Prezentace aplikace PowerPoint

1
Linear Programming
Cutting Stock Problem
_______________________________________________
____________________________ Operations
Research ? Jan Fábry
2
Linear Programming
Applications
Cutting Stock Problem
Finalproduct
Loss
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____________________________ Operations
Research ? Jan Fábry
3
Linear Programming
Applications
Cutting Stock Problem
_______________________________________________
____________________________ Operations
Research ? Jan Fábry
4
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
_______________________________________________
____________________________ Operations
Research ? Jan Fábry
5
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

_______________________________________________
____________________________ Operations
Research ? Jan Fábry
6
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)
_______________________________________________
____________________________ Operations
Research ? Jan Fábry
7
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
_______________________________________________
____________________________ Operations
Research ? Jan Fábry
8
Linear Programming
Applications
Cutting Stock Problem
Example John Starling Sons
Cutting patterns Decision variables
Board 1.1 m Board 1.1 m Board 1.1 m Board 1.1 m Board 1.4 m Board 1.4 m Board 1.4 m Board 1.4 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
Variable x1 x2 x3 x4 x5 x6 x7 x8 x9
Bird table Bird box
x10 x11
_______________________________________________
____________________________ Operations
Research ? Jan Fábry
9
Linear Programming
Applications
Cutting Stock Problem
Example John Starling Sons
Optimal solution
Board 1.1 m Board 1.1 m Board 1.1 m Board 1.1 m Board 1.4 m Board 1.4 m Board 1.4 m Board 1.4 m Board 1.4 m Bird table Bird box
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
_______________________________________________
____________________________ Operations
Research ? Jan Fábry
10

Linear Programming
Applications

• Production Process Models

• Blending Problems

• Marketing Research

Distribution Problems

• Portfolio Selection Problem

• Cutting Stock Problem

• Transportation Problem

• Assignment Problem

_______________________________________________
____________________________ Operations
Research ? Jan Fábry
11
Linear Programming
Transportation Problem
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____________________________ Operations
Research ? Jan Fábry
12
Linear Programming
Applications
Transportation Problem
Shipments
Sources
Destinations
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____________________________ Operations
Research ? Jan Fábry
13
Linear Programming
Applications
Transportation Problem
_______________________________________________
____________________________ Operations
Research ? Jan Fábry
14
Linear Programming
Applications
Transportation Problem
_______________________________________________
____________________________ Operations
Research ? Jan Fábry
15
Linear Programming
Applications
Transportation Problem
Example Star Chips, Inc.
WarehousesHumpolecPelhrimov
SubsidiariesBenešovJihlavaTábor
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____________________________ Operations
Research ? Jan Fábry
16
Linear Programming
Applications
Transportation Problem
Example Star Chips, Inc.
WarehousesHumpolecPelhrimov
SubsidiariesBenešovJihlavaTábor
Objective satisfy requirements with minimal
total shipping cost
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____________________________ Operations
Research ? Jan Fábry
17
Linear Programming
Applications
Transportation Problem
Example Star Chips, Inc.
Supply, demand, unit shipping cost
Benešov Jihlava Tábor Weekly supply
Humpolec 330 250 350 70
Pelhrimov 300 240 250 80
Weekly demand 45 60 35
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____________________________ Operations
Research ? Jan Fábry
18
Linear Programming
Applications
Transportation Problem
Example Star Chips, Inc.
Dummy destination
Benešov Jihlava Tábor Dummy Weekly supply
Humpolec 330 250 350 0 70
Pelhrimov 300 240 250 0 80
Weekly demand 45 60 35 10
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____________________________ Operations
Research ? Jan Fábry
19
Linear Programming
Applications
Transportation Problem
Example Star Chips, Inc.
Decision variables
Benešov Jihlava Tábor Dummy
Humpolec x11 x12 x13 x14
Pelhrimov x21 x22 x23 x24
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____________________________ Operations
Research ? Jan Fábry
20
Linear Programming
Applications
Transportation Problem
Example Star Chips, Inc.
Optimal solution
Benešov Jihlava Tábor Remain
Humpolec - 60 - 10
Pelhrimov 45 - 35 -
Total weekly cost 37 250 CZK
_______________________________________________
____________________________ Operations
Research ? Jan Fábry