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Bin packingFirst-fit algorithm
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2 35
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A B C D E F
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Each block will be fitted into the first bin that has room for it.Each block will be fitted into the first bin that has room for it.
Bin packingFirst-fit algorithm
A B C D E F
4
The first block fits into the first bin.The first block fits into the first bin.
1 23
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2 35
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Bin packingFirst-fit algorithm
A B C D E F
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The second block fits into the first bin.The second block fits into the first bin.
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2 35
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Bin packingFirst-fit algorithm
A B C D E F
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The third block will not fit in the first binThe third block will not fit in the first bin
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But there is room in the second bin.But there is room in the second bin.
Bin packingFirst-fit algorithm
The third bin is the first one the 5 will fit into.The third bin is the first one the 5 will fit into.
A B C D E F
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Bin packingFirst-fit algorithm
The second bin has room for the 3.The second bin has room for the 3.
Bin packingFirst-fit algorithm
A B C D E F
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The fourth bin is the first one with room for the 2.
The fourth bin is the first one with room for the 2.
Bin packingFirst-fit algorithm
A B C D E F
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The 3 also fits into the fourth bin.The 3 also fits into the fourth bin.
Bin packingFirst-fit algorithm
A B C D E F
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No room until the fifth bin for the 6.No room until the fifth bin for the 6.
Bin packingFirst-fit algorithm
A B C D E F
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The 3 has to start a new binThe 3 has to start a new bin
Bin packingFirst-fit algorithm
A B C D E F
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3 6
3 3 3 33
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Total usage is 6 bins.Total usage is 6 bins.
Bin Packing
Examples
of first-fit algorithm
Example 1Pack the following items
into bins of size 20:11, 7, 3, 4, 3, 5, 1, 8, 9
How many bins are required?
11, 7, 3, 4, 3, 5, 1, 8, 9 Bin size 20
7 538
39
4 1
Bins
11
11, 7, 3, 4, 3, 5, 1, 8, 9 Bin size 20
7 538
39
4 1
Bins
11
11, 7, 3, 4, 3, 5, 1, 8, 9 Bin size 20
7
538
39
4 1
Bins
11
11, 7, 3, 4, 3, 5, 1, 8, 9 Bin size 20
7
538
3
94 1
Bins
11
11, 7, 3, 4, 3, 5, 1, 8, 9 Bin size 20
7
538
3
94 1
Bins
11
11, 7, 3, 4, 3, 5, 1, 8, 9 Bin size 20
7
538
3
9
4
1
Bins
11
11, 7, 3, 4, 3, 5, 1, 8, 9 Bin size 20
7
5
3
8
3
9
4
1
Bins
11
11, 7, 3, 4, 3, 5, 1, 8, 9 Bin size 20
7 5
3
8
3
9
4
1
Bins
11
11, 7, 3, 4, 3, 5, 1, 8, 9 Bin size 20
7 5
3
8
3
9
4
1
Bins
11
11, 7, 3, 4, 3, 5, 1, 8, 9 Bin size 20
7 5
3
8
3
9
4
1
Bins
11
11, 7, 3, 4, 3, 5, 1, 8, 9 Bin size 20
7 5
3
83
9
4
1
Bins
11
11, 7, 3, 4, 3, 5, 1, 8, 9 Bin size 20
7 5
3
83
9
4
1
Bins
11
We need 3 bins.
Example 2Pack the following items into
bins of size 20:5, 8, 12, 4, 11, 17, 7, 9, 7
How many bins are required?
5, 8, 12, 4, 11, 17, 7, 9, 7Bin size 20
8
1711 912
74 7
Bins
5
5, 8, 12, 4, 11, 17, 7, 9, 7Bin size 20
8
1711 912
74 7
Bins
5
5, 8, 12, 4, 11, 17, 7, 9, 7Bin size 20
8
1711 912
74 7
Bins
5
5, 8, 12, 4, 11, 17, 7, 9, 7Bin size 20
8
1711 9
12
74 7
Bins
5
5, 8, 12, 4, 11, 17, 7, 9, 7Bin size 20
8
1711 9
12
74 7
Bins
5
5, 8, 12, 4, 11, 17, 7, 9, 7Bin size 20
8
1711 9
12
7
4
7
Bins
5
5, 8, 12, 4, 11, 17, 7, 9, 7Bin size 20
8
17
11
9
12
7
4
7
Bins
5
5, 8, 12, 4, 11, 17, 7, 9, 7Bin size 20
8
17
11
9
12
7
4
7
Bins
5
5, 8, 12, 4, 11, 17, 7, 9, 7Bin size 20
8
17
11
9
12
7
4
7
Bins
5
5, 8, 12, 4, 11, 17, 7, 9, 7Bin size 20
8 1711
9
12
7
4
7
Bins
5
5, 8, 12, 4, 11, 17, 7, 9, 7Bin size 20
8 1711
9
12
7
4 7
Bins
5
5, 8, 12, 4, 11, 17, 7, 9, 7Bin size 20
8 1711
9
12
7
4 7
Bins
5
5, 8, 12, 4, 11, 17, 7, 9, 7Bin size 20
8 1711
9
127
4 7
Bins
5
We need 5 bins.
