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Question 1173078: The Candido PILI Delicacies has 3 stores (A, B, C) which supplies it's 3 provincial customers (D, E, F) with their best selling product. Below is a table showing the capacity of each store, the requirements of each customer, and the cost for shipping to each customer.
Store D E F CAPACITY
A 30 40 10 150
B 20 10 50 100
C 70 20 20 250
Demand 200 160 140
Find the optimal strategy and minimum cost of transporting their best selling product from store to customer.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! This is a classic transportation problem, solvable using linear programming methods. We'll outline the steps to find the optimal strategy and minimum cost.
**1. Define Variables**
Let `x_ij` represent the number of units shipped from store `i` to customer `j`.
* `x_AD` = Units from Store A to Customer D
* `x_AE` = Units from Store A to Customer E
* `x_AF` = Units from Store A to Customer F
* `x_BD` = Units from Store B to Customer D
* `x_BE` = Units from Store B to Customer E
* `x_BF` = Units from Store B to Customer F
* `x_CD` = Units from Store C to Customer D
* `x_CE` = Units from Store C to Customer E
* `x_CF` = Units from Store C to Customer F
**2. Formulate the Objective Function**
The objective is to minimize the total transportation cost.
Minimize Z = 30`x_AD` + 40`x_AE` + 10`x_AF` + 20`x_BD` + 10`x_BE` + 50`x_BF` + 70`x_CD` + 20`x_CE` + 20`x_CF`
**3. Formulate the Constraints**
* **Supply Constraints (Store Capacities):**
* `x_AD` + `x_AE` + `x_AF` ≤ 150 (Store A Capacity)
* `x_BD` + `x_BE` + `x_BF` ≤ 100 (Store B Capacity)
* `x_CD` + `x_CE` + `x_CF` ≤ 250 (Store C Capacity)
* **Demand Constraints (Customer Requirements):**
* `x_AD` + `x_BD` + `x_CD` = 200 (Customer D Demand)
* `x_AE` + `x_BE` + `x_CE` = 160 (Customer E Demand)
* `x_AF` + `x_BF` + `x_CF` = 140 (Customer F Demand)
* **Non-negativity Constraints:**
* `x_ij` ≥ 0 for all `i` and `j`
**4. Solve the Linear Program**
This problem can be solved using various methods, including:
* **Software Solutions:**
* Excel Solver
* Dedicated Linear Programming software (e.g., LINGO, CPLEX)
* Online Linear Programming solvers
* **Transportation Algorithm:**
* Northwest Corner Rule
* Least Cost Method
* Vogel's Approximation Method (VAM)
* Then using the stepping stone or MODI method to reach optimality.
**Using a Solver (Example with Excel Solver)**
1. **Set up the spreadsheet:**
* Create a table representing the shipping quantities (`x_ij`).
* Create a table for the costs.
* Calculate the total cost using the objective function formula.
* Calculate the total shipped from each store and the total received by each customer.
2. **Use Solver:**
* Set the objective cell (total cost) to "Minimize."
* Set the changing variable cells (shipping quantities).
* Add the supply and demand constraints.
* Set the non-negativity constraint.
* Select a solving method (e.g., Simplex LP).
* Solve.
**Expected Outcome**
The solver will provide the optimal shipping quantities (`x_ij`) that minimize the total transportation cost.
**Important Notes**
* The total supply (150 + 100 + 250 = 500) equals the total demand (200 + 160 + 140 = 500), so this is a balanced transportation problem.
* The solution will provide the number of units to ship from each store to each customer.
* The objective function value will give the minimum total transportation cost.
By plugging the values that the solver provides into the objective function, you will arrive at the minimum cost.
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