Passing vehicle search (PVS): A novel metaheuristic algorithm

Accepted Manuscript

Passing Vehicle Search (PVS): A novel metaheuristic algorithm Poonam Savsani , Vimal Savsani PII: DOI: Reference:

S0307-904X(15)00702-7 10.1016/j.apm.2015.10.040 APM 10852

To appear in:

Applied Mathematical Modelling

Received date: Revised date: Accepted date:

4 February 2015 6 October 2015 21 October 2015

Please cite this article as: Poonam Savsani , Vimal Savsani , Passing Vehicle Search (PVS): A novel metaheuristic algorithm, Applied Mathematical Modelling (2015), doi: 10.1016/j.apm.2015.10.040

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Highlights A novel meta-heuristic optimization method is proposed It is based on the mathematics of the passing vehicle on a two-lane highway Performance is checked for challenging engineering design problems. Performance is also checked for challenging constrained benchmark problems

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Passing Vehicle Search (PVS): A novel metaheuristic algorithm

Poonam Savsani Pandit Deendayal Petroleum University

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Gandinagar, Gujarat, India.

Vimal Savsani* Pandit Deendayal Petroleum University

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Gandinagar, Gujarat, India.

*Corresponding Author E-mail: [email protected] [email protected]

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Contact: +91-79-2327-5484

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Abstract Vehicle passing mechanisms on two-lane highways are studied since the early decade of twentieth century for which many mathematical models are proposed by different researchers. We always learn from the surrounding which we experience, out of that driving vehicles on a road is not an exception. This motivates to apply the vehicle passing mechanism on a two-lane

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highway for the optimization problems. So, a new metaheuristic optimization algorithm, ‘Passing Vehicle Search (PVS)’, is proposed in the present work which considers the mathematics of the vehicle passing on a two-lane highway. Like other metaheuristic methods, PVS is also a population based method which requires an initial set of solutions to start with and it searches the optimum solution by following the mathematical characteristics of the vehicles overtaking on a

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two-lane highway. Simplified mathematical model is developed for the vehicles moving on twolane highways, which is further shaped to solve different optimization problems. The performance of PVS is investigated on different challenging engineering design optimization problems. The results show the effectiveness of PVS over other metaheuristics optimization

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algorithms.

Keywords: Passing vehicle search (PVS) algorithm, engineering design optimization,

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1. Introduction

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constrained optimization, meta-heuristics

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Optimization methods are broadly classified into deterministic methods and meta-heuristic methods. Deterministic methods are the classical methods like gradient based methods, sequential quadratic programming, geometric programming, sub-space trust region method, etc,

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which make use of specific mathematical rules at each iterations and will always give same solution under specified conditions. These types of methods may require gradient information, Hessian matrix information and starting point to start with. The effectiveness of the solution depends on the selection of the starting point. The main drawbacks of these classical methods are that it can only find the local optimum solution (fails for multi-modal problems), it is ineffective for highly constrained problems and also difficult to code. However, many of such methods are available in the optimization toolbox of Matlab which can be readily used to find the local

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optimum solutions. Due to these limitations, classical methods are not often used for the practical engineering optimization problems now-a-days. However, these methods are fast in finding the solutions compared to other meta-heuristics methods. Many research works are carried out since last 30 years for the development of new metaheuristic optimization algorithms and its exploration on diverse real life problems. Many names

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are suggested for such methods, though purpose remains the same, like, population based methods, heuristics methods, meta-heuristics methods, advanced optimization techniques, computational intelligence algorithms, non-tradition optimization techniques, clever algorithms (Xing and Gao, 2014) etc. It is logical to classify Genetic Algorithm (GA) (Holland, 1975), Ant Colony Optimization (ACO) (Dorigo, 1992), Particle Swarm Optimization (PSO)( Kennedy and

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Eberhart, 1995) and Artificial Immune Algorithms (AIA) (Farmer, 1986) as traditional metaheuristics because they are well recognized algorithms and lots of books, research papers and technical articles are available for the same (Xing and Gao, 2014). Furthermore, the algorithms developed after 1995, can be termed as non-traditional meta-heuristics optimization algorithms, out of which most of the algorithms are developed in the last decade, based on the different

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behavior of nature. These algorithms have drawn the attention of researchers and plenty of work is reported either based on the modifications or on the applications of these algorithms. It is very

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difficult to mention all such algorithms in this paper, and so a broad classification is briefly specified for some of these algorithms. Most of the algorithms developed are nature inspired

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based on the different behavior of flora and fauna. Some algorithms are developed based on the basic physics of the nature and also some are developed based on human related activities. So,

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these algorithms can be broadly classified as (a) Fauna based algorithms, (b) Flora based algorithms, (c) Physics based algorithms and (d) Human activity based algorithms. Some of the Fauna based algorithms include Bee inspired algorithms (Karaboga, 2005), Biogeography-based

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optimization algorithms (Simon, 2008), Bacteria inspired algorithms (Passino, 2002), Bat inspired algorithms (Yang, 2010), Cat optimization algorithms ( Chu and Tsai, 2007), Cuckoo search algorithms (Yang, 2009), Luminous insect based algorithms (Kridhnanand and Ghose, 2005; Yang, 2009), Fish inspired algorithms (Hersovoci et. al, 1998; Li and Qian, 2003), Frog based algorithms (Eusuff and Lansey, 2003), Rat inspired algorithms (Taherdangkoo et al, 2012), Cockroach inspired algorithms(Chen and Tang, 2010), Dove based algorithms (Su et al, 2009), Eagle based algorithms (Yang and Deb, 2010), Goose based algorithms (Liu et al, 2006),

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Monkey search algorithms( Mucherino, 2007) and Wolf colony inspired algorithms (Liu et al, 2011). Very few flora based optimization algorithms are developed which take in Invasive weed optimization algorithm (Mehrabian and Lucas, 2006) and Flower pollinating algorithm (Yang, 2012). Physics based algorithms include Charged system search algorithm (Kaveh and Talatahari, 2010), Electromagnetism based algorithm (Birbil, 2003), Gravitational search

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algorithm (Rashedi et al, 2009), Water drop algorithm (Shah-Hosseini, 2007) and Water cycle algorithm (Eskandar et al, 2012). Some of the algorithms are developed based on the mathematics of human based activities like Music inspired algorithms (Geem et al, 2001), Imperialist competitive algorithm (Atashpaz-Gargari, 2007), Harmony element algorithm (Rao et. al, 2009; Cui et al, 2009), Grenade explosion algorithm (Ahrari et al, 2009) and Teaching-

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learning based optimization (Rao et al, 2011). There are numerous applications of these algorithms in many area of research, out of which some includes, communication optimization, data mining, image processing, power system optimization, robot control, scheduling, engineering design optimization, stock market prediction, composite structures, expert system design, water resource management, cloud computing, inventory management, supply chain

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management, structure optimization, etc. As per ‘No Free Lunch (NFL)’ theorem (Wolpert & Macready, 1997), no single algorithm is best suited for all the optimization problems. So,

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continual research is required in investigating different algorithms, modifying existing algorithms (Wang et al, 2011; Wang et al., 2012; Wang et al., 2014) and developing new

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algorithms to suit real life applications with high efficacy. In this paper, a new meta-heuristic optimization algorithm, called ‘Passing vehicle search

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(PVS)’, is developed based on the passing behavior of a vehicle moving on a two-lane highway. This algorithm can be characterized under human activity based algorithms. The objective of this proposed algorithm, like other meta-heuristics, is to find the global solution or near optimal

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solutions for a given function. The performance of the proposed algorithm is tested on several challenging engineering design problems. The performance of PVS is checked based on the tendency of the algorithm to find best solutions in different runs, mean of solutions obtained in different runs, computational efforts, computational time, convergence, Friedman rank test (Joaquin et al, 2011) and Holm-Sidak multiple comparison test (Holm, 1979).

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The subsequent sections discusses the mathematical modeling for the passing vehicles, implementation steps of PVS for the optimization, performance of PVS on engineering design problems along with different constrained benchmark problems and conclusions at the last. 2. Mathematical modeling of passing vehicles on two-lane highways

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In the most developed and developing countries, two lane highways contribute to a major road networks. In the United States, as per the recent reports, two lane highways consists of over 65% of total urban and rural route (FHWA) and in India it consists of about 54% (NHAI). More than 60% of accident took place on rural two lane highways (Lamm et al., 2006). The essential thing in two lane vehicle passing is to have safe overtaking opportunities(passing) which depends on

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many complex inter related parameters such as availability of gaps in the opposing traffic stream, speed and acceleration characteristics of individual vehicles, traffic and driver characteristics, as well as road and weather conditions. Many studies are reported for the traffic simulation on a two lane highways, considering different factors like overtaking, platoons, delays, traffic quality and speed distribution ( Brilon, 1998; Erlander, 1971a, 1971b; Hammond,

Luttinen (2000; 2001a; 2001b; 2002).

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1941;Kallberg, 1980; McLean, 1989; Miller, 1963;Yeo, 1964; Ghods and Saccomanno , 2013;

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Passing on a two lane highway is done by using the lane reserved for oncoming vehicles, which occur, when passing demand and passing supply occur at a particular time and location. Passing

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is an effect of speed differences in the vehicles moving on the road and it occur when there is long enough headway by the oncoming vehicle and long enough sight distance for the overtaking

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vehicle. Passing also depends on drivers perspective as all the drivers have different desired speeds which again depends on factors like type of journey, type of vehicle used, whether conditions, road geometries, traffic conditions and driver’s personality (McLean, 1989, Luttinen,

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2001). Three major circumstances occur when a vehicle approach a slower vehicle; (a) it passes the slower vehicle ‘on a fly’ , (b) it follows slower vehicle till passing opportunity occurs and (c) it follows slower vehicle with no passing intention. If vehicles do not pass the slower vehicle, platoons begin to form. Platoons again affect the desired speed of the vehicle. So, mathematical theories of two-lane are difficult to establish and analyze (Luttinen, 2001)

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In this paper, a simplified mathematical model is formulated for passing vehicles on a two lane highways. Consider three vehicles on a two lane highways as shown in the Figure-1; Back Vehicle (BV), Front Vehicle (FV) and oncoming vehicle (OV), which are responsible for the passing mechanism. Passing only occurs if passing supply is more than passing demand. BV intends to pass FV, which is only possible if FV is slower than BV. If FV is faster than BV, then

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no passing is possible by BV. Moreover, passing depends on the position and speed of OV and also on the distance between BV, FV and OV and their velocities, which leads to different conditions as discussed below. Let,

y – Distance between FV and OV X1, X2, X3 – Distance from reference line

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x – Distance between BV and FV

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V1, V2, V3– Velocity of BV, OV and FV respectively

Assume three different vehicles (BV, FV and OV) on a two-lane highway possessing different

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velocities (V1, V2 and V3) at any particular time instance. Two primary conditions occurs based on the velocity of FV, i.e. FV is slower than BV (V1>V3) and vice versa. If FV is faster than BV , then no passing is possible and BV can move with its desired velocity. Passing is possible only if

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FV is slower than BV. In this situation also, overtaking is only possible, if the distance from the FV at which overtaking occurs(x1) is less than the distance travelled by OV (y-y1). So, following

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conditions arises for the selected vehicles.

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1. FV is slower than BV (V3 x1 (Secondary condition-1) (b) (y-y1) < x1 (Secondary condition -2)

2. FV is faster than BV (V3>V1) (Primary condition – 2)

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2.1. Primary Condition-1 This condition deals with the case of FV slower than BV, which is again divided into two subconditions; Secondary condition-1 and Secondary condition-2. Mathematical expressions for

2.1.1 Secondary Condition-1

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these conditions are explained as below:

Let x1 be the distance travelled by FV at which BV catches FV and thus passes it. Consider time equals to‘t’ for BV to catch FV.

