VERTICAL NETWORK ADJUSTMENT USING FUZZY GOAL PROGRAMMING

VERTICAL NETWORK ADJUSTMENT USING FUZZY GOAL PROGRAMMING Selçuk ALP 1, Erol YAVUZ, Nhat ERSOY 3 1 Vocatonal School, Yıldız Techncal Unversty, Turkey, Faculty of Engneerng & Archtecture, Okan Unversty, Turkey 3 Faculty of Cvl Engneerng, Yıldız Techncal Unversty, Turkey Correspondng author s e-mal: selcuk.alp@gmal.com ABSTRACT In engneerng, especally n Surveyng Engneerng, vertcal network adjustment s made to fnd out the defnte values of the unknowns and the measurements. Generally Least Square Method LS) s used for vertcal network adjustment. By ths study, FGP method, proposed as an alternatve method to LS Method for vertcal network adjustment, s eplaned wth an eample, and results are compared. Results found by usng these two methods, are close to each other. It s shown that FGP Method can be used n vertcal network adjustment. Only the level dfferences between the ponts have been used. Appromate heght values of the ponts that have been used n Least Square Method have not been used n Fuzzy Goal Programmng Method. Keywords: Fuzzy Goal programmng; Least Squares; Vertcal Network Adjustment 1. INTRODUCTION The am of the network adjustment s to fnd out the optmum values of the unknowns and the measurements, performed much more than needed. Through the senstveness and relance of the surveys, certan values and ther functons are set. In other words, the statstcal analyss of mathematcal model whch s used s made. Durng the network adjustment, measurements are accepted as normally dstrbuted. To perform ths am Least Squares LS) method of Gauss s appled. Ths method s wdely used n Surveyng Engneerng. FGP method has been proposed nstead of LS n ths research for network adjustment. FGP s based on GP formed by Charnes and Cooper [8, 9]. GP s wdely spread out by Ijr [16], Lee [19] and Ignzo [18]. The prncpal concept for lnear GP s to the orgnal multple objectves nto specfc numerc goal for each objectve. The objectve functon s than formulated and a soluton s sought whch mnmzes the weghted sum of devatons from ther respectve goal [7]. Nowadays, GP s one of the most mportant methods of Mult-Crtera Optmzaton methods. The man dea n GP s to determne goals for each constrant [30]. The GP s mult objectve programmng model based on the dstance functon concept where the decson-maker looks for the soluton that mnmzes the absolute devaton between the achevement level of the objectve and ts aspraton level [5]. GP s a common tool used n decson makng, but provdng crsp goals can be a problem for a decson maker. Snce Zadeh [39] proposed the concept of fuzzy sets, Bellman and Zadeh [6] have developed a basc framework for decson makng n a fuzzy envronment. Thereafter, research followed [3, 33] n whch Narasmhan [5] and Hannan [14] etended the fuzzy set theory to the feld of goal programmng [35]. A fuzzy programmng approach for lnear programmng problems wth several objectves was formed by Zmmerman [41]. Narasmhan [5] recommend a comple method for dealng wth the GP problem wth fuzzy goal and mentoned an approach to deal wth fuzzy prorty n 1980 [10].. LITERATURE REVIEW There have been a number of works on FGP. Samples are: Transportaton problem wth multple objectve functon [1, 1, 34, 40], producton 1

