An Application of Network Simplex Method for Minimum Cost Flow Problems

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BALKANJM 0 (0) -0 Contents lsts avalable at BALKANJM BALKAN JOURNAL OF MATHEMATICS journal homepage: www.balkanjm.

Avalable onlne October 0 Keywords: Network Optmzaton Mnmum Cost Network Flow Network Smplex Network Dual Feasble Soluton ABSTRACT Networks are more convenent for modelng because of ther smple

1 BALKANJM 0 (0) -0 Contents lsts avalable at BALKANJM BALKAN JOURNAL OF MATHEMATICS journal homepage: An Applcaton of Network Smplex Method for Mnmum Cost Flow Problems Ergun EROGLU *a a Istanbul Unversty Busness Admnstraton Faculty ARTICLE INFO Artcle hstory: Receved July 0 Accepted September 0 Avalable onlne October 0 Keywords: Network Optmzaton Mnmum Cost Network Flow Network Smplex Network Dual Feasble Soluton ABSTRACT Networks are more convenent for modelng because of ther smple mathematcal structure that can be easly represented wth a graph. Ths smplcty takes an advantage wth regard to algorthmc effcency. In ths paper, an mplementaton of network smplex algorthm s descrbed for solvng the mnmum cost network flow problem whch s one of the most fundamental and sgnfcant problems n the optmal desgn on a generalzed network wth the addtonal constrant. Network flow problem can be defned by a gven set of nodes and arcs wth known cost parameters for each arc and fxed external flow for each node. The optmzaton problem s to send flow from a set of supply nodes, through the arcs of a network, to a set of demand nodes, at mnmum total cost subject to the arc capacty constrants. The smplex algorthm appled to the network flow programmng problem. Network smplex method descrbes basc solutons for the network flow programmng problem and provdes procedures for computng the prmal and dual solutons assocated wth a gven bass to fnd the optmal soluton. 0 BALKANJM All rghts reserved.. Introducton Operatons research (OR) s the applcaton of scentfc methods, technques and tools to problems nvolvng the operatons of a system so as to provde those n control of the operatons wth optmum solutons to the problem. []. If one mentons about Operaton research, the word optmzaton comes to remnd. Optmzaton s to fnd the best value of the varables that make optmal the objectve functon satsfyng a set of constrants. It orgnated n the 90s, when George Dantzg used * Correspondngauthor: E-mal: eroglu@stanbul.edu.tr (E. Eroglu) BALKANJM All rghts reserved.

2 E. Eroglu/BALKANJM 0 (0) -0 mathematcal technques for generatng "programs" for mltary applcaton []. Optmzaton technology allows researchers to search for optmal solutons to complex busness, economcs, computer scence and engneerng problems. Optmzaton problems are often classfed as lnear or nonlnear dependng on whether the relatonshps n the problem are lnear or nonlnear wth respect to the varables []. Network optmzaton models have been most exctng developments n OR n recent years. They are wdely used n optmzaton problems whch have countless practcal applcatons n varous felds ncludng commodty transportaton, telecommuncaton systems, network desgn, resource plannng, schedulng, ralroad and hghway traffc plannng, electrcal power dstrbuton, project plannng, facltes locaton, resource management, and fnancal plannng and much more. The fundamental queston n network optmzaton s how to effcently transport some entty (commodty, product, electrcal power, vehcles, water etc.) from one pont to another n a network []. Network optmzaton s a specal type of lnear programmng model. Some specal types of network optmzaton models nclude: transportaton problems, assgnment problems, shortest path problems, mnmum spannng tree problems, maxmum flow problems, Chnese postman problem, knapsack problem and mnmum cost flow problems []. Transportaton problem was frst studed by a Russan mathematcan, L.V. Kantorovch, n a paper enttled Mathematcal Methods of Organzng and Plannng Producton (99) []. One of the most fundamental network flow problems s the Mnmum Cost Flow Problem (MCF). MCF Problem s to send flow from a set of supply nodes, through the arcs of a network, to a set of demand nodes, at mnmum total cost, and wthout volatng the lower and upper bounds on flows through the arcs. MCF often plays an mportant role n modelng operatons management problems such as producton and nventory plannng, supply chan management, mult echelon nventory plannng, capacty expanson, etc. MCF Problem and the Network Smplex Method (NSM) were ntally developed qute ndependently. The MCF has ts orgns n the formulaton of the classc transportaton type problem, whle NSM, as ts name suggests s deduced from the Smplex Method for solvng Lnear Programmng Problems [], []. Many network optmzaton models actually are specal types of lnear programmng problems. Therefore we begn to study of network flow problems wth a revew of lnear programmng (LP) problems. Let the number of decson varables s be, and the number of constrants be.mnmzaton of a LP Problem n prmal and dual mode takes the fallowng generc form [], []: Prmal Mode Dual Mode

