ABSTRACT The Minimization of Open Stacks Problem is a pattern sequencing problem that is based on the premise that the different items obtained from cutting patterns are piled in stacks in the work area until all items of the same size have been cut. Due to space limitations, it is gainful to find a sequence for the patterns that minimizes the number of open stacks. We have developed an integer programming model based on interval graphs that searches for an appropriate edge completion of the given graph of the problem, while defining a suitable coloring of its vertices

Isabel Cristina Lopes

J M Valério De Carvalho

CiteSeerX

 
Semana de Engenharia 2010 
Guimarães, 11 a 15 de Outubro 
 
 
 
THE MINIMIZATION OF OPEN STACKS PROBLEM 
 
Isabel Cristina Lopes*
#
 and J.M. Valério de Carvalho* 
*Department of Production and Systems, University of Minho 
#
ESEIG – Polytechnic Institute of Porto 
E-mail: cristinalopes@eu.ipp.pt 
 
 
KEYWORDS 
MOSP, Integer programming, Interval graphs. 
 
ABSTRACT 
The Minimization of Open Stacks Problem is a pattern 
sequencing problem that is based on the premise that the 
different items obtained from cutting patterns are piled 
in stacks in the work area until all items of the same size 
have been cut. Due to space limitations, it is gainful to 
find a sequence for the patterns that minimizes the 
number of open stacks. We have developed an integer 
programming model based on interval graphs that 
searches for an appropriate edge completion of the given 
graph of the problem, while defining a suitable coloring 
of its vertices. 
 
INTRODUCTION 
The Minimization of Open Stacks Problem (MOSP) is a 
problem that was first addressed by Yuen in 1991. It 
arose in the Australian flat glass industry, but it has 
applications in other cutting industries like steel tubes, 
paper, wooden panels, as well as in production planning 
(reducing the space needed for storage of clients’ open 
orders), and also in other fields such as VLSI Circuit 
Design with the Gate Matrix Layout Problem and PLA 
Folding, and in classical problems from Graph Theory 
such as Pathwidth, Modified Cutwidth and Vertex 
Separation. 
The MOSP is a NP-hard problem (Linhares and 
Yanasse, 2002)  which has been addressed in literature 
by many different methods, such as branch-and-bound 
techniques (Becceneri et al, 2004), heuristics, genetic 
algorithms, dynamic programming (Banda and Stuckey, 
2007) and integer programming (Baptiste, 2005).  
 
MINIMIZATION OF OPEN STACKS PROBLEM 
Consider, for instance, an industry that takes large 
wooden panels and cuts them in smaller panels of 
different sizes to meet clients' demands. A specification 
of how many subpanels of each size will be cut from a 
larger panel and where the cuts will be made defines a 
cutting pattern. Each cutting pattern can produce 
different items or just several copies of one same item. 
Usually a cutting machine processes just one cutting 
pattern at a time. The items already cut that are equal are 
piled in stacks by the machine. The stack of an item type 
remains near the machine if an item of that type will be 
cut from a forthcoming pattern. A stack is closed and 
removed from the work area only after all items of that 
size have been cut, and immediately before starting to 
proccess the next cutting pattern. As there are often 
space limitations around the cutting machines, there is 
danger of damages on the stacked items, difficulty in 
distinguishing similar items, and in some cases there is 
handling costs of removing the stack temporarily to the 
warehouse, it is advantageous to minimize the number 
of open stacks, and that can be done simply by finding 
an optimal sequence to process the cutting patterns. 
 
Most of the authors encounter the minimization of the 
number of open stacks problem (MOSP) while solving a 
two stage procedure: first, the classic problem of finding 
the best patterns to cut stock sheets of glass, paper or 
wood into smaller rectangular pieces is dealt, usually 
with an waste minimization purpose, and only after that 
comes the stage two, which is to determine the sequence 
in which those patterns should be cut, in order to 
minimize the number of open stacks. 
 
MOSP IN A GRAPH 
An instance of the MOSP can be associated with a graph 
having a vertex for each item that is cut and an edge 
between two vertices if the corresponding items are 
present in the same cutting pattern (Yanasse, 1997). A 
pattern with k different items will correspond to a clique 
of size k in the MOSP graph, because the k vertices must 
be connected to each other. 
The MOSP graph constructed in this way does not show 
explicitly the cutting patterns, so any model based on it 
will focus on sequencing the opening of the stacks rather 
than on sequencing the cutting patterns. That is not a 
problem, because it is possible to take a solution for the 
ordering of the vertices of the graph and construct a 
sequence for the corresponding cutting patterns. 
 
Interval Graphs 
There is a special type of graphs called interval graphs 
(Golumbic, 1980), that are undirected graphs such that 
its vertices can be put into a one-to-one correspondence 
with a set of intervals I of a linearly ordered set (like the 
real line) such that two vertices are connected by an 
edge if their corresponding intervals intersect. I is called 
an interval representation for the interval graph. 
 
Semana de Engenharia 2010 
Guimarães, 11 a 15 de Outubro 
 
 
 
By associating each open stack of a MOSP problem to 
an interval in the real line (the interval of time that the 
stack stays open), we can associate a solution of the 
MOSP to an interval representation of an interval graph. 
 
