THIS paper considers the following generalized linear

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1 Engneerng Letters, 25:3, EL_25_3_06 A New Branch and Reduce Approach for Solvng Generalzed Lnear Fractonal Programmng Yong-Hong Zhang, and Chun-Feng Wang Abstract In ths paper, for solvng generalzed lnear fractonal programmng (GLFP, a new branch-and-reduce approach s presented. Frstly, an equvalent problem (EP of GLFP s gven; then, a new lnear relaxaton technque s proposed; fnally, the problem EP s reduced to a sequence of lnear programmng problems by usng the new lnear relaxaton technque. Meanwhle, to mprove the convergence speed of our algorthm, two reducng technques are presented. The proposed algorthm s proved to be convergent, and some experments are provded to show ts feasblty and effcency. Index Terms Lnear relaxaton; Global optmzaton; Generalzed lnear fractonal programmng; Reducng technque; Branch and bound. I. INTRODUCTION THIS paper consders the followng generalzed lnear fractonal programmng (GLFP problem: n c p j x j +d mn GLFP n e j x j +f s.t. x D x R n Ax b}, where p 2, A R m n, b R m, c j, d, e j, f are arbtrary real numbers, D s bounded wth ntd, and n for x D, e j x j + f 0,,, p, j,, n. Among fractonal programmng, the problem GLFP s a specal class of optmzaton problems, whch has attracted the nterest of practtoners for many years. The man reason s that many applcatons n varous felds can be put nto the form GLFP, ncludng transportaton scheme, manufacturng, economc benefts, and mult-objectve bond portfolo [-5], etc. In addton, snce the problem GLFP may be not (quasconvex, t may has multple local optmal solutons, and many of whch fal to be globally optmal, that s, the problem GLFP posses sgnfcant theoretcal and computatonal dffcultes. So, t also has attracted the nterest of many researchers. Durng the past years, for x D, wth the assumpton that n n c j x j +d 0, e j x j +f > 0, many algorthms have Manuscrpt receved Feb. 26, 207. Ths paper s supported by the Natonal Natural Scence Foundaton of Chna (U40405, the Key Scentfc and Technologcal Project of Henan Provnce( ; the Youth Scence Foundaton of Henan Normal Unversty(203qk02; Henan Normal Unversty Natonal Research Project to Cultvate the Funded Projects ( Yong-Hong Zhang s wth College of Mathematcs and Informaton Scence, Henan Normal Unversty,Xnxang, , PR Chna; Henan Engneerng Laboratory for Bg Data Statstcal Analyss and Optmal Control, School of Mathematcs and Informaton Scences, Henan Normal Unversty e-mal:zhangyonghong09@26.com. Chun-Feng Wang s wth College of Mathematcs and Informaton Scence, Henan Normal Unversty, Xnxang, , PR Chna. been presented for solvng specal cases of problem GLFP. For example, when p, by usng varable transformaton, Charnes and Cooper proposed a method [6]. When p 2, Konno et al. proposed an effectve parametrc algorthm [7], whch can be used to solve large scale problem. When p 3, Konno and Abe developed a heurstc algorthm[8]. When p 3, through utlzng an updated objectve functon method, Btran and Novaes presented an approach, whch solves a sequence of lnear programmng problems dependng on updatng the local gradent of the fractonal objectve functon at successve ponts [9]. By usng an