LA ENSEÑANZA REAL

Table 4.3: Equivalence Test with Paired Data: Spatial-ABM travel time, Google Maps travel time

Method:

Test mean = mean of Spatial-ABM travel time Reference mean = mean of Google Maps travel time

Descriptive Statistics

Variable N Mean StDev

Modeltime 250 31.178 1.160

Googletime 250 31.518 2.865

Mean Difference

95% CI for

Difference StDev SE Equivalence

-0.340 2.799 0.177 (-0.633, 0)

Test

Null hypothesis: Difference ď -5 or Difference ě 5 Alternative hypothesis: -5 < Difference < 5

Alpha (α) level: 0.05

Null Hypothesis DF T-Value P-Value Difference ď -5 249 26.317 <10^-5

Difference ě 5 249 -30.166 <10^-5

With the results of these tests, it is confidently confirmed that the travel-time model is robust, consistent and reliable.

4.4 Spatial-ABM Optimization Model Verification

The verification on the optimization model involved:

• Parameter variability and comparison of results of several replications performed on the 55-node dataset and Lagos State dataset. As described in Section 3.8, the LNS spatial optimization algorithm returns a final set of endogenously selected facilities that is preceded by an initial feasible solution of the number of facilities.

With these tests, the clear understanding of the effect that the initial p value has on the convergence of the model results can be revealed, in terms of: the number of facilities selected and the consistency of convergence.

O. Olowofoyeku 4.4. SPATIAL-ABM OPTIMIZATION MODEL VERIFICATION

4.4.1 Initial Number of Facilities Variation on 55-Node Dataset

At initialization, the 55 nodes were fixed potential facility locations. Additional poten-tial HCFs were simulated and placed randomly in the environment so that the location of HCFs would not be restricted to the nodes. The number of facilities added were var-ied to verify the effect it has on the number of selected facilities, p. The candidate p-facility simulated were 50, 200, 500, and 600 in addition to the 55 fixed facilities. Us-ing a service distance D “ 11 and mandatory closeness distance S “ 15, 200 model simulation runs were performed for each set of initialization to:

1. endogenously determine the number of facilities that will ensure total coverage of health-care within D or S distance units

2. the consistency of the number of facilities selected 3. the effect of initial feasible solutions on the model output

(The node locations in the 55-node dataset are defined by latitude and longitude geographical coordinates, therefore no distance unit is considered in the literature.)

The results show that for each initial p value in the 200 model runs, the selected number of facilities are between the range of five and eight as shown in the line plots in Figure 4.3a. The results indicate that:

• Maximum number of facilities selected from the different initialization input was 8, and the minimum was 5. The highest value was from the highest number facilities input - 600.

• The median for all sets is 6.0 and all means range between 6.2 and 6.3.

• 50% of the output values for all initialization is between 6 and 7.

The model is therefore consistent and robust. Although, the outputs absolutely agree, the more the number of initial facilities, the longer it takes for the model to converge.

O. Olowofoyeku 4.4. SPATIAL-ABM OPTIMIZATION MODEL VERIFICATION

(a) selected number of facilities from varied initial input values

(b) average selected number of facilities from varying initial simulated can-didate facilities

Figure 4.3: 55-node dataset: selected number of facilities from varying initial candidate facili-ties

4.4.2 Initial Number of Facilities Variation on Lagos State Dataset

The same initial facilities variation test performed on the 55-node dataset was also car-ried out on the Lagos State dataset. This was necessitated by the rich geographic fea-tures that will enhance the considerations of additional criteria for solving the facility location problem. For example, facility location on water or an uninhabited zone is prohibitive, however, such spatial data is not available for the 55-node dataset. The population in the Lagos State dataset is also not represented on nodes, but as popula-tion density within the gridded plane. The initial value of p is the value obtained by dividing the population by the expected capacity of a facility, which for this research is 5,000. The number of facilities as initial feasible solution for the model had four variations by increasing the calculated value, p by 10%, up to 30% as: p, pp ` 10%pq, pp ` 20%pq, and pp ` 30%pq.

O. Olowofoyeku 4.4. SPATIAL-ABM OPTIMIZATION MODEL VERIFICATION

For example, if the calculated required value for p is 20, the initial number of facil-ities for the experiments will be varied as 20, 22, 24, 26 and each set will have 200 runs.

For this experiment, the required number of HCFs for the population was 29.

The line plots are shown in Figure 4.4. The set of runs that is initialized with the actual required number of facilities is designated 1. 2 is the set initialized with 10% of the required number of facilities, while 3 and 4 are initialized with 20% and 30% of the required number of facilities respectively.

The results show that the selected number of HCFs are between the range of 22 and 40 as shown in the line plots in Figure 4.4a. The box plots in Figure 4.4b shows that:

• for initialization with required p, minimum output of number of selected facilities is 22 and maximum is 35 with median value of 29.

• for initialization with additional 10% p, minimum output is 24 and maximum is 39 with median value of 30.

• for initialization with additional 20% p, minimum output is 23 and maximum is 40 with median value of 31.

• for initialization with additional 30% p, minimum output is 25 and maximum is 40 with median value of 32.

• The median of each set increases as the input value increases.

• 50% of the output values for initialization with additional 30% p is higher than the required number of facilities, p.

O. Olowofoyeku 4.4. SPATIAL-ABM OPTIMIZATION MODEL VERIFICATION

(a) selected facilities from varied initial value required for population

(b) average selected facilities from varied initial value required for population

Figure 4.4: Lagos State dataset: selected number of facilities from varying initial candidate facilities

The results are similar and consistent. However, in addition to the observations from experimenting with the 55-node dataset, the median value tends to be away from the required value of 29 (obtained from facility-population ratio) as the initial value increases.

4.4.3 Statistical Robustness of Coverage

The Maximal Covering Location Problem (MCLP) model covers a percentage of the population with a selected number of facilities within a specified service distance, D and returns the percentage of population that is not covered. The behaviour of the agents in this model was statistically tested with repeated simulations using four

re-O. Olowofoyeku 4.4. SPATIAL-ABM OPTIMIZATION MODEL VERIFICATION

gions in Lagos State - Badagri, Ikeja, Etiosa and Ibeju-Lekki, representing north, east, south and western parts of the state. For each region, four independent 100 simulations were carried out to produce 400 outputs and their results were statistically analysed. A box plot of each set was drawn to compare the output of percentage of population that is not covered (designated uncovered population in Figure 4.5) in the area of focus.

Figure 4.5: Coverage verification analysis

Having applied the covering model to different datasets with repeated runs, the results indicate that the model is robust and consistent in the selected number of HCFs and the population coverage.

The following observations were made from the experiments:

• Regardless of the value at initialization, the LNS algorithm is capable of recover-ing to produce expected outcomes.

• The initial value of p did not affect the final solution. The results show that the behaviour of the HCF agents at searching the neighbourhood and taking inde-pendent decisions to stay in suitable locations, including generating more HCF agents is reflected in the number of HCF agents selected. The graphs show clearly that the initial value of p did not affect the behaviour of the agents at replicating and finding suitable locations.

• Another interesting characteristic of the covering model is that a different number