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  • class__15-✍🏻 HandMade Excercise prep EXAM -LP -MathModels - Simplex Two-Stages + Transportation Problem/🇺🇸 Answers - English/Exercise_2-Answers

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‎class__15-✍🏻 HandMade Excercise prep EXAM -LP -MathModels - Simplex Two-Stages + Transportation Problem/🇺🇸 Answers - English/Exercise_2-Answers/2-answer_2-MD.md‎

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```markdown
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# 🚚 Transportation Problem Solution — Minimum Cost Method
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## 📋 Problem Statement
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Consider the transportation network given by the table below and determine the optimal transportation cost, starting the resolution using the Minimum Cost Method.
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---
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## 🧩 Problem Data
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### Supplies (Sources)
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| Source | Supply |
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|--------|--------|
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| S1 | 100 |
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| S2 | 800 |
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| S3 | 150 |
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| S4 | 400 |
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### Demands (Destinations)
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| Market | Demand |
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|--------|--------|
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| M1 | 700 |
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| M2 | 250 |
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| M3 | 500 |
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### Transportation Costs
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| | M1 | M2 | M3 |
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|--------|----|----|----|
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| **S1** | 5 | 2 | 8 |
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| **S2** | 6 | 3 | 7 |
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| **S3** | 4 | 3 | 6 |
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| **S4** | 8 | 6 | 4 |
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---
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## 📝 Step 1: Check for Balance
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- Total supply: \(100 + 800 + 150 + 400 = 1450\)
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- Total demand: \(700 + 250 + 500 = 1450\)
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**Since supply equals demand, the problem is balanced.**
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---
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## 📝 Step 2: Minimum Cost Method
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At each step, select the cell with the lowest available cost and allocate the maximum possible amount, updating supplies and demands accordingly.
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---
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### 🔢 Iteration 1
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- **Lowest cost:** 2 (S1 → M2)
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- Supply S1: 100 | Demand M2: 250
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- **Allocation:** \(x_{12} = 100\)
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Table after allocation:
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| | M1 | M2 | M3 | Remaining Supply |
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|--------|----|----|----|------------------|
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| **S1** | |100 | | 0 |
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| **S2** | | | | 800 |
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| **S3** | | | | 150 |
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| **S4** | | | | 400 |
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| **Remaining Demand** | 700 | 150 | 500 | |
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---
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### 🔢 Iteration 2
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- **Next lowest cost:** 3 (S2 → M2)
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- Supply S2: 800 | Demand M2: 150
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- **Allocation:** \(x_{22} = 150\)
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Table after allocation:
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| | M1 | M2 | M3 | Remaining Supply |
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|--------|----|----|----|------------------|
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| **S1** | |100 | | 0 |
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| **S2** | |150 | | 650 |
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| **S3** | | | | 150 |
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| **S4** | | | | 400 |
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| **Remaining Demand** | 700 | 0 | 500 | |
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---
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### 🔢 Iteration 3
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- **Next lowest cost:** 4 (S3 → M1)
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- Supply S3: 150 | Demand M1: 700
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- **Allocation:** \(x_{31} = 150\)
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Table after allocation:
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| | M1 | M2 | M3 | Remaining Supply |
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|--------|----|----|----|------------------|
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| **S1** | |100 | | 0 |
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| **S2** | |150 | | 650 |
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| **S3** |150 | | | 0 |
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| **S4** | | | | 400 |
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| **Remaining Demand** | 550 | 0 | 500 | |
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---
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### 🔢 Iteration 4
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- **Next lowest cost:** 4 (S4 → M3)
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- Supply S4: 400 | Demand M3: 500
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- **Allocation:** \(x_{43} = 400\)
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Table after allocation:
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| | M1 | M2 | M3 | Remaining Supply |
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|--------|----|----|----|------------------|
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| **S1** | |100 | | 0 |
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| **S2** | |150 | | 650 |
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| **S3** |150 | | | 0 |
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| **S4** | | |400 | 0 |
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| **Remaining Demand** | 550 | 0 | 100 | |
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---
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### 🔢 Iteration 5
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- **Next lowest cost:** 6 (S2 → M1)
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- Supply S2: 650 | Demand M1: 550
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- **Allocation:** \(x_{21} = 550\)
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Table after allocation:
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| | M1 | M2 | M3 | Remaining Supply |
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|--------|----|----|----|------------------|
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| **S1** | |100 | | 0 |
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| **S2** |550 |150 | | 100 |
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| **S3** |150 | | | 0 |
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| **S4** | | |400 | 0 |
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| **Remaining Demand** | 0 | 0 | 100 | |
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---
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### 🔢 Iteration 6
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- **Next lowest cost:** 7 (S2 → M3)
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- Supply S2: 100 | Demand M3: 100
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- **Allocation:** \(x_{23} = 100\)
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Final table:
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| | M1 | M2 | M3 | Remaining Supply |
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|--------|-----|-----|-----|------------------|
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| **S1** | | 100 | | 0 |
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| **S2** | 550 | 150 | 100 | 0 |
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| **S3** | 150 | | | 0 |
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| **S4** | | | 400 | 0 |
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| **Remaining Demand** | 0 | 0 | 0 | |
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---
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## 📊 Final Allocation Table
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| | M1 | M2 | M3 | Supply |
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|--------|-----|-----|-----|--------|
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| **S1** | - | 100 | - | 100 |
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| **S2** | 550 | 150 | 100 | 800 |
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| **S3** | 150 | - | - | 150 |
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| **S4** | - | - | 400 | 400 |
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| **Demand** | 700 | 250 | 500 | |
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---
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## 🧮 Total Cost Calculation
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\[
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\begin{align*}
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Z &= (100 \times 2) + (150 \times 3) + (550 \times 6) + (100 \times 7) + (150 \times 4) + (400 \times 4) \\
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&= 200 + 450 + 3300 + 700 + 600 + 1600 \\
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&= \boxed{6850}
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\end{align*}
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\]
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---
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## ✅ Final Answer
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The minimum transportation cost, using the Minimum Cost Method, is:
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\[
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\boxed{Z = 6850}
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\]
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---
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## 🖋️ Note
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The Minimum Cost Method provides a feasible initial solution but does not guarantee optimality.
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To verify optimality, it is recommended to apply methods such as:
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- **MODI Method (Modified Distribution Method)**
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- **Stepping Stone Method**
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```
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