A DUAL EXTERIOR POINT SIMPLEX TYPE ALGORITHM FOR THE MINIMUM COST NETWORK FLOW PROBLEM

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1 Yugoslav Journal of Operatons Research Vol 19 (2009), Number 1, DOI: /YUJOR G A DUAL EXTERIOR POINT SIMPLEX TYPE ALGORITHM FOR THE MINIMUM COST NETWORK FLOW PROBLEM George GERANIS Konstantnos PAPARIZZOS Angelo SIFALERAS Department of Appled Informatcs, Unversty of Macedona, gerans,paparrz,sfalera@uom.gr Receved: December 2007 / Accepted: May 2009 Abstract: A new dual smplex type algorthm for the Mnmum Cost Network Flow Problem (MCNFP) s presented. The proposed algorthm belongs to a specal exterorpont smplex type category. Smlarly to the classcal network dual smplex algorthm (NDSA), ths algorthm starts wth a dual feasble tree-soluton and reduces the prmal nfeasblty, teraton by teraton. However, contrary to the NDSA, the new algorthm does not always mantan a dual feasble soluton. Instead, the new algorthm mght reach a basc pont (tree-soluton) outsde the dual feasble area (exteror pont - dual nfeasble tree). Keywords: Operatons research, combnatoral optmzaton, mnmum cost network flow problem. 1. INTRODUCTION The Mnmum Cost Network Flow Problem (MCNFP) s the problem of fndng a mnmum cost flow of product unts, through a number of source nodes, snks and transshpment nodes. Other common problems, such as the shortest path problem, the transportaton problem, the transshpment problem, the assgnment problem etc., are the specal cases of the MCNFP. Such problems appear very frequently n dfferent technology sectors, lke the Informaton Technology, the Telecommuncatons, the Transportaton, the Resource Management, etc. Algorthms developed for the MCNFP can offer good solutons for such problems. A number of dfferent problems that can be solved by the followng MCNFP methods are descrbed n [1] and [9].

2 158 G. Gerans, K. Paparrzos, A.Sfaleras / Mnmum Cost Network The MCNFP can be easly transformed nto a Lnear Programmng Problem and well known general lnear programmng technques could be appled n order to fnd an optmal soluton. Such technques do not take advantage of some specal features met n the MCNFP. So, other specal Smplex-type algorthms have been developed, such as the Prmal Network Smplex Method and the Dual Network Smplex Method. There are also other non Smplex-type algorthms that can be used for the solvng of the same problem, as presented n [7], [2] and [15]. An exteror-pont prmal smplex-type algorthm for solvng the MCNFP has also been presented n [13]. Ths paper comes to present for the frst tme an exteror pont dual smplex-type algorthm for the MCNFP. The algorthm s named Dual Network Exteror Pont Smplex Algorthm (DNEPSA) for the MCNFP. It starts wth a dual feasble tree-soluton and, teraton by teraton, t produces new tree-solutons closer to an optmal soluton, reducng the problem s nfeasblty. Contrary to the Network Dual Smplex Algorthm, the tree-soluton at every teraton s not necessarly dual feasble but, after a number of teratons, the algorthm reaches a tree-soluton that s both prmal feasble and dual feasble and therefore t s optmal. Secton 2 gves the notaton that wll be used n ths paper and a short descrpton of the MCNFP. In Secton 3, the Dual Network Exteror Pont Smplex Algorthm (DNEPSA) s descrbed and the steps that have to be followed are descrbed n detal. An llustratve example presentng the algorthm step by step s gven n Secton 4. Fnally, Secton 5 gves some conclusons and plans for future work. 2. NOTATIONS AND PROBLEM STATEMENT In ths Secton we shall gve a short descrpton of the MCNFP. Let G=(N,A) be a drected network that conssts of a fnte set of nodes N and a fnte set of drected arcs A, that lnk together pars of nodes. Let n and m be the number of nodes and arcs respectvely. For each node N, there s an assocated varable b representng the avalable supply or demand at that node. A node s a supply node (source), f t s b > 0. On the other hand, t s a demand node (snk), f t s b < 0. Fnally, the node s a transshpment node n case t s b = 0. The total supply has to be equal to the total demand,.e. t has to be b = 0 (balanced network). N For every arc (,j) we have an assocated flow x that shows the amount of product unts transferred from node to node j and an assocated cost per unt value x. Therefore, the total cost s equal to cx and the MCNFP s the problem of fndng (, j) A a flow that mnmzes that total cost. We can have, for the flow x on arc (,j), a lower and an upper bound, l and u respectvely. Ths gves an addtonal constrant l x u for every arc (,j). In our case we consder t s l = 0 and u = +. In other words, our algorthm s appled to the uncapactated MCNFP. For every node t has to be

