A caterer has undertaken a contract for a series of 4 dinners, one each of 4 successive nights, to be given by an exclusive New York Club. The caterer must purchase napkins especially for these dinners because the club has requested a special type of napkin that has not been used before, and will not be used again. On each day, the number of meals served will be:

day 1: 600 meals,

day 2: 500 meals,

day 3: 800 meals,

day 4: 700 meals

One napkin is required for each meal on any given day. Two types of laundry service are available to the caterer. Regular service takes 2 days (e.g., laundry sent at the end of a meal on day 1 can be used again for a meal on day 3) and costs $2 per napkin. A faster service takes 1 day (e.g., laundry sent at the end of a meal on day 1 can be ready for use on the 2nd day's meal), and costs $4 per napkin. The price of napkin in the stores varies based on the demand in New York. The projected pricing scheme for the 4 days is as follows

day 1: $16,

day 2: $15,

day 3: $15.5,

day 4: $16

Further, space is a premium in New York City and hence any unused napkins need to be stored in a storage facility which charges a price of 60 cents per day per napkin. The storage applies to new napkins as well as napkins returning after laundry. The caterer wants to minimize the costs associated with purchasing, storing and laundering napkins [Used napkins not sent to laundry can be thrown away at no extra cost].

a) Set up a linear programming problem that assists the caterer with the napkin purchase decision, and the number that will be sent each day to the regular and to the fast laundry.

b) Provide a quick solution for the problem without using any optimizations. Discuss how you got the answer