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Stability achievement scalarization function for multiobjective nonlinear programming problems
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Applied Mathematical Modelling 32 (2008) 1044–1055 www.elsevier.com/locate/apm
Stability achievement scalarization function for multiobjective nonlinear programming problems Mohamed Abd El-Hady Kassem
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Department of Mathematics, Faculty of Science, Tanta University, Tanta, Egypt Received 1 April 2006; received in revised form 1 February 2007; accepted 27 February 2007 Available online 23 March 2007
Abstract In this paper, we present a method to determine the stability of nondominated criterion vectors using a modified weighted achievement scalarization metric. This method is based on the application of a particular objective function which scalarizes and parameterizes the original multiobjective nonlinear programming problem. Also, we show that this modified weighted achievement metric coincides with the metric introduced by Choo and Atkins [E.-U. Choo, D.R. Atkins, Proper efficiency in nonconvex multicriteria programming, Math. Oper. Res. 8 (1983) 467–470] and Kaliszewski [I. Kaliszewski, A modified weighted Tchebycheff metric for multiple objective programming, Comput. Oper. Res. 14 (1987) 315–323] in cases when sets of all criterion vectors are finite or polyhedral. Ó 2007 Elsevier Inc. All rights reserved. Keywords: Multiobjective nonlinear programming; Achievement scalarization function, nondominated solutions; Stability
1. Introduction In an earlier work, Steuer and Choo [1] introduced a procedure which samples the efficient set by computing the nondominated criterion vector that is closest to an ideal criterion vector according to a randomly weighted Tchebycheff metric for multiobjective programming and show that this metric coincides with the metric introduced by Choo and Atkins [2]. Kassem [3] introduced the notion of the stability set of the first kind for multiobjective nonlinear programming with fuzzy parameters. Kassem [4] introduced an interactive approach for solving vector optimization problems and determined the stability set of the first kind for this problems. Wierzbicki [5] presented methodological foundations, basic concepts and the notions of reference points and achievement functions, neutral and weighted compromise solutions. Korhonen [6] studied the problem of multiple objective programming support. The scalarization in vector optimization was introduced by Chankong and Haimes [7] and Jahn [8]. Li and Cheng [9] introduced the stability of multiobjective dynamic programming problems with fuzzy parameters in the objective functions and in the constraints.
*
Tel.: +20 403328272. E-mail address: [email protected]
0307-904X/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2007.02.028
M.A.E.-H. Kassem / Applied Mathematical Modelling 32 (2008) 1044–1055
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In this paper, we use a modified weighted achievement metric to obtain the set of nondominated solutions for multiobjective nonlinear programming problems and then find the stability for this solution. And we show that this modified weighted achievement metric (in cases when sets of all criterion vectors are finite or polyhedral) coincides with the metric introduced by both Kaliszewski [10,13] and Choo and Atkins [2]. Also, we study the stability of solutions to multiobjective nonlinear programming problems, using a modified weighted achievement metric and introduce an example to explain compute this stability of solutions. This paper also answer the remaining open question in [10]: if are there other forms augmentation of the (weighted) Tchebycheff metric which would have properties similar to that of the modified and augmented Tchebycheff metrics? 2. Problem formulation We consider the following 8 max > > > > > max > > < . MONLP : .. > > > > max > > > : subject to
multiobjective nonlinear programming (MONLP) problem: f1 ðxÞ ¼ z1 ; f2 ðxÞ ¼ z2 ;
fm ðxÞ ¼ zm ; x 2 X ¼ fx 2 Rn jgj ðxÞ 6 0;
j ¼ 1; 2; . . . ; kg;
where fi ðxÞ; i ¼ 1; 2; . . . ; m and gj ðxÞ; j ¼ 1; 2; . . . ; k are continuous and differentiable nonlinear functions and X Rn is bounded. We denote by Z Rm the set of all feasible criterion vectors, where Z is the image of X under ðf1 ðxÞ; . . . ; fm ðxÞÞ; and by N Z the set of all nondominated criterion vectors with respect to the nonnegative orthant Rmþ . Definition 1. z 2 Z is nondominated with respect to Rm þ if z 2 Z and zi P zi ; i ¼ 1; 2; . . . ; m; implies z ¼ z : 1 With this notation, f ðN Þ X is the set of nondominated solutions.
Definition 2. x 2 X is an efficient solution to the problem (MONLP) if and only if there does not exist another x 2 X such that fi ðxÞ P fi ðx Þ; i ¼ 1; 2; . . . ; m; with at least one strict inequality holds. To measure the distance between any f ðxÞ 2 Z and a certain reference vector f , Steuer and Choo [1] introduced the augmented weighted Tchebycheff metric k k jkf f ðxÞkj1 ¼ kf f ðxÞk1 þ qeT jf f ðxÞj;
where f i ¼ maxx2X fi ðxÞ þ Di ¼ maxz2Z zi þ Di ¼ zi þ Di ; Di > 0; i ¼ 1; 2; . . . ; m (f can be regarded as the posik tively perturbed utopia (ideal) point Pm z), kf f ðxÞk1 ¼ maxi fki j f i fi ðxÞ jg is the weighted TTchebycheff metn ric with k 2 X ¼ fk 2 R ki > 0; i¼1 ki ¼ 1g, q is a sufficiently small positive scalar, and e is the vector of T ones and j z j denotes ðj zi j; . . . ; j zm j Þ . Kaliszewski [10] introduced the modified weighted Tchebycheff metric k kf f ðxÞk1 q ¼ maxfki ½jf i fi ðxÞj þ qeT jf f ðxÞjg: i
Wierzbicki and Makowski [11] introduced the following basic form of an order consistent with the scalarizing function that has been widely used, called the achievement metric ( ) m X i fi ðxÞj j fj ðxÞj j f j f þe Sðf ; f ðxÞÞ ¼ max ; 16i6m jf i f i j jf j f j j j¼1 where f i ¼ maxx2X fi ðxÞ; i ¼ 1; 2; . . . m; and can be regarded as the positively perturbed utopia (ideal) point, f i ¼ minx2X fi ðxÞ; i ¼ 1; 2; . . . ; m:
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We now introduce a somewhat different measure, called the modified weighted achievement scalarization metric, as follows: " # m X jf i fi ðxÞj jf j fj ðxÞj S ðf ; f ðxÞÞ ¼ max ai þe ; 16i6m jf i f i j jf j f j j j¼1 a