plannng wth multple effcency crtera [7, 17], nput-output model for resource allocaton, portfolo management [7], operaton schedulng [13, 8] and equpment-purchasng problem [10] are only eamples of these applcatons. There have been a number of works on LS. Samples are: robust source localzaton [36], estmatng regonal deformaton [11], smoothng and dfferentaton [9] and global postonng []. The vertcal network adjustment has been performed by GP n research paper by Alp et all [4]. To see the applcaton n Surveyng Engneerng by the applcaton of FGP smlar results, whch are perceved n LS method, are observed. Ths shows that FGP may be used as an alternatve method n Surveyng Engneerng. FGP s epected to gve more effectve results than GP, because the amount of devaton s used n the adjustment, and the results are wthn the lmts of the amount of ths devaton. 3. LEAST SQUARE 3.1. Generalzed Least Square Generalzed least squares GLS) s a method for estmatng the unknown parameters n a lnear regresson model. The GLS s appled when the varances of the observatons are unequal, or when there s a certan degree of correlaton between the observatons. A set of N pars of observatons {Y, X } s used to fnd a functon relatng the value of the dependent varable Y) to the values of an ndependent varable X) In the standard formulaton, Wth one varable and a lnear functon, the predcton s gven by the followng equaton: Y a b Ths equaton nvolves two free parameters whch specfy the ntercept a) and the slope b) of the regresson lne. The least square method defnes the estmate of these parameters as the values whch mnmze the sum of the squares between the measurements and the model. Ths amounts to mnmzng the epresson: Y Y Y a bx ) 1) The estmaton of the parameters s gotten usng basc results from calculus and, specfcally, uses the property that a quadratc epresson reaches ts mnmum value when ts dervatves vansh. Takng the dervatve of wth respect to a and b and settng them to zero gves the followng set of equatons. Na bx Y 0 3) a and b b X a X Y X 0 4) The followng least square estmates of a and b are obtaned by solvng the normal equatons: a M bm Y X M Y : Denotng the means of Y M X : Denotng the means of X and b 5) Y MY X M X 6) X MX LS can be etended to more than one ndependent varable and to non-lnear functons [1]. 3.. Network Adjustment Accordng to Least Square Mathematcal model n network adjustment s formed of two consttuents. One of them s called functonal model and the other s called stochastc model [31]. These consttuents form the base of adjustment account. These models are formed before the adjustment. Geometrc heght dfference between the ponts A and B, B hab A dh 7) by addng orthometrc correcton to ths epresson db AB ) H AB hab d AB 8)

orthometrc heght dfferences are calculated. Heghts of the ponts H p ) are selected as unknowns n survey networks. Orthometrc heght dfferences are adjusted by the method of Least Square Adjustment. We can form the functonal model that s between the measured heght dfferences between the ponts P and P j and the adjusted heghts H and H j as below, Pvv m0 16) nu n: number of measured heght dfferences u: number of unknowns Mean error of heght dfference hj vj H j H 9) m m0 S 17) Accordng to ths model, the correcton equatons v j, formed for the heght dfferences h j vj dh dhj lj and the constant terms -l j ); lj H j H hj 10) 11) In ths equaton, H 0 and H j 0 are appromate heghts of the ponts P and P j. Stochastc, model that s formed by the defnton of weghts of the measurements, s; 1 Pj S j 1) Normal equatons, formed by correcton equatons formed by equaton 9) and the weghts calculated by stochastc model 1), are solved usng modernzed Gauss Algorthm. Adjustment unknowns dh and nvers weght matrcs Q hh of those are calculated. Adjusted heghts of the ponts are calculated usng below equaton.; H H dh 13) Adjusted values of heght dfferences are calculated usng below equaton h h v j j j h H H j j 14) 15) and result equatons are calculated usng above equaton Mean square root error value a posteror) of unt measurement. S : length of algnment km) Mean errors of adjusted heghts m m0 qh h 18) HJ qh h : dagonal term of matrcs Q hh A pror value of mean error of unt measurement that s obtaned from dfferences of departure and return measurements. Pdd S0 19) 4n Integrty of varance s calculaced by the help of epermental tests m T s 0 0 0) Meanng full of the results s tested choosng the possblty of error as α=%5 [0, 6]. 4. FUZZY GOAL PROGRAMMING MODEL Goal Programmng Method s not only a technque to mnmze the sum of all devatons, but also a technque to mnmze prorty devatons as much as possble [4]. The general form of a mult objectve programmng s as follows; ma/ mn Z C A b Z = z 1, z,, z k ) s vector of objectves, C s a KN) matr of constants, X s an N1) vector of decson varables, 1) 3