3 E. Eroglu/BALKANJM 0 (0) -0 9 max Z= N j a x N x 0 j j cx j b j j j mn G= M a y y M by c j j 0 For each constrant n the prmal we ntroduce a varable for the dual problem. For each varable n the prmal, we ntroduce a constrant n the dual. Dependng on whether the prmal constrant s an equalty or nequalty constrant, the correspondng dual varable s ether unrestrcted n sgn (URS) or restrcted n some way, respectvely. In addton, dependng on whether a varable n the prmal s unrestrcted n sgn or sgn constraned, we have an equalty or nequalty constrant, respectvely n the dual. There has been an example of an LP problem whch s wrtten n two dfferent modes lke prmal and dual. Prmal Equalty Constrant Inequalty Constrant Free Varable Nonnegatve Varable Prmal Problem max Z =c x c x Dual Free Varable Nonnegatve Varable Equalty Constrant Inequalty Constrant Dual Problem mn G = y b y b y b s. t : a x a x b / y s.: t a y a y a y c a x a x b / y a x a x b / y a y a y a y c y 0, y s URS, y 0 x, x 0 Prmal: Varables, Inequaltes Dual: Varables, Inequaltes In lnear programmng, the strong dualty theorem states that, when solutons to both the prmal and dual are equal to each other, then that s the optmal soluton of the lnear programmng problem []. N M c x = b y j j j Lnear programmng has a wde range of applcatons and extensons and can be appled to a wde varety of real world problems, ncludng network flow problems. One type of

4 0 E. Eroglu/BALKANJM 0 (0) -0 network flow problem s mnmum cost flow problem. MCF problems defne a specal class of lnear programs. The soluton algorthm descrbed n ths paper s based on the prmal smplex algorthm for lnear programmng. To determne optmalty condtons t s necessary to provde both the prmal and dual lnear programmng models for the network flow problem.. Mnmum Cost Flow Problem. Defnton and the Notaton The Mnmum Cost Flow Problem s to mnmze the total cost of flows along all arcs of a network, subject to conservaton of flow at each node, and upper and lower bounds on the flow along each arc. A network s a collecton of ponts, called vertces (nodes), and a collecton of lnes, called edges (arcs), connectng these ponts. Network topology s only one part of the graph. A network can be vsualzed by drawng the nodes as crcles and the arcs as lnes between them. For a drected network, the lnes are arrows pontng n the approprate drectons. For a gven network, whch s defned by a set of nodes, and a set of arcs connectng the nodes. b Node ( x j ; c j ) Arc b j j Node Fgure.Representaton of Node and Arc There are three types of nodes n a mnmum cost flow problem: supply node, demand node, and transshpment node. A supply node s defned as a node where the flow out of the node exceeds the flow nto the node. Smlarly, a demand node s where the flow nto the node exceeds the flow out of the node. A transshpment node s where the flow nto the node equals the flow out of the node. For example, a dstrbuton network would nclude the sources of the goods beng dstrbuted (supply nodes), the customers (demand nodes) and ntermedate storage facltes (transshpment nodes) []. b > 0 ( x j ; c j ) b j = 0 j ( x jk ; c jk ) b k < 0 k Fgure.Types of Nodes We wrte, j to say that there s an arc between nodes and. In a drected network, the arc, j s regarded as extendng from node to node j Typcally, a drected network model nvolves a flow or transportaton of somethng along the arcs, n the specfed drectons. In an undrected network, the arc, j just represents a connecton

