MODIFIED GEOMETRIC PROGRAMMING PROBLEM AND ITS APPLICATIONS

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1 J. Appl. Math. & Computing Vol ), No. 1-2, pp MODIFIED GEOMETRIC PROGRAMMING PROBLEM AND ITS APPLICATIONS SAHIDUL ISLAM AND TAPAN KUMAR ROY Abstract. In this paper, we propose unconstrained and constrained posynomial Geometric Programming GP) problem with negative or positive integral degree of difficulty. Conventional GP approach has been modified to solve some special type of GP problems. In specific case, when the degree of difficulty is negative, the normality and the orthogonality conditions of the dual program give a system of linear equations. No general solution vector exists for this system of linear equations. But an approximate solution can be determined by the least square and also max-min method. Here, modified form of geometric programming method has been demonstrated and for that purpose necessary theorems have been derived. Finally, these are illustrated by numerical examples and applications. AMS Mathematics Subject Classification: 90B05, 90C10, 65K18 Key words and phrases : Geometric programming, residual vector, least square method, max-min method, modified geometric programming problem. 1. Introduction Since late 1960 s, Geometric Programming GP) has been known and used in various fields like OR, Engineering sciences etc.). Duffin, Peterson and Zener [5] and Zener [14] discussed the basic theories on GP with engineering application in their books. Another famous book on GP and its application appeared in 1976 [3]. There are many references on applications and the methods of GP in the survey paper by Ecker [6]. They described GP with positive or zero degree Received June 13, Revised April 17, Corresponding author. This research was supported by C.S.I.R. junior research fellowship in the Department of Mathematics, Bengal Engineering College A Deemed University). This support is great fully acknowledged. c 2005 Korean Society for Computational & Applied Mathematics and Korean SIGGAM. 121

2 122 Sahidul Islam and Tapan Kumar Roy of difficulty. But there may be some problems on GP with negative degree of difficulty. Sinha et al. [13] proposed it theoretically. Abou-El-Ata and his group applied modified form of GP in inventory models [1], [7]). Authors [8] also applied MGP with negative degree of difficulty in an inventory model. In this paper we have proposed unconstrained / constrained GP and modified form of Geometric Programming MGP) problem with negative or positive integral degree of difficulty. The modified form of geometric programming method has been demonstrated and for that purpose necessary theorems have been derived. The least square and max-min methods are described to get approximate solutions of dual variables of GP/MGP problems with negative degree of difficulty. Finally, these are illustrated by numerical examples and applications. 2. Unconstrained problem 2.1. Geometric Programming GP) problem Primal program : Primal Geometric Programming PGP) problem is : Minimize gt) = C k m t α kj j 1) t j > 0, j =1, 2,...,m). Here C k > 0), k =1, 2,...,T 0 ) and α kj be any real number. It is an unconstrained posynomial geometric programming problem with the Degree of Difficulty DD) = T 0 m + 1). Dual program : Dual Programming DP) problem of 1) is : δ k =1, T 0 Maximize νδ) = α kj δ k =0, j =1, 2,...,m), δ k > 0, k =1, 2,...,T 0 ), T 0 Ck δ k ) δk 2) Normality condition) Orthogonality conditions) Positivity conditions)

3 Modified geometric programming and its applications 123 where δ =δ 1,δ 2,...,δ T0 ) T. Case I : For T 0 m+1, the dual program presents a system of linear equations for the dual variables, where the number of linear equations is either less than or equal to the dual variables. More or unique solution vector exist for the dual variable vectors [3]. Case II : For T 0 < m + 1, the dual program presents a system of linear equations for the dual variables, where the number of linear equations is greater than the number of dual variables. In this case generally no solution vector exists for the dual variables. However, one can get an approximate solution vector for this system using either the Least Square LS) or Max-Min MM) [4] method. These are applied to solve such a system of linear equations. Ones optimal dual variable vector δ are known, the corresponding values of the primal variable vector t is found from the following relations : m C k t α kj j = δ kν δ ), k =1, 2,...,T 0 ). 3) Taking logarithms in 3), T 0 log-linear simultaneous equations are transformed as δ α kj log t j ) = log k ν δ ) ), k =1, 2,...,T 0 ). 4) C k It is a system of linear equations in x j = log t j ) for j =1, 2,...,m. Since there are more primal variables t j than the number of terms T 0, many solutions t j may exist. So, to find the optimal primal variables t j, it remains to minimize the primal objective function with respect to reduced m T 0 0) variables. When m T 0 = 0 i.e. number of primal variables = number of log-linear equations, primal variables can be determined uniquely from log-linear equations Residual vector A system of linear equations say m equations and n unknown variables with m>n) takes the form α δ j = b i, i =1, 2,...,m) 5) i.e., Aδ = B, where A =[α ] m n,δ =[δ 1,δ 2,...,δ n ] T and B =[b 1,b 2,...,b m ] T. The residual vector is defined as R = Aδ B, 6) where R =[r 1,r 2,...,r m ] T.

