computational implementation of location problem models...

2012 SPRING ISEN 601 PROJECT

Computational Implementation of location problem models for medical

services Facility location problem for large-scale emergencies

Yeong In Kim, SooIn Choi

5/1/2012

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1. Intro

In this project, we have reviewed the paper, “A Modeling Framework for Facility Location of Medical

Services for Large-Scale Emergencies, 2007, H. Jia, F. Ordonez, and M. Dessouky” The paper is about

locating medical supplies for Large-Scale Emergencies, unlike previous research papers which address

common emergency situation.

We first briefly review the concepts of Emergency Medical Service (EMS) and Large-Scale EMS (LEMS),

and examine the modeling framework for each classified emergency. Then, we implement the provided

model for solutions in a solver using CPLEX, and analyze the results along the lines of analysis provided

in the paper.

2. Paper Review

2.1. Traditional models for Emergency Medical Service (EMS)

In general, the purpose of EMS facility location problems is to design the staffing levels and materials of

local emergency responders to deal with regular emergencies, such as household fires or vehicle

accidents. However, these solutions do not translate well to large-scale emergencies that have

tremendous magnitude and low frequency. Therefore, traditional EMS facility location models need to

be modified for a Large-scale Emergency Medical Service (LEMS) with consideration of facility location

objective, facility quantity, and service quality.

To be modified for LEMS, the traditional facility location models are first classified by eight of the most

common criteria:

1. Topological characteristics

2. Objectives

3. Solution methods

4. Features of facilities

5. Demand patterns

6. Supply chain type

7. Time horizon

8. Input parameters

Then, the traditional models are translated in the covering, P-median, and P-center models, in the way

of focusing mainly on the criteria of objectives.

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Before the modification for LEMS is considered, large- scale emergencies share similarities with regular

emergencies although they have many unique characteristics.

First, for the covering models which are the most widespread location models for emergency facility

location problems, the objective is to provide “coverage” to demand points. And, a demand point is

considered as covered only if a facility is available to serve the demand point within a distance limit.

Moreover, the covering problems are divided into two major parts: the Location Set Covering Problem

(LSCP) and the Maximal covering Location Problem (MCLP).

Next, for the P-median models, the objective is to determine the location of P facilities so as to minimize

the average (total) distance between demands and facilities.

Lastly, the P-center models, in contrast to the P-median models, attempts to minimize the maximum

distance from a demand point to a facility, which are referred to as the minimax model, and thus

addresses situations in which service inequity is more important than average system performance. In

addition, this model considers closest center assumption (CCA) and therefore, full coverage to all

demand points is always achieved. However, unlike the full coverage in the covering models, full

coverage in the P-center model requires only a limited number (P) of facilities.

2.1. Large-scale Emergency Medical Services (LEMS)

There are two main characteristics of Large-scale Emergency which distinguish it from other regular

emergencies:

(1) Sudden and high demand

(2) Low frequency.

These require redundant and dispersed placement of EMS facilities, which results medical supplies

mobilized and serviceability and survivability of facilities improved.

Other considerable aspects of LEMS are following:

(1) Potential demand categorized which differs from other regular emergencies

(2) Multiple types of facility quantity and quality

(3) The facility deployment strategies (Proactive, Reactive, and Hybrid type)

(4) The facility location objectives (minimizing unmet demands and life-loss)

(5) The eligibility in selection of facility sites.

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2.2 Modeling the formulation of LEMS

2.2.1. The general model

The general LEMS facility location model is proposed in consideration of the characteristics of large-scale

emergencies. The paper defines “𝐼” as demand points and ”𝐽” as possible facility locations.

( )

