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The Newsvendor Model: Lecture 10

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

- 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?

Table 1: The Demand Distribution

- 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

- Suppose MBPF starts with a potential order quantity of Qand 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 costB = /shirt and Overage costC = /shirt

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 …

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?

- 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}

- If D > Q, there will be stockout
- 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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How many papers should the newsboy buy?

- 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}

Marginal Analysis

- Detailed numerical calculations in MBPFinventory.xls show, as Qincreases:
- 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

- 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 Qobtained from (1)

(1)

- For MBPF Inc.
B =, C =

From MBPFinventory.xls,

Q should be

- The demand in the selling cycle can be characterized by a continuous random variableD with mean μ,standard deviationσ, and distribution function F (x)
- The optimal order quantity Q* is such that

(2)

Consider normal demands N(μ, σ2) 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)

µ

Q

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

- 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)

- Hedging factor zβis a function of the critical ratio β
β0.10.30 0.50 0.75 0.950.99

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
- zβσis called the safety stock

- 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

- 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 same
christmas

- 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

- 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?

- For each color (4 colors)slide 3
- Mean demand μ = 2,000; σ = 1500

- For each garment
- Sale price p = $90, Salvage values = $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?

- Recall the newsvendor model,
- We will also calculate the expected profit by

- 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 =

postponement

3000

- 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

- 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

How Much to Order

- Tradeoff between ordering and holding costs
Robustness and Square-root rule

- Tradeoff between setup time (capacity) and inventory cost

- Reorder point ROP = + IS = RL + zβσ
- Assuming demand is normally distributed:
- For given targetSL
ROP = + zβσ= NORMINV(SL, ,σ) = +NORMSINV(SL)·σ

- For givenROP
SL = Pr(DL ROP) = NORMDIST(ROP, ,.σ, True)

- For given targetSL
- Safety stock pooling (of n identical locations)

- Six basic reasons (functions) to hold inventory
- Total average inventory for one item
= Q/2 + zβσ Not own pipeline

= Q/2 + zβσ+RLOwn pipeline

- Managing multiple items
- ABC analysis: 80/20 rule, Pareto Chart

- 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