Development and evaluation of an integrated emergency response facility location model
JaeDong Hong^{1}, Yuanchang Xie^{2}, KiYoung Jeong^{3}
^{1}South Carolina State University, ^{2}University of Massachusetts Lowell, ^{3}University of HoustonClear Lake (UNITED STATES)
Received October 2011
Accepted February 2012
Hong, J.D., Xie, Y., & Jeong, K.Y. (2012). Development and evaluation of an integrated emergency response facility location model. Journal of Industrial Engineering and Management, 5(1), 421. http://dx.doi.org/10.3926/jiem.415

Abstract:
Purpose: The purpose of this paper is to propose and compare the performance of the “two” robust mathematical models, the Robust Integer Facility Location (RIFL) and the Robust Continuous Facility Location (RCFL) models, to solve the emergency response facility and transportation problems in terms of the total logistics cost and robustness.
Design/methodology/approach: The emergency response facilities include distribution warehouses (DWH) where relief goods are stored, commodity distribution points (CDP), and neighborhood locations. Authors propose two robust models: the Robust Integer Facility Location (RIFL) model where the demand of a CDP is covered by a main DWH or a backup CDP; the Robust Continuous Facility Location (RCFL) model where that of a CDP is covered by multiple DWHs. The performance of these models is compared with each other and to the Regular Facility Location (RFL) model where a CDP is covered by one main DWH. The case studies with multiple scenarios are analyzed.
Findings: The results illustrate that the RFL outperforms others under normal conditions while the RCFL outperforms others under the emergency conditions. Overall, the total logistics cost and robustness level of the RCFL outperforms those of other models while the performance of RFL and RIFL is mixed between the cost and robustness index.
Originality/value: Two new emergency distribution approaches are modeled, and evaluated using case studies. In addition to the total logistics cost, the robustness index is uniquely presented and applied. The proposed models and robustness concept are hoped to shed light to the future works in the field of disaster logistics management.
Keywords: emergency response, facility location, disaster recovery, emergency relief goods, spreadsheet model, facility disruptions

