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Fuzzy Inf. Eng. (2009) 1: 37-57 DOI 10.1007/s12543-009-0003-3 ORIGINAL ARTICLE Bing-yuan Cao Received: 16 November 2008/ Revised: 10 January 2009/ Accepted: 16 January 2009/ © Springer and Fuzzy Information and Engineering Branch of the Operations Research Society of China Abstract A rough posynomial geometric programming is put forward by the au- thor. This model is advantageous for us to consider questions not only from the quantity of aspect, but from the quality because it contains more information than a traditional geometric programming one. Here, a rough convex function concept is advanced in rough value sets on foundation of rough sets and rough convex sets. Be- sides, a knowledge expression model in rough posynomial geometric programming is established and so is a mathematical one. Thirdly, solution properties are studied in mathematical model of rough posynomial geometric programming, and antinomy of the more-for-less paradox is solved with an arithmetic in rough posynomial geomet- ric programming given, which can be changed into a rough linear programming after monomial rough posynomial geometric programming is solved. Finally, validity in model and algorithm is veriﬁed by examples. Keywords Rough set · Posynomial · Geometric programming · Rough number · Antinomy · Satisfactory solution 1. Introduction There exist a lot of incompatible phenomena, contradictory and conﬂicting, in the re- alistic world. If a transformation ﬁnds its way to them, these incompatible problems can be changed into compatible ones. Here we develop a rough geometric program- ming, built on the foundation of a classical geometric programming,which will be a useful tool to incompatible problems in realistic world. A geometric programming is a type of mathematical optimization problem char- acterized by objective and constraint functions with a special form. In fact, its name comes from the geometric-arithmetic mean inequality, playing an importation role in the early analysis of geometric programming. Bing-yuan Cao () School of Mathematics and Information Sciences, Guangzhou University, Guangzhou Higher Education Mega Center, Guangzhou 510006, P.R.China e-mail: caobingy@163.com 38 Bing-yuan Cao (2009) Suppose a crisp geometric programming to be (P) min g (x) s.t. g (x)1(1 i p), x > 0, then we call it a primal-type posynomial geometric programming, while (P ) min g (x) 1 0 s.t. g (x) 1(1 i p ) g (x) 1(p + 1 i p) x > 0 is called a reverse posynomial geometric programming, where g (x) = v (x)(0 i p) i ik k=1 are posynomials of x (i.e., it is a polynomial whose coeﬃcients are positive con- stants), whereas ⎪ γ ikl c x , (1 k J ;0 i p ), ⎪ ik i l=1 v (x) = ik ⎪ −γ ikl c x , (1 k J ; p + 1 i p), ⎩ ik i l=1 where c is a coeﬃcient, which is a non-negative real number, and an exponent γ ik ikl an arbitrary real number. In each item of reverse inequality v (x)(1 k J ; p + 1 ik i i p), we represent the exponent of x in this item by using −γ in stead of by using ikl γ , x = (x , x ,··· , x ) denotes a variable vector and ‘T’ a transpose symbol. ikl 1 2 m The dual programming of (P ) is [3][10] w w 00 ik p p J J i i a c 0k ik (D ) max d(w) = w a w 00 i=0 k=1 ik ik i=p +1 k=1 −w ik p p c a ik ik w −w i0 i0 w w i0 i0 ik i=1 i=p +1 s.t. w = w = 1, 00 0k k=1 Γ w = 0, w 0, where ⎛ ⎞ ⎜ ⎟ γ ··· γ ··· γ ⎜ ⎟ 011 01l 01m ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ··· ··· ··· ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ γ ··· γ ··· γ ⎜ 0J 1 0J l 0J m ⎟ 0 0 0 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ··· ··· ··· ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ Γ= ⎜ ⎟ ⎜ ⎟ ⎜ −1 −1 −1 ⎟ ⎜ ⎟ γ ··· γ ··· γ ⎜ ⎟ ⎜ p J 1 p J l p J m ⎟ p p p ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ −1 −1 −1 ⎜ ⎟ ⎜ −γ ··· −γ ··· −γ ⎟ ⎜ ⎟ p +1J 1 p +1J l p +1J m ⎜ p +1 p +1 p +1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ··· ··· ··· ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ −γ ··· −γ ··· −γ pJ 1 pJ l pJ m p p p Fuzzy Inf. Eng. (2009) 1:37-57 39 denotes structure of exponent to apiece term of variable x corresponding to an objec- tive function g (x), each constraint function g (x)(1 i p) and g (x)(p + 1 i p), 0 i i it is called an exponent matrix. It contains J = (J + J +···+ J ) row and m column. 