Optimal advance payment scheme involving fixed per-payment costs

The advance payment scheme is used when a payment larger than the billed amount is made. The extra payment is effectively a cash deposit for credit towards bills in future periods. This study presents a model for determining the optimal cash deposit amount when there is a fixed per-payment cost; the model balances the tradeoff between the fixed payment cost with the lost interest on the cash deposit. We use a renewal theory approach to obtain the long-term expected total cost per period when the bill amounts in a future period are deterministic, exponentially distributed, normally distributed, or Poisson occurrence based with a constant per occurrence charge. For the first two cases, we obtain explicit expressions for the optimal deposit amount; for the latter two, we give numerical procedures. An example in each case is presented, along with a comparison with the usual pay-each-period strategy. We conclude that the advance payment strategy offers significant cost savings when the per-payment cost is relatively large compared with the lost interest of one period's average bill amount; i.e. when interest rate is low or the average bill amount is small.