5. Next Monday, Concrete Co need to supply concrete to four different building sites. They prefer to use plants 1 and 2 for the production and delivery of the concrete, because they are quite close to the sites, but find it necessary to use plant 3 to satisfy the rest of the demand. The table below shows supply and demand amounts in tonnes, and cost coefficients in $ per tonne. Cij Plant 1 Plant 2 Plant 3 Demand Site 1 70 40 130 75 Site 2 Site 3 Site 4 40 20 50 50 60 210 90 23 25 30 80 19 Supply 80 55 7 Concrete Co wish to minimise their total cost. Treat this as a primal LP problem. (a) Show that Concrete Co have balanced their transportation problem. (b) Use the Least Cost Method to find a basic feasible solution to this primal problem. Verify that you have the correct number of basic variables. [Hint: You should look at 13 first. Then 12 and 21 are both contenders, but it does not matter what order you choose them in.] (c) Use complementary slackness conditions to find a dual solution that would be feasible if the primal solution were optimal. (d) Verify that your dual solution is not feasible. That is, find a non basic primal variable xij that is a candidate to enter the basis, and explain why it is a candidate. (e) Perform one iteration of the Transportation Simplex Method to achieve a better basic feasible solution. By how much does the total cost decrease? (f) Use duality to determine whether the new primal solution is optimal. Do not perform any further iteration(s).
5. Next Monday, Concrete Co need to supply concrete to four different building sites. They prefer to use plants 1 and 2 for the production and delivery of the concrete, because they are quite close to the sites, but find it necessary to use plant 3 to satisfy the rest of the demand. The table below shows supply and demand amounts in tonnes, and cost coefficients in $ per tonne. Cij Plant 1 Plant 2 Plant 3 Demand Site 1 70 40 130 75 Site 2 Site 3 Site 4 40 20 50 50 60 210 90 23 25 30 80 19 Supply 80 55 7 Concrete Co wish to minimise their total cost. Treat this as a primal LP problem. (a) Show that Concrete Co have balanced their transportation problem. (b) Use the Least Cost Method to find a basic feasible solution to this primal problem. Verify that you have the correct number of basic variables. [Hint: You should look at 13 first. Then 12 and 21 are both contenders, but it does not matter what order you choose them in.] (c) Use complementary slackness conditions to find a dual solution that would be feasible if the primal solution were optimal. (d) Verify that your dual solution is not feasible. That is, find a non basic primal variable xij that is a candidate to enter the basis, and explain why it is a candidate. (e) Perform one iteration of the Transportation Simplex Method to achieve a better basic feasible solution. By how much does the total cost decrease? (f) Use duality to determine whether the new primal solution is optimal. Do not perform any further iteration(s).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![5. Next Monday, Concrete Co need to supply concrete to four different building sites. They
prefer to use plants 1 and 2 for the production and delivery of the concrete, because they are
quite close to the sites, but find it necessary to use plant 3 to satisfy the rest of the demand.
The table below shows supply and demand amounts in tonnes, and cost coefficients in $ per
tonne.
Cij Site 1
70
40
130
75
Plant 1
Plant 2
Plant 3
Demand
Site 2
40
50
210
23
Site 3
20
30
80
19
Site 4
50
60
90
25
Supply
80
55
7
Concrete Co wish to minimise their total cost. Treat this as a primal LP problem.
(a) Show that Concrete Co have balanced their transportation problem.
(b) Use the Least Cost Method to find a basic feasible solution to this primal problem.
Verify that you have the correct number of basic variables.
[Hint: You should look at 13 first. Then 12 and 21 are both contenders, but it does
not matter what order you choose them in.]
(c) Use complementary slackness conditions to find a dual solution that would be feasible
if the primal solution were optimal.
(d) Verify that your dual solution is not feasible. That is, find a non basic primal variable
ï¿¡ that is a candidate to enter the basis, and explain why it is a candidate.
(e) Perform one iteration of the Transportation Simplex Method to achieve a better basic
feasible solution. By how much does the total cost decrease?
(f) Use duality to determine whether the new primal solution is optimal.
Do not perform any further iteration(s).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd84f55b1-6045-46a9-9610-9a34a878a241%2Fc37feaa9-fcdf-4471-b38a-de2b92a68948%2F2trpo0q_processed.png&w=3840&q=75)
Transcribed Image Text:5. Next Monday, Concrete Co need to supply concrete to four different building sites. They
prefer to use plants 1 and 2 for the production and delivery of the concrete, because they are
quite close to the sites, but find it necessary to use plant 3 to satisfy the rest of the demand.
The table below shows supply and demand amounts in tonnes, and cost coefficients in $ per
tonne.
Cij Site 1
70
40
130
75
Plant 1
Plant 2
Plant 3
Demand
Site 2
40
50
210
23
Site 3
20
30
80
19
Site 4
50
60
90
25
Supply
80
55
7
Concrete Co wish to minimise their total cost. Treat this as a primal LP problem.
(a) Show that Concrete Co have balanced their transportation problem.
(b) Use the Least Cost Method to find a basic feasible solution to this primal problem.
Verify that you have the correct number of basic variables.
[Hint: You should look at 13 first. Then 12 and 21 are both contenders, but it does
not matter what order you choose them in.]
(c) Use complementary slackness conditions to find a dual solution that would be feasible
if the primal solution were optimal.
(d) Verify that your dual solution is not feasible. That is, find a non basic primal variable
ï¿¡ that is a candidate to enter the basis, and explain why it is a candidate.
(e) Perform one iteration of the Transportation Simplex Method to achieve a better basic
feasible solution. By how much does the total cost decrease?
(f) Use duality to determine whether the new primal solution is optimal.
Do not perform any further iteration(s).
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