Efficient Algorithms on Multicommodity Flow over Time Problems with Partial Lane Reversals

. The multicommodity ﬂow problem arises when several diﬀerent commodities are transshipped from speciﬁc supply nodes to the corresponding demand nodes through the arcs of an underlying capacity network. The maximum ﬂow over time problem concerns to maximize the sum of commodity ﬂows in a given time horizon. It becomes the earliest arrival ﬂow problem if it maximizes the ﬂow at each time step. The earliest arrival transshipment problem is the one that satisﬁes speciﬁed supplies and demands. These ﬂow over time problems are computationally hard. By reverting the orientation of lanes towards the demand nodes, the outbound lane capacities can be increased. We introduce a partial lane reversal approach in the class of multi-commodity ﬂow problems. Moreover, a polynomial-time algorithm for the maximum static ﬂow problem and pseudopolynomial algorithms for the earliest arrival transshipment and maximum dynamic ﬂow problems are presented. Also, an approximation solution to the latter problem is obtained in polynomial-time.


Introduction
e multicommodity flow problems concern with the routing of various commodities through a network to the specific source-sink pairs. For instance, message routing in telecommunications, railway and vehicle routine in transportation, and production planning and logistics support in general and in an emergency can be modeled as a multicommodity flow problem. For details and implementations, we refer to [1][2][3][4].
e transportation network is considered as a network associated with the transshipment of distinct commodities where the supply points (origins), the demand points (destinations), and the junction of road segments constitute the nodes. e connections between the two nodes signify the arcs. On each arc, the capacity that limits the flow amount (i.e., transported commodities) and the travel time are allocated. e multicommodity flow problem can be classified into static and dynamic flow problems. e former one has been categorized as maximum, maximum concurrent, and minimum cost flow problems. e latter one has been categorized as maximum, maximum concurrent, quickest, and the earliest arrival multicommodity flow problems [5,6].
If we do not distinguish the flow in the multicommodity flow problem, then it becomes a single-commodity flow problem. e dynamic flow (also known as flow over time) problem is introduced by Ford and Fulkerson [7]. In connection to this problem, Gale [8] proposed a more general problem known as the earliest arrival flow problem. e earliest arrival flow problem having multiple sources and sinks with given supplies and demands is the earliest arrival transshipment problem. Generally, a solution to this problem does not exist in the case of multiple sinks.
However, it does always exists for multiple sources and a single-sink [9]. e multicommodity flow problem is more complex than their single-commodity part. Hall et al. [10] have shown that multicommodity flow over time is NP-hard even for series-parallel graphs or have only two commodities. Kappmeier [11] provided solutions of maximum multicommodity flow over time and multisource single-sink multicommodity earliest arrival transshipment problems using a time-expanded network within pseudopolynomialtime complexity.
Lane reversal implies flipping of arc orientations to amplify the flow and reduce the travel time by increasing its capacity. Flow models and strongly polynomial-time algorithms for two-terminal maximum and quickest flow problems were developed by Rebennack et al. [12]. e lane reversals are made at time zero and kept fixed afterward. e budget constraint problem with lane reversals is investigated in [13]. e optimal solutions of the earliest arrival flow in the two-terminal general network for both discrete and continuous-time settings can be found in [14][15][16]. Nath et al. [17] investigated the quickest contra-flowloc problem. By reversing the directions of arcs whenever necessary, a polynomial-time algorithm is presented by Pyakurel and Dhamala [18] for multisource single-sink earliest arrival transshipment. e major concern of partial lane reversals is to make use of the capacities of unused arcs in a network for other purposes. Pyakurel et al. [19] introduced the partial lane reversal strategy in which only essential arc capacities are used to extend the flow value. Dhamala et al. [20] investigated the quickest multicommodity flow problem with partial lane reversals. e saved capacities of unused arcs can be used for the logistic supports and facility location in emergency periods.
In this paper, we introduce the maximum static multicommodity flow (MSMCF), the maximum dynamic multicommodity flow (MDMCF), and the earliest arrival multicommodity transshipment (EAMCT) problems with partial lane reversals. We present efficient algorithms to solve these problems by reducing them into single-commodity flow problems and decomposing the flow along the paths. e paper is organized as follows. In Section 2, we provide some basic notations and models used in the article.
e MSMCF problem with partial lane reversals is introduced and an algorithm to solve the problem is presented in Section 3. e MDMCF problem with partial lane reversals is introduced in Section 4. In this section, we present algorithms to find a solution to this problem. e EAMCT problem and an algorithm to solve this problem is presented in Section 5. e paper is concluded in Section 6.