Example 3A wholesale greengrocer is packing the following items into crates
that will carry 24 kg:Item Weight of item (kg) Number of items
required
Carrots 9 1
Apples 6 2
Pears 7 1
Courgettes 3 1
Cauliflowers 14 1
Potatoes 20 1
Oranges 10 1
Leeks 6 1
Bananas 8 1
Tomatoes 5 3
Grapes 2 1
Onions 18 1
How many crates will the greengrocer need?
Crates (24 kg)
Item Weight of item (kg) Number of items required
Carrots 9 1
Apples 6 2
Pears 7 1
Courgettes 3 1
Cauliflowers 14 1
Potatoes 20 1
Oranges 10 1
Leeks 6 1
Bananas 8 1
Tomatoes 5 3
Grapes 2 1
Onions 18 1
Crates (24 kg)9 kg
Item Weight of item (kg) Number of items required
Carrots 9 1
Apples 6 2
Pears 7 1
Courgettes 3 1
Cauliflowers 14 1
Potatoes 20 1
Oranges 10 1
Leeks 6 1
Bananas 8 1
Tomatoes 5 3
Grapes 2 1
Onions 18 1
Crates (24 kg)9 kg, 6 kg
Item Weight of item (kg) Number of items required
Carrots 9 1
Apples 6 2
Pears 7 1
Courgettes 3 1
Cauliflowers 14 1
Potatoes 20 1
Oranges 10 1
Leeks 6 1
Bananas 8 1
Tomatoes 5 3
Grapes 2 1
Onions 18 1
Crates (24 kg)9 kg, 6 kg, 6 kg
Item Weight of item (kg) Number of items required
Carrots 9 1
Apples 6 2
Pears 7 1
Courgettes 3 1
Cauliflowers 14 1
Potatoes 20 1
Oranges 10 1
Leeks 6 1
Bananas 8 1
Tomatoes 5 3
Grapes 2 1
Onions 18 1
Crates (24 kg)9 kg, 6 kg, 6 kg, 7 kg
Item Weight of item (kg) Number of items required
Carrots 9 1
Apples 6 2
Pears 7 1
Courgettes 3 1
Cauliflowers 14 1
Potatoes 20 1
Oranges 10 1
Leeks 6 1
Bananas 8 1
Tomatoes 5 3
Grapes 2 1
Onions 18 1
Crates (24 kg)9 kg, 6 kg, 6 kg
7 kg
Item Weight of item (kg) Number of items required
Carrots 9 1
Apples 6 2
Pears 7 1
Courgettes 3 1
Cauliflowers 14 1
Potatoes 20 1
Oranges 10 1
Leeks 6 1
Bananas 8 1
Tomatoes 5 3
Grapes 2 1
Onions 18 1
Crates (24 kg)9 kg, 6 kg, 6 kg, 3 kg
7 kg
Item Weight of item (kg) Number of items required
Carrots 9 1
Apples 6 2
Pears 7 1
Courgettes 3 1
Cauliflowers 14 1
Potatoes 20 1
Oranges 10 1
Leeks 6 1
Bananas 8 1
Tomatoes 5 3
Grapes 2 1
Onions 18 1
Crates (24 kg)9 kg, 6 kg, 6 kg, 3 kg
7 kg
Item Weight of item (kg) Number of items required
Carrots 9 1
Apples 6 2
Pears 7 1
Courgettes 3 1
Cauliflowers 14 1
Potatoes 20 1
Oranges 10 1
Leeks 6 1
Bananas 8 1
Tomatoes 5 3
Grapes 2 1
Onions 18 1
Crates (24 kg)9 kg, 6 kg, 6 kg, 3 kg
7 kg, 14 kg
Item Weight of item (kg) Number of items required
Carrots 9 1
Apples 6 2
Pears 7 1
Courgettes 3 1
Cauliflowers 14 1
Potatoes 20 1
Oranges 10 1
Leeks 6 1
Bananas 8 1
Tomatoes 5 3
Grapes 2 1
Onions 18 1
Crates (24 kg)9 kg, 6 kg, 6 kg, 3 kg
7 kg, 14 kg, 20 kg
Item Weight of item (kg) Number of items required
Carrots 9 1
Apples 6 2
Pears 7 1
Courgettes 3 1
Cauliflowers 14 1
Potatoes 20 1
Oranges 10 1
Leeks 6 1
Bananas 8 1
Tomatoes 5 3
Grapes 2 1
Onions 18 1
Crates (24 kg)9 kg, 6 kg, 6 kg, 3 kg
7 kg, 14 kg
20 kg
Item Weight of item (kg) Number of items required
Carrots 9 1
Apples 6 2
Pears 7 1
Courgettes 3 1
Cauliflowers 14 1
Potatoes 20 1
Oranges 10 1
Leeks 6 1
Bananas 8 1
Tomatoes 5 3
Grapes 2 1
Onions 18 1
Crates (24 kg)9 kg, 6 kg, 6 kg, 3 kg
7 kg, 14 kg, 10 kg