𝑥1 = 𝑉3 𝑡 Distance travelled by BV in time t will be 𝑥 + 𝑥1 = 𝑉1 𝑡

=

𝑥+𝑥1 𝑉1

Hence, 𝑉𝑥

3 𝑥1 = 𝑉 −𝑉

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1

(1)

(2)

(3)

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𝑉3

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𝑥1

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Equating (1) and (2)

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So, distance travelled by FV in time t is given by:

(4)

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Now, Distance travelled by OV in time ‘t’ 𝑦1 = 𝑉2 𝑡

Substituting value of x1 in equation (1), we get

(5)

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𝑥

𝑡 = 𝑉 −𝑉 1

(6)

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Putting (6) in (5), we get 𝑉𝑥

2 𝑦1 = 𝑉 −𝑉 1

(7)

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Now, change in position of BV is given by: 𝐵𝑉𝑐1 = 𝑥 + 𝑥1

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Substitute value of x1 from equation (4) 𝑉

1 𝐵𝑉𝑐1 = 𝑥 (𝑉 −𝑉 )

(9)

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1

Substitute value of x corresponding to the reference line, which leads to: 𝑉

1 𝐵𝑉𝑐1 = (𝑋3 − 𝑋1 ) (𝑉 −𝑉 ) 1

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𝑉

1 𝑋1 + 𝐵𝑉𝑐1 = 𝑋1 + (𝑋3 − 𝑋1 ) (𝑉 −𝑉 ) 3

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1

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Change in the position for BV from reference line equals to:

2.1.2 Secondary condition – 2

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This condition is shown in Figure-1 for two different values of y-y1, in which one value is positive and other value is negative. In either of the situations, BV cannot overtake FV before OV

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crosses BV. Accident can be avoided, if BV does not change the lane till OV crosses BV. The distance at which BV and OV meets is somewhere in between the initial positions of BV and OV.

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So, the change in position of BV is given by: 𝐵𝑉𝑐2 = 𝑅(𝑥 + 𝑦) Where, R is a random number between 0 and 1. Change in the position for BV from reference line equals to:

(12)

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𝑋1 + 𝐵𝑉𝑐2 = 𝑋1 + 𝑅(𝑥 + 𝑦) = 𝑋1 + 𝑅(𝑋2 − 𝑋1 ) (13) 2.2. Primary condition-2 If FV is faster than BV, then it is not possible by the BV to pass FV. So, change in the position of

𝑋1 + 𝐵𝑉𝑐3 = 𝑋1 + 𝑅𝑥 = 𝑋1 + 𝑅(𝑋3 − 𝑋1 )

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the BV is:

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3. PVS for the optimization of a function

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To implement the above theory of passing vehicles for the optimization process, some correlation is required to be developed between the optimization terminology and the above theory. Let us assume set of solutions as different vehicles on a two lane highways. Value of objective function or fitness value represents vehicles velocity, i.e. solution with better fitness value is considered as a vehicle with high velocity. Design variables decide the position of the

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vehicle on a highway. So, PVS starts with a set of solutions called population of vehicles, out of which three vehicles (solutions) are selected at random. Among these three selected vehicles,

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current solution is correlated as BV and other two can be OV and FV randomly. Distance between the vehicles and their respective velocities are assigned based on the population size and its fitness value. After assigning distance and velocities, vehicles are checked for the conditions

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of passing. Based on the applied conditions, vehicles change their respective positions on the

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highways (changes the current solutions). Step-wise procedure for the implementation of PVS for the optimization of a given function is

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given in this section and PVS is explained with the aid of flow chart in Figure-2. Step 1: Define Parameters Population Size (PS), Stopping Criteria (like Number of Generations (NOG), Maximum FE, Error, etc), Number of Design Variables (DV), Bounds on Design Variables (LB, UB) Define function for the optimization Minimize f(X)

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Subject to X = { x1, x2,…xDV } Where, LBi ≤ xi ≤ UBi Step 2: Generate Initial Population

mathematically, it represents set of solutions.

𝑋𝑘…𝑃𝑆,𝑖…𝐷𝑉

𝑥1,1 𝑥2,1 =[ ⋮ 𝑥𝑃𝑆,1

𝑥1,2 𝑥2,2 ⋮ 𝑥𝑃𝑆,2

… …

𝑥1,𝐷𝑉 𝑥2,𝐷𝑉 ⋮ ] … 𝑥𝑃𝑆,𝐷𝑉

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Population in PVS represents number of vehicles on a two-lane highway and

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Step 3:

Select three vehicles (solutions), out of which one solution represents current vehicle (current solution) (BV) and two other solutions are randomly chosen.

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So, select r2 and r3 randomly as explained below

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Where, r1= current solution (BV), Xr1,i=1 to DV r2 ≠ (r1,0) = R1(PS), (OV), Xr2, i=1 to DV

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r3≠ (r1, r2, 0) = R2(PS), (FV), Xr3, i=1 to DV

Step 4:

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R1, R2 are random numbers  (0,1)

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Calculate Distance (D) and velocities (V) of the vehicles Distances between the vehicles are calculated based on the fitness value. Arrange the

population in an ascending order, Now, calculate the distance by using Equation-16. It is observed from the expressions that the value of distance is normalized to 1. For example if we consider a population size of 10 i.e. PS=10, then the normalized distance for the 3rd , 6th and 8th population member will be 0.3, 0.6 and 0.8 respectively, which ultimately indicates the distance from the datum. The velocity corresponding to the vehicle is calculated using Equation-17. For

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the vehicles considered above, the values of the velocities will be 0.7, 0.4 and 0.2 respectively. It can be noted from the values that the best solution among the three is assigned higher velocity (3rd solution is assigned 0.7 and 8th solution is assigned 0.2). 1

𝐷𝑘 = 𝑃𝑆 (𝑟𝑘 ) 𝑉𝑘 = 𝑅𝑘 (1 − 𝐷𝑘 )

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(16) (17)

Where, k=(1,2,3)and R is a random number  (0,1) Step 5: x1 and y1

using equation

𝑥 = |𝐷3 − 𝐷1 | 𝑦 = |𝐷3 − 𝐷2 |

(18) (19)

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x1, y1 using equation (4) and (7)

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Calculate distance x, y,

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Step 6: Update solution

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If V3
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If (y-y1)>x1

𝑛𝑒𝑤 𝑋𝑘…𝑃𝑆,𝑖…𝐷𝑉 = 𝑋𝑘,𝑖=𝐷1 + 𝑉𝑐𝑜 (𝑅𝑘 )(𝑋𝑘,𝐷1 − 𝑋𝑘,𝐷3 )

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Else

𝑛𝑒𝑤 𝑋𝑘…𝑃𝑆,𝑖…𝐷𝑉 = 𝑋𝑘,𝑖=𝐷1 + (𝑅𝑘 )(𝑋𝑘,𝐷1 − 𝑋𝑘,𝐷2 )

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End

Else

𝑛𝑒𝑤 𝑋𝑘…𝑃𝑆,𝑖…𝐷𝑉 = 𝑋𝑘,𝑖=𝐷1 + (𝑅𝑘 )(𝑋𝑘,𝐷3 − 𝑋𝑘,𝐷1 )

End

(22)

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Where, Ri  (0,1) , 𝑉

1 𝑉𝑐𝑜 = 𝑉 −𝑉 1

3

(23)

Step 7:

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Accept solution if it is better than the previous solution

Maintain diversity in the population by removing the duplicates as explained below

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For k = 1: 2: PS If Xk = Xk+1 i=rand*(DV) Xk+1,i =LBi + rand*(UBi - LBi) Endif Endfor Step 8:

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Repeat the procedure till the termination criteria is attained.

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4. Comparison of PVS with other meta-heuristics Like other meta-heuristics (GA, PSO, ABC, BBO, TLBO etc.), PVS is also a population based method, which implements, initial set of solutions for its working. All these algorithms are based

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on specific natural principles, like GA is based on the natural phenomena of evolution of species, PSO is based on the foraging behavior of swarm of birds, ABC is based on the foraging behavior

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of honey bee to find nectar, BBO is based on the theory of immigration and emigration of species from one place to the other, TLBO is based on the teaching-learning philosophy and so

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on. PVS uses vector difference between the solutions like PSO, DE, ABC, TLBO, CSA and FPA. It does not use the information of the best solution for updating the solutions like, PSO, CSA, FPA and TLBO. PVS uses the greedy selection like ABC, CSA and TLBO for accepting the solutions. Unlike ABC and TLBO, PVS do not require different phases for updating the solutions, like employed and onlooker bee phase in ABC and teaching and learning phase in TLBO. Like DE, PVS involve three solutions for updating the existing solutions, but however, the search expressions are different for both the algorithms. The unique feature of this algorithm

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is that it uses three different mathematical expressions for updating the solutions with different probabilities. Different algorithm requires different algorithm parameters for its proper working like, GA requires crossover probability and mutation probability, ABC requires number of employed bees, onlooker bees and limits, PSO requires learning factors, CSA requires probability and beta value. Only some algorithms are mentioned above as it is not possible to

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mention all the algorithms in this paper. These algorithms-parameters influence the performance of the algorithm and proper values of these parameters are essential to be tuned for particular applications. Sometimes, it is cumbersome to find appropriate value of these parameters and often user make a compromise by selecting some standard values from the literature. TLBO is the algorithm which do not requires any algorithm-parameter for its working. Like TLBO, PVS

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is also an algorithm-parameter free meta-heuristics optimization method, which can be considered as one of the effective features of this algorithm. 5. Performance of PVS

This section is projected to investigate the performance of PVS on 13 dissimilar challenging

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engineering design optimization problems. The performance of PVS is compared with the other well developed optimization techniques. PVS is coded in Matlab on an Intel® core™ i3 laptop

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with [email protected]. All the constrained problems are converted into unconstrained problems using static penalty method (Rao, 2002) approach. A penalty value is added to the objective function for each infeasible solution so that it will be penalized for violating the constraints. This

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method is popular because it is simple to apply. It require amount of penalty to be added and

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varies for different problems. An optimization problem is typically written as: 𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒 𝑓(𝑋), 𝑋 = {1,2 … 𝐷𝑉} 𝑆𝑢𝑏𝑗𝑒𝑐𝑡𝑒𝑑 𝑡𝑜: 𝑔𝑖 (𝑋) ≤ 0, 𝑖 = 1 … 𝑝 ℎ𝑖 (𝑋) = 0, 𝑖 = 𝑝 + 1 … 𝑁𝐶

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Where, p is the total number of nonlinear constraint and NC is total number of constraints. The constrained optimization problem can be converted in to unconstrained optimization problem using static penalty approach as follows: 𝑝

𝑁𝐶

𝑖=1

𝑖=𝑝+1

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𝑓(𝑋) = 𝑓(𝑋) + ∑ 𝑃𝑖 max{𝑔𝑖 (𝑋), 0} + ∑ 𝑃𝑖 max{|ℎ𝑖 (𝑋)| − 𝛿, 0} (25)

Where, Pi is a penalty factor which is generally assigned a large number. 6. Engineering design optimization problems:

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Ten different widely used engineering design problems (pressure vessel, spring, welded beam, speed reducer, bearing, multi-plate clutch, step-come pulley, robot gripper, hydrostatic thrust bearing, Belleville spring) and three additional challenging problems (4-stage gear box, stiffened welded shell and planetary gear box) are considered for the investigation. The characteristic of these problems are given in Table-1. The details of 10 widely used engineering problems are

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readily available in the literature (Rao et al, 2011; Eskander, 2012; Wang et al., 2009).

