Suppose r=150 and d=150. Which of the following actions (taken individually) would cause the new optimal solution to change? (Select all that apply.)

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BlackTom AI · Accepted Answer

We begin by restating the problem in our own words: with r = 150 and d = 150, which individual changes to the problem would cause the new optimal solution to differ from the current one? We are to assess each proposed action separately and explain why it would or would not change the optimal solution.

Option 1: Adding a constraint that says acres of Ryegrass must be at most 5.
- This adds a new upper bound on an existing decision variable (acres of Ryegrass). If the current optimal solution already uses 5 or fewer acres of Ryegrass, this new constraint is non-binding and would not alter the feasible set reachable by the current optimum; the optimal solution could remain the same. If, however, the current optimum uses more than 5 acres of Ryegrass, this constraint would cut off part of the current feasible region, potentially forcing a different mix of activities and a different optimum. Therefore, the impact is conditional: it would change the optimal solution only if Ryegrass acres in the current optimum exceed 5.

Option 2: Increasing hours available in baling by 1 hour.
- This modifies a resource bound for the baling activity. If, in the current optimum, baling hours are not fully utilized (i.e., baling is slack), increasing availability would not change the optimal solution because the binding constraints shaping the optimum remain the same. If baling hours are binding (the optimum uses all available baling hours), adding one hour expands the feasible region and could allow a higher objective value, potentially shifting the optimal solution to a different allocation. Thus, the effect depends on whether baling hours are binding at the current optimum.

Option 3: Removing the nonnegativity constraints.
- Nonnegativity constraints ensure decision variables cannot be negative. Removing them radically expands the feasible region, often allowing negative values that could improve the objective in nonstandard ways (depending on the objective and other constraints). In almost every practical LP, removing nonnegativity changes the solution because it invalidates the basic feasibility structure and often enables qualitative changes in which variables take positive values. Therefore, this action would change the optimal solution in virtually all reasonable models.

Option 4: Decreasing hours available in raking by 1 hour.
- Similar to Option 2, this tightens a resource constraint, but for the raking activity. If raking hours are not fully used in the current optimum, decreasing them would have no effect. If they are binding, reducing availability narrows the feasible region and can force a different optimal combination of activities. Hence, the change in the optimal solution depends on whether raking hours are binding in the present solution.

Summary of the conditional logic:
- Options 1, 2, and 4 can change the optimal solution depending on the current activity levels and which constraints are binding at the optimum. If the relevant constraint is non-binding, there is no change; if it is binding, the optimum may shift.
- Option 3 almost certainly changes the optimal solution because removing nonnegativity breaks the standard feasibility structure and generally yields a different optimal allocation.

In a concrete scenario, you would check the current optimal solution to see which constraints are binding (Ryegrass total, baling hours, raking hours) and then determine for Options 1, 2, and 4 whether the proposed changes would alter the binding status. If you have the specific current solution values, I can walk through which options would definitively change the optimum.

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Suppose r=150 and d=150. Which of the following actions (taken individually) would cause the new optimal solution to change? (Select all that apply.)

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