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Calibration of a Gravity–Opportunity Model by Global Optimization 49A i = ( ∑ jB j D j e −λW ije −βc ij) −1 (2)B j = ( ∑ iA i O i e −λW ije −βc ij) −1 (3)∑i j∑i jT ∗ijc ij = ∑ i jT ∗ijw ij = ∑ i jT ij c ij (4)T ij w ij . (5)These relations may be derived from the principle of the Maximum Likelihood applied to(1). The parameters β and λ are determined into a way that the results reproduce the averagecost of the observed trips and the average number observed in the intervenient opportunitiesper trip: (β, λ) minimizes the function⎛f(β, λ) = ⎝c − ∑ ijT ij (β, λ)T⎞2· c ij⎠⎛+ ⎝w − ∑ ijT ij (β, λ)T⎞2· w ij⎠(6)wherec =∑ijT ∗ij c ijIf Equations 4 – 5 are satisfied f(β, λ) = 0.T ∗ and w =∑ijT ∗ij w ijT ∗ . (7)3. A Numerical Method[7] has introduced an evolutionary version of an random perturbation algorithm based onthe gradient method. We present here a modification of these method, improved by the introductionof a Representation Formula established by [9]. The Representation is used inorder to generate the initial population. The experiments have established improvements inthe robustness and speed of convergence. A complete set of experiments with classical testfunctions is performed in [4].The Representation Formula reads as follows: let us consider a regular function f : R n → R,defined on a closed bounded not empty set S ⊂ R n , S ≠ φ. Assume that P is a probability onS having a strictly positive regular density. Then, we have:x ∗ =E(xg(λ, f(x)))limλ→+∞ E(g(λ, f(x))) , (8)where λ ∈ R + , g is a function conveniently chosen and E(•) denotes the mean. Equation8 may be interpreted as a weighted mean of x on S. The weights are connected to the valueof f(x) and, for g having suitable decreasing properties, the weights are smaller for pointscorresponding to higher values of the objective function. At the limit, the weights concentrateon the points corresponding to the global optimum. A possible choice for g, suggested by M.Pincus (see, for instance, [9]) is:g(λ, f(x)) = e −λ f(x) . (9)