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Extra info for Credibilistic Programming: An Introduction to Models and Applications (Uncertainty and Operations Research)
P ⎪ ⎩ E gi (x, ξ ) ≤ zero, i = 1, 2, . . . , n. five. four Fuzzy Simulation which will clear up the final entropy optimization versions, this part introduces a fuzzy simulation strategy to approximate the bushy entropy U : x → H f (x, ξ ) the place f is a true functionality, and ξ is a fuzzy vector. (5. 10) 114 five Entropy Optimization version Fig. five. 2 Entropy simulation with variable parameter N feel that ξ has a joint credibility functionality ν. Randomly generate vectors y i with credibilities νi for all i = 1, 2, . . . , N . in response to Definition five. 2, the entropy may be simulated because the numerical integration of functionality S(ν(x)). The process is indexed as follows. set of rules five. 1 (Fuzzy simulation for entropy) Step 1. Set h = zero and okay = zero. Step 2. Randomly generate y i with credibilities νi for all i = 1, 2, . . . , N . Step three. Calculate the minimal and greatest values a = min f (x, y i ) | 1 ≤ i ≤ N , b = max f (x, y i ) | 1 ≤ i ≤ N . Step four. Calculate sk = −νk ln νk − (1 − νk ) ln(1 − νk ). Step five. Set h → h + sk . If ok < N , set ok = okay + 1 and visit step four. Step 6. go back h(b − a)/N because the entropy. instance five. eight Taking ξ = (1, 2, 3), we learn the convergence of the entropy simulation set of rules via altering N from a hundred to 5000 with step of a hundred. the implications are illustrated via Fig. five. 2. it truly is proven that after N ≥ 2000, the simulated effects are strong and converge to the precise price H [ξ ] = 1. instance five. nine Taking N = 2000, we practice set of rules five. 1 on twenty fuzzy variables, together with equipossible fuzzy variables, triangular fuzzy variables, trapezoidal 5. five functions one hundred fifteen desk five. 1 Simulation effects on entropy Fuzzy variables Simulated worth certain worth Relative errors (0. zero, 1. zero) zero. 6931 zero. 6931 zero. 0000 (−1. zero, 2. zero) 2. 0794 2. 0794 zero. 0000 (3. five, 7. eight) 2. 9803 2. 9805 zero. 0001 27. 7233 27. 7259 zero. 0001 (−0. three, 1. eight, 2. three) 1. 3017 1. 3000 zero. 0013 (1. five, three. zero, four. 1) 1. 2833 1. 3000 zero. 0128 zero. 0123 (−50, −10) (10, 15, 20) four. 9387 five. 0000 (25, forty, 50) 12. 5234 12. 5000 zero. 0019 (1. zero, 2. zero, three. zero, four. zero) 1. 6843 1. 6931 zero. 0052 (3. 1, four. 2, four. five, 6. zero) 1. 5002 1. 5079 zero. 0051 (2. four, three. 6, three. 7, five. 6) four. 0464 four. 0763 zero. 0073 (10, 25, 30, forty five) 1. 5867 1. 6193 zero. 0201 N(1. five, 1. zero) 2. 5464 2. 5651 zero. 0073 N(2. five, 1. zero) 2. 5688 2. 5651 zero. 0014 N(3. 7, 2. 1) five. 4003 five. 3867 zero. 0025 N(6. zero, 1. three) three. 3310 three. 3346 zero. 0011 E(1. three) 1. 6476 1. 6673 zero. 0118 E(1. zero) 1. 2907 1. 2825 zero. 0063 E(3. five) four. 4391 four. 4889 zero. 0111 E(7. 6) nine. 7563 nine. 7474 zero. 0009 fuzzy variables, exponential fuzzy variables, and general fuzzy variables. We list the simulated effects through desk five. 1, and make comparisons with the precise values. The final column documents the relative mistakes (3. 25), which levels from zero. 00 % to two. 01 % and the typical errors is zero. fifty four percent. those effects suggest that the simulation set of rules can receive a really passable approximation for entropy. instance five. 10 during this instance, we take N = 2000, and practice set of rules five. 1 fifty instances at the triangular fuzzy variable ξ = (−0. three, 1. eight, 2. 3). the implications are recorded by way of desk five. 2. in comparison with the precise price H [ξ ] = 1.