Computational Implementation of location problem models for medical services

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1 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

2 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. Page 1 of 25

3 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 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. Page 2 of 25

4 2.2 Modeling the formulation of LEMS 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. = { f f l y s pl d h rw s I I { } J I J J = { = { 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 Specified Model 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 demand point. Then, these requirements are represented by the constraints below. from = 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: Page 3 of 25

5 I { } I J 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 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. Page 4 of 25

6 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., = ) 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. Demand Facility Table 1. Roadway distances between demand points and eligible facility sites West Hollywood Downtown LAX airport Port of LA Page 5 of 25 Port of Long Beach Disneyland Rowland Heights Site Site Site Site Site Site Site Demand Point (I) Table 2. Demand point characterization for dirty bomb emergency Population Occurrence Likelihood Impact Coefficient Weight Req. Quantity WH 76k high (0.7) miles DT 94k high (0.85) miles LAX 56k high (0.9) miles PLA 32k high (0.9) miles PLB 28k high (0.9) miles DL 34k med (0.5) miles RH 8k low (0.3) miles Req. Fac. Distance

7 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. Demand Point (I) Table 3. Demand point characterization for anthrax emergency Population Occurrence Likelihood Impact Coefficient Weight Req. Quantity WH 76k high (0.8) DT 94k high (0.85) LAX 56k high (0.8) PLA 32k high (0.4) PLB 28k high (0.4) DL 34k med (0.6) RH 8k low (0.3) 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. Page 6 of 25

8 Demand Point (I) Table 4. Demand point characterization for smallpox emergency Population Occurrence Likelihood Impact Coefficient Weight Req. Quantity WH 76k DT 94k LAX 56k PLA 32k PLB 28k DL 34k RH 8k 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 A Dirty Bomb Attack (Covering Problem) Based on those input parameters in the table 1 and 2, the covering model proposed in 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: Page 7 of 25

9 Figure 1. Input data for the traditional first coverage covering model Figure 2. The traditional first coverage covering model Page 8 of 25

10 Figure 3. Input data for the traditional multiple coverage covering model Figure 4. The traditional multiple coverage covering model Page 9 of 25

11 Figure 5. Input data for the proposed multiple coverage covering model Figure 6. The proposed multiple coverage covering model Page 10 of 25

12 Figure 7. Input data for the proposed first coverage covering model Figure 8. Input data for the proposed first coverage covering model Page 11 of 25

13 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 Page 12 of 25

14 Figure 10. The traditional first coverage p-center model Figure 11.. Input data for the traditional multiple coverage p-center model Page 13 of 25

15 Figure 12.. The traditional multiple coverage p-center model Page 14 of 25

16 Figure 13.. Input data for the provided multiple coverage p-center model Figure 14.. The provided multiple coverage p-center model Page 15 of 25

17 Figure 15. Input data for the provided first coverage p-center model Figure 16. The provided first coverage p-center model Page 16 of 25

18 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 Page 17 of 25

19 Figure 18. The traditional first coverage p-median model Figure 19. Input data for the traditional multiple coverage p-median model Page 18 of 25

20 Figure 20. The traditional multiple coverage p-median model Figure 21. Input data for the proposed multiple coverage p-median model Page 19 of 25

21 Figure 22. The proposed multiple coverage p-median model Figure 23. Input data for the proposed first coverage p-median model Page 20 of 25

22 Figure 24. The proposed first coverage p-median model Page 21 of 25

23 3.2 Solution Comparison 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 Since the total weighted demand is , 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 Dirty Bomb emergency (Covering) Solution (Site selection) 1, 4, 6, and 7 Traditional model Objective Value First Multiple coverage coverage Covered Covered Weighted Weighted Demand: Demand: 100% Covered population: 100% Solution (Site selection) 14.8% 1, 2, 3, and 6 Covered population: 21.3% Proposed model Objective Value First Multiple coverage coverage Covered Covered Weighted Weighted Demand: Demand: 99.6% 88.0% 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. Page 22 of 25

24 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 For the multiple coverage case, the suggested solutions give the objective values of , and 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 while the optimal solution in the paper is 1, 2, 3, and 7 with the objective value of 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 Anthrax emergency (P-center) Solution (Site selection) 1, 2, 5, and 6 Or 1, 2, 5, and 7 Traditional model Objective Value First Multiple coverage coverage Maximal Maximal Weighted Weighted Distance: Distance: Maximal Weighted Distance: Maximal Weighted Distance: Solution (Site selection) 1, 2, 3, and 6 Proposed model Objective Value First coverage Maximal Weighted Distance: Multiple coverage Maximal Weighted Distance: 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. Page 23 of 25

25 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 Smallpox emergency (P-median) Solution (Site selection) 1, 2, 5, and 7 Traditional model Objective Value First coverage Weighted total distance: 1740 miles Multiple coverage Weighted total distance: miles Solution (Site selection) 1, 2, 3, and 6 Proposed model Objective Value First coverage Weighted total distance: 1964 miles Multiple coverage Weighted total distance: 7528 miles 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. Page 24 of 25

26 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 Page 25 of 25

Rulemaking Hearing Rules of the Tennessee Department of Health Bureau of Health Licensure and Regulation Division of Emergency Medical Services

Rulemaking Hearing Rules of the Tennessee Department of Health Bureau of Health Licensure and Regulation Division of Emergency Medical Services Chapter 1200-12-01 General Rules Amendments of Rules Subparagraph