5 E. Eroglu/BALKANJM 0 (0) -0 between nodes and j An undrected network model may allow flows n ether drecton along an arc, or may not nvolve explct flows at all.. Bref Defntons about Network Graphs To understand a network flow model, some key terms must be defned. In an undrected graph arcs are unordered pars of nodes, for. In a drected graph arcs are ordered pars of nodes, j, for. We call a drected graph a network. Fgure.An undrected graph Fgure.A drected graph A walk s a lst of nodes,... K such that, for each k,..., K undrected graph and, or k k, k k A for a drected graph.,, k A for an A path s a walk where nodes are dstnct. A graph s connected f there s a path between any two nodes. A cycle s a walk where, K and also, k k for k,..., K and K. A graph s a cyclc f t does not contan any cycles. A tree s a connected a cyclc graph. ' A subnetwork s a graph ( N, A ') such that N A spannng tree s a subgraph wth N ' ' N and A ' A. N that s also a tree [0],[]. Some of the defntons are portrayed n graph n the followng fgures. k a) A Path b) A Chan c) A Crcut d) A cycle e) A tree Fgure. Representatons of Network Defntons.. Formulaton of the Mnmum Cost Flow Problem The LP model of the mnmum cost flow problem s shown below: mn N N cx j j j Subject to: x x b ( ) for all N :, j :, j j A k k A k

6 0 x u for all A j j E. Eroglu/BALKANJM 0 (0) -0 That the total net supply must equal zero can be seen by summng the flow balance equatons over all resultng n: N b 0 The problem s descrbed n matrx notaton as [0], []; mn cx, Ax=b and 0 l x u. where s a node-arc ncdence matrx havng a row for each node and a column for each arc. Note that there s one balance equaton for each node n the network. The flow varables x j have only,, coeffcents n these equatons[].. Illustratve Numercal Example We wll consder a numercal example for determnng the optmal flows of a network as shown below. In ths example, there s a network whch has nodes and 0 arcs. Node and node are supply nodes, node and node are transshpment nodes, and node, node, node and node are demand nodes. The unt costs of arcs and the supples and demands of the nodes are gven below. T cj T j c c c c c c c c c c c T b bb b b b b b b b T (-0) (0) $ $ (0) $ (-0) $ $ $ $ (0) (-0) $ (0) $ $ (-0) Fgure. Illustratve Network Example The objectve s to fnd the optmal flows along the arcs of network n order to mnmze the total cost subject to conservaton of flow at each node. The objectve functon and the

7 E. Eroglu/BALKANJM 0 (0) -0 constrants wll be composed as by usng decson varables (number of flows; j x x x x x x x x x x x The Objectve Functon, Zmn = x x x x x x x x x x Subject to: as; Node : Node : Node : Node : Node : Node : Node : Node : x x = 0 Dual Varables x x = 0 x x x x + x = 0 x x x x + x = 0 x x = -0 x x = -0 x = -0 x = -0 As stated earler, ths MCF problem s generally an nteger program, snce the decson varables x are restrcted to be zero or an nteger (upper bound).. j 0 xj u. s the node-arc ncdence matrx of a drected graph n our example. Table. Representaton of Node-Arc Incdence Matrx Node / Arc Node Node Node Node Node Node Node Node

8 E. Eroglu/BALKANJM 0 (0) -0 The matrx, node-arc ncdence matrx of a connected dgraph, has one row for each node of the network and one column for each arc. Each column of contans exactly two nonzero coeffcents: a " " and a " ". Thus, the columns of are gven by; aj e ej where and are unt vectors n, wth ones n the and postons. Clearly, the matrx does not have full rank, snce the sum of ts rows s the zero vectors [], [], []. Solvng a MCF problem wth the smplex algorthm, one has therefore that all the basc feasble solutons explored by the algorthm are spannng trees of the flow network. As t occurs for any LP, also n mn-cost flow problems one has feasble, nfeasble and degenerate bases. A bass s feasble f BxB xb B b. In ths case, t can be easly verfed solvng the system b, startng from a leaf of the spannng tree, and verfyng that x 0. [], []... Feasble Soluton and Optmalty To fnd the optmal soluton; frst, assume that we have a network wth n nodes, whch s a spannng tree. In order to show that the varables correspondng to the arcs n the tree consttute a bass, t s suffcent to show that the n tree varables are unquely determned []. In the smplex method, ths corresponds to settng the nonbasc varables to specfc values and unquely determnng the basc varables []. Therefore, the orgnal network wth nodes must have had n arcs. Next, we show that f an node connected subnetwork has n arcs and no loops, t s a spannng tree. In a general mnmum-cost flow model, a spannng tree for the network corresponds to a bass for the Smplex method. There must be at least two ends n the spannng tree snce t contans no loops. B b b b b b b b 0 0 b Fgure. Frst Step Soluton of Example In ths step Total Cost= =0 We do not know that, ths feasble soluton s optmal or not. We show how to effcently check the optmalty of a basc feasble soluton, usng complementarty slackness condtons to determne the optmalty of the problem. Therefore we have to control by

