OPSM 501: Operations Management

Week 10:

Supply Chain contracts

Newsvendor

Koç University Graduate School of BusinessMBA Program

Zeynep Aksinzaksin@ku.edu.tr

2

Hamptonshire Express

Anna has a degree from journalism & operations research

She has started a daily newspaper in her hometown She used a leased PC: lease cost $10 per day A local printer prints newspapers at 0.20 per copy Sales the next day between 6 am and 10 am Newsstand rental $30 per day Express sold to customers at $1 per copy Copies not sold by 10 am are discarded Anna estimates daily demand to be distributed

N(500,100)

3

Question 1

Optimal stocking quantity? Profit at this stocking quantity?

4

Ordering Level and Profits in Vertically Integrated Channel

h=1; Anna sells to market directly:i* = 584; E[Profit] = $331.33; Fill rate

98%

5

Improving demand through effort

After 6 weeks of operation, Anna thinks she can improve demand by adding a profile section

Experiments indicate that demand is a function of time she invests in preparing the section

She thinks D=500 +50 h

6

Question 2

How many hours should she invest daily in the creation of the profile section? Assume the opportunity cost of her time is $10 per hour.

Compare optimal profits to previous scenario

7

Optimal Level of Effort in Vertically Integrated Channel

Demand potential increases by 50 Expected profit increases by 0.8*50

h

h

h h+1

0 1 40

1 2 16.56

2 3 12.71

3 4 10.71

4 5 9.44

)1(*50*0.8 hh

i* = 684E[Profit] = 371.33

8

Delegating sales to Ralph

Anna is really busy, so asks Ralph to take-over the retailing portion of her job.

Ralph agrees to run the newsstand from 6 AM to 10 AM and pay the daily rent of $30

He estimates demand the next day based on viewing a copy of the paper the previous night at 10 PM

He buys the papers from Anna at $0.8 per copy Ralph is responsible for unsold copies at the end of the

day

9

Question 3

Assuming h=4 try to determine the optimal stocking quantity for Ralph?

Why is this quantity different than the one in Question 2? Now vary h in spreadsheet 3c which calculates the

optimal newsboy quantity for the differentiated channel, i.e. to maximize Ralph’s profit.

How would changing the transfer price from the current value of 0.8 impact Ann’s effort level and Ralph’s stocking decision? (Spreadsheet 3d)

Compare an integrated (centralized) firm to a differentiated (decentralized) one.

10

Ordering Level and Profits in Differentiated Channel

Case 1. h=4; Anna sells to market directly:i* = 684; E[Profit] = $371.33; Fill rate

98%

Case 2. h=4; Anna sells thru Ralph:i* = 516; E[Total Profit] = $322

Anna makes $260Ralph makes $ 62Fill rate 84%

Why??

11

Effect of Transfer (Wholesale) Price in Differentiated Channel

Breakdown of total profits (h=4)

0

50

100

150

200

250

300

350

400

transfer price

$ralph

anna

12

Optimal Effort in Decentralized Channel

Optimal effort level for Anna is h=2 (and not 4).

h=2 h=4

Stocking quantity: $487 $516

Anna’s profit: $262 $260

Ralph’s profit: $56 $ 62

Total profit: $318 $322

Fill rate: 83% 84%

Why??

13

Inefficiencies in a Differentiated Channel

Supplier chooses w, retailer chooses i* Retail ignores +ve effect of stocking one more

unit on supplier Supplier ignores +ve effect of cutting

wholesale price/increasing effort on retailer Supplier prices above marginal cost/exerts

low effort Retailer stocks less Supply chain profits shrink

14

Contracts

Specifies the parameters within which a buyer places orders and a supplier fulfills them

Example parameters: quantity, price, time, quality Double marginalization: buyer and seller make

decisions acting independently instead of acting together; both of them make a margin on the same sale – gap between potential total supply chain profits and actual supply chain profits results

Buyback contracts can be offered that will increase total supply chain profit

15

Returns policies

Rationale: set buyback price b so that (retailer cost structure

= supply cost structure)

