Time–Space Network

For our modeling approach, evacuation time is assumed to start as soon as the first evacuee arrives at the pickup point and ends when the issued warning is exhausted or the last evacuee is sheltered, whichever is earlier. The problem statement emphasizes two major variables to be modeled simultaneously, the time horizon and the space or location of all nodes including pickup points and shelters. The most important step in using our model is the construction of a time–

space network diagram (T–S network). The structure of the T–S network mainly comprises nodes and arcs with time and space attributes. The nodes for pickup points and shelters have both space attributes (P₁, P₂…for pickup points and S₁, S₂… for shelters) and time attributes (T₀, T₁, T₂… T_t). Pickup nodes are the supply nodes, as evacuees are collected at these points following some defined arrival pattern. In the case of multiple shelters, travel times from each shelter to pickup points will be different; therefore, a separate node needs to be modeled for each shelter in

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the T–S network. The third type of node, the sink node D, is the demand node where all evacuees are ultimately required to be moved. The sink node is further divided into two types: DL to model the people who are saved and DD to indicate the people who cannot be evacuated within the allowable warning time using available buses. For flood-prone areas, the primary requirement is to evacuate all the people out of the flood area. Shifting them to other designated shelters can be done at later stages, if required. Therefore, as far as node capacity limits are concerned, shelter nodes are also assumed to be unconstrained. Figure 3.1(a) shows a sample road network and Figure 3.1(b) explains the transformed T–S network diagram with nodes and arcs for each road.

The T–S network primarily consists of three sets of arcs: the first set is used for the bus, the second set is used for the evacuees, and the third set comprises sink arcs. The set of arcs used to model the buses has a flow capacity equal to the capacity of the bus used in the evacuation. To model different possibilities, bus arcs are further divided into two types: bus waiting arcs W_b and bus movement arcs M. A bus can either wait at a pickup point to pick up more evacuees arriving at that point or it can move to some other pickup point for this. Another possible bus movement is a trip to the shelter if the bus capacity limit is exhausted. A bus waiting arc is also provided at each shelter, in case it is required. Travel time on bus waiting arcs is equal to the difference between subsequent time periods of the node at which the bus is waiting. Travel time on bus movement arcs is equal to travel times between the nodes connected by these arcs for the real road network. The total number of bus arcs (bus waiting and bus moving) between any two nodes at any time period is limited by the number of buses allowed. For example, if we want to allow only one bus to move at a time between any two points, one movement arc will be used to model the movement between these two points. Furthermore, if we want to model multiple paths between any two points with different travel times, separate movement arcs for each path need to be modeled. The movement arcs in Figure 3.1(b) are drawn only for the first time period T0. A similar pattern is repeated for all time periods up to the last period T_t.

Shelte

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The second set of arcs, used for evacuees, is uncapacitated and consists of waiting arcs between subsequent time periods at the same pickup point, Wp. These are the arcs used to model a situation when all evacuees are not picked up from a point by the bus moving out of this point and the evacuees have to wait for the next bus.

The third set of uncapacitated sink arcs is further divided into two types. First, there are arcs connecting shelter nodes to the sink node DL at all time periods T1, T2 … Tt to indicate the people who are saved. The flows in these arcs are equated to the flows arriving at the shelter at that time period, and the travel times on these arcs are zero. Second, there are the arcs connecting pickup points at the last time period of evacuation time Tt to the sink node DD to indicate the people who cannot be evacuated. These uncapacitated arcs are assigned considerably higher travel times (approaching infinity) to restrict the movement of evacuees on these arcs and to maximize the number of evacuees saved indirectly.

P1 P2 P3 S

Sink Node DL

Sink Node DD

T0

T1

T2

T₃

Tt

Figure 3.1(b) Time–space network diagram

Bus movement arc Evac. waiting arc Bus waiting arc Evac. sheltered

Evac. died

35 3.3.2 Model Formulation

In this section, we describe the mixed integer linear programming formulation for the proposed SBED model. First, we define two decision variables;

(1) xij: a continuous integer variable that represents the flow of evacuees from a point i to a point j and

(2) y_ij: a binary variable that equals 1 if there is any flow of evacuees from point i to point j, and otherwise takes the value 0.

Now, the model formulation using these two decision variables will be as follows:

Minimize

(3.1) Subject to

(3.2) (3.3) (3.4) (3.5) (3.6)

(3.7) Where

t_ij = Travel time on arc ij

N = Set of shelter nodes and pickup nodes P = Set of pickup nodes

S = Set of shelter nodes

D = Set of sink nodes (D_L and D_D) A = Set of all types of arcs

M = Set of bus movement arcs for all types of bus movements, i.e., between pickup points, from pickup point to shelter, and from shelter to pickup points.

