Which of the following is NOT a question answered by standard sensitivity analysis information?

Question

BlackTom AI · Accepted Answer

Standard sensitivity analysis reports shadow prices (dual values), allowable increases and decreases for right-hand-side (RHS) values and objective-function coefficients, and reduced costs — which let you predict effects of small changes that stay within those allowable ranges.

1) Relevant knowledge points:
- Shadow price: the rate of change of the objective per unit change in a constraint’s RHS, valid only while the current basis remains optimal (i.e., for RHS changes within the allowable increase/decrease).
- Allowable increase/decrease (RHS): the range over which the shadow price is valid; outside it the optimal basis can change.
- Allowable increase/decrease (objective coefficient): the range over which the current optimal solution remains optimal; within that range you can compute objective changes directly.
- Reduced cost: tells whether a nonbasic variable can enter the basis and by how much objective must improve.

2) Step-by-step reasoning:
- If an RHS changes by an amount Δb within its allowable range, the change in optimal objective value = (shadow price) × Δb. Example: shadow price = 5 and Δb = +7 (within allowable increase), objective increases by 35.
- If a decision variable’s objective coefficient changes by Δc within its allowable range, the current solution stays optimal; the new objective value is original objective plus x* × Δc, where x* is the current value of that variable.
- If an RHS change is outside the allowable increase/decrease, the current basis may change, so the shadow price no longer applies. Standard sensitivity information does not tell you the new shadow price or the exact change in objective for such a large change—you must reoptimize.

3) Calculations (illustrative):
- Within range: objective change = shadow price × Δb. E.g., 5 × 7 = 35.
- Outside range: no valid calculation from standard sensitivity data.

4) Why the correct answer is correct:
The first option asks about the objective change when an RHS changes outside its allowable range. Standard sensitivity analysis explicitly provides valid objective-change information only for changes within allowable increases/decreases. For changes outside those ranges, the analysis cannot guarantee the same basis or shadow price, so it cannot answer the question.

5) Why the other options are incorrect:
- “All these can be answered...” is incorrect because, as explained, the first scenario cannot be answered without reoptimization.
- “By how much will the objective function value change if a decision variable's coefficient changes within the allowable increase and allowable decrease?” is answered by standard sensitivity (use allowable range and current x*).
- “If the right-hand-side value of a constraint changes, will the objective function value change?” can be answered using the shadow price (if shadow price = 0, no change for small changes within allowable range).

Conclusion: Therefore the statement that is NOT answered by standard sensitivity analysis is: “By how much will the objective function value change if the right-hand-side value of a constraint changes outside the allowable increase and allowable decrease?” This matches the provided answer.

Which of the following is NOT a question answered by standard sensitivity analysis information?单项选择题

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