INVENTORY SYSTEMS

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

Inventory CostsOrdering 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

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

Demand in period i, Di

Constant lead time, LCost 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

0for 0 ,

)}({

)0,max()0,max()}({

0

1

11

iQSX

YSsYIQ

QXY

DQXX

XpXhcQKsYIJ

i

iii

i

Lillii

iLiii

iiiii

Problem Formulation

n

ii

nSsJ

nJE

1,

1limmin

Period i

1 LiQ iD iX iY

Inventory Simulation in Excelhttp://ie.bilkent.edu.tr/~ie324/Inventory.xlsLead time: 1 periodDemand: Uniform over {0,1,2,3,….,10}

(Integer valued)

(s,S) Policy – Total Cost Structure

S s