machines. However, each operator performs better on some machines than on others. The shop has a contracted to do a big job that requires all 3 machines. The times required by the various operators to perform the required operations on each machine are summarized as follows: Operator Drill Press (min.) Lathe(min.) Grinder(min.)
1 22 18 35
2 41 30 28
3 25 36 18
Joe Henderson wants to assign one operator to each machine so that the total operating time for all 3 operators is minimized.
a. Formulate a linear programming model for this problem.
b. Solve the model by using the computer.
c. Joe’s brother, Fred, has asked him to hire his wife, Kelly, who is a machine operator. Kelly can perform each of the 3 required machine operations in 20 minutes. Should Joe hire his sister-in-law?
Brooks City has three consolidated high schools, each with a capacity of 1,200 students. The school board has partitioned the city into five busing districts north, south, east, west, and central each with different high school student populations. The three schools are located in the central, west, and south districts. Some students must be bused outside their districts, and the school board wants to minimize the total bus distance traveled by these students. The average distances from each district to the three schools and the total student population in each district are as follows:
Distance/Miles
District Central School West School South School Student Population
North 8 11 14 700
South 12 9 0 300
East 9 16 10 900
West 8 0 9 600
Central 0 8 12 500
The school board wants to determine the number of students to bus from each district to each school to minimize the total busing miles traveled.
a. Formulate a linear programming model for this problem.
b. Solve the model by using the computer.
The athletic boosters club for Beaconville has planned a 2-day fund-raising drive to purchase uniforms for all the local high schools and to improve facilities. Donations will be solicited during the day and night by telephone and personal contact. The boosters club has arranged for local college students to donate their time to solicit donations. The average donation from each type of contact and the time for a volunteer to solicit each type of donation are as follows: Average Donation ($) Average Interview Time (min.) Phone Personal Phone Personal Day 16 33 6 13 Night 17 37 7 19 The boosters club has gotten several businesses and car dealers to donate gasoline and cars for the college students to use to make a maximum of 575 personal contacts daily during the fundraising drive. The college students will donate a total of 22 hours during the day and 43 hours at night during the drive. THe president of the boosters club wants to know how many different types of donor contacts to schedule during the drive to maximize total donations. Formulate and solve an integer programming model for this problem. What is the difference in the total maximum value of donations between the integer and non-integer rounded-down solutions to this problem?
Harry and Melissa Jacobson produce handcrafted furniture in a workshop on their farm. They have obtained a load of 600 board feet of birch from a neighbor and are planning to produce round kitchen tables and ladder-back chairs during the next 3 months. Each table will require 30 hours of labor, each chair will require 18 hours, and between them they have a total of 480 hours of labor available. A table requires 40 board feet of wood to make, and a chair requires 15 board feet. A table earns the couple $575 in profit, and a chair earns $120 in profit. Most people who buy a table also want four chairs to go with it, so for every table that is produced, at least four chairs must also be made, although additional chairs can also be sold separately. Formulate and solve an integer programming model to determine the number of tables and chairs the Jacobsons should make to maximize profit.
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