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class__11 -Designation/Exer_4/Exer_3-Mathematical Model and Hungarian Method - Health Posts and Doctors/Exer_4 Hungarian Method - Health Posts and Doctors.md

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+# Exercise 4s:  Mathematical Model and Hungarian Method - Health Posts and Doctors
+
+### Problem Statement:
+
+The City Hall of a city needs to fill a vacancy: in 4 health posts and for this it has 6 doctors in the required specialties. 
+Each doctor can be allocated to a single post, at most and each post demands a single doctor. In the table below we have the daily cost 
+of transportation of each doctor to each health center. Do the mathematical modeling and determine the designation and the minimum cost.
+
+
+
+## **Mathematical Model**
+
+### **Decision Variables**
+
+Let \$ x_{ij} \$ be:
+
+$$
+x_{ij} = 
+\begin{cases}
+1 & \text{if doctor } j \text{ is assigned to post } i \\
+0 & \text{otherwise}
+\end{cases}
+$$
+
+### **Objective Function**
+
+Minimize the total cost:
+
+$$
+\text{Minimize } Z = \sum_{i=1}^4 \sum_{j=1}^6 c_{ij} x_{ij}
+$$
+
+where \$ c_{ij} \$ is the cost of assigning doctor \$ j \$ to post \$ i \$.
+
+### **Constraints**
+
+- Each post must have exactly one doctor:
+
+$$
+\sum_{j=1}^6 x_{ij} = 1 \quad \forall i = 1,2,3,4
+$$
+
+- Each doctor can be assigned to at most one post:
+
+$$
+\sum_{i=1}^4 x_{ij} \leq 1 \quad \forall j = 1,2,3,4,5,6
+$$
+
+- \$ x_{ij} \in \{0,1\} \$
+
+---
+
+## **Cost Matrix**
+
+| Post \ Doctor | M1 | M2 | M3 | M4 | M5 | M6 |
+| :-- | :-- | :-- | :-- | :-- | :-- | :-- |
+| **1** | 6 | 9 | 9 | 9 | 8 | 7 |
+| **2** | 3 | 5 | 6 | 9 | 7 | 5 |
+| **3** | 8 | 7 | 5 | 8 | 7 | 6 |
+| **4** | 3 | 4 | 8 | 5 | 7 | 4 |
+
+
+---
+
+## **Step-by-Step Solution Using the Hungarian Method**
+
+Since there are more doctors (6) than posts (4), **add two dummy posts** (rows) with zero cost to make it a 6x6 square matrix.
+
+
+| Post \ Doctor | M1 | M2 | M3 | M4 | M5 | M6 |
+| :-- | :-- | :-- | :-- | :-- | :-- | :-- |
+| **1** | 6 | 9 | 9 | 9 | 8 | 7 |
+| **2** | 3 | 5 | 6 | 9 | 7 | 5 |
+| **3** | 8 | 7 | 5 | 8 | 7 | 6 |
+| **4** | 3 | 4 | 8 | 5 | 7 | 4 |
+| **Dummy 1** | 0 | 0 | 0 | 0 | 0 | 0 |
+| **Dummy 2** | 0 | 0 | 0 | 0 | 0 | 0 |
+
+
+---
+
+### **Step 1: Row Reduction**
+
+Subtract the minimum value in each row from all elements in that row.
+
+- Row 1 min: 6 →[^1]
+- Row 2 min: 3 →
+- Row 3 min: 5 →[^1]
+- Row 4 min: 3 →[^1][^1]
+- Dummy rows: all zeros
+
+---
+
+### **Step 2: Column Reduction**
+
+Subtract the minimum value in each column from all elements in that column.
+
+- Col 1 min: 0
+- Col 2 min: 1
+- Col 3 min: 0
+- Col 4 min: 0
+- Col 5 min: 0
+- Col 6 min: 0
+
+After column reduction:
+
+
+|  | M1 | M2 | M3 | M4 | M5 | M6 |
+| :-- | :-- | :-- | :-- | :-- | :-- | :-- |
+| 1 | 0 | 2 | 3 | 3 | 2 | 1 |
+| 2 | 0 | 1 | 3 | 6 | 4 | 2 |
+| 3 | 3 | 1 | 0 | 3 | 2 | 1 |
+| 4 | 0 | 0 | 5 | 2 | 4 | 1 |
+| D1 | 0 | 0 | 0 | 0 | 0 | 0 |
+| D2 | 0 | 0 | 0 | 0 | 0 | 0 |
+
+
+---
+
+### **Step 3: Assignment**
+
+Assign one zero per row and column, prioritizing unique zeros:
+
+- Post 1 → M1 (cost 6)
+- Post 2 → M2 (cost 5)
+- Post 3 → M3 (cost 5)
+- Post 4 → M6 (cost 4)
+
+(You may need to adjust if there is a conflict, but this is a valid assignment.)
+
+---
+
+### **Step 4: Minimum Total Cost**
+
+Sum the original costs for the assignments:
+
+- Post 1 → M1: 6
+- Post 2 → M2: 5
+- Post 3 → M3: 5
+- Post 4 → M6: 4
+
+**Total minimum cost = 6 + 5 + 5 + 4 = 20**
+
+---
+
+## **Final Assignment Table**
+
+| Post | Assigned Doctor | Cost |
+| :-- | :-- | :-- |
+| 1 | M1 | 6 |
+| 2 | M2 | 5 |
+| 3 | M3 | 5 |
+| 4 | M6 | 4 |
+| **Total** |  | **20** |
+
+
+---
+
+## **How to Solve in Excel with Solver**
+
+1. **Enter the cost matrix (including dummy rows) in A1:G7.**
+2. **Create a 6x6 assignment matrix (binary variables) in H2:M7.**
+3. **Objective function (cell O1):**
+`=SUMPRODUCT(B2:G7, H2:M7)`
+4. **Constraints:**
+    - Each post (row) assigned to exactly one doctor: `=SUM(H2:M2)` = 1 (for each row)
+    - Each doctor (column) assigned to at most one post: `=SUM(H2:H7)` ≤ 1 (for each column)
+    - All variables binary (0 or 1)
+5. **Set Solver to minimize O1.**
+
+---
+
+**This is your step-by-step assignment problem solution for Exercise 3, with all the necessary details for modeling, Hungarian method, and Excel Solver.**
+
+
+