Sebasta Starz COMPROMISE HYPERSPHERE FOR STOCHASTIC DOMINANCE MODEL Abstract The am of the work s to preset a method of rakg a fte set of dscrete radom varables. The proposed method s based o two approaches: the stochastc domace model ad the compromse hypersphere. Moreover, a umercal llustrato of the method preseted s gve. Keywords Stochastc domace, compromse programmg, multple crtera optmzato. Itroducto Ths paper presets a method of rakg a fte set of dscrete radom varables. The method s based o oe of the multple crtera methods: the compromse hypersphere, Gass ad Roy [3]. The source of the compromse hypersphere s the compromse programmg, Chares ad Cooper [957], Zeley [98]. Adaptatos of the compromse hypersphere, optmzato wth radom varables, are based o stochastc domace, Levy [99]. The proposed method cossts of the followg steps: Step. Establsh feasble decsos ad correspodg radom varables. Step. Compute odomated radom varables the sese of stochastc domace. Step 3. Fd the compromse hypersphere. Step 4. Buld a rakg of odomated radom varables usg the compromse hypersphere. Our paper cossts of four sectos: Secto presets a descrpto ad propertes of the compromse hypersphere; Secto a model of stochastc domace s cosdered; Secto 3 presets the four steps of the method detal ad the umercal llustrato of the proposed algorthm s preseted secto 4. The paper cocludes wth remarks ad suggestos for further research.
3 Sebasta Starz. Compromse hypersphere The preseted method orgates the work of Gass ad Roy [3]. The am of ths method s to rak the fte set of odomated vectors y R,, y m R. I detal, the method looks as follows:. Solve the program: y, r =,..., m ( ) m max r d y, y, () where d: R R R deotes the dstace betwee two vectors. We deote the optmal soluto of () by of the cost fucto as m (). y, r ad the mmal value. Fd the rakg of the pots y, y,, y m based o the dstaces: r d( y, y ) =,, m. () I partcular, we look for the pot y closest to the hypersphere: m =,..., m r d( y, y ). (3) Remark Problem () s to fd a hypersphere wth the cetre y R ad the radus r R wth a mmal dstace from the set {y, y,, y m }. Remark I problem () oe ca use the well kow famly of metrcs l p :R R R as the fucto d wth the parameter p [, ]. The fucto l p :R R R s descrbed as follows:
COMPROMISE HYPERSPHERE... 33 l p ( yz, ) p p y z, p [, ) max yj zj, p= j=,..., j j = j= where y = (y,, y ) R, z = (z,, z ) R. Remark 3 I geeral, problem () s a complcated optmzato problem ad we use geetc algorthms to solve t, Koza [99, 994]. Remark 4 Problem (3) s to fd the pot closest to the hypersphere foud step. Problems () ad (3) are trval; t s eough to compare umbers, used step.. Stochastc domace I ths secto, we use the frst order stochastc domace, Shaked ad Shathkumar [993], Ogryczak ad Ruszczysk [999]. The relato of the frst order stochastc (FSD) domace s defed as follows: F x F x, ξ FSD ξ x R ( ) ( ) where F ξ (x)=p (ξ x) s the rght-cotuous cumulatve dstrbuto fucto of the radom varable ξ. We cosder the famly of dscrete radom varables {ξ : =,,, m}. Moreover, we assume that the followg set: X = {x R: {,,, m} P(ξ =x) > } s fte. It meas that we are able to eumerate the elemets of the set X the followg way: X = {x, x,..., x }, Where x s < x t for s < t. We call ξ * a odomated radom varable set Ω={ξ : =,,, m} the sese of FSD f ξ ξ
34 Sebasta Starz We buld the vector y the followg way: y = [ y, y,, ξ Ω ξ * FSD ξ F ξ F ξ *. coected wth dscrete radom varables ξ y ] = [ Fξ ( x ), Fξ ( x ),, F ( x ) ξ ]. I ths case the FSD relato has the followg form: ξ FSD ξ y y y y. Some addtoal aspects of FSD models oe ca fd papers by Ogryczak [] ad Ogryczak ad Romaszkewcz []. 3. Method of rakg The am of the proposed procedure s to choose a decso from a fte set of decsos. The returs of decsos are descrbed by meas of radom varables. The method s based o the stochastc order ad the compromse hypersphere method. The procedure looks as follows: Step. Establsh feasble decsos wth correspodg radom varables {ξ : =,,, m} ad the rght-cotuous cumulatve dstrbuto fucto. We obta: y, y,, y m, where y = [ y, y,, y ] = [ Fξ ( x ), Fξ ( x ),, F ( x ) ξ ]. Step. Compute odomated vectors the set {y, y,, y m } the sese of mmalzato,.e. y j s odomated f j j y y y y. We obta =,..., m y, y,, kp y (p m). The above vectors are coected wth the odomated radom varables the sese of FSD. Step 3. Solve problem () for y, y,, p y. p y ad cor- Step 4. Use values () to obta the rakg of y, y,, respodg odomated radom varables the sese of FSD.
