1. Class 18: Optimal Service Level: Newsvendor Problem
OPSM301 Spring 2012
Class 18:
Optimal Service Level: Newsvendor Problem

2. Optimal Service Level: The Newsvendor Problem
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3. Single Period Inventory Control
Examples:
Style goods
Perishable goods (flowers, foods)
Goods that become obsolete (newspapers)
Services that are perishable (airline seats)

4. Optimal Service Level and Accurate Response to Demand Uncertainty when you can order only once
South Face is planning to offer a special line of winter jackets, especially designed as gifts for the Christmas season. Each Christmas-jacket costs the company $250 and sells for $450. Any stock left over after Christmas would be disposed of at a deep discount of $195. Marketing had forecasted a demand of 2000 Christmas-jackets with a forecast error (standard deviation) of 500.
How many jackets should South Face order?
Demand forecast for Christmas jackets
18%
15.9%
16%
14.6%
14.6%
14%
11.6%
11.6%
12%
10%
7.8%
7.8% 8% 6%
4.5%
4.5% 4%
2.2%
2.2% 2%
0.9%
0.9%
0.5%
0.5% 0%
600
800
1000
1200
1400
1600
1800
2000
2200
2400
2600
2800
3000
3200
3400
Copyright © 2013 Pearson Education Inc. publishing as Prentice Hall
J.A. Van Mieghem/Operations/Supply Chain Mgt

5. In reality, you do not know demand for sure…
In reality, you do not know demand for sure… Impact of uncertainty if you order the expected Q = 2000
Cost
250
Sales Price
450
Discount Price
195
Order Size
2000
Demand
P(D)
Units Sold
Units Overstocked
Units Understocked
Profit
($000)
600
0.50%
1400 43
800
0.90%
1200 94
1000
2.20%
145
4.50%
196
7.80%
247
1600
11.60%
400
298
1800
14.60%
200
349
15.90%
2200
2400
2600
2800
3000
3200
3400
Expected
1804
198
350

6. What happens if you change your order level to hedge against uncertainty? Performance for all possible Q using Excel:

7. What happens if I order one more unit (on top of Q = 2000)?
Towards the newsvendor model: Suppose you placed an order of 2000 units but you are not sure if you should order more.
What happens if I order one more unit (on top of Q = 2000)?
Sell the extra unit with
probability …P(D>2000)
DP = 200 (MB=Cu)
Do not sell the extra unit with probability … P(D≤2000)
DP =-55 (-MC=-Co)
Expected profit from additional unit
E(DP) = P(D>2000) 200-P(D ≤ 2000) 55
So? ... Order more
or less than 2000?

8. Accurate Response to Risk: Balancing surplus and shortage risks
Demand matches
Supply
Shortage
Surplus
Capacity
Level
Demand
Do simply on BB:
Optimal if expected surplus = expected shortage cost. Thus: pCo = (1-p)Cu > p = Cu/(Co+Cu)
Do it for Palu: p = > Q =
Copyright © 2013 Pearson Education Inc. publishing as Prentice Hall
J.A. Van Mieghem/Operations/Supply Chain Mgt

9. The Value-maximizing Service Level The newsvendor formula
In general: raise service level (i.e., order an additional unit) if and only if
E(DP) = (1-SL)MB – SLMC > Sell Do not sell
Thus, optimal service level SL* (= Newsvendor formula)
Example: use formula for South Face Christmas order
SL* =
So how much should South Face order then?
How does this compare to forecasted
demand of 2000?
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10. Accurate response
Marginal benefit of stocking an additional unit = cost of underage=MB (e.g., retail price - purchase price)
Marginal cost of stocking an additional unit = cost of overage=MC (e.g., purchase price - salvage price)
Given an order quantity Q, increase it by one unit if and only if the expected benefit of being able to sell it exceeds the expected cost of having that unit left over.
At optimal Q,
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11. Example Mean demand=3.85 How much would you order? Demand Probability
Total 1.00

12. Single Period Inventory Control
Economics of the Situation Known:
1. Demand > Stock --> Underage (under stocking) Cost Cu = Cost of foregone profit, loss of goodwill
2. Demand < Stock --> Overage (over stocking) Cost Co = Cost of excess inventory
Co = 10 and Cu = 20
How much would you order? More than 3.85 or less than 3.85?

