A garden shop mixes two types of grass seed into a blend. Each type of grass has

Question

A garden shop mixes two types of grass seed into a blend. Each type of grass has been rated (per pound) according to its shade tolerance, ability to stand up to traffic, and drought resistance, as shown in the table below. Type I seed costs $3.20 per pound and Type II seed costs $4.30 per pound. If the blend needs to score at least 600 points for shade tolerance, 650 points for traffic resistance, and 900 points for drought resistance, how many pounds of each seed should be in the blend? The garden shop wants to minimize total cost of production.

Type I

Type II

Shade Tolerance

1.8

1.6

Traffic Resistance

2.3

1.3

Drought Resistance

3.6

4.5

Formulate a linear programming model for the above situation by determining

(a) The decision variables

(b) Determine the objective function. What does it represent?

(c) Determine all the constraints. Briefly describe what each constraint represents.

Note: Do NOT solve the problem after formulating.

Type I

Type II

Shade Tolerance

1.8

1.6

Traffic Resistance

2.3

1.3

Drought Resistance

3.6

4.5

Explanation / Answer

Ans a)

Let the amount(in pounds) of Type 1 seeds be X1 and the amount of Type 2 seeds be X2.

X1 and X2 are the decision variables

Ans b)

Cost per pound of type I seed = $3.20

Cost per pound of type II seed = $4.30

Objective function

Min Z = 3.2X1 + 4.3X2

It represents the total cost of the grass seeds.

Ans c)

Constraints

1.8X1 + 1.6X2 >= 600 (Shade tolerance constraint)

This constraint represents that the total shade tolerance points of Type 1 and Type 2 seeds must be greater than or equal to 600. Type 1 seeds contribute 1.8 points and type 2 seeds contribute 1.6 points per pound.

2.3X1 + 1.3X2 >= 650 (Traffic resistance constraint)

This constraint represents that the total traffic resistance points of Type 1 and Type 2 seeds must be greater than or equal to 650. Type 1 seeds contribute 2.3 points and type 2 seeds contribute 1.3 points per pound.

3.6X1 + 4.5X2 >= 900 (Drought resistance constraint)

This constraint represents that the total Drought resistance points of Type 1 and Type 2 seeds must be greater than or equal to 900. Type 1 seeds contribute 3.6 points and type 2 seeds contribute 4.5 points per pound.

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