ISE $310$ Project $2-$ Inventory Project

You may work with a partner and turn in a joint solution for Phases I & II$.$ For Phase III, the

technical work may be done with a partner, but each person must write a memo.

A local store uses an $($s$,$ S$)$ inventory policy for a particular product. Every Friday evening after

the store closes, the inventory level is checked. If the stock level for the product is greater than s$,$

no action is taken. Otherwise, if the stock on hand is less than or equal to s$,$ enough stock is

procured over the weekend so that, when the store reopens on Monday morning, the inventory

level is S$.$

Let $\{$Xn$,$ n $=1,2,...\}$ be the stock on hand when the inventory is checked on Friday evening of

week n and let $\{$Zn$,$ n $=1,2,...\}$ be the demand for product during week n$.$

Then for any $\backslash$omega $\backslash$epsi $,$

Xn $+1($

$\backslash$omega $)=$

Xn $($

$\backslash$omega $)$ Z n$+1($

$\backslash$omega $)$ if s $<$ Xn $($

$\backslash$omega $),$ Z n$+1($

$\backslash$omega $)<=$ Xn $($

$\backslash$omega $),$

S $$ Z n$+1($

$\backslash$omega $)$ if Xn $($

$\backslash$omega $)<=$ s$,$ Z n$+1($

$\backslash$omega $)<=$ S$,$

$0$ otherwise.

$$

$$

$$

$$

Let S $=5$ and assume each customer wants one unit of product. Assume that the number of

customers that arrive during a week has a Poisson distribution with parameter $\backslash$lambda $=6$ and that the

number of customers that arrive during one week is independent of the number that arrives in any

other week. $($i$.$e$.,$ Customers arrive at the store according to a Poisson process with rate $\backslash$lambda $=6$ per

week.$)$

Phase I $-70$ points $2$ April

A$)$ Show that $\{$Xn$,$ n $=1,2,...\}$ is a Markov chain. What is its state space? $(10$ points$)$

B$)$ For s$=2$ write the one$-$step transition matrix for $\{$Xn$,$ n $=1,2,...\}$ in terms of the variables $(15$

points$)$

pk $=$ P$\{$ exactly k customers arrive during the week $\},$ and

rk $=$ P$\{$ k or more customers arrive during the week $\}.$

What are the formulas for computing p k and r k$?$

C$)$ For s$=2$ if the starting inventory on Monday morning is $4$ units, what is the probability that a

shortage occurs by Friday afternoon? $(10$ points$)$

$($A shortage occurs if a customer arrives and there are no units available.$)$

D$)$ Write the form for the one$-$step transition matrix if the value of s were not known, i$.$e$.,$

In terms of pk and r k with s as a variable. $(15$ points$)$

E$)$ Write a program to generate $($i$.$e$.,$ fill in$)$ the one$-$step transition matrix for a passed value of s$.$

Print the one$-$step transition matrices for s $=2$ and s $=4.(20$ points$)$