Let's formulate the transportation problem as a **Linear Programming Model**. --- ### **1. Decision Variables** Let - \( x_{ij} \) = units shipped from **origin \( i \)** to **customer \( j \)** where \( i = 1,2,3 \) (origins), \( j = 1,2,3 \) (customers). So, the variables are: - \( x_{11}, x_{12}, x_{13} \): From Origin 1 to Customers 1,2,3 - \( x_{21}, x_{22}, x_{23} \): From Origin 2 to Customers 1,2,3 - \( x_{31}, x_{32}, x_{33} \): From Origin 3 to Customers 1,2,3 --- ### **2. Cost Table** | | Customer 1 | Customer 2 | Customer 3 | |-------------|------------|------------|------------| | **Origin 1**| $55 | $50 | $55 | | **Origin 2**| $70 | $60 | $60 | | **Origin 3**| $50 | $40 | $40 | --- ### **3. Objective Function** **Minimize total shipping cost:** \[ \begin{align*} \text{Min } Z =\ & 55x_{11} + 50x_{12} + 55x_{13} \\ & + 70x_{21} + 60x_{22} + 60x_{23} \\ & + 50x_{31} + 40x_{32} + 40x_{33} \end{align*} \] --- ### **4. Constraints** #### **A. Supply (Origin) Constraints** - **Origin 1:** \( x_{11} + x_{12} + x_{13} \leq 400 \) - **Origin 2:** \( x_{21} + x_{22} + x_{23} \leq 600 \) - **Origin 3:** \( x_{31} + x_{32} + x_{33} \leq 500 \) #### **B. Demand (Customer) Constraints** - **Customer 1:** \( x_{11} + x_{21} + x_{31} = 500 \) - **Customer 2:** \( x_{12} + x_{22} + x_{32} = 500 \) - **Customer 3:** \( x_{13} + x_{23} + x_{33} = 500 \) #### **C. Non-negativity** - \( x_{ij} \geq 0 \) for all \( i, j \) --- ## **Summary: LP Formulation** \[ \begin{align*} \text{Minimize } Z =\ & 55x_{11} + 50x_{12} + 55x_{13} \\ & + 70x_{21} + 60x_{22} + 60x_{23} \\ & + 50x_{31} + 40x_{32} + 40x_{33} \\ \text{subject to:} \\ & x_{11} + x_{12} + x_{13} \leq 400 \\ & x_{21} + x_{22} + x_{23} \leq 600 \\ & x_{31} + x_{32} + x_{33} \leq 500 \\ & x_{11} + x_{21} + x_{31} = 500 \\ & x_{12} + x_{22} + x_{32} = 500 \\ & x_{13} + x_{23} + x_{33} = 500 \\ & x_{ij} \geq 0 \text{ for all } i, j \end{align*} \] --- **Where:** - \( x_{ij} \) = units shipped from origin \( i \) to customer \( j \) - Objective: minimize total shipping cost - Subject to: supply, demand, and non-negativity constraints

Certainly! Here's a clear and concise explanation along with the final summary: --- ### **Explanation:** This problem is a classic transportation problem where a company (XYZ) needs to ship goods from three different origins to three different customers. The goal is to minimize the total shipping cost while satisfying customer demands and not exceeding origin capacities. **Key components:** - **Decision Variables:** Variables \( x_{ij} \) represent the number of units shipped from origin \( i \) to customer \( j \). - **Objective Function:** Minimize total shipping costs, calculated by multiplying the units shipped (\( x_{ij} \)) by their respective per-unit costs. - **Constraints:** - **Supply constraints:** The total units shipped from each origin cannot exceed its capacity. - **Demand constraints:** Each customer must receive exactly 500 units (since total demand is 3 × 500 = 1500 units). - **Non-negativity:** Shipping quantities cannot be negative. --- ### **Final LP Formulation:** \[ \boxed{ \begin{aligned} & \text{Minimize} \quad Z = 55x_{11} + 50x_{12} + 55x_{13} \\ & \quad + 70x_{21} + 60x_{22} + 60x_{23} \\ & \quad + 50x_{31} + 40x_{32} + 40x_{33} \\ \text{Subject to:} \\ & x_{11} + x_{12} + x_{13} \leq 400 \quad \text{(Origin 1 capacity)} \\ & x_{21} + x_{22} + x_{23} \leq 600 \quad \text{(Origin 2 capacity)} \\ & x_{31} + x_{32} + x_{33} \leq 500 \quad \text{(Origin 3 capacity)} \\ & x_{11} + x_{21} + x_{31} = 500 \quad \text{(Customer 1 demand)} \\ & x_{12} + x_{22} + x_{32} = 500 \quad \text{(Customer 2 demand)} \\ & x_{13} + x_{23} + x_{33} = 500 \quad \text{(Customer 3 demand)} \\ & x_{ij} \geq 0 \quad \text{for all } i,j \end{aligned} } \] --- ### **Summary:** This linear programming model aims to determine the optimal shipment quantities from each of the three origins to three customers to minimize the total transportation cost while satisfying supply capacities and customer demands. The variables, objective, and constraints together form a complete mathematical representation of the transportation problem, ready to be solved using LP methods such as the simplex algorithm or specialized transportation algorithms.