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Managing Uncertainty in the Supply Chain
TIØ4285 Produkjons- og nettverksøkonomi
Lecture 3
2
Agenda
• Classic Inventory models– Economic Order Quantity (aka Economic Lot Size)
– The (s,S) Inventory Policy
• Managing uncertainty– Risk Pooling
– Managing uncertainty with contracts
• Forecasting Methods
• Case The Newsboy Problem (The News Vendor Problem)
3
Classic inventory theory
4
The Economic Order Quantity
Assumptions:• Constant demand rate of D items per day• Fixed order quantity Q• Fixed cost K every time an order is placed• Inventory carrying cost h (holding cost) per unit and day• Lead time between placing an order an receipt of goods is zero• Initial inventory is zero• Planning horizon is infinite
5
Determining the EOQ
• Total inventory cost per cycle T
• Average cost per unit of time
• Cost minimizing quantity
2hTQK +
2KD hQQ
+
* 2KDQh
=
This is known as the Economic Order Quantity (EOQ)
Time
Inventory Level
Q
T Time
Inventory Level
Q
T
6
The (s,S) Inventory Policy, Introduction
Basic description of the (s,S) inventory policy:
Whenever out inventory level drops below a certain level, say s, we order (or produce) in order to increase the inventory level to S. Such a policy is referred to as an (s,S) or min max policy.
s is called the reorder point and S the order-up-to-level
7
The (s,S) Inventory Policy, Random Demand
Additional Assumptions:• Daily demand is random and normally distributed• Fixed cost K for placing an order• Inventory holding cost h is charged per item per unit time• If an order is placed, it arrives after the appropriate lead time• All orders that can’t be satisfied from stock are lost• A service level is defined as the probability that the retailer is not
stocking out during lead time
8
The (s,S) Policy with Random Demand
Additional information needed to calculate the
AVG = Average daily demandSTD = Standard deviation of daily demandL x STD2 = cumulative variance over timeL = Replenishment lead time
α = Service level (implying that the probability of stocking out is 1- α)
z = Safety factor associated with the service level (can be takenfrom tables, e. g. Table 3-2, p. 60 in SKS)
9
Time
S
s
Inventory Level Inventory position
Lead time L
Q
Time
S
s
Inventory Level Inventory position
Lead time L Time
S
s
Inventory Level Inventory position
Lead time L
Q
Reorder Point s and Order-up-to-level S
• Average demand during lead time
• Safety stock
z is taken from statistical table to ensure that the probability of a stock out during lead time is 1-α
• Reorder point s
L AVG⋅
z STD L⋅ ⋅
L AVG z STD L⋅ + ⋅ ⋅P{demand during lead time L AVG+z STD L} 1 α≥ × × × = −
10
• Recall from EOQ
• Order-up-to-level S
• Expected level of inventory
before receipt after receipt on average
Reorder Point s and Order-up-to-level S
S Q s= +
* 2K AVGQh⋅
=
z STD L⋅ ⋅ Q z STD L+ ⋅ ⋅2Q z STD L+ ⋅ ⋅
11
Example 3-7 (SKS p. 61) I
Define the inventory policy for a TV distributor given the following assumptions:
– Whenever an order is placed the distributor incurs a fixed cost of $4500 (independent of order size)
– The cost of the TV to the distributor is $250 and annual inventory holding cost is about 18% of the product cost
– Replenishment lead time is 2 weeks– Desired service level is 97%– Demand data for the last 12 months:
15698309246198151176287221100152200Sales
Aug.JulyJuneMayApr.Mar.Feb.Jan.Dec.Nov.Oct.Sept.Month
12
Example 3-7 (SKS p. 61) II
• Average monthly demand is 191.17, standard deviation of monthly demand is 66.53
• Lead time is two weeks, transform monthly demand data into weekly demand data:
• Table 3-2 (SKS, p. 60) determines the safety factor z for a 97% service level as 1.88
4.3
