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WHAT IS MULTIPLE CRITERIA ANALYSIS?. MCA describes any structured approach used to determine overall preferences among alternative options, where the options accomplish several objectives. - PowerPoint PPT Presentation
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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.
WHAT IS MULTIPLE CRITERIA ANALYSIS?
2
MULTIPLE CRITERIA DECISION MAKING (MCDM)
SITUASI PENGAMBILAN KEPUTUSAN:
1. Involving a single decision criteria ( SINGLE OBJECTIVE)2. Involves several conflicting objectives (MULTIPLE OBJECTIVE)
Analisis Pengambilan Keputusan:
1. A decision maker2. An array of feasible choices3. A well defined criteria, such as utility or profit: SINGLE or MULTIPLE
Multiple Criteria Decision Making (MCDM) merupakan suatu metode pengambilankeputusan yang didasarkan atas teori-teori, proses-proses, dan metode analitik yang
melibatkanketidakpastian, dinamika, dan aspek kriteria jamak.
Dalam metode optimasi konvensional,cakupan umumnya hanya dibatasi pada satu kriteria pemilihan (mono criteria),
dimanapemilihan yang diambil adalah pilihan yang paling memenuhi fungsi obyektif.
3
MULTIPLE CRITERIA DECISION MAKING (MCDM)
Economic vs. Technological Decisions.
Technological decision: a single criterionEconomic decision: a multiple criteria
Technological problems: Search and measurement
Scarce Economic Technological means problems problems
No scarce No problemsproblem
Several Single criteria criterion
4
MULTIPLE CRITERIA DECISION MAKING (MCDM)
Ilustrasi:1. Ke supermarket untuk MEMILIH produk sirup yang Paling
Murah2. Mencari pola tanam yang memaksimumkan the gross margin
(1) dan (2) : a technological problem
(2) Untuk menyelesaikannya: ONLY SEARCHES.
Decision Making does not reallyMulti-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 reallyMCDM 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.
MULTIPLE CRITERIA DECISION MAKING (MCDM)
Ilustrasi:
Pola tanam yang:Max gross marginMin Risk Conflicting objectivesMin Indebtedness
Solution this problem: Economic decision ………. Optimal solution
e.g. Development of a small rural region 1000 ha arable land:
Two crops: A and BWater requirement: 4000 and 5000 m3/haWater available : 4.200.000 m3Syarat rotasi tanaman: Luas tanam B <= luas tanam A
X1 = luas tanam AX2 = luas tanam B
X1 + X2 <= 10004000 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 A466.66
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 goalsMultiple 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.
10
MULTIPLE CRITERIA DECISION MAKING (MCDM)
TUJUAN GANDA DALAM PERTANIAN
Farm Level:Goals in agriculture DM:1. Maximum gross margin2. Minimum seasonal cash exposure3. Provision od stable employment for the permanent labor
Ranch planning:1. Red meat production2. Use of fossil fuel energy3. Profits
Land allocation problems:1. Money income2. Environmental benefits
FARM SYSTEM PROPERTIES AND PERFORMANCE CRITERIA
1. Productivity2. Profitability3. Stability4. Diversity5. Flexibility6. Time-dispersion7. Sustainability8. 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 : weightf(X): atributes function
MULTIPLE CRITERIA DECISION MAKING (MCDM)
TARGET = as aspiration levelan 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 marginObjective: Gross margin minimizeGoal : 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 IndeptednessSol I 200.000 500 50.000Sol II 200.000 600 50.000Sol III 300.000 700 60.000
DM wants:1. Gross margin,……….. As large as possible2. Labor and indeptedness ……….. As small as possible
MULTIPLE CRITERIA DECISION MAKING (MCDM)
Gross margin Labor Indeptedness
Sol IRendah Rendah Rendah ………. efisienSol II Rendah Tinggi Rendah ………. Tdk efisienSol 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)
DM preference for each of criteria ………….
(pembobotan)
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.