Preliminaries
e multicommodity flow problem consists of the shipping of distinct commodities from their respective origin nodes to corresponding destination nodes through a given network. Basic notations and mathematical formulations are given below.

Flow Models and Notations.
Let a dynamic network topology be given by N � (V, A, K, u, τ, d i , S + , S − , T) with finite sets of nodes V, arcs A, and commodities K � 1, 2, . . . , k { }, where |V| � n and |A| � m. Each commodity i ∈ K with demand d i is routed through a unique source-sink pair (s i , t i ). e sets S + and S − ⊂ V denote source and sink sets of all commodities, respectively. On each arc e � (v, w), the capacity function u: A ⟶ Z ≥0 restricts the flow of commodities and a nonnegative transit time function τ: A ⟶ Z ≥0 measures the time to transship the flow from the entry point v to the exit point w of arc e � (v, w). e time period T is given in advance which is denoted by T � 0, 1, . . . , T − 1 { } in discrete and T � [0, T) in continuoustime settings. A static network is a network besides the temporal dimension denoted by N � (V, A, K, u, d i , S + , S − ). Many nice properties developed primarily based on static network topology are building blocks for most of the realworld dynamic flow problems.

Static Multicommodity Flow.
A multicommodity flow ψ for the given static network N is the sum of all nonnegative static flows ψ i defined by the functions ψ i : A ⟶ R + for each commodity i satisfying where represents the excess flow of commodity i at node v. e sets } denote the outgoing arcs from node v and the incoming arcs to node v, respectively, such that A out (S − ) � ∅ and A in (S + ) � ∅, except in the lane reversals. e second condition of the constraints in (1) are flow conservation constraints for each commodity at intermediate nodes. e constraints in (2) are bundle constraints that limit the flow of each commodity on the arc. e network flow is called circulation if it satisfies flow conservation at all the nodes, i.e., excess i e maximum static multicommodity flow problem with k-commodities consist of ψ 1 , ψ 2 , . . . , ψ k single-commodity flows maximizing k i�1 |ψ i | satisfying constraints (1) and (2).
A polynomial-time solution of the MSMCF problem has been obtained by linear programming techniques (e.g., ellipsoid method or interior-point method). In this problem, each commodity i � 1, 2, 3, . . . , k has a unique source-sink pair (s i − t i ), and we wish to send the maximum flow from the source nodes s i to the sink nodes t i satisfying given bundle capacity u e on each arc e � (v, w). at is, we wish to maximize the sum of the flows over all the commodities.

Dynamic Multicommodity Flow.
For a given dynamic network N with constant transit time on arcs, a multicommodity flow over times ξ is a sum of flows defined by ξ i : A × T ⟶ R + satisfying constraints (4)- (6): where is the excess flow of commodity i on node v at time θ. Here, the second condition of the constraints in (4) are flow conservation constraints at time horizon T, whereas nonconservation of flow at intermediate time points θ ∈ T � 0, 1, . . . , T − 1 { } is represented by the constraints in (5). Similarly, the constraints in (6) are bundle constraints that limit the flow on each arc at each point of time. e aim is to maximize the amount of flow to satisfy the demand d i of each commodity i from s i to t i which is stated in the first and last conditions of (4). e earliest arrival multicommodity flow problem is to find a dynamic multicommodity flow of maximum value val max (ξ(θ)) for all time units θ ∈ 0, 1, { . . . , T − 1} defined by objective function: Instead of each time θ, if the maximization is considered for a given time θ � T − 1, it is defined as an MDMCF problem. e equality in (5)

