Mathematical Modeling and Case Study with Lingo Code

1. Mathematical Modelling
Basic Lingo Tutorial
MA Ilhami
adha@untirta.ac.id

2. Mathematical Modelling (Neumaier, 2003)
• A mathematical model is a description of a system using
mathematical concepts and language Mathematical modeling is the
art of translating problems from an application area into tractable
mathematical formulations whose theoretical and numerical analysis
provides insight, answers, and guidance useful for the originating
application.
• The creating of mathematical formulas to represent a real world
problem in mathematical terms
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3. Why Mathematical Model?
• is indispensable in many applications
• is successful in many further applications
• gives precision and direction for problem solution
• enables a thorough understanding of the system modeled
• prepares the way for better design or control of a system
• allows the efficient use of modern computing capabilities
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4. Mathematical Models
• First-Principle Model (Mechanistic Model): based on physical laws
(newton equation). Descriptive, explaining.
• Stochastic Model: based on distributions, averages (risk model, or
machine’s reliability model). Capable to deal with random
phenomena, hard to distinguish relations.
• Empirical/data Model: based on (historical) patterns, data. Not
explaining, relations based on reality.
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5. Mathematical Model Cycles
Specify the
problem
Set up a
metaphor
Formulate
mathematical
model
Solve
mathematical
problem
Interpret
solution
Compare with
reality
Use model to
explain, predict,
decide, or design
Real World
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6. 1. Specify the problem
• You are a new student of industrial engineering master degree of
UGM. You have two options, whether you are going to buy a used
motorcycle or using Gojek as your daily driver? Model it using a
mathematical model.
• The problem is.. “What is cheaper whether to buy a used motorcycle
of using Gojek as your daily driver?
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7. 2. Set a metaphor (ex. Rich Picture)
Gojek Rp. 10jt Bensin Service
Cuci
Motor
Tambal
Ban
Spare
part
P
Parking
Fee
For sale
?
Campus activity
Hangout
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8. 2. Set a metaphor (ex. Influence Diagram)
Total biaya transportasi lokal selama S2
Biaya
investasi
motor
X1 = beli motor seken
Biaya
operasion
al
Biaya
perawata
n
Biaya tak
terduga
Biaya
Parkir
Nilai jual
kembali
Biaya ke
kampus
via Gojek
X2 = pakai Gojek
Biaya
hangout
via Gojek
Harga motor
bekas
Frekuensi
isi bensin
Biaya
sekali isi
bensin
Frekuensi
service
Biaya
sekali
service
Frekuensi
cuci motor
Biaya per
cuci motor
Frekuensi
tambal ban
Biaya
tambal ban
Frekuensi
parkir
Biaya
sekali
parkir
Frekuensi ke
kampus
Biaya order
ke kampus
Frekuensi
main/jalan
Biaya
order main
Harga jual
kembali
Jumlah hari
selama s2
Biaya
cuci
motor
Frekuensi
ganti part
Biaya ganti
sparepart
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9. 3. Formulate Mathematical Model (Notations)
No Parameter Simbol Nilai Satuan
1 Jumlah Hari selama S2 d 730 Hari
2 Harga motor seken Hm 10000000 Rp
3 Biaya isi bensin Bm 20000 Rp/isi
4 Frekuensi isi bensin Fb 3 Hari/isi
5 Biaya sekali service Bs 150000 Rp/service
6 Frekuensi service Fs 2 bulan/service
7 Biaya sekali cuci motor Bc 15000 Rp/cuci
8 Frekuensi cuci motor Fc 1 minggu/cuci
9 Biaya tambal ban Bt 25000 Rp/tambal
10 Frekuensi tambal ban Ft 6 bulan/tambal
11 Biaya sparepart Bsp 500000 Rp/ganti
12 Frekuensi ganti sparepart Fsp 6 bulan/ganti
13 Biaya parkir Bp 2000 Rp/minggu
14 Frekuensi parkir Fp 2 1/minggu
15 Harga jual kembali motor Hjm 8000000 Rp
16 Biaya rutin ke kampus Gojek Bo 7000 Rp/order
17 Frekuensi ke kampus Gojek Fo 2 kali/hari
18 Biaya hangout via gojek Ho 7000 Rp/order
19 Frekuensi order hangout Hf 4 kali/minggu
No Variabel Keputusan Simbol Tipe
1 Keputusan membeli motor X1 Biner
2 Keputusan menggunakan Gojek X2 Biner
Model assumptions:
1. there are 365 days a year, and in two years the student will definitely be graduated
2. A motorcycle's price is Rp. 10,000,000.00 on average and its depreciation is one million rupiahs a year.
3. The motorcycle will be sold at the end of the study.
4. You will always pay Rp. 20,000.00 each time you fill the gasoline.
5. etc...
Model boundaries:
1. The student has two options only, which are buying a used motorcycle or using Gojek as a daily driver.
2. Etc…
No Fungsi Tujuan Simbol Tipe
1 Biaya investasi motor BIM Riil
2 Biaya operasional motor BOM Riil
3 Biaya perawatan BPM Riil
4 Biaya rutin lainnya BRM Riil
5 Biaya tak terduga BTM Riil
6 Biaya parkir BPR Riil
7 Pendapatan penjualan motor PPM Riil
8 Biaya order Gojek BOG Riil
9 Biaya hangout Gojek BHG Riil
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10. 3. Formulate Mathematical Model (Formulation)
Biaya operational (BOM) = X1*(d/Fb)*Bm
X1 = beli
motor seken
Biaya
operasional
Frekuensi isi
bensin (Fb)
Biaya sekali isi
bensin (Bm)
Jumlah hari
selama s2 (d)
Dimana:
d = jumlah hari selama s2 (2 tahun x 365 = 730 hari)
Fb = frekuensi isi bensi (3 hari sekali)
Bm = biaya sekali isi bensin (Rp. 20.000)
MA Ilhami

