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On Optimal Constraint Violation in Fuzzy Inequality Systems

On Optimal Constraint Violation in Fuzzy Inequality Systems Fuzzy Inf. Eng. (2012) 1: 3-11 DOI 10.1007/s12543-012-0097-x ORIGINAL ARTICLE On Optimal Constraint Violation in Fuzzy Inequality Systems M. Keyanpour· S. Ketabchi Received: 25 February 2011/ Revised: 13 October 2011/ Accepted: 10 February 2012/ © Springer-Verlag Berlin Heidelberg and Fuzzy Information and Engineering Branch of the Operations Research Society of China Abstract In this paper, we describe the technique for calculating minimum viola- tion of a system in fuzzy linear inequalities showing it is also an efficient violation. For this purpose, degree of inconsistency of a crisp system of linear inequalities is defined, and degree of feasibility and degree of consistency for a linear system with violation inequality are presented and then the minimum violation is calculated by solving a convex quadratic programming. The minimum violation of fuzzy lin- ear programming (FLP) is also computed with numerical examples illustrated by the obtained results and its practical implementation. Keywords Violation of constraint · Degree of feasibility · Degree of consistency · Fuzzy linear programming. 1. Introduction Fuzzy linear inequality system (FLIS) is a refinement of linear inequality system (LIS). We consider the FLIS for which all constraints are vague and they accept the violations. The violations are not distinct, and they are usually estimated. If the violations select incorrectly (we call this violation an inefficient violation), the system may have no solution. In this paper, we present a criterion for inconsistency of a system (IS), degree of feasibility and degree of consistency of a system. By solution of a quadratic convex programming, the minimum violations of FLIS are computed. We also show this violation have the minimum norm. There are various methods to the solution of FLP problem such as [1-8], but all of these methods assume the violation of the linear M. Keyanpour ()· S. Ketabchi() Department of Applied Mathematics, University of Guilan, Rasht, Iran Email: kianpour@guilan.ac.ir saed.ketabchi@gmail.com 4 M. Keyanpour · S. Ketabchi (2012) programming is on the hand. In this paper, a minimum violation of FLP problem is calculated by extending our method. Now we describe our notation: All vectors will be column vectors and we denote the n-dimensional real space by R , meaning A , and  the transpose of matrix A and Euclidean norm and∞ norm, respectively. A will denote the ith row of matrix A and for vector a ∈ R the plus function a is defined as (a ) = max{0, a}, i = + + i 1, 2,··· , n. 2. Fuzzy Set of Inconsistency of Crisp Systems In this section, inconsistency of a system is defined, and then we define a fuzzy set of inconsistency on adjunction matrixes such that if the grade of membership of an adjunction matrix [A, b], is zero, then the system Ax ≤ b (1) is completely consistent, and if this amount is one, then the system is completely inconsistent. m×n m Definition 2.1 Let A ∈ R and b ∈ R . The system Ax ≤ b is completely m×n inconsistence in x ∈ R if we have (Ax− b) > 0. We now consider the following quadratic programming: min||(Ax− b) || . (2) x∈R The objective function of this problem is convex piecewise quadratic and differen- tiable function and upper bounded by ||(−b) ||. ∗ ∗ ∗ Lemma 2.1 Let x be the solution of the Problem (2) and v = (Ax − b) . Then v = 0 if only if Ax ≤ b is consistence. m×n m Definition 2.2 Let A ∈ R and b ∈ R . We define the fuzzy set of inconsistency m×n m×(n+1) I, on adjunction matrixes, R , with the following membership function: ||(Ax− b) || m× (n+1) μ (A, b) = min for all (A, b) ∈ R . (3) x∈R ||(−b) || Lemma 2.2 The fuzzy set I is normal. Proof Since numerator and denominator is nonnegative in Relation (3), thusμ (A, b) 2 2 ≥ 0, on the other hand, min ||(Ax− b) || ≤||(−b) || , thusμ (A, b) ≤ 1. x + + 3. System of Linear Inequality with Constraints Violation In this section, we consider the system of inequalities with violations in constraints, i i T n as a x ≤ b, i = 1,··· , m, where a = [a ,··· , a ] ∈ R , b ∈ R and violations are v i i1 in i v ∈ R . The fuzzy set of feasibility of a constraints is defined, and then by using min i + operation of Zadeh [9], degree of consistency of a system is defined. Fuzzy Inf. Eng. (2012) 1: 3-11 5 Definition 3.1 Let x ∈ R and v ∈ R are given. The fuzzy set F = {(a,μ ˜ (a))| a ∈ 1 + x,v 1 F x,v (n+1) R } with the membership function ⎪ 1 1, a x ≤ b ⎨ 1 1 a x−b 1 1 μ (a) = μ (a , b ) = ˜ ˜ 1 ⎪ 1− , b ≤ a x ≤ b + v , v  0 F F 1 1 1 1 x,v x,v ⎪ 1 1 ⎪ 1 0, otherwise is called set of the degree of feasibility of x with violation v in inequality relation a x ≤ b . v 1 n+1 n It is obvious that the fuzzy set F is a set in R and for all x ∈ R and v ∈ R , x,v 1 + we have 0 ≤ μ (a , b ) ≤ 1. ˜ 1 x,v n T m Definition 3.2 Let x ∈ R and v = [v ,··· , v ] ∈ R be given. The fuzzy set F , 1 m x,v degree of feasibility x with violation v in system of Ax ≤ b is defined as follows: ˜ ˜ F (A, b) = F (a, b ), x,v x,v i i=1 where by using min operator Zadeh [9], its membership function shows as follows: μ (A, b) = minμ (a, b ). ˜ ˜ F F x,v x,v i=1 Lemma 3.1 Let A ,A be two arbitrary submatrices of A. Let b ,b , be two subvec- s s s s tors of b, and v ,v be two subvectors of v, corresponding to submatrices of A. Then s s we have: μ (A, b) = min{μ (A , b ),μ (A , b )}. ˜ ˜ ˜ s s s s F F F x,v x,vs x,v Lemma 3.2 Let x ∈ R . It is a solution of the system Ax ≤ b if and only if we have μ (A, b) = 1 for all v ∈ R . F + x,v In the following definition, we define a fuzzy set in the augmented matrixes, m×(n+1) R , which is expository of consistency of the relations Ax ≤ b. Definition 3.3 The violation v is called efficient violation for the system (1) if for all v > v , there exists x ∈ R such that μ (A, b) > 0. x,v Definition 3.4 Let v ∈ R be given. The fuzzy set C = {(A,μ (A))| A = (A, b) ∈ v C m×(n+1) R }, called set of degree of consistency inequality system with violation v”, is defined with membership function as follows: μ (A) = μ (A, b) = sup{μ (A, b) | x ∈ R }. (4) C C F v v x,v Lemma 3.3 In the above definition, we could replace “sup” with max operation. The proof is based on that the set {μ (A, b) | x ∈ R } is closed in R. x,v Theorem 3.1 The set C satisfies the following relations: m m×n m I. For all v ∈ R , A ∈ R , and b ∈ R we have 0 ≤ μ (A, b) ≤ 1. + v 6 M. Keyanpour · S. Ketabchi (2012) II. The system Ax ≤ b is feasible if and only if μ (A, b) = 1. III. Let A, A , A , b, b and b be the same defined in Lemma 3.1. We have: s s s s min{μ (A , b ),μ (A , b )}≤ μ (A, b) ≤ max{μ (A , b ),μ (A , b )}. C s s C s s C C s s C s s v v v v v ∗ ∗ ∗ Theorem 3.2 Let x be the solution of Problem (2). We define v = (Ax − b) , then: ∗ m I. v = 0 if and only if μ (A, b) = 1 for all v ∈ R . m C v + m ∗ II. For all v ∈ R such thatv < v , we haveμ (A, b) = 0. Proof By Theorem 3.1, this system is feasible if and only if μ (A, b) = 1 for all v ∈ R . To prove II, we consider the problem of the minimal correction, by changing the right hand side vector: min v x,v Ax ≤ b+ v. (5) ⎡ ⎤ ⎢ ⎥ ⎢ x ⎥ ⎢ ⎥ ⎢ ⎥ Let be an optimal solution of (5). According to the KKT optimality condi- ⎢ ⎥ ⎣ ⎦ tions, we have: v −λ = 0, (6) ∗ ∗ Ax ≤ b+ v , (7) T ∗ ∗ λ (Ax − b− v ) = 0, (8) where the vector λ ≥ 0 denotes the Lagrange multipliers. From (6), (7) and (8) we ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ∗ ∗ ⎢ ⎥ ⎢ ⎥ obtain v = (Ax − b) , and ⎢ ⎥ is an optimal solution to the Problem (2). Since ⎣ ⎦ the objective function of Problem (5) is strictly convex, v is unique (see [10]). Now ∗ m suppose thatv  > 0 , then the system Ax ≤ b+ v is not feasible for all v ∈ R such ∗ n i thatv < v , this means for all x ∈ R , there exists i(x) such thatμ (a, b ) = 0, ˜ i F(x,v) thus we obtainμ (A, b) = 0, and this completes the proof. Solutions to Problem (2) are called weak solutions. It is known from Theorem 3.2, weak solutions always exist even if μ (A, b) = 0. Also if μ (A, b) = 1, weak C C v v solutions are feasible points. Now suppose that A and A are two submatrices of A with (A x − b ) = 1 2 1 1 + ∗ ∗ 0, (A x − b) > 0, then by rearranging on v ,wehave 2 + ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ∗ ∗ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ v ⎥ ⎢ (A x − b ) ⎥ ⎢ 0 ⎥ 1 1 1 + ∗ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ v = = = . (9) ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ∗ ∗ ∗ v (A x − b ) A x − b 2 2 2 + 2 2 T T T m ∗ ∗ Let v = [v , v ] ∈ R be any vector, in which v > v , and consider v = v +λ(v− 2 2 1 2 + v ), whereλ> 0. Then obviously, we haveμ (A, b) > 0. By using these relations and Theorem 3.2, we have the following. ∗ ∗ ∗ T ∗ T T Corollary 3.1 Let x be the solution to Problem (2) and v = [(v ) , (v ) ] be 1 2 ∗ ∗ the efficient violation of System (1) and suppose that v and v are the same in (9). 