∑

∑ 𝐼

𝐼 𝐽

{ } 𝐽

𝐼 𝐽

= { f f l y s pl d h rw s

= { f f l y s rv s d m d p h rw s

= { f d m d p s v r d h rw s

𝑡ℎ 𝑜 𝑙𝑎𝑡 𝑜𝑛 𝑜𝑓 𝑑 𝑚𝑎𝑛𝑑 𝑜 𝑛𝑡

𝑡ℎ 𝑙 𝑘 𝑙 ℎ𝑜𝑜𝑑 𝑓𝑜𝑟 𝑑 𝑚𝑎𝑛𝑑 𝑜 𝑛𝑡 𝑡𝑜 ℎ𝑎𝑣 𝑎 𝑙𝑎𝑟𝑔 − 𝑠𝑐𝑎𝑙 𝑚 𝑟𝑔 𝑛𝑐𝑦 𝑠 𝑡 𝑎𝑡 𝑜𝑛 𝑘

𝑡ℎ 𝑚 𝑎𝑐𝑡 𝑐𝑜 𝑓𝑓 𝑐 𝑛𝑡 𝑓𝑜𝑟 𝑑 𝑚𝑎𝑛𝑑 𝑜 𝑛𝑡 𝑛𝑑 𝑟 𝑙𝑎𝑟𝑔 − 𝑠𝑐𝑎𝑙 𝑚 𝑟𝑔 𝑛𝑐𝑦 𝑠 𝑡 𝑎𝑡 𝑜𝑛 𝑘

∙ ∙ 𝑡ℎ 𝑑 𝑚𝑎𝑛𝑑 𝑎𝑡 𝑜 𝑛𝑡 𝑛𝑑 𝑟 𝑚 𝑟𝑔 𝑛𝑐𝑦 𝑠 𝑡 𝑎𝑡 𝑜𝑛 𝑘

𝑟 𝑑 𝑐𝑡 𝑜𝑛 𝑛 𝑠 𝑟𝑣 𝑐 𝑐𝑎 𝑎𝑏 𝑙 𝑡𝑦 𝑜𝑓 𝑓𝑎𝑐 𝑙 𝑡𝑦 𝑛𝑑 𝑟 𝑚 𝑟𝑔 𝑛𝑐𝑦 𝑠𝑐 𝑛𝑎𝑟 𝑜 𝑘

2.2.2 Specified Model

2.2.2.1. Covering

Since we are interested in covering the greatest amount of the demand that can be generated by an

emergency, the objective function is defined as

∑

Covering models might require that facilities should be located within specific distance from

demand point . Then, these requirements are represented by the constraints below.

= 𝑓 𝑑 𝐼 𝐽 ℎ 𝑟 𝑑 𝑠 𝑡ℎ 𝑠ℎ𝑜𝑟𝑡 𝑠𝑡 𝑑 𝑠𝑡𝑎𝑛𝑐

Then, if we set = { 𝑑 } be the set of eligible facility sites that can service demand point ,

then the problem can be formulated as follows:

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∑

∑

∑ 𝐼

{ } 𝐼 𝐽

2.2.2.2 P-Median

Both the P-median and P-center problems consider relaxed quality requirements for coverage. The

objective is to increase the accessibility and effectiveness of EMS facilities in response to an emergency

situation. The P-median problem for a given scenario 𝑘 can be stated as:

∑ 𝑑

∑

∑ 𝐼

𝐼 𝐽

{ } 𝐼 𝐽

2.2.2.3. P-Center

The objective of P-center problem is to minimize the maximum service distance for all demand points.

The service distance for demand point is defined as the average distance from demand point to its

nearest facilities. The P-center model for a given scenario k can be written as the following integer

linear program:

∑

∑ = 𝐼

𝐼 𝐽

∑ 𝑑 𝐼 𝑘

{ } 𝐼 𝐽

Note that the introduced model only considers the case of single quality of coverage.

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2.3. Illustrative Examples

In the paper, three illustrative examples are given for covering, p-center, and p-median problem. They

consider Los Angeles County for the examples, and only seven demand points and seven facility sites are

considered as possible locations. Also they allow only four local facilities to be opened due to resource

limitation (i.e., = ), and the reduction of facility capability caused by an emergency is ignored. (i.e.,

= )

2.3.1. A Dirty Bomb Attack. (Covering Problem)

The first emergency is a dirty bomb terrorist attack, which could have a severe local impact. In this case,

a proactive facility allocation is appropriate since EMS supplies such as preventive equipment and

anti-radioactive drugs are stored at certain facilities. The purpose of this strategy is to cover as much of

the population as possible with the required facility quantity and quality. In this aspect, covering model

is used to formulate this facility location problem. To evaluate parameters in this problem, the paper

provides two data tables.

Table 1. Roadway distances between demand points and eligible facility sites

Demand

Facility

West Hollywood

Downtown LAX

airport Port of LA

Port of Long Beach Disneyland

Rowland Heights

Site 1 5 4 10 25 27 33 33

Site 2 11 5 5 14 12 16 24

Site 3 4 5 10 31 30 33 35

Site 4 27 13 20 33 27 16 10

Site 5 28 18 10 4 4 20 36

Site 6 20 12 7 7 4 14 32

Site 7 30 20 17 12 8 8 27

Table 2. Demand point characterization for dirty bomb emergency

Demand Point (𝐼)

Population ( )

Occurrence Likelihood

( )

Impact Coefficient

( )

Weight ( )

Req. Quantity

( )

