Mixed Integer Optimisation (MIO)

Mixed Integer Optimisation takes discrete as well as continuous decisions into account during optimisation.

A major strength of gPROMS has always been the power of its steady-state and dynamic optimisation. The optimisation capabilities have now been extended to take into account integer or discrete decisions using Mixed Integer Optimisation (MIO).

MIO can be applied to both steady-state and dynamic gPROMS models. The latter may also involve discontinuous equations such as those described by gPROMS IF and CASE equations.

There are many applications in chemical engineering where discrete decisions need to be considered. Typically examples are decisions on equipment or process configuration, where the number of stages or equipment items, the inlet and outlet locations, or the restriction to standard equipment sizes are key considerations.

Steady-state MIO example - design of continuous distillation columns

Optimisation of feed and draw tray location

A typical MIO example is the steady-state design of a distillation column, where the number of trays and feed and side draw locations are key design decisions.

The MIO capability in gPROMS can be used to find the optimal configuration while simultaneously optimising continuous variables such as column diameter and reflux ratio.

The objective function used in such an application is typically an economic one that includes both capital and operating cost.

Dynamic MIO example - selection of optimal control schemes

The selection of the optimal control scheme for handling all anticipated disturbances to a system is another key decision in process engineering.

MIO can be used to select the best scheme while simultaneously providing optimal values for variables such as

Controller tuning parameters
Key equipment design parameters such as column diameter
Optimal operating (for example, startup) policy.

Steady-state MIO example - process synthesis

MIO can be used to determine the optimal process configuration from a number of proposed alternatives, taking into account Process synthesis - separation process selection

equipment and process constraints
product specification requirements
annualised capital and operating cost

The "separation options" in the diagram above can be different configurations of distillation columns, for example, or entirely different separation processes (for example flash units or membranes)

MIO - the economic benefits

MIO opens new opportunities in advanced areas such as the simultaneous design of processes and their control systems, potentially leading to significant economic benefits - as illustrated by the case study shown below.

	Existing design	Design using MIO
Feed location	14	8
Draw location	22	18
Q column I (MW)	19.5	14.7
Q column II (MW)	0.87	2.45
Capital cost M$/year	0.63	0.56
Operating cost M$/year	4.37	3.56
Total cost M$/year	5.00	4.12

Types of integer decision in gPROMS optimisation

The MIO facilities in gPROMS cater for the following discrete decision types:

Integer decisions - for example, the number of trays in a distillation column or the optimal number of reactors necessary to provide the required conversion and selectivity.
Binary decisions - for example, to determine whether an item of equipment needs to be included or not; or to obtain an optimal "bang-bang" profile for a time-varying on-off control.
Enumerated decisions - where the optimisation chooses an item out of a given set - for example, selecting a batch reactor out of a set of standard "off-the-shelf" sizes; or obtaining an optimal control profile for a "staged" heater, i.e. one that can operate only in certain distinct modes, e.g. OFF, LOW, MEDIUM or HIGH.
"Special Ordered Set of type 1 (SOS1)" decisions, a set of binary decision variables from which only one may have a value of 1. An example is feed tray location in a distillation column, where a number of locations may be proposed but only one may be selected by the optimiser.

Available