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 Programming2. Multiple Objectives ……… MOP: Multi Objective Program3. 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
d+ : 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 50003. Resources Uses:
Capital Year 1 550 400Year 2 200 175Year 3 300 250Year 4 325 200
4. Annual labor Prunning 120 180Harvest 400 450
5. Mesin pengolahan (jam/ha)35 35Ketersediaan sumberdaya:
1. Kapital tahun 1 : 15.000 tahun 2 s/d 4 : 7.000 per tahun
2. TK prunning : 4000 jam/ musimTK panen : 2000
3. Max. tractor hours : 10004. Periode panen dua macam tanaman berbeda.
GOAL PROGRAMMING: Farm Planning Model
Tujuan Usahatani:1. Maximize NPV2. Minimize pinjaman kapital selama 4 tahun3. Minimize TK musiman untuk prunning dan panen4. Minimize sewa traktor
(these are conflicting interests)
Strategi dengan Linear Programming biasa:1. NPV ------------- dimaksimumkan2. Tujuan lain --------- sebagai kendala sumberdaya3. Cash resources: Surplus tahun 1 dimasukkan sebagai tambahan tahun berikutnya
Max Z = f(X1,X2) = 6250 X1 + 5000 X2Subject to:
500X1 + 400X2 <= 15.000750X1 + 575X2 <= 22.0001050X1 + 825X2 <= 29.0001375X1 + 1025X2 <= 36.000120X1 + 180 X2 <= 4000400X1 <= 2000450X2 <= 200035X1 +35X2 <= 1000X1 >= 0X2 >= 0
Tujuan Usahatani:1. Maximize NPV2. Minimize pinjaman kapital selama 4 tahun3. Minimize TK musiman untuk prunning & panen4. 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 haX2 = 4.44 haNPV = 53.450
Tenaga kerja panen digunakan semuaSumberdaya lainnya tidak habis digunakan, ada sisa sumberdaya
Menurut LP ini optimal karena:1. Objectives yang diformulasikan sebagai kendala dipenuhi dulu sebelum NPV2. 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 kendalaRHS merupakan target yg dapat tercapai atau hanya dapat didekatiUntuk 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 …………….. g31050X1 + 825X2 + n4 – p4 = 29.000 …………….. g41375X1 + 1025X2 +n5 – p5 = 36.000 …………….. g5120X1 + 180 X2 + n6 – p6 = 4000 .…………….. g6400X1 + n7 – p7 = 2000 …………….. g7450X2 + n8 – p8 = 2000 …………….. g835X1 +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 perencanaanMenetapkan prioritas goals: Qi >>>> QjPrioritas tinggi dipenuhi lebih dahulu: Lexicographic order
e.g. Q1 : untuk g2, g3, g4, g5 adalah p2, p3, p4, p5Q2 : untuk g9 : p9Q3 : untuk g1: n1Q4 : 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.
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 …………….. g2750X1 + 575X2 + n3 – p3 = 22.000 …………….. g31050X1 + 825X2 + n4 – p4 = 29.000 …………….. g41375X1 + 1025X2 +n5 – p5 = 36.000 …………….. g5
Q4 120X1 + 180 X2 + n6 – p6 = 4000 .…………….. g6400X1 + n7 – p7 = 2000 …………….. g7450X2 + n8 – p8 = 2000 …………….. g8
Q2: 35X1 +35X2 + n9 – p9 = 1000 …………….. g9
Xi >= 0; nj >= 0, pj >= 0i = 1, 2j = 1, ……, 9
LGP : Optimum Solution
Optimum solution: X1 = 19.18 X2 = 9.38Deviation variable:
n1 = 33.250 p1 = 0n2 = 699 p2 = 0n3 = 2.221 p3 = 0n4 = 1.122 p4 = 0n5 = n6 = 0 p5 = p6 = 0n7 = 0 p7 = 5672n8 = 0 p8 = 2211n9 = 0 p9 = 0
Prioritas I (Q1) ---------------- g5 tercapaiPrioritas II (Q2) --------------- g9 tercapaiPrioritas IV (Q4) -------------- g6 tercapai
Dibandingkan dengan penyelesaian LP di atas, maka:NPV lebih tinggi
Sumberdaya ----------- habis dipakai, … kurangModal ------------------- ada sisa
LGP : Sensitivity Analysis
Kelemahan LGP: memerlukan banyak informasi dari Decision Maker, a.l.TargetWeightPriority orderedPreferences
Kalau informasi ini tidak ada, maka harus dilakukan analisis sensitivitas:Pengaturan kembali prioritasNilai-nilai targetPembobot
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 dipertukarkanSolusi II: Optimal untuk 12 dari 24 alternatif prioritasSolusi III: Kalau prioritas III digabungkan dengan prioritas IIDst.
