Determining Feasible Solution in Imprecise Linear Inequality Systems

Applied Mathematical Sciences, Vol. 2, 2008, no. 36, 1789 - 1797

Determining Feasible Solution in Imprecise Linear Inequality Systems A. A. Noura and F. H. Saljooghi Department of Mathematics Sistan and Balouchestan University, Zahedan, Iran f h [email protected] Abstract In this paper, we investigate imprecise linear inequality problems when there is no solution. In general, inconsistency of ordinary linear inequality model can be overcome by changes in the coefficients of constraints and right hand sides. Many of researches discuss infeasibility on crisp linear inequality, but few researchers have dealt with imprecise inequality. First, A method for solving imprecise linear inequality will be reviewed which based on the concept of degree of inequality holding true for two intervals, and then states of their solution will be examined. If system inequalities in the best state coefficients technology and right hand sides have solution and in the worst state coefficients technology and right hand sides haven’t solution, our objective is the determining greatest intervals of coefficients technology and right-hand sides so that system have solution. Indeed, the solutions which satisfy the linear inequalities with degree holding true are searched, such as have minimum relaxation. In sequence, numerical example is given for demonstrating our approach.

Mathematics Subject Classification: 90C05, 90C70 Keywords: interval linear inequality, degree holding inequality true, infeasibility

1

Introduction

Inconsistency in linear inequalities is a problem on many practical situations. In general, inconsistency of a model can be overcome by changes in the coefficients and right-hand sides of the constraints. Many works related to the problem of inconsistency in linear inequality and present some views on how it

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can be tackled. A classical approach to deal with infeasibility involving the adjustment of the right-hand side of the constraints to restore feasibility, the first time was given by Roodman in 1979 [5]. Vatolin [6] developed a broad theory on the study of inconsistent mathematical programming problems, presenting duality concepts for corrected systems. Many researches will have applied fuzzy set theory for relaxation in the original problem. For sample, Wang proposed a method to solve infeasible inequality problems by the concept of fuzzy relation [7]. Amaral [1] has studied the problem of the minimal correction of an infeasible set of linear inequalities accordingly to the Frobenius norm. His method fulfills the problem of correcting the matrix of coefficients A by A + H and vector b by b + p to minimize the Frobenius norm of [H, p]. Infeasibility in imprecise linear inequality is seldom discussed. We apply an approach for converting interval linear inequality to ordinal linear inequality. This approach uses degree of inequality holding true. In Section 2 we formulate the problem and the theoretical developments by regarding to the correction of inequalities system with degree of inequality holding true. Then in section 3 a special case of interval linear inequality which is infeasible will be solved. Finally in section 4 the behavior of the method over numerical example will be shown.

2

Solving imprecise inequality systems

Consider the following linear inequality system: n l u l u i = 1, . . . , m (1) j=1 [aij , aij ]xj ≤ [bi , bi ] j = 1, . . . , n xj ≥ 0 n This is the family of all systems i = 1, . . . , m (2) j=1 aij xj ≤ bi l u l u ∀ij whose data satisfy aij ∈ [aij , aij ] , bi ∈ [bi , bi ] In this inequality, technological coefficients and right hand sides are imprecise (interval). In this paper we will investigate feasibility or infeasibility of system (1). Definition 2.1 a system (1) is said to be weakly feasible if some systems (2) with data interval is feasible, and it is called strongly feasible if each system (2) with data interval is feasible.[2] The adaptive set of points to system (1) is called feasible region S. The following special cases are being considered from the system (1). The system (3) can be concluded if technological coefficients are lower bound and sources in upper bound(in the best state) . n l u i = 1, . . . , m (3) j=1 aij xj ≤ bi xj ≥ 0 j = 1, . . . , n Feasible region in this case called SI (Ideal space). Similarly, if the technological coefficients are in the upper bound and source in the lower bound then feasible region ,SA (Anti-ideal space),is resulted in the

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Determining feasible solution n

following form

j=1

auij xj ≤ bli xj ≥ 0

i = 1, . . . , m j = 1, . . . , n

(4)

