Posynomial Geometric Programming Problems with Multiple Parameters

JOURNAL OF COMPUTING, VOLUME 2, ISSUE 1, JANUARY 2010, ISSN 2151-9617 HTTPS://SITES.GOOGLE.COM/SITE/JOURNALOFCOMPUTING/ 84 Posynomial Geometric Programming Problems with Multiple Parameters A. K.Ojha and K.K.Biswal Abst ract — Geometric programming problem is a powerful tool for so lving some special type non-linear programming problems. It has a wide range of applications in optimization and engineeri ng for solving some complex optimization problems. Many applications of geometric programming ar e on engineering design problems where para meters are estimated using geometric programming. When the parameters in the problems are imprecise, the calculated objective val ue should be imprecise as well. In this paper we have developed a method to solve geometric programming problems where the exponent of the variables in the objective function, cost coefficient s and right hand side are mu ltiple parameters. The equivalent mathematical programming problems are formulated to find their corresponding value of t he objective function based on the duality theorem. By applying a variable separable technique the multi-c hoice m athematical programming problem is transformed into multiple one level geometric programming problem which produces multiple objec tive values that helps engi neers to handle m ore realistic engineering design problems. Index T erms — Duality theorem, Geometric programming, mult iple parameters, optimization,. Posynomial. ——————————  —————————— 1 I NTRODUCTION Various mathematical programmin g methods have been formulated to solve many challenging real world prob- lems. Since 1960 some authors [5] have cited that geomet- ric inequality helps to solve special type optimization problem which is known as geometric prog ramming. However Duffin, Peterson and Zener[8] laid the founda- tion stone to solve wide range of engineering problems by developing basic theories of geometric pr ogramming and its application in their text book. Geometric program- ming(GP) is a technique for solving polynomial type n on- linear programming p roblems. One of the remarka ble properties of Geometric programming is that a problem with highly nonlinear constraints can be stated equiva- lently with a dual program. If a primal problem is in po- synomial form then a global minimizing solution of the problem can be obtained by solving its corresponding dual maximizat ion problem be cause the dual constraints are linear, and linearly constrained programs a re gener- ally easier to solve than ones with nonlinear c onstraints. GP problem has a dual impact in the area of integrat ed circuit design[4,10,17] manufacturing sy stem design [8, 3],project manage ment[23], ma ximization of long run and short term profit[16], generalized geo metric program- ming problem with non positive variable s [24] and goal programming m odel [1]. Several algorith ms due to Beighter and Phillips[2], Fang et al.[9], Kortanek[12], Kor- tanek et al.