Question

Problem 3: Transition Matrix Problem (10 pts.) Consider an inventory system in which the sequence of events during each period is as follows: (1) We observe the number of units in inventory (call it i) at the beginning of the period. (2) Purchases: If i<1, 3 units are ordered from our supplier. If i> 2,0 units are ordered. Delivery of all ordered units is immediate (at the beginning of the current period i). (3) Demand: With probability 0.32, 0 units are demanded (sold) during the period; with probability 0.33, 1 unit is demanded (sold) during the period; with probability 0.35, 2 units are demanded (sold) during the period. (4) Inventory: We observe the number of units in inventory at the beginning of the next period (call it j) to see how many units we have from the previous period. Define a period's state to be the period's beginning inventory level. Determine the Transition Matrix that could be used to model this inventory system as a Markov Chain. Draw the Markov Chain as well.

Answer 1

The inventory levels are observed at the begining of the period. and the inventory on next day defents on the inventory of previous day, the amount ordered and demand during the previous day, so a morkov chain is formed (stationary) bez probabilities of demand does not change with time. to the transition matrix as follows the states are inventory levels at the begining of the period, which are o to 4. lo t = today - 2 3 4 I opool oo Ye Ye Ye o 1 10 13 HB / o o 1/ / Bo Too yo yo y The Proventory level mentioned at the stat of the day is Case ist if iso, hence is the placed order is of size 4-1 4-0 = 4 units The demand that can occurs in a day are of 0.1, 2 units, each with probability . so the table showing demand and corresponding inventery level for next day is given below - | Demand today. Inventory level tomorroid 0 2 occurs with probability &ß. for inventory levels as each demand of 2,8.4.

оссию Case-2 iza, if i=g, we do not order any more unite that is zelp writs are ordered. A demand of 0.1, 2 units on that day can so the table showing demand and corresponding inventory level is given below - Demand today I inventory level tomorrow Hence each demared occurs with probability of the. and we have inventory level of o. 1, 2 with probability