Contents lists available at ScienceDirect Computers & Industrial Engineering journal homepage: www.elsevier.com/locate/caie Nonlinear optimization problem subjected to fuzzy relational equations defined by Dubois-Prade family of t-norms Amin Ghodousian a, ⁎ , Marjan Naeeimi b , Ali Babalhavaeji b a Faculty of Engineering Science, College of Engineering, University of Tehran, P.O. Box 11365-4563, Tehran, Iran b Department of Engineering Science, College of Engineering, University of Tehran, Tehran, Iran ARTICLE INFO Keywords: Fuzzy relational equations Nonlinear optimization Genetic algorithm ABSTRACT In fuzzy set theory, triangular norms (t-norm for short) and triangular co-norms (t-conorm for short) play a key role by providing generic models for intersection and union operations on fuzzy sets. Various continuous and discontinuous t-norms have been proposed by many authors. Despite variation in the t-norms, most of the well- known continuous t-norms are Archimedean (for example, Frank, Yager, Hamacher, Sugeno-Weber and Schweizer-Sklar family). An interesting family of non-Archimedean continuous t-norms was introduced by Dubois and Prade. This paper is an attemp to study a nonlinear optimization problem whose constraints are formed as a special system of fuzzy relational equations (FRE). In this type of constraint, FREs are defined with max-Dubois- Prade composition. Firstly, we investigate the resolution of the feasible solutions set. Then, some necessary and sufficient conditions are presented to determine the feasibility or infeasibility of the solutions set. Also, some procedures are introduced for simplifying the problem. Since the feasible solutions sets of FREs are non-convex, conventional nonlinear programming methods may not be directly employed to solve the problem. Therefore, in order to overcome this difficulty, a genetic algorithm (GA) is designed based on some theoretical properties of the problem. It is shown that the proposed algorithm preserves the feasibility of new generated solutions. Moreover, a method is presented to generate feasible max-Dubois-Prade FREs as test problems. These test problems are used to evaluate the performance of our algorithm. Finally, the algorithm are compared with some related works. The obtained results confirm the high performance of the proposed algorithm in solving such nonlinear problems. 1. Introduction In this paper, we study the following nonlinear problem in which the constraints are formed as fuzzy relational equations defined with Dubois-Prade t-norm: = ∈ fx Aφx b x min () [0,1] n (1) where = … = … = × m n a I J A {1,2, , }, {1,2, , }, ( ) ij m n, ⩽ ≤ a 0 1 ij ∀ ∈ ∀ ∈ i I and j J ( ), is a fuzzy matrix, = ⩽ ⩽ × b b b ( ) ,0 1 im i 1 ∀ ∈ i I ( ), is an m-dimensional fuzzy vector, and “φ” is the max-Dubois-Prade composition, that is, = = φ xy T xy (,) (,) DP γ xy xyγ max{ , , } in which ∈ λ (0,1). If a i is the i’th row of matrix A, then problem (1) can be expressed as follows: = ∀ ∈ ∈ fx φax b i I x min () ( ,) , [0,1] i i n where the constraints mean: = = = ⎧ ⎨ ⎩ ⎫ ⎬ ⎭ = ∀ ∈ ∈ ∈ ∈ φax φa x T a x ax a xγ b i I ( ,) max { ( , )} max { ( , )} max max{ , ,} , i j J ij j j J DP γ ij j j J ij j ij j i especially, = T xy xy (,) min{ , } DP 0 and = T xy xy (,) DP 1 . The theory of fuzzy relational equations was first proposed by Sanchez (1976). He introduced a FRE with max-min composition and applied the model to medical diagnosis in Brouwerian logic. Nowadays, it is well-known that many issues associated with a body knowledge can be treated as FRE problems (Pedrycz, 2013). In addition to such ap- plications, FRE theory has been applied in many fields including fuzzy control, discrete dynamic systems, prediction of fuzzy systems, fuzzy decision making, fuzzy pattern recognition, fuzzy clustering, image compression and reconstruction, etc. Pedrycz (1983) categorized and extended two ways of the generalizations of FRE in terms of sets under discussion and various operations. Since then, many theoretical im- provements have been investigated and many applications have been https://doi.org/10.1016/j.cie.2018.03.038 Received 13 August 2017; Received in revised form 3 March 2018; Accepted 22 March 2018 ⁎ Corresponding author. E-mail addresses: a.ghodousian@ut.ac.ir (A. Ghodousian), m.naeeimi@ut.ac.ir (M. Naeeimi), ali.babalhavaeji@ut.ac.ir (A. Babalhavaeji). Computers & Industrial Engineering 119 (2018) 167–180 Available online 26 March 2018 0360-8352/ © 2018 Elsevier Ltd. All rights reserved. T