1 managing flow variability: safety inventory operations management session 23: newsvendor model
TRANSCRIPT
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Managing Flow Variability: Safety Inventory
Uncertain Demand
Uncertain Demand: What are the relevant trade-offs?
– Overstock
• Demand is lower than the available inventory
• Inventory holding cost
– Understock
• Shortage- Demand is higher than the available inventory
– Why do we have shortages?
– What is the effect of shortages?
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Managing Flow Variability: Safety Inventory
The Magnitude of Shortages (Out of Stock)
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Managing Flow Variability: Safety Inventory
What are the Reasons?
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Managing Flow Variability: Safety Inventory
What can be done to minimize shortages?
Better forecast
Produce to order and not to stock
– Is it always feasible?
Have large inventory levels
Order the right quantity
– What do we mean by the right quantity?
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Managing Flow Variability: Safety Inventory
Uncertain Demand
What is the objective?
– Minimize the expected cost (Maximize the expected profits).
What are the decision variables?
– The optimal purchasing quantity, or the optimal inventory
level.
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Managing Flow Variability: Safety Inventory
Optimal Service Level: The Newsvendor Problem
Cost of Holding Extra Inventory
Improved Service
Optimal Service Level under uncertainty
The Newsvendor ProblemThe decision maker balances the expected costs of ordering too much with the expected costs of ordering too little to determine the optimal order quantity.
How do we choose what level of service a firm should offer?
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Managing Flow Variability: Safety Inventory
News Vendor Model
Assumptions– Demand is random– Distribution of demand is known– No initial inventory– Set-up cost is equal to zero– Single period– Zero lead time– Linear costs:
• Purchasing (production)• Salvage value• Revenue• Goodwill
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Managing Flow Variability: Safety Inventory
Optimal Service Level: The Newsvendor Problem
Demand Probability of Demand100 0.02110 0.05120 0.08130 0.09140 0.11150 0.16160 0.2170 0.15180 0.08190 0.05200 0.01
Cost =1800, Sales Price = 2500, Salvage Price = 1700Underage Cost = 2500-1800 = 700, Overage Cost = 1800-1700 = 100
What is probability of demand to be equal to 130?What is probability of demand to be less than or equal to 140?What is probability of demand to be greater than 140?What is probability of demand to be equal to 133?
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Managing Flow Variability: Safety Inventory
Optimal Service Level: The Newsvendor Problem
Demand Probability of Demand100 0.002101 0.002102 0.002103 0.002104 0.002105 0.002106 0.002107 0.002108 0.002109 0.002
What is probability of demand to be equal to 116?What is probability of demand to be less than or equal to 116?What is probability of demand to be greater than 116?What is probability of demand to be equal to 113.3?
Demand Probability of Demand110 0.005111 0.005112 0.005113 0.005114 0.005115 0.005116 0.005117 0.005118 0.005119 0.005
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Managing Flow Variability: Safety Inventory
Optimal Service Level: The Newsvendor Problem
What is probability of demand to be equal to 130?What is probability of demand to be less than or equal to 145?What is probability of demand to be greater than 145?
Average Demand Probability of Demand100 0.02110 0.05120 0.08130 0.09140 0.11150 0.16160 0.2170 0.15180 0.08190 0.05200 0.01
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Managing Flow Variability: Safety Inventory
Compute the Average Demand
X P(x=X)100 0.02110 0.05120 0.08130 0.09140 0.11150 0.16160 0.2170 0.15180 0.08190 0.05200 0.01
N
1i
)( Demand Average ii XxPX
Average Demand = +100×0.02 +110×0.05+120×0.08 +130×0.09+140×0.11 +150×0.16+160×0.20 +170×0.15 +180×0.08 +190×0.05+200×0.01Average Demand = 151.6
How many units should I have to sell 151.6 units (on average)? How many units do I sell (on average) if I have 100 units?
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Managing Flow Variability: Safety Inventory
Suppose I have ordered 140 Unities.On average, how many of them are sold? In other words, what
is the expected value of the number of sold units?
