Analysis of the problem (1) The goal of this optimization problem is to maximize the benefits of securities recovery, and the decision to be made is the investment plan. That is, the distribution of the number of various securities to be purchased. Comprehensive consideration: specific securities purchase, capital restriction, average credit rating and average life span, according to the requirements of the topic, the problem can be solved by an optimization model composed of decision variables, decision objectives and constraints. The decision variables of the model are X 1, X2, X3, X4 and X5, which represent the value of buying A, B, C, D and E securities respectively. The unit is one million yuan. The objective function is to maximize the profit of bank managers under given conditions. Then, we can get from the table: max z = 0.043x1+0.027x2+0.025x3+0.022x4+0.045x5 (1). The constraint conditions are: x2+x3+x4 > = 4(2)x 1+X2+X3+X4+X5 & lt； = 10(3)6x 1+6x 2-4x 3-4x 4+36x 5 & lt； = 0(4)4x 1+ 10 x2-X3-2X4-3x 5 & lt； =0 (5) and X 1, X2, 3X, X4 and X5 are all negative. (1)(2)(3)(4)(5) The input LINDO of the linear programming model is as follows: max 0.043x1+0.027x2+0.025x3+0.022x4+0.045x5 stx2+x3+x4 >; = 4x 1+X2+X3+X4+X5 & lt； = 10 6x 1+6x 2-4x 3-4x 4+36x 5 & lt； = 0 4x 1+ 10 x2-X3-2X4-3x 5 & lt； =0 to end the solution and conduct sensitivity analysis, It is obtained that LP optimally finds the objective function value 1) 0.2983637 at step 0, and the variable value reduction cost x12.1818 0.000000x20.000000 0.030/kloc-0. 82x3 7.36363636 0.00000x4 0.000000 0.00000 0.002364 Iterations = 0 Constant range: OBJ coefficient range Variable current allows COEF to increase and decrease x10.043000 0.003500 0.01 3000 X2 0.027000 0.030 182 infinity X3 0.025000.0 17333 0.00 The highest after-tax income is RMB 2.98 million. Question (2) Problem Analysis According to the "shadow price" in (1), if the investment increases by 1 10,000 yuan, the income can increase by 0.0298 million yuan. Compared with the interest rate of 2.75%, the interest is 6.5438+0 million yuan, which should be borrowed. The right end of the second constraint is changed to 1 1 as a safe (1) model, and the solution is enough. Model solution: Securities A, C and E invested 2.4 million yuan, 8 10/00000 yuan and 5000 yuan respectively, with the highest return of 300700 yuan. (3) According to the allowable range of the target coefficient in the result of (1), the pre-tax income of securities A can be increased by 0.35%, so the pre-tax income of securities A can be increased. The pre-tax income of securities C has been reduced by 0. 1 12% (50% tax), so the pre-tax income of securities C can be reduced by 4.8%, so the investment should be changed.