Pattern-based ILP models for the one-dimensional cutting stock problem with setup cost (original) (raw)
Related papers
2010
Problem statement: One-dimensional cutting stock problem with discrete demands and capacitated planning objective is an NP hard problem. Approach: The mathematical model with column-generation technique by a branch-and-bound procedure and the heuristic based on the first fit decreasing method are proposed. Then, both approaches were compared and some characteristics were investigated such as upper-bound value, percentage above lower-bound value, computation time, and number of patterns. Results: The 24 instances were examined. The proposed heuristic provides the upper-bound value above the lower-bound around 0-16.78%. All upper-bound values from column-generation and integer programming are better than the proposed heuristic but all computation times are higher. Conclusion: The proposed heuristic has consistently high performance in computation times.
Combinatorial optimization modeling approach for one-dimensional cutting stock problems
modeling approach to one-dimensional cutting stock problem. The investigated problem seeks to determine the optimal length of the blanks and the optimum cutting pattern of each blank to meet the requirement for a given number of elements with different lengths. Blanks of particular type are offered with equal size in large quantities and the goal is to find such optimal length of blanks that leads to minimal overall trim waste. To achieve that goal a combinatorial optimization approach is used for modeling of one-dimensional cutting stock problem. Numerical example of real-life problem is presented to illustrate the applicability of the proposed approach. It is shown that numerical example can be solved for reasonable time by Lingo Solver and MS Excel Solver.
International Transactions in Operational Research, 2014
We address a 1-dimensional cutting stock problem where, in addition to trimloss minimization, cutting patterns must be sequenced so that no more than s different part types are in production at any time. We propose a new integer linear programming formulation whose constraints grow quadratically with the number of distinct part types and whose linear relaxation can be solved by a standard column generation procedure. The formulation allowed us to solve problems with 20 part types for which an optimal solution was unknown.
An integrated cutting stock and sequencing problem
European Journal of Operational Research, 2007
In this paper an integrated problem formulated as an integer linear programming problem is presented to find an optimal solution to the cutting stock problem under particular pattern sequencing constraints. The solution uses a Lagrangian approach. The dual problem is solved using a modified subgradient method. A heuristic for the integrated problem is also presented. The computational results obtained from a set of unidimensional instances that use these procedures are reported.
The Ordered Cutting Stock Problem
Decision Sciences, 2004
The one-dimensional cutting stock problem (CSP) is a classic combinatorial optimization problem in which a number of parts of various lengths must be cut from an inventory of standard-size material. The classic CSP ensures that the total demand for a given part size is met but ignores the fact that parts produced by a given cutting pattern may be destined for different jobs. As a result, applying the classic CSP in a dynamic production environment may result in many jobs being open (or partially complete) at any point in time-requiring significant material handling or sorting operations. This paper identifies and discusses a new type of one-dimensional CSP, called the ordered CSP, which explicitly restricts to one the number of jobs in a production process that can be open, or in process, at any given point in time. Given the growing emphasis on mass customization in the manufacturing industry, this restriction can help lead to a reduction in both in-process inventory levels and material handling activities. A formal mathematical formulation is provided for the new CSP model, and its applicability is discussed with respect to a production problem in the custom door and window manufacturing industry. A genetic algorithm (GA) solution approach is then presented, which incorporates a customized heuristic for reducing scrap levels. Several different production scenarios are considered, and computational results are provided that illustrate the ability of the GA-based approach to significantly decrease the amount of scrap generated in the production process.
Cutting stock with no three parts per pattern: Work-in-process and pattern minimization
Discrete Optimization, 2011
The Pattern Minimization Problem (PMP) consists in finding, among the optimal solutions of a cutting stock problem, one that minimizes the number of distinct cutting patterns activated. The Work-in-process Minimization Problem (WMP) calls for scheduling the patterns so as to maintain as few open stacks as possible. This paper addresses a particular class of problems, where no more than two parts can be cut from any stock item, hence the feasible cutting patterns form the arc set of an undirected graph G. The paper extends the case G = K n introduced in 1999 by McDiarmid. We show that some properties holding for G = K n are no longer valid for the general case; however, for special cases of practical relevance, properly including G = K n , quasi-exact solutions for the PMP and the WMP can be found: the latter in polynomial time, the former via a set-packing formulation providing very good lower bounds.
A two-objective mathematical model without cutting patterns for one-dimensional assortment problems
Journal of Computational and Applied Mathematics, 2011
This paper considers a one-dimensional cutting stock and assortment problem. One of the main difficulties in formulating and solving these kinds of problems is the use of the set of cutting patterns as a parameter set in the mathematical model. Since the total number of cutting patterns to be generated may be very huge, both the generation and the use of such a set lead to computational difficulties in solution process. The purpose of this paper is therefore to develop a mathematical model without the use of cutting patterns as model parameters. We propose a new, two-objective linear integer programming model in the form of simultaneous minimization of two contradicting objectives related to the total trim loss amount and the total number of different lengths of stock rolls to be maintained as inventory, in order to fulfill a given set of cutting orders. The model does not require pre-specification of cutting patterns. We suggest a special heuristic algorithm for solving the presented model. The superiority of both the mathematical model and the solution approach is demonstrated on test problems.