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 Acreage 1 500 2 800 3 700

JAWABAN USAHATANI LINEAR PROGRAMMING

Nama : Eka Krisdiana FitriNIM : 135040101111211Kelas : F Agribisnis 2013

Dik :

Jenis Produksi (bushel)/acre

Luas Lahan Jam kerja

Harga/bushel

Jagung 10 X1 4 3

Gandum 25 X2 10 4

maksimal 7 40

Jawaban ;Fungsi Tujuan:Memaksimalkan keuntunganΠ 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 ≤ 74 X1 + 10X2 ≤ 4010X1 ≥ 30

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)