Perform sensitivity analysis on the coefficient for Bowls. What is the allowable decrease for this objective coefficient?

Question

BlackTom AI · Accepted Answer

To approach this question, I will first restate what is being asked and then walk through the concept of sensitivity analysis for an objective coefficient in a linear programming model.

Question restatement: You are asked to perform sensitivity analysis on the coefficient for Bowls and determine the allowable decrease for this objective coefficient.

Conceptual background: In linear programming, the sensitivity (or post-optimal) analysis for an objective coefficient tells you how far you can change that coefficient without changing the optimal basis. The two relevant measurements are the allowable increase and the allowable decrease. These define a range around the current optimal value of the coefficient within which the same set of basic variables remains optimal, and the current optimal solution does not change.

General procedure to determine allowable decrease (conceptual steps):
- Step 1: Obtain the optimal solution from the solver, including the optimal objective value, the basic variables, and the current coefficient values.
- Step 2: Look at the sensitivity report (often labeled as Post-Optimality or Shadow Price/Reduced Cost analysis) for the coefficient of Bowls. Identify the reduced cost of nonbasic variables and the shadow price associated with any binding constraints if relevant. The key numbers for the coefficient are the allowable decrease and increase bounds.
- Step 3: The allowable decrease tells you how much you can decrease the Bowls coefficient before the optimal basis changes. It is calculated based on the ratio of the current objective coefficient to certain dual variables and the limits imposed by the constraints tied to the Bowls decision variable.
- Step 4: Compare the Bowls coefficient to its lower bound in the sensitivity report; the smallest amount you can drop the coefficient before a different variable enters (or leaves) the basis marks the allowable decrease.
- Step 5: If you need the exact figure, extract it from the solver’s sensitivity/solution report, which will present both the current coefficient value and its allowable decrease (and increase).

Application to the given data: The provided answer field indicates a numeric value of 11. Interpreting this in the standard sensitivity-analysis context, an allowable decrease of 11 means:
- You can reduce the Bowls coefficient by up to 11 units without changing the optimal basis.
- If you decrease the Bowls coefficient by more than 11, the current set of active constraints and basic variables could change, potentially shifting the optimal solution.
- This interpretation assumes standard LP formulations and a single objective with Bowls as one of the decision variables, and that the sensitivity data is read from the solver’s post-solution report.

Why there are no answer options to analyze: The provided data shows an empty list for answer_options, which means there are no multiple-choice alternatives to critique here. In such a case, the essential task is to understand the concept and apply it to the given coefficient, with the reported value serving as the bound derived from the model's sensitivity output.

Note on potential sources of confusion: If you expected to see both an allowable decrease and an allowable increase, both would be presented in the sensitivity report. Here, only the decrease bound is specified (11). Ensure you’re looking at the correct section of the report for Bowls and that the coefficient in question indeed corresponds to the Bowls decision variable in the model.

COMM_V 190 101-104 2025W1 2025W1 COMM 190 Midterm - Thursday, October 9, 2025 - Requires Respondus LockDown Browser

Perform sensitivity analysis on the coefficient for Bowls. What is the allowable decrease for this objective coefficient?

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