How to use it
- 1. Set the modelEdit the two objective rows and the resource constraints with the +/− steppers - both charts re-solve instantly.
- 2. Pick a methodThree classic ways to resolve the conflict: blend with a weight (weighted sum), cap one objective (ε-constraint), or rank them (lexicographic) - each has its own geometry.
- 3. Animate simplexPress the button: the marker starts at the origin and pivots to a better adjacent corner each iteration - that is all simplex does.
- 4. Read both spacesLeft: the feasible polygon (decision space). Right: every corner mapped to (Z1, Z2) - the red chain is the Pareto frontier.
Interactive studio
Objective 1 - profit Z1 (max)
Objective 2 - service Z2 (max)
Solution method
Fourth classic method - targets and deviations: Goal Programming Studio.
Resource A - assembly hours
Resource B - machine hours
Storage cap
The shaded polygon is every feasible plan; simplex only ever visits its corners. On the right, each corner is re-plotted by its two objective values - corners on the red chain are Pareto-optimal: you cannot improve one objective without giving up the other.
Corners - objective values & Pareto status
| Corner (X1, X2) | Z1 profit | Z2 service | Weighted Z | Status | Best for w in |
|---|
A corner is Pareto-optimal when no other corner beats it on both objectives at once. Every weight w between 0 and 1 selects some Pareto corner - the last column shows which weights select this one. Swipe the table sideways to see every column.
Try these experiments
- Slide w slowly from 1 to 0 and watch the star jump corner to corner along the Pareto frontier - it never visits a dominated corner.
- Find the critical weight where the optimum jumps: exactly there, two corners tie and the whole edge between them is optimal (alternate optima).
- Make the two objectives proportional (e.g. service = profit row doubled) - the Pareto frontier collapses to a single corner: no trade-off, no conflict.
- Raise the storage cap until it goes slack - the feasible polygon grows a corner and the frontier gains a new trade-off point.
- Switch to ε-constraint and sweep ε from 0 to max - the teal floor climbs, the polygon shrinks, and the optimum walks the whole frontier point by point.
- In lexicographic mode, flip the priority order - when the phase-1 optimum is a single corner, phase 2 has nothing left to decide; ties only matter on an optimal edge.
About this tool
The concept
With two objectives there is no single "best" plan - there is a Pareto frontier of plans where improving one objective must cost the other. The weighted-sum method turns each choice of weight w into an ordinary LP that simplex can solve.
How it works
We enumerate the feasible polygon's corners exactly, then run the simplex idea geometrically: pivot to a better adjacent corner until none improves. Weighted sum blends the objectives; ε-constraint turns one into a floor that trims the polygon; lexicographic runs simplex twice, the second pass confined to the first pass's optimal face.
Winning strategy
Read the objective-space chart first: count the Pareto corners, then use the weight slider to price the trade-off - the slope between two adjacent Pareto corners tells you how much service one dollar of profit costs. Dominated corners are never worth defending.