MS 401
Production and Service Systems Operations
Spring 2009-2010
Inventory Control – IV
Multiperiod Probabilistic Demand: (Q,R) Model
Slide Set #6
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Lot Size – Reorder Point Systems
• Infinite horizon model (similar to the EOQ model)
• Continuous review
– the level of on-hand inventory is know at all times
• Demand is random and stationary
– the mean rate of demand is know to be λ
•
•
•
•
•
Replenishment lead time τ is known and fixed
Setup cost K per order
Holding cost h per unit hold per unit time
Unit order cost c per item
Stock out cost p per unit of unsatisfied demand
– shortage cost, penalty cost
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Problem Setup
• Relevant random variable: Demand during lead time, D
– this is what we guard against
– let μ=E(D) and σ: standard deviation of D
–
• Decision variables
– Q: Lot size (order quantity)
– R: Reorder level
• Policy: When the on-hand inventory drops to R, place an
order of Q units which will arrive in τ units of time
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Actual On-hand Inventory Plot
Copyright © 2001 by The McGraw-Hill Companies, Inc
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Expected Inventory Level
R-μ+Q
Slope= -λ
Reorder
Point
R
Safety R-μ
Stock
Order
Arrival
Order
Arrival
Cycle, length
T=Q/ λ
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The Objective
• The objective is to find the (Q,R) values that minimize the
expected annual average cost function
• Three cost components
– holding cost
– setup cost
– penalty cost
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The Holding Cost
• The expected inventory level varies linearly between
– R-μ+Q and (where μ is the mean of the lead time demand D)
– R-μ (safety stock)
• Average holding cost per unit time
• This is only an approximation
– in reality, the true inventory level may be negative in which case
no holding charge should apply
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The Setup Cost
• Number of setups per unit time
= number of cycles per unit time = 1/T = λ/Q
• Average setup cost per unit time = K λ / Q
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The Penalty Cost
• Stock-outs occur if demand during leadtime (D) is greater
than the reorder level (R)
• Expected number of stock-outs (shortages) in a cycle:

n( R)  EmaxD  R,0   x  R f x dx
R
• Expected number of stock-outs per unit time:
n  R  n  R 

T
Q
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The Cost Function & Solution
• Objective: Minimize the expected average annual cost of
holding, setup and shortages
Q
GQ, R   h  R  
2
 K p nR 


Q
 Q
• The optimal solution (Q, R) is found by iteratively solving
the following two equations:
Q
2 K  pnR 
h
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Qh
1  F R  
p
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The Algorithm to Find (Q*, R*)
2 K  pnR 
Q
h
Equation (1)
Qh
1  F R  
p
Equation (2)
1. Select Q0=EOQ
2. Using equation (2) find R0. Let i=1
3. Using equation (1) find Qi
Using equation (2) find Ri.
If the differences Qi-Qi-1 and Ri-Ri-1 are small, stop;
otherwise let i=i+1 and continue step 3.
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Normally Distributed Demand Case
• Note that n(R) is not easy to calculate in general
• If demand is normally distributed, we have
R 
n( R)  L
  Lz 
  
• where L(z) is the standardiz normal loss function
– tabulated in Table A-4 of the Appendix of Nahmias
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(Q, R) Policy: Example
•
•
•
•
•
•
•
•
Harvey’s Specialty Shop sells is a popular mustard.
The mustard costs Harvey $10 a jar.
Replenishment lead time is 6 months.
Harvey uses a 20 percent annual interest rate and estimates the
loss of goodwill cost as $25 per jar in case of stockout.
Bookkeeping expenses for placing an order is $50.
During the six-month replenishment period, Harvey estimates
that he sells an average of 100 jars, and the standard deviation
of demand is 25.
Assume that demand is described by a normal distribution.
How should Harvey control the replenishment of the mustard?
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(Q, R) Policy: Example
• First, note that λ=200 (the annual demand)
2K
2 (50)(200)
Q0  EOQ 

