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Fuzzy disk for covering fuzzy points
European Journal of Operational Research 160 (2005) 560–573 www.elsevier.com/locate/dsw
Computing, Artificial Intelligence and Information Technology
Fuzzy disk for covering fuzzy points q Lushu Li, S.N. Kabadi, K.P.K. Nair
*
Faculty of Administration, University of New Brunswick, Fredericton, NB, Canada E3B 5A3 Received 16 May 2001; accepted 24 June 2003 Available online 23 September 2003
Abstract In this paper, we consider an important fuzzy version of the well known smallest covering circle problem which is also called the Euclidean 1-center problem. Here we are given a set of points in the plane whose locations are crisp. However, the requirement for coverage of each point is fuzzy as represented by its grade of membership. As such we qualify the points as fuzzy. In the above framework, we wish to find a fuzzy disk with minimum fuzzy area for covering the given fuzzy points. After developing a general model, certain special cases are investigated in detail. Polynomial algorithms are presented for the special cases. Ó 2003 Elsevier B.V. All rights reserved. Keywords: Location; Covering circle; Convex program; Polynomial algorithm; Fuzzy set
1. Introduction The classical smallest covering circle problem is defined as follows. Let Pi ¼ ðai ; bi Þ ði ¼ 1; 2; . . . ; nÞ be n given points in the Euclidean plane R2 . For any point P ¼ ðx; yÞ 2 R2 , let qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 dðP ; Pi Þ ¼ ðx ai Þ þ ðy bi Þ and H ðx; yÞ ¼ max dðP ; Pi Þ: 16i6n
Then the objective is to find a point P ¼ ðx ; y Þ in R2 so that H ðx ; y Þ is minimum. This smallest covering circle problem is also known as the Euclidean 1-center problem in the language of location theory, where one seeks to locate a point (a ‘‘facility’’) so as to minimize the largest distance between the facility and the given (‘‘demand’’) points. It has many practical applications in real world, such as the location of a Radar Station, Radio Transmitter, Hospital for Emergency Cases, Police Station or q This work was supported by research grants from the Natural Sciences and Engineering Research Council of Canada to S.N. Kabadi and K.P.K. Nair. * Corresponding author. Fax: +1-506-453-3561. E-mail address: [email protected] (K.P.K. Nair).
0377-2217/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0377-2217(03)00430-2
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Fire Station. Therefore it has been studied extensively by many authors and several algorithms have been developed (see, e.g. [2,11,12,16]. Among these MegiddoÕs algorithm [12] runs in linear time and it follows from [13] that in any fixed dimension the smallest ball enclosing n points can be found in OðnÞ time since the problem can be formulated as a convex quadratic programming problem [14]. In the classical smallest covering circle problem the points to be covered are crisp with respect to their locations in the plane and each point must be covered or enclosed by the smallest circle. To the best of our knowledge, there is only one paper [9] that incorporates fuzziness in the above problem. In [9] the fuzziness we introduced is entirely due to the fuzziness with respect to the locations of the points in the plane. Investigation of this problem resulted in two fuzzy models each of which has a polynomial algorithm. The contribution of this paper consists of introduction of another important type of fuzziness distinctly different from that in [9], development of a general model for covering fuzzy points by a fuzzy disk and investigation of special cases resulting in polynomial algorithms. The fuzziness considered arises from the fact that though the location of each point is crisp, its requirement to be covered is fuzzy as represented by its grade of membership. In this context we qualify the given points as fuzzy points. In the above framework, we wish to find a fuzzy disk with minimum fuzzy area for covering the given fuzzy points in the plane. We develop a general model of the above problem in Section 2, and in Section 3 certain special cases are investigated in detail. Polynomial algorithms for the special cases are presented in Section 4. Section 5 gives a numerical example, while Section 6 presents discussion. Relevant basic definitions and notations are given in the Appendix A. 2. Formulation of the smallest fuzzy disk problem The conventional, well-known smallest covering circle problem can be described as: given a finite set of points P ¼ fP1 ; P2 ; . . . ; Pn g R2 , where Pi ¼ ðai ; bi Þ, ði ¼ 1; 2; . . . ; nÞ, find a disk of minimum radius Dððx; yÞ; rÞ ¼ fðu; vÞjðu xÞ2 þ ðv yÞ2 6 r2 g R2 such that P Dððx; yÞ; rÞ. The optimal disk Dððx ; y Þ; r Þ is completely characterized by its center ðx ; y Þ and its radius r . We try to adapt this crisp model into its fuzzy version by using the fuzzy set framework (see Appendix A for basic definitions relevant for understanding the fuzzy models in this paper which can be found in [5,6]). e ¼ fðP1 ; lðP1 ÞÞ; ðP2 ; lðP2 ÞÞ; . . . ; ðPn ; lðPn ÞÞg 2 Thus, we assume that we are given a finite fuzzy subset P FðR2 Þ, where FðR2 Þ denotes the family of all fuzzy subsets of R2 and li ¼ lðPi Þ is the grade or degree of e Our focus is to find a minimal fuzzy disk D e membership of the crisp point Pi ¼ ðai ; bi Þ in the fuzzy subset P. e in the sense of fuzzy inclusion [4]. such that Pe D 2.1. Definitions of fuzzy disk and objective function A crisp disk Dððx; yÞ; rÞ is a crisp subset of R2 and it is characterized by its center C ¼ ðx; yÞ and radius r 2 2 in the formulation of Dððx; yÞ; rÞ ¼ fP 2 R2 j kP Ck 6 rg ¼ fðu; vÞ j ðu xÞ þ ðv yÞ 6 r2 g. Therefore e r~Þ as a fuzzy subset of R2 which is characterized by its fuzzy center (fuzzy point) e C; we define a fuzzy disk Dð 2 e C 2 FðR Þ and its fuzzy radius (non-negative fuzzy quantity) ~r 2 FðRþ Þ in a similar way, where FðR2 Þ and FðRþ Þ are the families of all fuzzy subsets of R2 and Rþ ¼ ½0; þ1Þ, respectively. e 2 FðR2 Þ have a membership function lC~ðx; yÞ, 8ðx; yÞ 2 R2 , and a non-negative fuzzy Let a fuzzy point C þ quantity ~ r 2 FðR Þ have a membership function lr~ðtÞ, 8t 2 Rþ . Then for any crisp point P ¼ ðu; vÞ 2 R2 , by the extension principle of fuzzy sets [18], the Euclidean distance on R2 can induce a fuzzy distance d~ðu;vÞ ¼ e 2 FðRþ Þ from the crisp point P ¼ ðu; vÞ to the fuzzy point C e and its membership function is given via: kP Ck ld~ðu;vÞ ðdÞ ¼ ðx;yÞ:
sup pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2
ðxuÞ þðyvÞ ¼d
flC~ðx; yÞg:
ð2:1Þ
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e 6 ~r. Corresponding to the crisp inequality kP Ck 6 r, we have to define then its fuzzy version kP Ck þ þ We can do this by inducing a mapping M~ : FðR Þ FðR Þ ! ½0; 1 from any crisp binary relation 2 f¼; >; <; P ; 6 g, using the extension principle of fuzzy sets [18] again, e BÞ e ¼ supfminflA~ ðxÞ; lB~ðyÞg j x; y 2 R; x yg: M~ð A;
ð2:2Þ
Here we use the minimum operation in the extension principle, which is considered as a standard operation in fuzzy world [18,19,21]. Of course, we can use other t-norms instead of the minimum operation and as such the final result will depend on the choice of the t-norm. It is possible to compare different t-norms in a particular application; however, it is not our focus in this paper. e 2 FðR2 Þ and non-negative fuzzy quantity ~r 2 FðRþ Þ are given, then for Hence, when the fuzzy point C 2 e r~Þ 2 ½0; 1, which is an analogue of the crisp inany crisp point P ¼ ðu; vÞ 2 R , we have M 6 ðkP Ck; equality kP Ck 6 r. Therefore, we have the following definition: e 2 FðR2 Þ and non-negative fuzzy quantity ~r 2 FðRþ Þ, we define a Definition 2.1. For given fuzzy point C e ~ e ~rÞ : R2 ! ½0; 1, via the following formulation e C; e C; fuzzy disk Dð rÞ as a fuzzy subset of R2 , i.e. a mapping Dð ~ðu;vÞ ; ~ ðd rÞ ¼ supfminfld~ðu;vÞ ðdÞ; lr~ðtÞg j d; t 2 Rþ ; d 6 tg lDð ~ rÞ ðu; vÞ ¼ M 6 ~ C;~ 8 82 99 3 < < == ¼ sup min 4 sup l ; ~ðx; yÞ5; lr~ðtÞ C pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ;; d 6t : 2 2 ðx;yÞ:
ð2:3Þ
ðxuÞ þðyvÞ ¼d
