Master Thesis Mathematical Modeling and Simulation On Fuzzy linear programming problems solved with Fuzzy decisive set method
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1 008:05 Mster Thess Mthemt Modeg d Smuto O Fuzzy er progrmmg proems soved wth Fuzzy desve set method Author shd Mehmood Thess for the degree Mster of Mthemt Modeg d Smuto 5 redt pots 5 ECTS redts Bekge Isttute of Tehoogy Shoo of Egeerg Deprtmet of Mthemts d See Supervsor: Prof. Eseth kus-adersso
2 Astrt I the thess there re two kds of fuzzy er progrmmg proems oe of them s er progrmmg proem wth fuzzy tehoog oeffets d the seod s er progrmmg proem whh oth the rght-hd sde d the tehoog oeffets re fuzzy umers. I sove the fuzzy er progrmmg proems wth fuzzy desve set method.
3 ACKNOWLEDGEMENT I the me of Ah who s the most grous d merfu. Frst d foremost I woud ke to thk my supervsor Dr. Eseth kus- Adersso for her gude d ptee through out of ths thess perod. Her supportve d kd tttude mkes t posse for me to ompete t. I kowedge her otrutos to ehe my kowedge o the suet. I m so thkfu to my freds espey Muhmmd Srfrz Iq for sprg ther tme to revew my thess. I m so grtefu pushers d uthors s ther setf terture heped me whe workg o my thess. Fy I must thk my fmy who hs provded ove d support through my study. Ther ove wys rems the key soure of motvto for me.
4 Cotets pges Itroduto 4-6 Fuzzy Desve Set Method. Ler Progrmmg Wth Fuzzy 7-0 Tehoog Coeffets. Lp Proems wth Fuzzy Tehoog 0-4 Coeffets d Fuzzy ght- hd-sde Numers. Empes Couso 5 eferees 0-4
5 Itroduto Before trodug fuzzy er progrmmg we w revew trdto er progrmmg LP. Ler progrmmg s ger method used to sove sets of er equtos. The form methodoogy ws deveoped roud 947. The purpose of er progrmmg s to fd optm soutos for systems whh re modeed y er equtos. I LP shrp ostrts re omed to mt the spe of fese soutos to the er proem eg posed. The vre dmesos of the system eg modeed ssume the form of vetor. The oetves of proem re so modeed wth er equtos. The erty of the ostrts d the oetves ees strght-forwrd souto methods. Vertes of the souto spe orrespod to optmzg vetors. The vetors re optmzg the sese tht o-zero er equtos of the system vres represetg the oetves heve ether mm or mm vues t the vertes of the fese souto spe. There re vrous wys to vsuze how the LP method heves optm souto for er system. Some proems re soved y mkg smpe d ovous hoes etwee sm set of optos. My mportt proems re more ompted; there e mutpe smuteous gos d demms d ompromses heret posse soutos. We requre frmework to detfy optm soutos. Costrts o tos d resoures hve to e detfed. Oetves hve to e defed. There hs to e wy of evutg the degree to whh defe oetves re met. est proems d oetves re ofte oy vguey defed. The proems gve s empes ove rge from trv to sute d ompe d re represettve of re fe proems. 5
6 Fuzzy er progrmmg FLP s refemet of er progrmmg LP whh ws deveoped se the etee-sevetes. The fuzzy oetve futo s hrterzed y ts memershp futo d so re the ostrts. Se we wt to stsfy optmze the oetve futo s we s the ostrt deso fuzzy evromet s defed ogy to o fuzzy evromet s the seeto of tvtes whh smuteousy stsfy the oetve futo d the ostrts. The retoshp etwee the ostrts d the oetve futo fuzzy evromet s therefore fuy symmetr.e. there s o oger dfferee etwee the former d tter. The deso mker my ot tuy wt to mmze or mmze the oetve futo. ther he mght wt to reh some sprto eves whh mght ot eve e defe rspy. Thus he mght wt to mprove the preset ost stuto osdery d so o. The ostrts mght e vgue oe of the foowg