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

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LESSON 17 INVENTORY MODELS (STOCHASTIC)INTRODUCT
ION 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

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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.

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Lot Size - Reorder Point (Q,R) Systems

• 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

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Lot Size - Reorder Point (Q,R) Systems

• 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.

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Lot Size - Reorder Point (Q,R) Systems

• 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.

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Lot Size - Reorder Point (Q,R) Systems
Whenever the inventory on hand hits R, a quantity
Q is ordered.
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Lot Size - Reorder Point (Q,R) Systems
Too high lead-time demand may cause stock-outs.
Safety stock reduces the chance of stock-outs.
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Lot Size - Reorder Point (Q,R) Systems
The reorder point is computed from the lead-time
demand and the safety stock.
Lead-Time Demand
Safety Stock
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Lot Size - Reorder Point (Q,R) Systems
Goal Find Q and R such that total annual
holding cost, orde- ring cost and stock-out cost
is minimized.
Lead-Time Demand
Safety Stock
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(Q,R) Policy Notation, Definition and Some
Formula
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(Q,R) Policy Notation, Definition and Some
Formula
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(Q,R) Policy Notation, Definition and Some
Formula
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(Q,R) Policy Notation, Definition and Some
Formula
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(Q,R) Policy Notation, Definition and Some
Formula
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(Q,R) Policy Notation, Definition and Some
Formula

• 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

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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.
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Example - Given A Q,R Policy, Find Cost
Find a. The safety stock

t


time,

Lead

lt

m

demand,

time
-
Lead

m
-



stock
Safety
R
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Example - Given A Q,R Policy, Find Cost
b. The average inventory level c. The
expected annual number of orders
Q
Q
m
-





stock
safety


inventory

Average
R
2
2

l


cycles

or

orders

of

number

annual

Expected
Q

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Example - Given A Q,R Policy, Find Cost
d. The probability of not stocking out during the
lead-time e. The expected number of units
stock-out per cycle
m
-
R


z
s

time
-
lead

during

out

stocking

not

of
y
Probabilit
(
)


786)
781-

pp.

4,
-
A
Table

(See


z
F
(
)
s

z
L
n

786)
781-

pp.

4,
-
A
Table

(See


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Example - Given A Q,R Policy, Find Cost
f. The fill rate g. The expected annual
number of shortages
n

-

b
1

rate,

fill

The
Q
l
n

shortages

of

number

annual

Expected
Q

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Example - Given A Q,R Policy, Find Cost
h.The holding cost per unit per year and penalty
cost per unit.



cost,

Holding
Ic
h

cost,
Penalty
p


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Example - Given A Q,R Policy, Find Cost
i. The average annual holding cost associated
with this policy.

regular

cost,

holding

Annual
hQ


2
stock
safety

cost,

holding

Annual

m
-

)
(
R
h

cost

holding

annual

Total
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Example - Given A Q,R Policy, Find Cost
j. The total annual cost associated with this
policy.
l
K


cost

ordering

Annual
Q
l
np


cost

out
-
stock

Annual
Q
l
l
np
K
hQ


m
-


cost

annual

Total
)
(
R
h
2
Q
Q

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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.)