Sensitivity & Scenario Analysis - Business Analytics

1. Which interpretation is most appropriate?
   Answer: The first $12$ units predict a $4{,}800$ increase, but the model should be re-solved to value the last $3$ units. Whenever you see a question involving shadow prices and constraint ranges in linear programming, your first instinct should be to check whether the proposed change falls within the allowable range — because shadow prices are only reliable predictors within that range. Here, the shadow price is $\$400$ per unit of warehouse capacity, and the allowable increase is $12$ units. This means the $\$400$ figure is valid for any addition up to $12$ units, predicting a profit gain of $12 \times 400 = \$4{,}800$. Beyond that threshold, the basis of the optimal solution changes — a different constraint becomes binding, and the shadow price may shift entirely. For the remaining $3$ units (units 13–15), the current model simply cannot tell you what happens without being re-solved. That's exactly what C captures: a reliable $\$4{,}800$ estimate for the first $12$ units, and an honest acknowledgment that the last $3$ require a new solution. A is wrong because it blindly applies the shadow price to all $15$ units ($15 \times 400 = \$6{,}000$), ignoring that the allowable range ends at $12$. This is the most common trap — students forget the range condition entirely. B is wrong because it correctly computes $\$4{,}800$ but then invents a rule that the extra $3$ units have zero value — the model doesn't say that; it simply says we don't know. D overcorrects by claiming nothing can be estimated, which discards the perfectly valid prediction for the first $12$ units. Your study tip: always split the proposed change into "within range" (shadow price applies) and "beyond range" (re-solve required). Never extend shadow price predictions past the allowable boundary.
2. Which plan should management select under the minimax-regret criterion?
   Answer: Plan B, because its maximum regret is $20$ monetary units Whenever you see a question about decision-making without reliable probabilities, think about regret-based reasoning. The minimax-regret criterion asks: for each plan, what is the worst-case "missed opportunity" — and which plan minimizes that worst case? To apply it, first build a regret table. For each scenario, find the best possible payoff, then subtract each plan's payoff from that best. Under strong markets, the best is $120$ (Plan A), so regrets are: A = $0$, B = $20$, C = $45$. Under moderate markets, the best is $95$ (Plan B): A = $15$, B = $0$, C = $10$. Under weak markets, the best is $70$ (Plan C): A = $40$, B = $15$, C = $0$. Now take each plan's maximum regret — A = $40$, B = $20$, C = $45$ — and choose the minimum. That's Plan B at $20$, making B the correct answer. Choice A is tempting because Plan A dominates in the strong scenario, but its weak-market regret of $40$ is too costly — you'd be picking the plan with the second-worst maximum regret. Choice C is outright wrong; Plan C has the highest maximum regret ($45$), the opposite of what minimax seeks. Choice D reflects a common misconception — minimax regret is specifically designed for situations without probabilities, so no probabilities are needed. Study tip: Always build the full regret table before comparing. A plan that looks attractive in one scenario can carry devastating regret in another — the maximum regret column is the only number that matters for this criterion.
3. What can the analyst conclude about whether the current optimal basis will remain optimal?
   Answer: It cannot be guaranteed by the sensitivity report because the combined percentages exceed $100\%$. Whenever a sensitivity analysis question involves simultaneous changes to multiple objective-function coefficients, you should immediately think about the 100% rule, not just the individual allowable ranges. The 100% rule states that if you make changes to multiple objective coefficients at once, the current optimal basis is guaranteed to remain optimal only if the sum of the percentages used of each coefficient's allowable range does not exceed 100%. Here, the first change uses $60\%$ of its allowable increase and the second uses $50\%$ of its allowable decrease. The combined percentage is $60\% + 50\% = 110\%$, which exceeds $100\%$. Therefore, the sensitivity report cannot guarantee the current basis remains optimal — the analyst must re-solve or use additional analysis. That makes C the correct answer. A is wrong because it confuses individual validity with simultaneous validity. Each change being within its own allowable range is a necessary but not sufficient condition when changes occur together. The 100% rule is precisely what closes this gap. B is wrong in its conclusion for the right reason. Yes, the percentages exceed 100%, but that does not mean the basis must change — it only means optimality can no longer be guaranteed. Exceeding 100% is a loss of certainty, not a proof of change. D is wrong because the direction of changes (one increase, one decrease) is irrelevant to the 100% rule. The rule applies regardless of direction; you sum the absolute percentages used. Study tip: Memorize the 100% rule as a two-step check — calculate each percentage used, sum them, and if the total exceeds $100\%$, the sensitivity report gives no guarantee. Direction never cancels out the percentages.
