Effectively Tackling Reinsurance Problems by Using Evolutionary and Swarm Intelligence Algorithms
Abstract
:1. Introduction
2. Black-Box Optimization Problems in Reinsurance
- is a search space, formed by feasible elements .
- is an objective function , to be optimized (maximized or minimized).
2.1. Problem 1: Excess of Loss Reinsurance
2.2. Problem 2: Stop-Loss Reinsurance
2.3. Problem 3: Threshold Proportional Reinsurance
3. Evolutionary-Based Algorithms
3.1. Evolutionary Algorithms: Evolutionary Programming
- Generate an initial population of μ individuals (solutions). Let t be a counter for the number of generations; set it to . Each individual is taken as a pair of real-valued vectors, , , where ’s are objective variables and ’s are standard deviations forGaussian mutations.
- Evaluate the fitness value for each individual (, ) (using the problem’s objective function).
- Each parent (, ), , then creates a single offspring (, ) as follows (j denotes components of the i-th vector):
- If , then , and if , then .
- Calculate the fitness values associated with each offspring (,.
- Conduct pairwise comparison over the union of parents and offspring: for each individual, p opponents are chosen uniformly at random from all the parents and offspring. For each comparison, if the individual’s fitness is better than the opponent’s, it receives a “win”.
- Select the μ individuals out of the union of parents and offspring that have the most “wins” to be parents of the next generation.
- Stop if the halting criterion is satisfied, and if not, set and go to Step 3.
3.2. Particle Swarm Optimization
4. Numerical Results
4.1. Results in Problem 1
Algorithm | Computation time (s) | ||
---|---|---|---|
EP | 0.57372111552 | 0.1200013 | 3.5 |
PSO | 0.5737211153 | 0.1200014 | 2.2 |
4.2. Results in Problem 2
Algorithm | Computation time (s) | ||
---|---|---|---|
EP | 0.0014486748 | 0.14 | 2.9 |
PSO | 0.0014486748 | 0.14 | 2.4 |
4.3. Results in Problem 3
Algorithm | b | Computation time (s) | |||
---|---|---|---|---|---|
Exponential | |||||
EP | 0.4980669653 | 1.0 | 0.7596477801 | 3.2688654179 | 9.6 |
PSO | 0.4980669664 | 1.0 | 0.7596477914 | 3.2688441178 | 8.5 |
Erlang(2,2) | |||||
EP | 0.415635 | 1.0 | 0.761572 | 1.9871 | 9.5 |
PSO | 0.415641 | 1.0 | 0.761564 | 1.9866 | 8.5 |
4.4. Discussion
Problem # | Optimal solution | Computation time (s) |
---|---|---|
Problem 1 | ; | 2100 |
Problem 2 | ; | 2840 |
Problem 3 (Exponential) | ; ; | 3700 |
Problem 3 (Erlang) | ; ; ; | 3850 |
5. Concluding Remarks
Acknowledgments
Conflicts of Interest
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Salcedo-Sanz, S.; Carro-Calvo, L.; Claramunt, M.; Castañer, A.; Mármol, M. Effectively Tackling Reinsurance Problems by Using Evolutionary and Swarm Intelligence Algorithms. Risks 2014, 2, 132-145. https://doi.org/10.3390/risks2020132
Salcedo-Sanz S, Carro-Calvo L, Claramunt M, Castañer A, Mármol M. Effectively Tackling Reinsurance Problems by Using Evolutionary and Swarm Intelligence Algorithms. Risks. 2014; 2(2):132-145. https://doi.org/10.3390/risks2020132
Chicago/Turabian StyleSalcedo-Sanz, Sancho, Leo Carro-Calvo, Mercè Claramunt, Ana Castañer, and Maite Mármol. 2014. "Effectively Tackling Reinsurance Problems by Using Evolutionary and Swarm Intelligence Algorithms" Risks 2, no. 2: 132-145. https://doi.org/10.3390/risks2020132