DergiPark: 245889trakyafbdBu makalede, Çok Amaçlı Doğrusal Kesirli Programlama Probleminin uygun bölgesindeki, her doğrusal kesirli amaç fonksiyonunu optimal yapan noktalarda, kesirli lineer amaç fonksiyonları Taylor serisine açılarak, Çok Amaçlı Doğrusal Kesirli Programlama Problemi, Çok Amaçlı Doğrusal Programlama Problemine dönüştürül-müştür. Daha sonra da, doğrusal amaç fonksiyonlarının ağırlıkları dikkate alınarak, ağırlıklı toplamı bulunmuştur. Ardından, tek amaçlı doğrusal programlama problemi elde edilmiştir. Bu doğrusal programlama probleminin optimal çözümü, çok amaçlı doğrusal kesirli programlama probleminin etkin, hatta, kuvvetli etkin çözümlerini belirlemektedir. Önerilen çözümün etkinliğini göstermek için, örnek uygulamalar yapılmış olup, örneklerin çözümünde WinQSB bilgisayar paket programı kullanılmıştır.In this paper, we have proposed a solution to Multi Objective Linear Fractional Programming Problem (MOLFPP) by expanding the order 1st Taylor polynomial series these objective functions at optimal points of each linear fractional objective functions in feasible region. MOLFPP reduces to an equivalent Multi Objective Linear Programming Problem (MOLPP). The resulting MOLPP is solved assuming that weights of these linear objective functions are equal and considering the sum of the these linear objective functions. The proposed solution to MOLFPP always yields efficient solution, even a strong-efficient solution. Therefore, the complexity in solving MOLFPP has reduced easy computational. To show the ability the proposed solution, three different numerical examples have been presented. The given examples are solved using optimization software WINQSB (Chang, 2001)