e and ~ e r~Þ, respectively. e C; and we will call C r the fuzzy center and fuzzy radius of the fuzzy disk Dð e ~rÞ is a crisp subset of R2 defined via e C; For each a 2 ½0; 1, the a-cut of fuzzy disk Dð if 0 < a 6 1; ~ rÞ ðu; vÞ P ag; ~ C;~ e r~Þ ¼ fðu; vÞ j lDð e C; ½ Dð a clfðu; vÞ j lDð ðu; vÞ > 0g; if a ¼ 0; ~ rÞ ~ C;~ e r~Þ is not a crisp disk, in general, but a union e C; where clðAÞ denotes the closure of a set A. Notice that ½ Dð a e ~rÞ are nested, i.e. a1 P a2 implies the e C; of crisp disks. Moreover, the a-cuts of fuzzy disk Dð e r~Þ ½ Dð e r~Þ . e C; e C; ½ Dð a1 a2 One way to define an optimal solution to the fuzzy disk problem is as follows. For each i 2 f1; 2; . . . ; ng, e ;~ define Si ¼ fPi 2 ½ DðX rÞlðPi Þ g. Obtain an optimal solution ððxi ; yi Þ; ri Þ to the classical smallest covering circle problem on crisp point set Si ; and assign it to membership value lðPi Þ. Define the resultant fuzzy set as the optimal solution to the given fuzzy disk problem. However, in this case, the resultant solution may not have any nice structure. If we are to define an objective that only stresses on fuzziness of the disk, then one way to do this is take into account the size of the fuzzy disk, i.e. the area of the fuzzy disk. Since the a-cuts e r~Þ , denoted e C; of a fuzzy disk are nested crisp subsets of R2 , we can calculate the area of each a-cut ½ Dð a e e e 0 Þ. Hence, the by Areað D a Þ, and form a closed range interval ½s1 ; s0 , where s1 ¼ Areað D 1 Þ and s0 ¼ Areað D e ~ e C; area of fuzzy disk Dð rÞ can be defined as a non-negative fuzzy quantity with support set ½s1 ; s0 as follows: e r~Þ be a fuzzy disk Dð e ~rÞ with its fuzzy center C e 2 FðR2 Þ and its fuzzy radius e C; e C; Definition 2.2. Let Dð þ þ e ~ r 2 FðR Þ. We define its area, Areað DÞ 2 FðR Þ, to be a non-negative fuzzy quantity with the membership function 8 if s > s0 ; < 0; e Þg; if s1 6 s 6 s0 ; supfa 2 ½0; 1 j s ¼ Areað½ D lAreaðDÞ ~ ðsÞ ¼ a : 1; if s < s1 :
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Hence, finding a minimal fuzzy disk in our problem means finding a fuzzy disk with minimal fuzzy area. It involves comparison of fuzzy quantities. The topic of ranking or ordering fuzzy quantities is very important in applying fuzzy set theory to optimization and decision-making problems. The problem of comparing fuzzy quantities has been widely investigated in publications. Many fuzzy ranking methods (FRM) can be found for instance in [1,8,10,20]. As elaborated in [8], most FRMs frequently used in literature can be defined using appropriate ranking functions. e B e 2 F ðRÞ be two fuzzy quantities. A simple method of comparison between them consists of the Let A; e > U ð BÞ, e ¼ U ð BÞ, e < U ð BÞ e U ð AÞ e U ð AÞ e are definition of a certain function U : F ðRÞ ! R such that U ð AÞ e e e e e e equivalent to A > B, A ¼ B, A < B, respectively. The function U is then called a ranking function on F ðRÞ e the ranking value of A. e and U ð AÞ e 2 FðRþ Þ, we assign ranking value U ðAreað DÞÞ e using commonly used In our problem, for each Areað DÞ measures, namely, its total integral value in continuous case and center of gravity in discrete case as defined below. e [10] (see Fig. 1), For the continuous fuzzy quantity Areað DÞ Z 1 1 e e U ðAreað DÞÞ ¼ Areað½ Da Þ da þ s1 ; ð2:4Þ 2 0 e and for the discrete fuzzy quantity Areað DÞ, X e ¼ e Þ: U ðAreað DÞÞ ai Areað½ D ð2:5Þ ai i
Hence, the objective of our problem becomes e Minimize U ðAreað DÞÞ:
ð2:6Þ
ðX~;~ rÞ2FðR2 ÞF ðRþ Þ
2.2. Statement of the fuzzy disk problem Based on the discussion above, our minimal fuzzy disk problem can then be stated as the following: e ¼ fðP1 ; l1 Þ; ðP2 ; l2 Þ; . . . ; ðPn ; ln Þg be a given finite discrete fuzzy subset of R2 with the grades of Let P e r~Þ, which is characterized e C; membership li , i ¼ 1; 2; . . . ; n. We want to find a fuzzy disk of minimal area Dð 2 þ e e e e by its fuzzy center C 2 FðR Þ and its fuzzy radius ~r 2 FðR Þ, such that P Dð C; ~rÞ. It is formulated as the follows: 8 e < Minimize U ðAreað DÞÞ; ~ rÞ2FðR2 ÞF ðRþ Þ ðC;~ ð2:7Þ : e Dð e ~ e C; subject to : P rÞ; e r~Þ is a fuzzy disk by Definition 2.1 and U ðAreað DÞÞ e C; e is given via Eq. (2.4) or Eq. (2.5). where Dð 2 e Recall that the given finite discrete fuzzy subset P of R is characterized by the mapping lP~ : R2 ! ½0; 1 with li 2 ð0; 1; if ðu; vÞ ¼ ðai ; bi Þ 2 P; lP~ ðu; vÞ ¼ 0; otherwise; 2 e e r~Þ means l ~ ðu; vÞ 6 lDð e and the fuzzy inclusion P Dð C; ~ rÞ ðu; vÞ for all ðu; vÞ 2 R . It is also equivalent to ~ C;~ P
e a ½ Dð e ~ e C; rÞa for 8a 2 ½0; 1. Therefore, model (2.7) can be rewritten as P 8 e < Minimize U ðAreað DÞÞ; ~ rÞ2FðR2 ÞF ðRþ Þ ðC;~
: subject to :
e ~ e C; Pi 2 ½ Dð rÞli ;
i ¼ 1; 2; . . . ; n:
ð2:8Þ
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e Fig. 1. Ranking value of a continuous fuzzy quantity Areað DÞ.
3. Special cases of the fuzzy disk problem In this section we present some special cases of the fuzzy disk problem described above and develop polynomial algorithms for these. 3.1. Continuous fuzzy center and continuous fuzzy radius e r~Þ, where the center C e 2 FðR2 Þ is a e C; Here we restrict ourselves to a subclass of the fuzzy disks Dð 2 continuous type fuzzy subset of R of a specific type, as described below and the radius r~ is a non-negative LR-fuzzy number. e 2 FðR2 Þ possesses its own crisp ‘‘center point’’ X ¼ ðx; yÞ and that its We assume that the fuzzy center C membership function is given by 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 2 ðs xÞ þ ðt yÞ A; lC~ðs; tÞ ¼ F @ bc where F ðÞ and bc > 0 are the reference (shape) function and spread parameter of fuzzy numbers, respectively. We also assume that the fuzzy radius ~r is a non-negative LR-type fuzzy number with the following membership function 8 < L rt ; t 6 r; ar lr~ðtÞ ¼ : R tr ; t > r; br where r > 0 is the central value of fuzzy number r~, LðÞ, RðÞ are the left and right reference (shape) functions and ar , br the left and right spread parameters of fuzzy number, respectively. Notice that for any ðu; vÞ 2 R2 , qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 min ðs xÞ þ ðt yÞ j ðs; tÞ : ðs uÞ þ ðt vÞ ¼ d ¼ d 2 þ q2 ðu; vÞ 2dqðu; vÞ ¼ jd qðu; vÞj; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where qðu; vÞ ¼ ðu xÞ2 þ ðv yÞ2 . Thus it follows from Eq. (2.1) and the monotonicity of F ðÞ that, for e between the crisp point P ¼ ðu; vÞ and fuzzy any crisp point P ¼ ðu; vÞ 2 R2 , the distance d~ðu;vÞ ¼ kP Ck e point C is a fuzzy number with the membership function
L. Li et al. / European Journal of Operational Research 160 (2005) 560–573
ld~ðu;vÞ ðdÞ ¼ ðs;tÞ:
sup pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2
ðsuÞ þðtvÞ ¼d
8 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 19 < ðs xÞ2 þ ðt yÞ2 = A ¼ F jd qðu; vÞj ; F@ : ; bc bc
565
d P 0:
2 e r~Þ defined via Eq. (2.3) e C; Hence the membership function lDð ~ rÞ : R ! ½0; 1 of the fuzzy disk Dð ~ C;~ becomes n o ^ jd qðu; vÞj ^ ~ðu;vÞ ; ~ ðd lr~ðtÞ lDð rÞ ¼ sup ld~ðu;vÞ ðdÞ lr~ðtÞ j d 6 t ¼ sup F ~ rÞ ðu; vÞ ¼ M 6 ~ C;~ bc d 6t ( 0 F qðu;vÞt ¼ R t0br ; if r < qðu; vÞ; bc r ¼ 1; if r P qðu; vÞ;
where t0 is the unique root of the following equation (see Fig. 2): qðu; vÞ t tr F ¼R : bc br
ð3:1Þ
In the above general setting, we define three special cases as follows. These three models have a common algorithm that is polynomial and it is given in Section 4. 3.1.1. Crisp center and continuous fuzzy radius e is a crisp point X ¼ ðx; yÞ. When taking F ð0Þ ¼ 1 and F ðtÞ ¼ 0 for t 6¼ 0, we get that the fuzzy center C e Then the unique root of Eq. (3.1) is t0 ¼ qðu; vÞ. In this case, the fuzzy disk DðX ; ~rÞ is a continuous fuzzy subset of R2 with the membership function ( R qðu;vÞr ; if r < qðu; vÞ; br lDð ~ rÞ ðu; vÞ ¼ ~ C;~ 1; if r P qðu; vÞ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where qðu; vÞ ¼ ðu xÞ2 þ ðv yÞ2 . So, for each a 2 ½0; 1, the a-cut qðu; vÞ r 2 e ½ DðX ; r~Þa ¼ ðu; vÞ 2 R j R Pa br qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 1 ð3:2Þ ¼ ðu; vÞ 2 R j ðu xÞ þ ðv yÞ 6 r þ br R ðaÞ ; is a crisp disk with the same center X ðx; yÞ, different radius rðaÞ ¼ r þ br R1 ðaÞ and the area e ;~ e Areað½ DðX rÞa Þ ¼ pr2 ðaÞ. By applying (2.4), we get the ranking value of the area of Dððx; yÞ; ~rÞ as
µ r(t)
µ
O
r
t0
µ d(u,v)(t)
ρ (u, v)
t
~ Fig. 2. Membership grade of fuzzy subset lD~0 ðC;~ ~Þ. ~ rÞ ðu; vÞ ¼ M ðdðu;vÞ ; r 6
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e ¼ 1p U ðAreað DÞÞ 2
Z
1
r2 ðaÞ da þ r2 :
0
Therefore, the following specific model can be derived from the model (2.8): 8 R1 2 > r2 þ 0 ½r þ br R1 ðaÞ da; < Minimize 2 þ 2 ðx;y;r;br Þ2R ðR Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > 2 2 : subject to : ðai xÞ þ ðbi yÞ 6 r þ br R1 ðli Þ; i ¼ 1; 2; . . . ; n:
ð3:3Þ
3.1.2. Fuzzy center and crisp radius When taking Lð0Þ ¼ Rð0Þ ¼ 1 and F ðtÞ ¼ RðtÞ ¼ 0 for t 6¼ 0, we get that the fuzzy radius ~r is a crisp non~ rÞ is a e C; negative number r > 0. Then the unique root of Eq. (3.1) is t0 ¼ r. In this case, the fuzzy disk Dð 2 continuous fuzzy subset of R with the membership function ( F qðu;vÞr ; if r < qðu; vÞ; bc lDð ~ ðu; vÞ ¼ ~ C;rÞ 1; if r P qðu; vÞ; So, for each a 2 ½0; 1, the a-cut ~ rÞ ¼ ðu; vÞ 2 R2 j F qðu; vÞ r P a e C; ½ Dð a bc qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n o 2 2 ¼ ðu; vÞ 2 R2 j ðu xÞ þ ðv yÞ 6 r þ bc F 1 ðaÞ : This is essentially in the same form as Eq. (3.2), and therefore the following model can be established by replacing R1 ðaÞ and br with F 1 ðaÞ and bc in the model (3.3), respectively: 8 R1 2 > r2 þ 0 ½r þ bc F 1 ðaÞ da; < Minimize 2 þ 2 ðx;y;r;bc Þ2R ðR Þ ð3:4Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > 2 2 : subject to : ðai xÞ þ ðbi yÞ 6 r þ bc F 1 ðli Þ; i ¼ 1; 2; . . . ; n: 3.1.3. Linear fuzzy center and linear fuzzy radius When taking F ðtÞ ¼ LðtÞ ¼ RðtÞ ¼ maxf0; 1 tg and bc > 0, ar > 0, br > 0, we get that the fuzzy radius ~ r is a triangular fuzzy number. Then Eq. (3.1) becomes t qðu; vÞ qðu; vÞ t ¼ ; bc br and its unique root is t0 ¼
