Exercise: Linear Programming
1. Farmer Jones must determine how many acres of corn and wheat to plant this
year. An acre of wheat yields 25 bushels of wheat and requires 10 hours of labor
per week. An acre of corn yields 10 bushels of corn and requires 4 hours of labor
per week. All wheat can be sold at $4 a bushel, and all corn can be sold at $3 a
bushel. Seven acres of land and 40 hours per week of labor are available.
Government regulations require that at least 30 bushels of corn be produced
during the current year. Let x1 = number of acres of corn planted, and x2 = number
of acres of wheat planted. Using these decision variable, formulate a Linear
Programming (LP) whose solution will tell Farmer Jones how to maximize the total
revenue from wheat and corn.
2. Answer these questions about Problem 1.
a. Is (x1 = 2, x2 = 3) in the feasible region?
b. Is (x1 = 4, x2 = 3) in the feasible region?
c. Is (x1 = 2, x2 = -1) in the feasible region?
d. Is (x1 = 3, x2 = 2) in the feasible region?
3. A farmer owner wants to know how many acres of three different crops to plant
on three different plots in order to maximize profit. The farmer’s tract of land
consists of 2000 acres. The farmers has subdivided the tract into three plots and
has contracted with three local farm families to operate the plots. The farm owner
has instructed each sharecropper to plant three crops: corn, peas and soybeans.
The size of each plot has been determined by the capabilities of each local farmer.
Plot sizes, crop restrictions, and profit per acre are given in the following tables.
Plot
1
2
3
Acreage
500
800
700
JAWABAN USAHATANI LINEAR PROGRAMMING
Nama
NIM
Kelas
Dik
: Eka Krisdiana Fitri
: 135040101111211
: F Agribisnis 2013
:
Jenis
Jam
kerja
4
Harga/bushel
Jagung
Produksi
Luas Lahan
(bushel)/acre
10
X1
Gandum
25
X2
10
4
7
40
maksimal
Jawaban
;
Fungsi Tujuan:
Memaksimalkan keuntungan
Π max = 3 C + 4 W
= 3 (X1.C) + 4(X2.W)
= 3 (X1.10) + 4(X2.25)
= 30 X1 + 100 X2
Fungsi kendala:
X1 + X2 ≤ 7
4 X1 + 10X2 ≤ 40
10X1 ≥ 30
3
Intepretasi:
Untuk mendapatkan keuntungan maksimal dan memenuhi peraturan pemerintah untuk
memproduksi minimal 30 bushels jagung, petani John harus menanami 7 hektar lahannya
dengan kombinasi jagung seluas 3 acres dan gandum seluas 2.8 acres. Sisa lahan yang tidak
ditanami yakni seluas 1.2 acres.
2. menentukan daerah feasible(layak) dan tidak feasible(tidak layak)
a) (x1 = 2, x2 = 3) = tidak feasible
b) (x1 = 4, x2 = 3) = tidak feasibletidak layak
c) (x1 = 2, x2 = -1) = tidak feasible karena x2 bernilai negatif. Sedangkan daerah
yang feasible (daerah arsiran warna ungu) terletak pada nilai positif
d) (x1 = 3, x2 = 2) = feasible (layak)
c) (x1 = 2, x2 = -1) = tidak feasible karena x2 bernilai negatif. Sedangkan daerah
yang feasible (daerah arsiran warna ungu) terletak pada nilai positif
d) (x1 = 3, x2 = 2) = feasible (layak)