WHAT IS MULTIPLE CRITERIA ANALYSIS? MCA describes any structured approach used to determine overall preferences among alternative options, where the options accomplish several objectives. In MCA, desirable objectives are specified and corresponding attributes or indicators are identified.

MULTIPLE CRITERIA DECISION MAKING (MCDM) SITUASI PENGAMBILAN KEPUTUSAN: Involving a single decision criteria ( SINGLE OBJECTIVE) Involves several conflicting objectives (MULTIPLE OBJECTIVE) Multiple Criteria Decision Making (MCDM) merupakan suatu metode pengambilan keputusan yang didasarkan atas teori-teori, proses-proses, dan metode analitik yang melibatkan ketidakpastian, dinamika, dan aspek kriteria jamak. Dalam metode optimasi konvensional, cakupan umumnya hanya dibatasi pada satu kriteria pemilihan (mono criteria), dimana pemilihan yang diambil adalah pilihan yang paling memenuhi fungsi obyektif. Analisis Pengambilan Keputusan: A decision maker An array of feasible choices A well defined criteria, such as utility or profit: SINGLE or MULTIPLE

MULTIPLE CRITERIA DECISION MAKING (MCDM) Economic vs. Technological Decisions. Technological decision: a single criterion Economic decision: a multiple criteria Technological problems: Search and measurement Scarce Economic Technological means problems problems No scarce No problems problem Several Single criteria criterion

MULTIPLE CRITERIA DECISION MAKING (MCDM) Ilustrasi: Ke supermarket untuk MEMILIH produk sirup yang Paling Murah Mencari pola tanam yang memaksimumkan the gross margin dan (2) : a technological problem Untuk menyelesaikannya: ONLY SEARCHES. Decision Making does not really Multi-Criteria Decision Making (MCDM) is the study of methods and procedures by which concerns about multiple conflicting criteria can be formally incorporated into the management planning process", as defined by the International Society on Multiple Criteria Decision Making

Decision Making does not really MCDM dapat dikelompokkan menjadi 2 kelompok besar, yakni Multiple Objective Decision Making (MODM) dan Multiple Attribute Decision Making (MADM). MADM menentukan alternatif terbaik dari sekumpulan alternatif (permasalahan pilihan) dengan menggunakan preferensi alternatif sebagai kriteria dalam pemilihan. MODM memakai pendekatan optimasi, sehingga untuk menyelesaikannya harus dicari terlebih dahulu model matematis dari persoalan yang akan dipecahkan. Decision Making does not really

MULTIPLE CRITERIA DECISION MAKING (MCDM) Ilustrasi: Pola tanam yang: Max gross margin Min Risk Conflicting objectives Min Indebtedness Solution this problem: Economic decision ………. Optimal solution e.g. Development of a small rural region 1000 ha arable land: Two crops: A and B Water requirement: 4000 and 5000 m3/ha Water available : 4.200.000 m3 Syarat rotasi tanaman: Luas tanam B <= luas tanam A X1 = luas tanam A X2 = luas tanam B X1 + X2 <= 1000 4000 X1 + 5000 X2 <= 4.200.000 -X1 + X2 <= 0 ………….. X2 <= X1

MULTIPLE CRITERIA DECISION MAKING (MCDM) X2 (ha) 4000X1+5000X2 = 4200000 -X1+X2 = 0 X1+X2=1000 A 466.66 E 200 B 0 466.66 800 C (1000) X1 (ha) Added value:A = 1000 /ha B = 3000/ha 1000X1 + 3000X2 = AE (Isovalue line) Employment: A = 500 HOK/ha 500X1 + 200X2 = CE (Iso employment line B = 200 HOK/ha

MULTIPLE CRITERIA DECISION MAKING (MCDM) Kriteria Nilai Tambah : Optimum solution: A(466.6 ; 466.6) Added value = 1.866.640 Kriteria Employment: Optimum solution: C(1000,0) ……employment = 500000 HOK Solusi Optimum: Garis ABC Optimum Point ? Multiple goals Multiple objectives The decision theory helps identify the alternative with the highest expected value (probability of obtaining a possible value).

