binary integer programming problem

A team of managers at a company has a $1.5 million budget for funding R&D projects with the objective of maximizing the estimated benefit of their project portfolio (i.e., the sum of the benefits of funded projects). Company analysts have provided the managers with a list of the costs and estimated benefits of seven project proposals, labeled A-G. The projects can be either fully funded or not funded at all. Project Cost (M$) Estimated benefit (M$) A 0.4 0.9 B 0.4 0.4 0.9 1.0 D 0.1 0.5 E 0.8 1.1 F 0.2 0.6 G 0.2 0.5 (a) The managers first apply a commonly used heuristic in which the projects are sorted in descending order of their estimated benefit-to- cost ratios (BCR), and then funded one-by-one until the budget is exhausted. Determine the portfolio selected based on this heuristic and compute its estimated benefit. (b) Formulate a binary integer programming problem to help the man- agers select the optimal R&D project portfolio, and solve it by using Excel Solver. (c) Solve the LP relaxation of the problem. Could this solution be of any use to the managers? (d) Modify the optimal solution of the LP relaxation into a binary solu- tion by rounding down the value of each decision variable. What is the estimated benefit of the resulting portfolio? (e) Modify the optimal solution of the LP relaxation into a binary so- lution by rounding the value of each decision variable to the nearest integer. Is the resulting portfolio feasible? A team of managers at a company has a $1.5 million budget for funding R&D projects with the objective of maximizing the estimated benefit of their project portfolio (i.e., the sum of the benefits of funded projects). Company analysts have provided the managers with a list of the costs and estimated benefits of seven project proposals, labeled A-G. The projects can be either fully funded or not funded at all. Project Cost (M$) Estimated benefit (M$) A 0.4 0.9 B 0.4 0.4 0.9 1.0 D 0.1 0.5 E 0.8 1.1 F 0.2 0.6 G 0.2 0.5 (a) The managers first apply a commonly used heuristic in which the projects are sorted in descending order of their estimated benefit-to- cost ratios (BCR), and then funded one-by-one until the budget is exhausted. Determine the portfolio selected based on this heuristic and compute its estimated benefit. (b) Formulate a binary integer programming problem to help the man- agers select the optimal R&D project portfolio, and solve it by using Excel Solver. (c) Solve the LP relaxation of the problem. Could this solution be of any use to the managers? (d) Modify the optimal solution of the LP relaxation into a binary solu- tion by rounding down the value of each decision variable. What is the estimated benefit of the resulting portfolio? (e) Modify the optimal solution of the LP relaxation into a binary so- lution by rounding the value of each decision variable to the nearest integer. Is the resulting portfolio feasible