Data Envelopment Analysis with metaheuristics Juan Aparicio 1 Domingo Giménez 2 José J. López-Espín 1 Jesús T. Pastor 1 1 Miguel Hernández University, 2 University of Murcia ICCS, Cairns, June 10, 2014 Aparicio, Giménez, López-Espín, Pastor () DEA with metaheuristics ICCS, Cairns, June 10, 2014 1 / 14
Outline 1 Introduction to DEA 2 Mix model of mathematical lineal programming 3 Metaheuristic Methods for Determining Closest Targets Experimental results 4 Conclusions and future works Aparicio, Giménez, López-Espín, Pastor () DEA with metaheuristics ICCS, Cairns, June 10, 2014 2 / 14
Introduction to DEA DEA (Data Envelopment Analysis) is a non-parametric technique to estimate the current level of efficiency of a set of entities. DEA also provides information on how to remove inefficiency through the determination of benchmarking information. Objective: Study DEA models based on closest to efficient targets, which are related to the shortest projection to the production frontier. Problem: Usually, these models have been solved with unsatisfactory methods since all of them are related in some sense to a combinatorial NP-hard problem. Possible solution: Metaheuristic algorithms. Aparicio, Giménez, López-Espín, Pastor () DEA with metaheuristics ICCS, Cairns, June 10, 2014 3 / 14
Object agent for the measure: DMU (Decision Making Unit). Each DMU j consumes m inputs, denoted as (x 1j,..., x mj ), to produce s outputs, denoted as (y 1j,..., y sj ). As usual, it is assumed that all DMUs operate in the same technological environment. 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). Aparicio, Giménez, López-Espín, Pastor () DEA with metaheuristics ICCS, Cairns, June 10, 2014 4 / 14
Mix model of mathematical lineal programming (Aparicio et al., 2007) t ik x ik max β k 1 m m i=1 s.t. β k + 1 s t + rk s r=1 = 1 (c.1) y rk β k x ik + n j=1 α jkx ij + t ik = 0 i (c.2) β k y rk + n j=1 α jky rj t + rk = 0 r (c.3) m i=1 ν ikx ij + s r=1 µ rky rj + d jk = 0 j (c.4) It must be solved n times, one for each DMU. 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) t ik 0 i (c.11) t + rk 0 r (c.12) d jk 0 j (c.13) α jk 0 j (c.14) Aparicio, Giménez, López-Espín, Pastor () DEA with metaheuristics ICCS, Cairns, June 10, 2014 5 / 14
We focus on some constraints: t ik x ik max β k 1 m m i=1 s.t. β k + 1 s t + rk s r=1 = 1 (c.1) y rk β k x ik + n j=1 α jkx ij + t ik = 0 i (c.2) β k y rk + n j=1 α jky rj t + rk = 0 r (c.3) m i=1 ν ikx ij + s r=1 µ rky 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) It must be solved n times, one for each DMU. b jk = 0, 1 (c.9) β k 0 (c.10) t ik 0 i (c.11) t + rk 0 r (c.12) d jk 0 j (c.13) α jk 0 j (c.14) Aparicio, Giménez, López-Espín, Pastor () DEA with metaheuristics ICCS, Cairns, June 10, 2014 6 / 14
Metaheuristic Methods for Determining Closest Targets Defining a Valid Solution: A solution is represented by a vector of real and binary values. Binary part: b 0k... b jk Real part: β k α 0k... α jk t 0k... t ik t + 0k... t + rk A valid chromosome satisfies the constraints (c.1, c.2, c.3, c.8, c.9, c.10, c.11, c.12, c14). Score: Value returned by the objective function. β k 1 m m t ik x ik i=1 Aparicio, Giménez, López-Espín, Pastor () DEA with metaheuristics ICCS, Cairns, June 10, 2014 7 / 14
Initialization Methods: Method 1. Random. Generate randomly b, β, α, t + rk and t ik Method 2. Heuristic. βk is randomly generated. α jk are generated randomly in different ranges depending on X and Y. FOR j := 1,..., n IF X k high AND Y k low α jk 0 IF X k low AND Y k high α jk Generate randomly 0.5 α jk 1 IF (X k low AND Y k low) OR (X k high AND Y k high) α jk Generate randomly 0 α jk 0.25 t + rk and t ik are deduced from (c.2) and (c.3). Aparicio, Giménez, López-Espín, Pastor () DEA with metaheuristics ICCS, Cairns, June 10, 2014 8 / 14
