Logical Development Of Vogel s Approximation Method (LD-VAM): An Approach To Find Basic Feasible Solution Of Transportation Problem

Transcription

1 INTERNATIONAL JOURNAL OF SCIENTIFIC & TECHNOLOGY RESEARCH VOLUME, ISSUE, FEBRUARY ISSN Logcal Development Of Vogel s Approxmaton Method (LD- An Approach To Fnd Basc Feasble Soluton Of Transportaton Problem Utpal Kant Das, Md. Ashraful Babu, Amnur Rahman Khan, Md. Abu Helal, Dr. Md. Sharf Uddn Abstract: The study n ths paper s to dscuss the lmtaton of Vogel s Approxmaton Method (VAM) and developed an mproved algorthm after resolvng ths lmtaton for solvng transportaton problem. Vogel s Approxmaton Method (VAM) s the more effcent algorthm to solve the transportaton problem but t has some lmtatons when hghest penalty cost appear n two or more row or column. For that case VAM does not gve any logcal soluton. In ths paper we stand a logcal approach for ths problem and developed an algorthm named by Logcal Development of Vogel s Approxmaton Method (LD-VAM) where feasble soluton from ths method are very close to optmal soluton more than VAM. Index Terms: Transportaton Problem (TP), LD-VAM, VAM, LPP,, Transportaton Cost. INTRODUCTION The Transportaton Problem s the specal class of Lnear Programmng Problem n the feld of Appled Mathematcs and also n Operaton Research. There are some exstng algorthms to solve the Transportaton Problem such as North West Corner Rule (NWC), Least Cost Method (LCM), and Vogel s Approxmaton Method (VAM) etc. where Vogel s Approxmaton Method (VAM) s known as more effcent algorthm to fnd the basc feasble soluton. But VAM has some lmtatons whch are dscussed n (.) and we fxed ths problem and developed a new algorthm n (.). To solvng Transportaton Problem frst a fall we have to formulate orgnal problem as a Lnear Programmng Problem (LPP) for m sources and n destnatons. Department of Computer Scence & Engneerng, IUBAT-Internatonal Unversty of Busness Agrculture and Technology, Uttara, Dhaka-, Bangladesh, Phone: , emal: [email protected] Department of Mathematcs, IUBAT-Internatonal Unversty of Busness Agrculture and Technology, Uttara, Dhaka-, Bangladesh, Ph: , Emal: [email protected]. Department of Mathematcs, Jahangrnagar Unversty, Savar, Dhaka-, Bangladesh, Phone: , emal: [email protected] Department of Mathematcs, IUBAT-Internatonal Unversty of Busness Agrculture and Technology, Uttara, Dhaka-, Bangladesh, Phone: , emal: [email protected] Department of Mathematcs, Jahangrnagar Unversty, Savar, Dhaka-, Bangladesh, Phone: , emal: [email protected] Each th source has a capacty of supply amount that s s and d j be the demand amount of each j th destnaton and c be the cost of th source to j th destnaton. The ultmate goal of solvng Transportaton Problem s to fnd the amount of product x whch wll be transfer from th source to j th destnaton so that total transportaton cost wll be mnmzed. The Lnear Programmng Problem (LPP) representng the Transportaton problem for m sources and n destnatons are generally gven as: Mnmze: Subject to: n j m x Z m n j c x x s, for,,... m x d, for j,,... n j, for all, j To applyng transportaton algorthm for solvng TP we have to make a Mathematcal model for ths LPP. METHODOLOGY: Intally we menton that there are some exstng methods for solvng transportaton problem such as North West Corner Rule (NWC), Least Cost Method (LCM), and Vogel s Approxmaton Method (VAM) etc. In ths secton we dscuss about Vogel s Approxmaton Method (VAM) and our proposed method Logcal Development of Vogel s Approxmaton Method (LD-VAM).. EXISTING ALGORITHM FOR VOGEL S APPROXIMATION METHOD ( The Vogel s Approxmaton Method (VAM) s an teratve procedure for computng a basc feasble soluton of a transportaton problem. Ths method s better than other two methods.e. North West Corner Rule (NWC) and Least cost Method (LCM), because the basc feasble soluton obtaned IJSTR