Req. Fac. Distance ( )

WH 76k high (0.7) 0.7 37.2 2 9 miles

DT 94k high (0.85) 0.8 64 3 8 miles

LAX 56k high (0.9) 0.9 45.4 3 10 miles

PLA 32k high (0.9) 0.8 23 2 10 miles

PLB 28k high (0.9) 0.8 20.2 2 12 miles

DL 34k med (0.5) 0.5 8.5 1 15 miles

RH 8k low (0.3) 0.3 0.72 1 15 miles

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2.3.2. An Anthrax Terrorist Attack (P-center Problem)

The second emergency is an anthrax terrorist attack. In this example, a reactive facility deployment

strategy is considered since the appropriate supplies need to be allocated to local facilities after

investigation of the specific anthrax. Another important feature of anthrax is that terrorists possibly

send infectious materials to multiple sites to increase the threats, which is difficult to predict where the

material will be sent. Therefore, to avoid the worst case, P-center model is appropriate for this example.

Similar to the preceding example, parameters that represent the attribute of the demand points are

given in table 1 and 3 below. Note that in both P-center and P-median problems, a facility quality

requirement at each demand point is not assigned because every demand point is allocated to the

nearest required number of facilities.

Table 3. Demand point characterization for anthrax emergency

Demand Point (𝐼)

Population ( )


( )

Impact Coefficient

( )

Weight ( )

Req. Quantity

( )

WH 76k high (0.8) 0.6 36.4 2

DT 94k high (0.85) 0.6 48.0 3

LAX 56k high (0.8) 0.7 31.4 2

PLA 32k high (0.4) 0.3 3.8 1

PLB 28k high (0.4) 0.3 3.4 1

DL 34k med (0.6) 0.5 10.2 1

RH 8k low (0.3) 0.3 0.72 1

2.3.3. A Smallpox Terrorist Attack (P-median problem)

The last emergency case in the paper is a terrorist attack using smallpox. This emergency is different

from the previous anthrax attack when considering the fact that the smallpox disease can spread much

faster than anthrax. Therefore, mass vaccination is needed due to the large number of first responders,

and the facilities have to be located both for storing the medical supplies at a local level and for

receiving and distributing the supplies from the federal government. This means that a hybrid facility

deployment strategy is suitable for this problem. By concentrating on the distribution of the supplies

from the federal government, the P-median model is used to minimize the total distance between all

the demand points and their service facilities. Table 4 shows the attributes and the required facility

quantity for each demand point. Note that the occurrence likelihood and the impact coefficient for all

demand points are set to 1 since all the population at each demand point must be served.

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Table 4. Demand point characterization for smallpox emergency

Demand Point (𝐼)

Population ( )


( )

Impact Coefficient

( )

Weight ( )

Req. Quantity

( )

WH 76k 1 1 76 3

DT 94k 1 1 94 4

LAX 56k 1 1 56 3

PLA 32k 1 1 32 2

PLB 28k 1 1 28 2

DL 34k 1 1 34 2

RH 8k 1 1 8 1

3. Computational Implementation and Solution comparison

In the paper, the authors solve the problem using AMPL/CPLEX. And then they compare solutions based

on the proposed model and the traditional model, which shows the benefits of the proposed model in

optimizing the objective values during the large-scale emergencies.

To verify the benefits of the proposed model, we implement both the traditional models and the

provided models for solutions in a solver. For better comparison with our result from the result in the

paper, we also use AMPL/CPLEX to find the optimal solutions to the problems.

3.1 Computational Implementation

3.1.1. A Dirty Bomb Attack (Covering Problem)

Based on those input parameters in the table 1 and 2, the covering model proposed in 2.2.2.1 could be

applied to determine the optimal facility location for medical supply storage. For the traditional covering

model, we use the model provided by Toregas et al (1971). The formulation of the traditional model is as

follow:

∑

∑ 𝐼

{ } 𝐽

AMPL/CPLEX code for this problem is as follow:

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Figure 1. Input data for the traditional first coverage covering model

Figure 2. The traditional first coverage covering model

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Figure 3. Input data for the traditional multiple coverage covering model

Figure 4. The traditional multiple coverage covering model

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Figure 5. Input data for the proposed multiple coverage covering model

Figure 6. The proposed multiple coverage covering model

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Figure 7. Input data for the proposed first coverage covering model

Figure 8. Input data for the proposed first coverage covering model

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3.1.2. An Anthrax Terrorist Attack (P-Center Problem)