Δ-Condensed Time-Expanded Graph.
e network topology N � (V, A, K, u, τ, d i , S + , S − , T) is considered. Fleischer and Skutella [21] introduced the Δ-condensed , where all transit times on arcs are multiples of Δ > 0 such that ⌈T/Δ⌉ is bounded by a polynomial in input size. e nodes and arcs in the Δ-condensed time-expanded network are defined as where S + ′ and S − ′ are the super terminals for each commodity as well as for the Δ-condensed time-expanded network, i.e., In this setting, for every arc corresponding to a discrete-time with a multiple of Δ, capacities are rescaled by Δu e . If arc transit times are not multiples of Δ, then transit times are rounded up to a multiple of Δ by τ e ′ � ⌈τ e /Δ⌉Δ and 0 ≤ τ e ′ − τ e < Δ for all arcs e ∈ A. If we take Δ � 1, then the Δ-condensed time-expanded network reduces to the classical time-expanded network.

Lane Reversal
Technique. For a given network N, the corresponding auxiliary network is denoted by N a � (V, A a , K, u a , τ a , d i , S + , S − , T) with undirected edges in . e capacity of the auxiliary arc is the sum of capacities of arcs e and e r such that u a e � u e + u e r , where u e � 0 if e ∉ A. e transit time of the auxiliary arc is Other network parameters are the same. e transformation of the multicommodity network with lane reversals is represented in Figures 1(a) and 1(b). e first, second, and third commodities are shipped through the paths s 1 − t 1 , s 2 − t 2 , and s 3 − t 3 , respectively. Example 1. Let us consider a multicommodity network, where s 1 , s 2 , and s 3 are the source nodes and t 1 , t 2 , and t 3 are the sink nodes as shown in Figure 1(a). e arcs between nodes v and w denoted by (v, w) and (w, v) represent the two way of road segments. e first and second numbers on the arcs represent capacity and transit time (cost) associated International Journal of Mathematics and Mathematical Sciences with the arcs. By adding two-way capacities of the arcs e and e r having the same transit time, an auxiliary network is formed with capacities u a and transit time τ a as shown in Figure 1(b).

Static Mutlicommodity Flow with Lane Reversals
In this section, we introduce the partial lane reversals on the static multicommodity flow problem that makes best utilization of arc capacities to optimize the solution. e procedure for lane reversals is as follows: If u a e − ψ e > 0, then the arc e r is reversed partially, and capacity of remaining arcs e r is saved.
(2) If ψ e − u e > 0 and u a e − ψ e � 0, then the arc e r is reversed completely.
(3) If ψ e − u e < 0 neither e nor e r is reversed.
e MSMCF problem with partial lane reversals sends the maximum flow from the sources s i to the corresponding sinks t i in the unique pair (s i , t i ) for each commodity i � 1, 2, . . . , k by saving the unused arc capacity.

Polynomial-Time Solution of MSMCF Problem.
General linear programming approach (e.g., ellipsoid method or interior-point method) solves the static multicommodity flow problem in polynomial-time. To solve the MSMCF problem with partial lane reversals (problem 1), we present Algorithm 1.

