Linear Programming
Linear Programming
Cutting Stock Problem
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Quantitative Methods of Management
Jan Fábry
Linear Programming
Applications
Cutting Stock Problem
Final product
Raw product
Loss
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Quantitative Methods of Management
Jan Fábry
Linear Programming
Applications
Cutting Stock Problem
Raw product
2 final products
Cutting patterns
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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
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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
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Quantitative Methods of Management
Jan Fábry
Linear Programming
Applications
Cutting Stock Problem
Example – John Starling & Sons
2 products: bird table bird box
Inputs: screws time
Special collection for the exhibition (with possible sales) in 20 days wooden boards
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
Price
Wooden board 30 cm
Wooden board 25 cm
Screw
Time
Wooden board 1.1 m
Bird table Bird box
260 CZK
1
1
8
30 min
-
570 CZK
2
4
16
60 min
-
Available
unknown unknown
3 000
8 hours per day
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
Pattern No 1
Board 1.1 m
2 3 4 5 6
Board 1.4 m
7 8 9
Board 30 cm 3 2
Board 25 cm -
Loss 20
2
-
1
3
5
-
4
10
4
-
20
3
2
-
2
3
5
1
4
10
-
5
15
Bird table
Bird box
Variable x
1 x
2 x
3 x
4 x
5 x
6 x
7 x
8 x
9 x
10 x
11
___________________________________________________________________________
Quantitative Methods of Management
Jan Fábry
Linear Programming
Applications
Cutting Stock Problem
Example – John Starling & Sons
Optimal solution x
1
Board 1.1 m x
2 x
3 x
4 x
5
Board 1.4 m x
6 x
7 x
8 x
9
Bird table
Bird box x
10 x
11
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
Cutting Stock Problem
Distribution
Problems
Transportation Problem
Assignment Problem
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Quantitative Methods of Management
Jan Fábry
Linear Programming
Transportation Problem
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Quantitative Methods of Management
Jan Fábry
Linear Programming
Applications
Transportation Problem
Shipments
Sources Destinations
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Quantitative Methods of Management
Jan Fábry
Sources
supply
supply
Linear Programming
Applications
Transportation Problem
Destinations
=
demand Balanced
=
demand Unbalanced
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Quantitative Methods of Management
Jan Fábry
Linear Programming
Applications
Transportation Problem
Sources
supply
supply
Dummy source
Destinations
>
demand
Dummy destination
<
demand
___________________________________________________________________________
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
Subsidiaries
Benešov
Jihlava
Tábor
Supply of potatoes 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
Humpolec
Pelhřimov
Weekly demand
Benešov Jihlava Tábor
330
300
45
250
240
60
350
250
35
Weekly supply
70
80
150
140
___________________________________________________________________________
Quantitative Methods of Management
Jan Fábry
Linear Programming
Applications
Transportation Problem
Example – Star Chips, Inc.
Dummy destination
Humpolec
Pelhřimov
Weekly demand
Benešov Jihlava Tábor Dummy
330
300
45
250
240
60
350
250
35
0
0
10
Weekly supply
70
80
150
___________________________________________________________________________
Quantitative Methods of Management
Jan Fábry
Linear Programming
Applications
Transportation Problem
Example – Star Chips, Inc.
Decision variables
Humpolec
Pelhřimov
Benešov
Jihlava
Tábor
Dummy x
11 x
21 x
12 x
22 x
13 x
23 x
14 x
24
___________________________________________________________________________
Quantitative Methods of Management
Jan Fábry
Linear Programming
Applications
Transportation Problem
Example – Star Chips, Inc.
Optimal solution
Humpolec
Pelhřimov
Benešov
Jihlava
Tábor
Remain
60 10
45 35 -
Total weekly cost 37 250 CZK
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Quantitative Methods of Management
Jan Fábry
