Z 3 7 /3 4 /7 /4 Weight/Durability Bike X Y Z X Y Z /3 3 2 /2 Prioritized decision criteria: Criteria Price Gears Weight Price Gears Weight /3 /5 3 /2 5 2 52

Analytical Hierarchy Process Example Problem Problem Solution ( of 4) Step : Develop normalized matrices and preference vectors for all the pairwise comparison matrices for criteria. Price Bike X Y Z Row Averages X Y Z 0.6667 0.2222 0. 0.6667 0.2222 0. Gear Action 0.6667 0.2222 0. 0.6667 0.2222 0..0000 Bike X Y Z Row Averages X Y Z 0.0909 0.2727 0.6364 0.0625 0.875 0.7500 0.026 0.795 0.779 0.0853 0.232 0.704.0000 53

Analytical Hierarchy Process Example Problem Problem Solution (2 of 4) Step continued: Develop normalized matrices and preference vectors for all the pairwise comparison matrices for criteria. Weight/Durability Bike X Y Z Row Averages X Y Z 0.4286 0.429 0.4286 0.5000 0.667 0.3333 0.4000 0.2000 0.4000 0.4429 0.698 0.3873.0000 Criteria Bike Price Gears Weight X Y Z 0.6667 0.2222 0. 0.0853 0.232 0.704 0.4429 0.698 0.3873 54

Analytical Hierarchy Process Example Problem Problem Solution (3 of 4) Step 2: Rank the criteria. Criteria Price Gears Weight Row Averages Price Gears Weight 0.6522 0.274 0.304 0.6667 0.2222 0. 0.6250 0.2500 0.250 0.6479 0.2299 0.222.0000 Price Gears Weight 0.6479 0.2299 0.222 55

Analytical Hierarchy Process Example Problem Problem Solution (4 of 4) Step 3: Develop an overall ranking. Bike X Bike Y Bike Z 0.6667 0.2222 0. 0.0853 0.232 0.704 0.4429 0.698 0.3837 0.6479 0.2299 0.222 Bike X score =.6667(.6479).0853(.2299).4429(.222) =.5057 Bike Y score =.2222(.6479).232(.2299).698(.222) =.238 Bike Z score =.(.6479).704(.2299).3873(.222) =.2806 Overall ranking of bikes: X first followed by Z and Y (sum of scores equal.0000). 56

End of chapter 57