2 Week

• proportionality

• additivity

• divisibility

• In a linear model, all relationships are proportional — meaning they change in direct proportion to the level of activity

• If you double the activity, you double its effect

  • 1 Basic → 5 hours, $108 profit

  • 2 Basics → 10 hours, $216 profit

⚠️ When proportionality doesn’t hold (nonlinear situations)

• economics of scale, chemistry

• the overall result is the sum of individual effects.

• That means:

  • The profit earned by each Basic doesn’t depend on how many XPs you make.

  • The hours used by each XP does not depend on how many Basics you produce.

  
  

The divisibility property says that the decision variables in a linear programming (LP) model can take any fractional (non-integer) value — not just whole numbers.

• In other words, we assume it’s possible to produce, use, or allocate partial units of something.

So the divisibility assumption makes linear programming fast and solvable.

Infeasability:

• “Your model’s rules fight each other — there’s no solution that fits all.”

• A model is infeasible when no solution satisfies all constraints at once.

• In other words, your restrictions contradict each other.

Reasons:

• You forgot to allow a variable to take zero (missing nonnegativity constraint).

  
  

Unboundedness:

• “Your model’s missing limits — the profit can grow forever.”

• A model is unbounded when the objective function can increase (or decrease) forever without hitting a constraint limit.

• That means there’s no upper (or lower) bound, Solver keeps improving the objective infinitely.

Reasons:

• Missing or incorrect upper bounds for variables or resources.

• Forgetting to restrict variables to nonnegative values (so Solver can push them to negative infinity).

  
  

• A blending problem happens when you need to mix several different inputs to produce one or more outputs with certain desired properties or costs.

• The goal is to find the best combination (blend) of inputs that meets all requirements at the minimum cost or maximum profit.

Only one coefficient (one cell in the objective) changes while all others stay fixed.

• When the selling price of gasoline increases — say from £75 → £80, this affects all variables that contribute to gasoline (crude oil1 and crude oil29), because their revenue coefficients both depend on that same price.