Naglaa Raga Said Assistant Professor of Operations. Egypt.

Transcription

1 Volue, Issue, Deceer ISSN: 77 8X Internatonal Journal of Adanced Research n Coputer Scence and Software Engneerng Research Paper Aalale onlne at: Optal Control Theory Approach to Sole Constraned Producton and Inentory Syste Hegazy Zaher * Professor of Matheatcal Statstcs, Insttute of Statstcal Studes and Research (ISSR), Caro Unersty, Egypt. Naglaa Raga Sad Assstant Professor of Operatons Research, (ISSR).Caro Unersty, Egypt. Taher Taha Zak PHD student, Dept. Of Operatons Research, (ISSR). Caro Unersty, Egypt. Astract: Ths paper proposes an optal control of producton nentory syste. The producton and nentory syste s soled usng optal control theory. We present fnte contnuous te lnear optal control odel to control dynac prce and producton flow rate to axze total reenue nus total cost. Where the deand s lnear wth prce, the producton flow rate of the product not exceeds the axu producton capacty rate. Total cost conssts of the anufacture holdng cost, uyer holdng cost and producton cost. Pontryagn Maxu Prncple s appled to fnd the controllers of the odel. Keywords: Producton Inentory Syste, Optal Control. Pontryagn Maxu Prncple I-Introducton Recently the optal control theory s used to sole proles of nentory and producton systes. Researchers are focusng on estatng the effect of the changes n the deand wth te n logstcs. Contnuous te optal control odels prode a conenent way for awareness the ehaor of systes where syste dynac plays a sgnfcant role. A lot of research s focusng on odels consderng all features of these proles. In partcular the ost attracte feature of these odels s to prode good schedulng producton and nentory polces n dersty of settngs. Moreoer they approxate well the fundaental stochastc of proles n a deternstc way. A contnuous te ethod has an excellent feature whch s not ntroducng any approxaton to the real settng; t prodes the accurate soluton of the syste. Lterature Reew: S.P.Seth, G. L. Thopson, () presented a odel n whch the factory produces a sngle hoogenous product and has a fnshed goods store. They assued that the odel s slar to HMMS odel. The odel s forulated and copletely soled usng optal control theory.the steady state soluton s otaned when horzon s nfnty. The optzaton prole s to nze total cost of the odel. M. A. Baten, A. A. Kal. (9) presented a odel n whch the nentory producton syste wth two paraeters Weull dstruted deteroraton tes n whch the nentory odel s consdered as lnear optal control prole and the odel s soled y pontryagn axu prncple, the soluton of the optal control prole s soled analytcally. Ynkuan Gu, Hongxa Zhang,, present a odel to control supply Chan. The an a of the odel s to ake the total cost of the supply chan nu. Ghodrat Alah Eaerd, et al () presented optal control of producton nentory syste consderng deteroratng tes. The an ojecte of ths study s to deterne nuotal producton and nentory costs. D. Iano,B. Sokolo () present a odel to analyze and to achee desgned econoc perforance n a actual te uncertan and dsturance executon enronent s tal and odern ssue n any supply chans. Ths study s the frst papers addressng the operate perspecte of the supply chan dynac doan. II- Assuptons and Notatons To estalsh the odel the followng notatons and assuptons are used. Consder a factory producng a certan product and hang hoogenous fnshng goods warehouse. The factory dstrutes the product to a uyer whch sells soe and stores the rest. The ojecte s to deterne the optal prce and optal producton rate to axze the total proft of oth the uyer and the endor. Inputs T : Te horzon Q : Vendor producton capacty h : Buyer holdng cost. h : Vendor holdng cost. c : The producton cost coeffcent, IJARCSSE All Rghts Resered Page 665

2 Zaher et al., Internatonal Journal of Adanced Research n Coputer Scence and Software Engneerng (), Deceer -, pp t a, : Coeffcents used for the product at te t n lnear relatonshp etween prce and deand d = a t P t û : Producton goal leel of the endor Î : Inentory goal leel of the uyer Î V : Inentory goal leel of the endor The eanng of nentory goal leel Î s that a safety stock that the copany wants to keep on hand. Slarly the eanng of producton goal leel û s explaned as ost effcent leel at whch t s requred to run the factory. Outputs t : Prce of one unt of the product at te t (control arale) P u t : Producton rate at te t (control arale.) I : Inentory leel (nuer of unts) of uyer at te t (state arale). I : Inentory leel (uer of unts) of endor at te t (state arale). The prole seeks to axze the reenue nus the nentory and productons costs. The ojecte functon of the odel can e wrtten as: Maxze T pt d t h ˆ I t I h ˆ c I t I u ˆ t u dt () Suject to I dt () I u t dt d = a t t P t t u t Q t,t (),T () (5) a( t) P t (6) ( t) u t, I t, I t, P t (7) Wth ntal condton I = I o, I = I o. We start y appled Pontryagn axu prncple [6],[] Haltonan functon H pt d t h I Lagrangan functon L H g t I ˆ III. The Matheatcal Model and Analyss h I t I ˆ c u t dt u t u d (8) f (9) Where,,,,,, satsfy the copleentary slackness condtons, u,, I,,, a Q u,, pt,, P t,, I, () Fro equaton (7) and equaton () t s clear that = = = = To otan the optal anufacturng (producton) rate and optal prce we dfferentate the Lagrange functon wth respect u t P t to respectely. L P, L u (), IJARCSSE All Rghts Resered Page 666

