Minimum Cost Noncrossing Flow Problem on Layered Networks

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1 Minimum Cot Noncroing Flow Problem on Layered Network İ. Kuban Altınel*, Necati Ara, Zeynep Şuvak, Z. Caner Taşkın Department of Indutrial Engineering, Boğaziçi Univerity, 44, Bebek, İtanbul, Turkey Abtract In thi work we focu on an extenion of the minimum cot flow problem in layered network. Feaible arc flow mut atify a pecific compatibility retriction in addition to flow balance and capacity retriction. Namely, at mot one of the croing arc i allowed to have poitive flow on it. Thi variant of the minimum cot flow problem, which we call the minimum cot noncroing flow problem, can frequently be encountered in real life. The determination of optimal temporal quay crane allocation to berthed veel in container terminal, and optimal train chedule through the tation on the ame railroad line are two example. We firt analyze the complexity of the problem and how that the noncroing flow problem i in fact NP-complete in a layered network. Then, we introduce mixed-integer linear programming formulation and dicu a polynomially olvable pecial cae. Next we how a ufficient condition for the exitence of a croing in an optimal olution, which can be ued for preproceing the arc in order to reduce the problem ize. Our computational experiment on a large tet et how that our preproceing algorithm can ignificantly reduce the number of arc. Keyword: Network flow, layered network, noncroing flow, integer programming.. Introduction The ordinary minimum cot flow problem (MCFP) i well-known and ha widepread application. It i alo faced a a relaxed ubproblem in olving many difficult combinatorial optimization problem []. Due to it pecial tructure, it can be olved efficiently, and many polynomial-time algorithm have been developed ever ince Ford and Fulkeron eminal work [8]. In thi work, we focu on the minimum cot noncroing flow problem (MCNFP), which i an extenion of the ordinary MCFP in layered network with additional compatibility contraint in conjunction with the flow balance, capacity, lower bound, and binary retriction. Layered graph and network provide effective modeling tool for the olution of ome difficult combinatorial optimization problem, a recently detailed and claified in []. They are often encountered in container terminal, epecially when the temporal allocation of the quay crane to load/unload the berthed veel according to their technical propertie [4, ], and when the cheduling of train through the tation on the ame railroad line i targeted. In general, a layered network provide a graphical tool to model the cheduling of flow with patial contraint. Compatibility contraint we conider belong to a pecial cla named a conflict, dijunctive, or excluionary ide contraint. They make MCNFP a relative of graph and network baed combinatorial optimization problem, which alo include conflict contraint. For example, tranportation problem with conflict contraint i tudied in [9, 0], and [], aignment problem with conflict Correponding author. Phone: + 90 () , Fax: + 90 () addree: altinel@boun.edu.tr (İ. Kuban Altınel*), aran@boun.edu.tr (Necati Ara), zeynep.arlan@boun.edu.tr (Zeynep Şuvak), caner.takin@boun.edu.tr (Z. Caner Taşkın) Preprint ubmitted to xxxx September 0, 08

2 contraint i tudied in [5], minimum panning tree problem with conflict contraint i tudied in [6, ] and [7], the hortet path problem with conflict contraint i tudied in [7], and latly maximum flow problem with conflict contraint i tudied in [6]. To the bet of our knowledge the firt four of the above mentioned tudie are the mot cloely related one to our. The problem they conider can be een imilar to our, ince their particular network tructure ha two layer. In the firt two, Sun [9] tudie the tranportation problem with conflict contraint (TPC), and propoe a Tabu Search heuritic. He alo develop a branch-and-bound (BB) algorithm in [0]. The third one i more theoretical and provide a complexity analyi of TPC; it i hown that even ome pecific cae are NP-hard []. In the lat one, the author aume unit flow capacitie and zero lower bound in addition to the two-layer network tructure, and propoe efficient heuritic for the aignment problem with conflict [5]. Thi paper conit of eight ection. We explain in the next ection a motivating real-life application of MCNFP, i.e. the quay crane cheduling problem. We introduce the notation and terminology in Section, and analyze the computational difficulty of MCNFP in Section 4. We propoe mixed-integer linear programming (MILP) formulation in Section 5, which i followed by Section 6, where we dicu a polynomially olvable realitic cae. Section 7 include reult that lead to a preproceing procedure ued in reducing the ize of the problem. The computational reult for the effect of preproceing on the performance of the formulation are reported in Section 8. Finally, concluding remark are provided in Section 9.. Flow cheduling with patial contraint Three important problem aociated with the management of eaide operation at container terminal are the berth allocation problem (BAP), crane aignment problem (CAP), and crane cheduling problem (CSP). Excellent urvey of the related work with a claification according to ome pecific attribute are provided in [,, 4, 5]. In general, BAP deal with the determination of optimal berthing time and poition of veel. It i poible to viualize a olution of BAP by mean of a time-berth diagram where the y-axi repreent the quay dicretized in berth ection which veel can occupy, and the x-axi repreent time period. A common aumption i that each berth ection i jut large enough to be occupied by only one quay crane. A ample time-berth diagram i given in Figure.. There are four veel and the rectangle repreent the area they cover on the time-berth diagram. CAP find the optimal number of crane aigned to the veel, and thu can be een a a pecial form of the optimal crane plitting problem [8]. The number within the parenthei in the rectangle are the crane number required per time period that guarantee the determined length of tay for the veel, and form a olution of CAP. For example, veel tay berthed at berth ection 4 8 for nine period, and demand three crane during period and four crane during period 4 9. CSP focue on aigning quay crane to optimal work place in each interval, given the berth location of the veel along the quay and number of the crane that hould erve them (i.e. the information diplayed in Figure. provide) with the objective of minimizing the total etup cot due to crane relocation on the berth over the planning horizon. An interval i the time that elape between two equential event. An event i a pecific veel activity capable to caue a change in the number of erving crane; it i an arrival or departure. More than one event can occur at the ame time. Thi i alo illutrated in Figure.. There are three interval. For example, the firt one tart with the arrival of veel,, and at time 0, and end with the departure of veel and, and arrival of veel 4 at time, which i alo the tarting time of the econd interval. Oberve that the number of crane in ervice during an interval cannot be larger than even, which i the total number of crane available in the terminal.