Theorem 2.2 Prove SA ⊆ S ⊆ SI . proof: ∀aij : alij ≤ aij ≤ auij (xj ≥ 0) ⇒ alij xj ≤ aij xj ≤ auij xj ⇒ n n  l aij xj ≤ nj=1 auij xj and bli ≤ bi ≤ bui j=1 aij xj ≤ j=1  xj ∈ SA ⇒ nj=1 auij xj ≤ bli   ⇒ nj=1 aij xj ≤ nj=1 auij xj ≤ bli ≤ bi i = 1, . . . , m  ⇒ nj=1 aij xj ≤ bi i = 1, . . . , m ⇒ xj ∈ S  also xj ∈ SA ⇒ nj=1 auij xj ≤ bli n  ⇒ j=1 alij xj ≤ nj=1 auij xj ≤ bli ≤ bui i = 1, . . . , m n l u i = 1, . . . , m ⇒ xj ∈ SI ⇒ j=1 aij xj ≤ bi corollary 2.1 If SA = φ then S = φ and SI = φ, means the system(1) is always feasible.(strongly feasible) corollary 2.2 If SI = φ then S = φ , means the system(1) is always infeasible. corollary 2.3 If SI = φ and SA = φ , then problem (1) will be feasible in some intervals and sometimes infeasible in the others.(weakly feasible) Several definitions exist how to determine the feasible region inequality (2). A classical approach to deal with infeasibility involving the adjustment of right hand side of the constraints to restore feasibility, in this approach technological coefficients are supposed deterministic. In approach Leon et al. [4],which discussed for linearly constrained problems ,with making perturb in the righthand side coefficients attained feasibility.the changing of right hand side of the constraints is performed according to value of duality variables and proportional to objective. The other definition given by Ishibuchi and Tanaka is based on the concept of degree of inequality holding true for two intervals.( this definition is foundation of our work). Definition 2.3 If A = [al , au ] is an interval number and a real number x , for relation A ≤ x , define degree holding true g(A ≤ x) as:[3] g(A ≤ x) =

⎧ ⎪ ⎨

0 x−al

u l ⎪ ⎩ a −a

1

x ≤ al al ≤ x ≤ au x ≥ au

(5)

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g(A ≤ x) = max{0, min ax−a u −al } l u In the same way, if B = [b , b ] is an interval number also, relation A ≤ B or A− B ≤ 0, therefore,C = [al − bu , au − bl ] ≤ 0 and the degree for g(C ≤ 0) holding u −al true is given as follows: g(C ≤ 0) = max{0, min au −bb l +b (6) u −al } According to above statement, the interval system (2) can be determined by the following theorem. Theorem 2.4 For a given degree of inequality holding true q, the interval constraint (2) can be transformed equally into the following crisp inequality n u l l u constraint: i = j=1 (qaij + (1 − q)aij )xj ≤ (1 − q)bi + qbi 1, . . . , m (7) Now if we consider degree of holding true qa for coefficients aij and qb for values bi , system (1) is converted to: n n l u l l u l i = 1, . . . , m j=1 aij xj + qa j=1 (aij − aij )xj ≤ bi + qb (bi − bi ) j = 1, . . . , n (8) xj ≥ 0 Feasible region of (8) can be transformed into SA , if qa = 1 and qb = 0, such as if qa = 0 and qb = 1 , feasible region (8) will be SI . For given qa and qb , feasible region (8) is called Sq . Regarding theorem (1), SA ⊆ Sq ⊆ SI . Theorem 2.5 If model (8) is feasible for qa and qb then for ever qa and qb with conditions qa ≤ qa and qb ≥ qb (with feasible region Sq ) is feasible. Proof: lets Sq = φ , means (8) is feasible, then ∀ qa ≤ qa ⇒ (auij − alij )qa ≤ (auij − alij )qa By multiplying above statement in xj ≥ 0 and summing statements: n n  l  n u l l qa nj=1 (auij − alij )xj a ij )xj ≤ j=1 aij xj + q j=1 (aij − a j=1 aij xj +   ∀ x ∈ Sq thus nj=1 alij xj +qa nj=1 (auij − alij )xj ≤ nj=1 alij xj +qa nj=1 (auij − alij )xj ≤ bli + qb (bui − bli ) so qb ≥ qb therefore bli + qb (bui − bli ) ≤ bli + qb (bui − bli ) n l  n u l l  u l ∀i j=1 aij xj + qa j=1 (aij − aij )xj ≤ bi + qb (bi − bi )    ⇒ x ∈ Sq means Sq ⊆ Sq , Since Sq = φ then Sq = φ . Now we discuss over special state for SI and SA .