Theorem 1.
e MSMCF with partial lane reversal problem can be solved using Algorithm 1 optimally.
Proof. First, we show that Algorithm 1 is feasible. We can compute maximum multicommodity flow either by using the ellipsoid or interior-point methods. erefore, Step 2 of the algorithm is feasible. Steps 1, 3, 4, 5, and 6 are feasible being transformation, decomposition, reversing arc, and saving arc capacities as in [19]. Hence, feasibility is shown. Now, we prove the optimality of the algorithm. On the transformed network N a , the MSMCF is computed iteratively using an efficient algorithm. As flow is send in a unique pair (s i , t i ) for each commodity i � 1, 2, . . . , k, the problem can be reduced to the static flow for a single commodity. In the static single-commodity flow, the maximum static flow on the transformed network is equivalent to the maximum static flow with partial lane reversals on the original network as in [19], and capacity of unused arcs is saved. us, the MSMCF in the transformed network is equal to the MSMCF with partial lane reversals for the original network. Proof.
e complexity of the algorithm is dominated by Steps 2 and 3. All other steps can be computed in linear time.
e maximum multicommodity flow problem is a linear programming problem, so the general linear programming technique (ellipsoid method or interior-point methods) solves the static multicommodity problem on the auxiliary network in polynomial-time in Step 2. is solution is equivalent to the maximum static multicommodity flow problem on the given network. e flow can be decomposed in O(mn) time in Step 3. Hence, the MSMCF with partial lane reversals can be computed in polynomial-time.
□ Example 2. If we consider network as in Example 1 without temporal dimension, then it becomes a static multicommodity network. We can send the flow on an auxiliary network as shown in Figure 1(b). e maximum static multicommodity flow before lane reversals can be calculated in Figure 1(a) and after lane reversals in Figure 2(a). e saved capacity of unused arcs is shown in Figure 2 e comparison of MSMCF before and after lane reversals is presented in Table 1  Proof. We know that the cut set C ⊆ A is the collection of disconnected and saturated arcs with the property that it disconnects sources and sinks. Let Val (C) be a value of cut which is the sum of the capacities of its individual arcs, i.e., Val (C) � e∈C u e . We know that the arcs e ∈ C have twoway orientations with asymmetric (or symmetric) capacities. If each arc e ∈ C has the symmetric capacity, the lane reversal reconfiguration of the multicommodity network expands the capacity of cut two times. Let C a ⊆ A a be the cut set in the auxiliary network, then its value becomes e minimum multicommodity cut is Moreover, we have that every optimal maximum static multicommodity ψ is less or equal to the minimum multicommodity cut in the auxiliary network. Consequently, the flow value can be increased up to double with lane reversals.
e reconfiguration of the multicommodity network with lane reversals is as follows: where P a is the collection of all paths.
By eorem 1, every optimal maximum static multicommodity flow in auxiliary network N a is a feasible maximum static multicommodity flow with lane reversals in original network N. is completes the proof.

Approximation Solution of MSMCF Problem.
Linear programming techniques provide polynomial-time solutions for multicommodity flow problems. However, in many applications, these problems are large in input size and can take a long time to solve using these techniques. Due to this, it is beneficial to develop approximation algorithms that provide solutions close to the optimal solution. As a consequence, an intense attempt was made to obtain an efficient approximate algorithm for the multicommodity flow problem.
Let a minimization (or a maximization) problem be X and let the optimal solution of the objective function be denoted by Opt(I) for an instance I ∈ X. Suppose ε > 0. An algorithm A is called a (1 + ε)(or (1 − ε)) approximation For a problem X, a polynomial-time approximation scheme (PTAS) is an approximation scheme with the running time polynomial in the input size of the problem. For a problem X, a fully polynomial-time approximation scheme (FPTAS) is an approximation scheme with running time polynomial in the input size of the problem as well as polynomial in 1/ϵ.
Fleischer [22] presented the first FPTAS for the maximum multicommodity flow problem that is independent of the number of commodities. It is faster than the best previous approximation schemes, whose running time is O(ε −2 m 2 ). To calculate the approximate solution, Fleischer uses path flow linear programming formulation of a maximum multicommodity flow problem. e dual of this linear program corresponds to the problem of assigning length to the edges of the network such that the length of the shortest path from s i to t i is at least one for each commodity.
Let us assume that l(e) and l(P) denote the length of edges and length of the shortest path from s i to t i for each i, respectively. Algorithm starts with l(e) � δ, ∀e, ψ � 0, and selects a path P ∈ P with l(P) < 1. e flow ψ(P) � ψ(P) + u is assigned along the path P where u � min e∈P u(e), and the length l(e) � l(e)(1 + (εu/u(e))), ∀e ∈ P, of each edge is updated. e process is repeated until there exists a path with l(P) < 1 otherwise return (ψ, l).
Based on this, we present Algorithm 2 to calculate approximate solution of MSMCF with partial lane reversals.
Example 1 with capacity on arcs only (cf. Figure 2(a)) is considered, and maximum multicommodity flow with lane reversals is calculated.
We choose the next path P 3 : In the same way, the next path is P 2 : s 2 − x − y − t 2 , l(p 2 ) < 1 and u � min e∈P 2 u e � 3, ψ(P 2 ) � 0 + 3 � 3. ere is no path from s 2 to t 2 . Since there is no anymore s i − t i path, the algorithm terminates, and maximum flow is 3 + 4 + 3 � 10.