1 Introduction
After emergency events such as natural disasters or terrorist attacks, it is critical through emergency response facilities to distribute for rapid recovery emergency supplies to the affected areas in a timely and efficient manner. The emergency response facilities considered in this paper include distribution warehouses (DWHs), where emergency relief goods are stored, intermediate response facilities termed Disaster Recovery Centers (DRCs), sometimes referred to as break of bulk points (BOBs), where emergency relief goods can be sent to the affected area in a timely manner for rapid recovery, and neighborhood locations in need of relief goods. The distribution of emergency supplies from these facilities to the affected areas must be done via a transportation network. Given the significance of transportation costs and the time involved in transporting the relief goods, the importance of optimally locating DWHs and BOBs in the transportation network is apparent.
Traditional facility location models, such as setcovering models, pcenter models, pmedian models, and fixed charge facility location problems (Dekle, Lavieri, Martin, EmirFarinas & Francis, 2005) implicitly assume that emergency response facilities will always be in service or be available, and each demand node is assumed to be satisfied by a supply facility as assigned by the optimization model. However, it is very likely that some emergency response facilities may be damaged or completed destroyed and cannot provide the expected services. When this happens, the demands of the affected areas will have to be satisfied by other facilities much farther away than the initially assigned facilities. This obviously will increase the distribution cost and time of relief goods. Compared to the priordisaster transportation costs minimized by the traditional facility location models, the actual or postdisaster transportation costs can be substantially higher. Thus, it is very important to take into account the postdisaster costs as well as the priordisaster costs in emergency response facility location modeling.
In light of the significant difference in siting between emergency response facilities and other types of facilities and the paucity of the research literature in this area, we propose a new emergency response facility location model that can better account for the uncertainty caused by the disruptions of critical infrastructure and that would minimize the postdisaster costs. Assuming that some DWHs might be unavailable after disastrous events, we compare the new model with a traditional facility location model based on case studies to demonstrate the developed model’s capability to better deal with the risks in emergency response caused by the disruptions of critical infrastructure.
2 Literature review
Facility location models have been extensively researched for decades. Dekle et al. (2005) develop a setcovering model and a twostage modeling approach to identify the optimal DRC sites. Their objective is to minimize the total number of DRCs, subject to each county’s residents being within a certain distance of the nearest DRC. Horner and Downs (2007) conduct a similar study to optimize BOB locations (in our paper, BOBs and DRCs are used interchangeably). As shown in Figure 1, emergency relief goods are shipped from central distribution warehouses to BOBs and distributed to victims of catastrophes. Given the number and locations of initial warehouses, Horner and Downs formulate the problem as a multiobjective integer programming. Two objectives are considered. The first objective is to minimize the transportation costs of servicing BOBs from warehouse locations, and the second one is to minimize the transportation costs between BOBs and neighborhoods in need of relief goods.
Snyder and Daskin (2005) develop a reliable facility location model based on the pmedian and the incapacitated fixedcharge location problem. They defined the extra transportation cost caused by the failure of one or more facilities as the “failure cost”. Obviously, adding additional facilities as backups would reduce the failure cost. However, this will increase the daytoday system operating cost. The main goal of their model is to find the best “tradeoff” between the operating cost and the expected failure cost of a facility location design. The developed model is solved by a Lagrangian relaxation algorithm. Berman, Krass and Menezes (2007) also develop a reliable facility location model based on the pmedian problem. In their research, each facility is assigned a failure probability. The objective is to minimize the expected weighted transportation cost and the expected penalty for certain customers not being served. The developed model has a nonlinear objective function and is difficult to solve by exact algorithms. These authors thus proposed a greedy heuristic for their model.
Figure 1. Distribution strategy for emergency relief goods (Horner & Downs, 2007)
Hassin, Ravi and Salman (2010) investigate a facility location problem considering the failures of network edges. Their goal is to maximize the expected demand that can be served after disastrous events. In their study, it is assumed that a demand node can be served by a facility if it is within a certain distance of the entity in the network that survived disaster. The failures of network edges are assumed to be dependent on each other. These authors formulate the problem as an exact dynamic programming model and develop an exact greedy algorithm to solve it. Eiselt, Gendreau and Laporte (1996) also propose a reliable model for optimally locating p facilities in a network that takes into account the potential failures of road network links and nodes. These authors develop a loworder polynomial algorithm to solve the proposed facility location model.
Li and Ouyang (2010) examined a continuous reliable incapacitated fixed charge location (RUFL) problem. They assume that facilities are subject to spatially correlated disruptions and have a locationdependent probability to fail during disastrous events. A continuum approximation (Langevin, Mbaraga & Campbell, 1996; Daganzo, 2005) approach is adopted to solve the developed model. The authors consider two methods to model the spatial correlation of disruptions, including positively correlated BetaBinomial facility failure.
Cui, Ouyang & Shen (2010) investigate a discrete reliable facility location design problem under the risk of disruptions. Their model considers a set of i customers and j facilities, with the goal of minimizing the sum of fixed facility and expected transportation costs. Similar to Snyder and Daskin (2005), Cui et al. (2010) assign each customer to multiple levels to ensure the robustness of the final facility location design. They also develop a Lagrangian relaxation algorithm to solve the proposed model.
Our research is built upon the work done by Horner and Downs (2007) and also motivated by the recent trend in facility location studies to consider the risk caused by critical infrastructure disruptions. Contrary to the onestage model developed by Horner and Downs and which optimized the location of BOBs only, we develop a twostage integrated facility location model that simultaneously optimizes the locations of DWHs and BOBs. In addition, we propose two robust models for the case of disasters.
The rest of this paper is organized as follows. In the next section, an integrated facility location model is introduced. Based on this integrated model formulation, robust integrated facility location models are proposed and described in detail. Following the description of the model formulations, case studies are conducted and the resulting analysis is presented. The last section summarizes the developed models and research findings. It also provides recommendations for future research directions.
3 Development of integrated facility location model
Let M be the set of all neighborhoods and potential distribution warehouse locations, indexed by m. We separate M into two sets: M={N, I}, where I denotes the set of potential distribution warehouse locations (indexed by i =1, 2, …,w) and N represents the set of neighborhoods (indexed by n =1, 2, …, p). In this research, we assume BOBs can be located at any neighborhoods and potential DWH locations, while DWH can be built at candidate DWH locations only. Based on these two assumptions, let J be the set of potential BOB locations indexed by , where j = 1, 2, …p, p+1, p+2, …p+i, …,p+w. Given this problem setting, we formulate the following integer quadratic programming (IQP) model that minimizes the total logistics cost, which is the sum of fixed facility costs and the transportation costs from DWHs to BOBs and between BOBs and neighborhoods/candidate DWH locations that are not selected:
(1) 
Subject to
(2) 

(3) 

(4) 

(5) 

(6) 

(7) 

(8) 

(9) 