MULTIPLE CRITERIA DECISION MAKING (MCDM) Site suitability assessment is inherently a multi-criteria problem. That is, land suitability analysis is an evaluation/decision problem involving several factors. In general, a generic model of site/land suitability can be described as: S = f (x1, x2,…, xn)) where S = suitability measure; x1., x2, …, xn = are the factors affecting the suitability of the site/land. MULTIPLE CRITERIA DECISION MAKING (MCDM)

TUJUAN GANDA DALAM PERTANIAN MULTIPLE CRITERIA DECISION MAKING (MCDM) TUJUAN GANDA DALAM PERTANIAN Farm Level: Goals in agriculture DM: 1. Maximum gross margin 2. Minimum seasonal cash exposure 3. Provision od stable employment for the permanent labor Ranch planning: 1. Red meat production 2. Use of fossil fuel energy 3. Profits Land allocation problems: 1. Money income 2. Environmental benefits FARM SYSTEM PROPERTIES AND PERFORMANCE CRITERIA 1. Productivity 2. Profitability 3. Stability 4. Diversity 5. Flexibility 6. Time-dispersion 7. Sustainability 8. Complementarity and environmental compatibility

MULTIPLE CRITERIA DECISION MAKING (MCDM) ATRIBUTES, OBJECTIVE, GOAL Atribute: Nilai DM yang berhubungan dengan realita objektif A = f(Xi) ……….. Xi = peubah keputusan e.g. Added value (economic yield) : V = 1000X1 + 3000X2 Employment : E = 500X1 + 200X2 Objective: direction of improvement of the attributes Maximization (or minimization) of the function of atributes Max f(x) : Max w1f1(X) + w2 f2(X) w : weight f(X): atributes function

MULTIPLE CRITERIA DECISION MAKING (MCDM) TARGET = as aspiration level an acceptable level of achievement for any one of the attributes GOAL: combining an attribute with a target 1000X1 + 2000X2 >= 2.000.000 atau X1 + X2 = 1000 Goal: f(X) >< t atau f(X) = t (target) Tipe I : gross margin, added value Tipe II : Limited resources…………. Air irigasi, tenaga kerja, kendala teknis, constraint

MULTIPLE CRITERIA DECISION MAKING (MCDM) Farm planning problem Atribute: Gross margin Objective: Gross margin minimize Goal : to achieve a gross margin of at least a certain target Kriteria : adalah atribut, objective, atau goal yang dianggap relevan dengan situasi pengambilan keputusan yang sedang dikaji MCDM = paradigma yang melibatkan beberapa atribute, objective atau goal. Criterion outcomes of decision alternatives can be collected in a table (called decision matrix or decision table) comprised of a set of columns and rows. The table rows represent decision alternatives, with table columns representing criteria. A value found at the intersection of row and column in the table represents a criterion outcome - a measured or predicted performance of a decision alternative on a criterion. The decision matrix is a central structure of the MCDA/MCDM since it contains the data for comparison of decision alternatives.

MULTIPLE CRITERIA DECISION MAKING (MCDM) GOAL and CONSTRAINT (KENDALA) Goal: RHS-nya = Target (dapat tercapai atau tidak tercapai) Constraint: RHS-nya harus terpenuhi eg. 1000X1 + 3000X2 >= 2.000.000 …. Bisa goal, bisa constraint Kalau sebagai GOAL, hanya didekati, sehingga ada simpangan positif atau negatif: 1000X1 + 3000X2 + n – p = 2.000.000 Dimana: n = simpangan negatif (d-) p = simpangan positif (d+) Goal function : f(X) + n – p = t (target)

MULTIPLE CRITERIA DECISION MAKING (MCDM) PARETO OPTIMALITY Efficient of Pareto Optimal solution: a feasible solution for which an increase in the value of one criterion can only be achieved by degrading the value of at least one other criterion e.g. Farm planning involving three criteria Gross margin Labor Indeptedness Sol I 200.000 500 50.000 Sol II 200.000 600 50.000 Sol III 300.000 700 60.000 DM wants: 1. Gross margin,……….. As large as possible 2. Labor and indeptedness ……….. As small as possible

