eksplorasi generic algebraic modeling system (gams) dalam

Ahmad Dinu Haq, Diah Chaerani, Isah AisahDepartement of Mathematics

Faculty of Mathematics and Natural Science

University of Padjadjaran

[email protected]

SOUTHEAST ASIAN MATHEMATICAL SOCIETY (SEAMS) SCHOOLON SPATIO TEMPORAL, DATA MINING AND OPTIMIZATION MODELING

UNPAD TRAINING CENTER, 9-19 AUG 2016

mailto:[email protected]

Water Sources

River Lake Reservoir Groundwater

Water

Model (Murray et al, 2012) :

Subject to

where

Minimize (1)it it

i t

p Z

; , (2)

; , , (3)

; , (4)

; , (5)

, , 0; ,

0; , ,

i

i

ikt it it it

k

ikt ikt

t

ijtl t it

l t j

t

ijtl it it

l t j

it it it

ijtl i

S X X d i t

S u i k t

Y X i t

Y Z X i t

X X Z i t

Y i j

, (6)

0; , ,ikt

t l

S i k t

i : index of area (also j)

k : index of water sources

t : index of time periods (l also)

: maximum supply of water in area i by source k in

period t

: demand for water in area i, period t

: set of areas that can be supplied by surplus in area i

: importance weight for area i, period t

: percentage of water surplus in period t permissible

to allocate to other areas/periods

: maximum number of years that water surplus could

be saved for future years

: utilized supply of water in area i by source k in

period t

: amount of water surplus in area i, period t

:amount of water deficit in area i, period t

: amount of water transferred from area i in period t

to area j in period l

: deficit after transfers in area i, period t

: population in area i

iktu

itd

i

itp

t

iktS

itX

itX

ijtlY

itZ

max 0, ikt itit

k it

u dp

pop

itpop

Allocation plot for water supply in Tarum Barat where k=1, i=1, j=2, dan t=12

Assumption:

• It is assumed that a water provider would not exceed 95 % of its surplus water supply in

any period, so αt = 0.95 in all periods

• a limit on the storage is assumed to be five years, so λ=5

Bendung

Curug

DI

Bendung

Beet

DI

Bendung

Cikarang

DI

Bendung

Bekasi

k

k

i

k

j

k

𝑖, 𝑡 1,1 1,2 1,3 1,4 1,5 1,6 1,7 1,8 1,9 1,10 1,11 1,12

𝑢𝑖 ,𝑡 44.71 44.21 57.25 26.83 18.87 25.04 37.22 57.61 76.48 89.20 58.55 38.06

𝑑𝑖,𝑡 114 114 114 114 114 114 114 114 114 114 114 114

𝑝𝑜𝑝𝑖 ,𝑡 199593 199593 199593 199593 199593 199593 199593 199593 199593 199593 199593 199593

