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The Newsvendor Model: Lecture 10

• Risks from stockout and markdown• The Newsvendor model• Application to postponement• Review for inventory management

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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?

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Demand 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 0.11

19 0.10

20 0.04

21 0.02

22 0.01

23 0.01

Total 1.00

Table 1: The Demand Distribution

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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

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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

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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 …

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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?

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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!

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The Clever Newsboy

How many papers should the newsboy

buy?

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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}

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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

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• 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)

CBBQ

)Demand(Prob.

BQ DemandProb.CQDemandProb. )](1[)(

The Critical Ratio

(1)

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Critical Ratio Solutions

• For MBPF Inc.B =, C =

From MBPFinventory.xls, Q should be

CB

BQDemandP )(

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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

CBBQ)F(Q*

)Demand(Prob. * (2)

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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. µ

Q

Prob.(demand≤Q)

CB

BQQF )(

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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)

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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 = 16.26 and = 2.48• From the normal table, we have z0.926 =Then Q* = Also from NORMINV(0.926, 16.26, 2.48)

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Hedging Factor and Safety Stock• Hedging factor zβ is a function of the critical ratio ββ 0.1 0.30 0.50 0.75 0.95 0.99 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

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• 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

Exercise: Christmas Trees

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Exercise: Christmas Trees • Mrs. Park’s past sales record

Sales 29 30 31 32 33 34 35 36 Prob. .05 .10 .15 .20 .20 .15 .10 .05

• 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 samechristmas

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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 forecastsA

B

A and B A

B

Point of delivery

dyeing fabricating

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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?

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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?

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Use of The Newsvendor Model• Recall the newsvendor model,

• We will also calculate the expected profit by

value salvage retailcostprice retail ratio critical

zQ *

506.2/)())(()( 2/** zespQspQcp

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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= Critical ratio =

Q*=Total profit =

3000

postponement

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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

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Worst off with Postponement• Option 2:

μ= 8000, σ= (45002+3x4502)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

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Review: Inventory ManagementHow Much to Order• Tradeoff between ordering and holding costs

Robustness and Square-root rule

• Tradeoff between setup time (capacity) and inventory cost

2//)(2)(,/2 **

HQQSRQTCRSHQTCHRSEOQQ

PTQPQP

se

RPPRT

Q s

min

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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)nzI sa

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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

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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

CB

BQ)Demand(Prob.