4 124 Sahidul Islam and Tapan Kumar Roy Least Square LS) method Approximate solutions for the system of linear equations 5) can be determined by LS method [9]. Find δ j j =1, 2,...,n), which minimize R T R = ri 2. Setting the partial derivatives of ri 2 with respect to δ j, and equating them to zero, following n normal equations are obtained : α 2 i1)δ 1 + α i1 α i2 )δ α i1 α in )δ n = α i1 b i ), α i2 α i1 )δ 1 + α 2 i2 )δ α i2 α in )δ n = α i2 b i ),... α in α i1 )δ 1 + α in α i2 )δ α 2 in )δ n = α in b i ). It is a symmetric positive definite system of equations in δ j and determines the components of dual variables δ j j =1, 2,...,n) Max-Min MM) method Approximate solutions for the system of linear equations 5) can also be determined by MM method. By this method one can obtain the dual variable vector δ for which the absolutely largest component of the residual vector R= Aδ B) becomes minimum. i.e., MinMax r 1, r 2,..., r m )) 8) α i1 δ 1 + α i2 δ α in δ n b i = r i, i =1, 2,...,m). δ j 0, j =1, 2,...,n). Considering an auxiliary variable r, the above problem 8) is equivalent to Minimize r 9) α i1 δ 1 + α i2 δ α in δ n b i r, α i1 δ 1 α i2 δ 2... α in δ n + b i r, δ j 0, j =1, 2,...,n). } i =1, 2,...,m) 7)

5 Modified geometric programming and its applications 125 Denoting δj r = δ j, j =1, 2,...,m) and 1 r = δ 0, 9) becomes a Linear Programming Problem LPP) as follows : Maximize δ 0 10) α i1 δ 1 + α i2δ α inδ n b iδ 0 1, } α i1 δ 1 α i2 δ 2... α in δ n + b i δ 0 i =1, 2,...,m) 1, δ j 0, j =0, 1, 2,...,n). After calculating δ 0,δ 1,δ 2,...,δ n from this LPP 10), approximate solutions of dual variables δ j can be obtained. So, applying either LS or MM method dual variables can be determined. Then corresponding optimal dual objective function of 2) is found. Example 1 Unconstrained GP problem). Min Zt) =t 1 t 2 t t 3 t1 1 t t 3 a) t 1,t 2,t 3 > 0. Here DD = ) = 1< 0). It is a polynomial geometric programming problem with negative degree of difficulty. Geometric programming method : ) δ1 ) δ2 ) δ Maximize νδ )= b) δ 1 δ 1 + δ 2 + δ 3 =1, δ 1 δ 2 =0, δ 1 δ 2 =0, 2δ 1 + δ 2 + δ 3 =0, δ 1,δ 2,δ 3 > 0. It is a system of 4 linear equations with 3 unknowns variables. Approximate solutions of this system of linear equations by LS method are δ1 =0.333,δ 2 = 0.333,δ3 =0.333 and corresponding optimal dual objective value i.e. ν δ )= So for primal decision variables, the following system of non-linear equations is found t 1 t 2 t 2 3 =2.1522, 2t 3 t 1 1 t 1 2 =2.1522, 5t 3 =2.1522, δ 2 δ 3

6 126 Sahidul Islam and Tapan Kumar Roy and the corresponding system of log-linear equations is x 1 + x 2 2x 3 = , x 3 x 1 x 2 = , c) x 3 = , where x i = logt i i =1, 2, 3). Solving this sytem of linear equations c), the optimal primal variables t i = ex i ) i =1, 2, 3) are obtained. These are given in Table 1. Non-Linear Programming NLP) method : From Karush - Kuhn - Tucker necessary conditions, we have t 2 t 2 3 2t 3 t 2 1 t 1 t 1 t 2 3 2t 3 t 1 1 t 2 2t 1 t 2 t t 1 1 t 1 2 =0 2 = = 0 Solving this system of non-linear equations, the optimal solutions t 1,t 2,t 3 are obtained. These are given in the Table 1. It is seen that GP method gives better result than NLP method [2]. Table 1 Optimal solutions of Example 1 Methods t 1 t 2 t 3 Z t ) GP NLP Modified geometric programming MGP) problem Primal program : A special type which is a separable function) of the Primal Geometric Programming PGP) problem is m Minimize gt) = g i t) = U ik t) = C ik t α ikj 11) t > 0, i =1, 2,...,n; j =1, 2,...,m). Here C ik > 0)k =1, 2,...,T 0 ) and α ikj are real number. It is an unconstrained posynomial geometric programming problem with DD= nt 0 nm + 1).