MULTIPLE CRITERIA DECISION MAKING (MCDM) Gross margin Labor Indeptedness Sol I Rendah Rendah Rendah ………. efisien Sol II Rendah Tinggi Rendah ………. Tdk efisien Sol III Tinggi Tinggi Tinggi ………. Optimal Pareto Bagaimana memilih di antara Sol I dan III ? It is an economic problem, ……. Preference of the DM for each of the three attributes Feasible solution ………….. Efficient or Not-efficient DM preference for each of criteria …………. (pembobotan)

(pembobotan) DM preference for each of criteria …………. The Goal programming is used for formulization of the problems which have multiple goals. Any farmlands usually have the ability of producing different crops. Multiple goals are considered for producing different crops in a high level of programming . In the linear goal programming cases, the goal is to reach the maximum output or to reach the minimum cost. We notice that the fulfillment of this goal is conditioned with some limitations like source, equipment, talents and capital. In the linear goal programming one goal is only purposed. DM preference for each of criteria …………. (pembobotan)

MULTIPLE CRITERIA DECISION MAKING (MCDM) Trade-off amongst decision making criteria Trade off between two criteria: fj(X’) – fj(X”) Tjk = ----------------------.. fj(X) dan fk(X) adalah dua fungsi tujuan fk(X’) – fk(X”) e.g. Trade-off antara margin dan labor untuk Sol III dan Sol I: T12 = (300.000 – 200.000) / (700-500) = 500 Setiap peningkatan labor 1 jam berakibat penurunan margin 500, Opportunity cost 1 jam labor = 500 unit marjin TRADE-OFF --------- OPPORTUNITY COST

MULTIPLE CRITERIA DECISION MAKING (MCDM) MCDM APPROACH 1. Multiple goals …………. GP : Goal Programming 2. Multiple Objectives ……… MOP: Multi Objective Program Multi Attributes Utility Theory (MAUT): Decision problems with a discrete number of feasible solutions Very strong assumptions about the preference of Decision Maker MOP : Efficient set of solutions Pareto Optimal Non-Pareto Optimal Feasible solution feasible solution Optimum Compromize Decision Maker Preferences

GOAL PROGRAMMING: GP GP : Simultaneous optimization of several goals . Minimized deviation d- : Goal 1 d+ : Goal 2 : Goal 3 Minimization process: 1. Lexicographic Goal Programming (LGP) 2. Weighted Goal Programming (WGP) LGP: Prioritas (p) goals Pembobot (w) , absolute weight …………. Deviasi Prioritas tinggi dupenuhi dulu, baru prioritas lebih rendah WGP: Relative weight Deviasi diberi pembobot sesuai dengan kepentingan relatif masing-masing goal

GOAL PROGRAMMING: Farm Planning Model Data Hipotetik: Usahatani. 1. Decision variables Pear tree (X1 ha) Peach tree (X2 ha) 2. NPV (Rp/ha) 6250 5000 3. Resources Uses: Capital Year 1 550 400 Year 2 200 175 Year 3 300 250 Year 4 325 200 4. Annual labor Prunning 120 180 Harvest 400 450 5. Mesin pengolahan (jam/ha) 35 35 Ketersediaan sumberdaya: 1. Kapital tahun 1 : 15.000 tahun 2 s/d 4 : 7.000 per tahun 2. TK prunning : 4000 jam/ musim TK panen : 2000 3. Max. tractor hours : 1000 4. Periode panen dua macam tanaman berbeda.

GOAL PROGRAMMING: Farm Planning Model Tujuan Usahatani: 1. Maximize NPV 2. Minimize pinjaman kapital selama 4 tahun 3. Minimize TK musiman untuk prunning dan panen 4. Minimize sewa traktor (these are conflicting interests) Strategi dengan Linear Programming biasa: 1. NPV ------------- dimaksimumkan 2. Tujuan lain --------- sebagai kendala sumberdaya 3. Cash resources: Surplus tahun 1 dimasukkan sebagai tambahan tahun berikutnya Max Z = f(X1,X2) = 6250 X1 + 5000 X2 Subject to: 500X1 + 400X2 <= 15.000 750X1 + 575X2 <= 22.000 1050X1 + 825X2 <= 29.000 1375X1 + 1025X2 <= 36.000 120X1 + 180 X2 <= 4000 400X1 <= 2000 450X2 <= 2000 35X1 +35X2 <= 1000 X1 >= 0 X2 >= 0

2. Minimize pinjaman kapital selama 4 tahun Tujuan Usahatani: 1. Maximize NPV 2. Minimize pinjaman kapital selama 4 tahun 3. Minimize TK musiman untuk prunning & panen 4. Minimize sewa traktor The goals of the problem are gross benefit, production costs, needed water, produced paddy, Urea fertilizer, Triple fertilizer, Potash fertilizer, Granule of stem borer, Dimicron of stem borer, Bieam Blast stem, Hynozan for blast disease, Cyvine pesticide, Botchlor herbicide and labor.

