Lexicographic Multi-objective Geometric Programming Problems

IJCSI International Journal of Computer Science Issues, Vol. 6, No. 2, 2009 ISSN (Online): 1694-0784 ISSN (Print): 1694 -0814 20 Lexicographic Multi-objective Geometric Programming Problems Dr.A. K. Ojha 1 and K. K. Biswal 2 1 School of Basic Sciences, IIT Bhuba neswar, Orissa, Pin-751013, India 2 Department of Mathematics , CTTC Bhubaneswar, B-36, Chandaka Industrial Area, B hubaneswar, Orissa, Pin-751024, India Abstract A Geomet ric programm ing (GP) is a t ype of mathematical problem charac terized by objective and constraint functions th at have a special form. Many methods ha ve been dev eloped to sol ve large scal e engineering d esign GP problem s. In this paper GP technique has been used to solve multi-objective GP problem as a vector optimization problem. The duality theory for lexi cographic geom etric program ming has been develope d to solve the pro blems with posy nomial in objectives and constraints. Keywords : Lexicographi c minimi zation, Ge ometric programming, duality theory , vector minimization and vector maximization. 1. Introduction In many real-life optimization problems, multiple objectives have been taken into accoun t, which may be related to the economical, social and environmental aspects of optimization problems. The multiple objectives are usually inco mmensurate and in conflict with one another. In general, a multiple objective optimization problem does not have a single solution that could opt imize all obje ctives sim ultaneously. It never search for o ptimal solution but for efficient solution that can b est suited compromise solution to all multiple objectives. Tho ugh the geometric programming t echnique due to D uffin et al.[3] helps to solve various types of nonlinear sing le objective posynomi al problems but there are very fe w work ha ve been made in this direction to solve multiple objectiv e GP problem s. Lexicograp hic optim ization approach is one such technique to handle multiple objective GP problem. Several mathematical and game theoretic applications of n onlinear lexi cographic optim izations are reported by Behringe r[1]. Im pressed upon the work of Behringe r, Nijkam p[6] applied the lexico graphic optimi zation technique in a land use probl em for industrial activities in a newly created industrial area in a Rhine-delta region near Rotterdam. Application of linear lexico graphic due to Isermann [4] and Turnovec[8] strengthen to handle scal ar valued optimization problems. Biswal[2] u sed fuzzy programmi ng[10] to sol ve multi-ob jective geom etric problem where as lexicographic ord er and duality has been studied by Martinez[5]. In this paper we have applied lexi cographic geometri c programm ing technique to solve special type of multi-objective optimi zation problem . The organization of the paper is as follows: Following introduction the definition of multi-objective g eometric programmi ng and lexicog raphic optim ization have been discussed in S ection-2 and 3 respectively. Definiti on of lexicogra phic geom etric program ming has been discussed in S ection 4 a nd the num erical exampl es have been incorporated in Sectio n 5. Finally the conclusion has been prese nted in Section 6. 