Inventory Problem

In a store with one type of item, orders are placed to the supplier at the end of each month if the inventory is less than smin. The ordering is designed to make the inventory go to smax.

Purpose of the simulation. Determine the best values of smin and smax to decrease all the overhead costs.

Example: smin =20 and smax = 40

Month  inventory   amount ordered

1             30                0

2            10                30

3           5 (backordered)  45

Costs-

Let Z = the number of items ordered, 
Y =the number of items backlogged 
 X= current inventory

(1)Ordering cost= set up cost of order + incremental cost*no. of items =30 +3Z 

(2) Holding cost= 1X per month  - warehousing

(3) Shortage costs- $5Y per month, where Y is the amount of backlog- bookkeeping and ill will of customer





Rules

(1) requests from customers( called demands) are filled immediately if they can be, else they are backlogged
The probability of a request for a given size is known.
The timing of the requests are also random variables with a known distribution

(2) There is a time lag between ordering an item from a supplier and its delivery between 1/2 month and a month that is randomly distributed.


(1) and (2) imply that there are 3 probability distributions we are dealing with
	(a) number of items requested
	(b) time between requests
	(3) time between ordering and delivery of order

Events

(1) arrival of an order
(2) demand(requests) from a customer
(4) beginning of month evaluation to see if more items must be ordered
(3) end of simulation








Example: Assume smin=20, smax =40
Let R = amount requested by customer
Let X= number of items in stock
Let Y = number of items backlogged
Let Z = number of items ordered
Initially x
X=30


Day	Event	R	X	Y	Z	type of cost	cost
0				30	0
10	demand	10	20	0		holding		30*10/30
20	demand	25    	0	5		holding		20*10/30
30	evaluation					backlog		5*5*10/30
						45	ordering		30+3*45
40	demand	5	0	10		backlog		5*5*10/30
45	arrival		35	0		backlog		10*5*5/30
50	demand	5	30			holding		35*5/30
60	evaluation		30			holding		30*10/30
75	demand	20	10			holding		30*15/30
90	evaluation		10		30	holding		10*15/30
							ordering		30 +3*30
100	demand	10	0			holding		10*10/30
110	arrival		30
120	end			30			holding		30*10/30