the magnitude of shortages (out of stock)
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1
Managing Flow Variability: Safety Inventory
The Newsvendor Problem
Ardavan Asef-Vaziri, Oct 2011
The Magnitude of Shortages (Out of Stock)
2
Managing Flow Variability: Safety Inventory
The Newsvendor Problem
Ardavan Asef-Vaziri, Oct 2011
What are the Reasons?
3
Managing Flow Variability: Safety Inventory
The Newsvendor Problem
Ardavan Asef-Vaziri, Oct 2011
Consumer Reaction
4
Managing Flow Variability: Safety Inventory
The Newsvendor Problem
Ardavan Asef-Vaziri, Oct 2011
Optimal Service Level: The Newsvendor Problem
Cost of ordering too much: holding cost, salvage Cost of ordering too
little: lost of sale, low service level
The 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?
5
Managing Flow Variability: Safety Inventory
The Newsvendor Problem
Ardavan Asef-Vaziri, Oct 2011
News Vendor Model; Assimptions Demand is random Distribution of demand is known No initial inventory Set-up cost is zero Single period Zero lead time Linear costs
Purchasing (production) Salvage value Revenue Goodwill
6
Managing Flow Variability: Safety Inventory
The Newsvendor Problem
Ardavan Asef-Vaziri, Oct 2011
Optimal Service Level: The Newsvendor Problem
Demand Probability of Demand100 0.02110 0.05120 0.08130 0.09140 0.11150 0.16160 0.20170 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 or equal to 140?What is probability of demand to be equal to 133?
0.090.02+0.05+0.08+0.09+0.11=
0.351-0.35+0.11= 0.76
0
P(R ≥ Q ) = 1-P(R ≤ Q)+P(R = Q) R is quantity of demandQ is the quantity ordered
7
Managing Flow Variability: Safety Inventory
The Newsvendor Problem
Ardavan Asef-Vaziri, Oct 2011
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 or equal to 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
0.0050.02+0.035 = 0.055
1-0.055+0.005 = 0.950
8
Managing Flow Variability: Safety Inventory
The Newsvendor Problem
Ardavan Asef-Vaziri, Oct 2011
Optimal Service Level: The Newsvendor Problem
What is probability of demand to be equal to 130?
Average Demand Probability of Demand100 0.02110 0.05120 0.08130 0.09140 0.11150 0.16160 0.20170 0.15180 0.08190 0.05200 0.01
0
What is probability of demand to be less than or equal to 145?
What is probability of demand to be greater than or equal to 145?
0.02+0.05+0.08+0.09+0.11 = 0.35
1-0.35 = 0.65P(R ≥ Q) = 1-P(R ≤ Q)
9
Managing Flow Variability: Safety Inventory
The Newsvendor Problem
Ardavan Asef-Vaziri, Oct 2011
Compute the Average Demand
Q i P( R =Q i )100 0.02110 0.05120 0.08130 0.09140 0.11150 0.16160 0.20170 0.15180 0.08190 0.05200 0.01
N
1i Demand Average )( ii QRPQ
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?
10
Managing Flow Variability: Safety Inventory
The Newsvendor Problem
Ardavan Asef-Vaziri, Oct 2011
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 R ≥ 140Prob(R ≥ 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 (Qi) 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 (R ≥ Qi) 1.00 0.98 0.93 0.85 0.76 0.65 0.49 0.29 0.14 0.06 0.01
11
Managing Flow Variability: Safety Inventory
The Newsvendor Problem
Ardavan Asef-Vaziri, Oct 2011
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 (Qi) 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 (R ≥ Qi) 1.00 0.98 0.93 0.85 0.76 0.65 0.49 0.29 0.14 0.06 0.01
12
Managing Flow Variability: Safety Inventory
The Newsvendor Problem
Ardavan Asef-Vaziri, Oct 2011
Cumulative Probabilities
Qi P(R=Qi) P(R<Qi) P(R≥Qi)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
13
Managing Flow Variability: Safety Inventory
The Newsvendor Problem
Ardavan Asef-Vaziri, Oct 2011
Number of Units Sold, Salvages
Qi P(R=Qi) P(R<Qi) P(R≥Qi) 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 0.20 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 Sold@700 Salvaged@-100
14
Managing Flow Variability: Safety Inventory
The Newsvendor Problem
Ardavan Asef-Vaziri, Oct 2011
Total Revenue for Different Ordering Policies
Qi P(R=Qi) P(R<Qi) P(R≥Qi) 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
15
Managing Flow Variability: Safety Inventory
The Newsvendor Problem
Ardavan Asef-Vaziri, Oct 2011
Denim Wholesaler; Marginal Analysis
The demand for denim is:
– 1000 with probability 0.10
– 2000 with probability 0.15
– 3000 with probability 0.15
– 4000 with probability 0.20
– 5000 with probability 0.15
– 6000 with probability 0.15
– 7000 with probability 0.10
Unit Revenue (p ) = 30Unit purchase cost (c )= 10Salvage value (v )= 5Goodwill cost (g )= 0
How much should we order?
