Project Management
PERT
___________________________________________________________________________
Quantitative Methods of Management
 Jan Fábry
Project Management
PERT
 Probabilistic time – 3 estimates
Optimistic
Pessimistic
Most likely
Expected
completion time
___________________________________________________________________________
Quantitative Methods of Management
 Jan Fábry
Project Management
PERT
 Activity time β-distribution
Most
Likely
Expected
Optimistic
0
aij
Pessimistic
mij μij
bij
Time
___________________________________________________________________________
Quantitative Methods of Management
 Jan Fábry
Project Management
PERT
 Expected completion time
 ij 
a ij  4 m ij  b ij
6
 Standard deviation
 ij 
b ij  a ij
6
___________________________________________________________________________
Quantitative Methods of Management
 Jan Fábry
Project Management
PERT
1) Using CPM ( tij = μij )
Critical
path
2) Expected duration of the project
M 

ij
( i , j ) CP
Variance of the project duration

2

  ij
2
( i , j ) CP
___________________________________________________________________________
Quantitative Methods of Management
 Jan Fábry
Project Management
PERT
Duration of the project - normal distribution
M – expected
duration
T – real
duration
σ – standard
deviation
M-σ
M
M+σ
T
___________________________________________________________________________
Quantitative Methods of Management
 Jan Fábry
Project Management
PERT
3) Probability analysis
probabilty of
completion time
within TD
desired time TD
desired time TD
z 
TD  M