4. Which conclusion is supported by this analysis?
   Answer: Customer price elasticity creates the greatest modeled profit variation over the specified ranges when other inputs are held constant. Whenever you see a question about sensitivity analysis, anchor yourself to what the technique actually does: it varies one input at a time across a specified range while holding all others constant, then measures the resulting output change. Critically, it assigns no probabilities and makes no claims about likelihood or joint effects. That's exactly why A is correct. The analysis shows that elasticity produces the largest modeled profit swing over management's chosen ranges when other inputs are fixed. That's a precise, limited claim — and it's exactly what one-way sensitivity analysis is designed to reveal. The word "modeled" and the phrase "specified ranges" are doing important work here; they keep the conclusion tightly scoped to what the method can actually support. B is wrong because sensitivity analysis says nothing about the probability that any input will deviate from its baseline. A variable can swing profits wildly if it moves, but still be very unlikely to move. Likelihood requires probability estimates, which this analysis explicitly lacks. C is wrong because the value of collecting additional data depends on decision context, cost of data collection, and how much uncertainty would actually be reduced — none of which sensitivity analysis addresses. High sensitivity to a variable doesn't automatically mean more data on it is worth pursuing. D is wrong because one-way sensitivity analysis tests inputs individually. It cannot rule out scenarios where two or more inputs move adversely at the same time. A joint adverse movement in competitor price and fulfillment cost could, in principle, exceed the elasticity effect — but this analysis simply wasn't designed to test that. Your study tip: when evaluating conclusions from sensitivity analysis, ask whether the claim requires probabilities, joint movements, or value-of-information reasoning. If yes, the method can't support it.
5. Why would scenario analysis generally be more informative than one-way sensitivity analysis for this decision?
   Answer: It evaluates coherent joint changes in related inputs, revealing combined effects that separate one-way changes may miss. When a question contrasts two analytical methods, ask yourself what each method does and doesn't capture. Sensitivity analysis and scenario analysis solve different problems, and recognizing that distinction is the key here. One-way sensitivity analysis moves a single input — say, fuel costs — while holding everything else fixed. This is useful for ranking individual risks, but it creates an artificial picture when inputs move together in the real world. In the retailer's case, an economic downturn simultaneously pushes order volume down and pushes failed-delivery rates and fuel costs up. These shifts are correlated by a common cause. Scenario analysis builds internally consistent "stories" — for example, a recession scenario where all three variables move together — so you see the combined, realistic impact on profitability. That is precisely why B is correct: scenario analysis evaluates coherent joint changes in related inputs, surfacing effects that separate one-way shifts would miss or understate. A is wrong because scenario analysis does not automatically generate objective probabilities. Probabilities must still be assigned by the analyst, often subjectively, so you cannot calculate expected profit without additional assumptions. C describes the opposite of what scenario analysis does. Isolating causal effects and removing correlations is the goal of controlled experimentation or regression analysis — not scenario analysis, which deliberately preserves correlations by grouping inputs into realistic bundles. D is wrong because scenario analysis does not optimize anything. It evaluates outcomes under specified conditions; finding the optimal plan requires a separate optimization step. Study tip: On exam questions comparing analytical tools, match each tool to its defining feature — sensitivity analysis isolates, scenario analysis combines. That pairing will steer you to the right answer quickly.