0 1 p Here w = (w , w ,..., w , w ,..., w ) is a J −dimensional variable vector 00 01 0J p1 pJ 0 p (J = 1 + J + ··· + J ), and w = w + w + ··· + w ; −w and −w denote 0 p i0 i1 i2 iJ ik i0 ik −w −w i0 ik a reversed direction inequality g (x)1 corresponding to factors ( ) and w i0 ik in the upper-right-corner exponent. For the sake of the assurance, objective function d(w) is deﬁned with continuity. We stipulate ik (w ) | = 1. ik w ik=0 Now, this problem will be solved by a rough set method, but the discussion here is limited only within rough posynomial geometric programming. In this paper, rough sets are introduced by discussing knowledge expression on model in Section 2, and a deﬁnition is discussed in the rough sets. Besides, a mathematical model is established in rough posynomial geometric programming in Section 3. In Section 4, a monomial rough posynomial geometric programming is discussed together with its arithmetic. First, any rough posynomial geometric programming is changed into a monomial one by using geometric inequality; then, the latter is turned into a rough linear pro- gramming by adopting rough transformation so that an optimal solution comes out to a rough linear programming after a solution is converted to an ordinary linear pro- gramming. Thereby, an optimal solution can be got to a rough posynomial geometric programming. And ﬁnally, an arithmetic is given. In Section 5, we validate the model to be eﬀective after numerical examples. The objective and constraint condition in geometric programming can be trans- formed a little by a certain experience (or a rule), or similarly, let the coeﬃcient a little ﬂuctuate, such that an incompatible monomial rough posynomial geometric programming is changed into a compatible one, and thereby a satisfactory solution is got to this problem. 2. Rough Sets and Its Model of Knowledge Expression An example is that drivers from Hongkong move on by left side, while the Chinese inlanders do that on the right side. If the former come into the inland, they will collide with vehicles by inlanders everywhere. What can be done with this incompatibility? Some persons consider a bridge in Huanggang of Shenzhen in China as a transform, by which the vehicles from Hongkong can automatically move on in the inland and so can the inlanders’ in Hongkong, so that the opposition problem is resolved. This is actually to say, we can begin with the self-contradict conﬂict phenomenon (bump vehicle), mine information and search methods creatively. That is, to change a disad- vantage factor into an avail one by the rough transform or an operation of the rough transform, (set up a method to a conversion bridge), turn an incompatible problem into a compatible one skillfully and provide the management and policy with a scien- tiﬁc thereunder and as well, which seems fantastic. To such problem, there are many 40 Bing-yuan Cao (2009) methods. Unfortunately, these methods fail to give a ﬁtting mathematic description, a result logically reasoned out represents that only a man-made fuzzy degree depend on subjective judgments or experiences. But the rough geometric programming is put forward in the paper, namely a ﬁx quantify model, it is an eﬀective method to an antinomy problem like the phenomenon mentioned above. It is fantastic that persons with idea of rough sets hope to mine information. We give a deﬁnition, ﬁrst, in the lower and the upper approximations by sets as follows [9][12]. Deﬁnition 1 Suppose U to be a universe, a relation deﬁned on U is called reﬂexive if each object x is similar to itself, i.e., x x; symmetric if x y ⇒ y x; transitive if x y, y z ⇒ x z for any x, y, z ∈ U. The similarity class of x, denoted by R(x), is the set of objects which are similar to x, R(x) = {y ∈ U|y x}. −1 Let R (x) be the class of objects to which x is similar, −1 R (x) = {y ∈ U|x y}. Then Deﬁnition 2 Let U be a universe, and X ⊂ U be a set representing a concept. Then its lower approximation is deﬁned by −1 B (X) = {x ∈ U|R (x) ⊂ X}; while the upper approximation is deﬁned by B (X) = R(x). x∈X That is, the lower approximation is a subset which contains the objects surely belong- ing to the set, whereas the upper one is a superset which contains the objects possibly belonging to the set. Theorem 1 Let U be a universe, and X be a set in U. Then we have B (X) ⊂ X ⊂ B (X). −1 Proof Let x ∈ B (X). Since the similar relation is reﬂexive, then x ∈ R (x). It −1 follows from R (x) ⊂ X that x ∈ X. Hence B (X) ⊂ X. Let x ∈ X. Since the similar relation is reﬂexive, we have x ∈ R(x). Thus + + x ∈ B (X) and X ⊂ B (X). The theorem holds. Deﬁnition 3 Let U be a universe, and X be a set in U. The boundary of X is deﬁned as B(X), written down as B(X) = B (X)− B (X). − Fuzzy Inf. Eng. (2009) 1:37-57 41 If B(X) = φ, then X is an ordinary set. If B(X) φ, then X is an approximate one and may be characterized by B (X) and B (X). Deﬁnition 4 The collection of all sets with the same lower and upper approximations is called a rough set, denoted by (B (X), B (X)). Deﬁnition 5 Suppose U to be a universe, X ⊆ U and R is an equivalence relation. If μ is approximately precision of X on (U, R) (an approximate space), and α is the expert’s evaluation on some object, it is an expert’s valuation or experiences, or result value got from the system