weekly demand data have been analyzed yielding the probability mass fnc. for weekly demad D as follows
Pr{D=0}=0.05, Pr{D=1}=0.1,
Pr{D=2}=0.2,Pr{D=3}=0.4,
Pr{D=4}=0.1,Pr{D=5}=0.1,Pr{D=6}=0.05.
Let X be the Markov chain where Xn is the inventory at the end of week n.
a) find transition matrix for X.
b) if at the end of week 1 there are 5 items in inventory, what is the probability that there will be 5 items in inventory at the end of week 3?
To find the transition matrix for X, we need to determine the probability of transitioning from one state to another. In this case, the states correspond to the inventory levels at the end of each week.
a) The transition matrix, denoted as P, can be constructed using the given probabilities. Since there are 7 possible inventory levels (0 to 6 items), the matrix will be a 7x7 matrix.
P = | p00 p01 p02 p03 p04 p05 p06 |
| p10 p11 p12 p13 p14 p15 p16 |
| p20 p21 p22 p23 p24 p25 p26 |
| p30 p31 p32 p33 p34 p35 p36 |
| p40 p41 p42 p43 p44 p45 p46 |
| p50 p51 p52 p53 p54 p55 p56 |
| p60 p61 p62 p63 p64 p65 p66 |
To calculate each cell, we use the probability mass function (PMF) values for the weekly demand. For example, p01 represents the probability of transitioning from inventory level 0 to level 1:
p01 = Pr{D=1} = 0.1
Similarly, p34 represents the probability of transitioning from inventory level 3 to level 4:
p34 = Pr{D=4} = 0.1
Using this method, we can fill in the entire transition matrix.
b) To find the probability of having 5 items in inventory at the end of week 3, given there are 5 items in inventory at the end of week 1, we need to calculate the probability using the transition matrix.
Let's denote the probability of having 5 items in inventory at the end of week 1 as P(X1=5) = p15. We want to find P(X3=5) given P(X1=5).
The probability of going from inventory level 5 to inventory level 5 in two steps (week 1 to week 3) can be calculated as follows:
P(X1=5, X3=5) = p15 * p54
However, since we are interested only in the inventory level at the end of week 3, we need to consider all possible inventory levels at the end of week 1. Hence, we sum up probabilities for all starting inventory levels (from 0 to 6):
P(X3=5) = Σp(X1=i) * p(i5), where i ranges from 0 to 6
Calculate this sum, substituting the appropriate p(i5) values from the transition matrix.
P(X3=5) = p01 * p15 + p11 * p25 + p21 * p35 + p31 * p45 + p41 * p55 + p51 * p65 + p61 * p75
Sum up the products to find the final probability.
Remember, the transition matrix P is specified by the given probabilities, and you can compute each cell accordingly using the provided PMF.