Sensitivity Analysis

The workshop's optimal plan — ten tables, twenty chairs, £1000 — was computed from numbers that are all slightly wrong. The £40 table profit is an estimate; the 30 carpentry hours might be 28 next week; a supplier's price could drift. A brittle answer that only holds for one exact set of inputs would be useless to a real manager. Sensitivity analysis asks the grown-up question: how much can the data change before my decision changes? It draws a comfort zone around the optimum, and it is where linear programming earns its keep in the boardroom.

Same workshop as before — \max\; 40x_1 + 30x_2 subject to x_1 + x_2 \le 30 (carpentry) and 2x_1 + x_2 \le 40 (finishing) — with optimum (10, 20) and shadow prices £20 and £10 from duality.

Ranging an objective coefficient

Suppose the profit on a table isn't really £40. As we nudge it, the iso-profit line tilts. For a while the same corner (10, 20) stays optimal — only the profit number changes, not the plan. The corner switches only when the objective's tilt matches a binding edge. That defines the allowable range for the coefficient. Slide the table profit and watch the optimum hold, then jump:

For the table coefficient the range is 30 \le c_1 \le 60: an allowable decrease of £10 (down from £40 to £30) and an allowable increase of £20 (up to £60). Anywhere in that band, still build 10 tables and 20 chairs — only the profit total moves. Cross a boundary and a different corner takes over (make only chairs below £30, only tables above £60). The chair coefficient has its own range, 20 \le c_2 \le 40.

Ranging a right-hand side — where the shadow price lives

The other lever is a resource limit. A shadow price is the slope of optimal profit against a right-hand side, and it stays constant only while the same set of constraints stays binding — that span is the RHS's allowable range. For carpentry, the £20/hr shadow price holds over 20 \le b_1 \le 40 hours:

Change	Current	Allowable range	Rate while it holds
Table profit c_1	£40	£30 – £60	plan fixed at (10, 20)
Chair profit c_2	£30	£20 – £40	plan fixed at (10, 20)
Carpentry hours b_1	30	20 – 40	shadow price £20/hr
Finishing hours b_2	40	30 – 60	shadow price £10/hr

So renting up to 10 extra carpentry hours reliably earns £20 each; the eleventh may earn less, because beyond 40 hours finishing becomes the binding bottleneck and carpentry's shadow price collapses.

Reduced cost: the price of a product you don't make

Now imagine a third product — a stool — needing 1 carpentry hour and 1 finishing hour, earning £25. Should the workshop make any? Its reduced cost compares its profit to the shadow-price value of the resources it would eat:

\text{reduced cost} = \underbrace{25}_{\text{profit}} - \big(\underbrace{1 \times 20}_{\text{carpentry}} + \underbrace{1 \times 10}_{\text{finishing}}\big) = 25 - 30 = -£5.

Negative, so a stool destroys £5 of profit versus using those hours on tables and chairs — leave it out (its optimal level is zero). The reduced cost also tells you the fix: the stool's profit would have to rise by £5, to £30, before it is worth making. For any product actually in the plan the reduced cost is zero — it exactly pays its way. Reduced cost is to variables what shadow price is to constraints.

Changing several things at once: the 100% rule

Every range above is one-at-a-time: it assumes all other data stay put. Change two coefficients together and the ranges can interact. The 100% rule is the safe test:

For each thing you change, take the fraction \dfrac{\text{actual change}}{\text{allowable change in that direction}}.
If the sum of those fractions is \le 100\%, the current basis stays optimal and the shadow prices / plan are unchanged — the individual conclusions still hold together.
If the sum exceeds 100%, the rule is silent: you cannot conclude, and must re-solve the LP to be sure.

Example: raise the table profit by £10 (that's 10/20 = 50\% of its allowable increase) and cut the chair profit by £5 (that's 5/10 = 50\% of its allowable decrease). The fractions sum to exactly 100%, so (10, 20) is still optimal. Push either change a hair further and you must re-solve.

A bare optimal plan says "do this." A sensitivity report says "do this, and here is how robust it is": which numbers you can ignore small errors in, which are on a knife-edge, what an extra hour of each resource is worth, and which idle product is closest to becoming worthwhile. That is the language of actual decisions under uncertainty — budgeting overtime, negotiating a supplier price, deciding which estimate is worth refining. Solvers print this report for free alongside the solution, and seasoned analysts read it first. The optimum tells you what to do today; the sensitivity report tells you how nervous to be about it.

The single commonest sensitivity-analysis blunder is treating each allowable range as independent and changing several inputs to the edges of their ranges at once. Each range is computed holding all other data fixed; simultaneous changes can knock the optimum off its corner even when every individual change looks "in range." Use the 100% rule for combined changes — and if the fractions add past 100%, don't guess, re-solve. Likewise, never push a right-hand side beyond its range and keep applying the old shadow price: past the boundary a different constraint binds and the marginal value has changed.