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 $3.00 per pound and Type II seed costs $4.10 per pound. If the blend needs to score at least 500 points for shade tolerance, 550 points for traffic resistance, and 800 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 1.6 1.4
Traffic Resistance 2.5 1.5
Drought Resistance 3.4 4.3

Formulate a linear programming model for the above situation by determining
(a) The decision variables.
(b) The objective function.
(c) All the constraints.
Note: Do NOT solve the problem after formulating.

Explanation / Answer

Dear Anonymous,

Let a be the variable which represents the amount of Type I seed in the blend, and b be the variable which expresses the amount of Type II seed in the blend. Then we can go straight to the first equation that is of interest to us, the cost function for the blend, let C stand for the cost, then

C = 3a + 4.10b.

It is this function that we want to minimize, hence this function is the objective function. The decision variables are the a and the b. Now it is clear that we cannot have negative quantities of seed in the blend, so we must have the following two constraints automatically:

a 0

b 0.

We know that the blend must score at least 500 points for the shade tolerance and that Type 1 seed rates 1.6 for shade tolerance and Type II rates 1.4 for shade tolerance, thus we have the following relationship:

1.6a + 1.4b 500.

Similarly, we know that Type 1 seed scores 2.5 for traffic resistance and Type II 1.5 for traffic resistance and that the blend has to rate at least 550 for traffic resistance which gives the following relationship:

2.5a + 1.5b 550.

Finally we know that Type 1 seed scores 3.4 for drought resistance and Type II 4.3 for drought resistance and that the blend must score at least 800 for drought resistance, thus we must have the following relationship:

3.4a + 4.3b 800.

Thus we have finished. Let me restate the problem in a more concise way:

Decision Variables

a: the pounds of Type I seed in blend.

b: the pounds of Type II seed in blend.

Objective(to be minimized)

C = 3a + 4.10b

Constraints

a 0

b 0

1.6a + 1.4b 500

2.5a + 1.5b 550

3.4a + 4.3b 800

I hope that this helps!!

David

Related Questions

Navigate

• Browse (All)
• Browse A
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.