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Matlab is a powerful tool for many engineering applications as it contains several ready to use toolboxes, from which ‘Optimization Toolbox’ and ‘Genetic Algorithm and Direct Search (GADS) toolbox’ are used to solve optimization problems. It is very strange to notice that no

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comparison of such toolboxes is reported in literature with other meta-heuristics. So, in this paper efforts are put to compare the results with the genetic algorithm(GA) and sequential

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quadratic programming (SQP) available in ‘GADS toolbox’ and “optimization toolbox’ respectively. SQP is a classical deterministic method which gives same solution (local optima) To determine appropriate starting point to obtain optimum

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for a particular starting point.

solution (global solution) is quite troublesome and so, different starting points are required to be investigated to build assurance in the solutions. In this paper, the results are obtained using 25 different starting points for SQP and 25 independent runs for the meta-heuristics algorithms. Following parameters are considered for GA and SQP in Matlab:

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For SQP: DiffMinChange = 0.01, DiffMaxChange = 0.1, TolX = 1e-100, TolFun = 1e-100, TolCon = 1e-100, MaxFunEvals = 25000, MaxIter= 25000. For

GA:

PopulationSize

=

50,

Generations

=

500,

CrossoverFcn

=

@crossoverarithmetic,SelectionFcn = @selectionroulette, StallGenLimit = 500,StallTimeLimit

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= 120,TolFun = 1e-100 In this paper, problems for pressure vessel design, spring design, welded beam design, speed reducer design, bearing design and multi-plate clutch design are solved using SQP, GA and PVS, whereas, problems for step-cone pulley design, robot gripper design, hydrostatic thrust bearing design, Belleville spring design, four stage gear box design, stiffened welded shell design and

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planetary gear design are solved using SQP, GA, CSA, FPA, GSA, BBO and PVS. 6.1 Pressure vessel design optimization problem

This problem is intended to minimize the total cost of a pressure vessel. The objective function is non-linear subjected to 3 linear and one nonlinear inequality constraints. There are 4 design

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variables, from which two are discrete and two are continuous. The first two discrete variables can attain value in the multiple of 0.065. The ratio of feasible solutions to search space (F/S) is

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approximately 0.40, which indicates the total feasible region in the search space. The ratio F/S is obtained using 100000 random solutions in the search space. The best known solution for this problem is f(X) = 6059.714335 at X = {0.8125, 0.4375, 42.0984456, 176.6365958}. At the

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optimum solution, there are two active constraints. This problem is attempted by many researchers using different optimization techniques and their variants. Some basic method and

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their variants considered in this paper for this problem includes, DE-PSO, TLBO, ABC, CVIPSO, BIANCA, DEC-PSO, BA, CSA, ISA, FFA, WCA, NM_PSO and MBA (Liu et al., 2010;

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Rao et. al., 2011; Akay &Karaboga, 2012; Montemurro et al., 2013; Mazhoud et al., 2013; Chun et al., 2013; Gandomi et al., 2013; Gandomi et al., 2013; Gandomi, 2014; Adil & Fehmi, 2015; Eskandar, 2012 ; Zahara & Kao, 2009; Ali et al, 2013 ). Some of the researchers have solved this problem using all the continuous design variables (Zahara & Kao, 2009; Eskandar, 2012; Ali et al, 2013). So, the results of PVS are obtained for both the cases; (a) with discrete design variable and (b) with continuous design variables. For PVS, population size of 50 is considered and stopping criteria as 42100 FE. Many researchers have used different function evaluations to

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investigate the performance of their proposed methods. So, the results obtained by using PVS at 5000, 15000, 20000 and 42100 FE are compared with the results of other algorithms. The results are mainly compared based on the mean solutions obtained in 25 different independent runs. The results are summarized in Table-2 and it is observed that PVS is capable of finding the global solution for the pressure vessel problem. The mean of solutions obtained in 25 runs at 5000 FE is

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better than ISA, at 15000 FE is better than CSA, at 20000 FE is better than BA. However, PVS has shown inferior results than TLBO and PSO-DE at 20000 and 42100 FE. Moreover, PVS has performed better than rest of the algorithms. PVS has outperformed WCA, MBA and NM-PSO in terms of mean and worst solutions by using continuous design variables. Moreover, it is also observed that SQP has failed in finding the feasible solution for 25 different starting points.

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Results obtained using GADS-toolbox is also not effective as it fails to find the global solution and its mean performance is also poor compared to the other methods. 6.2. Spring design optimization problem

The problem is defined to minimize the weight of a tension/compression considering constraints

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on minimum deflection, shear stress, surge frequency, limits on geometric dimensions. There are three continuous design variables; wire diameter (x1), the mean coil diameter (x2), and the

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number of active coils (x3). The optimum solution reported so far is f(X) = 0.012665 with X = {0.051749, 0.358179, 11.203763}. At this optimum solution there are 2 active constraints. Total feasible region accounts to nearly 1% of the search space. This problem is solved by many

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researchers using different optimization techniques like PSO-DE, TLBO, ABC, WCA, CVIPSO, BIANCA, BA, MBA, ISA and FFA. Many researchers have used different function

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evaluations to investigate the performance of their proposed methods. Thus, PVS is investigated using 2000, 8000, 20000 and 42100 FE considering the population size of 50. The results

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obtained in 25 independent runs are summarized in Table-3, from which it is observed that PVS is competent in finding the optimum solution for the spring design problem. For this problem also, GA and SQP have performed poorly compared to other meta-heuristics in obtaining the optimum solution and mean solutions. The minimum function evaluations of 2000 are reported by Eskandar (2012) using WCA. It is observed form the result that PVS fails to find the optimum solutions in 2000 function evaluations for 25 runs, but PVS has outperformed WCA in terms of mean result. For 8000 FE PVS is better than ISA and MBA and for 20000 FE PVS is better than

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BA and equivalent to TLBO. Also for 42100 FE PVS is nearly same to that of PSO-DE. PVS has performed better than rest of the algorithms in terms of quality of the solutions and computational efforts.

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6.3. Welded beam design optimization problem This problem is for the cost optimization of a welded beam considering four design variables for the height of weld (x1), length of weld (x2), height of beam (x3) and width of beam (x4). Constraints are considered for the shear stress, bending stress, buckling load on the bar, end deflection, and side constraints. This problem contain nearly 3.5% of feasible region in the

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search space, for which, the optimum value reported in the literature is f(X) = 1.724852 at X = {0.205730, 3.470489, 9.036624, 0.205730} with two active constraints. This problem is solved using several researches using different optimization techniques such as PSO–DE, TLBO, ABC, WCA, CVI-PSO, BIANCA, BAT, MBA, and FFA. This problem is solved considering the population size of 50 with 20000 and 50000 function evaluations. The results are summarized in

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Table-4, from which it can be observed that at 20000 FE PVS is better than BA and TLBO and at 50000 FE PVS have performed nearly same to that of FFA and MBA. The average performance

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(based on mean value) of PVS is better than rest of the algorithms. For this problem also SQP and GADS toolbox have shown inferior performance.

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6.4. Speed reducer problem

The objective of this problem is to minimize weight considering subjected to different

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constraints on bending stress, surfaces stress, transverse deflections of the shafts and stresses in the shafts. The 7 design variables considered are the face width(x1), module of teeth(x2), number

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of teeth in the pinion(x3), length of the first shaft between bearings(x4), length of the second shaft between bearings(x5) and the diameter of the first and second shafts respectively(x6,x7). The third variable (number of teeth) is an integer, while others are continuous. This problem contains nearly 0.4% of feasible region for which the optimum solution reported is f(X) = 2996.3481 at X = {3.49999, 0.6999, 17, 7.3, 7.8, 3.3502, 5.2866} with 3 active constraints. This problem is solved by considering two different range for the design variable ‘x5’ in the literature by various researchers ; (a) 7.8 ≤ x5 ≤ 8.3 (Design variable range-1) and (b) 7.3 ≤ x5 ≤ 8.3 (Design variable

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range-2). Problem with design variable-1 is solved with PVS using a population size of 50 with function evaluations as 5000, 20000 and 54350. The results are shown in Table-5. It is observed from the results that PVS has performed better than CSA with 5000 FE and nearly same as TLBO with 20000 FE. PVS have performed better than rest of the algorithms in terms of quality of solutions and computational effort.

Problem with design variable-2 is solved using 6000,

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15150 and 30000 FEs with a population size of 50. It is observed from the results that at 6000 FE PVS have performed better than MBA and at 15150 FE PVS have performed nearly same as WCA. At 30000 FE, PVS have performed nearly same as DELC. For this problem also GADS and optimization toolbox failed to attain the optimum solutions.

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6.5. Bearing design optimization problem

The objective of this problem is to maximize dynamic load carrying capacity considering 10 geometric design variables and 9 constraints based on assembly and geometric limitations. Out of 10 design variables, one design variable (number of balls in bearing) is required to attain integer value. This problem have nearly 1.5% of the feasible region and best known solution for

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this problem is f(X) = 81859.74 at X = { 21.42559, 125.7191, 11, 0.515, 0.515, 0.424266, 0.633948, 0.3, 0.068858, 0.799498} for which there are 4 active constraints. This problem is

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solved using various optimization techniques like GA, ABC, TLBO, WCA, MBA and MDDE. For PVS, FE are considered as 10000 and 20000 with a population size of 50. The results are summarized in Table-6, from which it is observed that PVS has done better than GA and ABC.

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Also, it has performed slightly better than TLBO, but results of PVS are inferior to MDDE in terms of mean solution. WCA and MBA has shown better results than all the algorithms but this

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result is not comparable as value of number of balls(x3) has to be an integer value, which is considered as continuous by WCA and MBA.

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6.6. Multi-plate disc clutch brake optimization problem This problem is intended to minimize weight of a multiple disc clutch brake, considering five discrete design variables as inner radius, outer radius, thickness of discs, actuating force, and number of friction surfaces. There are 8 different constraints based on geometry and operating conditions. The feasible region accounts to nearly 70% of the search space. The optimum solution for this problem is f(X) = 0.313656611, at X = {70, 90, 1, 810, 3} with one active

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constraints. The results of PVS are compared with other optimization techniques like ABC, TLBO and WCA, which are shown in Table-7. PVS is implemented by considering the population size of 20 with 600 and 1200 function evaluations. It is observed from the results that PVS has performed nearly same with TLBO and has shown inferior results than WCA and ABC for obtaining the mean solutions. However, PVS is capable of finding optimum solutions even at

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600 function evaluations in 25 independent runs. 6.7. Step-cone pulley

A step-cone pulley is to be designed for minimum weight with 5 design variables, out of which four design variables are for the diameters (d1, d2 d3 and d4) of each step and last one is for the width of the pulley (w), as shown in the Figure 3 . The design is subjected to 11 constraints out

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of which 3 are equality constraints and the rest are inequality constraints, which are used for the assurance of same belt length for all the steps, tension ratios, and power transmitted by the belt. The optimum solution reported so far in the literature is f(X) = 16.63450513 at X = {40, 54.76430219, 73.01317731, 88.42841977, 85.98624273} with 4 active constraints. This problem can be considered as one of the challenging optimization problem as the ratio of the feasible

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region to that of the total search space is nearly 0.0001 and it also contains 3 equality constraints, which increases the difficulty of the problem. This problem is solved by using PVS, CSA (Yang

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and Deb, 2009), FPA (Yang, 2012), GSA (Rashedi et al, 2009) and BBO (Simon, 2008) considering 15000 FEs with a population size of 50 and by using GA and SQP (from Matlab

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toolbox) considering 25000 FEs. The results are compared with the published results of TLBO,

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ABC and MDDE which are illustrated in Table-8 and 9.

6.8. Belleville Spring

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The problem is to minimize the weight of a Belleville spring considering 4 design variables like thickness of the spring, height of spring, external diameter of the spring and internal diameter of the spring, as shown in the Figure-4. Out of these design variables, thickness of the spring is a discrete variable and the rest are continuous variables. The design is subjected to constraints for compressive stress, deflection, height to deflection ratio, height to maximum height ratio, outer diameter, inner diameter, and slope of the spring. The feasible region is nearly 0.4% of the total search space from which the well-known solution if f(X) = 1.979674758 at X = {0.204143354,

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0.2, 10.03047329, 12.01}, with four active constraints. This problem is solved using PVS, CSA, FPA, GSA and BBO with a population size of 50 and FEs as 15000 and the results are shown in Table-8 and 9. 6.9. Robot gripper

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Robot gripper is designed to minimize the difference between maximum and minimum force applied by the gripper for the range of gripper end displacements, considering 7 continuous design variables for the geometric dimensions as shown in Figure-5. The design is subjected to six different geometric constraints which makes the feasible region near to 2.5% of the total search space. The optimum solution reported in the literature so far is f(X) = 4.247643634 at X = { 150, 150, 200, 0, 150, 100, 2.339539113}, with one active constraints. Moreover, the optimum

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solution occurs at the bound points of the design variables except last design variable, which is one of the unique features of this problem. This problem is solved for 25000 FEs and a population size of 50 by using PVS, CSA, FPA, GSA and BBO. The results are summarized in

6.10. Hydrostatic thrust bearing

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Table-8 and 9.

