# Optimise allocation to minimise variance

Background

I am trying to allocate customers $$C_i$$ to financial advisers $$P_j$$. Each customer has a policy value $$x_i$$. I'm assuming that the number of customers ($$n$$) allocated to each adviser is the same, and that the same customer cannot be assigned to multiple advisers. Therefore each partner will have an allocation of policy values like so:

$$P_1 = [x_1,x_2,x_3]$$, $$P_2 = [x_4,x_5,x_6]$$, $$P_3 = [x_7,x_8,x_9]$$

etc.

The Problem

After allocating customers, the average policy value is calculated for each adviser. I want to allocate customers to advisers in a way that minimises the variance in average policy values between each of the advisers.

What I have tried

My current algorithm randomly samples $$n$$ customers without replacement from the dataset and assigns each sample to an adviser. Once the allocations are made, the average policy value for each adviser is calculated and I compute the variance between each of the advisers. I repeat this over a defined number of iterations and return the allocation with the lowest variance. As expected, when dealing with larger volumes of advisers and customers this is not the most efficient process.

My Question

• Is there an algorithm that converges towards the optimal allocation rather than randomly allocating on every iteration?
• How can I frame this problem as an objective function to minimise the variance?
• Any links to implementations (preferably Python) would be appreciated if possible
• This feels like the setup to a good integer programming (IP) problem, but the variance adds nonlinearity that will be annoying. Are you tied to the idea of using variance as your measure of dispersion, or would you be happy using something else, say, minimize the difference between the smallest and largest policy values? – LarrySnyder610 May 23 at 18:51
• Minimising variance would be the most ideal, but if not possible then I'm open to other measures like the one you suggested – James May 23 at 21:01
• I would try formulating an IP using the measure I suggested and then trying others like variance once you have the first one working. – LarrySnyder610 May 23 at 21:06
• Thanks, I'll certainly look into IP. Could you provide any guidance on how to frame the objective function? – James May 24 at 8:51

• $$y_{ij} = 1$$ if we allocate customer $$C_i$$ to adviser $$P_j$$, $$0$$ otherwise
• $$z^\max =$$ the maximum policy value among all advisers
• $$z^\min =$$ the minimum policy value among all advisers
Then the formulation is: \begin{align} \text{minimize} \quad & z^\max - z^\min \\ \text{subject to} \quad & \sum_{i\in C} x_iy_{ij} \le z^\max \quad \forall j\in P \\ & \sum_{i\in C} x_iy_{ij} \ge z^\min \quad \forall j\in P \\ & \sum_{j\in P} y_{ij} = 1 \quad \forall i \in C \\ & \sum_{i\in C} y_{ij} = n \quad \forall j\in P \\ & y_{ij} \in \{0,1\} \quad \forall i \in C, j\in P \end{align} The objective function calculates the spread between the min and max policy value. The first constraint says that $$z^\max$$ has to be greater than or equal to each policy value; since the objective function encourages smaller values of $$z^\max$$, this means that $$z^\max$$ will equal the largest policy value. Similarly, the second constraint sets $$z^\min$$ equal to the smallest policy value. The third constraint says that each customer must be assigned to exactly one adviser. The fourth says that each adviser must have $$n$$ customers assigned to him/her. The last constraint says the $$y$$ variables must be binary.
If you're using Python, then I would implement this using the package PuLP, or one of the solver-specific packages such as gurobipy or docplex.