[13], Peterson[19], Rajgopal and Bricker [22] and Zhu and Kortanek[25] strengthen the solution of complicated Geometric programming problem f or the exact known value of cost and constraint coefficients. Sensitive analysis of variou s optimal solutions due to Dembo[6], Dinkle and Tretter [7] and Kyparisis [14] using Geometric programming technique simplifies certain en- gineering design problem in which some of the problem parameters are estimates of the actual value. There are certain problems in which some of the coefficients may not be presented in a precise manner. For example, in project management the time requir ed to complete the various activities in a rese arch and development project may be only known a pproximately. In order to determine the inventory policy of a novel technology product, the demand and supply quantities may be uncertain due to insufficient market information and are specified by ranges. If some parameters imprecise or uncertain, then the most liking values are usually adopted to make the conventional geometri c programming workable. This simplification might result in a derived result which is misleading. One way to manipulate imprecise parameters is via probability distributions. However, a probability distribution requi res constr uctions of prior predictable regularity or a posteri or frequency determination which may not be possible in certain cases. Uncertain parame- ters can be considered by applying interval estimates in- stead of single values. In the recent papers Liu [16] has studied the geometric prog ramming problems consider- ing the cost coefficient, constraint coefficients and the right hand sides are interval numbers where the derived objective values also lies in an interval. When the cost co- efficient, constraint coefficien ts and exponents of the deci- sion variable in the objective functions of the GP pro blem are multiple parameters the problem becoming more _____________________________ • Dr.A. K. Ojha, School of Basic Sciences, IIT Bhubaneswar, Orissa, Pin-751013, India. • K.K.Biswal Department of Mathematics, CT TC Bhubaneswar, B- 36, Chandaka Industrial Area, Bhubaneswar, Or issa, Pin-751024, India. JOURNAL OF COMPUTING, VOLUME 2, ISSUE 1, JANUARY 2010, ISSN 2151-9617 HTTPS://SITES.GOOGLE.COM/SITE/JOURNALOFCOMPUTING/ 85 complicated. In this paper we have developed methods for the solution of GP proble m when the cost, constraints, its right hand side and exponents are multiple parame- ters. For these multiple parame ters we construct multiple level mathematical programming models to find the value of the objective functi on. These results will provide the decision makers with more infor mation for making better decisions. The organization of this paper is as f ol- lows: Following introduction , mathematical formulations and methodology for solving geometric programming problems with multiple parameters have been discuss ed to find the respective objectiv e values of the problem in Section 2. Dual form of GPP has been discussed in Section 3. Some illustrative examples are given in Section 4 for understanding the problems and finally at Section 5 some conclusions are drawn from the discussion. 2 Mathematical Formulation A typical constrained posynomial geometric progra m- ming problem is presented as follows: ∏ ∑ = = = = n j a j T t t tj x C x g x Z 1 1 0 0 0 0 ) ( min subject to m i x C x g n j a j T t it i itj i ,..., 2 , 1 , 1 ) ( 1 1 = ≤ = ∏ ∑ = = (2.1) n j x j ,..., 2 , 1 , 0 = > The posynomial g 0 (x) is a objective function containing T 0 number of terms where as the posynomial g i (x), i = 1, 2,…,m contains T i terms with m inequality constraints. By the definition of posynomial all the co-efficients C it , i = 0,1, 2,…,m and t = 1, 2,…,T m are positive and the expo- nents a 0tj and a itj are arbitrary constants. Writing the right hand side of the geometri c programming problem given by (2.1) in more general form , we have ∏ ∑ = = n j a j T t t tj x C x 1 1 0 0 0 min such that m i b x C i n j a j T t it itj i ,..., 2 , 1 , 1 1 = ≤ ∏ ∑ = = (2.2) n j x j ,..., 2 , 1 , 0 = > where all b i are positive real numbers. If b i = 1 for all i then this modified geometric program becomes the origi- nal one given by(2.1). Considering C 0t ,C it , b i , a 0tj and a itj are the multiple values of the corresponding posynomial geomet ric program giv- en by (2.2)can be reduced in the following form restrictin g the number multiple parame ter as three number where middle one is the average of other two. ∏ ∑ = = n j a j T t t tj x C x 1 1 0 0 0 min such that m i b x C i n j a j T t it itj i ,..., 2 , 1 , 1 1 = ≤ ∏ ∑ = = (2.3) n j x j ,..., 2 , 1 , 0 = > . where { } {} {} U i M i L i i U it M it L it it U t M t L t t B B B b C C C C C C C C , , , , , , , , 0 0 0 0 = = = { } {} U itj M itj L itj itj U tj M tj L tj tj A A A a A A A a , , , , , 0 0 0 0 = = In order to find the objective values of the posynomia l function we will derive its corresponding values with respect to their counterpart of the param eters. Let us de- fine ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ = = ≤ ≤ ∈ ∈ ∈ = n j m i T t A a B b C c a b C S i itj itj i i it it ,..., 2 , 1 , ,..., 2 , 1 , 1 , , , : ) , , ( For each triplet S a b C ∈ ) , , ( we denote its correspond- ing objective Z of (2.3) by ) , , ( a b C Z Let U M L Z Z Z , , are the lower, middle and maximum values of ) , , ( a b C Z defined by { } S a b C a b C Z Z L ∈ = ) , , ( : ) , , ( min { } S a b C a b C Z Z U ∈ = ) , , ( : ) , , ( max are the values at the first and thir d parameter where { } S a b C a b C Z Z M ∈ = ) , , ( : ) , , ( is the value at middle parameter Now the above objectives U M L Z Z Z , , can be formu- lated as geometric programming as: ∏ ∑ = = ∈ = n j a j T t t L tj x C x S a b C Z 1 1 0 0 0 min ) , , ( min subject to m i b x C i n j a j T t it itj i ,..., 2 , 1 , 1 1 = ≤ ∏ ∑ = = (2.4) n j x j ,..., 2 , 1 , 0 = > ∏ ∑ = = ∈ = n j a j T t t U tj x C x S a b C Z 1 1 0 0 0 min ) , , ( max JOURNAL OF COMPUTING, VOLUME 2, ISSUE 1, JANUARY 2010, ISSN 2151-9617 HTTPS://SITES.GOOGLE.COM/SITE/JOURNALOFCOMPUTING/ 86 such that m i b x C i n j a j T t it itj i ,..., 2 , 1 , 1 1 = ≤ ∏ ∑ = = (2.5) n j x j ,..., 2 , 1 , 0 = > ∏ ∑ = = ∈ = n j a j T t t M tj x C x S a b C Z 1 1 0 0 0 min ) , , ( max such that m i b x C i n j a j T t it itj i ,..., 2 , 1 , 1 1 = ≤ ∏ ∑ = = (2.6) n j x j ,..., 2 , 1 , 0 = > Since b i in the model (2.4), (2.5), (2.6 ) may not be equal to the constant 1 then dividi ng the constraint coeffi- cients C it by b i i ∀ then it is transformed to the stan- dard form ∏ ∑ = = ∈ = n j a j T t t L tj x C x S a b C Z 1 1 0 0 0 min ) , , ( min such that () () m i x b C n j a j T t i it itj i ,..., 2 , 1 , 1 1 1 1 = ≤ ∏ ∑ = = − (2.7) n j x j ,..., 2 , 1 , 0 = > ∏ ∑ = = ∈ = n j a j T t t U tj x C x S a b C Z 1 1 0 0 0 min ) , , ( max such that () () m i x b C n j a j T t i it itj i ,..., 2 , 1 , 1 1 1 1 = ≤ ∏ ∑ = = − (2.8) n j x j ,..., 2 , 1 , 0 = > ∏ ∑ = = ∈ = n j a j T t t M tj x C x S a b C Z 1 1 0 0 0 min ) , , ( max Such that () () m i x b C n j a j T t i it itj i ,..., 2 , 1 , 1 1 1 1 = ≤ ∏ ∑ = = − (2.9) n j x j ,..., 2 , 1 , 0 = > Now our main object ive is to find the minimu m value of Z L and maximum value of Z U against all possible values on S and such that Z M should give the value of Z against all possible values of S which will approximately the be st possible values in between Z L and Z U To derive the minimum value of the model (2.4) against all possible values on S we can set C 0t to t C 0 to L t C 0 and exponent as L tj A 