equvalent transformaton and a lnearzaton technque, Shen and Wang proposed a branch and bound algorthm for solvng a sum of lnear ratos problem wth coeffcents [0]. Dur et al. ntroduced the algorthm (DMIHR and appled t to the class of fractonal optmzaton problem []. By adoptng monotonc transformaton technque, Shen et al. proposed a method to solve quadratc ratos fractonal program [2]. By usng the characterstcs of exponental and logarthmc functons, Wang et al. presented a twce lnearzaton technque [3]. Through usng sutable transformaton, Benson proposed a method, whch has a potental to solve GLFP by some well known technques [4]. By solvng an equvalent concave mnmum problem of the orgnal problem, Benson put forward a new branch and bound algorthm [5]. Under n n the assumpton that c j x j + d 0, e j x j + f 0, a branch and bound algorthm was developed [6]. Recently, two global optmzaton algorthms were proposed for solvng GLFP wth the only assumpton that e j x j + f n 0 [7,8]. The goal of ths research has two-fold. Frst, we present a new lnear relaxaton method. Second, n order to mprove the convergence speed of our algorthm, two reducng technques are proposed, whch can be used to elmnate the regon n whch the global mnmum soluton of GLFP does not exst. The man features of ths algorthm are ( the model consdered by ths paper has a more general form, whch only n request e j x j +f 0; (2 a new lnear relaxaton method s proposed n order to obtan a lower bound for the global optmal value of problem EP over parttoned subsets, whch s more convenent n the computaton than the methods [0,4,8]; (2 two reducng technques are presented, whch can be used to mproved the convergence speed of the proposed algorthm; (3 numercal experments are gven to show the feasblty and effcency of our algorthm. Ths paper s structured as follows. In Secton 2, an equvalent problem EP of problem GLFP s derved. In Secton 3, a new lnear relaxaton technque s presented for (Advance onlne publcaton: 23 August 207

2 Engneerng Letters, 25:3, EL_25_3_06 generatng the lnear relaxaton programmng problem LRP of problem EP, whch can provde a lower bound for the optmal value of problem EP. In order to mprove the convergence speed of our algorthm, two reducng technques are presented n Secton 4. In Secton 5, the global optmzaton algorthm s descrbed, and the convergence of ths algorthm s establshed. Numercal results are reported to show the feasblty and effcency of our algorthm n Secton 6. II. EQUIVALENT PROBLEM (EP OF GLFP n In problem GLFP, for x D, snce e j x j + f 0, by the ntermedate value theorem, we have n e j x j + f > 0 or n e j x j + f < 0. In addton, through usng the technques of [5,6], n c j x j + d 0 always can be satsfed. Therefore, wthout loss of generalty, we assume n that c j x j + d 0 always holds. For x D, let I + I n e j x j + f > 0,,, p}, n e j x j + f < 0,,, p}. Compute lj 0 mn x j, u 0 j x j(j x D x D,, n, and defne rectangle H 0 [l 0, u 0 ]. Furthermore, we compute y 0 n, y 0 n mne j lj 0,e ju 0 j }+f e j lj 0,e ju 0 j }+f,,, p, and construct rectangle Y 0 [y 0, y 0 ]. Through ntroducng new varables y (,, p, the problem GLFP can be converted nto an equvalent problem EP as follows: v(h 0 mnφ 0 (x, y p y ( n c j x j + d