In this problem, hydrostatic thrust bearing is required to be design for minimum power loss with

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four design variables for bearing step radius (R), recess radius (Ro), oil viscosity (μ) and flow rate (Q) as shown in the figure-6. The design is subjected to seven different constraints for load

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carrying capacity, inlet oil pressure, oil temperature rise, oil film thickness and physical constraints which make the feasible region nearly to 0.3% of the total search space. The optimum

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solution reported in the literature so far is f(X) = 1625.4427649821 at X = {5.95578050261541, 5.38901305194167, 0.00000535869726706299, 2.26965597280973}. This problem is also a challenging one due to its highly sensitive nature of the design variables. This problem is also

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solved for 25000 FEs with a population size of 50 by using PVS, CSA, FPA, GSA and BBO. Meta-heuristic methods have a need of specific algorithm-parameters for its effective working. These parameters hamper the quality of the solution and the searching ability of the algorithm. These parameters are problem specific and it may perhaps ensure that particular set of parameters work effectively on particular class of problems but fails on the other. So, these parameters are decided after conducting experiments with different values and the set of

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parameters which has shown better performance in the results are considered for the further investigation. In this work, the parameters considered for different algorithms are; for BBO, maximum immigration rate (Imax) =1, minimum immigration rate (Imin)=0, mutation coefficient (m)=0.05; for GSA,

gravitation constant (G0)=100, gravitation reduction factor (α)=20,

Normalizing factor (Fnorm)=2; for CSA, probability factor (pa)=0.25, levy flight factor (β)=1.5

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and for FPA, probability switch factor (pa)=0.25, levy flight factor (β)=1.5. It is observed form the results of Table-8 and 9 that for step-cone pulley design problem, PVS have performed nearly same as TLBO and ABC to attain the optimum solution. PVS has performed better than TLBO and ABC for the mean of function obtained in 25 runs. CSA and FPA have outperformed all the algorithms for the mean function value, but were unable to find

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the optimum solution. For robot gripper problem, except BBO and GSA, all algorithms have performed equally well to find the global solution. CSA has shown better results than all the algorithms to find the mean function value; however, PVS has ranked second. For hydrostatic thrust bearing, PVS is capable of finding the optimum solution in 25 runs; however CSA and FPA failed to attain the optimum solution. TLBO has outperformed all the algorithms to find the

Belleville spring, performance of

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mean of solutions, but it has computed this value using double function evaluations. For PVS, ABC and TLBO is nearly same. It can also be noted

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from the results that, GA and SQP have performed poorly for all the considered problems. However, SQP failed to find the feasible solutions for the robot gripper and hydrostatic bearing

problem.

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problem. SQP has reported only 8% of feasible solutions for the Belleville spring design

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The performance of PVS is evaluated considering three more challenging engineering design problems of planetary gear train, stiffened welded shell and 4-stage gear box, where all the

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design variables are discrete and the problem have many local optimum solutions in a very restricted feasible region. Detailed mathematical expressions for these problems are given in the Appendix.

6.11. Planetary gear train design optimization problem This problem is formulated by Simionescu et al., (2006), where the objective is to perform a gear-teeth number synthesis of an automatic planetary transmission as shown in the Figure-7,

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used in automobiles to minimize the maximum errors in the gear ratio. This problem considers 6 design variables based on the number of teeth in gears (N1, N2, N3, N4, N5, and N6) which can only take integer values. There are three more discrete design variables; number of planet gears (P) and module of gears (m1 and m2) which can only take specified discrete values. The problem is subjected to 11 constraints for different assembly and geometric restrictions, out of which one

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is equality constraint. The ratio of feasible region to the total search space is less than 0.0001 and the best solution reported so far in the literature is f(X)= 0.525 with X= {40, 21, 14, 19, 16, 69, 5, 2.25, 2.5}, with one active constraint 6.12. Stiffened welded shell design optimization problem

This problem is formulated by Jarmai et al. (2006) to optimize cost of a cylindrical shell member

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that is orthogonally stiffened by using ring stiffeners and stringers of halved I-section as shown in the Figure-8 . There are five design variables for shell thickness (t), number of longitudinal stiffeners (stringers) (ns), number of ring stiffeners (nr), box height (hr), and stringer stiffness height (h), which can only attain specified discrete values. The design is subjected to five different constraints which consider bucking and manufacturing restrictions. The feasible region

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is nearly 30% of the total search space for which the best value reported by Jarmai et al. (2006) is

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f(X) =55326.29 with X= {14, 27, 10, 250, 203} with no active constraints.

6.13. Four-Stage gear box

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This problem was introduced by Pomrehn and Papalambros (1995) with an objective to minimize the weight of a gear box. There are 22 design variables for the number of teeth, blank

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thickness and the positions of gear and pinion. All the design variables are discrete out of which 8 are integer variables. This problem is subjected to 86 different constraints for the strength of

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gears, contact ratio, size of gears, assembly of gears, pitch and kinematics, which makes the ratio of feasible region to the total search space less than 0.0001. The problem is challenging as there are many local solutions. The best solution reported in the literature is f(X)= 36.57 at X = {19 , 19, 20, 22, 38, 49, 42, 40, 3.175(1, 1, 1, 1,) 12.7(7, 4, 6, 3, 5, 7, 7, 4, 5, 7)}, with 25 active constraints.

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All these three problems are investigated by using PVS, CSA, FPA, GSA and BBO, considering 25000 FEs with a population size of 50. The results are illustrated in Table-10 and 11, where it is compared with the results from the literature obtained by using TLBO, ABC and Hybrid DE, BBO and GA. For planetary gear design problem, the performance of PVS and ABC is almost equal and it is better than all other algorithms. The performance of PVS is inferior to ABC, FPA

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and ABC-BBO for the mean of solutions; else PVS has shown better results than other algorithms. For stiffened welded shell design problem, all algorithm except GSA and BBO have performed poorly to find the optimum solution. The average performance of PVS, CSA and FPA are nearly same although better than other algorithms. For 4-stage gear box design problem, all algorithms have struggled to find the feasible solutions. For this particular problem, as all the

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algorithms failed to obtain even the feasible solutions, it will not be fair to compare the mean solutions. So, comparison is made based on the ability of the algorithm to find the feasible solutions. As observed from the results, PVS has shown better performance in finding the feasible solutions compared to other algorithms. PVS has shown better solution, though not optimum, compared to other algorithms. The results of CSA and FPA are nearly same to PVS.

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GA and SQP performed poorly for all these three challenging engineering design problems.

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As observed from the results, it is very difficult to identify the best performing algorithm, because different algorithms have performed diversely for each problem. Some algorithms have performed better on some cases and at a same time failed on the other. So, to quantify the

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algorithm performance and to rank them, statistical test called Friedman rank test is used (Joaquin et al, 2011). This test is performed on the best solutions and the mean solutions

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separately for CSA, FPA, ABC, TLBO, GSA, BBO and PVS. The results of the Friedman rank test are summarized in Table-12. It can be noted from the results that, the Friedman rank value is

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less for PVS followed by ABC and TLBO for obtaining the best solution (optimum solution). There is marginal difference between the values of TLBO, ABC, CSA and FPA. Rank is given based on the Friedman rank value, which considers PVS with a first rank. Friedman test based on the mean solution indicate that PVS is ranked first followed by CSA and FPA at the second position and TLBO at the third position. Though ranking is given to the algorithms, still it is difficult to identify the statistical significance among these algorithms. To check the statistical significance of the algorithms, Holm-Sidak multiple comparison test is employed (Holm, 1979).

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For this test, the considered algorithms are PVS, TLBO, ABC, CSA and FPA, because these algorithms have shown promising performance on the considered problems. The algorithms which are not considered for the test are GSA, BBO and GA due to their poor performance on the considered problems. The results of the Holm-Sidak test are given in Table-13, which shows the p-value of multiple pair wise comparison. Higher p-value indicates that there is no statistical

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difference in the results of both the algorithms. It is observed for the results that statistically TLBO, CSA and FPA are nearly same with PVS. There is more statistical difference in the results of PVS with ABC. As observed from the previous results shown in the Table-8 to 11, the performance of TLBO, CSA and FPA is nearly same and their higher p-value indicate less statistical difference among these algorithms. The best results along with their design variables

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obtained by using PVS are listed in Table-14 and 15.

It is also required to check the performance of any proposed optimization methods for the convergence of the algorithm and its computational efforts in terms of time consumed to run the algorithm. The convergence of all the algorithms vary with the problems and it will be clumsy to

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present the convergence for all the considered problems. So, a combined convergence strategy, considering different problems, is presented in this work. Firstly, the results are obtained for

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different independent runs considering same initial population (25 runs are considered in this work), for the specified FEs. The results are averaged over all the considered runs for each generation. These results are divided by the known optimum (best) solution for the particular

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problems. These values are normalized by dividing the previous obtained results with the maximum number. Now, all the values for each generation will lie between 0 and 1, as it is

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normalized to one. These values are used to plot the convergence graph which varies with generations (or function evaluations). In this paper, the convergence values are obtained by using

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PVS, CSA, TLBO, FPA, GSA and BBO, for the problems of step-cone, bearing, robot gripper, Belleville spring, welded stiffened shell and planetary gear, because for all these problems algorithms have obtained feasible solutions success-fully in each run. The convergence plot is given in Figure-9 for 25000 function evaluations considering the population size of 50. It can be noted from the figure-9 that the convergence of BBO and GSA is inferior compared to the other algorithms. At initial generations, convergence of TLBO is better than other algorithms. The convergence of PVS and CSA is nearly same and is better than FPA. The result of these

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algorithms converges with the increase in the generations for which the value is nearly same for all the algorithms (near about 10000 FEs), except BBO and GSA. The performance of the algorithm is also checked based on the computational efforts in terms of time consumed to operate the algorithm under specific conditions. Time taken by each algorithm (PVS, TLBO, CSA, FPA, GSA, BBO and GA-from Matlab GADS toolbox) for 25 runs is calculated for

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different engineering problems (step-cone, bearing, robot gripper, Belleville spring, welded stiffened shell and planetary gear) for 25000 function evaluations. These values are averaged for all the problems and are normalized to one with respect to the maximum value. These normalized time values are shown in Figure-10, from which it is observed that GA (from Matlab GADS toolbox) is computationally effective than other algorithms, this is but obvious as it is a

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commercially available tool. But GADS toolbox has reported inferior results compared to other algorithms. BBO is computationally effective and GSA is computationally poor than other algorithms, whereas computational time for TLBO and PVS is nearly same and it is better than CSA and FPA.

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7. Conclusions

A new meta-heuristics optimization method, called Passing Vehicle Search (PVS), is proposed in

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this work. It is shown that how ‘passing mechanism’ can be implemented to mould an algorithm for optimization. This algorithm is applied to 13 engineering design optimization problems and 13 challenging constrained benchmark functions. The performance of PVS is checked by taking

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into consideration, various aspects like ability to find optimum solutions, convergence of the algorithm and computational efforts and time. The effectiveness of PVS is check with the results

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obtained by many state-of-the art meta-heuristics. Though, PVS has performed better statistically on the considered set of problems in this work, it cannot be confirmed universally that PVS is

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better than other meta-heuristics as per ‘free lunch theorem’. It can be a future prospective for this algorithm to check its performance and suitability for a particular class of problems. The application of PVS on engineering design problems and constrained benchmark problems indicate that ‘passing vehicle’ theory can be applied successfully to practical optimization problems. The purpose of this paper is to open a scope for future work based on the proposed theory. There are many theories available for the vehicle passing behavior on a two-lane highway considering different approaches, which may possibly be applied to the proposed

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simplified model of passing vehicle. It is always required to have less computational time for any meta-heuristics and it can be noted from this work that PVS has shown good performance for computational time. It can be an interesting work to make PVS more computationally effective in future. Reference

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Liu, C, Yan, X, Liu, C, Wu, H. (2011). The wolf colony algorithm and its application. Chinese Journal of Electronics. 20, 212–216. Liu, JY, Guo, MZ, Deng, C. (2006). Geese PSO: An efficient improvement to particle swarm optimization. Computer Science. 33, 166–168. Luttinen, T. (2000). Level of service on Finnish two-lane highways. Transportation Research Circular E-C018: Fourth International Symposium on Highway Capacity.175-187 Luttinen, T. (2001). Percent time-spent-following as performance measure for two-lane highways. Transportation Research Record: Journal of the Transportation Research Board. 1776 (1), 52-59. Luttinen, T. (2001). Traffic flow on two-lane highways: an overview. TL Consulting EngineersResearch Report. Luttinen, T. (2002). Uncertainty in operational analysis of two-lane highways. Transportation Research Record: Journal of the Transportation Research Board. 1802(1), 105-114.