A s an MN) matr of constants, b s an M1) vector of constants [4]. Goal Programmng s a common tool used n decson makng, but provdng crsp goals can be a problem for decson makers [35]. Membershp Functon Lnear membershp functons are used n lterature and practce more than other types of membershp functons. For the above three types of fuzzy goals lnear membershp functons are defned and depcted as follows Fgure 1): Analytcal Defnton 1 f Gk ) gk U G ) f g G ) U k 1,... m 0 f Gk ) gk k k Z ) k k k k Uk gk 1 f Gk ) gk G ) L f L G ) g k m 1,... n 0 f Gk ) Lk k k Z ) k k k k gk Lk Z k ) 1 f Gk ) Lk Gk ) Lk f Lk Gk ) gk k n 1,... gk Lk Uk Gk ) f gk Gk ) U k Uk gk 0 f Gk ) Uk Fgure 1. Lnear membershp functon and Analytcal defnton The fuzzy programmng approach for handlng the mult-objectve problems was frstly ntroduced but Zmmermann [41]. Startng from goal programmng model 1) the adopted fuzzy verson accordng to Zmmermann s C Z s.. t A b ) Where and are the fuzzfcaton of and, respectvely. ) means essentally greater less) than. Snce Narasmhan [5] had presented the ntal FGP model and the soluton procedure, a few studes have proposed FGP models for mprovng the computatonal effcency. Hannan [14, 15] has ntroduced conventonal devaton varables nto the model, so that only a conventonal lnear programmng formulaton s requred; although, ths ncreases the number of varables n the formulaton [3]. Zmmerman type member functons are used n the approaches that have been developed for the soluton of Fuzzy goal programmng models. Zmmerman's trangular member functons for fuzzy nequaltes are as follows: 4

G k ) : k th fuzzy goal, b k : access value that s determned by decson maker for k th goal, d k1 : access value that s determned by decson maker for k th goal, d k : mamum postve devaton that s allowed from access value b k. 0 f Gk ) bk dk Gk ) bk 1 f bk Gk ) bk d k dk Gk ) bk k 3) bk Gk ) 1 f bk dk1 Gk ) b k d k1 0 f Gk ) bk dk1 0 f Gk ) bk d k Gk ) bk Gk ) bk k 1 f bk Gk ) bk dk 4) dk1 1 f Gk ) b k 0 f Gk ) bk d k bk Gk ) Gk ) bk k 1 f bk dk1 Gk ) bk 5) dk1 1 f Gk ) b k Goal Devatons, The goals for each objectve are consdered for each objectve functons namely under achevement and over achevement goals. Intally, the upper and lower bounds for each objectve functons are estmated and then the goals are ncluded as by addng the under achevement and removng the over achevement for each objectve on the left hand sde of the objectves as varables [34]. 5. APPLICATION 5.1. Problem The surveys made n levelng network to determne the heght of the ponts of the earth s gven n Table 1, and appromate elevatons s gven n Table. Table 1: Level Dfferences From Pont To Pont h k m) 1 1 0 6,78 1 5,116 3 3 3,553 4 3 4,944 5 4 0 5,40 6 0 11,907 7 3 0 8,364 8 3 1 1,575 9 4 6,500 As shown n Table 1, the measurement from pont 1 towards pont 0 was calculated and the value was determned as 6,78 m. Smlarly, the measurement calculated from pont towards pont 1, the value was determned as 5,116 m, and the measurement from pont to pont 3 the value was determned as 3,553 m. Other measurements are as shown n the above Table. 5

