All industries require decisions to be made with respect to multiple and potentially competing aims, amid uncertainty. Decision makers are often left wondering how they can balance the uncertainty whilst considering all scenarios to satisfy stakeholders’ needs.

In mathematics, this type of problem can be framed in the context of optimisation. Through careful selection of definitions of constraints and objectives, it is possible to use optimisation algorithms to support the decision maker in making effective decisions.

## Sequential decision making

The question of where to put another runway in the UK has been in the news again this month, with the Government’s Cabinet approving plans to build a third runway at Heathrow, one of the world’s busiest airports. The many years of debate and indecision testify to just how complicated it is to determine where best to add a runway that serves the UK’s interests, amid its many obligations to citizens, businesses and the wider environment.

This question is a good example of a more general problem arising when multiple stakeholders are involved, the consequences of the decision are multi-facetted and far reaching, or when there are simply multiple aspects to consider in order to produce a fair and balanced solution to a problem. The decisions often need to be made in a set order too, with a revaluation of the landscape made before the next decision as earlier choices will impact how a situation evolves. Furthermore, sources of uncertainty are abound, arising from factors that one has no or little control over (such as market movements or regulation) or factors where the decision maker may have some influence (such as organisational budgets).

It is worth noting that it is not just in the long-term planning of complex projects that decisions with competing aims or uncertainty occur. In finance, investors are naturally interested in investing in portfolios of stocks, bonds etc. that both gives a high expected return on investment, yet at the same time are low risk. Deciding on optimal portfolio compositions is a continually evolving process, with decisions needing to be frequently updated as new information on the movements of financial markets, and even geopolitical events, emerges.

## Scenario Analysis and Mathematical Optimisation

When faced with complex decisions that need to take account of uncertainty, a lot of value can be obtained by conducting a *scenario analysis*. This examines the solutions to a decision under a range of assumptions on the constraining factors surrounding the decision. Scenario analysis is a way of confronting uncertainty by considering numerous ‘what-if’ questions. As such, scenario analysis does not provide a pre-made recipe for arriving at the ‘correct’ decision; rather, it helps decision makers understand the options at their disposal and their consequences. Comparing a wide range of potential scenarios can provide stakeholders with confidence in the decision-making process and inform negotiations. In particular, when the problem and related data are large and too complex for a decision-maker to understand holistically, mathematical models are very well-placed to simplify such information to make it comprehensive yet helpful to stakeholders interpreting it.

When sufficient data for the constraints and decision criteria are available, optimisation can be developed within scenario analysis to better understand what the best decision might look like within a constrained scenario. In order to use mathematical optimisation, it is necessary to formulate two key aspects of a given decision making problem:

*The constraints in the problem, that put restrictions on the set of solutions to the decision; this is the way to define which solutions are viable.*Limitations on available resources, such as annual budgets, materials, output of work etc. are one source of constraints that can be quantifiably defined, as might be regulations that numerically cap certain behaviours. In scheduling problems, for example, the logical dependencies between tasks (statements like “Task A needs to be completed before task B can begin”) also define constraints that can be captured mathematically.*The criteria by which a decision is judged to be good or bad*. This is typically done by specifying metrics, or performance indicators, by which to assess a solution to the decision problem. For example, the overall monetary cost of implementing a decision will very often be something a decision maker would like to minimise, but other metrics may also be important, such as some measure of the risk imposed by a decision. In most cases, decisions need to be made that simultaneously optimise a mixture of performance indicators, such as keeping costs down while not exceeding certain risk thresholds.

In the context of decision making, the numerical measures by which a decision is judged are captured mathematically through the definition of *objective functions*, so that the goal of mathematical optimisation is to determine constraint-obeying solutions that maximise or minimise the objective functions. This mathematical approach can offer at least three key advantages:

*Optimisation algorithms typically work at a level of generality that can readily accommodate changes*to the constraints and objective functions, so that different scenarios with a variety of aims can be analysed without the need to reinvent the computational wheel each time. The use of software can leverage short run times to automatically run optimisation over a whole range of constraint and objective function configurations, which greatly facilitates scenario analysis.*The solutions constructed can be verified to be viable*, that is, to satisfy all the constraints supplied to it. This in itself can be valuable knowledge, especially in regulated industries where penalties apply for breaches to these constraints.*It is well suited to studying systems as a whole*, so that solutions obtained provide benefit across the whole system or across a large span of time. At the same time, there is the potential for granular levels of detail to be given when the available data supports it.

It is important to note that mathematical optimisation cannot make decisions for decision makers, but it can provide the means to process data in a way that gives insight into optimal decisions under different constraints and conditions. In particular, their numerical output provides a clear way to compare many potential future scenarios and provide analytical evidence to stakeholders to justify decisions.

However, the value they offer will always be tied to the assumptions and data they are combined with. Poorly formulated objective functions or inadequate collections of data/constraints can lead to serious disconnections between the model and reality, especially when unjustifiable assumptions are made. Optimisation should not be seen as a black box that produces “the” final solution, but rather as a tool to support the decision maker reach the best decision.

## Competing Objectives and Prioritisation

The framework described above based on mathematical optimisation offers various approaches to the question of finding optimal solutions when there are multiple and potentially competing objectives to consider. One approach is to conduct the optimisation with a single objective function that combines all the relevant metrics. For example, one could try converting risk into a monetary cost to determine an overall project cost to minimise both cost and overall risk for a potentially hazardous building project.

If converting all metrics into one common measurement is difficult or inappropriate, then it may be necessary to encapsulate each of them in their own objective functions and have each of them contribute to the optimisation separately. In some cases, such as decommissioning problems, there may be a clear prioritisation order for the different objective functions, which needs to be satisfied in a sequential ‘winner takes all’ approach. One successively optimises against objective functions down the priority chain (possibly allowing for a tolerance to afford more flexibility), eventually arriving at a schedule that is optimal according to the priorities assigned to each aim.

Alternatively, there may be no priority order, and instead a systematic or exploratory approach to understand the trade-offs between different objective functions is necessary. To do this, one assumes the values of all objective functions except for one and consider what optimal values the last can take, allowing the consideration of multiple scenarios. One user case of this approach is in considering financial portfolios by fixing the level of risk (first objective function), then optimising the level of potential financial returns (second objective function).

## Our conclusion

When sufficient data for constraints and assessment criteria are available, mathematical optimisation algorithms can be used as part of scenario analysis to give the decision maker insight into optimal ways forward, in the face of uncertainty and amid competing objectives. Their numerical output provides a clear way to compare a range of potential future scenarios and provide analytical evidence to stakeholders to justify critical decisions.