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Figure-1: Passing mechanism of a vehicle on a two-lane highways X2 X3 X1

y

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Initial Condition

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x1 BVC1 FV

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Primary Condition -1 OV

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Figure-2: Flow chart for Passing Vehicle Search (PVS) Input: Population size (PS), stopping criteria (NOG, FE, etc), optimization problem (f(X), number of design (DV) variables, bounds (UB, LB)

Generate initial population (Xki) and evaluates its objective function value f(Xki); Set Generation=1

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Evaluate for each solution: Select three solutions; r1- current solution, r2, r3 – randomly chosen solutions

Calculate distance (D1, D2, D3) and velocity (V1, V2 and V3) for these solutions

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If V3
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Calculate distance x,y,x1 and y1

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If (y-y1)>x1

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𝑛𝑒𝑤 𝑋𝑘…𝑃𝑆,𝑖…𝐷𝑉 = 𝑋𝑘,𝑖=𝐷1 + (𝑅𝑘 )(𝑋𝑘,𝐷3 − 𝑋𝑘,𝐷1 )

𝑛𝑒𝑤 𝑋𝑘…𝑃𝑆,𝑖…𝐷𝑉 = 𝑋𝑘,𝑖=𝐷1 + 𝑉𝑐𝑜 (𝑅𝑘 )(𝑋𝑘,𝐷1 − 𝑋𝑘,𝐷3 )

Xnew

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𝑛𝑒𝑤 𝑋𝑘…𝑃𝑆,𝑖…𝐷𝑉 = 𝑋𝑘,𝑖=𝐷1 + (𝑅𝑘 )(𝑋𝑘,𝐷1 − 𝑋𝑘,𝐷2 )

PT

Is f(Xnew)
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N

X=Xnew

X=X

Change duplicate solutions, if any

Is termination criterion reached? Y Output: f(X), {X}

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Figure -3: Step cone pulley (Rao et al, 2011)

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Figure – 4: Belleville spring (Rao and Savsani, 2012)

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Figure- 5 : Robot gripper (Rao et al, 2011)

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Figure- 6: Hydrostatic thrust bearing (Rao et al, 2011)

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Figure-7: Planetary gear train: 1 small sun gear; 2-3 broad planet gear; 4 large sun gear; 5 narrow

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planet gear; 6 ring gear (Lenchner and Naunheimer, 1999)

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Figure -8 : Stiffened welded cylindrical shell with stringer and ring stiffener acted by

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compression and external pressure (Jarmai et al., 2006)

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Figure-9: Convergence of different meta-heuristics

1

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TLBO CSA GSA FPA BBO PVS

0.9

0.7

0.6 0.5

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Normalized Function Value

0.8

0.4 0.3 0.2

0

5000

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15000

Function Evaluations

20000

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Figure-10: Comparison of different meta-heuristics based on computational efforts

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(normalized time)

1.2 1.00 1

0.84

0.8

0.86

0.70

0.6

0.41

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PVS

GADS-Toolbox

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CSA

FPA

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TLBO

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BBO

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Table-1: Characteristics of different engineering design problem No. of Continuous Discrete No. of design design design constraints variables variables variables 4 2 2 4

1

Pressure Vessel

2

Spring

3

3

0

4

3

Welded Beam

4

4

0

7

4

Speed reducer

7

6

1

11

5

Bearing

10

9

1

9

6

Multi plate clutch

5

0

7

Step cone Pulley

5

5

8

Robot gripper

7

7

9

Hydro static bearing

4

4

10

Belleville spring

4

3

11

Planetary gear train

12

Stiffened shell

13

4-stage gear box

F/S*

Objective

2

0.40

2

0.01

Minimize cost Minimize weight Minimize cost Minimize Weight Maximize Dynamic load carrying capacity Minimize Weight Minimize Weight Minimize difference in gripper force Minimize Power loss Minimize Weight Minimize error in gear ratio Minimize Cost Minimize Weight

2

0.035

3

0.004

4

0.015

5

8

1

0.700

0

11

4

0.000

0

7

1

0.025

0

7

3

0.003

1

7

4

0.004

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Active constraints

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Problems

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Sr. No.

0

9

11

0

0.000

5

0

5

5

0

0.300

0

22

86

26

0.000

PT

9

22

AC

CE

*F/S ratio between the feasible solutions in the search space(F) to the total search space(S)

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Table-2: Results for pressure vessel design optimization problem Algorithm

Best Mean Considering Mised-variables (Liu et al., 2010) 6059.714 6059.714 (Rao et. al., 2011) 6059.714 6059.714 (Akay &Karaboga, 2012) 6059.714 6245.308 (Montemurro et al., 2013) 6059.714 6292.123 (Mazhoud et al., 2013) 6059.938 6182.002 (Chun et al., 2013) 6059.714 6060.33 (Gandomi et al., 2013) 6059.714 6179.13 (Gandomi et al., 2013) 6059.714 6447.73 (Gandomi, 2014) 6059.714 6410.087 (Adil & Fehmi, 2015) 6059.714 6064.33 6059.714 6063.643 6059.714 6065.877 6059.714 6173.476 6062.244 6231.835

PSO-DE TLBO

NFS NFS Considering Continuous variables (Zahara & Kao, 2009) 5930.314 5946.79 (Eskandar, 2012) 5885.333 6198.617 (Ali et al, 2013) 5889.321 6200.64 5885.333 5885.409

CE AC

6059.714 NA

42,100 20,000

NA

30,000

6820.41 6447.325 6090.526 6318.95 6495.34 7332.84 6090.52 6090.526 6090.526 6410.09 6824.409

25,000 80,000 300,000 20,000 15,000 5,000 50,000 42,100 20,000 15,000 5,000

7545.54

25,000

AN US 6121.727 6854.328

PT

GA (Matlab GA-toolbox) SQP (Matlab Optimization toolbox)

M

PVS

ED

CVI-PSO BIANCA DEC-PSO BA CSA ISA FFA

NM-PSO WCA MBA PVS

Max_FE

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ABC

Worst

NFS

25,000

5960.056 6590.213 6392.5 5886.035

80,000 27,500 70650 20000

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Table-3: Results for spring design optimization problem Algorithm PSO–DE TLBO ABC

(Liu et al.) (Rao et. al., 2011) (Akay &Karaboga, 2012)

WCA

(Eskandar, 2012)

CVI-PSO

Best Mean Worst Max_FE 0.012665 0.012665 0.012665 42,100 0.012665 0.012666 NA 20,000 0.012709 NA 0.012746 0.012952 0.013013 0.015021

30,000 11,750 2000

(Montemurro et al., 2013)

0.012666

0.012731

0.012843

25,000

BIANCA

(Mazhoud et al., 2013)

0.012671

0.012681

0.012913

80,000

BA

(Gandomi et al., 2013)

0.012665

0.013501

0.016895

20,000

MBA

(Ali et al, 2013)

0.012665

0.012713

0.012900

7,650

ISA

(Gandomi, 2014)

0.012665

0.012799

0.013165

8000

FFA

(Adil & Fehmi, 2015)

0.012665 0.012665 0.012665 0.012665 0.012680 0.012671

0.012677 0.012665 0.012666 0.012670 0.012838 0.012683

0.000013 0.012665 0.012667 0.012710 0.013141 0.012693

50,000 42,100 20,000 8,000 2,000 25,000

0.012928

0.017951

0.032122

25,000

AN US

PVS

M

GA (Matlab GA-toolbox)

CE

PT

ED

SQP (Matlab Optimization toolbox)

AC

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0.012665 0.012665 0.012665

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Table-4: Results for welded beam design optimization problem

(Liu et al., 2010) (Rao et. al., 2011) (Akay &Karaboga, 2012)

WCA

(Eskandar, 2012)

CVI-PSO

(Montemurro et al., 2013)

1.724852 1.725124 1.727665

25,000

BIANCA

(Mazhoud et al., 2013)

1.724852 1.752201 1.793233

80,000

BA

(Gandomi et al., 2013)

1.7312 1.878656 2.345579

20,000

MBA

(Ali et al, 2013)

1.724853 1.724853 1.724853

47,340

FFA PVS

(Adil & Fehmi, 2015)

CE

Mean 1.724852 1.728447 1.741913 1.726427 1.73594

AN US

PT

ED

M

GA (Matlab GA-toolbox) SQP (Matlab Optimization toolbox)

AC

Best 1.724852 1.724852 1.724852 1.724856 1.724857

Worst Max_FE 1.724852 66,600 NA 20,000 NA 30,000 1.744697 46,450 1.801127 30,000

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Algorithm PSO–DE TLBO ABC

1.724852 1.724852 1.724852 1.724852 1.724852 1.724852 1.724852 1.724887 1.725056 2.026769 2.76033 3.162137

50,000 50000 20,000 25,000

2.124207 4.096846 5.493251

25,000

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Table-5: Results for speed-reducer design optimization problem

PVS GA (Matlab GAtoolbox) SQP (Matlab Optimization toolbox)

AC

CE

PT

ED

PVS

2996.627632

3008.465029 3083.238067 Design variable bounds-2 (Wang and Li, 2010) 2994.471066 2994.471066 (Eskandar, 2012) 2994.471066 2994.474392 (Ali et al, 2013) 2994.4824 2996.769 2994.47326 2994.7253 2994.471066 2994.483258 2994.471066 2994.472059

M

DELC WCA MBA

2996.415435

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PSO–DE ABC TLBO CSA FFA

Best Mean Worst Max_FE Design variable bounds-1 (Liu et al., 2010) 2996.348167 2996.348174 2996.348204 54,350 (Akay &Karaboga, 2012) 2997.058 2997.058 NA 30,000 (Rao et. al., 2011) 2996.34817 2996.34817 NA 20,000 (Gandomi et al., 2013) 3000.98 3007.1997 30090 5,000 (Adil & Fehmi, 2015) 2996.37 2996.51 2996.669 50,000 2996.4 2996.48271 2996.7123 5,000 2996.348165 2996.350001 2996.366218 20,000 2996.348165 2996.348165 2996.348165 54,350

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Algorithm

2997.57529

25,000

3160.475465

25,000

2994.471066 2994.505578 2999.65 2994.8327 2994.752395 2994.477593

30,000 15,150 6,300 6,000 15,150 30,000

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Table-6: Results for bearing design optimization problem

(Gupta et al, 2007) (Rao et. al., 2011) (Rao et. al., 2011) (Eskandar, 2012) (Ali et al, 2013)

MBA MDDE

Best Mean Worst Max_FE 81843.3 NA NA 225,000 81859.74 81496 78897.81 10,000 81859.74 81438.98 80807.85 20,000 85538.48 83847.16 83942.71 3950

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Algorithm GA4 ABC TLBO WCA

85535.9611 85321.4030 84440.1948 (Wenyin et al, 2014) 81858.83 81848.7 81701.18 81859.59 80803.57 78897.81 81859.74 81550 79834.79

PVS GA (Matlab GA-toolbox) SQP (Matlab Optimization toolbox)

81822.53

ED

M

AN US

74358.04 12% FS

PT CE AC

80588.51

79279.29

12% FS

15100 10,000 10,000 20,000 25000 25000

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Table-7: Results for multi-plate disc clutch brake optimization problem Algorithm

Best

Mean

Worst

Max_FE

ABC

(Rao et. al., 2011) 0.313657 0.324751 0.352864 600

TLBO

(Rao et. al., 2011) 0.313657 0.327166 0.392071 1200

WCA

(Eskandar, 2012)

0.313657 0.333652 0.352864 600

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PVS

0.313657 0.313657 0.313657 500

0.313657 0.328163 0.392071 1200 GA (Matlab GADS-toolbox)

0.313657 0.330712 0.401873 25000

SQP (Matlab Optimization

1.08682

AC

CE

PT

ED

M

AN US

toolbox)

1.939037 3.133503 25000

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Table-8: Results for step-cone pulley, robot gripper, hydrostatic thrust bearing and Belleville spring design optimization problem (Best solutions obtained in 25 runs)

Robot Gripper 25000 4.247644 4.247644 NA

BBO (2008)

48.95349

4.774746

CSA (2009)

16.72545

4.247644

Hydro Static Bearing 25000 1625.443 1625.443 1638.40

Belleville Spring 15000 1.979675 1.979675 1.979675

2573.441

2.081124

1713.029

1.981267

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Max_FE TLBO (Rao et al, 2011) ABC (Rao et al, 2011) MDDE(Wenyin et al, 2014)

Step-cone Pulley 15000 16.63451 16.63466 14.488

119.2703

4.598523

3093.021

2.400689

16.7416

4.24764

1837.852

1.984885

PVS

16.63496

4.247644

1625.443

1.979688

GA (Matlab GADS-toolbox)

21.79609

4.833581

2414.68

2.046987

SQP (Matlab Optimization toolbox)

58.3861

NFS

NFS

2.418122

AC

CE

PT

ED

M

AN US

GSA (2009)

FPA (2012)

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Table-9: Results for step-cone pulley, robot gripper, hydrostatic thrust bearing and Belleville spring design optimization problem (Mean solutions obtained in 25 runs)