20 kg
Item Weight of item (kg) Number of items required
Carrots 9 1
Apples 6 2
Pears 7 1
Courgettes 3 1
Cauliflowers 14 1
Potatoes 20 1
Oranges 10 1
Leeks 6 1
Bananas 8 1
Tomatoes 5 3
Grapes 2 1
Onions 18 1
Crates (24 kg)9 kg, 6 kg, 6 kg, 3 kg
7 kg, 14 kg
20 kg, 10 kg
Item Weight of item (kg) Number of items required
Carrots 9 1
Apples 6 2
Pears 7 1
Courgettes 3 1
Cauliflowers 14 1
Potatoes 20 1
Oranges 10 1
Leeks 6 1
Bananas 8 1
Tomatoes 5 3
Grapes 2 1
Onions 18 1
Crates (24 kg)9 kg, 6 kg, 6 kg, 3 kg
7 kg, 14 kg
20 kg
10 kg
Item Weight of item (kg) Number of items required
Carrots 9 1
Apples 6 2
Pears 7 1
Courgettes 3 1
Cauliflowers 14 1
Potatoes 20 1
Oranges 10 1
Leeks 6 1
Bananas 8 1
Tomatoes 5 3
Grapes 2 1
Onions 18 1
Crates (24 kg)9 kg, 6 kg, 6 kg, 3 kg
7 kg, 14 kg, 6 kg
20 kg
10 kg
Item Weight of item (kg) Number of items required
Carrots 9 1
Apples 6 2
Pears 7 1
Courgettes 3 1
Cauliflowers 14 1
Potatoes 20 1
Oranges 10 1
Leeks 6 1
Bananas 8 1
Tomatoes 5 3
Grapes 2 1
Onions 18 1
Crates (24 kg)9 kg, 6 kg, 6 kg, 3 kg
7 kg, 14 kg
20 kg, 6 kg
10 kg
Item Weight of item (kg) Number of items required
Carrots 9 1
Apples 6 2
Pears 7 1
Courgettes 3 1
Cauliflowers 14 1
Potatoes 20 1
Oranges 10 1
Leeks 6 1
Bananas 8 1
Tomatoes 5 3
Grapes 2 1
Onions 18 1
Crates (24 kg)9 kg, 6 kg, 6 kg, 3 kg
7 kg, 14 kg
20 kg
10 kg, 6 kg
Item Weight of item (kg) Number of items required
Carrots 9 1
Apples 6 2
Pears 7 1
Courgettes 3 1
Cauliflowers 14 1
Potatoes 20 1
Oranges 10 1
Leeks 6 1
Bananas 8 1
Tomatoes 5 3
Grapes 2 1
Onions 18 1
Crates (24 kg)9 kg, 6 kg, 6 kg, 3 kg
7 kg, 14 kg, 8 kg
20 kg
10 kg, 6 kg
Item Weight of item (kg) Number of items required
Carrots 9 1
Apples 6 2
Pears 7 1
Courgettes 3 1
Cauliflowers 14 1
Potatoes 20 1
Oranges 10 1
Leeks 6 1
Bananas 8 1
Tomatoes 5 3
Grapes 2 1
Onions 18 1
Crates (24 kg)9 kg, 6 kg, 6 kg, 3 kg
7 kg, 14 kg
20 kg, 8 kg
10 kg, 6 kg
Item Weight of item (kg) Number of items required
Carrots 9 1
Apples 6 2
Pears 7 1
Courgettes 3 1
Cauliflowers 14 1
Potatoes 20 1
Oranges 10 1
Leeks 6 1
Bananas 8 1
Tomatoes 5 3
Grapes 2 1
Onions 18 1
Crates (24 kg)9 kg, 6 kg, 6 kg, 3 kg
7 kg, 14 kg
20 kg
10 kg, 6 kg, 8 kg
Item Weight of item (kg) Number of items required
Carrots 9 1
Apples 6 2
Pears 7 1
Courgettes 3 1
Cauliflowers 14 1
Potatoes 20 1
Oranges 10 1
Leeks 6 1
Bananas 8 1
Tomatoes 5 3
Grapes 2 1
Onions 18 1
Crates (24 kg)9 kg, 6 kg, 6 kg, 3 kg
7 kg, 14 kg
20 kg
10 kg, 6 kg, 8 kg
Item Weight of item (kg) Number of items required
Carrots 9 1
Apples 6 2
Pears 7 1
Courgettes 3 1
Cauliflowers 14 1
Potatoes 20 1
Oranges 10 1
Leeks 6 1
Bananas 8 1
Tomatoes 5 3
Grapes 2 1
Onions 18 1
Crates (24 kg)9 kg, 6 kg, 6 kg, 3 kg
7 kg, 14 kg, 5 kg
20 kg
10 kg, 6 kg, 8 kg
Item Weight of item (kg) Number of items required
Carrots 9 1
Apples 6 2
Pears 7 1
Courgettes 3 1
Cauliflowers 14 1
Potatoes 20 1
Oranges 10 1
Leeks 6 1
Bananas 8 1
Tomatoes 5 3
Grapes 2 1
Onions 18 1
Crates (24 kg)9 kg, 6 kg, 6 kg, 3 kg
7 kg, 14 kg
20 kg, 5 kg
10 kg, 6 kg, 8 kg
