1. The Newsvendor Model: Lecture 10
Risks from stockout and markdown
The Newsvendor model
Application to postponement
Review for inventory management

2. Risks from Stockout and Markdown
MBPF designed a fancy garage FG to sell in the Christmas season
Each costs $3000 in materials and sales for $5500.
Unsold FG will be salvaged for $2800 each
All raw materials have to be purchased in advance
Based on market research, MBPF estimated the demand of FG to be between 10 and 23 and the probabilities are given in table 1
What should be the amount of raw materials to purchase for producing FG?

3. Table 1: The Demand Distribution
Probability 10
0.01 11
0.02 12
0.04 13
0.08 14
0.09 15
0.11 16
0.16 17
0.20 18 19
0.10 20 21 22 23
Total
1.00
Table 1: The Demand Distribution

4. U2’s Spring T-Shirt
U2 has a new premier T-Shirt for Spring05 in 4 colors
Hong Kong retail market has a 3 month season slide 23
The standard production method is to dye the fabric first and then make shirts with different colors.
The production cost is low but leadtime is long, at 3 months. So U2 needs to place order in December
The production and in-bound logistic cost is $30/shirt, and U2 will sell the shirt at $90/shirt
U2 does not sell its premier shirts at discount in Hong Kong market. After the season, U2 wholesales the shirts to a mainland company at $25/shirt

5. Marginal Cost and Marginal Benefit
Suppose MBPF starts with a potential order quantity of Q and considers adding an additional unit Q
- If this unit is sold, there is a benefit (profit)
B =
B is called marginal benefit or underage cost
- If this unit cannot be sold, there is a cost
C =
C is called marginal cost or overage cost
For U2, Underage cost B = /shirt and Overage cost C = /shirt

6. Fashion Goods
MBPF and U2’s have the so called “fashion goods” or newsvendor problem
Short selling season
Limited ordering opportunity
Uncertain demands
Newspapers, magazines, fish, meat, produce, bread, milk, high fashion …

7. One Ordering Chance
MBPF and U2 have only one chance to order (long) before the selling season
Too late to order when the selling starts
No more demand information before the sales
There is no way to predict demands accurately
MBPF keeps past sales record which can be useful
U2 also can forecast, but what are past sales data?

8. The Ordering Risks Suppose MBPF or U2 orders Q and demand is D
If D > Q, there will be stockout
The cost (risk) = B max {D –Q, 0}
If D ≤ Q, there will be overstocks
The cost (risk) = C max {Q – D, 0}
The (potential) stockout and markdown costs
In some industries, such as fashion industry, the total stockout and markdown cost is higher than the total manufacturing cost!

9. How many papers should the newsboy buy?
The Clever Newsboy ?
Extra
Read all about it
gfdhg gfhjsg fgdhgs fgashjg ghdgsh dsagh lzkfjh yteeorkjrorthjoi oier; oerzjlztrj ezrlkjzfsgj ;zrurtgfdgdggfg dsg fgsdf gf fgs gfd
How many papers should the newsboy buy?

10. The Newsvendor Model
We do not know for sure if it can be sold or not. Thus, we have to work with the expected marginal benefit and expected marginal cost
Expected marginal benefit = B·Prob.{ Demand > Q}
Expected marginal cost = C·Prob. { Demand ≤ Q}

11. Marginal Analysis
Detailed numerical calculations in MBPFinventory.xls show, as Q increases:
- The expected marginal benefit decreases;
- The expected marginal cost increases; and
Q= 19 is the largest value of Q at which the marginal benefit is still greater than the marginal cost
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

12. The Critical Ratio
Suppose Q can be continuous. Then, there is a Q at which the expected marginal benefit and cost are equal
We call B/(B+C)= β the critical ratio
What does (1) say?
The optimal order quantity Q* is smallest integer greater than the Q obtained from (1)
(1)

13. Critical Ratio Solutions
For MBPF Inc.
B =, C =
From MBPFinventory.xls,
Q should be

14. Newsvendor with Continuous Demands
The demand in the selling cycle can be characterized by a continuous random variable D with mean μ, standard deviation σ, and distribution function F (x)
The optimal order quantity Q* is such that
(2)

15. Normally Distributed Demands
Consider normal demands N(μ, σ 2) with distribution F (Q)
We then have
By this equation, we see that the critical ratio is the probability that the standard normal demand
Ds ≤(Q – μ)/σ.
Prob.(demand≤Q) Q

16. Solution For Normal Demands
Set (Q – μ)/σ= zβ.
Recall that there is a one-to-one correspondence between zβ and β, and they are completely tabulated in the normal table
We then have this simple solution:
Q* = μ + zβσ (3)

17. Solving Discrete Problems by Normal Approximation
Consider the product FG of MBPF Inc.
We use the normal distribution to approximate the demand distribution.
From MBPFinventory.xls:
µ = and  = 2.48
From the normal table, we have z0.926 =
Then Q* =
Also from NORMINV(0.926, 16.26, 2.48)