Supplier can use both w and b Supplier is bundling insurance with the

good

sr

sc

br

bw

16

Example

Breakdown of profits under a buyback scheme

050

100150200250300350400

buyback price

$ralph

anna

17

Reasons for return policies

Supplier is less risk averse than retailers

Supplier has a higher salvage value Safeguarding the brand Signalling information Avoiding brand switching

18

Costs of Return Policies

Extra transportation and handling Extra depreciation Getting the return rate wrong Retailer incentives

19

The case of books

Returns as in Hamptonshire Express… …However publisher has no control of return

quantities No control of inventory-shelf arrangements No control over private-label No control of retail price

Key difference: power has shifted from publisher to retailer

20

Video sales

Hollywood: video rentals and sales $20B business, and largest source of revenue

Rentals slipping– Competition from direct services– Customer dissatisfaction (20% cannot rent

video they want on a typical trip) What’s the problem? Bad forecasting?

Inefficient replenishment?

21

Revenue Sharing

Reduce wholesale price in return for a share of revenues

Encourages retailers to stock more $60 a tape

– $3/rental – rent each tape 20 times to break even

$9 a tape, studio receives half revenue– $3/rental – rent each tape 6 times to break

even Retailer stocks more

22

Revenue sharing

When does it work?– marginal cost of increasing inventory low– administrative burden low– for price-sensitive products

23

The Impact of Revenue Sharing

Blockbuster Instituted the “Go Home Happy” marketing initiative

Results– Store traffic went up– Market share 4th quarter 98 = 26%– Market share 2nd quarter 99 = 31%– Revenue in 2nd quarter 99: +17% from 98– Cash flow in 2nd quarter 99: +61% from 98

Supply

Sources:plantsvendorsports

RegionalWarehouses:stocking points

Field Warehouses:stockingpoints

Customers,demandcenterssinks

Production/purchase costs

Inventory &warehousing costs

Transportation costs Inventory &

warehousing costs

Transportation costs

Supply Chain Management: the challenge

Global optimization– Conflicting Objectives– Complex network of facilities– System Variations over time

Managing uncertainty– Matching Supply and Demand– Demand is not the only source of uncertainty

The newsvendor is all around us

Newspaper Apparel industry The flu shot

Recall Marks & Spencer

Expected demand Actual demand

Perfect forecast

Excess demandExcess stock

Recall Zara’a Approach to Demand uncertainty

Expected demand Actual demand

Small batches

Excess stock and unmet demand are avoided by stopping production when market saturates

Flu vaccine example

Each year’s flu vaccine is different: can’t produce ahead or keep from last year

Flu vaccine production requires growing strains: there is a lead time

Factories have limited capacity Demand is uncertain Need to commit to production before flu season starts Result: frequent shortage of vaccine or left overs at the

end of the season

The Newsvendor Model

Develop a Forecast: How did Anna come up with N(500, 100) for example?

11-30

O’Neill’s Hammer 3/2 wetsuit

Historical forecast performance at O’Neill

0

1000

2000

3000

4000

5000

6000

7000

0 1000 2000 3000 4000 5000 6000 7000

Forecast

Act

ual d

eman

d

.

Forecasts and actual demand for surf wet-suits from the previous season

Empirical distribution of forecast accuracy

Empirical distribution function for the historical A/F ratios.

