SERVICIOS SOCIALES COMARCALES

force

One aspect of GDCA that can be improved is allowing workers to make visits outside their working regions. Applying this will reduce the overall objective value because of the reduction in the number of unassigned visits. Making visits outside the working region was prevented during geographical decom- position because a sub-problem must have only workers who are available in sub-problem region. We list the number of visits and the number of workers

grouped by regions in Appendix B.

Originally, the full problem defines working region as a soft constraint so that assigning a worker to make visits outside its regions is allowed by having additional cost. Thus, this practice is valid to accommodate more visits.

Therefore, we intend to reduce the number of unassigned visits by allocat- ing visits to workers who are not available in the region. For this, sub-problems should have additional workers which are recruited from neighbouring re- gions. Ideally, using all workers in all regions should give the best possible outcome. Unfortunately, the MIP solver cannot handle a problem with such a large number of workers. Thus, number of workers from neighbour regions is added in order to match the total number of visits in a sub-problem.

A neighbour worker is defined by a neighbour score N(k, P)which is calcu-

lated by the number of visits the worker k can make and the distance from the worker departure location to the centre of the region. The function is presented in (4.3).

N(k, p) = d_k,c(p)+

∑

j∈T

(1−ηk_j) (4.3)

where c(p)is a location in the centre of sub-problem p and ηk_j is a binary qual-

ification parameter of worker k to visit j (ηk_j = 1 if worker k can make visit j,

ηk_j =0 otherwise). This scoring is only applied to workers who are not available

in the selected region, i.e. neighbour workers for sub-problem P. The workers

with the lowest score N(k, p)are added to the sub-problem until the total num-

ber of workers is equal to the total number of visits in the sub-problems.

Neighbour workers are added to sub-problems where number of workers is less than number of visits. We summarise steps to find additional workers below.

For each sub-problem,

n = |Tp| − |Kp|where Tpis a set of visits and Kpis a set of workers of the

sub-problem. If n >0 then do all following steps 2 - 5, otherwise does not

require additional worker and begin sub-problem solving (step 5).

2. Calculate neighbour score of workers who are not available in p, denoted

the set of these workers as K0p, using the function (4.3).

3. Sort workers in K0_p by their neighbour score from low to high value.

4. Add n lowest score workers to the worker set Kp.

5. Start solving the sub-problem and update worker’s unavailable period. After adding workers to a sub-problem, the method solves the sub-problem with conflict avoidance constraint and updates worker’s unavailable periods. Then the method tackles the next sub-problems in the ordering list.

The instances that require a neighbour workforce are instance sets D and F as presented in Table 4.6. For each instance, the table shows in columns two and seven, the number of regions that required additional workers. Columns three and eight give the average ratio between the number of available workers and the number of locations. Columns four and nine show the improvement ob- tained in the objective function value when using this process of adding neigh- bour workforce. The result shows that additional neighbour workforce is more beneficial to the set F instances for which the cost decreased by up to 75.63% from the solution without additional neighbour workforce. On average, the solution cost decreases by 39.55%.

On the other hand, some of the set D instances did not benefit from the ad- ditional workforce, which is an indication that such instances have the right number of workers for the demand. This experimental result suggests that in the set of F instances, the workforce might not be distributed well across regions according to the demanded visits, which then causes problems when

Table 4.6:Objective value improvement and average ratios between number of visits and number of workers for instance sets D and F. The second column shows the number of regions having not enough workers. The third column shows average workforce/locations ratio in regions that workers is less than visits. The forth column shows average decrease of the on objective function after having additional workers

Instance |A| |M| Ratio Decrease Instance |A| |M| Ratio Decrease

D-01 12 5 73.76% 41.17% F-01 44 13 39.51% 70.57% D-02 11 4 75.04% 50.78% F-02 45 18 40.92% 66.79% D-03 14 5 67.13% -5.87% F-03 53 22 31.87% 67.33% D-04 14 5 76.77% 0% F-04 46 17 40.09% 70.04% D-05 14 4 73.08% 0% F-05 58 19 37.97% 56.29% D-06 14 5 64.72% 0.10% F-06 43 13 44.58% 75.63% D-07 14 7 69.31% 7.34% F-07 63 23 33.73% 53.57%

#Regions is number of regions that workers is less than visits. Ratio is average of proportion between workers and locations.

Decrease is average of decreasing on objective function calculated by

(originalObj−addedWorkerObj) originalObj

|A|is a number of all regions,|M|is a number of regions that the number of workers is less

than the number of visits.

decomposing the problem by regions.