𝑝𝑖 ,𝑡 0.0003

47156

0.0003

49647

0.0002

84329

0.0004

36739

0.0004

7662

0.0004

45707

0.0003

84683

0.0002

82525

0.0001

87983

0.0001

24253

0.0002

77815

0.0003

80474

1,1 1,2 1,3

1,4 1,5 1,6

1,7 1,8 1,9

Minimum 0.000347156 0.000349647 0.000284329

0.000436739 0.00047662 0.000445707

0.000384683 0.000282525 0.000187983

Z Z Z

Z Z Z

Z Z Z

1,10 1,11 1,12 0.000124253 0.000277815 0.000380474Z Z Z

subject to

1,1,1 1,1 1,1 1,1

1,1,2 1,2 1,2 1,2

1,1,3 1,3 1,3 1,3

1,1,4 1,4 1,4 1,4

1,1,5 1,5 1,5 1,5

1,1,6 1,6 1,6 1,6

1,1,7 1,7 1,7 1,7

1,1,8 1,8 1,8 1,8

1,1,9 1,9 1,

S X X d

S X X d

S X X d

S X X d

S X X d

S X X d

S X X d

S X X d

S X X

9 1,9

1,1,10 1,10 1,10 1,10

1,1,11 1,11 1,11 1,11

1,1,12 1,12 1,12 1,12

d

S X X d

S X X d

S X X d

1,1,1 1,1,1

1,1,2 1,1,2

1,1,3 1,1,3

1,1,4 1,1,4

1,1,5 1,1,5

1,1,6 1,1,6

1,1,7 1,1,7

1,1,8 1,1,8

1,1,9 1,1,9

1,1,10 1,1,10

1,1,11 1,1,11

1,1,12 1,1,12

S u

S u

S u

S u

S u

S u

S u

S u

S u

S u

S u

S u

1,1,1,1 1,1,1,2 1,1,1,3 1,1,1,4 1,1,1,5 1,1,1,6

1,2,1,1 1,2,1,2 1,2,1,3 1,2,1,4 1,2,1,5 1,2,1,6 1,10.95

Y Y Y Y Y Y

Y Y Y Y Y Y X

1,1,2,2 1,1,2,3 1,1,2,4 1,1,2,5 1,1,2,6 1,1,2,7

1,2,2,2 1,2,2,3 1,2,2,4 1,2,2,5 1,2,2,6 1,2,2,7 1,20.95

Y Y Y Y Y Y

Y Y Y Y Y Y X

1,1,3,3 1,1,3,4 1,1,3,5 1,1,3,6 1,1,3,7 1,1,3,8

1,2,3,3 1,2,3,4 1,2,3,5 1,2,3,6 1,2,3,7 1,2,3,8 1,30.95

Y Y Y Y Y Y

Y Y Y Y Y Y X

1,1,4,4 1,1,4,5 1,1,4,6 1,1,4,7 1,1,4,8 1,1,4,9

1,2,4,4 1,2,4,5 1,2,4,6 1,2,4,7 1,2,4,8 1,2,4,9 1,40.95

Y Y Y Y Y Y

Y Y Y Y Y Y X

1,1,5,5 1,1,5,6 1,1,5,7 1,1,5,8 1,1,5,9 1,1,15,10

1,2,5,5 1,2,5,6 1,2,5,7 1,2,5,8 1,2,5,9 1,2,5,10 1,50.95

Y Y Y Y Y Y

Y Y Y Y Y Y X

1,1,6,6 1,1,6,7 1,1,6,8 1,1,6,9 1,1,6,10 1,1,6,11

1,2,6,6 1,2,6,7 1,2,6,8 1,2,6,9 1,2,6,10 1,2,6,11 1,60.95

Y Y Y Y Y Y

Y Y Y Y Y Y X

1,1,7,7 1,1,7,8 1,1,7,9 1,1,7,10 1,1,7,11 1,1,7,12

1,2,7,7 1,2,7,8 1,2,7,9 1,2,7,10 1,2,7,11 1,2,7,12 1,70.95

Y Y Y Y Y Y

Y Y Y Y Y Y X

1,1,8,8 1,1,8,9 1,1,8,10 1,1,8,11 1,1,8,12 1,1,8,13

1,2,8,8 1,2,8,9 1,2,8,10 1,2,8,11 1,2,8,12 1,2,8,13 1,80.95

Y Y Y Y Y Y

Y Y Y Y Y Y X

1,1,9,9 1,1,9,10 1,1,9,11 1,1,9,12 1,1,9,13 1,1,9,14

1,2,9,9 1,2,9,10 1,2,9,11 1,2,9,12 1,2,9,13 1,2,9,14 1,90.95

Y Y Y Y Y Y

Y Y Y Y Y Y X

1,1,10,10 1,1,10,11 1,1,10,12 1,1,10,13 1,1,10,14 1,1,10,15

1,2,10,10 1,2,10,11 1,2,10,12 1,2,10,13 1,2,10,14 1,2,10,15 1,100.95

Y Y Y Y Y Y

Y Y Y Y Y Y X

1,1,11,11 1,1,11,12 1,1,11,13 1,1,11,14 1,1,11,15 1,1,11,16

1,2,11,11 1,2,11,12 1,2,11,13 1,2,11,14 1,2,11,15 1,2,11,16 1,110.95

Y Y Y Y Y Y

Y Y Y Y Y Y X

