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Engineering Computer Science Operations Research : Applications and Algorithms

The possible weather conditions are given. The creation of a model for weather conditions based on the Markov chain is explained. Markov chain Markov chain is a special type of discrete-time stochastic process. In this process, the state of the system at any given time t+1 is based on the state of the system at time t. Possible states Each weather condition represents a state. Sunny days and cloudy days both represent two states of the Smalltown’s weather. Probabilities The probabilities of transition of one weather condition to other are as follows: It is given that the probability of a sunny day following a sunny day is 90% or 0.9. Therefore, the probability of a cloudy day following a sunny day is 10% or 0.1. Similarly, the given probability of a cloudy day following a cloudy day is 80% or 0.8. Therefore, the probability of a sunny day following a cloudy day is 20% or 0.2. Tabular format The probabilities of transition from one weather condition to the other weather condition is shown via a spreadsheet snapshot as follows. As seen from the spreadsheet snapshot, the probabilities of all the combinations of weather conditions are shown. Here, Weather represents a state. The spreadsheet snapshot shows the probability of transition from one state to another. Matrix The matrix representation of the tabular format of probabilities is as follows. P = ( 0.8 0.2 0.1 0.9 ) The matrix represents the probabilities in the form of a matrix. The values of probabilities are the same as calculated. Markov chain model The given probabilities of the transition of one weather condition to another based on the Markov chain is as follows. The Markov chain is as shown. The following observations can be made from the given model. The probability of a sunny day following a sunny day is 0.9. The probability of a cloudy day following a sunny day is 0.1. The probability of a cloudy day following a cloudy day is 0.8. The probability of a sunny day following a cloudy day is 0.2. Conclusion Hence, the weather of the Smalltown is modelled based on the Markov chain using the given data.

The possible weather conditions are given. The creation of a model for weather conditions based on the Markov chain is explained. Markov chain Markov chain is a special type of discrete-time stochastic process. In this process, the state of the system at any given time t+1 is based on the state of the system at time t. Possible states Each weather condition represents a state. Sunny days and cloudy days both represent two states of the Smalltown’s weather. Probabilities The probabilities of transition of one weather condition to other are as follows: It is given that the probability of a sunny day following a sunny day is 90% or 0.9. Therefore, the probability of a cloudy day following a sunny day is 10% or 0.1. Similarly, the given probability of a cloudy day following a cloudy day is 80% or 0.8. Therefore, the probability of a sunny day following a cloudy day is 20% or 0.2. Tabular format The probabilities of transition from one weather condition to the other weather condition is shown via a spreadsheet snapshot as follows. As seen from the spreadsheet snapshot, the probabilities of all the combinations of weather conditions are shown. Here, Weather represents a state. The spreadsheet snapshot shows the probability of transition from one state to another. Matrix The matrix representation of the tabular format of probabilities is as follows. P = ( 0.8 0.2 0.1 0.9 ) The matrix represents the probabilities in the form of a matrix. The values of probabilities are the same as calculated. Markov chain model The given probabilities of the transition of one weather condition to another based on the Markov chain is as follows. The Markov chain is as shown. The following observations can be made from the given model. The probability of a sunny day following a sunny day is 0.9. The probability of a cloudy day following a sunny day is 0.1. The probability of a cloudy day following a cloudy day is 0.8. The probability of a sunny day following a cloudy day is 0.2. Conclusion Hence, the weather of the Smalltown is modelled based on the Markov chain using the given data.

Operations Research : Applications and Algorithms

Operations Research : Applications and Algorithms

4th Edition

ISBN: 9780534380588

Author: Wayne L. Winston

Publisher: Brooks Cole

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Chapter 16, Problem 7RP

Chapter 17.2, Problem 2P

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Chapter 17 Solutions

Operations Research : Applications and Algorithms

Ch. 17.2 - Prob. 1P Ch. 17.6 - Prob. 11P Ch. 17 - Prob. 4RP

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