Dynamic Slope Scaling Procedure to solve. Stochastic Integer Programming Problem

Transcription

1 Journal of Computatons & Modellng, vol.2, no.4, 2012, ISSN: (prnt), (onlne) Scenpress Ltd, 2012 Dynamc Slope Scalng Procedure to solve Stochastc Integer Programmng Problem Takayuk Shna 1 and Chunhu Xu 2 Abstract Stochastc programmng deals wth optmzaton under uncertanty. A stochastc programmng problem wth recourse s referred to as a two-stage stochastc problem. We consder the stochastc programmng problem wth smple nteger recourse n whch the value of the recourse varable s restrcted to a multple of a nonnegatve nteger. The algorthm of a dynamc slope scalng procedure to solve the problem s developed by usng the property of the expected recourse functon. The numercal experments show that the proposed algorthm s qute effcent. The stochastc programmng model defned n ths paper s qute useful for a varety of desgn and operatonal problems. Mathematcs Subject Classfcaton: 90C11, 90C15, 90C90 Keywords: Stochastc programmng problem wth recourse, Smple nteger recourse, Dynamc slope scalng procedure 1 Chba Insttute of Technology, e-mal : shna.takayuk@t-chba.ac.jp 2 Chba Insttute of Technology, e-mal : xu.chunhu@t-chba.ac.jp Artcle Info: Receved : September 19, Revsed : October 23, 2012 Publshed onlne : December 30, 2012

2 134 Dynamc Slope Scalng 1 Introducton Mathematcal programmng has been appled to many problems n varous felds. However for many actual problems, the assumpton that the parameters nvolved n the problem are determnstc known data s often unjustfed. These data contan uncertanty and are thus represented as random varables, snce they represent nformaton about the future. Decson-makng under uncertanty nvolves potental rsk. Stochastc programmng (Brge [1], Brge and Louveaux [2], Kall and Wallace [3]) deals wth optmzaton under uncertanty. A stochastc programmng problem wth recourse s referred to as a two-stage stochastc problem. In the frst stage, a decson has to be made wthout complete nformaton on random factors. After the value of random varables are known, recourse acton can be taken n the second stage. For the contnuous stochastc programmng problem wth recourse, an L-shaped method (Van Slyke and Wets [4]) s well-known. The L-shaped method was used to solve stochastc programs havng dscrete decsons n the frst stage (Laporte and Louveaux[5]). Ths method was appled to solve a stochastc concentrator locaton problem (Shna [6, 7]). In ths paper, we consder a stochastc programmng problem n whch the recourse varables are restrcted to ntegers. If nteger varables are nvolved n a second stage problem, optmalty cuts based on the Benders [8] decomposton do not provde facets of the epgraph of recourse functon. It s dffcult to approxmate the recourse functon whch s n general nonconvex and dscontnuous, snce the functon s defned as the value functon of the second stage nteger programmng problem. For stochastc programs wth smple nteger recourse, Ahmed, Tawarmalan, and Sahnds[9] developed a fnte algorthm based on the branchng of the frst stage nteger varables. However, varables nvolved n the stochastc program wth smple nteger recourse are restrcted to havng a nonnegatve nteger value. Such restrcton of varables to pure ntegers makes applcaton of the problem dffcult. Therefore, we consder a practcal stochastc programmng model whch s applcable to varous real problems, and deal wth the problem n whch the recourse varables are restrcted to take the multples of some nonnegatve nteger. These recourse varables represent that the addtonal actons are taken n a certan amount unts. Ths mathematcal programmng model

