Strip packing problem

Given set ofN rectangles, the strip packing problem consists on placing them on a strip while minimizing the its length. The rectangles cannot overlap or be rotated. The height and length of each rectangle is known (Hi, Li;i∈N), and the strip has widthW29. Figure

A.1illustrates the strip packing problem.

The GDP formulation of this problem is as follows29,42: min lt s.t. lt≥xi+Li i∈N " Z_ij1 xi+Li ≤xj # ∨ " Z_ji1 xj+Lj ≤xi # ∨ " Z_ij2 yi−Hi ≥yj # ∨ " Z_ji2 yj −Hj ≥yi # i, j ∈N, i < j Z_ij1 _YZ_ji1 _YZ_ij2 _YZ_ji2 i, j ∈N, i < j 0≤xi ≤U B−Li i∈N Hi ≤yi ≤W i∈N Z_ij1, Z_ij2 ∈ {T rue, F alse} i, j ∈N, i6=j (5.6) 5.6. RESULTS 100

Figure 5.6: Illustration of the strip packing problem.

In (5.6), the objective is to minimize the length lt. The coordinates of the upper-left corner of rectangle i are represented with the variables (xi, yi). The global constraints

(lt ≥ xi + Li) enforce length of the strip corresponds to the largest xi + Li (i.e. the

coordinate of the top-left corner plus the length of the rectangle). There is a disjunction for every pair of rectanglesi, j ∈N, i < j. Each of the terms of the disjunction represents the relative position of rectangleiwith respect to rectanglej. The first term, corresponding toZ1

ij = T rue, represents rectanglei to the left of rectangle j. Zji1 = T rue represents

rectangleito the right of rectanglej. Z2

ij =T ruerepresents rectangleion top of rectangle

j. Finally,Z2

ji = T ruerepresents rectangleibelow rectanglej. The parameterU B is an

upper bound for the strip (e.g.U B =P

iLi).

The different algorithms were tested on 100 random instances of the strip packing problem. The range of values of the random parameters is as follows: N = 4,5;W = 5-7;Li = 1-

10;Hi = 2-5.

Figure5.7a shows that the HR disjunctive branch and bound performs better than the other three for this problem. The BM, random heuristic, and the proposed branch and bound perform similarly in terms of solution time. It is interesting to note that the proposed al- gorithm performs the third in terms of solution time. However, Figure5.7b shows that the

CHAPTER 5. LAGRANGEAN RELAXATION OF THE HR OF LINEAR GDP PROBLEMS AND ITS USE IN THE DISJUNCTIVE BRANCH AND BOUND

Figure 5.7: Performance curves for the different algorithm for: a) solution time, and b) number of nodes.

proposed algorithm requires the fewest number of nodes. Because some of the nodes re- quire the evaluation of more than one LP, the fewer number of nodes is not reflected in the solution time for this problem. As expected, the random heuristic and the HR disjunctive branch and bound require fewer nodes than the BM disjunctive branch and bound. The random heuristic algorithm requires similar number of nodes as the HR. However, it re- quires the evaluation of two LPs in many of the nodes, so the performance in terms of the solution time is worse (as shown in Figure5.7a). The heuristic that uses the Lagrangean relaxation requires fewer nodes than the random heuristic, which shows that the former is a better heuristic than randomly fixing variables. However, the improvement is small in this example, and hence the time spent in solving the Lagrangean relaxation at every node increases the total solution time. Note that for this problem the performance curves of all algorithms are not very different.

Figure5.8shows the performance curve of the number of nodes required to find the opti- mal solution and to find a feasible solution. Figure5.8a shows that the number of nodes required to find an optimal solution is smaller for the proposed algorithm than for the oth- ers. Figure5.8b shows that the first feasible solution can be almost always be found at the root node in the proposed and random algorithms. In particular, the proposed algorithm finds the first feasible solution at the root node in 94 instances; and in 4 instances it finds it at the first node. The improvement in finding optimal and feasible solutions can be better

Figure 5.8: Number of nodes required to find: a) the optimal solution, and b) the first feasible solution.

Table 5.2: Average of performance metrics for the different algorithms

Average of metric BM HR LAG RAN

Solution time (s) 172.6 93.0 167.4 143.9

Total number of nodes 1,546 859 763 843

Number of nodes to find optimal 1,258 447 229 364

Number of nodes to find feasible 493 214 2 66.5

appreciated in Table5.2. Table5.2shows an average of the metrics presented in Figures

5.7and5.8. Note that both algorithms with primal heuristic find the optimal and feasible solutions faster than the BM and HR disjunctive branch and bound. Out of the two algo- rithms with primal heuristic, the one that uses the Lagrangean relaxation finds the optimal and first feasible solution in fewer nodes.