Dynamic Multicommodity Flow with Partial Lane Reversals
In this section, we introduce the MDMCF with partial lane reversals by reverting the necessary arc capacities. Hall et al. [10] proved that multicommodity flow over time is NP-hard. ey provided the solution of the flow over time problem in a network having uniform path length (a network where the length of all paths is the same). e lane reversal strategy cannot be applied in case of uniform path length in general. It may violate the criterion of uniform path length after lane reversals as shown in example 3. Two algorithms to solve the MDMCF problem with lane reversals are presented in this section.
Example 3. Let us consider a multicommodity network where s 1 , s 2 , and s 3 are the source nodes and t 1 , t 2 , and t 3 are the sink nodes as shown in Figure 3(a). e first and second numbers on the arcs represent capacity and transit time associated with the arcs. e auxiliary network for reconfiguration is as shown in Figure 3(b). e network is of uniform path length before lane reversals and after the lane reversals, and a new path s 3 − x − y − t 3 is created that violates uniform path length. In general, it proves that the lane reversal strategy fails to satisfy the condition of uniform path length. e lane reversal technique can be applied in case of uniform path length by reversing only those arcs that do not violate uniform path length.

Pseudopolynomial Solution of MDMCF.
To deal with the maximum flow over time problem, Ford and Fulkerson [7] introduced the concept of time expansion. is well-known approach can be carried out in the case of the multicommodity flow over time problem. Kappmeier [11] has shown the equivalency between static multicommodity flow on a time-expanded network and dynamic multicommodity flow on the original network as given below. e MDMCF problem with partial lane reversals sends the maximum flow from the sources s i to the corresponding sinks t i in the unique pair of source and sink nodes (s i , t i ) for each commodity i � 1, 2, . . . , k, and for a given time by saving the unused arc capacity.
To solve the maximum multicommodity flow over time problem with partial lane reversals (problem 2), we design Algorithm 3. Proof. e theorem will be proved in three steps. In the first step, we show the feasibility.
Step 2 of the algorithm is welldefined because it transforms the given dynamic network flow problem into the static network flow problem on the time-expanded auxiliary network. e feasibility of other steps in the algorithm can be shown as similar to eorem 1. In the second step, we prove the optimality. Feasibility implies that any feasible solution of MDMCF on N a T is feasible to the MDMCF with lane reversal solution on network N. Dynamic multicommodity flow problem on network N reduces to a static multicommodity flow problem on N a T . By reducing the multicommodity to a single-commodity and decomposing the flow into the (s i − t i ) path, dynamic multicommodity flow solution can be obtained optimally on auxiliary network N a T . An optimal solution on N a T is equivalent to a feasible solution on N. e unused capacities of the arcs are saved by partial lane reversals in Step 5. At last, we show that the algorithm solves it in pseudopolynomial-time. We know that Step 3 is solved in O(mn) times, and Step 2 can be computed polynomially in the input size of the network, i.e., it depends on T. All other steps can be computed in linear time. us, Algorithm 3 solves the MDMCF problem with partial lane reversals on given network N optimally in pseudopolynomial-time complexity. (1) e given network is transformed by adding two-way capacities as N a � (V, A a , u a e , K, S + , S − ). (2) Compute the approximate MSMCF on the transformed network N a by using the algorithm by Fleischer [22].