3 Berth ection 8 4 Interval Interval Interval v () v () v 4 () v (4) v () Two departure and one arrival Departure Departure Time period v 4 () v (4) v (4) v (4) Figure.: Sample berth allocation and quay crane aignment for five veel A layered network repreentation decribing the ample ituation explained above i given in Figure.. Thi i a directed, layered, ingle ource, and ingle ink network. The only vertex of the firt layer i the uper ource, which repreent the terminal reource, with upply equal to the total number of available quay crane. Similarly, the lat layer conit of a ingle vertex. It i a uper ink with demand equal to the total number of available quay crane. The remaining vertice belong to internal layer and repreent the veel demanding crane at each time interval. Except the firt and lat layer, each one of L layer repreent a naphot of the berth during a time interval. In other word, layer l exit in the network if a veel arrive or depart changing the current naphot; it i then followed by a new one. Notice that the number of crane aigned to the berthed veel can change only between conecutive layer, ince only an arrival or a departure can caue uch a change. Thee event and the interval they repreent are alo depicted in Figure.. At each layer l, a berthed veel i repreented by a vertex, whoe demand i equal to the number of aigned crane. Thee vertice are called veel vertice, and they are ordered according to the poition of the veel berthed along the quay from the beginning to the end. There i a econd type of vertice below and above the veel vertice. They are called parking vertice and repreent the waiting area for the idle crane. In any layer, the number of parking vertice i one more than the number of veel vertice. To ummarize, by letting n l denote the number of berthed veel at the quay during interval l, there are n l veel vertice and n l + parking vertice. Hence, the total number of vertice in layer l i n l + and a vertex with an even index (i.e., 4,..., n l ) correpond to a veel vertex with a crane demand, while thoe with an odd index (i.e.,,..., n l + ) repreent parking vertice with finite capacitie. The network of Figure. i incomplete for the MCNFP formulation. A can be eaily oberved, the total demand i not equal to the total upply. Beide, the odd vertice are capacitated and they have to be appropriately preented. For thi purpoe, except the uper ource and uper ink t, we replace each vertex by two vertice connected with an arc, which i a known tranformation ued for capacitated vertice. A a reult, veel vertice, which are originally tranhipment ink, are replaced with two pure tranhipment vertice. Similarly, parking vertice, which are originally capacitated, are replaced with two uncapacitated pure tranhipment vertice. The detail of the tranformation can be found in the work by []. The flow on the arc of thi network correpond to crane relocation or movement from parking area to veel, from veel to veel (thi include the cae where a crane continue erving the ame veel or tart erving a new veel without an idle period), from veel to

4 t -4 - Figure.: A layered network decribing the ituation given in Figure. parking area in each time interval. The cot aociated with thee relocation are the unit flow cot. CSP become an ordinary minimum cot flow problem (MCFP) on the decribed layered network if crane croing i allowed. Unfortunately, thi i not poible in reality; quay crane are retricted to move on a rail and thu the relocation path cannot cro. In other word, it i not enough to olve the MCFP on the decribed layered network to determine an optimal crane chedule for the example of Figure., ince, for example, an optimal olution can include flow on arc (, 4) and (4, ) between the econd and third layer depicted in Figure.. A a conequence, it i poible to ay that CSP i in fact equivalent to a MCFP with additional patial contraint allowing only noncroing arc to have poitive flow value in an optimal olution, which make it a particular ubcla of MCNFP. MCNFP i a generalization of the CSP where we conider a flow problem with upplier and cutomer located on a line. The commodity flow i realized by mean of vehicle, which are retricted to move along a ingle track lane, and hence cannot pa each other a a patial retriction. Beide, upplier and cutomer have time varying operating characteritic. At a given time, ome of them can leave and/or new one can arrive, and can change their upplie/demand. The purpoe i to determine an optimal commodity flow chedule between them o that, the total ditribution cot i minimized. We will conider thi generalization in the ret of thi work.. Notation and terminology Let N = (V(N), A(N)) be a layered network coniting of L layer defined by the et V(N) of vertice and A(N) of arc. We define V l (N) a the et of vertice of layer l and n l it cardinality (i.e. n l = V l (N), l =,,..., L), 4

5 and aume that V (N) = {}, V L (N) = {t}, V(N) = L l= V l(n), n = n L = and n = V(N) = L l= n l. ha only outarc and t ha only inarc. Any arc (i, j) A(N) of the network i forward (i.e. tail i cloer to in the number of arc). There are neither backward arc, nor arc connecting two vertice at the ame layer. If we let A l(l+) (N) be the et of arc (i, j) having i V l (N) and j V l+ (N), then A(N) = L l= A l(l+)(n). We alo aume that A l(l+) (N) conit of all poible arc with tail in V l (N) and head in V l+ (N); i.e. A l(l+) (N) = {(i, j) : i V l (N), j V l+ (N)}. We conider a particular embedding of the network for vertex labeling: the vertice are located on the interection point of a grid where the vertical line repreent the layer and numbered from to n l at layer l tarting from the bottom to the top. A a conequence, if two arc (i, j ), (i, j ) A l(l+) (N) cro, then i > i and j < j and vice vera; they form a croing. Oberve that arc (, j) A (N) a well a arc (i, t) A (L )L (N) are noncroing. Any pair of path with at leat two ditinct arc that cro each other are aid to be croing. Notice that croing path are not necearily arc or vertex dijoint. Any dijoint pair of path may cro, and any two noncroing path may hare arc or vertice. The decribed layered network tructure i illutrated in Figure.. n l n l+ i j t Layer Layer Layer l Layer l+ Layer L- Layer L Figure.: A layered network with L layer Each arc (i, j) A(N) ha an aociated unit flow cot c i j. We alo aociate with each arc (i, j) A(N) a capacity u i j that denote the maximum amount of flow allowed on arc (i, j) and a lower bound l i j that denote the minimum amount that mut flow on arc (i, j). Each vertex i V(N) ha a number b i repreenting it upply/demand. If b i > 0, vertex i i a tranhipment upply vertex, if b i < 0 vertex i i a tranhipment demand vertex with a demand of b i, and if b i = 0 vertex i i a pure tranhipment vertex. In other word, vertex et V(N) can be expreed a V(N) = {, t} V + (N) V (N) V ± (N), where V + (N), V (N) and V ± (N) are repectively the ubet of tranhipment upply, tranhipment demand, and pure tranhipment vertice at layer l. Similarly, for every level l, V l (N) = Vl (N) V+ l (N) V± l (N), where V+ l (N), V l (N) and V± l (N) denoting the ubet of tranhipment upply, tranhipment demand and pure tranhipment vertice of layer l. Clearly, V (N) = V± (N) =, V+ (N) = V (N) = {}, and V L + (N) = V± L (N) =, V L (N) = V L(N) = {t}. We aume that L l= i V l (N) b i = 0, and l i j = 0 u i j for all (i, j) A(N) and they atify ufficient condition for the exitence of a feaible flow []. The function f : A(N) R i the flow function and aociate the variable f i j with arc (i, j). In the ordinary MCFP the goal i to determine a feaible flow with the minimum total cot. Recall that a flow f i feaible if it atifie flow balance equalitie at the vertice, lower and upper bound on the arc. In the claical theory, network 5

