OPSM 301 Operations Management

Class 17:

Inventory Management: the newsvendor

Koç University

Zeynep Aksinzaksin@ku.edu.tr

Single Period Inventory Control

Examples:– Style goods– Perishable goods (flowers, foods)– Goods that become obsolete (newspapers)– Services that are perishable (airline seats)

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

8

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:

Do we order more or less than the mean if:– Cu > Co ?– Cu < Co ?

*zQ

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 350. We make 10TL on every shirt we sell at the game, but lose 5TL on every shirt not sold. How many shirts should we make for the game?

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 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 350. We make 10 TL on every shirt we sell at the game, but lose 5 TL 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 2,400 + .4(350) = 2,540 shirts

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