The two stages involve making the decisions on the values of the two variables in order.At stage n,let the state s_n be the vector (R₁,R₂),where R_i is the slack in the i-th constraint,so s₁ = (13,9)and s₂ = (13 - 2 11ea84c6_c901_ca63_83dc_1992569ec41c_TB2462_11 ,9 - 11ea84c6_c901_ca64_83dc_650189c8fb77_TB2462_00 ).For stage 2 (the second variable),we have f₂ (R₁,R₂,x₂)= x₂,0 \(\le\) x₂\(\le\) min{R₁,R₂},so 11ea84c6_c901_f175_83dc_d5b564580514_TB2462_00 is easily solved,as shown in the following table. 11ea84c6_c901_f176_83dc_4b542b0e019d_TB2462_00 For stage 1 (the first variable),we now have f₁(13,9,x₁)= 11ea84c6_c902_1887_83dc_7954a5d9cfdf_TB2462_11 = 11ea84c6_c902_1888_83dc_e17f54016762_TB2462_11 .For 0 \(\le\) x₁ \(\le\) 2, 11ea84c6_c902_3f99_83dc_d1b0fc253086_TB2462_00 \(\implies\) f₁(13,9,x₁)= 11ea84c6_c902_3f9a_83dc_61f7810aa53f_TB2462_11 .Since 11ea84c6_c902_66ab_83dc_bdac37f38edb_TB2462_11 for 0 ≤ x₁ ≤ 2,we now have 11ea84c6_c902_66ac_83dc_6b683ccd2d61_TB2462_11 at x₁ = 2.For x₁ \(\le\) 2, 11ea84c6_c902_8dbd_83dc_0db9fabd8267_TB2462_11 \(\implies\) f₁(13,9,x₁)= 11ea84c6_c902_8dbe_83dc_cb4e668eae84_TB2462_11 Since 11ea84c6_c902_b4cf_83dc_2bb949c47aa7_TB2462_11 = 5 - 4x₁ < 0 for x₁ ≥ 0,it follows that 11ea84c6_c902_b4d0_83dc_9d415be54e18_TB2462_11 at x₁ = 2.Since f₁(13,9,x₁)is maximized at x₁ = 2 over both the range 0 ≤ x₁ ≤ 2 and the range x₁ ≥ 2,the maximum must be at x₁ = 2 over the entire range of x₁ ≥ 0.Consequently, 11ea84c6_c902_b4d1_83dc_75e652fe9298_TB2462_11 (13,9)= 11ea84c6_c902_dae2_83dc_5d8afae9009c_TB2462_11 at 11ea84c6_c902_dae3_83dc_a57d75d48423_TB2462_11 = 2.Therefore,since x₁ = 2 leaves a slack of R₁ = 13 - 2 11ea84c6_c903_01f4_83dc_4376dbeca1d1_TB2462_11 = 5 and R₂ = 9 - 11ea84c6_c903_01f5_83dc_dd46efd200f3_TB2462_11 = 5 in the two constraints,so 11ea84c6_c903_2906_83dc_f96d3c9256a1_TB2462_11 = min{R₁,R₂} = 5,the optimal solution for the overall problem is 11ea84c6_c903_2907_83dc_15f605169b8a_TB2462_11 = 2, 11ea84c6_c903_5018_83dc_75553210bff4_TB2462_11 = 5,with Z* = 15.

Consider the Following Integer Nonlinear Programming Problem

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