3

Weak feasibility of inequalities

. By attention the corollary 3.1, If SI = φ and SA = φ , then problem (1) can be feasible or infeasible for parts of intervals of coefficient and right-hand side, this is called weakly feasible. We will obtain intervals for qa and qb whereby problem (8) is feasible. On the other word, determination domain interval such as model (8) become strongly feasible.

Determining feasible solution

3.1

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Determination of minimum qb

If in system (8) be replaced qa = 0 , the system (9) is resulted. n l l u l i = 1, . . . , m (9) j=1 aij xj ≤ bi + qb (bi − bi ) The system (9) is feasible. Because if qb = 1 then feasible region (9) is converted to feasible region SI . Now we will reduce qb , in so that system (9) would remain feasible. In order to, consider the following model: qb∗ = min qb n l l u l s.t. i = 1, . . . , m j=1 aij xj ≤ bi + qb (bi − bi ) j = 1, . . . , n (10) qb ≥ 0 , xj ≥ 0 The linear programming (10) has a solution qb = 1, then is feasible. The optimal solution of (10) is less or equal 1 , therefore,0 ≤ qb∗ ≤ 1 . Theorem 3.1 The model (10) is infeasible for qb ≤ qb∗ , means the feasible region (9) is empty for qb ≤ qb∗ . Proof: It’s trivial. Since the model (10) is feasible for qb∗ , thus the model (9) is feasible. Therefore, for qb ≤ qb∗ and qa = 0 , the model (8) is infeasible. Theorem 3.2 The system(8)is infeasible for qb ≤ qb∗ and for all values qa . Proof:   qa ≥ 0 ⇒ alij ≤ alij + qa (auij − alij ) ⇒ nj=1 alij xj ≤ nj=1 [alij + qa (auij − alij )]xj Appositionally, if problem (8) is feasible, then n l u l l u l j=1 [aij + qa (aij − aij )]xj ≤ bi + qb (bi − bi ) Since qb ≤ qb∗ , thus bli + qb (bui − bli ) ≤ bli + qb∗ (bui − bli ) Regarding above statement and transitive relation, it will be resulted: n l l ∗ u l i = 1, . . . , m j=1 aij xj ≤ bi + qb (bi − bi ) which violate (11), therefore, theorem is valid. Now, consider the system (8) with qb = qb∗ . l u l l ∗ u l (12) j=1 [aij + qa (aij − aij )]xj ≤ bi + qb (bi − bi ) The inequality (12) for qa = 0 is feasible. To find maximum value for qa , so that the statement (12) is feasible, consider the following model: qa∗ = max qa n l u l l ∗ u l s.t. i = 1, . . . , m j=1 [aij + qa (aij − aij )]xj ≤ bi + qb (bi − bi ) j = 1, . . . , n (13) 0 ≤ qa ≤ 1 , xj ≥ 0 The model (13) is feasible for qa = 0, then has optimal solution,qa∗ , and 0 ≤ qa∗ ≤ 1. Thus the optimal solution is bounded. But this model is nonlinear and would be solved with nonlinear method. n

Theorem 3.3 The model (13) for qa > qa∗ is infeasible. Proof: It’s trivial,because otherwise contradicts qa∗ to be maximum.