2. Multi-objective ge ometric programming A multi-objective geometric programming problem can be defined as: Find x = (x 1 , x 2 ,… ,x n ) T so as to p k C x g ko tj k T t n j a j t k k x ,..., 2 , 1 , ) ( : min 1 1 0 0 0 = = ∑ ∏ = = (2.1) subject to IJCSI International Journal of Computer Science Issues, Vol. 6, No. 2, 2009 ISSN (Online): 1694-0784 ISSN (Print): 1694 -0814 21 m i C x g i itj T t n j d j it i x ,..., 2 , 1 , 1 ) ( 1 1 = ≤ = ∑ ∏ = = (2.2) n j x j ,..., 2 , 1 , 0 = > (2.3) where C k0t for all k and t are positive real numbers and d itj and a k0tj are real numbers for all i, k, t, j. T k0 =number of terms present in the k th objective function. T i =number of terms present in the i th constraint. In the abov e multi-ob jective geom etric prog ramming problem there are p num ber of m inimization type objective functions, m number of inequality type constraints and n number of strictly positive d ecision variables. Let us define {} ) ( ),...., ( ), ( ) ( 0 20 10 x g x g x g x F p = and {} 1 ) ( ,...., 1 ) ( , 1 ) ( ) ( 2 1 − − − = x g x g x g x G m Now the above optimization problem can be rewritten as: { } m n x G R x x F lex 0 ) ( , : ) ( : min ≤ ∈ (2.4) which is call ed lexic ographic ge ometri c programmi ng (LGP) problem. 3. Lexicographic optimization problem Let us denote n O an n-dim ensional zero vec tor and n m O × an m×n zero matrix. An inequality of the type ≥ x n O means ≥ x n O , but x ≠ n O A vector n R x ∈ is said to be lexicograp hically non negative if either x = n O or its fir st non-zero c omponent is positive and we denote it by n O x lex ≥ Similarly a non zero vector n R x ∈ is said to be lexicographically non positive if its first non zero component is negati ve and it i s denoted by n O x lex < An m×n matrix A is called lexicographically non negative if all its columns are lexico graphically non negative and we denote it as n m O A lex × ≥ In the simil ar manner we ca n define lexi cographically non positive and lex ico-graphically ne gative vector and its corresponding matrix. Let F be an p -dimensiona l vector valued function defined on n R and . n R X ⊂ A vector X x ∈ * is said to be a lexicographically minim al point of F with res pect to X if for any X x ∈ such that ) ( ) ( * x F x F lex ≤ The problem of findin g lexicographi cally minimum point of F with respect to X is called lexicograp hically minimizi ng problem which is denoted as : {} X x x F lex ∈ : ) ( : min (3.1) Simila rly a point X x ∈ is said to be lexicograph ically maximum point of F with res pect to X if ) ( ) ( x F x F lex ≥ and this problem is called lexicogra phically maximizi ng problem given by { } X x x F lex ∈ : ) ( : max (3.2) If we assume { } m n O x G R x X ≤ ∈ = ) ( : (3.3) then the lexic ographically minimizing p roblem can be defined as, { } m O x G x F lex ≤ ) ( : ) ( : min (3.4) A lexicographically mini mum point of F with respect to (3.3) is called an optimal solutio n to the problem given by (3.4). The problem given by (3.4) is cal led convex optimization if all the co mponents of F and G are convex functio ns. 4. Lexicographic geometric Programming The lexicographic multi-obj ective geometric programm ing defined by { } m n O x G R x x F lex ≤ ∈ ) ( , : ) ( : min where the fu nctions are defined by (2.1), (2. 