Consider the following integer nonlinear programming problem.Maximize Z = ,subject to ≥ 1, ≥ 1, ≥ 1,and , , are integers.Use dynamic programming to solve this problem.

Question

Consider the following integer nonlinear programming problem.Maximize Z =  ,subject to   ≥ 1,  ≥ 1,  ≥ 1,and  ,  ,  are integers.Use dynamic programming to solve this problem.

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The Answer is The three stages involve making the decisions on the values of the three variables in order.At stage n,let the state s_n be the remaining slack in the constraint

The three stages involve making the decisions on the values of the three variables in order.At stage n,let the state s<sub>n</sub> be the remaining slack in the constraint ,so s<sub>1</sub> = 10,s<sub>2</sub> = 10 - x<sub>1</sub>,and s<sub>3</sub> = s<sub>2</sub> - 2x<sub>2</sub>.Let .Then ,and for n = 1,2.For n = 3, For n = 2, For n = 1, Since = 2 leads to s<sub>2</sub> = 10 - = 8 and = 1,which then leads to s<sub>3</sub> = s<sub>2</sub> - 2 = 6 and = 2,it follows that (x<sub>1</sub>,x<sub>2</sub>,x<sub>3</sub>)= (2,1,2)is optimal with Z = 16.

,so s₁ = 10,s₂ = 10 - x₁,and s₃ = s₂ - 2x₂.Let

.Then

,and

for n = 1,2.For n = 3,

For n = 2,

For n = 1,

Since

= 2 leads to s₂ = 10 -

= 8 and

= 1,which then leads to s₃ = s₂ - 2

= 6 and

= 2,it follows that (x₁,x₂,x₃)= (2,1,2)is optimal with Z = 16.

Consider the Following Integer Nonlinear Programming Problem