Linear Programming
Assignment Problem
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Operations Research
 Jan Fábry
Linear Programming
Applications
Assignment Problem
employees
projects
jobs
service teams
doctors
Assignment „1 to 1“
jobs
managers
machines
cars
night shifts
Objective: maximize the effect of assignment
___________________________________________________________________________
Operations Research
 Jan Fábry
Linear Programming
Applications
Assignment Problem
Example – Prague Build, Inc.
 Excavating shafts for basements (Michle, Prosek, Radlice, Trója)
 Each excavation takes 5 days
 4 excavators stored in 4 separated garages (everyday‘s movement)
 One excavator to one destination
 Distances between garages and destinations
Objective: minimize total distance necessary for all movements
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Operations Research
 Jan Fábry
___________________________________________________________________________
Operations Research
 Jan Fábry
Linear Programming
Applications
Assignment Problem
Example – Prague Build, Inc.
Distances
Michle
Prosek
Radlice
Trója
Garage 1
5
22
12
18
Garage 2
15
17
6
10
Garage 3
8
25
5
20
Garage 4
10
12
19
12
___________________________________________________________________________
Operations Research
 Jan Fábry
Linear Programming
Applications
Assignment Problem
Example – Prague Build, Inc.
Decision variables
Michle
Prosek
Radlice
Trója
Garage 1
x11
x12
x13
x14
Garage 2
x21
x22
x23
x24
Garage 3
x31
x32
x33
x34
Garage 4
x41
x42
x43
x44
___________________________________________________________________________
Operations Research
 Jan Fábry
Linear Programming
Applications
Assignment Problem
Example – Prague Build, Inc.
Decision variables
xij =
1 if the excavator from the garage i
goes to the destination j
0 otherwise
Binary variable
___________________________________________________________________________
Operations Research
 Jan Fábry
Linear Programming
Applications
Assignment Problem
Example – Prague Build, Inc.
Optimal solution
Michle
Prosek
Radlice
Trója
Garage 1
1
0
0
0
Garage 2
0
0
0
1
Garage 3
0
0
1
0
Garage 4
0
1
0
0
___________________________________________________________________________
Operations Research
 Jan Fábry
Linear Programming
Applications
Assignment Problem
Example – Prague Build, Inc.
Optimal solution
1 movement
Michle
Prosek
Radlice
Trója
Garage 1
5 km
-
-
-
Garage 2
-
-
-
10 km
Garage 3
-
-
5 km
-
Garage 4
-
12 km
-
-
Minimal total distance
320 km
___________________________________________________________________________
Operations Research
 Jan Fábry
___________________________________________________________________________
Operations Research
 Jan Fábry
Network Models
___________________________________________________________________________
Operations Research
 Jan Fábry
Network Models
 Network
 Nodes
 Arcs
UNDIRECTED
DIRECTED
j
i
UNDIRECTED
NETWORK
j
i
DIRECTED
NETWORK
___________________________________________________________________________
Operations Research
 Jan Fábry
Network Models
 Path
Sequence of arcs in which the initial node of
each arc is identical with the terminal node of
the preceding arc.
3
7
5
___________________________________________________________________________
Operations Research
 Jan Fábry
Network Models
 Path
2
1
5
2
11

4
3
5
4
6
3
6
Open Path
___________________________________________________________________________
Operations Research
 Jan Fábry
Network Models
 Circuit (Cycle)
Path starting and ending in
the same node (closed path).
2
1
1
5
2
5
1
4
3
3
4
6
6
___________________________________________________________________________
Operations Research
 Jan Fábry
Network Models
 Connected Network
There is a path connecting every pair
of nodes in the network.
2
1
5
2
1
4
3
3
5
4
6
6
___________________________________________________________________________
Operations Research
 Jan Fábry
Network Models
 Unconnected Network
2
1
5
2
1
4
3
3
5
4
6
6
___________________________________________________________________________
Operations Research
 Jan Fábry
Network Models
 Tree
Connected network without any circuit.
2
3
3
2
4
4
6
3
4
5
4
1
Exactly 6 arcs (n-1)
Removing 1 arc
Adding 1 arc
7
Unconnected network
Circuit in the network
___________________________________________________________________________
Operations Research
 Jan Fábry
Network Models
 Tree
STAR
„CHRISTMAS“ TREE
SNAKE
___________________________________________________________________________
Operations Research
 Jan Fábry
Network Models
 Spanning Tree
Tree including all the nodes from the
original network.
