A full set of operations research knowledge points
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Chapter 9 Markov Analysis
1. The mathematical principle of Markov analysis
For the process of transitioning from one situation to another, if the process has a transition probability, and this transition probability can be calculated based on the immediately preceding situation, then this process is called a Markov process. The whole series of this conversion process is called a Markov chain.
Definition 1: Probability vector For
any vector, if its internal elements are non-negative numbers and the sum is equal to 1, then this vector is called a probability vector.
Definition 2: Probability matrix
In a matrix, each row is a probability vector, then this matrix is called a probability matrix.
Theorem 1: If A and B are both probability matrices, then the AB product is also a probability matrix. Similarly, the n-th power of A is also called a probability matrix.
Second, the steps of Markov analysis method
- Understand user needs, brand/brand conversion business conditions.
- Build transition probability matrix
- Calculate the possible future market share rate (market share)
- Determine the equilibrium conditions
3. Several conclusions of Markov analysis method
- The future development or evolution of many things is often dominated or influenced by the current situation of the thing.
- Markov has repeatedly found through research and experimentation that in the process of probability conversion of certain things, the result of the nth experiment is often determined by the result of the n-1th experiment.
- Markov process is a process of probability conversion
Fourth, Markov's requirements for analyzing the problem
- The order of the Markov analysis problem: The first-order Markov process only considers the selection of the previous period when determining the selection probability of the event period, and the second-order Markov process considers the first two when determining the selection probability of the time period. Choice of period
- Transition probability: the probability that a certain seller keeps, gains or loses consumers
- Transition probability matrix: arrange the transition probabilities into a matrix
- The determination
of future market share Suppose the market share of the first cycle is T1 and the transition probability matrix is P,
then the market share of the second cycle is T2 = T1 * P, and the market share of any cycle can be obtained by analogy. - The final (balanced) determination of market share The market share of
different sellers in the sales process changes every cycle. If the consumer's choice probability remains unchanged, then the market share will remain unchanged after a long period of conversion. , We call the final (balanced) market share
calculation method.
When the calculation method is finally balanced, we can derive the formula: T=TP, use this formula to list the linear equations, plus the characteristic of the probability vector T itself, that is, the non-negative sum is 1. Solve the unknowns.