MATH MODELING MARKOV CHAINS 3- Gournet Dining Services wants to determine the lo

Question

MATH MODELING MARKOV CHAINS

3- Gournet Dining Services wants to determine the long term demand for various caegories of food at lunchtime.Through careful data gathering the managment has determined that 4/5 of customers who purchased soup or salad meals will buy the same type next time.Only half of those who buy pizza ,pasta or burgers will buy the same type at the next purchase. 3/5 customers of those who buy mexican or asian food will get the same type next time . People who are switching types are eqally likely to buy any of the other choices

a)Set up the 3 state markov chains to model the buying behavior of the customers. Show the one step transition matrix

b)Assuming that someone is seen to be buying a hamburger what is the expected # of successive meals the same person will buy food in the same category before trying one of the others

Explanation / Answer

The 3 states are ss (soups and salads), ppb(pizza, pasta,burgers), ma(mexican, asian)

Given the probability of transitioning from 1 state to another, we know ss to ss as 4/5 and hence ss to ppb is same as ss to ma which is 1/10. since the total of ss to ss/ ppb/ ma is 1 and ss to ppb is as likely as ss to ma.

ppb to ppb is 1/2. ppb to ss is as likely as ppb to ma. therefore the other 2 probabilities from ppb are 1/4 each.

ma to ma is 3/5. ma to ss is as likely as ma to ppb. thefore the other 2 probabilities from ma are 1/5 each.

b)

let us say 25% were having ss and 25% were having ppb and 50% were having ma meals.

Then one transition can be done by multiplying the transition matrix with the current state matrix

Multipy

by

to get

b)

The return time will happen when the initial probabilities reach steady state probability.

the mean return time is calculated as 1/steady state probability.

Initially someone is having hamburger hence he is in ppb and prob(ppb) is 1 and prob(ss) and prob(ma) are 0.

multiply transition matrix by initial matrix ( 0 1 0 )

It wil give 1st state probability, multiply transition by 1st state to get 2nd state. then multiply transition by 2nd state. Repeat till you get nth state probabilities equal to n+1th state probabilities. this is steady state.

The steady state probabilities will be

the mean return time for hamburger is 1/0.210526 i.e. 4.75 or approximately 5.

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