Lecture 10. The Bin Packing Problem

The one dimensional bin packing problem is defined as follows. Given a set

1, … , of items and theirs weights ∈0,1, ∈. We wish to partition the set into minimal number of subsets ,,… , in such a way that

1, 1 .

The sets 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 is used 1 if item is in bin ; 0 otherwise 0 otherwise

min

s.t. ,1,…,;

1, 1, … , ;

,0,1, , 1, … , . 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.

Theorem 1. The existence of a polynomial time ‐approximation algorithm for any positive ε implies P = NP.

Proof. Let us consider the following NP‐complete problem. Given positive numbers ,…,. Is it possible to partition this set into two subsets , in such a way that We put and apply our ‐ approximation algorithm. If we get 2 bins, then answer is Yes, otherwise No. It is exact answer! „

Lecture 10 3 Strong Heuristic (FFD)

Rank the items by the weights:

≥ ≥ … ≥ and apply the First Fit strategy: − put the first item in the first bin; − at the step , we try to put item into the used bins and, if it is not possible, we put item into a new bin. ≤ 11 4 Theorem 2. 9 for all and there exist some instances for the bin packing problem with

≥ 11 . 9

Lecture 10 4 , 1 6 Hard Example 2, 6 12 1, … , 30 , 12 18 2, 18 30

¼ – 2ε ¼ – 2ε ¼ ε ¼ – 2ε ¼ 2ε ¼ ε ¼ –2ε ¼ 2ε ¼ ε ½ ε ½ ε ¼ –2ε

¼ –2ε ¼ 2ε ¼ ε

¼ –2ε 6 3 6m 2m 3m 9 FFD(L) = 11m

Lecture 10 5 Huge Reformulation

Given 1,…, is the set of items;

0 is the weight of item ;

0, integer, is the number of identical items

is the number of identical items in packing pattern . Find a partition of all items into a minimal number of bins.

Variables: 0, integer, is the number of bins for the pattern

min s.t. ,;

0, integer, . is the set of all possible patterns.

Lecture 10 6 LP‐Based Heuristic

Solve the linear programming relaxation

min s.t. ,;

0, . Put , . It is a feasible solution with deviation from the optimum at most ∑ ∑ where is the optimal solution for the LP model. Can we solve LP?

Lecture 10 7 The Column Generation Method

Let us consider a subset ′⊂ of patterns and assume that the following subproblem

min s.t.

, ; ≥ 0, ∈ ; has at least one feasible solution. Denote by the optimal solution to this subproblem.

Lecture 10 8 The Dual Problem

max

1, ;

0, . Denote by 0 its optimal solution. If

1, for \ ; (∗) then , ; 0, \ is the optimal solution for the LP problem.

Lecture 10 9 How to Check (∗)?

Let us consider the following knapsack problem:

max s.t. (capacity of bin) 1;

0, integer, .

If 1 then (∗) is satisfied.

If 1 then we have got a new pattern and include it in .

Lecture 10 10 The Framework of the Method

1. Select an initial subset . 2. Solve the subproblem for and its dual one, get , . 3. Solve the knapsack problem for and compute . 4. If 1 then STOP. 5. Include new pattern : ,, into subset and goto 2.

Surprise: As a rule, solution , is optimal for the bin packing problem. If it is not true, we have at most one additional bin only!

Lecture 10 11 Two‐Dimensional Packing Problem

Given: rectangles with size × , 1, … , . Find: a packing of the rectangles into a rectangle area with minimal square.

Rotations are forbidden × → min 6

2 5 It is guillotine solution.

1 3 4

Lecture 10 12 The Strip Packing Problem

Дано: rectangles with size ×,, and large strip with width . Find: a packing of rectangles into the strip with minimal length.

For 1 we have 2 5 one‐dimentional 4 bin packing problem

1 (NP‐hard) 3

Hometask. Design a linear integer programming model for the strip packing problem (with and without 90° rotations).

Lecture 10 13 The Two‐Dimensional Knapsack Problem

Given: rectangles with size × , profit for each rectangle, and the size of a vehicle × . Find: a subset of rectangles with maximal total profit which can be packed into the vehicle.

For , we have 5 6 4 the classical knapsack 9 7 problem 10

1 3 2 8

Hometask. Design a linear integer programming model for the two‐dimensional knapsack problem.

Lecture 10 14 The Two‐Dimensional Bin Packing Problem

Given: rectangles with size × and the size of a vehicle × . Find: a packing all rectangles into the minimal number of vehicles.

4 6 7

3

– 1 5 1 2

. . .

Hometask. Design LP‐based heuristic for the two‐dimensional bin packing problem.

Lecture 10 15