Method 3. Heuristic with local search. An extension of method 2 with an adjustment process for α jk. First, obtain βk, t 1k,...,t mk, t+ 1k,...,t+ sk, α 1k,...,α nk and b 1k,...,b nk as in Method 2. After that, adjust βk and α 1k,...,α nk : repeat while 1 i n and 1 r s such as t ik 0 or t+ rk 0 and the number of α jk 0 2 do if t ik 0 and t+ rk 0 then Increase β k end if if t ik 0 and t+ rk 0 then Decrease β k end if if t ik 0 or t+ rk 0 then Choose α jk no null randomly and make it equal to zero, decreasing the rest α jk in p and find the minimum α jk and modify its value in order to satisfy restrictions end if end while until a valid solution is obtained or Iterations MaxIter or the number of α jk no null is lower than 2 Aparicio, Giménez, López-Espín, Pastor () DEA with metaheuristics ICCS, Cairns, June 10, 2014 9 / 14
Method 4. Distributed metaheuristic. A genetic algorithm is used to produce valid solutions of the problem. The chromosomes are sets of α jk and β k which satisfy c.1, c.2, c.3, c.8, c.9, c.10 and c.14. The evaluation function is the sum of t + rk and t ik, and the solution with the highest score gives the best candidate. Aparicio, Giménez, López-Espín, Pastor () DEA with metaheuristics ICCS, Cairns, June 10, 2014 10 / 14
Experimental results Two objectives are pursued: To compare the effectiveness of the four methods proposed, studying the execution cost and the percentage of valid solutions. To study how the percentage of valid solutions decreases when the size of the problem increases. Comparison of the percentage of valid solutions with each generation method: Aparicio, Giménez, López-Espín, Pastor () DEA with metaheuristics ICCS, Cairns, June 10, 2014 11 / 14
Execution cost and percentage of valid solutions when the four methods of initiation are used, varying the problem size: size Method 1 Method 2 m n s time % val. time % val. 2 15 1 0.003 0.004 0.75 1.71 0.008 0.005 50.83 41.92 3 25 2 0.004 0.004 0.00 0.00 0.010 0.006 33.55 38.24 4 30 2 0.004 0.005 0.00 0.00 0.022 0.008 26.87 29.09 5 40 3 0.004 0.003 0.00 0.00 0.019 0.003 13.90 23.90 6 60 4 0.006 0.000 0.00 0.00 0.032 0.001 0.03 0.16 10 100 10 0.011 0.000 0.00 0.00 0.092 0.002 0.00 0.00 size Method 3 Method 4 m n s time % val. time % val. 2 15 1 26.423 51.440 82.08 38.58 17.244 3.566 10.58 3.12 3 25 2 6.722 16.025 90.05 30.46 21.283 0.801 0.80 0.83 4 30 2 0.223 0.584 100.00 0.00 29.521 10.540 1.57 1,20 5 40 3 13.125 20.640 73.90 43.40 18.187 1.209 0.00 0.00 6 60 4 2.066 1.132 34.74 44.07 103.698 0.185 0.18 0.46 10 100 10 8.426 3.235 32.65 41.44 302.316 0.713 0.00 0.00 Aparicio, Giménez, López-Espín, Pastor () DEA with metaheuristics ICCS, Cairns, June 10, 2014 12 / 14
Conclusions The application of heuristic and metaheuristic methods to obtain solutions for a mathematical programming model for Data Envelopment Analysis is studied. It is a first step to approximate the solution of the optimization problem. Four different methods to generate the sets of solutions were tested. The heuristic method with local search gives the best results. Aparicio, Giménez, López-Espín, Pastor () DEA with metaheuristics ICCS, Cairns, June 10, 2014 13 / 14
Future works Roadmap: Increment the number of valid solutions with hybrid metaheuristics: combination of local search with distributed metaheuristics. Analyze the application of other metaheuristics, and hyperheuristics on top of them. Inclusion of the methods in metaheuristics for the optimization problem with a reduced number of restrictions. Extend the methodology to include the remaining restriction. Aparicio, Giménez, López-Espín, Pastor () DEA with metaheuristics ICCS, Cairns, June 10, 2014 14 / 14