Parametrized Genetic Algorithms for NP-hard problems on Data - PowerPoint PPT Presentation

Data Envelopment Analysis Valid Solutions Genetic algorithm Hybrid metaheuristics Conclusions and future works Parametrized Genetic Algorithms for NP-hard problems on Data Envelopment Analysis Juan Aparicio 1 enez 2 Domingo Gim´ alez 1 Mart´ ın Gonz´ ın 1 Jos´ e J. L´ opez-Esp´ us T. Pastor 1 Jes´ 1 Miguel Hern´ andez University, 2 University of Murcia Spain The Santander Chair 2015 International Workshop of Efficiency and Productivity, Alicante, June 12, 2015 Genetic Algorithms for DEA Aparicio, Gim´ enez, Gonz´ alez, L´ opez-Esp´ ın, Pastor

Data Envelopment Analysis Valid Solutions Genetic algorithm Hybrid metaheuristics Conclusions and future works Outline 1 Data Envelopment Analysis 2 Valid Solutions 3 Genetic algorithm 4 Hybrid metaheuristics 5 Conclusions and future works Genetic Algorithms for DEA Aparicio, Gim´ enez, Gonz´ alez, L´ opez-Esp´ ın, Pastor

Data Envelopment Analysis Valid Solutions Genetic algorithm Hybrid metaheuristics Conclusions and future works • DEA ( D ata E nvelopment A nalysis): non-parametric technique to estimate the level of efficiency of a set of entities, DMU ( D ecision M aking U nit), all of them operating in the same technological environment. • Each DMU j consumes m inputs, denoted as ( x 1 j , ..., x mj ), to produce s outputs, denoted as ( y 1 j , ..., y sj ). • DEA also provides information on how to remove inefficiency through the determination of benchmarking information. • Objetive : the estimation of the production frontier and the technical efficiency of each DMU (the distance from each interior DMU to the boundary of the technology). Genetic Algorithms for DEA Aparicio, Gim´ enez, Gonz´ alez, L´ opez-Esp´ ın, Pastor

Data Envelopment Analysis Valid Solutions Genetic algorithm Hybrid metaheuristics Conclusions and future works Model of mathematical lineal programming (Aparicio et al., 2007) t − β k − 1 � m max ik i =1 m x ik s.t. t + β k + 1 � s rk = 1 ( c . 1) r =1 s y rk − β k x ik + � n j =1 α jk x ij + t − = 0 ∀ i ( c . 2) ik − β k y rk + � n j =1 α jk y rj − t + = 0 ∀ r ( c . 3) rk − � m i =1 ν ik x ij + � s r =1 µ rk y rj + d jk = 0 ∀ j ( c . 4) ν ik ≥ 1 ∀ i ( c . 5) µ rk ≥ 1 ∀ r ( c . 6) d jk ≤ Mb jk ∀ j ( c . 7) α jk ≤ M (1 − b jk ) ∀ j ( c . 8) b jk = 0 , 1 ( c . 9) β k ≥ 0 ( c . 10) ik ≥ 0 ∀ i ( c . 11) t − t + rk ≥ 0 ∀ r ( c . 12) d jk ≥ 0 ∀ j ( c . 13) α jk ≥ 0 ∀ j ( c . 14) It must be solved n times, one for each DMU. Genetic Algorithms for DEA Aparicio, Gim´ enez, Gonz´ alez, L´ opez-Esp´ ın, Pastor

Data Envelopment Analysis Valid Solutions Genetic algorithm Hybrid metaheuristics Conclusions and future works Approaches to the problem • Problem : combinatorial NP-hard problem, solved with unsatisfactory methods. • Exact solutions only for small problem sizes. • Possible solution: Metaheuristic algorithms . The main problem to apply metaheuristics is the difficulty of obtaining solutions satisfying all the constraints: • In 2014, reduced problems with less than 14 constraints. • Now, all the constraints and generation of a higher percentage of valid solutions, with a Genetic Algorithm , and improvement of the solutions with hybrid metaheuristics . Genetic Algorithms for DEA Aparicio, Gim´ enez, Gonz´ alez, L´ opez-Esp´ ın, Pastor

Data Envelopment Analysis Valid Solutions Genetic algorithm Hybrid metaheuristics Conclusions and future works Representation of solutions • A solution is represented by a vector of real and binary values. Binary part: b 0 k ... b jk t + t + Real part: β k α 0 k ... α jk t − ... t − ... 0 k 0 k ik rk satisfying the 14 constraints. • fitness: Value returned by the objective function. m t − β k − 1 � ik m x ik i =1 • Initialization: use of heuristics to generate valid solutions. Genetic Algorithms for DEA Aparicio, Gim´ enez, Gonz´ alez, L´ opez-Esp´ ın, Pastor

Data Envelopment Analysis Valid Solutions Genetic algorithm Hybrid metaheuristics Conclusions and future works First heuristic 1 Generate b jk ∀ j (c.9). Restrictions: number of b jk equal to 0, > s and < s + m . 2 Calculate the values of α jk and d jk ∀ j by means of a system of equations. 3 t + rk ∀ r and β k are generated to satisfy c.1, with a refinement process: Generate ∀ r , t + rk randomly between 0 and 1; Obtain β k using c.1. while β k ≤ 0 OR β k ≥ 1 do { Local Search on t + } if β k < 0 then Generate r randomly, and t + rk = t + rk / (2 . 0 + random (0 , 1 , 2)) else Generate r randomly, and t + rk = t + rk ∗ (2 . 0 + random (0 , 1 , 2)) end if Obtain β k using c.1. end while 4 α jk ∀ j are calculated using c.3 by solving the system of equations. 5 t − ik calculated using c.2. by solving the system of equations. 6 Finally, ν ik ∀ i generated randomly, µ rk ∀ r obtained from c.4, and number of d jk equal to 0 the same as number of α � = 0. Genetic Algorithms for DEA Aparicio, Gim´ enez, Gonz´ alez, L´ opez-Esp´ ın, Pastor

