Need for Health Centers Where to Place Health . . . What We Know What We Want Fuzzy Approach to Optimal Main Limitation and . . . Placement of Health Centers Objective Function: . . . Exact Formulation of . . . Juan Carlos Figueroa Garcia 1 , Carlos Franco 2 , and How Many Doctors . . . Vladik Kreinovich 3 Where to Actually . . . Home Page 1 Department of Industrial Engineering Universidad Distrital, Bogot´ a D.C, Colombia, filthed@gmail.com Title Page 2 School of Management, Universidad del Rosario ◭◭ ◮◮ Bogot´ a, Colombia, carlosa.franco@urosario.edu.co 3 Department of Computer Science, University of Texas at El Paso ◭ ◮ El Paso, Texas 79968, USA, vladik@utep.edu Page 1 of 25 Go Back Full Screen Close Quit
Need for Health Centers Where to Place Health . . . 1. Need for Health Centers What We Know • Many countries in the world have socialized medicine What We Want – in this sense, US is one of the few exceptions. Main Limitation and . . . Objective Function: . . . • In such countries, it is important to decide how to dis- Exact Formulation of . . . tribute the limited resources. How Many Doctors . . . • The objective is to best serve the population. Where to Actually . . . Home Page • In some case, all the patient needs is a regular general doctor; however, in many other cases: Title Page – the patient also needs to undergo some tests – blood ◭◭ ◮◮ test, X-ray, etc., ◭ ◮ – he/she may need to see a specialist, etc. Page 2 of 25 • It is more convenient for the patients if all the need Go Back medical professionals are placed at a single location. Full Screen • This is the main idea behind health centers. Close Quit
Need for Health Centers Where to Place Health . . . 2. Where to Place Health Centers? What We Know • Where are the best locations for these centers? What We Want Main Limitation and . . . • And, once we find these locations, what is the best way Objective Function: . . . to assign each patient to one of these centers? Exact Formulation of . . . • These are the problems that were raised in our previous How Many Doctors . . . paper. Where to Actually . . . Home Page • These are the problems that we deal with in this paper as well. Title Page ◭◭ ◮◮ ◭ ◮ Page 3 of 25 Go Back Full Screen Close Quit
Need for Health Centers Where to Place Health . . . 3. What We Know What We Know • Let X denote the area that we want to serve with What We Want health centers. Main Limitation and . . . Objective Function: . . . • Let ρ ( x ) denote the population density at geographic Exact Formulation of . . . location x , i.e., the number of people per unit area. How Many Doctors . . . • Once we know the population density, we can compute Where to Actually . . . the overall number of people in any given area A as Home Page � Title Page ρ ( x ) dx. A ◭◭ ◮◮ � • Let P = X ρ ( x ) dx denote the overall number of people ◭ ◮ in our area X . Page 4 of 25 Go Back Full Screen Close Quit
Need for Health Centers Where to Place Health . . . 4. What We Want What We Know • We need to decide how many health centers to place What We Want at different locations. Main Limitation and . . . Objective Function: . . . • Let h ( x ) denote the number of health centers per unit Exact Formulation of . . . area in the vicinity of a geographical location x . How Many Doctors . . . • Once we determine this density h ( x ), we can compute Where to Actually . . . the overall number of health centers in area A as Home Page � h ( x ) dx. Title Page A ◭◭ ◮◮ ◭ ◮ Page 5 of 25 Go Back Full Screen Close Quit
Need for Health Centers Where to Place Health . . . 5. Main Limitation and Objective Function What We Know • Our resources are limited: we can only build so many What We Want health centers. Main Limitation and . . . Objective Function: . . . • Let N denote the overall number of health centers that Exact Formulation of . . . � we can build, then X h ( x ) dx = N. How Many Doctors . . . • In the ideal world, every patient should be immediately Where to Actually . . . seen by a doctor. Home Page • In reality, it takes some time for a patient to reach the Title Page nearest health center. ◭◭ ◮◮ • The smaller this time, the better. ◭ ◮ • Thus, a reasonable objective function is the average Page 6 of 25 time that it takes for a patient to reach a doctor. Go Back • Let us describe this objective function in precise terms. Full Screen Close Quit