2 INTERNATIONAL JOURNAL OF SCIENTIFIC & TECHNOLOGY RESEARCH VOLUME, ISSUE, FEBRUARY ISSN by ths method s nearer to the optmal soluton. The exstng algorthm for Vogel s Approxmaton Method (VAM) s gven Step-: Indentfy the boxes havng mnmum and next to mnmum transportaton cost n each row and wrte the dfference () along the sde of the table aganst the correspondng row. Step-: Indentfy the boxes havng mnmum and next to mnmum transportaton cost n each column and wrte the dfference () along the sde of the table aganst the correspondng column. If mnmum cost appear n two or more tmes n a row or column then select these same cost as a mnmum and next to mnmum cost and penalty wll be zero. Step-: a. Indentfy the row and column wth the largest penalty, breakng tes arbtrarly. Allocate as much as possble to the varable wth the least cost n the selected row or column. Adjust the supply and demand and cross out the satsfed row or column. If a row and column are satsfes smultaneously, only one of them s crossed out and remanng row or column s assgned a zero supply or demand. b. If two or more penalty costs have same largest magntude, then select any one of them (or select most top row or extreme left column). Step-: a. If exactly one row or one column wth zero supply or demand remans uncrossed out, Stop. b. If only one row or column wth postve supply or demand remans uncrossed out, determne the basc varables n the row or column by the Least-Cost Method. c. If all uncrossed out rows or column have (remanng) zero supply or demand, determned the zero basc varables by the Least-Cost Method. Stop. d. Otherwse, go to Step-.. FINDING LIMITATIONS OF VOGEL S APPROXIMATION METHOD ( In VAM approach, Penaltes are determned by the dfference of two mnmum costs of each row and each column. The hghest penalty ndcates that among the two mnmum cost, one s the lowest and another s too hgh. In VAM algorthm select such row or column whch contans largest penalty to ensure the least cost n current teraton and avodng the chance of takng larger cost n next teraton. If hghest penalty cost appear n two or more row or column, the VAM algorthm selects any one of them (or select most top row or extreme left column) [(.) Step-(b)]. But t s not necessarly true that largest penalty always ensure the lowest cost because dfference of two pars of numbers can be equal when one of the par s smaller than another par. As an example, dfference of and and dfference of 7 and are equal but lowest numbers exst n second par. For that reason we can say that n VAM algorthm, f the lower par does not appear n the top most or extreme left poston then lowest cost wll not be selected n the current teraton so that total transportaton cost may not be mnmzed. For that case VAM may not be gven the lowest feasble soluton.. PROPOSED ALGORITHM FOR LOGICAL DEVELOPMENT OF VOGEL S APPROXIMATION METHOD (LD- From the above dscusson of the lmtatons of Vogel s Approxmaton Method (VAM) we fxed ths problem and proposed an mproved algorthm whch s named by Logcal Development of Vogel s Approxmaton Method (LD-VAM). In ths algorthm, we resolved the above dscussed problem of VAM.e, when largest penalty appear n two or more rows or columns then select that row or column among them whch contans the least cost and gve the maxmum possble allocaton. The algorthm of LD-VAM s gven Set s be supply amount of the th source and d j be the amount of demand of j th destnaton and c be the unt transportaton cost of th source to j th destnaton. Step-: Check: f s and d j then Stop. then f s d or f s d Step-: j j j j balance the transportaton problem addng dummy demand or supply. Step-: a. Identfy the smallest and next to smallest cost of each row and column and calculate the dfference between them whch s called by penalty. Set be the row penalty and be the column penalty. p c - c and p c -c h k j hj kj b. If smallest cost appear n two or more tmes n a row or column then select these same cost as a smallest and next to smallest cost and penalty wll be zero. p c - c = and p c -c Step-: Select max( p, p ) h k j hj kj j. Select lowest cost of that row or column whch has largest penalty and allocate maxmum possble amount x.e, mn s, d j. If the lowest cost appears n two or more cells n that row or column then choose the extreme left or most top lowest cost cell. Step-: If te occurs n the largest penaltes n some rows or columns then select that row or column whch contans least cost among them. Step-6: Adjust the supply and demand and cross out the satsfed row or column. If row and column are satsfed smultaneously then crossed out one of them and set zero supply or demand n remanng row or column. Step-7: a. If exactly one row or one column wth zero supply or demand remans uncrossed out, Stop. IJSTR