BBO (2008)

70.55566

5.257477

CSA (2009)

17.15562

4.465418

Hydro Static Bearing 25000 1797.708 1861.554 1759.10

Belleville Spring 15000 1.979687 1.995475 1.979675

3605.523

2.229694

1900.772

1.988929

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Robot Gripper 25000 4.937701 5.086611 NA

AN US

Max_FE TLBO (Rao et al, 2011) ABC (Rao et al, 2011) MDDE (Wenyin et al, 2014)

Step-cone Pulley 15000 24.01136 36.0995 16.7256

192.7157

7.146706

3913.656

2.666064

17.1318

4.71694

1930.239

1.990434

PVS

20.03032

4.636115

1832.492

1.983524

GA (Matlab GADS-toolbox)

60.07632

5.676179

3339.823

2.19501

4666.637

NFS

NFS

8% FS

AC

CE

PT

ED

SQP (Matlab Optimization toolbox)

M

GSA (2009)

FPA (2012)

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Table-10: Results for 4-stage gear box, welded stiffened shell and planetary gear train (Best solutions obtained in 25 runs)

Stiffened Welded Shell 25000 55724.831 55326.293 55326.293 55326.293 55326.29341 55724.831

Planetary Gear Box 25000 0.53 0.527814 0.52735 0.52735 0.525769 0.532222

CSA (2009)

37.289069

55326.293

0.526281

44.431698 37.40714

55944.771 55326.293

0.53623 0.52325

37.267472 NFS

55326.293 57165.985

0.525588 0.569018

NFS

NFS

NFS

PVS

AC

CE

PT

ED

M

GA (Matlab GADS-toolbox) SQP (Matlab Optimization toolbox)

AN US

GSA (2009) FPA (2012)

CR IP T

Max_FE ABC-GA (Rao and Savsani, 2012) ABC-DE (Rao and Savsani, 2012) ABC-BBO (Rao and Savsani, 2012) TLBO (Rao and Savsani, 2012) ABC(Rao and Savsani, 2012) BBO (2008)

4-Stage Gear Box 25000 55.494494 59.763563 46.623205 43.792433 49.836165 39.286301

best

ACCEPTED MANUSCRIPT

Table-11: Results for 4-stage gear box, welded stiffened shell and planetary gear train (Mean solutions obtained in 25 runs)

CSA

ED

PVS GA

(Matlab GADS-toolbox) (Matlab Optimization toolbox)

AC

CE

PT

SQP

55883.37 56783.8

0.527292 0.57247

31 % FS

55559.8

0.53167

19% FS 63% FS

56746 55590.9

0.73245 0.52978

88% FS

55531.2

0.53063

NFS

60271.6

32% FS

NFS

NFS

NFS

CR IP T

Planetary Gear Box 25000 0.53669 0.54479 0.52908 0.53371

M

GSA FPA

Stiffened Welded Shell 25000 56920.4 55852.8 55921.9 55667.4

AN US

Max_FE ABC-GA (Rao and Savsani, 2012) ABC-DE (Rao and Savsani, 2012) ABC-BBO (Rao and Savsani, 2012) TLBO (Rao and Savsani, 2012) ABC (Rao and Savsani, 2012) BBO

4-Stage Gear Box 25000 8% FS 24% FS 63% FS 53% FS 14% FS 76% FS

ACCEPTED MANUSCRIPT

Table-12: Friedman rank test results for engineering design problem PVS

GA

TLBO

ABC

CSA

FPA

24.5

23

GSA

BBO

50

44

50

1.00 1

2.78 7

14

49

1 1

3.50 6

22

ED PT CE AC

20.5

1.22 1.14 3 2 For Mean Solution 23 31 1.64 3

M

Friedman Rank Value Normalized Rank Value Rank

18

2.21 4

1.36 5

1.28 4

2.78 7

2.44 6

21

21

51

42

1.50 2

1.50 2

3.64 7

3.00 5

AN US

Friedman Rank Value Normalized Rank Value Rank

CR IP T

For Best Solution

ACCEPTED MANUSCRIPT

Table-13: Holm-Sidak test for engineering design problems p-value 0.03678 0.13381 0.14562 0.18356 0.40813 0.48621 0.51581 0.85681 0.89399 0.96235

CR IP T

Algorithms* 1-3 2-3 3-4 3-5 1-5 1-4 1-2 2-5 4-5 2-4

AC

CE

PT

ED

M

AN US

*1-PVS, 2-TLBO, 3-ABC, 4-CSA, 5-FPA

ACCEPTED MANUSCRIPT

Table-14: Best result obtained by PVS for pressure vessel, spring, speed reducer, bearing and multi-plate disc clutch brake Pressure Vessel

Spring

Welded

Speed reducer

Bearing

Beam

Multiplate disc clutch

Continuous

Range-1

Range-2

f(X)

6059.71434

5885.33277

0.01267

1.72485

2996.34817

2994.47107

81859.74121

0.31366

x1

0.81250

0.77817

0.05169

0.20573

3.50000

3.50000

21.42559

70.00000

x2

0.43750

0.38465

0.35680

3.47049

0.70000

0.70000

125.71906

90.00000

x3

42.09845

40.31962

11.28442

9.03662

17.00000

17.00000

11.00000

1.00000

x4

176.63660

200.00000

0.01267

0.20573

7.30000

7.30000

0.51500

980.00000 3.00000

x5

0.05171

x6

0.35721

x7

11.26005

AN US

Discrete

CR IP T

brake

x8 x9

AC

CE

PT

ED

M

x10

7.80000

7.71532

0.51500

3.35021

3.35021

0.40043

5.28668

5.28665

0.68016 0.30000 0.07999 0.70000

ACCEPTED MANUSCRIPT

Table-15: Best result obtained by PVS for step-cone pulley, Belleville spring, robot gripper,

Belleville

Robot

Hydrostatic

Planetary

Welded

4-stage

pulley

Spring

gripper

thrust bearing

gear

stiffened shell

gear box

f(X)

16.63451

1.97967

4.24764

1625.44386

0.52559

55326.29341

37.26747

x1

40.00000

0.20414

150.00000

5.95578

34

14

23

x2

54.76430

0.20000

150.00000

5.38901

25

27

23

x3

73.01318

10.03047

200.00000

0.00001

33

10

21

x4

88.42842

12.01000

0.00000

2.26966

32

250

21

x5

85.98624

150.00000

23

203

53

x6

100.00000

116

49

x7

2.31187

4

42

AN US

Step-cone

CR IP T

hydrostatic thrust bearing, planetary gear train, welded stiffened shell and 4-stage gear box

x8 x9 x10 x11 x12

M

x13 x14

ED

x15 x16 x17

PT

x18 x19

x21

AC

x22

CE

x20

2.5

42

1.75

3.175 3.175 3.175 3.175 76.2 38.1 76.2 76.2 76.2 88.9 63.5 50.8 88.9 50.8

ACCEPTED MANUSCRIPT

Appendix: Engineering design problem

CR IP T

(A.1) Planetary gear Minimize, 𝑓(𝑋) = 𝑚𝑎𝑥|𝑖𝑘 − 𝑖0𝑘 |, 𝑘 = {1,2, 𝑅} Where ,

AN US

𝑖1 = 𝑁6 /𝑁4 𝑖01 = 3.11 𝑖2 =

𝑁6 (𝑁1 𝑁3 +𝑁2 𝑁4 ) 𝑁1 𝑁3 (𝑁6 −𝑁4 )

𝑁2 𝑁6 ) 𝑁1 𝑁3

ED

𝑖𝑅 = − (

M

𝑖02 = 1.84

PT

𝑖0𝑅 = −3.11

X = {N1, N2, N3, N4, N5, N6, p, m1, m2}

CE

Subjected to:

AC

𝑔1 (𝑋) = 𝑚3 (𝑁6 + 2.5) ≤ 𝐷max 𝑔2 (𝑋) = 𝑚1 (𝑁1 + 𝑁2 ) + 𝑚1 (𝑁2 + 2) ≤ 𝐷max 𝑔3 (𝑋) = 𝑚3 (𝑁4 + 𝑁5 ) + 𝑚3 (𝑁5 + 2) ≤ 𝐷max

𝑔4 (𝑋) = |𝑚1 (𝑁1 + 𝑁2 ) − 𝑚3 (𝑁6 − 𝑁3 )| ≤ 𝑚1 + 𝑚3

ACCEPTED MANUSCRIPT

𝜋

𝑔5 (𝑋) = (𝑁1 + 𝑁2 ) sin (𝑝 ) − 𝑁2 − 2 − 𝛿22 ≥ 0 𝜋

𝑔6 (𝑋) = (𝑁6 − 𝑁3 ) sin (𝑝 ) − 𝑁3 − 2 − 𝛿33 ≥ 0 𝜋

2𝜋

CR IP T

𝑔7 (𝑋) = (𝑁4 + 𝑁5 ) sin (𝑝 ) − 𝑁5 − 2 − 𝛿55 ≥ 0 𝑔8 (𝑋) = (𝑁6 − 𝑁3 )2 + (𝑁4 + 𝑁5 )2 − 2(𝑁6 − 𝑁3 )(𝑁4 + 𝑁5 ) cos ( 𝑝 − 𝛽) ≥ (𝑁3 + 𝑁5 + 2 + 𝛿35 )2

Where, cos−1 ((𝑁6 − 𝑁3 )2 + (𝑁4 + 𝑁5 )2 − (𝑁3 + 𝑁5 )2 ) 2(𝑁6 − 𝑁3 )(𝑁4 + 𝑁5 )

𝑔9 (𝑋) = 𝑁6 − 2𝑁3 − 𝑁4 − 4 − 2𝛿34 ≥ 0 𝑔10 (𝑋) = 𝑁6 − 𝑁4 − 2𝑁5 − 4 − 2𝛿56 ≥ 0 𝑁6 −𝑁4 𝑝

= 𝑖𝑛𝑡𝑒𝑔𝑒𝑟

M

ℎ(𝑋) =

AN US

𝛽=

Where,

ED

Dmax = 220, p = (3,4,5) , m1 , m3 =(1.75, 2.0, 2.25, 2.5, 2.75, 3.0), δ22, δ33, δ55, δ35, δ56 = 0.5.

PT

17≤N1≤96, 14≤N2≤54, 14≤N3≤51, 17≤N4≤46, 14≤N5≤51, 48≤N6≤124, Ni =integer

CE

(A.2) Welded stiffened shell

AC

Minimize

𝑓(𝑋) = 𝐾𝑀 + ∑ 𝐾𝐹𝑖 + 𝐾𝑃 X={t, ns, nr, hr, h} Subjected to:

ACCEPTED MANUSCRIPT

𝑔1 (𝑋) = 𝜎𝑒 = √𝜎𝑎2 − 𝜎𝑎 𝜎𝑝 + 𝜎𝑝2 ≤

𝑔2 (𝑋) = 𝜎𝑒 ≤

𝑓𝑦1 √1 + 𝜆4𝑠

𝑓𝑦1 √1+𝜆4𝑝

CR IP T

𝑔3 (𝑋) = 𝐴𝑅𝑟𝑒𝑞 ≤ 𝐴𝑅 𝑔4 (𝑋) = 𝐼𝑅𝑟𝑒𝑞 ≤ 𝐼𝑅

Where, 𝐾𝑀 = 𝑘𝑀1 5𝜌𝑉1 + 𝑘𝑀1 𝜌𝑛𝑟 𝑉𝑅 + 𝑘𝑀2 𝜌𝑛𝑠 𝐴𝑠 𝐿 𝐾𝐹0 = 5𝑘𝐹 𝜃𝑒 𝜇

AN US

ℎ 2 (𝑅 − 2𝑟 ) 𝜋 𝑔5 (𝑋) = 𝑛𝑠 ≤ 𝑏 + 300

M

𝜇 = 6.8582513 − 4.527217𝑡 −0.5 + 0.009541996(2𝑅)0.5

ED

𝐾𝐹1 = 5𝑘𝐹 (𝜃√𝜅𝜌𝑉1 + (1.3)(0.1520𝐸 − 3)𝑡1.9358 2𝐿𝑠 , 𝜃 = 2, 𝜅 = 2

PT

𝐾𝐹2 = 𝑘𝐹 (𝜃√25𝜌𝑉1 + (1.3)(0.1520𝐸 − 3)𝑡1.9358 4(2𝑅𝜋)