Table : Appromate Elevatons Pont H 0 m) 0 * 40,500 1 46,780 51,900 3 48,360 4 45,400 Table shows the heghts of the ponts. The heght of pont 0) has been fed at 40, 500 m. The heghts of the other ponts have been measured and the appromate heghts, contanng measurement error values, have been determned. A software developed by Yavuz [38] s used to solve the problem accordng to LS adjustment and WnQSB s used to solve the problem accordng to FGP. 5. Usng the Least Square n Network Adjustment In Table 3, gven correcton equatons lsted below have been formed by usng data n Table 1. Table 3: Correcton Equatons v j -l j 1 v 1,0 +d 0 -d 1 +H 0 0-H 0 1- h 1,0 v,1 +d 1 -d +H 0 1-H 0 - h,1 3 v,3 +d 3 -d +H 0 3-H 0 - h,3 4 v 3,4 +d 4 -d 3 +H 0 4-H 0 3- h 3,4 5 v 4,0 +d 0 -d 4 +H 0 0-H 0 4- h 4,0 6 v,0 +d 0 -d +H 0 0-H 0 - h,0 7 v 3,0 +d 0 -d 3 +H 0 0-H 0 3- h 3,0 8 v 3,1 +d 1 -d 3 +H 0 1-H 0 3- h 3,1 9 v 4,0 +d 0 -d 4 +H 0 0-H 0 4- h 4,0 V j : Correctons d j : Correctons Unknows H 0 : Appomate elevatons h j : Elevaton Dfferences The table below shows the results of the Vertcal Network Adjustment equaton that has been solved accordng to Least Square Method. Table 4: LS Soluton Pont Value 0 40,500 1 46,881 5,003 3 48,454 4 45,503 5.3 Usng the Fuzzy Goal Programmng n Network Adjustment The am of network adjustment s to fnd out the optmum values by dstrbutng the correctons of the measurements n a proper way. To mplement ths am generally LS method s used. In ths study vertcal network adjustment s made by usng both LS and FGP methods. The heght values of 5 ponts n the network are accepted as decson varables X ) n network adjustment by FGP. By FGP model, values of decson varables, n other words adjusted heght values of ponts n network, are obtaned. 9 level dfferences surveyed between two ponts h j ) n network are accepted as goal values. The level dfferences between the two surveyed ponts are equalzed appromately to a measured level dfference accepted as the goal value. 9 goals are obtaned by addng the devaton varables to the equaltes and 1 absolute constrant are gven. The heght of the zero pont s accepted fed and the result s transferred as absolute constrant. 0 40, 500 6) In ths stuaton, goal programmng model of gven eperment, Goals, Goal1: 0 1 6, 78 Goal : 1 5,116 Goal 3: 3 3,553 Goal 4 : 4 3,944 Goal 5 : 0 4 5, 40 7) Goal 6 : 0 11,907 Goal 7 : 0 3 8,364 Goal 8 : 1 3 1,575 Goal 9: 4 6, 500 6

Absolute constrant 0 40, 500 8),,,, 0 9) 0 1 3 4 Zmmerman's trangular member functon and ma-mn approach of Bellman and Zadeh [6] are used n eperment of ths study. Goals that have been used n the model are accepted as fuzzy, thus goals have been taken as fleble. Tolerance value has been taken as 0, for goals. The goals are shown n Table 5. Ponts 1 1-0) -1) 3-3) 4 3-4) 5 4-0) 6-0) 7 3-0) 8 3-1) 9-4) Table 5: Goals of FGP Goals 0 1) 6,80 6,80 0 1) 1, 1 1 ) 5,10 5,10 1 ) 1, 1 3 ) 3,540 3,540 3 ) 1, 1 4 3),960,960 4 3) 1, 1 0 4) 4,900 4,900 0 4) 1, 1 0 ) 11, 400 11, 400 0 ) 1, 1 0 3) 7,860 7,860 0 3) 1, 1 1 3) 1,580 1,580 1 3) 1, 1 4 ) 6,500 6,500 4 ) 1, 1 When the mult-purpose fuzzy goal programmng equatons shown above 7), 8) and 9) are solved, the results are as shown n Table 6. Table 6: FGP Soluton Pont Value 0 40,500 1 46,880 5,010 3 48,460 4 45,500 5.4. Comparson between Least Square and Fuzzy Goal Programmng Takng the heght of pont 0) as fed, by the help of the measured heghts of 5 ponts; adjusted heghts are obtaned n both methods.the conclusons obtaned n both methods are gven n Table 7. Takng the heght of pont 0) as fed, by the help of the measured heghts of 5 ponts; adjusted heghts are obtaned n both methods.the conclusons obtaned n both methods are gven n Table 7. H 0 : appromate heghts of the ponts H E : adjusted heghts of the ponts wth LS H D : adjusted heghts of the ponts wth FGP 7