0 Hence the model (2.4) can be transformed to the form ∏ ∑ = = = n j A j T t L t L L tj x C x Z 1 1 0 0 0 min such that () m i x B C n j A j L i T t L it L itj i ,..., 2 , 1 , 1 1 1 1 = ≤ ∏ ∑ = − = (2.10) n j x j ,..., 2 , 1 , 0 = > Similarly to find the minimum value of the model (2.5) against all such possible values of S, then the ratio i it b C is minimum wh en the value of it C and i b are set to U it C and U i B for all i. Under this model (2.5) becomes ∏ ∑ = = = n j A j T t U t U U tj x C x Z 1 1 0 0 0 max such that () m i x B C n j A j U i T t U it U itj i ,..., 2 , 1 , 1 1 1 1 = ≤ ∏ ∑ = − = (2.11) n j x j ,..., 2 , 1 , 0 = > The minimum value of model (2.6) can be calculated at the corresponding midd le counter part of t he parameter by transforming in the form ∏ ∑ = = = n j A j T t M t M M tj x C x Z 1 1 0 0 0 min such that () m i x B C n j A j M i T t M it M itj i ,..., 2 , 1 , 1 1 1 1 = ≤ ∏ ∑ = − = ( 2 . 1 2 ) n j x j ,..., 2 , 1 , 0 = > JOURNAL OF COMPUTING, VOLUME 2, ISSUE 1, JANUARY 2010, ISSN 2151-9617 HTTPS://SITES.GOOGLE.COM/SITE/JOURNALOFCOMPUTING/ 87 3 Dual form of GPP Since model (2.5) is the conventional geometric pro- gramming problem then it can be solved directly by using primal based algorithm or dual based algorithm[19]. Me- thods due to Rajgopal and Bricker[22], Beightler and Phil- lips[2] and Duffin[8] projected in their analysis that the dual problem has the desirable features of being linearly constrained and having an objective function with struc- tural properties with more su itable solution. According to Liu[16] the model (2.5) can be transformed to the cor re- sponding dual geometric problem as () ( ) [] ∏∏ ∏ == = − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = 0 0 11 1 0 1 0 0 max T t m t t t w it i L i L it w t L t L i it t w w B C w C w Z such that 1 0 1 0 = ∑ = T t t w (3.1) ∑∑ == = = m i T t it itj i n j w a 11 ,...., 2 , 1 , 0 i t w it , , 0 ∀ ≥ () ( ) [] ∏∏ ∏ == = − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = 0 0 11 1 0 1 0 0 max T t m t t t w it i U i U it w t U t U i it t w w B C w C w Z such that 1 0 1 0 = ∑ = T t t w (3.2) ∑∑ == = = m i T t it itj i n j w a 11 ,...., 2 , 1 , 0 i t w it , , 0 ∀ ≥ () ( ) [] ∏∏ ∏ == = − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = 0 0 11 1 0 1 0 0 max T t m t t t w it i M i M it w t M t M i it t w w B C w C w Z such that 1 0 1 0 = ∑ = T t t w (3.3) ∑∑ == = = m i T t it itj i n j w a 11 ,...., 2 , 1 , 0 i t w it , , 0 ∀ ≥ The model (3.1),(3.2) and (3.3 ) are the usual dual problem and it can be solved using the method relating to the dual theorem. 4 Illustrative Examples To illustrate the methodology proposed in this paper for solving a GPP with multiple parameters of cost, con- straint coefficients and exponents of the decision vari- ables a few numerical examples are considered. Example:1 Let us consider th e geometric programming probl em which has the following mathematical form: () 3 2 1 2 1 1 3 ) 4 , 3 , 2 ( 2 ) 1 , 2 , 3 ( 1 40 40 30 , 20 , 10 : min t t t t t t t t t + + − − − − (4.1) subject to 1 ) 6 , 4 , 2 ( 1 3 ) 3 , 4 , 5 ( 2 2 2 2 1 ≤ + − − − − − − t t t t 0 , , 3 2 1 > t t t According to the model (3.1),(3.2) and (3.3) the problem can be transformed to its correspond dual program as () () 12 11 12 11 03 02 01 12 11 12 11 11 03 02 01 1 2 40 40 10 : max w w w w w w w L w w w w w w w w w Z + + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = subject to 1 03 02 01 = + + w w w (4.2) 0 2 3 11 03 02 01 = − + + − w w w w 0 5 2 2 12 11 03 02 01 = − − + + − w w w