3 INTERNATIONAL JOURNAL OF SCIENTIFIC & TECHNOLOGY RESEARCH VOLUME, ISSUE, FEBRUARY ISSN b. If only one row or column wth postve supply or demand remans uncrossed out, determne the basc varables n the row or column by the Least-Cost Method. c. If all uncrossed out rows or column have (remanng) zero supply or demand, determned the zero basc varables by the Least-Cost Method. Stop. d. Otherwse go to Step-. NUMERICAL ILLUSTRATION: Consder some specal types of Transportaton Problems where hghest penalty appear n two or more rows or columns and solve these usng Vogel s Approxmaton Method (VAM) and the proposed method Logcal Development of Vogel s Approxmaton Method (LD-VAM) and compare these results wth optmal soluton for measurng accuracy.. EXAMPLE-: Consder a Mathematcal Model of a Transportaton Problem n Table-.: D D D D D S Demand 6 9 Now we fnd the feasble soluton of ths problem usng LD- VAM and VAM respectvely n.. SOLUTION OF EXAMPLE- USING LOGICAL Iteraton-: Source Destnaton Row Supply D D D D D S Demand 6 9 In Iteraton-, the largest penalty appears n three tmes n S,, rows, but lowest cost appear n (, D) cell. As per LD-VAM algorthm allocate amount commodty n (, D) cell. Iteraton-: Row D D D D D S Demand 6 9 Iteraton-: Row D D D D D S Demand 6 Iteraton-: Row D D D D D S Demand 6 Iteraton-: Row D D D D D S Demand 6 6 IJSTR

4 INTERNATIONAL JOURNAL OF SCIENTIFIC & TECHNOLOGY RESEARCH VOLUME, ISSUE, FEBRUARY ISSN Iteraton-6: Row D D D D D S Demand 6 Iteraton-7: 6 9 Row D D D D D S Demand In Iteraton-6, only one row has remans wth postve supply and demand amount then as per LD-VAM algorthm allocates these by Least Cost Method. The fnal feasble soluton table s gven Table-.: D D D D D S Demand 6 9 Total Transportaton Cost (Usng LD- ( 6)+(9 )+( 8)+( )+(7 )+ (6 )+(8 )+( )=7.. SOLUTION OF EXAMPLE- USING VOGEL S APPROXIMATION METHOD ( Sou rce S Dem and Colu mn Pen alty D 9 Destnaton D D D Table-.: D Su ppl y Total Transportaton Cost (Usng Row ( ) () (8) (9) (7) (6) () (8) Observaton: We observed that VAM provdes feasble soluton for Example- s and LD-VAM provdes 7 whch s lower than VAM.. Example-: Consder a Mathematcal Model of a Transportaton Problem n Table-.: D D D D S Demand Now we fnd the feasble soluton of ths problem usng LD- VAM and VAM respectvely n below. IJSTR

5 INTERNATIONAL JOURNAL OF SCIENTIFIC & TECHNOLOGY RESEARCH VOLUME, ISSUE, FEBRUARY ISSN SOLUTION OF EXAMPLE- USING LOGICAL Table-.: soluton for Example- s 7 and LD-VAM provdes whch s lower than VAM.. EXAMPLE-: Consder a Mathematcal Model of a Transportaton Problem n Table-.: Sourc e S Destnaton D D D D Suppl y 6 7 D D D D D D6 S S S Demand SOLUTION OF EXAMPLE- USING LOGICAL Table-.: Dema nd Total Transportaton Cost (Usng LD- (7) () (9 ) ( ) () () ().. SOLUTION OF EXAMPLE- USING VOGEL S APPROXIMATION METHOD ( Table-.: D D D D D D6 S S S Demand Sour ce S Dem and Destnaton D D D D Total Transportaton Cost (Usng () () () () ( ) () (7) 7 Suppl y Observaton: We observed that VAM provdes feasble IJSTR Total transportaton Cost: () (6) (9 ) ( ) (7) () () (6 ) () () ( 6). EXAMPLE-: Consder a Mathematcal Model of a Transportaton Problem n Table-.: D D D D S Demand 8 6