4.3
Average monthly demandAverage weekly demand
Monthly standard deviationStandard deviation of weekly demand =
=
13
Example 3-7 (SKS p. 61) III
Calculate• Average demand during lead time
• Safety stock
• Reorder point
89.16 85.29 174.45 175L AVG z STD L⋅ + ⋅ ⋅ = + = ≈
32.0844.58Value
Standard deviation ofweekly demand
Average weeklydemand
Parameter
1.88 32.08 2 85.29z STD L⋅ ⋅ = ⋅ ⋅ =
2 44.58 89.16L AVG⋅ = ⋅ =
14
Example 3-7 (SKS p. 61) IV
• Weekly inventory holding cost
• Order quantity Q
• Order-up-to-level
• Average inventory level
0.18 250 0.8752⋅
=
2 2 4500 44.58 679.1 6790.87
K AVGQh⋅ ⋅ ⋅
= = = ≈
679 85.29 424.79 4252 2Q z STD L+ ⋅ ⋅ = + = ≈
679 175 854S Q s= + = + =
15
Introducing Variable Lead Times
With a normally distributed delivery lead time and
AVGL = Average lead time STDL = Standard deviation of lead time
• Reorder point s
• Order-up-to-level S
2 2 2s AVG AVGL z AVGL STD AVG STDL= ⋅ + ⋅ + ⋅
2 2 2S Q AVG AVGL z AVGL STD AVG STDL= + ⋅ + ⋅ + ⋅
16
Risk poolingFrom SKS, and additional example
17
Market Two
Risk Pooling
• Consider these two systems:
Supplier
Warehouse One
Warehouse Two
Market One
Market Two
Supplier WarehouseMarket One
18
Risk Pooling
• For the same service level, which system will require more inventory? Why?• For the same total inventory level, which system will have better service?
Why?• What are the factors that affect these answers?
19
Example: Demand variance of portfolio of n regions withstochastic demand D
If perfect correlation between the n regions (P=nD):
The variation of total demand is reduced if the regions arenegatively correlated. The more positively correlated, thehigher the variation.
The effect of centralized inventories comes from the fact thatpeaks does not appear simultaneuously in regions which arenot perfctly correlated.
Portfolio thinking
2( ) VAR (D)VAR P n=
n
ii=1
n n
i i ji=1 i=1
P= D
Var P= VAR (D ) + COV (D ,D )j i≠
∑
∑ ∑∑
20
Risk Pooling Example
• Compare the two systems:– two products– maintain 97% service level– $60 order cost– $.27 weekly holding cost– $1.05 transportation cost per unit in decentralized system, $1.10 in centralized
system– 1 week lead time
21
Risk Pooling Example
Week 1 2 3 4 5 6 7 8
Prod A,Market 1
33 45 37 38 55 30 18 58
Prod A,Market 2
46 35 41 40 26 48 18 55
Prod B,Market 1
0 2 3 0 0 1 3 0
Product B,Market 2
2 4 0 0 3 1 0 0
22
Risk Pooling Example
Warehouse Product AVG STD CV
Market 1 A 39.3 13.2 .34
Market 2 A 38.6 12.0 .31
Market 1 B 1.125 1.36 1.21
Market 2 B 1.25 1.58 1.26
23
Risk Pooling Example
Warehouse Product AVG STD CV s S Avg.Inven.
%Dec.
Market 1 A 39.3 13.2 .34 65 158 91
Market 2 A 38.6 12.0 .31 62 154 88
Market 1 B 1.125 1.36 1.21 4 26 15Market 2 B 1.25 1.58 1.26 5 27 15Cent. A 77.9 20.7 .27 118 226 132 26%Cent B 2.375 1.9 .81 6 37 20 33%
24
Risk Pooling:Important Observations• Centralizing inventory control reduces both safety stock and average
inventory level for the same service level.• This works best for
– High coefficient of variation, which reduces required safety stock.– Negatively correlated demand. Why?
• What other kinds of risk pooling will we see?
25
• Safety stock• Service level• Overhead costs• Customer lead time• Transportation costs
26
Managing uncertainty and the effect of Supply contracts
27
Contract design
• Pricing and volume discounts• Minimum and maximum purchase quantities• Delivery lead times• Product or material quality• Product return policies
28
Example
• Will show:– The effect of uncertainty on supply chain co-operation and risk sharing
– How to use contracts to manage risk
– When demand is uncertain simple co-ordination schemes based on single prices are not enough to ensure supply chain optimality
• From SKS Ch 3.
