Answer the following statement with TRUE or FALSE:

the correct way to implement a sensitivity analysis for the second coefficient of the objective function is to vary the second coefficient and the next one at the same time (eg.: coefficient #2 and #3).

Question

Answer the following statement with TRUE or FALSE:

the correct way to implement a sensitivity analysis for the second coefficient of the objective function is to vary the second coefficient and the next one at the same time (eg.: coefficient #2 and #3).

BlackTom AI · Accepted Answer

Sensitivity analysis for a single objective coefficient examines how much that single coefficient can change while keeping the current optimal basis and optimality of the solution.

1) Relevant concepts:
- In linear programming the objective function coefficients (c_j) determine the reduced costs (r_j = c_j - z_j) in the final tableau. 
- One-way (single-parameter) sensitivity analysis finds the allowable increase/decrease for one coefficient so that the current basis remains optimal. 
- Simultaneous (multi-parameter) changes are possible but require a different, more complex analysis (region-of-optimality in multi-dimensional space).

2) Step-by-step reasoning:
- Start from the optimal tableau and identify the current basic variables and their z-values. For a change in coefficient c2 only, let c2' = c2 + Δ. Update reduced cost for variable 2 (and for any nonbasic variables that depend on c2 through z-values) and find Δ such that all reduced costs for nonbasic variables remain ≥ 0 (for maximization) or ≤ 0 (for minimization). The allowable increase/decrease is derived directly from these inequalities.
- If instead you change both c2 and c3 simultaneously, you must solve inequalities in two variables (Δ2, Δ3) to find the region where all reduced costs satisfy optimality. That is not the standard single-parameter sensitivity procedure and is more complex.

3) Intermediate calculations (conceptual):
- For single change: r2' = (c2 + Δ) - z2; require r2' maintain sign constraints. Solve for Δ to get allowable interval.
- For two changes: you would have r2' = (c2 + Δ2) - z2 and r3' = (c3 + Δ3) - z3 plus other dependent r_j inequalities; solve a system of linear inequalities.

4) Why the correct answer is correct:
- The statement proposes varying coefficient #2 and #3 at the same time as the “correct way” for sensitivity of coefficient #2. That is incorrect because standard sensitivity analysis for a single coefficient varies only that coefficient; simultaneous variation is not the default method.

5) Why the other option is incorrect:
- “True” would imply that simultaneous variation of adjacent coefficients is the correct single-parameter method, which is false.

Conclusion: The correct answer is False, matching the provided answer.

Answer the following statement with TRUE or FALSE: the correct way to implement a sensitivity analysis for the second coefficient of the objective function is to vary the second coefficient and the next one at the same time (eg.: coefficient #2 and #3).True/False

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