EP(H 0 s.t. φ (x, y y ( n e j x j + f, I +, φ (x, y y ( n e j x j + f, I, x D H 0, y Y 0. The key equvalence result for problem GLFP and EP(H 0 s gven by the followng theorem. Theorem If (x, y s a global optmal soluton of problem EP(H 0, then x s a global optmal soluton of problem GLFP. Conversely, f x s a global optmal soluton of problem GLFP, then (x, y s a global optmal soluton of problem EP(H 0, where y,,, p. n e j x j +f Proof. From the defntons of problems GLFP and EP(H 0, the concluson can be obtaned easly, so t s omtted. By Theorem, n order to globally solve problem GLFP, we may solve problem EP(H 0 nstead. Therefore, n the followng we only consder how to solve the problem EP(H 0. III. LINEAR RELAXATION PROGRAMMING (LRP PROBLEM Let H x l x u} denote the ntal box H 0 or modfed box as defned for some parttoned subproblem n a branch and bound scheme. Ths secton wll show how to construct the problem LRP for problem EP over H. For convenence n expresson, let T c+ T c T e+ T e j c,j > 0, j,, n}, j c,j < 0, j,, n}, j e,j > 0, j,, n}, j e,j < 0, j,, n}. To derve the problem LRP of problem EP, for,, p, we frst compute y n, e j l j,e j u j }+f y n, and consder the term x j y n mne j l j,e j u j }+f the nterval [l j, u j ] and [y, y ]. Snce x j l j 0, y y 0, we have that s Furthermore, we have (x j l j (y y 0, x j y x j y l j y + l j y 0. x j y x j y + l j y l j y. ( Meanwhle, snce x j l j 0, y y 0, we have Furthermore, we can obtan (x j l j (y y 0, x j y x j y + l j y l j y. (2 Based on ( and (2, we can derve the problem LRP of problem EP. Towards ths end, frst, consder the objectve functon φ 0 (x, y, we have φ 0 (x, y p y ( n c j x j + d p ( p ( + φ l 0(x, y. c j x j y + c j (x j y + l j y l j y c j x j y + p d y c j (x j y + l j y l j y + p d y φ 0 (x, y p y ( n c j x j + d p ( p ( + φ u 0(x, y. c j x j y + c j (x j y + l j y l j y c j x j y + p d y c j (x j y + l j y l j y + p d y (Advance onlne publcaton: 23 August 207

3 Engneerng Letters, 25:3, EL_25_3_06 Then, consder the constrant functons φ (x, y,,, p. For I +, by usng ( and (2, we have φ (x, y n e j x j y + f y e j (x j y + l j y l j y (3 j T e+ + e j (x j y + l j y l j y + f y φ u (x, y. j T e For I, we have φ (x, y n e j x j y + f y e j (x j y + l j y l j y (4 j T e+ + e j (x j y + l j y l j y + f y φ l (x, y. j T e From the above dscusson, the lnear relaxaton programmng problem LRP can be establshed as follows, whch provdes a lower bound for the optmal value v(h of problem EP(H: LRP(H LB(H mn φ l 0(x, y s.t. φ u (x, y, I +, φ l (x, y, I, x D H, y Y [y, y]. Obvously, f H H H 0, then LB(H LB(H. IV. REDUCING TECHNIQUE To mprove the convergence speed of ths algorthm, we present two reducng technques, whch can be used to elmnate the regon n whch the global optmal soluton of problem EP(H 0 does not exst. Assume that UB and LB are the current known upper bound and lower bound of the optmal value v(h 0 of the problem EP(H 0. Let α j p cj y ξ, where ξ, f c j > 0, c j y, f c j < 0, β n c j l j + e, Λ p ( c j l j y + Ω p mnβ y, β y }, n c j l j y, γ k UB mnα j l j, α j u j } Ω Λ,,j k θ j p cj y η, where η, f c j > 0, c j y, f c j < 0, Λ 2 p ( c j l j y + c j l j y, Ω 2 p β y, β y }, τ k LB n,j k θ j l j, θ j u j } Ω 2 Λ 2, The reducng