no. 3
Sufficient condition for partial efficiency in a bicriteria nonlinear cutting stock problem
Arana-Jiménez, Manuel1 ; Salles Neto, L. L.2
1 Department of Statistics and Operational Research, Faculty of SSCC and Communication, University of Cádiz, Av. de la Universidad, Jerez 11406, Spain.
2 Departamento de Ciência e Tecnologia, Universidade Federal de São Paulo, Brazil.
RAIRO - Operations Research - Recherche Opérationnelle, Tome 51 (2017) no. 3, pp. 709-717

This work presents a sufficient criteria for partial efficient solutions of the cutting stock problem with two objectives. We consider two important objectives for an industry: number of processed objects (cost of raw materials) and number of different patterns (cost of setup). These optimality results are established through a new approach based on connections between discrete optimization and continuous vector optimization.

DOI : 10.1051/ro/2016058
Classification : 90C26, 90C29, 90C46
Keywords: Multiple objective programming, optimality conditions, continuous optimization, cutting stock problem

Arana-Jiménez, Manuel 1 ; Salles Neto, L. L. 2

1 Department of Statistics and Operational Research, Faculty of SSCC and Communication, University of Cádiz, Av. de la Universidad, Jerez 11406, Spain.
2 Departamento de Ciência e Tecnologia, Universidade Federal de São Paulo, Brazil. 
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     title = {Sufficient condition for partial efficiency in a bicriteria nonlinear cutting stock problem},
     journal = {RAIRO - Operations Research - Recherche Op\'erationnelle},
     pages = {709--717},
     year = {2017},
     publisher = {EDP Sciences},
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     number = {3},
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Arana-Jiménez, Manuel; Salles Neto, L. L. Sufficient condition for partial efficiency in a bicriteria nonlinear cutting stock problem. RAIRO - Operations Research - Recherche Opérationnelle, Tome 51 (2017) no. 3, pp. 709-717. doi: 10.1051/ro/2016058

M. Arana, A. Rufián, R. Osuna and G. Ruiz, Pseudoinvexity, optimality conditions and efficiency in multiobjective problems; duality. Nonlinear Anal. 68 (2008) 24–34. | MR | Zbl | DOI

A. Chinchuluun and P. Pardalos, A survey of recent developments in multiobjective optimization. Ann. Oper. Res. 154 (2007) 29-50. | MR | Zbl | DOI

A. Diegel, Cutting paper in Richards Bay: dynamic local and global optimization in the trim problem. Orion 3 (1988) 42–55.

A. Diegel, E. Montocchio, E. Walters, S. Schalkwyk and S. Naidoo, Setup minimising conditions in the trim loss problem. Eur. J. Oper. Res. 95 (1996) 631–640. | Zbl | DOI

M. Ehrgott and X. Gandibleux, A survey and annotated bibliography of multiobjective combinatorial optimization. OR Spektrum 22 (2000) 425–460. | MR | Zbl | DOI

S. Eilon, Optimizing the shearing of steel bars. J. Mech. Eng. Sci. 2 (1960) 129–142. | DOI

R. Golfeto, A. Moretti and L. Salles Neto, A genetic symbiotic algorithm applied to the cutting stock problem with multiple objectives. Adv. Model. Optim. 11 (2009) 473–501. | MR

R. Haessler, Controlling Cutting Pattern Changes in One-Dimensional Trim Problems. Oper. Res. 23 (1975) 483–493. | Zbl | DOI

C. Hardley, Optimal cutting of zinc-coated steel strip. Oper. Res. 4 (1976) 92–100.

J. Lee, In situ column generation for a cutting-stock problem. Comput. Oper. Res. 34 (2007) 2345–2358. | Zbl | DOI

L. Liu, X. Liu, J. Pei, W. Fan and P.M. Pardalos, A study on decision making of cutting stock with frustum of cone bars. Oper. Res. (2015) 1–18.

O.L. Mangasarian, Nonlinear Programming, McGraw Hill Book Company, New York (1969). | MR | Zbl

B.S. Mordukhovich, Variational analysis and generalized differentiation (I and II). Springer-Verlag Berlin Heidelberg, The Netherlands (2006). | MR | Zbl

R. Osuna, A. Beato and A. Rufián, Generalized Convexity in Multiobjective Programming. J. Math. Anal. Appl. 233 (1999) 205–220. | MR | Zbl | DOI

V. Pareto, Course d’economie politique. Rouge, Lausanne (1896).

S.D. Prestwich, A.O. Fajemisin, L. Climent and B. OSullivan, Solving a Hard Cutting Stock Problem by Machine Learning and Optimisation. European Conference on Machine Learning and Knowledge Discovery in Databases. IEEE (2015) 335–347.

L.L. Salles, A. Rufián, M. Arana and G.Ruiz, Weak efficiency in cutting stock problem. Anais do CNMAC 3 (2010) 187–193.

C. Sousa, E. Silva, M. Lopes and A. Ramos, The Cutting Stock Problem: A Case Study in a Manufacturer of Pet Vivaria. Oper. Res. Big Data 15 (2015) 221–228. | DOI

H. Stadler, A one-dimensional cutting stock problem in the aluminum industry and its solution. Eur. J. Oper. Res. 44 (1990) 209–223. | Zbl | DOI

S. Umetani, M. Yagiura and T. Ibaraki, One Dimensional Cutting Stock Problem with a Given Number of Setups: A Hybrid Approach of Metaheuristics and Linear Programming. J. Math. Model. Algorithms 5 (2006) 43–64. | MR | Zbl | DOI

G. Wascher, P. Carow and H. Muller, Entwicklung eines flexiben Verfahrens für Zuschneideprobleme in einem Kaltwalzwerk. Z. Oper. Res. 29 (1985) 209–230.

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