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where mi is as de ned in equation (1630) The formulation of equation (1630) implies that each particle has its own adaptive acceleration as a function of the slope of the search space at the current position of the particle Ratnaweera et al [706] builds further on a suggestion by Suganthan [820] to linearly adapt the values of c1 and c2 Suganthan suggested that both acceleration coe cients be linearly decreased, but reported no improvement in performance using this scheme [820] Ratnaweera et al proposed that c1 decreases linearly over time, while c2 increases linearly [706] This strategy focuses on exploration in the early stages of optimization, while encouraging convergence to a good optimum near the end of the optimization process by attracting particles more towards the neighborhood best (or global best) positions The values of c1 (t) and c2 (t) at time step t is calculated as c1 (t) = (c1,min c1,max ) t + c1,max nt t c2 (t) = (c2,max c2,min ) + c2,min nt (1636) (1637)

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A number of theoretical studies have shown that the convergence behavior of PSO is sensitive to the values of the inertia weight and the acceleration coe cients [136, 851, 863, 870] These studies also provide guidelines to choose values for PSO parameters that will ensure convergence to an equilibrium point The rst set of guidelines are obtained from the di erent constriction models suggested by Clerc and Kennedy [136] For a speci c constriction model and selected value, the value of the constriction coe cient is calculated to ensure convergence For an unconstricted simpli ed PSO system that includes inertia, the trajectory of a particle converges if the following conditions hold [851, 863, 870, 937]: 1>w> 1 ( 1 + 2 ) 1 0 2 (1638)

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Many salaried employees begin or stop work partway through a pay period, so the payroll staff must calculate what proportion of their salary has been earned This calculation also arises when a pay change has been made that is effective as of a date partway through the person s pay period To determine the amount of a partial payment, calculate the salaried employee s hourly rate and then multiply this rate by the number of hours worked A common approach for determining the hourly rate is to divide the total annual salary by 2,080 hours, which is the total number of work hours in a year

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and 0 w < 1 Since 1 = c1 U (0, 1) and 2 = c2 U (0, 1), the acceleration coe cients, c1 and c2 serve as upper bounds of 1 and 2 Equation (1638) can then be rewritten as 1 (1639) 1 > w > (c1 + c2 ) 1 0 2 Therefore, if w, c1 and c2 are selected such that the condition in equation (1639) holds, the system has guaranteed convergence to an equilibrium state The heuristics above have been derived for the simpli ed PSO system with no stochastic component It can happen that, for stochastic 1 and 2 and a w that violates the condition stated in equation (1638), the swarm may still converge The stochastic trajectory illustrated in Figure 166 is an example of such behavior The particle follows a convergent trajectory for most of the time steps, with an occasional divergent step Van den Bergh and Engelbrecht show in [863, 870] that convergent behavior will be observed under stochastic 1 and 2 if the ratio, ratio = is close to 10, where crit = sup | 05 1 < w, (0, c1 + c2 ] (1641) crit c1 + c2 (1640)

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day of each month, which amounts to 24 pay periods per year One employee, Stephanie Ortiz, has been hired partway through a pay period at an annual salary of $38,500 She starts work on the 20th of the month, and there are seven business days left in the pay period The payroll staff first determines her hourly rate of pay, which is $38,500 divided by 2,080 hours, or $1851 They then calculate the number of hours left in the pay period, which is eight hours a day times seven working days, or 56 hours Consequently, Ms Ortiz s pay for her first pay period will be $1851 times 56 hours, or $1,03656

It is even possible that parameter choices for which ratio = 05, may lead to convergent behavior, since particles spend 50% of their time taking a step along a convergent trajectory

Figure 131(a) illustrates the mutation operator of the DE as described in Section 1312 The optimum is indicated by x , and it is assumed that = 15 The crossover operator is illustrated in Figure 131(b) For this illustration the o spring consists of the rst element of the trial vector, ui (t), and the second element of the parent, xi (t)

Overtime is a pay premium of 50 percent of the regular rate of pay that is earned by employees on all hours worked beyond 40 hours in a standard work week This calculation can vary for individual states, so be sure to check with the local state

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