Maximizing Profit: Solving the Transportation Problem for BFE Company

[ajith]

The BFE company produces its own financial calculators at three plants for four wholesalers. The three plants will produce 60, 80 and 40 units, respectively, during the next time period. The company has made a commitment to sell 40 units to wholesaler 1, 60 units to wholesaler 2 and at least 20 units to wholesaler 3. Both wholesaler 3 and 4 also want to buy as many of the remaining units as possible. The net profit associated with shipping a unit from plant i for sale to wholesaler j is given by the following table:

View attachment 1265

Management wishes to know how many units to sell to wholesalers 3 and 4 and how many units to ship from each of the plants to each of the wholesalers to maximize profit.

(a) Find the optimal solution for this problem.

 

[MarkFL]

re: Transportation problem...

...

Does this mean you have no idea how to begin, or do you have any work or thoughts on how to begin that you wish to share, so our helpers know exactly where you are stuck?

 

[I like Serena]

re: Transportation problem...

The BFE company produces its own financial calculators at three plants for four wholesalers. The three plants will produce 60, 80 and 40 units, respectively, during the next time period. The company has made a commitment to sell 40 units to wholesaler 1, 60 units to wholesaler 2 and at least 20 units to wholesaler 3. Both wholesaler 3 and 4 also want to buy as many of the remaining units as possible. The net profit associated with shipping a unit from plant i for sale to wholesaler j is given by the following table:

https://www.physicsforums.com/attachments/1265

Management wishes to know how many units to sell to wholesalers 3 and 4 and how many units to ship from each of the plants to each of the wholesalers to maximize profit.

(a) Find the optimal solution for this problem.

Hi ajith, welcome to MHB! :)

As you may know, this is a linear optimization problem.
Such problems have a procedural plan to them.

1. Identify the decision variables.
2. Identify the objective function (maximize profit) and its relation to the decision variables.
3. Identify the constraints.
4. Use for instance Excel to find an optimal solution.

Are you aware of these steps?
And if so, how far did you get with them?

 

[ajith]

Re: Transportation problem...

Hi ajith, welcome to MHB! :)

As you may know, this is a linear optimization problem.
Such problems have a procedural plan to them.

1. Identify the decision variables.
2. Identify the objective function (maximize profit) and its relation to the decision variables.
3. Identify the constraints.
4. Use for instance Excel to find an optimal solution.

Are you aware of these steps?
And if so, how far did you get with them?

I did until here... After that a bit confusing...

This is correct? pls check.. :)

Maximize z = 80x₁₁ + 70x₁₂ + 50x₁₃ + 20x₁₄ + 50x₂₁ + 20x₂₂ + 10x₂₃ + 30x₂₄ +
+ 60x₃₁ + 40x₃₂ + 30x₃₃ + 50x₃₄Subject to:

x₁₁ + x₁₂ + x₁₃ + x₁₄ = 60
x₂₁ + x₂₂ + x₂₃ + x₂₄ = 80
x₃₁ + x₃₂ + x₃₃ + x₃₄ = 40

x₁₁ + x₂₁ + x₃₁ = 40
x₁₂ + x₂₂ + x₃₂ = 60
x₁₃ + x₂₃ + x₃₃ ≥ 20
x₁₄ + x₂₄ + x₃₄ ≤ 60 퓍ij ≥ 0 (i =1,2,3; j = 1,2,3,4)

So, how to find optimal solution ?

 

[I like Serena]

Re: Transportation problem...

I did until here... After that a bit confusing...

This is correct? pls check.. :)

Maximize z = 80x₁₁ + 70x₁₂ + 50x₁₃ + 20x₁₄ + 50x₂₁ + 20x₂₂ + 10x₂₃ + 30x₂₄ +
+ 60x₃₁ + 40x₃₂ + 30x₃₃ + 50x₃₄Subject to:

x₁₁ + x₁₂ + x₁₃ + x₁₄ = 60
x₂₁ + x₂₂ + x₂₃ + x₂₄ = 80
x₃₁ + x₃₂ + x₃₃ + x₃₄ = 40

x₁₁ + x₂₁ + x₃₁ = 40
x₁₂ + x₂₂ + x₃₂ = 60
x₁₃ + x₂₃ + x₃₃ ≥ 20
x₁₄ + x₂₄ + x₃₄ ≤ 60 퓍ij ≥ 0 (i =1,2,3; j = 1,2,3,4)

So, how to find optimal solution ?

That is... all correct.
And here I was thinking you had no clue how to approach the problem!

Btw, the last constraint x₁₄ + x₂₄ + x₃₄ ≤ 60 is not mentioned in the problem statement and furthermore redundant.

To find the optimal solution, I recommend Excel that has a "Solver" function that is dedicated to solve this type of problem.
Or if you want, you can also apply the Simplex algorithm yourself.

 

[ajith]

Re: Transportation problem...

That is... all correct.
And here I was thinking you had no clue how to approach the problem!

Btw, the last constraint x₁₄ + x₂₄ + x₃₄ ≤ 60 is not mentioned in the problem statement and furthermore redundant.

To find the optimal solution, I recommend Excel that has a "Solver" function that is dedicated to solve this type of problem.
Or if you want, you can also apply the Simplex algorithm yourself.

Yes... Can you show me the step to solve the problem?

 

[I like Serena]

Re: Transportation problem...

Yes... Can you show me the step to solve the problem?

Well, this is what you get if you put it into Excel.

View attachment 1269

That's it for today. Going to sleep now. (Sleepy)

 

[ajith]

Re: Transportation problem...

Well, this is what you get if you put it into Excel.

View attachment 1269

That's it for today. Going to sleep now. (Sleepy)

Thanks you for help Serena... So, how to do in manual way. Because in the exam, I need to show the step...That's why... (Wait)

 

[ajith]

This is correct step... please correct me if I'm wrong...

http://s5.postimg.org/g05yixzrr/math.png

I just wondering about the last constraint. So, it be like this:

Maximize z = 80x₁₁ + 70x₁₂ + 50x₁₃ + 20x₁₄ + 50x₂₁ + 20x₂₂ + 10x₂₃ + 30x₂₄ +
+ 60x₃₁ + 40x₃₂ + 30x₃₃ + 50x₃₄

Subject to:

x₁₁ + x₁₂ + x₁₃ + x₁₄ = 60
x₂₁ + x₂₂ + x₂₃ + x₂₄ = 80
x₃₁ + x₃₂ + x₃₃ + x₃₄ = 40

x₁₁ + x₂₁ + x₃₁ = 40
x₁₂ + x₂₂ + x₃₂ = 60
x₁₃ + x₂₃ + x₃₃ ≥ 20
x₁₄ + x₂₄ + x₃₄ = 0

 

[I like Serena]

x₁₄ + x₂₄ + x₃₄ = 0

No, that is not a constraint that is given.
To the contrary, the problem statement says: "Both wholesaler 3 and 4 also want to buy as many of the remaining units as possible."
Since x₁₄ + x₂₄ + x₃₄ is the amount sold to wholesaler 4, it should be as high as possible.

Due to the other constraints you can conclude that x₁₄ + x₂₄ + x₃₄ <= 60, but since that follows from the other constraints, there is no need to mention it explicitly.

Btw, the tableau that you showed suggests that you're supposed to find the solution in a particular way, which may not be the one I might show.
Do you perhaps have a worked example?

 

[ajith]

Do you perhaps have a worked example?

No... Can you show me, how to do??