7 Modified geometric programming and its applications 127 Dual program : According to Harari et al. [7], the Dual Programming DP) problem of 11) is as follows : n T 0 ) δik cik Maximize νδ) = 12) = 1 Normality conditions) α ikj =0, j =1, 2,...,m) Orthogonality conditions) > 0, k =1, 2,...,T 0 ), Positivity conditions) i =1, 2,...,n), where δ =δ 11,δ 12,...,,...,δ nt0 ) T. Case I : For nt 0 nm + n, the DP presents a system of linear equations for the dual variables. A solution vector for the dual variables exists [3]. Case II : For nt 0 <nm+ n, in this case generally no solution vector for the dual variables exists. However, using either the LS or MM method one can get an approximate solution vector for this system explained in & 2.1.3). Theorem If t is a feasible vector for the unconstrained PGP 11) and if δ is a feasible vector for the corresponding DP 12), then gt) n n νδ) Primal-Dual Inequality) Proof. The expression for gt) can be written as gt) = C ik m t α ikj. Here the weights are δ i1,δ i2,...,δ it0 and the positive terms are c i1 m t αi1j δ i1, c i2 m t αi2j δ i2,..., m c it0 t αit 0 j δ it0 for i =1, 2,...,n.

8 128 Sahidul Islam and Tapan Kumar Roy Now applying weighted A.M. - G.M. inequality, we get n m δ i1+δ i2+...+δ it0 ) m m c i1 t αi1j + c i2 t αi2j c it0 t αit 0 j δ i1 + δ i δ it0 ) n m c i1 t αi1j δ i1 δ i1 m c i2 δ i2 t αi2j δ i2... m c it0 t αit 0 j δ it0 δ it0 or or gt) ) n gt) n n T0 n T 0 m c ik t α ikj n ) T0 cik m T0 t α ikj δ ik [since from normality condition of 12) i.e., = 1 for i =1, 2,...,n] or = n T 0 ) n gt) n cik n ) δik m t T 0 [since from orthogonality condition of 12) i.e., i.e., This completes the proof. ) δik cik T0 α ikj α ikj =0, i =1, 2,...,n; j =1, 2,...,m)] gt) n n νδ).

9 Modified geometric programming and its applications 129 Theorem If t = t i1,...,t im ) for i = 2,...,n is a solution to the PGP 11), then the corresponding DP 12) is consistent. Moreover the vector δ =δi1,...,δ it 0 )i =1, 2,...,n) is defined by δik = u ikt ) g i t ), k =1, 2,...,T 0), where u ik t )=c ik m t α ikj which is the kth term of gt) for ith item is a solution for DP and equality holds in the primal-dual inequality i.e., gt )=n n νδ ). Proof. A typical term of gt) is u ik t) =c ik t α ik1 t i1 t α ik2 i2,...,t α ikm im, i =1, 2,...,n; k =1, 2,...,T 0). 13) So, u ik t) t = α ikj u ik. 14) t Since t =t i1,...,t im ) is a minimizer for gt) it follows that Using 14) we have gt ) t = u ik t ) t =0. 15) α ikj u ik t )=0. 16) Since g i t ) > 0, we can divide both sides of 16) by g i t ) to obtain uik t ) ) α ikj g i t =0. 17) ) Consequently, if we define δik = u ikt ) g i t ), then δ =δi1,...,δ it 0 ) satisfies the orthogonality condition for the DP and Also δik > 0. So the positivity condition is satisfied. Finally, T 0 δik = uik t ) ) g i t = g it ) ) g i t =1, i =1, 2,...,n) ) Hence the normality conditions hold. We conclude that the vector δ is feasible for the dual problem. So the DP is consistent.

