Problem 4: A store wants to reduce its inventory costs and has tasked you with building a mathematical model that describes its inventory system for one item. The store opens only on weekdays, and the demand during the week follows a Poisson distribution with mean 3 units. Note that during a week the store can sell at most the number of items it has in stock, even if demand is higher. At the end of the work week, the store manager checks the stock of the item and decides if she will place an order: if there are 2 or fewer units in stock, the manager places an order; and if there are more than 2 units in stock, the manager does not place an order. The orders are fulfilled before the start of the following week, and the number of units in an order is such that the store has 5 units in stock at the start of the week. The store has requested your help building a model that allows them to predict the number of units they'll have in stock at the end of any given week. a) Model this inventory system as a Markov chain. State your assumptions explicitly, if any. b) The store incurs in a cost of $2 per week for each unit in inventory at the end of the week, and of $5 if it has to place an order at the end of the week. At the end of this week the store had 5 units in stock, what is the expected cost that the store will have to pay next week?
Problem 4: A store wants to reduce its inventory costs and has tasked you with building a mathematical model that describes its inventory system for one item. The store opens only on weekdays, and the demand during the week follows a Poisson distribution with mean 3 units. Note that during a week the store can sell at most the number of items it has in stock, even if demand is higher. At the end of the work week, the store manager checks the stock of the item and decides if she will place an order: if there are 2 or fewer units in stock, the manager places an order; and if there are more than 2 units in stock, the manager does not place an order. The orders are fulfilled before the start of the following week, and the number of units in an order is such that the store has 5 units in stock at the start of the week. The store has requested your help building a model that allows them to predict the number of units they'll have in stock at the end of any given week. a) Model this inventory system as a Markov chain. State your assumptions explicitly, if any. b) The store incurs in a cost of $2 per week for each unit in inventory at the end of the week, and of $5 if it has to place an order at the end of the week. At the end of this week the store had 5 units in stock, what is the expected cost that the store will have to pay next week?
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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Transcribed Image Text:Problem 4: A store wants to reduce its inventory costs and has tasked you with building a
mathematical model that describes its inventory system for one item. The store opens only on
weekdays, and the demand during the week follows a Poisson distribution with mean 3 units. Note
that during a week the store can sell at most the number of items it has in stock, even if demand is
higher. At the end of the work week, the store manager checks the stock of the item and decides if
she will place an order: if there are 2 or fewer units in stock, the manager places an order; and if
there are more than 2 units in stock, the manager does not place an order. The orders are fulfilled
before the start of the following week, and the number of units in an order is such that the store
has 5 units in stock at the start of the week. The store has requested your help building a model
that allows them to predict the number of units they'll have in stock at the end of any given week.
a) Model this inventory system as a Markov chain. State your assumptions explicitly, if any.
b)
The store incurs in a cost of $2 per week for each unit in inventory at the end of the week,
and of $5 if it has to place an order at the end of the week. At the end of this week the store
had 5 units in stock, what is the expected cost that the store will have to pay next week?
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