Item Weight of item (kg) Number of items required
Carrots 9 1
Apples 6 2
Pears 7 1
Courgettes 3 1
Cauliflowers 14 1
Potatoes 20 1
Oranges 10 1
Leeks 6 1
Bananas 8 1
Tomatoes 5 3
Grapes 2 1
Onions 18 1
Crates (24 kg)9 kg, 6 kg, 6 kg, 3 kg
7 kg, 14 kg
20 kg
10 kg, 6 kg, 8 kg
5 kg
Item Weight of item (kg) Number of items required
Carrots 9 1
Apples 6 2
Pears 7 1
Courgettes 3 1
Cauliflowers 14 1
Potatoes 20 1
Oranges 10 1
Leeks 6 1
Bananas 8 1
Tomatoes 5 3
Grapes 2 1
Onions 18 1
Crates (24 kg)9 kg, 6 kg, 6 kg, 3 kg
7 kg, 14 kg
20 kg
10 kg, 6 kg, 8 kg
5 kg, 5 kg
Item Weight of item (kg) Number of items required
Carrots 9 1
Apples 6 2
Pears 7 1
Courgettes 3 1
Cauliflowers 14 1
Potatoes 20 1
Oranges 10 1
Leeks 6 1
Bananas 8 1
Tomatoes 5 3
Grapes 2 1
Onions 18 1
Crates (24 kg)9 kg, 6 kg, 6 kg, 3 kg
7 kg, 14 kg
20 kg
10 kg, 6 kg, 8 kg
5 kg, 5 kg, 5 kg
Item Weight of item (kg) Number of items required
Carrots 9 1
Apples 6 2
Pears 7 1
Courgettes 3 1
Cauliflowers 14 1
Potatoes 20 1
Oranges 10 1
Leeks 6 1
Bananas 8 1
Tomatoes 5 3
Grapes 2 1
Onions 18 1
Crates (24 kg)9 kg, 6 kg, 6 kg, 3 kg
7 kg, 14 kg, 2 kg
20 kg
10 kg, 6 kg, 8 kg
5 kg, 5 kg, 5 kg
Item Weight of item (kg) Number of items required
Carrots 9 1
Apples 6 2
Pears 7 1
Courgettes 3 1
Cauliflowers 14 1
Potatoes 20 1
Oranges 10 1
Leeks 6 1
Bananas 8 1
Tomatoes 5 3
Grapes 2 1
Onions 18 1
Crates (24 kg)9 kg, 6 kg, 6 kg, 3 kg
7 kg, 14 kg, 2 kg, 18 kg
20 kg
10 kg, 6 kg, 8 kg
5 kg, 5 kg, 5 kg
Item Weight of item (kg) Number of items required
Carrots 9 1
Apples 6 2
Pears 7 1
Courgettes 3 1
Cauliflowers 14 1
Potatoes 20 1
Oranges 10 1
Leeks 6 1
Bananas 8 1
Tomatoes 5 3
Grapes 2 1
Onions 18 1
Crates (24 kg)9 kg, 6 kg, 6 kg, 3 kg
7 kg, 14 kg, 2 kg
20 kg, 18 kg
10 kg, 6 kg, 8 kg
5 kg, 5 kg, 5 kg
Item Weight of item (kg) Number of items required
Carrots 9 1
Apples 6 2
Pears 7 1
Courgettes 3 1
Cauliflowers 14 1
Potatoes 20 1
Oranges 10 1
Leeks 6 1
Bananas 8 1
Tomatoes 5 3
Grapes 2 1
Onions 18 1
Crates (24 kg)9 kg, 6 kg, 6 kg, 3 kg
7 kg, 14 kg, 2 kg
20 kg
10 kg, 6 kg, 8 kg
5 kg, 5 kg, 5 kg, 18 kg
Item Weight of item (kg) Number of items required
Carrots 9 1
Apples 6 2
Pears 7 1
Courgettes 3 1
Cauliflowers 14 1
Potatoes 20 1
Oranges 10 1
Leeks 6 1
Bananas 8 1
Tomatoes 5 3
Grapes 2 1
Onions 18 1
Crates (24 kg)9 kg, 6 kg, 6 kg, 3 kg
7 kg, 14 kg, 2 kg
20 kg
10 kg, 6 kg, 8 kg
5 kg, 5 kg, 5 kg
18 kg
Item Weight of item (kg) Number of items required
Carrots 9 1
Apples 6 2
Pears 7 1
Courgettes 3 1
Cauliflowers 14 1
Potatoes 20 1
Oranges 10 1
Leeks 6 1
Bananas 8 1
Tomatoes 5 3
Grapes 2 1
Onions 18 1
Crates (24 kg)9 kg, 6 kg, 6 kg, 3 kg
7 kg, 14 kg, 2 kg
20 kg
10 kg, 6 kg, 8 kg
5 kg, 5 kg, 5 kg
18 kg
Item Weight of item (kg) Number of items required
Carrots 9 1
Apples 6 2
Pears 7 1
Courgettes 3 1
Cauliflowers 14 1
Potatoes 20 1
Oranges 10 1
Leeks 6 1
Bananas 8 1
Tomatoes 5 3
Grapes 2 1
Onions 18 1
So the greengrocer
needs 6 crates.
Example 4Pack the following items into
bins of size 16: 14, 2, 7, 3, 1, 9, 12, 4, 6
How many bins are required?