bc r þ br qðu; vÞ : b c þ br
~ ~ e C; In this case, the fuzzy disk Dð rÞ is also a continuous fuzzy subset of R2 with the membership function lDð ~ rÞ ðu; vÞ ¼ ~ C;~
1 qðu;vÞr ; bc þbr 1;
if r < qðu; vÞ; if r P qðu; vÞ:
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So, for each a 2 ½0; 1, the a-cut qðu; vÞ r 2 e Pa ½ DðX ; r~Þa ¼ ðu; vÞ 2 R j 1 b c þ br ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi q n o 2 2 ¼ ðu; vÞ 2 R2 j ðu xÞ þ ðv yÞ 6 r þ ð1 aÞðbc þ br Þ : This is also essentially in the same form as Eq. (3.2), and therefore the following model can be established by replacing R1 ðaÞ and br with ð1 aÞ and bð¼ bc þ br Þ in the model (3.3), respectively: 8
2 > 2 r þ 12 b 16 b2 ; < Minimize ðx;y;r;bÞ2R2 ðRþ Þ2 ð3:5Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > 2 2 : subject to : ðai xÞ þ ðbi yÞ 6 r þ ð1 aÞb; i ¼ 1; 2; . . . ; n: 3.2. Crisp center and general fuzzy radius e ; r~Þ with a crisp point X ðx; yÞ 2 R2 and a general non-negative fuzzy Now we consider the fuzzy disk DðX þ quantity ~ r 2 FðR Þ as its center and radius, respectively. From Eq. (2.3) we know that the membership e ; r~Þ becomes function of the fuzzy disk DðX lDðX sup lr~ðtÞ; ð3:6Þ ~ ;~ rÞ ðu; vÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi t:
ðxuÞ2 þðyvÞ2 6 t
e DðX e ; r~Þ , i ¼ 1; 2; . . . ; n. From Eq. (3.6) we know that e ;~ and the constraint P rÞ is equivalent to Pi 2 ½ DðX li e each ½ DðX ; r~Þli is a crisp disk with the same center X ðx; yÞ and different radius. Without loss of generality, let us assume that the given crisp points P1 ; P2 ; . . . ; Pn are ordered such that l1 P l2 P P ln . Let the optimal values of x and y be x and y respectively, and let the distances of the points P1 ; P2 ; . . . ; Pn from the center ðx ; y Þ be r1 ; r2 ; . . . ; rn respectively. Let ðr1 ; r2 ; . . . ; rk Þ be the maximal increasing sequence obtained as follows: r1 ¼ r1 ; in general, for any 1 6 i < k, let ri ¼ rj for some 1 6 j < n. Let u ¼ minfv : rv > rj ; v > jg. Then riþ1 ¼ rv . It is easy to see that the final optimal fuzzy radius has k distinct values r1 ; r2 ; . . . ; rk and for each 1 6 i 6 k, the membership grade of ri is lri ¼ lj , where j ¼ minfu : ru > ri g. e ~Þ is then Obviously, lr1 P lr2 P P lrk . The membership function lDðX ~ ;~ rÞ ðu; vÞ of the fuzzy disk DðX ; r of the form 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > lr1 ; if ðx uÞ2 þ ðy vÞ2 6 r1 ; > > qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > > > ; if r < ðx uÞ2 þ ðy vÞ2 6 r2 ; l > 1 r2 < .. .. ð3:7Þ lDðX ~ ;~ rÞ ðu; vÞ ¼ . . qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > > 2 2 > > lrk ; if rrðk1Þ < ðx uÞ þ ðy vÞ 6 rk ; > > qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > 2 2 : 0; if ðx uÞ þ ðy vÞ > rk ; e ;~ and the ranking value of the area of DðX rÞ is obtained from Eq. (2.5): K X e ¼p U ðAreað DÞÞ lij ri2j : j¼1
Therefore, we get the following model from the general model (2.8): 8 PK 2 < Minimize j¼1 lij rij ; ðX ;K;ri1 ;...;riK Þ2R2 Nn ½Rþ K
: subject to :
e ; r~Þ ; Pi 2 ½ DðX li
i ¼ 1; 2; . . . ; n:
e ;~ where Nn ¼ f1; 2; . . . ; ng and DðX rÞ is given by Eq. (3.7). A polynomial algorithm for solving (3.8) is included in Section 4.
ð3:8Þ
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4. Polynomial algorithms for the special cases of the fuzzy disk problem In this section, we develop polynomial algorithms for the special cases of different models of the the fuzzy disk problem discussed in the previous section. 4.1. Algorithm for the models in Section 3.1 Though we have three different models in Section 3.1, these can be treated in a combined fashion by defining the spread parameter b and the reference function GðÞ of fuzzy numbers. Here, b and GðÞ will take the place of br and RðÞ respectively in (3.3). Similarly, bc and F ðÞ in (3.4) will be replaced by b and GðÞ. Also in (3.5), GðtÞ ¼ maxf0; 1 tg. So we deal with the following general model which covers models (3.3)–(3.5): 8 R1 2 > r2 þ 0 ½r þ bG1 ðaÞ da; < Minimize ðx;y;r;bÞ2R2 ðRþ Þ2 ð4:1Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > 2 2 1 : subject to : ðai xÞ þ ðbi yÞ 6 r þ bG ðli Þ; i ¼ 1; 2; . . . ; n: The decision variables in this model are x; y; r > 0 and b, where b is the spread parameter of fuzzy number which measures the uncertainty of fuzzy number. When the spread is zero, the fuzzy number becomes a crisp number. As the spread increases, the fuzzy number becomes fuzzier. In practical applications, sometimes we are given the parameter b which represents the level of permissible violation of the ‘‘rigid’’ radius r. Hence, we shall first consider the case when b is fixed. When b is fixed it is trivial to see that our objective function is equivalent to minimizing r. We notice the fact that two crisp disks DðP ; rÞ ¼ fY 2 R2 j jY P k 6 rg and DðQ; qÞ at least touch each other if and only if kP Qk 6 r þ q. Thus, geometrically, the fuzzy disk problem formulated via (4.1) is to find a disk of minimum radius that at least touches each of the n given disks DðPi ; bG1 ðli ÞÞ, i ¼ 1; 2; . . . ; n. Therefore, an analog of MegiddoÕs method for solving the problem of the smallest ball containing balls [14] by the multidimensional search technique proposed in [13,14] can be employed to solve the model (4.1) in linear time. Refer to [7,13,14] for more details about the multidimensional search technique. Now let us consider the general case, when x; y; r > 0 and b, are our decision variables. Using the by now standard arguments (see [15]), it can be shown that the optimal crisp disk D ððx ; y Þ; r Þ is uniquely determined by two or three or four critical points. Here, by a critical point, we mean a point for which the corresponding inequality constraint is satisfied by the optimal solution as a strict equality. Using this observation, we get the following polynomial scheme for our problem: For each pair of distinct points, say i and j, solve the following system of equations to get values of x; y and r as functions of b: 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > ðx ai Þ2 þ ðy bi Þ2 ¼ r þ bG1 ðli Þ; > > > q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi < 2 2 ðx aj Þ þ ðy bj Þ ¼ r þ bG1 ðlj Þ; ð4:2Þ > > y bi > x ai > : ¼ : aj ai bj bi Substitute for r the corresponding function of b in the objective function of (4.1) to reduce it to a function of b, say gij ðbÞ. Minimizing gij ðbÞ over b is equivalent to finding an appropriate solution of the d equation db ðgij ðbÞÞ ¼ 0. It is easy to see that this reduces to finding an appropriate root bij of a polynomial rij ðbÞ of degree no more than 12 and with coefficients which are polynomial functions of ai , aj , bi , bj , G1 ðli Þ and G1 ðlj Þ. Now consider each triplet of distinct points, say i; j and k, and solve the following system of equations to get values of x, y and r as functions of b:
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8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > ðx ai Þ2 þ ðy bi Þ2 ¼ r þ bG1 ðli Þ; > > > < qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ðx aj Þ þ ðy bj Þ ¼ r þ bG1 ðlj Þ; > > > qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > : ðx a Þ2 þ ðy b Þ2 ¼ r þ bG1 ðl Þ: k k k
569
ð4:3Þ
As in the previous case, substitute for r the corresponding function of b in the objective function of (4.1) and reduce the problem of minimizing the resultant function of b to the problem of finding an appropriate root bijk of a polynomial rijk ðbÞ of degree no more than 12 and with coefficients which are polynomial functions of ai , aj , ak , bi , bj , bk , G1 ðli Þ, G1 ðlj Þ and G1 ðlk Þ. ^ that is feasible and has the Amongst all the roots of all the polynomials rij Õs and rijk Õs, find the one, say b, minimum of objective function value. Finding an approximate value of such a root b^ is precisely the problem solved in [3] by providing a polynomial algorithm. Now consider each quadruplet of distinct points and solve the corresponding four equations to get unique values of x, y, r and b. Consider the objective function values corresponding to those solutions among these that are feasible, and also the objective function value corresponding to b^ found earlier. Among these find the overall minimum to obtain an optimal solution to (4.1).