29
SnowTime Sporting Goods
• Fashion items have short life cycles, high variety of competitors• SnowTime Sporting Goods
– New designs are completed– One production opportunity– Based on past sales, knowledge of the industry, and economic conditions, the
marketing department has a probabilistic forecast– The forecast averages about 13,000, but there is a chance that demand will be
greater or less than this.
30
Supply Chain Time Lines
Jan 00 Jan 01 Jan 02
Feb 00 Sep 00 Sep 01
Design Production Retailing
Feb 01Production
31
SnowTime Demand Scenarios
Demand Scenarios
0%5%
10%15%20%25%30%
8000
1000
012
000
1400
016
000
1800
0
Sales
Pro
babi
lity
32
SnowTime 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
33
SnowTime Best Solution
• Find order quantity that maximizes weighted average profit.• Question: Will this quantity be less than, equal to, or greater than
average demand?
34
What to Make?
• Question: Will this quantity be less than, equal to, or greaterthan 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
35
SnowTime 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
36
SnowTime Expected Profit
Expected Profit
$0
$100,000
$200,000
$300,000
$400,000
8000 12000 16000 20000
Order Quantity
Prof
it
37
SnowTime Expected Profit
Expected Profit
$0
$100,000
$200,000
$300,000
$400,000
8000 12000 16000 20000
Order Quantity
Prof
it
38
SnowTime Expected Profit
Expected Profit
$0
$100,000
$200,000
$300,000
$400,000
8000 12000 16000 20000
Order Quantity
Prof
it
39
SnowTime: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?
40
SnowTime Demand Scenarios
Demand Scenarios
0%5%
10%15%20%25%30%
8000
1000
012
000
1400
016
000
1800
0
Sales
Pro
babi
lity
41
Probability of Outcomes
0%
20%
40%
60%
80%
100%
-300
000
-100
000
1000
0030
0000
5000
00
Cost
Prob
ability
Q=9000Q=16000
42
Manufacturer Manufacturer DC Retail DC
Stores
Manufacturer Fixed Production Cost =$100,000
Variable Production Cost=$35
Selling Price=$125
Salvage Value=$20
Wholesale Price =$80
Supply ContractsWhen we add the manufacturer
• No fixed cost for distributor• (otherwise equal to previous example)
43
Demand Scenarios
Demand Scenarios
0%5%
10%15%20%25%30%
8000
1000
012
000
1400
016
000
1800
0
Sales
Pro
babi
lity
44
Distributor Expected Profit
Expected Profit
0
100000
200000
300000
400000
500000
6000 8000 10000 12000 14000 16000 18000 20000
Order Quantity
45
Distributor Expected Profit
Expected Profit
0
100000
200000
300000
400000
500000
6000 8000 10000 12000 14000 16000 18000 20000
Order Quantity
46
Supply Contracts (cont.)
• Distributor optimal order quantity is 12,000 units• Distributor expected profit is $470,000• Manufacturer profit is $440,000• Supply Chain Profit is $910,000
–IS there anything that the distributor and manufacturer can do to increase the profit of both?
