An Approach for Solving Multi-Objective Linear Fractional Programming Problem and It s Comparison with Other Techniques

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1 International Journal of Scientific and Innovative Mathematical Reearch (IJSIMR) Volume 5, Iue 11, 2017, PP 1-5 ISSN X (Print) & ISSN (Online) DOI: wwwarcjournalorg An Approach for Solving Multi-Objective Linear Fractional Programming Problem and It Comparion with Other Technique Huma Akhtar 1*, Geeta Modi 2 *1 Reearch Scholar Govt MVM Bhopal, India, 2 Profeor and Head, Department of Mathematic, Govt MVM Bhopal, India, *Correponding Author: Huma Akhtar, Reearch Scholar Govt MVM Bhopal, India, Abtract: Thi paper preent Harmonic Average and Advanced Harmonic Average technique to olve multiobjective linear fractional programming problem (MOLFPP) to ingle objective linear fractional programming problem (SOLFPP) and uggeted an algorithm for it olution The propoed method can be illutrated with the help of numerical example The numerical reult in thi paper indicate that Advanced Harmonic Average technique i better than other technique (uch a Chandra Sen, Mean, Median, Geometric Mean, New Geometric Average and Harmonic average) Keyword: Linear Fractional Programming Problem (LFPP), Harmonic Average and Advanced Harmonic Average Technique 1 INTRODUCTION Linear Fractional Programming deal with that cla of mathematical programming problem in which the relation among the variable are linear; the contraint relation mut be in linear form and the objective function to be optimized mut be a ratio of two linear function uch a profit/cot, actual cot/ tandard cot, output/employee, etc and it i applied to different dicipline uch a production planning, financial and corporative planning, health care and hopital planning A tudy of multi-objective linear programming problem (MOLPP) i introduced in [2] which ugget an approach to et up multi-objective function under the limitation o that the optimum value of individual problem wa greater than zero Sulaiman and Sadiq tudied the Multi-objective function by uing mean and median technique [4] Alo Sulaiman and Salih tudied the multi-objective fractional programming problem by uing mean and median technique [5] Nahar Samun et al uggeted a new geometric average technique to optimize the objective function where a ingle objective function i developed from multi-objective function [1] In 2016 Sulaiman et al uggeted a new technique by uing Harmonic mean of the value of objective function for olving Multi-objective linear programming problem [3] In order to extend thi work, we have defined MOLFPP and ugget an algorithm to olve linear factional programming problem for multi-objective function by uing Harmonic Average and Advanced Harmonic Average technique The reult i compared with different technique uch a Chandra Sen, Mean & Median, Arithmetic Average Geometric Average and New Geometric Average The Advanced Harmonic Average technique give better reult than all thoe technique 2 MATHEMATICAL FORM OF LFPP The mathematical form of LFP problem i given a follow: c Max Z = T X+α d T X+β AX b = X 0 International Journal of Scientific and Innovative Mathematical Reearch (IJSIMR) Page 1

2 Where X, c and d are n 1 vector, b i an m 1 vector, c T, d T denote tranpoe of vector, A i an m n matrix and, are calar 3 MULTI-OBJECTIVE LINEAR FRACTIONAL PROGRAMMING PROBLEM Multi-Objective function that are the ratio of two linear objective function are aid to be MOLFPP which can be defined a: Max z 1 = c 1 T X+α 1 d 1 T X+β1 Max z 2 = c 2 T X+α 2 d 2 T X+β2 Max z r = c r T X+α r d r T X+βr Min z r+1 = c r+1 T X+α r+1 d r+1 T X+βr+1 (31) Min z = c T X+α d T X+β AX = b (32) X 0 (33) Where b i an m-dimenional vector of contant, X i an n-dimenional column vector of deciion variable, r i number of objective function to be maximized, i the number of objective function to be maximized and minimized and (-r) i the number of objective function that i minimized A i an m n matrix of contant, all vector are aumed to be column vector unle tranpoed(t) c i, d i (where i = 1,2,,) are n-dimenional vector of contant, α i, β i (where i = 1,2,,) are calar 4 SOLVING MOLFPP BY USING THE FOLLOWING TECHNIQUES 41 Harmonic Average Technique:- Step1: Solve each objective function by uing implex technique Step2: Check the feaibility of the olution obtained in tep1, if it i feaible then go to tep3, otherwie ue dual implex technique to remove infeaibility Step3: Aign a name to the optimum value of each objective function Max z i ay φ i, i = 1,2, r and Min z i ay φ i, i = r+1, r+2,, Step4: Calculate Harmonic Average Hav 1 = Hav φ i, i =1,2, r and Hav 2 = Hav φ i, i = r+1, r+2,, Step5: Optimize the combined objective function under the ame contraint (32) & (33) a follow: Max Z = r Max z i i=1 - Hav 1 Min z i i=r+1 (411) Hav 2 42 Advanced Harmonic Average (AH av )Technique:- Step1, Step2, Step3 are the ame a given in algorithm (41) Step4: Select m 1 = min φ i, i = 1, 2,,r and m 2 = max φ i, i= r+1,, then calculate International Journal of Scientific and Innovative Mathematical Reearch (IJSIMR) Page 2