4.2. Algorithm for the model in Section 3.2 Now we develop a polynomial algorithm for model (3.8). We order the given crisp points P1 ; P2 ; . . . ; Pn such that l1 P l2 P P ln . As we discussed in the previous section, for any center C ¼ ðx; yÞ 2 R2 , the optimal fuzzy radius of the fuzzy disk and hence the optimal objective function value depends on the ordering of the distances of the points P1 ; P2 ; . . . ; Pn from C. Let ri ðx; yÞ ¼ dðC; Pi Þ, ði ¼ 1; 2; . . . ; nÞ. We perform the following procedure to select a maximum subsequence ði1 ¼ 1; i2 ; . . . ; ik Þ of ð1; 2; . . . ; nÞ such that ri1 ðx; yÞ < ri2 ðx; yÞ < < riK ðx; yÞ: Selection Procedure FOR ði ¼ 1; i 6 n 1; i þ þÞ { FOR ðj ¼ i þ 1; i 6 n; j þ þÞ { IF rj ðx; yÞ 6 ri ðx; yÞ THEN delete rj ðx; yÞ } } Then the undeleted values i1 ¼ 1; i2 ; . . . ; ik give the desired subsequence. The corresponding optimal fuzzy radius rðx; yÞ has values ri1 ðx; yÞ; ri2 ðx; yÞ; . . . ; riK ðx; yÞ with membership grades li1 ; li2 ; . . . ; liK resPK 2 pectively and the objective function is j¼1 ½rij ðx; yÞ lij (here ri1 ðx; yÞ ¼ r1 ðx; yÞ). The objective function thus depends on the subsequence ði1 ; i2 ; . . . ; ik Þ which in turn depends on the center ðx; yÞ. We shall therefore tessellate the plane so that for all the points ðx; yÞ in any region the subsequence obtained is the same. For every pair Pi ; Pj 2 fP1 ; P2 ; . . . ; Pn g, i 6¼ j, find perpendicular bisector of the line segment joining the
two points. Do it for every one of the n2 ¼ nðn1Þ pairs of points. These nðn1Þ lines divide the plane into at 2 2
M MðMþ1Þ nðn1Þ 4 most 2 þ 1 3 ¼ Oðn Þ convex polygonal regions, where M ¼ 2 . For each one of these regions, the ordering of distances of P1 ; P2 ; . . . ; Pn from any point ðx; yÞ in the region is the same and consequently the subsequence ði1 ; i2 ; . . . ; ik Þ is the same. (See Fig. 3 for an illustration for n ¼ 3.)
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L23 r1 < r 2 R1 : r 2 < r 3 r1 < r 3
P1
L12
r1 < r 2 R2 : r 2 > r 3 r1 > r 3
L31
r1 < r 2 R3 : r 2 > r 3 r1 > r 3
r1 > r 2 R6 : r 2 < r 3 r1 < r 3
P3 P2 r1 > r 2 R5 : r 2 < r 3 r1 < r 3
r1 > r 2 R4 : r 2 > r 3 r1 > r 3
Fig. 3. An illustration of convex polygonal regions for n ¼ 3. The maximum subsequences: R1 : r1 < r2 < r3 ; R2 : r1 < r2 ; R3 : r1 < r2 ; R4 : r1 ; R5 : r1 < r3 ; R6 : r1 < r3 .
2 Therefore, PKl over each2 one of these convex, polygonal regions of R , say Rl , the objective function Zl ðx; yÞ ¼ j¼1 ½rij ðx; yÞ lij is convex and thus we can minimize it over the convex, polygonal region, to get a local optimal solution ðxl ; yl Þ, using any of the known convex optimization methods in say OðhÞ time. The best among these Oðn4 Þ local optima is the overall optimal solution to our problem. We thus have an Oðn4 hÞ scheme to solve model (3.8).
5. A numerical example In general, optimal solution for the fuzzy version of the problem may vary significantly from that for the classical (crisp) case. Instead of illustrating this for each case separately, we give an example only for the case of the model in Section 3.2. e ¼ fðP1 ¼ ð0; 0Þ; l1 ¼ 0:9Þ; ðP2 ¼ ð4; 0Þ; l2 ¼ 0:2Þ; ðP3 ¼ ð0; 3Þ; l3 ¼ 0:1Þg. Thus, we consider the data P The three perpendicular bisectors of the line segments joining the three pairs of points divides the plane into six regions, as shown in Fig. 4. Region I: The objective function is Z1 ðx; yÞ ¼ 0:9ðr1 ðx; yÞÞ2 þ 0:2ðr2 ðx; yÞÞ2 ¼ 0:9ðx2 þ y 2 Þ þ 0:2ðx2 8x þ 16 þ y 2 Þ ¼ 1:1x2 þ 1:1y 2 1:6x þ 3:2: This is minimized over the region I by the point ðx1 ¼ 118 ; y1 ¼ 32Þ; and the corresponding minimum objective function value is 3.518. For the other regions, the relevant results are given below: Region II: 2
Z2 ðx; yÞ ¼ 0:9ðr1 ðx; yÞÞ ¼ 0:9x2 þ 0:9y 2 : Optimum point is ðx2 ¼ 2; y2 ¼ 32Þ; minimum objective function value is 5.625.
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Fig. 4. A numerical example.
Region III: Z3 ðx; yÞ ¼ 0:9ðr1 ðx; yÞÞ2 ¼ 0:9x2 þ 0:9y 2 : Optimum point is ðx3 ¼ 2; y3 ¼ 32Þ; minimum objective function value is 5.625. Region IV: 2
2
Z4 ðx; yÞ ¼ 0:9ðr1 ðx; yÞÞ þ 0:1ðr3 ðx; yÞÞ ¼ 0:9ðx2 þ y 2 Þ þ 0:1ðx2 þ y 2 6y þ 9Þ ¼ x2 þ y 2 0:6y þ 0:9: Optimum point is ðx4 ¼ 2; y4 ¼ 0:3Þ; minimum objective function value is 4.63. Region V: 2
2
Z5 ðx; yÞ ¼ 0:9ðr1 ðx; yÞÞ þ 0:2ðr2 ðx; yÞÞ þ 0:1ðr3 ðx; yÞÞ
2
¼ 0:9ðx2 þ y 2 Þ þ 0:2ðx2 8x þ 16 þ y 2 Þ þ 0:1ðx2 þ y 2 6y þ 9Þ ¼ 1:2x2 þ 1:2y 2 1:6x 0:6y þ 4:1:
9 Optimum point is x5 ¼ 23 ; y5 ¼ 150 ; minimum objective function value is 2.631. 25 Region VI: 2
2
Z6 ðx; yÞ ¼ 0:9ðr1 ðx; yÞÞ þ 0:2ðr2 ðx; yÞÞ ¼ 1:1x2 þ 1:1y 2 1:6x þ 3:2: Optimum point is ðx6 ¼ 118 ; y6 ¼ 0Þ; minimum objective function value is 2.618. The overall optimal solution is thus, center: ðx ¼ 118 ; y ¼ 0Þ, fuzzy radius: 118 with membership grade 0.9 and 36 with membership grade 0.2. The corresponding minimum objective function value is 2.618. 11 It should be noted that for the classical version of this problem with input data, the crisp points P1 ¼ ð0; 0Þ; P2 ¼ ð4; 0Þ, and P3 ¼ ð0; 3Þ, the optimal solution has center ð2; 32Þ, which is significantly different than the solution to the fuzzy version of the problem. 6. Discussion The problem considered in this paper has strong motivation from possible practical applications. For example, each crisp point Pi may represent a community or entity such as a school, hospital, manufacturing facility or vital installation of some type in a region. In this case, the grade of membership li may represent the population density or importance weight of the ‘‘demand’’ point Pi . Each point thus has a fuzzy
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requirement of coverage in the sense of protection or service of some type such as quality improvement of air, water, etc. in the location. The fuzzy requirement of coverage of each point is represented by its grade of membership. The coverage (protection or service) is given by a center whose domain is defined by the covering disk. Because of the fuzziness of coverage demand, the disk is a fuzzy one. The special cases of the general model investigated in this paper and the resultant polynomial algorithms facilitate finding optimal solutions efficiently in certain cases and provide insight in general context. As such, they are useful in both applications and further research. As revealed by the numerical example, the optimal solution in the fuzzy version could be significantly different from that of the corresponding crisp version. No doubt, further research would be needed to crystallize practical applications in real world situations [17]. Acknowledgements We thank the anonymous referees for their many helpful comments which improved the presentation of the paper. Appendix A. Basic definitions of fuzzy set and fuzzy number e of X is defined by a membership function Definition A.1. Let X be a universe of discourse. A fuzzy subset A e fA~ : X ! ½0; 1. The function value fA~ðxÞ represents the grade of membership of x in A. e denoted by A e a , is defined by Definition A.2. For each a 2 ½0; 1, the a-cut set of fuzzy set A, e a ¼ fx 2 X j lA~ðxÞ P ag: A e is a fuzzy set defined on R whose membership function lA~ðxÞ, x 2 R, is Definition A.3. A fuzzy number A (1) convex, (2) normalized, i.e. there exists a x0 2 R such that lA~ðx0 Þ ¼ 1, and (3) piecewise continuous. e is called an LR-fuzzy number if its membership function is defined by Definition A.4. A fuzzy number A 8 ax > > < L aA ; for x < a; aA > 0; lA~ðxÞ ¼ 1; for a 6 x 6 a; > > : R xa ; for x > a; bA > 0; bA e i.e. the function from ½0; þ1Þ to [0, 1] satisfying where LðtÞ and RðtÞ are the reference functions of A, Lð0Þ ¼ Rð0Þ ¼ 1 and continuous strictly decreasing on that part of ½0; þ1Þ on which they are positive, aA e is represented by four parameters and bA are the spread parameters. Symbolically, the LR-fuzzy number A e a; a; aA and bA , and it is denoted by A ¼ ða; a; aA ; bA ÞLR . Details about LR-fuzzy numbers and their arithmetic operations can be found in [5]. References [1] G. Bortolan, R. Degani, A review of some methods for ranking fuzzy subsets, Fuzzy Sets and Systems 5 (1985) 1–20. [2] R.K. Chakraborty, P.K. Chaudhuri, Note on geometrical solutions for some minimax location problems, Transportation Sciences 15 (1981) 164–166.