Wb = Waiting arcs for bus

δ_i= Supply of evacuees at pickup point C = Capacity of the bus.

36 B = Total number of available buses

T0 = Start time for evacuation

Tt = Last possible time to evacuate or the warning time

In the model formulation described above, equation (3.1) states the objective function to minimize the sum of the product of travel times and flows (evacuees) for all possible arcs while fulfilling the side constraints. Constraint (3.2) is a typical flow conservation constraint for the evacuees. Constraint (3.3) is bus capacity constraint; this constraint also ensure a relation between the variables for flow of evacuees x and the binary variable y, i.e., if the flow in an arc is zero, the binary variable representing the bus movement also gets the value zero. Constraint (3.4) limits the number of buses used in the evacuation to be less than or equal to the number allocated. This constraint is applied by limiting the number of bus arcs at the first time period T₀, which is the start of evacuation, and the last time for evacuation Tt. Constraint (3.5) is a flow conservation constraint for buses, and ensures that the number of buses entering a node at any time must be equal to the number of buses leaving that node at the same time. The last two constraints (3.6) and (3.7) set the two decision variables as an integer variable and a binary variable, respectively.

The proposed model and T–S network structure completely explain the requirements stated in the problem statement. We shall now discuss these requirements individually. The given warning time is divided into equal time periods starting from T0 to Tt. Time T0 indicates the start of evacuation and time Tt indicates the last possible time for evacuation or the warning time.

Evacuation completion is observed from resulted arc flows after the model run. If the evacuation of all evacuees with the specified number of buses is completed before the modeled warning time, zero flow will be observed in sink arcs DD as well as in other arcs (for both evacuees and buses) for time periods later than the evacuation completion time. Next, the flexibility of route choice is modeled on the basis of existing road network connectivity of pickup points and shelter. All possible bus movements from any node are modeled by bus movement arcs in the T–

S network, allowing for route choice flexibility. This flexibility of route adds to the complexity of the analyzed network by increasing exponentially the number of bus movement arcs, and is hence feasible only for fewer pickup points. Additionally, the non-availability of any flooded/damaged link after a lapse of known time can be modeled accordingly through bus movement arcs. The next model requirement is multiple trips of buses used in the evacuation,

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which is achieved by modeling shelter nodes identical to pickup point nodes in the T–S network.

The flows in bus movement arcs from shelter to pickup points are forced to be equal to zero.

Furthermore, any requirement of fixing a particular start point for some or all the buses can be fulfilled by modifying constraint (3.4) slightly and setting the total number of bus movements (waiting and moving) from a specified pickup point equal to the desired number of buses starting their trip from that point. A separate provision for bus waiting times at some or all the pickup points will increase evacuation times accordingly. However, bus waiting arcs are provided in the T–S network so that a bus can wait for evacuees and it does so if required for optimization of results. Last but not least, modeling of other delays for loading/unloading times or fueling times is not incorporated directly into the model. However, these times can be considered by incorporating them into travel times between pickup points or from pickup points to shelters.

Occasionally, poor preparation of a T–S network structure may need post-processing of the model results before its application to a real world scenario. This fact is illustrated with a hypothetical road network of three pickup points encircled by a shelter, as shown in Figure 3.2.

The evacuees with their arrival patterns are also shown (each of the five evacuees arrive at all three points for five time periods). The travel times between all pickup point and from pickup points to shelter are the same and are equal to the time interval modeled in the T–S network preparation. For an input of 3 buses, each with a capacity of 25 people and maximum evacuation time of 5 time periods, the model output of bus trips is shown on the same figure by bus movement arcs.

Hypothetical road network

Figure 3.2 Model output for 3-buses, 3-points, and 5-time periods P1 P2 P3

S S

S

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By looking at the input, a better solution for this case seems to delay all three buses until the time period T4—one bus at each pickup point—and then evacuate all 25 arrived evacuees to the shelter in a single trip by each bus. However, according to the model output, each of the buses pick up 5, 10, and 10 evacuees at times T0, T2, and T4, respectively, from each pickup point and move them to the shelter in three trips. The reason for such an output lies in the objective function defined in equation (3.1) and the prepared T–S network. For the case when travel times to the shelter and modeled waiting time intervals of the T–S network structure are the same or even comparable, the model prefers to shift evacuees to the shelter rather than wait for the next time period. This shifting of evacuees in three trips costs lesser in the defined objective function value than when keeping all the evacuees waiting until the next/last time, and then evacuating them to the shelter in single trip.

3.4 Explanation through a Case Study