COMPROMISE HYPERSPHERE... 35 4. Example Step. Let us cosder a set of seve dscrete radom varables: ξ, {,,, }. The probabltes characterzg these radom varables are preseted table. Table Descrpto of radom varables ξ ξ ξ 3 ξ 4 ξ 5 ξ 6 ξ 7 ξ 8 ξ 9 ξ P(ξ =).3.4.....4. P(ξ =)..4.5.3..6.4 P(ξ =)..3.4..4..4.4 P(ξ =3).5.6...4.3... Vectors y bult for the radom varables cosdered are preseted table. Table Vectors y for cosdered radom varables y y y 3 y 4 y 5 y 6 y 7 y 8 y 9 y.3.4.....4..4.4.5.5.5.3.7.4.5.5.4.8.9.6.7.8.8.9 Step. Compute the odomated vectors the set {y, y,, y }. The odomated vectors are as follows: {y, y, y 3, y 4, y 5, y 6, y 7 }. We deote the set of dces of the odomated vectors by N,.e.: N = {,, 3, 4, 5, 6, 7}. Step 3. By solvg problem () wth the set {y : N} ad d = l : m max y, r N 4 ( j j) r y y j=,
36 Sebasta Starz we obta the followg optmal soluto: y = (.7388;.55;.5;.39575), r = 3.6347 ad the mmal value of the cost fucto: Step 4. By solvg problem (3) m ( ) =.98. m N r 4 j = ( y j y j ), we obta values (as dstaces betwee pots ad the hypersphere) show Table 3. Moreover, Table 3 presets the rakg based o these values. 4 ( y j y j ) j = Rakg for d = l Table 3 y y y 3 y 4 y 5 y 6 y 7 r.9.798.98.678.98.858.98 Rakg 4 5 5 3 5 Coclusos ad further research I ths paper we have proposed a method of rakg dscrete radom varables. We have used two approaches: the stochastc domace ad the compromse hypersphere. I future, the followg aspects of the preseted method are worth studyg: comparg wth other methods of radom varables rakg, the case of cotuous radom varables, a teractve verso of the method, aalyss of the method for dfferet metrcs d, applcatos to real lfe problems. Refereces Chares A., Cooper W.W. (957): Goal Programmg ad Multple Objectve Optmzato. Europea Joural of Operatoal Research,, pp. 39-45.
COMPROMISE HYPERSPHERE... 37 Gass S.I., Roy P.G. (3): The Compromse Hypersphere for Multobjectve Lear Programmg. Europea Joural of Operatoal Research, 44, pp. 459-479. Koza J.R. (99): Geetc Programmg. Part. MIT Press, Cambrdge, MA. Koza J.R. (994): Geetc Programmg. Part. MIT Press, Cambrdge, MA. Levy H. (99): Stochastc Domace ad Expected Utlty: Survey ad Aalyss. Maagemet Scece, 38, pp. 553-593. Ogryczak W., Ruszczyńsk A. (999): From Stochastc Domace to Mea-Rsk Models: Semdevatos as Rsk Measures. Europea Joural of Operatoal Research, 6, pp. 33-5. Ogryczak W., Romaszkewcz A. (): Welokryterale podejśce do optymalzacj portfela westycj. W: Modelowae preferecj a ryzyko. Wydawctwo Akadem Ekoomczej, Katowce, pp. 37-338. Ogryczak W. (): Multple Crtera Optmzato ad Decsos uder Rsk. Cotrol ad Cyberetcs, 3, pp. 975-3. Shaked M., Shathkumar J.G. (993): Stochastc Orders ad ther Applcatos. Academc Press, Harcourt Brace, Bosto. Zeley M. (98): Multple Crtera Decso Makg. McGraw-Hll, New York.