technques are derved as n the followng theorems. Theorem 2 For any subrectangle H H 0 wth H j [l j, u j ], f there exsts some ndex k, 2,, n} such that > 0 and γ k < u k, then there s no globally optmal soluton of problem EP(H 0 over H ; f < 0 and γ k < l k, for some k, then there s no globally optmal soluton of problem EP(H 0 over H 2, where H (Hj n H, wth Hj Hj, j k, ( γ k, u k ] H k, j k, H 2 (Hj 2 n H, wth Hj 2 Hj, j k, [l k, γ k H k, j k. proof Frst, we show that for all x H, φ 0 (x, y > UB. Consder the kth component x k of x. Snce x k ( γ k, u k ], t follows that γ k < x k u k. From > 0, we have γ k < x k. For all x H, by the above nequalty and the defnton of γ k, t mples that that s UB n,j k UB < n,j k mnα j l j, α j u j } Ω Λ < x k, mnα j l j, α j u j } + x k + Ω + Λ n α j x j + p β y + Λ φ l 0(x, y. Thus, for all x H, we have φ 0 (x, y φ l 0(x, y > UB v(h 0,.e. for all x H, φ 0 (x, y s always greater than the optmal value v(h 0 of the problem EP(H 0. Therefore, there can not exst globally optmal soluton of problem EP(H 0 over H. For all x H 2, f there exsts some k such that < 0 and γ k < l k, from arguments smlar to the above, t can be derved that there s no globally optmal soluton of problem EP(H 0 over H 2 Theorem 3 For any subrectangle H H 0 wth H j [l j, u j ], f there exsts some ndex k, 2,, n} such that θ k > 0 and τ k > θ k l k, then there s no globally optmal soluton of problem EP(H 0 over H 3 ; f θ k < 0 and τ k > θ k u k, for some k, then there s no globally optmal soluton of problem EP(H 0 over H 4, where H 3 (Hj 3 n H, wth Hj 3 Hj, j k, [l k, τ k θk H k, j k, H 4 (Hj 4 n H, wth Hj 4 Hj, j k, ( τ k θk, u k ] H k, j k. proof Frst, we show that for all x H 3, φ 0 (x, y < LB. Consder the kth component x k of x. By the assumpton and the defntons of θ k and τ k, we have l k x k < τ k θ k. Note that θ k > 0, we have τ k > θ k x k. For all x H 3, by the above nequalty and the defnton of τ k, t mples that LB > n,j k θ j l j, θ j u j } + θ k x k + Ω 2 + Λ 2 n θ j x j + p β y + Λ 2 φ u 0(x, y φ 0 (x, y. (Advance onlne publcaton: 23 August 207

4 Engneerng Letters, 25:3, EL_25_3_06 Thus, for all x H 3, we have v(h 0 LB > φ 0 (x, y. Therefore, there can not exst globally optmal soluton of problem EP(H 0 over H 3. For all x H 4, f there exsts some k such that θ k < 0 and τ k < θ k u k, from arguments smlar to the above, t can be derved that there s no globally optmal soluton of problem EP(H 0 over H 4 V. ALGORITHM AND ITS CONVERGENCE In ths secton, based on the former results, we present the branch and bound algorthm to solve problem EP(H 0. Ths method need to solve a sequence of lnear relaxaton programmng problems over parttoned subsets of H 0 n order to fnd a global optmal soluton. A. Branchng rule Durng each teraton of the algorthm, the branchng process creates a more refned partton that cannot yet be excluded from further consderaton n searchng for a global optmal soluton for problem EP(H 0, whch s a crtcal element n guaranteeng convergence. Ths paper chooses a smple and standard bsecton rule, whch s suffcent to ensure convergence snce t drves the ntervals shrnkng to a sngleton for all the varables along any nfnte branch of the branch and bound tree. Consder any node