Allocation and transparent optimization

A household with one car may choose who uses it by urgency, by whose trip takes least time, or by a rule that each person gets a turn. None of those rules is identical to maximizing the number of trips. Scarce bottle supply raises a similar question. A planner should show who receives less under each rule, then let the authorized policy govern the choice instead of burying that choice inside a spreadsheet.

There are twenty four good bottles available before all due dates. Group A requests eighteen and has contribution six currency units per bottle. Group B requests twelve and has contribution ten. Shipments cannot exceed demand and cannot be negative. Pure contribution maximization gives twelve to B and twelve to A, worth one hundred ninety two. A rule that protects at least fifteen for A gives fifteen to A and nine to B, worth one hundred eighty. The policy costs twelve units of contribution in this case.

A proportional allocation based on demand shares gives A twenty four times eighteen divided by thirty, or fourteen point four. B receives nine point six. Bottles are indivisible, so a rounding rule is required. Giving floors first leaves fourteen for A and nine for B, with one bottle remaining. Awarding the remainder to the largest fractional remainder yields fourteen A and ten B. Its contribution is one hundred eighty four. This is a transparent rule, not a mathematical proof that every customer considers it fair.

Service goals can be hard constraints, weighted scores, or ordered priorities. A hard minimum of fifteen A has a different meaning from a preference for A. If the requested minimum rises above available supply, the model should flag infeasibility rather than silently weaken the rule. Record who may change a priority, which orders it covers, and how customers are informed. Do not treat revenue, contribution, strategic importance, and lateness penalties as interchangeable amounts unless their definitions and units are explicit.

A map of a garden can show where a shed is allowed to stand: away from the fence, inside the plot, and clear of a path. The allowed area is the overlap of all those rules. A two-variable planning model works similarly. Each capacity rule removes choices. Once the feasible area is visible, a value rule chooses among the remaining points. This is easier to audit than a single unexplained output number.

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 two x plus y at most eight and x plus two y at most eight. Maximize contribution five x plus four y. 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 and zero, three and zero, three and two, eight thirds and eight thirds, and zero and four. The intersection of the resource boundaries solves two x plus y equals eight and x plus two y equals eight, giving x and y each equal eight thirds. 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.

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 and three, which violates both resource constraints.

In your lab, keep all twenty candidate pairs in view, even the ones that fail. Test each against both resource rules before you compare value. Then change one limit and repeat the check. A small model gives you a chance to see every choice. Use that chance to explain why the best whole bottle plan can differ from the best plan with fractions, and why a service rule may change the result again.

1. Define the case

   Original card shows the two lesson titles, fictional business, and planning goal.

   • Northstar case
   • Explicit scope
   • Check the units
2. Read the inputs

   Original table highlights the first lesson’s supplied quantities with unit and timing labels.

   • Given quantities
   • Usable dates
   • Stated limits
3. Trace the first check

   Original arithmetic flow follows the first lesson’s worked calculation to its checked answer.

   • Calculate load
   • Find the gap
   • State the result
4. Add the second view

   Original paired cards show how the second lesson adds a new constraint or decision policy.

   • New condition
   • Same discipline
   • No hidden assumptions
5. Compare scenarios

   Original three-column table uses the lesson lab baseline, stress case, and response.

   • Baseline
   • Stress case
   • Feasible response
6. Release the decision

   Original decision card links result, uncertainty, owner, and next verification gate.

   • Explain tradeoffs
   • Name the owner
   • Verify before release