ð1Þ
where f i ¼ maxx2X fi ðxÞ þ Di ; i ¼ 1; 2; . . . m; f i ¼ minx2X fi ðxÞ; i ¼ 1; 2; . . . ; m, a 2 K ¼ fa 2 Rm j ai > 0; Pm i¼1 ai ¼ 1g; e is a sufficiently small positive scalar. Later in the paper we show that this metric coincides (up to scalar multiplication) with both the metric introduced by both Choo and Atkins [2] and Kaliszewski [10]. By the definition of f i , f i fi ðxÞ > 0 for all f ðxÞ 2 Z: Thus, in all formula we can drop the absolute value sign. An element f ðxÞ 2 Z is said to define the isoquant of S a ðf ; f ðxÞÞ if and only if
ai ¼ f f ðxÞ i
i
f i f i
þe
1 Pm
f j fj ðxÞ j¼1 f j f j
2 m X 4 i¼1
31 1 Pm fj fj ðxÞ5 ; f i fi ðxÞ þ e j¼1 f j f j f i f i
i ¼ 1; 2; . . . ; m:
Then, 2 m X S a ðf ; f ðxÞÞ ¼ 4
31
f i fi ðxÞ i¼1 f f i i
1 Pm fj fj ðxÞ5 : þ e j¼1 fj f j
The following figure represents the isoquants of the corresponding metric restricted to the set e ff ðxÞ 2 Rm j fi ðxÞ 6 f i ; i ¼ 1; 2; . . . ; mg, where h ¼ tan1 1þe (see [10,11]). θ
f (x)
.f
θ
Theorem 1. Let Z be finite, I Z ¼ fi j f ðxi Þ 2 Zg; I N ¼ fi j f ðxi Þ 2 N g and f ðxp Þ 2 N : Then f ðxp Þ uniquely solves ( " #) m X i fi ðxÞ f ðxÞ f f j j p min S a ðf ; f ðxÞÞ ¼ min max ai þ ep ; f ðxÞ2Z f ðxÞ2Z 16i6m fi fi f j f j j¼1 where ep is any positive real number satisfying the inequality
ep <
min
8 minj > > p >0
> Z nfpg > Pm j2I > fl ðxj Þfl ðxp Þ >0 : l¼1
f l f l
and the fapi g is specified as
f i fi
Pm
l¼1
fi ðxp Þfi ðxj Þ9 > f i f i >
fl ðxj Þfl ðxp Þ f l f l
> = > > > ;
;
ðIÞ
M.A.E.-H. Kassem / Applied Mathematical Modelling 32 (2008) 1044–1055
api
1 ¼ m P f i fi ðxp Þ þe f i f i
j¼1
f j fj ðxp Þ f j f j
2 m X 4 i¼1
f i fi ðxp Þ f i f i
þe
31
1 Pm
f j fj ðxp Þ j¼1 f j f j
5 :
Proof. Suppose f ðxj Þ 2 Z; f ðxj Þ 6¼ f ðxp Þ. There exists at least one index i such that fi ðxp Þ fi ðxj Þ fi ðxp Þ > fi ðxj Þ ) > : fi fi fi fi Then for all such i we have m m X X fi ðxp Þ fl ðxp Þ fi ðxj Þ fl ðxj Þ p p þ e > þ e f i f i f l f l f i f i f l f l l¼1 l¼1
and therefore m m X X fi ðxp Þ fl ðxp Þ fi ðxj Þ fl ðxj Þ p p þ e 66 ¼ þ e : f i f i f l f l f i f i f l f l l¼1 l¼1
To see this let us observe that it is obviously true for m X fl ðxp Þ fl ðxj Þ P 0: f l f l l¼1
Suppose now that m X fl ðxj Þ fl ðxp Þ > 0: f l f l l¼1
Then
i:
minj p
fi ðx Þfi ðx Þ >0 f i f i
m X fi ðxp Þ fi ðxj Þ fl ðxj Þ fl ðxp Þ p > e f i f i f l f l l¼1
is valid if min
ep <
f ðxp Þf ðxj Þ i: i f fi >0 i i
Pm
l¼1
fi ðxp Þfi ðxj Þ f i f i
:
fl ðxj Þfl ðxp Þ f l f l
We assume that ep satisfies min
ep <
min Pm j2Ifl ðxZ jnfpg; Þfl ðxp Þ l¼1
f l f l
f ðxp Þf ðxj Þ i: i f fi >0 i i
Pm
l¼1
>0
with fapi g specified as in (*) 2 m X a min S ðf ; f ðxÞÞ ¼ 4 f ðxÞ2Z
i¼1
fi ðxp Þfi ðxj Þ f i f i
fl ðxj Þfl ðxp Þ f l f l
f i fi ðxp Þ f i f i
þe
;
1 Pm p
f l fl ðxp Þ l¼1 f l f l
1047
31 5 :
ðÞ
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Indeed, by the assumption about the value of ep we have guaranteed that for each j 2 I Z n fpg there exists an i such that m m X X fi ðxp Þ fl ðxp Þ fi ðxj Þ fl ðxj Þ p p þ e > þ e f i f i f l f l f i f i f l f l l¼1 l¼1
or equivalently m m X X f i fi ðxp Þ f l fl ðxp Þ f i fi ðxj Þ f l fl ðxj Þ p p þ e < þ e : f i f i f l f l f i f i f l f l l¼1 l¼1
Hence, api
"
# " # m m j X f i fi ðxp Þ p X f l fl ðxp Þ f l fl ðxj Þ p f i fi ðx Þ a p p þe þe ¼ S ðf ;f ðx ÞÞ < ai f i f i f l f l fifi f l f l l¼1 l¼1 " # m f i fi ðxj Þ p X f l fl ðxj Þ a j 6 max api þe l f l ¼ S ðf ;f ðx ÞÞ; j 2 I Z n fpg; i fi fi f l¼1 h
which completes the proof.
Lemma 1. As the previous proof, Theorem 1 holds with min
e0p <
min Pm j2Ifl ðxN jnfpg; Þfl ðxp Þ l¼1
f l f l
f ðxp Þf ðxj Þ i: i f fi >0 i i
Pm
l¼1
>0
fi ðxp Þfi ðxj Þ f i f i
fl ðxj Þfl ðxp Þ f l f l
:
The proof is similar as in Theorem 1. Theorem 2. Let Z be finite and 9 8 9> 8 > > > fi ðxl Þfi ðxj Þ > > > = < => < f i f i : e < min min min P m fs ðxj Þfs ðxl Þ>> l2I N > > j2I N nflg f ðxl Þf ðxj Þ > > ; : i: i f fi >0 s f s P s¼1 > > f j l m fs ðx Þfs ðx Þ i i ; : s¼1
f s f s
>0
Pm
¼ 1g such that f ðxp Þ solves the
i ¼ 1; 2; . . . ; m;
ðIIIÞ
Then f ðxp Þ 2 N if and only if there exist a a 2 K ¼ fa 2 Rm j ai > 0; following problem: min subject to
ðIIÞ
i¼1 ai
g;
" # m X f i fi ðxÞ f j fj ðxÞ þe g P ai ; fi fi f j f j j¼1
f i ðxÞ 2 Z: Proof. First, follows from Theorem 1 and Lemma 1 when fai g is defined as the fapi g in (*). f ðxp Þ is then a unique solution of the problem. Conversely, suppose that f ðxp Þ 62 N minimizes the above problem for some a 2 K. With f ðxp Þ dominated there exists a f ðxq Þ such that f ðxq Þ P f ðxp Þ; f ðxq Þ 6¼ f ðxp Þ. Since f i fi ðxq Þ f i fi ðxp Þ 6 f i f i fi fi
M.A.E.-H. Kassem / Applied Mathematical Modelling 32 (2008) 1044–1055
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and for any e > 0 m m X X f j fj ðxq Þ f j fj ðxp Þ e < e ; f j f j f j f j j¼1 j¼1 f ðxq Þ implies a smaller objective function value than f ðxp Þ. However, this contradicts the assumption about f ðxp Þ. Thus, f ðxp Þ 2 N . h Remark 1. We note from Theorems 1 and 2 that, if Z is polyhedral I Z ¼ fi j f ðxi Þ is an extreme point of Z}, I N ¼ fi 2 I Z j f ðxi Þ is nondominated}, and e is defined as in (II). Then f ðxp Þ 2 N if and only if there exists a a 2 K such that f ðxp Þ solves the problem (III). 3. Related results Let us assume that ðgp ; xp Þ be an optimal solution to the modified weighted achievement problem (III) with all the inequality constraints of (III) active . Then the intersection of the m active inequality constraints in the objective function space can be obtained by solving the following simultaneous equation: m X f i fi ðxp Þ f j fj ðxp Þ gp þ e ¼ p; ai f i f i f j f j j¼1 which can be written in the form m m X X fi ðxp Þ fj ðxp Þ gp f i f j þ e ¼ þ e p: f j f j ai f i f i f j f j f i f i j¼1 j¼1 Hence, this equation can be written in the following matrix form: 3 2 m P 3 2 f j f 1 gp 2 f1 ðxp Þ 3 1þe e e e e 1 f 1 þ e j f j ap f f 6 1 7 j¼1 f 1 f1 6 e 7 6 1þe e e e 7 76 f ðxp Þ 7 6 6 7 m P f j f2 gp 7 2 7 6 e 7 6 6 ap 7 e 1þe e e 76 f2 f2 7 6 f2 f 2 þ e 6 f j f j 2 7 76 6 7¼6 j¼1 6 e 7: 7 6 6 e e 1 þ e e 7 76 ... 7 6 6 7 . .. .. 74 .. .. .. .. 6 .. 7 5 6 p 5 4 . 7 6 fm ðx Þ . . . . . m 5 4 P p f fm g j f m fm þe ap e e e e 1 þ e f m f m f j f j j¼1