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75

A/F ratio

Prob

abili

ty

Product description Forecast Actual demand Error* A/F Ratio**JR ZEN FL 3/2 90 140 -50 1.56EPIC 5/3 W/HD 120 83 37 0.69JR ZEN 3/2 140 143 -3 1.02WMS ZEN-ZIP 4/3 170 163 7 0.96HEATWAVE 3/2 170 212 -42 1.25JR EPIC 3/2 180 175 5 0.97WMS ZEN 3/2 180 195 -15 1.08ZEN-ZIP 5/4/3 W/HOOD 270 317 -47 1.17WMS EPIC 5/3 W/HD 320 369 -49 1.15EVO 3/2 380 587 -207 1.54JR EPIC 4/3 380 571 -191 1.50WMS EPIC 2MM FULL 390 311 79 0.80HEATWAVE 4/3 430 274 156 0.64ZEN 4/3 430 239 191 0.56EVO 4/3 440 623 -183 1.42ZEN FL 3/2 450 365 85 0.81HEAT 4/3 460 450 10 0.98ZEN-ZIP 2MM FULL 470 116 354 0.25HEAT 3/2 500 635 -135 1.27WMS EPIC 3/2 610 830 -220 1.36WMS ELITE 3/2 650 364 286 0.56ZEN-ZIP 3/2 660 788 -128 1.19ZEN 2MM S/S FULL 680 453 227 0.67EPIC 2MM S/S FULL 740 607 133 0.82EPIC 4/3 1020 732 288 0.72WMS EPIC 4/3 1060 1552 -492 1.46JR HAMMER 3/2 1220 721 499 0.59HAMMER 3/2 1300 1696 -396 1.30HAMMER S/S FULL 1490 1832 -342 1.23EPIC 3/2 2190 3504 -1314 1.60ZEN 3/2 3190 1195 1995 0.37ZEN-ZIP 4/3 3810 3289 521 0.86WMS HAMMER 3/2 FULL 6490 3673 2817 0.57* Error = Forecast - Actual demand** A/F Ratio = Actual demand divided by Forecast

Normal distribution tutorial

All normal distributions are characterized by two parameters, mean = and standard deviation =

All normal distributions are related to the standard normal that has mean = 0 and standard deviation = 1.

For example:– Let Q be the order quantity, and (, ) the parameters of the normal demand

forecast.– Prob{demand is Q or lower} = Prob{the outcome of a standard normal is z or

lower}, where

– (The above are two ways to write the same equation, the first allows you to calculate z from Q and the second lets you calculate Q from z.)

– Look up Prob{the outcome of a standard normal is z or lower} in the Standard Normal Distribution Function Table.

orQ

z Q z

11-34

-

0.0020

0.0040

0.0060

0.0080

0.0100

0.0120

0.0140

0.0160

0.0180

0 25 50 75 100 125 150 175 200

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

-100 -75 -50 -25 0 25 50 75 100

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

-4 -3 -2 -1 0 1 2 3 4

Converting between Normal distributions

Start with = 100,= 25, Q = 125

Center the distribution over 0 by subtracting the mean

Rescale the x and y axes by dividing by the standard deviation

125

100125

Q

z

11-35

Start with an initial forecast generated from hunches, guesses, etc. – O’Neill’s initial forecast for the Hammer 3/2 = 3200 units.

Evaluate the A/F ratios of the historical data:

Set the mean of the normal distribution to

Set the standard deviation of the normal distribution to

Using historical A/F ratios to choose a Normal distribution for the demand forecast

Forecast

demand Actual ratio A/F

Forecast ratio A/F Expected demand actual Expected

Forecast ratios A/F of deviation Standard

demand actual of deviation Standard

11-36

O’Neill’s Hammer 3/2 normal distribution forecast

3192320099750 . demand actual Expected

118132003690 . demand actual of deviation Standard

O’Neill should choose a normal distribution with mean 3192 and standard deviation 1181 to represent demand for the Hammer 3/2 during the Spring season.

Product description Forecast Actual demand Error A/F RatioJR ZEN FL 3/2 90 140 -50 1.5556EPIC 5/3 W/HD 120 83 37 0.6917JR ZEN 3/2 140 143 -3 1.0214WMS ZEN-ZIP 4/3 170 156 14 0.9176

… … … … …ZEN 3/2 3190 1195 1995 0.3746ZEN-ZIP 4/3 3810 3289 521 0.8633WMS HAMMER 3/2 FULL 6490 3673 2817 0.5659Average 0.9975Standard deviation 0.3690

11-37

Empirical vs normal demand distribution

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0 1000 2000 3000 4000 5000 6000

Quantity

Prob

abili

ty

.

