Suppose that SocialNet considers the following limitation on its investment decisions: the total number of selected tests must be less than 3 unless they pay a public relations firm to help communicate the value of the tests to users. If they pay the firm, any number of tests is allowed.

SocialNet introduces a new binary variable X that is equal to 0 if the public relations firm is not hired, and equal to 1 if it is. Which of the following is a linear constraint that correctly represents the new limitation?

SocialNet is a social network company that seeks to improve its user experience and engagement. The company has identified five A/B test projects to consider running on its platform during the next quarter. Table 6 shows the estimated costs (in $ thousands) incurred by each A/B test project during each month of the next quarter. For example, Test 1 requires financing of $50,000 in month 1,$30,000 in month 2 , and $20,000 in month 3 . Table 6: Monthly Costs (in \$ thousands) for Each A/B Test Project The corresponding estimated net present values of revenue uplifts (in \$ thousands) for each project are shown in Table 7. For example, Test 1 has an estimated net present value of revenue uplift of $120,000. The revenue uplift is only realized at the end of the test. Table 7: Estimated Net Present Values of Revenue Uplifts (\$ thousands) for Each A/B Test Project In addition to financing requirements, each project also requires a certain number of users to allocate to different treatment arms. Table 8 shows the number of users each project would require in each of the three months of the quarter. Table 8: Monthly User Allocation Requirements for Each A/B Test Project For every project, you either fully invest in it over all three months or do not invest in it at all. On a monthly basis, SocialNet has a budget of $100,000 to use for A/B tests and can allocate a maximum of 10,000 users to different treatment arms. This means that, in every month, - The total cash required by the selected projects cannot exceed $100,000; - The total user allocation requirements of selected projects cannot exceed 10,000 users. For example, SocialNet may elect to run Test 1 and Test 2 , since this choice will not violate the budget or user allocation availability. From the perspective of staying under budget, the total cash requirements of $90,000 in month 1 and $80,000 in month 2 do not exceed the budget of $100, 000 , etc. From the perspective of having sufficient users to allocate, the total number of users required in each of the three months (5,500,6,500, and 7,000 users, respectively) does not exceed the monthly allocation limit of 10,000 users. SocialNet seeks to determine which A/B test projects to select to maximize the total estimated net present value of the revenue uplift of the selected portfolio of projects, while satisfying the monthly budget and user allocation constraints. In order to analyze this problem, the analyst working for SocialNet introduces the following decision variables: Ti=1 if test project i is selected, and Ti=0 otherwise, i=1,,5