Flow Improvement in Evacuation Planning with Budget Constrained Switching Costs

Many large-scale natural and human-created disasters have drawn the attention of researchers towards the solutions of evacuation planning problems and their applications. )e main focus of these solution strategies is to protect the life, property, and their surroundings during the disasters. With limited resources, it is not an easy task to develop a universally accepted model to handle such issues. Among them, the budget-constrained network flow improvement approach plays significant role to evacuate the maximum number of people within the given time horizon. In this paper, we consider an evacuation planning problem that aims to shift a maximum number of evacuees from a danger area to a safe zone in limited time under the budget constraints for network modification. Different flow improvement strategies with respect to fixed switching cost will be investigated, namely, integral, rational, and either to increase the full capacity of an arc or not at all. A solution technique on static network is extended to the dynamic one. Moreover, we introduce the static and dynamic maximum flow problems with lane reversal strategy and also propose efficient algorithms for their solutions. Here, the contraflow approach reverses the direction of arcs with respect to the lane reversal costs to increase the flow value. As an implementation of an evacuation plan may demand a large cost, the solutions proposed here with budget constrained problems play important role in practice.


Introduction
People are living under the threats of different natural and human-created disasters, such as hurricanes, floods, wildfires, or chemical spills. Many disasters are uncertain and unavoidable, but their effects can be minimized with efficient evacuation planning. But, the development of efficient models and algorithms for these planning problems is always challenging. To find efficient transportation routes during the evacuation, network flow models have been widely used. An evacuation network is interpreted by using a directed graph where the intersections of roads are represented by nodes, road segments between nodes are represented by arcs, and routes between two nodes are taken as directed paths. e places where evacuees are gathered and start to move at risk are considered as source nodes, and the safe destinations where they are supposed to arrive are sink nodes. Each node has a nonnegative integer capacity which bounds the maximum possible flow amount through it. Every arc has a cost or a transit time assigned to it. A flow in the network is considered as the evacuees or the vehicles carrying evacuees.
e network flow problems are to find maximum flow or minimum cost flow in a given network. e maximum dynamic flow problem (MDF) in two-terminal networks is solved polynomially using a static minimum cost flow solution [1]. A flow maximization seeks to send as much flow amount as possible within a time bound. A large number of researchers have studied different flow models for various objectives such as the earliest arrival flow to maximize the flow at every possible time; quickest flow to shift given amount of flow in minimum time; lexicographically maximum flow to maximize the flow in given priority order; and quickest transshipment problem to satisfy given demand and supply in minimum time. ese dynamic flow models have widely been used in solving several evacuation planning problems. Static flow solutions are the building blocks for dynamic flow solutions. Usually, the evacuation plans should respect the given time bound that may be continuous or discrete. e authors in [2] show that approximate continuous time solution can be obtained by applying the natural transformation to a discretized time solution. Most of these models, except that of cost minimization problems themselves, consider the travel time on an arc as only the cost on it and do not take care of any additional costs occurred during the evacuation plans. ey aim to fulfill the respective objectives on fixed network topology. For details, we refer to [3] and [4] and the references therein.
Different types of network modification problems exist in the literature. Generally, the original network is assumed to be not modifiable in the sense that capabilities or costs remain fixed as in the given network. However, this assumption is not valid in many real evacuation scenarios. For example, the capacity of an arc can also be increased up to some limit subject to some capacity incremental cost. For this, a fixed budget can be distributed to increase capacities in the network such that the network topology is modified and an objective, for instance, the flow, with respect to new capacities is maximized. ere are three variants of this improvement strategy that deal with rational, integral, and either of the all possible or not at all capacity values in [5]. e first two variants are polynomial time solvable, while the last one is NP-hard even in the cases of bipartite and series-parallel graphs.
is third variant called the 0/1 maximum flow improvement strategy is equivalent to the maximum flowfixed cost problem which is a bicriteria optimization problem where the flow has to be maximized under the budget constraint. A fully polynomial time approximation scheme for series-parallel graphs is presented. e network modification problems that relate to arc-based improvement and node-based upgrading models are also investigated.
Contraflow increases the outbound road capacities by reversing the direction of arcs towards the safe destinations. It increases the flow value and decreases the evacuation time by reducing the congestion in an emergency or rush-hour traffic management. e arc reversals are performed on the existing networks with permissible lane reversals without any additional costs. e authors in [6,7] prove that the contraflow problems for general networks are NP-hard. e former have presented different heuristics for multiterminal network, and the latter have polynomially solved maximum dynamic contraflow (MDCF) and quickest contraflow problems in the case of two-terminal networks, respectively. By introducing the contraflow approach, different evacuation planning problems are efficiently solved in [8,9]. e earliest arrival transshipment contraflow in multisource networks and with zero transit time in multisink networks has been polynomially solved. e authors in [10] investigate the quickest contraflow problem with constantand load-dependent transit times. e authors in [11] have developed a class of contraflow algorithms and performed computational experiments. e technique of lane reversals is beneficial for other purposes, for example, crossing eliminations, logistic supports, and use of emergency vehicles and facility locations. e contraflow approach with crossing elimination, facility location-allocation, and partial lane reversal strategies are introduced in [12][13][14], respectively. e third approach makes use of nonreversed arcs in contraflow for supporting facilities and emergency logistics. Multimodel integrated contraflow for uncertain arrivals of evacuees in an evacuation region with a low mobility population is presented in [15]. Considering the influence of intersections, an improved critical-road model has been investigated to find the optimal contraflow links [16].
For provided limited resources, it is not possible to select all arc reversals as demanded by the optimal contraflows. In this paper, we investigate contraflow problems with fixed budget constraint distributed to the arc reversals. e total given budget allows us to reverse only a certain percentage of arcs in a given network. We introduce the maximum dynamic flow improvement problem (MDFIP) and also the maximum contraflow improvement problems in both static and dynamic networks. en, we propose polynomial time algorithms to solve these problems. To the best of our knowledge, this is the first attempt to incorporate the issues of arc reversal costs on contraflow problems subject to the given total budget constraint. As arc reversals require a lot of costs at emergency period, this approach is more practicable in implementing the contraflow algorithms. e paper is organized as follows. e network flow models are given in Section 2. e solutions on static and dynamic flow improvement problems are presented in Sections 3.1 and 3.2, respectively. e contraflow models and their solution procedures with unit switching costs on arcs for the flow improvements are proposed in Section 4. Section 5 concludes the paper.

