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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

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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 Aparicio1 Domingo Gim´ enez2 Mart´ ın Gonz´ alez1 Jos´ e J. L´

  • pez-Esp´

ın1 Jes´ us T. Pastor1

1Miguel Hern´

andez University, 2University 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´

  • pez-Esp´

ın, Pastor

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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´

  • pez-Esp´

ın, Pastor

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Data Envelopment Analysis Valid Solutions Genetic algorithm Hybrid metaheuristics Conclusions and future works

  • DEA (Data Envelopment Analysis): non-parametric technique to

estimate the level of efficiency of a set of entities, DMU (Decision Making Unit), all of them operating in the same technological environment.

  • Each DMUj consumes m inputs, denoted as (x1j, ..., xmj), to

produce s outputs, denoted as (y1j, ..., ysj).

  • 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´

  • pez-Esp´

ın, Pastor

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Data Envelopment Analysis Valid Solutions Genetic algorithm Hybrid metaheuristics Conclusions and future works

Model of mathematical lineal programming (Aparicio et al., 2007)

max βk − 1

m

m

i=1 t−

ik

xik

s.t. βk + 1

s

s

r=1 t+

rk

yrk

= 1 (c.1) −βkxik + n

j=1 αjkxij + t− ik

= ∀i (c.2) −βkyrk + n

j=1 αjkyrj − t+ rk

= ∀r (c.3) − m

i=1 νikxij + s r=1 µrkyrj + djk

= ∀j (c.4) νik ≥ 1 ∀i (c.5) µrk ≥ 1 ∀r (c.6) djk ≤ Mbjk ∀j (c.7) αjk ≤ M(1 − bjk) ∀j (c.8) bjk = 0, 1 (c.9) βk ≥ 0 (c.10) t−

ik ≥ 0

∀i (c.11) t+

rk ≥ 0

∀r (c.12) djk ≥ 0 ∀j (c.13) αjk ≥ 0 ∀j (c.14)

It must be solved n times, one for each DMU.

Genetic Algorithms for DEA Aparicio, Gim´ enez, Gonz´ alez, L´

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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

  • btaining solutions satisfying all the constraints:
  • In 2014, reduced problems with less than 14 constraints.
  • Now, all the constraints and generation of a higher percentage
  • f valid solutions, with a Genetic Algorithm,

and improvement of the solutions with hybrid metaheuristics.

Genetic Algorithms for DEA Aparicio, Gim´ enez, Gonz´ alez, L´

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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: b0k ... bjk Real part: βk α0k ... αjk t−

0k

... t−

ik

t+

0k

... t+

rk

satisfying the 14 constraints.

  • fitness: Value returned by the objective function.

βk − 1 m

m

  • i=1

t−

ik

xik

  • Initialization: use of heuristics to generate valid solutions.

Genetic Algorithms for DEA Aparicio, Gim´ enez, Gonz´ alez, L´

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Data Envelopment Analysis Valid Solutions Genetic algorithm Hybrid metaheuristics Conclusions and future works

First heuristic

1 Generate bjk ∀j (c.9). Restrictions: number of bjk equal to 0, > s and

< s + m.

2 Calculate the values of αjk and djk ∀j by means of a system of equations. 3 t+

rk ∀r and βk are generated to satisfy c.1, with a refinement process:

Generate ∀r, t+

rk randomly between 0 and 1; Obtain βk using c.1.

while βk ≤ 0 OR βk ≥ 1 do {Local Search on t+} if βk < 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 βk using c.1. end while 4 αjk ∀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, νik ∀i generated randomly, µrk ∀r obtained from c.4, and number

  • f djk equal to 0 the same as number of α = 0.

Genetic Algorithms for DEA Aparicio, Gim´ enez, Gonz´ alez, L´

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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 bjk ∀j generated as in heuristic one; values α generated randomly. 2 αjk ∀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 xik < n

j=1 αjkxij

then j0 / 1

m

m

i=1 xij0 − 1 s

s

i=1 yij0 = maxj=1,...,n{ 1 m

m

i=1 xij − 1 s

s

i=1 yij}

αj0k = αj0k ∗ 0.95 end if end for for r = 1, . . . , s do j0 / . . . αj0k = αj0k ∗ 1.05 end for ∀j adjust αjk with a similar refinement method. Adjust βk 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 βk satisfy c.2., c.3., c.11. and c.12. 4 νik ∀i, µrk ∀r and djk ∀j as in the first method.