p
___________________________________________________________________________
Quantitative Methods of Management
 Jan Fábry
Project Management
PERT
3) Probability analysis
completion
time
desired probability p
desired probability p
zp
T D  M  z p
___________________________________________________________________________
Quantitative Methods of Management
 Jan Fábry
Project Management
PERT
Example - Music DM, Inc.
Activity
Description
aij
mij
bij
μij
Predecessors
A
Songs Selection
11
15
19
15
None
B
Mastering
7
8
9
8
A
C
PM Elaborating
5
6
7
6
A
D
Customers Analysis
5
7
9
7
A
E
PM Production
2
3
10
4
C, D
F
PM to PH
3
4
11
5
E
G
Customers Selection
2
3
4
3
D
___________________________________________________________________________
Quantitative Methods of Management
 Jan Fábry
Project Management
PERT
Example - Music DM, Inc.
Activity
Description
aij
mij
bij
μij
Predecessors
H
Make CD Copies
8
11
20
12
B, D
I
Data to PH
2
3
4
3
G
J
Laser Print
6
8
16
9
F, I
K
Mailing
6
8
10
8
H, J
___________________________________________________________________________
Quantitative Methods of Management
 Jan Fábry
Project Management
PERT
Example - Music DM, Inc.
H
Critical path:
9
4
B
1
A
D
2
C
3
D2
G
7
I
D1
5
K
10
J
8
F
E
6
___________________________________________________________________________
Quantitative Methods of Management
 Jan Fábry
Project Management
PERT
Example - Music DM, Inc.
Critical activities
Activity
Description
aij
mij
bij
μij
σij
A
Songs Selection
11
15
19
15
8/6
D
Customers Analysis
5
7
9
7
4/6
E
PM Production
2
3
10
4
8/6
F
PM to PH
3
4
11
5
8/6
J
Laser Print
6
8
16
9
10/6
K
Mailing
6
8
10
8
4/6
___________________________________________________________________________
Quantitative Methods of Management
 Jan Fábry
Project Management
PERT
Example - Music DM, Inc.
Expected duration of the project
Variance of the project duration
Std. deviation of the project duration
M = 48
2 =9
=3
___________________________________________________________________________
Quantitative Methods of Management
 Jan Fábry
Project Management
PERT
Example - Music DM, Inc.
Duration of the project
45
48
51
Time (days)
___________________________________________________________________________
Quantitative Methods of Management
 Jan Fábry
Project Management
PERT
Example - Music DM, Inc.
Probability analysis
Desired time TD = 45
z
45  48
3
 1
p  0 . 1587
___________________________________________________________________________
Quantitative Methods of Management
 Jan Fábry
Project Management
PERT
Example - Music DM, Inc.
Probability analysis
Desired probability = 0.95
zp = 1.645
T D  48  1 . 645 ( 3 )  52 . 935
___________________________________________________________________________
Quantitative Methods of Management
 Jan Fábry
Inventory Models
___________________________________________________________________________
Quantitative Methods of Management
 Jan Fábry
Inventory Models
Inventory
 Stored for use in future
 Examples of inventories
- raw material
- finished goods
- semi-finished products
- spare parts
___________________________________________________________________________
Quantitative Methods of Management
 Jan Fábry
Inventory Models
Reasons for Maintaining Inventories
 Fluctuating demand
 Delayed supply
 Savings on ordering cost
 Benefits on large quantities
 Inflation
___________________________________________________________________________
Quantitative Methods of Management
 Jan Fábry
Inventory Models
Management
 How much to order?
 When to order?
 Objective – minimize total cost
___________________________________________________________________________
Quantitative Methods of Management
 Jan Fábry
Inventory Models
Partial Inventory Cost
 Ordering & setup cost – fixed charge
frequency of ordering
cost
 Unit purchasing cost – variable cost
quantity discounts
___________________________________________________________________________
Quantitative Methods of Management
 Jan Fábry
Inventory Models
Partial Inventory Cost
 Holding & carrying cost – variable cost
- storage cost (space)
- store keeping operations (movement)
- insurance & taxes
- interest (invested capital )
- oportunity cost
- spoilage or obscolescence
___________________________________________________________________________
Quantitative Methods of Management
 Jan Fábry
Inventory Models
Partial Inventory Cost
 Shortage (stockout) cost – variable cost
- idled production (idled machine)
- placing special expensive expediting order
- loss of customers due to late deliveries
___________________________________________________________________________
Quantitative Methods of Management
 Jan Fábry
Inventory Models
Inventory Terminology
 Inventory level
Available size of the inventory
 Demand & depletion
Rate of demand  Depletion rate
___________________________________________________________________________
Quantitative Methods of Management
 Jan Fábry
Inventory Models
Inventory Terminology
 Reordering
Reorder point = signal inventory level
Lead time
Time
Placing
order
Receiving shipment
&
Replenishment
___________________________________________________________________________
Quantitative Methods of Management
 Jan Fábry
Inventory Models
Inventory Terminology
Shortage !!
 Shortage
Receiving
shipment
Lead time
Time
Optimal
reorder date
Inventory level = 0
Placing
order
___________________________________________________________________________
Quantitative Methods of Management
 Jan Fábry
Inventory Models
Inventory Terminology
 Safety stock
Buffer being built to prevent shortage
In case of probabilistic demand
___________________________________________________________________________
Quantitative Methods of Management
 Jan Fábry
Inventory Models
Inventory Models
 Deterministic models
Rate of demand is known with certainty
 Probabilistic models
Probabilistic demand (random variable)
___________________________________________________________________________
Quantitative Methods of Management
 Jan Fábry
Inventory Models
Demand Classification
Demand
Deterministic
Static
Dynamic
Probabilistic
Stationary
Nonstationary
Increasing Level of Mathematical Difficulty
Probability
RateRate
of demand
density
ofdensity
demand
function
- function
known
- known
–with
unchanhged
with
certainty
certainty
over time
Probability
– varies
in time
- not- constant
constant
___________________________________________________________________________
Quantitative Methods of Management
 Jan Fábry
Inventory Models
Deterministic Models
 Basic economic order quantity model
(EOQ model)
 EOQ model with back orders allowed
(with planned shortages)
 Economic production lot size model
 EOQ model with quantity discounts
___________________________________________________________________________
Quantitative Methods of Management
 Jan Fábry
Inventory Models
Probabilistic Models
 Probabilistic model with continuous
demand
 Single-period decision model
___________________________________________________________________________
Quantitative Methods of Management
 Jan Fábry