GOAL PROGRAMMING: Farm Planning Model Solusinya: X1 = 5 ha X2 = 4.44 ha NPV = 53.450 Tenaga kerja panen digunakan semua Sumberdaya lainnya tidak habis digunakan, ada sisa sumberdaya Menurut LP ini optimal karena: 1. Objectives yang diformulasikan sebagai kendala dipenuhi dulu sebelum NPV 2. Setiap solusi yang layak harus memenuhi fungsi kendala Pendekatan tujuan tunggal dengan banyak fungsi kendala seperti ini lazimnya menghasilkan solusi yang tidak memuaskan, sehibngga muncullah pendekatan MULTIPLE CRITERIA GOALS PROGRAMMING

The role of d+ and d- in GP Dalam model GP, formula ketidak-samaan seperti di atas dianggap sebagai goal (g) dan bukan sebagai kendala RHS merupakan target yg dapat tercapai atau hanya dapat didekati Untuk setiap fungsi goal diberi dua macam variabel ( n dan p) untuk mengubahnya menjadi persamaan: 6250X1 + 5000X2 + n1 – p1 = 200.000 …………… g1 500X1 + 400X2 + n2 – p2 = 15.000 …………….. g2 750X1 + 575X2 + n3 – p3 = 22.000 …………….. g3 1050X1 + 825X2 + n4 – p4 = 29.000 …………….. g4 1375X1 + 1025X2 +n5 – p5 = 36.000 …………….. g5 120X1 + 180 X2 + n6 – p6 = 4000 .…………….. g6 400X1 + n7 – p7 = 2000 …………….. g7 450X2 + n8 – p8 = 2000 …………….. g8 35X1 +35X2 + n9 – p9 = 1000 …………….. g9 DM --------------- to maximize NPV Simpangan negatif (n) : Under achievement of goal Simpangan positif (p) : Goal has surpassed (Over achievement) n = d- p = d+ d- = 0, atau d+ = 0, atau d- = d+ = 0 Min Σ di- + di+ ------------- Min Σ ni + pi : Tujuan GP: minimize deviation

LGP : Lexicographic Goal Programming DM: Mendefine semua tujuan (goal) yang relevan dengan situasi perencanaan Menetapkan prioritas goals: Qi >>>> Qj Prioritas tinggi dipenuhi lebih dahulu: Lexicographic order e.g. Q1 : untuk g2, g3, g4, g5 adalah p2, p3, p4, p5 Q2 : untuk g9 : p9 Q3 : untuk g1: n1 Q4 : untuk g6, g7, g8: p6, p7, p8 Min A = [ (p2+p3+p4+p5), (p9), (n1), (p6+p7+p8)] …… The achievement - function System stability refers to the absence or minimization of year-to-year fluctuations in either production or value of output. (The latter also implies either stability in input costs, yields and prices or counterbalancing movements in these influences on value of output.) Where conditions are favourable, price and production instability can often be countered by more careful activity selection (e.g., of drought-tolerant varieties, pest-immune crops); by diversification of activities; by seeking greater flexibility in product use or disposal; by multiple cropping over both space and time; and by increasing on-farm storage capacity and post-harvest handling efficiency.

Profit is usually but not necessarily measured in money terms. DM: Mendefine semua tujuan (goal) yang relevan dengan situasi perencanaan There are basically two major farm-operating objectives, profit maximization on market-oriented farms and household sustenance on subsistence-oriented farms. By profit maximization is meant maximization of net gain measured as total benefit less total cost. Profit is usually but not necessarily measured in money terms.

