Advanced mathematical decision-support for industrial production, inventory, warehousing, energy, logistics, quality, maintenance and uncertainty.
All headline indicators update continuously. Click Simple beside any input or result to see a plain-language explanation.
| Industrial flow | Mathematical layer |
|---|---|
| Raw materials | Stochastic demand + inventory control |
| Production | Capacity + OEE + dynamic state model |
| Warehouse | Capacity + queue/flow model |
| Utilities | Load dynamics + energy economics |
| Distribution | Fleet + stochastic travel/cycle model |
| Uncertainty | Monte Carlo / SDE / Markov chain |
| Optimization | Nonlinear constrained optimization |
Enter normal operating numbers. The advanced uncertainty model is in the Mathematics Lab.
Stress the industrial system with changes in demand, capacity, supply lead time and transport disruption.
Interpretation: \(X_t\) is the industrial state (inventory, WIP, machine health, queue length, etc.). \(\mu\) is the predictable drift. \(\Sigma\) describes random variation. \(W_t\) is a vector Wiener process. This is a stochastic differential equation (SDE).
The engine demonstrates constrained nonlinear optimization with non-negativity bounds. In a production implementation, \(x\) can represent production allocation, inventory targets, energy dispatch or transport allocation.
For a multi-server station, this uses the Erlang-C structure as a baseline. Multiple service classes can be evaluated side-by-side. When \(\rho\ge1\), the ideal steady-state queue is unstable.
States can represent Healthy, Degraded and Failed equipment. The transition matrix describes probabilities of moving between states in one time step.
A dynamic inventory balance: replenishment enters, demand leaves, and \(\delta I\) represents proportional loss, decay or leakage. Euler integration is shown here; higher-order solvers can be used in a calibrated implementation.
Fourier decomposition represents periodic demand as a combination of harmonics. It is useful when demand has weekly, monthly or seasonal cycles.
Laplace transforms convert differential equations into algebraic equations in \(s\). This first-order transfer function can represent a simplified response of inventory, temperature, pressure, flow or production rate.
The Jacobian measures how strongly outputs change when inputs change. The engine uses central finite differences around the current operating point.
Real analysis provides the language for limits, continuity, convergence and stability. Here we numerically inspect a cost function and estimate a local Lipschitz slope.
Production sequencing can be viewed as permutations of jobs. The symmetric group \(S_n\) contains all permutations of \(n\) jobs, with composition as the operation and the identity permutation as the neutral element.
Topology studies properties preserved under continuous deformation. For an industrial network, a useful applied analogy is connectivity: whether factories, warehouses and utilities remain connected under failures.