wys. The sg mght ot e met the strt mthemt sese ut smer votos mght we e epte. Ths hppe f the ostrts represet sprto eves s metoed ove or f for ste the ostrts represet sesory requremets tste oor sme et. whh ot dequtey e ppromted y rsp ostrt. Of ourse the oeffets of the vetors or or of the mtr A tsef hve fuzzy hrter ether euse they re fuzzy ture or euse perepto of them s fuzzy. The roe of the ostrts e dfferet from tht ss er progrmmg where the voto of y sge ostrt y y mout reders the souto fese. The deso mker mght ept sm votos of dfferet ostrts. Fuzzy er progrmmg offers umer of wys for those types of vgueess [7]. 6
7 Approh d Methods I fuzzy deso mkg proems the oept of mmzg deso ws proposed y Bem d deh []. Ths oept ws dopted to proems of mthemt progrmmg y Tk et.[9]. mmerm [0] preseted fuzzy pproh to mutoetve er progrmmg proems. Fuzzy er progrmmg proem wth fuzzy oeffets ws formuted y Negot [] d ed roust progrmmg. Duos d Prde [] vestgted er fuzzy ostrts. Tk d As [4] so proposed formuto of fuzzy progrmmg wth fuzzy ostrts d gve method for ts souto whh ses o equty retos etwee fuzzy umers. Shoheg [] osdered the fuzzy er progrmmg proem wth fuzzy ostrts d defuzzfted t y frst determg upper oud for the oetve futo. Further the so-oted rsp proem soved y the fuzzy desve set method hs ee trodued y Skw d Y []. Frst osder er progrmmg proems whh oy tehoog oeffets re fuzzy umers d the er progrmmg proems whh oth tehoog oeffets d rghthd sde umers re fuzzy umers. The frst proem s overted to equvet rsp proem. There ests proem of fdg pot whh stsfes the ostrts d the go the go e ompre to oetve futo wth the mmum degree. The de of the pproh s due to Bem d deh [] The rsp proems oted y suh mer e o-er eve o-ove where the o-erty rses ostrts. For sovg these proems we utze the fuzzy desve set method s trodued y Skw d Y []. I ths method omto wth the seto method d the smpe method of er progrmmg s used to ot fese souto. 7
![Tk d As [4] so proposed formuto of fuzzy progrmmg wth fuzzy ostrts d gve method for ts souto whh ses o equty retos etwee fuzzy umers.](/docs-images/48/18360236/images/page_7.jpg "Shoheg [] osdered the fuzzy er progrmmg proem wth fuzzy ostrts d defuzzfted t y frst determg upper oud for the oetve futo.")
8 . Ler progrmmg wth fuzzy tehoog oeffets m suet 0 to m t est oe > 0.. Assumptos s futo of the foowg fuzzy set. µ d 0 / d f f f < < d d Where ϵ d d > 0 for m For defuzzfto of ths proem we frst fuzzfy the oetve futo. Ths s doe y utg the ower d upper ouds of the optm vues frst. The ouds of the optm vues z d z u re oted y sovg the stdrd er progrmmg proems. 8
9 m suet to m.. d m 0. d.. The oetve futo tkes vues z d z etwee d d. Let ed ower d upper ouds of the optm vues. tehoog oeffets z m z z d z u m z z. The z d Assumpto. The er rsp proems.. d.. hve fte optm vues. I ths se the fuzzy set G of optm vues whh s suset of s defed s see Kr d Yu [5] zu re 9
10 / 0 < < u u u G f f f µ..4 The fuzzy set of the th ostrt C whh s suset of m s defed y < < < u d d / 0 µ..5 Aordg to the defto of the fuzzy deso proposed y Bem d deh [] we ot the fuzzy deso set D hrterzed y the memershp futo stted s the mmum over the fuzzy go G-set d some ostrts C. Provded tht the go d the ostrts e over the sme uverse spe the futo s formzed due to the formu m m C G D µ µ µ..6 0
![.5 Aordg to the defto of the fuzzy deso proposed y Bem d deh [] we ot the fuzzy deso set D](/docs-images/48/18360236/images/page_10.jpg "hrterzed y the memershp futo stted s the mmum over the fuzzy go G-set d some ostrts C.")
