Proceedings of the 2018 IISE Annual ConferenceK. Barker, D. Berry, C. Rainwater, eds.

A Hybrid Approach for Humanitarian Evacuation Before and Aftera Hurricane

Hayford Adjavor, Yong Wang*Department of Systems Science and Industrial Engineering, Binghamton University, NY

13902.*Corresponding author: yongwang@binghamton.edu

Abstract

Humanitarian Relief chain (HRC) has continued to receive extensive attention in the field of operationsresearch. Thus, this paper investigates the evacuation and rescue efforts during hurricane Katrina, and itsimpact on the city of New Orleans, Louisiana. The problem is first formulated as a multi-depot multi-traveling salesperson problem (MmTSP) which is an NP-hard problem. We use techniques from the field ofnetwork science to validate and justify the optimal number of evacuation centers. A Geographic InformationSystem (GIS) tool was then used to provide a visual structure of the affected area network. Genetic Algo-rithm was subsequently used to implement the MmTSP mathematical formulation to find feasible solutionsto the problem. A major finding of this work is the potential presented by this hybrid strategy for futureground-breaking investigations in this growing field of humanitarian logistics and supply chain.

Keywordshumanitarian logistics, network science, GIS, genetic algorithm

1. Introduction & Related WorkEvacuating people and distributing relief items to areas affected by natural and/or human-made disastersis of critical importance to stakeholders in the field of humanitarian relief supply chain (HRC). The urgentneed to evacuate people and deliver items make transportation (air, road, rail, marine) a crucial part of therelief management process. The need for such services during the pre-disaster planning (known as prepared-ness & response phase), and post-disaster (termed recovery & clean-up phase) cannot be overemphasized.These phases of disaster management are similar to those recognized by [1]; " ... mitigation, preparednessresponse, and recovery".However, the role of transportation is sometimes ‘trivialized’. When it is not marginalized, the goal haslargely remained cost mitigation. The work by [2] supports this position. Other works have focused ontravel times, but such efforts have also been done within the context of penalties embedded in the mathe-matical model. This gives the impression that cost was more important. We argue that human lives are morevaluable during disasters than the cost element. As [3] noted, “... the summation of logistic and deprivationcosts are usually not accounted for in the mathematical model". The goal here is to continue the discourseon the role of transportation in HRC. Hence, this work combines the ‘power’ of network science techniqueslike degree & degree distribution and, that of operations research (Genetic Algorithm) to address the prob-lem.Literature exist in the social sciences, engineering and operations research (OR) and to a small extent, thenatural science on the subject matter. The past two decades have witnessed increased research of the topicin engineering & OR. Researches in HRC have focused on evacuation, dispatch of humanitarian workers,distribution of relief items/supplies, and vehicle routing needs. Other researchers like [2, 3], have advocated

Adjavor and Wang

equity." The work of [4] have focused on the classification of techniques in analyzing data in HRC. Al-though the work of [4] seems thorough, one could hardly find a strategy like the one proposed by this work.Furthermore, their approach focused more on minimizing delivery or evacuation cost. [5] acknowledgedthat, “...disaster prevention, protection, and reconstruction are the major areas of focus to reduce human suf-fering and damage from disasters. A key point in that work is their multi-objective programming techniquefor designing relief delivery systems; minimizing total cost, minimizing total travel time, and maximizingminimal satisfaction during the planning period were featured.

2. Affected Area in PerspectiveFigure 1 presents the transportation network formed by plotting the coordinates data of the areas affected byHurricane Katrina. It captures the shelters, and areas impacted by Katrina. According to the website of [6],the population of New Orleans, Louisiana was 455,188 in 2005; but this dropped to 230,172 after the storm.Katrina was responsible for the drastic reduction.

Figure 1: Katrina affected & evacuation map. Source: created with ArcGIS.

2.1 Problem Statement & AssumptionsThe problem of concern in this investigation is how to evacuate people from disaster affected areas to safelocation (depots or shelters). And, simultaneously, distribute relief items to affected people/areas in an ef-fective, and judicious manner without invoking penalties into the objective function of the model. HurricaneKatrina is arguably the most expensive disaster in recent history, hence deserves attention. In the wake ofthe disaster, 18 shelters were setup to service roughly 20 affected areas. Were the 18 shelters necessary?how many drivers would have been efficient for moving people and distributing relief items? The followingassumptions are made: 1) road transportation is the major form of movement of any kind. 2) transportcapacity is limitless; 3) there would be no penalty awarded to the objective function; 4) there is scarcity ofresources, but human life is priceless; 5) the decision to establish 18 shelters was made scientifically.

3. Proposed ApproachThe problem in question is formulated as a multi-depot multiple traveling salesman problem (MmTSP)which is a variation of the classic traveling salesman problem (TSP). The work of [7] is relevant in thisregard. There are two variants of the MmTSP as acknowledged by [7]; the “fixed depot" and the “non-fixeddepot" cases. In the former case, the traveling salesmen do not necessarily have to return to the stating depot,but do in the latter case. This work, adopts the former case based on assumption 4.

