3. Suppose a local post office requires the following number of workers on the different days...
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3. Suppose a local post office requires the following number of workers on the different days of the week. Day Number of Day workers Number of workers Mon 11 Fri 20 Tue 12 Sat Wed Thu 13 14 Sun 10 10 The post-office may hire regular workers, who could start on any day of the week, work for 5 consecutive days, and take the next two days off. The cost of hiring a regular worker is $100 per week. (a) Formulate an IP to find a schedule (that is, how many regular workers should start working on each day of the week) that meets the post-office's requirements and minimizes its total cost. (b) Now, suppose that in addition to hiring regular workers, the post office may also hire part-time workers, who are available on a daily contract every day of the week. The cost of hiring a part-time worker is $25 per day for Mon-Fri and $40 per day for Sat-Sun. Modify your IP in (a) so that it captures this new detail. (c) How do you think the optimum values of the two IPs you presented in (a) and (b) compare to each other? (Hint: Don't actually solve the two IPs.) 1.For each of the following statements, indicate if it is TRUE or FALSE, and justify your answer with a proof or a counterexample. (a) Let A be a matrix and x be a vector such that Ax 0. Then either A 0 and x 0, or A 0 and x 0. (b) Let A be a matrix and x be a nonzero vector. If Ax = 0, then the row vectors of A are linearly dependent. (c) Let A be an n n matrix, and b,c are vectors in Rn. If Ax = b has a unique solution, then Ax = c also has a unique solution. (d) Suppose the system Ax = b has infinitely many solutions. Then there exists a nonzero vector x such that Ax = 0. 3. Suppose a local post office requires the following number of workers on the different days of the week. Day Number of Day workers Number of workers Mon 11 Fri 20 Tue 12 Sat Wed Thu 13 14 Sun 10 10 The post-office may hire regular workers, who could start on any day of the week, work for 5 consecutive days, and take the next two days off. The cost of hiring a regular worker is $100 per week. (a) Formulate an IP to find a schedule (that is, how many regular workers should start working on each day of the week) that meets the post-office's requirements and minimizes its total cost. (b) Now, suppose that in addition to hiring regular workers, the post office may also hire part-time workers, who are available on a daily contract every day of the week. The cost of hiring a part-time worker is $25 per day for Mon-Fri and $40 per day for Sat-Sun. Modify your IP in (a) so that it captures this new detail. (c) How do you think the optimum values of the two IPs you presented in (a) and (b) compare to each other? (Hint: Don't actually solve the two IPs.) 1.For each of the following statements, indicate if it is TRUE or FALSE, and justify your answer with a proof or a counterexample. (a) Let A be a matrix and x be a vector such that Ax 0. Then either A 0 and x 0, or A 0 and x 0. (b) Let A be a matrix and x be a nonzero vector. If Ax = 0, then the row vectors of A are linearly dependent. (c) Let A be an n n matrix, and b,c are vectors in Rn. If Ax = b has a unique solution, then Ax = c also has a unique solution. (d) Suppose the system Ax = b has infinitely many solutions. Then there exists a nonzero vector x such that Ax = 0.
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