2
1
5
2
1
4
3
3
5
4
6
6
___________________________________________________________________________
Operations Research
 Jan Fábry
Network Models
 Evaluated Network
Values
- distance
- time
- cost
- capacity
yij
Arcs
i
j
Nodes
i
yi
j
yj
___________________________________________________________________________
Operations Research
 Jan Fábry
Network Models
Basic Network Applications
 Shortest Path Problem
 Traveling Salesperson Problem (TSP)
 Minimal Spanning Tree
 Maximum Flow Problem
Project Management
 Critical Path Method (CPM)
 Program Evaluation Review Technique (PERT)
___________________________________________________________________________
Operations Research
 Jan Fábry
Network Models
Shortest Path Problem
___________________________________________________________________________
Operations Research
 Jan Fábry
Network Models
Shortest Path Problem
Shortest path between 2 nodes
2
14
1
1
18
5
2
23
12
30
10
4
15
25
3
5
3
4
15
6
16
6
___________________________________________________________________________
Operations Research
 Jan Fábry
Network Models
Shortest Path Problem
Shortest Paths Between All Pairs of Nodes
1
2
3
4
5
6
1
2
3
4
14
24
26
14
10
12
24
10
15
26
12
15
-
32
18
28
23
40
26
16
15
5
6
32
40
18
26
28
16
23
15
30
30
-
___________________________________________________________________________
Operations Research
 Jan Fábry
Network Models
Traveling Salesperson
Problem
___________________________________________________________________________
Operations Research
 Jan Fábry
Network Models
Traveling Salesperson Problem (TSP)
2
Home
city 1 11
14
18
5
2
23
12
30
10
4
15
25
3
5
3
Shortest tour
4
15
6
16
6
110 km
___________________________________________________________________________
Operations Research
 Jan Fábry
Network Models
Minimal Spanning Tree
___________________________________________________________________________
Operations Research
 Jan Fábry
Network Models
Minimal Spanning Tree
Example - Exhibition
 Exhibition area with 9 locations that need electricity power
 Use cable for extensions
 Price of cable = 10 CZK / 1 m
Objective: minimize the cost of all the extensions
___________________________________________________________________________
Operations Research
 Jan Fábry
Network Models
Minimal Spanning Tree
Example - Exhibition
85
2
2
88
Power
7
7
76
1
90
75
60
63
9
5
55
80
3
3
43
8
74
68
52
9
70
5
40
61
54
8
35
10
10
4
4
120
6
___________________________________________________________________________
Operations Research
 Jan Fábry
71
6
Network Models
Minimal Spanning Tree
Optimum
490 m
Example - Exhibition
4 900 CZK
2
2
7
7
Power
61
76
1
9
9
5
5
40
3
52
43
8
3
60
55
68
8
35
10
4
4
10
6
6
___________________________________________________________________________
Operations Research
 Jan Fábry
Network Models
Maximum Flow Problem
Input
Capacited
network
Source
7
9
Gas
Fluid
Traffic
Information
People
3
Sink
Output
___________________________________________________________________________
Operations Research
 Jan Fábry
Network Models
Maximum Flow Problem
UNDIRECTED
ARC
i
DIRECTED
ARC
j
i
j
Flow
Flow  Capacity
___________________________________________________________________________
Operations Research
 Jan Fábry
Network Models
Maximum Flow Problem
Mathematical Model
 Flow through each arc  Capacity of the arc
 Quantity flowing out = Quantity flowing into
(except the source and the sink)
 Total flow into the source = 0
 Total flow out of the sink = 0
 Total flow out of the source = Total flow into the sink
___________________________________________________________________________
Operations Research
 Jan Fábry
Network Models
Maximum Flow Problem
Example – White Lake City
 The city is situated on the edge of a small lake
 To minimize disruptive effects of possible flood
 Reconstruction of drain system
 2 alternatives - Northern Channel & Southern Channel
Objective: maximizing the quantity of water being pumped in one hour
___________________________________________________________________________
Operations Research
 Jan Fábry
Northern
Channel
740
2
900
800
300
270
370
1100
5
8
410
700
3
280
7
550
400
300
4
720
130
6
220
Lake
1510
Reservoir
780
2
6
1050
800
470
3
520
230
370
800
1400
840
4
8
250
660
290
700
5
420
7
Southern
Channel
___________________________________________________________________________
Operations Research
 Jan Fábry
Northern Channel
610
1 930 m3
610
22
55
100
1
1100
Optimum
300
410
700
33
88
300
76
220
390
300
44
1500
9
220
67
130
___________________________________________________________________________
Operations Research
 Jan Fábry
Southern Channel
Optimum
2 450 m3
780
22
66
780
470
1050
470
33
9
200
1
800
400
1400
840
44
88
250
210
55
360
150
7
___________________________________________________________________________
Operations Research
 Jan Fábry