in dynamic model simulation, calling [(1−μ)α, (1+μ)α] an approximately precision rough number [11], where μ can be used by the type denoting |B (X)| μ = , |B (X)| here |X| denotes a radix number in set X, B (X) is called a lower approximation; B (X) is called an upper approximation. Obviously, this kind of rough number is an interval value, that is, it only needs ﬁnding approximation number [(1−μ)α, (1+μ)α], and we are deemed to get a roughly right result. As for two of arbitrarily rough numbers x and y, they all mean two interval values. Therefore, as to the operation of x and y, they are two interval operations substantially. As for operations of interval by addition, minus, multiplication and division , there follows a deﬁnition below. Deﬁnition 6 Suppose x ∈ [x , x ] and y ∈ [y , y ], then 1 2 1 2 1) Addition. x+ y ∈ [[x , x ]+ [y , y ]] = [x + y , x + y ]. 1 2 1 2 1 1 2 2 2) Minus. x− y ∈ [[x , x ]− [y , y ]] = [x − y , x − y ]. 1 2 1 2 1 2 2 1 3) Multiplication. x∗ y ∈ [min{x y , x y , x y , x y }, max{x y , x y , x y , x y }]. 1 1 1 2 2 1 2 2 1 1 1 2 2 1 2 2 1 1 1 4) Division. If 0 [y , y ], then ∈ [ , ] and 1 2 y y y 2 1 x x x x x x x x x 1 1 2 2 1 1 2 2 ∈ [min{ , , , }, max{ , , , }]. y y y y y y y y y 2 1 2 1 2 1 2 1 Example 1 If x = [−2, 3], then x = x(x − 2) = [−2, 3]∗ ([−2, 3]∗ [−2, 3]− 2). Because x 0, hence [−2, 3] ∗ [−2, 3] = [0, 9]. The constant 2 goes thought the internal degenerates to a point, so there is [0, 9]− 2 = [−2, 7], hence x = x(x − 2) = [−2, 3]∗ [−2, 7] = [−21, 28]. Therefore, to arbitrary x ∈ [−2, 3], the internal calculations can guarantee x ∈ [−21, 28]. We give deﬁnition in rough sets by a connection degree as follows. 42 Bing-yuan Cao (2009) Deﬁnition 7 Suppose that x is an arbitrary point on real region R = (−∞,+∞), X =< a, b > is an arbitrary interval on real region, then we call a+ b 1 ρ(x, X ) = |x− |− (b− a) 2 2 a distance of point x with interval X . Deﬁnition 8 Suppose X =< a, b >, X =< c, d >, X ⊆ X, and there is not a common 0 0 endpoint, then the rough connection function of x on X and X is ρ(x, X ) K(x) = , (1) D(x, X , X) where sign < a, b > denotes the interval which can contain the endpoint a (or b), or it can not either contain the endpoint a (or b); ρ denotes distance of point x with interval, and ρ(x, X)−ρ(x, X ), x X ⎨ 0 0 D(x, X , X) = 0 ⎪ −1, x ∈ X denotes the position value of point x with regard to the interval. Deﬁnition 9 [1][2] Let U be a universe and K(x) ∈ (−∞,+∞) be a real number. If for arbitrary element x ∈ U, such that there exists K(x) corresponding to it. Then we call ( ) ( ) B = { x, y |x ∈ U, y = K x ∈ (−∞,+∞)} rough sets on universe U, where y = K : U → [0, 1], ( ) x → K x a rough connection function in A and K (x) a rough connection degree in x on A, B = {x|x ∈ U, K (x) 0} a positive region in B, B = {x|x ∈ U, K (x) 0} a negative region in B, and J = {x|x ∈ U, K(x) = 0} a zero boundary in B. We write R(U ) as all rough sets on U. We will abandon an algebra operation sign in geometric programming [7], and constitute a model of knowledge expression, then the incompatible programming problem mentioned above becomes a quaternary form M = (U, P, C, Q), (2) Fuzzy Inf. Eng. (2009) 1:37-57 43 where U denotes universe, P denotes substantiality, C denotes character and Q aﬁx quantify relation between P with regard to C. Moreover this quality value relation is i m ikl denoted by exponent polynomial g ¯ (x) = c ¯ x . i ik k=1 l=1 Deﬁnition 10 Given the knowledge expression system M = (U, P, C, Q), for every subset X ⊆ U and non-distinct relation B, the lower and the upper approximation sets of X [8] are deﬁned by B (X) = {Y|(Y ∈ U|ind(B)∧ Y ⊆ X)}; − i i i B (X) = {Y|(Y ∈ U|ind(B)∧ Y ∩ X φ)}, i i i where U|ind(B) = {X|(X ⊆ U ∧∀x∀y∀b(b(x) = b(y))) is a partition of non-distinct relation B about U, and it is an equivalence relation. If (2) is implemented by rough transformation T , then we have TM = (TU, TP, TC, TQ). (3) We call (3) a transformation model knowledge expression of rough geometric pro- gramming, and it is a compatible model. If (2) is a compatible one, we can then use a classical method to the problem. If (2) is an incompatible one or an opposite one, we will adopt diﬀerent transformation T , and convert (2) to (3)−type compatible model. Model (2) is such an ideal tool in an actual decision. In the knowledge expression model of rough geometric programming, the substantiality with characteristic trans- formation contains persons’ creative thoughts, which needs a combination of nature determination and ﬁxed amount together, basically by the former description. As for the relation between substantiality and characteristic, it is an abstract model in knowledge expression. Deﬁnition 11 [1][2] Let B be rough sets and T be a transform. Then we call B (T ) = {x|x ∈ U, K (x) 