The material of Simplified Swarm Optimization (SSO)
SSO was originally designed by Yeh in 2009 (Yeh, 2009), and called discrete Particle Swarm Optimization (DPSO) to overcome the drawback of PSO in discrete problem initially. SSO is an emerging populationbased stochastic optimization method (Yeh 2009, Yeh et al. 2009, 2012a, 2012b, 2013). It belongs to the category of Swarm Intelligence methods; it is also an evolutionary computation method.
In SSO, a solution is encoded as a finitelength string. Each solution has a fitness value, which is determined by the fitness function to be optimized. Like most soft computing techniques, SSO is also initialized with a population of random solutions inside the problem space and then searches for optimal solutions by updating generations. Analogous to PSO, each solution moves toward its best previous solution (i.e., pBest) and toward the best solution in the entire swarm during each generation (i.e., gBest) in SSO. Both pBest and gBest are adopted directly from PSO. However, a random movement is added to SSO to maintain population diversity and enhance the capacity of escaping from a local optimum (Yeh 2009, Yeh et al. 2009, 2012a, 2012b, 2013). Thus, each solution is a compromise among the current solution, the pBest (as a local search), the gBest (as a global search), and a random movement.
Let X_{t,i}=(X_{t,i}_{,1}, X_{t,i,2}, ..., X_{t,i,Nvar}) be the ith solution at generation t, x_{t,i,j} be the jth variable of X_{t,i}, P_{i}=(p_{i,}_{1}, p_{i,}_{2},…, p_{i}_{,Nvar}) be the pBest of the ith solution, P_{gBest} be the solution with the best fitness function value among all pBests. Let c_{g}, c_{p},c_{w}, and c_{r} be the parameters represent the probabilities of the new variable value generated from G, pBest, and a random number in SSO, respectively, where c_{g}+c_{p}+c_{w}+c_{r}=1. The detailed steps of SSO are described is based only on the following simple mathematically modeling after c_{g}, c_{p}, and c_{w }are given:
? pgBest , j
x ? ? pi , j
if ρ ?[0, Cg ? cg )
if ρ ?[Cg , C p ? Cg ? c p )
t ?1,i , j ? x
if ρ ?[C , C
? C ? c )
? t ,i , j
p w p w
?? x
if ρ ?[Cw , 1)
where r is a random variable generated from [0,1], t=1, 2, ..., Ngen (the number of generations), i=1, 2, …, Nsol (the number of solutions), j=1, 2, …, Nvar (the number of variables), x is a random easible value between l_{i }(the lower bound) and u_{i} (the upper bound), i.e., l_{i}≤x≤u_{i}.
The detailed steps of SSO are described in the following:
PROCEDURE SSO
STEP 0. Generate X_{0,i}=P_{i} randomly, calculate F(X_{0,i}), find gBest, and let t=1, where i=1,2,…, Nsol. STEP 1. Let i=1. STEP 2. Update X_{t}_{1,i} to X_{t}_{,i} based on the avove equation and calculate F(X_{t}_{,i}). STEP 3. If F(X_{t}_{,i}) is better than F(P_{i}), then let P_{i}=X_{t}_{,i}. Otherwise, go to STEP 5. STEP 4. If F(P_{i}) is better than F(P_{gBest}), then let gBest=i. STEP 5. If i<Nsol, let i=i+1 and go to STEP 2. STEP 6. If t=Ngen and/or CPU time are met, then halt; otherwise let t=t+1 and go back to STEP 1.
The above update procedure in SSO is explained in the following example.
Example: Let C_{g}=.4, C_{p}=.7, C_{w}=.9, and X_{3,2}=(3,3,3,0,0,0,0,0). The X_{4,2} obtained from X_{3,2} using the above updated mechanism of SSO based on the related elements of P_{2}=(4,2,3,2,1,1,1,1), P_{gBest}=(2,4,2,1,1,0,2,1), and r are listed in below.
i