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[3] R. Chandrasekaran, A. Tamir, Algebraic optimization: The Fermat–Weber location problem, Mathematical Programming 46 (1990) 219–224. [4] A. Charnes, W.W. Cooper, Chance-constrained programming, Management Sciences 6 (1959) 73–79. [5] D. Dubois, H. Prade, Fuzzy Sets and Systems: Theory and Applications, Academic Press, New York, 1980. [6] D. Dubois, H. Prade, Possibility Theory, Plenum, New York, 1988. [7] W.E. Dyer, On a multidimensional search technique and its application to the Euclidean one-center problem, SIAM Journal on Computing 15 (1986) 725–738. [8] A. Gonzalez, A study of the ranking function approach through mean values, Fuzzy Sets and Systems 35 (1990) 29–41. [9] L. Li, S.N. Kabadi, K.P.K. Nair, Fuzzy versions of the covering circle problem, European Journal of Operational Research 137 (2002) 93–109. [10] T.S. Liou, M.J. Wang, Ranking fuzzy numbers with integral value, Fuzzy Sets and Systems 50 (1992) 247–255. [11] J. Elzinga, D.W. Hearn, Geometrical solutions for some minimax location problems, Transportation Sciences 6 (1972) 379–394. [12] N. Megiddo, Linear time algorithms for linear programming in R3 and related problems, SIAM Journal on Computing 12 (1983) 759–776. [13] N. Megiddo, Linear programming in linear time when the dimension is fixed, Journal of the Association for Computing Machinery 31 (1984) 114–127. [14] N. Megiddo, On the ball spanned by balls, Discrete and Computational Geometry 4 (1989) 605–610. [15] K.P.K. Nair, R. Chandrasekaran, Optimal location of a single service center of certain types, Naval Research Logistics Quarterly 18 (1971) 503–509. [16] M.T. Shamos, D. Hoey, Closest-point problems, in: Proceedings of the 16th Annual IEEE Symposium on Foundations of Computer Sciences, IEEE, New York, 1975, pp. 151–162. [17] J.L. Verdegay, Applications of fuzzy optimization in operation in operational research, Control Cybernetics 13 (1984) 229–239. [18] L.A. Zadeh, Fuzzy sets, Information and Control 8 (1965) 338–353. [19] L.A. Zadeh, Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets and Systems 1 (1978) 3–28. [20] Q. Zhu, E.S. Lee, Comparison and ranking fuzzy numbers, in: J. Kacprzyk, M. Fedrizzi (Eds.), Fuzzy Regression Analysis, Omnitech Press and Physica-Verlag, Wurzburg, 1992, pp. 21–44. [21] H.J. Zimmermann, Fuzzy Set Theory and Its Applications, Kluwer Academic Publishers, Boston, Dordrecht, London, 1996.
Computing, Artificial Intelligence and Information Technology
Fuzzy disk for covering fuzzy points q Lushu Li, S.N. Kabadi, K.P.K. Nair
*
Faculty of Administration, University of New Brunswick, Fredericton, NB, Canada E3B 5A3 Received 16 May 2001; accepted 24 June 2003 Available online 23 September 2003
Abstract In this paper, we consider an important fuzzy version of the well known smallest covering circle problem which is also called the Euclidean 1-center problem. Here we are given a set of points in the plane whose locations are crisp. However, the requirement for coverage of each point is fuzzy as represented by its grade of membership. As such we qualify the points as fuzzy. In the above framework, we wish to find a fuzzy disk with minimum fuzzy area for covering the given fuzzy points. After developing a general model, certain special cases are investigated in detail. Polynomial algorithms are presented for the special cases. Ó 2003 Elsevier B.V. All rights reserved. Keywords: Location; Covering circle; Convex program; Polynomial algorithm; Fuzzy set
1. Introduction The classical smallest covering circle problem is defined as follows. Let Pi ¼ ðai ; bi Þ ði ¼ 1; 2; . . . ; nÞ be n given points in the Euclidean plane R2 . For any point P ¼ ðx; yÞ 2 R2 , let qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 dðP ; Pi Þ ¼ ðx ai Þ þ ðy bi Þ and H ðx; yÞ ¼ max dðP ; Pi Þ: 16i6n
Then the objective is to find a point P ¼ ðx ; y Þ in R2 so that H ðx ; y Þ is minimum. This smallest covering circle problem is also known as the Euclidean 1-center problem in the language of location theory, where one seeks to locate a point (a ‘‘facility’’) so as to minimize the largest distance between the facility and the given (‘‘demand’’) points. It has many practical applications in real world, such as the location of a Radar Station, Radio Transmitter, Hospital for Emergency Cases, Police Station or q This work was supported by research grants from the Natural Sciences and Engineering Research Council of Canada to S.N. Kabadi and K.P.K. Nair. * Corresponding author. Fax: +1-506-453-3561. E-mail address: [email protected] (K.P.K. Nair).
0377-2217/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0377-2217(03)00430-2
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Fire Station. Therefore it has been studied extensively by many authors and several algorithms have been developed (see, e.g. [2,11,12,16]. Among these MegiddoÕs algorithm [12] runs in linear time and it follows from [13] that in any fixed dimension the smallest ball enclosing n points can be found in OðnÞ time since the problem can be formulated as a convex quadratic programming problem [14]. In the classical smallest covering circle problem the points to be covered are crisp with respect to their locations in the plane and each point must be covered or enclosed by the smallest circle. To the best of our knowledge, there is only one paper [9] that incorporates fuzziness in the above problem. In [9] the fuzziness we introduced is entirely due to the fuzziness with respect to the locations of the points in the plane. Investigation of this problem resulted in two fuzzy models each of which has a polynomial algorithm. The contribution of this paper consists of introduction of another important type of fuzziness distinctly different from that in [9], development of a general model for covering fuzzy points by a fuzzy disk and investigation of special cases resulting in polynomial algorithms. The fuzziness considered arises from the fact that though the location of each point is crisp, its requirement to be covered is fuzzy as represented by its grade of membership. In this context we qualify the given points as fuzzy points. In the above framework, we wish to find a fuzzy disk with minimum fuzzy area for covering the given fuzzy points in the plane. We develop a general model of the above problem in Section 2, and in Section 3 certain special cases are investigated in detail. Polynomial algorithms for the special cases are presented in Section 4. Section 5 gives a numerical example, while Section 6 presents discussion. Relevant basic definitions and notations are given in the Appendix A. 2. Formulation of the smallest fuzzy disk problem The conventional, well-known smallest covering circle problem can be described as: given a finite set of points P ¼ fP1 ; P2 ; . . . ; Pn g R2 , where Pi ¼ ðai ; bi Þ, ði ¼ 1; 2; . . . ; nÞ, find a disk of minimum radius Dððx; yÞ; rÞ ¼ fðu; vÞjðu xÞ2 þ ðv yÞ2 6 r2 g R2 such that P Dððx; yÞ; rÞ. The optimal disk Dððx ; y Þ; r Þ is completely characterized by its center ðx ; y Þ and its radius r . We try to adapt this crisp model into its fuzzy version by using the fuzzy set framework (see Appendix A for basic definitions relevant for understanding the fuzzy models in this paper which can be found in [5,6]). e ¼ fðP1 ; lðP1 ÞÞ; ðP2 ; lðP2 ÞÞ; . . . ; ðPn ; lðPn ÞÞg 2 Thus, we assume that we are given a finite fuzzy subset P FðR2 Þ, where FðR2 Þ denotes the family of all fuzzy subsets of R2 and li ¼ lðPi Þ is the grade or degree of e Our focus is to find a minimal fuzzy disk D e membership of the crisp point Pi ¼ ðai ; bi Þ in the fuzzy subset P. e in the sense of fuzzy inclusion [4]. such that Pe D 2.1. Definitions of fuzzy disk and objective function A crisp disk Dððx; yÞ; rÞ is a crisp subset of R2 and it is characterized by its center C ¼ ðx; yÞ and radius r 2 2 in the formulation of Dððx; yÞ; rÞ ¼ fP 2 R2 j kP Ck 6 rg ¼ fðu; vÞ j ðu xÞ þ ðv yÞ 6 r2 g. Therefore e r~Þ as a fuzzy subset of R2 which is characterized by its fuzzy center (fuzzy point) e C; we define a fuzzy disk Dð 2 e C 2 FðR Þ and its fuzzy radius (non-negative fuzzy quantity) ~r 2 FðRþ Þ in a similar way, where FðR2 Þ and FðRþ Þ are the families of all fuzzy subsets of R2 and Rþ ¼ ½0; þ1Þ, respectively. e 2 FðR2 Þ have a membership function lC~ðx; yÞ, 8ðx; yÞ 2 R2 , and a non-negative fuzzy Let a fuzzy point C þ quantity ~ r 2 FðR Þ have a membership function lr~ðtÞ, 8t 2 Rþ . Then for any crisp point P ¼ ðu; vÞ 2 R2 , by the extension principle of fuzzy sets [18], the Euclidean distance on R2 can induce a fuzzy distance d~ðu;vÞ ¼ e 2 FðRþ Þ from the crisp point P ¼ ðu; vÞ to the fuzzy point C e and its membership function is given via: kP Ck ld~ðu;vÞ ðdÞ ¼ ðx;yÞ:
sup pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2
ðxuÞ þðyvÞ ¼d
flC~ðx; yÞg:
ð2:1Þ
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e 6 ~r. Corresponding to the crisp inequality kP Ck 6 r, we have to define then its fuzzy version kP Ck þ þ We can do this by inducing a mapping M~ : FðR Þ FðR Þ ! ½0; 1 from any crisp binary relation 2 f¼; >; <; P ; 6 g, using the extension principle of fuzzy sets [18] again, e BÞ e ¼ supfminflA~ ðxÞ; lB~ðyÞg j x; y 2 R; x yg: M~ð A;
ð2:2Þ