3 G. Gerans, K. Paparrzos, A.Sfaleras / Mnmum Cost Network 159 x x = b j (, j) A ( j,) A because the outgong flow must be equal to the ncomng flow plus the node s supply. Therefore, the MCNFP can be formulated as follows: mnmze cx (2.1) (, j) A subject to x x = b N (2.2) j (, j) A ( j,) A x 0, ( ) A (2.3) Snce t s b = 0, by usng formula (2.2), t comes out that N ( x x ) = 0 N (, j) A ( j,) A j That means that constrants (2.2) are lnearly dependent and we could arbtrarly drop out one of them. Of course, a problem lke ths could be solved usng standard wellknown Lnear Programmng algorthms, but these algorthms do not take advantage of some specal features met n the MCNFP. There s a set of dual varables w, one for every node, and a number of reduced cost varables w, one for every drected arc. These are the varables used for the formulaton of the dual problem. Network smplex-type algorthms start from a basc tree-soluton and compute vectors x, w, s consstng of varables x, w and w respectvely. If for a tree-soluton T, t s x 0 for every arc (, j) T, then that soluton s sad to be prmal feasble. If for a tree-soluton T, t s s 0 for every arc (, j) T then t s sad to be dual feasble. A soluton beng both prmal feasble and dual feasble represents an optmal soluton. Prmal network smplex-type algorthms start from a prmal feasble soluton, whle dual network smplex-type algorthms, lke the algorthm descrbed here, start from a dual feasble soluton. 3. ALGORITHM DESCRIPTION The Dual Network Exteror Pont Smplex Algorthm (DNEPSA for short), starts from a dual feasble basc tree-soluton T and, after a number of teratons, t comes to a tree-soluton that s both prmal feasble and dual feasble and therefore, t s optmal. In contrary to the exstng dual network smplex-type algorthms, the tree-solutons formed durng the teratons are not necessarly always dual feasble but they can be both prmal nfeasble and dual nfeasble. So, DNEPSA s a smplex-type algorthm startng from a dual soluton that reaches an optmal soluton by followng a route consstng of solutons that do not belong to the prmal feasble area.

4 160 G. Gerans, K. Paparrzos, A.Sfaleras / Mnmum Cost Network There are dfferent technques that can be used n order to fnd a startng dual feasble tree-soluton. An algorthm that can construct a dual feasble tree-soluton for the generalzed network problem (and also for pure networks) s descrbed n [8] and an mproved verson of the algorthm s presented n [11], whch gves a dual feasble soluton that s closer to an optmal soluton. Gven the startng dual feasble tree-soluton, t s easy, startng from the leaf nodes, to compute the flows x for all the basc arcs (,j) n that tree (step 1). It s also very easy to compute values for the dual varables w from the equatons: w w = c, for every arc (, j) T (3.1) In equatons (3.1) we have n-1 equatons and n varables, so, we can choose one of the dual varables (e.g. w ) and set t equal to an arbtrary value (e.g. 0). Then, t s easy to compute the values of the rest of the dual varables. In order to calculate the reduced costs s for the non-basc arcs (,j), we can use the followng equatons: s = c + w wj, for every arc (, j) T (3.2) whle t s s = 0 for all the basc arcs. Next, the algorthm creates a set, named I_, that contans the basc arcs (,j) havng negatve flow,.e. t s x < 0. It also creates set I + contanng the rest of the arcs. If t s I_=, ths means that the tree-soluton s feasble and therefore t s optmal (step 2). After ths, the algorthm consders the non-basc arcs (, j) T. When such an arc (,j) s added to the basc tree-soluton T, then a cycle C s created. That cycle may contan arcs of I_ havng the same orentaton as (,j) and others havng the opposte orentaton. For every non-basc arc, let d be the dfference between the number of arcs n I_ havng the same orentaton n cycle as (,j) mnus the number of them havng the opposte orentaton. J_ conssts of those non-basc arcs (,j) havng d > 0 (step 3). After creatng set J_, we have to choose amongst ts arcs, the one that wll be the enterng arc (g,h). Ths s the arc of J_ that gves the mnmum rato s / d (step 4). Next, the algorthm has to fnd the leavng arc (k,l). Ths s done by checkng the cycle C formed after addng the enterng arc (g,h) to the tree T. For the arcs of I_ havng the same orentaton n C as (g,h), we choose arc (k 1,l 1 ) that corresponds to the mnmum absolute flow. Smlarly, for the arcs of I + havng orentaton opposte to the enterng arc (g,h), we choose arc (k 2,l 2 ) that corresponds to the mnmum absolute flow. If we denote θ 1 and θ 2 these two mnmum values, we decde the leavng arc by comparng θ 1 aganst θ 2. If t s θ 1 θ 2, then the leavng arc s (k,l) = (k 1,l 1 ), otherwse t s (k,l) = (k 2,l 2 ). At ths pont, the algorthm has come to a new basc tree-soluton T. The same process has to be repeated untl the algorthm reaches to an optmal soluton.