1 2 Fuzzy Inf. Eng. (2012) 1: 3-11 7 T T T m ∗ Then for every v = [v , v ] ∈ R for which v > v , we have μ (A, b) > 0, where 2 2 C + v 1 2 ∗ ∗ v = v +λ(v− v ) andλ> 0. 4. Application to Fuzzy Linear Programming In this section, we consider two different models in FLP, where we find minimum violation. First model is called ordinary linear programming (OLP) and second is called penalty linear programming model (PLP). Every model can be applied in a special situation. 4.1. OLP Model In this model, we consider the following programming problem: min z = c x s.t : Ax ≤ b, (10) Bx ≤ d, x ≥ 0, m×n k×n m k n m where A ∈ R , B ∈ R , b ∈ R , d ∈ R , C ∈ R , v ∈ R , and the constraint Bx ≤ d, x ≥ 0 are hard constraints and it is assumed that the set{x|Bx ≤ d, x ≥ 0} is nonempty. It assumes that z is an upper bounded of the objective function, we consider the following problem: c x ≤ z , v u Ax ≤ b, (11) Bx ≤ d, x ≥ 0. To determine z , we can solve the following problem min z = c x s.t : A x ≤ b , (12) 1 1 Bx ≤ d, x ≥ 0, where A is submatrix of A and b subvector of b with (A x − b ) = 0. 1 1 1 1 + T T T T T T Let A = [c, A ] , b = [z , b ] , and v ˆ = [v , v ] . Then System (11) is converted u 0 to the following fuzzy system: ˆ ˆ Ax ≤ b, v ˆ Bx ≤ d, (13) x ≥ 0. It is obvious that: ˆ ˆ ∀x : x ≥ 0, Bx ≤ d ⇒ μ (x) = μ (A, b), D ˜ x,v ˆ 8 M. Keyanpour · S. Ketabchi (2012) where μ is the fuzzy decision proposed by Bellman and Zadeh [1]. The optimal fuzzy decision is the solution to the following problem [1]: max{μ (x)|∀x : x ≥ 0, Bx ≤ d} = max{μ (A, b)|∀x : x ≥ 0, Bx ≤ d}. D ˜ x,v ˆ T T T T T T T T T T T T ˘ ˘ Let A = [c, A , B , I ] , b = [z , b , d , 0 ] and v ˘ = [v , v , 0 , 0 ] . The u 0 system is converted to the following fuzzy system: ˘ ˘ Ax ≤ b. (14) v ˘ Now by using of our procedure, we can find the minimum violation for v , v .We consider the following constraints quadratic programming problem to compute the violations v , v: min ||(Ax− b) || x∈R Bx ≤ d, x ≥ 0. It is noted that if the set {x : Bx ≤ d, x ≥ 0} is infeasible, we could consider violations for some of the constraints such that the new system will be feasible. 4.2. PLP Model In this method, we consider a penalty for increasing of violation and the following problem: min z = c x+ c v (15) i i i=1 s.t : Ax ≤ b, (16) x ≥ 0, (17) where c , i = 1,··· , n gives a real positive number as the penalty of ith violation, v . i i It assumes that z is an upper bounded of the objective function, we consider the following problem: c x+ c v ≤ z , (18) i i v u i=1 Ax ≤ b, (19) x ≥ 0, (20) and the continuing is similar to Subsection (4.1). 5. Numerical Results To illustrate all the results presented in Sections 2, 3, and 4, we consider the following example (see [11]). Fuzzy Inf. Eng. (2012) 1: 3-11 9 Example 5.1 We consider the following system: ⎡ ⎤ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ −0.10433318 −0.3349605 −2.24440190 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ −2.31759372 −2.0354161 0.7579334 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ −0.67781831 0.6546597 ⎥ ⎢ x ⎥ ⎢ 0.4302541 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ Ax = ≤ = b. ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 1.05241872 −0.4327864 ⎥ x ⎢ ⎥ 2.5746725 ⎢ 2 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 0.01449416 −1.93122220⎥ ⎢ −2.6003448 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎣ ⎦ ⎦ 0.24375548 0.5536801 0.4284550 I. By solving the unconstrained minimization Problem (2), we obtain the optimal ∗ T solution x = [1.0371, 1.8348] . The objective function is ∗ 2 ∗ (Ax − b)  = 3.8156, v = , 1.5216, 0, 0.67938, 0, 0, 0.84021 ∗ ∗ also v = 1.5216, 0.67938, 0.84021 and v  = 1.7395. We have μ (A, b) = Dc 0.50641. This means X = {x ∈ R : Ax ≤ b} = ∅. Now consider the fuzzy system with violation v, Ax ≤ b + v, where v = 0.4, 0.6, 0.02, 0, 0, 0.2 ,but v = 0.44766 and v < v , then from Theorem 3.2, degree of consistency of Ax ≤ b with violation v is zero, i.e., μ (A, b) = 0. II. For the OLP model, we consider the following problem: min z = c x s.t : Ax ≤ b, x ≥ 0, where c = [1 1] and A, b are the same in Example (5.1.I) By solving (12), we obtain the upper bounded of the objective function to Problem T T T T ˆ ˆ (11). It is z = 1.3465 and by considering A = [c, A , −I] , b = [z , b , 0] , and u u T T v ˆ = [v , v ] our system convert to the following system ˆ ˆ Ax ≤ b. (21) v ˆ To obtain the optimal violation vector of System (21), we solve the unconstrained problem ˆ ˆ min||(Ax− b) || . (22) x∈R ∗ ∗ 2 ˆ ˆ We obtain vˆ = 0.26549,v ˆ  = 1.8457,(Ax ˆ − b)  = 3.4067 and 0 + ∗ T v ˆ = [0.26549, 1.7683, 0, 0.25608, 0, 0.026019, 0.37825, 0, 0] . This implies that x ˆ is the solution to the problem. III. We consider Problem (10) where c, z , A, and b are the same in Example ⎡ ⎤ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ 1−1⎥ ⎢ 1⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ (5.1.II) and B = , d = . ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ 00 0 10 M. Keyanpour · S. Ketabchi (2012) By solving the unconstrained problem: ˘ ˘ min||(Ax− b) || , (23) x∈R ∗ T we obtain v ˘ = [0.26549, 1.7683, 0, 0.25608, 0, 0.026019, 0.37825, 0, 0, 0] , ∗ 2 ∗ ∗ T ˘ ˘ (Ax ˘ − b)  = 3.4067, v˘ = 0.26549, v ˘  = 1.8457 and x ˘ = [0.27691, 1.3351] . + 0 This means that: ˆ ˆ ∀x : x ≥ 0, Bx ≤ d ⇒ μ (x) = μ (A, b). D F x,v ˆ IV. For the PLP model, we consider the following problem: min z = c x+ c v i i i=1 s.t : Ax ≤ b, x ≥ 0 , T 6 T where c = [1, 1] , c = [1× 10 , 1] and A, b are the same in Example (5.1.I). We see that z = 1.3465 can be considered as an upper bounded of the objective ⎡ ⎤ ⎢ c c ⎥ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ z ⎥ ⎢ ⎥ ⎢ u ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ A 0 ⎢ ⎥ ⎢ ⎥ T T ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ function to this problem. Suppose that A = , b = b and v = [v , v ] , ⎢ ⎥ ⎢ ⎥ 0 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ −I 0 ⎢ ⎥ ⎣ ⎦ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ 0 −I then our system is converted to the following system, Ax ≤ b. (24) By solving the unconstrained problem, min||(Ax− b) || , (25) x∈R ∗ T we obtain x = [−2.5832e− 008, 1.9111e+ 000, −1.5646e− 006, 1.0000e+ 000] , ∗ −012 −06 v = 2.5260× 10 , 1.5216, 0, 0.067938, 0, 0, 0.84021, 0, 0, 2.5253× 10 , 0 , ∗ ∗ −012 2 v = 2.5260 × 10 , (Ax − b)  = 3.0259, andv  = 1.7395. 0 + This implies that x is the solution to the problem. 6. Conclusion Here we have formulated an optimal constraint violation problem in fuzzy inequal- ity with a convex programming problem and derived the optimality conditions for this problem from the Karush-Kuhn-Tucker (KKT) system. Then the method is ex- tended to find the minimum violation of fuzzy linear programming problem. Besides numerical examples are solved to show the efficiency of the method. Acknowledgments Fuzzy Inf. Eng. (2012) 1: 3-11 11 The authors are extremely thankful to the referees for their helpful comments and suggestions which led to the improvement of the originally submitted version of this work. The Matlab code of this paper is available for everyone who wants to test the method. Please contact us by e-mail. References 1. Bellmann R E, Zadeh L A (1970) Decision making in fuzzy environment. Management Science 17: 141-164 2. Fang S C, Hu C F, Wang H F, Wu S Y (1999) Linear programming with fuzzy coefficients in con- straints. Computers & Mathematics with Applications 37(10): 63-76 3. Maleki H R, Tata M, Mashinchi M (2000) Linear programming with fuzzy variables. Fuzzy Sets and Systems 109: 21-33 4. Nakahara Y (1998) User oriented ranking criteria and its application to fuzzy mathematical program- ming problems. Fuzzy Sets and Systems 94: 275-286 5. Ramik J R, Rommelfanger H (1996) Fuzzy mathematical programming based on some new inequality relations. Fuzzy Sets and Systems 81: 77-87 6. Buckley J J (1995) Joint solution to fuzzy programming problems. Fuzzy Sets and Systems 72: 215-220 7. Tong S (1994) Interval number and fuzzy number linear programming. Fuzzy Sets and Systems 66: 301-306 8. Julien B (1994) A extension to possibilistic linear programming. Fuzzy Sets and Systems 64: 195- 9. Zadeh L A (1965) Fuzzy sets. Information and Control 8: 338-353 10. Golikov A I, Evtushenko Y G (2003) Theorems of the alternative and their applications in numerical methods. Computational Mathematics and Mathematical Physics 43(3): 338-358 11. Amaral P, Terosset M W, Barahona P (2000) Correcting an inconsistent set of linear inequalities by nonlinear programming. Technical Report, Department of Computational and Applied Mathematics, Rice University, Houston, TX 77005: 1-27 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Fuzzy Information and Engineering Taylor & Francis

On Optimal Constraint Violation in Fuzzy Inequality Systems

Keyanpour, M.; Ketabchi, S.
