operations reasearch 1

The Problem A decision has been made by the management of a private school that one of it's middle schools is to be closed at the end of the year. Therefore, all of next year's middle school stu- dents (sixth grade, seventh grade, and eighth grade) are to be assigned to the three remaining middle schools. School-1 has a capacity of 900 students, school-2 capacity is 110 students, and school-3 capacity is 1000 students. The school provides buses as a transportation for students. Students has to be transported from six residential areas to each of the three re- maining schools. The yearly cost per student of busing from each of the six residential areas to each of the schools is shown in the table below. In the table, o indicates that busing is not needed and a dash indicates an infeasible assignment. School management required that each grade must constitute between 30 and 36 percent of each school's population. For ex- ample the percentage of sixth grade students in school-1 must be between 30 and 36 percent. Your job is to assist the school manager in determining how many students in each area should be assigned to each school such that the busing cost is minimized. You need to formulate and solve a linear program for this problem. Total Number of Students Number of Students Number of Students Number of Students Area Who Should Join 6th Who Should Join 7th Who Should Join 8th Grade Grade Grade 1 146 171 235 2 222 168 210 3 165 176 209 4 100 112 5 195 170 135 6 153 125 171 450 600 550 350 500 450 Busing Cost per Student School-1 School School $300 0 5700 $400 $500 $600 $300 $200 $200 $500 $0 $400 $500 $300 0 Tasks to be completed for the Project Task-1: Formulate an LP for this problem Clearly define each component in your model using mathematical equations and in words. If any of your equations is nonlinear, you need to linearize it. Please number each equation. Make sure to include the following in your LP model with description of each: Decision variables with proper indexing if needed. Objective function Constraints (functional and sign restriction)