Empirical distribution function (diamonds) and normal distribution function withmean 3192 and standard deviation 1181 (solid line)

11-38

Demand Scenarios for a Jacket

Demand Scenarios

0%5%

10%15%20%25%30%

Sales

P

robabili

ty

Costs

Production cost per unit (C): $80 Selling price per unit (S): $125 Salvage value per unit (V): $20 Fixed production cost (F): $100,000 Q is production quantity, D demand

Profit =Revenue - Variable Cost - Fixed Cost + Salvage

Best Solution

Find order quantity that maximizes weighted average profit.

Question: Will this quantity be less than, equal to, or greater than average demand?

What to Make?

Question: Will this quantity be less than, equal to, or greater than average demand?

Average demand is 13,100 Look at marginal cost Vs. marginal profit

– if extra jacket sold, profit is 125-80 = 45– if not sold, cost is 80-20 = 60

So we will make less than average

Scenarios

Scenario One:– Suppose you make 12,000 jackets and demand ends

up being 13,000 jackets.– Profit = 125(12,000) - 80(12,000) - 100,000 = $440,000

Scenario Two:– Suppose you make 12,000 jackets and demand ends

up being 11,000 jackets.– Profit = 125(11,000) - 80(12,000) - 100,000 + 20(1000) = $

335,000

44

Scenarios and their probabilitiesDemand

Pro

du

ctio

n q

uan

tity

8000 10000 12000 14000 16000 1800011% 11% 28% 22% 18% 10%

5,000 $125,000.00 $125,000.00 $125,000.00 $125,000.00 $125,000.00 $125,000.00 $125,000

5,500 $147,500.00 $147,500.00 $147,500.00 $147,500.00 $147,500.00 $147,500.00 $147,500

6,000 $170,000.00 $170,000.00 $170,000.00 $170,000.00 $170,000.00 $170,000.00 $170,000

6,500 $192,500.00 $192,500.00 $192,500.00 $192,500.00 $192,500.00 $192,500.00 $192,500

7,000 $215,000.00 $215,000.00 $215,000.00 $215,000.00 $215,000.00 $215,000.00 $215,000

7,500 $237,500.00 $237,500.00 $237,500.00 $237,500.00 $237,500.00 $237,500.00 $237,500

8,000 $260,000.00 $260,000.00 $260,000.00 $260,000.00 $260,000.00 $260,000.00 $260,000

8,500 $230,000.00 $282,500.00 $282,500.00 $282,500.00 $282,500.00 $282,500.00 $276,725

9,000 $200,000.00 $305,000.00 $305,000.00 $305,000.00 $305,000.00 $305,000.00 $293,450

9,500 $170,000.00 $327,500.00 $327,500.00 $327,500.00 $327,500.00 $327,500.00 $310,175

10,000 $140,000.00 $350,000.00 $350,000.00 $350,000.00 $350,000.00 $350,000.00 $326,900

10,500 $110,000.00 $320,000.00 $372,500.00 $372,500.00 $372,500.00 $372,500.00 $337,850

11,000 $80,000.00 $290,000.00 $395,000.00 $395,000.00 $395,000.00 $395,000.00 $348,800

11,500 $50,000.00 $260,000.00 $417,500.00 $417,500.00 $417,500.00 $417,500.00 $359,750

12,000 $20,000.00 $230,000.00 $440,000.00 $440,000.00 $440,000.00 $440,000.00 $370,700

12,500 -$10,000.00 $200,000.00 $410,000.00 $462,500.00 $462,500.00 $462,500.00 $366,950

13,000 -$40,000.00 $170,000.00 $380,000.00 $485,000.00 $485,000.00 $485,000.00 $363,200

13,500 -$70,000.00 $140,000.00 $350,000.00 $507,500.00 $507,500.00 $507,500.00 $359,450

14,000 -$100,000.00 $110,000.00 $320,000.00 $530,000.00 $530,000.00 $530,000.00 $355,700

