A company has three manufacturing plants (in Atlanta, Tulsa, and Springfield) that produce a product that is then shipped to one of four distribution centers. The three plants can produce 13, 18, and 12 truckloads of product each week, respectively. Each distribution center needs 10 truckloads of product each week. The shipping costs per truckload between the plants and distribution centers are given in the table. The company needs to determine how much to ship from each plant to each distribution center and would like to minimize total shipping costs.

 

Distribution Center

Plant A B C D

Atlanta $800 $1,300 $400 $700

Tulsa $1,100 $1,400 $600 $1,000

Springfield $600 $1,200 $800 $900

 

4) To accomplish the minimum total shipping cost, which plant(s) produce(s) maximum amounts?

Select one or more:

A. Atlanta

B. Tulsa

C. Springfield

 

5) To which distribution center(s) does Springfield plant have to send the products in order to keep the total shipping cost of the company minimum?

 

Select one or more:

A. Distribution Center B

B. Distribution Center D

C. Distribution Center A

D. Distribution Center C

Here, The Given Data is :

Step 1 :

Step 2 :

Now,

Here, Assume the Decision Variables be x_ij defines the allotment of i^th plant to j^th Distribution Center.

where,

i=1 for Atlanta, 2 for Tulsa, 3 for Springfield

and j = A,B,C,D

and c_ij defines the unit truckload cost incurred in transporting the products from i^th plant to j^th Distribution Center.

where,

i=1 for Atlanta, 2 for Tulsa, 3 for Springfield

and j = A,B,C,D

Step 3 :

Here,

The Objective Function is to minimize the shipping cost :

So, 

Objective Function = MIN ∑(x_ij*c_ij)

Where x_ij is the shipping allotment and c_ij is the cost.

and

i=1 for Atlanta, 2 for Tulsa, 3 for Springfield

and j = A,B,C,D

Step 4 :

Constraints:

The firm should allocate no more than supply ship no more than capacity. The demand also has to be met.

Hence,

Capacity for Atlanta, Tulsa and springfield are 13,18and 12 respectively,

Therefore,

x_1A + x_1B + x_1C + x_1D <= 13

x_2A + x_2B + x_2C + x_2D <= 18

x_3A + x_3B + x_3C + x_3D <= 12

and Each DC needs 10 truckloads per week,

Therefore,

x_1A + x_2A + x_3A = 10

x_1B + x_2B + x_3B = 10

x_1C + x_2C + x_3C = 10

Step 5 :

Now, To solve the above LP, Construct the excel sheet as follows :

Step 6 :

Now, use the the excel solver on objective function cell with available constraints as follows :

Step 7 :

Hence, output excel sheet will be :

Step 8 :

and the required shipping plan is as follows :

Step 9 :

Hence,

Answer (5) :

From the output shipping plan, we can see that Tulsa plant produces maximum amount to accomplish the minimum shipping cost.

Step 10 :

Answer (6) :

From the output shipping plan, we can see that, Springfield plant to send the products to Distribution center A and to Distribution center B to minimize the total shipping cost of company.

Please see the attached file for the complete solution