International Journal of Mathematics Trends and Technology (IJMTT) Volume 52 Number 9 December 2017

Transcription

1 Iteratioal Joural of Mathematics Treds ad Techology (IJMTT) Volume 5 Number 9 December 7 Optimal Solutio of a Degeerate Trasportatio Problem Reea.G.patel, Dr.P.H.Bhathawala Assistat professor, Departmet of applied sciece ad Humaities,S.P.C.E,Visagar,North Gujarat. Professor ad Head (Retired) Veer Narmad South Gujarat Uiversity, Surat ABSTRACT: The Trasportatio Problem is criticaltool for real life problem. Mathematically it is a applicatio of Liear Programmig problem. At the poit whe the aalysts are doig some work o Trasportatio problem has a typical iquiry that, how we ca way to deal with the optimality of Trasportatio problem. Optimality gives us the optimal route that prompts the either most etreme beefit or least aggregate cost whichever is required. Sice last umerous years, there was so much research has bee improved the situatio for No-Degeerate Trasportatio problem, however here we are acquaitig the ew approach to get the optimality whe the Trasportatio problem facig the degeeracy.so, here i this paper, the algorithm tries to clarify the optimal solutio of Degeerate Trasportatio Problem, or close to the optimal solutio. KEY WORDS: IBFS, Degeeracy, optimality, L-Shape of Trasportatio problem, Degrees of freedom for optimality. INTRODUCTION: Trasportatio problem is eceptioally powerful crucial part of liear programmig problem which ca be coected for required sources of supply to correspodig destiatio of demad, with the ed goal that the aggregate trasportatio cost ought to be limited. The essetial phase of ay trasportatio problem as Iitial Basic Feasible Solutio is gotte by ay of the techiques as North West Corer strategy, Least Cost Method, Vogel's Approimatio techique, ad the remaiig ad most vital work is to be for optimality of the Trasportatio problem is Verified by MODI. The Trasportatio Problem was first settled by F.L. Hitchcock. At that poit after T. C. Koopmas worked agai o the hypothesis of F.L. Hitchcock i the followig paper-().these two research work are etremely useful i the developmet of trasportatio techiques. The liear programmig with fuzzy umbers ad its optimal solutio preseted by Bazarra, Jarvis ad Sherali i 99. O the other had Lai ad Huag i 99, accepted the circumstaces i which all the parameters are fuzzy umber. Additioally they have utilized the triagular possibility distributio o fuzzy umbers. Sice most recet couple of years prior, i, Swarup, Gupta ad Moha disclosed the strategy to reach up to the sesitivity aalysis or post optimality aalysis of the differet parameters i the liear programmig problems. I corporate divisio so may producers took after the Optimizatio basics as ofte as possible i the liear programmig problem for ay kid of real world problem. For this imperative reaso, it is eceptioally pivotal to build up the ew methodologies that ca prompt the model to "best fit" i to the real world as much as possible. Here we have built up aother way to deal with the optimal solutio or close to the optimal solutio. Additioally, the ew calculatio depicted here gives the best way for the optimality with stepwise methodology with umerical cases for the justificatio. MATHEMATICAL ASPECTS RELATED TO TRANSPORTATION PROBLEM: Balaced Trasportatio Problem: A Trasportatio Problem is said to be balaced trasportatio problem if total umber of supply is same as total umber of demad. Ubalaced Trasportatio Problem: A Trasportatio Problem is said to be ubalaced trasportatio problem if total umber of supply is ot same as total umber of demad. Objective Fuctio: It is a liear fuctio of decisio variables epressig the objective of the decisio maker. Feasible solutio: Ay solutio Xij is said to be a feasible solutio of a trasportatio problem if it satisfies the costraits. ISSN: Page 55