Fuzzy Information and Engineering , Volume 4 (1): 9 – Mar 1, 2012
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Fuzzy Inf. Eng. (2012) 1: 3-11 DOI 10.1007/s12543-012-0097-x ORIGINAL ARTICLE On Optimal Constraint Violation in Fuzzy Inequality Systems M. Keyanpour· S. Ketabchi Received: 25 February 2011/ Revised: 13 October 2011/ Accepted: 10 February 2012/ © Springer-Verlag Berlin Heidelberg and Fuzzy Information and Engineering Branch of the Operations Research Society of China Abstract In this paper, we describe the technique for calculating minimum viola- tion of a system in fuzzy linear inequalities showing it is also an efficient violation. For this purpose, degree of inconsistency of a crisp system of linear inequalities is defined, and degree of feasibility and degree of consistency for a linear system with violation inequality are presented and then the minimum violation is calculated by solving a convex quadratic programming. The minimum violation of fuzzy lin- ear programming (FLP) is also computed with numerical examples illustrated by the obtained results and its practical implementation. Keywords Violation of constraint · Degree of feasibility · Degree of consistency · Fuzzy linear programming. 1. Introduction Fuzzy linear inequality system (FLIS) is a refinement of linear inequality system (LIS). We consider the FLIS for which all constraints are vague and they accept the violations. The violations are not distinct, and they are usually estimated. If the violations select incorrectly (we call this violation an inefficient violation), the system may have no solution. In this paper, we present a criterion for inconsistency of a system (IS), degree of feasibility and degree of consistency of a system. By solution of a quadratic convex programming, the minimum violations of FLIS are computed. We also show this violation have the minimum norm. There are various methods to the solution of FLP problem such as [1-8], but all of these methods assume the violation of the linear M. Keyanpour ()· S. Ketabchi() Department of Applied Mathematics, University of Guilan, Rasht, Iran Email: kianpour@guilan.ac.ir saed.ketabchi@gmail.com 4 M. Keyanpour · S. Ketabchi (2012) programming is on the hand. In this paper, a minimum violation of FLP problem is calculated by extending our method. Now we describe our notation: All vectors will be column vectors and we denote the n-dimensional real space by R , meaning A , and  the transpose of matrix A and Euclidean norm and∞ norm, respectively. A will denote the ith row of matrix A and for vector a ∈ R the plus function a is defined as (a ) = max{0, a}, i = + + i 1, 2,··· , n. 2. Fuzzy Set of Inconsistency of Crisp Systems In this section, inconsistency of a system is defined, and then we define a fuzzy set of inconsistency on adjunction matrixes such that if the grade of membership of an adjunction matrix [A, b], is zero, then the system Ax ≤ b (1) is completely consistent, and if this amount is one, then the system is completely inconsistent. m×n m Definition 2.1 Let A ∈ R and b ∈ R . The system Ax ≤ b is completely m×n inconsistence in x ∈ R if we have (Ax− b) > 0. We now consider the following quadratic programming: min||(Ax− b) || . (2) x∈R The objective function of this problem is convex piecewise quadratic and differen- tiable function and upper bounded by ||(−b) ||. ∗ ∗ ∗ Lemma 2.1 Let x be the solution of the Problem (2) and v = (Ax − b) . Then v = 0 if only if Ax ≤ b is consistence. m×n m Definition 2.2 Let A ∈ R and b ∈ R . We define the fuzzy set of inconsistency m×n m×(n+1) I, on adjunction matrixes, R , with the following membership function: ||(Ax− b) || m× (n+1) μ (A, b) = min for all (A, b) ∈ R . (3) x∈R ||(−b) || Lemma 2.2 The fuzzy set I is normal. Proof Since numerator and denominator is nonnegative in Relation (3), thusμ (A, b) 2 2 ≥ 0, on the other hand, min ||(Ax− b) || ≤||(−b) || , thusμ (A, b) ≤ 1. x + + 3. System of Linear Inequality with Constraints Violation In this section, we consider the system of inequalities with violations in constraints, i i T n as a x ≤ b, i = 1,··· , m, where a = [a ,··· , a ] ∈ R , b ∈ R and violations are v i i1 in i v ∈ R . The fuzzy set of feasibility of a constraints is defined, and then by using min i + operation of Zadeh [9], degree of consistency of a system is defined. Fuzzy Inf. Eng. (2012) 1: 3-11 5 Definition 3.1 Let x ∈ R and v ∈ R are given. The fuzzy set F = {(a,μ ˜ (a))| a ∈ 1 + x,v 