Model LGP nya: Min A = [ (p2+p3+p4+p5), (p9), (n1), (p6+p7+p8) ] Subjected to: Q3 : 6250X1 + 5000X2 + n1 – p1 = 200.000 …………… g1 Q1 500X1 + 400X2 + n2 – p2 = 15.000 …………….. g2 750X1 + 575X2 + n3 – p3 = 22.000 …………….. g3 1050X1 + 825X2 + n4 – p4 = 29.000 …………….. g4 1375X1 + 1025X2 +n5 – p5 = 36.000 …………….. g5 Q4 120X1 + 180 X2 + n6 – p6 = 4000 .…………….. g6 400X1 + n7 – p7 = 2000 …………….. g7 450X2 + n8 – p8 = 2000 …………….. g8 Q2: 35X1 +35X2 + n9 – p9 = 1000 …………….. g9 Xi >= 0; nj >= 0, pj >= 0 i = 1, 2 j = 1, ……, 9

LGP : Optimum Solution Optimum solution: X1 = 19.18 X2 = 9.38 Deviation variable: n1 = 33.250 p1 = 0 n2 = 699 p2 = 0 n3 = 2.221 p3 = 0 n4 = 1.122 p4 = 0 n5 = n6 = 0 p5 = p6 = 0 n7 = 0 p7 = 5672 n8 = 0 p8 = 2211 n9 = 0 p9 = 0 Prioritas I (Q1) ---------------- g5 tercapai Prioritas II (Q2) --------------- g9 tercapai Prioritas IV (Q4) -------------- g6 tercapai Dibandingkan dengan penyelesaian LP di atas, maka: NPV lebih tinggi Sumberdaya ----------- habis dipakai, … kurang Modal ------------------- ada sisa

LGP : Sensitivity Analysis Kelemahan LGP: memerlukan banyak informasi dari Decision Maker, a.l. Target Weight Priority ordered Preferences Kalau informasi ini tidak ada, maka harus dilakukan analisis sensitivitas: Pengaturan kembali prioritas Nilai-nilai target Pembobot Alternatif strategi perencanaan --------------- SKENARIO MISALNYA: Mengubah kembali prioritas Dalam contoh di atas ada 4 prioritas, maka permutasinya ada 4 ! = 4x3x2x1 = 24 macam kombinasi .

LGP : Solusi Enam macam solusi di antaranya adalah sbb: SOLUSI X1 X2 NPV g7+g8 g9 g2 g5 I 19.18 9.38 33.250 7.893 0 0 II 5 4.44 146.55 0 0 0 III 0 35.12 24.400 16.125 229 0 IV 28.57 0 21.437 9.428 0 3.284 V 0 40 0 19.20 400 5000 VI 32 0 0 10.800 120 8000 Solusi I: Kalau urutan dari dua prioritas pertama saling dipertukarkan Solusi II: Optimal untuk 12 dari 24 alternatif prioritas Solusi III: Kalau prioritas III digabungkan dengan prioritas II Dst.

LGP : Pengubahan nilai target dari beberapa goal, misalnya: Kalau target g1 diturunkan menjadi 166.775, maka solusi optimum tidak berubah, tetapi kalau diturunkan lagi, maka nilai NPV akan merosot dan simpangan dari g6, g7, g8 menurun Kalau target g9 dikurangi, maka solusi optimum berubah, NPV menurun Kalau g9 ditingkatkan, maka solusi optimum dapat berubah dan NPV naik 3. Kalau target g6, g7, g8 berubah, maka: Nilai solusi optimum tidak berubah Simpangan berubah terhadap g6, g7, g8.

WGP : Weighted Goal Programming Semua goals masuk ke dalam fungsi tujuan komposit Simpangan diberi pembobot sesuai dengan kepentingan relatif dari masing-masing goal Misalnya: g2, g3, g4, dan g5, sebagai rigid constraint yang harus dipenuhi, ……………. Sebagai kendala (constraint) g1, g6, g7, g8, dan g9, sebagai goals, ada lima macam simpangan yang perlu pembobotan Target NPV = 175.600 …………. Max NPV sesuai dg cash-flow - constraint Variabel fungsi tujuan: mencerminkan persentase simpangan dari target, bukan simpangan absolut. Model: Minimize the sum of the percentage deviations from targets