2) and (2.3 ). Now we will prove the following theorem for the existence of uniqu e optimal solution o f the lexicographi c optimi zation GP problem . Theorem: - If the primal problem of the geometric programmi ng is consiste nt and its d ual program has a maximizing point with strictly positive components t hen the primal pr oblem of geom etric program ming has a unique opti mal solut ion if and only if the rank of its exponent matrix is equal to the number of columns. Proof : - From the duality theorem due to Duffin[3], we know that for each optimization point x * for the primal problem t here exist a maximi zing point w * of dual program whi ch is given by the f ollowing eq uations. . ,.... 2 , 1 , ,... 2 , 1 ), ( 0 * * 0 1 0 0 k t k n j a j t k T t p k w v w x C tj k = = = ∏ = (4.1) IJCSI International Journal of Computer Science Issues, Vol. 6, No. 2, 2009 ISSN (Online): 1694-0784 ISSN (Print): 1694 -0814 22 . ,.... 2 , 1 , ,... 2 , 1 , ) ( * * 1 i i it n j d j it T t m i w w x C itj = = = ∏ = λ (4.2) Taking logari thm on both sides of equat ion (4.1) an d (4.2) we have 0 0 * 0 2 1 ,.... 2 , 1 , ,.... 2 , 1 , ) ( ln ) .... ln( 0 2 0 1 0 k t k t k a n a a T t p k C w v w x x x tn k t k t k = = = . ,.... 2 , 1 , ,.... 2 , 1 , ) ( ln ) .... ln( * * 2 1 2 1 i i it it a n a d T t m i w C w x x x itn it it = = = λ which can be expressed as k0 k0t n k0tn 2 k0t2 1 k0t1 T ..., 2, 1, t 2,....p; 1, k ); ln( ln x a . ln x a ln x a = = = + … + + β (4.3) And 2,...T 1, t 2,....m; 1, i ); ln( ln x d .... ln x d ln x d i it n itn 2 it2 1 it1 = = = + + + β ( 4 . 4 ) where 0 0 * * 0 ,.... 2 , 1 , ) ( k t k kot t k T t C w v w = = β and . ,...., 2 , 1 , ,.... 2 , 1 , ) ( * * m i T t w C w i i it it it = = = λ β Now by substituting z j = ln x j ; j = 1, 2,…., n 0 0 0 ,....., 2 , 1 , ,.... 2 , 1 , ) ln( k t k t k T t p k = = = γ β and i it it T t m i ,...., 2 , 1 , ,.... 2 , 1 , ln = = = γ β the equations (4.3) and (4 .4) reduces to 0 0 0 2 2 0 1 1 0 ,..., 2 , 1 , ... k t k n tn k t k t k T t z a z a z a = = + + + γ (4.5) and i it n itn it it T t m i z d z d z d ,..., 2 , 1 , ,..., 2 , 1 , ... 2 2 1 1 = = = + + + γ (4.6) If we assume ∑ = = m i i T T 1 then we will have T number of equations and n varia bles, which is exactly the dimension of the expo nent matri x. Writing the equations in the matrix form we have γ = Az (4 .7) where A is the exponent matrix of dimension T×n and T ≥ n. From the basic knowl edge of linear algebra the system of e quations (4.7) has a unique sol ution if a nd only if the rank of the matrix A is equal to the number of its columns. With the substitu tion n j x z or x e j j j z j ,..., 2 , 1 , ln = = = Our lexicogr aphic optimizati on problem can be expressed as. { } m z G z F lex 1 ) ( : ) ( : min ≤ (4.8) where F(z) is an p-di mensional vector valued funct ion with ∑ = ∑ = = 0 1 0 1 0 0 ) ( k n j j tj k T t z a t k k e C z g , k=1,2,…. ,p and ∑ = ∑ = = i n j j itj T t z d it i e C z g 1 1 ) ( , i=1,2,….,m Due to the m onotonicity of logarit hm function t he problem (4.8) can be e xpressed as: { } m O z G z F lex ≤ ) ( ln : ) ( ln : min ( 4 . 9 ) with { } ) ( ln ),...., ( ln ), ( ln ) ( ln 0 20 10 z g z g z g z F p = and { } ) ( ln ),...., ( ln ), ( ln ) ( ln 2 1 z g z g z g z G m = Introduci ng new varia bles ∑ = ∑ = = ∑ = 0 1 0 1 0 1 0 0 k n j j tj k n j j z tj k a T t z a t k t k kot e C e C w , k=1,2,…,p (4.10) A n d ∑ = ∑ ∑ = = = i n j j itj n j j itj T t z d it z d it ki kit e C e C u w 1 1 1 , k=1,2,….p, i= 1,2,….,m (4.11) After suitable transforma tion [3 ] the dual problem associated with the lexicographic geom etric programming problem can be obtain ed. ) ( : max w V lex such that 1 0 1 0 = ∑ = k T t t k w , k=1,2,….,p (4.12) m i n j p k d w a w ki T t m i T t itj kit tj k t k ,...., 2 , 1 , ,...., 2 , 1 , ,...., 2 , 1 , 0 0 11 1 0 0 = = = = + ∑∑ ∑ == = IJCSI International Journal of Computer Science Issues, Vol. 6, No. 2, 2009 ISSN (Online): 1694-0784 ISSN (Print): 1694 -0814 23 ( 4 . 