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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.
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