14,500 -$130,000.00 $80,000.00 $290,000.00 $500,000.00 $552,500.00 $552,500.00 $340,400

15,000 -$160,000.00 $50,000.00 $260,000.00 $470,000.00 $575,000.00 $575,000.00 $325,100

15,500 -$190,000.00 $20,000.00 $230,000.00 $440,000.00 $597,500.00 $597,500.00 $309,800

16,000 -$220,000.00 -$10,000.00 $200,000.00 $410,000.00 $620,000.00 $620,000.00 $294,500

16,500 -$250,000.00 -$40,000.00 $170,000.00 $380,000.00 $590,000.00 $642,500.00 $269,750

Average Profit

Expected Profit

Expected Profit

Expected Profit

$0

$100,000

$200,000

$300,000

$400,000

8000 12000 16000 20000

Order Quantity

Pro

fit

Expected Profit

Expected Profit

$0

$100,000

$200,000

$300,000

$400,000

8000 12000 16000 20000

Order Quantity

Pro

fit

Expected Profit

Expected Profit

$0

$100,000

$200,000

$300,000

$400,000

8000 12000 16000 20000

Order Quantity

Pro

fit

Important Observations

Tradeoff between ordering enough to meet demand and ordering too much

Several quantities have the same average profit Average profit does not tell the whole story

Question: 9000 and 16000 units lead to about the same average profit, so which do we prefer?

Probability of Outcomes

0%

20%

40%

60%

80%

100%

Cost

Pro

ba

bilit

y

Q=9000

Q=16000

Key Insights from this Model

The optimal order quantity is not necessarily equal to average forecast demand

The optimal quantity depends on the relationship between marginal profit and marginal cost

Fixed cost has no impact on production quantity, only on whether to produce or not

As order quantity increases, average profit first increases and then decreases

As production quantity increases, risk increases. In other words, the probability of large gains and of large losses increases

Example

Mean demand=3.85 How much would you order?

Demand Probability

1 0.10

2 0.15

3 0.20

4 0.20

5 0.15

6 0.10

7 0.10

Total 1.00

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?

Incremental AnalysisProbability Probability Incremental

Incremental that incremental that incremental Expected

Demand Decision unit is not needed unit is needed Contribution

1 First 0.00 1.00 -10(0.00)+20(1.00)

=20

2 Second 0.10 0.90 -10(0.10)+20(0.90)

=17

3 Third 0.25 0.75 12.5

4 Fourth 0.45 0.55 6.5

5 Fifth 0.65 0.35 0.5

6 Sixth 0.80 0.20 -4

7 Seventh 0.90 0.10 -7

Co = 10 and Cu = 20

Generalization of the Incremental Analysis

ChancePoint

Stock n-1

DecisionPoint

Stock n

Base Case

nth unit needed

nth unit not needed

Pr{Demand n}

Pr{Demand n-1}

Cash FlowCu

-Co

0

Generalization of the Incremental Analysis

ChancePoint

Stock n-1

DecisionPoint

Stock n

Base Case

Expected Cash FlowCu Pr{Demand n} -Co Pr{Demand n-1}

Generalization of the Incremental Analysis

Order the nth unit ifCu Pr{Demand n} - Co Pr{Demand n-1} >= 0

or

Cu (1-Pr{Demand n-1}) - Co Pr{Demand n-1} >= 0

or

Cu - Cu Pr{Demand n-1} -Co Pr{Demand n-1} >= 0

or

Pr{Demand n-1} =< Cu /(Co +Cu)

Then order n units, where n is the greatest number that satisfies the above inequality.

Incremental Analysis

IncrementalDemand Decision Pr{Demand n-1} Order the unit?1 First 0.00 YES

2 Second 0.10 YES3 Third 0.25 YES4 Fourth 0.45 YES5 Fifth 0.65 YES6 Sixth 0.80 NO -7 Seventh 0.90 NO

Cu /(Co +Cu)=20/(10+20)=0.66

Order quantity n should satisfy:P(Demand n-1) Cu /(Co +Cu)< P(Demand n)

Order Quantity for Single Period, Normal Demand

Find the z*: z value such that F(z)= Cu /(Co +Cu)

Optimal order quantity is: *zQ

Transform

X = N(mean,s.d.) to

z = N(0,1)

z = (X - mean) / s.d.

