Consider the problem min z=10x1 + 8x_{2},s.t.c: x_{1} + 2x_{2} >= b_{1} 2x_{1} - x_{2} >= b_{2} x_{1} + 3x_{2} >= b_{3}where b1,b2,b3 are constants.For specific values  of b1, b2, b3. The optimal solutio isIs To y_{7} y_{8} Saln.

Basic

y

0

Yé y." Y3 y_{11}

Ys

0

1715-15-1-15

45

1-

a

- 2/5 1/5 0

45

Y2"

y_{1}

oe

0

C-215 0 y_{5} 2/5 0 29/5

z_{j} - c_{j}

where a,b,c,d, e are constants. Determine

(1) The values of bi, be, by that yield the givenoptimal soln

(ii) the optimal dual soln &

(iii) the values of a, b & e in the optimal tableare.

The given problem is a linear programming problem that involves minimizing the objective function \( z = 10x_1 + 8x_2 \), subject to the following constraints:

\[
x_1 + 2x_2 \geq b_1
\]
\[
2x_1 - x_2 \geq b_2
\]
\[
x_1 + 3x_2 \geq b_3
\]
Where \( b_1 \), \( b_2 \), and \( b_3 \) are constants. You're also asked to interpret information regarding the optimal solution based on the table (which seems to have values in a simplex tableau form).

### Steps to Solve:

#### Part (i): Determine the values of \( b_1 \), \( b_2 \), and \( b_3 \) that yield the given optimal solution

- To find these values, we can examine the constraints. The values of \( b_1 \), \( b_2 \), and \( b_3 \) are derived based on the final simplex tableau, but this part of the problem requires the specific values provided in the simplex tableau (which I couldn't fully interpret from the text as it appears incomplete).

#### Part (ii): The optimal dual solution

- The dual problem involves assigning variables (dual variables) to the constraints. Since this is a minimization problem, each constraint corresponds to a dual variable in a maximization problem.

#### Part (iii): The values of \( a \), \( b \), and \( e \) in the optimal tableau

- The simplex tableau contains information about the basic variables and the coefficients that help determine the optimal values. Here, \( a \), \( b \), and \( e \) are likely constants that relate to the final tableau, and their values come from the optimal condition of the dual problem.

### Next Steps

I will need more specific details from the simplex tableau to correctly determine \( a \), \( b \), \( e \), and the dual solution. Could you provide more clear values or upload an image for better analysis?

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Here are some follow-up questions you might explore:
1. What is the dual formulation of this problem?
2. How do you interpret the tableau in the simplex method?
3. Can you explain how the constraints impact the feasible region?
4. What is the relationship between primal and dual solutions?
5. How do you verify the optimality of a solution in linear programming?

**Tip**: In linear programming, the final simplex tableau provides both primal and dual solutions, and complementary slackness can be used to verify their correctness.

Consider the problem min z=10x1 + 8x_{2},s.t.c: x_{1} + 2x_{2} >= b_{1} 2x_{1} - x_{2} >= b_{2} x_{1} + 3x_{2} >= b_{3} where b1, b2, b3 are constants. For specific values of b1, b2, b3. The optimal solution is To y_{7} y_{8} Saln. Basic y 0 Yé y." Y3 y_{11} Ys 0 1715-15-1-15 45 1- a - 2/5 1/5 0 45 Y2" y_{1} oe 0 C-215 0 y_{5} 2/5 0 29/5 z_{j} - c_{j} where a, b, c, d, e are constants. Determine (1) The values of bi, be, by that yield the given optimal solution (ii) the optimal dual solution & (iii) the values of a, b & e in the optimal table.

Explore how to solve the linear programming problem with the objective function z = 10x1 + 8x2 using the Simplex method. Learn how to determine the values of constants b1, b2, and b3, find the optimal dual solution, and interpret the simplex tableau for constants a, b, and e. Suitable for undergraduate students.

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