m
Denoting the left-hand matrix by B, its determinant becomes detðBÞ ¼ 1 þ me 6¼ 0. Also, denoting the adjugate matrix of B by ½~ bij , we have ~ bii ¼ 1 þ ðm 1Þe; ~bji ¼ e: Therefore, the solution to this simultaneous matrix equation is represented by !!, m m X X gp f j f l p ~ fi ðx Þ ¼ þe p bji det B; fj fj f l f l ai j¼1
l¼1
g . which implies that fi ðxÞ ¼ fiffi i a ð1þmeÞ i
Lemma 2. Assume: (1) Z is finite, f ðxp Þ 2 N and f ðxq Þ 2 Z such that f ðxp Þ 6¼ f ðxq Þ. (2) 8 2 31 > > m > > 6P 7 1 > > 4 f f ðxp Þ 1P 5 ; f i ðxp Þ 6¼ f i 8i; m m < fi fi ðxp Þ p P f j fj ðxp Þ f j fj ðxp Þ i i p i¼1 þe þe f i f i f i f i f j f j f j f j api ¼ j¼1 j¼1 > > > > 1; f i ðxp Þ ¼ f i ; > > : 0; f i ðxp Þ 6¼ f i ; but 9j 3 fj ðxp Þ ¼ f j ;
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(3) bp be the minimal objective function value of the problem min
b;
subject to b P
" api
# m X f i fi ðxp Þ f j fj ðxp Þ þe ; f i f i f j f j j¼1
i ¼ 1; 2; . . . ; m;
where e is any positive real number. Then there does not exist a f ðxq Þ 6¼ f ðxp Þ such that f ðxq Þ 2 Z lies in " ! bp f i p m ; þ1 ; Wðb Þ ¼ ff ðxÞ 2 R jfi ðxÞ 2 p f i f i ai ð1 þ meÞ when api > 0; i ¼ 1; 2; . . . ; mg. Proof Case 1 : fi ðxp Þ 6¼ f i 8i: Substituting for api in each of the m-constraints " # m p X f j fj ðxp Þ p f i fi ðx Þ b P ai þe ; i ¼ 1; 2; . . . ; m; f i f i f j f j j¼1 we have "
m X f i fi ðxp Þ f j fj ðxp Þ bP þ e f i f i f j f j j¼1
Thus p
b ¼
api
#1 ;
1 6 i 6 k:
" # m X f i fi ðxp Þ f j fj ðxp Þ þe ; f i f i f j f j j¼1
1 6 i 6 k:
Or m m X X fi ðxp Þ fj ðxp Þ bp f i f j þ e ¼ þ e : f j f j ap f i f i f j f j f i f i j¼1 j¼1
This equation can be written as in the above matrix form which implies to !!, m m X X bp f j f l p ~ fi ðx Þ ¼ þe p bji det B; fj fj f l f l ai j¼1 l¼1 where det B ¼ 1 þ me; ~ bii ¼ 1 þ ðm 1Þe; ~ bji ¼ e and then we have p b fi ; 1 6 i 6 m: p fi ðxp Þ ¼ a ð1 þ meÞ fi fi i Since f ðxp Þ is nondominated, there does not exist a q 6¼ p such that f ðxq Þ 2 Wðbp Þ: Case 2 : there exist j such that fj ðxp Þ ¼ f j . With there being only one j such that fj ðxp Þ ¼ f j , api ¼ 0 for all i 6¼ j: Thus bp ¼ 0 and ( " !) f j p m Wðb Þ ¼ f ðxÞ 2 R jfj ðxÞ 2 ; þ1 : fj fj Since f ðxp Þ is the member of Z for which the jth component is greater than or equal to f j , there does not exist a q 6¼ p such that f ðxq Þ 2 Wðbp Þ. h
M.A.E.-H. Kassem / Applied Mathematical Modelling 32 (2008) 1044–1055
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Suppose now that Z is composed of at least two objective vectors (as for example, in multiobjective programming problems with objective valued functions fi ; i ¼ 1; 2; . . . ; mÞ. We introduce the following parameter: d ¼ max i
jfi ðxÞ fi ðx0 Þj : jf i fi j
max0
f ðxÞ;f ðx Þ2N
By assumptions, d is finite and d > 0. It is easy now to verify that in the case of Theorem 2 we may take 1 . e < dðm1Þ1 1 1 1 Denoting e ¼ c1 so c1 < dðm1Þ1 that is c 1 > dðm 1Þ 1 then c > dðm 1Þ:We can now write the modified weighted achievement metric as ( " #) m X i fi ðxÞ f ðxÞ f f j j S a ðf ; f ðxÞÞ ¼ max ai þe 16i6m fi fi f j f j j¼1 ( " #) m 1 X f i fi ðxÞ f j fj ðxÞ ¼ max ai þ i c 1 j¼1 f j f j fi fi ( " #) m 1 f i fi ðxÞ X f j fj ðxÞ max ai ðc 1Þ ¼ þ c1 i fi fi f j f j j¼1 ( " #) 1 f i fi ðxÞ X f j fj ðxÞ max ai c ¼ þ c1 i fi fi f j f j j6¼i ( " !# ) 1 f f ðxÞ max ai M ¼ i c1 f f i
where M is a m m matrix defined as c if i ¼ j; ðMÞij ¼ 1 if i 6¼ j: The last form of the modified weighted achievement metric is exactly that introduced in [10]. In general, the modified weighted achievement metric can be rewritten as ( " !# ) f f ðxÞ max ai Bmm ; i f f i
where Bmm is a m m matrix, which is defined in the above simultaneous matrix equation in the form: ð1 þ eÞ if i ¼ j; ðBmm Þij ¼ e if i 6¼ j: 1 By elementary calculations, Bmm B1 mm ¼ I where I is the identity matrix, we have the inverse matrix Bmm which has the following form: ( 1þðm1Þe if i ¼ j; 1 1þme ðBmm Þij ¼ e if i 6¼ j: 1þme
Now we are able to show that the modified weighted achievement metric is (up to scalar multiplication) the metric introduced by Choo and Atkins [2] and Kaliszewski [10]. Choo and Atkins [2] defined 1 kf f ðxÞkk;# 1 ¼ maxfki j ½I # ðf f ðxÞÞi jg; i
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M.A.E.-H. Kassem / Applied Mathematical Modelling 32 (2008) 1044–1055
where I# is a m m matrix 1 if i ¼ j; ðI # Þij ¼ # if i 6¼ j; where # satisfies the inequality 1 < # < 0: The limit for # was chosen to ensure positive values of all entries of 2m I 1 # . e 1þme Let # ¼ 1þðm1Þe and as e is positive # is negative. Then I # ¼ 1þðm1Þe I 1 e and ( " !# ) m X f j fj ðxÞ f f ðxÞ S a# ðf ; f ðxÞÞ ¼ max ai I 1 þe # 16i6m f j f j f f j¼1 i ( " !# ) 1 m X 1 þ me f ðxÞ f f ðxÞ f j j I 1 þe ¼ max ai 16i6m 1 þ ðm 1Þe e f j f j f f j¼1 ( " !# ) i m X 1 þ ðm 1Þe f f ðxÞ f j fj ðxÞ Ie þe ¼ max ai i 1 þ me f j f j f f j¼1 ( " !# ) i m X 1 þ ðm 1Þe f f ðxÞ f j fj ðxÞ max ai I e þe ¼ 16i6m 1 þ me f j f j f f j¼1 i
1 þ ðm 1Þe a S ðf ; f ðxÞÞ: ¼ 1 þ me With the above equivalence we can make use of the result of Choo and Atkins [2] and Kaliszewski [10] concerning properly nondominated elements of N. If Z is finite or polyhedral, all elements of N are properly nondominated. Choo and Atkins [2] proved that for any positive e, each nondominated element of Z determined by minimizing S a ðf ; f ðxÞÞ over Z satisfies the following inequality: 1 1 þ ðm 1Þe ¼ ; # e where L > 0 is a positive scalar. This formula permits the necessity of determining the bounds on e as in Theorem 2 and Remark 1 to be avoided. Instead of that, we can use any positive number e provided that no element of N for which L P 1þðm1Þe is of our interest. e In the following section, we obtain the range of the positive number e that corresponding the same solution for the modified weighted achievement scalarization convex programming problem. L<