0, K (Tx) 0} . + a positive rough region in B, B (T ) = {x|x ∈ U, K(x) 0, K(Tx) 0} . − a negative rough region in B; B (T ) = {x|x ∈ U, K (x) 0, K (Tx) 0} a positive rough stabilization region in B, and B (T ) = {x|x ∈ U, K(x) 0, K(Tx) 0} a negative rough stabilization region in B. 44 Bing-yuan Cao (2009) Deﬁnition 12 Given the rough element, it is quaternary form M = (U , P , C , Q ), 0 0 0 0 0 if U is transformed to U, P to P, C to C, and Q to Q, then they are called a 0 0 0 0 universe transform, substantiality transform, character transform and quantify value transform for objects in rough element M , respectively, written down as T U = U, T P = P, T C = C, T Q = Q. (4) U 0 P 0 C 0 Q 0 0 0 0 0 If (4) holds, then this transform is called the replacement transform of rough element M , written down as T M = (T , T , T , T )(U , P , C , Q ) M 0 U P C Q 0 0 0 0 0 0 0 0 0 = (T U , T P , T C , T Q ) U 0 P 0 C 0 Q 0 0 0 0 0 = (U, P, C, Q). Example 2 (Transform bridge problem in Huanggang of Shenzhen in China) Suppose quaternary form to be 3-ary one R = (P, C, Q) because the universe is the same, then R = (Chinese inland, traﬃc rules, driving at right side) = (P , C, Q ), 1 1 1 R = (Hongkong, traﬃc rules, driving at left side) = (P , C, Q ), 2 2 2 r = (come-and-go auto of Chinese inland and Hongkong, runtime, 24 hours/day), then this is a conﬂicting problem in R and R under the condition r: 1 2 (R R ) ↑ (r). 1 2 Let z = (L , C, unilateralism moving from inland to Hongkong) 1 1 z = (L , C, unilateralism moving from Hongkong to inland), 2 2 and take L = L L . Then L is a link part with N and N , where L is a bridge 1 2 1 2 1 orbit connecting right half highway in inland with the left half one in Hongkong, L is a bridge orbit connecting right half highway in Hongkong with the left one in inland, denotes integrable. Again we take rough transform: T R = R = (N L , C, f (ω)), T R = R = (N L , C, f (ω)), 1 1 1 1 1 2 2 2 2 2 1 2 where moving right,ω = inland, f (ω) = 1 ⎪ unilateral moving to Hongkong,ω = orbit up in a bridge, moving left,ω = Hongkong, f (ω) = 2 ⎪ unilateral moving to inland,ω = orbit below in a bridge, such that (R R ) ↓ (r), 1 2 Fuzzy Inf. Eng. (2009) 1:37-57 45 that is, R and R coexist under the condition r, where L = L L is a connecting 1 2 1 2 transition thing of R and R . That is each one goes his or her own way, each gets 1 2 what he or her needs, and it is compatible. 3. Rough Posynomial Geometric Programming Now we induce the rough posynomial geometric programming. Deﬁnition 13 Suppose X =< a, b >,X =< c, d >,X ⊆ X, and there is not a common 0 0 point. If J J m i i ikl g ¯ (x) = v ¯ (x) = c ¯ x (0 i p) i ik ik k=1 k=1 l=1 is a rough posynomial deﬁned on D⊂R , respectively, it is a rough exponent polyno- mial for x(> 0), and c ¯ (> 0) denotes a rough coeﬃcient. Then the rough connection ik function of g ¯ (x) on X , Xis i 0 ρ(¯ g (x), X ) i 0 K (x) = K (¯ g (x)) = , (5) i i D(¯ g (x), X , X) i 0 where sign < a, b > denotes the interval which can contain the point a (or b), or it might not contain the point a (or b);ρ denotes the distance of g ¯ (x) with interval, and ρ(¯ g (x), X)−ρ(¯ g (x), X ), g ¯ (x) X ⎨ i i 0 i 0 D(¯ g (x), X , X) = 0 0 ⎪ −1, g ¯ (x) ∈ X i 0 denotes the position value of g ¯ (x) with regard to the interval. For the sake of a simple computation, we might as well suppose the following. A rough connection function of objective function in (P)is 0, if g ¯ (x) z , ⎪ 0 0 M − g ¯ (x) 0 0 K (x) = , if z < g ¯ (x) z + d , (6) 0 ⎪ 0 0 0 0 M − m ⎪ 0 0 1, if g ¯ (x) > z + d , 0 0 0 where d 0 is a maximum ﬂexible index of g ¯ (x) and z an objective value. 0 0 0 The rough connection function of constraint functions in (P)is 0, if g ¯ (x) > 1+ d, ⎪ i i M − g ¯ (x) ⎨ i i K (x) = 1− , if 1 < g ¯ (x) 1+ d, (7) i ⎪ i i M − m i i 1, if g ¯ (x) 1, (1 i p), where d ∈R denotes a maximum ﬂexible index of g ¯ (x). i i Deﬁnition 14 Suppose g ¯ (x) to be a rough objective function deﬁned on R, and g ¯ (x)(1 i p) to be a rough constraint function deﬁned on R , then we call i 46 Bing-yuan Cao (2009) (P) min g ¯ (x) s.t. g ¯ (x) b (1 i p), i i x > 0, a particular form of rough posynomial geometric programming, where g ¯ (x) = c ¯ i ik k=1 ikl x is a rough posynomial, c ¯ , b are the rough numbers, and it is deﬁned by ik i l=1 [(1−α)μ, (1+α)μ]. Here, ‘min’ denotes minimum value taken for objective function, the rough con- nection function of (P) is deﬁned by (6) and (7). We deﬁne symbol “ ” as a ﬂexible version of at a ‘certain degree’ [3][4], or approximately less than or equal to. The dual programming of (P) is [3][4] p p c ¯ ik w i0 ik (D) max d(w) = ( ) w i0 i=0 k=1 ik i=1 s.t. w = 1, Γ w = 0, w 0, where ⎛ ⎞ ⎜ ⎟ ⎜ γ ··· γ ··· γ ⎟ 011 01l 01m ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ . . . ⎜ ⎟ ⎜ . . . ⎟ ⎜ ⎟ ⎜ . . . ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ γ ··· γ ··· γ ⎜ ⎟ 0J 1 0J l 0J m 0 0 0 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ Γ= ⎜ ⎟ ⎜ ⎟ ··· ··· ··· ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ γ ··· γ ··· γ ⎜ ⎟ p11 p1l p1m ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ··· ··· ··· ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ γ ··· γ ··· γ pJ 1 pJ l pJ m p p p denotes the structure of an exponent to apiece term of variable x corresponding to an objective function g ¯ (x) and each constraint function g ¯ (x)(1 i p), respectively, 0 i called exponent matrices. It contains J = (J + J +···+ J ) row and m column, J is 0 1 p the sum of apiece term in g ¯ (x)(0 i p), w = (w ,..., w ,..., w ,..., w ) is i 01 0J p1 pJ 0 p a J−dimensional variable vector, and w = w = w + w +···+ w (0 i p) i0 ik i0 i1 iJ k=1 is the sum of each dual variables corresponding to an objective function g ¯ (x)(i = 0) or constraint function g ¯ (x)(1 i p). For the sake of the assurance objective function, d(w) is deﬁned with continue, we stipulate ik (w ) | = 1. ik w ik=0 Fuzzy Inf. Eng. (2009) 1:37-57 47 In order to study rough posynomial geometric programming, we give some prop- erties. Deﬁnition 15 Let K ∈R(U ) and∀x , x ∈ U,∀λ ∈ [0, 1], we have 1 2 ) ( ) ( ) K (λx + (1−λ) x K x K x . 1 2 1 2 Then we call K a convex rough set. Otherwise, we call K a non-convex one. Deﬁnition 16 We call K = {x|K (x) α}, K = {x|K (x)>α} an α−cut and strong α−cut in rough subsets K, respectively, whereα ∈ (−1, 0]. Theorem 2 Let K ∈R(U ). K is a convex rough set ⇔∀α ∈ −1, 0], K is an ordinary convex set. Proof “ ⇒ ”If K is a convex rough set, then ∀x , x ∈ K , and α ∈ (−1, 0],λ ∈ 1 2 α [0, 1], we have K (λx + (1−λ) x ) K (x ) K (x )>α. 1 2 1 2 And then, λx + (1−λ) x ∈ K , therefore, K is an ordinary convex set. 1 2 α α “ ⇐ ”If α ∈ (−1, 0], K is an ordinary convex set, and ∀x , x ∈ U . Let α = α 1 2 ( ) ( ) K x K x , then K(x ) α, K(x ) α. Therefore x , x ∈ K .As K is also 1 2 1 2 1 2 α α convex, ∀λ ∈ [0, 1], we haveλx + (1−λ) x ∈ K , that is 1 2 α K (λx + (1−λ) x ) K (x ) K (x ). 1 2 1 2 Hence K is a convex rough set from Deﬁnition 15. Deﬁnition 17 Let g ¯ (x)(0 i p) be rough functions deﬁned on D⊂R ,K, K be i i the rough value sets deﬁned on range D of g ¯ (x). Then we call g ¯ (x) a rough constraint i i with regard to K (x) when K (x) = K (¯ g (x)). i i i But when ⎪ 0, g ¯ (x) < 0, ⎨ i K (x) = 1, g ¯ (x) 0, the rough constraint of g ¯ (x) with regard to K is an ordinary inequality constraint. i i Deﬁnition 18 Let K, K (0 i p) be a convex rough set deﬁned on U with K (x) = i i K (¯ g (x)), and for all α ∈ [0, 1], K = {x|K (x) α} iα i (resp. K = {x|K (x) α}) iα i represent a convex set. Then g ¯ (x) is called a rough convex (resp. concave) function with respect to K . i 48 Bing-yuan Cao (2009) Deﬁnition 19 If we deﬁne K (x)(0 i p) in (6) and (7) as a strongly quasi-convex (resp. quasi-concave) function of x, then g ¯ (x) stands for a strongly rough convex (resp. concave) function about K . Deﬁnition 20 Suppose that K (x), K (0 i p) is a strictly quasi-convex (resp. quasi-concave) function of x deﬁned on U with K (x) = K (¯ g (x)), then for all of i i α ∈ [0, 1], Kiα = {x|K (x)<α} (resp. K = {x|K (x)>α}) iα represents a convex set and g ¯ (x) is called a strictly rough convex (resp. concave) function with respect to K . If −g ¯ (x) means a convex (resp. strictly quasiconvex, or strongly quasiconvex) function, then g ¯ (x) is a concave (resp. strictly quasiconcave, or strongly quasicon- cave) function. Deﬁnition 21 If K, K (0 i p) denotes a strongly quasiconvex (resp. strongly 1 2 1 2 quasiconcave) function, then ∀x , x ∈ U, x x andλ ∈ [0, 1], 1 2 1 2 K (λx + (1−λ)x ) < max{K (x ), K (x )} i i i 1 2 1 2 (resp. K (λx + (1−λ)x ) > min{K (x ), K (x )}). i i i We deduce a general mathematical deﬁnition in rough posynomial geometric pro- gramming as follows. If S is a rough set deﬁned on D ⊂R , g ¯ (x) is a rough constraint set with regard to K (1 i p) and S (x) = K (x) = min K (¯ g (x)) (x), i i 1ip 1ip then we call S (x) a rough compatible solution set. Deﬁnition 22 Let K be a rough set deﬁned on D⊂R, m = inf g ¯ (x), M sup g ¯ (x), 0 0 0 0 0 x∈D x∈D ∗ ∗ ∗ ∗ ∗ K , K be a rough value sets of g ¯ (x) and if K (x) = K (¯ g (x)), and K (¯ g (x)) be 0 0 0 0 0 0 0 0 strict monotone descent on [m , M ] with satisfaction of K (¯ g (x)) = 0 at g ¯ (x) M. 0 0 0 0 Then we call K rough satisfactory point sets of objective function g ¯ (x) deﬁned on D. Deﬁnition 23 Let H (x) be a rough set deﬁned on R. If there exists a rough optimal point set K of g ¯ (x), such that ∗ ∗ H (x) = min[K (x), S (x)], then H (x) is called a rough optimal point set of g ¯ (x) on a rough compatible solution set S , and max H (x) is called a rough posynomial geometric programming of g ¯ (x) x∈R ∗ ∗ with regard to H (x), where S (x) = min K (x). 