10 130 Sahidul Islam and Tapan Kumar Roy Also Now we have g i t ) T 0 = u 1k t ) δ 1k = u 2kt ) δ 2k =...= u nkt ) δ nk, k =1, 2,...,T 0 ). gt ) n g ) i t ) = n = = = T 0 u1k t ) δ 1k T 0 ) δ c1k n T 0 δ 1k cik δ ik T0 δ 1k+δ 2k +...+δ nk ) ) δ 1k c 2k 1k u 2k t ) δ 2k ) δ δ 2k ) δ ik = νδ ) ) δ 2k... cnk 2k... unk t ) δ nk ) δ nk δ nk ) δ nk i.e., gt )=n n νδ ). Hence the equality holds in the Primal-Dual inequality. This implies that δ is an optimal dual variable of the DP with components δik = u ikt ) g i t ), i =1, 2,...,n; k =1, 2,...,T 0). This compltes the proof. We can get dual variables which optimize ν δ ) such that g i t) u ik t) = = n n νδ) n following theorem 2.2.2)

11 Modified geometric programming and its applications 131 i.e., So, c ik m = n νδ). u ik t) = n νδ). t α ikj = n νδ), i =1, 2,...,n; k =1, 2,...,T 0 ). 18) Taking logarithms in 18), the linear simultaneous equations are transformed as δ ) n ik νδ) α ikj logt ) = log. 19) c ik It is a system of linear equations in x = Logt ). Since there are more primal variables t than the number of terms nt 0, many solutions t may exist. So, to find the optimal primal variables t, it remains to minimize the primal objective function with respect to reduced nm nt 0 0) variables. When nm nt 0 =0, primal variables can be determined uniquely from log-linear equations. Example 2 Unconstrained MGP problem). Minimize Zt) =t 11 t 21 t t 31t 1 11 t t 31 + t 12 t 22 t t 32t 2 22 t t 32 t > 0, i =1, 2, ; j =1, 2, 3). This problem can be written as Minimize Zt) = C ik t α ikj 20) t > 0, i =1, 2; j =1, 2, 3), where C 11 = 1,C 12 =1,C 13 =2,C 21 =3,C 22 =5,C 23 =4, α 111 = 1,α 211 =1,α 121 = 2, α 221 =1,α 131 =1,α 231 = 3, α 112 = 1, α 212 = 1, α 122 =1,α 222 = 2,α 132 = 2,α 232 =1, α 113 = 0,α 213 =0,α 123 =1,α 223 =0,α 133 =0,α 233 =1. Dual Programming of MGP 20) is as follows : 2 3 Maximize vδ) = ) δik cik

12 132 Sahidul Islam and Tapan Kumar Roy δ 11 + δ 12 + δ 13 =1, δ 21 + δ 22 + δ 23 =1, δ 11 δ 12 =0, 2δ 11 + δ 12 + δ 13 =0, δ 11 2δ 12 =0, δ 21 δ 22 =0, δ 21 2δ 22 =0, 3δ 21 + δ 22 + δ 23 =0 where δ =δ 11,δ 12,δ 13,δ 21,δ 22,δ 23 ) T. There is a system of eight linear equations with six unknown variable. Applying LS method the above system of linear equations reduces to δ 11 δ 12 =0, 2δ 11 + δ 12 + δ 13 =0, δ 11 + δ 12 + δ 13 =1, δ 21 2δ 22 =0, 3δ 21 + δ 22 + δ 23 =0, δ 21 + δ 22 + δ 23 =1, δ 11,δ 12,δ 13,δ 21,δ 22,δ 23 > 0. Approximate solutions of this sytem of linear equations are δ11 = 0.333, δ12 =0.333, δ13 =0.333, δ21 = 0.25, δ22 =0.125, δ 23 =0.625 and optimal dual objective value i.e., ν δ )= The value of the objective function is gt )= = So the Theorem is verified. And following system of non-linear equations gives optimal primal variables t 11 t 21 t 2 31 =4.3856, t 12 t 22 t 2 32 =3.2925, 2t 31 t11 1 t 1 21 =4.3856, 2t 32 t12 1 t 1 22 =1.6463, 5t 31 =8.2313, 5t 32 = Solving this system of non-linear equations, optimal solutions are determined. The optimal solutions by MGP of this problem are given in the Table 2. Here optimal solutions of Example 2 by NLP are also given in the Table 2. It is seen that MGP method gives better than NLP method, which is expected.