4. Stability set of the first kind Here, we assume the problem (MONLP) is stable, therefore the problem (III) is also stable [12]. Definition 3. The stability set of the first kind to the problem (III) (which determined the subset of the parametric ða; eÞ-space that has the same corresponding optimal solution) corresponding to the optimal solution ðxp ; gp Þ is defined by T ðxp ; gp Þ ¼ fða; eÞ 2 Rmþ1 jðxp ; gp Þis a solution of problem ðIIIÞg: The Kuhn–Tucker conditions of the problem (III) at a point xp take the following form: " # m m X X X ogj ðxp Þ 1 ofi ðxp Þ 1 ofl ðxp Þ ui ai þe mj ¼ 0; c ¼ 1; . . . ; n; þ oxc f i f i oxc f l f l oxc j2J i¼1 l¼1 m X i¼1
ui ¼ 1;
ð2Þ ð3Þ
M.A.E.-H. Kassem / Applied Mathematical Modelling 32 (2008) 1044–1055
# m X f i fi ðxp Þ f l fl ðxp Þ 6 0; g þ ai þe fi fi f l f l l¼1
1053
"
mj gj ðxp Þ ¼ 0;
i ¼ 1; 2; . . . ; m;
ð4Þ
j 2 J f1; 2; . . . ; kg;
ð5Þ
p
gj ðx Þ < 0; j 62 J ; ( " #) m X f i fi ðxp Þ f l fl ðxp Þ ui g þ ai þe ¼ 0; fi fi f l f l l¼1 ui P 0; mj P 0;
ð6Þ i ¼ 1; 2; . . . ; m;
ð7Þ
i ¼ 1; 2; . . . ; m; j 2 J:
ð8Þ ð9Þ
The determination of the stability set of the first kind, T ðxp ; gp Þ, depends only on whether any of the variables ui ; i ¼ 1; 2; . . . ; m and any of the variables mj ; j ¼ 1; 2; . . . ; k which solves the Eqs. (2), (3), (8) and (9) are positive or zero. Let ui ¼ 0; i 2 I f1; 2; . . . ; mg; ui > 0; i 62 I and mj ¼ 0; j 2 J f1; 2; . . . ; kg;
mj > 0; j 62 J ;
solve Eqs. (2), (3), (8) and (9) then in order that the other Kuhn–Tucker conditions Eqs. (4)–(7) are satisfied, we must have " # m X f i fi ðxp Þ f l fl ðxp Þ p g ¼ ai þe ; i 62 I; fi fi f l f l l¼1 " # m X f i fi ðxp Þ f l fl ðxp Þ p g P ai þe ; i 2 I: fi fi f l f l l¼1 Let us consider the following sets: A ¼ fI j ui ¼ 0; i 2 I; ui > 0; i 62 I and mj ¼ 0; j 2 J ; mj > 0; j 62 J solve the Eqs. (2), (3), (8) and (9)}, ( " # m X f i fi ðxp Þ f l fl ðxp Þ p p mþ1 p ; i 62 I; T I ðx ; g Þ ¼ ða; eÞ 2 R jg ¼ ai þe fi fi f l f l l¼1 " # ) m p X i fi ðxp Þ f ðx Þ f f l l gp P ai þe ; i2I : ð10Þ fi fi f l f l l¼1 It is clear from the definition of the stability set of the first kind that [ T I ðxp ; gp Þ: T ðxp ; gp Þ ¼
ð11Þ
I2A
The set T I ðxp ; gp Þ can be takes the following form: 8 " #" #1 m < gp f i fi ðxp Þ X f l fl ðxp Þ p p mþ1 ; T I ðx ; g Þ ¼ ða; eÞ 2 R je ¼ : ai fi fi f l f l l¼1 9 " #" #1 m = gp f i fi ðxp Þ X f l fl ðxp Þ e6 ; i 2 I : ; ai fi fi f l f l l¼1 Example. Let us consider the following MONLP problem: max
2
ðx1 3Þ þ x2 ¼ z1 ; 2
max x1 þ ðx2 4Þ ¼ z2 ; subject to x1 þ x2 6 5; x1 P 0;
x2 P 0:
i 62 I;
ð12Þ
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M.A.E.-H. Kassem / Applied Mathematical Modelling 32 (2008) 1044–1055
First we solve the following problems: 2
z1 ¼ ðx1 3Þ þ x2 ;
(i) max
subject to x1 þ x2 6 5; x1 P 0; x2 P 0: The solution is z1 ¼ max z1 ¼ 14 at the point ð0; 5Þ. 2
z1 ¼ ðx1 3Þ þ x2 ; (ii) min subject to x1 þ x2 6 5; x1 P 0; x2 P 0: The solution is z1 ¼ min z1 ¼ 2 at the point (3, 2). 2 (iii) max z2 ¼ x1 þ ðx2 4Þ ; subject to x1 þ x2 6 5;
x1 P 0;
x2 P 0:
The solution is z2 ¼ max z2 ¼ 21 at the point (5, 0). 2
(iv) min
z2 ¼ x1 þ ðx2 4Þ ;
subject to x1 þ x2 6 5; x1 P 0; x2 P 0: The solution is z2 ¼ min z2 ¼ 0:75 at the point (0.5, 4.5). For assumption a1 ¼ 0:4; a2 ¼ 0:6 and e ¼ 0:3 the equivalent problem (III) takes the following form: min g; subject to 2
2
0:73128 0:04316ðx1 3Þ 0:04316x2 0:00592x1 0:00592ðx2 4Þ 6 g; 2
2
1:01892 0:03852x1 0:03852ðx2 4Þ 0:01494x2 0:01494ðx1 3Þ 6 g; x1 þ x2 6 5; x1 P 0;
x2 P 0:
By using the Kuhn–Tucker conditions (2)–(9), we obtain g ¼ 0:121112 at the point (0, 4.8). From the Eqs. (10)–(12), the stability set of the first kind takes the form T ð0; 4:8Þ ¼ fða; eÞ 2 R3 ja1 e P 0:99501633a2 þ ea2 g: 5. Conclusion This paper presents an approach to compute the stability of efficient solutions to multiobjective nonlinear problems, using a modified weighted achievement scalarizing metric. The method is based on a particular objective function which scalarizes and parameterizes the multiobjective nonlinear problem. Also, we show how to determine nondominated criterion vectors (and then nondominated solutions, if necessary) by the modified weighted achievement metric in cases when sets of all criterion vectors are finite or polyhedral, and we show that this modified weighted achievement metric coincides with the metric introduced by both Choo and Atkins [2] and Kaliszewski [10].
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Acknowledgement The author is very grateful to the Editor-in-Chief, Professor Mark Cross, and the anonymous referees for their insightful and constructive comments and suggestions that have led to an improved version of this paper. References [1] E.R. Steuer, E.-U. Choo, An Interactive weighted Tchebycheff Procedure for Multiple objective programming, Math. Program. 26 (1983) 326–344. [2] E.-U. Choo, D.R. Atkins, Proper efficiency in nonconvex multicriteria programming, Math. Oper. Res. 8 (1983) 467–470. [3] M. Kassem, Decomposition of the fuzzy parametric space in multiobjective nonlinear programming problems, Eur. J. Oper. Res. 101 (1997) 204–219. [4] M. Kassem, Interactive stability of vector optimization problems, Eur. J. Oper. Res. 134 (2001) 616–622. [5] P.A. Wierzbicki, Reference point Methods in vector optimization and Decision Support, Interim Report, IR-98-017/April (1998), International Institute for Applied Systems Analysis, Laxenburg, Austria. [6] P. Korhonen, Multiple objective programming support, Interim Report, IR-98-010/March 1998, International Institute for Applied Systems Analysis, Laxenburg, Austria. [7] V. Chankong, Y.Y. Haimes, Multiobjective Decision Making: Theory and Methodology, North-Holland, New York, 1983. [8] J. Jahn, Scalarization in vector optimization, Math. Program. 29 (1984) 203–218. [9] D. Li, C. Cheng, Stability on multiobjective dynamic programming problems with fuzzy parameters in the objective functions and in the constraints, Eur. J. Oper. Res. 158 (2004) 678–696. [10] I. Kaliszewski, A modified weighted Tchebycheff metric for multiple objective programming, Comput. Oper. Res. 14 (1987) 315–323. [11] P.A. Wierzbicki, M. Makowski, Multi-objective optimization in negotiation support, WP-92-007/1992, International Institute for Applied Systems Analysis, Laxenburg, Austria. [12] R. Rockafellar, Duality and stability in extremal problems involving convex functions, Pac. J. Math. 21 (1967) 167–181. [13] I. Kaliszewski, Norm scalarization and proper efficiency in vector optimization, Found. Control Eng. 11 (1986) 117–131.