Preliminaries
Consider a directed dynamic network N � (V, A, u e (t),  e maximum static flow model is to maximize Objective (1) satisfying Constraints (2)-(4): where Outgoing flow from the sources, conservation of flow in intermediate nodes, and entering flow to the sinks are, respectively, shown in Constraint (2). Constraint (4) represents bounds of flows on arcs.

Maximum Dynamic Flow Model.
e dynamic network flow model with discrete time setting satisfies Constraints (5)- (8): Constraint (8) ensures that the flow does not enter arc e at time t if it will have to leave the arc after the given time horizon. e maximum dynamic flow that can enter the arc e within each integral time step t is bounded by the time varying capacity u e (t); this is ensured by Constraint (7). Flow conservation conditions are ensured in Constraint (5). Flow value at T is defined in (9) and is to be maximized for the MDF: Multiterminal network for single commodity flow can be reduced to the standard two-terminal network by introducing one virtual source node and one virtual sink node. Virtual arcs connect the new source to true sources and true sinks to the new sink. e transit times of these virtual arcs are zero. e capacities of arcs connecting to the virtual source with all other sources are bounded by the capacities of these sources. e capacities of arcs connecting to virtual sink from true sinks are bounded by the capacities of these sinks. If θ e � 0 for all e ∈ A and T � 0, then the formulated problem reduces to the classical maximum flow problem on a static network.

Time-Expanded Flow Model.
e dynamic network flow problem in N � (V, A, u e (t), θ e , T) can be reduced to a static network flow problem in the time-expanded network N T � (V T , A T , u T ), which is a static representation of the dynamic network. Construction of time-expanded network is as follows: Relation (11) e conservation and capacity constraints are given by (13) and (14), respectively. Equation (15) represents the feasibility of supplies and demands. is constraint implies that the total supply is equal to the total demand: e feasibility of minimum cost flow problem can be determined by solving a maximum flow problem [1,18]. For this, one introduces a supersource node s * , a supersink node If the maximum flow saturates all the source arcs, the minimum cost flow problem is feasible; otherwise, it is infeasible.