In this problem, we use parameters given in the table 1 and 3. For the traditional covering model, we

use the model provided by Garfinkel et al. (1977) with slight modification. The formulation we use for

the traditional model is as follow:

∑ =

𝐼 𝐽

∑ 𝐼

= ∑ 𝑑 𝐼

𝐼 𝑘

{ } 𝐼 𝐽

𝑡ℎ 𝑎𝑣 𝑟𝑎𝑔 𝑑 𝑠𝑡𝑎𝑛𝑐 𝑓𝑟𝑜𝑚 𝑎 𝑑 𝑚𝑎𝑑 𝑜 𝑛𝑡 𝑡𝑜 𝑓𝑎𝑐 𝑙 𝑠 𝑡ℎ𝑎𝑡 𝑠 𝑟𝑣 AMPL/CPLEX code for this problem is as follow:

Figure 9. Input data for the traditional first coverage p-center model

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Figure 10. The traditional first coverage p-center model

Figure 11. . Input data for the traditional multiple coverage p-center model

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Figure 12. . The traditional multiple coverage p-center model

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Figure 13. . Input data for the provided multiple coverage p-center model

Figure 14. . The provided multiple coverage p-center model

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Figure 15. Input data for the provided first coverage p-center model

Figure 16. The provided first coverage p-center model

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3.1.3. A Smallpox Terrorist Attack (P-Median Problem)

Parameters for this emergency case are given in the table 1, and 4. For the traditional model, we use the

model provided by ReVelle and Swain (1970). To properly implement the traditional model in a solver,

the traditional model is used with slight modification.

∑ 𝑑

∑ =

∑ = 𝐼

𝐼 𝐽

{ } 𝐼 𝐽

AMPL/CPLEX code for this problem is as follow:

Figure 17. Input data for the traditional first coverage p-median model

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Figure 18. The traditional first coverage p-median model

Figure 19. Input data for the traditional multiple coverage p-median model

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Figure 20. The traditional multiple coverage p-median model

Figure 21. Input data for the proposed multiple coverage p-median model

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Figure 22. The proposed multiple coverage p-median model

Figure 23. Input data for the proposed first coverage p-median model

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Figure 24. The proposed first coverage p-median model

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3.2 Solution Comparison

3.2.1. A Dirty Bomb Attack (Covering Problem)

The optimal result based on the traditional covering model suggests locating only 3 facilities at sites 1, 4,

and 6. Considering that such a Large-scale Emergency require redundant placement of facilities, we

arbitrarily locate one more facility at site 7. Then, the located facilities can cover 100% population and

100% weighted demand as well. However, when multiple coverage constraints are considered, the

suggested solution gives only 21.3% coverage of population and 14.8% coverage of weighted demand.

The reason for significant decrease in multiple coverage case is because the solution provided for first

coverage problem cannot satisfy the required facility quantity at demand points except “Port of Long

Beach”, “Disneyland”, and “Rowland Heights”.

For the proposed model, the optimal solution is given as site 1, 2, 3, and 6 to maximize the coverage of

weighted demand with the objective value of 175.18. Since the total weighted demand is 199.02, the

solution gives the 88.0% (175.18/199.02) coverage of weighted demand. Instead of weighted demand,

when we consider coverage of population with the given solution, it gives 87.8% coverage of population.

For first coverage case, the solution based on the proposed model provides 99.6% (198.22/199.02)

coverage of weighted demand and provides 97.6% coverage of population, because the solution gives

no facility that can serve “Rowland Heights”

Table 5 Solution comparison for covering

Models/ Solutions

Emergencies

Traditional model Proposed model

Solution (Site selection)

Objective Value Solution

(Site selection)

Objective Value

First coverage

Multiple coverage

First coverage

Multiple coverage

Dirty Bomb emergency (Covering)

1, 4, 6, and 7

Covered Weighted Demand:

100%


14.8% 1, 2, 3, and 6


99.6%


88.0%

Covered population:

100%

Covered population:

21.3%

Covered population:

97.6%

Covered population:

87.8%

From the comparison of the traditional model and the provided model, we find that the solution based

on proposed model provides a significant improvement in the amount of the population covered with

the required quantity of facilities while incurring only a minor less amount of first coverage.

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3.2.2. An Anthrax Terrorist Attack (P-Center Problem)

When we implement the traditional model for the first coverage case, the model suggests locating

facilities at 1, 2, 5, and 7, which slightly differ from the solution (1, 2, 5, and 6) provided in the paper.

However, both solutions have the same objective value as 191.76. For the multiple coverage case, the

suggested solutions give the objective values of 431.46, and 335.58 respectively which are more than 75%

larger than the first coverage distance. The reason for this significant increase is that the given solution

allocates the largest demand point “Downtown” to a facility which is located far away from

“Downtown”.

