Linear Programming

Assignment

  1. The garden center at a local home improvement store must allocate its space. In particular, it must decide how many square feet to devote to each of two types of products: plants (P≥0, measured in square feet) and supplies (S≥0, measured in square feet). The garden center earns $0.10 profit (per day) for each square foot of space devoted to plants and $0.05 (per day) for each square foot of space devoted to supplies. The garden center faces the following constraints:
    • The garden center can devote no more than 10 thousand square feet to these products (combined).
    • Based upon past experience, the amount of space devoted to supplies should be at least three times as large as that devoted to plants.

Type the following sentences, deleting the blanks and replacing them with numbers (rounding all numbers to the nearest integer).

“In order to maximize profit, the garden center should devote _____ square feet to plants and ______ square feet to supplies. If it makes this choice, its daily profit will be $______.”

  1. The manager of a local television station is trying to decide the best daily mix of the two types of programs that the station airs – network shows (N≥0, measured in hours) and local shows (L≥0, measured in hours). He is trying to maximize the station’s advertising revenue. Each hour of network programming (on average) earns $60,000 of advertising revenue. Each hour of local programming earns (on average) $16,000 of advertising revenue. The station faces the following constraints:
  • The station can’t broadcast more than 24 hours of programming each day.
  • Network programming costs (on average) $50,000 per hour, while local programming costs (on average) $10,000 per hour. The station’s budget allows for spending up to $500,000 per day on programming.

Type the following sentences, deleting the blanks and replacing them with numbers (rounding all numbers to one decimal):

“In order to maximize advertising revenue, the station should devote _____ hours to network programming and ______ hours to local programming. If it makes this choice, its advertising revenue will be $______.”

  1. Wal-Mart, Target and other retailers are implementing schedule-optimization systems to cut their labor costs. Suppose that Wal-Mart hires two types of workers, full-time and part-time. Let F be the number of full-time hours used in a typical day. Let P be the number of part-time hours used. (Assume that fractional hours can be used.) Part-time workers cost $10 per hour and full-time workers cost $18 an hour. Wal-Mart’s goal is to minimize labor costs. Suppose that a particular Wal-Mart faces the following constraints:
    • Full-time workers supervise part-time workers. In particular at least one full-time hour is needed for every four part-time hours.
    • Each worker (whether full-time or part-time) can serve 25 customers per hour, and the store must be able to serve at least 1,000 customers.

Type the following sentences, deleting the blanks and replacing them with numbers (rounding all numbers to the nearest integer):

“In order to minimize labor costs, the Wal-Mart should hire _____ hours of full-time workers and ______ hours of part-time workers. If it makes this choice, its labor costs will be $_____.”

  1. Maximize

W= 4x1 +0x2 – x3

           Subject to

x1 + x2+ x3 ≤6

                                                                        x1 – x2 + x3 ≤10

                                                                        x1 – x2 – x3 ≤4

                                    x1 ,x2 ,x3 ≥ 0

  1. A company manufactures three products X,Y and Z. Each product requires machine time and finishing time as shown in the following table:

Machine time                                     Finishing time

x 1 hr. 4 hr.
y 2hr. 4 hr.
z 3hr. 8 hr.

The numbers of hours of machine time and finishing time available per month are 900 and 5000, respectively. The unit profit on X,Y, and Z is $6,$8, and $12, respectively. What is the maximum profit per month that can be obtained