Application of Sub Matrixes for Phase Process Optimization of Linear Programming

Transcription

1 VIII Iteratoa Coferece Heavy achery-h 24, Zatbor, Jue 24, B B.43 Appcato of Sub atres for Phase Process Optzato of Lear Prograg a Korevć, Vada Grovć, Nead Koarevć 2, Zvoo Petrovć, oe Raovć Facuty of echaca ad Cv Egeerg Kraevo, Uversty of Kraguevac, Kraevo, Serba 2 Facuty of echaca Egeerg, Uversty of Begrade, Begrade, Serba The probe of optzato of a utphase producto process by the ear prograg ethod s very frequet aufacturg practce. Istead of a cassca souto to the probe, by creatg costrat equatos per producto phase, the paper proposes the ethodoogy of appcato of sub atres for sovg the cope atheatca ode of the probe a atr for. The ethod s ustrated the eape of a optzato process of aufacturg ad asseby of a hydrauc vave for reguato of pressure ad fow, whch s teded for staato o hydrauc bar feeders for CNC aches. Probe s soved by atlab software pacage. Keywords: Phase process, ear prograg, optzato, sub atres. INTRODUCTION I cotrast to the sge-phase producto process whch the fa products are drecty produced fro se-fshed products, utphase process ca be dvded to severa sge-phase processes, terreated ad codtoed by the gve costrats. The uber of eves of utphase process breadow s detered by obectve codtos such as techoogy producto process ad asseby of the product, athough breadow ca soetes be aso due to subectve decsos [] [2]. The paper [2] dscusses prograg of utphase processes that ca be atheatcay soved by ear prograg ethod. The procedure of forg a atheatca ode to optze utphase process s preset the eape of aufacturg ad asseby of hydrauc vaves for reguatg pressure ad fow, whch s desged for staato o hydrauc feeders fro rod atera for CNC aches. The tas cossts of the foowg procedure [2]: t s ecessary to progra the ut-phase producto process order to produce the opta aout of cotro vaves fro avaabe quatty of se-fshed product ad stadard parts whe at the sae te proft fro the sae of the vave s au. aret tatos the aout of product paceet do ot est. 2. ATHEATICAL ODEL The atheatca ode ts atr for reads: It s ecessary to aze the obectve fucto:. a F(X) d X () wth satsfyg the costrats wth respect to avaabe quattes: X B (2) ad the o-egatvty codto: X (3) For the gve eape [2], the atrces, X ad B read: 26, 24 38, 733 2, 255 6,456 53,36 38,42 22, B X *Correspodg author: Dosteeva 9, 36 Kraevo, Serba, oarevc.@fv.g.ac.rs

2 B.44 VIII Iteratoa Coferece Heavy achery-h 24, Zatbor, Jue 24 Fg.. Asseby structure of the product 4,888 3,459,646,29,97,534,9, It ca be oted that the atr of coeffcets, the atr of varabes X ad the atr of costrats V cosst of severa sub atres. I the gve eape, the atr s of deso p q ad s coposed of p q sub atres. If the foowg ars are troduced: the uber of dfferet se-fshed products the uber of stadard parts the uber of parts fabrcated frst phase the uber of sub-assebes fabrcated secod phase the uber of a assebes fabrcated thrd phase the uber of products s the uber of phases a atr М ca be preset the foowg shape: q М p. Koarevć, V. Grovć, N. Koarevć, Zv. Petrovć,. Raovć

3 VIII Iteratoa Coferece Heavy achery-h 24, Zatbor, Jue 24 B.45 where atr deso are p s+ q s The geeratg of sub atres of atr ca be perfored the foowg aer: a) The forato of the frst cou of atr М fro ts sub atres s as foows Sub atr М s of deso, sub atr М2 s of deso, sub atr М3 s of deso, sub atr М4 s of deso, sub atr М5 s of deso. For preseted eape [2] these atres are: 4,888 3, 459,646,29, 97,534,9, [ ] b) The forato of the secod cou of atr М fro ts sub atres for gve eape s as foows [ ] ; ; c) The forato of the thrd cou of atr М fro ts sub atres for gve eape s as foows 3 ; 23 ; 33 ; 43 ; 53 [ ] ; d) The forato of the fourth cou of atr М fro ts sub atres for gve eape s as foows 4 ; 24 ; 34 ; 2 44 ; 54 [] ; Appcato of Sub atres for Phase Process Optzato of Lear Prograg