11 After dptg..6 to the purpose of fdg the memershp degree of the optm fuzzy deso we determe souto of the proem s m µ > 0 D m m µ > 0 G m µ C..7 osequety the proem.. eomes to the foowg optmzto proem mλ µ G 0 λ µ λ C 0 λ < < m..8 I whh λ s prmeter from [0 ]. By usg..4 d..5 the proem..8 e wrtte s m λ λ 0 λ d 0 0 < < m 0 λ...9 Note tht the ostrts proem..9 otg the ross produt terms λ re ot ove. Therefore the souto of ths proem requres the spe pproh dopted for sovg geer oove optmzto proems.
![m µ C..7 osequety the proem.. eomes to the foowg optmzto proem mλ µ G 0 λ µ λ C 0 λ < < m..8 I whh λ s prmeter from [0 ].](/docs-images/48/18360236/images/page_11.jpg "By usg..4 d..5 the proem..8 e wrtte s m λ λ 0 λ d 0 0 < < m 0 λ...9 Note tht the ostrts proem.")
12 . Lp proems wth fuzzy tehoog oeffets d Fuzzy ght- hd-sde umers. m 0 m.. Where t est oe > 0 Assumpto. d re fuzzy umers wth the foowg er Memershp futos: µ d / d 0 f f f < < d d d µ p / 0 p f f f < < p p Where X ϵ d > 0 for defuzzfto of the proem.. we frst ute the ower d upper ouds of the optm vues. The optm vues z d z u e defed y sovg the foowg stdrd er progrmmg proems for whh we ssume tht they hve the fte optm vues.
13 0 m m d.. d 0 m m p.. d 0 m m p d..4 d 0 m 4 m..5
14 Let z m z z z z 4 d z u m z z z z 4 The oetve futo tkes vues etwee z d zu whe tehoog oeffets tke vues etwee d d d the rght-hd sde umers tke vues etwee d p. The fuzzy set of optm vues G whh s suset of Is defed y µ G 0 / u f f f < < u u The fuzzy set of the th ostrt C whh s suset of µ C 0 / d p f f f s defed y < <..6 d d p. p..7 4
15 By usg the method of defuzzfto s for the proem..8 the proem.. s redued to the foowg rsp proem: mλ λ 0 λd 0 λp 0 0 λ. < < m..8 Note tht the proem..8 s so o ove progrmmg proem smr to the proem..9. The gorthm of the fuzzy desve set method Ths method s sed o the de tht for fed vue ofλ the proems..9 d..8 re er progrmmg proems. Otg the optm * souto λ to the proems..9 d..8 s equvet to determg the mmum vue of λ for whh the fese set s oempty. The gorthm of ths method for the proem..9 d..8 s s foows. Agorthm Step Set λ d test whether fese set stsfyg the ostrts of the Proem..9 ests or ot y usg the optmty rtero of the smpe method. If fese set ests set λ otherwse set λ 0 d λ d go to the et step. Step For the vue of λ λ λ / updted the vue of λ d λ usg the seto method s foows: λ λ f fese set s oempty for λ λ λ f fese set s empty for λ Cosequety for eh λ test whether fese set of the proem..9 ests or ot y usg the optmty rtero of the Smpe method d * determe the mmum vue λ stsfyg the ostrts of the proem..9 5
16 . Empe. Sove the optmzto proem [8]. m Tke fuzzy prmeters s d L L L L 4 L4 5 L5 s used y Shoheg[]. Suppose tht e.g. L mes fuzzy set wth pek d d spred. The memershp futo of L s L 4 0 for for for < 4 > 4 6
![Shoheg[]. Suppose tht e.g. L mes fuzzy set wth pek d d spred.](/docs-images/48/18360236/images/page_16.jpg "The memershp futo of L s L 4 0 for for for < 4 > 4 6")