subproblem dentfed by rectangle H x R n l j x j u j, j,, n} H 0. The branchng rule s as follows: ( let k argl j u j j,, n}; ( let π k (l k + u k /2; ( let H x R n l j x j u j, j k, l k x k π k }, H 2 x R n l j x j u j, j k, π k x k u k }. Through usng ths branchng rule, the rectangle H s parttoned nto two subrectangles H and H 2. B. Branch and bound algorthm Based upon the results and operatons gven above, ths subsecton summarzes the basc steps of the proposed algorthm. Let LB(H k be the optmal functon value of LRP over the subrectangle H H k, and (x k, y k be an element of the correspondng argmn. Algorthm statement Step. Choose ϵ 0. Fnd an optmal soluton x 0 and the optmal value LB(H 0 for problem LRP(H wth H H 0. Set LB 0 LB(H 0, y 0,,, p, n e j x 0 j +f and UB 0 φ 0 (x 0, y 0. If UB 0 LB 0 ϵ, then stop: (x 0, y 0 and x 0 are ϵ-optmal solutons of problems EP(H 0 and GLFP, respectvely. Otherwse, set Q 0 H 0 }, F, k, and go to Step 2. Step 2. Set LB k LB k. Subdvde H k nto two subrectangles H k,, H k,2 va the branchng rule. Set F F H k }. Step 3. Set t t +. If t > 2, go to Step 5. Otherwse, contnue. Step 4. If LB(H k,t > UB k, set F F H k,t }, and go to Step 3. Otherwse, set y k,t n e j x k,t j + f,,, p. Let UB k mnub k, φ 0 (x k,t, y k,t }. If UB k φ 0 (x k,t, y k,t, set x k x k,t, y k y k,t, go to Step 3. Step 5. Set F F H Q k UB k LB(H}, Q k H H (Q k H k,, H k,2 }, H / F }. Step 6. Set LB k mnlb(h H Q k }. Let H k be the subrectangle whch satsfes that LB k LB(H k. If UB k LB k ϵ, stop, (x k, y k and x k are global ϵ-optmal solutons of problems EP(H 0 and GLFP, respectvely. Otherwse, set k k +, and go to Step 2. C. Convergence analyss In ths subsecton, we gve the global convergence propertes of the above algorthm. Theorem 4 The above algorthm ether termnates fntely wth a globally ϵ-optmal soluton, or generates an nfnte sequence (x k, y k } of teraton such that along any nfnte branch of the branch and bound tree, whch any accumulaton pont s a globally optmal soluton of problem EP(H 0. Proof When the algorthm s fnte, by the algorthm, t termnates at some step k 0. Upon termnaton, t follows that UB k LB k ϵ. From Step and Step 6 n the algorthm, a feasble soluton (x k, y k for the problem EP(H 0 can be found, and the followng relaton holds By Secton 3, we have φ 0 (x k, y k LB k ϵ. LB k v(h 0. Snce (x k, y k s a feasble soluton of problem EP(H 0, φ 0 (x k, y k v(h 0. Taken together above, t mples that v(h 0 φ 0 (x k, y k LB k + ϵ v(h 0 + ϵ, and so (x k, y k s a global ϵ-optmal soluton to the problem EP(H 0 n the sense that v(h 0 φ 0 (x k, y k v(h 0 + ϵ. If the algorthm s nfnte, then an nfnte sequence (x k, y k } wll be generated. Snce the feasble regon of EP(H 0 s bounded, the sequence (x k, y k } must be has a convergence subsequence. Wthout loss of generalty, set lm k (xk, y k (x, y, then we have lm k yk y n e j x j +f lm ( n c j x k j + d n c j x j + d. k, (Advance onlne publcaton: 23 August 207

