Matheatcs Letters 208; 4(: -5 http://www.scecepublshggroup.co/j/l do: 0.648/j.l.208040. ISSN: 2575-503X (Prt; ISSN: 2575-5056 (Ole A Iovatve Algorthc Approach for Solvg Proft Maxzato Probles Abul Kala Azad, Mosharraf Hossa 2 Departet of Matheatcs, Rajshah Goveret Cty College, Rajshah, Bagladesh 2 Departet of IPE, Rajshah Uversty of Egeerg ad Techology, Rajshah, Bagladesh Eal address: To cte ths artcle: Abul Kala Azad, Mosharraf Hossa. A Iovatve Algorthc Approach for Solvg Proft Maxzato Probles. Matheatcs Letters. Vol. 4, No., 208, pp. -5. do: 0.648/j.l.208040. Receved: Deceber 0, 207; Accepted: Deceber 25, 207; Publshed: Jauary 9, 208 Abstract: The ew algorthc techque developed ths artcle to solve the proft axzato probles usg trasportato algorth of Trasportato Proble (TP has three basc parts; frst covertg the axzato proble to the zato proble, secod forattg the Total Opportuty Table (TOT fro the coverted Trasportato Table (TT, ad last allocatos of profts usg the Row Average Total Opportuty Value (RATOV ad Colu Average Total Opportuty Value (CATOV. The curret algorth cosders the average of the cell values of the TOT alog each row detfed as RATOV ad the average of the cell values of the TOT alog each colu detfed as CATOV. Allocatos of profts are started the cell alog the row or colu whch has the hghest RATOVs or CATOVs. The Ital Basc Feasble Soluto (IBFS obtaed by the curret ethod s better tha soe other falar ethods whch s dscussed ths paper wth the three dfferet szed exaples. Keywords: TP, TT, TOT, RATOV, CATOV, IBFS. Itroducto Trasportato probles lear prograg show the atheatcal optal ways to get the solutos of trasportato probles. Trasportato algorth s a effectve tool to get optal proft. Soe of the usual algorths to solve the trasportato probles are North West Corer (NWC Method, Matrx Ma Method, ad Vogel s Approxato Method (VAM. Afterwards ay researchers provde ay helpful algorths to get IBFS of trasportato probles. Soe of the ethods ad algorths that the curret work has goe through P. Pada ad G. Nataraja s A New Approach for Solvg Trasportato Probles wth Mxed Costrats []; A Iovatve Method for Solvg Trasportato Proble [2] by N. M. Deshukh; Modfed Vogel s Approxato Method for Ubalace Trasportato Proble [3] by N. Balakrsha; Serder Korukoglu ad Serka Ball s A Iproved Vogel s Approxato Method (IVAM for the Trasportato Proble [4]; Harvey H. Shore s The Trasportato Proble ad the Vogel s Approxato Method [5]; A odfcato of Vogel s Approxato Method through the use of Heurstcs [6] by D. G. Shshak, J. A. Kaslk ad T. D. Barelay; A. R. Kha s A Re-soluto of the Trasportato Proble: A Algorthc Approach [7]; A ew approach for fdg a Optal Soluto for Trasportato Probles by V. J. Sudhakar, N. Arusakar, ad T. Karpaga [8]; Krca ad Satr s A Heurstc for Obtag a Ital Soluto for the Trasportato Proble [9]. Md. Arul Isla et al. [0] calculate the Dfferece Idcators by takg the dfferece of the largest ad the ext largest cell value of each row ad each colu of the TOT for the allocato of uts of the TT to detere axu proft. The above etoed algorths cted ths artcle are beefcal to fd the IBFS to solve proft axzato objectve. The curret research also ads a useful algorth whch gves a better IBFS the proft axzato probles. 2. Algorth The developed algorth the curret work volves three parts:
2 Abul Kala Azad ad Mosharraf Hossa: A Iovatve Algorthc Approach for Solvg Proft Maxzato Probles. Covertg the axzato proble to the zato proble Coverso of the proble 2. Algorth for Total Opportuty Table (TOT 3. Algorth for allocato Subtract the hghest proft fro the other profts the TT ad place the subtracted values a ew a TT. I ths way the axzato proble becoes the zato proble. Algorth for TOT Subtract the sallest etry fro each of the eleets of every row of the ew TT ad place the o the rght-top Step of correspodg eleets. Apply the sae operato o each of the colus ad place the o the rght-botto of the correspodg Step 2 eleets. Step 3 For the TOT whose etres are the suato of rght-top ad rght-botto eleets of Steps ad 2. Algorth for Allocato Step Step 2 Step 3 Step 4 Step 5 Step 6 Place the average of total opportuty values of cells of the TOT the rght sde alog each row the frst bracket detfed as Row Average Total Opportuty Value (RATOV ad the average of total opportuty values of cells the below alog each colu the frst bracket detfed as Colu Average Total Opportuty Value (CATOV. Idetfy the hghest eleet aog the RATOVs ad CATOVs, f there are two or ore hghest eleets; choose the hghest eleet alog whch the sallest valued eleet s preset. If there are two or ore sallest eleets, choose ay oe of the arbtrarly. Allocate x, b j o the left top of the sallest etry the (, j th of If If a < bj, leave the th row ad readjust b j as a > bj b / j = bj a., leave the jth colu ad readjusta as a = a b, leave ether th row or j-th colu but ot both. / j. If a = bj Repeat Steps to 4 utl the r requreet satsfed. Calculate p = = p j x j the TT correspodg to the basc cells of TOT., P beg the axu proft where p j s the proft ut of -th row ad j-th colu of 3. Nuercal Exaples Exaple 0 Three products P, P 2, ad P 3 are produced three aches M, M 2, ad M 3 ad ther proft args are gve the followg table. Table. Tabular For of the Proble. Capacty M 290 280 300 200 M 2 250 270 230 200 M 3 350 370 380 200 Dead 50 300 50 600 We wat to axze the proft by the curret algorth. Soluto We subtract the axu proft 380 fro other profts of Table 2. Mzg the Proble. M 90 00 80 200 M 2 30 0 50 200 M 3 30 0 0 200 50 300 50 600 The row dffereces ad colu dffereces fro the lowest row ad the lowest colu Table 3. Row-Colu Dfferece Table. M 90 0 60 00 20 90 800 80 200 M 2 30 20 00 00 00 50 40 50 200 M 3 30 30 0 00 0 00 0 200 50 300 50 600 The TOT s: Table 4. Total Opportuty Table. M 70 0 80 200 M 2 20 00 90 200 M 3 30 0 0 200 50 300 50 600
Matheatcs Letters 208; 4(: -5 3 The allocatos wth the help of RATOVs ad CATOVs Table 5. Allocatos TOT. M 50 70 0 50 80 200 M 2 20 200 00 90 200 M 3 30 00 0 00 0 200 ( b j 50 300 50 600 73.3 73.3 90 CATOV 50 60 40-60 40 86.6 86.6 95 RATOV 36.6 - - 3.3 3.3 5 The allocatos the orgal TT Table 6. Allocatos the Orgal Proble. M 50 290 280 50 300 200 M 2 250 200 270 230 200 M 3 350 00 370 00 380 200 50 300 50 600 The axu proft s = = 290 50 + 300 50 + 270 200 + 370 00 + 380 00 = 87500 Exaple 02 Three products P, P 2, ad P 3 are produced four aches M, M 2, M 3, ad M 4, ad ther proft args are gve the followg table. Table 7. Tabular For of the Proble. Capacty M 0 5 2 50 M 2 6 9 20 30 M 3 2 3 7 20 M 4 23 2 25 60 Dead 80 70 0 60 We wat to axze the proft by the curret algorth. Soluto We subtract the axu proft 25 fro other profts of Table 8. Mzg the Proble. M 5 0 3 50 M 2 9 6 5 30 M 3 4 2 8 20 p j x j M 4 2 23 0 60 80 70 0 60 The row dffereces ad colu dffereces fro the lowest row ad the lowest colu Table 9. Row-Colu Dfferece Table. M 5 5 3 0 0 0 33 3 50 M 2 9 4 7 6 6 50 5 30 M 3 4 0 2 2 8 2 8 4 8 20 M 4 20 2 223 3 00 0 60 80 70 0 60 The TOT s: Table 0. Total Opportuty Table. M 8 0 6 50 M 2 3 7 5 30 M 3 2 0 32 20 M 4 2 36 0 60 80 70 0 60 The allocatos wth the help of RATOVs ad CATOVs Table. Allocatos TOT. M 8 50 0 6 50 M 2 3 20 7 0 5 30 M 3 20 2 0 32 20 M 4 60 2 36 0 60 ( b 80 70 0 60 j CATOV RATOV 3.2 5.7 3.2 3.2 5.7-7.3 5.3-0 5 -.3 9 9 9 7.6 24 - - 4.6 6 6 6 2.6 9 9 - The allocatos the orgal TT Table 2. Allocatos the Orgal Proble. M 0 50 5 2 50 M 2 6 20 9 0 20 30 M 3 20 2 3 7 20 M 4 60 23 2 25 60 80 70 0 60 The axu proft s = 2930 = = 5 50 + 9 20 + 20 0 + 2 20 + 23 60 Exaple 03 Four products P, P 2, P 3, ad P 4 are produced three aches M, M 2, ad M 3 ad ther proft args are gve the followg table. p j x j