Here we use the minimum operation in the extension principle, which is considered as a standard operation in fuzzy world [18,19,21]. Of course, we can use other t-norms instead of the minimum operation and as such the final result will depend on the choice of the t-norm. It is possible to compare different t-norms in a particular application; however, it is not our focus in this paper. e 2 FðR2 Þ and non-negative fuzzy quantity ~r 2 FðRþ Þ are given, then for Hence, when the fuzzy point C 2 e r~Þ 2 ½0; 1, which is an analogue of the crisp inany crisp point P ¼ ðu; vÞ 2 R , we have M 6 ðkP Ck; equality kP Ck 6 r. Therefore, we have the following definition: e 2 FðR2 Þ and non-negative fuzzy quantity ~r 2 FðRþ Þ, we define a Definition 2.1. For given fuzzy point C e ~ e ~rÞ : R2 ! ½0; 1, via the following formulation e C; e C; fuzzy disk Dð rÞ as a fuzzy subset of R2 , i.e. a mapping Dð ~ðu;vÞ ; ~ ðd rÞ ¼ supfminfld~ðu;vÞ ðdÞ; lr~ðtÞg j d; t 2 Rþ ; d 6 tg lDð ~ rÞ ðu; vÞ ¼ M 6 ~ C;~ 8 82 99 3 < < == ¼ sup min 4 sup l ; ~ðx; yÞ5; lr~ðtÞ C pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ;; d 6t : 2 2 ðx;yÞ:
ð2:3Þ
ðxuÞ þðyvÞ ¼d
e and ~ e r~Þ, respectively. e C; and we will call C r the fuzzy center and fuzzy radius of the fuzzy disk Dð e ~rÞ is a crisp subset of R2 defined via e C; For each a 2 ½0; 1, the a-cut of fuzzy disk Dð if 0 < a 6 1; ~ rÞ ðu; vÞ P ag; ~ C;~ e r~Þ ¼ fðu; vÞ j lDð e C; ½ Dð a clfðu; vÞ j lDð ðu; vÞ > 0g; if a ¼ 0; ~ rÞ ~ C;~ e r~Þ is not a crisp disk, in general, but a union e C; where clðAÞ denotes the closure of a set A. Notice that ½ Dð a e ~rÞ are nested, i.e. a1 P a2 implies the e C; of crisp disks. Moreover, the a-cuts of fuzzy disk Dð e r~Þ ½ Dð e r~Þ . e C; e C; ½ Dð a1 a2 One way to define an optimal solution to the fuzzy disk problem is as follows. For each i 2 f1; 2; . . . ; ng, e ;~ define Si ¼ fPi 2 ½ DðX rÞlðPi Þ g. Obtain an optimal solution ððxi ; yi Þ; ri Þ to the classical smallest covering circle problem on crisp point set Si ; and assign it to membership value lðPi Þ. Define the resultant fuzzy set as the optimal solution to the given fuzzy disk problem. However, in this case, the resultant solution may not have any nice structure. If we are to define an objective that only stresses on fuzziness of the disk, then one way to do this is take into account the size of the fuzzy disk, i.e. the area of the fuzzy disk. Since the a-cuts e r~Þ , denoted e C; of a fuzzy disk are nested crisp subsets of R2 , we can calculate the area of each a-cut ½ Dð a e e e 0 Þ. Hence, the by Areað D a Þ, and form a closed range interval ½s1 ; s0 , where s1 ¼ Areað D 1 Þ and s0 ¼ Areað D e ~ e C; area of fuzzy disk Dð rÞ can be defined as a non-negative fuzzy quantity with support set ½s1 ; s0 as follows: e r~Þ be a fuzzy disk Dð e ~rÞ with its fuzzy center C e 2 FðR2 Þ and its fuzzy radius e C; e C; Definition 2.2. Let Dð þ þ e ~ r 2 FðR Þ. We define its area, Areað DÞ 2 FðR Þ, to be a non-negative fuzzy quantity with the membership function 8 if s > s0 ; < 0; e Þg; if s1 6 s 6 s0 ; supfa 2 ½0; 1 j s ¼ Areað½ D lAreaðDÞ ~ ðsÞ ¼ a : 1; if s < s1 :
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Hence, finding a minimal fuzzy disk in our problem means finding a fuzzy disk with minimal fuzzy area. It involves comparison of fuzzy quantities. The topic of ranking or ordering fuzzy quantities is very important in applying fuzzy set theory to optimization and decision-making problems. The problem of comparing fuzzy quantities has been widely investigated in publications. Many fuzzy ranking methods (FRM) can be found for instance in [1,8,10,20]. As elaborated in [8], most FRMs frequently used in literature can be defined using appropriate ranking functions. e B e 2 F ðRÞ be two fuzzy quantities. A simple method of comparison between them consists of the Let A; e > U ð BÞ, e ¼ U ð BÞ, e < U ð BÞ e U ð AÞ e U ð AÞ e are definition of a certain function U : F ðRÞ ! R such that U ð AÞ e e e e e e equivalent to A > B, A ¼ B, A < B, respectively. The function U is then called a ranking function on F ðRÞ e the ranking value of A. e and U ð AÞ e 2 FðRþ Þ, we assign ranking value U ðAreað DÞÞ e using commonly used In our problem, for each Areað DÞ measures, namely, its total integral value in continuous case and center of gravity in discrete case as defined below. e [10] (see Fig. 1), For the continuous fuzzy quantity Areað DÞ Z 1 1 e e U ðAreað DÞÞ ¼ Areað½ Da Þ da þ s1 ; ð2:4Þ 2 0 e and for the discrete fuzzy quantity Areað DÞ, X e ¼ e Þ: U ðAreað DÞÞ ai Areað½ D ð2:5Þ ai i
Hence, the objective of our problem becomes e Minimize U ðAreað DÞÞ:
ð2:6Þ
ðX~;~ rÞ2FðR2 ÞF ðRþ Þ
2.2. Statement of the fuzzy disk problem Based on the discussion above, our minimal fuzzy disk problem can then be stated as the following: e ¼ fðP1 ; l1 Þ; ðP2 ; l2 Þ; . . . ; ðPn ; ln Þg be a given finite discrete fuzzy subset of R2 with the grades of Let P e r~Þ, which is characterized e C; membership li , i ¼ 1; 2; . . . ; n. We want to find a fuzzy disk of minimal area Dð 2 þ e e e e by its fuzzy center C 2 FðR Þ and its fuzzy radius ~r 2 FðR Þ, such that P Dð C; ~rÞ. It is formulated as the follows: 8 e < Minimize U ðAreað DÞÞ; ~ rÞ2FðR2 ÞF ðRþ Þ ðC;~ ð2:7Þ : e Dð e ~ e C; subject to : P rÞ; e r~Þ is a fuzzy disk by Definition 2.1 and U ðAreað DÞÞ e C; e is given via Eq. (2.4) or Eq. (2.5). where Dð 2 e Recall that the given finite discrete fuzzy subset P of R is characterized by the mapping lP~ : R2 ! ½0; 1 with li 2 ð0; 1; if ðu; vÞ ¼ ðai ; bi Þ 2 P; lP~ ðu; vÞ ¼ 0; otherwise; 2 e e r~Þ means l ~ ðu; vÞ 6 lDð e and the fuzzy inclusion P Dð C; ~ rÞ ðu; vÞ for all ðu; vÞ 2 R . It is also equivalent to ~ C;~ P
e a ½ Dð e ~ e C; rÞa for 8a 2 ½0; 1. Therefore, model (2.7) can be rewritten as P 8 e < Minimize U ðAreað DÞÞ; ~ rÞ2FðR2 ÞF ðRþ Þ ðC;~
: subject to :
e ~ e C; Pi 2 ½ Dð rÞli ;
i ¼ 1; 2; . . . ; n:
ð2:8Þ
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e Fig. 1. Ranking value of a continuous fuzzy quantity Areað DÞ.
3. Special cases of the fuzzy disk problem In this section we present some special cases of the fuzzy disk problem described above and develop polynomial algorithms for these. 3.1. Continuous fuzzy center and continuous fuzzy radius e r~Þ, where the center C e 2 FðR2 Þ is a e C; Here we restrict ourselves to a subclass of the fuzzy disks Dð 2 continuous type fuzzy subset of R of a specific type, as described below and the radius r~ is a non-negative LR-fuzzy number. e 2 FðR2 Þ possesses its own crisp ‘‘center point’’ X ¼ ðx; yÞ and that its We assume that the fuzzy center C membership function is given by 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 2 ðs xÞ þ ðt yÞ A; lC~ðs; tÞ ¼ F @ bc where F ðÞ and bc > 0 are the reference (shape) function and spread parameter of fuzzy numbers, respectively. We also assume that the fuzzy radius ~r is a non-negative LR-type fuzzy number with the following membership function 8 < L rt ; t 6 r; ar lr~ðtÞ ¼ : R tr ; t > r; br where r > 0 is the central value of fuzzy number r~, LðÞ, RðÞ are the left and right reference (shape) functions and ar , br the left and right spread parameters of fuzzy number, respectively. Notice that for any ðu; vÞ 2 R2 , qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 min ðs xÞ þ ðt yÞ j ðs; tÞ : ðs uÞ þ ðt vÞ ¼ d ¼ d 2 þ q2 ðu; vÞ 2dqðu; vÞ ¼ jd qðu; vÞj; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where qðu; vÞ ¼ ðu xÞ2 þ ðv yÞ2 . Thus it follows from Eq. (2.1) and the monotonicity of F ðÞ that, for e between the crisp point P ¼ ðu; vÞ and fuzzy any crisp point P ¼ ðu; vÞ 2 R2 , the distance d~ðu;vÞ ¼ kP Ck e point C is a fuzzy number with the membership function
L. Li et al. / European Journal of Operational Research 160 (2005) 560–573
ld~ðu;vÞ ðdÞ ¼ ðs;tÞ:
sup pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2
ðsuÞ þðtvÞ ¼d
8 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 19 < ðs xÞ2 þ ðt yÞ2 = A ¼ F jd qðu; vÞj ; F@ : ; bc bc
565
d P 0:
2 e r~Þ defined via Eq. (2.3) e C; Hence the membership function lDð ~ rÞ : R ! ½0; 1 of the fuzzy disk Dð ~ C;~ becomes n o ^ jd qðu; vÞj ^ ~ðu;vÞ ; ~ ðd lr~ðtÞ lDð rÞ ¼ sup ld~ðu;vÞ ðdÞ lr~ðtÞ j d 6 t ¼ sup F ~ rÞ ðu; vÞ ¼ M 6 ~ C;~ bc d 6t ( 0 F qðu;vÞt ¼ R t0br ; if r < qðu; vÞ; bc r ¼ 1; if r P qðu; vÞ;
where t0 is the unique root of the following equation (see Fig. 2): qðu; vÞ t tr F ¼R : bc br
ð3:1Þ
In the above general setting, we define three special cases as follows. These three models have a common algorithm that is polynomial and it is given in Section 4. 3.1.1. Crisp center and continuous fuzzy radius e is a crisp point X ¼ ðx; yÞ. When taking F ð0Þ ¼ 1 and F ðtÞ ¼ 0 for t 6¼ 0, we get that the fuzzy center C e Then the unique root of Eq. (3.1) is t0 ¼ qðu; vÞ. In this case, the fuzzy disk DðX ; ~rÞ is a continuous fuzzy subset of R2 with the membership function ( R qðu;vÞr ; if r < qðu; vÞ; br lDð ~ rÞ ðu; vÞ ¼ ~ C;~ 1; if r P qðu; vÞ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where qðu; vÞ ¼ ðu xÞ2 þ ðv yÞ2 . So, for each a 2 ½0; 1, the a-cut qðu; vÞ r 2 e ½ DðX ; r~Þa ¼ ðu; vÞ 2 R j R Pa br qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 1 ð3:2Þ ¼ ðu; vÞ 2 R j ðu xÞ þ ðv yÞ 6 r þ br R ðaÞ ; is a crisp disk with the same center X ðx; yÞ, different radius rðaÞ ¼ r þ br R1 ðaÞ and the area e ;~ e Areað½ DðX rÞa Þ ¼ pr2 ðaÞ. By applying (2.4), we get the ranking value of the area of Dððx; yÞ; ~rÞ as
µ r(t)
µ
O
r
t0
µ d(u,v)(t)
ρ (u, v)
t
~ Fig. 2. Membership grade of fuzzy subset lD~0 ðC;~ ~Þ. ~ rÞ ðu; vÞ ¼ M ðdðu;vÞ ; r 6
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e ¼ 1p U ðAreað DÞÞ 2
Z
1
r2 ðaÞ da þ r2 :
0
Therefore, the following specific model can be derived from the model (2.8): 8 R1 2 > r2 þ 0 ½r þ br R1 ðaÞ da; < Minimize 2 þ 2 ðx;y;r;br Þ2R ðR Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > 2 2 : subject to : ðai xÞ þ ðbi yÞ 6 r þ br R1 ðli Þ; i ¼ 1; 2; . . . ; n:
ð3:3Þ
3.1.2. Fuzzy center and crisp radius When taking Lð0Þ ¼ Rð0Þ ¼ 1 and F ðtÞ ¼ RðtÞ ¼ 0 for t 6¼ 0, we get that the fuzzy radius ~r is a crisp non~ rÞ is a e C; negative number r > 0. Then the unique root of Eq. (3.1) is t0 ¼ r. In this case, the fuzzy disk Dð 2 continuous fuzzy subset of R with the membership function ( F qðu;vÞr ; if r < qðu; vÞ; bc lDð ~ ðu; vÞ ¼ ~ C;rÞ 1; if r P qðu; vÞ; So, for each a 2 ½0; 1, the a-cut ~ rÞ ¼ ðu; vÞ 2 R2 j F qðu; vÞ r P a e C; ½ Dð a bc qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n o 2 2 ¼ ðu; vÞ 2 R2 j ðu xÞ þ ðv yÞ 6 r þ bc F 1 ðaÞ : This is essentially in the same form as Eq. (3.2), and therefore the following model can be established by replacing R1 ðaÞ and br with F 1 ðaÞ and bc in the model (3.3), respectively: 8 R1 2 > r2 þ 0 ½r þ bc F 1 ðaÞ da; < Minimize 2 þ 2 ðx;y;r;bc Þ2R ðR Þ ð3:4Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > 2 2 : subject to : ðai xÞ þ ðbi yÞ 6 r þ bc F 1 ðli Þ; i ¼ 1; 2; . . . ; n: 3.1.3. Linear fuzzy center and linear fuzzy radius When taking F ðtÞ ¼ LðtÞ ¼ RðtÞ ¼ maxf0; 1 tg and bc > 0, ar > 0, br > 0, we get that the fuzzy radius ~ r is a triangular fuzzy number. Then Eq. (3.1) becomes t qðu; vÞ qðu; vÞ t ¼ ; bc br and its unique root is t0 ¼
bc r þ br qðu; vÞ : b c þ br
~ ~ e C; In this case, the fuzzy disk Dð rÞ is also a continuous fuzzy subset of R2 with the membership function lDð ~ rÞ ðu; vÞ ¼ ~ C;~
1 qðu;vÞr ; bc þbr 1;
if r < qðu; vÞ; if r P qðu; vÞ:
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So, for each a 2 ½0; 1, the a-cut qðu; vÞ r 2 e Pa ½ DðX ; r~Þa ¼ ðu; vÞ 2 R j 1 b c þ br ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi q n o 2 2 ¼ ðu; vÞ 2 R2 j ðu xÞ þ ðv yÞ 6 r þ ð1 aÞðbc þ br Þ : This is also essentially in the same form as Eq. (3.2), and therefore the following model can be established by replacing R1 ðaÞ and br with ð1 aÞ and bð¼ bc þ br Þ in the model (3.3), respectively: 8
2 > 2 r þ 12 b 16 b2 ; < Minimize ðx;y;r;bÞ2R2 ðRþ Þ2 ð3:5Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > 2 2 : subject to : ðai xÞ þ ðbi yÞ 6 r þ ð1 aÞb; i ¼ 1; 2; . . . ; n: 3.2. Crisp center and general fuzzy radius e ; r~Þ with a crisp point X ðx; yÞ 2 R2 and a general non-negative fuzzy Now we consider the fuzzy disk DðX þ quantity ~ r 2 FðR Þ as its center and radius, respectively. From Eq. (2.3) we know that the membership e ; r~Þ becomes function of the fuzzy disk DðX lDðX sup lr~ðtÞ; ð3:6Þ ~ ;~ rÞ ðu; vÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi t:
ðxuÞ2 þðyvÞ2 6 t
e DðX e ; r~Þ , i ¼ 1; 2; . . . ; n. From Eq. (3.6) we know that e ;~ and the constraint P rÞ is equivalent to Pi 2 ½ DðX li e each ½ DðX ; r~Þli is a crisp disk with the same center X ðx; yÞ and different radius. Without loss of generality, let us assume that the given crisp points P1 ; P2 ; . . . ; Pn are ordered such that l1 P l2 P P ln . Let the optimal values of x and y be x and y respectively, and let the distances of the points P1 ; P2 ; . . . ; Pn from the center ðx ; y Þ be r1 ; r2 ; . . . ; rn respectively. Let ðr1 ; r2 ; . . . ; rk Þ be the maximal increasing sequence obtained as follows: r1 ¼ r1 ; in general, for any 1 6 i < k, let ri ¼ rj for some 1 6 j < n. Let u ¼ minfv : rv > rj ; v > jg. Then riþ1 ¼ rv . It is easy to see that the final optimal fuzzy radius has k distinct values r1 ; r2 ; . . . ; rk and for each 1 6 i 6 k, the membership grade of ri is lri ¼ lj , where j ¼ minfu : ru > ri g. e ~Þ is then Obviously, lr1 P lr2 P P lrk . The membership function lDðX ~ ;~ rÞ ðu; vÞ of the fuzzy disk DðX ; r of the form 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > lr1 ; if ðx uÞ2 þ ðy vÞ2 6 r1 ; > > qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > > > ; if r < ðx uÞ2 þ ðy vÞ2 6 r2 ; l > 1 r2 < .. .. ð3:7Þ lDðX ~ ;~ rÞ ðu; vÞ ¼ . . qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > > 2 2 > > lrk ; if rrðk1Þ < ðx uÞ þ ðy vÞ 6 rk ; > > qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > 2 2 : 0; if ðx uÞ þ ðy vÞ > rk ; e ;~ and the ranking value of the area of DðX rÞ is obtained from Eq. (2.5): K X e ¼p U ðAreað DÞÞ lij ri2j : j¼1
Therefore, we get the following model from the general model (2.8): 8 PK 2 < Minimize j¼1 lij rij ; ðX ;K;ri1 ;...;riK Þ2R2 Nn ½Rþ K
: subject to :
e ; r~Þ ; Pi 2 ½ DðX li
i ¼ 1; 2; . . . ; n:
e ;~ where Nn ¼ f1; 2; . . . ; ng and DðX rÞ is given by Eq. (3.7). A polynomial algorithm for solving (3.8) is included in Section 4.
ð3:8Þ
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4. Polynomial algorithms for the special cases of the fuzzy disk problem In this section, we develop polynomial algorithms for the special cases of different models of the the fuzzy disk problem discussed in the previous section. 4.1. Algorithm for the models in Section 3.1 Though we have three different models in Section 3.1, these can be treated in a combined fashion by defining the spread parameter b and the reference function GðÞ of fuzzy numbers. Here, b and GðÞ will take the place of br and RðÞ respectively in (3.3). Similarly, bc and F ðÞ in (3.4) will be replaced by b and GðÞ. Also in (3.5), GðtÞ ¼ maxf0; 1 tg. So we deal with the following general model which covers models (3.3)–(3.5): 8 R1 2 > r2 þ 0 ½r þ bG1 ðaÞ da; < Minimize ðx;y;r;bÞ2R2 ðRþ Þ2 ð4:1Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > 2 2 1 : subject to : ðai xÞ þ ðbi yÞ 6 r þ bG ðli Þ; i ¼ 1; 2; . . . ; n: The decision variables in this model are x; y; r > 0 and b, where b is the spread parameter of fuzzy number which measures the uncertainty of fuzzy number. When the spread is zero, the fuzzy number becomes a crisp number. As the spread increases, the fuzzy number becomes fuzzier. In practical applications, sometimes we are given the parameter b which represents the level of permissible violation of the ‘‘rigid’’ radius r. Hence, we shall first consider the case when b is fixed. When b is fixed it is trivial to see that our objective function is equivalent to minimizing r. We notice the fact that two crisp disks DðP ; rÞ ¼ fY 2 R2 j jY P k 6 rg and DðQ; qÞ at least touch each other if and only if kP Qk 6 r þ q. Thus, geometrically, the fuzzy disk problem formulated via (4.1) is to find a disk of minimum radius that at least touches each of the n given disks DðPi ; bG1 ðli ÞÞ, i ¼ 1; 2; . . . ; n. Therefore, an analog of MegiddoÕs method for solving the problem of the smallest ball containing balls [14] by the multidimensional search technique proposed in [13,14] can be employed to solve the model (4.1) in linear time. Refer to [7,13,14] for more details about the multidimensional search technique. Now let us consider the general case, when x; y; r > 0 and b, are our decision variables. Using the by now standard arguments (see [15]), it can be shown that the optimal crisp disk D ððx ; y Þ; r Þ is uniquely determined by two or three or four critical points. Here, by a critical point, we mean a point for which the corresponding inequality constraint is satisfied by the optimal solution as a strict equality. Using this observation, we get the following polynomial scheme for our problem: For each pair of distinct points, say i and j, solve the following system of equations to get values of x; y and r as functions of b: 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > ðx ai Þ2 þ ðy bi Þ2 ¼ r þ bG1 ðli Þ; > > > q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi < 2 2 ðx aj Þ þ ðy bj Þ ¼ r þ bG1 ðlj Þ; ð4:2Þ > > y bi > x ai > : ¼ : aj ai bj bi Substitute for r the corresponding function of b in the objective function of (4.1) to reduce it to a function of b, say gij ðbÞ. Minimizing gij ðbÞ over b is equivalent to finding an appropriate solution of the d equation db ðgij ðbÞÞ ¼ 0. It is easy to see that this reduces to finding an appropriate root bij of a polynomial rij ðbÞ of degree no more than 12 and with coefficients which are polynomial functions of ai , aj , bi , bj , G1 ðli Þ and G1 ðlj Þ. Now consider each triplet of distinct points, say i; j and k, and solve the following system of equations to get values of x, y and r as functions of b:
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8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > ðx ai Þ2 þ ðy bi Þ2 ¼ r þ bG1 ðli Þ; > > > < qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ðx aj Þ þ ðy bj Þ ¼ r þ bG1 ðlj Þ; > > > qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > : ðx a Þ2 þ ðy b Þ2 ¼ r þ bG1 ðl Þ: k k k
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ð4:3Þ
As in the previous case, substitute for r the corresponding function of b in the objective function of (4.1) and reduce the problem of minimizing the resultant function of b to the problem of finding an appropriate root bijk of a polynomial rijk ðbÞ of degree no more than 12 and with coefficients which are polynomial functions of ai , aj , ak , bi , bj , bk , G1 ðli Þ, G1 ðlj Þ and G1 ðlk Þ. ^ that is feasible and has the Amongst all the roots of all the polynomials rij Õs and rijk Õs, find the one, say b, minimum of objective function value. Finding an approximate value of such a root b^ is precisely the problem solved in [3] by providing a polynomial algorithm. Now consider each quadruplet of distinct points and solve the corresponding four equations to get unique values of x, y, r and b. Consider the objective function values corresponding to those solutions among these that are feasible, and also the objective function value corresponding to b^ found earlier. Among these find the overall minimum to obtain an optimal solution to (4.1).