17 > > p p d d Frst sove foowg two su-proems: M d M Optm soutos of these su proems re: By usg these optm vues the proem e redued to foowg equvet o-er progrmmg proem: 7
18 M λ λ 0 λ λ λ Tht s M λ λ λ 0 λ λ λ λ 4 λ 5 λ. Let sove. y usg the fuzzy desve set method For λ 4 0 Se fese set s empty tkg λ 0 d λ the ew vue of λ 0/ / s pped. For λ /
19 Se fese set s empty tkg λ 0 d λ / the ew vue of λ 0// /4 s pped. For λ /4 0.5 the proem e wrtte s Se fese set s oempty tkg λ /4 d λ / the ew vue of λ /4// /8 s pped. For λ / Se fese set s empty tkg λ /4 d λ /8 the ew vue of λ /4/8/ 5/6 s pped. For λ 5/ Se fese set s oempty tkg λ 5/6 d λ /8 the ew vue of λ 5/6/8/ / s pped. For λ /
20 Se fese set s oempty tkg λ / d λ /8 the ew vue of λ //8/ /64 s pped. For λ / Se fese set s oempty tkg λ /64 d λ /8 the ew vue of λ /64/8/ 47/8 s pped. For λ 47/ Se fese set s empty tkg λ /64 d λ 47/8 the ew vue of λ /6447/8/ 9/56 s pped. For λ 9/ Se fese set s empty tkg λ /64 d λ 9/56 the ew vue of λ /649/56/ 85/5s pped. For λ 85/
21 Se fese set s empty tkg λ /64 d λ 85/5 the ew vue of λ /6485/5/ 69/04 s pped. For λ 69/ Se fese set s empty tkg λ /64 d λ 69/04 the ew vue of λ /6469/04/ 77/048 s pped. For λ 77/ Se fese set s oempty tkg λ 77/048d λ 69/04 the ew vue of λ 77/04869/04/ 475/4096 s pped. For λ 475/ Se fese set s oempty tkg λ 475/4096 d λ 69/04 the ew vue of λ 475/409669/04/ 95/89 s pped. For λ 95/
22 Se fese set s empty tkg λ 475/4096 d λ 95/89 the ew vue of λ 475/409695/89/ 590/684 s pped. For λ 590/ Se fese set s oempty tkg λ 590/684 d λ 95/89 the ew vue of λ 590/68495/89/ 80/768 s pped. For λ 80/ Se fese set s oempty tkg λ 80/768 d λ 95/89 the ew vue of λ 80/76895/89/ 80/768 s pped. For λ 607/ Se fese set s oempty tkg λ 607/6556 d λ 95/89 the ew vue of λ 607/655695/89/ 475/07 s pped. For λ 475/
23 Se fese set s oempty tkg λ 475/07 d λ 95/89 the ew vue of λ 475/0795/89/ 944/644 s pped. For λ 944/ Se fese set s oempty tkg λ 944/644 d λ 95/89 the ew vue of λ 944/64495/89/ 8886/5488 s pped. * For λ 8886/ Frst set the vue of λ put the λ vue. equto fter sovg the equto you see we hve ew futo wrtte eow the. ow frst sove tht futo wth smpe er progrmmg method f the souto vue s ess the wth rght hd sde vue the fese set s empty tkg the λ o d λ or f the vue s greter the fese set s oempty tkg λ d λ 0 ths proess otue ut we ot the optm vue of λ. Chek the dfferee etwee the prevous md d the et md d ompre the soute dfferee to ury ostt ε most equ to zero. For ste f ε0.000 the < Thus the st md o.60 e treted s optm. For updte the λ use ths formu λ λ λ /
24 Empe. Sove the optmzto proem M. Tke fuzzy prmeters s; L d L L L L L > > p p d d Frst sove foowg two su-proems; M d 4
25 0 8 9 M Optm soutos of these su proems re; d d d BY usg these optm vues the proem e redued to foowg equvet o-er progrmmg proem; M λ λ λ λ λ 5
26 Tht s M λ λ λ 0 λ.6 6.5λ λ λ 6 λ 6 λ 0.4 Let sove.4 y usg the fuzzy desve set method For λ Se fese set s empty tkg λ 0 d λ the ew vue of λ 0/ / s pped. For λ / Se fese set s empty tkg λ 0 d λ / the ew vue of λ 0// /4 s pped. For λ /