11. 3. Formulate Mathematical Model (Formulation)
X1 = beli
motor seken
Biaya
perawatan
Frekuensi service
rutin (Fs)
Biaya sekali
service rutin (Bs)
Jumlah hari
selama s2 (d)
Biaya perawatan (BPM) = X1*(d/(Fs*30))*Bs
Dimana:
d = jumlah hari selama s2 (2 tahun x 365 = 730 hari)
Fs = frekuensi service (2 bulan sekali)
Bs = biaya sekali service (Rp. 150.000,-)
MA Ilhami

12. 3. Formulate Mathematical Model (Formulation)
Biaya dan Pendapatan Terkait Motor:
Biaya operational (BOM) = X1*(d/Fb)*Bm
Biaya investasi motor (BIM) = X1*Hm
Biaya perawatan (BPM) = X1*(d/(Fs*30))*Bs
Biaya rutin (BRM) = X1*((d/7)/Fc)*Bc
Biaya tak terduga (BTM) = X1*((d/30)/(Fsp))*Bsp + X1*((d/30)/(Ft))*Bt
Biaya parkir (BPR) = X1*((d/7)*Fp)*Bp
Penjualan penjualan motor (PPM) = X1*Hjm
Biaya dan Pendapatan Terkait Gojek:
Biaya order Gojek (BOG) = X2*d*Fo*Bo
Biaya hangout Gojek (BHG) = X2*d*Ho*Ho
Objective Function = X1*(d/Fb)*Bm + X1*Hm + X1*(d/(Fs*30))*Bs +
X1*((d/7)/Fc)*Bc + X1*((d/30)/(Fsp))*Bsp + X1*((d/30)/(Ft))*Bt +
X1*((d/7)*Fp)*Bp - X1*Hjm + X2*d*Fo*Bo + X2*(Hf*d/7)*Ho
Objective Function = BOM + BIM + BPM + BRM + BTM + BPR – PPM + BOG + BHG
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13. 3. Formulate Mathematical Model (Constraints)
Choice Constraint: only one choice may be selected X1 or X2
X1 + X2 = 1
Binary Constraint: the decision variables are binary.
X1, X2 Є {0,1}
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14. 4. Solve The Mathematical Problem (Using Lingo)
Declaration of indexes or set of indexes
Declaration of parameters
Declaration of the objective
function followed by constraints
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15. 4. Solve The Mathematical Problem (Using Lingo)
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16. 4. Solve The Mathematical Problem (Using Lingo)
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17. 5. Interpret the solution
Objective function reflects the cost we spend if the
decision variable is selected (which is Rp. 12.802.260)
The selected decision variable is X1 = 1 (we should by
used motor cycle)
MA Ilhami

18. Let’s complicate the problem a little bit …
Model assumptions:
1. There are 365 days a year, and in two years the student will definitely be graduated
2. A motorcycle's price is Rp. 10,000,000.00 on average and its depreciation is 1.2 million rupiahs a year.
3. The motorcycle will be sold at the end of the study.
4. You will always pay Rp. 20,000.00 each time you fill the gasoline.
5. etc...
So the decisions become…
Some parameters have changed…
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19. Excel Dataset (Using Define Name)
MA Ilhami

20. The Modified Lingo Code…
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21. The Solutions (in the orange table)
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22. Sensitivity Analysis
• Think several parameters that have significant effects on the decision
or objective function.
• Let say: Hjm (motor cycle’s selling price after 2 years) or Bm (Gasoline
price).
• Try to change the parameters’ value and draw a graphic of
parameters and objective function.
MA Ilhami

23. Sensitivity of Hjmt(t = 1)
Hjmt (t=1) 8,800,000 8,810,000 8,820,000 8,830,000 8,835,000 8,840,000
BELI MOTOR
Bbesin - - - - 2,433,333 2,433,333
Bservice - - - - 912,500 912,500
Bcuci - - - - 782,143 782,143
Bpart - - - - 1,013,889 1,013,889
Btambal - - - - 50,694 50,694
Bparkir - - - - 208,571 208,571
Bmotor
GOJEK
Borderkampus 10,220,000 10,220,000 10,220,000 10,220,000 5,110,000 5,110,000
Bordermain 2,920,000 2,920,000 2,920,000 2,920,000 1,460,000 1,460,000
Hitung OF 13,140,000 13,140,000 13,140,000 13,140,000 13,136,131 13,131,131
X11 - - - - 1 1
X12 1 1 1 1 - -
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