When I can sell all 140 units? I can sell all 140 units if x ≥ 140Prob(x ≥ 140) = 0.76The expected number of units sold –for this part- is(0.76)(140) = 106.4Also, there is 0.02 probability that I sell 100 units 2 unitsAlso, there is 0.05 probability that I sell 110 units5.5Also, there is 0.08 probability that I sell 120 units 9.6Also, there is 0.09 probability that I sell 130 units 11.7106.4 + 2 + 5.5 + 9.6 + 11.7 = 135.2
Deamand (X) 100 110 120 130 140 150 160 170 180 190 200Porbability 0.02 0.05 0.08 0.09 0.11 0.16 0.20 0.15 0.08 0.05 0.01Prob(x ≥ X) 1.00 0.98 0.93 0.85 0.76 0.65 0.49 0.29 0.14 0.06 0.01
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Suppose I have ordered 140 Unities.On average, how many of them are salvaged? In other words,
what is the expected value of the number of salvaged units?
0.02 probability that I sell 100 units. In that case 40 units are salvaged 0.02(40) = .80.05 probability to sell 110 30 salvaged 0.05(30)= 1.5 0.08 probability to sell 120 20 salvaged 0.08(20) = 1.60.09 probability to sell 130 10 salvaged 0.09(10) =0.9 0.8 + 1.5 + 1.6 + 0.9 = 4.8
Total number Sold 135.2 @ 700 = 94640Total number Salvaged 4.8 @ -100 = -480Expected Profit = 94640 – 480 = 94,160
Deamand (X) 100 110 120 130 140 150 160 170 180 190 200Porbability 0.02 0.05 0.08 0.09 0.11 0.16 0.20 0.15 0.08 0.05 0.01Prob(x ≥ X) 1.00 0.98 0.93 0.85 0.76 0.65 0.49 0.29 0.14 0.06 0.01
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Cumulative Probabilities
X P(x=X) P(x<X) P(x≥X)100 0.02 0 1110 0.05 0.02 0.98120 0.08 0.07 0.93130 0.09 0.15 0.85140 0.11 0.24 0.76150 0.16 0.35 0.65160 0.2 0.51 0.49170 0.15 0.71 0.29180 0.08 0.86 0.14190 0.05 0.94 0.06200 0.01 0.99 0.01
Probabilities
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Number of Units Sold, Salvages
X P(x=X) P(x<X) P(x≥X) Sold Salvage100 0.02 0 1 100 0110 0.05 0.02 0.98 109.8 0.2120 0.08 0.07 0.93 119.1 0.9130 0.09 0.15 0.85 127.6 2.4140 0.11 0.24 0.76 135.2 4.8150 0.16 0.35 0.65 141.7 8.3160 0.2 0.51 0.49 146.6 13.4170 0.15 0.71 0.29 149.5 20.5180 0.08 0.86 0.14 150.9 29.1190 0.05 0.94 0.06 151.5 38.5200 0.01 0.99 0.01 151.6 48.4
Probabilities Units
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Sold@700 Salvaged@-100
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Total Revenue for Different Ordering Policies
X P(x=X) P(x<X) P(x≥X) Sold Salvaged Sold Salvaged Total100 0.02 0 1 100 0 70000 0 70000110 0.05 0.02 0.98 109.8 0.2 76860 20 76840120 0.08 0.07 0.93 119.1 0.9 83370 90 83280130 0.09 0.15 0.85 127.6 2.4 89320 240 89080140 0.11 0.24 0.76 135.2 4.8 94640 480 94160150 0.16 0.35 0.65 141.7 8.3 99190 830 98360160 0.2 0.51 0.49 146.6 13.4 102620 1340 101280170 0.15 0.71 0.29 149.5 20.5 104650 2050 102600180 0.08 0.86 0.14 150.9 29.1 105630 2910 102720190 0.05 0.94 0.06 151.5 38.5 106050 3850 102200200 0.01 0.99 0.01 151.6 48.4 106120 4840 101280
Probabilities Units Revenue
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Managing Flow Variability: Safety Inventory
Example 2: Denim Wholesaler
The demand for denim is:
– 1000 with probability 0.1
– 2000 with probability 0.15
– 3000 with probability 0.15
– 4000 with probability 0.2
– 5000 with probability 0.15
– 6000 with probability 0.15
– 7000 with probability 0.1
Unit Revenue (r ) = 30Unit purchase cost (c )= 10Salvage value (v )= 5Goodwill cost (g )= 0
Cost parameters:
How much should we order?Session 23 Operations Management 19
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Example 2: Marginal Analysis
Marginal analysis: What is the value of an additional unit?