13. Example Mean demand=3.85 How much would you order? Demand Probability
Total 1.00

14. Incremental Analysis Co = 10 and Cu = 20
Probability Probability Incremental
Incremental that incremental that incremental Expected
Demand Decision unit is not Sold unit is Sold Contribution
1 First (0.00)+20(1.00)
=20
2 Second (0.10)+20(0.90)
=17
3 Third
4 Fourth
5 Fifth
6 Sixth
7 Seventh
Co = 10 and Cu = 20

15. Generalization of the Incremental Analysis
Cash Flow Cu
-Co
nth unit needed
Pr{Demand  n}
Stock n-1
Decision
Point
Stock n
Base Case
Chance
Point
nth unit not needed
Pr{Demand < n}

16. Generalization of the Incremental Analysis
Chance
Point
Stock n
Decision
Base Case
Expected Cash Flow
Cu Pr{Demand  n} -Co Pr{Demand < n}
P(selling)
P(not selling)

17. Generalization of the Incremental Analysis
Order the nth unit if
Cu P(selling it) > Co P(not selling it)
Cu (1-P(not selling))> Co P(not selling) or
Cu (1-P(not selling)) - Co P(not selling) >= 0
P(not selling) < Cu /(Co +Cu)
P(Demand<=n-1) < Cu /(Co +Cu)
Then order n units, where n is the greatest number that satisfyies the above inequality.
Order quantity n should satisfy:
P(Demand  n) >= Cu /(Co +Cu) > P(Demand<= n-1)
Optimal order quantity= Smallest n that satisfies this

18. Single-Period Inventory Model Formulas
Find the smallest value Q such that
P(D<=Q)=F(Q)>= Cu/Co+Cu 8

19. Incremental Analysis Order quantity n should satisfy:
Demand Decision Pr{Demand <= n} Order nth??
1 First Yes
2 Second Yes
3 Third Yes
4 Fourth Yes
5 Fifth Yes
6 Sixth No -
7 Seventh 1
Cu /(Co +Cu)=20/(10+20)=0.66
Order quantity n should satisfy:
Cu /(Co +Cu)  P(Demand  n)
What if the demand has continuous distribution?
P(Demand<=n)= F(n)= Cu /(Co +Cu)

20. Order Quantity for Single Period, Normal Demand
Find the z*: z value such that F(z)= Cu /(Co +Cu)
Optimal order quantity is:
Do we order more or less than the mean if:
Cu > Co ?
Cu < Co ?

21. Example 1: Single Period Model
Our college basketball team is playing in a tournament game this weekend. Based on our past experience we sell on average 2,400 shirts with a standard deviation of We make $10 on every shirt we sell at the game, but lose $5 on every shirt not sold. How many shirts should we make for the game?
Cu = $10 and Co = $5; P ≤ $10 / ($10 + $5) = .667
Z(.667) = .4 (from standard normal table or using NORMSINV() in Excel)
therefore we need 2, (350) = 2,540 shirts

22. Example 2: Finding Cu and Co
A textile company in UK orders coats from China. They buy a coat from 250€ and sell for 325€. If they cannot sell a coat in winter, they sell it at a discount price of 225€. When the demand is more than what they have in stock, they have an option of having emergency delivery of coats from Ireland, at a price of 290.
The demand for winter has a normal distribution with mean 32,500 and std dev 6750.
How much should they order from China??
Cu=75-35=40
Co=25
F(z)=40/(40+25)=40/65=0.61z=0.28
 q= *6750=34390

23. Example 3: Airline ticket sale problem
The ticket price for a NY-Chicago flight is $200. Each plane can hold up to 100 pasengers. Usually, some of the passengers who have purchased tickets for a flight fail to show up (no-shows). To protect against no-shows, the airline will try to sell more than 100 tickets for each flight (over-booking). According to law, if a passenger with a ticket cannot be boarded, a compensation of 100$ should be paid. Past data indicated that the number of no-shows for this route is normally distributed with a mean of 20 and a standard deviation of 5. Anybody who does not show-up receives a 200$ refund. How many tickets should the airline sell to maximize revenues less compensation costs?
Cu=200, Co=100
NORMSINV(0.66)=0.41
Optimal # overbooked= *5 =22