14, 2, 7, 3, 1, 9, 12, 4, 6Bin size 16
2
9
1 47 6
3
12
Bins
14
14, 2, 7, 3, 1, 9, 12, 4, 6Bin size 16
2
9
1 47 6
3
12
Bins
14
14, 2, 7, 3, 1, 9, 12, 4, 6Bin size 16
2
9
1 47 6
3
12
Bins
14
14, 2, 7, 3, 1, 9, 12, 4, 6Bin size 16
2
9
1 4
7
63
12
Bins
14
14, 2, 7, 3, 1, 9, 12, 4, 6Bin size 16
2
9
1 4
7
63
12
Bins
14
14, 2, 7, 3, 1, 9, 12, 4, 6Bin size 16
2
9
1 4
7
6
3
12
Bins
14
14, 2, 7, 3, 1, 9, 12, 4, 6Bin size 16
2
9
1
4
7
6
3
12
Bins
14
14, 2, 7, 3, 1, 9, 12, 4, 6Bin size 16
29
1
4
7
6
3
12
Bins
14
14, 2, 7, 3, 1, 9, 12, 4, 6Bin size 16
2
9
1
4
7
6
3
12
Bins
14
14, 2, 7, 3, 1, 9, 12, 4, 6Bin size 16
2
9
1
4
7
6
3
12
Bins
14
14, 2, 7, 3, 1, 9, 12, 4, 6Bin size 16
2
9
1
4
7
6
3
12
Bins
14
14, 2, 7, 3, 1, 9, 12, 4, 6Bin size 16
2
9
1
4
7
6
3
12
Bins
14
14, 2, 7, 3, 1, 9, 12, 4, 6Bin size 16
2
9
1
4
7
6
3
12
Bins
14
14, 2, 7, 3, 1, 9, 12, 4, 6Bin size 16
2
9
1
4
7
6
3
12
Bins
14
We need 4 bins.
Example 5A music store is ordering musical instruments. They are packed into boxes
and delivered by a lorry. Each box can hold 20 kg.
Instrument Weight of instrument (kg)
Number of each instrument ordered
Guitars 15 2
Trumpets 2 3
Saxophones 7 2
Violins 1 1
Trombones 2 3
Flutes 1 2
Bassoons 4 4
Tubas 18 1
Cymbals 1 2
How many boxes will be needed to deliver all the instruments?
Boxes (20 kg)
Instrument Weight of instrument (kg)
Number of each instrument ordered
Guitars 15 2
Trumpets 2 3
Saxophones 7 2
Violins 1 1
Trombones 2 3
Flutes 1 2
Bassoons 4 4
Tubas 18 1
Cymbals 1 2
Boxes (20 kg)
Instrument Weight of instrument (kg)
Number of each instrument ordered
Guitars 15 2
Trumpets 2 3
Saxophones 7 2
Violins 1 1
Trombones 2 3
Flutes 1 2
Bassoons 4 4
Tubas 18 1
Cymbals 1 2
Boxes (20 kg)15 kg
Instrument Weight of instrument (kg)
Number of each instrument ordered
Guitars 15 2
Trumpets 2 3
Saxophones 7 2
Violins 1 1
Trombones 2 3
Flutes 1 2
Bassoons 4 4
Tubas 18 1
Cymbals 1 2
Boxes (20 kg)15 kg, 15 kg
Instrument Weight of instrument (kg)
Number of each instrument ordered
Guitars 15 2
Trumpets 2 3
Saxophones 7 2
Violins 1 1
Trombones 2 3
Flutes 1 2
Bassoons 4 4
Tubas 18 1
Cymbals 1 2
Boxes (20 kg)15 kg
15 kg
Instrument Weight of instrument (kg)
Number of each instrument ordered
Guitars 15 2
Trumpets 2 3
Saxophones 7 2
Violins 1 1
Trombones 2 3
Flutes 1 2
Bassoons 4 4
Tubas 18 1
Cymbals 1 2
Boxes (20 kg)15 kg
15 kg
Instrument Weight of instrument (kg)
Number of each instrument ordered
Guitars 15 2
Trumpets 2 3
Saxophones 7 2
Violins 1 1
Trombones 2 3
Flutes 1 2
Bassoons 4 4
Tubas 18 1
Cymbals 1 2
Boxes (20 kg)15 kg, 2 kg
15 kg
Instrument Weight of instrument (kg)
Number of each instrument ordered
Guitars 15 2
Trumpets 2 3
Saxophones 7 2
Violins 1 1
Trombones 2 3
Flutes 1 2
Bassoons 4 4
Tubas 18 1
Cymbals 1 2
Boxes (20 kg)15 kg, 2 kg, 2 kg
15 kg
Instrument Weight of instrument (kg)
Number of each instrument ordered
Guitars 15 2
Trumpets 2 3
Saxophones 7 2
Violins 1 1
Trombones 2 3
Flutes 1 2
Bassoons 4 4
Tubas 18 1
Cymbals 1 2
Boxes (20 kg)15 kg, 2 kg, 2 kg, 2 kg
15 kg
Instrument Weight of instrument (kg)
Number of each instrument ordered
Guitars 15 2
Trumpets 2 3
Saxophones 7 2
