Column Generation Technique for Solving Two-Dimensional Cutting Stock Problems: Method of Stripe Approach

Abstract

Abstract. We consider two-dimensional cutting stock problems where single rectangular stocks have to be cut into some smaller rectangular so that the number of stocks needed to satisfy the demands is minimum. In this paper we focus our study to the problem where the stocks have to be cut with guillotine cutting type and fixed orientation of finals. We formulate the problem as an integer programming, where the relaxation problem is solved by column generation technique. New pattern generation is formulated based on method of stripe. In obtaining the integer solution, we round down the optimal solution of the relaxation problem and then we derive an extra mix integer programming for satisfying the unmet demands. The optimal solution of the original problem is the combination of the round-down solution and the optimal solution of the extra mix integer programming.A numerical example of the problem is given in the end of this paper.

Penyelesaian Masalah Cutting Stock Dengan Pengelasan Menggunakan Teknik Column Generation

“…Jika reduced cost bernilai negative, kolom baru y akan masuk sebagai basis. Kolom baru akan terus dibangkitkan selama kondisi optimal model master belum tercapai [4].…”

“…Masalah cutting stock dengan pengelasan yang sudah diselesaikan dengan teknik column generation memperoleh solusi optimal berupa bilangan yang tidak bulat. Untuk memperoleh solusi bilangan bulat diperlukan model tambahan, yaitu model Mixed Integer Programming (MIP) [4]. Misalkan 𝐹 himpunan final yang belum terpenuhi.…”

“…Jika solusi pembulatan diperoleh dari pembulatan solusi program linear, maka dapat mengakibatkan solusi hasil pembulatan bukan merupakan solusi fisibel. Sejumlah peneliti telah melakukan penelitan agar dapat memberikan solusi berupa bilangan bulat [4,5,6].…”

“…Masalah cutting stock dengan pengelasan tersebut diselesaikan menggunakan teknik column generation. Model integer programming dilibatkan untuk memenuhi kekurangannya sehingga diperoleh solusi bilangan bulat [4].…”

An introduction to the two‐dimensional rectangular cutting and packing problem

“…A CG‐based algorithm for the SLOPP with guillotine cuts using mixed integer programming to obtain an integer solution was proposed by Novianingsih et al. (2012). The repeated constrained column generation is a CG‐based algorithm for the two‐staged SSSCSP proposed by Cui and Zhao (2013) that solves the sub‐problems as unconstrained ones, then, repeatedly, the residual problems are also solved through CG solving the (constrained) sub‐problem until all demand is satisfied.…”