Take Management Science Exam and compare your solution. From the course Management Science at Technische Universität München (TUM).
A farming family owns 200 hectares and has 50000forinvestment.Eachmembercanprovide2500workinghoursduringtheyear.Incomeincashcancomefromthreecropsandfromcows.Eachcowneeds0.2hectaresoflandand100humanhoursofpersonalwork.Eachcowwillgive1000 income each year for the family. The existing stable is sufficient for up to 32 cows. The estimated hours of personal work and the income per cultivated hectare for the three types of crop are given in Table 1.
Soybeans Corn
Oats
Working hours (per hectare)
25
60
15
Net annual income ($ per hectare and year) 600
800
400
Table 1: Work and income per cultivated hectare for crops
The family, which consists of four persons, wants to determine how much land should be cultivated for each crop type and how many cows should be kept to maximize the annual net profit.
A farming family owns 200 hectares and has 50000forinvestment.Eachmembercanprovide2500workinghoursduringtheyear.Incomeincashcancomefromthreecropsandfromcows.Eachcowneeds0.2hectaresoflandand100humanhoursofpersonalwork.Eachcowwillgive1000 income each year for the family. The existing stable is sufficient for up to 32 cows. The estimated hours of personal work and the income per cultivated hectare for the three types of crop are given in Table 1.
Soybeans Corn
Oats
Working hours (per hectare)
25
60
15
Net annual income ($ per hectare and year) 600
800
400
Table 1: Work and income per cultivated hectare for crops
The family, which consists of four persons, wants to determine how much land should be cultivated for each crop type and how many cows should be kept to maximize the annual net profit.
A farming family owns 200 hectares and has 50000forinvestment.Eachmembercanprovide2500workinghoursduringtheyear.Incomeincashcancomefromthreecropsandfromcows.Eachcowneeds0.2hectaresoflandand100humanhoursofpersonalwork.Eachcowwillgive1000 income each year for the family. The existing stable is sufficient for up to 32 cows. The estimated hours of personal work and the income per cultivated hectare for the three types of crop are given in Table 1.
Soybeans Corn
Oats
Working hours (per hectare)
25
60
15
Net annual income ($ per hectare and year) 600
800
400
Table 1: Work and income per cultivated hectare for crops
The family, which consists of four persons, wants to determine how much land should be cultivated for each crop type and how many cows should be kept to maximize the annual net profit.
A planner has to determine the production quantities for two different products where the production process consists of three stages with limited capacity. The problem can be modeled as follows where each constraint corresponds to the capacity of one production stage.
max 3x1 + 2x2
s.t.
2x1 + x2 ≤ 100 (1)
x1 + x2 ≤ 80 (2)
x1 ≤ 40 (3)
x1, x2 ≥ 0
The simplex algorithm is applied to determine an optimal solution and it stops with the following system of equations:
x1 = 20 - x3 + x4
x2 = 60 + x3 - 2x4
x5 = 20 + x3 - x4
Z = 180 - x3 - x4
Determine the optimal values of the decision variables and the optimal value of the objective function.
How large is the optimal value of the objective function if the third constraint is changed to x1 ≤ 30? Explain your answer in one sentence.
A linear program is already transformed in the following tableau in order to apply the Simplex Algorithm.
_ X1 X2 X3 X4 X5 X6
X4 3 -2 2 100 15
X5 -1 2 1 0 1 0 3
X6 1 -1 1 0 0 1 3
Z -2 4 -1 0 0 0
Select the pivot column and the corresponding pivot row!
How do you recognize in the final table whether there are an infinite number of optimal solutions?
Consider the following problem
max 2x1 - x2
s.t.
3x1 - 2x2 + x3 ≤ 15
-x1 + x2 + 2x3 ≤ 3
x1 - x2 + x3 ≤ 3
x1, x2, x3 ≥ 0.
The application of the simplex algorithm leads to the following final tableau:
x1 X2 X3 x4 X5 X6
X1 1 0 1 2 0 2 1
X2 0 1 5 1 3 0 2 4
X6 0 0 2 0 1 1 6
Z 0 0 2 1 1 0 18
Perform a sensitivity analysis with respect to the coefficient in the objective function of x₁. In which interval can the coefficient change without a change of the mix of the basic and non-basic variables?