Correctons related to measured level dfferences, receved by both methods have been obtaned n mm senstvty. Level dfferences for both methods are obtaned by addng the correcton amount to the levelng measurements. For both methods; comparng level dfferences between 7 dfferent measured ponts and adjusted heghts of the ponts, the result control of the adjustment n both methods s done. The comparsons obtaned n both methods are gven n Table 8. Pont Appromate H0 m) Table 7: Levelng Adjustment wth LS and FGP Dfferences LS FGP LS-FGP Correctons Adjusted HE m) Correctons Adjusted HD m) m mm 0 40,500 0,000 40,500 0,000 40,500 0,000 0 1 46,780-0,101 46,881-0,100 46,880-0,001-1 51,900-0,103 5,003-0,110 5,010 0,007 7 3 48,360-0,094 48,454-0,100 48,460 0,006 6 4 45,400-0,103 45,503-0,100 45,500-0,003-3 Table 8: Comparson between LS and FGP Results Dfferences LS FGP LS-FGP) Level Dfferences H0 m) Correctons Adjusted HE m) 1-0) 6,80-0,101 6,381-0,100 6,380 0,001 1-1) 5,10-0,00 5,1-0,010 5,130-0,008-8 -3) 3,540-0,009 3,549-0,010 3,550-0,001-1 3-4),960 0,009,951 0,000,960 0,009 9 4-0) 4,900-0,103 5,003-1,100 5,000 0,003 3-0) 11,400-0,103 11,503-0,110 11,510-0,007-7 3-0) 7,860-0,094 7,954-0,100 7,960-0,006-6 3-1) 1,580 0,007 1,573 0,000 1,580-0,007-7 -4) 6,500 0,000 6,500-0,010 6,510-0,010-10 Correctons Adjusted HE m) m mm 6. CONCLUSION The functonal and the stochastc models of the adjustment whch are used are approprate accordng to LS approach. Close values to the values obtaned by LS method are receved by usng FGP method whch s a mnmzng of devaton of the targets whose ams are determned. 8

The hghest dfference between the results of two methods s 10 mm. In FGP method, dfferent from LS method, wthout usng the appromate heght values of the ponts, but usng only the level dfferences between two ponts, the heght of the ponts are calculated close to the heght values obtaned accordng to LS. In addton, the study made by FGP has been more effectve results than the work prevously carred out wth GP. In the case of heght values are unknown or can not be measured, n ths study, t has been shown that FGP can be used. REFERENCES [1] Abd, H.010). The method of least squares Stdues n Salknd, N.J.; Dougherty, D.M.; Frey, B. eds.) Encyclopeda of Research Desgn) pp. 705-708. Sage, Thousand Oaks CA). [] Adhkary, K. R. 00). Global Postonng System on Cadastral Survey of Nepal. Nepal: ACRS. [3] Alp, S. & Yavuz, E. & Ersoy, N. 011). Usng Lnear Goal Programmng n Surveyng Engneerng for Vertcal Network Adjustment,. Internatonal Journal of the Physcal Scences. 68), 198-1987. [4] Alp, S. 008). Establshment and Implementaton of a Model for Urban Bus Publc Transportaton System Usng Lnear Goal Programmng Method. PhD Thess. Marmara Unversty, Insttute of Socal Scences: Istanbul. [5] Aoun, B. & Martel J.M. & Hassane A. 009). Fuzzy Goal Programmng Model: An Overvew of the Current State-of-the Art. Journal of Mult-Crtera Decson Analyss. 16, 149-161. [6] Bellman, R.E. & Zadeh, L.A.1970). Decson-makng n a fuzzy envronment. Management Scences, 171), 35-336. [7] Belmokadddem, M. 009). Applcaton of a Fuzzy Goal Programmng Approach wth Dfferent Importance and Prortes to Aggregate Producton Plannng. Journal of Appled Quanttatve Methods, 43), 317-331. [8] Charnes; A. & Cooper W.W. 1961). Management Models and Industral Applcatons of Lnear Programmng. New York: Wley & Sons. [9] Charnes, A. & Cooper, W.W. & Ferguson, R.O.1955). Optmal Estmaton of Eecutve Compensaton by Lnear Programmng. Management Scence. 