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$Using Dynamic Programming To Solve Problem Essay Paper Quiz 5) Consider the following nonlinear maximization problem with the nonlinear constraint (1), and suppose we want to solve it via dynamic programming. max0.25x3+y2+4.5z S.t. xyz=4x,y,z∈{1,2,4} a) Clearly define stages, states, actions at each stage, and state transition function (relation between Sn and Sn+1)⋅(+20 points) b) Use dynamic programming to solve this problem by constructing the usual tables. State the optimal solution. (+80 points) Using Dynamic Programming To Solve Problem Essay Paper ORDER YOUR PAPER NOW Expert Answer This solution was written by a subject matter expert. It's designed to help students like you learn core concepts. Step-by-step 1st step All steps Answer only Step 1/2 Defining stages, states, actions at each stage, and state transition function. Explanation for step 1 In order to solve this problem using dynamic programming, we need to define a sequence of stages, states at each stage, actions that can be taken at each stage, and a state transition function that describes how the state at each stage is related to the state at the next stage Using Dynamic Programming To Solve Problem Essay Paper. The stages of the dynamic programming solution can be defined as follows: Stage 1: At this stage, we consider the value of x. Stage 2: At this stage, we consider the value of y. Stage 3: At this stage, we consider the value of z. At each stage, the state is defined as the current values of x, y, and z. The actions at each stage are the possible values of x, y, and z, which are {1, 2, 4}. The state transition function describes the relationship between the state at stage n and the state at stage n+1. In this case, the state transition function would be defined as follows: Using Dynamic Programming To Solve Problem Essay Paper Sn+1(x, y, z) = Sn(x', y', z') where (x', y', z') is the state at stage n, and (x, y, z) is the state at stage n+1. Therefore, the state transition function describes how the values of x, y, and z at stage n+1 are related to the values of x', y', and z' at stage n. Step 2/2 b) Using dynamic programming to solve this problem by constructing the usual tables. State the optimal solution. Explanation for step 2 Here, we will be using dynamic programming to solve the problem. Final answer (b) To solve this problem using dynamic programming, we can construct a table with the stages as rows and the states as columns. We will fill in the table with the maximum value of the objective function (0.25x^3+y^2+4.5z) that can be achieved at each stage, given the current state. We can start by filling in the table for stage 1, which corresponds to the values of x. We consider all possible values of x and the corresponding values of y and z that satisfy the constraint xyz = 4. The table for stage 1 would look like this: Using Dynamic Programming To Solve Problem Essay Paper x y z 0.25x^3+y^2+4.5z 1 4 1 0.25 + 16 + 4.5 2 2 1 2 + 4 + 4.5 4 1 1 16 + 1 + 4.5 We can then move on to stage 2 and consider the values of y, given the values of x and z that we have already determined. The table for stage 2 would look like this: x y z 0.25x^3+y^2+4.5z 1 1 1 0.25 + 1 + 4.5 1 2 1 0.25 + 4 + 4.5 1 4 1 0.25 + 16 + 4.5 2 1 1 2 + 1 + 4.5 2 2 1 2 + 4 + 4.5 4 1 1 16 + 1 + 4.5 Finally, we can consider the values of z at stage 3, given the values of x and y that we have already determined. The table for stage 3 would look like this: Using Dynamic Programming To Solve Problem Essay Paper ORDER YOUR PAPER NOW x y z 0.25x^3+y^2+4.5z 1 1 1 0.25 + 1 + 4.5 1 2 1 0.25 + 4 + 4.5 1 4 1 0.25 + 16 + 4.5 2 1 1 2 + 1 + 4.5 2 2 1 2 + 4 + 4.5 2 1 2 2 + 1 + 9 2 2 2 2 + 4 + 9 4 1 1 16 + 1 + 4.5 4 1 2 16 + 1 + 9 4 1 4 16 + 1 + 18 To find the optimal solution using dynamic programming, we can take the maximum value from the final table, which is 16 + 1 + 18 = 35, and the corresponding state (x, y, z) = (4, 1, 4). This means that the optimal solution for the nonlinear maximization problem with the nonlinear constraint xyz = 4 and the possible values of x, y, and z being {1, 2, 4} is achieved when x = 4, y = 1, and z = 4, with a maximum value of 0.254^3 + 1^2 + 4.54 = 35 Using Dynamic Programming To Solve Problem Essay Paper$