4 Abul Kala Azad ad Mosharraf Hossa: A Iovatve Algorthc Approach for Solvg Proft Maxzato Probles Table 3. Tabular For of the Proble. P 4 ( M 4 5 6 7 0 M 2 7 5 6 8 3 M 3 8 6 5 5 7 Dead 9 3 8 0 40 Capacty a We wat to axze the proft by the curret algorth. Soluto We subtract the axu proft 8 fro other profts of Table 4. Mzg the Proble. P 4 M 4 3 2 0 M 2 3 2 0 3 M 3 0 2 3 3 7 9 3 8 0 40 The row dffereces ad colu dffereces fro the lowest row ad the lowest colu Table 5. Row-Colu Dfferece Table. P 4 M 4 3 4 3 2 2 0 0 0 M 2 3 3 2 2 0 00 0 3 M 3 0 0 0 22 0 3 3 3 3 3 7 9 3 8 0 40 The TOT s: Table 6. Total Opportuty Table. P 4 M 7 3 0 M 2 2 4 2 0 3 M 3 0 2 4 6 7 ( b j 9 3 8 0 40 The allocatos wth the help of RATOVs ad CATOVs Table 7. Allocatos TOT. P 4 M 7 5 3 5 0 M 2 2 4 3 2 0 0 3 M 3 9 0 8 2 4 6 7 ( b j 9 3 8 0 40 3 3 2.3 2.3 CATOV - 3 2.3 2.3-3.5.5 0.5 - -.5 0.5 RATOV 3.6.6 2 2 2 3 4 - - The allocatos the orgal TT Table 8. Allocatos the Orgal Proble. P 4 M 4 5 5 5 6 7 0 M 2 7 5 3 6 0 8 3 M 3 9 8 8 6 5 5 7 9 3 8 0 40 The axu proft s = 273 = p x = 5 5 + 6 5 + 6 3 + 8 0 + 8 9 + 6 8 4. Coparso of Results The curret algorth etoed the artcle gves optal or ear optal proft. However, a coparso of the developed work wth the three exstg covetoal ethods s preseted case of the three above exaples. Table 9. Assesset of the Developed Algorth. Methods Solutos Exaple Exaple 2 Exaple 3 Curret Method 87500 2930 273 North-West Corer Method 87000 290 92 Matrx Ma Method 87000 280 273 VAM 85500 2930 273 Optal Soluto 87500 2930 273 5. Cocluso The developed ethod cosders all the opportutes of the cell values of the TOT by takg averages of the cell values. O the other had, soe other ethods take soe of the cell values oly (e. the lowest ad the ext lowest, the hghest ad the lowest etc.. The results or outcoes of the preset algorth are optal or ear optal solutos whle several exaples were tested. Refereces [] P. Pada ad G. Nataraja, A New Approach for Solvg Trasportato Probles wth Mxed Costrats, Joural of Physcal Sceces, Vol. 4, 200, 53-6, 200. [2] N. M. Deshukh, A Iovatve Method for Solvg Trasportato Proble, Iteratoal Joural of Physcs ad Matheatcal Sceces ISSN: 2277-2 (Ole, 202. [3] N. Balakrsha, Modfed Vogel s Approxato Method for Ubalace Trasportato Proble, Appled Matheatcs Letters 3(2, 9,,990. [4] Serdar Korukoglu ad Serka Ball, A Iproved Vogel s Approxato Method for the Trasportato Proble, Assocato for Scetfc Research, Matheatcal ad Coputatoal Applcato Vol. 6 No. 2, 370-38, 20. j j
Matheatcs Letters 208; 4(: -5 5 [5] H. H. Shore, The Trasportato Proble ad the Vogel s Approxato Method, Decso Scece (3-4, 44-457, 970. [6] D. G. Shshak, J. A. Kaslk ad T. D. Barelay, A odfcato of Vogel s Approxato Method through the use of Heurstcs, Ifor 9,259-263, 98. [7] Aur Raha Kha, A Re-soluto of the Trasportato Proble: A Algorthc Approach Jahagragar Uversty Joural of Scece, Vol. 34, No. 2, 49-62, 20. [8] V. J. Sudhakar, N. Arusakar, T. Karpaga, A ew approach for fd a Optal Soluto for Trasportato Probles, Europea Joural of Scetfc Research 68 254-257, 202. [9] O. Krca ad A. Satr, A Heurstc for Obtag a Ital Soluto for the Trasportato Proble, Joural of Operatoal Research Socety, Vol. 4, No. 9, pp. 865-87, 990. [0] Md. Arul Isla et al., Proft Maxzato of a Maufacturg Copay: A Algorthc Approach, J. J. Math. ad Math. Sc., Vol. 28, 29-37, 203.