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corollary 3.1 The system (8) for qa = qa∗ and qb = qb∗ is feasible, and according to theorem 2.4 for each qa ≤ qa∗ and qb ≥ qb∗ is always feasible.

3.2

Determination of maximum qa

Consider the system (8) with qb = 1 n l u l u i = 1, . . . , m j=1 [aij + qa (aij − aij )]xj ≤ bi j = 1, . . . , n (14) xj ≥ 0 This model for qa = 0 is feasible. Now we want to find maximum value of qa , so that the model (14) would stay feasible. In order to, use of the model (15):  qa∗ = max qa n l u l u s.t. i = 1, . . . , m j=1 [aij + qa (aij − aij )]xj ≤ bi j = 1, . . . , n (15) 0 ≤ qa ≤ 1 , xj ≥ 0  Since qa = 0 is a feasible solution for (15) and 0 ≤ qa∗ ≤ 1 , then the above  model is feasible and bounded. If qa∗ is optimal solution of the model (15),  the model will be infeasible for qa > qa∗ , therefore, feasible region of (14) is ∗ empty. (Otherwise,qa can not be maximal).  Since the model (15) for qa ≤ qa∗ is feasible, and also qa∗ is a feasible solution  for (15), thus qa∗ ≤ qa∗ . The model (8) is infeasible for all qa > qa∗ and all values for qb . consider the model (8) with qa = qa∗ . n l ∗ u l l u l i = 1, . . . , m j=1 [aij + qa (aij − aij )]xj ≤ bi + qb (bi − bi ) xj ≥ 0 j = 1, . . . , n (16) A solution for (16) is qb = 1 . We look for minimum value qb , so that model (16) be stayed feasible. For this purpose, we apply the model (17) as follow:  qb∗ = min qb n l ∗ u l l u l s.t. i = 1, . . . , m j=1 [aij + qa (aij − aij )]xj ≤ bi + qb (bi − bi ) j = 1, . . . , n (17) qb ≥ 0 , xj ≥ 0 The model (17) has a feasible solution qb = 1 , therefore 0 ≤ qb∗ ≤ 1 . Since  (17) for qb ≥ qb∗ is feasible, then (16) is also feasible. 



Theorem 3.4 The system (8) for qb < qb∗ and qa ≥ qa∗ is infeasible. Proof: n n n  ∗ n l u l l u l j=1 aij xj + qa j=1 (aij − aij )xj ≤ j=1 aij xj + qa j=1 [aij − aij )]xj ∗ Appositionally, if problem (8) is feasible for qb < qb , then the right hand side of above statement isn’t greater than bli +qb (bui −bli ) , therefore violate minimum     qb∗ , thus the model (8) for qa = qa∗ and qb ≤ qb∗ is infeasible. If qa ≤ qa∗ and ∗ qb < qb then the model (8) has solution. n n n  ∗ n l u l l u l j=1 aij xj + qa j=1 [aij − aij )]xj ≤ j=1 aij xj + qa j=1 [aij − aij )]xj  ⇒ bli + qb∗ (bui − bli ) ≤ bli + qb (bui − bli )

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Determining feasible solution 

Also if qb ≥ qb∗ , we show the system (8) is infeasible. Since if doesn’t mean n  l xj + qa nj=1 (auij − alij )xj ≤ bli + qb (bui − bli ) qb ≤ 1 ⇒ bli + qb (bui − bli ) ≤ ij j=1 a   bui ⇒ nj=1 alij xj + qa nj=1 (auij − alij )xj ≤ bui This contradicts first suppose. 



Therefore the model (8) for ∀qb ≥ qb∗ and qa ≤ qa∗ is feasible. The above discussion and feasible and infeasible regions are shown in table 1.

0 .. . qa∗

.. .  qa∗ .. . 1



0 · · · qb∗ φ φ F

· · · qb∗ F F

··· 1 F F

φ φ

φ φ

F F

F F

F F

F F

F F

φ φ

φ φ

φ

φ

F F

F F

F F

φ φ

φ φ

φ φ

φ φ

φ φ

φ φ

φ φ

F: feasible , φ: infeasible





In the distance of qa∗ < qa < qa∗ and qb∗ < qb < qb∗ , the problem will be feasible stair-like.

