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By Charles G. Moore

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Rk (x) ≥ λ, k = 1, 2, · · · , m, N bmin ≤ j aij xi ≤ bmax , j j = 1, · · · , P. t. Rik xi , k = 1, 2, · · · , m Rk = xi = 1, i=1 xmin i ≤ xi ≤ xmax , i i = 1, · · · , n. t. λ wk Rk ≥ λ, k = 1, 2, · · · , m, n xi = 1, i=1 xmin i ≤ xi ≤ xmax , i i = 1, · · · , n. t. µRk (x) ≥ λ, x,λ n k = 1, · · · , m, xi = 1, i=1 xmin i ≤ xi ≤ xmax , i i = 1, · · · , n. 3 Conclusion In this chapter, we introduce Ramaswamy’s Model. Ramaswamy (1998) gave a numerical example in which the investor is only allowed to hold government bonds and plain vanilla options, and only two scenarios are assumed: “bullish” and “bearish”.

These new modifications could provoke other disagreements. (b) Because of the risk in this case, one think that a fuzzy multi-objective decision approach would be the most appropriate. The following results justify that the algorithm is well defined. 1 If problem (AP) is feasible, then φmin ≥ 1/k, where k denotes the number of non-null components of vector (R, S). Proof. It suffices to construct a feasible solution for the dual of (AP) with objective value 1/k, because this provides us with a lower bound for the optimal value of (AP).

T. max min{0, t ⎩ ⎭ j=1 n+1 xi = 1, i=1 0 ≤ xi ≤ ui , n+1 j=1 i = 1, 2, · · · , n + 1, laj + lbj βj − αj + 2 6 xj ≥ E(ˆl0 ). where w0 is a given constant representing the tolerance level of risk, and E(ˆl0 ) is the required level of fuzzy turnover rate. (P3-1), (P3-2) and (P3-3), (P3-4) can be used interchangeably to generate the efficient frontier of portfolios. In this sequel, we only discuss (P3-3) and (P3-4). To solve (P3-3) and (P3-4), we consider the following transformation. t. max t n (rit − ri )xi } , t = 1, 2, · · · , T min{0, ≤ w0 , i=1 n+1 ki |xi − x0i | ≤ xn+2 , i=1 n+1 xi = 1, i=1 0 ≤ xi ≤ ui , i = 1, 2, · · · , n + 1, n+1 βj − αj laj + lbj + )xj ≥ E(ˆl0 ).

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