F(z) = Prob( N(0,1) < z)

Transform back, knowing z*:

X* = mean + z*s.d.

The Standard Normal Distributionz 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

0.0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.53590.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.57530.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.61410.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.65170.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.68790.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.72240.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.75490.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.78520.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.81330.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.83891.0 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.86211.1 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.88301.2 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.90151.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.91771.4 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.93191.5 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.94411.6 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.95451.7 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.96331.8 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.97061.9 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.97672.0 0.9772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.98172.1 0.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.98572.2 0.9861 0.9864 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.98902.3 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.99162.4 0.9918 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.99362.5 0.9938 0.9940 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.99522.6 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.99642.7 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0.99742.8 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.99812.9 0.9981 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.99863.0 0.9987 0.9987 0.9987 0.9988 0.9988 0.9989 0.9989 0.9989 0.9990 0.99903.1 0.9990 0.9991 0.9991 0.9991 0.9992 0.9992 0.9992 0.9992 0.9993 0.99933.2 0.9993 0.9993 0.9994 0.9994 0.9994 0.9994 0.9994 0.9995 0.9995 0.99953.3 0.9995 0.9995 0.9995 0.9996 0.9996 0.9996 0.9996 0.9996 0.9996 0.9997

F(z)

z0

Example

z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.090.0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.53590.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.57530.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.61410.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.65170.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.68790.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.72240.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.75490.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.78520.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.81330.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.83891.0 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.86211.1 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.88301.2 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.90151.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.91771.4 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.93191.5 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.94411.6 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.95451.7 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.96331.8 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.97061.9 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.97672.0 0.9772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.98172.1 0.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.98572.2 0.9861 0.9864 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.98902.3 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.99162.4 0.9918 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.99362.5 0.9938 0.9940 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.99522.6 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.99642.7 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0.99742.8 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.99812.9 0.9981 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.99863.0 0.9987 0.9987 0.9987 0.9988 0.9988 0.9989 0.9989 0.9989 0.9990 0.99903.1 0.9990 0.9991 0.9991 0.9991 0.9992 0.9992 0.9992 0.9992 0.9993 0.99933.2 0.9993 0.9993 0.9994 0.9994 0.9994 0.9994 0.9994 0.9995 0.9995 0.99953.3 0.9995 0.9995 0.9995 0.9996 0.9996 0.9996 0.9996 0.9996 0.9996 0.9997

If we want tohave cum. probability of95%z=1.64

For demand:Mean=20std dev=10

Then:Q=20 + 1.64*10=36.4

Example: Anna’s first stocking decision

Cu = 1-0.2=0.8 and Co = 0.2; P ≤ 0.8 / (0.8 + 0.2) = .8

Z.8 = .84 (from standard normal table or using NORMSINV() in Excel)

therefore Anna needs 500 + .84(100) = 584 papers

Example: What about Ralph’s first stocking decision?

Anna sets h=4 D=N(500+50*2, 100) Cu = 1-0.8=0.2 and Co = 0.8; P ≤ 0.2 / (0.8 +

0.2) = .2

Z0.2 = - Z0.8 =-0.84 (from standard normal table or using NORMSINV() in Excel)

therefore Anna needs 600 - 0.84(100) = 516 papers

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??

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=40Co=25F(z)=40/(40+25)=40/65=0.61z=0.28 q=32500+0.28*6750=34390

Example 3: 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 profit, for this order size.

Demand Probability500 0.10600 0.2700 0.2800 0.2900 0.101000 0.101100 0.10

P(D<=n-1)00.10.30.50.70.80.9

Cu=20 Co=10P(D<=n-1)<=20/30=0.66

<=0.66 q=800

For q=800:E(units sold)=710E(profit)=13,300