Problem 1. Optimization problem formulation (4 points, Excel not needed) You are the partner of a consulting firm. You have five project opportunities in the beginning of the year from four clients A (2 projects), B, C and D. For each project opportunity, you have estimated its professional fees, the probability of winning the project, the consulting staff needed in terms of consultant days, and the level of expertise support needed from experts. The estimates are described in the following table: Opportunity A-P1 A-P2 B-P1 C-P1 D-P1 Professional Fees ($000) 400 500 600 700 1,000 Probability of winning 0.50 0.40 0.35 0.45 0.30 Consulting Staffing Requirement (days) 150 220 240 280 300 Expertise Support Needed (days) 10 5 12 8 10 Your consulting team has a total capacity of 1,000 consultant days and you can get 30 days of the expert's support from the expertise ter. a. Formulate an optimization problem to help you select which project opportunities to pursue to maximize your expected income which is 10% of the total professional fees. Define your variables, show your objective function and the necessary constraints (1 point) b. Suppose project A-P2 is a follow-up project of A-P1, which means you cannot do A-P2 if you don't do A-P1, what constraint do you need to add to the formulation? (1 point) C. If client A and client B are competitors, which means if you do a project for A then you cannot do a project for B and vice versa, what constraint(s) do you need to add? (1 point) d. Suppose that you have only four (4) project managers and each of them can only do one project, what constraint will you need to add to the formulation? (1 point)