4

Numerical example

Consider interval inequality system as follows: ⎧ ⎪ ⎪ ⎪ ⎨

[2, 3]x1 + [1, 3]x2 + [−2, 1]x3 ≤ [2, 6] [1, 4]x1 + [−1, −0.5]x2 + [3, 5]x3 ≤ [−6, 5] ⎪ [−2, 1]x1 + [−3, 0]x2 + [1, 2]x3 ≤ [−2, −1] ⎪ ⎪ ⎩ xj ≥ 0 j = 1, 2, 3 qa∗

In accordance with the models (10) and (13), minimum qb∗ = 0.267 also   = 0 , and if qa∗ = 0.85 then qb∗ = 1.

qa∗ = 0 0.1 0.2 0.7 ∗ qa = 0.85 .. . qa = 1

0 · · · qb∗ = 0.267 φ φ F φ φ φ φ φ φ φ φ φ φ φ φ φ φ

φ φ

φ φ

0.3 0.4 F F F F φ F φ φ φ φ φ φ

φ φ



0.5 0.8 qb∗ = 1 F F F F F F F F F F F F φ φ F φ φ

φ φ

φ φ

Having used the above-mentioned example, the approach is going to be compared with the approach represented in Leon’s article [3].

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Constraint

qa = 0

constraint constraint constraint qa = 0.5 constraint constraint constraint qa = 1 constraint constraint constraint

1 2 3 1 2 3 1 2 3

RHS in

RHS in Leon’s model

RHS in interval model

infeasible state

for feasible state

for feasible state

2 6 2 2 6 2 2 6 2

4 4 2 5.499 2.0623 -1.9377 19 3.166 0.1667

3.068(qb = 0.267) 3.063 1.733 3.068(qb = 0.42) 1.38 1.58 10(qb = 2) 16 0

The determine of the right-hand side values to get a feasible answer in the represented model in this paper carries less distance comparing to Leon’s model.According above example,less changing in right-hand side can attain a feasible solution, in other hand, greater interval of right-hand side causes feasible solution for problem.

5

conclusion

This investigation proposed a method to solve infeasibility imprecise inequality problems by the concept of degree of inequality holding true. The discussions are accomplished on solution of linear inequalities and usually infeasibility has been removed with the use of change coefficients or right hand sides. But few works have studied infeasibility in imprecise linear inequality. In this paper, we have reviewed a method to solve interval inequalities, and then have inspected states of solution. In the later section, we have respected the problem while it was weakly feasible, and by proposed approach,the greatest intervals of technological coefficients and right-hand sides are computed to attain strongly feasible solution.

References [1] P. Amaral, P. Barahona, ”Connections between the total least squares and the correction of an infeasible system of linear inequalities”, Linear Algebra and its Applications , 395 (2005), 191 - 210. [2] M. Fiedler, J.Nedoma,J.Ramik , J.Rohn, K.Zimmermann, ”Linear optimization problems with inexact data”, Springer Science + Business Media, 2006.

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Determining feasible solution

[3] M. Gen, R.Cheng, ”genetic algorithms and engineering design”, Wiley, New York, 1997. [4] T. Leon, V. Liern, ”A fuzzy method to repair infeasibility in linearly constraint problems”, Fuzzy Sets and Systems , 122 (2001) 237-243 [5] G.M. Roodman,” Post-infeasibility analysis ming”,Manage. Sci. 25 (1979) 916- 922.

in

linear

program-

[6] A.A. Vatolin, ”Parametric approximation of inconsistent systems of linear equations and inequalities”, Seminarber., Humboldt-Univ. Berlin, Sekt. Math. 81 (1986) 145-154. [7] H.F. Wang, H.L. Liao ”Theory and Methodology resolution on the infeasibility of variational inequality ”, European Journal of Operational Research 106 (1998) 198-203. Received: April 6, 2008