5 G. Gerans, K. Paparrzos, A.Sfaleras / Mnmum Cost Network 161 The algorthm s steps are descrbed below. The symbols and used here, denote arcs that have the same orentaton or opposte orentaton, respectvely. Algorthm DNEPSA 1. Start wth a dual feasble tree T. Compute the flow varables x for all the basc arcs, the dual varables w and the reduced cost varables s, by applyng formulas (3.1) and (3.2). 2. Set I _ = {(, j) T : x < 0} and I+ = {(, j) T : x 0} If t s I _ =, then the current tree-soluton T s an optmal soluton. 3. Set J _ = (, j) T : f added to T, n the cycle created, t s d = p n > 0 where { } p s the number of arcs n and n s the number of arcs n 4. Choose arc ( g, h) J _, where t s sgh s mn{ : (, j) J_} d = d gh Arc (g,h) s the enterng arc. 5. Compute values θ 1 and θ 2 where: kl 11 { } θ 1 = x = mn x (, j) I _ and(, j) ( g, h) { } θ kl = x = mn x (, j) I and(, j) ( g, h) If 1 2 I _ havng the same orentaton as (,j) I _ havng the opposte orentaton. θ θ then the leavng arc s ( k,1) = ( k1), otherwse the leavng arc s ( k,1) = ( k,1 ) For the new tree-soluton T, compute the flows x and the dual problem varables w and s. Repeat the process from step 2. The algorthm s presented n more detal n the next Secton, where an llustratve example s gven.

6 162 G. Gerans, K. Paparrzos, A.Sfaleras / Mnmum Cost Network 4. AN ILLUSTRATIVE EXAMPLE We ll now gve an llustratve, step by step, example where the algorthm presented wll be appled to a MCNFP. Fgure 1 shows a network G = (N, A), consstng of 6 nodes and 12 arcs. Next to each node there s a value showng the node s supply (negatve values mean demands). For every arc n A, the cost per product unt flow s also shown. Fgure 4.1: The graph G = (N,A) where DNEPSA s appled We wll apply below the algorthm s steps to fnd a soluton for the MCNFP as appled to graph G. The algorthm fnds an optmal soluton after 3 teratons. Iteraton 1 Step-1. In order to start, the algorthm needs an ntal dual feasble basc tree-soluton. Fgure 2 below shows such a dual feasble tree. Such an ntal soluton can be obtaned by usng exstng technques, as t was sad n Secton 2.

7 G. Gerans, K. Paparrzos, A.Sfaleras / Mnmum Cost Network 163 Fgure 4.2: The ntal dual feasble tree The tree shown s a dual feasble tree, snce t s s 0 for every arc (,j). Ths can be easly verfed by solvng equatons (3.1), whch take the followng form: w w 5 w1 = 16 5 w2 = 57 w5 w3 = 107 w w 5 w4 = 32 5 w6 = 1 By settng w 1 equal to 0, we have w 5 = 16, w 2 = 41, w 3 = 91, w 4 = 16 and w 6 = 15. By applyng equatons (3.2) we have: s16 = c16 + w1 w6 = = 26 s26 = c26 + w2 w6 = ( 41) 15 = 48 s36 = c36 + w3 w6 = ( 91) 15 = 24 s46 = c46 + w4 w6 = 84 + ( 16) 15 = 53 s53 = c53 + w5 w3 = ( 91) = 178 s64 = c64 + w6 w4 = ( 16) = 31 s63 = c63 + w6 w3 = ( 91) = 149 whle, for all the basc arcs t s s = 0.