11), 138 151. [10] Chen Y.L. & Chen L.H. & Huang C.Y. 009). Fuzzy Goal Programmng Approach To Solve The Equpments- Purhasng Problem of An FMS. Internatonal Journal of Industral Engneerng, 164), 70-81. [11] Dong, D. & Herrng T.A. & Kng R.W. 1998). Estmatng Regonal Deformaton from a Combnaton of Space and Terrestral Geodetc Data. Journal of Geodesy, 71), 00-14. [1] El-Wahed, W.F.A. & Lee, S.M. 006). Interactve Fuzzy Goal Programmng for Mult-Objectve Transportaton Problems. The Internatonal Journal of Management Scence, 341), 158-166. [13] Gocoechea, A. & Hansen D.R., & Ducksten, L. 198). Mult-objectve Decson Analyss wth Engneerng and Busness Applcatons, New York: Wley & Sons. [14] Hannan, E.L. 1981). Lnear programmng wth multple goals. Fuzzy Sets and Systems, 61), 35-48. [15] Hannan, E.L. 1981). On Fuzzy Goal Programmng. Decson Scences, 13), 5-531. [16] Ijr, Y. 1965). Management Goals and Accountng for Control. Amsterdam: North-Holland. [17] Ignzo, J.P. 198). Lnear Programmng n Sngle and Multple Objectve Systems. New Jersey: Prentce- Hall. [18] Ignzo, J.P. 1976). Goal Programmng and Etensons. Lengton: Lengton Books. [19] Lee, S.M. 197). Goal Programmng for Decson Analyss. Phladelpha: Auerback. [0] Leck, A. 004). GPS Satellte Surveyng. Thrd Edton. New Jersey: John Wley&Sons Inc. [1] Lohgaonkar, M.H. & Bajaj, V.H. & Jadhav V.A. 010). Addtve Fuzzy Multple Goal Programmng Model for Unbalanced Mult-Objectve Transportaton Problem. Internatonal Journal Machne Intellgence, 1), 9-34. [] Loren, R. & Benson D. B. 1999). Introducton to Strng Feld Theory, nd Edton. New York: Sprnger- Verlag. 9

[3] Luhandjula, M.K. 1987). Multple objectve programmng problems wth possblstc coeffcents. Fuzzy Sets and Systems, 11), 135-145. [4] Mohamed, R.H. 1997). The Relatonshp between goal programmng and fuzzy programmng. Fuzzy Sets and Systems 891), 15-. [5] Narasmhan, R. 1980). Goal programmng n a fuzzy envronment. Decson Scences, 171), 35-336 [6] Peng, J. & Chun, G. & Zang, H. 006). An Aggregate Constrant Method for Inequalty- Constraned Least Squares Problems. Journal of Geodesy, 791), 705-713. [7] Sharma, H.P. & Sharma, D.K. & Jana R.K. 009). Credt Unon Portfolo Management - An Addtve Fuzzy Goal Programmng Approach. Internatonal Research Journal of Fnance and Economcs, 301), 18-9. [8] Stadler, W. 1979). A survey of multobjectve optmzaton of the vector mamum problem. Journal of Optmzaton Theory and Applcatons. 91), 1-5. [9] Stener, J. & Termona, Y. & Deltour, J. 197). Smootng and Dfferentaton of Data by Smplfed Least Square Procedure. Analytcal Chemstry, 4411), 1906-1909. [30] Steuer, R.E. 1986) Multple Crtera Optmzaton : Theory, Computaton and Applcaton. New Jersey: John Wley&Sons Inc. [31] Teunssen, P.J.G. 1999). Adjustment Theory. Holland: Delft Unversty Press:. [3] Twar, R.N. & Dharmar, S. & Rao J.R. 1987). Fuzzy Goal Programmng An Addtve Model. Fuzzy Sets and Systems. 41), 7-34. [33] Tong, R.M. & Bonssone, P.P. 1980). A lngustc approach to decson-makng wth fuzzy sets. IEEE Transportaton Systems. 101), 660-716. [34] Venkatasubbaah, K. & Acharyulu S.C. & Chanda Moul K.V.V. 011). Fuzzy Goal Programmng Method for Solvng Mult-Objectve Transportaton Problems. Global Journal Research n Engneerng, 113), 4-10. [35] Wang, H.F. & Fu C.C. 1997). A Generalzaton of Fuzzy Goal Programmng wth Preemptve Structure. Computers Operatons Research, 49), 819-87. [36] Weng Y. & Xao, W. & Xe, L. 011). Total Least Squares Method for Robust Source Localzaton n Sensor Networks Usng TDOA Measurements. Internatonal Journal of Dstrbuted Sensor Networks. 011. [37] Wdrow, B. & Steams, S. d. 1995). Adaptve Sgnal Processng. New Jersey: Prentce Hall. Englewood. [38] Yavuz, E. 00). Stochastc Model Search n Horzontal Control Network. PhD Thess. Istanbul Techncal Unversty, Insttute of Scence and Technology: Istanbul. [39] Zadeh, L.A. 1965). Fuzzy Sets. Informaton and Control. 81), 338-353. [40] Zangabad, M. & Malek, H. R. 007). Fuzzy Goal Programmng for Mult-objectve Transportaton Problems. Journal of Appled Mathematcs and Computng, 41-), 449-460. [41] Zmmermann, H. J. 1978). Fuzzy Programmng and Lnear Programmng wth Several Objectve Functons. Fuzzy Sets and Systems, 11), 45-55. 10