 100
h
(0.2)(10)
Q0 h (100)(2)
1  F R0  

 0.04
p (25)(200)
FromT ableA - 4 (Nahmias),we find z  1.75
T hisresultsin R 0  z    144
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(Q, R) Policy: Example
Next,we find Q1 
Hcnce, Q1 
2 K  pnR0 
h
2(200)50  (25)(0.405)
 110
2
Q1  110 and Q 0  100 are not close enough.Hence we cont inueit erat ions.
We find R 1  143and Q 2  111
Next wefind R 2  143
Because bot h Q 2 and R 2 are wit hin one unit of Q1 and R 1 , we st op.
Hence,t heopt imalvaluesare (Q*,R*)  (111,143)
Harveyshould order111 jars each t imehis invent oryhit s143 jars.
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(Q, R) Policy: Example (cont.)
• Determine the following for the example stated above:
– the safety stock
– the average annual holding, setup and penalty costs
associated with the inventory control of the mustard
– the average time between placement of orders
– the proportion of order cycles in which no stock out
occurs
– the proportion of demands that are not met
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Service Levels
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Service Levels in (Q, R) Systems
• Managers often have difficulty in determining the stock
out cost p because it includes intangible components such
as the loss of goodwill cost
• A common substitute is the service level
• There are number of different definitions of service level,
all related to the probability that the demand is met
• We will discuss two commonly used service levels:
– type-1 Service ()
– type-2 Service ()
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Type 1 Service Level
• Type-1 Service Level (): The probability of not stocking out
during lead time
• Equivalent to
• Computation of (Q, R) values:
– determine R value that satisfies F(R) = 
– set Q =EOQ
• A Type-1 Service objective is appropriate when a shortage
has the same consequence independent of its time or amount
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Type 2 Service Level
• Type-2 Service Level (): The proportion of demand met
from on-hand stock (fill rate)
– i.e.
• Expected number of stock outs per cycle: n(R)
• Expected demand during a cycle: Q
– why?
• Therefore, n(R)/Q is the proportion of demand NOT met
from on hand stock in each cycle
• Hence, R is determined from n(R)/Q=1- 
• Q is set as the EOQ
– not optimal but a good approximation
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Type 1 vs. Type 2 Service Levels
Scenario 1
Order
Cycle
Demand
Scenario 2
Stocks
Order
Cycle
Demand
Stocks
1
1,000
990
1
1,000
1,000
2
2,000
1,990
2
2,000
2,000
3
4,000
3,990
3
4,000
0
4
2,000
1,990
4
2,000
2,000
5
1,000
990
5
1,000
1,000
Total
10,000
9,950
Total
10,000
6,000
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Type 1 vs. Type 2 Service Levels
• Type-2 (fill rate) assumes as if there is a cost of
disappointing each customer
– and, the customers are treated as equal
• Type-1 (in-stock rate) assumes as if there is a high cost of
disappointing even one customer
– no credit for partial fulfillment in a given period
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Recall the Harvey’s Specialty Shop
• Suppose Harvey feels uncomfortable with the stock-out cost
concept and decides to use a service level of 98%
• Type-1: =0.98. From Table A-4 (Nahmias) we obtain z=2.05
hence
• Type-2:  =0.98. In this case again, use EOQ as the Q. We need
to solve n(R)=EOQ*(1-  ) to determine R
n(R)=100*(1-0.98)=2=L(z)*  hence
Therefore R=100+25*1.02 = 126.
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Important Notes
• Type-1 and type-2 service levels are used as a substitute to
“penalty cost”
– hence, if either of the service levels is specified, we do not need a
penalty cost to solve the problem
• Type-1 service: Q*=EOQ. Hence, no iteration to find Q*
• Type-2 service: EOQ is an approximation to Q*. We
assume that the approximation is good enough, hence we
will not iterate to find Q*
– ignore the section in p.257 of Nahmias
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Scaling of Lead Time Demand: Example
• Weekly demand for a certain type of automotive spark plug is
normally distributed with mean 34 and standard deviation 12.
Procurement lead time is six weeks. Determine the lead time
demand distribution.
• The demand over the lead time is normally distributed with
– mean: μ=6*34=204
– standard deviation: 12* sqrt(6)=29.39
Caution!