4 B.46 VIII Iteratoa Coferece Heavy achery-h 24, Zatbor, Jue 24 Oe ca ote the foowg: sub atres М2, М3 и М4 are zero atres because phase processes acheveet of subassebes, a assebes ad products s ot possbe ust fro se-fshed products; sub-atres М4, М5 и М52 are aso zero atrces because t s ot possbe to acheve part fro sub-assebes ad assebes, or s t possbe to acheve sub-asseby fro a asseby; sub-atres 3, 42 ad 53 are dagoa atrces where the eeets o the a dagoa have a vaue of -, so they represet a scaar atrces, e. whch are egatve of 3 [ a] [ a] [ a] [ ] [ ] [ ] E a E a 42 E a 53 The etoed atres ca be defed soey o,,,,, ad s vaues, whe the other sub atres of a atr ca be fored o the bass of data for each phase of deveopet. The atr X s a atr of cous ad w have as ay sub atres as phases the process that s. r s, so there are 4 sub atres the gve eape: X X 2 X, X 3 X 4 r The uber of eeets of sub atres are: for atr X the uber of eeets s equa to the uber of parts, for atr X2 the uber of eeets s equa to the uber of sub-assebes, for atr X3 the uber of eeets s equa to the uber of a assebes, ad for atr X4 the uber of eeets s equa to the uber of products. Sub atres of atr X for gve eape are: X 5 ; X2 ; X [ ] ; X [ ] The atr В s aso cou atr ad cossts of cs+ cou atres: B B2 B B 3 B4 B 5 c The uber of eeets of sub atres s: the uber of eeets of the atr B s equa to the uber of se-fshed products, the uber of eeets of the atr B2 s equa to the uber of stadard parts, the uber of eeets of the atr B3 s equa to the uber of parts, the uber of eeets of the atr B4 s equa to the uber of sub-assebes, the uber of eeets of the atr B5 s equa to the uber of a assebes. Sub atres B3, B4, B5, are zero atres, whe sub atr B represets avaabe quatty of sefshed products ad sub atr B2 represet avaabe aout of stadard parts. For gve eape these sub atres are: 5 26, , , B 2 B 6, B3 53, ,42 22,696 7 ; ; ; B4 B ; [ ] 5 3. PROBLE SOLVING For sovg the utphase probe of ear prograg wth sub atres the software pacage ATLAB s used. For data fro the etoed eape progra code s wrtte the foowg for: 7; 8; 8; 3; ; ; s4; [ ; ; ; ; ;.... Koarevć, V. Grovć, N. Koarevć, Zv. Petrovć,. Raovć

5 VIII Iteratoa Coferece Heavy achery-h 24, Zatbor, Jue 24 B.47.9 ;....99;] 2zeros(,) 3-eye() 4zeros(,) 5zeros(,) 2zeros(,) 22[ ;... ;... ;... ;... ;... ;... ;... ;] 32[ ;... ;... ;... ;... ;... ;... ;... ;] 42-eye() 52zeros(,) 3zeros(,) 23[;;;;;;;] 33[;;;;;;;] 43[;;] 53-eye() 4zeros(,) 24[;;;;;;2;] 34[;;;;;;;] 44[;;] 54[] [ 2 3 4; ; ; ; ;] B[26.24;38.733;2.255;6.456;53.36;38. 42;22.696] B2[5;6;5;7;5;6;;7] B3zeros(,) B4zeros(,) B5zeros(,) B[B;B2;B3;B4;B5] F[ ]' [,fva,etfag,output]prog(- F,,B,[],[],[],[],[]) After the sth terato the foowg souto s obtaed:.e+3 * fva e+3 etfag output teratos: 6 agorth: 'arge-scae: teror pot' cgteratos: Therefore, based o the avaabe quatty of sefshed products ad stadard parts t s possbe to produce 56 uts of a product, where the proft of 5.6, of oetary uts s acheved. 4. CONCLUSION The paper [2] presets the ethodoogy of a atheatca odeg of the phase process of ear prograg atr for. Ue the tradtoa way of atheatca odeg where for each phase equatos of restrctos for certa category of recourses are defed, by appyg ths ethodoogy t s possbe to defe the atres Х, ad В fro the suary tabe, whch defes the forg of the product phases, whch sgfcaty reduces errors occurrece the atheatca odeg process. For utphase probes ear prograg that have ay costrats ad varabes, atres Х, М ad В ca be epressed by sub atres whch spfes the process of atheatca odeg because the aorty of sub atres ca be defed autoatcay. Due to ted space, ths paper, four phase producto probe s aayzed o practca eape oy wth costrats of the avaabe quattes of sefshed products ad stadard parts. For sovg the probe atlab software pacage s used. Rea practca probes are ore cope because products are ore cope, requrg a greater uber producto phases, ad atheatca odes have ore resource costrats such as: abor restrcto, the avaabe capacty of aches, aret costrats, the avaabe oetary fud ad etc. Further authors efforts w be focused o the software sovg of the probe wth uted uber of phases ad arge uber of dfferet types of costrats. ACKNOWLEDGEENT The authors woud e to epress ther grattude to the stry of Educato ad Scece of the Repubc of Serba for ther support to ths research through the proect TR372. Appcato of Sub atres for Phase Process Optzato of Lear Prograg

6 B.48 VIII Iteratoa Coferece Heavy achery-h 24, Zatbor, Jue 24 REFERENCES [] Vada A., Prea ateatčh etoda u eoo, Iforator, Zagreb, 98. [2] Koarevć., Grovć V., Radčevć B., Petrovć Zv., ode For Optzato Of Phase Processes By The Lear Prograg ethod, 36th Iteratoa Coferece o Producto Egeerg, Kopao, 23, pp6-68 [3] Raovć., Leara agebra teora atrca earh operatora, Aadesa sao, Beograd, 27. [4] auzć Z., Deterate, atrce, vetor, aatča geoetra, Građevsa ga, Beograd, Koarevć, V. Grovć, N. Koarevć, Zv. Petrovć,. Raovć