(10) 
where,
a_{i}: fixed cost for contructing and operating DWH_{i};
b_{j}: fixed cost for contructing and operating BOB_{j};
B_{j}: 1 if neighborhood j is selected as a BOB, 0 otherwise (decision variable);
d_{ij}: distance between DWH_{i} and BOB_{j};
d_{im}: distance between DWH_{i} amd location m;
d_{jm}: distance between BOB_{j}_{ }and location m;
D_{B}: maximum number of BOBs can be built (set to 5);
D_{m}: demand of location (can be either neighborhood or DWH) m;
D_{w}: maximum number of DWHs can be built (set to 3 in this study);
k_{i}: maximum number of BOBs a DWH must handle (set to 1 in this study);
K_{i}: maximum number of BOBs a DWH can handle (set to 5 in this study);
L_{j}: minimum number of neighborhoods a BOB needs to cover (set to 2);
U_{j}: maximum number of neighborhoods a BOB can cover (set to 6);
W_{i}: 1 if a candidate warehouse i is selected, 0 otherwise (decision variable);
x_{ij}: 1 if BOB_{j} is covered by DWH_{i}, 0 otherwise (decision variable);
y_{jm}: 1 if location m is covered by CDP_{j}, 0 otherwise (decision variable).
Since the main purpose of this paper is to demonstrate how the proposed model works, we further simplify the objective function by excluding the fixed cost terms for BOBs and for DWH. Also, the numbers of BOBs and DWHs to be built are prespecified. For realworld applications, once the real data are available, such restrictions can be readily relaxed to generate meaningful results. In this paper, we use the following simplified objective function for the simultaneous optimization of DWH and BOB locations.
(11) 
Constraints (2) require that at most D_{W} DWHs can be constructed; D_{W} is provided by the user.
Constraints (3) ensure that the potential DWH location will not be selected simultaneously as both DWH and BOB.
Constraints (4) ensure that if a potential DWH location i is not selected (i.e., W_{i}=0) (its demand must be satisfied by a BOB).
Constraints (5) make certain that each neighborhood is assigned to exactly one BOB.
Constraints (6) limit the minimum and maximum number of BOBs to be served by each DWH.
Constraint (7) ensure that DWHs only supply the selected BOBs, not all candidate BOBs.
Constraints (8) limit the total number of selected BOBs to be less than or equal to a userspecified number, D_{B}.
Constraints (9) ensure that neighborhoods or unselected DWH locations can only be assigned to the candidate BOBs that are finally selected.
Constraints (10) ensure that each selected candidate BOB must cover a minimum number of L_{j} neighborhoods and can only cover a maximum of U_{j} neighborhoods. Hereafter, this newly introduced model given by Equations (2)(11) is referred to as the Integrated Facility Location (IFL) model.
4 Development of robust optimization models
A property of the IFL model is that the optimal plan generated by it may not be optimal after disastrous events. If a DWH becomes unavailable after the disaster, BOBs assigned to this DWH need to be reassigned to other adjacent DWHs with extra capacity. Then the postdisaster logistics cost may become much larger than the predisaster optimal cost. To reduce postdisaster logistics cost, one potential solution is to require each BOB to be covered by a backup DWH as well as a main DWH. To do that, we solve the IFL model after changing the righthandside of Equation (4) to be 2 from 1 and find the optimal DWH and BOB locations, denoted by W_{i}^{*2} and B_{j}^{*2}. We call this model the Robust Integer Facility Location (RIFL) model. Note that the robust model would minimize the postdisaster cost, not the predisaster cost. To find the predisaster cost for the RIFL model, we solve for the optimal coverage of BOBS and neighbors, x_{ij}^{*} and y_{jm}^{*}, after setting the RHS of Equation (4) back to be 1, with the W_{i}^{*2} and B_{j}^{*2} fixed.
An alternative way of developing the robust model is to add the capacity constraints of candidate DWHs in a disasterprone area. For instance, if a DWH has a high probability of being damaged in disastrous events, one can specify that all BOBs assigned to this DWH can only have up to certain percentages of their demand satisfied by it. This strategy would avoid putting all eggs in one basket and improve the robustness of the model. In fact, if a DWH is partially damaged due to disaster, this model would be useful. Now, let x_{ij} be a continuous decision variable between 0 and 1, denoting the fraction of BOB_{j}’s demand satisfied by DWH_{i}. Then, the following capacity constraint is added to the IFL model:
(12) 
Where, C_{i}: maximum fraction of BOB’s demand that can be satisfied by DWH_{i}
For candidate DWHs with a high probability of damage or shutdown during disastrous events, C_{i} would take relatively smaller values, whereas for DWHs in stable and safe areas, C_{i} would take larger values. By making x_{ij} a continuous decision variable, the robust facility location model becomes a mixed integer quadratic programming (MIQP) problem, which can be linearized by defining a new decision variable as follows:
z_{ijm} = x_{ij }· y_{jm}, 
(13) 
Where z_{ijm} denotes the fraction of neighborhood m’s demand satisfied by DWH_{i} via BOB_{j}. Then solving this robust facility location problem is equivalent to solving the following mixed integer linear programming (MILP) problem:

(14) 
Subject to equations 2, 3, 4, 5, 7, 8, 9 and 10;
(15) 

(16) 

(17) 

(18) 
We call the above model the Robust Continuous Facility Location (RCFL) model. Note that if C_{i}=1, for all i, the RCFL model is equivalent to the IFL model and produces exactly the same solutions. To find the predisaster cost for the RCFL model, we solve the RCFL model by adjusting C_{i}, such that the postdisaster cost is minimized. Then with W_{i}^{*} and B_{j}^{*} obtained for the minimum postdisaster cost fixed and C_{i}=1, for all i, we solve the RCFL model again and the resulting total cost will be the predisaster cost.
5 Case study and observations
The integrated model and two robust models can be solved by a variety of optimization software packages, such as LINDO, LINGO, or GAMS. However, coding the developed MILP model using these tools may not be an easy task, since so many decision variables and constraints are involved. Recently, many researchers and practitioners are paying significant attention to Microsoft Excel spreadsheetbased optimization modeling because of its nonalgebraic approach. Several powerful software packages based on the Excel spreadsheet model, such as Solver, What’s Best, CPLEX, etc., make Excel spreadsheetbased modeling attractive. In this paper, a CPLEX for Microsoft Excel AddIn is used to solve the proposed MILP model.
To evaluate the developed MILP model, we conduct a case study using cities in South Carolina. 20 cities are selected as neighborhoods and 5 cities among neighborhoods, with Charleston, Columbia, Florence, Greenville, and Orangeburg considered as candidate sites for DWHs, as shown in Figure 2. All neighborhoods are candidate locations for BOBs. Tables 1(a), 1(b) and 1(c) show the distances (in miles) between any two neighborhoods. Also shown in Table 1(c) are the demands (in thousands) for all neighborhoods. These demands are hypothetical values proportional to each neighborhood’s year 2000 population and can be readily replaced by true demand data for realworld applications. Based on these input data, an Excel Spreadsheet model is developed.
Figure 2. Candidate Warehouses, BOBs, and Neighborhoods
We solve the three models, IFL, RIFL, and RCFL. To show how robust the RIFL and RCFL models are, two scenarios are considered. The first (normal) scenario assumes that all candidate DWHs remain available after disastrous events, whereas the second considers the shutdown/unavailability of a DWH. Hereafter, these scenarios are referred to as normal and shutdown scenarios, respectively. For normal scenario, we evaluate and present the results of facility location and transportation scheme as shown in Tables 2(a), 2(b) and 2(c). From the results under normal scenario in Tables 2(a), 2(b) and 2(c), we see that all three models include Columbia and Charleston as DWHs. Thus, it would be interesting to see what would happen if one of DWHs is unavailable and to compare the postdisaster costs of the three models. We select DWH Columbia to be unavailable after disaster, evaluate the three models, and present the results in Tables 2(a), 2(b) and 2(c), under the shutdown scenario.
Note that in Tables 2(a), 2(b) and 2(c), we assume that Columbia, the unavailable DWH for the shutdown case, can still cover the Columbia area and consequently is not assigned to any BOB. We call this Case I. But, more likely, the unavailable DWH after disaster can’t even operate for its own area. Thus, it might be necessary for the affected area to be assigned to a BOB. We call this situation Case II.
No. 
Neighborhoods 
Aiken 
Anderson 
Augusta 
Beaufort 
Camden 
Clemson 
Clinton 
1 
Aiken 
0.00 
99.69 
16.98 
121.37 
86.19 
120.42 
69.85 
2 
Anderson 
99.69 
0.00 
92.34 
246.70 
148.32 
18.05 
50.04 
3 
Augusta 
16.98 
92.34 
0.00 
127.63 
128.68 
110.82 
81.00 
4 
Beaufort 
121.37 
246.70 
127.63 
0.00 
166.79 
271.49 
181.15 
5 
Camden 
86.19 
148.32 
128.68 
166.79 
0.00 
169.48 
87.01 
6 
Clemson 
120.42 
18.05 
110.82 
271.49 
169.48 
0.00 
63.95 
7 
Clinton 
69.85 
50.04 
81.00 
181.15 
87.01 
63.95 
0.00 
8 
Conway 
186.02 
253.07 
228.30 
188.83 
110.14 
264.79 
190.71 
9 
Georgetown 
206.74 
269.41 
224.91 
137.08 
113.48 
247.81 
226.23 
10 
Greenwood 
55.53 
39.50 
62.00 
167.60 
102.90 
56.53 
26.97 
11 
Hilton Head 
152.40 
277.66 
158.59 
41.02 
198.23 
239.42 
196.77 
12 
Myrtle Beach 
207.12 
266.99 
225.29 
202.69 
124.06 
262.91 
204.74 
13 
Rock Hill 
124.47 
120.98 
142.64 
206.76 
71.32 
120.00 
65.57 
14 
Spartanburg 
142.14 
60.36 
160.32 
225.30 
125.80 
59.19 
35.54 
15 
Sumter 
112.39 
172.26 
130.57 
125.70 
29.34 
168.17 
104.57 
16 
Charleston 
162.96 
226.73 
207.56 
70.32 
146.74 
