This document discusses inventory systems and the (s,S) inventory control policy. It provides information on inventory costs including ordering, holding, and shortage costs. The (s,S) policy is optimal for this cost structure, where an order is placed to bring the inventory position up to level S when it falls to or below level s. The document also provides the inputs, state variables, and formulation for this inventory control problem. An Excel simulation of inventory with this policy is referenced.

1. INVENTORY SYSTEMS

2. Assume periodic review, i.i.d. random demands,
constant (possibly non-zero) lead times and full
backlogging.
When to order?
How much to order?
Inventory Systems

3. Inventory Costs
Ordering Cost
 setup cost for placing an order, K
 per-unit ordering cost, c
 order amount, Qi
Holding (Storage) Cost
 per-period per-unit of inventory, h
 inventory level, Xi
Shortage (Unsatisfied Demand) Cost
 per-period per-unit of inventory, p
otherwise0
0>+ ii QcQK
)0,max( iXp −
)0,max( iXh
(s, S) inventory control policy is optimal for this cost structure
IGLEHART, 1963

4. When the inventory position (inventory level +
quantity on order) falls to or below the level s,
place an order to bring the inventory position up
to S.
(s,S) Ordering Policy

5. Demand in period i, Di
Constant lead time, L
Cost Information (K, c, h, p)
Inventory control policy parameters (s, S) – decision variables
Before ordering
 Inventory level at period i, Xi: (On hand) – (Backorders)
 Inventory position at period i, Yi: (Inventory level) + (On order)
Order quantity at period i, Qi
Inputs and State Variables

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Problem Formulation
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7. Inventory Simulation in Excel
http://ie.bilkent.edu.tr/~ie324/Inventory.xls
Lead time: 1 period
Demand: Uniform over {0,1,2,3,….,10} (Integer valued)

8. (s,S) Policy – Total Cost Structure
S s