Need for Health Centers Where to Place Health . . . 6. Objective Function: Towards a Formal Descrip- What We Know tion What We Want • The time t ( x ) that it takes for a patient at location x Main Limitation and . . . to reach the nearest health center can be computed as Objective Function: . . . Exact Formulation of . . . t ( x ) = d ( x ) v ( x ) , where: How Many Doctors . . . Where to Actually . . . • d ( x ) is the distance from location x to the nearest Home Page health center, and Title Page • v ( x ) is the average transportation speed in the vicin- ◭◭ ◮◮ ity of the location x . ◭ ◮ • The speed v ( x ) is usually: Page 7 of 25 – smaller in the city center, Go Back – slightly larger in the suburbs, and Full Screen – even larger outside the city limits. Close Quit
Need for Health Centers Where to Place Health . . . 7. Objective Function (cont-d) What We Know • Let m ( x ) denote the maximum distance m ( x ) that it What We Want takes for points around x to reach a doctor. Main Limitation and . . . Objective Function: . . . • This distance corresponds to the case when the location Exact Formulation of . . . is at the edge of the zone allocated to this center. How Many Doctors . . . • So, it is attained at the edge of a disk of radius m ( x ) Where to Actually . . . served by this center. Home Page • In this circle, there is exactly one health center. Title Page • Based on the density h ( x ) of health centers, we can ◭◭ ◮◮ estimate the number of health centers in this disk area: ◭ ◮ � h ( x ) dx ≈ h ( x ) · ( π · m ( x ) 2 ) . Page 8 of 25 Go Back • We know that this value is 1, since there is only one Full Screen health center in this disk area. Close Quit
Need for Health Centers Where to Place Health . . . 8. Objective Function (cont-d) What We Know • So, we conclude that h ( x ) · ( π · m ( x ) 2 ) = 1, i.e., that What We Want Main Limitation and . . . 1 m ( x ) = . Objective Function: . . . � π · h ( x ) Exact Formulation of . . . • What is the average distance d ( x ) from a center of the How Many Doctors . . . disk of radius m ( x ) to a point on this disk? Where to Actually . . . Home Page • For all the points at distance r from the center, this distance is r . Title Page • The area of the small vicinity of this disk is 2 π · t dr . ◭◭ ◮◮ • Thus, the average distance can be computed as ◭ ◮ � m ( x ) Page 9 of 25 1 π · ( m ( x )) 2 · 2 1 3 · π · ( m ( x )) 2 = r · (2 π · r dr ) = π · ( m ( x )) 2 · Go Back 0 2 2 1 Full Screen 3 · m ( x ), so d ( x ) = . 3 · √ π · � h ( x ) Close Quit
Need for Health Centers Where to Place Health . . . 9. Objective Function (cont-d) What We Know • So, t ( x ) = d ( x ) 2 1 What We Want v ( x ) = . 3 · √ π · � h ( x ) · v ( x ) Main Limitation and . . . Objective Function: . . . • This is the time that it takes for each patient to reach Exact Formulation of . . . the health center. How Many Doctors . . . • The average time that it takes all the patients to reach Where to Actually . . . the health center can be then computed as Home Page 1 2 ρ ( x ) � � Title Page ρ ( x ) · t ( x ) dx = dx. 3 · √ π · P · P · � h ( x ) · v ( x ) X X ◭◭ ◮◮ • Now, we are ready to formulate the problem in precise ◭ ◮ terms. Page 10 of 25 Go Back Full Screen Close Quit
Need for Health Centers Where to Place Health . . . 10. Exact Formulation of the Problem and Its So- What We Know lution What We Want • We know the functions ρ ( x ) and v ( x ). Main Limitation and . . . Objective Function: . . . • Based on this knowledge, we need to find the function � Exact Formulation of . . . h ( x ) that, under constraint h ( x ) dx = P , minimizes How Many Doctors . . . 2 ρ ( x ) � dx. 3 · √ π · P · Where to Actually . . . � h ( x ) · v ( x ) X Home Page • Multiplying all the value of the objective function by Title Page the same constant does not change which value is larger. ◭◭ ◮◮ • Thus, minimizing the above objective function is equiv- ◭ ◮ alent to minimizing a simpler expression Page 11 of 25 ρ ( x ) � . � Go Back h ( x ) · v ( x ) X Full Screen • To solve this problem, we can use the Lagrange multi- plier method. Close Quit
Recommend
More recommend