WGP : Minimize: n1 W1 ------------------ x 100/1 + 175.600 p6 4000 p7 W3 ------------------ x 100/1 + p8 W4 = --------------- x 100/1 + 2000 p9 W5 = -------------- x 100/1 1000 Subjected to:

Semua goals masuk ke dalam fungsi tujuan komposit Profit maximization measured in money terms can generally be taken as the planning objective on large commercial farms and estates, but this is increasingly constrained by external factors such as labour laws, health and safety regulations, and national policies to produce crops which will generate foreign exchange or serve as a basis for local industrialization. Internal constraints can also exist on such farms and take the form of management jealousy in protecting the 'mark' of their product even when production of lower quality produce might yield more profit, and spending more than the necessary amount of money on estate upkeep to maintain estate appearance and status. Profit maximization measured in money terms can also be the primary objective of some small independent specialized and small dependent specialized farms. Semua goals masuk ke dalam fungsi tujuan komposit

WGP : Subject to: 500X1 + 400X2 <= 15.000 6250X1 + 5000X2 +n1 – p1 = 175.000 120X1 + 180 X2 +n6 – p6 = 4000 400X1 + n7 - p7 = 2000 450X2 + n8 – p8 = 2000 35X1 +35X2 + n9 – p9 = 1000 X1 , X2 >= 0 nj, pj >= 0 j = 1 and j = 6, ……, 9 Dimana: w1, …………, w5 = pembobot bagi simpangan deviasi Pembobot ini dapat sama, atau dapat berbeda nilainya Misalnya: Petani lebih mementingkan pendapatan atau penghasilannya daripada sewa TK dan sewa traktor

GP : A critical assessment of GP Penerapannya harus dilandasi oleh logika ilmiah yang kuat dan benar Lima situasi dimana GP tidak bagus: 1. Apabila solusi optimal dengan menggunakan GP identik dengan solusi optimal yang diperoleh dnegan LP biasa 2. Trade-off antar goal dalam prioritas tertentu dapat dilakukan, tetapi trade-off lintas prioritas tidak dapat dilakukan 3. Kepekaan GP untuk menghasilkan situasi optimal -------- inferior 4. Maksimisasi dari “Achievement Function” dari GP tidak sama dengan “optimizing the utility function” dari decision maker 5. Apabila prioritas terlalu banyak.

GP : A critical assessment of GP

Some extension of GP : LGP & WGP Fractional GP: Apabila beberapa goals (misalnya struktur biaya usahatani) harus diintroduksi sebagai ratios atau sebagai fractional goals Minmax GP : Minimize the maximum of deviations Achievement of all goals must be greater than or equal to their targets e.g. Min. d ………………. max deviations s.t. nj <= d fj(X) + nj – pj = tj ………….. (target) X € F ……….. (feasible set)

MOP: Multiple Objective Programming DM a multiple objective environment the define goals mungkin tidak ada MOP Membedakan antara: Solusi layak yang Pareto Optimal, Solusi layak yang Non Pareto Optimal Konsep tradisional tentang optimal diganti dengan idea efisiensi dan / atau Non-dominansi

MOP: Multiple Objective Programming Multiobjective programming formally permits formulations where: solutions are generated which are as consistent as possible with target levels of goals; solutions are identified which represent maximum utility across multiple objectives; or c) solution sets are developed which contain all nondominated solutions. Multiple objectives can involve such considerations as leisure, decreasing marginal utility of income, risk avoidance, preferences for hired labor, and satisfaction of desirable, but not obligatory, constraints.

Approximation of the MOP Problem MOP: Problem optimasi simultan beberapa objektif yang menghadapi seperangkat kendala (biasanya linear) Mencoba mengidentifikasi “the set” yang mengandung solusi efisien (non-dominated dan Pareto Optimal) To generate the efficient set: Eff. Z(X) = [ Z1(X), Z2(X), …………. Zq(X) ] Subject to: X € F Eff ………….. Mencari solusi efisien F ………… Feasible set

unweighted multiple objectives; MOP: Problem optimasi simultan beberapa objektif yang menghadapi seperangkat kendala (biasanya linear) We will use "multiple objective programming" to refer to any mathematical program involving more than one objective regardless of whether there are goal target levels involved. For example: goal programming has been used to refer to multiple objective problems with target levels; b). multiobjective programming has been used to refer to only the class of problems with weighted or unweighted multiple objectives; c) vector maximization has been used to refer to problems in which a vector of multiple objectives are to be optimized; d) risk programming has been used to refer to multiobjective problems in which the objectives involve income and risk.

