CRUZ ROJA

To summarise, this chapter presents a decomposition method to solve the home health care problem. Problem decomposition is made by geographical region. The approach avoids having conflicting assignments by solving sub-problems in sequences. Each sub-problem solution gives only a part of a working path. A full working path is then built from multiple parts during the combining sub- problem solutions step. Finally, adding neighbour workforce is applied as an extension of this method. The idea is to add other workers from neighbour regions to take unassigned visits.

There are three main studies presented in this chapter: permutation study, strategies study, and neighbour workforce study. The permutation study aims to find the best outcome of applying GDCA method as it searches on every possible sub-problem permutation order. The strategies study compares order-

ing rules and finds the best ordering rule for general uses. Finally, neighbour workforce is used to improve the solution quality.

The permutation study shows that GDCA is able to find optimal solution. However, the condition depends on a defined geographical region. Further- more, the sub-problem sequence used indeed affects the solution quality as shown in A-04 where the solution gap ranged from 3.41% to 80%. This is the main reason to select sequences which provide a low objective function.

The strategies study finds effective ordering rules which could give a higher quality solution. The study is crucial as using permutation is very limited due to the number of permutations growing exponentially. Thus, the study tests six ordering strategies. The result suggests ordering the sub-problem by the number of visits gives the lowest average objective function and consumes less computational time. Furthermore, the result is compared back to the permuta- tion study to find differences to the best possible outcome of decomposition method. It shows strategies could match the best permutation on two instances while the rest has slight differences up to 4% of relative gap.

Neighbour workforce is an extension to the GDCA which focuses on im- proving solution quality. The test only applies to instance sets D and F as the number of workers in the sub-problems in these instances is less than the num- ber of required visits. The study shows the extension is able to reduce objective function down to 75% of the original value.

From these studies, we have seen that this approach is able to find a feasible solution especially on instance sets D and F where solving them as a whole problem is impossible. However, we have seen high objective value on several test instances. The reasons behind this could be due to the approach of avoiding conflicts. Thus, in the next chapter, we introduce a potential alternative for dealing with conflicts.

Chapter 5

Decomposition with Conflict Repair

In the previous chapter, geographical decomposition with conflict avoidance (GDCA) was shown to have potential for solving the larger problem instances. However, we can see that the solution quality depends on having the right sub- problem solving sequence. In fact, because it was not found that a particular sequence dominated the others, this indicates that finding the right sequence would not be be practical. Therefore, we propose a sequence free decom- position technique that not only takes less parameters by removing solving sequences but also does not require conflict avoidance constraints (4.1) - (4.2), which is not required by the main problem definition. Later in this chapter, we propose a geographical decomposition with conflict repair (GDCR) and then present an improved version of GDCA, a repeated decomposition with conflict repair (RDCR). These algorithms aim to solve the home healthcare problem presented in Section 2.5.

The content of this chapter is to be appear in:

• Wasakorn Laesanklang and Dario Landa-Silva. Decomposition Techniques

with Mixed Integer Programming and Heuristics to Solve Home Health- care Planning Problems. Annals of Operations Research, Online First, 2016..

5.1

Repairing Process in the Literature

The term “Repairing” meaning to correct infeasible solutions has been used mostly in the context of evolutionary algorithms [8]. A repairing method in genetic algorithms recombines an infeasible solution to generate a feasible one [105]. For a scheduling problem, a systematic repair approach was proposed us- ing a bias heuristic to tackle schedules with excessive work-in-progress [130]. An iterative heuristic repairing method had been proposed in the scheduling problem as part of an automated scheduling and rescheduling system [131]. Ba- sically, the method relaxes some constraints when constructive methods found difficulty in completing a feasible solution. The iterative repairing processes in this method is applied iteratively until the solution quality is satisfactory. Another repairing technique was implemented to support the local search al- gorithm for tackling the job-shop scheduling problem [98]. The use of this re- pairing technique allows local search moves to continue its search when the move finds an infeasible solution.

Applying a repairing process is not commonly known for mathematical pro- gramming based decomposition methods because most approaches, i.e. Bend- ers’ decomposition, only generates a solution from the feasible region. There- fore, solutions obtained by Benders’ decomposition method do not need to be repaired. However, the solution solved by the decomposition approaches pro- posed in this chapter may result in an infeasible solution when using the full model. The proposed decomposition approaches use the MIP solver to solve every sub-problem which is generated from decomposing the full problem. Thus, a solution to the sub-problem is feasible only for the sub-problem, but when combining all sub-problem solutions, the combined solution becomes in- feasible as sub-problems are solved independently. As a result, we use conflict- ing assignments repair to fix solutions provided by the decomposition stage.

Monday, 08 December 2014 Valid Conflict Valid paths Conflict paths Monday, 08 December 2014 Monday, 08 December 2014 Unassigned Unassigned Valid paths Unassigned

Valid paths December 2014Monday, 08

Valid Conflict Valid Conflict Greedy heuristic Sp lit w o rk fo rc e Solve sub-problem Collect valid paths Collect Conflict paths C re at e n e w s u b -p ro b lem fo r e a ch w o rkf o rc e Solve sub-problem Solve sub-problem Collect Valid path C ol le ct U na ss ign ed t as ks Split task by regions

Figure 5.1:Illustrating the Geographical Decomposition with Conflict Repair Approach.