Research Article Extension of Wolfe Method for Solving Quadratic Programming with Interval Coefficients

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1 Hidawi Joural of Applied Matheatics Volue 2017, Article ID , 6 pages Research Article Extesio of Wolfe Method for Solvig Quadratic Prograig with Iterval Coefficiets Syaripuddi, 1 Herry Suprajito, 2 ad Fatawati 2 1 Departet of Matheatics, Faculty of Matheatics ad Natural Scieces, Mulawara Uiversity, Kapus Guug Kelua, Jl. Barog Togkok, Saarida 1068, Idoesia 2 Departet of Matheatics, Faculty of Sciece ad Techology, Airlagga Uiversity, Kapus C Uair, Jl.Mulyorejo,Surabaya60115,Idoesia Correspodece should be addressed to Fatawati; fata47uair@gail.co Received 10 April 2017; Accepted 1 August 2017; Published 14 Septeber 2017 Acadeic Editor: Frak Werer Copyright 2017 Syaripuddi et al. This is a ope access article distributed uder the Creative Coos Attributio Licese, which perits urestricted use, distributio, ad reproductio i ay ediu, provided the origial work is properly cited. Quadratic prograig with iterval coefficiets developed to overcoe cases i classic quadratic prograig where the coefficiet value is ukow ad ust be estiated. This paper discusses the extesio of Wolfe ethod. The exteded Wolfe ethod ca be used to solve quadratic prograig with iterval coefficiets. The extesio process of Wolfe ethod ivolves the trasforatio of the quadratic prograig with iterval coefficiets odel ito liear prograig with iterval coefficiets odel. The ext step is trasforig liear prograig with iterval coefficiets odel ito two classic liear prograig odels with special characteristics, aely, the optiu best ad the worst optiu proble. 1. Itroductio Quadratic prograig is a special for of oliear prograig which has special characteristics; that is, the objective fuctio is i quadratic fors ad costrait fuctios areliearfor[1].althoughthequadraticprograigis part of oliear prograig, the copletio is still adoptig soe liear prograig proble solvig ethods, oe of which is the Wolfe ethod. This ethod trasfors the quadratic prograig proble ito a liear prograig proble. Wolfe [2] odified the siplex ethod to solve quadratic prograig proble by addig coditios of the Karush-Kuh-Tucker (KKT) ad chagig the objective fuctio of quadratic fors ito a liear for. Iterval quadratic prograig is a developet of the classic quadratic prograig that utilizes iterval aalysis theory developed by Moore [3]. This developet ai is to accoodate cases which cotai the ucertaity, that is, whe the data value is ukow for certai, but the data lies withi a iterval where the values of the upper liit ad lower liit are kow. The special characteristics of the iterval quadratic prograig proble are the coefficiets of the objective fuctio ad costrait fuctios are i the iterval for. Research o quadratic prograig with iterval coefficiets has bee coducted by Liu ad Wag [4]. However, the coefficiets of quadratic fors i the objective fuctio o the developed odel are ot i the iterval for yet. Furtherore,LiadTia[5]geeralizedtheodeli[4] by assuig that the quadratic coefficiets of the objective fuctio are i the iterval for. Refereces [4, 5] used the duality theory to create a ethod of solvig the quadratic prograig with iterval coefficiets. Quadratic prograig odel with iterval coefficiets is trasfored ito two classic quadratic prograig odels with the special characteristics, called the best optiu ad worst optiu proble. A copletio ethod was developed based o the ethod of solvig the liear prograig with iterval coefficiets that have bee discussed by soe researchers [6 10]. This paper will discuss the extesio of Wolfe ethod to solve quadratic prograig with iterval coefficiets. Therefore, this article will focus o how to trasfor the quadratic prograig with iterval coefficiets ito liear

