The EPA currently funds water quality improvement projects via a priority-ranking scheme using the "The Funding Line Approach." In this approach, each project is assigned a dimensionless priority number based on the water quality improvement it achieves. Projects with higher priority numbers are considered better than lower-priority-number projects. A funding cost is associated with each project, and a total amount of money is available for all projects chosen in a particular funding cycle. The procedure to determine which projects receive funding is to take the first-, second-, third-, and so on, priority projects in order, adding their costs as they are placed on the list, until the sum of costs of the projects chosen exceeds the budget. Then the last project added is removed from the list, and the remaining projects from the top of the priority list down to the funding line are those chosen in this funding cycle.

C_j: cost of the jth project

p_j: priority points associated with the jth project

B:budget available for this funding cycle

n:number of projects

This is an example of a knapsack problem.

How would the math program for this problem look? If the objective function will be to maximizes priority points achieved subject to a budget limit. Also could you please explain to me why using math programming and an exact solution approach (e.g. branch and bound) will not produce a worse solution than the Funding Line Approach and why math programming and an exact solution approach is likely to produce a better solution.