1 F x,v (n+1) R } with the membership function ⎪ 1 1, a x ≤ b ⎨ 1 1 a x−b 1 1 μ (a) = μ (a , b ) = ˜ ˜ 1 ⎪ 1− , b ≤ a x ≤ b + v , v  0 F F 1 1 1 1 x,v x,v ⎪ 1 1 ⎪ 1 0, otherwise is called set of the degree of feasibility of x with violation v in inequality relation a x ≤ b . v 1 n+1 n It is obvious that the fuzzy set F is a set in R and for all x ∈ R and v ∈ R , x,v 1 + we have 0 ≤ μ (a , b ) ≤ 1. ˜ 1 x,v n T m Definition 3.2 Let x ∈ R and v = [v ,··· , v ] ∈ R be given. The fuzzy set F , 1 m x,v degree of feasibility x with violation v in system of Ax ≤ b is defined as follows: ˜ ˜ F (A, b) = F (a, b ), x,v x,v i i=1 where by using min operator Zadeh [9], its membership function shows as follows: μ (A, b) = minμ (a, b ). ˜ ˜ F F x,v x,v i=1 Lemma 3.1 Let A ,A be two arbitrary submatrices of A. Let b ,b , be two subvec- s s s s tors of b, and v ,v be two subvectors of v, corresponding to submatrices of A. Then s s we have: μ (A, b) = min{μ (A , b ),μ (A , b )}. ˜ ˜ ˜ s s s s F F F x,v x,vs x,v Lemma 3.2 Let x ∈ R . It is a solution of the system Ax ≤ b if and only if we have μ (A, b) = 1 for all v ∈ R . F + x,v In the following definition, we define a fuzzy set in the augmented matrixes, m×(n+1) R , which is expository of consistency of the relations Ax ≤ b. Definition 3.3 The violation v is called efficient violation for the system (1) if for all v > v , there exists x ∈ R such that μ (A, b) > 0. x,v Definition 3.4 Let v ∈ R be given. The fuzzy set C = {(A,μ (A))| A = (A, b) ∈ v C m×(n+1) R }, called set of degree of consistency inequality system with violation v”, is defined with membership function as follows: μ (A) = μ (A, b) = sup{μ (A, b) | x ∈ R }. (4) C C F v v x,v Lemma 3.3 In the above definition, we could replace “sup” with max operation. The proof is based on that the set {μ (A, b) | x ∈ R } is closed in R. x,v Theorem 3.1 The set C satisfies the following relations: m m×n m I. For all v ∈ R , A ∈ R , and b ∈ R we have 0 ≤ μ (A, b) ≤ 1. + v 6 M. Keyanpour · S. Ketabchi (2012) II. The system Ax ≤ b is feasible if and only if μ (A, b) = 1. III. Let A, A , A , b, b and b be the same defined in Lemma 3.1. We have: s s s s min{μ (A , b ),μ (A , b )}≤ μ (A, b) ≤ max{μ (A , b ),μ (A , b )}. C s s C s s C C s s C s s v v v v v ∗ ∗ ∗ Theorem 3.2 Let x be the solution of Problem (2). We define v = (Ax − b) , then: ∗ m I. v = 0 if and only if μ (A, b) = 1 for all v ∈ R . m C v + m ∗ II. For all v ∈ R such thatv < v , we haveμ (A, b) = 0. Proof By Theorem 3.1, this system is feasible if and only if μ (A, b) = 1 for all v ∈ R . To prove II, we consider the problem of the minimal correction, by changing the right hand side vector: min v x,v Ax ≤ b+ v. (5) ⎡ ⎤ ⎢ ⎥ ⎢ x ⎥ ⎢ ⎥ ⎢ ⎥ Let be an optimal solution of (5). According to the KKT optimality condi- ⎢ ⎥ ⎣ ⎦ tions, we have: v −λ = 0, (6) ∗ ∗ Ax ≤ b+ v , (7) T ∗ ∗ λ (Ax − b− v ) = 0, (8) where the vector λ ≥ 0 denotes the Lagrange multipliers. From (6), (7) and (8) we ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ∗ ∗ ⎢ ⎥ ⎢ ⎥ obtain v = (Ax − b) , and ⎢ ⎥ is an optimal solution to the Problem (2). Since ⎣ ⎦ the objective function of Problem (5) is strictly convex, v is unique (see [10]). Now ∗ m suppose thatv  > 0 , then the system Ax ≤ b+ v is not feasible for all v ∈ R such ∗ n i thatv < v , this means for all x ∈ R , there exists i(x) such thatμ (a, b ) = 0, ˜ i F(x,v) thus we obtainμ (A, b) = 0, and this completes the proof. Solutions to Problem (2) are called weak solutions. It is known from Theorem 3.2, weak solutions always exist even if μ (A, b) = 0. Also if μ (A, b) = 1, weak C C v v solutions are feasible points. Now suppose that A and A are two submatrices of A with (A x − b ) = 1 2 1 1 + ∗ ∗ 0, (A x − b) > 0, then by rearranging on v ,wehave 2 + ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ∗ ∗ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ v ⎥ ⎢ (A x − b ) ⎥ ⎢ 0 ⎥ 1 1 1 + ∗ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ v = = = . (9) ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ∗ ∗ ∗ v (A x − b ) A x − b 2 2 2 + 2 2 T T T m ∗ ∗ Let v = [v , v ] ∈ R be any vector, in which v > v , and consider v = v +λ(v− 2 2 1 2 + v ), whereλ> 0. Then obviously, we haveμ (A, b) > 0. By using these relations and Theorem 3.2, we have the following. ∗ ∗ ∗ T ∗ T T Corollary 3.1 Let x be the solution to Problem (2) and v = [(v ) , (v ) ] be 1 2 ∗ ∗ the efficient violation of System (1) and suppose that v and v are the same in (9). 