248.36 
170.50 
17 
Columbia 
56.41 
116.50 
75.10 
134.16 
34.69 
128.22 
61.20 
18 
Florence 
132.44 
192.92 
136.00 
150.80 
50.43 
201.61 
137.72 
19 
Greenville 
150.96 
31.00 
120.94 
234.12 
134.62 
30.10 
41.61 
20 
Orangeburg 
53.75 
135.02 
76.00 
83.91 
62.98 
161.39 
97.82 
Table 1(a). Distances (in miles) between Neighborhoods
No. 
Neighborhoods 
Conway 
Georgetown 
Greenwood 
Hilton Head 
Myrtle Beach 
Rock Hill 
Spartanburg 
1 
Aiken 
186.02 
206.74 
55.53 
152.40 
207.12 
124.47 
142.14 
2 
Anderson 
253.07 
269.41 
39.50 
277.66 
266.99 
120.98 
60.36 
3 
Augusta 
228.30 
224.91 
62.00 
158.59 
225.29 
142.64 
160.32 
4 
Beaufort 
188.83 
137.08 
167.60 
41.02 
202.69 
206.76 
225.30 
5 
Camden 
110.14 
113.48 
102.90 
198.23 
124.06 
71.32 
125.80 
6 
Clemson 
264.79 
247.81 
56.53 
239.42 
262.91 
120.00 
59.19 
7 
Clinton 
190.71 
226.23 
26.97 
196.77 
204.74 
65.57 
35.54 
8 
Conway 
0.00 
36.62 
218.67 
193.54 
14.03 
186.15 
223.24 
9 
Georgetown 
36.62 
0.00 
247.64 
157.04 
34.76 
232.88 
258.84 
10 
Greenwood 
218.67 
247.64 
0.00 
183.21 
232.70 
89.97 
59.39 
11 
Hilton Head 
193.54 
157.04 
183.21 
0.00 
191.40 
210.80 
231.61 
12 
Myrtle Beach 
14.03 
34.76 
232.70 
191.40 
0.00 
200.16 
237.25 
13 
Rock Hill 
186.15 
232.88 
89.97 
210.80 
200.16 
0.00 
61.93 
14 
Spartanburg 
223.24 
258.84 
59.39 
231.61 
237.25 
61.93 
0.00 
15 
Sumter 
80.81 
79.19 
116.18 
138.17 
94.56 
87.32 
130.47 
16 
Charleston 
97.41 
60.92 
191.91 
104.98 
97.34 
186.88 
205.42 
17 
Columbia 
140.20 
123.04 
72.81 
142.64 
146.75 
67.33 
93.13 
18 
Florence 
53.11 
68.54 
165.08 
170.49 
67.14 
96.09 
170.14 
19 
Greenville 
231.03 
266.62 
51.09 
234.53 
244.49 
89.80 
29.09 
20 
Orangeburg 
124.74 
105.96 
95.52 
102.33 
138.49 
108.05 
129.92 
Table 1(b). Distances (in miles) between Neighborhoods (continued)
No. 
Neighborhoods 
Sumter 
Charleston 
Columbia 
Florence 
Greenville 
Orangeburg 
Demand (in 1000s) 
1 
Aiken 
112.39 
162.96 
56.41 
132.44 
150.96 
53.75 
29 
2 
Anderson 
172.26 
226.73 
116.50 
192.92 
31.00 
135.02 
26 
3 
Augusta 
130.57 
207.56 
75.10 
136.00 
120.94 
76.00 
196 
4 
Beaufort 
125.70 
70.32 
134.16 
150.80 
234.12 
83.91 
13 
5 
Camden 
29.34 
146.74 
34.69 
50.43 
134.62 
62.98 
8 
6 
Clemson 
168.17 
248.36 
128.22 
201.61 
30.10 
161.39 
12 
7 
Clinton 
104.57 
170.50 
61.20 
137.72 
41.61 
97.82 
9 
8 
Conway 
80.81 
97.41 
140.20 
53.11 
231.03 
124.74 
12 
9 
Georgetown 
79.19 
60.92 
123.04 
68.54 
266.62 
105.96 
9 
10 
Greenwood 
116.18 
191.91 
72.81 
165.08 
51.09 
95.52 
23 
11 
Hilton Head 
138.17 
104.98 
142.64 
170.49 
234.53 
102.33 
48 
12 
Myrtle Beach 
94.56 
97.34 
146.75 
67.14 
244.49 
138.49 
32 
13 
Rock Hill 
87.32 
186.88 
67.33 
96.09 
89.80 
108.05 
72 
14 
Spartanburg 
130.47 
205.42 
93.13 
170.14 
29.09 
129.92 
37 
15 
Sumter 
0.00 
106.14 
43.41 
39.28 
150.20 
56.99 
41 
16 
Charleston 
106.14 
0.00 
114.54 
109.92 
214.24 
75.98 
121 
17 
Columbia 
43.41 
114.54 
0.00 
79.49 
100.91 
40.83 
130 
18 
Florence 
39.28 
109.92 
79.49 
0.00 
177.93 
90.34 
38 
19 
Greenville 
150.20 
214.24 
100.91 
177.93 
0.00 
137.71 
62 
20 
Orangeburg 
56.99 
75.98 
40.83 
90.34 
137.71 
0.00 
13 
Table 1(c). Distances (in miles) between Neighborhoods (continued) and Demands
To further investigate the effects of the shutdown of DWHs and to see the performance of the robust models, we consider various shutdown scenarios, present the resulting costs for both cases in Table 3, and compare the results for the three models.
As expected, the total transportation cost (TTC) for each model increases under the shutdown scenario and the increase in TTC are also reported in Tables 2(a), 2(b), 2(c) and 3. For the IFL model, the TTC goes from $47,451.54 to 69,995.04, a 47.5% increase. We observe that, on average, two robust models, RIFL and RCFL, outperform than the nonrobust IFL model under the shutdown scenario, though they underperform under the normal scenario.
Now, we propose a performance measure index, which is called a robustness index (RI) to show how much the results from each model are robust enough to cover the diverse scenarios in terms of cost minimization. Although there are many definitions of robustness, we adopt the one from Dong (2006) as “the extent to which the network is able to perform its function despite some damage done to it, such as the removal of some of the nodes and/or link in a network.” In this paper, each model’s performance may be evaluated by comparing it with the best performing model in terms of average TTC and its standard deviation. Hence we propose the following robustness index (RI):
RI for a model g is defined as