47
Manufacturer Manufacturer DC Retail DC
Stores
Fixed Production Cost =$100,000
Variable Production Cost=$35
Selling Price=$125
Salvage Value=$20
Wholesale Price =$80
Supply Contracts
48
Retailer Profit (Buy Back=$55)
0
100,000
200,000
300,000
400,000
500,000
600,000
6000
7000
8000
9000
1000
011
000
1200
013
000
1400
015
000
1600
017
000
1800
0
Order Quantity
Reta
iler P
rofit
49
Retailer Profit (Buy Back=$55)
0
100,000
200,000
300,000
400,000
500,000
600,000
6000
7000
8000
9000
1000
011
000
1200
013
000
1400
015
000
1600
017
000
1800
0
Order Quantity
Reta
iler P
rofit
$513,800
50
Manufacturer Profit (Buy Back=$55)
0
100,000
200,000
300,000
400,000
500,000
600,000
6000
7000
8000
9000
1000
011
000
1200
013
000
1400
015
000
1600
017
000
1800
0
Production Quantity
Man
ufac
ture
r Pro
fit
51
Manufacturer Profit (Buy Back=$55)
0
100,000
200,000
300,000
400,000
500,000
600,000
6000
7000
8000
9000
1000
011
000
1200
013
000
1400
015
000
1600
017
000
1800
0
Production Quantity
Man
ufac
ture
r Pro
fit $471,900
52
Manufacturer Manufacturer DC Retail DC
Stores
Fixed Production Cost =$100,000
Variable Production Cost=$35
Selling Price=$125
Salvage Value=$20
Wholesale Price =$80
Supply Contracts
53
Retailer Profit (Wholesale Price $70, RS 15%)
0
100,000
200,000
300,000
400,000
500,000
600,000
6000
7000
8000
9000
1000
011
000
1200
013
000
1400
015
000
1600
017
000
1800
0
Order Quantity
Reta
iler P
rofit
54
Retailer Profit (Wholesale Price $70, RS 15%)
0
100,000
200,000
300,000
400,000
500,000
600,000
6000
7000
8000
9000
1000
011
000
1200
013
000
1400
015
000
1600
017
000
1800
0
Order Quantity
Reta
iler P
rofit
$504,325
55
Manufacturer Profit (Wholesale Price $70, RS 15%)
0
100,000
200,000
300,000
400,000
500,000
600,000
700,000
6000
7000
8000
9000
1000
011
000
1200
013
000
1400
015
000
1600
017
000
1800
0
Production Quantity
Man
ufac
ture
r Pro
fit
56
Manufacturer Profit (Wholesale Price $70, RS 15%)
0
100,000
200,000
300,000
400,000
500,000
600,000
700,000
6000
7000
8000
9000
1000
011
000
1200
013
000
1400
015
000
1600
017
000
1800
0
Production Quantity
Man
ufac
ture
r Pro
fit
$481,375
57
Supply Contracts
Strategy Retailer Manufacturer TotalSequential Optimization 470,700 440,000 910,700 Buyback 513,800 471,900 985,700 Revenue Sharing 504,325 481,375 985,700
58
Manufacturer Manufacturer DC Retail DC
Stores
Fixed Production Cost =$100,000
Variable Production Cost=$35
Selling Price=$125
Salvage Value=$20
Wholesale Price =$80
Supply Contracts
59
Supply Chain Profit
0
200,000
400,000
600,000
800,000
1,000,000
1,200,000
6000
7000
8000
9000
1000
011
000
1200
013
000
1400
015
000
1600
017
000
1800
0
Production Quantity
Supp
ly C
hain
Pro
fit
60
Supply Chain Profit
0
200,000
400,000
600,000
800,000
1,000,000
1,200,000
6000
7000
8000
9000
1000
011
000
1200
013
000
1400
015
000
1600
017
000
1800
0
Production Quantity
Supp
ly C
hain
Pro
fit
$1,014,500
61
Supply Contracts
Strategy Retailer Manufacturer TotalSequential Optimization 470,700 440,000 910,700 Buyback 513,800 471,900 985,700 Revenue Sharing 504,325 481,375 985,700 Global Optimization 1,014,500
62
Supply Contracts: Key Insights
• Effective supply contracts allow supply chain partners to replace sequential optimization by global optimization
• Buy Back and Revenue Sharing contracts achieve this objective through risk sharing
63
Other Contracts
• Quantity Flexibility Contracts– Supplier provides full refund for returned items as long as the number of
returns is no larger than a certain quantity• Sales Rebate Contracts
– Supplier provides direct incentive for the retailer to increase sales by means of a rebate paid by the supplier for any item sold above a certain quantity
64
Forecasting(Hillier & Lieberman Ch 20, SKS Ch 3)
65
Forecasting
Basic Principles of Forecasting:
• The forecast is always wrong
• The longer the forecast horizon, the worse the forecast
• Aggregate forecasts are more accurate
66
Forecasting Methods
Forecasting methods can be split into four general categories:
• Judgment methods,
• Market research methods,
• Causal methods, and
• Time-series methods
67
Judgment Methods
• Panels of expertsA group of experts is assembled in order to agree upon a forecast. It is assumed that sharing information and communicating allows for reaching a consensus.