1 2 Fuzzy Inf. Eng. (2012) 1: 3-11 7 T T T m ∗ Then for every v = [v , v ] ∈ R for which v > v , we have μ (A, b) > 0, where 2 2 C + v 1 2 ∗ ∗ v = v +λ(v− v ) andλ> 0. 4. Application to Fuzzy Linear Programming In this section, we consider two different models in FLP, where we find minimum violation. First model is called ordinary linear programming (OLP) and second is called penalty linear programming model (PLP). Every model can be applied in a special situation. 4.1. OLP Model In this model, we consider the following programming problem: min z = c x s.t : Ax ≤ b, (10) Bx ≤ d, x ≥ 0, m×n k×n m k n m where A ∈ R , B ∈ R , b ∈ R , d ∈ R , C ∈ R , v ∈ R , and the constraint Bx ≤ d, x ≥ 0 are hard constraints and it is assumed that the set{x|Bx ≤ d, x ≥ 0} is nonempty. It assumes that z is an upper bounded of the objective function, we consider the following problem: c x ≤ z , v u Ax ≤ b, (11) Bx ≤ d, x ≥ 0. To determine z , we can solve the following problem min z = c x s.t : A x ≤ b , (12) 1 1 Bx ≤ d, x ≥ 0, where A is submatrix of A and b subvector of b with (A x − b ) = 0. 1 1 1 1 + T T T T T T Let A = [c, A ] , b = [z , b ] , and v ˆ = [v , v ] . Then System (11) is converted u 0 to the following fuzzy system: ˆ ˆ Ax ≤ b, v ˆ Bx ≤ d, (13) x ≥ 0. It is obvious that: ˆ ˆ ∀x : x ≥ 0, Bx ≤ d ⇒ μ (x) = μ (A, b), D ˜ x,v ˆ 8 M. Keyanpour · S. Ketabchi (2012) where μ is the fuzzy decision proposed by Bellman and Zadeh [1]. The optimal fuzzy decision is the solution to the following problem [1]: max{μ (x)|∀x : x ≥ 0, Bx ≤ d} = max{μ (A, b)|∀x : x ≥ 0, Bx ≤ d}. D ˜ x,v ˆ T T T T T T T T T T T T ˘ ˘ Let A = [c, A , B , I ] , b = [z , b , d , 0 ] and v ˘ = [v , v , 0 , 0 ] . The u 0 system is converted to the following fuzzy system: ˘ ˘ Ax ≤ b. (14) v ˘ Now by using of our procedure, we can find the minimum violation for v , v .We consider the following constraints quadratic programming problem to compute the violations v , v: min ||(Ax− b) || x∈R Bx ≤ d, x ≥ 0. It is noted that if the set {x : Bx ≤ d, x ≥ 0} is infeasible, we could consider violations for some of the constraints such that the new system will be feasible. 4.2. PLP Model In this method, we consider a penalty for increasing of violation and the following problem: min z = c x+ c v (15) i i i=1 s.t : Ax ≤ b, (16) x ≥ 0, (17) where c , i = 1,··· , n gives a real positive number as the penalty of ith violation, v . i i It assumes that z is an upper bounded of the objective function, we consider the following problem: c x+ c v ≤ z , (18) i i v u i=1 Ax ≤ b, (19) x ≥ 0, (20) and the continuing is similar to Subsection (4.1). 5. Numerical Results To illustrate all the results presented in Sections 2, 3, and 4, we consider the following example (see [11]). Fuzzy Inf. Eng. (2012) 1: 3-11 9 Example 5.1 We consider the following system: ⎡ ⎤ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ −0.10433318 −0.3349605 −2.24440190 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ −2.31759372 −2.0354161 0.7579334 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ −0.67781831 0.6546597 ⎥ ⎢ x ⎥ ⎢ 0.4302541 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ Ax = ≤ = b. ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 1.05241872 −0.4327864 ⎥ x ⎢ ⎥ 2.5746725 ⎢ 2 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 0.01449416 −1.93122220⎥ ⎢ −2.6003448 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎣ ⎦ ⎦ 0.24375548 0.5536801 0.4284550 I. By solving the unconstrained minimization Problem (2), we obtain the optimal ∗ T solution x = [1.0371, 1.8348] . The objective function is ∗ 2 ∗ (Ax − b)  = 3.8156, v = , 1.5216, 0, 0.67938, 0, 0, 0.84021 ∗ ∗ also v = 1.5216, 0.67938, 0.84021 and v  = 1.7395. We have μ (A, b) = Dc 0.50641. This means X = {x ∈ R : Ax ≤ b} = ∅. Now consider the fuzzy system with violation v, Ax ≤ b + v, where v = 0.4, 0.6, 0.02, 0, 0, 0.2 ,but v = 0.44766 and v < v , then from Theorem 3.2, degree of consistency of Ax ≤ b with violation v is zero, i.e., μ (A, b) = 0. II. For the OLP model, we consider the following problem: min z = c x s.t : Ax ≤ b, x ≥ 0, where c = [1 1] and A, b are the same in Example (5.1.I) By solving (12), we obtain the upper bounded of the objective function to Problem T T T T ˆ ˆ (11). It is z = 1.3465 and by considering A = [c, A , −I] , b = [z , b , 0] , and u u T T v ˆ = [v , v ] our system convert to the following system ˆ ˆ Ax ≤ b. (21) v ˆ To obtain the optimal violation vector of System (21), we solve the unconstrained problem ˆ ˆ min||(Ax− b) || . (22) x∈R ∗ ∗ 2 ˆ ˆ We obtain vˆ = 0.26549,v ˆ  = 1.8457,(Ax ˆ − b)  = 3.4067 and 0 + ∗ T v ˆ = [0.26549, 1.7683, 0, 0.25608, 0, 0.026019, 0.37825, 0, 0] . This implies that x ˆ is the solution to the problem. III. We consider Problem (10) where c, z , A, and b are the same in Example ⎡ ⎤ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ 1−1⎥ ⎢ 1⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ (5.1.II) and B = , d = . ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ 00 0 10 M. Keyanpour · S. Ketabchi (2012) By solving the unconstrained problem: ˘ ˘ min||(Ax− b) || , (23) x∈R ∗ T we obtain v ˘ = [0.26549, 1.7683, 0, 0.25608, 0, 0.026019, 0.37825, 0, 0, 0] , ∗ 2 ∗ ∗ T ˘ ˘ (Ax ˘ − b)  = 3.4067, v˘ = 0.26549, v ˘  = 1.8457 and x ˘ = [0.27691, 1.3351] . + 0 This means that: ˆ ˆ ∀x : x ≥ 0, Bx ≤ d ⇒ μ (x) = μ (A, b). D F x,v ˆ IV. For the PLP model, we consider the following problem: min z = c x+ c v i i i=1 s.t : Ax ≤ b, x ≥ 0 , T 6 T where c = [1, 1] , c = [1× 10 , 1] and A, b are the same in Example (5.1.I). We see that z = 1.3465 can be considered as an upper bounded of the objective ⎡ ⎤ ⎢ c c ⎥ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ z ⎥ ⎢ ⎥ ⎢ u ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ A 0 ⎢ ⎥ ⎢ ⎥ T T ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ function to this problem. Suppose that A = , b = b and v = [v , v ] , ⎢ ⎥ ⎢ ⎥ 0 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ −I 0 ⎢ ⎥ ⎣ ⎦ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ 0 −I then our system is converted to the following system, Ax ≤ b. (24) By solving the unconstrained problem, min||(Ax− b) || , (25) x∈R ∗ T we obtain x = [−2.5832e− 008, 1.9111e+ 000, −1.5646e− 006, 1.0000e+ 000] , ∗ −012 −06 v = 2.5260× 10 , 1.5216, 0, 0.067938, 0, 0, 0.84021, 0, 0, 2.5253× 10 , 0 , ∗ ∗ −012 2 v = 2.5260 × 10 , (Ax − b)  = 3.0259, andv  = 1.7395. 0 + This implies that x is the solution to the problem. 6. Conclusion Here we have formulated an optimal constraint violation problem in fuzzy inequal- ity with a convex programming problem and derived the optimality conditions for this problem from the Karush-Kuhn-Tucker (KKT) system. Then the method is ex- tended to find the minimum violation of fuzzy linear programming problem. Besides numerical examples are solved to show the efficiency of the method. Acknowledgments Fuzzy Inf. Eng. (2012) 1: 3-11 11 The authors are extremely thankful to the referees for their helpful comments and suggestions which led to the improvement of the originally submitted version of this work. The Matlab code of this paper is available for everyone who wants to test the method. Please contact us by e-mail. References 1. Bellmann R E, Zadeh L A (1970) Decision making in fuzzy environment. Management Science 17: 141-164 2. Fang S C, Hu C F, Wang H F, Wu S Y (1999) Linear programming with fuzzy coefficients in con- straints. Computers & Mathematics with Applications 37(10): 63-76 3. Maleki H R, Tata M, Mashinchi M (2000) Linear programming with fuzzy variables. Fuzzy Sets and Systems 109: 21-33 4. Nakahara Y (1998) User oriented ranking criteria and its application to fuzzy mathematical program- ming problems. Fuzzy Sets and Systems 94: 275-286 5. Ramik J R, Rommelfanger H (1996) Fuzzy mathematical programming based on some new inequality relations. Fuzzy Sets and Systems 81: 77-87 6. Buckley J J (1995) Joint solution to fuzzy programming problems. Fuzzy Sets and Systems 72: 215-220 7. Tong S (1994) Interval number and fuzzy number linear programming. Fuzzy Sets and Systems 66: 301-306 8. Julien B (1994) A extension to possibilistic linear programming. Fuzzy Sets and Systems 64: 195- 9. Zadeh L A (1965) Fuzzy sets. Information and Control 8: 338-353 10. Golikov A I, Evtushenko Y G (2003) Theorems of the alternative and their applications in numerical methods. Computational Mathematics and Mathematical Physics 43(3): 338-358 11. Amaral P, Terosset M W, Barahona P (2000) Correcting an inconsistent set of linear inequalities by nonlinear programming. Technical Report, Department of Computational and Applied Mathematics, Rice University, Houston, TX 77005: 1-27

Journal

Fuzzy Information and Engineering – Taylor & Francis

Published: Mar 1, 2012

Keywords: Violation of constraint; Degree of feasibility; Degree of consistency; Fuzzy linear programming

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