13 Modified geometric programming and its applications 133 Table 2 Optimal solutions of Example 2 Methods t 11 t 21 t 31 t 12 t 22 t 32 Z t ) MGP NLP Constrained problem 3.1. Geometric Programming GP) problem Primal program : g r t) = Minimize g 0 t) = T r C rk +T r 1 m m C 0k t α rkj j 1 t α 0kj j 21) r =1, 2,...,l) t j > 0, j =1, 2,...,m); where C 0k > 0)k =1, 2,...,T 0 ), C rk > 0) and α rkj k =1, 2,...,1+T r 1,...,T r ; r =0, 1, 2,...,l; j =1, 2,...,m) are real numbers. It is a constrained posynomial PGP problem. The number of terms in each posynomial constraint function varies and it is denoted by T r for each r = 0, 1, 2,...l. Let T = T 0 + T 1 + T T l be the total number of terms in the primal program. The Degree of Difficulty is DD) = T m + 1). Dual program : The dual programming of 21) is as follows : l T r ) δrk crk Maximize νδ) = δ 0k =1, r=0 δ rk T r Normality condition) δ rs δrk 22) s=1+t r 1

14 134 Sahidul Islam and Tapan Kumar Roy l T r α rkj δ rk =0, j =1, 2,...,m), Orthogonality conditions) r=0 δ rk > 0, r =0, 1, 2,...,l; k =1, 2,...,T r ). Positivity conditions) Case I. For T m+1, the dual program presents a system of linear equations for the dual variables. A solution vector exists for the dual variables [3]. Case II. For T < m + 1, in this case generally no solution vector exists for the dual variables. However, one can get an approximate solution vector for this system using either LS or MM method discussed earlier. The solution procedure of this GP problem is same as the unconstrained GP problem. Example 3 Constrained GP problem). Minimize Zt) =2Ot 1 3 t Ot 6 2 t2 4 t 3 1t t1 1 t4 2t 2 3t t 1,t 2,t 3,t 4 > 0. It is a constrained posynomial geometric programming problem with degree of difficulty 1. This problem is also solved by GP and NLP as discussed before and optimal solutions are shown in Table 3. Table 3 Optimal solutions of Example 3 Methods t 1 t 2 t 3 t 4 Z t ) GP NLP Modified Geometric Programming MGP) problem Primal program : Minimize g 0 t) = g r t) = t > 0, g i0 t) = T r m C rik k=t r 1+1 m C 0ik t α rikj 1, i =1, 2,...,n; j =1, 2,...,m) t α 0ikj 23) r =1, 2,...,l)

15 Modified geometric programming and its applications 135 where C ik > 0) and α rikj k =1, 2,...,T r 1 +1,...,T r ) are real number. An associated Convex Geoemetric Programming CGP) problem is m Minimize h 0 x) = h i0 x) = c 0ik e α 0ikj x 24) h r x) 1= T r k=t r 1+1 c rik e m α rikjx 1 0, x > 0i =1, 2,...,n; j =1, 2,...,m). r =1, 2,...,l) Moreover, because t = e x is a strictly increasing function, the GP and CGP are equivalent in the sense that t =t i1,t i2,...,t im ) is a solution of GP 23) if and only if x =x i1,x i2,...,x im ) is a solution of CGP 24) where t = ex. Dual program : According to Harari & Ata [7], the dual programming of 23) is n T r ) δik n ) δik cik Maximize νδ) = δ sk r =0, 1, 2,...,l) 25) α 0ikj + l T r r=1 k=t r 1+1 s=1 = 1 Normality conditions) α rikj =0j =1, 2,...,m) Orthogonality conditions) > 0, k =1, 2,...,T r 1 +1,...,T r ; r =1, 2,...,l) Positivity conditions) for i =1, 2,...,n, where δ =δ 11,δ 12,δ 1k,...,δ nt0 ) T. Case I. For nt 0 nm+n, the dual program presents a system of linear equations for the dual variables. A solution vector exists for the dual variables [3]. Case II. For nt 0 <nm+ n, in this case generally no solution vector exists for the dual variables. However, one can get an approximate solution vector for this system using either the LS or MM method. The solution procedure of this MGP problem is same as the unconstrained MGP problem. Example 4 Constrained MGP problem). Minimize Zt) =2Ot 1 31 t Ot 6 21 t t 1 32 t t 6 22 t2 42

16 136 Sahidul Islam and Tapan Kumar Roy t 3 11 t t 1 11 t4 21 t2 31 t t3 12 t t 1 12 t4 22 t2 32 t t 11,t 21,t 31,t 41 t 12,t 22,t 32,t 42 > 0. It is solved by MGP and also NLP method, the optimal results are given in Table 4. Table 4 Optimal solutions of Example 4 Methods t 11 t 21 t 31 t 41 t 12 t 22 t 32 t 42 Z t ) MGP NLP Theorem If t is a feasible vector for the constrained PGP 23) and if δ is a feasible vector for the corresponding DP 25), then g 0 t) n n νδ) Primal-Dual Inequality). Proof. The expression for g 0 t) can be written as m c 0ik g 0 t) = t α 0ikj 26) We can apply the weighted A.M. G.M. inequality to this new expression for g 0 t) and obtain n T0 m gt) n T 0 c 0ik t α 0ikj or ) n g0 t) n n ) T0 C0ik m T0 t α 0ikj δ ik