THE MODEL 
We have developed an integer programming formulation 
for the MOSP (Lopes and Carvalho, 2010) based on 
properties of interval graphs. Given an instance of the 
problem, we first build the MOSP graph G=(V,E) as 
explained previously. This model consists in finding out 
which edges should be added to the original MOSP 
graph G=(V,E) in order to get an interval graph 
H=(V,E F) that minimizes the maximum number of 
simultaneously open stacks.  
According to the sequence in which the patterns are 
proccessed, there may be more or less open stacks 
simultaneously. Each arc of the future interval graph 
means that, for a period of time, the two stacks (the 
respective vertices of the arc) will remain both open. 
The initial graph contains only the arcs that must be 
there, in any possible sequence in which the patterns can 
be proccessed. The rest of the arcs that are added later to 
the graph will differ according to the sequence of the 
patterns. It is the choice of these arcs that defines which 
are the other simultaneously open stacks.  
Our model is based on inequalities from a known IP 
model for the linear ordering problem, and new 
inequalities inspired on a characterization of interval 
graphs that uses a linear ordering of the vertices (Olariu, 
1991) that corresponds to an ordering of the maximal 
cliques. The value of the optimum of the MOSP is equal 
to the size of the biggest clique in the solution graph H 
and, because interval graphs are perfect graphs, it is 
equal to the chromatic number of the graph, which is the 
number of colors needed to assign to the vertices of the 
graph such that there are no two adjacent vertices of the 
same color. We also use in the model inequalities 
derived from the representatives formulation for the 
vertex coloring problem (Campêlo et al, 2004) to count 
the number of different colors and find the number of 
simultaneously open stacks. 
 
CONCLUSIONS  
The integer programming model that was developed for 
the MOSP has already shown some promising results; 
furthermore it is being strengthened with symmetry 
breaking inequalities and lower bounds. This 
formulation has also been applied to similar problems in 
pattern sequencing and in graph theory.  
 
ACKNOWLEDGEMENTS 
This work was financially supported by an FCT grant 
SFRH/BD/32151/2006 and an IPP grant. 
 
REFERENCES 
M. G. Banda and P. J. Stuckey. 2007. “Dynamic programming 
to minimize the maximum number of open stacks.” 
Informs Journal On Computing, 19, 607-617. 
P. Baptiste. 2005. “Simple MIP formulations to minimize the 
maximum number of open stacks.” In Constraint 
Modeling Challenge (Edinburgh, Scotland, Jul 31). IJCAI, 
9-13. 
J. C. Becceneri, H. H. Yanasse, and N. Y. Soma. 2004 “A 
method for solving the minimization of the maximum 
number of open stacks problem within a cutting process.” 
Computers & Operations Research, 31, No.14, 2315-
2332. 
M. Campêlo, R. Corrêa, and Y. Frota. 2004. “Cliques, holes 
and  the vertex coloring polytope.” Information 
Processing Letters, 89, No.4, 159-164. 
M. C. Golumbic. 1980. Algorithmic graph theory and perfect 
graphs. Academic Press, New York. 
A. Linhares and H. H. Yanasse. 2002. “Connections between 
cutting-pattern sequencing, VLSI design, and flexible 
machines.” Computers & Operations Research, 29, No.12, 
1759-1772. 
I.C. Lopes and V. Carvalho. 2010. “Using Interval Graphs in 
an Order Processing Optimization Problem”, Lecture 
Notes in Engineering and Computer Science: Proceedings 
of The World Congress on Engineering 2010, WCE 2010, 
(London, U.K., 30 June - 2 July), IAENG, 1722-1728. 
S. Olariu. 1991. “An optimal greedy heuristic to color interval 
graphs.” Information Processing Letters, 37, No.1, 21-25. 
H. H. Yanasse. 1997. “On a pattern sequencing problem to 
minimize the maximum number of open stacks.” 
European Journal of Operational Research, 100, 454-
463. 
B. J. Yuen. 1991 “Heuristics for sequencing cutting patterns.” 
European Journal of Operations Research, 55, No.2 
(Nov), 183-190. 
 
AUTHOR BIOGRAPHIES 
ISABEL CRISTINA LOPES was 
born in Porto, Portugal and went to 
the University of Porto, where she 
obtained her Master degree in 
Mathematics. She is a teacher at 
ESEIG Polytechnic Institute of Porto 
since 2000 and she is finishing her 
PhD in the field of Operations 
Research at the University of Minho. Her e-mail address 
is: cristinalopes@eu.ipp.pt. 
J. M. VALÉRIO DE CARVALHO 
is Professor of Operations Research in 
the University of Minho, and 
researcher of the Systems Enginee-
ring, Optimization and Operations 
Research (SEOOR) Group of the 
Algoritmi Research Center. His main 
research interests include large scale 
integer programming, and applications in cutting and 
packing problems, scheduling and network design. 


THE MINIMIZATION OF OPEN STACKS PROBLEM

http://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=77BF532C07566DEF45E7FAC78150C6BF?doi=10.1.1.1053.9601&rep=rep1&type=pdf

THE MINIMIZATION OF OPEN STACKS PROBLEM

Abstract

Similar works

Full text

Available Versions

CiteSeerX