LGP :
Pengubahan nilai target dari beberapa goal, misalnya:
1. 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
2. Kalau target g9 dikurangi, maka solusi optimum berubah, NPV menurunKalau g9 ditingkatkan, maka solusi optimum dapat berubah dan NPV naik
3. Kalau target g6, g7, g8 berubah, maka:Nilai solusi optimum tidak berubahSimpangan 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:
n1W1 ------------------ x 100/1 +
175.600
p6W2 ------------------ x 100/1 +
4000
p7W3 ------------------ x 100/1 +
p8
W4 = --------------- x 100/1 + 2000
p9W5 = -------------- 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.
WGP :
Subject to:500X1 + 400X2 <= 15.000750X1 + 575X2 <= 22.0001050X1 + 825X2 <= 29.0001375X1 + 1025X2 <= 36.000
6250X1 + 5000X2 +n1 – p1 = 175.000120X1 + 180 X2 +n6 – p6 = 4000400X1 + n7 - p7 = 2000450X2 + n8 – p8 = 200035X1 +35X2 + n9 – p9 = 1000X1 , X2 >= 0 nj, pj >= 0j = 1 and j = 6, ……, 9
Dimana: w1, …………, w5 = pembobot bagi simpangan deviasiPembobot ini dapat sama, atau dapat berbeda nilainyaMisalnya: 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 -------- inferior4. Maksimisasi dari “Achievement Function” dari GP tidak sama dengan
“optimizing the utility function” dari decision maker5. Apabila prioritas terlalu banyak.
38
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 deviationss.t. nj <= dfj(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:
1. solutions are generated which are as consistent as possible with target levels of goals;
2. solutions are identified which represent maximum utility across multiple objectives; or
3. 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 € FEff ………….. Mencari solusi efisienF ………… Feasible set
43
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:
a) 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.
44
MOP :
Misalnya : Petani mempunyai tua tujuan:
1. Memaksimumkan NPV investasinya dalam pengembangan kebun2. 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 X2Z2(X) : - 400 X1 – 450 X2
Subject to:550X1 + 400X2 <= 15.000750X1 + 575X2 <= 22.0001050X1 + 825X2 <= 29.0001375X1 + 1025X2 <= 36.000120X1 + 180 X2 <= 400035X1 +35X2 <= 1000X1 + X2 >= 10 X >= 0
45
MOP :
X2
1375X1 + 1025X2 = 36000
35X1 + 35X2 = 1000
D
C E X1 + X2 >= 10
F
120X1 + 180X2 = 4000
A BX1
Feasible set of F adalah Poligon ABCDE
Deskripsi untuk kelima titik ekstrim adalah sbb:
MOP :
Titik Peubah Keputusan Fungsi TujuanEkstrim X1 X2 Z1(NPV) Z2(jam kerja sewaan)
A 10 0 62.500 4.000B 26.18 0 163.625 10.472C 19.18 9.38 166.775 11.893D 0 22.22 111.111 10.000E 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 tujuanABC --------------- 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 – 62.500T 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 11.893Jam 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 ……….. 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 € FZj (X) >= Lj j = 1, 2, ……., k-1, ……k+1, …., q
Zk(X) : tujuan yang dioptimalkanLj : 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 MethodNilai 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 11.89323.59 3.47 164.788 11.000 11.00026.05 0 163.713 10.500 10.50026.18 0 163.625 10.472 10.47225.0 0 156.25 10.000 10.00022.50 0 140.625 9.000 9.00020.00 0 125.000 8.000 8.00017.50 0 109.375 7.000 7.00015.00 0 93.750 6.000 6.00012.50 0 78.125 5.000 5.00011.25 0 70.312 4.500 4.50010.00 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 € FW >= 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 = 166.755Labor = 6000 jamTractor = 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 <= 36.000X1 + X2 >= 10120X1 + 180X2 <= 4000400X1 + 450X2 + n1 – p1 = 600035X1 + 35X2 + n2 – p2 = 10006250X1 + 5000X2 + n3 – p3 = 166.755X >= 0 n >= 0 p >= 0
…………dsst…….www//marno.lecture.ub.ac.id