w w 0 12 03 01 = − + − w w w (4.3) 0 , , , , 12 11 03 02 01 ≥ w w w w w () () 12 11 12 11 03 01 12 11 12 11 11 03 2 02 01 1 6 40 40 30 : max w w w w w w w U w w w w w w w w w Z + + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = subject to 1 03 02 01 = + + w w w (4.4) 0 2 11 03 02 01 = − + + − w w w w 0 3 2 4 12 11 03 02 01 = − − + + w w w w w 0 12 03 01 = − + − w w w 0 , , , , 12 11 03 02 01 ≥ w w w w w JOURNAL OF COMPUTING, VOLUME 2, ISSUE 1, JANUARY 2010, ISSN 2151-9617 HTTPS://SITES.GOOGLE.COM/SITE/JOURNALOFCOMPUTING/ 88 () () 12 11 12 11 03 02 01 12 11 12 11 11 03 02 01 1 4 40 40 20 : max w w w w w w w M w w w w w w w w w Z + + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = subject to 1 03 02 01 = + + w w w (4.5) 0 2 2 11 03 02 01 = − + + − w w w w 0 4 2 3 12 11 03 02 01 = − − + + w w w w w 0 12 03 01 = − + − w w w 0 , , , , 12 11 03 02 01 ≥ w w w w w Using LINGO the dual solution of th e optimal objectiv e values of Z can be obtaine d as: Z L =125.9045, for * 01 w = 0.1410885; * 02 w = 0.5767344; * 03 w = 0.2821770; * 11 w = 0.2178230; * 12 w =0.1410885, Z M = 194.9390, for * 01 w = 0.1456535; * 02 w = 0.5266261; * 03 w = 0.3277204; * 11 w =0.2815197; * 12 w = 0.1820669; Z U = 296.2627, for * 01 w = 0.1537485; * 02 w = 0.4362556; * 03 w =0.40999 59; * 11 w = 0.3462515; * 12 w = 0.2562475 In the constrained geometric progr am ming problem , the dual optimal solu- tions * w provide weights of the terms in the constraints of the trans- formed primal pr oblem. The correspondi n g primal solution of t he geometric programm ing problem is obtained for L Z as * 1 t = 1.305470; * 2 t = 1.390561; * 3 t =0.4892672, for M Z as * 1 t = 1.804540; * 2 t = 1.422246; * 3 t = 0.6223022 and for U Z as * 1 t =2.380155; * 2 t =1.35740; * 3 t = 0.9398071 The derived optimal solution s for the lower, middle and up- per parts of the multiple parameters are the best possible values as represented by Liu[16]. In the next example we shall set the right hand side of the constrained as the multiple pa rameter. Example:2 Let us consider the geometric programming problem with multiple parameters in objective functi on: () 2 3 ) 1 , 2 , 3 ( 2 2 1 1 4 3 1 2 ) 2 , 3 , 4 ( 1 ) 7 , 5 , 3 ( 3 , 2 , 1 : min − − − − − − − − − − + x x x x x x x x (4.6) such that ( ) 5 , 4 , 3 ) 3 , 5 . 2 , 2 ( 1 3 1 1 3 3 1 ≤ + − − x x x x 1 ) 4 , 5 . 3 , 3 ( 4 2 2 1 2 4 ) 3 , 2 , 1 ( 3 1 2 ≤ + − − − − − x x x x x x 1 x , 2 x , 3 x , 0 4 > x Now the corresponding dual pr ogram is formulated as fol- lows: () () () () 22 21 22 21 12 11 12 11 02 01 22 21 22 21 12 11 12 11 02 01 3 1 3 1 3 2 3 1 : max w w w w w w w w w w L w w w w w w w w w w w Z + + + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = (4.7) subject to 1 02 01 = + w w 0 2 3 2 4 22 12 11 02 01 = + − + − − w w w w w 0 3 22 21 02 01 = + − − − w w w w 0 2 21 12 11 02 01 = − − + − w w w w w 0 2 22 21 01 = + − − w w w 0 , , , , , 22 21 12 11 02 01 ≥ w w w w w w () () () () 22 21 22 21 12 11 12 11 02 01 22 21 22 21 12 11 12 11 02 01 4 1 5 1 5 3 7 3 : max w w w w w w w w w w U w w w w w w w w w w w Z + + + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = (4.8) 1 02 01 = + w w 0 2 3 2 2 22 12 11 02 01 = + − + − − w w w w w 0 22 21 02 01 = + − − − w w w w 0 3 2 21 12 11 02 01 = − − + − w w w w w 0 2 22 21 01 = + − − w w w 0 , , , , , 22 21 12 11 02 01 ≥ w w w w w w JOURNAL OF COMPUTING, VOLUME 2, ISSUE 1, JANUARY 2010, ISSN 2151-9617 HTTPS://SITES.GOOGLE.COM/SITE/JOURNALOFCOMPUTING/ 89 () () () () 22 21 22 21 12 11 12 11 02 01 22 21 22 21 12 11 12 11 02 01 5 . 