Solving the linear multiple choice knapsack problem with two objectives: profit and equity

Proof of Proposition 2 For clarity, we treat each option separately. Option A: Assume that the application of Option A is stopped at some point before an associated stopping condition is encountered. At that point, the increasing and decreasing slopes of all sets whose cost is not modified by the application of this option remain unchanged. Moreover, the upper sets remain upper, their decreasing slopes remain unchanged, and the increasing slope of the increasing set remains unchanged, too. Therefore, the ∆P ∆f ratios of Options A and D remain unchanged, too. Since Option A was selected for application at the beginning of the iteration, its ratio was not greater than the ratio of Option C, i.e., a−mo ≤ an− em m+ n =⇒ am+an−m2o−mno ≤ an−em =⇒ a+e ≤ mo+no =⇒ o ≥ a+ e m+ n . (1)