1,1,12,12 1,1,12,13 1,1,12,14 1,1,12,15 1,1,11,16 1,1,11,17

1,2,12,12 1,2,12,13 1,2,12,14 1,2,12,15 1,2,12,16 1,2,12,17 1,120.95

Y Y Y Y Y Y

Y Y Y Y Y Y X

1,1,6,1 1,1,6,2 1,1,6,3 1,1,6,4 1,1,6,5 1,1,6,6

1,2,6,1 1,2,6,2 1,2,6,3 1,2,6,4 1,2,6,5 1,2,6,6 1,6 1,6

Y Y Y Y Y Y

Y Y Y Y Y Y Z X

1,1,7,2 1,1,7,3 1,1,7,4 1,1,7,5 1,1,7,6 1,1,7,7

1,2,7,2 1,2,7,3 1,2,7,4 1,2,7,5 1,2,7,6 1,2,7,7 1,7 1,7

Y Y Y Y Y Y

Y Y Y Y Y Y Z X

1,1,8,3 1,1,8,4 1,1,8,5 1,1,8,6 1,1,8,7 1,1,8,8

1,2,8,3 1,2,8,4 1,2,8,5 1,2,8,6 1,2,8,7 1,2,8,8 1,8 1,8

Y Y Y Y Y Y

Y Y Y Y Y Y Z X

1,1,9,4 1,1,9,5 1,1,9,6 1,1,9,7 1,1,9,8 1,1,9,9

1,2,9,4 1,2,9,5 1,2,9,6 1,2,9,7 1,2,9,8 1,2,9,9 1,9 1,9

Y Y Y Y Y Y

Y Y Y Y Y Y Z X

1,1,10,5 1,1,10,6 1,1,10,7 1,1,10,8 1,1,10,9 1,1,10,10

1,2,10,5 1,2,10,6 1,2,10,7 1,2,10,8 1,2,10,9 1,2,10,10 1,10 1,10

Y Y Y Y Y Y

Y Y Y Y Y Y Z X

1,1,11,6 1,1,11,7 1,1,11,8 1,1,11,9 1,1,11,10 1,1,11,11

1,2,11,6 1,2,11,7 1,2,11,8 1,2,11,9 1,2,11,10 1,2,11,11 1,11 1,11

Y Y Y Y Y Y

Y Y Y Y Y Y Z X

1,1,12,7 1,1,12,8 1,1,12,9 1,1,12,10 1,1,12,11 1,1,12,12

1,2,12,7 1,2,12,8 1,2,12,9 1,2,12,10 1,2,12,11 1,2,12,12 1,12 1,12

Y Y Y Y Y Y

Y Y Y Y Y Y Z X

Plot of water provider: The model:

Bendung

Curug

DI

Walahar

DI

Salamdarma

DI Macan

DI Gadung

k i

j

𝑖, 𝑡 1,1 1,2 1,3 1,4 1,5 1,6 1,7 1,8

𝑢𝑖 ,𝑡 285 285 285 285 285 285 285 285

𝑑𝑖 ,𝑡 93.94 165.20 133.33 127.53 183.19 239.27 220.70 60.36

𝑝𝑜𝑝𝑖,𝑡 41609 41609 41609 41609 41609 41609 41609 41609

𝑝𝑖 ,𝑡 0.0045917

95

0.0028791

85

0.0036451

25

0.0037845

18

0.0024468

26

0.0010990

41

0.0015453

39

0.005398

832

Plot of water provider: The model:

Bendung

Curug

DI

Walahar

DI

Jengkol

DI

Barugbug

k i

j

𝑖, 𝑡 1,1 1,2 1,3 1,4 1,5 1,6 1,7 1,8

𝑢𝑖 ,𝑡 285 285 285 285 285 285 285 285

𝑑𝑖 ,𝑡 93.94 165.20 133.33 127.53 183.19 239.27 220.70 60.36

𝑝𝑜𝑝𝑖,𝑡 41609 41609 41609 41609 41609 41609 41609 41609

𝑝𝑖 ,𝑡 0.0045917

95

0.0028791

85

0.0036451

25

0.0037845

18

0.0024468

26

0.0010990

41

0.0015453

39

0.005398

832

• The General Algebraic Modeling

System (GAMS) is specifically designed

for modeling linear, nonlinear and mixed

integer optimization problems.

• Using GAMS, data are entered only

once in familiar list and table form.

• Models are described in concise

algebraic statements

The Main Function:

To build mathematics algorithm

model and solve the model by giving the

optimal solution

Inputs Outputs

Sets Echo Print

Data Reference Maps

Variable Equation Listing

Equation Status Reports

Model dan Solve Results

The Basic Components of GAMS

How to Install GAMS?