6 6-6 5 t (a) Arc flow 4 t t 4 t (b) Flow path of an equivalent path and circuit flow (c) Flow circuit of an equivalent path and circuit flow Figure.: Two way to expre a flow in a network flow problem can be equivalently formulated by either defining flow on arc (i.e. arc flow) or directed path and circuit (i.e. path and circuit flow). Thi i a conequence of the flow decompoition theorem [8], which eventually enable the (unique) repreentation of a path and circuit flow a nonnegative arc flow, and (not necearily unique) repreentation of a nonnegative arc flow a a path and circuit flow. An example i provided in Figure.. Notice that there i alway a flow path connecting a ource vertex to a ink vertex. In relation to the flow problem we conider in thi work, namely MCNFP, there i no circuit involved in thi decompoition becaue of the (directed) layered tructure of the network, and thu an arc flow can be repreented a a path flow, and vice-vera. Figure. provide an example for thi particular ituation. Then it i poible to ay that an (arc) flow on N = (V(N), A(N)) i noncroing if and only if all path of the equivalent path flow are noncroing, ince an arc (i, j) with poitive flow (i.e. f i j > 0) appear on at leat one path with poitive flow on it, and if it i croed by another arc, then there exit another path having poitive flow on it with an arc croing arc (i, j). For example, in Figure., the path t and t are croing ince they both have poitive flow (i.e. and unit of flow repectively) and arc (, ) and (, ) are croing. In hort, we can refer to MCNFP a the MCFP with noncroing flow path, and a flow path i a directed path with poitive flow on it. For the MCNFP, the directed path from a ource vertex to a demand vertex with poitive flow on it arc, namely flow path, mut be noncroing in addition to flow balance, lower and upper bound retriction in order to be feaible. We ay uch flow i feaible and alo noncroing. In other word, an optimal olution of MCNFP i a noncroing flow with the minimum total cot, which i feaible with repect to the mentioned balance, lower and upper bound retriction. Clearly, MCFP i the 6

7 relaxation of MCNFP obtained by relaxing patial compatibility retriction that do not allow arc croing on the flow path. t t t t t t t (a) Arc flow (b) An equivalent path flow Figure.: Two way to expre a flow in a layered network 4. The difficulty of the minimum cot noncroing flow problem We firt define the deciion problem aociated with the MCNFP and it variant with retricted total flow cot for the flow path (MCNFP-RC) in the following. MCNFP Intance: A layered network N = (V(N), A(N)) with L Z + layer each of which having n l Z + vertice l =,,..., L except the firt and lat one: they conit of ingle vertice, namely a ource for layer l = and a ink t for layer l = L. There i a upply/demand b i Z + for every vertex i V(N) atifying i V + (N) b i = i V (N) b i. For each arc (i, j) A(N) there i a capacity u i j Z +, and unit flow cot c i j Z +. There i alo a given number C Z +. Quetion: I there a noncroing arc flow with total cot le than C? MCNFP-RC Intance: The ame a the MCNFP intance. Quetion: I there a noncroing arc flow with total cot le than C for every flow path? Propoition. MCNFP-RC i NP-complete for with B Z +. B, if l = (i.e. i = ) b il = B, if l = L (i.e. i L = t) 0, otherwie, Proof. i. MCNFP-RC NP: If a flow i given, it feaibility can be checked in polynomial time. Checking the feaibility of the flow with repect to the flow balance contraint and bound can be done in O( A(N) ) time. 7

8 Checking whether there i a croing require at mot O( V(N) L) time. A a conequence of the flow decompoition theorem [8] given a nonnegative arc flow it i poible to generate all flow path in O( V(N) + A(N) ) time and the number of flow path i O( V(N) + A(N) ) in the wort cae. Thu, checking whether or not the total cot of each path i le than C take O(( V(N) + A(N) ) A(N) ) = O( A(N) ) time. Therefore, there i a polynomial time certificate checking algorithm and MCNFP-RC NP. ii. MCNFP-RC i hard (Reduction from the et partitioning problem): The Set Partitioning Problem (SPP), which i known to be NP-complete [9], reduce polynomially to MCNFP-RC. The SPP deal with the following quetion: Given a et S of V element with value v Z +, v =,,..., V and v S v = D, i there a ubet S S uch that v S v = v S \S v = D? An intance of MCNFP-RC correponding to an arbitrary SPP intance can be the complete layered network N = (V(N), A(N)) with a. L = V + layer, b. n l =, l =,,..., L ; n = n L = vertice at each layer having a upply/demand, if l = (i.e. i = ) b il =, if l = L (i.e. i L = t) 0, otherwie c. C = D + (V + ), d. unit flow cot D, if l = ; i l+ =,, if l = L ; i L =,, if l =,,..., L ; i l =, i l+ = c il i l+ =, if l =,,..., L ; i l =, i l+ = l +, if l =,,..., L ; i l = i l+ = l +, if l =,,..., L ; i l = i l+ = e. lower bound l il i l+ = 0 and capacitie u il i l+ =, if l =,,..., L ; i l =, i l+ =, if l =,,..., L ; i l =, i l+ = u il i l+ Z +, otherwie, Figure 4. illutrate the network obtained after thi tranformation for S = {,, }, V =, S = {, }, L = + = 6. The number on the arc are the unit cot. For vertex numbering we ue the previouly mentioned convention. Two noncroing flow path atifying the total flow cot retriction C = D + 4 are preented uing dahed arc. Oberve that the path have unit flow on them, atify balance equalitie, lower bound and capacity retriction, and cot retriction. Furthermore, they are noncroing. The firt of the two path preent ubet S (path ) and the econd one the ubet S \S (path ). They 8

