The XYZ Construction company won a bidding to supply cements to three different horizontal (road) projects. Project 1, 2, and 3 require 100 bags, 170 bags, and 180 bags, respectively. It has three warehouses: A, B, and C. The available bags per warehouse are as follows: 120 bags for A, 180 bags for B, and 150 bags for C. It will cost P50/bag to transport from warehouse A to Project 1, P20/bag to Project 2, P40/bag to Project 3; warehouse B cost P30, P50, and P40, respectively for Project 1, 2 and 3; P40, P30, P20 for warehouse C. Assign number of bags from each warehouse to its corresponding Project that will minimize the cost of transportation.
To minimize the cost of transportation, we need to assign the bags in the warehouses to their corresponding projects in the most cost-effective way. We can do this by finding the minimum cost for each project.
For Project 1:
- From Warehouse A: We can transport a maximum of 120 bags. The cost is 120 * P50/bag = P6000.
- From Warehouse B: We can transport a maximum of 100 bags. The cost is 100 * P30/bag = P3000.
- From Warehouse C: We can transport a maximum of 100 bags. The cost is 100 * P40/bag = P4000.
The minimum cost for Project 1 is P3000 from Warehouse B.
For Project 2:
- From Warehouse A: We can transport a maximum of 170 bags. The cost is 170 * P20/bag = P3400.
- From Warehouse B: We can transport a maximum of 170 bags. The cost is 170 * P50/bag = P8500.
- From Warehouse C: We can transport a maximum of 170 bags. The cost is 170 * P30/bag = P5100.
The minimum cost for Project 2 is P3400 from Warehouse A.
For Project 3:
- From Warehouse A: We can transport a maximum of 150 bags. The cost is 150 * P40/bag = P6000.
- From Warehouse B: We can transport a maximum of 180 bags. The cost is 180 * P40/bag = P7200.
- From Warehouse C: We can transport a maximum of 180 bags. The cost is 180 * P20/bag = P3600.
The minimum cost for Project 3 is P3600 from Warehouse C.
Therefore, the optimal assignment of bags from each warehouse to its corresponding project is as follows:
- Project 1: 100 bags from Warehouse B
- Project 2: 170 bags from Warehouse A
- Project 3: 180 bags from Warehouse C
To minimize the cost of transportation, we can use the Assignment Problem approach. We'll create a matrix to represent the costs of transporting cement bags from each warehouse to each project. Then, we'll use the Hungarian Algorithm or the Assignment Algorithm to find the optimal allocation of bags.
Step 1: Create a cost matrix
- Create a 3x3 matrix representing the costs of transporting bags from each warehouse to each project.
- Fill in the matrix with the corresponding transportation costs.
Project 1 Project 2 Project 3
A P50 P20 P40
B P30 P50 P40
C P40 P30 P20
Step 2: Use the Hungarian Algorithm or the Assignment Algorithm
- Apply the Hungarian Algorithm to find the optimal assignment of bags that minimizes the overall transportation cost.
- The algorithm will assign each warehouse to a project in such a way that the total cost is minimized.
Step 3: Analyze the optimal allocation
- Once you've applied the algorithm, you will find the optimal allocation of bags from each warehouse to each project.
- The optimal allocation will minimize the total cost of transportation.
For example, the optimal allocation might be:
- Warehouse A: 100 bags to Project 1
- Warehouse B: 70 bags to Project 2, 110 bags to Project 3
- Warehouse C: 0 bags to Project 1, 0 bags to Project 2, 70 bags to Project 3
By following this allocation, the XYZ Construction company can minimize the cost of transportation while supplying the required number of bags to each project.
To minimize the cost of transportation, we need to assign the bags from each warehouse to their corresponding projects in the most cost-effective way. We can solve this problem using a mathematical optimization technique called the transportation problem.
Here's how we can approach this problem step by step:
Step 1: Define the decision variables.
Let's define the decision variables as follows:
- Let X1A, X1B, X1C represent the number of bags assigned from warehouse A, B, and C to Project 1, respectively.
- Let X2A, X2B, X2C represent the number of bags assigned from warehouse A, B, and C to Project 2, respectively.
- Let X3A, X3B, X3C represent the number of bags assigned from warehouse A, B, and C to Project 3, respectively.
Step 2: Formulate the objective function.
The objective is to minimize the total cost of transportation. We can express this as:
Minimize Cost = 50*(X1A + X2A + X3A) + 20*(X1B + X2B + X3B) + 40*(X1C + X2C + X3C)
Step 3: Define the constraints.
We need to ensure that the number of bags assigned from each warehouse to each project does not exceed the available number of bags in each warehouse and the required number of bags for each project. The constraints can be written as:
X1A + X1B + X1C <= 120 (Warehouse A capacity)
X2A + X2B + X2C <= 180 (Warehouse B capacity)
X3A + X3B + X3C <= 150 (Warehouse C capacity)
X1A + X2A + X3A = 100 (Project 1 requirement)
X1B + X2B + X3B = 170 (Project 2 requirement)
X1C + X2C + X3C = 180 (Project 3 requirement)
Step 4: Solve the linear programming problem.
We can use an optimization solver or software to solve this linear programming problem. The solution will provide the number of bags to be assigned from each warehouse to each project to minimize the transportation cost.
The optimal solution will look something like this:
X1A = 20, X1B = 0, X1C = 100 (Warehouse A to Project 1)
X2A = 0, X2B = 120, X2C = 60 (Warehouse B to Project 2)
X3A = 80, X3B = 50, X3C = 0 (Warehouse C to Project 3)
In this solution, the total cost of transportation would be 50*(20+0+80) + 20*(0+120+50) + 40*(100+60+0) = 7800 units of currency (P).
Therefore, to minimize the cost of transportation, the XYZ Construction company should assign 20 bags from Warehouse A, 120 bags from Warehouse B, and 80 bags from Warehouse C to Project 1, Project 2, and Project 3, respectively.