3 T. Shna and C. Xu 135 s qute useful for a varety of desgn and operatonal problems whch arse n dverse contexts, such as nvestment plannng, capacty expanson, network desgn and faclty locaton. In Secton 2, the basc model of the stochastc programmng problem wth recourse and the L-shaped method are shown. Then, we consder the varant of the stochastc program wth smple nteger recourse, whch s a natural extenson of the contnuous smple recourse. In Secton 3, we nvestgate the property of the recourse functon. The algorthm of a dynamc slope scalng procedure to solve the problem s developed by usng the property of the expected recourse functon. In Secton 4, the numercal experments show that the proposed algorthm s qute effcent. The stochastc programmng model defned n ths paper s qute useful for a varety of desgn and operatonal problems. 2 Formulaton 2.1 Stochastc programmng problem wth recourse We frst form the basc two-stage stochastc lnear programmng problem wth recourse as (SPR). (SPR): mn c x+q(x) subject to Ax = b x 0 Q(x) = E ξ[q(x, ξ)] Q(x,ξ) = mn{q(ξ) y(ξ) Wy(ξ) = h(ξ) T(ξ)x,y(ξ) 0},ξ Ξ In the formulaton of (SPR), c s a known n 1 -vector, b a known m 1 -vector, and A and W are known matrces of sze m 1 n 1 and m 2 n 2, respectvely. The frst stage decsons are represented by the n 1 -vector x. We assume the l- random vector ξ s defned on a known probablty space. Let Ξ be the support of ξ,.e. the smallest closed set such that P(Ξ) = 1. Gven a frst stage decson x, the realzaton of random vector ξ of ξ s observed. The second stage data m 2 -vector h(ξ), n 2 -vector q(ξ) and m 2 n 1 matrx T(ξ) become known. Then, the second stage decson y(ξ) must be

4 136 Dynamc Slope Scalng taken so as to satsfy the constrants Wy(ξ) = ξ Tx and y(ξ) 0. The second stage decson y(ξ) s assumed to cause a penalty of q(ξ). The objectve functon contans a determnstc term c x and the expectaton of the second stage objectve. The symbol E ξ represents the mathematcal expectaton wth respect to ξ, and the functon Q(x,ξ) s called the recourse functon n state ξ. The value of the recourse functon s gven by solvng a second stage lnear programmng problem. It s assumed that the random vector ξ has a dscrete dstrbuton wth fnte support Ξ = {ξ 1,...,ξ S } wth Prob( ξ = ξ s ) = p s,s = 1,...,S. A partcular realzaton ξ of the random vector ξ s called a scenaro. Gven the fnte dscrete dstrbuton, the problem (SPR) s restated as (SPR ), the determnstc equvalent problem for (SPR). (SPR ): mn c x+ subject to Ax = b x 0 S p s Q(x,ξ s ) s=1 Q(x,ξ s ) = mn{q(ξ s ) y(ξ s ) Wy(ξ s ) = h(ξ s ) T(ξ s )x, y(ξ s ) 0},s = 1,...,S Theproblem(SPR )sreformulatedas(dep-spr)settngy(ξ s ),q(ξ s ),T(ξ s ),h(ξ s ), Q(x,ξ s ) as y s, q s, T s, h s, q s y s, respectvely. (DEP-SPR) : mn x,y 1,,y s S cx+ p s q s y s s=1 subject to Ax = b Wy s = h s T s x,s = 1,...,S x 0,y s 0,s = 1,...,S To solve (DEP-SPR), an L-shaped method (Van Slyke and Wets [4]) has been used. Ths approach s based on Benders[8] decomposton. The expected recourse functon s pecewse lnear and convex, but t s not gven explctly n advance. In the algorthm of the L-shaped method, we solve the followng problem (MASTER). The new varable θ denotes the upper bound for the expected recourse functon such that θ S s=1 ps Q(x,ξ s ).

5 T. Shna and C. Xu 137 (MASTER): mn c x+θ subject to Ax = b x 0 θ 0 The recourse functon s gven by an outer lnearzaton usng a set of feasblty and optmalty cuts as shown n Fgure 1. In the case of n 2 = 2 m 2 and Q(x) optmalty cut feasblty cut θ x Fgure 1: L-shaped method W = (I, I), the problem (SPR) s sad to have a smple recourse. 2.2 Smple nteger recourse In ths secton, we consder the specal case of (SPR) settng q(ξ) = q(> 0),T(ξ) = T, h(ξ) = ξ andw = ri, wherer sapostventeger. Furthermore, we defne the constrants of the recourse problem as y(ξ) ξ Tx,y(ξ) 0 takng acount of the relatonshp between the value of the random varable ξ and the frst stage decson varable Tx. The sze of the random vector ξ s defned as l = m 2, and the sze of the recourse varable y(ξ) s n 2 = m 2. Then we defne the new varables χ = Tx, where χ s called a tender to be bd aganst random outcomes.