Applied Mathematical Modelling 32 (2008) 1044–1055 www.elsevier.com/locate/apm
Stability achievement scalarization function for multiobjective nonlinear programming problems Mohamed Abd El-Hady Kassem
*
Department of Mathematics, Faculty of Science, Tanta University, Tanta, Egypt Received 1 April 2006; received in revised form 1 February 2007; accepted 27 February 2007 Available online 23 March 2007
Abstract In this paper, we present a method to determine the stability of nondominated criterion vectors using a modified weighted achievement scalarization metric. This method is based on the application of a particular objective function which scalarizes and parameterizes the original multiobjective nonlinear programming problem. Also, we show that this modified weighted achievement metric coincides with the metric introduced by Choo and Atkins [E.-U. Choo, D.R. Atkins, Proper efficiency in nonconvex multicriteria programming, Math. Oper. Res. 8 (1983) 467–470] and Kaliszewski [I. Kaliszewski, A modified weighted Tchebycheff metric for multiple objective programming, Comput. Oper. Res. 14 (1987) 315–323] in cases when sets of all criterion vectors are finite or polyhedral. Ó 2007 Elsevier Inc. All rights reserved. Keywords: Multiobjective nonlinear programming; Achievement scalarization function, nondominated solutions; Stability
1. Introduction In an earlier work, Steuer and Choo [1] introduced a procedure which samples the efficient set by computing the nondominated criterion vector that is closest to an ideal criterion vector according to a randomly weighted Tchebycheff metric for multiobjective programming and show that this metric coincides with the metric introduced by Choo and Atkins [2]. Kassem [3] introduced the notion of the stability set of the first kind for multiobjective nonlinear programming with fuzzy parameters. Kassem [4] introduced an interactive approach for solving vector optimization problems and determined the stability set of the first kind for this problems. Wierzbicki [5] presented methodological foundations, basic concepts and the notions of reference points and achievement functions, neutral and weighted compromise solutions. Korhonen [6] studied the problem of multiple objective programming support. The scalarization in vector optimization was introduced by Chankong and Haimes [7] and Jahn [8]. Li and Cheng [9] introduced the stability of multiobjective dynamic programming problems with fuzzy parameters in the objective functions and in the constraints.
*
Tel.: +20 403328272. E-mail address: [email protected]
0307-904X/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2007.02.028
M.A.E.-H. Kassem / Applied Mathematical Modelling 32 (2008) 1044–1055
1045
In this paper, we use a modified weighted achievement metric to obtain the set of nondominated solutions for multiobjective nonlinear programming problems and then find the stability for this solution. And we show that this modified weighted achievement metric (in cases when sets of all criterion vectors are finite or polyhedral) coincides with the metric introduced by both Kaliszewski [10,13] and Choo and Atkins [2]. Also, we study the stability of solutions to multiobjective nonlinear programming problems, using a modified weighted achievement metric and introduce an example to explain compute this stability of solutions. This paper also answer the remaining open question in [10]: if are there other forms augmentation of the (weighted) Tchebycheff metric which would have properties similar to that of the modified and augmented Tchebycheff metrics? 2. Problem formulation We consider the following 8 max > > > > > max > > < . MONLP : .. > > > > max > > > : subject to
multiobjective nonlinear programming (MONLP) problem: f1 ðxÞ ¼ z1 ; f2 ðxÞ ¼ z2 ;
fm ðxÞ ¼ zm ; x 2 X ¼ fx 2 Rn jgj ðxÞ 6 0;
j ¼ 1; 2; . . . ; kg;
where fi ðxÞ; i ¼ 1; 2; . . . ; m and gj ðxÞ; j ¼ 1; 2; . . . ; k are continuous and differentiable nonlinear functions and X Rn is bounded. We denote by Z Rm the set of all feasible criterion vectors, where Z is the image of X under ðf1 ðxÞ; . . . ; fm ðxÞÞ; and by N Z the set of all nondominated criterion vectors with respect to the nonnegative orthant Rmþ . Definition 1. z 2 Z is nondominated with respect to Rm þ if z 2 Z and zi P zi ; i ¼ 1; 2; . . . ; m; implies z ¼ z : 1 With this notation, f ðN Þ X is the set of nondominated solutions.
Definition 2. x 2 X is an efficient solution to the problem (MONLP) if and only if there does not exist another x 2 X such that fi ðxÞ P fi ðx Þ; i ¼ 1; 2; . . . ; m; with at least one strict inequality holds. To measure the distance between any f ðxÞ 2 Z and a certain reference vector f , Steuer and Choo [1] introduced the augmented weighted Tchebycheff metric k k jkf f ðxÞkj1 ¼ kf f ðxÞk1 þ qeT jf f ðxÞj;
where f i ¼ maxx2X fi ðxÞ þ Di ¼ maxz2Z zi þ Di ¼ zi þ Di ; Di > 0; i ¼ 1; 2; . . . ; m (f can be regarded as the posik tively perturbed utopia (ideal) point Pm z), kf f ðxÞk1 ¼ maxi fki j f i fi ðxÞ jg is the weighted TTchebycheff metn ric with k 2 X ¼ fk 2 R ki > 0; i¼1 ki ¼ 1g, q is a sufficiently small positive scalar, and e is the vector of T ones and j z j denotes ðj zi j; . . . ; j zm j Þ . Kaliszewski [10] introduced the modified weighted Tchebycheff metric k kf f ðxÞk1 q ¼ maxfki ½jf i fi ðxÞj þ qeT jf f ðxÞjg: i
Wierzbicki and Makowski [11] introduced the following basic form of an order consistent with the scalarizing function that has been widely used, called the achievement metric ( ) m X i fi ðxÞj j fj ðxÞj j f j f þe Sðf ; f ðxÞÞ ¼ max ; 16i6m jf i f i j jf j f j j j¼1 where f i ¼ maxx2X fi ðxÞ; i ¼ 1; 2; . . . m; and can be regarded as the positively perturbed utopia (ideal) point, f i ¼ minx2X fi ðxÞ; i ¼ 1; 2; . . . ; m:
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M.A.E.-H. Kassem / Applied Mathematical Modelling 32 (2008) 1044–1055
We now introduce a somewhat different measure, called the modified weighted achievement scalarization metric, as follows: " # m X jf i fi ðxÞj jf j fj ðxÞj S ðf ; f ðxÞÞ ¼ max ai þe ; 16i6m jf i f i j jf j f j j j¼1 a
ð1Þ