3 1 4 1 4 5 . 2 5 2 : max w w w w w w w w w w M w w w w w w w w w w w Z + + + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = (4.9) subject to 1 02 01 = + w w 0 2 3 2 3 22 12 11 02 01 = + − + − − w w w w w 0 2 22 21 02 01 = + − − − w w w w 0 2 2 21 12 11 02 01 = − − + − w w w w w 0 2 22 21 01 = + − − w w w 0 , , , , , 22 21 12 11 02 01 ≥ w w w w w w The optimal dual and primal soluti on of the geometric program is as follows. Dual Z L = 47.47193 for * 01 w = 0.833333; * 02 w = 0.166666; * 11 w = 0. 9236831E - 07; * 12 w = 0; * 21 w =0.5; * 22 w = 1.83333 the corresponding primal optimal variables ar e * 1 x = 0.3205667; * 2 x = 1.481980; * 3 x = 1.064722; * 4 x =1.719745 . Dual Z M = 44.53226 for * 01 w = 0.8571429; * 02 w = 0.1428571; * 11 w = 0.1360334E 06; * 12 w = 0; * 21 w = 0.2857142; * 22 w = 1.428571 and its corresponding primal optimal variable s are * 1 x =0.2482863; * 2 x = 3.164843; * 3 x = 1.128281; * 4 x =1.220395. Dual Z U = 23.22874 for * 01 w =0.8751308; * 02 w = 0.1248692; * 11 w =0.5231659E03; * 12 w =0.2513079; * 21 w = 0.1248 692; * 22 w =1.124869 and its corresponding prim al optimal variables are * 1 x = 0.1314416; * 2 x = 60.08292; * 3 x = 1.524756; * 4 x =0.2167734 This example demonstrates that the objective values remains in the required range when th e parameters are multiple in nature. 5 Conclusions From 1960 geometric programming problem has under- gone several changes. In most of the engineering pr ob- lems the parameters are conside red as deterministic. In this paper we have discussed the problems with multiple parameters. In the above discussed two examples it is understood that the value of the objective remain within the range for the multiple parameters in exponent, cost and constrained. In the second example the objective val- ues are in decreasing order due to the exponent of the decision variable are consid ered in decreasing order. Geometric programming has already shown its powe r in practice in the past. In many real w orld geometric pro- gramming proble m the parameters may not be known precisely due to insufficient informations and hence this paper will help the wider applications in the field of en- gineering problems. 6 Acknowledgment s The authors are very much thankful to the anonymous referees for their valuable suggest ion for the improve- ment of the paper. 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[25] J.Zhu, K.O.Kort anek and S.Huang: Controlled dual perturba- tions for central path traject ories in geometric programming. Euro- pean Journal of Operational Rese arch 73(1992)524-531. Dr.A.K.Ojha: Dr.A.K.Ojha received a Ph.D(mathematics) from Utkal University in 1997. Currently he is an Asst.Prof. in Mathematics at I. I.T. Bhubaneswar, India. He is performing research in Nural Network, Geomet ric Pro- gramming, Genetical Algorithem, and Particle Swar m Optimization. He has served more than 27 years in differ- ent Govt. colleges in the state of Orissa. He has published 22 research papers in different journals and 7 books for degree students such as: Fortran 77 Programming, A text book of modern alge bra, Fundamentals of N umerical Analysis etc. K.K.Biswal: Mr.K.K.Biswal received a M.Sc.(Mathematics) from Utkal University in 1996. C ur- rently he is a lecturer in Mathematics at CTTC, Bhu- baneswar, India. He is performing res earch works in Geometric Progra mming. He is served more than 7 years in different colleges in the state of Orissa. He has pub- lished 2 research papers in different journals.