Güzel, Nuran

Sivri, Mustafa

English

Trakya University Academic Open Access System

http://fenbil.trakya.edu.tr/tujs Trakya Univ J Sci, 6(2): 91-98 , 2005 ISSN  1305-6468 DIC: 172NGIT620512050106     Research Article / Araştırma Makalesi  TAYLOR SERIES SOLUTION OF MULTI OBJECTIVE LINEAR FRACTIONAL PROGRAMMING PROBLEM   Nuran GÜZEL1*,Mustafa SİVRİ2 Yıldız Teknik Üniversitesi, Fen Edebiyat Fakültesi, Matematik Bölümü, 34210 Davutpaşa-Esenler / İstanbul *Corresponding Author: E-mail: nguzel@yildiz.edu.tr    Alınış : 12.02.2005 Kabul Ediliş : 11.05.2005    Abstract: In this paper, we have proposed a solution to Multi Objective Linear Fractional Programming Problem (MOLFPP) by expanding the order 1st Taylor polynomial series these objective functions at optimal points of each linear fractional objective functions in feasible region. MOLFPP reduces to an equivalent Multi Objective Linear Programming Problem (MOLPP). The resulting MOLPP is solved assuming that weights of these linear objective functions are equal and considering the sum of the these linear objective functions.  The proposed solution to MOLFPP always yields efficient solution, even a strong-efficient solution. Therefore, the complexity in solving MOLFPP has reduced easy computational.     To show the ability the proposed solution, three different numerical examples have been presented. The given examples are solved using optimization software WINQSB.(Chang, 2001)  Keywords: Multi Objective Linear Fractional Programming Problem (MOLFPP), Multi Objective Linear Programming Problem (MOLPP), Taylor series, WinQSB   Çok Amaçlı Doğrusal Kesirli Programlama Probleminin Taylor Serisiyle Çözümü  Özet: Bu makalede, Çok Amaçlı Doğrusal  Kesirli Programlama Probleminin uygun bölgesindeki, her doğrusal kesirli  amaç fonksiyonunu optimal yapan noktalarda, kesirli lineer amaç fonksiyonları Taylor serisine açılarak, Çok Amaçlı Doğrusal Kesirli  Programlama Problemi, Çok Amaçlı Doğrusal Programlama Problemine dönüştürül-müştür. Daha sonra da, doğrusal amaç fonksiyonlarının ağırlıkları dikkate alınarak, ağırlıklı toplamı bulunmuştur. Ardından, tek amaçlı doğrusal programlama problemi elde edilmiştir. Bu doğrusal  programlama probleminin optimal çözümü, çok amaçlı doğrusal kesirli programlama probleminin etkin, hatta, kuvvetli etkin çözümlerini belirlemektedir.     Önerilen çözümün etkinliğini göstermek için, örnek uygulamalar yapılmış olup, örneklerin çözümünde WinQSB bilgisayar paket programı kullanılmıştır.  Anahtar Kelimeler: Çok Amaçlı Doğrusal Kesirli Programlama Problemi, Çok Amaçlı Doğrusal Programlama Problemi, Taylor serisi, WinQSB    Introduction      In the modelling of the real word problems like financial and corporate planning, production planning, marketing and media selection, university planning and student admissions, health care and hospital planning, air force maintenance units, bank branches, etc. frequently may be faced up with decision to optimise dept/equity ratio, profit/cost, inventory/sales, actual cost/standard cost, output/employee, student/cost, nurse/patient ratio etc. respect to some constraints (Lai and Hwang, 1996).  Nuran GÜZEL, Mustafa SİVRİ  92    In the literature, different approaches appear to solve different models of Linear Fractional Programming Problem (LFPP). Because, fractional programming solves more efficiently the above problems respect to Linear Programming Problem (LPPP).     When some of the studies have achieved solution methods (Lai and Hwang, 1996), (Charnes and Cooper, 1962), (Zionts, 1968), (Chakraborty and Gupta, 2002), others have concentrated on applications (Lai and Hwang, 1996), (Munteanu and Rado, 1960), (Gilmore and Gomory, 1963), (Sengupta et all, 2001), (Chakraborty and Gupta, 2002). In these papers are discussed in details LFPP. It is showed that LFPP can be optimised easily. But, in the great scale decision problems, there is more than one objective, which must be satisfied at the same time as possible. However, most of these are linear fractional objectives. It is difficult to talk about the optimal solutions of these problems. The solutions searched for these problems are weak efficient or strong efficient. If required, one compromise solution can be reached by the affection of the models with the decision makers (DMs).     