Static Flow Improvements.
In the maximum static flow improvement problem (MSFIP), an additional nonnegative number U e , for each e ∈ A, is given so that the capacity u e International Journal of Mathematics and Mathematical Sciences can be increased with some nonnegative cost up to the upper bound U e ≥ u e . e improvement capacity function with nonnegative unit cost b e is defined as I: A ⟶ Q ≥0 . e objective of the problem is to maximize the flow from the sources to sinks by increasing the capacities of arcs within the budget restriction where incremental cost is to be accepted to increase arc capacity. Flow improvement problems (16)- (22) have been formulated as in [5]: Constraint (19) controls the arc flows, Constraint (20) limits capacity increment, and Constraint (21) bounds the total capacity incremental cost B with b as the unit cost. By denoting F st to be the sum of flow out from the source that enter into the sink, the objective function equals Problem 1. e MSFIP with capacity improvement cost is maximum static flow problems (16)- (22), where capacity of arcs with unit costs can be increased up to specified limit bounded by the improvement cost.
e MSFIP with continuous improvement strategy, in which the improvement function I(e) takes any rational values respecting the upper bound, can be solved optimally in polynomial time.
e integral MSFIP that takes only integral improvements can be transformed into a budget constraint minimum cost flow problem in polynomial time.
Assume that the unit cost b e of increasing the capacity of arc e is a nonnegative number and also assume that the optimal flow improvement corresponding to the arc e is I * e � max 0, f * e − u e , where f * is the optimal improved maximum flow. Otherwise, the strategy would waste the cost. Such a behavior can be modeled by a flow cost c e defined as By definition, c e is a piecewise-linear convex function. In this case, each e ∈ A is replaced by two parallel arcs e 0 and e 1 to make it linear. e capacities u e and linear costs c e of these arcs are set as is construction is valid from the convexity of cost function. us, the improved maximum flow value F * st can be obtained by solving minimum cost flow with cost at most B by the method of binary search in [0, nU max ].
Theorem 1 (see [5]). The integral MSFIP can be solved optimally in polynomial time by O(log(nU max )) minimum cost flow computations in a directed network with 2m arcs, where U max � max U e : e ∈ A is the maximum capacity.
Instead of performing a binary search on the interval [0, nU max ], the interval can be searched only in multiplicative steps of 1 + ϵ, where ϵ > 0 is a fixed accuracy parameter. e value F st ′ found by this modified binary search satisfies F st ′ ≥ (F * st /(1 + ϵ)).
Each of the minimum cost flow computations have to be carried out in a graph with O(gm) arcs, where g is the maximum number of breakpoints occurring in the piecewise-linear cost functions. Furthermore, one can solve the problem in strongly polynomial time applying Megiddo's parametric search [19]. e same search can be applied to solve the rational MSFIP, too. e flow improvement strategy, either to increase the capacity of each arc to its maximum capacity or leave the capacity of arc unchanged, is NP-hard. is 0/1 MSFIP is equivalent to the maximum flow problem with fixed cost on arcs. For given nonnegative capacity u e and cost c e on each e ∈ A, the latter problem asks to find a subset A * of A such that e∈A * c e ≤ B and the source-sink flow is maximized. But, the decision variant of this problem is NP-complete. We state the following.
Theorem 3 (see [5]). The maximum flow problem with fixed cost on arcs is NP-hard even on series-parallel and bipartite graphs.
e pseudo-polynomial time algorithm for the maximum fixed cost flow problem on series-parallel graphs is presented. ey are converted into a fully polynomial time approximation scheme by the scaling technique.
Theorem 4 (see [5]). When the problems are restricted to series-parallel graphs, the maximum flow problem with fixed cost on arcs and 0/1 MSFIP can be solved with fully polynomial time approximation scheme.

Dynamic Flow Improvements.
is section extends the MSFIP to maximum dynamic flow improvement problem (MDFIP) considering the time factor. Let I(t) and b(t) be capacity improvement and improvement unit cost functions in dynamic network, respectively. e proposed MDFIP can be formulated as follows: subject to is is an extension of MDF where Budget Constraint (32) has been imposed to bound the capacity improvement cost, Constraint (31) deals with the maximum possible improvement, and Constraint (30) deals with the feasibility of flow with improved capacity.
Let (v(t), w(t + θ e )) ∈ A T and let I e(t) be the capacity improvement on arc e ∈ A; then, the capacity improvement function could be defined as I T e(t) � I (v(t),w(t+θ e )) � I e (t), for all e(t) ∈ A T . Each dynamic network can be transformed into the corresponding time-expanded network for a given time horizon. e maximum dynamic flow in the given network is equal to the maximum flow in time-expanded network [1].