You can download GAMS on this website

www.gams.com/download/

http://www.gams.com/download/

For Example:

Max 109 90 115

subject to

100 (land)

6 4 8 500 (labor)

, , 0 (nonnegativity)

corn wheat peanut

corn wheat peanut

corn wheat peanut

corn wheat peanut

X X X

X X X

X X X

X X X

the number of the land area devoted to corn production

= the number of the land area devoted to wheat production

= the number of the land area devoted to peanut production

corn

wheat

peanut

X

X

X

The optimal solution

50, 50, 0

9950

corn wheat peanutX X X

Z

SCALARS d1 /nilai_numerik/, d2

/nilai_numerik/, d3

/nilai_numerik/, ...

u1 /nilai_numerik/, u2 /nilai_numerik/, u3

/nilai_numerik/, ...;

VARIABLES F Fungsi Objektif

Xs1, Xs2, Xs3, ...

Xd1, Xd2, Xd3, ...

Z1, Z2, Z3, ...

S1, S2, S3, ...

Y1, Y2, Y3, ...

Ya, Yb, Yc, ...;

POSITIVE VARIABLES

Xs1, Xs2, Xs3, ...

Xd1, Xd2, Xd3, ...

Z1, Z2, Z3, ...

S1, S2, S3, ...

Y1, Y2, Y3, ...;

EQUATIONS FungsiObjektif meminimumkan

defisit air

Kendala1a (kendala untuk persamaan (2.4))

Kendala2a

Kendala3a

.

.

.

Kendala1b (kendala untuk persamaan (2.5))

Kendala2b

Kendala3b

.

.

.

Kendala1c (kendala untuk persamaan (2.6))

Kendala2c

Kendala3c

.

.

.

Kendala1d (kendala untuk persamaan (2.7))

Kendala2d

Kendala3d

.

.

.

;

FungsiObjektif.. F =e= p1*Z1 + p2*Z2 + p3*Z3 + ... ;

Kendala1a.. S1-Xs1+Xd1 =e= d1;



.

.

.

Kendala1b.. S1 =l= u1;



.

.

.

Kendala1c.. Y1+Y2+Y3+... =l= alpha*Xs1;

Kendala2c.. ... =l= alpha*Xs2;

Kendala2c.. ... =l= alpha*Xs2;

.

.

.

Kendala1d.. Ya+Yb+Yc+... +Z1 =e= Xd1;

Kendala2d.. ... +Z2 =e= Xd2;

Kendala3d.. ... +Z3 =e= Xd3;

.

.

.

MODEL Alokasi_Air /all/;

SOLVE Alokasi_Air USING LP MINIMIZING F;

Optimal Solution for Each Numeric Experiment

Variable I II III

F 0 0 0

44.71 93.94 93.94

44.21 165.20 165.20

57.25 133.33 133.33

26.83 127.53 127.53

18.87 183.19 183.19

25.04 239.27 239.27

37.22 220.70 220.70

57.61 60.36 60.36

76.48 - -

Variable I II III

89.20 - -

58.55 - -

38.06 - -

25.04 239.27 239.27

37.22 220.70 220.70

57.61 60.36 60.36

76.48 - -

89.20 - -

58.55 - -

38.06 - -

1,1X

1,2X

1,3X

1,4X

1,5X

1,6X

1,7X

1,8X

1,9X

1,10X

1,11X

1,12X

1,1,6,1Y

1,1,7,2Y

1,1,8,3Y

1,1,9,4Y

1,1,10,5Y

1,1,12,7Y

1,1,11,6Y

Bazara, M.S. dan Jarvis, J.J. 1977. Linear Programming and Network Flows. Canada: John Wiley

& Sons, Inc.

Broke, A., Kendrick, D., Meeraus, A., dan Raman, R. 1998. GAMS A User’s Guide. Washington:

GAMS Development Coorporation.

Bussieck, M.R., dan Meeraus, A. 2004. General Algebraic Modeling System. Dalam Josef

Kallrath (Ed.), Modelling Languages in Mathematical Optimization (hlm. 137-157). United

States of America: Springer US.

Chaerani, D., Ruchjana, B.N., Dewanto, S.P., Abdullah, A.S., Rejito, J., Rosandi, Y., dan

Dharmawan, I.A. 2016. Determining The Robust Counterpart of Uncertain Spatial

Optimization Model for Water Supply Allocation Problem. Application of Mathematics in

Industry and Life AIP Conference Proceedings, 1716: 020002-1 - 020002-9.

Chattopadhyay, D. 1999. Application of General Algebraic Modeling System to Power System

Optimization. IEEE Transactions on Power Systems, 14 (1): 15-22.

Cressie, N.A.C. 1993. Statistics for Spatial Data. United States of America: John Wiley & Sons,

Inc.

