Bina Nusantara Mata Kuliah: K0194-Pemodelan Matematika Terapan Tahun : 2008 Aplikasi Model Markov Pertemuan 22:

Bina Nusantara Learning Outcomes Mahasiswa akan dapat menghitung matriks transisi dan perhitungan keseimbangan..

Bina Nusantara Outline Materi: Matriks Transisi Kondisi Keseimbangan Aplikasi model Markov Contoh penerapan

Bina Nusantara Andrei Markov ( ) Paul Eherenfest: Diffusion model, early 1900s Statistical interpretation of the second law of thermodynamics: The entropy of a closed system can only increase. Proposed the “Urn Model” to explain diffusion. Albert Einstein, 1905 Realized Brownian motion would provide a magnifying glass into the world of the atom. Brownian motion has been extensively modeled by Markov Chains

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Osmosis Particles are separated by a semi- permeable membrane, which they can pass through in either direction. There are N+1 states, given by the number of white molecules inside. Suppose that there are N black particles inside the membrane, and N white particles outside the membrane. Each second, one random molecule goes from outside the membrane to inside, and vice versa.

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5 molecules

Bina Nusantara N molecules

Bina Nusantara N molecules

Bina Nusantara If this process runs for a while, an interesting question is: How much time, on average, is the process in each state? A Markov chain with transition matrix P is said to be regular if some power of P has all positive entries for some n. In a regular Markov chain, it is possible to get from any state to any other state in n steps. The Markov chain for our osmosis process is regular. Even starting with all black particles inside, if a white particle entered at every step, then the process would pass from zero white inside through all possible states.

Bina Nusantara Contoh Aplikasi Persoalan peralihan merk adalah persoalan yang sering terjadi dalam dunia perdagangan yang disebabkan oleh karena para pelanggan pindah dari satu merk kepada merk yang lain. Peralihan para pelanggan timbul karena iklan, promosi tertentu, harga, ketidakpuasan dan lain sebagainya. Analisis rantai Markov yang dipakai untuk menyelesaikan persoalan ini telah banyak digunakan sebagai alat diagnosis untuk mengusulkan rencana-rencana strategi pasar. Tabel berikut ini menggambarkan tingkah laku peralihan merk dari 1000 pelanggan untuk 4 merk dagang, masing masing merk 1,2,3 dan 4 sbb:

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