x_{3,2,i}

p_{2,i}

g_{i}

r

x_{4,2,i}

Remark

1

3

4

2

.991

1^{*}

C_{w}<r

2

3

2

4

.375

4

r<C_{g}

3

3

3

2

.428

3

C_{g}<r<C_{p}

4

0

2

1

.195

1

r<C_{g}

5

0

1

1

.698

1

C_{g}<r<C_{p}

6

0

1

0

.875

0

C_{p}<r<C_{w}

7

0

1

2

.428

1

C_{g}<r<C_{p}

8

0

1

1

.195

0

C_{p}<r<C_{w}

^{*} A random integer generated in [l_{1}=0, u_{1}=4].
The advantages of SSO are simplicity, efficiency, and flexibility (Yeh 2009, 2012a, 2012b, 2013). The update mechanism of SSO is much simpler than those of other major soft computing techniques, like PSO (which must calculate both the velocity and position functions), the Genetic Algorithm (which requires such genetic operations as crossover and mutation), the Estimation Distribution of Algorithm (which has difficulty building an appropriate probability model, and the Immune System Algorithm (which does not consider the interaction of variables. Using the simple approaches above, the diversity of a population can be efficiently maintained. Additionally, SSO has been proven effective in exploring large and complex spaces in many optimization problems (Yeh 2009, Yeh et al. 2009, 2012a, 2012b, 2013).
P.S.
The original SSO concept:
Yeh, W.C., Study on Quickest Path Networks with Dependent Components and Apply to RAP, 2008/08/01~2011/07/31, NSC972221E007099MY3 (Individual & Basic Research), Multiyear Research Project, "Distinguished Scholars Research Project" granted by National Science Council, Taiwan.
The 1st submitted SSO journal paper:
Yeh, W.C. 2009. A TwoStage Discrete Particle Swarm Optimization for the Problem of Multiple MultiLevel Redundancy Allocation in Series Systems. Expert Systems with Applications, 36, 91929200.
The 1st accepted SSO journal paper:
Yeh, W.C., Chang, W.W., and Chung, Y.Y., “A New Hybrid Approach for Mining Breast Cancer Pattern Using Discrete Particle Swarm Optimization and Statistical Method”, Expert Systems with Applications, Vol. 36, No. 4, 2009/5, pp. 82048211.


