this is linear programming HWit needs correct answers and explanation on file. For the graph answers, you draw it by using electronic paper and pencil from iPad or tablet PC like that and put on the file please.HOMEWORK 4

You must show your problem solving process. A solution only cannot get any credits.

1. Consider the following linear programming problem:

Maximize:

P = 12×1 + 10×2 subject

to:

5×1 + 3×2 ≤ 15

x1 + x2 ≤ 4

x1 ≥ 0, x2 ≥ 0

1.1 Using slack variables, convert the system of inequalities (i-system) into a system of

equations (esystem).

1.2 Using the table method, fill in the following table.

x1

x2

0

0

0

s1

s2

P = 12×1 + 10×2

Feasible? Yes or No

0

0

0

0

0

0

0

0

0

1.3 What is a maximum value of P? What are the values of x1 and x2 that lead to the

maximum value of P?

1

2. Use the simplex method to solve the problem:

Maximize:

P = 3×1 + 2×2 subject

to:

5×1 + 2×2 ≤ 20

3×1 + 2×2 ≤ 16

x1 ≥ 0, x2 ≥ 0

2

3. A farmer has 640 acres to plant in corn and soybeans. Each acre of corn requires 45

labor-hours and each acre costs $100 in seed and fertilizer. Each acre of soybeans

requires 60 labor hours and each acre costs $80 in seed and fertilizer. The farmer

estimates that he has 36,000 hours of labor and $60,000 capital to spend. The farmer

estimates that each acre planted in corn will yield a profit of $120 and each acre planted

in soybeans will yield a profit of $100. Under these conditions, how many acres of corn

and how many acres of soybeans should the farmer plant to maximize his profit? Note:

Use the simplex method to solve the problem.

3

4. Consider the following linear programming problem:

Minimize: C = 13×1 + 12×2

subject to:

½ x1 – x2 ≥ 2

x1 + x2 ≥ 7

x1 ≥ 0, x2 ≥ 0

4.1 Form the matrix A, using the coefficients and constants in the problem constraints

and objective function.

4.2 Find AT.

4.3 State the dual problem.

4.4 Use the simplex method to solve the dual problem.

4

4.5 Read the solution of the minimization problem from the bottom row of the final

simplex tableau.

5. Use the big M method to solve problem:

Maximize:

P = 6×1 + 2×2

subject to:

x1 + 2×2 ≤ 20

2×1 + x2 ≤ 16

x1 + x2 ≥ 9

x1 ≥ 0, x2 ≥ 0

5

6. A company manufactures outdoor furniture consisting of regular chairs, rocking chairs, and

chaise lounges. Each piece of furniture passes through three different production departments:

fabrication, assembly, and finishing. Each regular chair takes 1 hour to fabricate, 2 hours to

assemble, and 3 hours to finish. Each rocking chair takes 2 hours to fabricate, 2 hours to

assemble, and 3 hours to finish. Each chaise lounge takes 3 hours to fabricate, 4 hours to

assemble, and 2 hours to finish. There are 2,500 labor-hours available in the fabrication

department, 3,000 labor-hours available in the assembly department, and 3,500 labor-hours

available in the finishing department. The company makes a profit of $17 on each regular

chair, $24 on each rocking chair, and $31 on each chaise lounge. How many chairs of each

type should the company produce in order to maximize profit? What is the maximum profit?

Note: Use the simplex method.

6

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