2 Iteratioal Joural of Mathematics Treds ad Techology (IJMTT) Volume 5 Number 9 December 7 Basic Feasible Solutio: Ay solutio Xij is said to be a feasible solutio of a trasportatio problem if it satisfies the costraits. The feasible solutio is said to be basic feasible solutio if the umber of oegative allocatios is equal to (m+-) while satisfyig all rim requiremets, i.e., it must satisfy requiremet ad availability costrait. There are three ways to get basic feasible solutio. Iitial basic feasible solutio: The iitial solutio obtaied by ay of the three methods must satisfy the followig coditios: ) The solutio must be feasible, i.e. it must satisfy all the supply ad demad costraits (also called rim coditios) ) The umber of positive allocatios must be equal to (m+-), where m is the umber of s ad is the umber of colums. No - Degeerate Basic Feasible Solutio:A basic feasible solutio is said to be o-degeerate if it has eactly (m+-) positive allocatios i the Trasportatio Problem. If the allocatios are less tha the required umber of (m+-) the it is called the Degeerate Basic Feasible Solutio. The this type of solutio is ot easy to modify because at this stage it is impossible to draw a closed loop for each occupied cell. Thus degeeracy eeds to be removed for the improvemet i the obtaied solutio. Thus the degeeracy occurs at two differet stages. ) At iitial Basic feasible solutio, i which the umber of occupied cells may be less tha (m+-). ) At ay stage while movig towards optimal solutio, i which two or more occupied cells may become uoccupied simultaeously. MATHEMATICAL BACKGROUND: Let us cosider the stadard balaced trasportatio problem with m sources A i (with supplies a i ), i I={,,...,m} ad destiatios B j (with demads b j ), j J={,,,...,}. If X ij = the umber of load uits movig from A i to B j, the feasible solutio () ad set of feasible solutios (X) is: X={/ j J X ij = a i, i I; i I X ij = b j, j J; X ij i, j ; a i = b j }. m Mathematically the problem ca be stated as miimize z = i= j = C ij ij subject to j = ij = a i ; for i =,,.., m (supply costraits) Ad i= ij = b j For j=,... (demad costraits) X ij for all i & j. A trasportatio problem is said to be balaced if the total supply from all sources equals to the total m demads i all destiatios i.e. i= a i = j = b j, otherwise it is called the ubalaced trasportatio problem. Trasportatio Problem: Origis (i) Destiatios (j) Supply (a i )... X X... X a C C C X X... X a C C C X C X C X C a M X m X m... X m a m C m C m C m Demad (b j ) b b... b a i = b j ISSN: Page 5

3 Iteratioal Joural of Mathematics Treds ad Techology (IJMTT) Volume 5 Number 9 December 7 First, let us covert the costraits of the trasportatio problem ito our stadard matri form for a liear programmig problem, A=B. X=,...,,...,,..., ] [, m B= a,..., a, b..., b ] [ m, If the Costraits are writte as,,... a... a, =... a, m m = m m = b,... m = b, () Now the equivalet reduced echelo form of the above system () is as follows,. a, b... m b... m b m m b a =, m m m m =a m () The Degree of Freedom for Optimality:- () Costruct the Trasportatio matri alog with supply ad demad equatios of order (m+) (m) correspodig to the system-(). () Fid its reduced echelo form of the matri which correspods to system-(). () Now check the pivot elemets ( wise) of this matri with correspodig allocatios of the s i the simple trasportatio matri of order (m). () Now the umber of allocatios must follow the relatio betwee the pivot elemets i the matri of order (m+) (m) ad the correspodig allocatios of s i simple Trasportatio matri of order (m) with the degree of freedom for optimality (m-) as per the simple trasportatio matri, such that (m-) umber of allocatios must be from (m+-) pivots from the matri of order (m+) (m). ALGORITHM: Step: Costruct the trasportatio matri from the give trasportatio problem. Step: Fid a IBFS usig ay oe of the method as NWCM, LCM, VAM. Step:If the degeeracy is foud,i.e. (umber of allocatios are less tha the (m+-). So for resolutio of degeeracy at iitial solutio, apply a egligible quatity close To zero to oe or more (if required) to uoccupied cells the it will coverted to m+- Occupied cells. This small quatity is deoted by( ).this small quatity will either affect the total Cost or the supply ad demad values. I a miimizatio trasportatio problem,allocate the ( ) to uoccupied cells which has miimum trasportatio cost, while i maimizatio problems allocate the ( ) to uoccupied cells which has maimum trasportatio cost. I some of the problems must be added i oe of the uoccupied cells that may help i the followig algorithm. () ISSN: Page 57