where f i ¼ maxx2X fi ðxÞ þ Di ; i ¼ 1; 2; . . . m; f i ¼ minx2X fi ðxÞ; i ¼ 1; 2; . . . ; m, a 2 K ¼ fa 2 Rm j ai > 0; Pm i¼1 ai ¼ 1g; e is a sufficiently small positive scalar. Later in the paper we show that this metric coincides (up to scalar multiplication) with both the metric introduced by both Choo and Atkins [2] and Kaliszewski [10]. By the definition of f i , f i fi ðxÞ > 0 for all f ðxÞ 2 Z: Thus, in all formula we can drop the absolute value sign. An element f ðxÞ 2 Z is said to define the isoquant of S a ðf ; f ðxÞÞ if and only if
ai ¼ f f ðxÞ i
i
f i f i
þe
1 Pm
f j fj ðxÞ j¼1 f j f j
2 m X 4 i¼1
31 1 Pm fj fj ðxÞ5 ; f i fi ðxÞ þ e j¼1 f j f j f i f i
i ¼ 1; 2; . . . ; m:
Then, 2 m X S a ðf ; f ðxÞÞ ¼ 4
31
f i fi ðxÞ i¼1 f f i i
1 Pm fj fj ðxÞ5 : þ e j¼1 fj f j
The following figure represents the isoquants of the corresponding metric restricted to the set e ff ðxÞ 2 Rm j fi ðxÞ 6 f i ; i ¼ 1; 2; . . . ; mg, where h ¼ tan1 1þe (see [10,11]). θ
f (x)
.f
θ
Theorem 1. Let Z be finite, I Z ¼ fi j f ðxi Þ 2 Zg; I N ¼ fi j f ðxi Þ 2 N g and f ðxp Þ 2 N : Then f ðxp Þ uniquely solves ( " #) m X i fi ðxÞ f ðxÞ f f j j p min S a ðf ; f ðxÞÞ ¼ min max ai þ ep ; f ðxÞ2Z f ðxÞ2Z 16i6m fi fi f j f j j¼1 where ep is any positive real number satisfying the inequality
ep <
min
8 minj > > p >
> Z nfpg > Pm j2I > fl ðxj Þfl ðxp Þ >0 : l¼1
f l f l
and the fapi g is specified as
f i fi
Pm
l¼1
fi ðxp Þfi ðxj Þ9 > f i f i >
fl ðxj Þfl ðxp Þ f l f l
> = > > > ;
;
ðIÞ
M.A.E.-H. Kassem / Applied Mathematical Modelling 32 (2008) 1044–1055
api
1 ¼ m P f i fi ðxp Þ þe f i f i
j¼1
f j fj ðxp Þ f j f j
2 m X 4 i¼1
f i fi ðxp Þ f i f i
þe
31
1 Pm
f j fj ðxp Þ j¼1 f j f j
5 :
Proof. Suppose f ðxj Þ 2 Z; f ðxj Þ 6¼ f ðxp Þ. There exists at least one index i such that fi ðxp Þ fi ðxj Þ fi ðxp Þ > fi ðxj Þ ) > : fi fi fi fi Then for all such i we have m m X X fi ðxp Þ fl ðxp Þ fi ðxj Þ fl ðxj Þ p p þ e > þ e f i f i f l f l f i f i f l f l l¼1 l¼1
and therefore m m X X fi ðxp Þ fl ðxp Þ fi ðxj Þ fl ðxj Þ p p þ e 66 ¼ þ e : f i f i f l f l f i f i f l f l l¼1 l¼1
To see this let us observe that it is obviously true for m X fl ðxp Þ fl ðxj Þ P 0: f l f l l¼1
Suppose now that m X fl ðxj Þ fl ðxp Þ > 0: f l f l l¼1
Then
i:
minj p
fi ðx Þfi ðx Þ >0 f i f i
m X fi ðxp Þ fi ðxj Þ fl ðxj Þ fl ðxp Þ p > e f i f i f l f l l¼1
is valid if min
ep <
f ðxp Þf ðxj Þ i: i f fi >0 i i
Pm
l¼1
fi ðxp Þfi ðxj Þ f i f i
:
fl ðxj Þfl ðxp Þ f l f l
We assume that ep satisfies min
ep <
min Pm j2Ifl ðxZ jnfpg; Þfl ðxp Þ l¼1
f l f l
f ðxp Þf ðxj Þ i: i f fi >0 i i
Pm
l¼1
>0
with fapi g specified as in (*) 2 m X a min S ðf ; f ðxÞÞ ¼ 4 f ðxÞ2Z
i¼1
fi ðxp Þfi ðxj Þ f i f i
fl ðxj Þfl ðxp Þ f l f l
f i fi ðxp Þ f i f i
þe
;
1 Pm p
f l fl ðxp Þ l¼1 f l f l
1047
31 5 :
ðÞ
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M.A.E.-H. Kassem / Applied Mathematical Modelling 32 (2008) 1044–1055
Indeed, by the assumption about the value of ep we have guaranteed that for each j 2 I Z n fpg there exists an i such that m m X X fi ðxp Þ fl ðxp Þ fi ðxj Þ fl ðxj Þ p p þ e > þ e f i f i f l f l f i f i f l f l l¼1 l¼1
or equivalently m m X X f i fi ðxp Þ f l fl ðxp Þ f i fi ðxj Þ f l fl ðxj Þ p p þ e < þ e : f i f i f l f l f i f i f l f l l¼1 l¼1
Hence, api
"
# " # m m j X f i fi ðxp Þ p X f l fl ðxp Þ f l fl ðxj Þ p f i fi ðx Þ a p p þe þe ¼ S ðf ;f ðx ÞÞ < ai f i f i f l f l fifi f l f l l¼1 l¼1 " # m f i fi ðxj Þ p X f l fl ðxj Þ a j 6 max api þe l f l ¼ S ðf ;f ðx ÞÞ; j 2 I Z n fpg; i fi fi f l¼1 h
which completes the proof.
Lemma 1. As the previous proof, Theorem 1 holds with min
e0p <
min Pm j2Ifl ðxN jnfpg; Þfl ðxp Þ l¼1
f l f l
f ðxp Þf ðxj Þ i: i f fi >0 i i
Pm
l¼1
>0
fi ðxp Þfi ðxj Þ f i f i
fl ðxj Þfl ðxp Þ f l f l
:
The proof is similar as in Theorem 1. Theorem 2. Let Z be finite and 9 8 9> 8 > > > fi ðxl Þfi ðxj Þ > > > = < => < f i f i : e < min min min P m fs ðxj Þfs ðxl Þ>> l2I N > > j2I N nflg f ðxl Þf ðxj Þ > > ; : i: i f fi >0 s f s P s¼1 > > f j l m fs ðx Þfs ðx Þ i i ; : s¼1
f s f s
>0
Pm
¼ 1g such that f ðxp Þ solves the
i ¼ 1; 2; . . . ; m;
ðIIIÞ
Then f ðxp Þ 2 N if and only if there exist a a 2 K ¼ fa 2 Rm j ai > 0; following problem: min subject to
ðIIÞ
i¼1 ai
g;
" # m X f i fi ðxÞ f j fj ðxÞ þe g P ai ; fi fi f j f j j¼1
f i ðxÞ 2 Z: Proof. First, follows from Theorem 1 and Lemma 1 when fai g is defined as the fapi g in (*). f ðxp Þ is then a unique solution of the problem. Conversely, suppose that f ðxp Þ 62 N minimizes the above problem for some a 2 K. With f ðxp Þ dominated there exists a f ðxq Þ such that f ðxq Þ P f ðxp Þ; f ðxq Þ 6¼ f ðxp Þ. Since f i fi ðxq Þ f i fi ðxp Þ 6 f i f i fi fi
M.A.E.-H. Kassem / Applied Mathematical Modelling 32 (2008) 1044–1055
1049
and for any e > 0 m m X X f j fj ðxq Þ f j fj ðxp Þ e < e ; f j f j f j f j j¼1 j¼1 f ðxq Þ implies a smaller objective function value than f ðxp Þ. However, this contradicts the assumption about f ðxp Þ. Thus, f ðxp Þ 2 N . h Remark 1. We note from Theorems 1 and 2 that, if Z is polyhedral I Z ¼ fi j f ðxi Þ is an extreme point of Z}, I N ¼ fi 2 I Z j f ðxi Þ is nondominated}, and e is defined as in (II). Then f ðxp Þ 2 N if and only if there exists a a 2 K such that f ðxp Þ solves the problem (III). 3. Related results Let us assume that ðgp ; xp Þ be an optimal solution to the modified weighted achievement problem (III) with all the inequality constraints of (III) active . Then the intersection of the m active inequality constraints in the objective function space can be obtained by solving the following simultaneous equation: m X f i fi ðxp Þ f j fj ðxp Þ gp þ e ¼ p; ai f i f i f j f j j¼1 which can be written in the form m m X X fi ðxp Þ fj ðxp Þ gp f i f j þ e ¼ þ e p: f j f j ai f i f i f j f j f i f i j¼1 j¼1 Hence, this equation can be written in the following matrix form: 3 2 m P 3 2 f j f 1 gp 2 f1 ðxp Þ 3 1þe e e e e 1 f 1 þ e j f j ap f f 6 1 7 j¼1 f 1 f1 6 e 7 6 1þe e e e 7 76 f ðxp Þ 7 6 6 7 m P f j f2 gp 7 2 7 6 e 7 6 6 ap 7 e 1þe e e 76 f2 f2 7 6 f2 f 2 þ e 6 f j f j 2 7 76 6 7¼6 j¼1 6 e 7: 7 6 6 e e 1 þ e e 7 76 ... 7 6 6 7 . .. .. 74 .. .. .. .. 6 .. 7 5 6 p 5 4 . 7 6 fm ðx Þ . . . . . m 5 4 P p f fm g j f m fm þe ap e e e e 1 þ e f m f m f j f j j¼1
