TIPO DE INSTRUMENTO Nº (porcentaje)
6.1. EL INSTITUTO DE GRANADA. CREACIÓN Y DESARROLLO HISTÓRICO
1. First the spatial datasets are prepared in GIS to be integrated into the ABM. These datasets are:
a) the administrative boundary of the region where HCFs are to be placed b) the locations of existing HCF
c) the no-coverage feature obtained from the accessibility measure d) the population density map
e) point features of schools, markets and religious facilities
f) water body dataset such as rivers, lakes, lagoon, ocean and swamp that are forbidden locations for HCFs
g) boundary map of committed zones that are also forbidden locations for HCFs which may be airports, stadiums, or military cantonments
2. The second step is to prepare the model. Since NetLogo does not relate with spatial data directly, all the spatial data will be converted to agents - turtles, links and patches. The turtles are point features, links are line features, and patches are area features. The projection and transformation data is uploaded to the ABM so that all analyses will be in the same coordinate system. In the ABM:
a) The GIS features will be re-drawn in the model so that agents can be created from them.
b) Each agent will adopt the properties of the spatial data it represents.
c) The patches that intersect the demographic feature will adopt the popula-tion density value of the grid that intersects it. This is achieved using the
"gis:apply-coverage" primitive.
d) The area property value of the no-coverage polygon feature adopted by the ABM is divided by the total number of patches contained in the no-coverage feature. This gives the area of a square patch. The population per patch is therefore the population density multiplied by the patch area.
e) The intersection of different area features with patches are assigned different colours to distinguish them from other area features. This colour becomes the attribute of the patch that is unique to the feature it represents.
f) Where any forbidden zone intersects a patch, the population value within that patch is set to zero to indicate that no population resides there.
g) Population in a set of patches is the total population of each patch that repre-sents the area of interest.
3.8 Large Neighbourhood Search for Location-Allocation
Several heuristics or meta-heuristics algorithms exist to solve covering problems, such as Tabu Search, Simulation Annealing (SA), Genetic Algorithm (GA), and Greedy
O. Olowofoyeku 3.8. LARGE NEIGHBOURHOOD SEARCH FOR LOCATION-ALLOCATION
Randomized Adaptive Search Procedure (GRASP) (De Smith et al., 2018). Although such algorithms may or may not proffer globally optimal solutions, they are simple and easy to implement and can be used to quickly obtain a feasible solution for the cover-ing problem. The Large Neighbourhood Search (LNS) meta-heuristic (described in Section 2.7) has been shown to find near optimal and sometimes optimal solutions to optimization problems very quickly and it is simple to implement. It also has the ad-vantage of easily recovering from a poor initial solution (Bruglieri et al., 2018; Demir et al., 2012). The LNS meta-heuristics offer a local search that makes it possible to choose a locally optimum solution at each step of the optimization which may eventu-ally lead to a globeventu-ally optimum solution. This algorithm is therefore employed for this research.
An approximate solution is first found and further improvements are made with local search in the search space of the neighbourhood by using a destroy and repair technique. The search space is the space of all possible solutions for the specific prob-lem, and the neighbourhood space is the available space after a relocation of a current solution. For this research, search algorithm ensures:
1. full or partial coverage of HCFs within a desired service distance with MCLP 2. full coverage with PCLP within endogenous maximum distance
3. the population assigned to a HCF does not exceed the HCF’s threshold carrying capacity
4. budget constraints are considered by distinguishing HCFs into standard and low-cost facility
5. suitability conditions are met for locating HCFs.
The LNS algorithm takes an initial feasible solution S as input, and computes an initial best solution Sbin the neighbourhood of S through a destroy and repair method.
Sb is returned as the current solution S. A temporary solution Stcan be the solution Sb or be discarded, depending on if it is accepted or not (Algorithm 2.1). Different solutions are obtained with either local search or greedy function. The local search explores the current solution to provide a better viable solution.
