Lot size reorder point systems

(Q, R) system

Introduction

Generalize EOQ model with reorder point R for the case where demand is stochastic

Multi-period newsboy problem was not realistic for 2 reasons:

– No ordering cost– No lead time

(Q,R) system with stochastic demand are common in practice

Form the basis of many commercial inventory systems

Changes in Inventory Over Time for Continuous-Review (Q, R) System

Fig. 5-5

(Q,R) inventory system

The systems is continuous review Demand is random and stationary Fixed lead time Cost involved

– K: ordering cost– h: holding cost per unit per unit time– c: cost per item– p: shortage cost per unit of unsatisfied demand

Inventory Model

Decision variables: Q and R Costs

– Holding cost– Set up (ordering cost)– Penalty (shortage) cost– Proportional ordering cost (cost of items ordered)

Holding cost

λτ

R- λτ

Q + R- λτQ + R - λτ

R- λτ

Q/2 + R - λτ

Penalty cost

x

Expected number of shortages

R

dxxfRxRn )()()(

Total cost function

Q

Rnp

Q

KR

QhRQG

)(

2),(

Holding cost Ordering cost Shortage cost

Necessary conditions for optimality

))(1()()('

0)('

0)(

2 22

RFdxxfRn

Q

Rnph

R

G

Q

Rnp

Q

Kh

Q

G

R

Optimal solution

p

QhRF

h

RpnKQ

)(1

)]([2

Service Level in (Q,R) systems

Difficult to determine an exact value of p A substitute for penalty cost is a service level Two types of service level are considered

– Type 1 service level– Type 2 service level

Type 1 service level

In this case we specify the probability of no shortage in the lead time

Symbol is used to represent this probability In this case

– Determine R to satisfy the equation F(R) = – Set Q = EOQ

Interpretation of

•The proportion of cycles in which no shortage occurs

•Appropriate when a shortage occurrence has the same consequence regardless of its time or amount

•Not how service level is interpreted in most applications

•Different items have different cycle lengths this measure will not be consistent among different products making the choice of alpha difficult

Type 2 service level

Measures the proportion of demands that are met from stock

Symbol β is used to represent this proportion n(R)/Q is the average fraction of demands

that stock out each cycle n(R)/Q = 1 - β

Approximate solution with Type 2 service level constraint

Set Q= EOQ Find R to solve n(R)=EOQ(1 – β)