9 E. Eroglu/BALKANJM 0 (0) -0 fndng node potentals and then compute reduced costs for all nontree arcs, usng these node potentals. The dual of the mn-cost flow problem s the followng: max b T T A cb Actually, can be vewed as the lnear programmng dual varable (dual soluton) correspondng to the flow conservaton constrant of node []. Wth respect to a gven potental functon, the reduced cost of an s defned as cj j cj [] For the arcs of bass n our example, the node potentals are: Arc - c, Arc - c, Arc - c, Arc - c, 0 Arc - c, Arc - c, Arc - c 0,,,,, 0,,, 0 After fndng the node potentals of the arcs of bass, we have to determne the reduced costs for the nonbasc arcs. A feasble flow s an optmum flow f and only f t admts no negatve cost augmentng cycle. If all of the reduced costs for the nonbasc arcs are computed as postve number, the feasble soluton s optmal. We now compute reduced costs for all nontree arcs, usng node potentals founded before: Reduced costs for the nonbasc arcs usng equaton below. ( c c or c c ) [], [], [], [0], [] j j j j j j Arc - c Arc - c 0 Arc - c 0

10 E. Eroglu/BALKANJM 0 (0) -0 If examned the reduced costs, you wll see that the reduced cost of Arc - s postve therefore we have to choose the Arc - as enterng arc. Now we have to decde to determne the leavng arc. To be able to understand better the topc of leavng arc, we have to consttute a cycle by addng Arc -. In ths cycle, f the flow on Arc - s then the arcs of bass wll be: Arc -, Arc - and Arc -, The mnmum value of the should be 0 for nonnegatvty constrants for the flows. So, the flows wll be lke; Arc -, Arc - and Arc - The flow on Arc - s 0, therefore Arc - s a leavng arc Fgure.Smple Illustraton of Leavng and Enterng arc 0 In our example the enterng arc must be Arc -. Leavng arc s Arc (-0) (-0) (0) (-0) (0) (-0) (0) 0 (-0) (0) 0 0 (-0) 0 (-0) 0 (-0) Fgure 9. Last Step Soluton of Example (Optmal) In ths step, Total Cost= =0 We now have to fnd node potentals agan for Basc Nodes: Arc - c, Arc - c, Arc - c, Arc - c, Arc - c,

11 E. Eroglu/BALKANJM 0 (0) -0 Arc - c, Arc - c 0,,,,,,,, 0 Reduced Costs for Nonbasc Nodes [], [], [], []: Arc - c Arc - c Arc - c If reexamned the reduced costs, you wll see that all of them are zero or negatve. So, the last flow of drected graph s optmal. If one of the reduced costs of an arc s zero, t shows that there s an alternatve optmal soluton for the example.. Case Study for Network Smplex Soluton A company dstrbutes ts product whch has manufactured n three dfferent ctes. These ctes are called supply nodes. In addton to three ctes, there are 9 ctes whch use ths company s product. These ctes are called demand nodes. Also ctes have no demands, they are only transshpment nodes. So the problem has nodes and arcs between these nodes. The unt transportaton costs for these arcs are gven below. The objectve of the company s to determne the optmal flows n order to mnmze the total cost of ts transportaton. For ths mnmum cost flow problem, the data and the drected graph are gven below: T b bb b b b b b b b b b b b 9 b T Arcs Unt Costs ($) Arcs Unt Costs ($) Arcs Unt Costs ($)