Suppose the wholesaler purchases 1000 units
What is the value of the 1001st unit?
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Example 2: Marginal Analysis
Wholesaler purchases an additional unit
Case 1: Demand is smaller than 1001 (Probability 0.1)
– The retailer must salvage the additional unit and losses $5 (10 – 5)
Case 2: Demand is larger than 1001 (Probability 0.9)
– The retailer makes and extra profit of $20 (30 – 10)
Expected value = -(0.1*5) + (0.9*20) = 17.5
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Example 2: Marginal Analysis
What does it mean that the marginal value is positive?
– By purchasing an additional unit, the expected profit
increases by $17.5
The dealer should purchase at least 1,001 units.
Should he purchase 1,002 units?
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Example 2: Marginal Analysis
Wholesaler purchases an additional unit
Case 1: Demand is smaller than 1002 (Probability 0.1)– The retailer must salvage the additional unit and losses $5 (10 – 5)
Case 2: Demand is larger than 1002 (Probability 0.9)– The retailer makes and extra profit of $20 (30 – 10)
Expected value = -(0.1*5) + (0.9*20) = 17.5
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Example 2: Marginal Analysis
Assuming that the initial purchasing quantity is between 1000 and
2000, then by purchasing an additional unit exactly the same
savings will be achieved.
Conclusion: Wholesaler should purchase at least 2000 units.
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Managing Flow Variability: Safety Inventory
Example 2: Marginal Analysis
Wholesaler purchases an additional unit
Case 1: Demand is smaller than 2001 (Probability 0.25)– The retailer must salvage the additional unit and losses $5 (10 – 5)
Case 2: Demand is larger than 2001 (Probability 0.75)– The retailer makes and extra profit of $20 (30 – 10)
Expected value = -(0.25*5) + (0.75*20) = 13.75
What is the value of the 2001st unit?
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Example 2: Marginal Analysis
Why does the marginal value of an additional unit decrease, as
the purchasing quantity increases?
– Expected cost of an additional unit increases
– Expected savings of an additional unit decreases
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Example 2: Marginal Analysis
We could continue calculating the marginal values
Demand ProbabilityCumlative Probability
Expected marginal cost
Expected marginal savings Marginal Value
1000 0.1 0.1 0.5 18.0 17.502000 0.15 0.25 1.3 15.0 13.753000 0.15 0.4 2.0 12.0 10.004000 0.2 0.6 3.0 8.0 5.005000 0.15 0.75 3.8 5.0 1.256000 0.15 0.9 4.5 2.0 -2.507000 0.1 1 5.0 0.0 -5.00
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Managing Flow Variability: Safety Inventory
Example 2: Marginal Analysis
What is the optimal purchasing quantity?
– Answer: Choose the quantity that makes marginal value: zero
Quantity
Marginal value
1000 2000 3000 4000 5000 6000 7000 8000
17.5
13.75
10
5
1.3
-2.5
-5
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Managing Flow Variability: Safety Inventory
Net Marginal Benefit:
Net Marginal Cost:
MB = p – c
MC = c - v
MB = 30 - 10 = 20
MC = 10-5 = 5
Analytical Solution for the Optimal Service Level
Suppose I have ordered Q units.
What is the expected cost of ordering one more units?
What is the expected benefit of ordering one more units?
If I have ordered one unit more than Q units, the probability of not selling that extra unit is if the demand is less than or equal to Q. Since we have P( D ≤ Q).
The expected marginal cost =MC× P( D ≤ Q)
If I have ordered one unit more than Q units, the probability of selling that extra unit is if the demand is greater than Q. We know that P(D>Q) = 1- P( D≤ Q).