16
Managing Flow Variability: Safety Inventory
The Newsvendor Problem
Ardavan Asef-Vaziri, Oct 2011
Marginal Analysis
Marginal analysis: What is the value of an additional unit ordered?
Suppose the wholesaler purchases 1000 units
What is the value of having the 1001st unit? Marginal Cost: The retailer must salvage the
additional unit and losses $5 (10 – 5). P(R ≤ 1000) = 0.1Expected Marginal Cost = 0.1(5) = 0.5
17
Managing Flow Variability: Safety Inventory
The Newsvendor Problem
Ardavan Asef-Vaziri, Oct 2011
Marginal Analysis
Marginal Profit: The retailer makes and extra profit of $20 (30 – 10)P(R > 1000) = 0.9 Expected Marginal Profit= 0.9(20) = 18MP ≥ MCExpected Value = 18-0.5 = 17.5By purchasing an additional unit, the expected profit increases by
$17.5
The retailer should purchase at least 1,001 units.
18
Managing Flow Variability: Safety Inventory
The Newsvendor Problem
Ardavan Asef-Vaziri, Oct 2011
Marginal Analysis
Should he purchase 1,002 units?Marginal Cost: $5 salvage P(R ≤ 1001) = 0.1 Expected Marginal Cost = 0.5Marginal Profit: $20 profit P(R >1002) = 0.9 18 Expected Marginal Profit = 18Expected Value = 18-0.5 = 17.5
Conclusion:
Wholesaler should purchase at least 2000 units.
Assuming that the initial purchasing quantity is between 1000 and 2000, then by purchasing an additional unit exactly the same savings will be achieved.
19
Managing Flow Variability: Safety Inventory
The Newsvendor Problem
Ardavan Asef-Vaziri, Oct 2011
Marginal Analysis
Marginal analysis: What is the value of an additional unit ordered?
Suppose the retailer purchases 2000 units
What is the value of having the 2001st unit? Marginal Cost: The retailer must salvage the
additional unit and losses $5 (10 – 5). P(R ≤ 2000) = 0.25Expected Marginal Cost = 0.25(5) = 1.25
20
Managing Flow Variability: Safety Inventory
The Newsvendor Problem
Ardavan Asef-Vaziri, Oct 2011
Marginal Analysis
Marginal Profit: The retailer makes and extra profit of $20 (30 – 10)P(R > 2000) = 0.75 Expected Marginal Profit= 0.75(20) = 15MP ≥ MCExpected Value = 15-1.25 = 13.75By purchasing an additional unit, the expected profit increases by
$13.75
The retailer should purchase at least 2,001 units.
21
Managing Flow Variability: Safety Inventory
The Newsvendor Problem
Ardavan Asef-Vaziri, Oct 2011
Marginal Analysis
Should he purchase 2,002 units?Marginal Cost: $5 salvage P(R ≤ 2001) = 0.25 Expected Marginal Cost = 1.25Marginal Profit: $20 profit P(R >2002) = 0.75 Expected Marginal Profit = 15Expected Value = 18-0.5 = 13.75
Conclusion:
Wholesaler should purchase at least 3000 units.
Assuming that the initial purchasing quantity is between 2000 and 3000, then by purchasing an additional unit exactly the same savings will be achieved.
22
Managing Flow Variability: Safety Inventory
The Newsvendor Problem
Ardavan Asef-Vaziri, Oct 2011
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
Demand ProbabilityCumulative Probability
Expected Marginal Cost
Expected Marginal Profit
Expected Marginal Value
1000 0.10 0.1 0.50 18 17.502000 0.15 0.25 1.25 15 13.753000 0.15 0.40 2.00 12 10.004000 0.20 0.60 3.00 8 5.005000 0.15 0.75 3.75 5 1.256000 0.15 0.90 4.50 2 -2.507000 0.10 1.00 5.00 0 -5.00
23
Managing Flow Variability: Safety Inventory
The Newsvendor Problem
Ardavan Asef-Vaziri, Oct 2011
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
24
Managing Flow Variability: Safety Inventory
The Newsvendor Problem
Ardavan Asef-Vaziri, Oct 2011
Marginal Profit:
Marginal Cost:MP = p – c
MC = c - v
MP = 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 the probability demand to be less than or equal to Q.