6 INTERNATIONAL JOURNAL OF SCIENTIFIC & TECHNOLOGY RESEARCH VOLUME, ISSUE, FEBRUARY ISSN SOLUTION OF EXAMPLE- USING LOGICAL Table-.: D D D D S Demand 8 Total transportaton Cost: EXAMPLE-: Consder a Mathematcal Model of a Transportaton Problem n Table-.: Source Destnatons D D D D D D6 D7 Supply S S Demand SOLUTION OF EXAMPLE- USING LOGICAL 9 Total transportaton Cost: RESULT ANALYSIS: In above two examples and some other examples, we observed that feasble solutons usng Logcal Development of Vogel s Approxmaton Method (LD-VAM) are lower than Vogel s Approxmaton Method (VAM) where some of these are exactly equal to the optmal soluton and some are very close to optmal soluton. The comparson table of these solutons s gven Transportat on Problem Problem Sze Table-6: Methods LD-VAM VAM Optmal Soluton Example- 7 7 Example- 7 Example Example- 6 6 Example CONCLUSION: In ths paper we fnd a lmtaton of Vogel s Approxmaton Method (VAM) and developed an mproved algorthm by resolvng ths lmtaton named Logcal Development of Vogel s Approxmaton Method (LD-VAM) for solvng Transportaton Problem. From the above examples and other transportaton problems, LD-VAM provdes the lower feasble soluton than VAM whch are very close to optmal soluton and sometmes equal to optmal soluton. REFERENCES [] Hamdy A. Taha. Operaton Research: An Introducton, Eghth Edton, ISBN-: , [] P. Rama Murthy. Operaton Research, Second Edton, ISBN (): [] Hamdy A Taha. TORA Optmzng System Software. Table-.: Source Destnatons Supply D D D D D D6 D7 S S Demand IJSTR [] Hller, F. S. and G. J. Leberman. 99. Introducton to Operatons Research, 6th ed. New York: McGraw- Hll, Inc. [] A. Edward Samuel and M. Venkatachalapathy. Modfed Vogel s Approxmaton Method for Fuzzy Transportaton Problems, Appled Mathematcal Scences, Vol.,, no. 8, [6] Nagraj Balakrshnan,Tulane Unversty. Modfed Vogel s Approxmaton Method for the Unbalanced Transportaton Problem. Appl. Math. Lett. Vol., No., pp. S, 99 [7] M.A. Hakm, An Alternatve Method to fnd Intal Basc Feasble Soluton of a Transportaton problem. Annals 7

7 INTERNATIONAL JOURNAL OF SCIENTIFIC & TECHNOLOGY RESEARCH VOLUME, ISSUE, FEBRUARY ISSN of Pure and Appled Mathematcs, Vol., No.,, pp. -9, ISSN: 79-87X (P), (onlne) [8] Ramakrshnan, G.S., An mprovement to Goyal's modfed VAM for the unbalanced transportaton problem, Journal of Operatonal Research Socety, 9(6) (988) 69-6 [9] Md. Amrul Islam, Amnur Rahman Khan, M. Sharf Uddn and M. Abdul Malek. Determnaton of Basc Feasble Soluton of Transportaton Problem: A New Approach. Jahangrnagar Unversty Journal of Scence, Vol., No., pp. 8, ISSN - 89 (). [] Korukoğlu, S. and S. Ball. An Improved Vogel s Approxmaton Method for the transportaton problem. Mathematcal and Computatonal Applcatons, Vol. 6, No., pp. 7-8, [] Amnur Rahman Khan. A Re-soluton of the Transportaton Problem: An Algorthmc Approach. Jahangrnagar Unversty Journal of Scence, Vol., No., pp. 9-6 ISSN -89 (). [] Mathrajan, M. and B. Meenaksh, Expermental Analyss of some Varants of Vogel s Approxmaton Method, Asa-Pacfc Journal of Operatonal Research () () 7-6. [] Shmshak, D.G., J.A. Kaslk and T.D. Barclay, A modfcaton of Vogel's approxmaton method through the use of heurstcs, INEOR, 9 (98) 9-6. [] Renfeld, N.V. and W.R. Vogel, Mathematcal Programmng, Prentce-Hall, Englewood Glffs, New Jersey (98). IJSTR 8