Violins 1 1
Trombones 2 3
Flutes 1 2
Bassoons 4 4
Tubas 18 1
Cymbals 1 2
Boxes (20 kg)15 kg, 2 kg, 2 kg
15 kg, 2 kg
Instrument Weight of instrument (kg)
Number of each instrument ordered
Guitars 15 2
Trumpets 2 3
Saxophones 7 2
Violins 1 1
Trombones 2 3
Flutes 1 2
Bassoons 4 4
Tubas 18 1
Cymbals 1 2
Boxes (20 kg)15 kg, 2 kg, 2 kg
15 kg, 2 kg
Instrument Weight of instrument (kg)
Number of each instrument ordered
Guitars 15 2
Trumpets 2 3
Saxophones 7 2
Violins 1 1
Trombones 2 3
Flutes 1 2
Bassoons 4 4
Tubas 18 1
Cymbals 1 2
Boxes (20 kg)15 kg, 2 kg, 2 kg, 7 kg
15 kg, 2 kg
Instrument Weight of instrument (kg)
Number of each instrument ordered
Guitars 15 2
Trumpets 2 3
Saxophones 7 2
Violins 1 1
Trombones 2 3
Flutes 1 2
Bassoons 4 4
Tubas 18 1
Cymbals 1 2
Boxes (20 kg)15 kg, 2 kg, 2 kg
15 kg, 2 kg, 7 kg
Instrument Weight of instrument (kg)
Number of each instrument ordered
Guitars 15 2
Trumpets 2 3
Saxophones 7 2
Violins 1 1
Trombones 2 3
Flutes 1 2
Bassoons 4 4
Tubas 18 1
Cymbals 1 2
Boxes (20 kg)15 kg, 2 kg, 2 kg
15 kg, 2 kg
7 kg
Instrument Weight of instrument (kg)
Number of each instrument ordered
Guitars 15 2
Trumpets 2 3
Saxophones 7 2
Violins 1 1
Trombones 2 3
Flutes 1 2
Bassoons 4 4
Tubas 18 1
Cymbals 1 2
Boxes (20 kg)15 kg, 2 kg, 2 kg
15 kg, 2 kg
7 kg, 7 kg
Instrument Weight of instrument (kg)
Number of each instrument ordered
Guitars 15 2
Trumpets 2 3
Saxophones 7 2
Violins 1 1
Trombones 2 3
Flutes 1 2
Bassoons 4 4
Tubas 18 1
Cymbals 1 2
Boxes (20 kg)15 kg, 2 kg, 2 kg
15 kg, 2 kg
7 kg, 7 kg
Instrument Weight of instrument (kg)
Number of each instrument ordered
Guitars 15 2
Trumpets 2 3
Saxophones 7 2
Violins 1 1
Trombones 2 3
Flutes 1 2
Bassoons 4 4
Tubas 18 1
Cymbals 1 2
Boxes (20 kg)15 kg, 2 kg, 2 kg, 1 kg
15 kg, 2 kg
7 kg, 7 kg
Instrument Weight of instrument (kg)
Number of each instrument ordered
Guitars 15 2
Trumpets 2 3
Saxophones 7 2
Violins 1 1
Trombones 2 3
Flutes 1 2
Bassoons 4 4
Tubas 18 1
Cymbals 1 2
Boxes (20 kg)15 kg, 2 kg, 2 kg, 1 kg
15 kg, 2 kg
7 kg, 7 kg
Instrument Weight of instrument (kg)
Number of each instrument ordered
Guitars 15 2
Trumpets 2 3
Saxophones 7 2
Violins 1 1
Trombones 2 3
Flutes 1 2
Bassoons 4 4
Tubas 18 1
Cymbals 1 2
Boxes (20 kg)15 kg, 2 kg, 2 kg, 1 kg
15 kg, 2 kg
7 kg, 7 kg
Instrument Weight of instrument (kg)
Number of each instrument ordered
Guitars 15 2
Trumpets 2 3
Saxophones 7 2
Violins 1 1
Trombones 2 3
Flutes 1 2
Bassoons 4 4
Tubas 18 1
Cymbals 1 2
Boxes (20 kg)15 kg, 2 kg, 2 kg, 1 kg
15 kg, 2 kg, 2 kg
7 kg, 7 kg
Instrument Weight of instrument (kg)
Number of each instrument ordered
Guitars 15 2
Trumpets 2 3
Saxophones 7 2
Violins 1 1
Trombones 2 3
Flutes 1 2
Bassoons 4 4
Tubas 18 1
Cymbals 1 2
Boxes (20 kg)15 kg, 2 kg, 2 kg, 1 kg
15 kg, 2 kg, 2 kg, 2 kg
7 kg, 7 kg
Instrument Weight of instrument (kg)
Number of each instrument ordered
Guitars 15 2
Trumpets 2 3
Saxophones 7 2
Violins 1 1
Trombones 2 3
Flutes 1 2
Bassoons 4 4
Tubas 18 1
Cymbals 1 2
Boxes (20 kg)15 kg, 2 kg, 2 kg, 1 kg
15 kg, 2 kg, 2 kg
7 kg, 7 kg, 2 kg
Instrument Weight of instrument (kg)
Number of each instrument ordered
Guitars 15 2
Trumpets 2 3