There exist several methodologies to solve multi objective linear fractional programming problem (MOLFPP) in the literature. Most of these methodologies are computationally burdensome (Chakraborty and Gupta, 2002). Kornbluth and Steuer (1981), Y.J.Lai and C.L.Hwang (1996) have developed an algorithm for solving the MOLFPP for all weak-efficient vertices of the feasible region. Nykowski, Z.Zolkiewski (1978) and Dutta et all (1992)  have proposed a compromise procedure for MOLFPP. Choo and  Atkins (1982) have given an analysis of the bicriteria LFPP. M.K.Luhandjula (1984) solved MOLFPP using a fuzzy approach (Lai and Hwang, 1996). He used linguistic approach to solve MOLFPP by introducing linguistic variables to represent linguistic aspirations of the DM. Dutta et all (1992) modified the linguistic approach of Luhandjula  to solve MOLFPP.     In this paper, we have proposed a solution to MOLFPP using the 1st order Taylor polynomial series at optimal point of each linear fractional objective function in feasible region. LFPP has been transformed into a more simplified structure MOLPP by using Taylor polynomial series and an efficient, even a strong-efficient solution has been proposed by using the approach.    Linear Fractional Programming Problem (LFPP)   The general format of Linear Fractional Programming Problem may be written as:        .,          0         s.t     R  ,RA R, dx, c, xb, AxxdxcMaxmxnnTTTT∈∈∈≥=++βαβα                                                       (1) Definition:  The two Mathematical programming problems  (i)   Max   subject to , ( )F x x∈Δ(ii)  Max , subject to will be said to be equivalent if only if is a one one map q(.) of the feasible set of (i), onto the feasible set of (ii), such that F(x)=G(q(x)) for all ( )G x ,x∈Δx∈Δ ( Chakraborty and Gupta, 2002).  Taylor Series Solution Of Multi Objective Linear Fractional Programming Problem  93 Multiple Objective Linear Fractional Programming Problem (MOLFPP)               A MOLFPP may be written following as: (x)}(x),…, Z(x), Z {ZMax Z(x) = k21  s.t.  { },0,, ≥≤∈=∈ xbAxRxXx n               (2) with  ., mxnn RARb ∈∈)()()(xDxNxdxcxZiiiiiii =++=βα,                                                    .,, RandRdc iinii ∈∈ βαLet be *( )( )( )i i ii ii i ic x N xMax Z x Zd x D xαβ+= = =+ and Miniiiiiiii ZxDxNxdxcxZMin ==++=)()()(βα.    Model Development          In this paper, we consider the Multi Objective Linear Fractional Programming Problem (MOLFPP)   mxnnnkiRARbwithxbAxRxXxtsxZxZxZMaxxMaxZ∈∈≥≤∈=∈=,}0,,{..)}(),...,(),({)( 21             (3) and )()()(xDxNxdxcxZiiiiiii =++=βα where .  ,0)(,,, ixDRandRdc iiinii ∀>∈∈ βα      Let be maximum value ith objective function ,*)( ii ZxZMax = ,i∀ on the feasible region it occurs when for i=1,...,k. ),,...,,( **2*1*iniii xxxx =Suppose that ( )Z x*x*ix)(xand all of its partial derivates of order less than or equal n+1 are continuous on the feasible region X,  By expanding the 1st order Taylor polynomial series for objective function  about , objective function  is obtained from where function is called the 1st Taylor polynomial in n variables about  and is the remainder term associated with . We have  .X∈)(1 xPi)(xZi )(xZi )()()( 11 xRxPxZ iii +=*ix )(1 xRi1Pi* ** * *1 1 1 2 21 2** 21( ) ( )( ) ( ) ( ) [( ) ( )( )... ( ) ] ( )i i i ii i i i i ii in inZ x Z xZ x P x Z x x x x xx xZ xx x Ox∂ ∂≅ = + − + − +∂ ∂∂+ − +∂h )()()[()()()( 21***1 hOxxZxxxZxPxZnj jiiijjiiii +∂∂−+=≅ ∑= for i=1,...,k.                                          (4) Trakya Univ J Sci, 6(2), 91-98 , 2005   Nuran GÜZEL, Mustafa SİVRİ  94Where  is order of the maximum error. This polynomial gives an accurate approximation to  when )( 2ho)(xZi x  is close to .  *xBy replacing)()()(xDxNxdxcxZiiiiiii =++=βα by ])()[()()()(1***1 ∑= ∂∂−+=≅nj jiiijjiiii xxZxxxZxPxZ  in MOLFPP (3), all of the objective functions , i=1,2,...,k, become the 1st order linear functions as  )(xZi,1jnjji xae ∑=+ ,, Rae ji ∈  .  So, MOLFPP (3) reduces the following MOLPP (5): ,...,2,1 ki =⎭⎬⎫⎩⎨⎧+=+=+= ∑ ∑ ∑= = =njnjnjjkjkkjjjj xaexZxaexZxaexZMax1 1222111 )(...,,)(,)(1  { },0,, ≥≤∈=∈ xbAxRxXx n               (5) with  ., mxnn RARb ∈∈        If we assume that the weights of objective functions in problem (5) are equal, then problem (5) is written as follows: ⎭⎬⎫⎩⎨⎧++++∑ ∑ ∑ ∑= = = =njnjnjkiijkjjjjj exaxaxaMax1 1 1 121 ...  { },0,, ≥≤∈=∈ xbAxRxXx n                          (6) with  ., mxnn RARb ∈∈       In problem (6), set X is non-empty convex set having feasible points. The optimal solution of problem (6) gives the efficient solution of   MOLFPP (5). Because, weights of objective functions that are expanded Taylor series are equal and in (6) is considered the weighted objective function.   