We have lookup tables for Standard Normal distribution ( =0, =1) Convert to Standard Normal!
F (Q) P ( D Q)
4 0.444 9
D Q P ( D Q ) P ( D Q ) P P( Z z ) ( z ) Safety Standard normal
Z
zstock
Expected demand
Find z from the lookup table Q = z +
Standard Normal Distribution - 1 Density function ( z )
e
z2 2
2 Symmetric around the mean Area under the entire curve 1 P(Z )
1
Standard Normal Distribution - 2 Density function ( z )
e
z2 2
2 Symmetric around the mean Area under the entire
We will use ɸ(z) and F(z) interchangeably
curve 1 P(Z )
F(z)=P(Z≤z)
z
Standard Normal Distribution - 2 Density function ( z )
e
z2 2
2 Symmetric around the mean Area under the entire
We will use ɸ(z) and F(z) interchangeably
curve 1 P(Z )
F(z)=P(Z≤z)
F(-z)=P(Z≤-z)=1-F(z)
-z
z
2
Example - INFORMS P( Z z ) ( z ) Area under the curve 0.444 z -0.14 Q* z 2000(0.14) 5000 Q* 4720
z = - 0.14
Example - INFORMS Why is Q* co=50 z=0.21
Q* Q*
Standard deviation
Standard deviation
Example – Fashion Bags Input
Unit cost c = $28.50 Selling price p = $150 Salvage value s = $20 Cost of inventory = $0.40 for each dollar tied up in inventory at the end of the season
7
Example – Fashion Bags Input
Unit cost c = $28.50 Selling price p = $150 Salvage value s = $20 Cost of inventory = $0.40 for each dollar tied up in inventory at the end of the season
Computed input:
Holding cost per bag cu = co =
F (Q)
h=
cu cu co
Example – Fashion Bags Input
Unit cost c = $28.50 Selling price p = $150 Salvage value s = $20 Cost of inventory = $0.40 for each dollar tied up in inventory at the end of the season
Computed input:
Holding cost per bag cu = p – c = $121.5 co = c – s + h = $19.9
F (Q)
h = (0.40)(28.50) = $11.4
cu 121.5 0.86 cu co 121.5 19.9
8
Example – Fashion Bags – Normally distributed demand Demand ~ Normal( 150, 20) F (Q) 0.86 z 1.08
Area under the curve= Critical ratio = 0.86
Q* z (20)(1.08) 150 Q * 172
z = 1.08
Example – Fashion Bags – Uniformly distributed demand
Demand Uniform between 50 and 250 =150
Same expected demand as in Normal distribution
0.86 (Q * 50) Q* 222
1 200
Uniform density
Area under the curve = Critical ratio = 0.86
1/200
Q 250
50 Q*= 222
9
Example – Fashion Bags
Even though both the Normal and the Uniform distributions have the same mean (=150), why did we get different quantities?
Normal distribution Q*=172 Uniform distribution Q*=222
Example – Fashion Bags
Even though both the Normal and the Uniform distributions have the same mean (=150), why did we get different quantities?
Normal distribution Q*=172 Uniform distribution Q*=222
Because of the variance (equivalently, standard deviation) and the shape of the distribution !!!
Normal =20 Uniform =57.7
Normal
Uniform
10
Class exercise Georgia Tech bookstore must decide how many 2005 calendars to order. Each calendar costs the bookstore $2 and is sold for $4.50. After January 1, any unsold calendars are returned to the publisher for a refund of 75 cents per calendar. The number of calendars sold by January 1 follows the probability distribution below. How many calendars should be ordered to maximize expected profit? # of calendars sold Probability
100
150
200
250
300
0.30
0.20
0.30
0.15
0.05
Class exercise Q: number of calendars ordered D: demand D≤Q Cost = 2Q – 4.5D - 0.75(Q-D) = 1.25Q - 3.75D D≥Q Cost = 2Q – 4.5Q = -2.5Q cu = 4.5 – 2 = $2.5 co = 2 – 0.75 = $1.25 Critical ratio = (2.5)/(2.5+1.25) = 2/3 Q=? # of calendars sold
100
150
200
250
300
Probability
0.30
0.20
0.30
0.15
0.05
F(Q)=P(D≤Q)
0.30
0.50
0.80
0.95
1.00
11
Summary - Newsvendor
Single period Depending on the relationship between the cost of shortage or excess inventory, we may order more or less than expected demand Optimal order quantity
Higher variability may cause an increase or a decrease in the optimal order quantity
As increases, Q* will deviate more from the mean
Inventory Control - Demand Variability Constant/Stationary Variable/Non-Stationary Uncertainty Stochastic Deterministic
increases as shortage cost increases decreases as holding cost increases
Economic Order Quantity (EOQ) – Tradeoff between fixed cost and holding cost
Lot size/Reorder level (Q,R) or (s,S) models – Tradeoff between fixed cost, holding cost, and shortage cost
Aggregate Planning – Planning for capacity levels given a forecast
