Lecture 10. The Bin Packing Problem
The one dimensional bin packing problem is defined as follows. Given a set 𝐿 =
{1, … , 𝑛} of items and theirs weights 𝑤𝑖 (0,1), 𝑖𝐿. We wish to partition the
set 𝐿 into minimal number 𝑚 of subsets 𝐵1 , 𝐵2 , … , 𝐵𝑚 in such a way that
∑ 𝑤𝑖 ≤ 1, 1 ≤ 𝑗 ≤ 𝑚.
𝑖ϵ𝐵𝑗
The sets 𝐵𝑗 we will call bins.
In other words, we wish to pack all items in a minimal number of bins.
It is NP-hard problem in the strong sense.
Lecture 10
1
Mathematical Model
Decision variables:
1 if bin 𝑗 is used
𝑦𝑗 = {
;
0 otherwise
𝑥𝑖𝑗 = {
𝑛
1 if item 𝑖 is in bin 𝑗
0 otherwise
min ∑ 𝑦𝑗
𝑗=1
𝑛
s.t.
∑ 𝑤𝑖 𝑥𝑖𝑗 ≤ 𝑦𝑗 , 𝑗 = 1, … , 𝑛;
𝑖=1
𝑛
∑ 𝑥𝑖𝑗 = 1, 𝑖 = 1, … , 𝑛;
𝑗=1
𝑦𝑖 , 𝑥𝑖𝑗 ϵ{0,1}, 𝑖, 𝑗 = 1, … , 𝑛.
Can we find the optimal solution for the linear programming relaxation
in polynomial time?
Lecture 10
2
Bad News
 There is much symmetry in the model.
 The problem is hard to approximate.
3
Theorem 1. The existence of a polynomial time ( − 𝜀)-approximation algorithm
2
for any positive  implies P = NP.
Proof. Let us consider the following NP-complete problem. Given 𝑛 positive
numbers 𝑎1 , … , 𝑎𝑛 . Is it possible to partition this set into two subsets 𝐴1 , 𝐴2 in
such a way that ∑𝑖∈𝐴1 𝑎𝑖 = ∑𝑖∈𝐴2 𝑎𝑖 ?
1
𝑎𝑖
2
𝑐
We put 𝐶 = ∑𝑛
𝑖=1 𝑎𝑖 , 𝑤𝑖 =
3
, 𝑖 = 1, … , 𝑛, and apply our (2 − 𝜀)-
approximation algorithm. If we get 2 bins, then answer is Yes, otherwise No.
It is exact answer! 
Lecture 10
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Strong Heuristic (FFD)
Rank the items by the weights:
𝑤1  𝑤2  …  𝑤𝑛
and apply the First Fit strategy:
 put the first item in the first bin;
 at the step 𝑘, we try to put item 𝑘 into the used bins and, if it is not possible,
we put item 𝑘 into a new bin.
Theorem 2. 𝐹𝐹𝐷(𝐿)  11
𝑂𝑃𝑇(𝐿) + 4 for all 𝐿 and there exist some instances
9
for the bin packing problem with
𝐹𝐹𝐷(𝐿)  11
𝑂𝑃𝑇(𝐿).
9
Lecture 10
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1
Hard Example
𝐿 = {1, … , 30 𝑚}
2
1
𝑤𝑖 =
4
1
4
1
+ 𝜀,
1 ≤ 𝑖 ≤ 6𝑚
+ 2𝜀, 6𝑚 < 𝑖 ≤ 12𝑚
+ 𝜀, 12𝑚 < 𝑖 ≤ 18𝑚
{ 4 − 2𝜀, 18𝑚 < 𝑖 ≤ 30𝑚
¼ – 2
¼ – 2
¼ + 
¼ – 2
½ +
¼ + 2
¼ + 2
¼ + 
½ +
¼ + 
¼ + 2
6𝑚
3𝑚
𝑂𝑃𝑇(𝐿) = 9𝑚
Lecture 10
¼ + 
6m
¼ – 2
¼ – 2
¼ – 2
¼ – 2
2m
3m
FFD(L) = 11m
5
Huge Reformulation
Given 𝐿 = {1, … , 𝑛} is the set of items;
𝑤𝑖 > 0 is the weight of item 𝑖;
𝑛𝑖 > 0, integer, is the number of identical items 𝑖
𝑎𝑖𝑗 is the number of identical items 𝑖 in packing pattern 𝑗.
Find
a partition of all items into a minimal number of bins.
Variables:
𝑥𝑗 ≥ 0, integer, is the number of bins for the pattern 𝑗
min ∑ 𝑥𝑗
𝑗∈𝐽
s.t.
∑ 𝑎𝑖𝑗 𝑥𝑗 ≥ 𝑛𝑖 , 𝑖 ∈ 𝐿;
𝑗∈𝐽
𝑥𝑗 ≥ 0, integer, 𝑗 ∈ 𝐽.
𝐽 is the set of all possible patterns.
Lecture 10
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LP-Based Heuristic
Solve the linear programming relaxation
min ∑ 𝑥𝑗
𝑗∈𝐽
s.t.
∑ 𝑎𝑖𝑗 𝑥𝑗 ≥ 𝑛𝑖 , 𝑖 ∈ 𝐿;
𝑗∈𝐽
𝑥𝑗 ≥ 0, 𝑗 ∈ 𝐽.
