Write a technical report to explain what you do for the following three parts of the project. Provide your computer codes in the Appendix section of the report. You may use any programming language for the coding tasks $($MS Excel and ARENA are not programming languages!$).$ Your report should include an Introduction chapter to summarize what you do for the project and separate chapters for each part. In each chapter, please explain your aims and methods$/$algorithms in detail and provide the pseudo$-$codes of the algorithms in the main body of the text.

PART I Inventory Control in Discrete Time $($Periodic Review$)$

As the manager of a book store you control the stock of a particular stochastic modelling book which can be ordered weekly from the supplier. At the end of each week, after you closed the store on Saturdays, you control the inventory. If there is no book in the stock, then you order $4$ books. Otherwise, you do not order at all. An order you give arrive to the store early on Monday. The store is closed on Sundays. If the book is out of stock when a customer demands it$,$ you lose that customer.

The relevant cost components for the system are as follows:

$$ Fixed ordering costs: $10$ TL per order

$$ Variable ordering cost: $8$ TL per unit ordered.

$$ Holding cost $($inventory carrying cost$)$: $3$TL for each unit on hand at the end of the period.

$$ Lost sales cost: $25$ TL for each lost demand.

First, select a probability mass function for the size of the weekly demand which has the sample space $\{0,1,2,3,4\}.$ Then, write a computer code to numerically calculate the long$-$run distribution of the inventory level and the long$-$run average system cost via simulation. Compare the numerical results with the theoretical solutions. Sketch how the error $($deviation from the theoretical results$)$ changes with the length of the simulation. Please also explain how you obtain the theoretical solutions.

PART II Inventory Control in Continuous Time $($Continuous Review$)$

In this part, suppose you control the system continuously. Whenever the inventory level hits zero you order $4$ books. Furthermore, for this part, customer arrival process will be modeled as a Poisson process. That is$,$ the times between two consecutive customer arrivals are independent and identically distributed Exponential random variables. All the other conditions $($zero lead$-$time, the cost structure, lost sales, etc.$)$ are the same with Part I.

First, select a customer arrival rate and a probability mass function for the demand size of an arriving customer. The demand size should have the sample space $\{0,1,2,3,4\}.$ Then, write a computer code to numerically calculate the long$-$run distribution of the inventory level and the long$-$run average system cost via simulation. Compare the numerical results with the theoretical solutions. Sketch how the error $($deviation from the theoretical results$)$ changes with the length of the simulation. Please also explain how you obtain the theoretical solutions.

PART III Parameter Optimization for the $(,)$ Policy

In Parts I and II the described ordering policy is of $(,)$ type where $=0,=4.$ Describe and apply a $($numerical$)$ optimization method to find the best values of $$ and $$ for the settings you considered in Parts I and II$.$