CE

2 𝐾𝐹3 = 𝑛𝑟 𝑘𝐹 (3√3𝜌𝑉𝑅 + (1.3)(0.3394𝐸 − 3)𝑎𝑤𝑟 4𝜋(𝑅 − ℎ𝑟 ))

AC

2 𝐾𝐹4 = 𝑘𝐹 (3√𝑛𝑟 + 1𝜌(5𝑉1 + 𝑛𝑟 𝑉𝑅 ) + (1.3)(0.3394𝐸 − 3)𝑎𝑤𝑟 𝑛𝑟 4𝑅𝜋) 2 𝐾𝐹5 = 𝑘𝐹 (3√(𝑛𝑟 + 𝑛𝑠 + 1)𝜌(5𝑉1 + 𝑛𝑟 𝑉𝑅 + 𝑛𝑠 𝐴𝑠 𝐿) + (1.3)(0.3394𝐸 − 3)𝑎𝑤𝑠 𝑛𝑠 2𝐿)

𝐾𝑃 = 𝑘𝑝 (2𝑅𝜋𝐿 + 2𝑅𝜋(𝐿 − 𝑛𝑟 ℎ𝑟 ) + 2𝑛𝑟 𝜋ℎ𝑟 (𝑅 − ℎ𝑟 ) + 4𝜋𝑛𝑟 ℎ𝑟 (𝑅 −

ℎ𝑟 ) + 𝑛𝑠 𝐿(ℎ1 + 2𝑏)) 2

ACCEPTED MANUSCRIPT

𝑉1 = 2𝑅𝜋𝑡𝐿𝑠 , 𝑉𝑅 = 2𝜋𝛿𝑟 ℎ𝑟2 (𝑅 − ℎ𝑟 ) + 4𝜋𝛿𝑟 ℎ𝑟2 (𝑅 −

ℎ𝑟 2

),

𝑁

𝐹 𝜎𝑎 = 2𝑅𝜋𝑡 ,

𝑡𝑒 = 𝑡 + 𝑆=

𝐴𝑠 𝑆

CR IP T

𝑒

,

2𝑅𝜋 𝑛𝑠 𝑝 𝑅

𝐴𝑅

𝛼=𝐿

𝑒0 𝑡

AN US

𝐹 𝜎𝑝 = 𝑡(1+𝛼) ,

,

𝐿𝑟 = 𝑛

𝐿

M

𝐿𝑒0 = min(𝐿𝑟 , 𝐿𝑒𝑟 = 1.56√𝑅𝑡), ,

𝜎𝑒 = √𝜎𝑎2 − 𝜎𝑎 𝜎𝑝 + 𝜎𝑝2 𝜎𝑝

𝐸𝑎𝑠

𝑓𝑦 1.1

𝐸𝑝𝑠

)

CE

𝑓𝑦1 =

+𝜎

PT

𝜎𝑎

𝜆2𝑠 = 𝑓𝑦1 /𝜎𝑒 (𝜎

ED

𝑟 −1

𝑡 2

𝜋2 𝐸

AC

𝜎𝐸𝑎𝑠 = 𝑐𝑎𝑠 12(1−𝜐2 ) (𝑠) , 𝜌𝑎𝑠 𝜉𝑎𝑠 2

𝑐𝑎𝑠 = 𝜓𝑎𝑠 √1 + ( 𝑠2

𝑧𝑎𝑠 = 𝑅𝑡 √1 − 𝜐 2 , 𝜉𝑎𝑠 = 0.702𝑧𝑎𝑠

𝜓𝑎𝑠

) ,

ACCEPTED MANUSCRIPT

−0.5

𝑅

𝜌𝑎𝑠 = 0.5 (1 + 150𝑡) 𝑐𝑝𝑠 𝜋 2 𝐸 𝑡 2

(𝑠) ,

10.92

𝑐𝑝𝑠 = 𝜓𝑝𝑠 √1 + (

𝜌𝑝𝑠 𝜉𝑝𝑠 𝜓𝑝𝑠

2

) ,

CR IP T

𝜎𝐸𝑝𝑠 =

,

𝑠

𝜉𝑝𝑠 = 1.04 (𝐿 ) √𝑧𝑝𝑠 , 𝑟

𝑧𝑝𝑠 = 𝑧𝑎𝑠 , 2 𝑠 2

AN US

𝜓𝑝𝑠 = (1 + (𝐿 ) ) , 𝑟

𝜎𝑝

+𝜎

𝐸𝑎𝑝

𝑐𝑎𝑝 𝜋 2 𝐸 10.92

𝑡

2

) (𝐿 ) , 𝑟

𝜌𝑎𝑝 𝜉𝑎𝑝

CE

𝜉𝑎𝑝 = 0.702𝑧𝑎𝑝 ,

𝜓𝑎𝑝

𝐿2𝑟

𝑅𝑡

AC

𝑧𝑝𝑝 = 𝑧𝑎𝑝 , 𝜓𝑎𝑝 =

1+𝛾𝑠 𝐴 1+ 𝑠

,

𝑠𝑒 𝑡

𝛾𝑠 =

10.92𝐼𝑠𝑒𝑓 𝑠𝑡 3

2

) ,

PT

𝑐𝑎𝑝 = 𝜓𝑎𝑝 √1 + (

𝑧𝑎𝑝 = 0.9539

),

ED

𝜎𝐸𝑎𝑝 = (

𝐸𝑝𝑝

,

𝑠𝐸 = 1.9𝑡√𝐸/𝑓𝑦 ,

M

𝜎𝑎

𝜆2𝑝 = 𝑓𝑦1 /𝜎𝑒 (𝜎

ACCEPTED MANUSCRIPT

𝑖𝑓 𝑠𝐸 ≤ 𝑠 → 𝑠𝑒 = 𝑠𝐸 , 𝑒𝑙𝑠𝑒 𝑠𝑒 = 𝑠 ℎ1 + 𝑡 + 𝑡𝑓 ℎ1 ℎ1 𝑡 𝑡 ( + ) + 𝑏𝑡 ( ) 𝑤 𝑓 2 4 2 2 𝑍𝐺 = ℎ 𝑡 𝑠𝑒 𝑡 + 𝑏𝑡𝑓 + 12 𝑤

𝜎𝐸𝑝𝑝 =

2

𝑡

ℎ1 𝑡𝑤 2

𝑐𝑝𝑝 (𝜋 2 𝐸) 10.92

2

𝑡

(𝐿 ) , 𝑟

𝜌𝑝𝑝 𝜉𝑝𝑝

𝑐𝑝𝑝 = 𝜓𝑝𝑝 √1 + (

𝜓𝑝𝑝

2

) ,

M

𝜉𝑝𝑝 = 1.04√𝑧𝑝𝑝 ,

ED

𝜓𝑝𝑝 = 2(1 + √1 + 𝛾𝑠 ),

1

= 42𝜀,

1

,

34.17189

AC

𝛿𝑟 =

CE

𝜀 = √235/𝑓𝑦 ,

PT

𝑡𝑟 = 𝛿𝑟 ℎ𝑟 , 𝛿𝑟

2

( 4 + 2 − 𝑧𝐺 ) + 𝑏𝑡𝑓 (

𝐴𝑅 = 3ℎ𝑟 𝑡𝑟 = 3𝛿𝑟 ℎ𝑟2 , 2

𝐴𝑅𝑟𝑒𝑞 = (𝑍 2 + 0.06) 𝐿𝑟 𝑡 , 𝐿2

𝑍 = 0.9539 𝑅𝑡𝑟 ,

ℎ1 +𝑡+𝑡𝑓 2

2

− 𝑧𝐺 ) ,

AN US

𝐴𝑠 = 𝑏𝑡𝑓 +

ℎ1 𝑡𝑤 ℎ1

CR IP T

3 𝑡

ℎ

𝑤 𝐼𝑠𝑒𝑓 = 𝑠𝑒 𝑡𝑧𝐺2 + ( 21 ) (12 )+

ACCEPTED MANUSCRIPT

𝐿𝑒 = min (𝐿𝑟 , 2(1.56√𝑅𝑡))

𝐼𝑅 =

𝛿𝑟 ℎ𝑟4 6

, 2

ℎ

𝑡

𝐼𝑅𝑟𝑒𝑞 = 𝐼𝑎 + 𝐼𝑝 , 𝐼𝑎 =

𝐴 𝜎𝑎 𝑡(1+ 𝑠 )𝑅04 𝑠𝑡

500𝐸𝐿𝑟

,

𝑝𝐹 𝑅𝑅02 𝐿𝑟 3𝐸

(2 +

AN US

𝑅0 = 𝑅 − (ℎ𝑟 − 𝑦𝐸 ), 𝐼𝑝 =

2

+ 2𝛿𝑟 ℎ𝑟2 ( 2𝑟 − 𝑦𝐸 ) + 𝛿𝑟 ℎ𝑟2 𝑦𝐸2 + 𝐿𝑒 𝑡 (ℎ𝑟 + 2 − 𝑦𝐸 ) ,

CR IP T

𝑦𝐸 =

𝑡 2 3𝛿𝑟 ℎ𝑟2 +𝐿𝑒 𝑡

𝐿𝑒 𝑡(ℎ𝑟 + )+𝛿𝑟 ℎ𝑟3

3𝐸𝑦𝐸 𝛿0 𝑓𝑦 2 𝑅0 ( −𝜎𝑝 ) 2

),

𝛿0 = 0.005𝑅,

M

𝑎𝑤𝑠 = 0.4𝑡𝑤 ,

ED

𝑎𝑤𝑟 = 0.4𝑡𝑟

𝑡𝑓 = √33.20534 + (6.701288𝐸 − 4)ℎ2

PT

𝑏 = √5851.785 + (1.671844𝐸 − 2)ℎ2 log(ℎ)

CE

𝑡𝑤 = √15.62577 + (4.358947𝐸 − 5)ℎ2 log(ℎ)

AC

ℎ1 = ℎ − 2𝑡𝑓

ACCEPTED MANUSCRIPT

𝐿𝑠 = 3000, 𝜌 = 7.85𝐸 − 6, 𝑁𝑓 = 5.4𝐸7, 𝑝𝐹 = 1.5, 𝐿 = 15000, 𝑅 = 1850, 𝑓𝑦 = 355, 𝐸 = 2.1𝐸5, 𝜐 = 0.3, 𝑘𝑚1 = 𝑘𝑚2 = 𝑘𝑓 = 1, 𝑘𝑝 = 14.4𝐸 − 6, 𝜌𝑝𝑠 = 0.6, 𝜓𝑎𝑠 = 4, 𝜌𝑎𝑝 = 0.5, 𝜌𝑝𝑝 = 0.6 , 4 ≤ t, ns , nr ≤ 40,

CR IP T

130≤hr≤510,

h = (152, 203, 254, 305, 356, 406, 457, 533, 610, 686, 762, 838, and 914) ,

AN US

t, ns , nr =integer, hr varies in the step of 10.

