Revenue as a non-linear function of price
R = p x q with q = a - b p
=> R = a p - b p^2
Peak at p = a / (2b)
Any pricing question where quantity falls with price. The p^2 term makes it non-linear, and the negative coefficient makes it concave, so the peak is a global maximum.
- p
- Price, the decision variable
- a
- Demand at a price of zero
- b
- Units of demand lost per rupee of price
Convex cost with a ratio term
TC(Q) = (D / Q) x S + (Q / 2) x H
Minimum at Q* = sqrt( 2 D S / H )
At Q*, ordering cost = holding cost
The standard example of a non-linear but convex model. Useful as a check: if the two cost components are not equal, you are not at the optimum.
- D
- Annual demand in units
- S
- Cost of placing one order
- H
- Cost of holding one unit for one year
- Q
- Order quantity, the decision variable
Engine choice rule
linear everywhere -> Simplex LP (global, exact)
smooth non-linear -> GRG Nonlinear (local; global if convex/concave)
kinked or discontinuous -> Evolutionary (no guarantee)
Before every Solver run. Take the strongest engine the model allows, and reformulate a kink away if you can.
- smooth
- Has a gradient everywhere: products, powers, ratios, exponentials
- kinked
- Contains IF, ABS, MAX, MIN or VLOOKUP