• Delphi methodA group of experts makes independent forecasts. The results of all forecasts is presented to the experts and they are asked to refine their forecasts based on the now available knowledge. The process is repeated a specified number of times or until theexpert’s forecast are within a certain margin.
68
Market Research Methods
• Market testingGroups of potential customers are assembled and tested for theirresponse to products. Their response is extrapolated to the whole market and used for estimating demand.When new models are launched in the automotive industry, potential customers are assembled to check their response to the planned design long before production starts.
• Market surveyCustomer response is estimated by gathering data from various potential customer groups. Data is usually collected through interviews, telephone-based surveys, written surveys, etc.
69
Causal Methods
Causal methods try to generate forecast based on data other than the data being predicted.
Using a causal forecasting method, you may try to forecast sales for the next three months based on• inflation• GNP• advertising/marketing activities• weather• etc.
70
Time-Series Methods
Time-series based methods use a variety of past data to estimate future data. Examples of time-series methods:
• Last-Value forecasting• Averaging forecast• Moving average• Exponential smoothing• Methods for data with trends• Methods for seasonal data• etc.
71
Eample of time series
Demand
0
20
40
6080
100120
1 3 5 7 9 11 1 3 5 7 9 11
Month
Volu
me
72
Last-Value forecasting
• Variance is large because of small sample size (1 data point)• Worth considering if
– Constant-level is not likely– World is changing so fast that most of the previous observations are
irrelevant– Often called the naive method (by statisticians)– May be the only relevant value (under rapid changes)
1t tF x+ =
73
Averaging Forecasting
• Excellent if the process is stable
• Normally limited to young processes
11
ti
ti
xFt+
=
=∑
74
Moving Average
• Very simple forecasting method
• Calculating the average of previous demand over a given number of time periods
• Every past demand point is weighted equally
• Determining the number of time periods n to calculate the average demand is very important
• Only recent history and multiple observations
11
ti
ti t n
xFn+
= − +
= ∑
75
Exponential Smoothing
• α- smoothing constant (between 0 and 1)• Forecast is a weighted average of the previous forecast and the last
demand
• More recent value receive more weight than past values
• Similar to the Moving Average Forecast, but tracks changes in the process faster
• Lags behind a continuous trend
• Choice of the smoothing constant α (weight of the last demand) is important
( )( )
1
1
1
1
t t t
it t t i
i
F x F
F x x
α α
α α α
+
+ −
= + −
= + −∑
76
77
Methods for Data with Trends
• Regression Analysis,tries to fit a straight line into data points. Thus, it’s identifying linear trends.
• Holt’s Method,combines the concepts of exponential smoothing with the ability to capture linear trends.
78
Example
79
Box-Jenkins method
80
Box-Jenkins
• Box-Jenkins is a methodology for identifying, estimating, and forecasting ARIMA models
• Requires a large amount of data (historical observations of a variable)• Alternative name: ARIMA method• Iterative process
81
ARIMA models
• Describes a stochastic process or a model of one • Explains the variable Y based only on the history of Y
– No other explanatory variables– Not derived from economic theory
• Stands for "autoregressive integrated moving- average" – Made up of sums of autoregressive and moving-average components
• An ARIMA process is stationary – mean, variance and autocorrelation structure do not change over time
82
ARIMA components
• AR (Autoregressive)– Process that can be described by a weighted sum of its previous values and a
white noise error
– AR(1):
– AR(p):
• MA (Moving Average)– Process that can be described by a constant plus a moving average of the
current and past error terms
– MA(1):
– MA(q):
1t t tY Yα θ−= +
1
n
t p t p tp
Y Yα θ−=
= +∑
t tY µ βθ= +
1
n
t q t qq
Y µ β θ −=
= +∑
83
ARIMA components (2)
• ARMA (p, q)– May be non-stationary (integrated)
• Non- constant mean, variance or covariance– The d-differences of a time series that are integrated of order d is
stationary• 1- differences of Y: Yt – Yt-1
• ARIMA (p, d, q)– p AR terms, integrated of order d and q MA terms
84
White noise error
• Uncorrelated random error term with zero mean and constant variance• Often assumed to be normally distributed