The expected marginal benefit = MB× [1-Prob.( D ≤ Q)]
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Managing Flow Variability: Safety Inventory
As long as expected marginal cost is less than expected marginal profit we buy the next unit. We stop as soon as: Expected marginal cost ≥ Expected marginal profit
MC×Prob(D ≤ Q*) ≥ MB× [1 – Prob(D ≤ Q*)]
MB
MB MCProb(D ≤ Q*) ≥
Analytical Solution for the Optimal Service Level
MB = p – c = Underage Cost = Cu
MC = c – v = Overage Cost = Co
ou
u
CC
c
MCMB
MBQDP
)( *
vp
cp
vccp
cp
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Marginal Value: The General Formula
P(D ≤ Q*) ≥ Cu / (Co+Cu)
Cu / (Co+Cu) = (30-10)/[(10-5)+(30-10)] = 20/25 = 0.8
Order until P(D ≤ Q*) ≥ 0.8
P(D ≤ 5000) ≥ = 0.75 not > 0.8 still order
P(D ≤ 6000) ≥ = 0.9 > 0.8 Stop
Order 6000 units
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Demand ProbabilityCumlative Probability Marginal Value
1000 0.1 0.1 0.5 18.0 17.502000 0.15 0.25 1.3 15.0 13.753000 0.15 0.4 2.0 12.0 10.004000 0.2 0.6 3.0 8.0 5.005000 0.15 0.75 3.8 5.0 1.256000 0.15 0.9 4.5 2.0 -2.507000 0.1 1 5.0 0.0 -5.00
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Managing Flow Variability: Safety Inventory
Analytical Solution for the Optimal Service Level
In Continuous Model where demand for example has Uniform or Normal distribution
MCMB
MBQDP
)( *
ou
u
CC
c
vp
cp
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Managing Flow Variability: Safety Inventory
Marginal Value: Uniform distribution
Suppose instead of a discreet demand of
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Demand ProbabilityCumlative Probability Marginal Value
1000 0.1 0.1 0.5 18.0 17.502000 0.15 0.25 1.3 15.0 13.753000 0.15 0.4 2.0 12.0 10.004000 0.2 0.6 3.0 8.0 5.005000 0.15 0.75 3.8 5.0 1.256000 0.15 0.9 4.5 2.0 -2.507000 0.1 1 5.0 0.0 -5.00
Pr{D ≤ Q*} = 0.80
We have a continuous demand uniformly distributed between
1000 and 7000
1000 7000
How do you find Q?
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Managing Flow Variability: Safety Inventory
Marginal Value: Uniform distribution
l=1000 u=7000
?
u-l=6000
1/60000.80
Q-l = Q-1000
(Q-1000)*1/6000=0.80Q = 5800
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Managing Flow Variability: Safety Inventory
Type-1 Service Level
What is the meaning of the number 0.80?
F(Q) = (30 – 10) / (30 – 5) = 0.8
– Pr {demand is smaller than Q} =
– Pr {No shortage} =
– Pr {All the demand is satisfied from stock} = 0.80
It is optimal to ensure that 80% of the time all the demand is
satisfied.
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Managing Flow Variability: Safety Inventory
Marginal Value: Normal Distribution
Suppose the demand is normally distributed with a mean of 4000
and a standard deviation of 1000.
What is the optimal order quantity?
Notice: F(Q) = 0.80 is correct for all distributions.
We only need to find the right value of Q assuming the normal
distribution.
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Managing Flow Variability: Safety Inventory
Marginal Value: Normal Distribution
0
0.00005
0.0001
0.00015
0.0002
0.00025
0.0003
0.00035
0.0004
0.00045
0 1000 2000 3000 4000 5000 6000 7000 8000
Series1
4841
Probability of excess inventory
Probability of shortage
0.80
0.20
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Managing Flow Variability: Safety Inventory
Type-1 Service Level
Recall that:
F(Q) = Cu / (Co + Cu) = Type-1 service level
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Managing Flow Variability: Safety Inventory
Type-1 Service Level
Is it correct to set the service level to 0.8?