(A.3) 4-stage gear box Minimize

ED

Where, i =(1, 2, 3, 4)

M

4

2 2 𝑏𝑖 𝑐𝑖2 (𝑁𝑝𝑖 + 𝑁𝑔𝑖 ) 𝜋 𝑓=( )∑ 2 1000 𝑖=1 (𝑁𝑝𝑖 + 𝑁𝑔𝑖 )

Subject to:

2

PT

2𝑐1 𝑁𝑝1 (𝑁𝑝1 + 𝑁𝑔1 ) 366000 𝜎𝑁 𝐽𝑅 𝑔1 = ( + )( )≤ 2 𝜋𝜔1 𝑁𝑝1 + 𝑁𝑔1 0.0167𝑊𝐾𝑜 𝐾𝑚 4𝑏1 𝑐1 𝑁𝑝1 2

AC

CE

366000𝑁𝑔1 2𝑐2 𝑁𝑝2 (𝑁𝑝2 + 𝑁𝑔2 ) 𝜎𝑁 𝐽𝑅 𝑔2 = ( + )≤ )( 2 𝜋𝜔1 𝑁𝑝1 𝑁𝑝2 + 𝑁𝑔2 0.0167𝑊𝐾𝑜 𝐾𝑚 4𝑏2 𝑐2 𝑁𝑝2

2

𝜎𝑁 𝐽𝑅 366000𝑁𝑔1 𝑁𝑔2 2𝑐3 𝑁𝑝3 (𝑁𝑝3 + 𝑁𝑔3 ) 𝑔3 = ( + )≤ )( 2 𝜋𝜔1 𝑁𝑝1 𝑁𝑝2 𝑁𝑝3 + 𝑁𝑔3 0.0167𝑊𝐾𝑜 𝐾𝑚 4𝑏3 𝑐3 𝑁𝑝3

ACCEPTED MANUSCRIPT

2

𝜎𝑁 𝐽𝑅 366000𝑁𝑔1 𝑁𝑔2 𝑁𝑔3 2𝑐4 𝑁𝑝4 (𝑁𝑝4 + 𝑁𝑔4 ) 𝑔4 = ( + )≤ )( 2 𝜋𝜔1 𝑁𝑝1 𝑁𝑝2 𝑁𝑝3 𝑁𝑝4 + 𝑁𝑔4 0.0167𝑊𝐾𝑜 𝐾𝑚 4𝑏4 𝑐4 𝑁𝑝4

3

2

3

2

CR IP T

2𝑐1 𝑁𝑝1 (𝑁𝑝1 + 𝑁𝑔1 ) 366000 𝜎𝐻 𝑠𝑖𝑛𝜑𝑐𝑜𝑠𝜑 𝑔5 = ( + )≤( ) ( ) )( 2 2 𝜋𝜔1 𝑁𝑝1 + 𝑁𝑔1 4𝑏1 𝑐1 𝑁𝑔1 𝑁𝑝1 𝐶𝑝 0.0334𝑊𝐾𝑜 𝐾𝑚 366000𝑁𝑔1 2𝑐2 𝑁𝑝2 (𝑁𝑝2 + 𝑁𝑔2 ) 𝜎𝐻 𝑠𝑖𝑛𝜑𝑐𝑜𝑠𝜑 𝑔6 = ( + )≤( ) ( ) )( 2 2 𝜋𝜔1 𝑁𝑝1 𝑁𝑝2 + 𝑁𝑔2 4𝑏2 𝑐2 𝑁𝑔2 𝑁𝑝2 𝐶𝑝 0.0334𝑊𝐾𝑜 𝐾𝑚 3

2

366000𝑁𝑔1 𝑁𝑔2 2𝑐3 𝑁𝑝3 (𝑁𝑝3 + 𝑁𝑔3 ) 𝜎𝐻 𝑠𝑖𝑛𝜑𝑐𝑜𝑠𝜑 𝑔7 = ( + )≤( ) ( ) )( 2 2 𝜋𝜔1 𝑁𝑝1 𝑁𝑝2 𝑁𝑝3 + 𝑁𝑔3 4𝑏3 𝑐3 𝑁𝑔3 𝑁𝑝3 𝐶𝑝 0.0334𝑊𝐾𝑜 𝐾𝑚 3

2

366000𝑁𝑔1 𝑁𝑔2 𝑁𝑔3 2𝑐4 𝑁𝑝4 (𝑁𝑝4 + 𝑁𝑔4 ) 𝜎𝐻 𝑠𝑖𝑛𝜑𝑐𝑜𝑠𝜑 + )≤( ) ( ) )( 2 2 𝜋𝜔1 𝑁𝑝1 𝑁𝑝2 𝑁𝑝3 𝑁𝑝4 + 𝑁𝑔4 4𝑏4 𝑐4 𝑁𝑔4 𝑁𝑝4 𝐶𝑝 0.0334𝑊𝐾𝑜 𝐾𝑚

AN US

𝑔8 = (

2

2

𝑔17−20 = 𝑑min ≤ (𝑁

2𝑐𝑖 𝑁𝑝𝑖 𝑁𝑝𝑖 + 𝑁𝑔𝑖 2𝑐𝑖 𝑁𝑔𝑖 𝑁𝑝𝑖 +𝑁𝑔𝑖

ED

𝑔13−16 = 𝑑min ≤

M

𝑠𝑖𝑛𝜑(𝑁𝑝𝑖 + 𝑁𝑔𝑖 ) 𝑠𝑖 𝑛2 𝜑 1 1 𝑠𝑖 𝑛2 𝜑 1 1 + + ( ) + 𝑁𝑔𝑖 √ + +( ) − ≥ 𝐶𝑅min 𝜋𝑐𝑜𝑠𝜑 4 𝑁𝑝𝑖 𝑁𝑝𝑖 4 𝑁𝑔𝑖 𝑁𝑔𝑖 2

PT

𝑔9−12 = 𝑁𝑝𝑖 √

+2)𝑐

𝑔21 = 𝑥𝑝1 + ( 𝑁 𝑝1+𝑁 1 ) ≤ 𝐿max 𝑔1

CE

𝑝1

(𝑁 +2)𝑐

𝑔22−24 = (𝑥𝑔(𝑖−1) + ( 𝑁 𝑝𝑖+𝑁 𝑖)

AC

𝑝𝑖

𝑔25 = −𝑥𝑝1 +

(𝑁𝑝1 +2)𝑐1 𝑁𝑝1 +𝑁𝑔1

𝑔26−28 = (−𝑥𝑔(𝑖−1) +

𝑔𝑖

𝑖=2,3,4

≤ 𝐿max

≤0

(𝑁𝑝𝑖 +2)𝑐𝑖 𝑁𝑝𝑖 +𝑁𝑔𝑖

) 𝑖=2,3,4

≤0

ACCEPTED MANUSCRIPT

(𝑁𝑝1 + 2)𝑐1 ≤ 𝐿max 𝑁𝑝1 + 𝑁𝑔1

𝑔30−32 = (𝑦𝑔(𝑖−1) +

𝑔33 = −𝑦𝑝1 +

(𝑁𝑝𝑖 +2)𝑐𝑖 𝑁𝑝𝑖 +𝑁𝑔𝑖

(𝑁𝑝1 +2)𝑐1 𝑁𝑝1 +𝑁𝑔1

𝑔34−36 = (−𝑦𝑔(𝑖−1) + 𝑔37−40 = 𝑥𝑔𝑖 +

) 𝑖=2,3,4

≤0

(𝑁𝑝𝑖 +2)𝑐𝑖 𝑁𝑝𝑖 +𝑁𝑔𝑖

(𝑁𝑔𝑖 +2)𝑐𝑖 𝑁𝑝𝑖 +𝑁𝑔𝑖

≤ 𝐿max

)

CR IP T

𝑔29 = 𝑦𝑝1 +

≤0

𝑖=2,3,4

≤ 𝐿max

(𝑁 +2)𝑐 𝑝𝑖

AN US

𝑔41−44 = (−𝑥𝑔𝑖 + ( 𝑁 𝑔𝑖+𝑁 𝑖) ≤ 0 𝑔𝑖

(𝑁 +2)𝑐

𝑔45−48 = (𝑦𝑔𝑖 + ( 𝑁 𝑔𝑖+𝑁 𝑖) ≤ 𝐿max 𝑝𝑖

𝑔𝑖

(𝑁𝑔𝑖 +2)𝑐𝑖

𝑔49−52 = (−𝑦𝑔𝑖 + ( 𝑁

𝑝𝑖 +𝑁𝑔𝑖

) ≤0

M

𝑔53−56 = (0.945𝑐𝑖 − 𝑁𝑝𝑖 − 𝑁𝑔𝑖 )(𝑏𝑖 − 5.715)(𝑏𝑖 − 8.255)(𝑏𝑖 − 12.70)(−1) ≤ 0

ED

𝑔57−60 = (0.646𝑐𝑖 − 𝑁𝑝𝑖 − 𝑁𝑔𝑖 )(𝑏𝑖 − 3.175)(𝑏𝑖 − 8.255)(𝑏𝑖 − 12.70)(+1) ≤ 0 𝑔61−64 = (0.504𝑐𝑖 − 𝑁𝑝𝑖 − 𝑁𝑔𝑖 )(𝑏𝑖 − 3.175)(𝑏𝑖 − 5.715)(𝑏𝑖 − 12.70)(−1) ≤ 0

PT

𝑔65−68 = (0𝑐𝑖 − 𝑁𝑝𝑖 − 𝑁𝑔𝑖 )(𝑏𝑖 − 3.175)(𝑏𝑖 − 5.715)(𝑏𝑖 − 8.255)(+1) ≤ 0 𝑔69−72 = (𝑁𝑝𝑖 + 𝑁𝑔𝑖 − 1.812𝑐𝑖 )(𝑏𝑖 − 5.715)(𝑏𝑖 − 8.255)(𝑏𝑖 − 12.70)(−1) ≤ 0

CE

𝑔73−76 = (𝑁𝑝𝑖 + 𝑁𝑔𝑖 − 0.945𝑐𝑖 )(𝑏𝑖 − 3.175)(𝑏𝑖 − 8.255)(𝑏𝑖 − 12.70)(+1) ≤ 0

AC

𝑔77−80 = (𝑁𝑝𝑖 + 𝑁𝑔𝑖 − 0.646𝑐𝑖 )(𝑏𝑖 − 3.175)(𝑏𝑖 − 5.715)(𝑏𝑖 − 12.70)(−1) ≤ 0 𝑔81−84 = (𝑁𝑝𝑖 + 𝑁𝑔𝑖 − 0.504𝑐𝑖 )(𝑏𝑖 − 3.175)(𝑏𝑖 − 5.715)(𝑏𝑖 − 8.255)(+1) ≤ 0 𝑔85 = 𝜔min ≤

𝜔1 (𝑁𝑝1 𝑁𝑝2 𝑁𝑝3 𝑁𝑝4 ) (𝑁𝑔1 𝑁𝑔2 𝑁𝑔3 𝑁𝑔4 )

ACCEPTED MANUSCRIPT

𝑔86 =

𝜔1 (𝑁𝑝1 𝑁𝑝2 𝑁𝑝3 𝑁𝑝4 ) (𝑁𝑔1 𝑁𝑔2 𝑁𝑔3 𝑁𝑔4 )

≤ 𝜔max

Where, 2

CR IP T

2

𝑐𝑖 = √(𝑥𝑔𝑖 − 𝑥𝑝𝑖 ) + (𝑦𝑔𝑖 − 𝑦𝑝𝑖 )

CRmin = 1.4, dmin = 25.4, ø =20o , W = 55.9, JR = 0.2 , KM = 1.6, K0 = 1.5, Lmax = 127, σH = 3290, σN = 2090, ω1 = 5000, ωmin = 245, ωmax = 255, Cp=464, 𝑥𝑝1 , 𝑦𝑝1 , 𝑥𝑔𝑖 , 𝑦𝑔𝑖 = (12.7, 25.4, 38.1, 50.8, 63.5, 76.2, 88.9, 101.6, 114.3)

AN US

𝑏𝑖 = (3.175, 5.715, 8.255, 12.7) 7 ≤ 𝑁𝑝𝑖 , 𝑁𝑔𝑖 ≤ 76, Npi, Ngi = integer

AC

CE

PT

ED

M

X={Np1, Ng1, Np2, Ng2…..b1, b2…..xp1, xg1, xg2….yp1, yg1, yg2….yg4}

ACCEPTED MANUSCRIPT

Abbreviation Artificial Bee Colony

ACO

Ant colony optimization

AIA

Artificial immune algorithm

BA

Bat Algorithm

BBO

Biogeography based optimization

BIANCA

Multi-population GA

BV

Back vehicle

CSA

Cuckoo search algorithm

CVI-PSO

Constraint Violation with Interval arithmetic PSO

DE

Differential Evolution

DEC-PSO

Diversity-Enhanced Constrained PSO

DELC

Differential evolution level comparison

DV

Design variable

FE

Function evaluations

FFA

Fire Fly Algorithm

FHWA

Federal Highway Administration

FPA

Flower pollinating algorithm

FS

Feasible solution

FV

Front vehicle

GA

Genetic algorithm

GADS

Genetic algorithm and direct search

ISA

AN US

M

ED

PT

AC

LB

CE

GSA

CR IP T

ABC

Gravitation search algorithm Interior Search Algorithm Lower bound

MBA

Mine Blast Algorithm

MDDE

Multi-member Diversity-based DE

NC

Number of constraints

NFS

No feasible solution

NHAI

National highway authority of India

NM–PSO

Nelder–Mead particle swarm optimization

ACCEPTED MANUSCRIPT

Number of generations

OV

Oncoming vehicle

PS

Population size

PSO

Particle swarm optimization

PSO–DE

Particle swarm optimization-differential evolution

PVS

Passing vehicle search

SQP

Sequential quadratic programming

TLBO

Teaching-learning based optimization

UB

Upper bound

WCA

Water cycle algorithm

AC

CE

PT

ED

M

AN US

CR IP T

NOG