The Data Analytics blog

The word optimisation is used quite loosely and can relate to many different areas. For example, there is search engine optimisation (getting your website pages to the top of online search results), process optimisation (making existing processes more efficient), code optimisation (making your code run more efficiently) and then there is mathematical optimisation.

In this blog post, we'll be focusing on mathematical optimisation: what it is, how it can be applied in making more optimal business decisions at a customer level, and specifically how it's applied in credit risk. **And you can even try using optimisation yourself - using an optimisation tool we've shared in this post - to see the various scenarios resulting from your decisions**.

**Scroll to the bottom to try it for yourself!**

## The foundation of Mathematical Optimisation: Data Analytics and decisions

Making the best possible decision is crucial to the success of any business and can really give you the competitive edge. This is particularly relevant if data can be used to help you make more objective, scientific and, ultimately, better decisions. This is one of the main drivers for the recent push to utilise all that can be utilised within the data analytics universe.

The types of decisions that can be made can also be placed on a spectrum – ranging from high level strategic insights ("Sales for duvets are lagging, so let’s clear them out by putting them on sale.") to much more pin-point in nature ("Do not give Joe an increase on his revolving limit this month, but give Jane an increase of $50."). High level insights are considered *Descriptive *in nature and typically come out of business intelligence projects, where the likes of Qlik or Tableau can provide rich insights.

**Read about the 4 Types of Data Analytics**

When one moves into the more advanced record-level decisioning, the complexity and value can go up dramatically. The complexity comes in by way of decision systems that need to be created, often combining multiple predictive models. These predictive models need to be carefully created, deployed and tested before they can be trusted to be used in making key business decisions. A good example of a decision system is a profit model, which can contain predictive components such as the propensity of each individual to respond to an offer, the associated cost of writing the facility, future sales or interest and possible future bad debt.

An added complexity is when a given action can itself affect a customer’s behaviour. For instance, if you offer a customer a $20 increase in their credit limit, surely their behaviour would be different if a $100 increase was offered, particularly around take-up and downstream bad debt. Modelling this behaviour needs to be part of the decision system, and getting this right requires some out-the-box thinking and actual experiences from use cases.

In the scenario where no business constraints are applied, it is a very simple exercise to determine what actions should be taken to maximise the profit number – just sort by predicted profit and push that account or lead through the most profitable route. But business is not always so simple. What if there are constraints that will limit how much you can push through each channel? Examples are numerous:

- You have certain budgetary constraints,
- Your service provider can only process so many SMS’s per day,
- You only have a limited number of call centre agents at any given time.

These constraints need to “somehow” be met, but it is the application of these constraints that can be analytically difficult to achieve, especially if you have levers that are pulling against one another.

One needs an engine that can squeeze out as much juice as possible, given the various operational constraints. One of the most effective ways of achieving this is through the well-established mathematical optimisation methodology.

## What is Mathematical Optimisation?

**So, what is mathematical optimisation?** In the simplest of terms, mathematical optimisation selects the __best__ decision from a list of possible decisions that ensures that the specified criteria are met.

If I've done a bad job defining it for you, well then, here's the Wikipedia definition in brief: "*Mathematical Optimisation is the selection of a best element (with regard to some criterion) from some set of available alternatives."*

## Examples of Mathematical Optimsation in Business

To help convey how mathematical optimisation can be used in business, I've listed some real life examples of mathematical optimisation below, which you may already be familiar with. Each time, mathematical optimisation answers specific questions to ensure the optimal outcome of a decision:

**Portfolio management:**what stocks should a portfolio manager pick to maximise future yield whilst staying within the fund’s prescribed level of risk (Beta) given limited funds?**Stock level management:**what should the business order and when to minimise the overall cost of stock but still meet the required supply SLAs?**Traffic light management:**when should each traffic light change colour to minimise the overall vehicle traffic time for the town?**Hotel pricing:**what is the optimal price for each room that will maximise occupancy while taking into account room availability but staying within a range of prices and taking into account estimated take-up for the range of possible prices?**Credit risk:**numerous questions can be answered. We'll look at three scenarios below.

As you can see, mathematical optimisation is already widely used to optimise business outcomes, maxmise efficiency and increase profitability.

For a more comprehensive overview to the different types of optimisation that can be encountered, here is a good resource: Types of Optimization Problems.

## Mathematical Optimisation in Credit Risk

Let's now look specifically at how mathematical optimisation is applied in credit risk.

Once the required decision framework has been constructed, the optimisation framework can be incorporated. Depending on the application, one can then observe the recommended action at an account level, i.e. which account should go where. This will ensure that you will be maximising whatever it is you are looking for, but staying within your predefined business constraints. These constraints can be anything, as long as they form a part of the underlying decision system.

This approach is not limited to pure operational optimisation, but can incorporate complex action-effect models that cater for the different way in which we humans behave under a particular action. We understand the relationship between human behaviour and data and can bring in this dimension where appropriate.

We have put together three interactive examples - **using the Optimisation Tool further down** - showing how optimisation can be used to refine a credit lender's account level decisions.

**1. Optimisation of credit limit increases**

- The challenge is to identify at an account level what credit limit increase should be offered (if at all) that will maximise predicted profit, staying within business constraints:
- Overall bad rate
- Overall bad debt provision value
- Total number of limit increases
- Total value of limit increases

- A credit provider is able to model the profit at an account level, taking into account utilisation, margin on spend, bad debt and take-up.
- If you change the business constraints, e.g. opening up or reducing the allowable bad debt number, the credit limit recommendations will change accordingly. See the Optimisation Tool - the account level outcomes can be seen in the table below the tree map.
- In the Optimisation Tool, the colours refer to the size of the loan, the size of the blocks refer to the portion increase as a percentage of the total population. A zero increase obviously refers to a recommendation of no increase for that customer.

**2. Optimisation of channel**

- Given predicted response rates by channel type (SMS, USSD, letter, outbound calls) and the associated costs by channel, what is the optimal channel in order to minimse the overall cost, ensuring a minimum number of activated accounts?
- Using the Optimisation Tool below, if you reduce the cost by channel, the proportion for that channel should increase, as can be observed in the treemap. Similarly, if you change the target activated accounts, the mix will change in order to meet this new target with the least cost.

**3. Optimisation of EDC allocation**

- A credit provider works with 4 external debt collectors (EDCs). Each EDC is able to collect different amounts for each account, which is encapsulated in a predictive score. Each account therefore has 4 separate propensity-to-pay scores.
- From these scores, the predicted amounts in monetary terms can be estimated.
- Each EDC will have a different commission structure – simplistically a % of payments in this case.
- Since we have costs and estimated revenues, we can calculate the profit for each account across all 4 EDCs – this forms the basis of the decision framework on top of which optimisation can be applied.
- For this example, the applied business constraints are total number of accounts that the business is comfortable handing over to EDCs and the total commission cost.
- Changing the commission amounts per EDC changes the distribution, as demonstrated in the dynamic treemap – e.g. decreasing EDC1 commission will push up the profit for this EDC and the optimisation tool will push more accounts through EDC1

## Mathematical Optimisation Tool: Try it yourself!

Use the tool below to try mathematical optimisation yourself and see the result of different scenarios in credit risk:

We've helped large retail credit lenders in Africa and the Middle East make use of mathematical optimisation to improve their profitability. If you'd be interested in finding out more, please get in touch!

### About the Author, Robin Davies

Robin Davies is the Head of Product Development at Principa. Robin’s team packages complex concepts into easy-to-use products that help our clients to lift their business in often unexpected ways.