1 3 ) , 1 ∑ = = i T i kit ki w u k=1,2,….,p, i= 1,2,….,m (4.14) where ) ( w V k is the p-dim ensional vect or valued function of the fo rm ∏ ∏∏ ∑ = == = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = m i u ki w m i T t kit it t k T t kot t k k ki kit i k u w d w C w V 1 11 0 1 0 0 ) ( , k=1,2,….,p (4.15) According to the theory of ge ometric pro gramming problem (4.12) is call ed the normali ty conditions, (4.13) is called orthogonality co ndition and (4 .15) is the dual function. The number 1 1 1 0 − − + = ∑ ∑ = = n T T d m k i p k k is the degree of diffic ulty. As the problem with zero degree of difficulty is easily solvable then the dual problem can be solved to get the maximi zing vector * w . Since the vector * w is the unique solut ion to the dual const raints, it is also the maximi zing vector fo r the dual problem . Using this dual op timizing vector the op timal solution * x to the primal probl em can be deter mined by using follo wing relati onships. ) ( .... * 0 * 0 2 1 0 0 2 0 1 0 t k k t k a n a a t k w V w x x x C tn k t k t k = , k=1,2,….,p (4.16 ) , .... * * 2 1 2 1 ki it d n d d it u w x x x C itn it it = k=1,2,…, p; i=1,2,… ,m ( 4 . 1 7 ) where ∑ = = i T i kit ki w u 1 * * , k=1,2,…., p (4.18) 5. Numerical Example Example: Let us c onsider a numerical example { } 1 2 1 1 5 3 3 2 1 1 20 2 3 1 2 1 1 10 ) ( , ) ( ) ( : min − − − − − − − − + = = = x x x x x x g x x x x g x F lex subject to 10 3 2 2 3 2 1 ≤ + x x x x x 0 , , 2 3 2 1 3 1 > ≤ x x x x x Solution Proc edure of LGP problems . Step 1: At first the objective functions of the multi-objectiv e GP problem are ranked accordi ng their priority. Let us assume that the first objective function is in priority on e i.e. P 1 and the second o bjective functi on in priorit y 2 i.e. P 2 and so on, and p th objective function in priority p i.e P p . Step 2: Then first objective fun ction g 10 (x) is minimized subject to all the original constraints. Let th e minimum of the first objective be ) 1 ( 10 g at x (1) . Then we m ove to step 3. Step 3: Then the second object ive function g 20 (x) is minimi zed subject to the or iginal constraints with one addition al constraint, i.e. ≥ ) ( 10 x g ) 1 ( 10 g Let the minimum value of th e second objective function be ) 2 ( 20 g at x (2) . Then we move to next step. Step 4: In step 4 third objecti ve function g 30 (x) is minimized subject to the or iginal constraint with two add itional constraints, i.e. ≥ ) ( 10 x g ) 1 ( 10 g ≥ ) ( 20 x g ) 2 ( 20 g Same procedure is repeated for all t he objective functions. Step 5: Finally, the last objectiv e function is minimized su bject to all the original cons traints plus addition al p-1 constraints. ≥ ) ( 10 x g ) 1 ( 10 g , ≥ ) ( 20 x g ) 2 ( 20 g , …. ≥ − ) ( 0 , 1 x g p ) 1 ( 0 , 1 − − p p g Let A be the exponent matrix. ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ − − − = 1 0 1 1 1 0 2 1 1 2 1 1 A The rank of the matrix is 2 which is less than the number of columns. The dual program of the function ) ( 10 x g is as IJCSI International Journal of Computer Science Issues, Vol. 6, No. 2, 2009 ISSN (Online): 1694-0784 ISSN (Print): 1694 -0814 24 () 112 111 211 112 111 101 112 111 211 112 111 101 5 . 