8 164 G. Gerans, K. Paparrzos, A.Sfaleras / Mnmum Cost Network For every node n the graph, t s x x = b k j (, k) N ( j,) N Ths happens because for every node the outgong flow has to be equal to the ncomng flow plus the node s supply. That s, the followng equatons have to be satsfed: node1: x 15 = 3 node 2: x 25 = 5 node3: x 35 = 3 node 4: x 45 = 3 node 4: x 45 = 3 node5: x x x x x = 6 node6: x 65 = By solvng the above equatons we have x 15 = 3, x 25 = 5 x 45 = 3 x 65 = 8. Ths s not a feasble tree snce we found some negatve flows (t s x 65 < 0 ) Step-2 It s I_ = {(6,5)}, snce t s x 65 = 8< 0 and the remanng arcs form I +. Step-3 If we add the non-basc arc (1,6) to the basc tree, a cycle C s created. In ths cycle, arc (1,6) has the same orentaton as (6,5) whch belongs to I_. So, (1,6) J_. By checkng n a smlar way all the non-basc arcs, t comes out that J_ = {(1,6),(2,6),(3,6),(4,6)} wth d16 = d26 = d36 = d46 = 1. Step-4 For the arcs n J_ t s s 16 = 26, s 26 = 48, s 36 = 48, and s 46 = 53. It s s d = = mn{,,, } So arc (g,h)=(3,6) s the enterng arc. Step-5 After addng the enterng arc (3,6) to the basc tree, a cycle C s created, as shown n Fgure 3.

9 G. Gerans, K. Paparrzos, A.Sfaleras / Mnmum Cost Network 165 Fgure 4.3: The cycle created after addng the enterng arc Arc (6,5) belongs to I_ and has the same orentaton as the enterng arc (3,6). So, t s θ 1 = x65 = 8. Arc (3,5), on the other hand, belongs to I + and does not have the same orentaton as the enterng arc. So, t s θ 2 = x35 = 3. We have θ 1 > θ 2 whch means that arc (3,5) s the leavng arc. Step-6 The tree shown n Fgure 4 s now the new basc tree. Fgure 4.4: The new basc tree-soluton

10 166 G. Gerans, K. Paparrzos, A.Sfaleras / Mnmum Cost Network For the new tree t s x 15 = 3, x 25 = 5, x 36 = 3, x 36 = 3, x 45 = 3 and x 65 = 5. Ths s not a feasble tree, so the process has to be contnued. By usng formulas (3.1) and (3.2), the same way as for the startng basc tree, we fnd that s 16 = 26, s 26 = 48, s 35 = 24, s 46 = 53, s 53 = 202, s 64 = 31 and s 63 = 173. Ths new basc tree s not a dual feasble tree-soluton. Iteraton 2 Step-2 It s I_ = {(6,5)}, snce t s x 65 = -5 < 0 and the remanng arcs form I +. Step-3 By checkng, as n the frst teraton, what happens when the non-basc arcs are added to the basc tree and ther orentaton, we fnd that t s J_={(1,6),(2,6),(4,6),(5,3)} wth d 16 = d 26 = d 46 = d 53 = 1. Step-4 For the arcs n J_ t s s16 = 26, s26 = 48, s46 = 53 and s53 = 202. It s s d = = mn{,,, } So arc (g,h)=(1,6) s the enterng arc. Step-5 After addng the enterng arc (1,6) to the tree, a cycle C s created as shown n Fgure 5. Fgure 4.5: The basc tree after addng the enterng arc Arc (6,5) belongs to I_ and has the same orentaton as the enterng arc (1,6). So, t s θ 1 = x65 = 5. Arc (1,5), on the other hand, belongs to I + and does not have the same orentaton as the enterng arc. So, t s θ 2 = x15 = 3. We have θ1 > θ2 so, arc (1,5) s the leavng arc. Step-6 The tree, shown n Fgure 6, forms now the new basc tree-soluton.

11 G. Gerans, K. Paparrzos, A.Sfaleras / Mnmum Cost Network 167 Fgure 4.6: The new basc tree-soluton For the new tree, t s x16 = 3, x25 = 5, x36 = 3, x45 = 3 and x65 = 2.e. t s not a feasble tree. It s also: s = 26, s = 48, s = 24, s = 53, s = 202, s = 31 and s = Iteraton 3 Step-2 It s I_ = {(6,5)}, snce t s x 65 = 2< 0 and I + contans the remanng arcs. Step-3 Smlarly, as n the prevous teratons, we fnd that J_={(2,6),(4,6),(5,3)} and d26 = d46 = d53 = 1. Step-4 For the arcs n J_ t s s26 = 48, s46 = 53 and s53 = 202. It s s d = = mn{,, } So, arc (g,h)=(2,6) s the enterng arc. Step-5 If we add arc (2,6) to the tree, a cycle C s created, as shown n Fgure 7.

