Knapsack problem

The following optimization problem is called the knapsack problem:

with each x _j equal to 0 or 1, with all (a _j, c _j, b) usually taken to be positive integers. The name is due to interpreting the problem as one in which a camper has a knapsack which can carry up to b pounds. The camper has a choice of packing up to n items, with x _j = 1 if the item is packed and x _j = 0 if the item is not packed. Item j weighs a _j pounds. Each item has a “value” c _j to the camper if it is packed. The camper wishes to choose that collection of items having the greatest total value subject to the weight condition. The knapsack problem arises in many applications such as selecting a set of projects and as a subproblem of other problems. It can be solved by dynamic programming or by integer-programming methods. If the x _j are ordered such that c ₁/a ₁ ≥ c ₂/a ₂ ≥ ⃛ ≥ c _n/a _n and the integer restrictions on the variables are replaced by 0 ≤ x _j≤ 1, then an optimal solution to the relaxed problem is to...