Approximate Solution of MDMCF.
A time-expanded network is a well-known technique to solve flow over time problems, but it has the drawback of a large blow-up of its size. By reducing the size of the time-expanded network, an efficient algorithm is presented. is reduction technique is known as condensation in the setting of a time-expanded network, and the network is known as the Δ-condensed time-expanded network. If we take Δ � 1, then the Δ-condensed time-expanded network reduces to the classical time-expanded network. To, solve Problem 2 in fully polynomial-time, we present Algorithm 4.

Theorem 4. Algorithm 4 provides approximate solution of the MDMCF problem with partial lane reversals.
Proof. First, we prove the feasibility.
Step 1 of the algorithm is well-defined as it transforms the given dynamic network flow into the static network flow on the Δ-condensed auxiliary network. e feasibility of other steps of the algorithm is similar to eorem 1.
Next, we prove the optimality. Feasibility implies that any approximate optimal solution of MDMCF with lane reversals on network N is also a feasible approximate solution to the MDMCF on N Δa . Dynamic multicommodity flow problem on network N reduces to a static multicommodity flow problem on N Δa . By reducing the multicommodity to a single commodity and decomposing it into the (s i − t i ) path, an approximate dynamic multicommodity flow solution can be obtained optimally on auxiliary network N Δa . An approximate optimal solution on N Δa is a feasible solution on N. e unused capacities of the arcs by partial lane reversals are saved in Step 5. us, an approximate MDMCF solution with lane reversals on each arc of the given network N can be computed optimally. Proof.
e complexity of Algorithm 4 is dominated by Steps 2 and 3.
Step 3 is solved in O(mn) time. An approximate solution of a static multicommodity flow problem on the Δ-condensed auxiliary network is obtained by Fleischer [22] in O(ε −2 m 2 ) time complexity in Step 2. e Δ-condensed auxiliary network contains (n 2 /ε 2 ) nodes and (mn/ε 2 ) arcs having polynomial-time complexity. Remaining steps can be solved in linear O(m) time. So, the problem can be computed in fully polynomial-time.
By scaling the capacities and transit times on arcs given in Figure 1  { } is considered. e earliest arrival multicommodity flow problem is a feasible multicommodity flow over time ξ with time horizon T of a maximum value of ξ(θ) for each θ. is problem becomes EAMCT if it is a feasible multicommodity transshipment over time ξ with time horizon T such that ξ fulfills all supplies and demands d i and value of ξ(θ) is maximal at every point in time θ ∈ 0, 1, . . . , T − 1 { }. Kappmeier [11] proved that the EAMCT exists in the case of multiple sources and a single-sink. ey observed that it is possible to reduce the problem of the multicommodity to the single-commodity in the single sourcesink case. If all commodities commence in the same origin and have the same destination, then it can simply consider all commodities as one, compute the earliest arrival transshipment, and split the flow up into the commodities.

