PROBLEM -1. where S. C basis x. 0, for entering

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1 ISSN: ISO 9:8 Certified Volume 4 Iue 8 February 5 Optimum Solution of Linear Programming Problem by New Method Putta aburao; Supriya N. Khobragade and N.W.Khobragade Department of Mathematic RTM Nagpur Univerity Nagpur 44. btract In thi paper new alternative method for imple method ig M method and dual imple method are introduced. Thee method are eay to olve linear programming problem. Thee are powerful method. It reduce number of iteration and ave valuable time by kipping calculation of net evaluation. Key word: Linear programming problem optimal olution imple method alternative method. I. INTRODUCTION Khobragade et al. [ 64] uggeted an alternative approach to olve linear programming problem. In thi paper an attempt ha been made to olve linear programming problem (LPP) by new method which i an alternative for imple method. Thi method i different from Khobragade et al. [ 64] Method. II. N LTERNTIVE LGORITHM FOR SIMPLEX METHOD To find optimal olution of any LPP by an alternative method for imple method algorithm i given a follow: Step (). Check objective function of LPP i of maimization or minimization type. If it i to be minimization type then convert it into a maimization type by uing the reult: Min. = Ma.. Step (). Check whether all (RHS) are nonnegative. If any i negative then multiply the correponding equation of the contraint by(). Step (). Epre the given LPP in tandard form then obtain initial baic feaible olution. Step (4). Select ma vector. C j ij ij for entering Step (5). Chooe greatet coefficient of deciion variable. (i) If greatet coefficient i unique then element correponding to thi row and column become pivotal (leading) element. (ii) If greatet coefficient i not unique then ue tie breaking technique. Step (6). Ue uual imple method for thi table and go to net tep. Step (7). Ignore correponding row and column. Proceed to tep 5 for remaining element and repeat the ame procedure until an optimal olution i obtained or there i an indication for unbounded olution. Step (8). If all row and column are ignored then current olution i an optimal olution. PROLEM III. SOLVED PROLEMS Ma. Z 5 Subject to the contraint: 5 8. SOLUTION: We have the contraint = 5 = = 8 where S S S are lack variable. New Simple Table. C bai 5 8 /5 /5 /5 /5 6 4/5 /5 9/7 5/4 /4 5 8/7 4/7 /7 5/7 /4 5/4 Since all row and column are ignored hence an optimum baic feaible olution ha been reached. Optimum olution i 8/ 7 5/ 7 and ma. Z 85/ 7. PROLEM Minimum Z 7 5 Subject to the contraint:

2 SOLUTION. We have the contraint 5 = = 8 4 = Where are lack variable. New Simple table. ISSN: ISO 9:8 Certified Volume 4 Iue 8 February 5 Since all row and column are ignored hence an opt. baic feaible olution ha been reached. C bai 4 5 C bai / / / / / / 5/ / / 7 7/4 4 /4 7 /4 /4 5 5/4 8 /4 5 7/4 7/6 /4 /6 7 /4 / /4 /4 5 5/4 9/4 7/4 5 6/59 4/59 / /59 /59 4/59 Since all row and column are ignored hence an optimum baic feaible olution ha been reached. Optimum olution i 4 5 and ma. Z. Min Z (ma Z) PROLEM Maimize Z Subject to the contraint: 4 5. SOLUTION: We have the contraint 4 5 New Simple Table. Optimum olution i and ma. Z 9. PROLEM 4 Minimum Z Subject to the contraint: SOLUTION. We have the contraint where are lack variable. New imple table. C bai / /4 / /4 5/ 8 /4 4

