Linear Programming
Cutting Stock Problem
___________________________________________________________________________
Quantitative Methods of Management
 Jan Fábry
Linear Programming
Applications
Cutting Stock Problem
Final
product
Raw
product
Loss
___________________________________________________________________________
Quantitative Methods of Management
 Jan Fábry
Linear Programming
Applications
Cutting Stock Problem
Raw
product
2 final
products
Cutting
patterns
___________________________________________________________________________
Quantitative Methods of Management
 Jan Fábry
Linear Programming
Applications
Cutting Stock Problem
 Raw products
- pipes
- roles of paper or textile
- bars
- wooden sticks or bars
- steel plates
 Objective
 minimization of total loss or number of cut raw products
 maximization of number of assembled products
___________________________________________________________________________
Quantitative Methods of Management
 Jan Fábry
Linear Programming
Applications
Cutting Stock Problem
 Decision variables ~ patterns
The value of the variable determines the
number of the raw products being cut/sliced
according to corresponding pattern
 Finding of all possible cutting patterns
___________________________________________________________________________
Quantitative Methods of Management
 Jan Fábry
Linear Programming
Applications
Cutting Stock Problem
Example – John Starling & Sons
 2 products:
bird table
bird box
 Inputs:
wooden boards
screws
time
 Special collection for the exhibition (with possible sales)
in 20 days
Objective: maximize total revenue (all production will be sold)
___________________________________________________________________________
Quantitative Methods of Management
 Jan Fábry
Linear Programming
Applications
Cutting Stock Problem
Example – John Starling & Sons
Bird table
Bird box
Available
Price
260 CZK
570 CZK
-
Wooden board 30 cm
1
2
unknown
Wooden board 25 cm
1
4
unknown
Screw
8
16
3 000
Time
30 min
60 min
8 hours per day
Wooden board 1.1 m
-
-
500
Wooden board 1.4 m
150
___________________________________________________________________________
Quantitative Methods of Management
 Jan Fábry
Linear Programming
Applications
Cutting Stock Problem
Example – John Starling & Sons
Cutting patterns & Decision variables
Board 1.1 m
Board 1.4 m
Pattern No
1
2
3
4
5
6
7
8
9
Board 30 cm
3
2
1
-
4
3
2
1
-
Board 25 cm
-
2
3
4
-
2
3
4
5
Loss
20
-
5
10
20
-
5
10
15
Bird Bird
table box
Variable
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
___________________________________________________________________________
Quantitative Methods of Management
 Jan Fábry
Linear Programming
Applications
Cutting Stock Problem
Example – John Starling & Sons
Optimal solution
Board 1.1 m
Bird Bird
table box
Board 1.4 m
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
0
65
0
0
48
0
0
0
102
0
160
Total revenue
91 200 CZK
___________________________________________________________________________
Quantitative Methods of Management
 Jan Fábry
Linear Programming
Applications
 Production Process Models
 Blending Problems
 Marketing Research
 Portfolio Selection Problem
Distribution
Problems
 Cutting Stock Problem
 Transportation Problem
 Assignment Problem
___________________________________________________________________________
Quantitative Methods of Management
 Jan Fábry
Linear Programming
Transportation Problem
___________________________________________________________________________
Quantitative Methods of Management
 Jan Fábry
Linear Programming
Applications
Transportation Problem
Shipments
Sources
Destinations
___________________________________________________________________________
Quantitative Methods of Management
 Jan Fábry
Linear Programming
Applications
Transportation Problem
Sources
Destinations
 supply
=
 demand
Balanced
 supply
=
 demand
Unbalanced
___________________________________________________________________________
Quantitative Methods of Management
 Jan Fábry
Linear Programming
Applications
Transportation Problem
Sources
 supply
Destinations
>
 demand
Dummy destination
 supply
<
 demand
Dummy source
___________________________________________________________________________
Quantitative Methods of Management
 Jan Fábry
Linear Programming
Applications
Transportation Problem
Example – Star Chips, Inc.
Warehouses
Humpolec
Pelhřimov
Benešov
Humpolec
Tábor
Jihlava
Pelhřimov
Subsidiaries
Benešov
Jihlava
Tábor
___________________________________________________________________________
Quantitative Methods of Management
 Jan Fábry
Linear Programming
Applications
Transportation Problem
Example – Star Chips, Inc.
Warehouses
Humpolec
Pelhřimov
Supply of potatoes
Subsidiaries
Benešov
Jihlava
Tábor
Requirements
Objective: satisfy requirements with minimal total shipping cost
___________________________________________________________________________
Quantitative Methods of Management
 Jan Fábry
Linear Programming
Applications
Transportation Problem
Example – Star Chips, Inc.
Supply, demand, unit shipping cost
Benešov
Jihlava
Tábor
Weekly
supply
Humpolec
330
250
350
70
Pelhřimov
300
240
250
80
Weekly demand
45
60
35
150
140
___________________________________________________________________________
Quantitative Methods of Management
 Jan Fábry
Linear Programming
Applications
Transportation Problem
Example – Star Chips, Inc.
Dummy destination
Benešov
Jihlava
Tábor
Dummy
Weekly
supply
Humpolec
330
250
350
0
70
Pelhřimov
300
240
250
0
80
Weekly demand
45
60
35
10
150
___________________________________________________________________________
Quantitative Methods of Management
 Jan Fábry
Linear Programming
Applications
Transportation Problem
Example – Star Chips, Inc.
Decision variables
Benešov
Jihlava
Tábor
Dummy
Humpolec
x11
x12
x13
x14
Pelhřimov
x21
x22
x23
x24
___________________________________________________________________________
Quantitative Methods of Management
 Jan Fábry
Linear Programming
Applications
Transportation Problem
Example – Star Chips, Inc.
Optimal solution
Benešov
Jihlava
Tábor
Remain
Humpolec
-
60
-
10
Pelhřimov
45
-
35
-
Total weekly cost
37 250 CZK
___________________________________________________________________________
Quantitative Methods of Management
 Jan Fábry