Since we have P( R ≤ Q).
The expected marginal cost =MC× P( R ≤ Q)
25
Managing Flow Variability: Safety Inventory
The Newsvendor Problem
Ardavan Asef-Vaziri, Oct 2011
Analytical Solution for the Optimal Service LevelIf I have ordered one unit more than Q units, the probability of selling that extra unit is the probability of demand to be greater than Q.
We know that P(R > Q) = 1- P(R ≤ Q).
The expected marginal benefit = MB× [1-Prob.( r ≤ Q)]
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.
26
Managing Flow Variability: Safety Inventory
The Newsvendor Problem
Ardavan Asef-Vaziri, Oct 2011
MC×Prob(R ≤ Q*) ≥ MP× [1 – Prob( R ≤ Q*)]
Analytical Solution for the Optimal Service Level
MP = p – c = Underage Cost = Cu
MC = c – v = Overage Cost = Co
ou
u
CCc
MCMPMPQRP
)( *
vpcp
vccpcp
MCMPMP
MBMB MCProb(R ≤ Q*) ≥
27
Managing Flow Variability: Safety Inventory
The Newsvendor Problem
Ardavan Asef-Vaziri, Oct 2011
Marginal Value: The General Formula
P(R ≤ Q*) ≥ Cu / (Co+Cu)
Cu / (Co+Cu) = (30-10)/[(10-5)+(30-10)] = 20/25 = 0.8
Order until P(R ≤ Q*) ≥ 0.8
P(R ≤ 5000) ≥ = 0.75 not > 0.8 still order
P(R ≤ 6000) ≥ = 0.9 > 0.8 Stop
In Continuous Model where demand for example has Uniform or Normal distribution
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
MCMPMPQRP
)( *
ou
u
CCc
vpcp
28
Managing Flow Variability: Safety Inventory
The Newsvendor Problem
Ardavan Asef-Vaziri, Oct 2011
Type-1 Service Level
What is the meaning of the number 0.80?
80% of the time all the demand is satisfied.
– Probability {demand is smaller than Q} =
– Probability {No shortage} =
– Probability {All the demand is satisfied from stock} = 0.80
29
Managing Flow Variability: Safety Inventory
The Newsvendor Problem
Ardavan Asef-Vaziri, Oct 2011
Marginal Value: Uniform distribution
Suppose instead of a discreet demand of
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{r ≤ Q*} = 0.80
We have a continuous demand uniformly distributed between 1000 and 7000
1000 7000
How do you find Q?
30
Managing Flow Variability: Safety Inventory
The Newsvendor Problem
Ardavan Asef-Vaziri, Oct 2011
Marginal Value: Uniform distribution
l=1000 u=7000
?
u-l=6000
1/60000.80
Q-l = Q-1000
(Q-1000)/6000=0.80Q = 5800
31
Managing Flow Variability: Safety Inventory
The Newsvendor Problem
Ardavan Asef-Vaziri, Oct 2011
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.
P(z ≤ Z) = 0.8 Z= 0.842
Q = mean + z Standard Deviation 4000+841 = 4841
32
Managing Flow Variability: Safety Inventory
The Newsvendor Problem
Ardavan Asef-Vaziri, Oct 2011
Marginal Value: Normal Distribution
00.00005
0.00010.00015
0.00020.00025
0.00030.00035
0.00040.00045
0 2000 4000 6000 8000
4841
Probability of
excess inventoryProbability of
shortage
0.80
0.20
Given a service level, how do we calculate z?From our normal table orFrom Excel Normsinv(service level)
33
Managing Flow Variability: Safety Inventory
The Newsvendor Problem
Ardavan Asef-Vaziri, Oct 2011
Additional Example
Your store is selling calendars, which cost you $6.00 and sell for $12.00 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?
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
CCCQRP
34
Managing Flow Variability: Safety Inventory
The Newsvendor Problem
Ardavan Asef-Vaziri, Oct 2011
Additional Example - Solution
By convention, for the continuous demand distributions, the results are rounded to the closest integer.
2533.06.0)(**
QQZP
Look for P(x ≤ Z) = 0.6 in Standard Normal table or
for NORMSINV(0.6) in excel 0.2533
63533.62)2533.0(10602533.0* Q
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.
