The fractional multidimensional knapsack problem: solution and uniqueness

Every solution to the fractional multidimensional knapsack problem admits a cost–benefit interpretation: for each solution, it is possible to associate a price to each dimension such that an object is placed (not placed) in the knapsack if its cost under the associated price system is strictly less (greater) than its benefit. In particular, an object is fractionally placed in the knapsack only if its cost equals its benefit. The characterization is sharper than what is implied by strong duality. Practical conditions are then provided that guarantee the solution is unique. The results yield a simple criterion for when an incremental addition to the collection of objects placeable in the knapsack is worthwhile.