At initialization, the value of a number of facilities to cover the population that is not covered is first obtained from facility to population density. If a fraction is obtained, the value is rounded up to be the initial value. This value becomes the approximate required number of facilities that will provide coverage for the uncovered demand.
There locations will be randomly placed within the no-coverage boundary. The opti-mal or near-optiopti-mal location of potential facilities will be found based on a local search, that considers maximal distance that is expected to be between a facility and the pop-ulation, population within newly defined catchment, closeness of potential facility to one another, and closeness to amenities.
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Further elimination and addition strategies will be done by considering the num-ber of HCFs within the catchment of each HCF. The catchment will be an average distance computed from the bounding circles from the result of the initial catchment model in that zone. The population within the catchment will be calculated and com-pared with the number of agents from the HCF-population ratio. Additional HCF will be created within the catchment if required. Another suitability analysis is the HCF proximity to other amenities such as schools, religious facilities and markets, which are expected to be within the catchment. No facility is allowed within the catchment of an existing real-life facility. The objective is to locate a HCF on a suitable place and allocate a population to the HCF. Hence the name location-allocation.
A final solution is returned after all suitability analysis have been done. The out-put of the MCLP provides the percentage of the population that cannot reach a facility within the threshold distance or time. The relaxed form assigns the set of uncovered population to the closest facility which also includes existing facilities within the re-gion, by relaxing the maximum distance between demand and supply.
Three different sets of candidate facility locations that will be returned are those that meet all conditions, those that are without amenities in their catchment, and those whose population size is below the threshold carrying capacity.
The LNS algorithm is broken down to six phases for the sake of this research.
These are:
• Initialization phase
• Construction phase
• Destruction phase
• Repair phase
• Sorting phase
• Improvement and Stopping phase
These phases are described in the next sections. Figure 3.8 shows the flow chart of the phases.
3.8.1 Initialization
At initialization, the required number of facilities is obtained from a facility to popula-tion ratio. An estimate number of facility agents to be simulated can be the approximate required for the population or greater than the required. Let N “ t1, 2, 3, ..., nu be the set of initial facility locations for the under-served population, and M “ t1, 2, 3, ..., mu be the initial value of facility locations to be selected. An initial value of facility loca-tions can be selected to be m ą n or m “ n or m ă n. The number of HCF agents will
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Initialization:
determine initial feasible solu-tion with greedy random variable
Construction phase:
*separate facilities occupying the same geographic location
*regenerate additional facilities within radius
Destruction phase:
*elliminate facilities on location with no population
*elliminate facilities outside the geo-graphic boundary of the targe region
Repair phase:
*regenerate more facilities in catchment with population greater than capacity
Sorting phase:
*sort facilities to classes of meet all criteria;
requires amenities; low-cost establishment
*determine percentage of population not covered
Improvement phase:
*link demand not covered to closest facilities Figure 3.8: LNS Phases
eventually change using the principle of subtraction and addition to remove or add facilities as they meet certain criteria.
The population density map is imported into the ABM and the square patches adopt the number of population per size of the patch. The population will be calcu-lated by summing up the population in all patches within the no-coverage dataset that represents the under-served area, obtained from the results of the catchment model (Equation 3.30). The total population on all patches that intersect the no-coverage fea-ture are the uncovered demands. It is assumed that people will not reside in water and committed parcels such as airports and stadiums. Therefore, where such spatial feature intersects a patch, the population density property of the patch is assigned a
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zero value.
Population within a geographic area “
n
ÿ
i“n
P P (3.30)
Where:
P P = total population in a patch that intersects no-coverage feature n= number of patches
M agentset of the required number of facilities are created and randomly placed to occupy any location within the uncovered area. The required number of facilities will depend on the %coverage chosen by the user based on a budget constraint. To effectively manage and utilize available resources, policy makers can choose from dif-ferent alternatives to the coverage of healthcare services by manipulating the carrying capacity of the HCF. The optimal carrying capacity suggested by (WHO, 2013) is one HCF to 5,000 population. This is taken to be 100% setting in this research. The facility density can be increased to 7,500 giving a 75% setting for ratio 1/7,500, or 10,000 to give 50% setting for ratio 1/10,000. The ratio is a choice to be made by the decision maker or user as an opportunity to minimize the number of facilities to be built. This is a user-defined variable in the model.
