This continuous-time optimization used in phases 1–3 seeks to maximize profit and minimize grade transitions (and associated waste material production) while observing scheduling constraints. The objective function is formulated as follows:

where

${T}_{M}$ is the makespan,

n is the number of products,

m is the number of slots (

$m=n$ in these cyclic schedules),

${z}_{i,s}$ is the binary variable that governs the assignment of product

i to a particular slot

s,

${t}_{s}^{s}$ is the start time of the slot

s,

${t}_{s}^{f}$ is the end time of slot

s,

${\Pi}_{i}$ is the per unit price of product

i,

${W}_{\tau}$ is an optional weight on grade transition minimization,

${\tau}_{s}$ is the transition time within slot

s,

${c}_{storage,i}$ is the per unit cost of storage for product

i, and

${\omega}_{i}$ represents the amount of product

i manufactured,

and where

q is the production volumetric flow rate and

${\tau}_{s}$ is the transition time between the product made in slot

$s-1$ and product

i made in slot

s. The time points must satisfy the precedence relations:

which require that a time slot be longer than the corresponding transition time, impose the coincidence of the end time of one time slot with the start time of the subsequent time slot, and define the relationship between the end time of the last time slot (

${t}_{n}^{f}$) and the total makespan or horizon duration (

${T}_{M}$).

Products are assigned to each slot using a set of binary variables,

${z}_{i,s}\in \left\{0,1\right\}$, along with constraints of the form:

which ensure that one product is made in each time slot and each product is produced once.

The makespan is fixed to an arbitrary horizon for scheduling. Demand constraints restrict production from exceeding the maximum demand (

${\delta}_{i}$) for a given product, as follows:

The continuous-time scheduling optimization requires transition times between steady-state products (${\tau}_{{i}^{\prime}i}$) as well as transition times from the current state to each steady-state product if initial state is not at steady-state product conditions (${\tau}_{{0}^{\prime}i}$).

Transition times are estimated using NMPC via the following objective function:

where

${W}_{sp}$ is the weight on the set point for meeting target product steady-state,

${W}_{\Delta u}$ is the weight on restricting manipulated variable movement,

${W}_{u}$ is the cost for the manipulated variables,

u is the vector of manipulated variables,

${x}_{sp}$ is the target product steady-state, and

${x}_{0}$ is the start process state from which the transition time is being estimated. The transition time is taken as the time at which and after which

$|x-{x}_{sp}|<\delta $, where

$\delta $ is a tolerance for meeting product steady-state operating conditions. This formulation harnesses knowledge of nonlinear process dynamics in the system model to find an optimal trajectory and minimum time required to transition from an initial concentration to a desired concentration. This method for estimating transition times also effectively captures the actual behavior of the controller selected, as the transition times are estimated by a simulation of actual controller implementation. This work uses

${W}_{\Delta u}=0$ and

${W}_{u}=0$.