Numerical Examples  Example 1. Let us consider a MOLFPP with two objectives as follows: ⎭⎬⎫⎩⎨⎧+++=+++−=1257)(,323)(2121221211 xxxxxZxxxxxZMax  s.t.  121 ≥− xx  1532 21 ≤+ xx                          (7)                                   31 ≥x   .0, 21 ≥xx   It is observed that for each x in the feasible region. ,0,0 21 ≥< ZZ.1115)0;5.7( =,2314)6.2;6.3( 21−= MaxMax ZZ   Taylor Series Solution Of Multi Objective Linear Fractional Programming Problem  95      By expanding the 1st order Taylor polynomial series for objective functions  about points (3.6; 2.6) and (7.5; 0) in feasible region, respectively are obtained from )()( 21 xZandxZ41.02835.02599.0)()6.2;6.3()6.2()6.2;6.3()6.3(2314)(2112121111−+−=∂∂−+∂∂−+−=xxxZxZxxZxxZ          (8) 3282.104486.0004722.0)()0;5.7()0()0;5.7()5.7(1115)(2122221212+−=∂∂−+∂∂−+=xxxZxZxxZxxZ  So, all fractional objectives are transformed to the linear objectives. The obtained MOLPP is equivalent to the following LPP when weights of objective functions are equal: 9182.023864.0255178.0)}()({ 2121 ++−=+ xxxZxZMax  s.t.  121 ≥− xx                (9) 1532 21 ≤+ xx                                                                                                                 31 ≥x   .0, 21 ≥xx                  Optimal solution of problem (9) is at point (3,2) and maximum value is 0.6299. The point (3,2) is efficient solution of the given original problem in the feasible region. The solution for original problem is obtained as .2023,85,2,3 2121 =−=== ZZxx    Example 2. Let us consider a MOLFPP with three objectives as follows:  Max ⎭⎬⎫⎩⎨⎧+++=+++=+++−=2324)(,1257)(,323)(212132121221211 xxxxxZxxxxxZxxxxxZ  s.t.  121 ≥− xx  1532 21 ≤+ xx                                                         (10) 1 29 9x x+ ≥  31 ≥x   .0, 21 ≥xx   It is observed that  for each x in the feasible region. 0,0,0 321 ≥≥< ZZZ1714)6.2;6.3(,2314)6.2;6.3(1 =−=Max ZZ ,1723)2.0;2.7( 32 =MaxMax Z    Trakya Univ J Sci, 6(2), 91-98 , 2005   Nuran GÜZEL, Mustafa SİVRİ  96By expanding the 1st order Taylor polynomial series for objective functions  about points (3.6; 2.6), (7.5;0) and (3.6; 2.6) in feasible region, respectively are obtained the following linear objective functions: )()(),( 321 xZandxZxZ41.02835.02599.0)( 211 −+−= xxxZ , 3167.10456.000629.0)( 212 −−= xxxZ ,                                    (11) 72662.008996.003806.0)( 213 ++−= xxxZ  . So, the obtained MOLPP is equivalent to the following LPP when weights of objective functions are equal: 1 2 3 1 2{ ( ) ( ) ( )} 0.29167 0.32786 1.63332Max Z x Z x Z x x x+ + = − + +  s.t. 121 ≥− xx  1532 21 ≤+ xx                                                        (12) 1 29 9x x+ ≥  31 ≥x   .0, 21 ≥xx   The solution to the above LPP is obtained as 4372.1)6.2;6.3)}(()()({ 321 =++ xZxZxZMax . The solution for original problem is obtained as ;  .1714,121139,2314,6.2,6.3 32121 ==−=== ZZZxx     Example 3.  Let us consider a MOLFPP with three objectives as follows:  Max ⎭⎬⎫⎩⎨⎧+−−+=+−−=++−+−=163173)(,15)(,364)(1213221221211 xxxxZxxxxZxxxxxZ  s.t.   41 ≤x       42 ≤x                 (13) 1 2 7x x+ ≤  321 ≤+− xx  1 2 3x x− ≤   .0, 21 ≥xx    It is observed that 0,0,0 321 <<< ZZZ  for each x in the feasible region. 21)3,4(,1)1,4(,131)4,1( 321 −=−=−= MaxMaxMax ZZZ   At these corresponding points, expanding Taylor series, we have 1 2 3 1 2.58358 .33284 3.6167Z Z Z x x+ + = + −   Taylor Series Solution Of Multi Objective Linear Fractional Programming Problem  97The given MOLFPP problem is written as: s.t     (14)        The solution to the above LP is obtained as Max1 2 3 1{ } .58358 .3328Max Z Z Z x+ + = + 24 3.6167x −  41 ≤  x42 ≤x             1 2x x+ 7≤  321 ≤+− xx  1 2 3x x− ≤.0, 21 ≥xx   1 2 3{ }(4,3) .2867Z Z Z+ + = −  . The solution for original problem is obtained as 1 2 1 2 314, 3,x x Z −= = = 1, 1, .6 2Z Z= − = −     onclusion er, we have proposed a solution to Multi Objective Linear Fractional Programming Problem eferences KRABORTY M., GUPTA S, Fuzzy mathematical programming for multi objective  linear  Nav. ar fractional programming, J.Optim. Theory O JR, Multiple objective linear fractional programming-A fuzzy set proach to the cutting stock tive Linear Fractional Programming, ltiple Objective Decision Making, Springer,1996. ai economice la cuptoarecle de WSKI I., ZOLKISKI Z., A compromise procedure for the multiple objective  C    In this pap(MOLFPP) using Taylor polynomial series. With the help of the order 1st Taylor polynomial series at optimal points of each linear fractional objective function in feasible region, Multi Objective Linear Fractional Programming Problem reduces to an equivalent Multi Objective Linear Programming Problem (MOLPP). 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Çok Amaçlı Doğrusal Kesirli Programlama Probleminin Taylor Serisiyle Çözümü

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Çok Amaçlı Doğrusal Kesirli Programlama Probleminin Taylor Serisiyle Çözümü

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