3 AH av = 2 m 1 m 2 m 1 + m 2 Step5: Optimize the combined objective function under the ame contraint (32) & (33) a: r i=r+1 ) Max Z = ( i=1 Max z i Min z i (421) AH av 5 NUMERICAL EXAMPLE 51 Example Max z 1 = 3x 1 2x 2 Max z 2 = 9x 1+3x 2 Max z 3 = 3x 1 5x 2 Min z 4 = 6x 1+2x 2 Min z 5 = 3x 1 x 2 Solution: After finding the value of each of individual objective function, the reult are given below: Table 1 i φ i x i Hav 1 Hav 2 AH av 1 3/2 (1,0) 27/20 2 9/2 (1,0) 1 3 3/4 (1,0) 4-3/2 (1,0) 3/2 5-3/2 (1,0) i) Harmonic Average Technique:- Max Z = r Max z i i=1 - Hav 1 Min z i i=r+1 Hav 2 Max Z = 126x 1 10x 2 9x 1 +9x 2 +9 Hence the optimal olution i: Max Z = 7, x 1 = 1, x 2 = 0 ii) Advanced Harmonic Average (AH av ) Technique:- r i=r+1 ) Max Z = ( i=1 Max z i Min z i where AH AH av = 2 m 1 m 2 av Max Z = 39x 1 3x 2 m 1 + m 2 International Journal of Scientific and Innovative Mathematical Reearch (IJSIMR) Page 3

4 Hence the optimal olution i: Max Z = 975, x 1 = 1, x 2 = 0 6 COMPARISON OF THE NUMERICAL RESULTS Comparion of the numerical reult which are obtained from the example 51 i hown in the following table2: Table 2 Technique Example 61 Chandra Sen Technique 5 Mean Technique 5 Median Technique 65 Arithmetic Mean 5 New Arithmetic Average 8665 Geometric Mean 5931 New Geometric Average Harmonic Average 7 Advanced Harmonic Average 975 In the above table, it i clear that the reult obtained in example 51 when uing advanced harmonic average technique i better than other reult 7 CONCLUSION In thi paper, we have defined Harmonic Average and Advanced Harmonic Average technique and then compare Advanced Harmonic Average technique with other technique namely Chandra Sen, Mean & Median, Arithmetic Mean & New Arithmetic Average, Geometric Mean & Advanced Geometric Average and Harmonic Average technique The comparion of thee technique are baed on the value of the objective function After olving the numerical example, we found that MaxZ which obtained by our technique(advanced Harmonic average technique) i better than other technique(chandra Sen, mean & median, arithmetic mean & new arithmetic average, geometric mean & new geometric average and harmonic average technique) REFERENCES [1] Nahar Samun and Alim Md Abdul, A new geometric average technique to olve multi-objective linear fractional programming problem and comparion with new arithmetic average technique, IOSR Journal of Mathematic, Vol 13, No 3, PP 39-52, 2017 [2] Sen Chandra, A new approach for multi-objective rural development planning, The Indian Economic Journal, Vol 30, No 4, PP 91-96, 1983 [3] Sulaiman, NA and Mutafa RB, Uing harmonic mean to olve multi-objective linear programming problem, American Journal of Operation Reearch, Vol 6, No 1, PP 25-30, 2016 [4] Sulaiman NA and Sadiq GW, Solving the linear multi-objective programming problem; uing mean and median value, Al-Rafiden Journal of Computer Science and Mathematical, Univerity of Moul, Vol 3, No 1, PP 69-83, 2006 [5] Sulaiman, NA & alih A D, Uing mean and median value to olve linear fractional multi-objective programming problem, Zanco Journal for Pure and Applied Science, Salahaddin-Erbil Univerity, Vol 22, No5, 2010 International Journal of Scientific and Innovative Mathematical Reearch (IJSIMR) Page 4

AUTHOR S BIOGRAPHY Mi Huma Akhtar i a reearch cholar in Department of Mathematic, Govt MVM Bhopal, Madhya Pradeh, India under the Maulana Azad National Fellowhip (MANF) for Minority Community She ha

$the National Seminar Her area of reearch i optimization of fractional function and their application Dr Geeta Modi i working a a Profeor & Head of the Department Mathematic, Govt MVM Bhopal, Madhya$

5 AUTHOR S BIOGRAPHY Mi Huma Akhtar i a reearch cholar in Department of Mathematic, Govt MVM Bhopal, Madhya Pradeh, India under the Maulana Azad National Fellowhip (MANF) for Minority Community She ha 28 year teaching experience before awarding MANF for Minority Community She ha been Qualified MP SET-2017 and he ha publihed 02 reearch paper in International Journal and 01 reearch paper preent in the National Seminar Her area of reearch i optimization of fractional function and their application Dr Geeta Modi i working a a Profeor & Head of the Department Mathematic, Govt MVM Bhopal, Madhya Pradeh, India She ha 34 year of teaching Experience Preently he i the chairman of central board of tudie (Mathematic) Govt of Madhya Pradeh, India She i the VC nominee member of board of tudie IEHE Bhopal Madhya Pradeh India, and he i Ex-Chairman of board of tudie (Mathematic) Barkatullah Univerity Bhopal She ha publihed more than 50 article in National and International Journal 13 reearch cholar awarded Ph D under her uperviion and 06 reearch cholar regitered now Citation: H Akhtar, G Modi, " An Approach for Solving Multi-Objective Linear Fractional Programming Problem and It Comparion ", International Journal of Scientific and Innovative Mathematical Reearch, vol 5, no 11, p 1-5, 2017, Copyright: 2017 Author Thi i an open-acce article ditributed under the term of the Creative Common Attribution Licene, which permit unretricted ue, ditribution, and reproduction in any medium, provided the original author and ource are credited International Journal of Scientific and Innovative Mathematical Reearch (IJSIMR) Page 5

A Proposed New Average Method for Solving Multi-Objective Linear Programming Problem Using Various Kinds of Mean Techniques

Mathematics Letters 2018; 4(2): 25-33 http://www.sciencepublishinggroup.com/j/ml doi: 10.11648/j.ml.20180402.11 ISSN: 2575-503X (Print); ISSN: 2575-5056 (Online) A Proposed New Average Method for Solving