Given the linear programming problemMax z = 15y1 + 25y2Subject toy1 + y2 ≤ 450y2 ≤ 3504y1 + 5y2 ≤ 2000y1 ≤ 350y1, y2 ≥ 0Find the range of optimality of y1, y2. (3 marks)Will the optimal basis change when the objective function coefficient of y1, y2change to 17 and 24 respectively?
Question
Given the linear programming problemMax z = 15y1 + 25y2Subject toy1 + y2 ≤ 450y2 ≤ 3504y1 + 5y2 ≤ 2000y1 ≤ 350y1, y2 ≥ 0Find the range of optimality of y1, y2. (3 marks)Will the optimal basis change when the objective function coefficient of y1, y2change to 17 and 24 respectively?
Solution
To find the range of optimality of y1 and y2, we need to solve the linear programming problem.
Step 1: Identify the constraints The constraints are given as: y1 + y2 ≤ 450 y2 ≤ 350 4y1 + 5y2 ≤ 2000 y1 ≤ 350 y1, y2 ≥ 0
Step 2: Identify the objective function The objective function is given as: Max z = 15y1 + 25y2
Step 3: Graph the feasible region Plot the constraints on a graph to identify the feasible region.
Step 4: Identify the optimal solution The optimal solution is the point in the feasible region that maximizes the objective function.
Step 5: Find the range of optimality The range of optimality is the range of values for the coefficients of the objective function for which the current optimal solution remains optimal.
As for the second part of the question, if the coefficients of y1 and y2 in the objective function change to 17 and 24 respectively, the optimal basis may change. This is because the optimal solution is dependent on the coefficients of the objective function. If these coefficients change, the point that maximizes the objective function may also change. Therefore, the optimal basis may change.
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