(19) 
where AVG(lambda) and STD(lambda) stand for average and standard deviation of each model lambda’s cost under given scenarios and alpha denotes the weight between the average and the standard deviation. Note that as RI for the model becomes closer to 1, the more robust the model would be. And RI can be used to decide the rank of each model in terms of robustness. We calculate RI for the three models for all possible shutdown scenarios and present them in Table 3. We calculate three different RIs RI for a normal scenario and for Case I and Case II under the shutdown scenario, and an overall RI for both cases with the assumption that all individual scenarios have the same weight. As the RI values indicate, the IFL is most efficient under normal scenario, whereas the RIFL and RCFL seem to be the most robust for Case II and for Case I, respectively, under shutdown scenario. That is, on average, these robust models generate a slightly higher TTC for the normal scenario, but produce a lower TTC for the shutdown case than IFL.
Model 
IFL 

Scenario 
Normal 
Shutdown 
DWH Selected 
1. Charleston 2. Columbia 3. Greenville 
1. Charleston 3. Greenville 
BOBs covered by (DWH #) 
1. Beaufort (1) 2. Aiken (2) 3. Sumter(2) 4. Anderson (3) 5. Spartanburg (3) 
1. Beaufort (1) 2. Aiken (3) 3. Sumter(1) 4. Anderson (3) 5. Spartanburg (3) 
Neighborhoods Assigned to (BOB) 
•(Beaufort), Hilton Head •(Aiken), Augusta, Orangeburg •(Sumter), Camden Conway, Florence, Georgetown, MyrtleBeach •(Anderson), Clemson, Greenwood •(Spartanburg) Clinton, Rock Hill

