**ASSIGNMENT**

The warehouse capacity of a given product is of 3 items at most. The daily demand for this product is given by the following probability distribution: p(0) = 0.3; p(1) 0.25, p(2)1.25, p(3+)=0.2, where 3+ means 3 or more. When the remaining items at the end of a day are less than 2, an order is made at the start of the next day to replenish the level at 3 items (to be available at the start of the next day).

1. Provide a transition probability matrix on the inventory level of this product at the start of each day, starting with 3 items.

2. Show how to calculate the probability of depleting the entire stock of 3 items in two days?

3. Show how to calculate the expected time between two consecutive orders?

4. Show how to calculate the long-run fraction of time the stock is empty?

*Question 3:* Rework of failed items in a given production facility alternates between two processes. After each operation, the quality control manager evaluates whether rework necessitates to be done again in the same process or in the other one. Eventually, the quality control manager decides on whether the defective item should be classified as good item or scrap. In either case, no rework is made afterward. Past experience shows that the transitions of a defective item over process 1, process 2, “good”, or “scrap” are described by the following transition matrix in the respective order.

1/5 2/5 1/5 1/5 1/6 1/2 1/6 1/6 P= 0 0 1 0 0 0 0 1 1. The daily average of defective items is 25. All items must start by process

How many end up “good” items in the average?

The average rework time in each process is of 2 hours. How long it takes in the average for an item in rework?

The cost of rework is 15 Dollars per hour in process I and 10 Dollars per hour in process

What is the expected daily loss due to reworking defective items that end up scraps?