3.1 Validation & Justification of Selected Shelters/DepotsAs stated prior, the dataset used consists of eighteen evacuation centers in addition to the twenty affectedareas. The puzzle, however, is how those 18 depots were chosen?. Assumption 5 provides the basis onwhich to proceed with this validation and justification. To achieve this, the average degree k measure of all

Adjavor and Wang

38 nodes was computed using Equation 1 below:

Ktotal =N(N−1)

2(1)

The degree of a node, a common metric used in network science, which indicates the total number of linksconnected to the i-th shown in Equation (2) was computed. ki is the degree of node i; N is the total numberof nodes in the graph; and ai j is the component of the adjacency matrix of the graph. Also, p probabilityand γ denote nodes in the network. Degree distribution concept of [8], another network science technique,was then used to classify the Katrina network using Equation (3) by [8] below.

ki =N

∑j

ai j (2)

p(k)∼ k−γ (3)It should be noted, however, that, no attempt was made here to implement the power law degree distributionof [8] since that is not the focus of the paper. This was simply used to help validate and justify the requirednumber of depots. Hence, Equations (5) and (6) were used for that purpose. ArchGIS (A GeographicInformation System Software package) was used to develop the road network map of the impacted areas.Figure 1 is the end product.

3.2 Mathematical Formulation & Genetic AlgorithmThe graph G = (V,A) expressed in Figure 1 represents the network structure of Hurricane Katrina; theevacuation centers (depots or shelters) and affected areas. V is the set of nodes (depots and affected areas)and A is the set of 2-tuples of nodes (represented by i and j in the adopted mathematical model).

Mathematical Model The mathematical model of [7, 9–11]as expressed below, best represent the prob-lem of this paper. The 2-index decision variable xi j approach proposed by [7, 11] is also best suited for thisproblem. Table 1 shows the parameters used in the mathematical model ( integer linear programming):

xi j =

{1, if arc (i,j) is visited.

0, otherwise(4)

Table 1: Parameters and variable description

Paramter/Variable Meaning

di j distance from i to jui number of nodes visited on traveler’s pathL max. no. of nodes a salesman visitedK min. no. of nodes visited by a salesmanD set of depot nodesV ’ set of affected customers or locations to be evacuatedV the union set of sets D and V ’

mi the number of salesmen initially present at depot im total number of salesmen in the network of graph G

Minimize: ∑(i, j)∈A

di jxi j (5)

Adjavor and Wang

Subject to: ∑j∈V ’

xi j = mi, ∀i ∈ D (6)

∑i∈V ’

xi j = m j, ∀ j ∈ D (7)

∑i∈V

xi j = 1, ∀ j ∈V ’(8)

∑j∈V

xi j = 1, ∀i ∈V ’(9)

ui +(L−2) ∑k∈D

xki−∑k∈D

xik 6 L−1, ∀i ∈V ’(10)

ui + ∑k∈D

xki +(2−K) ∑k∈D

xik > 2, ∀i ∈V ’(11)

xki + xik 6 1, K ∈ D, ∀i ∈V ’ (12)ui−u j +Lxi j +(L−2)xi j 6 L−1, i 6= j, ∀i, j ∈V ’ (13)

xi j ∈ {0,1} ∀i, j ∈V (14)

As [7] noted, Equations (6) and (7) ensure that exactly mi salesmen depart from depot i and exactly mi

salesmen return to depot i. Equations (8) and (9) ensure that exactly one tour enters each node and exactlyone tour exits each node. Constraints Equations (10) and (11) impose bound condition on the number ofnodes a salesman visits. Equation (12) prohibits salesmen from serving only a single customer and/oraffected area. Finally, Equations (13) and (14) are the sub-tour elimination constraints which are standardto most TSP and VRP problems.

Genetic Algorithm GA is a population based algorithm used in the search of near-optimal solutions inNP-Hard optimization problems; TSP and its variations fall in this category. Hence, this formulation of ourproblem as MmTSPs is well-suited to be solved with GA. [12] discussed ‘operators’ in GA formulation forthe mTSP case. They proposed a different approach to encode the solution of the problem. The work of [13]also focused on the design of routes of various depot multiple traveling salesman problem (VmTSP) and theuse of GA. The GA for MmTSP case in this work was designed following the standard form, however themutation probability was not well suited. As [12] acknowledged, parameter tuning must ‘play’ greater roleto get the appropriate outcome.

4. Discussion of Results/AnalysisThe result from implementing Equation (1) is first noted. As earlier stated, the degree of node is a simpleyet an effective metric to show the connectivity of a node within a network. Based on this metric, the initial38 nodes (affected areas and depots), have a total of 703 edges in the network. The in-degree and out-degreemeasures were then computed. Subsequently, the degree distribution was calculated per Equations (2 &3). This helped to identify the most ‘important’ nodes in the network. After comparing the average degreedistribution, a threshold of 10 links was identified per a nodes. Hence, for a node to be chosen as a depotand/or shelter, it must have at least 10 in-degree and out-degrees. Based on the afore-stated threshold, theinitial 18 depots were scaled back to 16. The implication is that, the 20 affected areas could have beenequally, efficiently, and judiciously serviced by the 16 shelters without issues. The next important resultstems from implementation of the mathematical model described prior using the Genetic Algorithm. Thevalues in Table 2 are then adjusted in a way best described as parameter tuning and sensitivity analysis toachieve desire results per suggestions noted by [12].