0 10 1 10 1 1 ) ( : max w w w w w w w w w w w w w V + + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = Subject to: 1 101 = w 0 211 111 101 = + + − w w w 0 112 111 101 = + + − w w w 0 2 211 112 111 101 = + + + − w w w w The solution of the dual pr ogram gives 1 101 = w , ... 666 . 0 111 = w , 3333334 . 0 112 = w , 3333334 . 0 211 = w and the mean value of dual v(w * ) = 0.15. Using t he primal dual relationship we have the op timal solution of g 10 (x) are x 1 * =0.9086967, x 2 * = 1.514494, x 3 * = 2.200954 and its primal optimal solution is 0.15. Similarly the dual program of g 20 (x) is defined as, () 212 211 211 212 211 202 201 212 211 211 212 211 202 201 5 . 0 10 1 10 1 1 1 ) ( : max w w w w w w w w w w w w w w w V + + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = Subject to: 1 202 201 = + w w 0 221 211 202 201 = + + − − w w w w 0 3 221 211 202 201 = + + − − w w w w 0 2 5 221 212 211 201 = + + + − w w w w The solution of t he problem gives its optim al value 1 * 2 10 4316470 . 0 ) ( − × = w V with * 201 w = 0.6666667, * 202 w = 0.3333…, * 211 w = 1, * 212 w = 1.3333…, * 221 w = 0 Using prim al dual relations hip we have x 1 = 3.020273 , x 2 = 23.01163, x 3 = 0.2483217 6. Conclusions In this chapt er solution procedure of lexicogra phic GP has been presented. Unless th e objective functions ranked properl y, lexicographi c solution m ay not be acceptable to a design engine er. If there are p num ber of objective func tions one m ay formulate p! no of ways priority, which is a very difficult task fo r a design engineer. Als o sometimes m ore than one objective functions remain in a pr iority. Unless the priority is proper solutio n of a real life probl em gives som e abnormal result. To set the proper rankin g, met hod of Analytic Hierarchy Proces s (AHP) m ay be adopted. Two popular met hod of AHP by Saaty[7] nam ely row- column Addp otion m ethod and Eigen-val ue method i s used to find the proper ranking (weights) of th e objective func tion. If the weig hts of the objecti ve function can be estimated, then using the weights multi- objective GP problem can be converted to a single objective GP problem and so lved. References [1] Behringer,F.A ., Lexicographic quasiconcave multiobject ive program -ming, Zeits chrift fier operati on research 21, 103-116, 1977. [2] Biswal,M.P., Fuzzy Programming technique t o solve multi-object ive geomet ric programm ing proble m, Fuzzy sets and systems 51,67-71 , 1992. [3] Duffin R.J.,Peterson E.L.and Zener C.M., Programming Theory and Application, W iley, New York. ,196 7 [4] Iserma nn,H., Linear l exicograp hic optimizat ion, OR-spektrum 4,223 -228. ,19 82 [5] Martinez-Legaz,J .E., Lexicographi c order and duality in multiob jectve programming , European Journal of operation al research 33, 342-348 , 1988. [6] Nijkamp,P. Environmental policy analysis in operational methods and models, Wiley, New York. ,1980 [7] Saaty T.L., A Scaling Method for Priorities in Hierarchical Structures, Journal of Mathematical pshycholo gy, 15, 234-2 81, 1977 [8] Turnovec F., Le xicographi c optimization problems in production schedu ling in optimization , Theory, methods, Applications, Vol-I Dum Tech- niky CSVTS Praha, Prague, 295- 306, 1985 [9] Zimmerm ann,H.J. ,Fuzzy programmi ng and linear programming with several objective functi ons. Fuzzy sets and system s 1, 46-55. ,1 978 [10] Zimm ermann,H.J . Fu zzy set theory and its applications, 2nd ed Kluwer Academic ,publishers, Dordrecht-Bos ton, 1990.