9 + D/ D/ t l = l = l = l = 4 l = 5 l = 6 Figure 4.: Noncroing path correponding to an SPP intance are not necearily dijoint; arc (, ) between layer 5 and 6 i travered by both path. The noncroing flow path repreentation of et S and S i not unique. They can be repreented uing two other path a well. For example path for S and path for S \S. Notice that, thi time arc (, ) between layer 5 and 6 i on both path. What mut be done now i to how that S ha a ubet S uch that v S v = v S \S v = D if and only if there i a feaible flow with noncroing flow path each having at mot C = D + (V + ) total flow cot. Firt, uppoe that S ha a ubet S uch that v S v = v S \S v = D. Then, it i poible to generalize the path tructure of Figure 4. o that the firt path include the element of S and the econd path the element of S \S. A can be noticed, thee flow path are noncroing, atify flow balance equalitie, lower bound and D capacity retriction, and each ha a total flow cot D + (V + ) (i.e. + v S v + (V + ) = D + (V + ) = D + v S \S v + (V + )). Thu, if the et S ha a ubet S uch that v S v = v S/S v = D, it i poible to contruct two noncroing flow path each with a total flow cot equal to D + (V + ). Converely, uppoe that we are given a flow feaible with repect to flow balance equalitie, lower bound, capacity retriction and having only noncroing flow path each with a cot le than D + (V + ). Firt of all for the decribed MCNFP-RC intance there can be at mot two, t-flow path ince exactly two unit of flow ha to be ent out of ource. Single, t-flow path (with two unit of flow on it) i not poible becaue the total cot of the one with the mallet total cot i D + (V + ), which i larger than the retriction D + (V + ). Hence, two ditinct flow path have to tart at ource. Beide, they mut atify the cot retriction (i.e. each ha a total cot of at mot D + (V + )). Conider the arc (i l, i l+ ) uch that i l = i l+ =, for l =,,..., L. Thi i the pair of arc with cot v +, v S. Then, at leat one of thee two arc i l = i l+ =, mut appear on one of thee two path for each l =,,..., L. Otherwie, there i a croing becaue of the network tructure and unit upper bound on arc (i l, i l+ ) with i l = and i l+ =, and i l = and i l+ = for l =,,..., L (i.e. in cae there i one which i miing on both path) or one of the path ha cot larger than D + (V + ) (i.e. one of them can be travered by both path), which i a contradiction. In hort, there are two flow path each having unit flow on it and partitioning the arc with cot v + v S (i.e. thee arc appear exactly on one of them) and thu the um of the total cot i equal to D + v S v + (V + ) = (D + (V + )). Thi implie that each flow path atifie fully it total cot retriction D + (V + ), ince each ha a total flow cot le than D + (V + ) with grand total exactly equal to (D + (V + )). Finally, one of the flow path cannot include all of them (i.e. the et S entirely) becaue thi reult in a total cot of D + v S v +(V +) = D+(V +) > D+(V +). Let S be the et of thee arc and S be it ubet included in the firt path. Then, other path would travere the arc in S /S. Recall that each one of thee path ha 9

10 total cot D + (V + ). Therefore, D + v S v + (V + ) = D + (V + ) and D + v S \S v + (V + ) = D + (V + ), which implie that v S v = v S \S v = D D = D. Thi tranformation can be done in O(V) time. Propoition. MCNFP i NP-complete when with B Z +. B, if l = (i.e. i = ) b il = B, if l = L (i.e. i L = t) 0, otherwie, Proof. i. MCNFP NP: Firt of all any certificate of MCNFP can be checked in polynomial time imilar to MCNFP-RC. Hence, MCNFP NP. ii. MCNFP i hard (Reduction from MCNFP-RC): Conider any arbitrary intance of MCNFP-RC with unit cot c i j Z +, (i, j) A(N), upply/demand b i Z +, i V(N) and capacitie u i j Z +, (i, j) A(N) a aumed in the aertion, and cot retriction D Z + for the flow path. To generate a particular MCNFP intance we keep the ame layered network tructure of a MCNFP-RC intance, but modify unit cot, arc capacitie, upplie and demand. We chooe B, if l = (i.e. i = ) b i l = B, if l = L (i.e. i L = t) 0, otherwie, with B = B + (L ) = B +. We et the unit cot all equal to λ, arc capacitie to u i j = u i j +, and the retriction C = λ (L )B. Here λ Z + i larger enough than BD. Firt uppoe that MCNFP-RC ha a noncroing flow f Q A(N) + atifying flow balance, arc capacity contraint and cot retriction on the flow path P k = (V(P k ), A(P k )) k =,,..., K. Here, K i the number of noncroing flow path and K V(N) + A(N) a a coneqeunce of the flow decompoition theorem [8]. However, for thi particular cae, due to the integrality of the flow and network tructure K B. Since each flow path atifie the cot retriction, (i, j) A(P k ) c i j f k D k =,,..., K, and conequently K c i j f k KD. k= (i, j) A(P k ) Here, f k i the amount of poitive flow ent through the kth flow path. Then, c i j f i j KD () (i, j) A(N) 0

