Optimal allocation of limited resources among discrete risk-reduction options

Abstract

This study exposes a critical weakness of the (0-1) knapsack dynamic programming approach, widely used for optimal allocationof resources. The (0-1) knapsack dynamic programming approach could waste resources on insignificant improvements andprevent the more efficient use of the resources to achieve maximum benefit. Despite the numerous extensive studies, this criticalshortcoming of the classical formulation has been overlooked. The main reason is that the standard (0-1) knapsack dynamicprogramming approach has been devised to maximise the benefit derived from items filling a space with no intrinsic value. While this is an appropriate formulation for packing and cargo loading problems, in applications involving capital budgeting,this formulation is deeply flawed. The reason is that budgets do have intrinsic value and their efficient utilisation is just asimportant as the maximisation of the benefit derived from the budget allocation.

Accordingly, a new formulation of the (0-1) knapsack resource allocation model is proposed where the weighted sum of thebenefit and the remaining budget is maximised instead of the total benefit. The proposed optimisation model produces solutionssuperior to both – the standard (0-1) dynamic programming approach and the cost-benefit approach.

On the basis of common parallel-series systems, the paper also demonstrates that because of synergistic effects, sets includingthe same number of identical options could remove different amount of total risk. The existence of synergistic effects doesnot permit the application of the (0-1) dynamic programming approach. In this case, specific methods for optimal resourceallocation should be applied. Accordingly, the paper formulates and proves a theorem stating that the maximum amount ofremoved total risk from operations and systems with parallel-series logical arrangement is achieved by using preferentiallythe available budget on improving the reliability of operations/components belonging to the same parallel branch. Improvingthe reliability of randomly selected operations/components not forming a parallel branch leads to a sub-optimal risk reduction.The theorem is a solid basis for achieving a significant risk reduction for systems and processes with parallel-series logicalarrangement.