17 Modified geometric programming and its applications 137 Again g r t) can be written as g r t) = [using normality condition] n T 0 ) δik C0ik m T0 = t α 0ikj. 27) T r k=t r 1+1 m c rik t α rikj. 28) Applying weighted A.M. G.M. inequality in 28) we have n δ ik m g r t) n T r c rik and g r t)) n n T r k=t r 1+1 k=t r 1+1 crik ) δik m t t α rikj Tr k=t r 1 +1 α rikj n ) δik δ sk, r =1, 2,...,l). n Using 1 g r t)) r =1, 2,...,l) [since g r t) 1r =1, 2,...,l)] n T r ) δik n ) δik crik m Tr k=t 1 δ sk t r 1 +1 α rikj, 29) k=t r 1+1 s=1 r =1, 2,...,l). Multiplying 27) and 29) we have ) n g0 t) n T r ) δik n ) δik Cik δ sk n δ ik s=1 m T0 t α l 0ikj + Tr r=1 k=t r 1 +1 α rikj 30) r =0, 1, 2,...,l). Using Orthogonality conditions the inequality 30) becomes ) n g0 t) n T r ) δik n ) δik cik δ sk, r =0, 1, 2,...,l) n s=1 s=1

18 138 Sahidul Islam and Tapan Kumar Roy i.e., This compltes the proof. g 0 t) n n νδ). Theorem Suppose that the constrained PGP 23) is super-consistent and that t is a solution for GP. Then the corresponding DP 25) is consistent and has a solution δ which satisfies g 0 t )=n n νδ ) and u ik t ) δik = g 0 t ), i =1, 2,...,n; k =1, 2,...,T 0) λ ir δ )u ik t ), i =1, 2,...,n; k = T r 1 +1,...,T r ; r =1, 2,...,l). Proof. Since GP is super-consistent, so is the associated CGP. Also since GP has a solution t =t i1,t i2,...,t im ), the associated CGP has a solution x = x i1,x i2,...,x im ) given by x = lnt. According to the Karush-Kuhn-Tucker K-K-T) conditions, there is a vector λ =λ i1,...,λ il ) such that λ ir 0, 31) λ ir h irx ) 1) = 0, 32) h i0 x ) l + λ h ir x ) ir =0. x x r=1 33) Because t = e x for i =1, 2,...,n; j =1, 2,...,m; it follows that for r =0, 1, 2,...,l h ir x) = h irx) t = g irt) e x. x t x t So, condition 33) is equivalent to g i0 t ) l + λ g ir t ) ir = 0 t t r=1 34) since e x > 0 but t > 0. Hence 34) is equivalent to t g i0 t ) + t l r=1 Now the terms of g ir t) are of the form u ik t) =c rik λ irt g ir t ) =0. 35) t n t α rikj.

19 Modified geometric programming and its applications 139 It is clear that t g ir t ) T r = α rikj u ik t ), i =1, 2,...,n; j =1, 2,...,n; r =1, 2,...,l). t k=t r 1+1 So, 35) implies to l α 0ikj u ik t )+ T r r=1k=t r 1+1 If we divide the last equation by g i0 t )= u ik t ), we obtain T r Define the vector δik by δik = α 0ikj u ik t ) g i0 t ) + λ irα rikj u ik t )=0, i =1, 2,...,n; j =1, 2,...,m). l T r r=1 k=t r 1+1 α rikj λ ir u ikt ) g i0 t ) u ik t ) g i0 t ), i =1, 2,...,n; k =1, 2,...,T 0) =0. λ ir u ik t ) g i0 t, i =1, 2,...,n; k = T r 1 +1,...,T r ; r =1, 2,...,l). ) Note that δik > 0i =1, 2,...,n; k =1, 2,...,T 0) and that for each r 1, either δik > 0 for all k with T r 1 +1 k T r or δik = 0 for all k with T r k T r according as the corresponding Karush-Kuhn-Tucker multipliers λ ir, i =1, 2,...,n; r =1, 2,...,l) is positive or zero. Also observe that vector δ satisfies all of the m exponents constraint equations in DP as well as the constraint T 0 δik = u ik t ) g i0 t ) = g i0t ) g i0 t ) =1. 36) Therefore, δ =δ i1,...,δ it 0 ) is a feasible vector for DP. Hence the DP is consistent. The Karush-Kuhn-Tucker multiplers λ ir are related to the corresponding λ irδ ) in DP as follows : T r T r λ ir δ )= δik = λ u ik t ) ir g i0 t ) = λ ir g ir t ) g i0 t, i =1, 2,...,n; r =1, 2,...,l). )