3 ISSN: ISO 9:8 Certified 4 4/5 /5 / 5 /5 / / /5 8/5 7/5 /5 4/5 4/5 8/5 /5 / / / / Since all row and column are ignored an optimum baic feaible olution ha been reached. Hence olution i /5 4/5 / and Min Z (ma Z) 84/ 5 IV. LTERNTIVE LGORITHM FOR IGM METHOD To find optimal olution of any LPP by an alternative method for igm method algorithm i given a follow: Step (). Check objective function of LPP i of maimization or minimization type. If it i to be minimization type then convert it into a maimization type by uing the reult: Min. = Ma.. Step (). Check whether all (RHS) are nonnegative. If any i negative then multiply the correponding equation of the contraint by. Step (). Epre the given LPP in tandard form then obtain initial baic feaible olution. If baic olution i nonfeaible due to the contraint of the type and then we add artificial variable to the correponding contraint in tandard form. ign very large value for maimization and for minimization in objective function. Step (4). Select ma C j ij Volume 4 Iue 8 February 5 ij for entering vector. Step (5). Chooe greatet coefficient of deciion variable. (i) If greatet coefficient i unique then variable correponding to thi column become incoming variable. (ii) If greatet coefficient i not unique then ue tie breaking technique. Step (6). Compute the ratio with. Chooe minimum ratio then variable correponding to thi row i outgoing variable. The element correponding to incoming variable and outgoing variable become pivotal (leading) element. Step (7). Ue uual imple method for thi table and go to net tep. Step (8). Ignore correponding row and column. Proceed to tep 5 for remaining element and repeat the ame procedure until an optimal olution i obtain or there i an indication for unbounded olution. Step (9). If all row and column are ignored then current olution i an optimal olution. PROLEM 5 Ma Z 6 4 Subject to: 4. SOLUTION: We have the contraint 4. Where are lack variable and i artificial variable. Simple table: C bai 4 M 6 4 7/ / 8 / / M 5 / / 4 6 /7 /7 6 6 /7 /7 M 9 /7 /7 Since all row and column are ignored hence an optimum olution ha been reached. Therefore optimum olution i: 6 6 ; Ma Z 6 PROLEM 6 Min Z Subject to:

4 ISSN: ISO 9:8 Certified Volume 4 Iue 8 February 5 SOLUTION: We have the contraint Step (4). Chooe mot negative then variable correponding to thi row become outgoing variable. Select the mot negative C 4 6 j ij ij of then variable correponding to thi column become incoming variable. The element correponding to incoming variable and. outgoing variable i pivotal (leading) element. where are urplu and lack variable repectively Step (5). Ue uual imple method for thi table and go to net tep. and are artificial variable. Step (6). Ignore correponding row and column. Proceed Simple table: to tep 4 for remaining element and repeat the ame C procedure until an optimal olution i obtained in finite bai number tep or there i an indication of the noneitence of a feaible olution. M M 6 4 M 5/ /4 / 4/ / / / / / /5 /5 /5 /5 6/5 /5 4/5 /5 6/5 /5 /5 4/5 Since all row and column are ignored hence an optimum olution ha been reached. Therefore optimum olution i: /5 6 / 5 ; Min Z /5 V. LTERNTIVE LGORITHM FOR DUL SIMPLEX METHOD To find optimal olution of any LPP by an alternative method for dual imple method algorithm i given a follow: Step (). The objective function of the LPP mut be maimize. If it i minimize then convert it into maimize by uing the reult: Min. = Ma.. Step (). Convert all contraint into by multiplying the correponding equation of the contraint by. Step (). Convert inequality contraint into equality by addition of lack variable and obtain an initial baic olution. Epre the above information in the form of a table known a dual imple table. Step (7): If all row and column are ignored then current olution i an optimal olution. PROLEM 7 Minimize Z= Subject to 4 8 SOLUTION: We have the contraint 4 Initial imple table C bai / / / 6 / / / / / / Since all olution. are poitive current olution i an optimal 6

