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Algorithm 175 MMAS Algorithm with Periodic Use of the Global-Best Path Initialize parameters , , , nk , p, min , max , f ; t = 0, max (0) = max , min (0) = min ; Place all ants, k = 1, , nk ; ij (t) = max (0), for all links (i, j); x+ (t) = , f (x+ (t)) = 0; repeat if stagnation point then for each link (i, j) do Calculate ij (t) using equation (1728); ij (t + 1) = ij (t) + ij (t); end end for each ant k = 1, , nk do xk (t) = ; repeat Select next node j with probability de ned in equation (176); xk (t) = xk (t) {(i, j)}; until full path has been constructed; Compute f (xk (t)); end (t mod f = 0) (Iteration Best = false) : (Iteration Best = true); if Iteration Best = true then Find iteration-best: x+ (t) = xk (t) : f (xk (t)) = mink =1,,nk {f (xk (t))}; Compute f (x+ (t)); end else Find global-best: x = xk (t) : f (xk (t)) = mink =1,,nk {f (xk (t))}; Compute f (x); if f (x) < f (x+ (t)) then x+ (t) = x; f (x+ (t)) = f (x); end end for each link (i, j) x+ (t) do Apply global update using equation (1719); end Constrict ij (t) to be in [ min (t), max (t)] for all (i, j); x+ (t + 1) = x+ (t); f (x+ (t + 1)) = f (x+ (t)); t = t + 1; Update max (t) using equation (1726); Update min (t) using equation (1727); until stopping condition is true; Return x+ (t) as the solution;

Form 1040-ES, Estimated Tax for Individuals Otherwise, you may owe additional tax Two earners/two jobs If you have a working spouse or more than one job, figure the total number of allowances you are entitled to claim on all jobs using worksheets from only one Form W-4 Your withholding usually will be most accurate when all allowances are claimed on the Form W-4 for the highest paying job and zero allowances are claimed on the others Nonresident alien If you are a nonresident alien, see the Instructions for Form 8233 before completing this Form W-4 Check your withholding After your Form W-4 takes effect, use Pub 919 to see how the dollar amount you are having withheld compares to your projected total tax for 2003 See Pub 919, especially if your earnings exceed $125,000 (Single) or $175,000 (Married) Recent name change If your name on line 1 differs from that shown on your social security card, call 1-800-772-1213 for a new social security card

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The max-min approach was also applied to ACS, referred to as MMACS In this case, the larger the value of r0 (refer to equation (1717)), the tighter the pheromone concentration limits have to be chosen A lower max / min ratio has to be used for larger r0 to prevent ants from too often preferring links with high pheromone concentrations Even though pheromone concentrations are clamped within a de ned range, stagnation still occurred, although less than for AS To address this problem, a pheromone smoothing strategy is used to reduce the di erences between high and low pheromone concentrations At the point of stagnation, all pheromone concentrations are increased proportional to the di erence with the maximum bound, ie ij (t) ( max (t) ij (t)) (1728)

Form W-4 (2003)

Using this smoothing strategy, stronger pheromone concentrations are proportionally less reinforced than weaker concentrations This increases the chances of links with low pheromone intensity to be selected as part of a path, and thereby increases the exploration abilities of the algorithm Algorithm 175 summarizes MMAS, where the iteration-best path is used as default to update pheromones and the global-best is used periodically at frequency f

Gambardella and Dorigo [213, 300] developed a variant of ACS in which the local update rule was inspired by Q-learning [891] (also refer to 6) In Ant-Q, the pheromone notion is dropped to be replaced by Ant-Q value (or AQ-value) The goal of Ant-Q is to learn AQ-values such that the discovery of good solutions is favored in probability Let ij (t) denote the AQ-value on the link between nodes i and j at time step t Then the transition rule (action choice rule) is given by (similarly to equation (1717)) j=

Purpose Complete Form W-4 so that your employer can withhold the correct Federal income tax from your pay Because your tax situation may change, you may want to refigure your withholding each year Exemption from withholding If you are exempt, complete only lines 1, 2, 3, 4, and 7 and sign the form to validate it Your exemption for 2003 expires February 16, 2004 See Pub 505, Tax Withholding and Estimated Tax Note: You cannot claim exemption from withholding if: (a) your income exceeds $750 and includes more than $250 of unearned income (eg, interest and dividends) and (b) another person can claim you as a dependent on their tax return Basic instructions If you are not exempt, complete the Personal Allowances Worksheet below The worksheets on page 2 adjust your withholding allowances based on itemized

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