International Journal of Mathematics and Mathematical Sciences
Proof.
Considering the flow over time network N � (V, A, K, u, τ, d i , S + , t, T) having multiple sources (ksources) and a single-sink t with k-commodities. Defining new supplies and demands d ′ by To compute an earliest arrival transshipment for the instance defined by the network N, we first build the timeexpanded network with supplies and demands and calculate minimum cost flow by considering transit time as the cost using the successive shortest path algorithm. is yields multicommodity flow over time corresponding to the static multicommodity flow.
We split the path for each commodity i ∈ K into commodity-dependent paths and select a source s ∈ ∪ i∈K S i + with supply d i s > 0. An s − t path P with a flow value ξ P > 0 is chosen, and ξ ′i P � min ξ p , d i s is set as the flow value. en, we update the residual demand d i s � d i s − ξ ′i P and flow value ξ P � ξ P − ξ ′i P and continue the process until there is no any source in S i + with positive supply according to Kappmeier [11]. e above construction creates a feasible solution of the EAMCT problem. We have P∈P st ξ P � i∈K d i v by the definition of new demand for each source. ξ ′i P � min ξ p , d i s assures that no path capacity is violated as no more supply is sent exceeding the capacity. e flow value is reduced until arc capacities are zero, for any path P, and the constructed flow obeys the capacity constraint due to i∈K ξ ′i P � ξ P . Hence, feasible flow satisfying all demands is ξ i P for all i. e flow obtained is the earliest arrival flow for a single-commodity setting. us, a path decomposition of the earliest arrival transshipment is computed in network N. Selection of path flow for each    Proof. e theorem will be proved in two steps. In the first step, we show the feasibility of the algorithm, and in the second step, we show the optimality. Steps 1 and 4 of the algorithm are feasible according to Rebennack et al. [12]. e feasibility of Step 2 is verified by using the EAMCTalgorithm (Kappmeier [11]). Step 3 of the algorithm is feasible being decomposition of flow along paths and cycles and assures that there is no cycle. e feasibility of Step 5 of the algorithm is due to Pyakurel et al. [19]. us, Algorithm 5 is feasible. To prove the optimality, we compute the EAMCT solution on the auxiliary network using the algorithm by Kappmeier [11]. From the feasibility of the algorithm, we conclude that every feasible EAMCT solution on the auxiliary network N a is equivalent to the EAMCT solution with Input: given a multicommodity flow network N � (V, A, K, u, τ, d i , S + , t, T) Output: the EAMCT with partial lane reversals (1) e given network is transformed to auxiliary network by adding two-way capacities in N a � (V, A a , K, u a , τ a , d i , ∪ i∈K S i + , t, T) as u a e � u e + u e r , τ a e � τ e , if e ∈ A, τ e r , otherwise. (2) Compute the EAMCT on the transformed network N a by using algorithm by Kappmeier [11].
(3) Decompose the flow along the s i − t paths and cycles and remove flows in cycles ∀i ∈ K. (4) Reverse e r (θ) ∈ A T up to the capacity ψ a (θ) − u e iff ψ a (θ) > u e , u e replaced by 0 whenever e(θ) ∉ A T where ψ e � k i�1 ψ i e and u e � k i�1 u i e . (5) For each e(θ) ∈ A T , if e r (θ) is reversed, s c (e r (θ)) � u a e − ψ e (θ) and s c (e) � 0. If neither e nor e r is reversed, s c (e(θ)) � u e − ψ e (θ) > 0, where s c (e) is the saved capacity of e. (6) Transform the solution to the original network.  Computation of the EAMCT after lane reversals is shown in Figure 7(b). Table 3 represents the value of EAMCTon each path with partial lane reversals.

Conclusions
One of the major problems in operational research is transshipping several kinds of commodities (goods) in underlying network topology, respecting capacity constraints on the arcs. Maximizing the sum of flow of all commodities in the specified period, it turns to maximum multicommodity flow over time. Multicommodity flow over time is computationally hard. A time-expanded network is a technique to solve flow over time problems, but it has pseudopolynomial-time complexity. By shrinking the size of the network, a Δ-condensed time-expanded network is introduced without changing flow values too much, and it provided an efficient approximation scheme. By flipping the orientation of lanes, the capacity of the lanes will be increased that amplifies the flow value and reduces the time horizon. Partial lane reversal strategy reverses only necessary arc capacities and saves the capacity of unused arcs that can be used in case of emergency.
In this paper, we introduce a partial lane reversal strategy in static multicommodity flow, multicommodity flow over time, and earliest arrival multicommodity transshipment problems and provide algorithms to solve these problems. e first problem is solved in polynomial-time, the second problem is solved in pseudopolynomial-time as well as polynomial-time approximately, and the third problem is solved in pseudopolynomial-time complexity.
Data Availability e authors have not used any additional data in this article.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.