35
Managing Flow Variability: Safety Inventory
The Newsvendor Problem
Ardavan Asef-Vaziri, Oct 2011
Additional Example - Solution
Qnew = 1.2 * 63 = 75.6 ~ 76 units
)6.1()10
6076()76()( *
ZPZPRPQRP
Look for P(Z ≤ 1.6) = 0.6 in Standard Normal table
or for NORMSDIST(1.6) in excel 0.9452
1012
212129452.0)( * cc
vccpcp
CCCQRP
ou
u
55.2452.912 cc
36
Managing Flow Variability: Safety Inventory
The Newsvendor Problem
Ardavan Asef-Vaziri, Oct 2011
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:
Qi 4 5 6 7 8 9 10 11 12 13P(R=Qi) 0.04 0.06 0.16 0.18 0.2 0.1 0.1 0.08 0.04 0.04
37
Managing Flow Variability: Safety Inventory
The Newsvendor Problem
Ardavan Asef-Vaziri, Oct 2011
Additional Example
a) How many units are sold if we have ordered 7 units
There is 0.18 + 0.20 + 0.10 + 0.10 + 0.08 + 0.04 + 0.04 = 0.74
There is 0.74 probability that demand is greater than or equal to 7.
There is 0.16 probability that demand is equal to 6.
There is 0.06 probability that demand is equal to 5.
There is 0.04 probability that demand is equal to 4.
The expected number of units sold is
0.74(7) + 0.16 (6) + 0.06 (5) + 0.04 (4) = 6.6
Qi 4 5 6 7 8 9 10 11 12 13P(R=Qi) 0.04 0.06 0.16 0.18 0.2 0.1 0.1 0.08 0.04 0.04
38
Managing Flow Variability: Safety Inventory
The Newsvendor Problem
Ardavan Asef-Vaziri, Oct 2011
Additional Example
b) How many units are salvaged?
7-6.6 = 0.4. Alternatively, we can compute it directly
There is 0.74 probability that we salvage 7 – 7 = 0 units
There is 0.16 probability that we salvage 7- 6 = 1 units
There is 0.06 probability that we salvage 7- 5 = 2 units
There is 0.04 probability that we salvage 7-4 = 3 units
The expected number of units salvaged is
0.74(0) + 0.16 (1) + 0.06 (2) + 0.04 (3) = 0.4 and 7-0.4 = 6.6 sold
Qi 4 5 6 7 8 9 10 11 12 13P(R=Qi) 0.04 0.06 0.16 0.18 0.2 0.1 0.1 0.08 0.04 0.04
39
Managing Flow Variability: Safety Inventory
The Newsvendor Problem
Ardavan Asef-Vaziri, Oct 2011
Additional Example
c) Compute the total profit if we order 7 units.
Out of 7 units, 6.6 sold, 0.4 salvaged.
P = 75, c= 25, v=10.
Profit per unit sold = 75-25 = 50
Cost per unit salvaged = 25-10 = 15
Total Profit = 6.6(50) + 0.4(15) = 333 - 9 = 324
40
Managing Flow Variability: Safety Inventory
The Newsvendor Problem
Ardavan Asef-Vaziri, Oct 2011
Additional Example
d) Compute the expected Marginal profit of ordering the 8th unit.
MP = 75-25 = 50
P(R ≥ 8) = 0.2 + 0.1 + 0.1 + 0.08 + 0.04 + 0.04 = 0.56
Expected Marginal profit = 0.56(50) = 28
d) Compute the expected Marginal cost of ordering the 8th unit.
MC = 25 – 10 = 15
P(R ≤ 7) = 1-0.56 = 0.44
Expected Marginal cost = 0.44(15) = 6.6
Qi 4 5 6 7 8 9 10 11 12 13P(R=Qi) 0.04 0.06 0.16 0.18 0.2 0.1 0.1 0.08 0.04 0.04
41
Managing Flow Variability: Safety Inventory
The Newsvendor Problem
Ardavan Asef-Vaziri, Oct 2011
Additional Example - Solution
Overage Cost = Co = Unit Cost – Salvage = 0.25 – 0.1 = 0.15Underage Cost = Cu = Selling Price – Unit Cost = 0.75 – 0.25 = 0.50
77.0)(
77.015.050.0
50.0*)(
*
QRP
CCCQRP
ou
u ProbabilityCumulative Probability
Qi P(R=Qi) F(Qi)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 1.00
Q* = 10
e) What is the optimum order quantity for Mac to minimize his cost?
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