4.2. Algorithm for the model in Section 3.2 Now we develop a polynomial algorithm for model (3.8). We order the given crisp points P1 ; P2 ; . . . ; Pn such that l1 P l2 P P ln . As we discussed in the previous section, for any center C ¼ ðx; yÞ 2 R2 , the optimal fuzzy radius of the fuzzy disk and hence the optimal objective function value depends on the ordering of the distances of the points P1 ; P2 ; . . . ; Pn from C. Let ri ðx; yÞ ¼ dðC; Pi Þ, ði ¼ 1; 2; . . . ; nÞ. We perform the following procedure to select a maximum subsequence ði1 ¼ 1; i2 ; . . . ; ik Þ of ð1; 2; . . . ; nÞ such that ri1 ðx; yÞ < ri2 ðx; yÞ < < riK ðx; yÞ: Selection Procedure FOR ði ¼ 1; i 6 n 1; i þ þÞ { FOR ðj ¼ i þ 1; i 6 n; j þ þÞ { IF rj ðx; yÞ 6 ri ðx; yÞ THEN delete rj ðx; yÞ } } Then the undeleted values i1 ¼ 1; i2 ; . . . ; ik give the desired subsequence. The corresponding optimal fuzzy radius rðx; yÞ has values ri1 ðx; yÞ; ri2 ðx; yÞ; . . . ; riK ðx; yÞ with membership grades li1 ; li2 ; . . . ; liK resPK 2 pectively and the objective function is j¼1 ½rij ðx; yÞ lij (here ri1 ðx; yÞ ¼ r1 ðx; yÞ). The objective function thus depends on the subsequence ði1 ; i2 ; . . . ; ik Þ which in turn depends on the center ðx; yÞ. We shall therefore tessellate the plane so that for all the points ðx; yÞ in any region the subsequence obtained is the same. For every pair Pi ; Pj 2 fP1 ; P2 ; . . . ; Pn g, i 6¼ j, find perpendicular bisector of the line segment joining the
two points. Do it for every one of the n2 ¼ nðn1Þ pairs of points. These nðn1Þ lines divide the plane into at 2 2
M MðMþ1Þ nðn1Þ 4 most 2 þ 1 3 ¼ Oðn Þ convex polygonal regions, where M ¼ 2 . For each one of these regions, the ordering of distances of P1 ; P2 ; . . . ; Pn from any point ðx; yÞ in the region is the same and consequently the subsequence ði1 ; i2 ; . . . ; ik Þ is the same. (See Fig. 3 for an illustration for n ¼ 3.)
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L23 r1 < r 2 R1 : r 2 < r 3 r1 < r 3
P1
L12
r1 < r 2 R2 : r 2 > r 3 r1 > r 3
L31
r1 < r 2 R3 : r 2 > r 3 r1 > r 3
r1 > r 2 R6 : r 2 < r 3 r1 < r 3
P3 P2 r1 > r 2 R5 : r 2 < r 3 r1 < r 3
r1 > r 2 R4 : r 2 > r 3 r1 > r 3
Fig. 3. An illustration of convex polygonal regions for n ¼ 3. The maximum subsequences: R1 : r1 < r2 < r3 ; R2 : r1 < r2 ; R3 : r1 < r2 ; R4 : r1 ; R5 : r1 < r3 ; R6 : r1 < r3 .
2 Therefore, PKl over each2 one of these convex, polygonal regions of R , say Rl , the objective function Zl ðx; yÞ ¼ j¼1 ½rij ðx; yÞ lij is convex and thus we can minimize it over the convex, polygonal region, to get a local optimal solution ðxl ; yl Þ, using any of the known convex optimization methods in say OðhÞ time. The best among these Oðn4 Þ local optima is the overall optimal solution to our problem. We thus have an Oðn4 hÞ scheme to solve model (3.8).
5. A numerical example In general, optimal solution for the fuzzy version of the problem may vary significantly from that for the classical (crisp) case. Instead of illustrating this for each case separately, we give an example only for the case of the model in Section 3.2. e ¼ fðP1 ¼ ð0; 0Þ; l1 ¼ 0:9Þ; ðP2 ¼ ð4; 0Þ; l2 ¼ 0:2Þ; ðP3 ¼ ð0; 3Þ; l3 ¼ 0:1Þg. Thus, we consider the data P The three perpendicular bisectors of the line segments joining the three pairs of points divides the plane into six regions, as shown in Fig. 4. Region I: The objective function is Z1 ðx; yÞ ¼ 0:9ðr1 ðx; yÞÞ2 þ 0:2ðr2 ðx; yÞÞ2 ¼ 0:9ðx2 þ y 2 Þ þ 0:2ðx2 8x þ 16 þ y 2 Þ ¼ 1:1x2 þ 1:1y 2 1:6x þ 3:2: This is minimized over the region I by the point ðx1 ¼ 118 ; y1 ¼ 32Þ; and the corresponding minimum objective function value is 3.518. For the other regions, the relevant results are given below: Region II: 2
Z2 ðx; yÞ ¼ 0:9ðr1 ðx; yÞÞ ¼ 0:9x2 þ 0:9y 2 : Optimum point is ðx2 ¼ 2; y2 ¼ 32Þ; minimum objective function value is 5.625.
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Fig. 4. A numerical example.
Region III: Z3 ðx; yÞ ¼ 0:9ðr1 ðx; yÞÞ2 ¼ 0:9x2 þ 0:9y 2 : Optimum point is ðx3 ¼ 2; y3 ¼ 32Þ; minimum objective function value is 5.625. Region IV: 2
2
Z4 ðx; yÞ ¼ 0:9ðr1 ðx; yÞÞ þ 0:1ðr3 ðx; yÞÞ ¼ 0:9ðx2 þ y 2 Þ þ 0:1ðx2 þ y 2 6y þ 9Þ ¼ x2 þ y 2 0:6y þ 0:9: Optimum point is ðx4 ¼ 2; y4 ¼ 0:3Þ; minimum objective function value is 4.63. Region V: 2
2
Z5 ðx; yÞ ¼ 0:9ðr1 ðx; yÞÞ þ 0:2ðr2 ðx; yÞÞ þ 0:1ðr3 ðx; yÞÞ
2
¼ 0:9ðx2 þ y 2 Þ þ 0:2ðx2 8x þ 16 þ y 2 Þ þ 0:1ðx2 þ y 2 6y þ 9Þ ¼ 1:2x2 þ 1:2y 2 1:6x 0:6y þ 4:1:
9 Optimum point is x5 ¼ 23 ; y5 ¼ 150 ; minimum objective function value is 2.631. 25 Region VI: 2
2
Z6 ðx; yÞ ¼ 0:9ðr1 ðx; yÞÞ þ 0:2ðr2 ðx; yÞÞ ¼ 1:1x2 þ 1:1y 2 1:6x þ 3:2: Optimum point is ðx6 ¼ 118 ; y6 ¼ 0Þ; minimum objective function value is 2.618. The overall optimal solution is thus, center: ðx ¼ 118 ; y ¼ 0Þ, fuzzy radius: 118 with membership grade 0.9 and 36 with membership grade 0.2. The corresponding minimum objective function value is 2.618. 11 It should be noted that for the classical version of this problem with input data, the crisp points P1 ¼ ð0; 0Þ; P2 ¼ ð4; 0Þ, and P3 ¼ ð0; 3Þ, the optimal solution has center ð2; 32Þ, which is significantly different than the solution to the fuzzy version of the problem. 6. Discussion The problem considered in this paper has strong motivation from possible practical applications. For example, each crisp point Pi may represent a community or entity such as a school, hospital, manufacturing facility or vital installation of some type in a region. In this case, the grade of membership li may represent the population density or importance weight of the ‘‘demand’’ point Pi . Each point thus has a fuzzy
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requirement of coverage in the sense of protection or service of some type such as quality improvement of air, water, etc. in the location. The fuzzy requirement of coverage of each point is represented by its grade of membership. The coverage (protection or service) is given by a center whose domain is defined by the covering disk. Because of the fuzziness of coverage demand, the disk is a fuzzy one. The special cases of the general model investigated in this paper and the resultant polynomial algorithms facilitate finding optimal solutions efficiently in certain cases and provide insight in general context. As such, they are useful in both applications and further research. As revealed by the numerical example, the optimal solution in the fuzzy version could be significantly different from that of the corresponding crisp version. No doubt, further research would be needed to crystallize practical applications in real world situations [17]. Acknowledgements We thank the anonymous referees for their many helpful comments which improved the presentation of the paper. Appendix A. Basic definitions of fuzzy set and fuzzy number e of X is defined by a membership function Definition A.1. Let X be a universe of discourse. A fuzzy subset A e fA~ : X ! ½0; 1. The function value fA~ðxÞ represents the grade of membership of x in A. e denoted by A e a , is defined by Definition A.2. For each a 2 ½0; 1, the a-cut set of fuzzy set A, e a ¼ fx 2 X j lA~ðxÞ P ag: A e is a fuzzy set defined on R whose membership function lA~ðxÞ, x 2 R, is Definition A.3. A fuzzy number A (1) convex, (2) normalized, i.e. there exists a x0 2 R such that lA~ðx0 Þ ¼ 1, and (3) piecewise continuous. e is called an LR-fuzzy number if its membership function is defined by Definition A.4. A fuzzy number A 8 ax > > < L aA ; for x < a; aA > 0; lA~ðxÞ ¼ 1; for a 6 x 6 a; > > : R xa ; for x > a; bA > 0; bA e i.e. the function from ½0; þ1Þ to [0, 1] satisfying where LðtÞ and RðtÞ are the reference functions of A, Lð0Þ ¼ Rð0Þ ¼ 1 and continuous strictly decreasing on that part of ½0; þ1Þ on which they are positive, aA e is represented by four parameters and bA are the spread parameters. Symbolically, the LR-fuzzy number A e a; a; aA and bA , and it is denoted by A ¼ ða; a; aA ; bA ÞLR . Details about LR-fuzzy numbers and their arithmetic operations can be found in [5]. References [1] G. Bortolan, R. Degani, A review of some methods for ranking fuzzy subsets, Fuzzy Sets and Systems 5 (1985) 1–20. [2] R.K. Chakraborty, P.K. Chaudhuri, Note on geometrical solutions for some minimax location problems, Transportation Sciences 15 (1981) 164–166.
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