Genetic Algorithms for DEA Aparicio, Gim´ enez, Gonz´ alez, L´

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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 26.4251.44 8235.58 33.2110.82 7218.12 0.090.02 1000.00 3 25 2 6.7216.03 9030.46 72.8915.56 2420.97 0.880.68 962.85 4 30 2 0.220.16 1000.00 89.8418.63 1621.13 0.881.74 951.49 5 40 3 13.1320.64 7443.40 116.3912.86 1.62.49 27.2242.38 929.07 6 60 4 2.011.13 3544.07 117.2614.15 0.060.10 93.4670.08 5335.57

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´

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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

  • ne of these types.

1 Only β is considered. The mean of β1 and β2 of the two ascendants is

  • btained and randomly perturbed. The values of t−

ik and t+ rk are

recalculated so that constraints c.1, c.2 and c.3 are fulfilled. 2 Values of t+, t−, ν, µ or d are crossed. In each combination only 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´

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Data Envelopment Analysis Valid Solutions Genetic algorithm Hybrid metaheuristics Conclusions and future works

Comparison with CPLEX

Fitness Time (logarithmic scale)

5 10 15 20 25 30 0.1 0.2 0.3 0.4 0.5 0.6 0.7 m=4,n=30, s=3 CPLEX crossover 1 crossover 2 crossover 3 iterations fitness

  • Small problems: solutions with GA close to those with

CPLEX.

  • Large problems: CPLEX impracticable.

Genetic Algorithms for DEA Aparicio, Gim´ enez, Gonz´ alez, L´

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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) SS1 = Combine(SS,ParamCom) SS2 = Improve(SS1,ParamImp) S = Include(SS2,ParamInc) end while Different values of the Metaheuristic parameters ⇒ different metaheuristics and hybridizations.

Genetic Algorithms for DEA Aparicio, Gim´ enez, Gonz´ alez, L´

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Data Envelopment Analysis Valid Solutions Genetic algorithm Hybrid metaheuristics Conclusions and future works

Applications

Successfully applied to:

  • Signal filter design.
  • Electricity consumption in exploitation of water-wells.
  • Estimation of parameters in kinetic reactions.
  • Maximum diversity problem.
  • p-hub problem.
  • Tasks-to-processors assignation.
  • Simultaneous Equation Models.
  • Molecules docking.

Genetic Algorithms for DEA Aparicio, Gim´ enez, Gonz´ alez, L´

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Data Envelopment Analysis Valid Solutions Genetic algorithm Hybrid metaheuristics Conclusions and future works

Metaheuristics in the experiments

And Hyperheuristic implemented with the same parameterized scheme and searches for the best combination of metaheuristic parameters in the scheme.

Genetic Algorithms for DEA Aparicio, Gim´ enez, Gonz´ alez, L´

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Data Envelopment Analysis Valid Solutions Genetic algorithm Hybrid metaheuristics Conclusions and future works

Comparison of fitness

m=2 s=1 N=50 m=3 s=2 N=30 m=4 s=2 N=28 m=4 s=3 N=20 m=5 s=3 N=20 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 CPLEX Hiperheuristic SS GA GR

Tipo de problema Promedio Fitness Problem size Mean fitness

Genetic Algorithms for DEA Aparicio, Gim´ enez, Gonz´ alez, L´

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Data Envelopment Analysis Valid Solutions Genetic algorithm Hybrid metaheuristics Conclusions and future works

Conclusions

  • Application of Genetic algorithms and hybrid metaheuristics

for a mathematical programming model for Data Envelopment Analysis.

  • The results of previous works are improved: all the constraints

are considered, and larger number of valid solutions are generated.

  • Small problems: metaheuristics give fitness values close to the
  • ptimum,

and hyperheuristics can be used to obtain satisfactory hybrid metaheuristics.

  • Metaheuristics can be applied for large problems, for which

huge execution times make exact methods impracticable.

Genetic Algorithms for DEA Aparicio, Gim´ enez, Gonz´ alez, L´

  • pez-Esp´

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Data Envelopment Analysis Valid Solutions Genetic algorithm Hybrid metaheuristics Conclusions and future works

Future works

  • Improvement of heuristics to generate valid solutions.
  • Hybridization of metaheuristics and exact methods,

and exploration of the use of heuristics in CPLEX.

  • Improvement of the hyperheuristic.
  • Parallelism to reduce the high execution time of

metaheuristics, and specially of hyperheuristics.

Genetic Algorithms for DEA Aparicio, Gim´ enez, Gonz´ alez, L´

  • pez-Esp´

ın, Pastor