Data Envelopment Analysis Valid Solutions Genetic algorithm Hybrid metaheuristics Conclusions and future works Second heuristic used to recalculate non valid solutions after the first heuristic 1 b jk ∀ j generated as in heuristic one; values α generated randomly. 2 α jk ∀ j modified to satisfy c.1, c.2., c.3., c.11. and c.12. { Local Search on α } for i = 1 , . . . , m do if x ik < � n j =1 α jk x ij then j 0 / 1 � m i =1 x ij 0 − 1 � s i =1 y ij 0 = max j =1 ,..., n { 1 � m i =1 x ij − 1 � s i =1 y ij } m s m s α j 0 k = α j 0 k ∗ 0 . 95 end if end for for r = 1 , . . . , s do j 0 / . . . α j 0 k = α j 0 k ∗ 1 . 05 end for ∀ j adjust α jk with a similar refinement method. Adjust β k to satisfy c.11. and c.12. Obtain t + rk ∀ r and t − ik ∀ i using c.2. and c.3. 3 Similar refinement (LS) to do β k satisfy c.2., c.3., c.11. and c.12. 4 ν ik ∀ i , µ rk ∀ r and d jk ∀ j as in the first method. Genetic Algorithms for DEA Aparicio, Gim´ enez, Gonz´ alez, L´ opez-Esp´ ın, Pastor

Data Envelopment Analysis Valid Solutions Genetic algorithm Hybrid metaheuristics Conclusions and future works Percentage of valid solutions size 9 constraints - ICCS14 13 constraints - ICAC14 14 constraints m n s time (sec) % val. time (sec) % val. time (sec) % val. 2 15 1 26 . 42 51 . 44 82 35 . 58 33 . 21 10 . 82 72 18 . 12 0 . 09 0 . 02 100 0 . 00 3 25 2 6 . 72 16 . 03 90 30 . 46 72 . 89 15 . 56 24 20 . 97 0 . 88 0 . 68 96 2 . 85 4 30 2 0 . 22 0 . 16 100 0 . 00 89 . 84 18 . 63 16 21 . 13 0 . 88 1 . 74 95 1 . 49 5 40 3 13 . 13 20 . 64 74 43 . 40 116 . 39 12 . 86 1 . 6 2 . 49 27 . 22 42 . 38 92 9 . 07 6 60 4 2 . 01 1 . 13 35 44 . 07 117 . 26 14 . 15 0 . 06 0 . 10 93 . 46 70 . 08 53 35 . 57 Now higher percentage of valid solutions and for all the constraints ⇒ apply metaheuristics to improve solutions. Genetic Algorithms for DEA Aparicio, Gim´ enez, Gonz´ alez, L´ opez-Esp´ ın, Pastor

Data Envelopment Analysis Valid Solutions Genetic algorithm Hybrid metaheuristics Conclusions and future works • Initialization : with the heuristics. • End Condition : a maximum number of iterations or a maximum number without improving the best solution. • Selection : valid solutions are selected for combination. Non-valid solutions are substituted for new valid solutions. • Crossover Individual with components of six types, each combination works with one of these types. 1 Only β is considered. The mean of β 1 and β 2 of the two ascendants is ik and t + obtained and randomly perturbed. The values of t − rk are recalculated so that constraints c.1, c.2 and c.3 are fulfilled. Values of t + , t − , ν , µ or d are crossed. In each combination only 2 parameters of one type randomly selected, with middle point combination. 3 Combination of the previous crossovers. All the parameters are candidates, and one is randomly selected. • Mutation : each individual a 10% probability of being mutated. One parameter is selected randomly, and new values are randomly generated. Genetic Algorithms for DEA Aparicio, Gim´ enez, Gonz´ alez, L´ opez-Esp´ ın, Pastor

Data Envelopment Analysis Valid Solutions Genetic algorithm Hybrid metaheuristics Conclusions and future works Comparison with CPLEX Fitness Time (logarithmic scale) m=4,n=30, s=3 0.7 0.6 0.5 CPLEX 0.4 crossover 1 fitness crossover 2 0.3 crossover 3 0.2 0.1 0 0 5 10 15 20 25 30 iterations • Small problems: solutions with GA close to those with CPLEX. • Large problems: CPLEX impracticable. Genetic Algorithms for DEA Aparicio, Gim´ enez, Gonz´ alez, L´ opez-Esp´ ın, Pastor

Data Envelopment Analysis Valid Solutions Genetic algorithm Hybrid metaheuristics Conclusions and future works Parameterized scheme Initialize( S , ParamIni ) while not EndCondition( S , ParamEnd ) do SS = Select( S , ParamSel ) SS 1 = Combine( SS , ParamCom ) SS 2 = Improve( SS 1, ParamImp ) S = Include( SS 2, ParamInc ) end while Different values of the Metaheuristic parameters ⇒ different metaheuristics and hybridizations. Genetic Algorithms for DEA Aparicio, Gim´ enez, Gonz´ alez, L´ opez-Esp´ ın, Pastor