Saxophones 7 2
Violins 1 1
Trombones 2 3
Flutes 1 2
Bassoons 4 4
Tubas 18 1
Cymbals 1 2
Boxes (20 kg)15 kg, 2 kg, 2 kg, 1 kg
15 kg, 2 kg, 2 kg
7 kg, 7 kg, 2 kg, 2 kg
Instrument Weight of instrument (kg)
Number of each instrument ordered
Guitars 15 2
Trumpets 2 3
Saxophones 7 2
Violins 1 1
Trombones 2 3
Flutes 1 2
Bassoons 4 4
Tubas 18 1
Cymbals 1 2
Boxes (20 kg)15 kg, 2 kg, 2 kg, 1 kg
15 kg, 2 kg, 2 kg
7 kg, 7 kg, 2 kg, 2 kg
Instrument Weight of instrument (kg)
Number of each instrument ordered
Guitars 15 2
Trumpets 2 3
Saxophones 7 2
Violins 1 1
Trombones 2 3
Flutes 1 2
Bassoons 4 4
Tubas 18 1
Cymbals 1 2
Boxes (20 kg)15 kg, 2 kg, 2 kg, 1 kg
15 kg, 2 kg, 2 kg, 1 kg
7 kg, 7 kg, 2 kg, 2 kg
Instrument Weight of instrument (kg)
Number of each instrument ordered
Guitars 15 2
Trumpets 2 3
Saxophones 7 2
Violins 1 1
Trombones 2 3
Flutes 1 2
Bassoons 4 4
Tubas 18 1
Cymbals 1 2
Boxes (20 kg)15 kg, 2 kg, 2 kg, 1 kg
15 kg, 2 kg, 2 kg, 1 kg
7 kg, 7 kg, 2 kg, 2 kg
Instrument Weight of instrument (kg)
Number of each instrument ordered
Guitars 15 2
Trumpets 2 3
Saxophones 7 2
Violins 1 1
Trombones 2 3
Flutes 1 2
Bassoons 4 4
Tubas 18 1
Cymbals 1 2
Boxes (20 kg)15 kg, 2 kg, 2 kg, 1 kg
15 kg, 2 kg, 2 kg, 1 kg
7 kg, 7 kg, 2 kg, 2 kg,1 kg
Instrument Weight of instrument (kg)
Number of each instrument ordered
Guitars 15 2
Trumpets 2 3
Saxophones 7 2
Violins 1 1
Trombones 2 3
Flutes 1 2
Bassoons 4 4
Tubas 18 1
Cymbals 1 2
Boxes (20 kg)15 kg, 2 kg, 2 kg, 1 kg
15 kg, 2 kg, 2 kg, 1 kg
7 kg, 7 kg, 2 kg, 2 kg,1 kg
Instrument Weight of instrument (kg)
Number of each instrument ordered
Guitars 15 2
Trumpets 2 3
Saxophones 7 2
Violins 1 1
Trombones 2 3
Flutes 1 2
Bassoons 4 4
Tubas 18 1
Cymbals 1 2
Boxes (20 kg)15 kg, 2 kg, 2 kg, 1 kg
15 kg, 2 kg, 2 kg, 1 kg
7 kg, 7 kg, 2 kg, 2 kg,1 kg, 4 kg
Instrument Weight of instrument (kg)
Number of each instrument ordered
Guitars 15 2
Trumpets 2 3
Saxophones 7 2
Violins 1 1
Trombones 2 3
Flutes 1 2
Bassoons 4 4
Tubas 18 1
Cymbals 1 2
Boxes (20 kg)15 kg, 2 kg, 2 kg, 1 kg
15 kg, 2 kg, 2 kg, 1 kg
7 kg, 7 kg, 2 kg, 2 kg,1 kg
4 kg
Instrument Weight of instrument (kg)
Number of each instrument ordered
Guitars 15 2
Trumpets 2 3
Saxophones 7 2
Violins 1 1
Trombones 2 3
Flutes 1 2
Bassoons 4 4
Tubas 18 1
Cymbals 1 2
Boxes (20 kg)15 kg, 2 kg, 2 kg, 1 kg
15 kg, 2 kg, 2 kg, 1 kg
7 kg, 7 kg, 2 kg, 2 kg,1 kg
4 kg, 4 kg
Instrument Weight of instrument (kg)
Number of each instrument ordered
Guitars 15 2
Trumpets 2 3
Saxophones 7 2
Violins 1 1
Trombones 2 3
Flutes 1 2
Bassoons 4 4
Tubas 18 1
Cymbals 1 2
Boxes (20 kg)15 kg, 2 kg, 2 kg, 1 kg
15 kg, 2 kg, 2 kg, 1 kg
7 kg, 7 kg, 2 kg, 2 kg,1 kg
4 kg, 4 kg, 4 kg
Instrument Weight of instrument (kg)
Number of each instrument ordered
Guitars 15 2
Trumpets 2 3
Saxophones 7 2
Violins 1 1
Trombones 2 3
Flutes 1 2
Bassoons 4 4
Tubas 18 1
Cymbals 1 2
Boxes (20 kg)15 kg, 2 kg, 2 kg, 1 kg
15 kg, 2 kg, 2 kg, 1 kg
7 kg, 7 kg, 2 kg, 2 kg,1 kg
4 kg, 4 kg, 4 kg, 4 kg
Instrument Weight of instrument (kg)
Number of each instrument ordered
Guitars 15 2
Trumpets 2 3
Saxophones 7 2
Violins 1 1
Trombones 2 3
Flutes 1 2
Bassoons 4 4