Problem 2.
e MDFIP with capacity improvement cost is the MDF, where capacities of arcs can be increased up to a specified limit accepting the improvement cost.

Theorem 5.
e integral and continuous flow improvement problems in dynamic network can be solved optimally in pseudo-polynomial time.
Proof. Let us transform the given integral MDFIP in the dynamic network into the MSFIP in the corresponding static network assuming and improvement cost b T e(t) . As the integral maximum flow improvement problem in static network can also be solved optimally (cf. eorem 1), the integral MDFIP can also be solved optimally.
With similar arguments, the continuous MDFIP can be solved optimally in pseudo-polynomial time.
□ e dynamic improvement and dynamic improvement cost have been considered in (25)-(32). While constructing time-expanded network in the proof of eorem 5, they are transformed into static improvement and static improvement cost by applying the transformations I T e(t) ≔ I (v(t),w(t+θ e )) � I e (t)and b T e(t) ≔ b e (t) ∀ e(t) ∈ A T , respectively. If one consider improvement and its cost both static in (25)-(32), the time-expanded network copies the arc with the same improved capacity for every arc of the dynamic network. ese modifications ensure that each copy of the same arc is improved by International Journal of Mathematics and Mathematical Sciences the same amount of capacity and the improvement cost of an arc is counted only once for all of its copies.

Flow Improvements with Arc Switching Costs
Contraflow approach increases the flow value by reversing the directions of arcs towards the sinks as a flow towards the sources is neither preferred nor expected in emergencies.
is concept without any reversal cost is first incorporated in [20] and analytically studied in [7], where the arc is reversed with its full capacity or left as it is. However, a reversal may require some switching costs. Different contraflow models and solution procedures with switching costs are considered throughout this section. Figure 1 explains how contraflow works in time invariant network, where there is no arc reversal cost. Given network N � (V, A, u e , θ e , T), the contraflow uses an auxiliary network N � (V, A, u, θ, T), where the set of arcs A contains e if e � (v, w) ∈ A or e r � (w, v) ∈ A.
e capacity and symmetric transit time functions are considered as u e ≔ u e + u e r and θ e ≔ θ e for e ∈ A θ e r else ∀e ∈ A, respectively. Other parameters of the given network remain the same.

Maximum Static Flow
Problem. e maximum static contraflow problem (MSCFP) introduced in [7] maximizes the source-sink flow value, where directions of arcs have been reversed without considering the reversal costs. In this section, we introduce budget constraint MSCFP and propose its solution procedure for a two-terminal network. e switching cost contraflow (SCCF) finds a feasible flow in N � (V, A, u e ), where the arc directions can be reversed accepting switching costs whenever flow can be improved. e static SCCF problem has a similar structure as the minimum concave-cost network flow problem [21]. ese problems ask to find a feasible flow while minimizing the total cost. e total cost is the sum of concave-costs due to the use of arcs by feasible flow. It can be assumed that the concave-cost per arc consists of a fixed cost, whenever this particular arc is used, and a variable cost depends on the amount of flow. Fixing the variable cost to zero leads to a special problem called minimum cost fixed flow (MCFF) problem. e improvement strategy function I is a 0/1 decision if additional capacity is used or not, independent of how much additional capacity is used. An integral flow improvement strategy is considered throughout this section.

Problem 3.
e MSCFP with switching cost is the maximum flow problem in the static network where arcs can be reversed accepting some switching costs and total cost is restricted by a given budget. is problem is denoted by MSCFPWSC (Algorithm 1).