4 Iteratioal Joural of Mathematics Treds ad Techology (IJMTT) Volume 5 Number 9 December 7 Step: The, skip or omit the miimum -cost cells oly from uoccupied cells (o basic variables) from the trasportatio matri. Step: 5 Assig +θ to the et miimum cost cell from uoccupied cells ad start to make a loop with occupied cells if possible, otherwise move to the et to et cell from uoccupied cells. The fid θ = mi( θ)ad add that mi (-θ) value at +θ ad subtract that mi (-θ) value from ( θ). Step: Cotiue this process uless ad util the loop made cotais at least cells from L-shape of the matri havig (m+-) pivots of the system-. The, fid the cost of the matri. If we observe that this trasportatio cost is less tha the cost obtaied i Step: the apply the Test for Optimality. Otherwise, go to Step-. Step: 7Repeat Step-, ad 5 util The Test for Optimality is satisfied. The Test for Optimality: () All miimum cosecutive -cost cells are allocated i the simple trasportatio matri. () All miimum cosecutive -cost cells are allocated i the simple trasportatio matri with the degree of freedom for optimality (m-). () At least oe of miimum cosecutive -cost cells are allocated i the simple trasportatio matri with the degree of freedom for optimality (m-). Step: ow the total miimum cost is calculated as sum of the product of cost ad correspodig allocated m value of Supply/demad. I.e. total cost = i= j = C ij ij. NUMERICAL EXAMPLES: ) A maufacturer wats to ship loads of his product as show below. The matri gives the kilometres from sources of supply to the destiatios, D D D D D 5 Supply S 5 S 7 7 S 9 Demad 5 5 The shippig cost is Rs. per load per km. what shippig schedule be used i order to miimize the total trasportatio cost? Solutio: Here the give problem is ubalaced Trasportatio Problem. So the iitial basic feasible solutio is developed after makig the balace trasportatio problem by addig dummy is give by, D D D D D 5 Supply S 5 5 S 7 7 S 5 9 Dummy Demad 5 5 Here at this stage, we get the Degeeracy as the IBFS does ot have required umber of m+- =+5-= occupied cells, therefore the IBFS is degeerate. So for removig the degeeracy, apply the to uoccupied cell (S, D 5 ) which has the miimum trasportatio cost amog the uoccupied cells, as the give problem is for miimizatio problem. D D D D D 5 Supply S 5 5 S 7 7 S 5 9 ISSN: Page 5

5 Iteratioal Joural of Mathematics Treds ad Techology (IJMTT) Volume 5 Number 9 December 7 Dummy Demad 5 5 Now by our ew algorithm, we get the optimal solutio as give below, D D D D D 5 Supply S 5 S 7 7 S 9 Dummy Demad 5 5 Total Cost obtaied by ew method is as follows, Total Miimum Cost = ((*)+(*)+(*)+(*)+(*)+(*)+(*))* = Rs. 9 CONCLUSION:The fudametal poit of this paper is to reach up to the optimal solutio of a Trasportatio problem particularly whe it eperieces to the Degeeracy with well-ordered process. The above algorithm gives the optimal or close to the optimal solutio for a Degeerate Trasportatio Problem with less umber of steps with accuracy of makig the decisio of optimality. The future etet of this method is that the decisio maker put some coceivable augmetatios of less umber of steps cotrast with this algorithm. Cosequetly this method is most capable method to this preset real world issues. ACKNOWLEDGEMENTS: I might wat to offer my true thaks to Dr.Bhavi. S. Patel, Assistat Professor, Applied Sciece ad Humaities Departmet at Sakalchad Patel College of Egieerig, Visagar for his priceless directio, collaboratio, cosistet cosolatio, bolster, supportive remarks, sagacious proposals ad suggestios. I geuiely value his regarded directio ad cosolatio, his kowledge ad compay at the time of crisis would be remembered lifelog. REFERENCES: ) Bazaraa M.S., Jarvis J.J., Sherali H.D., 99. Liear Programmig ad Network Flows, Joh Weily, Secod Editio, New York. ) Lai Y.J., Hwag C.L., 99. Mathematical Programmig Methods ad Applicatios, Spriger, Berli. ) Swarup K., Gupta P.K., Moha M.,. Operatios Research, Sulta Chad ad Sos, New Delhi. ) Chares, Cooper (95). The Steppig-Stoe method for eplai liear programmig. Calculatio i trasportatio problems. Maagemet Sciece ()9-9 5) Datzig G.B (9).Liear Programmig ad Etesios, New Jersey: Priceto Uiversity Press. ) Hitchcock FL (9). The distributio of a product from several sources to Numerous Localities, Joural of Mathematical Physics ) Koopmas TC (97). Optimum Utilizatio of the Trasportatio system proceedig of the Iteratioal statistical coferece, Washigto D.C. ) A Chares, W.W. Cooper ad A. Hederso. A Itroductio to Liear Programmig, Wiley, New York, 95. 9) Taha. H. A., Operatio s Research Itroductio, Pretice Hall Of Idia (PVT), New Delhi,. ) Sharma J.K., Operatios Research Theory ad Applicatios, Macmilla Idia (LTD), New Delhi, 5. ) Sudhakar V.J., Aruasakar.N., Karpagam.T., A New Approach for fidig a optimal solutio for Trasportatio problems, Europea Joural of Scietific Research,ISSN 5-X Vol. No.(), pp 5-57 ) Hadley. G., Liear Programmig, Uiversity of Chicago, Addiso-Wesley Publishig Compay, Ic. Readig, Massachusetts. Palo Alto. Lodo ISSN: Page 59