MOP : Misalnya : Petani mempunyai tua tujuan: 1. Memaksimumkan NPV investasinya dalam pengembangan kebun 2. Meminimumkan jumlah jam kerja TK-upahan dalam panen. Kendala luas kebun minimum 10 ha Modelnya adalah: Eff. Z(X) = [ Z1(X), Z2(X) ] Dimana: Z1(X) : 6250 X1 + 5000 X2 Z2(X) : - 400 X1 – 450 X2 Subject to: 550X1 + 400X2 <= 15.000 750X1 + 575X2 <= 22.000 1050X1 + 825X2 <= 29.000 1375X1 + 1025X2 <= 36.000 120X1 + 180 X2 <= 4000 35X1 +35X2 <= 1000 X1 + X2 >= 10 X >= 0

MOP : X2 1375X1 + 1025X2 = 36000 35X1 + 35X2 = 1000 D C E X1 + X2 >= 10 F 120X1 + 180X2 = 4000 A B X1 Feasible set of F adalah Poligon ABCDE Deskripsi untuk kelima titik ekstrim adalah sbb:

MOP : Titik Peubah Keputusan Fungsi Tujuan Ekstrim X1 X2 Z1(NPV) Z2(jam kerja sewaan) A 10 0 62.500 4.000 B 26.18 0 163.625 10.472 C 19.18 9.38 166.775 11.893 D 0 22.22 111.111 10.000 E 0 10 50.00 4.500 Kelima titik ekstrim tersebut melahirkan kima titik ekstrim baru dalam “RUANG TUJUAN”

NOP : Z2: Jam kerja TK 12.000 C’ 10.000 D’ F B’ 5000 E’ Ideal point A’ 70.000 110.000 170.000 Z1 = NPV A’B’C’ -------------- the efficient set dalam ruang tujuan ABC --------------- the efficient set dalam ruang peubah

MOP : The efficient set: Merupakan kurva transformasi yang mengukur hubungan antara dua macam atribut Slope dari garis A’B’ dan B’C’ mencerminkan trade-off (opportunity cost) di antara ke dua atribut Trade off antara NPV dan jam kerja di sepanjang A’B’ adalah: 163.625 – 62.500 T A’B’ = ---------------------------- = 25.28 rp/jam 10.472 – 4.000 Setiap jam kerja menghasilkan NPV = 25.28 Besarnya opportunity cost ini menjadi pertimbangan dalam menentukan pilihan oleh Decision Maker.

Matriks pay-off dalam MOP : Matriks pay-off untuk dua tujuan: NPV Jam kerja sewaan NPV 166.755 11.893 Jam kerja sewaan 62.500 4.000 Baris I : Maks NPV (166.755) sesuai dengan TK-sewaan 11.893 Baris II : TK-sewa minimum (4000 jam) sesuai dg NPV=62.500 Konflik antara tujuan NPV dan tujuan TK-sewaan: Max NPV menghasilkan TK-sewa yang tinggi (300%) Min TK-sewa menghasilkan NPV rendah (50%) Elemen dalam diagonal utama matriks pay-off disebut IDEAL-POINT (SOLUSI dimana SEMUA TUJUAN mencapai NILAI OPTIMUMNYA)

Kalau ada konflik di antara tujuan, maka ideal point ……… Kalau ada konflik di antara tujuan, maka ideal point ……….. TIDAK FEASIBLE Kebalikan dari Ideal Point adalah “Anti Ideal” atau “Nadir Point” . Perbedaan antara Ideal Point dan Nadir Point, merupakan kisaran nilai dari fungsi tujuan The decision theory is descriptive when it shows how people take decisions, and prescriptive when it tells people how they should take decisions.