24. Example 4: Single Period Inventory Management Problem
Manufacturing cost=60TL, Selling price=80TL, Discounted price (at the end of the season)=50TL Market research gave the following probability distribution for demand. Find the optimal q, expected number of units sold for this orders size, and expected fill rate and expected profit, for this order size.
Demand Probability
P(D<=n)
0.1
0.3
0.5
0.7
0.8
0.9 1
Cu=20 Co=10
P(D<=n )>= 20/30=0.66
>= 0.66
q=800
For q=800:
E(units sold)=710
E(fill rate)=94%
E(profit)=13,300
E(SL)=70%

25. More on Example 4: Would you accept the advertising campaign? Why ?
Your advertising agency proposes an advertisement campaign for your
product at a cost of 500 M TL
Your advertising agency estimates that the demand distribution will change
as follows as a result of the campaign:
Demand (Units)
Probability
700
0.3
800
900
0.2
1000
0.1
1100
Would you accept the advertising campaign? Why ?

26. Solution: Calculate the expected profit with the new demand:
Optimal order quantity changes- increases to 900
E(sales)=810
E(overage)=90
E(profit)=15300 > 500
So accept the campaign

27. Chapter 7
Learning Objectives
Service level is an economic tradeoff between cost of under and over stocking
Good model for Accurate Response for “fashion” goods
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28. Improving Supply Chain Performance:
Improving Supply Chain Performance: The Effect of Pooling/Centralization
Notes:
Copyright © 2013 Pearson Education Inc. publishing as Prentice Hall
J.A. Van Mieghem/Operations/Supply Chain Mgt

29. South Face’s Internet restructuring: Centralized inventory management
There are 9 stores.
Weekly demand per store = 59 jackets/ week
with standard deviation = 30 / week
Holding cost: H = $ 50 / jacket, year
Fixed order cost: S = $ 2,200 / order
Supply lead time: L = 4 weeks
Desired cycle service level F(z*) = 95%. sL=2x30=60
Safety Stock per store=zsL=1.64x60=98.4
Total Safety Stock for South Face=9x98.4= (decentralized case)
South Face now is considering restructuring to an Internet store.
Avg. lead-time demand m =4x4x59=944
Stdev. lead-time demand s =9xsL=3x60=180
Thus, total safety stock =1.64x180= (centralized/pooled case)
Question: What if the demands are correlated?
In general, if the demand in different locations are independent, Icsafety= N z sL
Copyright © 2013 Pearson Education Inc. publishing as Prentice Hall
J.A. Van Mieghem/Operations/Supply Chain Mgt

30. Principle of Aggregation and Pooling Inventory
Chapter 7
Principle of Aggregation and Pooling Inventory
Pooling Inventory: Available inventory is shared among various sources of demand
Pooling makes aggregation(centralization) possible even without physical centralization
Different centralization concepts
Physical Centralization
Information Centralization
Specialization
Commonality
Postponement
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31. Postponement (Delayed Differentiation)
The practice of delaying part of a process in order to reduce the need for safety inventory
Benetton and HP Printers applied this strategy successfully
Reduction in safety inventory due to:
Better forecast
Aggregation of demand before differentiation

32. Learning Objectives: Centralization/pooling
Chapter 7
Learning Objectives: Centralization/pooling
Centralization reduces safety stocks (pooling) and cycle stocks (economies of scale)
Can offer better service for the same inventory investment or same service with smaller inventory investment.
Different methods to achieve pooling efficiencies:
Physical centralization, Information centralization, Specialization, Commonality, Postponement/late customization.
Cost savings are proportional to square root of # of locations pooled.
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33. Levers for Reducing Safety Inventory
Reduce demand variability through improved forecasting
Reduce replenishment lead time
Reduce variability in replenishment lead time
Pool safety inventory for multiple locations or products
Exploit product substitution
Use common components
Postpone product differentiation until closer to the point of actual demand