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“Inventory Position” Concept
• We checked “on hand inventory” level when making the
ordering decision
• When lead times are long, this can cause problems
• We might need to place a second order before an order
arrives
• For such cases, checking the “inventory position” is better:
Inventory position = Inventory on hand + (inventory on
order)
• Inventory position never falls before R
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Periodic Ordering Systems: (s,S) Policy
• (Q,R) model assumes that
– the inventory level is monitored continuously
– and we can place an order anytime we need
• In practice,
– the firm might know its inventory position only at certain time
intervals (say, each Friday evening, after an inventory count)
– or the firm might order only at certain times (due to supplier’s
rules etc.)
• In such cases, it is not possible to use the (Q,R) model
• The correct model to use: (s,S) model
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Periodic Ordering Systems: (s,S) Policy
• (s,S) policy:
– Check your inventory position every T time units
– If the position is less than s, order (S-s) units
– If the position is at least s, do not order
• Note that the order quantities might be different at each order
• Determining the optimal (s,S) values is difficult. A good
approximation:
– compute the (Q,R) values as if using a (Q,R) model
– set s=R, S=Q+R
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Multiproduct Systems: ABC Analysis
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Multiproduct Systems: A-B-C Analysis
•
•
There is a trade-off between the cost of controlling the
system and the potential benefits from that control
Typically
– the top 20% of the items account for the 80% of the annual dollar
value of sales
– the next 30% for the next 15% value of sales
– remaining 50% for 5% value of sales
•
To use ABC analysis:
1.
2.
3.
4.
select criterion for ranking (for example, annual sales value)
rank items on basis of criterion
calculate cumulative percentages
set up classes around break points
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The Distribution of Inventory by Value
Copyright © 2001 by The McGraw-Hill Companies, Inc
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Example A-B-C Analysis
vi
Di
Item
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
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80
514
19
2442
6289
128
1541
4
25
2232
2
1
6
12
101
715
1
35
1
422.3
54.07
0.65
16.11
4.61
0.63
2.96
22.05
5.01
2.48
4.78
38.03
9.01
25.89
59.5
20.78
2.93
19.52
28.88
Divi
33,784.00
27,791.98
12.35
39,340.62
28,992.29
80.64
4,561.36
88.20
125.25
5,535.36
9.56
38.03
54.06
310.68
6,009.50
14,857.70
2.93
683.20
28.88
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Example A-B-C Analysis: Sorted by $ Usage
Item
Di
4
1
5
2
16
15
10
7
18
14
9
20
8
6
13
12
19
3
11
17
vi
Divi
2442
16.11
39340.62
80
422.3
33784.00
6289
4.61
28992.29
514
54.07
27791.98
715
20.78
14857.70
101
59.5
6009.50
2232
2.48
5535.36
1541
2.96
4561.36
35
19.52
683.20
12
25.89
310.68
25
5.01
125.25
4
29.86
119.44
4
22.05
88.20
128
0.63
80.64
6
9.01
54.06
1
38.03
38.03
1
28.88
28.88
19
0.65
12.35
2
4.78
9.56
1
2.93
2.93
Total
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Cum. % of CostxUsage
Example A-B-C Analysis: Cumulatives Graph
100
80
60
40
20
0
Proportion of SKUs
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Multiproduct Systems: Exchange Curves
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Exchange Curve Example
• Consider a deterministic system consisting of n products
with varying demand rates λi and item values ci
• Assume that EOQ is used for each item’s replenishment
• The same K and I value for all items
• We may treat K/I as an aggregate policy parameter
– If K/I is large, lot sizes and average investment in inventory will be
large
– If K/I is small, the number of annual replenishments will be high
• An exchange curve allows us to see this trade-off easily
• Also, we do not need to know the “exact” values of K and I
together
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Exchange Curve Example
The number of annual replenishments:
The average value of on-hand inventory:
The multiplication of these two
expressions is equal to:
1 n

i ci 


2  i 1

2
The exchange curve makes it easy to see
the trade-offs
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