85
The Box-Jenkins Methodology
1. Identification of the model(Choosing p, d and q)
2. Parameter estimation of the chosen model
3. Diagnostic checking(Are the residuals white noise?)
4. Forecasting
If yes – go to step 4If no – go to step 1
86
2. Parameter estimation
• Estimate the value of the p AR and the q MA parameters– Simple problems: use least squares method– Otherwise: nonlinear (in parameter) estimation methods– Done by several statistical packages
• (SPSS, MiniTab, S- Plus, EasyReg,…..)
87
3. Diagnostic checking
• Test if the estimated model fits the historical data• In particular: test if the residuals are white noise
– If not – the model is not appropriate and a new one must be estimated– If they are – the model may be good and used for forecasting
88
4. Forecasting
• Use the model to forecast new values• Can be used as a test of the goodness of the model
– Out-of-sample tests– Cross validation
89
Conclusions
• Requires no economic theory÷ Can not make use of any economic theory+ Can not misuse economic theory/ make false assumptions
• Usually provides accurate forecasts• Requires a large amount of data• Only needs data for one variable
90
Regretion analysys• Assume that a company’s demand
fucntion ca be written on the form
• Y er dependent variable
•Are examples on independent variables.
• We wish to estimateBased on historical data
1 1 2 2 3 4 rY A B x B x B I B P= + + + +
1 2, , , rx x I P
1 2 3 4, , , ,A B B B B
91
Population regression curve
• Line shows the value of E(Y|X)
• The line is fitted so that squared deviation is minimized
Sales
Advertising
1 1Y A B x= +
92
93
Explained variance and theconstant of determination
•Variance in dependent variable :
and
Explained variance :
Unexplained variance:
2
1( )
n
ii
Y Y=
−∑
ˆ ˆ( ) ( )i i i iY Y Y Y Y Y− = − + −
2
1
ˆ( )n
ii
Y Y=
−∑
2
1
ˆ( )n
i ii
Y Y=
−∑
94
Explained variance and theconstant of determination
2
2
2 1
2
1
Variation not explained by regression 1Total variation
ˆ( )1
( )
n
i iin
i ii
R
Y YR
Y Y
=
=
= −
−= −
−
∑
∑
95
Methods for Seasonal Data
Seasonal data has to be accounted for.
1. Calculate seasonal factor
2. Calculate seasonally adjusted values
3. Select time-series forecasting method4. Apply this method to the seasonally adjusted data5. Multiply this forecast with the seasonal factor in order to get the next
actual value
average for the periodSeasonal factoroverall average
=
actual valueSeasonal adjusted valueseasonal factor
=
96
Example on Forecasting Seasonal Demand (HL p. 1019) I
965043
794933
682223
699213
872442
778432
706422
725712
826641
656931
646521
680911
Call VolumeQuarterYear
CCW's Average Daily Call Volume
97
Example on Forecasting Seasonal Demand (HL p. 1019) IICalculating seasonal factors
8880 1.187529
=
6784 0.907529
=
7434 0.997529
=
30117 75294
=
7019 0.937529
=
Average=
Total=30117
88804
74343
67842
70191
Seasonal FactorThree-Year AverageQuarter
98
Example on Forecasting Seasonal Demand (HL p. 1019) III
817896501.1843802979490.9933758068220.9023751869920.9313701282741.1842786377840.9932784970640.9022780372570.9312700582661.1841663565690.9931718364650.9021732268090.9311
Call VolumeCall VolumeFactorQuarterYearSeasonally AdjustedActualSeasonal
99
Stochastic inventory modelscase: The newsboy problem
100
The Newsboy Problem
Short problem description:
You are selling newspapers. Each morning you go and buy the amount of newspapers you think you are going to sell in the course of the day. Unfortunately, you don’t know exactly how many newspapers you are going to sell, your demand is uncertain. In addition, you have no chance of buying additional newspapers if you realize you face a higher demand than what you anticipated. So all demand you cannot satisfy from stock is lost. In the other case – you have bought too many newspapers – you will not be able to sell them later, because no one will be interested in yesterday’s news. (But you can sell the papers with a loss as recycling material.)