For the proposed model, the optimal solution is given as 1, 2, 3, and 6 with the objective value of 223.72

while the optimal solution in the paper is 1, 2, 3, and 7 with the objective value of 235.2. When the first

coverage case is considered with the given solution, the objective value is the same as the solution of

traditional model (191.76).

Table 6 Solution comparison for p-center

Models/ Solutions

Emergencies




(Site selection)

Objective Value

First coverage

Multiple coverage

First coverage

Multiple coverage

Anthrax emergency (P-center)

1, 2, 5, and 6

Or

1, 2, 5, and 7

Maximal Weighted Distance:

191.76


335.58 1, 2, 3, and 6


191.76


223.72 Maximal

Weighted Distance:

191.76


431.46

This implies that the proposed model gives better service quality other than the traditional model, which

results the less life and economic loss in such an emergency.

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3.2.3. A Smallpox Terrorist Attack (P-Median Problem)

The traditional model recommends placing facilities at site 1, 2, 5, and 7 with the objective value 1740

miles for the first coverage problem. For the multiple coverage problem, the traditional model gives the

objective value of 11,018 miles based on the given solution. On the other hand, the provided model

suggests the optimal solution as site 1, 2, 3, and 6 with objective value 7,528. Based on the given

solution 1, 2, 3, and 6 the proposed model obtains 1,964 miles as the objective value for the first

coverage problem. Notice that this result differs from the objective value (2,040 miles) in the paper.

When using the traditional model to locate the facilities, we only consider the first coverage case.

Therefore, the sites 1, and 2 are selected since they are relatively close to the largest-weighted demand

points, “West Hollywood”, “Downtown”, and “LAX airport”. Similarly, the site 5 and 7 are selected for

“Port of LA”, “Port Long beach”, and “Disneyland” which have the largest weight among the remaining.

Since the weight of “Rowland Heights” has only small contribution to the objective value, the distance

between “Rowland Heights” and a facility is less considered.

Unlike the traditional model, the proposed model in the paper considers the multiple coverage problem.

The model suggests 1, 2 and 3 to be selected since those facilities are close to the largest weighted

demand points satisfying the number of facilities they require. Note that distances from the sites 6 to

the demand points are relatively short compare to remaining facility sites. Therefore, the model reduces

the objective value by selecting site 6 instead of the others.

Table 7 Solution comparison for p-median

Models/ Solutions

Emergencies




(Site selection)

Objective Value

First coverage

Multiple coverage

First coverage

Multiple coverage

Smallpox emergency (P-median)

1, 2, 5, and 7

Weighted total

distance: 1740 miles

Weighted total


1, 2, 3, and 6

Weighted total


Weighted total


From the comparison of the solutions of the two models, we verify the benefit of the proposed model.

Even though the traditional model suggests somewhat better objective value in the first coverage case,

the proposed model has much advantage of reducing the weighted total distance by 46.4% when the

multiple facility coverage requirement is considered.

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4. Conclusion

The primary goal of this project is to implement the model provided by a research paper, “A Modeling

Framework for Facility Location of Medical Services for Large-scale Emergencies”, in a solver. To that

end, we have solved three different emergency examples by implementing the traditional models and

the proposed models using AMPL/CPLEX.

While conducting this project, there were some difficulties. Since the reference in the paper does not

provide specific formulas for the computational implementation, we, in some cases, have formulated

the models based on what we have learned in the class. But, the results were very similar to the results

in the paper.

As a result, we verified the advantages of the proposed model in consideration of the multiple coverage

cases. For the first coverage case, there was only a slight disadvantage of the proposed model compared

to the traditional one. Therefore, it turned out to be reasonable to make a decision to locate facilities

based on the proposed model overall. In addition, we have learned how location problems can be

applied to the reality and have also obtained knowledge about “Maximal covering model” which is not

covered in the class.

5. Reference

[1] H. Jia, F. Ordonez, and M. Dessouky, A Modeling Framework for Facility Location of Medical Services

for Large-Scale Emergencies, IIE Transactions, 2007

[2] Toregas, Ralph Swain, Charles ReVelle and Lawrence Bergman, The Location of Emergency Service

Facilities, INFORMS, 1971

[3] Garfinkel, R. S., Neebe, A. W. and Rao, M. R., The m-center problem: Minimax facility location.

Management Science, 1977

[4] ReVelle, C. and Swain, R.W., Central facilities location. Geographical Analysis, 1970

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