MOP : The Constraint Method Ide dasar metode ini adalah: 1. Mengoptimalkan salah satu tujuan, sedangkan tujuan-tujuan lainnya dianggap “RESTRAINTS” 2. Efficient set diperoleh dengan jalan “parameterizing” RHS dari tujuan-tujuan yang dianggap sebagai RESTRAINTS Misalnya: Problematik MOP dengan fungsi tujuan: Max Zk (X) Subject to: X € F Zj (X) >= Lj j = 1, 2, ……., k-1, ……k+1, …., q Zk(X) : tujuan yang dioptimalkan Lj : RHS, divariasi secara parametrik http://www.environment.fhwa.dot....rces.asp

MOP : The Constraint Method Misalnya: NPV ditetapkan sebagai tujuan yang harus dioptimalkan, maka aplikasi metode Constraint ini menghasilkan LP parametrik sbb: Max. 6250X1 + 5000X2 (NPV) Subject to: X € F (technical constraints) 400X1 + 450X2 <= L1 ( hours of labor) Nilai L1 beragam antara 4000 – 11.893 jam/ha Dengan jalan parameterizing L1 untuk nilai-nilai antara 4000 – 11.893 akan diperoleh the efficient set.

MOP : The Constraint Method Nilai L1 beragam dalam kisaran 4000 – 11.893 jam/ha Aproksimasi efficient set ----------- Titik ekstrim sbb: X1 X2 Z1 Z2 RHS (L1) 19.18 9.38 166.755 11.893 11.893 23.59 3.47 164.788 11.000 11.000 26.05 0 163.713 10.500 10.500 26.18 0 163.625 10.472 10.472 25.0 0 156.25 10.000 10.000 22.50 0 140.625 9.000 9.000 20.00 0 125.000 8.000 8.000 17.50 0 109.375 7.000 7.000 15.00 0 93.750 6.000 6.000 12.50 0 78.125 5.000 5.000 11.25 0 70.312 4.500 4.500 10.00 0 62.500 4.000 4.000 parameterizing

MOP : The Weighting Method Ide dasar metode ini adalah: Mengkombinasikan semua tujuan menjadi satu fungsi tujuan tunggal Setiap fungsi tujuan diberi pembobot , kemudian baru dijumlahkan (+) The efficient set diperoleh melalui variasi parametrik dari pembobot. Misalnya: Problem MOP dengan q-tujuan yang harus dimaksimumkan: Max W1Z1(X) + W2Z2(X) + ………. + WqZq(X) Subject to: X € F W >= 0 Model LP parametriknya sbb: Max W1(6250X1 + 5000X2) + W2(-400X1 – 450X2) Subject to: X € F (kendala teknis) W1 >= 0, W2 >= 0

Dengan menetapkan : W1 + W2 = 1 dan memvariasikannya secara parametrik, maka diperoleh: Untuk: 0.4 <= W1 <= 1 Titik optimalnya C atau C’ 0 <= W2 <= 0.6 Untuk: 0.1 <= W1 <= 0.4 Titik optimalnya B atau B’ 0.6 <= W2 <= 0.9 Untuk: 0 <= W1 <= 0.1 Titik optimalnya A atau A’ 0.9 <= W2 <= 1.0 W (pembobot): preferensi pengambil keputusan terhadap masing-masing tujuan, bukan menyatakan kepentingan dari masing-masing tujuan W merupakan parameter yang dapat divariasikan secara sistematik untuk menghasilkan “EFFICIENT SET”

MULTIGOAL PROGRAMMING Metode ini berada di antara GP dan MOP Metode ini bekerja meminimumkan SIMPANGAN Misalnya: Max NPV = 166.755 Labor = 6000 jam Tractor = 1000 jam Model: Eff. Z(n,p) = [ Z1(n,p), Z2(n,p), Z3(n,p) ] Dimana: Z1(n,p) = p1 Z2(n,p) = p2 Z3(n,p) = p3 Subject to: 1375X1 + 1925X2 <= 36.000 X1 + X2 >= 10 120X1 + 180X2 <= 4000 400X1 + 450X2 + n1 – p1 = 6000 35X1 + 35X2 + n2 – p2 = 1000 6250X1 + 5000X2 + n3 – p3 = 166.755 X >= 0 n >= 0 p >= 0

…………dsst……. www//marno.lecture.ub.ac.id