18. Hedging Factor and Safety Stock
Hedging factor zβ is a function of the critical ratio β β zβ
When B < C (cost of lost sale < cost of overstock), overstock is more damaging and we order (zβσ) less than the expected demand
When B>C, lost sales is more damaging and we order zβσ more
When B=C, the impact of overstock and lost sales are the same, the best strategy is order the expected demand
zβσ is called the safety stock

19. Exercise: Christmas Trees
Mrs. Park owns a convenience store in Toronto
Each year, she sells Christmas trees from Dec. 3 to Dec. 24
She needs to order the trees in September
In the season, she sells a tree for $75
After Dec. 24, an unsold tree is salvaged for $15
Her cost is $30/tree inclusive

20. Exercise: Christmas Trees
Mrs. Park’s past sales record
Sales
Prob
Please give: (1) Critical ratio; (2) Hedging factor; and (3) Safety stock
Suppose Mrs. Park’s regular profit margin is $70, $30, or $10, and all else remain the same. Do the same
christmas

21. Postponement
Delay of product differentiation until closer to the time of the sale
All activities prior to product differentiation require aggregate forecasts which are more accurate than individual product forecasts
Point of delivery A B A
A and B B
dyeing
fabricating

22. Benefits of Postponement
Individual product forecasts are only needed close to the time of sale – demand is known with better accuracy (lower uncertainty)
Results in a better match of supply and demand
Valuable in e-commerce – time lag between when an order is placed and when customer receives the order (this delay is expected by the customer and can be used for postponement)
Question: Is postponement always good? What is the main factor(s) that determines the benefits of postponement?

23. Computing Value of Postponement for U2
For each color (4 colors) slide 3
Mean demand μ = 2,000; σ = 1500
For each garment
Sale price p = $90, Salvage value s = $25
Production cost using Option 1 (long leadtime) c = $30
Production cost using Option 2 (uncolored thread) c = $32
What is the value of postponement?

24. Use of The Newsvendor Model
Recall the newsvendor model,
We will also calculate the expected profit by

25. The Value of Postponement
Option 1: μ= 2000 and σ= 1500
Critical ratio =
Q*=
Profit from each color =
Total profit =
Option 2: μ= 8000 and σ=
postponement
3000

26. Value of Postponement with Dominant Product
Dominant color: μ=6,200, σ= 4500
Other three colors: μ= 600, σ= 450
Critical ratio =
Option 1:
Q*1= profit =
Q*2= profit =
Total expected profit =
postponement

27. Worst off with Postponement
Option 2:
μ= 8000, σ= ( x4502)1/2 =
Critical ratio =
Q*=
Profit =
Postponement allows a firm to increase profits and better match supply and demand if the firm produces a large variety of products whose demands are not positively correlated and are of about the same size
4567
postponement

28. Review: Inventory Management
How Much to Order
Tradeoff between ordering and holding costs
Robustness and Square-root rule
Tradeoff between setup time (capacity) and inventory cost

29. When to Order Reorder point ROP =  + IS = RL + zβσ
Assuming demand is normally distributed:
For given target SL
ROP =  + zβσ= NORMINV(SL, ,σ) =  +NORMSINV(SL)·σ
For given ROP
SL = Pr(DL  ROP) = NORMDIST(ROP, ,.σ, True)
Safety stock pooling (of n identical locations)
Relation between order quantity and reorder point.
Suppose your fixed ordering cost is not high while holding cost is relatively high and leadtime is relatively long. What may happen if you use the ROP policy as we defined? You may have a reorder cycle shorter than the leadtime. What is wrong with this?
You may order more frequently and receive order during a leadtime for another order. When an order arrives, the inventory level may still be lower than the ROP and you may not see ROP and place an new order. You may let the safety stock be depleted and run shortage. Is the ROP system wrong or do we really have such a serious problem?
In order to simplify the discussion, our textbook and many other business school textbooks have not pointed out that we in fact should monitor a different state variable (inventory level) which is called the inventory position: inventory on-hand + inventory on-order – backorder.
The computation of ROP is still the same. However, when an order is placed, the inventory position is raised immediately to ROP+Q.
In this case, even the order you just placed has not yet arrive, the inventory position may reduces from above ROP to ROP and another order can be placed during the leadtime of the previous order. Thus there might be multiple orders outstanding at the same time.
When order quantity is greater than average leadtime demand, on average, there should be only one order outstanding and the inventory position is usually equal to the on-hand inventory level.
For us, you should be aware of this subtlety, but you are not required to use inventory position in homework or midterm.

30. Managing System Inventory
Six basic reasons (functions) to hold inventory
Total average inventory for one item
= Q/2 + zβσ Not own pipeline
= Q/2 + zβσ+RL Own pipeline
Managing multiple items
- ABC analysis: 80/20 rule, Pareto Chart

31. Newsvendor
Stockout and markdown are major risks for inventory decisions
The critical ratio balances the stockout cost and the markdown cost:
- when B>C, we add a positive safety stock because stockout is more damaging;
- when B<C, we add a negative safety stock
Safety stock is used to hedge the risks
Q* = μ + zβσ