LESSON 17: INVENTORY MODELS (STOCHASTIC) INTRODUCTION TO THE Q , R SYSTEMS

Outline

• Multi-Period Models – Lot size-Reorder Point (Q, R) Systems– Notation, Definition and Some Formula– Example: Given a Q, R Policy, Find Cost

LESSON 17: INVENTORY MODELS (STOCHASTIC)INTRODUCTION TO THE Q,R SYSTEMS

Lot Size - Reorder Point (Q,R) Systems

• In the simple EOQ model, demand is known and fixed. However, often demand is random. The lot size-reorder point (Q, R) systems allow random demand.

• There are two decision variables in a (Q, R) system:– Order quantity, Q and– Reorder point, R

• The Q, R policy is as follows:– When the level of on-hand inventory hits reorder

point, R place an order with lot size Q.


• In the simple EOQ model, R is the demand during the lead time.

• However, in presence of random demand, R usually includes a safety stock, in addition to the expected demand during the lead time. So,

Reorder point, R = lead-time demand + safety stock


• In the simple EOQ model, only holding cost and ordering costs are considered.

• In presence of random demand, the demand may sometimes be too high and exceed the inventory on hand. The result is stock-out.

• For each unit of shortage, a penalty cost p is charged. See Lesson 16 for more information on penalty cost.

Penalty cost = p per unit.


• The goal of a lot size-reorder point system is to find Q and R so that the total annual holding cost, ordering cost and stock-out cost is minimized.

• The current lesson only covers how to compute cost from a given policy.

• The next three lessons address the question how to find optimal Q and R so that the total annual cost is minimized.


Whenever the inventory onhand hits R, a quantity Q is ordered.


Too high lead-time demand may cause stock-outs. Safety stock reduces the chance ofstock-outs.


Safety Stock

Lead-Time Demand

The reorder point is computedfrom the lead-time demand and the safety stock.


Safety Stock

Lead-Time Demand

Goal: Find Q and R such that total annual holding cost, orde-ring cost and stock-out cost is minimized.

τ

τ

τ

p

y

y

demand time-lead of deviation standard

demand annual of deviation standard

demand time-lead mean

yearin time lead

unit per cost out-stock

(Q,R) Policy Notation, Definition and Some Formula


zLn

zL

zF

z

zF

Rz

cycle per units out-stock

786781- pp. 4,- ATable from available

function, loss edstandardiz the

time-lead during out stocking not ofy Probabilit

786781- pp. 4,- ATable from available

of left the on area the

curve, normal the under area cumulative the

,


Pro

babi

lity

L ead -t im e D em and

zF Area

R

R

z

zF1-

Area

P ro b(sto ck o u t )

P ro b(no sto ck o u t )

Q

n

Q

RQQ

R

Q

R

shortages of number annual Expected

cycles or orders of number annual Expected

stocksafety inventory Average

stocksafety inventory, Average

regular inventory, Average

stockSafety

22

2


Q

np

Q

KRh

hQ

Q

np

Q

K

Rh

hQ

)(2

)(2

cost annual Total

cost out-stock Annual

cost ordering Annual

stocksafety cost, holding Annual

regular cost, holding Annual


• Type 1 service– Type 1 service level, is the probability of not

stocking out during the lead time.

– F(z) is available from Table A-4, pp. 781-786• Type 2 service

– Type 2 service level is measured by fill rate, which is the proportion of demands that are met from stock

Q

n1

R

zzF ,


Example - Given A Q,R Policy, Find Cost

Annual demand for number 2 pencils at the campus store is normally distributed with mean 2,000 and standard deviation 300. The store purchases the pencils for 10 cents and sells them for 35 cents each. There is a two-month lead time from the initiation to the receipt of an order. The store accountant estimates that the cost in employee time for performing the necessary paper work to initiate and receive an order is $20, and recommends a 25 percent annual interest rate for determining holding cost. The cost of a stock-out is the cost of lost profit plus an additional 20 cents per pencil, which represents the cost of loss of goodwill. Currently, a (Q,R) system with Q = 1500, R = 500 is used.

Find

a. The safety stock


stockSafety

demand, time-Lead

time, Lead

R

b. The average inventory level

c. The expected annual number of orders


stocksafety inventory Average22

RQQ

Q

cycles or orders of number annual Expected

d. The probability of not stocking out during the lead-time

e. The expected number of units stock-out per cycle


786)781- pp. 4,- ATable (See

time-lead during out stocking not ofy Probabilit

zF

Rz

786)781- pp. 4,- ATable (See

zLn

f. The fill rate

g. The expected annual number of shortages


1 Q

n rate, fill The

shortages of number annual Expected

Q

n

h.The holding cost per unit per year and penalty cost per unit.


cost,Penalty

cost, Holding

p

Ich

i. The average annual holding cost associated with this policy.


)(

2

cost holding annual Total

stocksafety cost, holding Annual

regular cost, holding Annual

Rh

hQ

j. The total annual cost associated with this policy.


)(2

Q

np

Q

KRh

hQ

Q

np

Q

K

cost annual Total

cost out-stock Annual

cost ordering Annual

READING AND EXERCISES

Lesson 17

Reading:

Section 5.4, pp. 259-262 (4th Ed.), pp. 250-254 (5th Ed.)

Exercise:

13b (use the result of 13a), p. 271 (4th Ed.), p. 261 (5th Ed.)

LESSON 17: INVENTORY MODELS (STOCHASTIC) INTRODUCTION TO THE Q , R SYSTEMS

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