Hiller, F.S. and Lieberman, G.J. 2001. Introduction to Operation Research (7th ed.). New York:

The McGraw Hill.

McCarl, B.A. 2002. GAMS User Guide, (Online),

http://www.gams.com/help/topic/gams.doc/userguides/mccarl/mccarlgamsuserguide.pdf,

(diakses 22 November 2015).

McKinney, D.C. dan Savitsky, A.G. 1999. Basic Optimization Models for Water and Energy

Management, (Online), http://www.ce.utexas.edu/prof/mckinney/ce385d/papers/GAMS-

Tutorial.pdf, (diakses 01 September 2015).

Murray, A.T., Gober, P., Anselin, L., Rey, S.J., Sampson, D., Padegimas, P.D., dan Liu, Y. 2012.

Spatial Optimization Models for Water Supply Allocation. Water Resour Manage, 26 (8):

2243-2257.Nazir, M. 2003. Metode Penelitian. Jakarta: Ghalia Indonesia.

Nazir, M. 2003. Metode Penelitian. Jakarta: Ghalia Indonesia.

Peraturan Direksi Perum Jasa Tirta II Nomor : 1/7 /PRT/2014 tentang Rencana Pokok

Penyediaan dan Penggunaan Air untuk Tanam Padi Rendeng Musim Tanam 2014/2015,

Tanam Padi Gadu dan Tanam Palawija Musim Tanam 2015 serta Kebutuhan Air untuk Air

Minum, Industri, dan Perkebunan Tahun 2014/2015 di Daerah Irigasi Jatiluhur. 2014. Jawa

Barat: Perum Jasa Tirta II.

Prahasta, E. 2009. Sistem Informasi Geografis: Konsep-konsep Dasar (Perspektif Geodesi &

Geomatika). Bandung: Penerbit Informatika.

Rao, S.S. 2009. Engineering Optimization Theory and Practice. New Jersey. John Wiley and

Sons, Inc.

http://www.gams.com/help/topic/gams.doc/userguides/mccarl/mccarlgamsuserguide.pdf

http://www.ce.utexas.edu/prof/mckinney/ce385d/papers/GAMS-Tutorial.pdf

Ruchjana, B.N. 2002. Suatu Model Generalisasi Space-Tine Autoregresi dan Penerapannya pada

Produksi Minyak Bumi. Disertasi tidak diterbitkan. Bandung: Institut Teknologi Bandung.

http://gams.com , di akses 21 Maret 2016.

http://gams.com/docs/intro.htm, diakses 21 Maret, 2016.

Gambar 1. Air. Sumber: http://3.bp.blogspot.com/-

0mnzymm9NKw/VXZk6bPpscI/AAAAAAAAATQ/2ThIlCbGLN0/s1600/kuncuran%2Bair

%2Bbersih.jpg

Gambar 2. Sungai. Sumber: http://cdn2.bigcommerce.com/n-

d57o0b/ghzhrhp/product_images/uploaded_images/upper-rogue-river.jpg

Gambar 3. Danau. Sumber: https://s-media-cache-

ak0.pinimg.com/originals/13/d7/fa/13d7fa57ee79fc8ca474262b7010a6cf.jpg

Gambar 4. Waduk. Sumber: http://www.pu.go.id/images/uploads/gallery/photo/Waduk-Jatibarang-

Semarang-Di-Operasionalkan_0.jpg

Gambar 5. Air Tanah. Sumber:

http://media.nationalgeographic.co.id/daily/640/0/201601071642923/b/foto-kualitas-air-

tanah-jakarta-sudah-kritis.jpg

http://gams.com/

http://gams.com/docs/intro.htm

http://3.bp.blogspot.com/-0mnzymm9NKw/VXZk6bPpscI/AAAAAAAAATQ/2ThIlCbGLN0/s1600/kuncuran+air+bersih.jpg

http://cdn2.bigcommerce.com/n-d57o0b/ghzhrhp/product_images/uploaded_images/upper-rogue-river.jpg

https://s-media-cache-ak0.pinimg.com/originals/13/d7/fa/13d7fa57ee79fc8ca474262b7010a6cf.jpg

http://www.pu.go.id/images/uploads/gallery/photo/Waduk-Jatibarang-Semarang-Di-Operasionalkan_0.jpg

http://media.nationalgeographic.co.id/daily/640/0/201601071642923/b/foto-kualitas-air-tanah-jakarta-sudah-kritis.jpg

eksplorasi generic algebraic modeling system (gams) dalam

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