Tubas 18 1
Cymbals 1 2
Boxes (20 kg)15 kg, 2 kg, 2 kg, 1 kg
15 kg, 2 kg, 2 kg, 1 kg
7 kg, 7 kg, 2 kg, 2 kg,1 kg
4 kg, 4 kg, 4 kg, 4 kg
Instrument Weight of instrument (kg)
Number of each instrument ordered
Guitars 15 2
Trumpets 2 3
Saxophones 7 2
Violins 1 1
Trombones 2 3
Flutes 1 2
Bassoons 4 4
Tubas 18 1
Cymbals 1 2
Boxes (20 kg)15 kg, 2 kg, 2 kg, 1 kg
15 kg, 2 kg, 2 kg, 1 kg
7 kg, 7 kg, 2 kg, 2 kg,1 kg, 18 kg
4 kg, 4 kg, 4 kg, 4 kg
Instrument Weight of instrument (kg)
Number of each instrument ordered
Guitars 15 2
Trumpets 2 3
Saxophones 7 2
Violins 1 1
Trombones 2 3
Flutes 1 2
Bassoons 4 4
Tubas 18 1
Cymbals 1 2
Boxes (20 kg)15 kg, 2 kg, 2 kg, 1 kg
15 kg, 2 kg, 2 kg, 1 kg
7 kg, 7 kg, 2 kg, 2 kg,1 kg
4 kg, 4 kg, 4 kg, 4 kg,18 kg
Instrument Weight of instrument (kg)
Number of each instrument ordered
Guitars 15 2
Trumpets 2 3
Saxophones 7 2
Violins 1 1
Trombones 2 3
Flutes 1 2
Bassoons 4 4
Tubas 18 1
Cymbals 1 2
Boxes (20 kg)15 kg, 2 kg, 2 kg, 1 kg
15 kg, 2 kg, 2 kg, 1 kg
7 kg, 7 kg, 2 kg, 2 kg,1 kg
4 kg, 4 kg, 4 kg, 4 kg
18 kg
Instrument Weight of instrument (kg)
Number of each instrument ordered
Guitars 15 2
Trumpets 2 3
Saxophones 7 2
Violins 1 1
Trombones 2 3
Flutes 1 2
Bassoons 4 4
Tubas 18 1
Cymbals 1 2
Boxes (20 kg)15 kg, 2 kg, 2 kg, 1 kg
15 kg, 2 kg, 2 kg, 1 kg
7 kg, 7 kg, 2 kg, 2 kg,1 kg, 1 kg
4 kg, 4 kg, 4 kg, 4 kg
18 kg
Instrument Weight of instrument (kg)
Number of each instrument ordered
Guitars 15 2
Trumpets 2 3
Saxophones 7 2
Violins 1 1
Trombones 2 3
Flutes 1 2
Bassoons 4 4
Tubas 18 1
Cymbals 1 2
Boxes (20 kg)15 kg, 2 kg, 2 kg, 1 kg
15 kg, 2 kg, 2 kg, 1 kg
7 kg, 7 kg, 2 kg, 2 kg,1 kg, 1 kg
4 kg, 4 kg, 4 kg, 4 kg
18 kg
Instrument Weight of instrument (kg)
Number of each instrument ordered
Guitars 15 2
Trumpets 2 3
Saxophones 7 2
Violins 1 1
Trombones 2 3
Flutes 1 2
Bassoons 4 4
Tubas 18 1
Cymbals 1 2
Boxes (20 kg)15 kg, 2 kg, 2 kg, 1 kg
15 kg, 2 kg, 2 kg, 1 kg
7 kg, 7 kg, 2 kg, 2 kg,1 kg, 1 kg
4 kg, 4 kg, 4 kg, 4 kg,1 kg
18 kg
Instrument Weight of instrument (kg)
Number of each instrument ordered
Guitars 15 2
Trumpets 2 3
Saxophones 7 2
Violins 1 1
Trombones 2 3
Flutes 1 2
Bassoons 4 4
Tubas 18 1
Cymbals 1 2
Boxes (20 kg)15 kg, 2 kg, 2 kg, 1 kg
15 kg, 2 kg, 2 kg, 1 kg
7 kg, 7 kg, 2 kg, 2 kg,1 kg, 1 kg
4 kg, 4 kg, 4 kg, 4 kg,1 kg
18 kg
Instrument Weight of instrument (kg)
Number of each instrument ordered
Guitars 15 2
Trumpets 2 3
Saxophones 7 2
Violins 1 1
Trombones 2 3
Flutes 1 2
Bassoons 4 4
Tubas 18 1
Cymbals 1 2
5 boxes are
needed.
![Approximation schemes Bin packing problem. Bin Packing problem Given n items with sizes a 1,…,a n (0,1]. Find a packing in unit-sized bins that minimizes](https://img.pdfslide.us/doc/110x75/56649e7a5503460f94b7a5ef/approximation-schemes-bin-packing-problem-bin-packing-problem-given-n-items.jpg)
![arXiv:1708.05930v1 [cs.AI] 20 Aug 2017 · 2017-08-22 · bin packing problem with conflicts ( [Khanafer et al., 2010], [Gendreauet al., 2004]) and bin packing problem with frag-ile](https://img.pdfslide.us/doc/110x75/5f01f9717e708231d401f427/arxiv170805930v1-csai-20-aug-2017-2017-08-22-bin-packing-problem-with-coniicts.jpg)