Expert Answer

Using Dynamic Programming To Solve Problem Essay Paper

Quiz 5) Consider the following nonlinear maximization problem with the nonlinear constraint (1), and suppose we want to solve it via dynamic programming.

max 0.25 x^{3} + y^{2} + 4.5 z S.t. x yz = 4 x, y, z \in {1, 2, 4}

a) Clearly define stages, states, actions at each stage, and state transition function (relation between

S_{n}

and

S_{n + 1}) \cdot (+ 20

points) b) Use dynamic programming to solve this problem by constructing the usual tables. State the optimal solution. (+80 points) Using Dynamic Programming To Solve Problem Essay Paper

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Expert Answer

Step-by-step

1st step

All steps

Answer only

Step 1/2

Defining stages, states, actions at each stage, and state transition function.

Explanation for step 1

In order to solve this problem using dynamic programming, we need to define a sequence of stages, states at each stage, actions that can be taken at each stage, and a state transition function that describes how the state at each stage is related to the state at the next stage Using Dynamic Programming To Solve Problem Essay Paper.

The stages of the dynamic programming solution can be defined as follows:

Stage 1: At this stage, we consider the value of x.

Stage 2: At this stage, we consider the value of y.

Stage 3: At this stage, we consider the value of z.

At each stage, the state is defined as the current values of x, y, and z. The actions at each stage are the possible values of x, y, and z, which are {1, 2, 4}.

The state transition function describes the relationship between the state at stage n and the state at stage n+1. In this case, the state transition function would be defined as follows: Using Dynamic Programming To Solve Problem Essay Paper

Sn+1(x, y, z) = Sn(x', y', z')

where (x', y', z') is the state at stage n, and (x, y, z) is the state at stage n+1.

Therefore, the state transition function describes how the values of x, y, and z at stage n+1 are related to the values of x', y', and z' at stage n.

Step 2/2

b) Using dynamic programming to solve this problem by constructing the usual tables. State the optimal solution.

Explanation for step 2

Here, we will be using dynamic programming to solve the problem.

Final answer

(b) To solve this problem using dynamic programming, we can construct a table with the stages as rows and the states as columns. We will fill in the table with the maximum value of the objective function (0.25x^3+y^2+4.5z) that can be achieved at each stage, given the current state.

We can start by filling in the table for stage 1, which corresponds to the values of x. We consider all possible values of x and the corresponding values of y and z that satisfy the constraint xyz = 4. The table for stage 1 would look like this: Using Dynamic Programming To Solve Problem Essay Paper

x	y	z	0.25x^3+y^2+4.5z
1	4	1	0.25 + 16 + 4.5
2	2	1	2 + 4 + 4.5
4	1	1	16 + 1 + 4.5

We can then move on to stage 2 and consider the values of y, given the values of x and z that we have already determined. The table for stage 2 would look like this:

x	y	z	0.25x^3+y^2+4.5z
1	1	1	0.25 + 1 + 4.5
1	2	1	0.25 + 4 + 4.5
1	4	1	0.25 + 16 + 4.5
2	1	1	2 + 1 + 4.5
2	2	1	2 + 4 + 4.5
4	1	1	16 + 1 + 4.5

Finally, we can consider the values of z at stage 3, given the values of x and y that we have already determined. The table for stage 3 would look like this: Using Dynamic Programming To Solve Problem Essay Paper

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x	y	z	0.25x^3+y^2+4.5z
1	1	1	0.25 + 1 + 4.5
1	2	1	0.25 + 4 + 4.5
1	4	1	0.25 + 16 + 4.5
2	1	1	2 + 1 + 4.5
2	2	1	2 + 4 + 4.5
2	1	2	2 + 1 + 9
2	2	2	2 + 4 + 9
4	1	1	16 + 1 + 4.5
4	1	2	16 + 1 + 9
4	1	4	16 + 1 + 18

To find the optimal solution using dynamic programming, we can take the maximum value from the final table, which is 16 + 1 + 18 = 35, and the corresponding state (x, y, z) = (4, 1, 4). This means that the optimal solution for the nonlinear maximization problem with the nonlinear constraint xyz = 4 and the possible values of x, y, and z being {1, 2, 4} is achieved when x = 4, y = 1, and z = 4, with a maximum value of 0.254^3 + 1^2 + 4.54 = 35 Using Dynamic Programming To Solve Problem Essay Paper

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