Description 55 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) ISSN (Print) 2313-4410, ISSN (Online) 2313-4402 © Global Society of Scientific Research and Researchers http://asrjetsjournal.org/ Golinski’s Speed Reducer Problem Revisited Using Genetic Algorithm Rashid Mehmood a*, Muhammad Hassan Qazi b, Huneza Ata c, Rohail Zaheer d a ,b,c,dMS Student, Institute of Space Technology, P.O Box 2750, Islamabad 44000, Pakistan aEmail: Rashidmehmood28@hotmail.com bEmail: maverick.syed29@gmail.com cEmail: hunezaata@gmail.com dEmail: rz.dhothar@gmail.com Abstract There are many constraints that constitute the Gear Box Design. Golinski’s Speed reducer problem is studied and the weight of the Gear box is minimized. This is done by considering certain design variables of the Gear box, applying Optimization algorithms that provide results which agree upon a resulting weight that is minimum satisfying the concerned design variables. Keywords: Gear box; Design Optimization; Aircraft; Propeller shaft; Genetic Algorithm; Sensitivity Analysis; NASA. 1. Introduction There are many cases of Multidisciplinary Optimization problems that have been set as bench mark problems. Golinski’s Speed reducer is one of the most studied bench mark problems of the NASA Langley Multidiscliplinary design Optimization (MDO) test suite. It was modeled by Golinski to minimize the weight of the Gear box. This gear box can be used in a light airplane between the engine and the propeller for allowing each of them to rotate at its efficient speed. ------------------------------------------------------------------ * Corresponding author. http://asrjetsjournal.org/ American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 16, No 1, pp 55-65 56 2. Problem statement Minimize the weight of gearbox although satisfying a number of constraints constituted by gear and shaft design. 3. Literature Review Many researchers have worked on the Golinski’s speed reducer problem with different approaches. Some of which are highlighted as follows. Tapabrata Ray [1] worked on this problem. He used Particle Swarm Optimization and reported a feasible solution as all constraints were satisfied. Kuang and his colleagues [2] worked on this problem using Taguchi search method. He reported the solution of the problem but the solution is not feasible because it pertains constraint violation. Rao [10] worked on this problem using sequential search method. He reported the solution of this problem but the solution is not feasible because it pertains two constraint violations. Similarly prior to the above mentioned works, other researchers also proposed solutions. For example Li and Papalambros [3], Azarm and Li [4], and the MDO test suite of NASA also presented solutions to this problem. However all of them were not feasible solutions. Other optimization approaches have also been applied by many researchers to find feasible solutions. For example: Rania Hassan, Babak Cohenim and Olivier de Weck [5] applied Particle Swarm Optimization and Genetic algorithm Optimization to this problem and presented a comparison of results obtained. Luis Vicente Santana-Quintero and Carlos A. Coello Coello [6] applied their algorithm (based on Differential evolution) on this problem and studied the results. Xiaoping Du [7] treated this problem as a probabilistic design problem. He applied Sequential optimization and Reliability assessment method to this problem and assessed the results. Other optimization approaches have also been applied by many researchers to find feasible solutions. For example: Rania Hassan, Babak Cohenim and Olivier de Weck [6] applied Particle Swarm Optimization and Genetic algorithm Optimization to this problem and presented a comparison of results obtained. Luis Vicente Santana-Quintero and Carlos A. Coello Coello [7] applied their algorithm (based on Differential evolution) on this problem and studied the results. Xiaoping Du [8] treated this problem as a probabilistic design problem. He applied Sequential optimization and Reliability assessment method to this problem and assessed the results. 