The accompanying table reflects the number of high-school-age students living in each sector and the distance in miles from each sector to each school Distance to School Sector School in Sector School in Sector School in Sector B ? E A 7 5 6 B 0 4 12 ? 4 0 7 D 5 4 7 E 12 7 0 Total Number of Students 800 600 400 1,000 700 3,500 Each high school has a capacity of 1,200 students You have been asked to develop a linear programming model so as to minimize the total number of student miles traveled by bus. Decision variable X, Number of students living in sector i traveling to school located in sector The number of decision variables for the model = 15 a) The objective function, for the LP model Minimize Z +5 V +6 Y+ Save The Hills County, Michigan, supondent education is responsible for another school his county Homecognizes the need to bus continumber of students for con A-El the county are beyond walking distance to a school. The contender portion the county in geographic sectors as he temps to establish a son that wil me the total ber of de wed by bus. Here at det happens to live in a consector and is op the high school in that actor, there is no need to busca he can wok to school. The the shore located in sectors and The accompanying table molects the number of high-school-age students vingineach sector and the distance from cach sector to each school Distance to School Sector School in Sector School in Sector School in Sector Number of C Students 0 300 4 12 000 4 400 D 5 1 1000 12 7 0 700 To too Hach high school of 1.200 student A DOG 0 The Hills County, Michigan, superintendent of education is responsible for assigning students to the high school in This county He recognizes the need to bus a certain ber of students for several sectors, A-E of the county are beyond walking distance to a school. The superintendent on the county into geographic sector settempts to establish a plan that will me the sotalromber of student mes veled by bus. He also recognizes that student happens to live in a stain sector and is stigned to the righ school in that cor there is no need to be him because he can walk to school. The two school rocodin sectors. Cand The accompanying tables the number of high-school-age students living in each octor and the data in stronach sector to each school Distance to School Sector School in Sector School in Sector School in Sector Number of C E Students 100 000 150 1.000 12 700 SYD 12 O Each co200

A linear programming computer package is needed.

EZ-Windows, Inc., manufactures replacement windows for the home remodeling business. In January, the company produced 15,000 windows and ended the month with 9,000 windows in inventory. EZ-Windows’ management team would like to develop a production schedule for the next three months. A smooth production schedule is obviously desirable because it maintains the current workforce and provides a similar month-to-month operation. However, given the sales forecasts, the production capacities, and the storage capabilities as shown, the management team does not think a smooth production schedule with the same production quantity each month possible.

	February	March	April
Sales forecast	15,000	16,500	20,000
Production capacity	14,000	14,000	18,000
Storage capacity	6,000	6,000	6,000

The company’s cost accounting department estimates that increasing production by one window from one month to the next will increase total costs by $1.00 for each unit increase in the production level. In addition, decreasing production by one unit from one month to the next will increase total costs by $0.65 for each unit decrease in the production level. Ignoring production and inventory carrying costs, formulate a linear programming model that will minimize the cost of changing production levels while still satisfying the monthly sales forecasts. (Let F = number of windows manufactured in February, M = number of windows manufactured in March, A = number of windows manufactured in April, I₁ = increase in production level necessary during month 1, I₂ = increase in production level necessary during month 2, I₃ = increase in production level necessary during month 3, D₁ = decrease in production level necessary during month 1, D₂ = decrease in production level necessary during month 2, D₃ = decrease in production level necessary during month 3, s₁ = ending inventory in month 1, s₂ = ending inventory in month 2, and s₃ = ending inventory in month 3.)

Min = ______

s.t.

February Demand =________

March Demand =_________

April Demand =_________

Change in February Production =_________

Change in March Production =__________

Change in April Production =__________

February Production Capacity = _________

March Production Capacity = __________

April Production Capacity = _________

February Storage Capacity = _________

March Storage Capacity = _________

April Storage Capacity = _________

Find the optimal solution.

(F, M, A, I₁, I₂, I₃, D₁, D₂, D₃, s₁, s₂, s₃) = (____________)

Cost = $___________