6. Holding volume and the manual-process assumptions constant, by approximately how much can the automated process's contribution margin per unit decrease before the recommended process changes?
   Answer: $1.43$ monetary units per unit When two processes compete, the smarter question isn't just "which is better now?" but "at what point does the recommendation flip?" This is a sensitivity analysis problem — you're finding how much a key assumption can move before the current winner loses its advantage. Start by calculating each process's operating profit at 14,000 units. The automated process earns $25 \times 14{,}000 - 180{,}000 = 170{,}000$. The manual process earns $15 \times 14{,}000 - 60{,}000 = 150{,}000$. Automated wins by $20{,}000$. Now, let the automated contribution margin per unit drop by $d$. The new automated profit becomes $(25 - d) \times 14{,}000 - 180{,}000$. Set this equal to the manual profit and solve for the breakeven point: $(25 - d)(14{,}000) - 180{,}000 = 150{,}000$ $350{,}000 - 14{,}000d - 180{,}000 = 150{,}000$ $170{,}000 - 14{,}000d = 150{,}000$ $14{,}000d = 20{,}000 \implies d \approx 1.43$ So the automated process can lose approximately **$1.43$ monetary units per unit before the two processes tie — confirming answer B**. Answer A ($0.71$) results from mistakenly dividing the profit gap by $28{,}000$ (doubling the volume), a arithmetic setup error. Answer C ($2.14$) comes from dividing $30{,}000$ — the fixed-cost difference — by 14,000 instead of the actual profit gap. Answer D ($8.57$) reflects dividing the full contribution margin difference ($10$) by the contribution margin ratio, a conceptual mismatch. On sensitivity questions, always anchor yourself: find the current advantage in total dollars, then ask how much the variable must shift to erase that advantage.
7. What is the most defensible interpretation of the resulting prescription?
   Answer: The prescription is not robust because a plausible Channel X value falls below the $0.38$ switching threshold. Whenever you see a question about choosing between two options based on estimated values, you should immediately ask: how confident are we in those estimates? This question tests sensitivity analysis and robustness of prescriptions — the idea that a recommendation is only as strong as its stability across plausible input values. Here, Channel X has a point estimate of $0.42$, but the A/B test interval runs from $0.34$ to $0.50$. Channel Y is fixed at $0.38$. The switching threshold — the value at which you'd prefer Y over X — is exactly $0.38$. Notice that $0.34$, the lower bound of Channel X's interval, falls below $0.38$. This means a perfectly plausible realization of Channel X's true value would flip the decision in favor of Channel Y. The prescription to choose X is therefore not robust, making C the correct answer. Answer A is wrong because it treats the point estimate gap of $0.04$ as conclusive evidence. A small gap combined with a wide interval is precisely when robustness breaks down, not when it holds. Answer B is wrong on two levels: "closeness to the midpoint" is not a meaningful decision criterion, and Y being near X's midpoint ($0.42$) does not make Y preferred — it just confirms the estimates are similar. Answer D is wrong because sensitivity analysis does not require symmetric or equal-width intervals on both sides. You can absolutely assess robustness when only one estimate has uncertainty. Study tip: When an interval straddles the switching threshold, the recommendation is never robust — regardless of where the point estimate sits. Train yourself to check whether competing values fall inside the confidence interval.