m
Denoting the left-hand matrix by B, its determinant becomes detðBÞ ¼ 1 þ me 6¼ 0. Also, denoting the adjugate matrix of B by ½~ bij , we have ~ bii ¼ 1 þ ðm 1Þe; ~bji ¼ e: Therefore, the solution to this simultaneous matrix equation is represented by !!, m m X X gp f j f l p ~ fi ðx Þ ¼ þe p bji det B; fj fj f l f l ai j¼1
l¼1
g . which implies that fi ðxÞ ¼ fiffi i a ð1þmeÞ i
Lemma 2. Assume: (1) Z is finite, f ðxp Þ 2 N and f ðxq Þ 2 Z such that f ðxp Þ 6¼ f ðxq Þ. (2) 8 2 31 > > m > > 6P 7 1 > > 4 f f ðxp Þ 1P 5 ; f i ðxp Þ 6¼ f i 8i; m m < fi fi ðxp Þ p P f j fj ðxp Þ f j fj ðxp Þ i i p i¼1 þe þe f i f i f i f i f j f j f j f j api ¼ j¼1 j¼1 > > > > 1; f i ðxp Þ ¼ f i ; > > : 0; f i ðxp Þ 6¼ f i ; but 9j 3 fj ðxp Þ ¼ f j ;
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M.A.E.-H. Kassem / Applied Mathematical Modelling 32 (2008) 1044–1055
(3) bp be the minimal objective function value of the problem min
b;
subject to b P
" api
# m X f i fi ðxp Þ f j fj ðxp Þ þe ; f i f i f j f j j¼1
i ¼ 1; 2; . . . ; m;
where e is any positive real number. Then there does not exist a f ðxq Þ 6¼ f ðxp Þ such that f ðxq Þ 2 Z lies in " ! bp f i p m ; þ1 ; Wðb Þ ¼ ff ðxÞ 2 R jfi ðxÞ 2 p f i f i ai ð1 þ meÞ when api > 0; i ¼ 1; 2; . . . ; mg. Proof Case 1 : fi ðxp Þ 6¼ f i 8i: Substituting for api in each of the m-constraints " # m p X f j fj ðxp Þ p f i fi ðx Þ b P ai þe ; i ¼ 1; 2; . . . ; m; f i f i f j f j j¼1 we have "
m X f i fi ðxp Þ f j fj ðxp Þ bP þ e f i f i f j f j j¼1
Thus p
b ¼
api
#1 ;
1 6 i 6 k:
" # m X f i fi ðxp Þ f j fj ðxp Þ þe ; f i f i f j f j j¼1
1 6 i 6 k:
Or m m X X fi ðxp Þ fj ðxp Þ bp f i f j þ e ¼ þ e : f j f j ap f i f i f j f j f i f i j¼1 j¼1
This equation can be written as in the above matrix form which implies to !!, m m X X bp f j f l p ~ fi ðx Þ ¼ þe p bji det B; fj fj f l f l ai j¼1 l¼1 where det B ¼ 1 þ me; ~ bii ¼ 1 þ ðm 1Þe; ~ bji ¼ e and then we have p b fi ; 1 6 i 6 m: p fi ðxp Þ ¼ a ð1 þ meÞ fi fi i Since f ðxp Þ is nondominated, there does not exist a q 6¼ p such that f ðxq Þ 2 Wðbp Þ: Case 2 : there exist j such that fj ðxp Þ ¼ f j . With there being only one j such that fj ðxp Þ ¼ f j , api ¼ 0 for all i 6¼ j: Thus bp ¼ 0 and ( " !) f j p m Wðb Þ ¼ f ðxÞ 2 R jfj ðxÞ 2 ; þ1 : fj fj Since f ðxp Þ is the member of Z for which the jth component is greater than or equal to f j , there does not exist a q 6¼ p such that f ðxq Þ 2 Wðbp Þ. h
M.A.E.-H. Kassem / Applied Mathematical Modelling 32 (2008) 1044–1055
1051
Suppose now that Z is composed of at least two objective vectors (as for example, in multiobjective programming problems with objective valued functions fi ; i ¼ 1; 2; . . . ; mÞ. We introduce the following parameter: d ¼ max i
jfi ðxÞ fi ðx0 Þj : jf i fi j
max0
f ðxÞ;f ðx Þ2N
By assumptions, d is finite and d > 0. It is easy now to verify that in the case of Theorem 2 we may take 1 . e < dðm1Þ1 1 1 1 Denoting e ¼ c1 so c1 < dðm1Þ1 that is c 1 > dðm 1Þ 1 then c > dðm 1Þ:We can now write the modified weighted achievement metric as ( " #) m X i fi ðxÞ f ðxÞ f f j j S a ðf ; f ðxÞÞ ¼ max ai þe 16i6m fi fi f j f j j¼1 ( " #) m 1 X f i fi ðxÞ f j fj ðxÞ ¼ max ai þ i c 1 j¼1 f j f j fi fi ( " #) m 1 f i fi ðxÞ X f j fj ðxÞ max ai ðc 1Þ ¼ þ c1 i fi fi f j f j j¼1 ( " #) 1 f i fi ðxÞ X f j fj ðxÞ max ai c ¼ þ c1 i fi fi f j f j j6¼i ( " !# ) 1 f f ðxÞ max ai M ¼ i c1 f f i
where M is a m m matrix defined as c if i ¼ j; ðMÞij ¼ 1 if i 6¼ j: The last form of the modified weighted achievement metric is exactly that introduced in [10]. In general, the modified weighted achievement metric can be rewritten as ( " !# ) f f ðxÞ max ai Bmm ; i f f i
where Bmm is a m m matrix, which is defined in the above simultaneous matrix equation in the form: ð1 þ eÞ if i ¼ j; ðBmm Þij ¼ e if i 6¼ j: 1 By elementary calculations, Bmm B1 mm ¼ I where I is the identity matrix, we have the inverse matrix Bmm which has the following form: ( 1þðm1Þe if i ¼ j; 1 1þme ðBmm Þij ¼ e if i 6¼ j: 1þme
Now we are able to show that the modified weighted achievement metric is (up to scalar multiplication) the metric introduced by Choo and Atkins [2] and Kaliszewski [10]. Choo and Atkins [2] defined 1 kf f ðxÞkk;# 1 ¼ maxfki j ½I # ðf f ðxÞÞi jg; i
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where I# is a m m matrix 1 if i ¼ j; ðI # Þij ¼ # if i 6¼ j; where # satisfies the inequality 1 < # < 0: The limit for # was chosen to ensure positive values of all entries of 2m I 1 # . e 1þme Let # ¼ 1þðm1Þe and as e is positive # is negative. Then I # ¼ 1þðm1Þe I 1 e and ( " !# ) m X f j fj ðxÞ f f ðxÞ S a# ðf ; f ðxÞÞ ¼ max ai I 1 þe # 16i6m f j f j f f j¼1 i ( " !# ) 1 m X 1 þ me f ðxÞ f f ðxÞ f j j I 1 þe ¼ max ai 16i6m 1 þ ðm 1Þe e f j f j f f j¼1 ( " !# ) i m X 1 þ ðm 1Þe f f ðxÞ f j fj ðxÞ Ie þe ¼ max ai i 1 þ me f j f j f f j¼1 ( " !# ) i m X 1 þ ðm 1Þe f f ðxÞ f j fj ðxÞ max ai I e þe ¼ 16i6m 1 þ me f j f j f f j¼1 i
1 þ ðm 1Þe a S ðf ; f ðxÞÞ: ¼ 1 þ me With the above equivalence we can make use of the result of Choo and Atkins [2] and Kaliszewski [10] concerning properly nondominated elements of N. If Z is finite or polyhedral, all elements of N are properly nondominated. Choo and Atkins [2] proved that for any positive e, each nondominated element of Z determined by minimizing S a ðf ; f ðxÞÞ over Z satisfies the following inequality: 1 1 þ ðm 1Þe ¼ ; # e where L > 0 is a positive scalar. This formula permits the necessity of determining the bounds on e as in Theorem 2 and Remark 1 to be avoided. Instead of that, we can use any positive number e provided that no element of N for which L P 1þðm1Þe is of our interest. e In the following section, we obtain the range of the positive number e that corresponding the same solution for the modified weighted achievement scalarization convex programming problem. L<
4. Stability set of the first kind Here, we assume the problem (MONLP) is stable, therefore the problem (III) is also stable [12]. Definition 3. The stability set of the first kind to the problem (III) (which determined the subset of the parametric ða; eÞ-space that has the same corresponding optimal solution) corresponding to the optimal solution ðxp ; gp Þ is defined by T ðxp ; gp Þ ¼ fða; eÞ 2 Rmþ1 jðxp ; gp Þis a solution of problem ðIIIÞg: The Kuhn–Tucker conditions of the problem (III) at a point xp take the following form: " # m m X X X ogj ðxp Þ 1 ofi ðxp Þ 1 ofl ðxp Þ ui ai þe mj ¼ 0; c ¼ 1; . . . ; n; þ oxc f i f i oxc f l f l oxc j2J i¼1 l¼1 m X i¼1