•(Beaufort), Hilton Head •(Aiken), Orangeburg •(Sumter), Camden Conway, Florence, Georgetown, MyrtleBeach •(Anderson),August Clemson, Greenwood •(Spartanburg) Clinton, Rock Hill

(CDB,CBN) TTC 
($29116, $18,335) $47,451 (A) 
($36,889, $33,105) $69,995 (B) 
Increase (B)(A) 
$22,543 

CDB: Cost from DWHs to BOBs, 1^{st} Term in Eq. (12). CBN: Cost from BOBs to Neighbors, 2^{nd} Term in Eq. (12). TTC= CDB+CBN Table 2(a). Results comparison for normal/shutdown scenarios for three models 

Model 
RIFL 

Scenario 
Normal 
Shutdown 
DWH Selected 
1. Charleston 2. Columbia 3. Orangeburg 
1. Charleston 3. Orangeburg 
BOBs covered by (DWH #) 
1. Beaufort (1) 2. Camden(2) 3. Sumter (2) 4. Clinton (2) 5. Aiken (3) 
1. Beaufort (1) 2. Camden(3) 3. Sumter (3) 4. Clinton (3) 5. Aiken (3) 
Neighborhoods Assigned to (BOB) 
•(Beaufort), HiltonHead •(Camden), Rock Hill •(Sumter), Conway, Florence, Georgetown, MyrtleBeach •(Clinton),Anderson, Clemson, Greenwood, Spartanburg, Greenville, •(Aiken), Augusta

•(Beaufort), Hilton Head •(Camden), Rock Hill •(Sumter), Conway, Florence, Georgetown, MyrtleBeach •(Clinton), Anderson, Clemson, Spartanburg, Greenville, •(Aiken), Augusta, Greenwood

(CDB,CBN) TTC 
($35,231, $23,216) $58,448 (A) 
($44,462, $23,873) $68,335 (B) 
Increase (B)(A) 
$9,887 

CDB: Cost from DWHs to BOBs, 1^{st} Term in Eq. (12). CBN: Cost from BOBs to Neighbors, 2^{nd} Term in Eq. (12). TTC= CDB+CBN Table 2(b). Results comparison for normal/shutdown scenarios for three models (continued) 

Model 
RCFL 

Scenario 
Normal 
Shutdown 
DWH Selected 
1. Charleston 2. Columbia 3. Greenville 
1. Charleston 3. Greenville 
BOBs covered by (DWH #) 
1. Beaufort (1) 2. Georgetown(1) 3. Aiken (2) 4. Anderson (3) 5. Spartanburg (3) 
1. Beaufort (1) 2. Georgetown(1) 3. Aiken (3) 4. Anderson (3) 5. Spartanburg (3) 
Neighborhoods Assigned to (BOB) 
•(Beaufort), HiltonHead •(Georgetown), Conway, MyrtleBeach, Sumter, Florence •(Aiken), Augusta, Camden, Orangeburg •(Anderson), Clemson, Greenwood •( Spartanburg) Clinton, Rock Hill

•(Beaufort), HiltonHead •(Georgetown), Conway, MyrtleBeach, Sumter, Florence •(Aiken), Orangeburg •(Anderson), August, Clemson, Greenwood •( Spartanburg), Camden Clinton, Rock Hill

(CDB,CBN) TTC 
($31,531, $19,992) $51,523 (A) 
($30,303, $35,079) $65,383 (B) 
Increase (B)(A) 
$13,860 

CDB: Cost from DWHs to BOBs, 1^{st} Term in Eq. (12). CBN: Cost from BOBs to Neighbors, 2^{nd} Term in Eq. (12). TTC= CDB+CBN 
Table 2(c). Results comparison for normal/shutdown scenarios for three models (continued)
Shutdown Scenario 
Model 