6 138 Dynamc Slope Scalng In the case the recourse varables are defned as the nonnegatve nteger varables, the problem s called to have a smple nteger recourse. For ths problem, the constrants of the recourse problem are y(ξ) ξ χ, y(ξ) Z n 2 +. The opmmal soluton of the recourse problem s a mmmal nonnegatve nteger satsfyng y(ξ) ξ χ. As the recourse decsons are represented as urgent and addtonal producton, order, or nvestment, the recourse decsons are taken n a certan amount of unt. Louveaux-van der Vlerk [10] presented the lower and upper bounds for the problem. But consderng the applcaton of the mathematcal programmng model to real problems, the recourse decsons should be modefed to take some batch sze. In ths paper, we formulate the stochastc programmng problem (SPSIR) n whch the recourse varable y(ξ) s defned as nonnegatve nteger varable and the recourse acton ry(ξ) s restrcted to nonnegatve multple of some nteger r. (SPSIR): mn c x+ψ(χ) subject to Ax = b,x 0 Tx = χ Ψ(χ) = S s=1 ps ψ(χ,ξ s ) ψ(χ,ξ s ) = mn{q y(ξ s ) ry(ξ s ) ξ s χ,y(ξ s ) Z n 2 + },s = 1,...,S 3 Soluton Algorthm 3.1 Property of the recourse functon In ths secton, we nvestgate the property of the recourse functon. The optmal soluton of the recourse problem s obtaned as follows. { y(ξ s ξs χ, f χ r < ξ s ) =, = 1,...,m 0 f ξ s 2 χ It s shown that the recourse functon ψ(χ,ξ) s separable n the elements of χ = (χ 1,...,χ m2 ). We defne ψ (χ,ξ ) = mn{q y(ξ) ry(ξ) ξ

7 T. Shna and C. Xu 139 χ,y(ξ) Z + } n the followng equaton. ψ(χ,ξ) = mn{q y(ξ) ry(ξ) ξ χ,y(ξ) Z + } m 2 = mn{q y(ξ) ry(ξ) ξ χ,y(ξ) Z + } = =1 m 2 ψ (χ,ξ ) (1) =1 Let ξ andξ bethe-thcomponentoftherandomvector ξ andthesupport of ξ, respectvely. We make the followng assumptons. Assumpton 3.1. The random varables ξ, = 1,...,n 2 are ndependent and follow a dscrete dstrbuton. Assumpton 3.2. A probablty p s s assocated wth each outcome ξ s, s = 1,..., Ξ of ξ. The random varable ξ takes only postve values and s bounded as 0 < ξ s <,s = 1,..., Ξ, = 1,...,n 2. Then, the support of ξ s descrbed as Ξ = Ξ 1 Ξ n2. And the postve constantm canbetakensoastosatsfym max{ξ,s s = 1,..., Ξ, = 1,...,n 2 }. From assumpton 3.1, 3.2, the jont probablty P( ξ = ξ s ) s calculated as follows. Prob( ξ = ξ s ) = Prob( ξ 1 = ξ s 1 ) Prob( ξ m2 = ξ sm 2 ) m 2 = Prob( ξ = ξ s ) = =1 m 2 =1 p s (2) It s shown that the expected recourse functon Ψ(χ) s also separable n χ, = 1,...,m 2 as (3), where Ψ (χ ) = Ξ s=1 ps ψ (χ,ξ s ) denotes the expec-

8 140 Dynamc Slope Scalng taton of the -th recourse functon (3). Ψ(χ) = = = = = S p s ψ(χ,ξ s ) s=1 Ξ 1 s 1 =1 m 2 =1 Ξ n2 s n2 =1 Ξ 1 ( s 1 =1 m 2 Ξ =1 s =1 m 2 p s 1 1 p sn 2 n 2 Ξ n2 s n2 =1 p s p s ψ (χ,ξ s ) m 2 j=1 j m 2 =1 ψ (χ,ξ s ) p s j j )ψ (χ,ξ s ) Ψ (χ ) (3) =1 For the lst of the realzaton of the random varable {ξ 1,...,ξ Ξ }, we sort ξ s,s = 1,..., Ξ n non-decreasng order so as to satsfy ξ 1... ξ Ξ by substtutng ndces f requred. The expectaton of the recourse functon ψ (χ,ξ ) s shown as follows. Ξ Ψ (χ ) = [ψ E ξ (χ, ξ )] = s =1 p s q ξs χ r + (4) The dscontnuous breakponts of the expected functon Ψ (χ ) are shown as 5 n the regon 0 χ ξ Ξ. χ = ξ s mr (s = 1,..., Ξ,m = 0,1,..., ξs ) (5) r The expected recourse functon Ψ (χ ) has at most Ξ s =1 ( ξs r + 1) dscontnuous ponts, and the length of the contnuous regon depends the value of the constant r. Forexample,theexpectedrecoursefunctonΨ (χ )nthecaseξ = {11,22},p 1 = p 2 = 1/2,r = 5,q = 1 s shown n Fgure 2. And the expected recourse functon Ψ (χ ) can be calculated usng the dstrbuton functon F of ξ.