5 6 Min Z PROLEM 8 ISSN: ISO 9:8 Certified Volume 4 Iue 8 February 5 ; Ma Z* ;. Maimize Z Subject to: 4 6 SOLUTION: We have the contraint 4 6 Simple table: C bai 6 4 / 5/4 /4 / /4 /4 / 5/4 /4 /5 /5 /5 6/5 /5 4/5 Since all are poitive current olution i an optimal olution. X = /5 X = 6/5 MX Z = IX. CONCLUSION lternative method for imple method ig M method and dual imple method have been derived to obtain the olution of linear programming problem. The propoed algorithm have implicity and eae of undertanding. Thee reduce number of iteration and improve the optimum olution in mot of the cae. Thee method ave valuable time a there i no need to calculate the net evaluation ZjCj. REFERENCES []. Mr Lokhande K. G. Khobragade N. W. Khot P. G.: Simple Method: n lternative pproach International Journal of Engineering and Innovative Technology Volume Iue P: 4648 (). []. Khobragade N. W. and Khot P. G.: lternative pproach to the Simple MethodI ulletin of Pure and applied Science Vol. (E) (No.); P. 54 (4). []. Khobragade N. W. and Khot P. G.: lternative pproach to the Simple MethodII cta Ciencia Indica Vol. IM No. 65 India (5). [4]. Sharma S. D.: Operation Reearch Kedar Nath Ram Nath R. G. Road Meerut5 (U.P.) India. [5]. Ga S. I.: Linear Programming /e McGrawHill Kogakuha Tokyo (969). [6]. Ghadle K.P; Pawar T.S and Khobragade N.W (): Solution of Linear Programming Problem by New pproach Int. J. of Engg. nd Information Technology vol. Iue 6 pp.7 [7]. Khobragade N.W Lamba N.K and Khot P. G (9): lternative pproach to Revied Simple Method Int. J. of Pure and ppl. Math. vol. 5 No [8]. Khobragade N.W Lamba N.K and Khot P. G (): lternative pproach to Wolfe Modified Simple Method for Quadratic Programming Problem Int. J. Latet Trend in Math. vol. No. pp. 94. [9]. Mr. Vaidya N.V and Khobragade N.W (): Optimum olution to the imple method n alternative approach Int. Journal of Latet Trend in Math (accepted) UK. [].Mr.Vaidya N.V and Khobragade N.W (): Solution of Game problem uing New pproach Int. J. of Engg. nd Information Technology vol. Iue 5 pp.886. [].Mr Lokhande K.G; Khobragade N.W and Khot P. G (): lternative pproach to Linear Fractional Programming Int. J. of Engg. nd Information Technology vol. Iue 4 pp.697. [].Khobragade N.W Lamba N.K and Khot P. G (): Solution of LPP by KKL Method Int. J. of Engg. nd Information Technology vol. Iue 4 pp.44. [].Khobragade N.W Lamba N.K and Khot P. G (): Solution of Game Theory Problem by KKL Method Int. J. of Engg. nd Information Technology vol. Iue 4 pp.555. [4].Mr. N.V Vaidya and Khobragade N.W (4): pproimation algorithm for optimal olution to the linear programming problem Int. Journal of Math in Operational Reearch Vol.6 No. pp

6 ISSN: ISO 9:8 Certified UTHOR IOGRPHY Volume 4 Iue 8 February 5 Dr. N.W. Khobragade for being M.Sc in tatitic and Math he attained Ph.D in both ubject. He ha been teaching ince 986 for 8 year at PGTD of Math RTM Nagpur Univerity Nagpur and uccefully handled different capacitie. t preent he i working a Profeor. chieved ecellent eperience in Reearch for 5 year in the area of oundary value problem (Thermoelaticity in particular) and Operation Reearch. Publihed more than 8 reearch paper in reputed journal. Fourteen tudent awarded Ph.D Degree and i tudent ubmitted their thei in Univerity for award of Ph.D Degree under their guidance. Supriya Khobragade Student of M.E final in Computer Science R..I.T College Nerul Navi Mumbai. Putta aburao for being M.Sc Mphil in Math he ha been teaching ince for 4 year at P Siddhartha College of rt and Sci Vijaywada (.P). 8