This forms the initial feasible solution.
3.8.2 Construction Phase
In the construction phase, three processes are involved. The first process is to place the facilities created at initialization phase randomly at different locations to form the initial feasible solution.
At initial potential HCF placement at different locations, there are no conflicts of interest and HCF can be located anywhere without considering if any other agent is occupying the same location. Since the HCFs are randomly placed, there is a possibil-ity of some occupying the same location. If such situation occurs, one of them takes a decision to remain on the location and asks others to move until they each find a location that is not occupied by another HCF.
In the second process, the constraints of having not more than one facility in a location is enforced by separating any overlapping facilities. Each of the newly located facilities will examine its location to find out if there is any other facility sharing the same location with it. If this is so, this facility retains its position and the other facility or facilities are instructed to move to another location within the no-coverage area where no other facility is located. This forms a new feasible solution.
In the third process, if the configuration of facility placement is such that there is no other facility outside the catchment boundary of a HCF, within a neighbourhood
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of at most three-quarters of the catchment radius, it implies some demands will be outside health coverage. Further densification is done by creating additional facilities within the specified distance neighbourhood of the affected HCFs. The choice of three-quarters outside the catchment is to reduce catchment overlap while still ensuring that as much demand is covered as possible. This forms another new feasible solution.
The algorithm for this processes is shown in Algorithm 3.2.
Algorithm 3.2 LNS Algorithm Construction Phase
1: randomly place facilities S
2: for all facilities on j do
3: if there is another facility on same location then
4: tell others to relocate to another location without a facility
5: end if
6: end for
7: Sb “ S; make this initial feasible solution
8: for all facilities S do
9: create service area
10: if there is no facility in neighbourhood up to three-quarter of the catchment ra-dius then
11: regenerate another facility in that radius
12: Sb“ S;
13: end if
14: end for
15: return Sb
3.8.3 Destruction Phase
The destruction phase consists principally of removing facilities from the existing so-lution. Such agent takes a decision to leave the population of HCFs through a process called agent death. That is the agent ceases to exist. The removed set of facilities do not have to be stored in the memory, because agents can be created and re-inserted. The selection of the destruction operators is governed by suitability constraint variables.
At initial potential HCF placement, HCF can be located anywhere in the search space and the suitability of the location is not considered. The destruction is based upon the following considerations:
1. Outside boundary removal: The neighbourhood of the specified boundary is searched by each agent within the search space. For this removal operation, elimination is done for HCF from the initial solution that is not contained within the polygon boundary of the no-coverage dataset and the administrative boundary of the tar-get region. As HCFs are placed on the centre of patches that intersect the polygon feature, some agents may fall outside the expected boundary. An illustration is shown in Figure 3.9. When agents are created, they are placed at the centre of
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the patch. An agent may actually be on the patches (shaded) that intersect the polygon boundary, however it is outside the polygon boundary.
boundary
patches intersected by boundary layer healthcare facility agent
Figure 3.9: Patches and Agent location
2. Uninhibited location removal: In another destruction exercise, a suitability analy-sis is done by each HCF to evaluate if it is on a water body or on a committed land space such as airport, stadium and military cantonment regarded as not habitable. HCF that is on such uninhabited location is removed from the current solution.
3. Neighbourhood-based destruction: In this process, each facility checks within the neighbourhood of its catchment and instructs any other facility within that neigh-bourhood to cease to exist, including facilities outside a distance less than three-quarters of its catchment radius. This is to reduce catchment overlap. The search space is a circular neighbourhood that is defined by the average radius of the minimum bounding circle and is a relaxation on the network travel-distance.