12 168 G. Gerans, K. Paparrzos, A.Sfaleras / Mnmum Cost Network Fgure 4.7: The basc tree after addng the enterng arc Arc (6,5) belongs to I_ and has the same orentaton as the enterng arc (2,6). So, t s θ 1 = x65 = 2. Arc (2,5), on the other hand, belongs to I + and does not have the same orentaton as the enterng arc. So t s θ 2 = x 25 = 5. It s θ 1 θ 2 so arc (6,5) s the leavng arc. Step-6 Therefore, the new tree shown n Fgure 8 s now the basc tree. Fgure 4.8: The new basc tree-soluton For ths tree t s x 16 = 3, x 36 = 3, x 45 = 3, x 26 = 2 and x 25 = 3,.e. t s a prmal feasble tree-soluton and also a dual feasble tree snce t s s = 22, s = 24, s = 5, s = 154, s = 79, s = 48 and s = 173. Therefore, we have

13 G. Gerans, K. Paparrzos, A.Sfaleras / Mnmum Cost Network 169 found an optmal soluton. Ths s detected mmedately by our algorthm snce t s Ι _ = and the algorthm stops. 5. CONCLUSIONS AND FUTURE WORK Frst of all, we plan to present all the necessary mathematcal proof of correctness of the proposed algorthm n a future work. DNEPSA s based on a set of lemmas and theorems, whch were omtted n ths paper, due to paper length. Furthermore, there are specal mproved data structures that could be used n order to mprove the performance of DNEPSA. Such data structures nclude dynamc trees and Fbonacc heaps, as descrbed n [10], [16] and [6]. There are also other useful data structures and technques, as presented n [3] and [5], whch obtan very good performance, by applyng varous operatons on graphs and trees. There are several state-of-the-art algorthm mplementatons that can be appled to the MCNFP. For example, mplementatons lke RELAX IV, NETFLO and MOSEK demonstrate very good performance results. Some of these mplementatons are descrbed n [4] and [14]. At ths pont, DNEPSA has already been developed (n C programmng language) and tested thoroughly aganst a bg varety of problems. Therefore, t wll be very nterestng to compare DNEPSA aganst some of the well known MCNF algorthms Fnally, DNEPSA wll be ncorporated nto the Network Optmzaton sute WebNetPro whch s descrbed n [13]. Ths way, WebNetPro s capabltes wll be extended by addng new algorthms nto ths web sute. REFERENCES [1] Ahuja, R., Magnant, T., Orln, J., and Reddy, M., Applcatons of network optmzaton, Handbooks of Operatons Research and Management Scence, (1995) [2] Ahuja, R., and Orln, J., Improved prmal smplex algorthms for shortest path, assgnment and mnmum cost flow problems, Massachusetts Insttute of Technology, Operatons Research Center, Massachusetts Insttute of Technology, Operatons Research Center, Workng Paper OR , (1988). [3] Al, A.I., Helgason, R.V., Kennngton, J.L., and Lall, H.S., Prmal smplex network codes: State-of-the-art mplementaton technology, Networks, 8 (4) (1978) [4] Bertsekas, D. P., and Tseng, P., RELAX-IV: A Faster verson of the RELAX code for solvng mnmum cost flow problems, Techncal Report, Massachusetts Insttute of Technology, Laboratory for Informaton and Decson Systems, [5] Eppsten, D., Clusterng for faster network smplex pvots, Proceedngs of the 5 th Annual ACM-SIAM Symposum on Dscrete Algorthms, [6] Fredman, M., and Tarjan, R., Fbonacc heaps and ther uses n mproved network optmzaton algorthms, Journal of the ACM, 34 (3) (1987) [7] Glover, F., Karney, D., and Klngman, D., Implementaton and computatonal comparsons of prmal, dual and prmal-dual computer codes for mnmum cost network flow problems, Networks, 4 (3) (1974) [8] Glover, F., Klngman, D., and Naper, A., Basc dual feasble solutons for a class of generalzed networks, Operatons Research, 20 (1) (1972) [9] Glover, F., Klngman, D., and Phllps, N., Network Models n Optmzaton and Ther Applcatons n Practce, Wley Publcatons, 1992.

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