what is multiple criteria analysis?

1

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.

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 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), dimanapemilihan yang diambil adalah pilihan yang paling

memenuhi fungsi obyektif.

3


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


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 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

5

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.

6


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

7


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

8


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).

9


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


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

11


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

Productivity is primarily a measure of the relative suitability of a system or activity in a particular

agro-ecological environment.

On commercial farms it is an indicator of relative efficiency of resource use and management

performance. It is an underlying condition for profitability but should not necessarily be taken

as a desirable attribute or objective in itself.

On non-commercial farms, productivity is a necessary condition for achieving family

sustainability - but only to a limit.

12


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

Profit is normally measured in money terms as gross financial revenue minus total financial cost

per period.

Note, however, that it may - if need be - also be assessed subjectively in qualitative terms as net gain, i.e., as total benefit less total cost however

measured.

Such an approach might be used in assessing the performance of subsistence farms having no

significant market interaction, leading to qualitative assessment of a system as, e.g., profitable or not

profitable.

13


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.

http://rfptemplates.technologyevaluation.com/What-is-a-Decision-Matrix.html

14


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)

15


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 50.000Sol 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

16


Gross margin Labor IndeptednessSol 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)

17

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.

18


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

The actual decision boils down to selecting "a good choice" from a number of available choices. Each choice represents a decision alternative. In the multi-criteria decision-making (MCDM) context, the selection is facilitated by evaluating

each choice on the set of criteria.

The criteria must be measurable - even if the measurement is performed only at the nominal scale (yes/no;

present/absent) and their outcomes must be measured for every decision alternative. Criterion outcomes provide the

basis for comparison of choices and consequently facilitate the selection of one, satisfactory choice.

19


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

20

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

21

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.

22


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

23

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.

24


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

25

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 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 …………….. g2750X1 + 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

26

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.

27

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.

28

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

29

LGP : Optimum Solution

Optimum solution: X1 = 19.18 X2 = 9.38

Deviation 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

30

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

.

31

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.

32

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.

33

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

34

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:

35

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.

36

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

37

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.

38

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)

39

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-dominansiMultiobjective programming formally permits formulations

where:a) solutions are generated which are as consistent as

possible with target levels of goals; b) 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.

40

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

Sustainability is meant the capacity of a system to maintain its productivity/profitability at a satisfactory level over a long or indefinite time period regardless

of year-to-year fluctuations (i.e., of its short-term instability).

In an agricultural production context, sustainability is relevant to farming systems of whatever

composition, but not necessarily to the individual production phases of short-term crops.

The concept involves the evaluation of farm activities and systems in terms of their (interrelated)

ecological, economic and socio-cultural sustainability over long time periods of many years.

41

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.

42

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 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

43

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:

44

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”

45

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

46

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.

47

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)

48

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.

49

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

50


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.

51


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 11.89323.59 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

52

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

53

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”

54

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

what is multiple criteria analysis?

Documents

such as utility or profit

memaksimumkan the gross

a and bwater requirement

aspek kriteria jamak