River Formation Dynamics Revisited with Proposed New Developments 79

4.3 Minimization using River Formation Dynamics Scheme

4.3.5 River Formation Dynamics Revisited with Proposed New Developments 79

Even though the current river formation dynamics scheme is capable of determining an optimal solution, when compared with other evolutionary methods, the ability to find a global optimum solution in all the scenarios is comparatively weak. The reason for such weak perfor- mance is due to the lack of sufficient tuning of parameters (e.g., longitudinal slope, stochastic flow ) at the end of the search and also the lack of efficient utilization of certain key factors, such as transverse slope, sediment transport rate, etc. during analysis. Therefore, better real- ization of water flow is necessary to find an optimum solution accurately and efficiently. Con- sidering it, we propose to improve the performance of RFD by incorporating a new stochastic flow model based on sediment transport rate affected by transverse slope characteristics.

4.3.6 Proposed Model Equation

Similar to the Brownian motion of water flow, we consider the search toward optimum solution in RFD to be guided by a stochastic variable model. In view of this, we formulate a model centered around the volume of water flowing at certain instant of time. As the volume of water flow is often affected by the amount of sediment being transported, we consider volume of the sediment deposited at the bottom being transported per unit width per unit time as a decision variable during formulation of model equation. At first, we establish the equation governing transverse (cross) slope and flow velocity (u) in two different directions, i.e., downstream (s) and transverse (n), while considering the case of downstream elevated channel bed. A channel bed is shown in Figure 4.2 having mild transverse slope (Sn) with uniform sized sediment materials. In this regard, volume of sediment transport rate per unit width per unit time in boths andn directions (i.e,q_s andq_n, respectively) can be related to flow velocity using (4.11) [116],

q_n q_s = u_n

u_s +βS_n, (4.11)

where, β ∈ [1,2]is a dimensionless parameter. As water drops choose to flow in different paths, both longitudinal and transverse slopes play a major role in making a decision, i.e., choosing a path. If flow varies in both s and n directions, probability of the water drop traversing in a particular direction described described in (4.8) can be rewritten as (4.12),

P r(x₁, x₂) =







Sf(x1,x2)

∑deg(xi)

xk=1 Sf(xi,xk) if xj ∈N(xi), 0 if x_j ∈/ N(x_i),

(4.12)

where,S_f denotes the resultant slope and|S_f|=√

S_l²+S_n²;S_lrepresents longitudinal slope insdirection and,Sl(x1, x2) = ^h_x^2−h1

2−x1 andSn(x1, x2) = ^h_x^4−h2

3−x1. As the effect of change inqn

to the flow of water drops is minimal,qscan be considered as a decision variable for function evaluation during optimization. Further, with drops traveling in direction of decreasing alti- tude values, both slopesS_landS_n start to decline as drops tend to reach flat surface (sea). In this regard, the volume of sediment transport rate per unit width per unit time in downstream

4.3 Minimization using River Formation Dynamics Scheme

direction, qs, can be considered to be minimum at sea level. Therefore, we consider qs as a decision variable to be updated in each time step for a new function evaluation. As qs is changed in each time step, it can be modeled by (4.13),

q_s=q_s+δq_s, (4.13)

where, δq_s denotes the change in volume of sediment transport rate per unit width per unit time in sdirection. To demonstrate the existence of any trend in (4.13), a parametric study is performed in section 4.3.7, which supports the effectiveness of the model described in (4.13). When a drop chooses certain probabilityP rto follow a path,qschanges. Therefore, a weighted random positive seed value,εis added toq_sin each iteration as a token for choosing a better path probabilistically as described in Algorithm 12. Further, for each decision variable (sayq_s), there exists a separateεvalue which changes in each iteration. AsΥ, whereε∈Υ, is evaluated for all decision variables (for allq_s), which choose paths probabilistically to move in descent or ascent direction, it can be considered as a random vector in a multidimensional space spanned by the decision variables. Therefore, it is necessary to estimate the weighted mean vector, µε and weighted covariance matrix, Σε for suitable estimation of ε for each decision variable.

h 1

h 3

h 2 h ⁴

x 1 x ₂

x 3 θ

n s

ω

Figure 4.2:A slice of lengthdx= (x1−x2)along a river for the formulation longitudinal and trans- verse slope. The notations are: water depths at different locationsh_i, wherei= 1,2,3,4, longitudinal slope angleθ, transverse slopew, downstream directions, normal directionn.

Further, if the drop discovers an improved pattern (minimum sediment transport rate) in an iteration, the correspondingq_svalue is stored and represented as regional sediment deposition rate per unit time per unit width, q_s^rsed of that iteration. The weighted difference between q_s^rsed and the current q_s value is added stochastically to the current q_s value to search for

Algorithm 12:Procedure for evaluation ofε

fori←−1topopSizedo forj←−1tonumV ardo

cij ←−ϕ

/* Sf denotes final slope */

Sfij←−√

S_l²+S_n²

/* N denotes no. of possible paths */

P r_ij←−(Sf_ij)/(∑N x=1Sf_ij)

Generate random numberr1between0and1 ifr₁< P r_ijthen

c_ij←−c_ij+ 1 end

c_ij ←−c_ij−r₁×c_ij end

end

/* µ_ε denotes weighted mean vector of ε */

µε←− ^∑∑^popSizei=1popSize^{∑numV arj=1 cijP rij} i=1

∑numV ar j=1 c_ij

/* Σ_ε denotes weighted covariance matrix of ε */

Σ_ε←− ^{∑popSizei=1 ∑}∑^{numV arj=1}popSize^{cij(P rij−µε)T(P rij−µε)} i=1

∑_{numV ar}

j=1 cij

an optimum solution in the regional direction. Moreover, with more number of water drops flowing in different directions, the weighted difference between best pattern found among all the drops,q_s^lsed(large scale sediment deposition) and individual currentqsvalue is also added to the current q_s value. These additions to the sediment’s movement motivate the drops to search in two dimensional space. Considering these additions, change in volume of sediment transport rate per unit width per unit time,δq_sin (4.13) can be expressed as (4.14),

δq_s =ε+w₁×(q_s^rsed−q_s) +w₂ ×(q_s^lsed−q_s), (4.14) wherew₁ andw₂ denote random positive coefficients. Random weighting of control param- eters (q_s^rsed andq_s^lsed) in (4.14) can induce spurious trend during movement of sediments in (4.13) and can lead to explosion, similar to “drunkard’s walk” of control parameters in tra- ditional stochastic algorithms [68, 117]. It is shown in [68, 117] that the random variation of these positive coefficients can be constricted to provide a search domain for stochastic pro- cesses. However, these analyses rely on the assumption that the time series are stationary, i.e., the covariances or means of local best points are assumed constant to evaluate a stable point

4.3 Minimization using River Formation Dynamics Scheme

for coefficients, e.g.,w1 andw2. As random values ofw1 andw2 can cause explosion [117], the occurrence of such adverse situation during optimization analysis renders the stationarity assumption inefficient [68]. Moreover, this assumption of stationarity during evaluation of certain control parameters with few other conditions during analysis do not provide much so- lace to experimental analysis of stochastic model equation described in (4.13). Therefore, in this paper, we consider to maintain a scheduled bound for bothw₁andw₂coefficients through stability analysis [117].