27 Se fese set s oempty tkg λ /4 d λ / the ew vue of λ /4 // /8 s pped. For λ / Se fese set s empty tkg λ /4 d λ /8 the ew vue of λ /4 /8/ 5/6 s pped. For λ 5/ Se fese set s oempty tkg λ 5/6 d λ /8 the ew vue of λ 5/6 /8/ / s pped. For λ / Se fese set s oempty tkg λ / d λ /8 the ew vue of λ / /8/ /64 s pped. For λ /
28 Se fese set s empty tkg λ / d λ /64 the ew vue of λ / /64/ 45/8 s pped. For λ 45/ Se fese set s oempty tkg λ 45/8 d λ /64 the ew vue of λ 45/8 /64/ 45/8 s pped. For λ 9/ Se fese set s oempty tkg λ 9/56 d λ /64 the ew vue of λ 9/56 /64/ 8/5 s pped. For λ 8/ Se fese set s empty tkg λ 9/56 d λ 8/5 the ew vue of λ 9/56 8/5/ 65/04 s pped. For λ 65/
29 Se fese set s empty tkg λ 9/56 d λ 65/04 the ew vue of λ 9/56 65/04/ 79/048 s pped. For λ 79/ Se fese set s empty tkg λ 9/56 d λ 79/048 the ew vue of λ 9/56 79/048/ 457/4096 s pped. For λ 457/ Se fese set s empty tkg λ 9/56 d λ 457/4096 the ew vue of λ 9/56 457/4096/ 9/89 s pped. For λ 9/ Se fese set s oempty tkg λ 9/89 d λ 457/4096 the ew vue of λ 9/89 457/4096/ 587/684 s pped. For λ 587/
30 Se fese set s empty tkg λ 9/89 d λ 587/684 The ew vue of λ 9/89 587/684/ 65/768 s pped. For λ 65/ Se fese set s oempty tkg λ 65/768 d λ 587/684 The ew vue of λ 65/ /684/ 07/6556 s pped. For λ 07/ Se fese set s empty tkg λ 65/768 d λ 07/6556 The ew vue of λ 65/768 07/6556/ 07/6556 s pped. * For λ 466/ We ot the optm vue of λ t λ * y usg the fuzzy desve set method. Couso The oetve of ths thess s to fd the optm souto of the proem usg the fuzzy desve set method. The method s sed o the trsformto of fuzzy er progrm to equvet o-fuzzy form otg prmeter λ. By utzg the seto method we oud oth the oetve futo d the ostrs whe fdg the optm vue of the prmeter λ whh wrrts the o-emptess of the souto set. 0
31 eferees Azmov A.Y. Gsmov.N.: O wek ougy wek Su dfferets d duty wth zero-gp o-ove optmzto Iterto Jour of Apped Mthemts Vo Duos D Prde H System of er fuzzy ostrts Fuzzy Sets d Systems Bem.E. deh L.A Deso-mkg fuzzy evromet Mgemet See B4-B64. 4 Kett O. Or M. Equvet formutos of oer teger proems for effet optmzto Mgemet See Vo. 6 No Kr G.J. Yu B. Fuzzy Sets d Fuzzy Log-Theory d Apptos Prete-H I p. 6 L Y-J. Hwg C-L. Fuzzy Mthemt Progrmmg Leture Notes Eooms d Mthemt Systems Sprger-Verg 99 0p. 7 Fuzzy ogs d er progrmmg fd optm souto y Bre Hse Opertos reserh eghth edto y Hmdy A.Th. 9 Tk H. Okud T. As K O fuzzy mthemt progrmmg J. Cyerets mmerm H.J.: Fuzzy mthemt progrmmg Comput. & Ops. es. Vo. 0 No Negot C.V.: Fuzzess mgemet OPSA/TIMS Mm 970. Skw M. Y H. Itertve deso mkg for mut-oetve Ler frto progrmmg proems wth fuzzy prmeters
32 Cyerets Systems Shoheg T. Iterv umer d Fuzzy umer er progrmmg Fuzzy Sets d Systems Tk H. As K. Fuzzy er progrmmg proems wth fuzzy Numers Fuzzy Sets d Systems Sovg fuzzy er progrmmg proems wth peewse er Memershp futos y the determto of rsp mmzg Deso y S. Efft d H. Asy.
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