4. Problem formulation There are seven design variables in this problem (as shown in Figure 1): 1x = Width of the Gear face, American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 16, No 1, pp 55-65 57 2x = Teeth module 3x = Number of teeth of pinion 4x = Length of shaft 1 between bearings 5x = Length of shaft 2 between bearings 6x = Diameter of shaft 1 7x = Diameter of shaft 2 The variable bounds for the problem are as follows: 2.6≤ 1x ≤ 3.6 0.7≤ 2x ≤ 0.8 17≤ 3x ≤ 28 7.3≤ 4x ≤ 8.3 7.3≤ 5x ≤ 8.3 2.9≤ 6x ≤ 3.9 5.0≤ 7x ≤ 5.9 These parameters form a constrained optimization problem given in Equation 1), This problem is subject to the following constraints: These constraints are symbolized as 1G , 2G , 3G , 4G , 5G , 6G , 7G , 8 9,G G , 24 25,G G respectively. Constraints 10G and 11G are side constraints of 1x , constraints 12G and 13G are side constraints of 2x , 14G and 15G are side constraints of 3x , 16G and 17G are side constraints of 4x , 18G and 19G are side constraints of 5x , 20G and 21G are side constraints of 6x and the constraints 22G and 23G are side constraints of 7x . American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 16, No 1, pp 55-65 58 1 2 1 2 1 2 1 1 2 3 1 2 2 1 2 3 1 1 3 4 2 3 4 6 1 1 3 4 2 3 5 7 1 1 2 6 3 4 2 3 6 1 1 2 6 3 5 2 3 7 2 3 2 1 1 2 6 27 1 397.5 1 1.93 1 1.93 1 (745 ) 16.9 10 110.0 1 (745 ) 157.5 10 85.0 1 / 40 1.0 5 / 1.0 /12 1.0 (1.5 [ ] /[ ] [ ] /[ ] x x x x x x x x x x x x x x x x x x x x x x x x x x x x x − − − − − − − − − − − − − − − − ≤ ≤ ≤ ≤ + × ≤ + × ≤ ≤ ≤ ≤ + 1 4 1 7 5 1.9) 1 (1.1 1.9) 1 x x x − − ≤ + ≤ 5. Optimization algorithms 5.1 Genetic Algorithm Genetic algorithm has been used to find a feasible solution of this problem. Genetic algorithm is a search algorithm developed by John Holland in 1970. GA (Genetic Algorithm) is based on the Darwinian theory of evolution, “Survival of the fittest “. GA are search algorithms that imitate natural selection and natural genetic behavior. They combine survivors of the fittest among structures, with structured yet randomized information exchange to form a search algorithm. The basic GA is composed of a fitness function, a selection technique, a reproduction (cross over) and mutation operators with fixed probabilities. A genetic loop is formed which is shown in the figure [2]. The complete functional loop of Genetic Algorithm is shown in the figure [3]. There are advantages of using GA for this problem are that GA is an intelligent random search method it searches in a feasible search area. The structure of functionality of GA allows a broader search in an area with feasible solutions. 5.2 Fmincon FMINCON is a constrained function minimization method. It searches for the best solution in a space around the starting point hence it searches a local minima. Therefore it has been used as a hybrid function for this problem by providing it with the function value obtained by GA, as an initial or starting value to further enhance the solution obtained by GA. The optimization flow diagram of the hybrid optimization approach is shown in Figure [4]. American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 16, No 1, pp 55-65 59 5.3 Fminmax This function starts with an initial estimate and minimizes the largest function value (this large value is the worst case) of a set of multi-variable functions. Since this problem consists of multivariable functions, so FMINMAX is used to find a feasible solution. 6. Discussion and Comparison of results In this work, the results of Rao, Kuang et.al NASA test suite and Ray are analyzed. Before starting optimization, the sensitivity analysis using Latin Hypercube sampling is done. After optimization, the robustness of obtained results is checked. The pre-optimization sensitivity and robustness of result is compared to those of Rao, Kuang and Ray. Table 1: Starting variables for Optimization x1 x2 x3 x4 x5 x6 x7 Fval 2.87 0.73 18.73 7.86 7.76 3.04 5.18 3087 Genetic Algorithm Optimization tool Conditions 7. Pre-optimization sensitivity analysis The pre-optimization sensitivity analysis