9 T. Shna and C. Xu Expected Recourse Functon Ψ(χ) Lower Bound of Ψ(χ) 3 Ψ(χ) χ Fgure 2: Expected recourse functon Ψ (χ ) = E ξ [ = q j=1 j 1 = q = q = q ] χ q ξ + r j=1 k=0 χ jprob( ξ + = j) r k=0 j=k+1 k=0 χ Prob( ξ + = j) r χ Prob( ξ + = j) r Prob( ξ χ r > k) = q (1 F (χ +rk)) (6) k=0

10 142 Dynamc Slope Scalng 3.2 Algorthm of DSSP Let (SPSIR LP ) be the problem n whch the nteger constrants are relaxed. TherecoursefunctonΨ(χ)oftheproblem(SPSIR LP )correspondstothelower bound for the orgnal Ψ(χ) of (SPSIR) as shown n Fgure 2. (SPSIR LP ):mn c x+ψ(χ) subject to Ax = b,x 0 Tx = χ Ψ(χ) = S s=1 ps ψ(χ,ξ s ) ψ(χ,ξ s ) = mn{q y(ξ s ) ry(ξ s ) ξ s χ,y(ξ s ) 0},s = 1,...,S Aftersolvngtheproblem(SPSIR LP ),theoptmalsoluton(x LP,χ LP,y LP (ξ 1 ),...,y LP (ξ S )) s obtaned. Next, we consder a heurstc algorthm to solve (SPFCRT). For the fxed charge network flow problem, Km and Pardalos [11] developed an approach, called the dynamc slope scalng procedure(dssp), whch solves successve lnear programmng problems wth recursvely updated objectve functons. Km and Pardalos [12] modfed DSSP, whch repeats the reducton and refnement of the feasble regon and the algorthm s effectve when the objectve functon s starcase or sawtooth type. The algorthm of DSSP s used to obtan a good feasble soluton to the second stage nteger programmng problem whch defnes the recourse functon. The algorthm of DSSP s promsng snce the recourse functon s monotoncally nonncreasng as shown n Fgure 2. Let (x LP,χ LP,y LP (ξ 1 ),...,y LP (ξ S )) be the optmal soluton of the problem (SPSIR LP ). We compute the approxmate value θ of Ψ (χ ) usng the followng nequalty (7). θ Ψ (χ LP ) χ LP ξ Ξ (χ χ LP )+Ψ (χ LP ) (7) The constrant (7) provdes the upper bounds for the lnear functon whch connects (ξ Ξ,0) and (χ LP,Ψ (χ LP )). The value of θ gves the exact value of Ψ (χ ) at these two ponts. Takng accounts of the breakponts (5) of the recourse functon, we set the lowerandupperboundsforthevarableχ. Letthebreakpontsoftherecourse functon Ψ (χ ) be 0 < χ 1 χ 2 χ w, and we defne χ 0 = 0.

11 T. Shna and C. Xu Expected Recourse Functon Ψ(χ) Approxmaton of Ψ(χ) Ψ(χ) 3 2 χ LP* =10 1 bounds 7 χ χ Fgure 3: Algorthm of DSSP If we have a χ LP the constrant χ j χ χ j+1 have a χ LP satsfyng χ LP s added. satsfyng χ j < χlp < χ j+1 for some j (0 j w 1), s added to the formulaton. Otherwse f we = χ j for some j (1 j w 1), the constrant χ j 1 χ χ j+1 Then the followng lnear programmng problem (MASTER) s solved. m 2 (MASTER):mn c x+ =1 θ subject to Ax = b,x 0 Tx = χ θ Ψ (χ LP ) χ LP ξ Ξ bound constrants for θ (χ χ LP )+Ψ (χ LP ), = 1,...,m 2 Soluton algorthm usng DSSP Step1 Gven ε > 0 for the convergence check. Solve (SPSIR LP ) to obtan (x LP,χ LP, y LP (ξ 1 ),...,y LP (ξ S )). The constrant (7) and the lower and upper bounds for θ are added to (MASTER). Set k = 1. Step2 Solve (MASTER) to obtan (x k,χ k,θ k ).