Results of the initial parameter settings from 30 experiments show the best found solution produce an eco-nomic value of 278,712.45 miles and route assignments sequence;[[0, 21, 34, 1], [1, 24, 0], [2, 30, 14], [3,19, 4], [4, 22, 3], [5, 17, 25, 11], [6, 28, 31, 13],[7, 20, 35, 10], [8, 33, 2], [9, 16, 9], [10, 23, 9], [11, 6], [12,

Adjavor and Wang

Table 2: Initial parameters of GA code

ItemNo. Description Assigned

Value

1. Crossover Probability 100 %2. Mutation Probability n/a3. Population Size 1004. Max. No. Generation 10005. No. of Affected Areas 206. No. of Depots 167. No. of Salesmen 16

18, 32, 3], [13, 9], [14, 3],[15, 27, 26, 29, 7]] for each of the 16 salesmen; hence the first route set - [0, 21,34, 1] - in the list are the cities assigned to the first salesperson, the second set for the next salesperson, etc.In the beginning of the experiment, the genetic algorithm’s search for solution ’jump’ around displayinga high economic value of 306, 408.05 miles. The average fitness value then decreases as the model runincreases and the search ’stabilizes’; an indication of the solution improvement and convergence as Figure2 reveals.

Figure 2: Model solution convergence. Source: from GA run.

5. Conclusions & Future WorkDrawing a ‘hard’ and any authoritative conclusions about the model’s efficacy may sound a little prematureat this point. Nonetheless, the results discussed above indicate huge future prospects of applying GeneticAlgorithm and network science techniques to solving the MmTSP non-fixed problem. A prospect that theresearchers will continue to explore. Consequentially, the following have been identified for future research:1) a comparison of the GA performance to other meta-heuristics like ant colony. 2) an implementation ofthe mutation operator. As intimated, this was not considered in this work. 3) a validation of the model usingan exact and/or traditional technique, and 4) a comparison study involving the fixed case of MmTSP, andthe non-fixed scenario.

References[1] Berkoune, D., Renaud, J., Rekik, M., and Ruiz, A., 2012, “Transportation in disaster response opera-

tions," Socio-Economic Planning Sciences, 46(1):23-32.

Adjavor and Wang

[2] Huang, M., Smilowitz, K., and Balcik, B., 2012, “Models for relief routing: Equity, efficiency andefficacy," Transportation Research Part E: Logistics and Transportation Review, 48(1):2-18.

[3] Holguin-Veras, J., Perez, N., Jaller, M., Van, W., Aros-Vera, N., and Luk, F., 2013, “On the appropriateobjective function for post-disaster humanitarian logistics models," Journal of Operations Management,31(5):262-280.

[4] Ozdamar, L., and Ertem, M. A., 2015, “Models, solutions and enabling technologies in humanitarianlogistics," European Journal of Operational Research, 244(1):55-65.

[5] Tzeng, G.-H., Cheng, H.-J., and Huang, T. D., 2007, “Multi-objective optimal planning for design-ing relief delivery systems," Transportation Research Part E: Logistics and Transportation Review,43(6):673-686.

[6] United States Census Bureau, 2018, Census data mapper - geography, U.S. Census Bureau, 2018.(Accessed on 01/05/2018).

[7] Kara, I., and Bektas, T., 2006, “Integer linear programming formulations of multiple salesman problemsand its variations," European Journal of Operational Research, 174(3):1449-1458.

[8] Barabasi, A. L., and Albert, R., 1999, “Emergence of scaling in random networks," Science,286(5439):509-512.

[9] Anand, J., Kulkarni, R. V., and Tai, K., 2010, “Probability collectives: A multi-agent approach forsolving combinatorial optimization problems," Applied Soft Computing, 10(3):759-771.

[10] Kulkarni, R.V., and Bhave, P. R., 1985, “Integer programming formulations of vehicle routing prob-lems," European Journal of Operational Research, 20(1):58-67.

[11] Kara, I., Gilbert, L., and Bektas, T., 2004, “A note on the lifted miller-tucker-zemlin subtour elimina-tion constraints for the capacitated vehicle routing problem," European Journal of Operational Research,158 (3):793-795.

[12] Kiraly, A., and Abonyi, J., 2011, “Optimization of multiple traveling salesmen problem by a novelrepresentation based genetic algorithm," Studies in Computational Intelligence, 366:241-269.

[13] Abbasi, E., Garakani, M., and Abvali, M., 2014, “Designing routes of various depot multiple travelingsalesman problem by using of genetic algorithm," Journal of Applied Business and Finance Researches,3(2):55-65.