5 Engneerng Letters, 25:3, EL_25_3_06 Furthermore, we have lm UB k lm φ 0(x k, y k φ 0 (x, y, k k lm LB k lm ( p ( k k + n c j x j +d n e j x j +f p c j (x k j y + l jy k l jy c j (x k j y + l j y k l jy + p d y k φ 0 (x, y. Therefore, we have lm (UB k LB k 0. k Ths mples that (x, y s a global optmal soluton of problem EP(H 0. By Theorem, x s a global optmal soluton of problem GLFP. VI. NUMERICAL EXPERIMENTS In ths secton, to verfy the performance of the proposed algorthm, some numercal experments are reported, and compared wth several latest algorthms[0,3,7-20]. The algorthm s mplemented by Matlab 7., and all test problems are carred out on a Pentum IV (3.06 GHZ mcrocomputer. The smplex method s appled to solve the lnear relaxaton programmng problems. The results of problems -7 are summarzed n Table I, where the followng notatons have been used n row headers: ϵ: convergence error; Iter: number of algorthm teratons. Table II summarzes our computatonal results of Example 8. For ths test problem, ϵ s set to e 3. In Table II, Ave.Iter represents the average number of teratons; Ave.Tme stands for the average CPU tme of the algorthm n seconds, whch are obtaned by randomly runnng our algorthm for 0 test problems. Example [8] 0.9 x + 2x x 4x ( 0. 4x 3x x + x s.t. x + x 2.5, x x 2 0, 0 x, 0 x 2. Example 2 [3,8,9] 4x + 3x 2 + 3x x + 4x x 2 + 3x x + 4x 2 + 5x x + 2x 2 + 5x x + 5x 2 + 5x x + 2x 2 + 4x x 2 + 4x s.t. 2x + x 2 + 5x 3 0, x + 6x 2 + 3x 3 0, 5x + 9x 2 + 2x 3 0, 9x + 7x 2 + 3x 3 0, x 0, x 2 0, x 3 0, x 4 0. Example 3 [20] x + 2x mn 3x 4x x 3x x + x s.t. x + x 2.5, x x 2 0, 0 x, 0 x 2. Example 4 [9,20] 3x + 5x 2 + 3x x + 4x 2 + 5x x + 4x x + 3x 2 + 2x x + 2x 2 + 4x x + 4x 2 + 3x s.t. 6x + 3x 2 + 3x 3 0, 0x + 3x 2 + 8x 3 0, x 0, x 2 0, x 3 0. Example 5 [0] 63x 8x x + 26x x + 26x x + 73x x + 73x x + 3x x + 3x x 8x s.t. 5x 3x 2 3,.5 x 3. Example 6 [0] 3x + 4x x + 5x 2 + 4x x + 5x 2 + 3x x + 5x 2 + 4x x + 2x 2 + 4x x + 3x 2 + 3x x 2 + 4x x 2 + 3x s.t. 6x + 3x 2 + 3x 3 0, 0x + 3x 2 + 8x 3 0, x 0, x 2 0, x 3 0. Example 7 [7] 37x + 73x x + 3x x 6x x + 26x s.t. 5x 3x 2 3,.5 x 3. Example 8 mn s.t. n c p j x j +d n e j x j +f x D x R n Ax b}, where the elements of the matrx A R m n, c j, e,j R are randomly generated n the nterval [0,]. All constant terms of denomnators and numerators are the same number, whch randomly generated n [,00]. The elements of b R m are equal to. Ths agrees wth the way random numbers are generated n [3]. TABLE I: Computatonal results of Examples -7 Example ϵ Methods Optmal soluton Optmal value Iter e-9 [8] (0.0, e-9 ours (0.0, e-6 [3] (.,.365e-5,.35e e-5 [20] (0.003, 0.0, e-9 [8] (., 0.0, e-9 ours (., 0.0, e-8 [20] (0.0, e-8 ours (0.0, e-5 [9] (0.0,.6725, e-8 [20] (0.0, , e-8 ours (0.0, , e-6 [0] (3.0, e-6 ours (3.0, e-6 [0] (-.838e-6, , e-6 ours (0.0, , e-4 [7] (3.0, e-4 ours (3.0, From Table I, t can be seen that, for Examples -7, our algorthm can determne the global optmal soluton effectvely than that of the references [0,3,7-20]. (Advance onlne publcaton: 23 August 207

6 Engneerng Letters, 25:3, EL_25_3_06 TABLE II: Computatonal results of Example 8 (p, m, n Ave.Tme Ave.Iter (5,30, (5,50, (5,00, (0,30, (0,50, (0,00, [9] Y. G. Pe, D. T. Zhu, Global optmzaton method for mzng the sum of dfference of convex functons ratos over nonconvex regon, Journal of Appled Mathematcs and Computng, vol. 4, no., pp 53-69, 203. [20] H. J. Jao, A branch and bound algorthm for globally solvng a class of nonconvex programmng problems, Nonlnear Analyss: Theory, Methods