Optimal Reward Allocation with Multi-Dimensional Customer Scoring: A Lagrangian Approach

Source abstract

Aim: This paper addresses a constrained resource allocation problem in which rewards are distributed to ranked recipients under multiple scoring criteria. The problem is motivated by retail loyalty programmes but is formulated generally. Study Design: A formal mathematical model is developed using constrained optimisation with a geometric decay structure for three ranked recipients. Three scoring criteria are evaluated: diversity-spending, frequency-spending, and three-dimensional composite. Place and Duration of Study: The research was conducted at the Department of Mathematics and the Department of Cooperative Economics and Management Studies, Auchi Polytechnic, Auchi, from April to September 2026. Methodology: Lagrangian methods are used to solve the optimisation problem. The budget constraint is shown to bind at the optimum. Substituting the binding constraint reduces the problem to a one-dimensional optimisation over the decay factor ρ, yielding a quadratic equation whose coefficients depend only on the top-three scores. Three propositions are derived and proved. Numerical simulations with 10,000 customers drawn from a trivariate log-normal distribution illustrate the model. Result: The optimal decay factor depends on the relative score gaps among the top three customers. The three-dimensional criterion achieves the highest efficiency index compared to two-dimensional alternatives (and). The efficiency advantage increases as the budget tightens. Conclusion: The model provides a closed-form solution requiring only transaction data already captured by point-of-sale systems. Including category diversity improves reward efficiency, particularly under budget constraints. Limitations include the log-normal distributional assumption, the treatment of customers as independent agents, and the absence of behavioural response. The log-normal assumption can be tested empirically using the Shapiro-Wilk test and Q-Q plots. Future research could calibrate the simulation parameters using actual transaction data and extend the framework to multi-retailer competition.