2 2 Joural of Applied Matheatics prograig with iterval coefficiets. Furtherore, the liear prograig with iterval coefficiets which has bee obtaied fro the trasforatio will be solved by usig the ethod i [8]. Thispaperisorgaizedasfollows.Sectio2discusses iterval arithetic operatios. I Sectio 3, a geeral for of liear prograig with iterval coefficiets is stated. I Sectio 4, a geeral for of quadratic prograig with iterval coefficiets is stated. Extesio of Wolfe ethod as oe ethod of solvig the quadratic prograig with iterval coefficiets is discussed i Sectio 5, whereas Sectio 6 discusses uerical exaples, ad Sectio 7 provides soe cocludig rearks. 2. Iterval Arithetic The basic defiitio ad properties of iterval uber ad iterval arithetic ca be see at Moore [3], Alefeld ad Herzberg [11], ad Hase [12]. Defiitio 1. Aclosedrealitervalx =[x I,x S ] deoted by x is a real iterval uber which ca be defied copletely by x = [x I,x S ] = {x R x I x x S ; x I,x S R}, (1) Defiitio 2. Arealitervaluberx = [x I,x S ] is called degeerate, if x I =x S. Defiitio 3. Let x =[x I,x S ] ad y =[y I,y S ];the 1. x +y=[x I +y I,x S +y S ] (additio), 2. x y =[x I,x S ][y I,y S ]=[x I,x S ]+[y S,y I ]= [x I y S,x S y I ] (subtractio), 3. x y=[i{x I y I,x I y S,x S y I,x S y S },ax{x I y I,x I y S, x S y I,x S y S }] (ultiplicatio), 4. x/y =x(1/y) =[x I,x S ][1/y S,1/y I ], 0 y(divisio). 3. Liear Prograig with Iterval Coefficiets where x I ad x S arecalledifiuadsupreu,respectively. Thegeeralforofliearprograigwithitervalcoefficiets is defied as follows: Maxiize z = [c ji,c js ]x j (2a) [a iji,a ijs ] x j [b ii,b is ],,2,..., (2b) x j 0,,2,...,, (2c) where x j R, [c ji,c js ], [b ii,b is ],ad[a iji,a ijs ] I(R). The odel i (2a) (2c) is solved by eas of trasforig the liear prograig with iterval coefficiets ito two classic liear prograig odels with the special characteristics, aely, the best optiu ad the worst optiu probles. The best optiu proble has properties of best versio o the objective fuctio ad axiu feasible area o the costrait fuctio. O the other had, the worst optiu proble has a characteristic that it is the worst versio of the objective fuctio ad the iiu feasible area of the costrait fuctio. Chieck ad Raada [8] provide a rule to deterie thebestadworstoptiuprobleialiearprograig proble with iterval coefficiets. The costraits of liear prograig with iterval coefficiets which have a iequality sig ( ) i(2b)havecharacteristicsofthe axiu feasible area ad the iiu feasible area which is give by the followig theore. Theore 4 (Chieck ad Raada [8]). Supposethatwe have a iterval iequality give [a ji,a js ]x j [b I,b S ], where x j 0.The, a jix j b S is axiu feasible area ad a jsx j b I is iiu feasible area. The objective fuctio of liear prograig with iterval coefficiets for the case of axiizig (2a) has characteristics of the best versio ad the worst versio of the objective fuctio is expressed i the followig theore. Theore 5 (Chieck ad Raada [8]). If z = [c ji, c js ]x j is the objective fuctio for x j 0,the c jsx j c jix j,where c jsx j isthebestversiooftheobjective fuctio ad c jix j is the worst versio of the objective fuctio. 4. Quadratic Prograig with Iterval Coefficiets The geeral for of quadratic prograig with iterval coefficiets itroduced by Li ad Tia [5] is defied as follows: Maxiize z = [c ji,c js ]x j [q jki,q jks ]x j x k (3a) [a iji,a ijs ]x j [b ii,b is ],,2,..., (3b) x j 0,,2,...,, (3c)

