An Opt + 1 algorithm for the cutting stock problem with a constant number of object lengths

In the cutting stock problem we are given a set $T=T_1,T_2, \ldots, T_d$ of object types, where objects of type $T_i$ have integer length $p_i > 0$. Given a set $\mathcal{O}$ of $n$ objects containing $n_i$ objects of type $T_i$, for each $i = 1, \ldots, d$, the problem is to pack $\mathcal{O}$ into the minimum number of bins of capacity $\beta$. In this paper we consider the version of the problem in which the number $d$ of different object types is constant and we present an algorithm that computes a solution using at most $OPT+1$ bins, where $OPT$ is the value of an optimum solution

High Multiplicity Strip Packing

In the two-dimensional high multiplicity strip packing problem (HMSPP), we are given k distinct rectangle types, where each rectangle type Ti has ni rectangles each with width 0 \u3c wi and height 0 \u3c hi The goal is to pack these rectangles into a strip of width 1, without rotating or overlapping the rectangles, such that the total height of the packing is minimized. Let OPT(I) be the optimal height of HMSPP on input I. In this thesis, we consider HMSPP for the case when k = 3 and present an OPT(I) + 5/3 polynomial time approximation algorithm for it. Additionally, we consider HMSPP for the case when k = 4 and present an OPT(I) + 5/2 polynomial time approximation algorithm for it

High Multiplicity Strip Packing Problem With Three Rectangle Types

The two-dimensional strip packing problem (2D-SPP) involves packing a set R = {r1, ..., rn} of n rectangular items into a strip of width 1 and unbounded height, where each rectangular item ri has width 0 \u3c wi ≤ 1 and height 0 \u3c hi ≤ 1. The objective is to find a packing for all these items, without overlaps or rotations, that minimizes the total height of the strip used. 2D-SPP is strongly NP-hard and has practical applications including stock cutting, scheduling, and reducing peak power demand in smart-grids. This thesis considers a special case of 2D-SPP in which the set of rectangular items R has three distinct rectangle sizes or types. We present a new OPT + 5/3 polynomial-time approximation algorithm, where OPT is the value of an optimum solution. This algorithm is an improvement over the previously best OPT + 2 polynomial-time approximation algorithm for the problem