1ip Fuzzy Inf. Eng. (2009) 1:37-57 49 For arbitrary K ∈ (−1, 0] in (1) or (5), after action on transformation T , T K ∈ [0, 1], a rough posynomial geometric programming is changed into a rough posyno- mial geometric programming. Let D ⊂R be convex sets. If an objective function g ¯ (x) is a rough convex function (resp. a strange rough convex function) with regard to K [2], constraint (x)(1 i p) is a rough convex function (resp. a strange rough convex functions g ¯ function) with regard to K , then we call (P) a rough convex geometric programming (resp. a strange rough convex geometric programming) problem with regard to g ¯ (x). There appear three kinds of rough posynomial geometric programming. a) Type (R , X, C). The objective function is clear, but the constraint set is rough. b) Type (R , X, C). The constraint set is perfectly ﬁxed and the objective function is rough. ¯ ¯ c) Type (R , X, C). The objective function and the constraint set all are rough. Following next is that only a) is considereed. p ∗ ∗ Deﬁnition 24 Let X ⊂R be a convex rough set. If g ¯ (x) and g ¯ (x) denote rough 0 i ∗ ∗ convex (resp. strongly rough convex) functions with respect to K and K (1 i p), respectively, then max H (x) represents a rough convex (resp. a strongly rough x∈R convex) programming with respect to g ¯ (x). Theorem 3 A rough posynomial geometric programming (P) can be turned into a rough convex geometric programming. Proof Take replacement of x = exp{z}(1 l m), then l l J J i m i γ z ikl l ikl l=1 g ¯ (x) = c ¯ x = c ¯ e i ik ik k=1 l=1 k=1 = G (z) (0 i p), hence (P) is changed into ¯ ¯ (G) minG (z) s.t. G (z) b (1 i p). i i It is easy to prove that G (0 i p) is a rough convex function of z, In fact, because exp{ γ z} > 0, ikl l dz l=1 exp{ γ z} denotes a strictly convex function about z , such that for arbitrary λ ∈ ikl l l l=1 50 Bing-yuan Cao (2009) [0, 1], we have exp{ γ [(1−λ)¯ u +λv ¯ ]} ikl l l l=1 m m (1−λ)exp{ γ u ¯ }+λexp{ γ v ¯ }. ikl l ikl l l=1 l=1 Therefore, exp{ γ z} is a convex function of z. ikl l l=1 2 1 2 Again, because of rough numbers c ¯ ∈ (0, c ) ⊂ [c , c ]; by applying (6) and i i i −1 ∗ (7), we know K (α) > 0(k > 0). Therefore, G (z) denotes a convex function of i i z. From the arbitrary choice of α on [0,1], we know G (z) denotes a rough convex programming with respect to z and the theorem is proved. Deﬁnition 25 Let (P) be a rough posynomial geometric programming. If there exists ∗ ∗ ∗ ∗ ∗ x , such that H (x ) = max H (x), then we call x a rough satisfactory solution in x∈R (P). Theorem 4 For arbitrary i, G (z) is a rough convex function, and a rough strict local satisfactory minimal solution to (G) is its rough global satisﬁed minimal solution as well. Proof Because (G) is rough convex, so is (G ) for arbitrary α ∈ [0, 1](here (G ) α α (G) ). But (G ) is an ordinary convex programming. This theorem is proved by α α Theorem 1.2.1 in [10] and Theorem 1. 4. Monomial Rough Posynomial Geometric Programming and Its Calculation Let g ¯ (x) = x . Then the rough posynomial geometric programming (P) is obviously 0 0 changed into min x −1 s.t. x g ¯ (x) 1, g ¯ (x) b (1 i p), i i x , x > 0, which, with only one more variable in it than in (P), still belongs to a rough posyn- omial geometric programming and is obviously equivalent to (P). We might as well let the objective function in (P)be g ¯ (x) = x . 0 1 Practically, we can estimate the extent where optimal solutions exist. Therefore, a rough posynomial geometric programming with a variable subject to lower and upper bounds min x (8) s.t. g ¯ (x) b (1 i p) i i 0 < x x x L U Fuzzy Inf. Eng. (2009) 1:37-57 51 can be considered, where, x and x represent two positive vectors. L U Theorem 5 Let K = K (¯ g (x))(1 i p) be a continuous and strictly nondecreasing i i function. If v ¯ > 0,ε > 0(1 k J ;0 i p) with ε = 1. Then ik ik i ik k=1 g ¯ (x, y) g ¯ (x). (9) i i Proof ∀y > 0, we suppose v ¯ (y) ik ε = (1 k J ;0 i p), ik i g ¯ (y) where ikl v ¯ (y) = c ¯ y , ik ik ik l=1 J J m i i ikl g ¯ (y) = v ¯ (y) = c ¯ y . i ik ik ik k=1 k=1 l=1 Obviously, ikl c ¯ y ik ik J J i i l=1 ε = ik J m k=1 k=1 ikl c ¯ y ik ik k=1 l=1 J m ikl c ¯ y ik ik k=1 l=1 = = 1. J m ikl c ¯ y ik ik k=1 l=1 We suppose ik J J i i v ¯ (x) ik γ ikl g ¯ (x, y) = = c ¯ x , i i k=1 ik k=1 where ik J J i i c ¯ ik c ¯ = ,γ = γ ε i il il ik k=1 ik k=1 and ikl g ¯ (x) = c ¯ x . i ik k=1 l=1 Under the presumption of the theorem and by means of Lemma 2.3.1 in [3], we have J J i i ik (¯v (x)) ε v ¯ (x). (10) ik ik ik k=1 k=1 Applying (10), then ⎛ ⎞ J m ik ⎜ ⎟ c ¯ ⎜ ⎟ ik γ ⎜ ikl⎟ ⎜ ⎟ g ¯ (x, y) = ⎜ x ⎟ ⎝ ⎠ ik k=1 l=1 ⎛ ⎞ J m ⎜ ⎟ ⎜ c ¯ ⎟ ik ⎜ γ ⎟ ikl ⎜ ⎟ ε x ⎜ ⎟ ik ⎝ ⎠ ik k=1 l=1 J m ikl = c ¯ x = g ¯ (x). ik i k=1 l=1 52 Bing-yuan Cao (2009) Therefore, g ¯ (x, y) g ¯ (x). i i And (9) is proved. We call (9) a rough geometric inequality. Specially, g ¯ (y, y) = g ¯ (y). i i Now, we change (P) into a monomial rough posynomial geometric programming for solution by means of inequality (9), such that a rough optimal solution in (P) can be obtained without any diﬃculty. Let F = {x|g ¯ (x, x ) b (1 i p), x x x } for arbitrariness x > 0. Then i k i L U k solve min x (11) k−1 s.t. x ∈ F by taking k = 1. Because g ¯ (x, x ) is a rough monomial, a rough geometric programming is changed i k into a monomial rough one. In this paper, only a monomial posynomial rough geometric programming is in- troduced. In (P), given J = J =··· = J = 1, it becomes 0 1 p 0l min c ¯ x l=1 (12) il s.t. c ¯ x b (1 i p), i i l=1 x > 0, i.e., (12) is a monomial posynomial rough geometric programming. We research equivalence form in (P) only by studying an equivalence form in (12). Theorem 6 Let a monomial posynomial rough geometric programming be (12). Then (12) is equivalent to a rough linear program min r z 0l l l=1 (13) s.t. γ z + ln c ¯ ln b (1 i p), il l i i l=1 or max (−r z ) 0l l l=1 s.t. γ z ln b − ln c ¯ (1 i p), il l i i l=1 Fuzzy Inf. Eng. (2009) 1:37-57 53 and its rough satisfactory minimal solution is the same with (12). Proof Take replacement of z = ln x (1 l m). We might as well take c ¯ = 1, l l 0 r Z il l l=1 then g ¯ (x) = c ¯ e . Thereby (12) is changed into i i r Z 0l l l=1 min e r Z (14) il l l=1 ¯ s.t. c ¯ e b, i i (1 i p), so (14) is a rough convex set from Theorem 3, containing a rough satisfactory minimal solution again by Theorem 4. But (12) ⇔ (14), however (14) ⇔ (13), such that the second conclusion holds in the theorem. Similarly, we can acquire the dual programming of (12) as max (¯c ) i=1 s.t. w = 1, γ w = 0(1 l n), il i i=0 w 0, which is equivalent to max (log c ¯ )w l l l=1 s.t. γ w = −γ (1 l m), lm l 0l l=1 w 0, or min (− log c ¯ )w l l l=1 s.t. γ w = −γ (1 l m), lm l 0l l=1 w 0. Therefore, we conclude the algorithm of rough posynomial geometric program- ming (P) as follows. Algorithm 1 Change (P) into a variable limited by lower and upper bounds for programm (8). 0 0 0 0 ˜ ˆ ¯ 2 ∀x > 0, let F = {x|˜¯ g (x, x ) b(1 i p), x x x }. From (9), we know i L U 1 0 ¯ ¯ a rough feasible solution set is F ⊂ F in (8) and take k = 1. 54 Bing-yuan Cao (2009) 3 Find (8). Solution to (8) can be got only by ﬁnding its equivalent program (11). Obviously, constraint functions of (11) are rough monomials, called a monomial rough posynomial geometric programming of variable subject to lower and upper bounds, and is equivalent to rough linear programming of variable subject to lower and upper bounds from [4]. There are many solution methods to (11), we obtain a rough optimal solution to (8) from rough linear programming. If no rough optimal solution exists in (11), nor does it in (8). It ends. Otherwise we turn to 4 for help. 0 0 4 If (11) has a rough optimal solution, one of the methods in 3 can be used to get a solution and the rough optimal one is denoted by x . 0 k k k 5 If g ˜ (x ) = max g ˜ (x ) 1, then x is the rough optimal solution to (8). It i(k) 1ik i ends. Otherwise we turn to 6 for help. 6 Let k k ˜ ˆ C = {x|g ¯ (x, x ) b, 0 < x x x }, i(k) L U k k−1 k k−1 k ˜ ˜ ˜ ˜ ˜ F = F C min{F (x), C (x)}. Use k as k+ 1 and then back to 3 . 5. Numerical Example Example 3 Consider a posynomial geometric program 6 3 min x x x x 1 2 3 4 −1 −1 2 s.t. x x x x = b , 1 1 2 3 4 2 −1 (15) x x x x = b , 2 3 2 1 4 2 3 x x x x = b , 1 2 3 3 4 x x x x > 0. 1 2 3 4 Let z = ln x (l = 1, 2, 3, 4). Then problem (15) is turned into l l min f (z) = (z + z + 6z + 3z ) 1 2 3 4 s.t. z − z − z + 2z = ln b , 1 2 3 4 1 (16) 2z + z + 2z − z = ln b , 1 2 3 4 2 z + z + 2z + 3z = ln b . 1 2 3 4 3 6 4 If we take b = e, b = e , b = e , the optimal solution is z = 2, z = 0, z = 1, 1 2 3 1 2 3 2 7 4 and the optimal value is f (2, 0, 1) = 8. But if b = e , b = e , b = e are taken, 1 2 3 the optimal solution is z = 3, z = 1, z = 0. However, the optimal value reduces, 1 2 3 it is f (3, 1, 0) = 4. This is strange, the ‘more-for-less paradox’ takes place, which is called antinomy in mathematics [5]. Fuzzy Inf. Eng. (2009) 1:37-57 55 Example 4 Find a rough posynomial geometric program 6 3 min x x x x 1 2 3 4 −1 −1 2 s.t. x x x x = b , 1 1 2 3 4 2 −1 (17) x x x x = b , 2 3 2 1 4 2 3 x x x x = b , 1 2 3 3 4 x x x x > 0, 1 2 3 4 ¯ ¯ where b (i = 1, 2, 3) are rough numbers, i.e., b → T (b ) = b and T is a rough i i 1 i i 1 transformation. (17) is turned into rough linear program min f (z) = (z + z + 6z + 3z ) 1 2 3 4 s.t. z − z − z + 2z = ln b , 1 2 3 4 1 (18) 2z + z + 2z − z = ln b , 1 2 3 4 2 z + z + 2z + 3z = ln b , 1 2 3 4 3 correspondingly. Then the rough constant ln b by rough transformation T is changed, i.e., let T ln b = {ln(b + dα)} i i i (i=1,2,3) (i=1,2,3) = (ln(b + d α), ln(b + d α), ln(b + d α)) 1 1 2 2 3 3 1−d α 6+d α 4+d α 1 2 3 = (ln e , ln e , ln e ). We might as well take b = 1, b = 6, b = 4, d = d = d = 1, then (18) is changed 1 2 3 1 2 3 into min (z + z + 6z + 3z ) 1 2 3 4 s.t. z − z − z + 2z 1+α, 1 2 3 4 2z + z + 2z − z 6+α, 1 2 3 4 z + z + 2z + 3z 4+α, 1 2 3 4 and its solution is −1 z = B ln b(α) ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 01 −1⎟ ⎜ 1+α⎟ ⎜ 2 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ = −23 −4 6+α = −3α . ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 1 −23 4+α 1+ 2α Since (18) does not request the solution to be non-negative, then z is an optimal solution to the problem. Put it into the objective function, we obtain f (2,−3α, 1+ 2α) = 2− 3α+ 6+ 12α+ 0 = 8+ 9α. 56 Bing-yuan Cao (2009) If only when α 0, antinomy fails to appear in (18), thereby, it can never take place in (17). However the option value of α(α ∈ (0, 1)) is decided according to the actual circumstance. We consider the method in the dual programming, and suppose the problem con- sidered has already been changed into a rough linear programming (17) as follows. Example 5 Find max f (x) = 100x + 40x + 60x 1 2 3 s.t. 4x + x + 2x c ¯ , 1 2 3 1 3x ++4x c ¯ , 1 3 2 2x + x + x c ¯ , 1 2 3 3 2x + 2x c ¯ , 2 3 4 x , x , x , x > 0, 1 2 3 4 where c ¯ = T {b }, c ¯ = T {b }, c ¯ = T {b }, c ¯ = T {b } is rough constraint constant, 1 1 1 2 1 2 3 1 3 4 1 4 b , b , b , b is a constraint constant in a prime linear programming. 1 2 3 4 Because its dual programming is max g(y) = c ¯ y + c ¯ y + c ¯ y + c ¯ y 1 1 2 2 3 3 4 4 s.t. 4y + 3y + 2y 100, 1 2 3 y++y + 2y 40, 1 3 4 2y + 4y + y + 2y 60, 1 2 3 4 y , y , y , y > 0, 1 2 3 4 if we take 3−λ 11− 3λ 14− 3λ c ¯ = 4−λ, c ¯ = , c ¯ = , c ¯ = , 1 2 3 4 2 4 4 then through ﬁnding solution, we get the following. 8 18 ∗ T At λ ∈ [ , ], the optimal solution is y = (20, 0, 0, 10) and the optimal value 3 5 is g ¯ = 115−λ . ∗ T At λ ∈ [0, ], the optimal solution is y = (20, 0, 10, 15) and the optimal value is g ¯ = 105−λ . Because λ can be freely ﬁxed at close interval [a, b], we might as well take λ = a+ b a+ ( )(k = 1, 2), thereby 8 18 8 1 47 8 18 λ = + ( − ) = ∈ [ , ], 3 5 3 2 15 3 5 and 8 1 4 8 λ = 0+ ( − 0) = ∈ [0, ], 3 2 3 3 13 1 ∗ ∗ and then we have optimal value g ¯ = 111 and g ¯ = 103 . 1 2 15 2 Fuzzy Inf. Eng. (2009) 1:37-57 57 Hence, for the diﬀerent λ ∈ [0, 1], the diﬀerent objective coeﬃcient c ¯ can be acquired. And decision makers can adjust objectively coeﬃcient basis to the practical circumstance, with acquirement of a satisﬁed solution expected. 6. Conclusions Many phenomena are described by means of geometric programming in the realistic world. However, usually there exist no optimal solutions in many problems, even feasible solutions can not be found, either [6]. We conclude this kinds of problems as incompatible geometric programming, and it is studied by using theory in a rough set. The writer here believes that the method developed in the paper will be of signiﬁcant use in engineering, designs and economic administration in the future. Acknowledgments Thanks to the support by National Natural Science Foundation of China (No. 70771030 and No. 70271047) and Project Science Foundation of Guangzhou University. References 1. Cai, W. (1987) The matter element analysis, Guangdong High Teach Publishing Hose 220-235 2. Cao, B.Y. (1990) Extension convex set. Acta Science Naturalium Univ. Norm. Hunanenist, 13: 18-24 3. Cao, B.Y. (2002) Fuzzy geometric programming. Kluwer Academic Publishers, Dordrecht 4. Cao, B.Y. (1993) Fuzzy geometric programming (I). Fuzzy Sets and Systems 53: 135-154 5. Cao, B.Y. (1991) A method of fuzzy set for studying linear programming ‘contrary theory’. J.of Hunan Educational Inst. (Natural Sci.) 9: 17-22 6. Cao, B.Y. (2008) Optimal models and methods with fuzzy quantities. Springer-Verlag, Heidelberg, (to appear) 7. Wang, G.Y. (2001) Rough sets theory and knowledge obtain, Xi’an Jiaotong University Press 8. Liu, B.D. (2002) Theory and practice of uncertain programming, Springer-Verlag Company, Physica- Verlag, Heidelberg 9. Pawlak, Z (1982) Rough sets, International Journal of Information and Computer Sciences 11: 341- 10. Wu, F., Yuan, Y.Y. (1982) Geometric programming: Math. Prac. & Theory 1-2 11. Wang, Q.Y., Liu, Q. (1996) Approximate accuracy rough number and its application in DSS. Mini- Micro Systems 17: 60-66 12. Slowinski, R., Vanderpooten, D, A generalized deﬁnition of rough approximations based on similar- ity, IEEE Transactions on Knowledge and Data Engineering 12: 331-336
Fuzzy Information and Engineering – Taylor & Francis
Published: Mar 1, 2009
Keywords: Rough set; Posynomial; Geometric programming; Rough number; Antinomy; Satisfactory solution
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