Decision Variables:

x1: Number of gyms to be constructed

x2: Number of athletic fields to be constructed

x3: Number of tennis courts to be constructed

x4: Number of swimming pools to be constructed

Constraints:

Demand Goals Constraints:

x17

x210

x38

x412

Cost Constraint:

80,000x1+24,000x2+15,000x3+40,000x4600,000

Land Constraint:

4x1+8x2+3x3+5x450

Usage Constraint:

1,500x1+3,000x2+500x3+1,000x420,000

Weighted Percentage Deviation for Goals:

D1: Underachievement of demand goals (weighted by expected usage)

D2: Overachievement of the grant money goal (weighted and made more undesirable)

D3: Overachievement of the land usage goal (weighted and made more undesirable)

D4: Underachievement of the usage goal (weighted by expected usage)

Objective Function:

Minimize: Z=D1+D2+D3+D4

The Goal Programming model aims to determine the optimal values of x1, x2, x3, and x4 that minimize the weighted deviation from the goals while satisfying the constraints. The solution to this model will provide the best allocation of resources to construct each type of facility while considering both the achievement of goals and the undesirable deviations.

The department has established the following goals. listed in order of their priority: (1) The department wants to spend the total grant because any amount not spent must be returned to the government. (2) The department wants the facilities to be used by a total of at least 20,000 people each week. (3) The department wants to avoid having to secure more than the 50 acres of land already located. (4) The department would like to meet the demands of the city council for new facilities. However, this goal should be weighted according to the number of people expected to use each facility. a. Formulate a goal programming model to determine how many of each type of facility should be constructed to best achieve the city's goals. b. Solve this model by using the computer so that the solution values are integers.The Bay City Parks and Recreation Department has received a federal grant of $600,000 to expand its public recreation facilities. City council representatives have demanded four different types of facilities-gymnasiums, athletic fields, tennis courts, and swimming pools. In fact, the demand by various communities in the city has been for 7 gyms, 10 athletic fields, 8 tennis courts, and 12 swimming pools. Each facility costs a certain amount, requires a certain number of acres, and is expected to be used a certain amount, as follows: Expected Usage Facility Cost Required Acres (people/week) Gymnasium $80.000 1.500 Athletic field 24,000 3,000 Tennis court 15,000 500 Swimming pool 40.000 1,000 The Parks and Recreation Department has located 50 acres of land for construction (although more land could be located, if necessary). 2