University professor
Dr.-Ing. Bernd Hellingrath
Chair for Information Systems and Sup-
ply Chain Management
Leonardo-Campus 3
48149 Münster
Tel. +49 251 83-38000
Fax +49 251 83-38009
Tutorial Operations Management
Inventory Management (Part 2)
Suggested Solutions
Hint: Please round off to two digits.
Exercise 1: Newsvendor Model with Discrete Demand Distribution
a)
The Newsvendor Model takes into account variable ordering costs, carrying costs as well as
underage costs. The carrying costs, respectively the costs for overages, result from the costs
per unit minus the return price or residual value if there are any (co = c − v) . The underage
costs respectively the costs for shortages result from opportunity costs that are the losses of
profit caused by the underage costs (cu = r − c). In the Newsvendor Model, the demand is
regarded for only one period. Furthermore, the demand is assumed to be stochastic.
b)
18% 17%
16%
16% 15%
14%
14%
12%
10%
Frequency
10% 9%
8%
8%
6% 5%
4% 3% 3%
2%
0%
10 20 30 40 50 60 70 80 90 100
Demand [pieces]
, 2
c)
The costs of the given order quantities are calculated by multiplying the expected overage
with the costs for overages and multiplying the expected underages with the costs for under-
ages. Exemplary calculation for S=40:
co = c − v = 2.00 − 0.50 =1.50 [€/piece]
cu = r − c = 2.50 − 2.00 = 0.50 [€/ piece]
y f(y) S-y Z(|S-y|) Z(|S-y|)*f(y)
10 3% 30 45 1.35
Overage costs
20 5% 20 30 1.50
30 8% 10 15 1.20
40 10% 0 0 0.00
50 15% -10 5 0.75
60 17% -20 10 1.70
70 16% -30 15 2.40 Underage costs
80 14% -40 20 2.80
90 9% -50 25 2.25
100 3% -60 30 0.90
Z(S=40) 14.85
Therefore the expected costs of the different order quantities are calculated as:
S [pieces] Z(S) [€]
10 24.45
20 20.05
30 16.65
40 14.85
50 15.05
60 18.25
70 24.85
80 34.65
90 47.25
100 61.65
In this case, an order quantity of S=40 [pieces] is the optimal solution.
Dr.-Ing. Bernd Hellingrath
Chair for Information Systems and Sup-
ply Chain Management
Leonardo-Campus 3
48149 Münster
Tel. +49 251 83-38000
Fax +49 251 83-38009
Tutorial Operations Management
Inventory Management (Part 2)
Suggested Solutions
Hint: Please round off to two digits.
Exercise 1: Newsvendor Model with Discrete Demand Distribution
a)
The Newsvendor Model takes into account variable ordering costs, carrying costs as well as
underage costs. The carrying costs, respectively the costs for overages, result from the costs
per unit minus the return price or residual value if there are any (co = c − v) . The underage
costs respectively the costs for shortages result from opportunity costs that are the losses of
profit caused by the underage costs (cu = r − c). In the Newsvendor Model, the demand is
regarded for only one period. Furthermore, the demand is assumed to be stochastic.
b)
18% 17%
16%
16% 15%
14%
14%
12%
10%
Frequency
10% 9%
8%
8%
6% 5%
4% 3% 3%
2%
0%
10 20 30 40 50 60 70 80 90 100
Demand [pieces]
, 2
c)
The costs of the given order quantities are calculated by multiplying the expected overage
with the costs for overages and multiplying the expected underages with the costs for under-
ages. Exemplary calculation for S=40:
co = c − v = 2.00 − 0.50 =1.50 [€/piece]
cu = r − c = 2.50 − 2.00 = 0.50 [€/ piece]
y f(y) S-y Z(|S-y|) Z(|S-y|)*f(y)
10 3% 30 45 1.35
Overage costs
20 5% 20 30 1.50
30 8% 10 15 1.20
40 10% 0 0 0.00
50 15% -10 5 0.75
60 17% -20 10 1.70
70 16% -30 15 2.40 Underage costs
80 14% -40 20 2.80
90 9% -50 25 2.25
100 3% -60 30 0.90
Z(S=40) 14.85
Therefore the expected costs of the different order quantities are calculated as:
S [pieces] Z(S) [€]
10 24.45
20 20.05
30 16.65
40 14.85
50 15.05
60 18.25
70 24.85
80 34.65
90 47.25
100 61.65
In this case, an order quantity of S=40 [pieces] is the optimal solution.