12 E. Eroglu/BALKANJM 0 (0) -0 b b b b b b b b b b b b b b 9 b Fgure 0.Network graph of company s product flows To fnd the optmal soluton, network smplex method s appled to a feasble dual soluton. As the tool, Mcrosoft Excel 00 Solver s used for solvng ths model. The results are gven n table below. Arcs Flow Arcs Flow Arcs Flow Total Mnmum Transportaton (Transshpment) Cost s $ for the company. The optmal arc flows are shown n fgure below. [Bold arcs are bass (tree) and dscrete arcs are nonbasc (nontree)] Fgure.Optmal Soluton, Bold Arcs (Basc), Others (Nonbasc)

13 . Concluson E. Eroglu/BALKANJM 0 (0) -0 9 Networks are very mportant subclass of lnear programs that are ntutve, easy to solve and useful for modelng busness problems. Networks provde a useful way to thnk about problems even f there are addtonal constrants or varables that preclude use of networks for modelng the whole problem. In ths paper, an mplementaton of network smplex algorthm s descrbed for solvng the mnmum cost network flow problem. The MCF problem plays a fundamental role n network flow theory and has a wde range of applcatons. For fndng the optmal flows, NSM dual feasble soluton has been used for the network problem showng all calculatons on an llustratve example. Fnally, a bgger network problem s solved usng the same method and the results are shown n the end. The results show that f the sze of network s moderate, NSM s useful for solvng network flows and fndng optmal cost.

14 0 References E. Eroglu/BALKANJM 0 (0) -0 [] Churchman, CW., Ackoff, R.L and Arnoff, E.L., Introducton to Operatons Research Hardcover, January, 99. [] Dantzg, George B., Lnear Programmng and Extensons, Sprnger Inc, 9. [] Bradley, S.P., Hax, A.C. and T.L. Magnant, Appled Mathematcal Programmng, Addson-Wesley, 9. [] Ahuja, R. K., Magnant, T. L. and J. B. Orln, Network Flows: Theory, Algorthms and Applcatons New Jersey, Prentce Hall, Englewood Clffs, NJ, 99. [] Bertsmas, D. and J.N. Tstskls, Introducton to Lnear Optmzaton, Athena Scentfc, 99. [] Kantorovch, L.V. Mathematcal Methods n the Organzaton and Plannng of Producton, translated n Management Scence (90), Volume:, p: -, 99. [] Ahuja, R.K., Magnant, T.L., and J.B. Orln, Network Flows n Handbooks n Operatons Research and Management Scence, Volume: Edted by G.L. Nemhauser (Elsever Scence Publcatons B.V., North Holland), 99. [] Ahuja, R.K. and J. Orln, Improved Prmal Smplex Algorthms for Shortest Path, Assgnment and Mnmum Cost Flow Problems, Massachusetts Insttute of Technology, Operatons Research Center, Workng Paper, pp. 9, 9. [9] Kral, Z. and P. Kovacs, Effcent mplementatons of mnmum cost flow algorthms, Acta Unv. Sapentae, Informatca,, p: -, 0. [0] Vanderbe, Robert J., Lnear Programmng: Foundatons and Extensons, Kluwer Academc Publshers, Second Edton, 00. [] Goldberg, A.V., An Effcent Implementaton of a Scalng Mnmum-Cost Flow Algorthm, Journal of Algorthms, (), p: 9, 99. [] Morrson, Davd R. Sauppe, Jason J. and Sheldon H. Jacobson, A Network Smplex Algorthm for the Equal Flow Problem on a Generalzed Network, INFORMS Journal on Computng, December, 0. ] Goldberg, A. V., Grgorads, M. D. and R. E. Tarjan, Use of dynamc trees n a network smplex algorthm for the maxmum flow problem, Mathematcal Programmng, 0, p: -90, 99. [] Bazaraa, Mokhtar S., Jarvs, John J. and Hanf D. Sheral, Lnear Programmng and Network Flows, Thrd Edton, John Wley & Sons Inc, Hardcover, 00. [] Gerans, G., Paparrzos K. and A. Sfaleras, On a dual network exteror pont smplex type algorthm and ts computatonal behavor, RAIRO-Operatons Research, p: -, 0. [] Hller, Frederck S., Leberman and J. Gerald, Introducton to Operatons Research, McGraw-Hll, New York, 00. [] Amberg, A., Domscheke, W. and S. Braunschweg, (99). Capactated Mnmum Spannng Trees: Algorthms usng ntellgent search, Combnatoral optmzaton: Theory and Practce,, p: 9-9, 99.

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