A small linear model you can check by hand

Understand the constraint

Let x be units of plain bottle A and y be units of decorated bottle B. For a deliberately small model, each A uses two machine minutes and one labor minute; each B uses one machine minute and two labor minutes. The capacities are eight machine minutes and eight labor minutes. Demand caps are x at most three and y at most four. Nonnegative choices satisfy 2x+y≤8 and x+2y≤8. Maximize contribution 5x+4y. These small artificial times make the geometry easy to inspect.

For the continuous relaxation, enumerate boundary intersections and retain only feasible ones. The feasible vertices are zero/zero, three/zero, three/two, eight thirds/eight thirds, and zero/four. The intersection of the resource boundaries solves 2x+y=8 and x+2y=8, giving x=y=8/3. Objective values at the five vertices are zero, fifteen, twenty three, twenty four, and sixteen. Thus twenty four is the continuous optimum. Fractions are useful as an upper bound but cannot be shipped as whole bottles.

Interpret the result

For integer bottles, enumerate x from zero through three and y from zero through four. Keep a pair only when both resource inequalities hold. For each x, the best y is four, three, three, and two respectively because B has positive contribution. The corresponding values are sixteen, seventeen, twenty two, and twenty three. The best integer mix is three A and two B, value twenty three. Blindly rounding both continuous quantities upward gives three/three, which violates both resource constraints.

Create one row for each resource or stock position and one column for each time bucket. Put the unit in the row label and keep inputs separate from calculated results. Begin with the smallest case you can verify by hand, then extend the same equations across periods. Give every receipt a usable date and every requirement a due date. Preserve the baseline before changing an assumption so the effect of an action is visible.

Validate and explain

Use a second check that follows the physical story. Count what enters, what is used, what waits, and what leaves. A plan that balances in total can still fail at an earlier cutoff. Mark any unavailable input as an assumption rather than filling it with a convenient zero. A zero means a known absence; an unknown value means the promise remains conditional. This difference belongs in both the model and the decision note.

A calculation becomes useful when it changes a choice. Compare the baseline with at least two scenarios, identify the first limiting condition in each, and explain what additional resource would actually change delivered output. Keep a service measure alongside any cost or value measure. State whether the choice maximizes value, protects a commitment, or reduces the worst shortfall. A different goal can produce a different sensible answer from the same facts. Finish by testing one deliberately wrong proposal, such as a receipt used before arrival or a quantity above a resource ceiling. Your table should make that error visible. Save the rejected proposal with a short explanation, because it teaches a reviewer how the checks work. The eight hours include rebuilding, scenario work, writing, and teach-back; they are self-paced estimates rather than measured completion times. Faster readers should deepen the checks instead of merely rereading the text.

Build, test and explain a plan

Rebuild all 20 integer pairs x=0..3 and y=0..4, recording both loads, feasibility, and contribution. Repeat the full table for machine capacity 9 with labor 8, then labor capacity 9 with machine 8. Keep all other data unchanged. Finally impose x≥2 and y≥3 and identify feasible integer pairs under base capacities. Sketch the continuous feasible polygon and explain its upper-bound role. The eight-hour estimate covers an original workbook or paper model, scenario calculations, checks, and communication; it is not experimentally validated.

Use paper or a spreadsheet. The estimates below include the study and worked example already on this page.

Your deliverables

• A reproducible planning table with inputs, formulas, units, and checks.
• A scenario comparison with service and resource consequences.
• A decision memo and short teach-back outline.

Keep these three ideas

• Enumerate feasible corner points and distinguish continuous from integer choices.
• Keep quantities, resource units, and due times explicit.
• Recheck all constraints after a change, and report remaining uncertainty.

Common mistakes

• Treating capacity in a later time bucket as available before the due gate.
• Calling an expected or conditional quantity a guaranteed shipment.