Xiangyong Kong*; Liqun Gao and Haibin Ouyang, “Solving the redundancy allocation problem with multiple strategy choices using a new simplified particle swarm optimization”, Reliability Engineering & System Safety, Vol. 144, 2015/12, pp. 147158. (DOI: 10.1016/j.ress.2015.07.019)
J.H. Lee, WeiChang Yeh, and M.C. Chuang,.“Web page classification based on a simplified swarm optimization”, Applied Mathematics and Computation, Vol. 270, 2015/11/01, pp. 1324.
ChiaLing Huang, “A particlebased simplified swarm optimization algorithm for reliability redundancy allocation problems”, Reliability Engineering & System Safety, Vol. 142, pp. 221230. (DOI: 10.1016/j.ress.2015.06.002)
WeiChang Yeh, ChhyMing Lai*, “Accelerated Simplified Swarm Optimization with Exploitation Search Scheme for Data Clustering”, PLOS ONE, Vol. 10, No. 9, 2015/09/08, (DOI: 10.1371/journal.pone.0137246).
WeiChang Yeh*, “An improved simplified swarm optimization”, KnowledgeBased Systems, Vol. 82, 2015/7/31, pp. 6069.
WeiChang Yeh*, “Orthogonal Simplified Swarm Optimization for the SeriesParallel Redundancy Allocation Problem with a Mix of Components”, KnowledgeBased Systems, Vol. 64, 2014/7, pp. 1–12.
Changseok Bae, Noorhaniza Wahid, Yuk Ying Chung, WeiChang Yeh, Neil Bergmann, and Zhe Chen, “Effective Audio Classification Algorithm Using Swarmbased Optimization”, International Journal of Innovative Computing, Information and Control, Vol. 10, No. 1, 2014/2, pp. 151–167.
WeiChang Yeh, YunChih Ke, PoChun Chang, YuanMing Yeh, and Vera Chung, “Forecasting Wind Power in the Mai Liao Wind Farm based on the MultiLayer Perceptron Artificial Neural Network Model with Improved Simplified Swarm Optimization”, International Journal of Electrical Power & Energy Systems, Vol. 55, 2014/02/28, pp. 741748.
Rasoul AzizipanahAbarghooee, “A new hybrid bacterial foraging and simplified swarm optimization algorithm for practical optimal dynamic load dispatch”, International Journal of Electrical Power & Energy Systems, Vol. 49, 2013/07, pp. 414429.
Rasoul AzizipanahAbarghooee, Taher Niknam, Masihallah Gharibzadeh and Faranak Golestaneh, “Robust, fast and optimal solution of practical economic dispatch by a new enhanced gradientbased simplified swarm optimisation algorithm”, IET Generation Transmission & Distribution, Vol. 7, No. 6, 2013/06, pp. 620635.
Neil Bergmann, Yuk Ying Chung, Xiangrui Yang, Zhe Chen, WeiChang Yeh, Xiangjian He, and Raja Jurdak, “Using Swarm Intelligence to Optimize the Energy Consumption for Distributed Systems”, Modern Applied Science, Vol. 7, 2013, No. 6, pp. 5966.
WeiChang Yeh*, “New ParameterFree Simplified Swarm Optimization for Artificial Neural Network training and its Application in the Prediction of Time Series”, IEEE Transactions on Neural Networks and Learning Systems, Vol. 24 , No. 4, 2013/04, pp. 661665.
WeiChang Yeh*, “Simplified Swarm Optimization in Disassembly Sequencing Problems with Learning Effects”, Computers & Operations Research, Vol. 39, No. 9, 2012/9, pp. 21682177.
Yuk Ying Chung and Noorhaniza Wahid, “A hybrid network intrusion detection system using simplified swarm optimization (SSO)”, Applied Soft Computing, Vol. 12, No. 9, 2012/09, pp. 30143022.
WeiChang Yeh*, “Novel Swarm Optimization for Mining Classification Rules on Thyroid Gland Data”, Information Sciences, Vol. 197, 2012/8, pp. 6576.
WeiChang Yeh, ChenMin Lin, and ShangChia Wei, “Disassembly Sequencing Problems with Stochastic Processing Time using Simplified Swarm Optimization”, International Journal of Innovation, Management and Technology, Vol. 3, No. 3, 2012/6, pp.226231.
Changseok Bae, WeiChang Yeh, Noorhaniza Wahid, Yuk Ying Chung, and Yao Liu, “A New Simplified Swarm Optimization (SSO) using Exchange Local Search Scheme”, International Journal of Innovative Computing, Information and Control, Vol. 8, No. 6, 2012/6, pp. 43914406.
TsungJung Hsieh, HsiaoFen Hsiao*, and WeiChang Yeh, “Mining Financial Distress Trend Data Using Penalty Guided Support Vector Machines based on Hybrid of Particle Swarm Optimization and Artificial Bee Colony Algorithm”, Neurocomputing, Vol. 82, No. 1, 2012/4, pp. 196206.
WeiChang Yeh*, “Optimization of the Disassembly Sequencing Problem on the Basis of Selfadaptive Simplified Swarm Optimization”, IEEE Transactions on Systems Man CyberneticsSystems, Vol. 42, No. 1, 2012/1, pp. 250261.
Changseok Bae, WeiChang Yeh and Yuk Ying Chung, “Simpli?ed Swarm Optimization for Life Log Data Mining”, IT CONVERGENCE AND SERVICES, Lecture Notes in Electrical Engineering, 2011, Vol. 107, No. 6, pp. 583589.
Mohd Afizi Mohd Shukran, Yuk Ying Chung, WeiChang Yeh, Noorhaniza Wahid,
Ahmad Mujahid Ahmad Zaidi, “Image Classification Technique using Modified Particle Swarm Optimization”, Modern Applied Science, Vol. 5, No. 5, 2011/10, pp. 150164.
Manimaran, V. Selladurai, WeiChang Yeh, and M. Sivakumar, “Particle Swarm Optimisation for FixedCharge Transportation Problem in a Multistage Supply Chain Network”, International Journal of Logistics Systems and Management, Vol. 9, No. 3, 2011/9, pp. 328350.
WeiChang Yeh*, WeiWen Chang, and ChuanWei Chiu, “A Simplified Swarm Optimization for Discovering the Classification Rule Using Microarray Data of Breast Cancer”, International Journal of Innovative Computing, Information and Control, Vol.7, No.5A, 2011/5, pp. 22352246.
Changseok Bae, WeiChang Yeh*, YukYing Chung, and SinLong Liu, “Feature Selection with Intelligent Dynamic Swarm and Rough Set”, Expert Systems with Applications, Vol. 37, No. 10, 2010/10, pp. 70267032.
WeiChang Yeh, YiCheng Lin, Yuk Ying Chung, and MingChang Chih, “A Particle Swarm Optimization Approach Based on Monte Carlo Simulation for Solving the Complex Network Reliability Problem”, IEEE Transactions on Reliability, Vol. 59, No. 1, 2010/3, pp. 212221.
WeiChang Yeh*, “A TwoStage Discrete Particle Swarm Optimization for the Problem of Multiple MultiLevel Redundancy Allocation in Series Systems”, Expert Systems with Applications, Vol. 36, No. 5, 2009/7, pp. 91929200.
WeiChang Yeh*, WeiWen Chang, and Yuk Ying Chung, “A New Hybrid Approach for Mining Breast Cancer Pattern Using Discrete Particle Swarm Optimization and Statistical Method”, Expert Systems with Applications, Vol. 36, No. 4, 2009/5, pp. 82048211.
HerShing Wang, WeiChang Yeh, PeiChiao Huang*, WeiWen Chang, “Using Association Rules and Particle Swarm Optimization Approach for Part Change”, Expert Systems with Applications, Vol. 36, No. 4, 2009/5, pp. 81788184.
WeiChang Yeh*, “A Simple Hybrid Particle Swarm Optimization”, INTECH Open Access Publisher 2008. 

WeiChang Yeh, TsungJung Hsieh, "Solving reliability redundancy allocation problems using an artificial bee colony algorithm", Computers & Operations Research, Vol. 38, No. 11, pp.14651473, 2011/11/30. 

Yeh, W.C., "Orthogonal simplified swarm optimization for the series–parallel redundancy allocation problem with a mix of components", KnowledgeBased Systems, Vol.64, pp.1–12, 2014/7. 

WeiChang Yeh, "An Improved Simplified Swarm Optimization", KnowledgeBased Systems, Vol. 82, pp. 6069, 2015 