3 Zaher et al., Internatonal Journal of Adanced Research n Coputer Scence and Software Engneerng (), Deceer -, pp at t t () () P t = u t = û + To otan (the adjont equatons) L =, L = () I I = h I t (5) = h I t (6) Fro equatons (), (), (), (5) we can get the followng syste of lnear dfferental equatons. I = I = û = = I I t t t t t a (7) at t t + t (8) c h (9) h ˆ () I These equatons can e wrtten n atrx for as followng: I I h h c I I t V t t t a a uˆ h h () Wth the oundary condtons I I T T I I () The equaton (.) can e expressed n the atrx for as x Ax () Matrx A has four egenalues,, and four egenectors. The egenalues can e otaned frohe deternant of characterstc atrx A I. h h c h c h h () B B AD A Where h A, B = h h, D h h c c The correspondng characterstc roots of atrx A, there exst non zero ectors such that A I are egenectors., IJARCSSE All Rghts Resered Page 667 (5) where

4 Zaher et al., Internatonal Journal of Adanced Research n Coputer Scence and Software Engneerng (), Deceer -, pp c (6) h h Fro aoe we can get t h t h h h t h t The general soluton conssts of hoogeneous soluton araton of paraeters technque. xh partcular soluton x.of the syste we wll use the p x h c e ce ce ce (7) x p Le Le Le Le t t t t (8) Let the atrx a s the nerse of atrx a t L e a a a a L e = a a a a a uˆ L e a a a a L e a a a a h h Fro equaton (9) we get L, L, L, L then y usng ntegraton we get L, L, L, L The coplete optal soluton x = ( x h ) ( I I t t t t c e ce c e ce Le Le Le Le x p ) s gen y ; (9) () Nuercal Exaple A nuercal exaple s gen for two dfferent states of deand and prce rates. Tale presents the alues of syste paraeters and ntal states whch are used n nuercal exaples. Tale Paraeters Value c $, IJARCSSE All Rghts Resered Page 668

5 Zaher et al., Internatonal Journal of Adanced Research n Coputer Scence and Software Engneerng (), Deceer -, pp h $ h $ û 5 Î Î 5 I 5 o I 5 o.5 a a +t T 5 Where c, h, h are the producton cost, the uyer holdng cost and the endor holdng cost per unt product. Respectely û, Î, Î are the producton goal rate, nentory goal leel of the uyer and nentory goal leel of the endor. we suarze the alues of the optal anufacture rate and optal prce and nentory leel of uyer and nentory leel of endor at the end of plannng horzon te (T) and ojecte functon n tale. Tale Suary of result at t =T = 5 Deand rate The results Constant a= Lnear a= t I I 8 u (5) 5 5 P (5) Holdng cost of uyer Holdng cost of endor 7 7 Producton cost J (optal) 6 5 IV. Conclusons Ths study has descred the soluton of nentory producton syste usng Pontryagn axu prncple. The prce control polcy and producton control polcy has axzed the ojecte functon that easure the proft of oth uyer and endor. Ths odel can e extended n any ways.for exaple transportaton cost, order cost, and shortage cost of oth uyer and endor. Also ths odel can extend to nclude ultple uyers, ultple endors and ult-products to represent constrant supply chan prole. References. B. C. Kuo, Autoatc Control Systes. Thrd Edton Prentce Hall Inc, Englewood Clffs, New Jersey. ISBN ,975. B. D. O. Anderson and J. B. Moore, Optal Control: Lnear Quadratc Methods, Prentce Hall, Englewood Clffs, NJ, 99.. E. Adda, G.Peraks, a Nonlnear Contnuous Te Optal Control Model of Dynac Prcng and Inentory Control wth No Backorders. Wley Interscence, Wley perodcals,inc 7. D. Lerzon, Calculus of Varatons and Optal Control Theory, Polna Ben-sra,9. 5. D. Atherton, Control Engneerng, Darek Atherton &Ventus pulshng Aps ISBN , D.S. Ndu, Optal Control Systes. By CRC Press LLC. ISBN ,. 7. D. Iano, B. Sokolo, Control and Syste- Theoretc Identfcaton of the Supply Chan Dynacs Doan for Plannng, Analyss and Adaptaton of Perforance under Uncertanty, European Journal of Operatonal Research, () -,. 8. G. A. Eaerd, M. S. Kar and M. Safee, Applcaton of Optal Control Theory to Adjust the Producton Rate of Deteroratng Inentory Sste. Mddle East journal of Scentfc Research (); 56-5, ISSN99-9,. 9. K. Suraanan, J. B. Rawlngs, C. T. Maraelas, J.F. Cerrllo, Integraton of Control Theory and Schedulng Methods for Supply Chan Manageent,Coputer and Checal Engneerng 5 ()-,., IJARCSSE All Rghts Resered Page 669

6 Zaher et al., Internatonal Journal of Adanced Research n Coputer Scence and Software Engneerng (), Deceer -, pp M..A. Baten, A. A.Kal, An Optal Control Approach to Inentory Producton Syste wth Weull Dstruted Deteroraton, Journal of Matheatcs and Statstcs 5 ():6-, 9.. P.C. Yang,H.M Wee, A Collaorate Syste wth Perssle Delay n Payent for Deteroratng Ites, Matheatcal and Coputer Modelng (6) 9-, 6... R. V. Dullpat. Matla an: Introducton wth Applcatons. New Age Internatonal (p) ISBN S.P. and G. L. Thopson,Optal Control Theory, Applcatons to Manageent Scence and Econocs. nd Edn., Sprnger, USA., Inc. ISBN ,... Y. Benhadd, L. Tadj, M. Bounkhel, Optal Control of Inentory Systes wth Deteroratng Ites and Dynac Costs, Appled Matheatcs 9- ISBN Y.Gu and H. Zhang, An Optal Inentory Control Model for Supply Chan Shortage Constrants, Journal of Coputers Vol.6 No 9,., IJARCSSE All Rghts Resered Page 67