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Mata Kuliah
Tahun
: K0194-Pemodelan Matematika Terapan
: 2008
Model MARKOV
Pertemuan 21:
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Learning Outcomes
• Mahasiswa akan dapat menjelaskan pengertian
dan mengerjakan masalah markov dalam
berbagai contoh aplikasi..
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Outline Materi:
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Pengertian Analisis Markov
Karakteristik Markov
Pengambilan Keputusan Markov
Contoh penerapan
Pengertian
• Analisis Markov adalah suatu cara menganalisis perangai
beberapa variabel yang sedang beredar dalam usaha
menduga perangai mendatang dari variabel yang sama.
• Analisis Markov telah digunakan dengan berhasil terhadap
berbagai macam situasi keputusan antara lain
penyelidikan dan dugaan terhadap perangai konsumen
perihal kesetia-annya terhadap “merk” tertentu dan
peralihan mereka dari satu merk ke merk lainnya,
perubahan sikap pelanggan dari “pembayaran lang-sung”
ke pembayaran terlambat 30 hari “atau” pembayaran
terlambat 60 hari “hingga” hutang buruk/kredit macet.
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Proses Markov
Adapun proses model rantai Markov, dapat dilakukan dengan
langkah-langkah :
• Menyusun Matriks Probabilitas Transisi
Probabilitas transisi adalah proba-bilitas suatu merk tertentu (atau
penjual) akan tetap menguasai para pelanggannya.
• Menghitung kemungkinan Market Share di waktu yang akan datang
Perhitungan market share di periode waktu kedua dapat diperoleh
dengan mengalikan martiks probabilitas transisi dengan market share
pada periode pertama.
• Menentukan Kondisi Equilibrium
Kondisi equilibrium tercapai bila tidak ada pesaing yang mengubah
matriks probabilitas transisi. Peng-gunaan matriks probabilitas transisi
dapat menggambarkan kondisi-kon-disi equilibrium.
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A Markov Chain is a mathematical model for a process
which moves step by step through various states.
In a Markov chain, the probability that the
process moves from any given state to any other
particular state is always the same, regardless of
the history of the process.
A Markov chain consists of states and
transition probabilities
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.
Each transition probability is the probability of
moving from one state to another in one step.
The transition probabilities are independent of
the past, and depend only on the two states
involved. The matrix of transition probabilities
is called the transition matrix.
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0 1 2 3 4
P(Homer wins) = .4
P(Homer loses) = .6
Homer and Marge both start with $2
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If P is the transition
matrix for a Markov Chain,
then the nth power of P
gives the probabilities of
going from state to state in
exactly n steps.
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If the vector v represents the initial state,
then the probabilities of winding up in the
various states in exactly n steps are exactly
v times the nth power of P .
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When they both start with $2, the probability that Homer
is ruined is 9/13.
If Homer starts with $ x and Marge starts with $ N-x, and
P(Homer wins) = p,
P(Homer loses) = q, then the probability Homer is ruined is
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Suppose you be on red in roulette.
P(win) = 18/38 = 9/19; P(lose) = 10/19.
Suppose you and the house each have $10
Now suppose you have $ 10
and the house has $20
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Now suppose you and the house each have $100.
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