8. Using the simultaneous right-hand-side change rule, what is the best estimate of the change in optimal profit?
   Answer: An increase of $1{,}680$ monetary units, because the combined proportional change is within $100\%$ When a linear program has multiple binding constraints, sensitivity analysis gives you shadow prices and allowable ranges for each constraint individually. The simultaneous right-hand-side change rule (also called the 100% rule) lets you check whether those shadow prices remain valid when several right-hand sides change at once: divide each proposed change by its allowable range in that direction, sum the ratios, and if the total stays at or below 100%, you can safely apply all shadow prices simultaneously. Here, the machine hours increase by 8 out of an allowable increase of 20, giving a ratio of $8/20 = 0.40$, or 40%. Labor decreases by 6 out of an allowable decrease of 12, giving $6/12 = 0.50$, or 50%. The combined ratio is $40\% + 50\% = 90\% \leq 100\%$, so both shadow prices hold. The estimated profit change is: $\Delta \text{Profit} = (300)(+8) + (120)(-6) = 2{,}400 - 720 = +1{,}680$ This confirms D as correct. Choice A misinterprets shadow prices as costs rather than marginal profit contributions — shadow prices represent value added, not expenses incurred. Choice B incorrectly checks each constraint in isolation and discards the machine-hours impact, which is not how the rule works. Choice C simply ignores the negative labor effect entirely, which overstates the benefit and violates the logic of the analysis. Your study tip: always check the 100% rule before applying shadow prices to simultaneous changes. If the sum of ratios exceeds 100%, the shadow prices may no longer be valid and the estimate is unreliable.
9. By how many percentage points can the high-demand probability decrease before the two launch strategies become equally attractive on expected monetary value?
   Answer: $5$ percentage points When comparing two strategies using expected monetary value (EMV), the break-even probability is the point where both strategies yield identical expected returns. Whenever a question asks how much a probability can shift before two options become equally attractive, set their EMV expressions equal and solve. Let $p$ represent the probability of high demand. The EMV formulas are: - Aggressive: $300p + (-100)(1-p) = 400p - 100$ - Conservative: $180p + 80(1-p) = 100p + 80$ Setting them equal to find the break-even probability: $400p - 100 = 100p + 80$ $300p = 180$ $p = 0.60$ So the two strategies are equally attractive at $60\%$. Since the current probability is $65\%$, the probability can decrease by $65\% - 60\% = 5$ percentage points — confirming D is correct. Choice A (3.75 points) likely results from a setup error, such as misreading the payoffs or forgetting to account for the low-demand outcomes correctly when forming the EMV expressions. Choice B (20 points) is a significant overestimate — this might come from comparing only the high-demand payoffs ($300 - 180 = 120$ vs. $100 - 80 = 20$) without properly solving the system. Choice C (15 points) may result from an arithmetic error mid-calculation, such as subtracting coefficients incorrectly. A useful habit: always write out both full EMV expressions before solving. Break-even analysis questions reward systematic setup — rushing to compare single payoffs without accounting for all outcomes is the most common trap here.
10. What annual contribution should the analyst use for the recession scenario?
    Answer: $493{,}000$ monetary units after jointly applying all three scenario assumptions Scenario analysis in business analytics requires you to apply all relevant changes simultaneously to their respective drivers — not mix and match, and not apply percentages to an already-derived figure. Here's how to build the recession contribution correctly. Start by adjusting each input: revised demand = $100{,}000 \times (1 - 0.15) = 85{,}000$ units; revised price = $20 \times (1 - 0.05) = 19.00$; revised variable cost = $12 \times (1 + 0.10) = 13.20$. The revised contribution margin per unit is $19.00 - 13.20 = 5.80$. Total contribution = $85{,}000 \times 5.80 = \mathbf{493{,}000}$, confirming answer A. Each distractor reflects a specific analytical error. B applies the revised unit economics ($5.80$ per unit) but forgets to adjust demand, using the baseline $100{,}000$ units: $100{,}000 \times 5.80 = 580{,}000$ — it ignores the volume impact entirely. C misapplies the percentages by operating on baseline contribution ($100{,}000 \times 8 = 800{,}000$) rather than on the underlying drivers, then adjusting: applying $-15\%$, $-5\%$, and $+10\%$ directly to $800{,}000$ conflates margin-level and driver-level effects, producing $581{,}400$ — a methodologically flawed shortcut. D applies only the demand decline to baseline contribution ($85{,}000 \times 8 = 680{,}000$), ignoring that price and cost conditions also change in the recession. The key study tip: in scenario analysis, always trace changes back to their source drivers — volume, price, and cost — recalculate each, then rebuild the metric from scratch. Applying percentages to derived figures like contribution is a classic exam trap.

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