is done using Latin Hypercube Sampling with 300 samples. It was observed that 1x was the most effective variable among all design variable. On the basis of the sensitivity result, the six samples which gave minimum constraint values were sorted out and their average value was taken. This average value was set as initial guess for optimization. This value is 3087. Population Type Vector Size Creation Function Scaling Selection Double 20 Constraint Dependent Rank Stochastic Uniform Mutation Crossover Migration Initial Penalty Factor Constraint Dependent Scattered Forward 10 100 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 16, No 1, pp 55-65 60 The initial variables are changed now, these are shown in table [1] and are set as starting variables for the optimization. The Genetic algorithm is hybridized with FMINCON. The results are shown in tables [2] and [3]. 8. Robustness of Solution The solution obtained 3040.63 kg is perturbed +1%. Then again LHS (Latin Hypercube Sampling) is used. Same is done for the results of Rao, Kuang et.al NASA test suite and Tapabrata Ray. The variation in function values and sum of constraints is noticed for all results. The result of the comparison is shown as in table [4]. 9. Conclusion The Speed Reducer problem is revisited to minimize the weight of the Gearbox using Optimization techniques. The Genetic Algorithm is utilized and then it is hybridized by FMINCON and FMINMAX separately to get the best results. The following conclusions can be drawn from this work: 1. The table [4] clearly shows that the result obtained in the present work is feasible and robust than the other results. 2. The variations and deviations in function values is an acceptable one provided that all constraints are satisfied. The product of function value and Constraint value is a good indicator of feasibility of results. 3. The obtained value can be further minimized by a different approach. Particle swarm optimization is an attractive option for future work on this problem. Figure 1: The Speed Reducer labelled with the Design variables American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 16, No 1, pp 55-65 61 Table 2: Strategies employed and Resulting Function values SR# Strategy Changes x1 x2 x3 x4 x5 x6 x7 FUNCTION VALUE 1 DEFAULT OPTIONS 3.5 1 17 7.3 7.71843216 3.3502512 5.2866879 3040.824577 2 POPULATION SIZE=100 3.5 1 17 7.3 7.71531991 3.35021467 5.28665446 3040.629933 3 SELECTION=REMAINDER 3.5 1 17 7.3 7.71733339 3.35027333 5.28666176 3040.706076 4 SELECTION=TOURNAMENT 3.5 1 17 7.3 7.71723781 3.35021469 5.28665532 3040.684092 5 SELECTION=ROULETTE 3.5 1 17 7.3 7.71659944 3.35024189 5.28665572 3040.673446 6 MIGRATION=BOTH 3.5 1 17 7.3 7.72648597 3.35021467 5.28665852 3040.944651 7 REPRODUCTION:CO=57.26% 3.5 1 17 7.3 7.71541222 3.35021467 5.2866545 3040.632527 8 REPRODUCTION:CO=50.42% 3.5 1 17 7.300004 7.71537178 3.35021466 5.28666058 3040.638033 9 HYBRID FUNC: FMINCON 3.5 1 17 7.3 7.71531991 3.35021467 5.28665446 3040.629928 Table 3: Effect on Constraints Cons1 Cons2 Cons3 Cons4 Cons5 Cons6 Cons7 Con8 Con9 Con10 Con11 Sum of cons - 0.0754 - 0.2194 - 0.1682 - 0.9074 0.1284 - 0.0387 - 0.6892 0.0822 -0.615 - 0.1705 - 0.0198 - 2.6931 - 0.2088 - 0.3254 - 0.1879 - 0.9041 0.2003 0.0671 - 0.6677 0.1541 -0.639 - 0.1606 - 0.0135 - 2.6854 - 0.1401 - 0.2628 -0.186 - 0.8961 0.172 - 0.0063 - 0.6741 0.1965 - 0.6518 - 0.1643 - 0.0596 - 2.6725 0.0116 - 0.1617 - 0.3489 - 0.9046 - 0.0188 0.0654 - 0.6874 0.16 - 0.6408 - 0.1517 0.0083 - 2.6688 - 0.0832 - 0.4006 - 0.3473 - 0.9176 0.1755 0.0352 - 0.6007 0.3639 - 0.6945 - 0.1579 - 0.0354 - 2.6626 - 0.1384 - 0.4044 -0.24 - 0.9064 0.2199 0.022 -0.609 0.3461 - 0.6905 - 0.1813 - 0.0736 - 2.6556 - 0.1322 - 0.3318 - 0.2615 - 0.9079 0.0984 0.0821 - 0.6373 0.3425 - 0.6896 - 0.1891 - 0.0232 - 2.6496 - 0.1913 - 0.3682 - 0.2779 - 0.9204 0.2243 0.0241 - 0.6256 0.4161 - 0.7058 -0.153 -0.007 - 2.5855 - 0.0839 - 0.3327 - 0.3823 - 0.9193 0.1014 0.0862 - 0.6231 0.4236 - 0.7073 - 0.1492 0.009 - 2.5776 - 0.0245 - 0.2432 - 0.2001 - 0.9192 0.0821 0.0296 - 0.6679 0.1764 - 0.6458 - 0.1921 