Shouldn’t we aim to provide 100% serviceability?
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Managing Flow Variability: Safety Inventory
Type-1 Service Level
What is the optimal purchasing quantity?
0
0.00005
0.0001
0.00015
0.0002
0.00025
0.0003
0.00035
0.0004
0.00045
0 1000 2000 3000 4000 5000 6000 7000 8000
Series15282
Probability of excess inventory
Probability of shortage
0.90
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Managing Flow Variability: Safety Inventory
Type-1 Service Level
How do you determine the service level?
For normal distribution, it is always optimal to have:
Mean + z*Standard deviation
µ + z
The service level determines the value of Z
z is the level of safety stock
+z is the base stock (order-up-to level)
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Managing Flow Variability: Safety Inventory
Type-1 Service Level
Given a service level, how do we calculate z?
From our normal table or
From Excel
– Normsinv(service level)
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Managing Flow Variability: Safety Inventory
Additional Example
Your store is selling calendars, which cost you $6.00 and sell for
$12.00 You cannot predict demand for the calendars with certainty.
Data from previous years suggest that demand is well described by a
normal distribution with mean value 60 and standard deviation 10.
Calendars which remain unsold after January are returned to the
publisher for a $2.00 "salvage" credit. There is only one opportunity
to order the calendars. What is the right number of calendars to
order?
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Additional Example - Solution
MC= Overage Cost = Co = Unit Cost – Salvage = 6 – 2 = 4
MB= Underage Cost = Cu = Selling Price – Unit Cost = 12 – 6 = 6
6.046
6)( *
ou
u
CC
CQDP
By convention, for the continuous demand distributions, the results are rounded to the closest integer.
Session 23 Operations Management 45
2533.06.0)(**
QQZP
Look for P(Z ≤ z) = 0.6 in Standard Normal table or for
NORMSINV(0.6) in excel 0.2533
63533.62)2533.0(10602533.0* Q
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Managing Flow Variability: Safety Inventory
Additional Example - Solution
Suppose the supplier would like to decrease the unit cost in order to
have you increase your order quantity by 20%. What is the minimum
decrease (in $) that the supplier has to offer.
Qnew = 1.2 * 63 = 75.6 ~ 76 units
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)6.1()10
6076()76()( *
ZPzPDPQDP
Look for P(Z ≤ 1.6) = 0.6 in Standard Normal table or for
NORMSDIST(1.6) in excel 0.9452
10
12
212
129452.0)( * cc
vccp
cp
CC
CQDP
ou
u
55.2452.912 cc
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Managing Flow Variability: Safety Inventory
Additional Example
On consecutive Sundays, Mac, the owner of your local newsstand, purchases a
number of copies of “The Computer Journal”. He pays 25 cents for each copy and
sells each for 75 cents. Copies he has not sold during the week can be returned to
his supplier for 10 cents each. The supplier is able to salvage the paper for
printing future issues. Mac has kept careful records of the demand each week for
the journal. The observed demand during the past weeks has the following
distribution:
What is the optimum order quantity for Mac to minimize his cost?
Quantity Q 4 5 6 7 8 9 10 11 12 13Probability p(D=Q) 0.04 0.06 0.16 0.18 0.20 0.10 0.10 0.08 0.04 0.04
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Additional Example - Solution
Overage Cost = Co = Unit Cost – Salvage = 0.25 – 0.1 = 0.15
Underage Cost = Cu = Selling Price – Unit Cost = 0.75 – 0.25 = 0.50
77.0)(
77.015.050.0
50.0*)(
*
QDP
CC
CQDP
ou
uProbability
Cumulative Probability
Q p(D=Q) F(Q)4 0.04 0.045 0.06 0.106 0.16 0.267 0.18 0.448 0.20 0.649 0.10 0.74
10 0.10 0.8411 0.08 0.9212 0.04 0.9613 0.04 1.00The critical ratio, 0.77, is between Q = 9 and Q = 10.
Remember from the marginal analysis explanation that the results are rounded up. Because at 9 still it is at our benefit to order one more.
So Q* = 10