3 Joural of Applied Matheatics 3 where x j R, [c ji,c js ], [q iji,q ijs ], [b ii,b is ], [a iji,a ijs ] I(R), [q jki,q jks ]x j x k is egative seidefiite, ad I(R) are the set of all iterval ubers i R. The odel as show i (3a) (3c) is a geeralizatio of the odel i [4]. The coefficiet of the objective fuctio ad costraits of the quadratic prograig with iterval coefficiets odel i (3a) (3c) have a iterval for. The idea to solve the odel is the extesio of Wolfe ethod. This ethod focuses o how to trasfor the quadratic prograig with iterval coefficiets i (3a) (3c) ito liear prograig with iterval coefficiets i (2a) (2c). Furtherore, liear prograig with iterval coefficiets obtaied fro the trasforatio is doe usig the ethod i [8]. Extesio of Wolfe ethod is the ai result i this paper. The fudaetal differece betwee the extesios of Wolfe ethod ad the ethod i [5] is, o the extesio of Wolfe ethod, quadratic prograig odel with iterval coefficiets is trasfored ito liear prograig with iterval coefficiets, while, i [5], the odel of quadratic prograig with iterval coefficiets is aitaied. 5. Extesio of Wolfe Method Wolfe ethod is oe ethod for solvig quadratic prograig probles by eas of trasforig the quadratic prograig probles ito a liear prograig proble. Wolfe[2]odifiedthesiplexethodtosolvequadratic prograig probles by addig a requireet Karush- Kuh-Tucker (KKT) ad chagig the quadratic objective fuctio ito a liear objective fuctio. The extesio of Wolfe ethod is used to solve quadratic prograig proble with iterval coefficiets. Steps of extesio of Wolfe ethod are declared as follows. For of Lagrage fuctio for the proble i (3a) (3c) is L (x, y, r, λ, μ)= λ i ( [c ji,c js ]x j [q jki,q jks ]x j x k ([a iji,a ijs ]x j [b ii,b is ]+y i 2 )) μ j (x j +r j 2 ), where λ i, i = 1,2,...,, μ j, j = 1,2,...,,areLagrage ultipliers ad L(x, y, r, λ, μ) is Lagrage fuctio with iterval coefficiets. Local iiu poits of the fuctio L were obtaied by the first partial derivatives of the fuctio L with respect (4) to the variables ad equatig to zero (KKT ecessary coditios) (see [13, 14]). =[c x ji,c js ]+ [q jki,q jks ]x k i [a iji,a ijs ]λ i +μ j =0,,2,...,, (5a) y i =2λ i y i =0,,2,...,, (5b) r i =2μ j r j =0,,2,...,, (5c) λ i = [a iji,a ijs ]x j +y i 2 [b ii,b is ] =0,,2,...,, (5d) =x μ j r 2 j =0,,2,...,, (5e) i x j,λ i,μ j,y i,r j 0,,2,...,,,2,...,. (5f) Results siplificatio of (5a) (5f) is [q jki,q jks ]x k + =[c ji,c js ],,2,...,, [a iji,a ijs ]λ i μ j (6a) [a iji,a ijs ]x j +s i =[b ii,b is ],,2,...,, (6b) x j,λ i,μ j,s i 0,,2,...,,,2,..., (6c) ad satisfies the copleetary coditios, μ j x j =0,,2,...,, λ i s i =0,,2,...,. (6d) Add artificial variables V j,,2,...,,i(6a)foraiitial basis, as follows: [q jki,q jks ]x k + =[c ji,c js ]. [a iji,a ijs ]λ i μ j + V j Furtherore, create a liear prograig with iterval coefficiets, where the objective fuctio is to iiize the uber of artificial variables V j,,2,...,,adcostrait is (7), (6b), (6c), ad (6d) obtaied fro ecessary coditios of KKT. (7) Miiize z = V 1 + V 2 + +V (8a)