How many newspapers do you buy every morning?
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The Newsboy Problem, Formulation by Rudi and PykeWe change notation here:
W - cost of buying a newspaperR - selling price of the newspaper (R > W)S - return price (salvage value) of a newspaper (S < W)
Q - number of newspapers bought in the morningD - stochastic demand
In addition, we will need the following definition:
00
if otherwise
y yy+ ≥=
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The Newsboy Problem, Reformulation by Rudi and Pyke• Define expected profit as function of Q
• With
and
we reformulate Πr(Q)
and call Gr(Q) the cost of uncertainty
( ) min( , ) ( )r Q E R D Q S Q D WQ+ Π = + − −
( )
( ) ( ) ( )( ) ( )( )
r
r
G Q
Q R W ED E R W D Q W S Q D+ + Π = − − − − + − −
min( , ) ( )D Q D D Q += − −
( ) ( )Q D D Q Q D+ += − − + −
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The Newsboy Problem, Reformulation by Rudi and Pyke• The first term
represents expected number of units short times the opportunity cost per unit short. Thus, define the unit underage cost Cu as:
• The second term
represents expected number of left over units times the opportunity cost per left over unit. Thus, define the unit overage cost Co as:
• Rewrite Gr(Q) as
( )( )R W D Q +− −
oC W S= −
( )( )W S Q D +− −
uC R W= −
( ) ( ) ( )r u oG Q E C D Q C Q D+ + = − + −
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The Newsboy Problem, Reformulation by Rudi and Pyke
In order to determine the order quantity that results in the highest expected profit, derive Gr(Q) and find Q such that the derivative equals zero.
• Derivative of the first term of Gr(Q):
• Derivative of the second term of Gr(Q):
• The derivative of Gr(Q):
Pr( )uC D Q− >
( ) Pr( ) Pr( )r u oG Q C D Q C Q D′ = − > + >
Pr( )oC Q D>
( ) ( ) ( )r u oG Q E C D Q C Q D+ + = − + −
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The Newsboy Problem, Reformulation by Rudi and Pyke• With Pr(D>Q) = 1 – Pr(D<Q), we can rewrite G’r(Q) as
• Equating this derivative to zero, we get the following Newsboy optimality condition
• This allows to find the optimal order quantity Q*
[ ]( ) 1 Pr( ) Pr( )r u oG Q C D Q C D Q′ = − − < + <
Pr( ) u
u o
CD QC C
< =+
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Introducing the Concept of Real Options into the Newsboy Problem
The Newsboy problem is now extended by the concept of Real Options. It becomes more clear if you do not think of newspapersnow, but of this year’s favorite Christmas present, this summer’s most popular fashion item, etc.
Ahead of the selling season, the retailer buys q call options at unit cost c. Each of these options gives the retailer the right to buy one unit of the finished product at exercise price x after observing customer demand.The idea is for the manufacturer to induce the retailer to buy more finished products – thus being able to capture more sales – by sharing some of the risk of demand uncertainty with the retailer.
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Introducing the Concept of Real Options into the Newsboy Problem• Expected profit
• Reformulate πr(q) according to the regular Newsboy problem
• Express expected opportunity cost gr(q) with overage and underage cost
• Optimality Condition
( )
( ) ( ) ( )( ) ( )
r
r
g q
q R x c ED E R x c D q c q Dπ + + = − − − − − − + −
*( ) ( )min( , )r q R x D q cqπ = − −
( ) ( ) ( )r u og q E k D q k q D+ + = − + −
Pr( ) u
u o
kD qk k
< =+
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• What can you say about the manufacturers profit– Without real options?– With real options?