Theorem 6. Algorithm 1 solves Problem 3 optimally in polynomial time for integral reversed capacity.
Proof.
Step 1 constructs an auxiliary network from the given network which is feasible.
Step 2 defines the cost function for the improved capacity which is convex and nonlinear, whereas Step 3 is to linearize the cost function defined in Step 2. us, both these steps are feasible.
Step 4 finds the optimal flow F * st in N � (V, A, u e ) using the binary search method in [0, nU max ], and Step 5 reverses the direction of arcs according to the direction of the flow obtained in Step 4. So, Step 4 and Step 5 are also feasible. Hence, the algorithm is feasible.
To restrict on budget, this algorithm only reverses the required capacity of the arc so that I e � max 0, f e − u e → , where f e represents the flow through the arc e. Cost for the reversed capacity is defined in Step 2 which is not linear but convex and is linearized in Step 3, as in [5,22]. en, MSCFP with switching cost is equivalent to the budget constraint maximum flow improvement problem with integral improvement which is equivalent to the budget constraint minimum cost flow problem. us, Problem 3 is equivalent to the budget constraint minimum cost flow problem. Suppose F * st be the optimal integral flow achievable on budget B which can be obtained by the method of binary search in [0, nU max ] [5].
Step 1 constructs an auxiliary network in linear time and is similar for Step 2 and Step 3. □ Algorithm 1 reverses the direction of arcs with integral capacity; thus, the improvement is integral. is implies that the optimal flow of the problem is integral. For this, the binary search algorithm can test only for the integral flow values of the interval. Instead of applying binary search in the interval, one can apply Megiddo's parametric search [19] to extract the solution in Step 4 of Algorithm 1. e maximum value F st ′ satisfying the budget constraint lies in 1, 1 + ϵ, . . . , (1 + ϵ) k , where k � log 1+ϵ (nU max ) and ϵ > 0 is a fixed accuracy parameter. is modified binary search finds new value such that F st ′ ≥ (F * st /(1 + ϵ)). If such modification is applied in Step 4 of Algorithm 1, a (1 + ϵ) approximation for budget constrained maximum contraflow can be obtained in O(log log 1+ϵ (nU max )) minimum cost flow computations, where ϵ > 0 is a fixed accuracy parameter.  Figure 2(b)) applying the contraflow approach for which B � 37 is required for the complete arc reversals which is more than the given budget.
is means that the upper bound of the budget constraint contraflow is 37. e budget constraint contraflow can be found applying Algorithm 1. e optimal budget constraint flow in the auxiliary network in Figure 1 [11,19]. e middle value is 15. Its minimum cost is 17 as the reversed arcs are (x, s), (t, x), and (x, y) with capacities 4, 4, and 1, respectively. Here, we reverse the arc (x, y) only with capacity 1 as the budget should not be wasted. e resulting network flow is shown in Figure 2(c). Next, the possible upper value should be searched in [16,19]. Let us check the cost for 17. For this, the minimum cost is 27 which exceeds the given budget, and thus, it is infeasible. Similarly, the minimum cost is checked for 16 units of flow which is 22. Again, this exceeds the budget. Hence, the budget constraint contraflow is 15 which is shown in Figure 2(c) for which 17 units of budget have been used. Still, 3 units of budget remain unused but they are insufficient.

Theorem 7.
e fixed switching cost contraflow problem is equivalent to 0/1-MSFIP where all data to be positive integral [7]. Since the 0/1 maximum flow improvement is equivalent to the maximum fixed cost flow problem, this implies that the FSCCF is NP-hard to solve ( eorem 3). e fixed cost for arc reversals makes the problem NP-hard, even in the static case. e static contraflow algorithm is 'blind' for the arc reversal decisions. Adding a time component to FSCCF makes it practically even more difficult to solve. Based on the above results, the following theorems are proposed.

Theorem 8.
e FSCCF problem is NP-hard even in seriesparallel graphs. Proof.
e FSCCF problem is equivalent to 0/1-MSFIP ( eorem 7). As the maximum flow problem with fixed cost on arcs is equivalent to 0/1-maximum flow improvement [5], then by eorem 3, it can be claimed that the FSCCF problem is NP-hard even in series-parallel graphs.

□ Theorem 9.
ere is a fully polynomial time approximation scheme for the FSCCF problems when the problems are restricted to series-parallel graphs.
Proof. As in proof of eorem 8, the FSCCF problem is equivalent to 0/1-MSFIP and the maximum flow problem with fixed costs on arcs. Since all hardness and approximation results of these problems will be carried over to the FSCCF problem so that a fully polynomial time approximation scheme can be obtained for the FSCCF problems in series-parallel graphs ( eorem 4).

□
As the fixed switching cost contraflow problem is equivalent to the 0/1 maximum flow improvement which is equivalent to the minimum cost fixed flow problem, an approximation solution can be found by using the cost-totime ratio approach [5,18].
is can be obtained by assuming the capacities u e and unit cost c e /u e .

Maximum Dynamic Flow Problem.
e notion of MDF is introduced and solved in [7] by reversing the arcs at time zero with zero cost. In this section, reversal cost is considered to maximize the dynamic contraflow subject to budget constraint which is an extension of MSCFP from static to dynamic networks.

Problem 4.
e MDCF problem with switching cost is the MDF where arcs can be reversed accepting switching cost such that the total cost is subject to the budget constraint. It is denoted by MDCFPWSC.