20 140 Sahidul Islam and Tapan Kumar Roy The Karush-Kuhn-Tucker condition 32) becomes λ ir g irt ) 1) = 0. 37) So, we get λ ir g irt )=λ ir. Therefore, for r =1, 2,...,l and k = T r 1 +1,...,T r, we see that δ ik = λ ir u ikt ) g i0 t ) = λ ir g irt )u ik t ) g i0 t ) = λ ir δ )u ik t ), 38) The fact that δ is feasible for DP and t is feasible for GP implies that g 0 t ) n n νδ ) because of Primal-Dual Inequality. Moreover, the values of δik i =1, 2,...,n; r =1, 2,...,l; k =1, 2,...,T r 1+ 1,...,T r ) are precisely those that force equality in the Arithmetic-Geometric Mean Inequalities that where used to obtain the Duality Inequality. Finally, equation 37) shows that either g ir t ) = 1 or λ ir = 0i = 1, 2,...,n; r = 1, 2,...,l) and equation 38) shows that λ ir = 0 if and only if λ irδ )=0, i =1, 2,...,n; r =1, 2,...,l). This means that the value of δik actually force equality in the Primal-Dual Inequality. This completes the proof. 4. Application 4.1. GP problem Grain-box problem) Problem 1a. It has been decided to shift grain from a warehouse to a factory in an open rectangular box of length x 1 meters, width x 2 meters and height x 3 meters. The bottom, sides and the ends of the box cost $80, $10 and $20/m 2 respectively. It costs $1 for each round trip of the box. Assuming that the box will have no salvage value, find the minimum cost of transporting 80 m 3 of grains.

21 Modified geometric programming and its applications 141 As stated, this problem can be formulated as the unconstrained GP problem 80 Minimize +40x 2 x 3 +20x 1 x 2 +80x 1 x 2, x 1 x 2 x 3 39) where x 1 > 0, x 2 > 0, x 3 > 0. It is an unconstrained posynomial geometric programmiing problem. The optimal dimension of the box are x 1 =1m, x 2 =1/2 m, x 3 =2m and minimum total cost of this problem is $ 200. Problem 1b. Supposed that we now consider the following variant of the above problem similar discussion has done Duffin, Peterson and Zener [1] in their book). It is required that the sides and bottom of the box should be made from scrap materials but only 4 m 2 of these scrap materials are available. This variation of the problem leads us to the following constrained posynomial GP problem : Minimize 80 x 1 x 2 x 3 +40x 2 x 3 2x 1 x 3 + x 1 x 2 4, where x 1 > 0, x 2 > 0,x 3 > 0. 40) Solving this constrained GP problem, we have the minimum total cost $ and optimal dimensions of the box are x 1 =1.58 m, x 2 =1.25 m, x 3 =0.63 m MGP problem Multi-grain-box problem) Problem 2a. Suppose that to shift grains from a warehouse to a factory in a finite number say n) of open rectangular boxes of lengths x 1i meters widths x 2i meters, and heights x 3i meters i =1, 2,...,n). The bottom, sides and the ends of the each box cost $a, $b i and $c i /m 2 respectively. It cost $1 for each round trip of the boxes. Assuming that the boxes will have no salvage value, find the minimum cost of transporting d i m 3 of grain. As stated, this problem can be formulated as an unconstrained MGP problem ) d i Minimize gx) = + a i x 1i x 2i +2b i x 1i x 3i +2c i x 2i x 3i, x 1i x 2i x 3i where x 1i > 0, x 2i > 0, x 3i > 0 i =1, 2,...,n). 41)