An algorithm of the destruction procedure is presented in Algorithm 3.3. At ini-tialisation of this phase, a feasible solution of the immediate past procedure serves as input. Another improved solution is returned following a partial destruction.
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Algorithm 3.3 LNS Algorithm Destruction Phase
1: Violation of geographical location constraint
2: for all facilities S do
3: if your location is outside neighbourhood boundary then
4: destroy
5: end if
6: end for
7: Sb “ S;
8: for all facilities S do
9: if your location is on a location with no population then
10: destroy
In this phase the partially destroyed solution will be repaired by enforcing the demand constraints. New facilities are added to the existing list of facilities by agents "giving birth" to other agents. An agent can reproduce another agent that will inherit all of its parent’s properties, but a different identity. The number of new facilities to be cre-ated may not be the same number that was destroyed. This depends on the additional
"births" required to meet the objectives.
The first insertion will enforce capacity constraints in the main algorithm. Each facility will evaluate the population within its catchment using a summing up of the population on the patches inside its catchment. If the population is more than the ex-pected capacity and the excess population value is more than a predefined percentage, additional facilities will be generated inside that catchment. The newly created facil-ities will be randomly placed within the catchment of the parent facility. The target percentage is introduced to reduce cost. For example, if there are 8,000 population in the service area of a facility and the capacity is 5,000, one new facility will be created if the population is in excess by over 50%. This relaxation is introduced to minimize the number of under-served facilities. It is a variable factor, depending on the budget capacity.
The newly inserted facilities search their locational suitability. If there be any vi-olation of a geographical locational constraint, they move to another location, but still within the catchment of their parent facility.
A second repair is to identify the facilities whose population size is below the target value. These set of facilities are retained to serve their community, but will be termed low-cost facilities. They will be recommended to be established with reduced
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resource cost.
The repair phase therefore serves a cost-control objective. The repair algorithm is shown in Algorithm 3.4.
Algorithm 3.4 LNS Algorithm Repair Phase
1: Capacity constraint
2: for all facilities S do
3: calculate demand in your catchment
4: if demand is larger than your capcity then
5: regenerate (additional facilities “ required facility ´ 1) within your catchment
6: for all facilities generated do
7: if you are on same geographical location with another then
8: relocate within the catchment
9: else if your location is on location with no population then
10: relocate within the catchment
11: else
12: stay within the catchment
13: end if
14: end for
15: else if demand is below target then
16: form a list of low-cost facilities
17: else
18: all conditions are met by you
19: end if
20: end for
21: Sb “ S;
22: return Sb
3.8.5 Sorting Phase
This sorting phase sorts the selected facilities by the facilities with other institutional facilities in their catchments, facilities serving population below the threshold pop-ulation capacity, and facilities with no institutional facilities within their catchments.
It separates low-cost facilities, where the cost is defined as the cost of establishing a facility in relation to the population to be served by the facility.
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Algorithm 3.5 LNS Algorithm Sorting Phase
1: Sort facilities by close amenities and establishment cost
2: for all facilities S do
3: if you have no amenities within your catchment then
4: form a list of "no-amenities" facilities
5: end if
6: end for
7: for all facilities S do
8: if facility status is low-cost then
9: change state
10: end if
11: end for
12: Sb “ S;
13: for all facilities S do
14: if your location is on location with no population then
15: leave the solution
16: else
17: remain on your location
18: end if
19: end for
20: Sb “ S;
21: return Sb
22: calculate percentage with no-coverage
3.8.6 Improvement and Stopping Phase - PCLP
The capacity and distance constraints are relaxed in this phase. Uncovered populations are linked to any closest facilities to them. The percentage population uncovered will be calculated from:
UP% “ pU P
IP q ˚ 100% (3.31)
Where:
U P = total population not within the catchment of the new facilities IP = initial total population that intersects the no-coverage feature
U P = total population not within the catchment of the new facilities IP = initial total population that intersects the no-coverage feature