11 follow, ince f i j = K k= {P k :(i, j) A(P k )} f k, where f i j i the flow on arc (i, j). At thi point we have to how the following claim. Claim. K f k = (L )B k= (i, j) A(P k ) Proof. Firt of all, K k= (i, j) A(P k ) f k = K k= A(P k ) f k. becaue of the pecial tructure of the network N = (V(N), A(N)) (i.e. layered network with forward arc) every feaible arc flow can be repreented a a path flow having exactly L arc. Beide, each flow path with poitive flow on it connect a ource vertex to a ink vertex [8]. Hence, K k= (i, j) A(P k ) f k = K K A(P k ) f k = (L ) f k. k= In addition, the um of the flow over the flow path i equal to the um of upplie, which i equal to the negative of the um of the demand, namely to B. In other word K k= f k = i V + (N) b i = i V (N) b i = B, which complete the proof. Then, for unit cot c i j = c i j + λ i j with λ i j Z +, λ = max (i, j) A(N) {λ i j } = BD and λ = min (i, j) A(N) {λ i j } the total flow cot become (i, j) A(N) c i j f i j = = (i, j) A(N) K c i j f i j + k= (i, j) A(P k ) KD + λ K c i j f k + (i, j) A(N) k= (i, j) A(P k ) = KD + λ(l )B BD + λ(l )B = λ[(l )B + ] ( = λ(l ) B + ) L λ(l )(B + ) = λ(l )B λ (L )B. K k= λ i j f i j k= (i, j) A(P k ) f k λ i j f k The econd term of the fourth expreion follow from the econd term of the third expreion a conequence of claim. The fifth expreion follow from the fourth ince B K. We alo ue the definition λ = BD and the fact that λ λ. Notice that, K C = λ (L )B = λ f k. k= (i, j) A(P k ) In other word thi upper bound C i the total cot of an arc flow f, with path flow f k k =,,..., K on the flow path P k k =,,..., K, for the ame network tructure with B and u i j a defined previouly, and unit flow cot et to λ. Notice that it i poible to obtain f by increaing the flow f on one of the flow path P k k =,,..., K,

12 ay f p on path P p by one unit and keeping the remaining one the ame, i.e. by etting f p = f p +, f k = f k for k p. Thi i a feaible olution of the particular MCNFP intance we have created, i.e. a noncroing flow feaible with repect to the flow balance equalitie and capacity retriction, with total cot equal to C. Converely, uppoe that the particular MCNFP intance ha a noncroing flow with total cot not larger than C. Let P k = (V(P k ), A(P k )), k =,,..., K be the correponding K flow path of a feaible flow f of the particular MCNFP intance, which alo atifie total cot retriction. Let alo f k k =,,..., K be the path flow correponding to thee K flow path. Hence, K C = λ (L )B = λ = = = = K k= f k k= (i, j) A(P k ) c i j f k (i, j) A(P k ) c i j f i j (i, j) A(N) (i, j) A(N) K k= K k= K k= K k= c i j f i j + (i, j) A(P k ) c i j f k + (i, j) A(P k ) c i j f k + (i, j) A(N) (i, j) A(P k ) c i j f k + λ K k= K k= K λ i j f i j (i, j) A(P k ) λ i j f k (i, j) A(P k ) λ i j f k f k k= (i, j) A(P k ) (i, j) A(P k ) c i j f k + λ(l )B. The firt inequality i a conequence of our election of λ. For example etting λ = c with c = max (i, j) A(N) {c i j } i a poibility. The lat equality i a conequence of claim, ince it can be hown that K k= (i, j) A(P k ) f k = (L )B imilarly. Hence, we can write K k= (i, j) A(P k ) c i j f k C λ(l )B = λ (L )B λ(l )B = (L )B (λ λ), which become after etting K k= (i, j) A(P k ) c i j f k D λ = λ D (L )B For example for λ = c and λ = BD it i poible to et λ = c D (L )B,

13 provided that c D BD + (L )B in order to have λ λ, which make C λ(l )B D. Alo for c = αc + λ with c = max (i, j) A(N) {c i j }, α = (L )B, and λ = BD we have c D. In hort, (i, j) A(P k ) c i j f k D k =,,..., K follow, ince c i j 0 and f k > 0 (i, j) A(P k ), implying (i, j) A(P k ) c i j f k 0, k =,,..., K. Therefore, it i poible to obtain a feaible olution f of MCNFP-RC uing the flow path of the noncroing arc flow f by imply decreaing the flow on one of the flow path P k k =,,..., K, ay f p on path P p by one unit and keeping the remaining one the ame, i.e. by etting f p = f p, f k = f k for k p. Finally, thi tranformation can be done in L l= (n ln l+ ) + L l= n l = O( V(N) + V(N) ), which i polynomial and the proof i complete. The next two propoition follow directly form Propoition and Propoition. Propoition. MCNFP-RC i NP-complete for general demand upply/demand, i.e. b i Z + for every vertex i V(N) atifying i V + (N) b i = i V (N) b i. Proof. In Propoition we have hown that a retriction of MCNFP-RC i NP-complete. It i obtained by etting with B Z +. B, if l = (i.e. i = ) b il = B, if l = L (i.e. i L = t) 0, otherwie, Propoition 4. MCNFP i NP-complete for general demand upply/demand, i.e. b i Z + for every vertex i V(N) atifying i V + (N) b i = i V (N) b i. Proof. In Propoition we have hown that a retriction of MCNFP i NP-complete. It i obtained by etting with B Z +. B, if l = (i.e. i = ) b i l = B, if l = L (i.e. i L = t) 0, otherwie, 5. Formulation It i poible to formulate MCNFP a a mixed-integer linear programming problem (MILP) by allowing only noncroing arc to have poitive flow. In other word, for each arc (i, j) A l(l+) (N), if there i a poitive flow

14 on (i, j), i.e. if f i j > 0, then f pq = 0 for all (p, q) A l(l+) (N) with either p i and j + q n l+, or i + p n l and q j. Obviouly, f i j = 0 if f pq > 0 for one of uch (p, q) A l(l+) (N). In addition to the flow variable f i j we introduce binary deign variable x i j A l(l+) (N) to model thi. x i j i et to if f i j > 0. Beide, if x i j = then f pq = 0 for all (p, q) uch that either p i and j + q n l+, or i + p n l and q j. Thi allow u to define a lit S i j of arc incompatible (i.e. croing) with arc (i, j) A l(l+) (N) a S i j = {(p, q) A l(l+) (N) : p i, j + q n l+ ; i + p n l, q j } l =,,..., L. () Then, we obtain the following MILP fomulation for MCNFP. MCNFP: min.t. L c j f j + c i j f i j + c it f it () (, j) A (N) (, j) A (N) (i, j) A l(l+) (N) l= (i, j) A l(l+) (N) (i,t) A L L (N) f, j = b (4) f i j f ji = b i i V l (N); l =,,..., L (5) ( j,i) A (l )l (N) (i,t) A (L )L (N) f it = b t (6) 0 f i j u i j x i j (i, j) A l(l+) (N); l =,,..., L (7) x pq + x i j (p, q) S i j ; (i, j) A l(l+) (N); l =,,..., L (8) x i j {0, } (i, j) A l(l+) (N); l =,,..., L. (9) Without contraint (8) and (9), and with u i j intead of u i j x i j in contraint (7) the formulation i the one of ordinary minimum cot flow problem on the layered network illutrated in Figure.. We call contraint (8) and (9) compatibility contraint; flow can only be ent through only noncroing arc. Another equivalent formulation of MCNFP i obtained by replacing inequalitie (8) with x pq + S i j x i j S i j (i, j) A l(l+) (N); l =,,..., L. (0) (p,q) S i j A can be noticed, thi formulation give a weaker LP bound ince inequalitie (0) are obtained by aggregating inequalitie (8) over the lit S i j for each arc (i, j). 6. A polynomially olvable pecial cae Let u aume that the network N = (V(N), A(N)) i not only layered but alo complete (i.e. all arc between the vertice of layer l and l + exit) and the unit cot are nonnegative, ymmetric and additive for i j, and 4