22 142 Sahidul Islam and Tapan Kumar Roy In particular here we assume that the transporting d i m 3 of grains by the three different open rectangular boxes whose bottom, sides and the ends of each box costs are given in the Table 5. It is solved and optimal results are shown in the Table 6. Table 5 Input data for Problem 2a ith Box a i $/m 2 ) b i $/m 2 ) c i $/m 2 ) d i m 3 ) i= i= Table 6 Optimal results of Problem 2a ith Box x 1i meter) x 2i meter) x 3i meter) g x )$) i= i= Problem 2b. Suppose that we now consider the following variant of the above problem. It is required that the sides and bottom of the boxes should be made from scrap materials but only wm 2 of these scrap materials are available. This variation of the problem leads us to the following constrained modified geometric program : ) d i Minimize fx) = +2c i x 2i x 3i x 1i x 2i x 3i 2x 1i x 3i + x 1i x 2i ) w, where x 1i > 0, x 2i > 0, x 3i > 0 i =1, 2,...,n). 42) In particular here we assume transporting d i m 3 of grains by the three different open rectanggular boxes. The end of each box cost is c i $/m 2 and amount of the transportin grains by three open rectangular boxes are d i m 3. Input data of this problem is given in the Table 7. It is a constrained posynomial MGP problem. Optimal results of this problem are shown in Table 8.

23 Modified geometric programming and its applications 143 Table 7 Input data for Problem 2b ith Box c i $/m 2 ) d i m 3 ) wm 2 ) i= i= Table 8 Optimal results of Problem 2b ith Box x 1i meter) x 2i meter) x 3i meter) g x )$) i= i= Conclusion We have discussed unconstrained constrained GP and MGP problem with negative or positive integral degree of difficulty. The LS and MM methods are described to solve GP/MGP problem with negative degree of difficulty. Here, modified form of geometric programming method has been demonstrated and for that purpose necessary theorems have been derived. This technique can be applied to solve the different decision making problems like in inventory [10] and other areas). Acknowledgement The authors wish to acknowledge the helpful comments and suggestions of the referee s. References 1. M. O. Abou-El-Ata and K. A. M. Kotb, Multi-item EOQ inventory model with varying holding cost under two restrictions a geometric programming approach, Production Planning & Control 85) 1997), H. Basirzadeh, A. V. Kayyad and Effati, An approach for solving nonlinear problems, J. Appl. Math. & Computing 92) 2002), C. S. Beightler and D. T. Philips, Applied Geometric Programming, Wiley, New York, G. S. Beightler, D. T. Phillips and D. J. Wilde, Foundations of Optimization, Prentice-Hall, Englewood Cliffs, NJ, 1979.

24 144 Sahidul Islam and Tapan Kumar Roy 5. R. J. Duffin, E. L. Peterson and C. M. Zener, Geometric Programming Theory and Applications, Wiley, New York, J. Ecker, Geometric programming : methods, computations and applications, SIAM Rev. 223) 1980), A. M. A. Hariri and M. O. Abou-El-Ata, Multi-item EOQ inventory model with varying holding cost under two restrictions a geometric programming approach, Production Planning & Control 85) 1997), Sahidul Islam and T. K. Roy, An economic production quantity model with flexibility and reliability consideration and demand dependent unit cost under a constraint : Geometric Programming Approach, Proceedings of National Symposium Department of Mathematics, University of Kalyani, India, 21-22th March, 2002, C. L. Lawson and R. J. Hanson, Solving Least Squares Problems, Prentice Hall, Inc., Englewood Cliffs, NJ, A. K. Pal and B. Mandal, An EOQ Model fo r Deteriorating Inventory with Alternating Demand Rate, Korean J. Comput. & Appl. 42) 1997), A. L. Peressini, F. E. Sullivan and J. J. Jr. Uhl, The Mathematics Nonlinear Programming, Springer-Verlag, New York, F. Scheid, Numerical Analysis, McGraw-Hill, New York, S. B. Sinha, A. Biswas and M. P. Biswal, Geometric programming problems with negative degrees of difficulty, European Journal of Operational Research, ), C. Zener, Engineering Design by Geometric Programmin, Wiley, Sahidul Islam received his B. Sc. Hons.) from Rampurhat College under University of Burdwan and M. Sc. at Jadavpur University. In 2003, he received a Junior Research Fellow from Council of Scientific and Industrial Research at Bengal Engineering College Deemed University). His research interests focus on some optimization problem in information and Fuzzy systems and related OR models. Some papers are published and accepted in Proceedings of National Seminars. T. K. Roy is a senior lecturer ini Mathematics at Bengal Engineering College Deemed University). His areas of research are Fuzzy set theory, Applications of OR in information and Fuzzy Systems. He has published papers in various international and national journals including European Journal of Operational Research, Computer and Operation Research, Production Planning and Control, OPSEARCH. Some papers are also published and accepted in Proceedings of international and national Seminars. Department of Mathematics, Bengal Engineering College Deemed University), Howrah , West Bengal, India sahidul@yahoo.ca roy t k@yahoo.co.in