**Question 3**

A company is using linear programming to determine the optimal production mix for two products, A and B. The objective is to maximize profit. The original linear programming problem is as follows:

Maximize Z = 3A + 5B

subject to:

(1) 2A + B ≤ 10;

(2) A + 3B ≤ 12;

(3) A, B ≥ 0.

The company has found the optimal solution to be A = 3.6, B = 2.8, and the maximum profit Z = 24.8.

Now, the company is considering making changes to the problem. They anticipate the following changes:

**(a)**The profit from product A may change from $3 to $4.

**(b)**The profit from product B may change from $5 to $6.

**(c)**The availability of resource in Constraint (1) may change from 10 units to 12 units.

**(d)**The availability of resource in Constraint (2) may change from 12 units to 10 units.

One change will be carried out at one time. That is, when the company makes one change, the rest will remain the same. Determine how each of these changes would impact the optimal solution.

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**Question 4**

**(a)**By using the Simplex Method, solve the following linear programming:

**(b)**Formulate the dual programming of the above linear programming and determine the optimal objective value of the dual programming.

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