15 c i j = 0 for i = j. Namely, c ii = 0, () c i j 0, () c i j = c ji, () j c i j = c k(k+). (4) k=i Notice that (4) i valid if i < j. Otherwie we can interchange the limit of the ummation and apply () a a conequence of ymmetry. Recall that for a pair of croing arc (i, j ) and (i, j ), i < i and j < j. Beide, there are ix poible ordering of thee four vertice according to the convention we ue for numbering the vertice (i.e. vertex label denote their order from bottom in their layer): i. i < i j < j iii. i j < j i v. j i < i j ii. j < j i < i iv. j i < j i vi. i j < i j. Thee ix cae are illutrated in Figure 6. with ix naphot from two conecutive layer of a layered network. Horizontal line repreent the inequalitie of the ordering. Strict inequalitie are reflected with additional node below or underneath of the tail/head of the croing arc. Solid arc repreent the croing, wherea dahed one repreent their compatible equivalent. Oberve that, if the flow conervation i atified and there i one unit of flow on each one of the croing (olid) arc before the correction, there mut be one unit of flow on the new (dahed) arc and zero unit of flow on the croing arc in order to correct the croing and guarantee flow balance equation at the ame time. The next lemma how uch change doe not increae total flow cot. Propoition 5. The unit correction cot i nonincreaing under aumption () (4) of the unit flow cot. Proof. We will evaluate the cot of one unit of flow on arc (i, j ) and (i, j ) (i.e. f i j = f i j = and f i j = f i j = 0) with the cot of one unit of flow on arc (i, j ) and (i, j ) (i.e. f i j = f i j = 0 and f i j = f i j = ), namely c i j + c i j with c i j + c i j for the ix poible croing. i. i < i j < j : c i j + c i j = c i i + c i j + c j j + c i j = c i j + c i j ii. j < j i < i : c i j + c i j = c i i + c i j + c j j + c i j = c i j + c i j iii. i j < j i : c i j + c i j = c i j + c j j + c i j + c j j = c i j + c i j + c j j iv. j i < j i : c i j + c i j = c i j + c j j + c i j + c j j = c i j + c i j + c j j v. j i < i j : c i j + c i j = c i i + c i j + c i i + c i j = c i j + c i j + c i i vi. i j < i j : c i j + c i j = c i j + c j i + c i j + c i j = c i j + c i j + c i j. 5

16 i. i < i j < j ii. j < j i < i j i i j i j i j Layer l Layer l+ Layer l Layer l+ iii. i j < j i iv. j i < j i i i j j j i i Layer l Layer l+ j Layer l Layer l+ v. j i < i j vi. i j < i j j j i i i j j Layer l Layer l+ i Layer l Layer l+ Figure 6.: Six poible croing Then, a a conequence of Propoition 5, it i poible to how that correcting the croing in an optimal alternate olution of MCFP reult in an optimal noncroing flow. Propoition 6. If the unit cot atify aumption () (4), then the MCFP ha an optimal olution with no croing arc with poitive flow. 6

17 Proof. Conider an optimal olution f of the MCFP and croing arc (i, j ) and (i, j ), which mean fi, j > 0 and fi, j > 0, and either i < i and j < j or i < i and j < j. Without lo of generality we can aume that i < i and j < j. It i poible to correct the croing by a imple operation and adjut the flow on the correponding arc without harming it feaibility. If f i j f i j > 0, then add new arc (i, j ) and (i, j ) with flow fi j, adjut the flow on arc (i, j ) by ubtracting fi j, and finally delete arc (i, j ). However, if fi j > fi j, then operate imilarly by adding new arc (i j ) and (i j ) with flow fi, j, adjut the flow on arc (i, j ) by ubtracting f i j, and finally delete arc (i j ). Thee operation are illutrated in Figure 6.. The croing repreented by olid arc i corrected by replacing them with dahed arc. Oberve that flow balance i preerved at vertice i, i, j, j. Conequently, only cae (i) and (ii) or cae (iv)-(vi) repectively with c j j = 0, c i j = 0, c i i = 0 and c i j = 0 can occur in an optimal olution of the MCFP, ince otherwie it i poible to create a new feaible flow with one fewer croing and maller total flow cot after implementing the above operation, which contradict the optimality of flow f. Therefore, the elimination of the croing in an optimal olution of the MCFP reult in an alternative optimal olution with no croing. i f i j j i f i j j f i j (f i j f i j ) f i j f i j (fi j - f i j ) f i j f i j i j f i j i j Layer l Layer l+ f i j f i j (a) Layer l Layer l+ f i j > f i j (b) Figure 6.: Two poible correction Notice that Propoition 6 ha an implicit aumption a well: the two correction operation are implementable, which may not be poible if (i, j ) or (i, j ) are miing in the network, and/or there i not enough reidual capacity on both of them. However, in cae the complete layered network i uncapacitated (i.e. u i j =, (i, j) A(N)) they can be applied to correct the croing. Propoiton 5 and Propoition 6 have alo ome implication when arc have finite capacitie. Thi i tated with the following two corollarie. Corollary. For poitive (i.e. c i j > 0, i j, (i, j) A(N)), ymmetric and additive cot, and c i j = 0 for i = j, and finite upper bound, if an optimal olution of the MCFP ha croing of one of the type (iii) - (vi), then f i j = u i j and f i j = u i j. Proof. Aume that an optimal olution f of the MCFP ha a croing coniting of arc (i, j ) and (i, j ). A a conequence of the poitivity aumption of unit cot and unit cot comparion given in Propoition 5, 7