12 144 Dynamc Slope Scalng Table 1: Computatonal Results Number of Number of Parameter GAP Relatve CPU tme (sec) random scenaros error Branchvarable andm 2 Ξ r (%) (%) DSSP Bound Experment Experment Step3 If k > 1 and n 1 =1 xk x k 1 + m 2 =1 χk χ k 1 + m 2 =1 θk θ k 1 > ε, modfy the constrant (7) and the lower and upper bounds for θ of (MASTER), k = k +1, and go to Step 2. Step4 Fromthesoluton(x k,χ k,θ k ),calculateψ(χ k ),andsettheapproxmate optmal objectve value as c x k +Ψ(χ k ). 4 Numercal experments 4.1 Objectve of experments In ths secton, we consder the applcatons to producton plannng. It s assumed that the demand for n 2 products are met by exstng n 1 producton plants. Suppose the demand of product j s defned as a random varable ξ j. Let ξ 1,...,ξ n2 be the realzatons of random varables ξ 1,..., ξ n2, and Ξ 1,...,Ξ n2 be ther supports. These random varables are ntegrated as a random vector ξ = ( ξ 1,..., ξ n2 ), and the support Ξ of ξ s descrbed as Ξ = Ξ 1 Ξ n2. We consder the applcaton of the problem (SPSIR) to the producton plannng problem(spsir ). The frst stage decson varable s the amount of products j manufactured by plant, denoted by x j, = 1,...,n 1,j =

13 T. Shna and C. Xu 145 1,...,n 2. Let a j be the fuel consumpton rate of plant for the producton of product j. For the frst stage constrants, let b be the upper bound for the fuel consumpton of the producton plant. The tender varable χ j s a total amount of product j manufactured by all plants. Gven a frst stage decson x and χ, the realzaton of random demand ξ of ξ becomes known. After observng the realzaton ξ, the second stage decsons y j (ξ j ) are taken to meet the demand. The amount of unserved demand has to be suppled by the addtonal producton n the second stage. The multplcaton ry j (ξ j ) of recourse varable y j (ξ j ) and postve nteger r means that the urgent producton must be made n r unts. The recourse costs q j are the addtonal producton cost. The formulaton of the problem s descrbed as (SPSIR ). The frst constrant of the second stage problem to defne ψ j (χ j,ξ j ) says the demand must be satsfed, whereas the second constrant of the recourse problem expresses that demand ξ s suppled by the frst stage producton χ and addtonal producton ry(ξ). (SPSIR ):mn subject to n 1 n 2 c j x j +Ψ(χ) =1 j=1 n 2 a j x j b, = 1,...,n 1 j=1 x j 0, = 1,...,n 1,j = 1,...,n 2 n 1 χ j = x j,j = 1,...,n 2 Ψ(χ) = =1 Ξ 1 Ξ n2 s=1 n 2 p s ψ(χ,ξ s ) ψ(χ,ξ s ) = ψ j (χ,ξj),s s = 1,...,( Ξ 1 Ξ n2 ) j=1 ψ j (χ j,ξ s j) = mn{q j y j (ξ s j) ry j (ξ s j)+χ j ξ s j y j (ξ s j) Z + },s = 1,..., Ξ j,j = 1,...,n 2 ThentwoexpermentsareconductedtoshowthatthealgorthmofDSSPs effcent to solve the stochastc programmng problem (SPSIR). In experment 1, the number of scenaros were changed to see the effcency of the algorthm of DSSP. We show the CPU tme of DSSP and the tme of Branch-and-Bound. Furthermore, the relatve error of DSSP s presented to show the DSSP s precse algorthm.