IFL 
RIFL 
RCFL 

Normal 
Shutdown 
Normal 
Shutdown 
Normal 
Shutdown 

Case I 
Case II 
Case I 
Case II 
Case I 
Case II 

DWH 1 
$47,451 
$51,345 
$70,000 
$58,448 
$59,277 
$77,372 
$47,451 
$51,345 
$70,000 
DWH 2 
$47,451 
$69,995 
$85,883 
$58,448 
$68,335 
$81,033 
$51,523 
$65,383 
$81,271 
DWH 3 
$47,451 
$58,017 
$65,573 
$58,448 
$59,046 
$60,316 
$47,500 
$56,265 
$63,834 
DWHs 1 & 2 
$47,451 
$85,958 
$130,222 
$58,448 
$69,164 
$100,523 
$56,716 
$80,770 
$125,034 
DWHs 2 & 3 
$47,451 
$107,307 
$142,028 
$58,448 
$117,534 
$139,085 
$52,478 
$101,848 
$135,829 
DWHs 1 & 3 
$47,451 
$61,911 
$88,849 
$58,448 
$62,940 
$82,306 
$48,550 
$58,001 
$84,141 
AVG 
$47,451 
$72,422 
$97,093 
$58,448 
$72,716 
$90,106 
$50,703 
$68,935 
$93,252 
STD 
0 
$20,824 
$31,742 
0 
$22,376 
$27,203 
$3,617 
$19,107 
$29,851 
RI 
1 
0.934 
0.892 
0.811 
0.900 
1 
0.468 
1 
0.938 
Overall AVG 
$72,321 
$73,756 
$70,996 

Overall STD 
$29,305 
$23,289 
$26,392 

Overall RI 
0.888 
0.981 
0.941 

*AVG and STD stand for average and standard deviation, respectively. *Alpha is set to 0.5 for RI. DWH 1: Charleston for all models. DWH 2: Columbia for all models. DWH 3: Greenville for IFL and RCFL, Orangeburg for RIFL 
Table 3. Comparison between integrated and two robust models
For Case I under the shutdown scenario, RIFL generates the highest TTC among the three models for the normal scenario and generates a slightly lower TTC than IFL. For the same weight between the average and the standard deviation, i.e., alpha=0.5, the overall RI also indicates that RIFL has the highest robustness, followed by RCFL and IFL in this order. The threshold value for alpha, denoted by turns out to be 0.7586. It implies that for RCFL seems to be the most robust model, followed by RIFL and IFL.
From Table 3, we recommend that the proposed robust models, RIFL and RCFL, be used for optimally locating DWHs under the risk of disruptions. As discussed previously, transport of relief goods happens mostly after disaster. Therefore, for siting emergency response facilities, it would be more important to minimize the postdisaster cost rather than the predisaster cost and to better consider the unavailability of emergency facilities. The example provided here clearly demonstrates that the proposed robust facility location models can well suit the needs of siting emergency response facilities.
6 Summary and conclusions
In this paper, we develop an IFL (Integrated Facility Location) model and propose two robust models and compare them with a nonrobust IFL. For the RCFL (Robust Continuous Facility Location) model, we introduce a continuous variable, defined in Equation (13), to denote the capacity constraint on a candidate DWH in disasterprone areas, so that it can only partially satisfy the demand of BOBs. We formulate the problem as a mixed integer linear programming model and solve it using CPLEX for Microsoft Excel AddIn. For the RIFL (Robust Integer Facility Location) model, we set the constraint requiring each BOB to be served by multiple DWHs (two DWHs in this paper) on the IFL model, which requires each BOB to be served by one DWH. We propose a performance measure index to show how well the models perform after disaster, RI, defined in (19). Using numerical examples, we show that the two robust models, RIFL and RCFL, yield emergency response facility location plans of slightly higher TTCs (total transportation cost) than the IFL model under normal situations. However, they generate more robust facility location plans in the sense that they can perform better when some of the selected DWHs are shut down after disaster and these unavailable DWHs can’t distribute emergency supplies to the affected areas (Case II).
The purpose of establishing emergency response facilities is for distributing relief goods after disaster. Therefore, when evaluating the efficiency and robustness of emergency response facility location plans, more weight should be given to their postdisaster performance. The resulting RIFL and RCFL models are designed in a robust manner such that they can better address scenarios with failures of key transportation infrastructure. Case studies are conducted to demonstrate the developed model’s capability to deal with uncertainties in transportation networks. Thus, the developed robust models can help federal and local emergency response officials develop efficient and robust disaster relief plans.
For future research, it would be necessary to develop a robust model when both a DWH and a BOB could be unavailable in the shutdown scenario. In addition, we implicitly assume that each DWH always carries enough inventories of emergency relief goods, so that for the shutdown scenario the other DWH(s) can ship enough relief goods to the extra BOBs. Thus, it would be also interesting to include the constraint on the capacity of DWHs in any proposed model.
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