18 c i j + c i j > c i j + c i j, and a a conequence of correction operation given in Propoition 6, the new flow i till feaible ince thi operation conerve flow balance at vertice i, i, j, j and ha maller total cot, which contradict the optimality of f. Hence, thi operation mut have been blocked, which i poible only if f i j = u i j and f i j = u i j. Corollary. For poitive (i.e. c i j > 0, i j, (i, j) A(N)), ymmetric and additive cot, and c i j = 0, for i = j, and finite upper bound the croing of arc (i, j ) and (i, j ) can be corrected by one of the two operation given in Propoition 6 if fi j < min{u i j, u i j } for fi j fi j or if fi j < min{u i j, u i j } for fi j > fi j. Proof. Directly follow from the definition of the correction operation. 7. Reducing the number of croing An optimal olution of the MCFP relaxation defined by () (6), which i obtained after dropping compatibility contraint (7) and (8), and replacing u i j x i j with u i j in (9), can have croing. The efficiency of any exact olution algorithm can be improved if ome of the potential croing can be detected and deleted in advance. The following propoition and it corollary provide a tool in thi direction. Propoition 7. An arc (p, q) A l(l+) (N) i croed by an arc i. (r, ) A l(l+) (N) with r > p and < q in an optimal olution f of the MCFP if f ji > { j V l (N): j p} {( j,i) A l(l+) (N):i<q} {i V l+ (N):i<q} b i ii. (r, ) A l(l+) (N) with r < p and > q in an optimal olution f of the MCFP if f ji > { j V l (N): j p} {( j,i) A l(l+) (N):i>q} {i V l+ (N):i>q} b i Proof. We only how part (i), ince the proof of part (ii) i imilar. Conider the flow balance equation of the vertice of layer l + with demand b i (i.e. the et Vl+ (N)) and and add them ide by ide for vertice i < q to obtain {i V l+ (N):i<q} b i = {i Vl+ (N):i<q} (i, j) A l+l+ (N) f i j i Vl+ (N):i<q ( j,i) A l(l+) (N) The econd ummation on the right hand ide can be plit into two for arc ( j, i) A l(l+) (N) repectively for j > p and j p which reult in {i V l+ (N):i<q} b i = {i Vl+ (N):i<q} (i, j) A l+l+ (N) f i j {i Vl+ (N):i<q} {( j,i) A l(l+) (N): j p} f ji f ji. {i Vl+ (N):i<q} {( j,i) A l(l+) (N): j>p} Notice that the firt two term on the right hand ide repreent the difference between the total outflow from the demand vertice of layer l + which are below vertex q, and the total inflow to the ame vertice from the vertice f ji. 8

19 of layer l which are below vertex p including p a well. Then, {i V l+ (N):i<q} b i {i Vl+ (N):i<q} {( j,i) A l(l+) (N): j p} { j V l (N):i<q} {( j,i) A l(l+) (N): j p} follow ince f i j 0 for all (i, j) A l+l+(n), and { j V l (N):i<q} {( j,i) A l(l+) (N): j p} Therefore, if the condition of the aertion i true, then 0 > {i V l+ (N):i<q} b i + f ji { j V l (N):i<q} {( j,i) A l(l+) (N): j p} f ji f ji {i Vl+ (N):i<q} {( j,i) A l(l+) (N): j>p} {i Vl+ (N):i<q} {( j,i) A l(l+) (N): j>p} {i Vl+ (N):i<q} {( j,i) A l(l+) (N): j p} f ji f ji. {i Vl+ (N):i<q} {( j,i) A l(l+) (N): j>p} f ji f ji f ji and 0 < {i Vl+ (N):i<q} {( j,i) A l(l+) (N): j>p} f ji follow conequently. Hence, there mut exit an arc (r, ) {( j, i) A l(l+) (N) : j > p, i < q} with f r > 0. Corollary. An arc (p, q) A l(l+) (N) with poitive flow cannot exit in an optimal olution of MCNFP if one of the following condition hold. i. ii. { j V l (N): j p} { j V l (N): j p} {( j,i) A l(l+) (N):i<q,i V l+ (N)} min{u ji, b i } > {( j,i) A l(l+) (N):i>q,i V l+ (N)} min{u ji, b i } > {i V l+ (N):i<q} b i {i V l+ (N):i>q} b i Proof. Directly follow from Propoition 7 a a conequence of the fact that 0 f ji min{u ji, b i } for ( j, i) A l(l+) (N), i Vl+ (N). Firt of all, notice that thi rule i related to the atifaction of the total demand of a ubet of vertice in V l+ (N). Beide, although it provide a ufficient condition for an arc to be croed, all poible croing cannot be prevented in an optimal olution of the MCFP relaxation by the condition decribed in Corollary. Neverthele, it can reduce the number of croing by deleting arc in the network. An arc (p, q) A l(l+) (N) i croed by another arc (i, j) A l(l+) (N) with p < i n l and j < q n l+ if the total demand aociated with vertice j < q cannot be atified by the total inflow to them from vertice i p a tated in cae (i) of Corollary. Cae (ii) deal with the ituation that (p, q) i croed by an arc (i, j) with i < p and j < q n l+. It i alo poible to tate upply verion of Propoition 7 and Corollary. We give their tatement in the following without proof for the ake of completene, ince their proof are very imilar and can be done by rewording the argument for the upplie intead of the demand. Propoition 8. An arc (p, q) A l(l+) (N) i croed by an arc 9