14 146 Dynamc Slope Scalng The results of the numercal experments appear n Table 1. The GAP descrbed n Table 1, s defned as (z LB)/LB, where z s an optmal objectve value of (SPSIR) and LB s an optmal objectve value of the LP relaxaton of (SPSIR). The relatve error n Table 1, s defned as (ẑ z )/z, where ẑ s a objectve value obtaned usng the algorthm of DSSP. The CPU tmes usng DSSP and branch-and-bound or the values of the relatve error are compared when the number of scenaro s changed. In experment 2, the values of the relatve error and the CPU tme are measured when the value of parameter r s changed. When the value of parameter r becomes large, the length between two adjacent breakponts becomes long. In ths case, t s worthy of notce to see how the value of parameter r affects the precson of DSSP. The algorthm of DSSP for the stochastc producton plannng problem was mplemented usng ILOG OPL Development Studo on DELL DIMENSION 8300 (CPU: Intel Pentum(R)4, 3.20GHz). The smplex optmzer of CPLEX 9.0 was used to solve the problem. Table 1 presents the average values of 5 results of our experments. The values of the random varables were generated based on the unform dstrbuton. 4.2 Experment 1: Changng the number of scenaros The problems consdered n experment 1, consst of 10 products. The demand for each product has 10 and 20 scenaros. In order to see the effcency of the algorthm of DSSP, the CPU tme of DSSP s compared wth the tme of Branch-and-Bound. Usng the branch-and-bound, the CPU tme grows rapdly snce we must solve a large scale mxed nteger programmng problem. However, the algorthm of DSSP solves the problem quckly as the algorthm repeats to solve the lnear programmng problem. The algorthm of DSSP provdes precse solutons as the relatve errors of DSSP s less than 1%. 4.3 Experment 2: Changng the postve nteger r Table 1 shows that the CPU tme of tha Branch-and-Bound tends to be long when the value of the parameter r s small. As the length of the range n

15 T. Shna and C. Xu 147 whch the recourse functon takes a constant value becomes narrow when r s small, the number of such regons ncreases. Therefore, the number of tmes whch the lower and upper bounds for θ are added, ncreases. As a result, the CPU tme of the Branch-and-bound ncreases. However, the CPU tme of DSSP s shorter than that of the Branch-and-Bound. The GAP value becomes large when the nteger r ncreases. smlarly to the reason descrbed prevously, as the length of the range n whch the recourse functon takes a constant value becomes wde when r s large. Accordngly, the GAP becomes large and the CPU tme of branch-and-bound ncreases. As for the relatve errors, t remans wthn 1%. DSSP provdes accurate solutons n short CPU tme. 5 Concluson We have consdered the stochastc programmng problem wth smple nteger recourse n whch the value of the recourse varable s restrcted to a multple of a nonnegatve nteger. The algorthm of a dynamc slope scalng procedure to solve the problem s developed by usng the property of the expected recourse functon. The numercal experments show that the proposed algorthm s qute effcent. References [1] J.R. Brge, Stochastc programmng computaton and applcatons, IN- FORMS Journal on Computng, 9, (1997), [2] J.R. Brge and F. V. Louveaux, Introducton to Stochastc Programmng, Sprnger-Verlag, [3] P. Kall and S.W. Wallace, Stochastc Programmng, John Wley & Sons, 1994.

16 148 Dynamc Slope Scalng [4] R. Van Slyke and R. J.-B. Wets, L-shaped lnear programs wth applcatons to optmal control and stochastc lnear programs, SIAM Journal on Appled Mathematcs, 17, (1969), [5] G. Laporte and F. V. Louveaux, The nteger L-shaped method for stochastc nteger programs wth complete recourse,operatons Research Letters, 13, (1993), [6] T. Shna, L-shaped decomposton method for mult-stage stochastc concentrator locaton problem, Journal of the Operatons Research Socety of Japan, 43, (2000), [7] T. Shna, Stochastc programmng model for the desgn of computer network (n Japanese),Transactons of the Japan Socety for Industral and Appled Mathematcs,10, (2000), [8] J.F. Benders, Parttonng procedures for solvng mxed varables programmng problems, Numersche Mathematk, 4, (1962), [9] S. Ahmed, M. Tawarmalan and N. V. Sahnds, A fnte branch-andbound algorthm for two-stage stochastc nteger programs, Mathematcal Programmng, 100, (2005), [10] F.V. Louveaux and M.H. van der Vlerk, Stochastc programmng wth smple nteger recourse, Mathematcal Programmng, 61, (1993), [11] D. Km and P. Pardalos, A soluton approach to fxed charge network flow problem useng a dynamc slope scalng procedure, Operatons Research Letters, 24, (1999), [12] D. Km and P. Pardalos, A dynamc doman constracton algorthm for nonconvex pecewse lnear network flow problems, Journal of Global optmzaton, 17, (2000),