4 4 Joural of Applied Matheatics, [q jki,q jks ]x k + =[c ji,c js ],,2,..., [a iji,a ijs ]λ i μ j + V j (8b) [a iji,a ijs ]x j +s i =[b ii,b is ],,2,..., (8c) x j,λ i,μ j,s i, V j 0,,2,...,,,2,..., (8d) satisfyig the copleetary coditios, μ j x j =0, λ i s i =0,,2,...,,,2,...,, (8e) where V j 0is artificial variable. The odel as show i (8a) (8e) is liear prograig with iterval coefficiets which is added by copleetary coditios. This odel is the result of the trasforatio fro the quadratic prograig with iterval coefficiets odel by extesio of Wolfe ethod. The ext step, liear prograig with iterval coefficiets odel i (8a) (8e), was solved by trasforig ito two liear prograig cases with the special characteristics, aely, the best ad the worst optiu proble. The trasforatio process ca be writte i Algorith 6 as follows. Algorith 6. (1) Give a quadratic prograig proble with iterval coefficiets i (3a) (3c), Extesio of Wolfe ethod is based o (3a) (3c) equivalet to the liear prograig with iterval coefficiets i (8a) (8e). (2) Use Theores 4 ad 5 for trasforig the liear prograig with iterval coefficiets i (8a) (8e) ito two classic liear prograig odels with special characteristics; aely, (a) the best optiu proble is q jks x k + Miiize z S = V 1 + V 2 + +V (9a), a iji λ i μ j + V j =c js,,2,..., (9b) satisfyig the copleetary coditios, μ j x j =0, λ i s i =0,,2,...,,,2,...,, (b) the worst optiu proble is q jki x k + (9e) Miiize z I = V 1 + V 2 + +V (10a), a ijs λ i μ j + V j =c ji, a ijs x j +s i =b ii, x j,λ i,μ j,s i, V j 0,,2,...,,2,...,,2,...,,,2,..., satisfyig the copleetary coditios, μ j x j =0, λ i s i =0,,2,...,,,2,...,. (10b) (10c) (10d) (10e) (3) The optiu value of the quadratic prograig with iterval coefficiets is obtaied by cobiig the optiu value fro the worst ad the best optiu proble; that is, z =[z I,z S ]. Algorith 6 shows that the best ad the worst optiu proble are liear prograig odels added by copleetary coditios. Thus, both probles ca be solved by siplex ethod. 6. Nuerical Exaple Cosider the followig exaple of quadratic prograig with iterval coefficiets i the joural Li ad Tia [5]. Miiize z = [10, 6] x 1 + [2, 3] x 2 + [1, 1] x 1 x 2 (11a) a iji x j +s i =b is, (9c) + [4, 10] x [10, 20] x2 2,2,..., [1, 2] x 1 +3x 2 [1, 10] (11b) x j,λ i,μ j,s i, V j 0,,2,...,,,2,..., (9d) [2, 8] x 1 + [4, 6] x 2 [4, 6] x 1,x 2 0. (11c) (11d)

5 Joural of Applied Matheatics 5 Table 1: Two classic liear prograig odels with special characteristics. The best optiu proble The worst optiu proble (1) Classic liear prograig odel (2) Classic liear prograig odel Miiize z S = V 1 + V 2 Miiize z I = V 1 + V 2 8x 1 x 2 +λ 1 2λ 2 μ 1 + V 1 = 10 20x 1 +x 2 +2λ 1 +8λ 2 μ 1 + V 1 =6 x 1 20x 2 3λ 1 4λ 2 +μ 2 + V 2 =2 x 1 40x 2 3λ 1 6λ 2 +μ 2 + V 2 =3 x 1 +3x 2 +s 1 =10 2x 1 +3x 2 +s 1 =1 2x 1 +4x 2 +s 2 =6 8x 1 +6x 2 +s 2 =4 x i,λ i,μ i,s i, V i 0,,2 x i,λ i,μ i,s i, V i 0,,2 satisfyig copleetary coditios: satisfyig copleetary coditios: λ 1 s 1 =0, λ 2 s 2 =0,adμ 1 x 1 =0, μ 2 x 2 =0 λ 1 s 1 =0, λ 2 s 2 =0,adμ 1 x 1 =0, μ 2 x 2 =0 Solutio: z S = 6.25, x 1 =1,25,adx 2 =0. Solutio: z I = 0.9, x 1 = 0.3,adx 2 =0 Accordig to Li ad Tia [5], for the solutio of the odel i (11a) (11c), the best optiu proble is z S = 0.9, x 1 = 0.3, ad x 2 =0,theworstoptiuprobleisz I = 6.25, x 1 = 1, 25, adx 2 =0,adtheoptiuvalueisz =[z I,z S ]= [6.25, 0.9]. This paper presets oly the axiizatio proble so that ay iiizatio proble will be coverted ito axiizatio proble, the siple procedure to covert a iiizatio proble to a axiizatio proble ad vice versa. Siply ultiply the objective fuctio of a iiizatio proble by 1covertigititoaaxiizatioproble adviceversa. Maxiize z = [6, 10] x 1 + [3, 2] x 2 + [1, 1] x 1 x 2 + [10, 4] x [20, 10] x 2 2 (12a) [1, 2] x 