20 i. (r, ) A l(l+) (N) with r > p and < q in an optimal olution f of the MCFP if f ji < {i V l+ (N):i q} {( j,i) A l(l+) (N): j>p} { j V l + (N): j>p} b j ii. (r, ) A l(l+) (N) with r < p and > q in an optimal olution f of the MCFP if f ji < {i V l+ (N):i q} {( j,i) A l(l+) (N):k<p} { j V l + (N): j<p} b j Corollary 4. An arc (p, q) A l(l+) (N) with poitive flow cannot exit in an optimal olution of MCNFP if one of the following condition hold. i. ii. min{u ji, b j } < {i V l+ (N):i q} {( j,i) A l(l+) (N): j>p, j V l + (N)} { j V l + (N): j>p} min{u ji, b j } < {i V l+ (N):i q} {( j,i) A l(l+) (N): j<p, j V l + (N)} { j V l + (N): j<p} b j b j Thi time, thi rule i related to the atifaction of the total upply of a ubet of vertice in V l (N). An arc (p, q) A l(l+) (N) i croed by another arc (i, j) A l(l+) (N) with p < i n l and j < q n l+ if the total outflow from vertice p < i n l to vertice q j n l+ a tated a cae (i) in Corollary 4. Cae (ii) deal with the ituation that (p, q) i croed by an arc (i, j) with i < p n l and q < j n l+. A a reult, the efficiency of an exact olution algorithm may increae becaue the network ize i maller. The preproceing proce that Corollary ugget can be tated formally a Algorithm given below. The proce Corollary 4 ugget i very imilar and the correponding algorithm i not included for the ake of brevity. 8. Computational reult We have realized a et of computational experiment on a large et of randomly generated tet intance in order to ae the trength of the relaxation (i.e. LP relaxation of the formulation and MCFP relaxation) and the value of the preproceing cheme. A NETGEN-like [] intance generator, which exploit the layered tructure of the network, ha been developed for generating tet intance. After etting the number of layer L and the maximum number of vertice in a layer n max, the vertex number of vertice for layer,,..., L are generated uniformly within [, n max ]. Layer and L have a ingle vertex, namely and t. Arc are obtained by connecting the vertice of the adjacent layer, and the croing one are determined according to the vertex numbering convention we ue. Then, a keleton, which guarantee a feaible noncroing flow, i contructed and it arc are aigned large enough unit cot in order to prevent them from participating in an optimal olution. intance are generated with 0,,,..., 0 layer; three intance for each value. n max i et to 5, 6, 7, 8, 9, and 0 arbitrarily, and exactly one intance i generated for each combination. The propertie of the tet intance are reported in Table. The firt column include the intance number. Column -5 lit the baic tructural propertie; thee are the number of layer, maximum number of vertice at each layer, number of 0

21 Algorithm Preproceing for MCNFP Input: A layered network N = (V(N), A(N)), arc capacitie u and unit flow cot c; Output: A preproceed network; :begin : for l =,,..., L do : for all arc (p, q) A l(l+) (N) uch that p n l, q do 4: D = 0, C = 0 5: for j =,,..., q do 6: if b j < 0 then 7: D D + b j 8: for all arc (i, j) A l(l+) (N) do 9: if i p then 0: C C + min{u i j, b j } : end if : end for : end if 4: end for 5: end for 6: if C < D then 7: Delete (p, q) 8: ele 9: D = 0, C = 0 0: for j = q +, q +,..., n l+ do : if b j < 0 then : D D + b j : for all arc (i, j) A l(l+) (N) do 4: if i p then 5: C C + min{u i j, b j } 6: end if 7: end for 8: end if 9: end for 0: if C < D then : Delete (p, q) : end if : end if 4: end for 5:end

22 Table : Propertie of the generated tet intance Intance Number L n max V(N) A(N) Croing Arc Pair Number Max. Num. Arc Arc Denity (%) Max. Num. Pair Croing Arc Pair Denity (%) ,75 4, , ,04 9,94 9, , ,78 6, , ,74 8, , ,08 9, , ,704 8, , ,706 9, , ,867 0, , ,009 7,796, , ,469, , ,540, , ,56 9, , ,4 8, , ,49, , ,78 5,66 6, , ,566 0, , ,004 4,5, , ,95 46,7 0, , ,68, , ,59 4, , ,59 5, , ,97 8, , ,06 4, , ,89, , ,467 7, , ,70 7, , ,085 5,747 0, , , 7,8 0, , ,40 6, , ,60 44,6 5,5 4.56,, ,5 5, , ,79, , ,60 0, ,75.9 vertice and arc. Column 6 include the number of croing arc pair. The value given in column 7 and 9 are the maximum poible number of arc and arc pair in the network repectively. They are equal to V(N) ( V(N) ) and A(N) ( A(N) )/. They are ued to calculate the arc and croing arc pair denitie reported in column 8 and 0, which are obtained by dividing the element of column 4 by the element of column 7 and the element of column 5 by the element of column 9. The computation are carried out on worktation with Intel Xeon CPU E5-687W0.0 GHz proceor and 64.0 GB RAM, and operating within Microoft Window 7 Profeional environment. The program are coded in C ++. The CPU time and objective value are obtained uing CPLEX.6 with default option on. 8.. Formulation and relaxation We tart our experiment with the MILP formulation and their relaxation. Baed on the average lited in the lat row of Table, we can ay that the econd formulation (i.e. ()-(7), (9), (0)) i the mot efficient one in term of relaxation, although the firt formulation (i.e. () - (9)) give 0.6% higher lower bound. Thi i probably becaue it LP relaxation can be olved fater, which mean a fater proce of the node of the branch-and-bound tree. The weaket lower bound belong to the MCFP relaxation. Although their computation require the olution of MCFP, it can be done efficiently uing one of the known algorithm(e.g. [0]). A a reult, a very efficient branch-and-bound algorithm can be developed by taking advantage of thi. 8.. The effect of preproceing In order to judge the effect of the preproceing, we compare the CPU time of both formulation and relaxation with preproceing. We prefer not to report preproceing time ince it take le than 0.00 econd for

23 Table : Formulation and relaxation Intance Firt Formulation Second Formulation MCFP Relaxation No. z Optimum CPU (ec.) CPU Relaxation (ec.) Lower Bound CPU Optimum (ec.) CPU Relaxation (ec.) Lower Bound CPU Relaxation (ec.) Lower Bound, , , ,7.0 7,697, , , ,.0 0, , , , , , , , , , , ,7.0 6, , , , ,60, , , , , , , , , , , , , , , ,088.0, , , ,85.0 6, , , , , , , , ,07, , , ,78.0 5, ,69.5, , , , , , , ,77, ,94.7, , , , , , ,44.0 9, , , , , , , , , , , ,8.0 7, , , ,85.0 5,77, , , , , ,957., , , , , , , , , , , , , , , , , , , , , , , , , , ,690.0, , , , , , , , , , , ,48.0 Average, , , ,66.4