and Applcatons, vol. 70, no. 2, pp 3-23, From Table II, the computatonal results show that our algorthm performs well on the test problems, and can solve them n a reasonable amount of tme. Meanwhle, we fnd that, wth the sze of the problem becomng large, the average number of teratons and the average CPU tme do not ncrease quckly. The results n Tables I and II show that our algorthm s both feasble and effcent. REFERENCES [] H. Konno, H. Watanabe, Bond portfolo optmzaton problems and ther applcatons to ndex trackng: a partal optmzaton approach, Journal of the Operatons Research Socety of Japan, vol. 39, no.3, pp , 996. [2] J. E.,Falk, S. W., Palocsay, Optmzng the sum of lnear fractonalfunctons,n Recent Advancesn Global Optmzaton, Prnceton Unversty Press,Prnceton, NJ, USA, 992. [3] R. Horst, P. M. Pardalos, N. V. Thoa, Introducton to Global Optmzaton, Kluwer Academc Publshers, Dordrecht, Netherlands, 2nd edton, [4] X. J. Wang, S. H. Cho, Optmsaton of stochastc multtem manufacturng for shareholder wealth msaton, Engneerng Letters, Vol. 2, no. 3, pp 27-36, 203. [5] T. Hasuke, H. Katagr, Senstvty analyss for random fuzzy portfolo selecton model wth nvestor s subjectvty, IAENG Internatonal Journal of Appled Mathematcs, vol. 40, no.3, pp 85-89, 200. [6] A. Charnes, W. W. Cooper, Programs wth lnear fractonal functons, Naval Research Logstcs Quarterly, vol. 9, no. 3-4, pp 8-96, 962. [7] H. Konno, Y. Yajma, T. Matsu, Parametrc smplex algorthms for solvng a specal class of nonconvex mnmzaton problems, Journal of Global Optmzaton, vol., no., pp 65-8, 99. [8] H. Konno, N. Abe. Mnmzaton of the sum of three lnear fractonal functons, Journal of Global Optmzaton, vol. 5, no. 4, pp , 999. [9] G. R. Btran, A. G. Novaes, Lnear programmng wth a fractonal objectve functon, Operatons Research, vol. 2, no., pp 22-29, 973. [0] P. P. Shen, C. F. Wang. Global optmzaton for sum of lnear ratos problem wth coecents, Appled Mathematcs and Computaton, vol. 76, no., pp , [] M. Dur, C. Khompatraporn, Z. B. Zabnsky, Solvng fractonal problems wth dynamc multstart mprovng ht-and-run, Annals of Operatons Ressearch, vol. 56, no., pp 25-44, [2] P. P. Shen, Y. C. Chen,Y. Ma, Solvng sum of quadratc ratos fractonal programs va monotonc functon, Appled Mathematcs and Computaton, vol. 22, no., pp , [3] Y. J. Wang, P. P. Shen, Z. A. Lang, A branch-and-bound algorthm to globally solve the sum of several lnear ratos, Appled Mathematcs and Computaton, vol. 68, no., pp 89-0, [4] H. P. Benson, On the global optmzaton of sums of lnear fractonal functons over a convex set, Journal of Optmzaton Theory and Applcaton, vol. 2, no., pp 9-39, [5] H. P. Benson, Solvng sum of ratos fractonal programs va concave mnmzaton, Journal of Optmzaton Theory and Applcaton, vol. 35, no., pp -7, [6] Y. J, K. C. Zhang, S. J. Qu, A determnstc global optmzaton algorthm, Appled Mathematcs and Computaton, vol. 85, no., pp , [7] C. F. Wang, P. P. Shen, A global optmzaton algorthm for lnear fractonal programmng, Appled Mathematcs and Computaton, vol. 204, no., pp , [8] H. J. Jao, S. Y. Lu, A practcable branchand bound algorthm forsum of lnear ratos problem, European Journal of Operatonal Research, vol. 243, no. 3, pp , 205. (Advance onlne publcaton: 23 August 207