6. (Modified from Winston's text): Each hour from 10am to 7pm, Bank One receives checks and...

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6. (Modified from Winston's text): Each hour from 10am to 7pm, Bank One receives checks and must process them. Its goal is to process all the checks the same day they are received (by 8pm). The bank has 13 check-processing machines, each of which can process up to 500 checks per hour. It takes one worker to operate each machine. Bank One hires both full-time and part-time workers. Full-time workers work 10am-6pm, 11am-7pm, or noon- 8pm and are paid $160 per day. Part-time workers work either 10am-3pm. 11am-4pm, noon-5pm, 1pm-6pm, 2pm-7pm, or 3pm-8pm and are paid $75 per day. The number of checks received each hour are given in the table below. In the interest of maintaining continuity, Bank One believes it must have at least one full-time worker at all times during the day. Develop a cost-minimizing work schedule that processes all checks by 8pm. Time Checks Received 10am 5000 11am 4000 Noon 3000 1pm 4000 2pm 2500 3pm 3000 4pm 4000 5pm 4500 6pm 4500 7pm 3000 Formulate an in integer program to schedule the workers needed during the day. Parame- terize the problem to handle any number of checks received - you do not necessarily need to parameterize for an arbitrary number of starting periods; in practice, on any given day, the number of time periods will remain constant but the expected number of checks received will vary. Note 1: this schedule is not cyclic and therefore does not require use of the modulus function that we discussed in class. Note 2: checks received at 10am can be processed at any point during the day, i.e, we can have an inventory of checks. Hint: this is a mix of a scheduling problem where we assign workers, an inventory problem where checks arrive and leave our system, and a fairly simple production problem where the number of checks we process in an hour is bounded by the workers available from our schedule (and the number of machines available). 6. (Modified from Winston's text): Each hour from 10am to 7pm, Bank One receives checks and must process them. Its goal is to process all the checks the same day they are received (by 8pm). The bank has 13 check-processing machines, each of which can process up to 500 checks per hour. It takes one worker to operate each machine. Bank One hires both full-time and part-time workers. Full-time workers work 10am-6pm, 11am-7pm, or noon- 8pm and are paid $160 per day. Part-time workers work either 10am-3pm. 11am-4pm, noon-5pm, 1pm-6pm, 2pm-7pm, or 3pm-8pm and are paid $75 per day. The number of checks received each hour are given in the table below. In the interest of maintaining continuity, Bank One believes it must have at least one full-time worker at all times during the day. Develop a cost-minimizing work schedule that processes all checks by 8pm. Time Checks Received 10am 5000 11am 4000 Noon 3000 1pm 4000 2pm 2500 3pm 3000 4pm 4000 5pm 4500 6pm 4500 7pm 3000 Formulate an in integer program to schedule the workers needed during the day. Parame- terize the problem to handle any number of checks received - you do not necessarily need to parameterize for an arbitrary number of starting periods; in practice, on any given day, the number of time periods will remain constant but the expected number of checks received will vary. Note 1: this schedule is not cyclic and therefore does not require use of the modulus function that we discussed in class. Note 2: checks received at 10am can be processed at any point during the day, i.e, we can have an inventory of checks. Hint: this is a mix of a scheduling problem where we assign workers, an inventory problem where checks arrive and leave our system, and a fairly simple production problem where the number of checks we process in an hour is bounded by the workers available from our schedule (and the number of machines available).

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