ui ¼ 1;
ð2Þ ð3Þ
M.A.E.-H. Kassem / Applied Mathematical Modelling 32 (2008) 1044–1055
# m X f i fi ðxp Þ f l fl ðxp Þ 6 0; g þ ai þe fi fi f l f l l¼1
1053
"
mj gj ðxp Þ ¼ 0;
i ¼ 1; 2; . . . ; m;
ð4Þ
j 2 J f1; 2; . . . ; kg;
ð5Þ
p
gj ðx Þ < 0; j 62 J ; ( " #) m X f i fi ðxp Þ f l fl ðxp Þ ui g þ ai þe ¼ 0; fi fi f l f l l¼1 ui P 0; mj P 0;
ð6Þ i ¼ 1; 2; . . . ; m;
ð7Þ
i ¼ 1; 2; . . . ; m; j 2 J:
ð8Þ ð9Þ
The determination of the stability set of the first kind, T ðxp ; gp Þ, depends only on whether any of the variables ui ; i ¼ 1; 2; . . . ; m and any of the variables mj ; j ¼ 1; 2; . . . ; k which solves the Eqs. (2), (3), (8) and (9) are positive or zero. Let ui ¼ 0; i 2 I f1; 2; . . . ; mg; ui > 0; i 62 I and mj ¼ 0; j 2 J f1; 2; . . . ; kg;
mj > 0; j 62 J ;
solve Eqs. (2), (3), (8) and (9) then in order that the other Kuhn–Tucker conditions Eqs. (4)–(7) are satisfied, we must have " # m X f i fi ðxp Þ f l fl ðxp Þ p g ¼ ai þe ; i 62 I; fi fi f l f l l¼1 " # m X f i fi ðxp Þ f l fl ðxp Þ p g P ai þe ; i 2 I: fi fi f l f l l¼1 Let us consider the following sets: A ¼ fI j ui ¼ 0; i 2 I; ui > 0; i 62 I and mj ¼ 0; j 2 J ; mj > 0; j 62 J solve the Eqs. (2), (3), (8) and (9)}, ( " # m X f i fi ðxp Þ f l fl ðxp Þ p p mþ1 p ; i 62 I; T I ðx ; g Þ ¼ ða; eÞ 2 R jg ¼ ai þe fi fi f l f l l¼1 " # ) m p X i fi ðxp Þ f ðx Þ f f l l gp P ai þe ; i2I : ð10Þ fi fi f l f l l¼1 It is clear from the definition of the stability set of the first kind that [ T I ðxp ; gp Þ: T ðxp ; gp Þ ¼
ð11Þ
I2A
The set T I ðxp ; gp Þ can be takes the following form: 8 " #" #1 m < gp f i fi ðxp Þ X f l fl ðxp Þ p p mþ1 ; T I ðx ; g Þ ¼ ða; eÞ 2 R je ¼ : ai fi fi f l f l l¼1 9 " #" #1 m = gp f i fi ðxp Þ X f l fl ðxp Þ e6 ; i 2 I : ; ai fi fi f l f l l¼1 Example. Let us consider the following MONLP problem: max
2
ðx1 3Þ þ x2 ¼ z1 ; 2
max x1 þ ðx2 4Þ ¼ z2 ; subject to x1 þ x2 6 5; x1 P 0;
x2 P 0:
i 62 I;
ð12Þ
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First we solve the following problems: 2
z1 ¼ ðx1 3Þ þ x2 ;
(i) max
subject to x1 þ x2 6 5; x1 P 0; x2 P 0: The solution is z1 ¼ max z1 ¼ 14 at the point ð0; 5Þ. 2
z1 ¼ ðx1 3Þ þ x2 ; (ii) min subject to x1 þ x2 6 5; x1 P 0; x2 P 0: The solution is z1 ¼ min z1 ¼ 2 at the point (3, 2). 2 (iii) max z2 ¼ x1 þ ðx2 4Þ ; subject to x1 þ x2 6 5;
x1 P 0;
x2 P 0:
The solution is z2 ¼ max z2 ¼ 21 at the point (5, 0). 2
(iv) min
z2 ¼ x1 þ ðx2 4Þ ;
subject to x1 þ x2 6 5; x1 P 0; x2 P 0: The solution is z2 ¼ min z2 ¼ 0:75 at the point (0.5, 4.5). For assumption a1 ¼ 0:4; a2 ¼ 0:6 and e ¼ 0:3 the equivalent problem (III) takes the following form: min g; subject to 2
2
0:73128 0:04316ðx1 3Þ 0:04316x2 0:00592x1 0:00592ðx2 4Þ 6 g; 2
2
1:01892 0:03852x1 0:03852ðx2 4Þ 0:01494x2 0:01494ðx1 3Þ 6 g; x1 þ x2 6 5; x1 P 0;
x2 P 0:
By using the Kuhn–Tucker conditions (2)–(9), we obtain g ¼ 0:121112 at the point (0, 4.8). From the Eqs. (10)–(12), the stability set of the first kind takes the form T ð0; 4:8Þ ¼ fða; eÞ 2 R3 ja1 e P 0:99501633a2 þ ea2 g: 5. Conclusion This paper presents an approach to compute the stability of efficient solutions to multiobjective nonlinear problems, using a modified weighted achievement scalarizing metric. The method is based on a particular objective function which scalarizes and parameterizes the multiobjective nonlinear problem. Also, we show how to determine nondominated criterion vectors (and then nondominated solutions, if necessary) by the modified weighted achievement metric in cases when sets of all criterion vectors are finite or polyhedral, and we show that this modified weighted achievement metric coincides with the metric introduced by both Choo and Atkins [2] and Kaliszewski [10].
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Acknowledgement The author is very grateful to the Editor-in-Chief, Professor Mark Cross, and the anonymous referees for their insightful and constructive comments and suggestions that have led to an improved version of this paper. References [1] E.R. Steuer, E.-U. Choo, An Interactive weighted Tchebycheff Procedure for Multiple objective programming, Math. Program. 26 (1983) 326–344. [2] E.-U. Choo, D.R. Atkins, Proper efficiency in nonconvex multicriteria programming, Math. Oper. Res. 8 (1983) 467–470. [3] M. Kassem, Decomposition of the fuzzy parametric space in multiobjective nonlinear programming problems, Eur. J. Oper. Res. 101 (1997) 204–219. [4] M. Kassem, Interactive stability of vector optimization problems, Eur. J. Oper. Res. 134 (2001) 616–622. [5] P.A. Wierzbicki, Reference point Methods in vector optimization and Decision Support, Interim Report, IR-98-017/April (1998), International Institute for Applied Systems Analysis, Laxenburg, Austria. [6] P. Korhonen, Multiple objective programming support, Interim Report, IR-98-010/March 1998, International Institute for Applied Systems Analysis, Laxenburg, Austria. [7] V. Chankong, Y.Y. Haimes, Multiobjective Decision Making: Theory and Methodology, North-Holland, New York, 1983. [8] J. Jahn, Scalarization in vector optimization, Math. Program. 29 (1984) 203–218. [9] D. Li, C. Cheng, Stability on multiobjective dynamic programming problems with fuzzy parameters in the objective functions and in the constraints, Eur. J. Oper. Res. 158 (2004) 678–696. [10] I. Kaliszewski, A modified weighted Tchebycheff metric for multiple objective programming, Comput. Oper. Res. 14 (1987) 315–323. [11] P.A. Wierzbicki, M. Makowski, Multi-objective optimization in negotiation support, WP-92-007/1992, International Institute for Applied Systems Analysis, Laxenburg, Austria. [12] R. Rockafellar, Duality and stability in extremal problems involving convex functions, Pac. J. Math. 21 (1967) 167–181. [13] I. Kaliszewski, Norm scalarization and proper efficiency in vector optimization, Found. Control Eng. 11 (1986) 117–131.