Put 𝑥𝑗 = ⌈𝑥𝑗∗ ⌉, 𝑗 ∈ 𝐽. It is a feasible solution with deviation from the optimum
at most
𝜀=
∑𝑗∈𝐽(⌈𝑥𝑗∗ ⌉ − 𝑥𝑗∗ )
∑𝑗∈𝐽 𝑥𝑗∗
where 𝑥𝑗∗ is the optimal solution for the LP model.
Can we solve LP?
Lecture 10
7
The Column Generation Method
Let us consider a subset 𝐽 𝐽 of patterns and assume that the following subproblem
min ∑ 𝑥𝑗
𝑗∈𝐽′
s.t.
∑ 𝑎𝑖𝑗 𝑥𝑗 ≥ 𝑛𝑖 , 𝑖 ∈ 𝐿;
𝑗∈𝐽′
𝑥𝑗  0,
𝑗𝐽′ ;
has at least one feasible solution.
Denote by 𝑥𝑗∗ the optimal solution to this subproblem.
Lecture 10
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The Dual Problem
max ∑ 𝑛𝑖 𝜆𝑖
𝑖∈𝐿
∑ 𝑎𝑖𝑗 𝜆𝑖 ≤ 1, 𝑗 ∈ 𝐽′ ;
𝑖∈𝐿
𝜆𝑖 ≥ 0, 𝑖 ∈ 𝐿.
Denote by 𝜆∗𝑖 ≥ 0 its optimal solution. If
∑ 𝑎𝑖𝑗 𝜆∗𝑖 ≤ 1, for 𝑗 ∈ 𝐽 \ 𝐽′ ;
()
𝑖∈𝐿
then
𝑥𝑗∗ ,
𝑗 ∈ 𝐽′;
𝑥̅𝑗 = {
0, 𝑗 ∈ 𝐽 \ 𝐽′
is the optimal solution for the LP problem.
Lecture 10
9
How to Check ()?
Let us consider the following knapsack problem:
𝛼 = max ∑ 𝜆∗𝑖 𝑦𝑖
𝑖∈𝐿
s.t.
∑ 𝑤𝑖 𝑦𝑖 ≤ 1 ;
(capacity of bin)
𝑖∈𝐿
𝑦𝑖 ≥ 0, integer, 𝑖 ∈ 𝐿.
If 𝛼 ≤ 1 then () is satisfied.
If 𝛼 > 1 then we have got a new pattern and include it in 𝐽′ .
Lecture 10
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The Framework of the Method
1. Select an initial subset 𝐽′ ⊂ 𝐽.
2. Solve the subproblem for 𝐽′ and its dual one, get 𝑥𝑗∗ , 𝜆∗𝑖 .
3. Solve the knapsack problem for 𝜆∗ and compute 𝛼.
4. If 𝛼 ≤ 1 then STOP.
5. Include new pattern 𝑗0 : 𝑎𝑖𝑗0 = 𝑦𝑖∗ , 𝑖 ∈ 𝐿, into subset 𝐽′ and goto 2.
Surprise: As a rule, solution 𝑥𝑗 = ⌈𝑥𝑗∗ ⌉, 𝑗 ∈ 𝐽 is optimal for the bin packing
problem. If it is not true, we have at most one additional bin only!
Lecture 10
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Two-Dimensional Packing Problem
Given: 𝑛 rectangles with size 𝑤𝑖  𝑙𝑖 , 𝑖 = 1, … , 𝑛.
Find: a packing of the rectangles into a rectangle area with minimal square.
Rotations are forbidden
𝑦
𝐿  𝑀  min
6
2
𝑊
1
It is guillotine solution.
5
3
4
𝐿
Lecture 10
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The Strip Packing Problem
Дано: 𝑛 rectangles with size 𝑤𝑖 𝑙𝑖 , 𝑖 ∈ 𝐿, and large strip with width 𝑊.
Find: a packing of rectangles into the strip with minimal length.
𝑦
2
4
𝑊
1
For 𝑙𝑖 = 1 we have
one-dimentional
bin packing problem
(NP-hard)
5
3
𝑥
Hometask. Design a linear integer programming model for the strip packing
problem (with and without 90° rotations).
Lecture 10
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The Two-Dimensional Knapsack Problem
Given: 𝑛 rectangles with size 𝑤𝑖  𝑙𝑖 , profit 𝑐𝑖 for each rectangle, and the size
of a vehicle 𝑊  𝐿.
Find: a subset of rectangles with maximal total profit which can be packed into
the vehicle.
5
For 𝑙𝑖 = 𝐿, we have
6
4
the classical knapsack
problem
9
7
𝑊
10
1
2
3
8
𝐿
Hometask. Design a linear integer programming model for the two-dimensional
knapsack problem.
Lecture 10
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The Two-Dimensional Bin Packing Problem
Given: 𝑛 rectangles with size 𝑤𝑖  𝑙𝑖 and the size of a vehicle 𝑊  𝐿.
Find: a packing all rectangles into the minimal number of vehicles.
4
6
7
3
1
𝑛– 1
5
𝑛
2
...
Hometask. Design LP-based heuristic for the two-dimensional bin packing problem.
Lecture 10
15