1 +3x 2 [1, 10] (12b) [2, 8] x 1 + [4, 6] x 2 [4, 6] x 1,x 2 0. (12c) (12d) We apply the extesio of Wolfe ethod for trasforig quadratic prograig with iterval coefficiets odel i ((12a) (12d)) ito liear prograig with iterval coefficiets odel. We have Miiize z=v 1 + V 2 (13a) [8, 20] x 1 + [1, 1] x 2 + [1, 2] λ 1 + [2, 8] λ 2 μ 1 + V 1 = [6, 10] (13b) [1, 1] x 1 + [20, 40] x 2 + [3, 3] λ 1 + [4, 6] λ 2 μ 2 + V 2 = [3, 2] (13c) [1, 2] x 1 +3x 2 [1, 10] (13d) [2, 8] x 1 + [4, 6] x 2 [4, 6] x 1,x 2 0. (13e) (13f) We apply Algorith 6 for trasforig liear prograig with iterval coefficiets odel i ((13a) (13f)) ito two classic liear prograig odels with special characteristics, aely, the best optiu ad the worst optiu proble. The result of the trasforatio is show i Table 1. So, the optiu value of the quadratic prograig with iterval coefficiets is obtaied by cobiig the optiu value fro the worst ad the best optiu proble; that is, z =[z I,z S ] = [0.9, 6.25].Thissolutiogivesthesae value as obtaied by Li ad Tia [5]. 7. Coclusio This paper presets a extesio of Wolfe ethod. The extesio of Wolfe ethod perfored by trasforig the quadratic prograig with iterval coefficiets odel ito liear prograig with iterval coefficiets odel. Furtherore, liear prograig with iterval coefficiets odel is trasfored ito two classic liear prograig odels usig Algorith 6. The extesio of Wolfe ethod has a particular beefit: the fial odel is liear prograig. Hece, it ca be solved by the siplex ethod. Coflicts of Iterest The authors declare that there are o coflicts of iterest regardig the publicatio of this paper. Refereces [1] F. S. Hillier ad G. J. Liebera, Itroductio to operatios research, Holde-Day, Ic., Oaklad, Calif., Third editio, 1980.

6 6 Joural of Applied Matheatics [2] P. Wolfe, The siplex ethod for quadratic prograig, Ecooetrica,vol.27,pp ,1959. [3] R. E. Moore, Iterval Aalysis, Pretice-Hall, Eglewood Cliffs, NJ, USA, [4] S.-T. Liu ad R.-T. Wag, A uerical solutio ethod to iterval quadratic prograig, Applied Matheatics ad Coputatio,vol.189,o.2,pp ,2007. [5] W. Li ad X. Tia, Nuerical solutio ethod for geeral iterval quadratic prograig, Applied Matheatics ad Coputatio,vol.202,o.2,pp ,2008. [6] S. C. Tog, Iterval uber ad fuzzy uber liear prograigs, Fuzzy Sets ad Systes. A Iteratioal Joural i Iforatio Sciece ad Egieerig, vol.66,o.3,pp , [7] K. Raada, Liear Prograig with Iterval [Msc. thesis], Carleto Uiversity, Ottawa, Otario, [8] J. W. Chieck ad K. Raada, Liear prograig with iterval coefficiets, Joural of the Operatioal Research Society,vol.51,o.2,pp ,2000. [9] D. Kuchta, A odificatio of a solutio cocept of the liear prograig proble with iterval coefficiets i the costraits, Cetral Europea Joural of Operatios Research (CEJOR),vol.16,o.3,pp ,2008. [10] H. Suprajito ad I. B. Mohd, Iterval liear prograig, i Proceedigs of ICOMS-3,Bogor,Idoesia,2008. [11] G. Alefeld ad J. Herzberger, Itroductio to Iterval Coputatios, Acadeic Press, New York, NY, USA, [12] E. Hase, Global optiizatio usig iterval aalysis, vol.165 of Moographs ad Textbooks i Pure ad Applied Matheatics, Marcel Dekker, Ic., New York, [13] J. Zhag, Optiality coditio ad wolfe duality for ivex iterval-valued oliear prograig probles, Joural of Applied Matheatics,vol.2013,ArticleID641345,2013. [14] H.-C. Wu, The Karush-KUH-Tucker optiality coditios i a optiizatio proble with iterval-valued objective fuctio, Europea Joural of Operatioal Research, vol.176, o. 1, pp , 2007.

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