0.0302 - 2.5743 - 0.1469 - 0.4634 - 0.1727 - 0.9309 0.2297 0.0719 - 0.5804 0.3636 - 0.6944 -0.22 0.0203 - 2.5233 -0.107 -0.292 - 0.1632 - 0.9161 0.2334 0.0314 - 0.6625 0.1803 -0.647 - 0.1636 0.0126 - 2.4936 0.0168 - 0.1602 - 0.3403 -0.889 0.074 0.0832 - 0.6783 0.2623 - 0.6699 - 0.1318 - 0.0444 - 2.4778 - 0.0468 - 0.1754 - 0.1756 - 0.9317 0.1329 - 0.0473 - 0.6712 0.3861 - 0.6994 -0.182 0.0609 - 2.3495 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 16, No 1, pp 55-65 62 - 0.0406 - 0.1718 -0.051 - 0.8893 0.1888 0.0469 - 0.6905 0.1587 - 0.6404 - 0.1874 - 0.0442 - 2.3209 - 0.0919 - 0.2672 - 0.0778 - 0.8943 0.2298 0.0281 - 0.6443 0.4555 - 0.7137 - 0.2053 - 0.0796 - 2.2608 0.0259 - 0.1681 - 0.1573 -0.894 0.2082 0.0215 - 0.6709 0.3144 -0.683 - 0.1659 - 0.0589 - 2.2281 0.0149 - 0.2087 - 0.0105 - 0.9134 0.2119 - 0.0004 - 0.6552 0.3828 - 0.6987 - 0.2206 - 0.0172 -2.115 0.0154 - 0.1272 - 0.0958 - 0.8845 0.1587 0.0615 - 0.6702 0.4711 - 0.7168 - 0.2001 - 0.0723 - 2.0601 Table 4: Robustness of Solution Results by Variation in function value Variation in constraint violation sum Product of Fv and Cv Constra- ints violated X- optimal Perturb -ation in X- optimal Total Variation in X- optimal Present work (GA+Fminc on) Hybrid and Fminimax 145.6312328 0.13960393 2 20.330692 72 0 3040 1% 4.79% Rao^3 130.9404375 0.13471720 1 17.639929 21 2 2987.298 5 1% 4.38% Kuanget 143.4972434 0.15628103 5 22.425897 66 1 2876.117 6 1% 4.98% NASA 150.6812575 0.15502371 9 23.359168 93 2 2985.151 9 1% 5.04% Tapabrata Ray 156.2475078 0.15757876 24.621288 45 0 2996.232 2 1% 5.20% Figure 2: Basic Genetic Algorithm cycle American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 16, No 1, pp 55-65 63 Figure 3: Complete Genetic Algorithm cycle applied for the Speed reducer problem Figure 4: Optimization flow diagram with Hybrid optimization approach. American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 16, No 1, pp 55-65 64 2 2 1 2 3 3 2 2 3 3 1 6 7 6 7 2 2 4 6 5 7 ( ) 0.7854 (3.3333 14.9334 43.0934) 1.5079 ( ) 7.477( ) 0.7854( ) f x x x x x x x x x x x x x x = + − − + + + + + (1) Acknowledgements The authors wish to acknowledge the support provided for this research by Dr. Qasim Zeeshan. References [1] Golinski’s speed reducer problem revisited. Tapabrat Ray, AIAA Journal, vol. 41, Issue 3, 2003. p.556- 558 [2] Ku, K. J., Rao, S. S. and Chen, L., 1998, “Taguchi-Aided Search Method for Design Optimization of Engineering Systems,” Engineering Optimization, Vol. 30, p. 1-23 [3] Li, H. L., and Papalambros, P. A., “Production System for Use of Global Optimization Knowledge,” Journal of Mechanisms, Transmissions and Automation in Design. [4] Azarm, S., and Li, W. C., “Multi-Level Design Optimization Using Global Monotonicity Analysis,” Journal of Mechanisms, Transmissions and Automation in Design Vol. 111, 259-263 [5] Rania Hassan, Babak Cohanim and Olivier de Weck, (2004), “A Comparison of Particle Swarm Optimization and the Genetic Algorithm”, AIAA [6] Vicente Santana-Quintero and Carlos A. Coello Coello . “An Algorithm Based on Differential Evolution for Multi-Objective Problems”, International Journal of Computational Intelligence Research. ISSN 0973-1873 Vol.1, No.2 (2005), p. 151–169 [7] Xiaoping Du,” Sequential Optimization and Reliability Assessment Method for Efficient Probabilistic Design”, J. Mech. Des Volume 126(issue 2), 225-233 (May 05, 2004) [8] Mostafa Z. Ali.”Using cultural algorithms to solve optimization problems with a social fabric approach”. [9] David Edward Goldberg, Genetic algorithms in Search, Optimization, and Machine Learning, Addison-Wesley Publishing Company,1989. [10] Singiresu S. Rao, Engineering Optimization, 3rd edition, John Wiley & Sons, 1996. [11] Venkataraman.P. Applied optimization with MATLAB programming, John Wiley & Sons, 2002. American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 16, No 1, pp 55-65 65 [12] Ross Baldick, Applied optimization Formulation and Algorithms for Engineering Systems, Cambridge University Press, 2006. 1. Introduction 2. Problem statement 5. Optimization algorithms Acknowledgements