From f3b913745ec6afb2b99bfe8f3d5c8639a47f36bd Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Fabiana=20=F0=9F=9A=80=20=20Campanari?=
<113218619+FabianaCampanari@users.noreply.github.com>
Date: Sat, 10 May 2025 18:14:02 -0300
Subject: [PATCH] Update README.md
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Content-Type: text/plain; charset=UTF-8
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Signed-off-by: Fabiana 🚀 Campanari <113218619+FabianaCampanari@users.noreply.github.com>
---
README.md | 152 ++++++++++++++++++++++++++++++++++--------------------
1 file changed, 96 insertions(+), 56 deletions(-)
diff --git a/README.md b/README.md
index b737993..2cddbb1 100644
--- a/README.md
+++ b/README.md
@@ -1527,44 +1527,26 @@ The **Assignment Problem** aims to allocate *n* tasks to *n* agents (machines, w
### [**Step 2](): Subtract Column Minimums**
-#### Subtract the minimum value in each column from all elements in that column]().
+### Problem Recap
-- Col 1 min: 0 → [0, 0, 3]
-- Col 2 min: 0 → [2, 2, 0]
-- Col 3 min: 1 → [0, 0, 1]
-
-
-
-#### [**Matrix after column subtraction:**]()
-
-| | M1 | M2 | M3 |
-|---------|----|----|----|
-| Task 1 | 0 | 2 | 0 |
-| Task 2 | 0 | 2 | 0 |
-| Task 3 | 3 | 0 | 1 |
-
-
-
-### [**Step 3](): Assignment (Cover Zeros)**
-
-- Cover all zeros using the minimum number of lines (rows or columns).
-- Assign tasks to machines where possible (one zero per row/column).
-
-**Optimal Assignment:**
-- Task 1 → Machine 1 (cost 2)
-- Task 2 → Machine 3 (cost 2)
-- Task 3 → Machine 2 (cost 2)
+- **3 tasks** must be assigned to **3 machines**.
+- Each task can be done by any machine, but with different costs.
+- Each task must be assigned to exactly one machine, and each machine to exactly one task.
+- **Goal:** Minimize total assignment cost.
+### Cost Table
-### ***Total Minimum Cost = [2 + 2 + 2 = 6]()***
-
-
+| | Machine 1 | Machine 2 | Machine 3 |
+|---------|-----------|-----------|-----------|
+| Task 1 | 2 | 4 | 3 |
+| Task 2 | 1 | 3 | 2 |
+| Task 3 | 5 | 2 | 4 |
-## 2. Excel Solver Step-by-Step
+---
-### **A. Excel Table Setup**
+## Step 1: Set Up the Excel Spreadsheet
-#### 1. **Cost Table (A1:D4)**
+### 1. Enter the Cost Matrix
| | B | C | D |
|-----|------|------|------|
@@ -1573,7 +1555,9 @@ The **Assignment Problem** aims to allocate *n* tasks to *n* agents (machines, w
| T2 | 1 | 3 | 2 |
| T3 | 5 | 2 | 4 |
-#### 2. **Decision Variables Table (F1:I4)**
+- Place this table in cells **B2:D4**.
+
+### 2. Create the Decision Variable Table
| | G | H | I |
|-----|------|------|------|
@@ -1582,42 +1566,98 @@ The **Assignment Problem** aims to allocate *n* tasks to *n* agents (machines, w
| T2 | x21 | x22 | x23 |
| T3 | x31 | x32 | x33 |
-Each cell is 0 or 1 (to be filled by Solver).
+- Place this table in **G2:I4**.
+- These cells will be filled with 0 or 1 by the Solver (1 = assigned, 0 = not assigned).
-
+### 3. Calculate the Total Cost
-#### 3. **Objective Function (K2)**
+In cell **K2**, enter:
-
-```
+```bash
=SUMPRODUCT(B2:D4, G2:I4)
```
-#### 4. **Row Constraints (One task per machine)**
+This formula multiplies each assignment by its cost and sums the total.
-- J2: `=SUM(G2:I2)` (should be 1)
-- J3: `=SUM(G3:I3)` (should be 1)
-- J4: `=SUM(G4:I4)` (should be 1)
+### 4. Add Row and Column Sums for Constraints
-
+#### Row Sums (Each Task Assigned Once)
-#### 5. **Column Constraints (One machine per task)**
+- In **J2**: `=SUM(G2:I2)`
+- In **J3**: `=SUM(G3:I3)`
+- In **J4**: `=SUM(G4:I4)`
+
+#### Column Sums (Each Machine Assigned Once)
+
+- In **G5**: `=SUM(G2:G4)`
+- In **H5**: `=SUM(H2:H4)`
+- In **I5**: `=SUM(I2:I4)`
+
+---
+
+## Step 2: Configure Excel Solver
+
+1. **Go to**: Data > Solver
+2. **Set Objective**:
+ - Set **K2** (total cost) to **Minimize**.
+3. **By Changing Variable Cells**:
+ - Select **G2:I4**.
+4. **Add Constraints**:
+ - **J2:J4 = 1** (each task assigned once)
+ - **G5:I5 = 1** (each machine assigned once)
+ - **G2:I4 = binary** (only 0 or 1 allowed)
+5. **Choose Solving Method**:
+ - Use "Simplex LP" or "GRG Nonlinear" (either works for this size).
+6. **Click Solve**.
+
+---
+
+## Step 3: Solution Example
+
+After running Solver, you should get a solution like:
+
+| | M1 | M2 | M3 | Row Sum |
+|-----|----|----|----|---------|
+| T1 | 1 | 0 | 0 | 1 |
+| T2 | 0 | 0 | 1 | 1 |
+| T3 | 0 | 1 | 0 | 1 |
+|Col Sum| 1| 1 | 1 | |
+
+- **Task 1 → Machine 1** (cost 2)
+- **Task 2 → Machine 3** (cost 2)
+- **Task 3 → Machine 2** (cost 2)
+
+**Total minimum cost:** 6
+
+---
+
+## Excel Table and Formula Summary
+
+| | M1 | M2 | M3 | Row Sum |
+|-----|------|------|------|---------|
+| T1 | G2 | H2 | I2 | J2 |
+| T2 | G3 | H3 | I3 | J3 |
+| T3 | G4 | H4 | I4 | J4 |
+|Col Sum|G5 | H5 | I5 | |
+
+- **Total Cost:** `=SUMPRODUCT(B2:D4, G2:I4)`
+- **Row Sums:** `=SUM(G2:I2)`, etc.
+- **Column Sums:** `=SUM(G2:G4)`, etc.
+
+---
+
+## Result (in English)
+
+**The optimal assignment is:**
+- Task 1 to Machine 1 (cost 2)
+- Task 2 to Machine 3 (cost 2)
+- Task 3 to Machine 2 (cost 2)
+
+**Total minimum cost:** 6
-- G5: `=SUM(G2:G4)` (should be 1)
-- H5: `=SUM(H2:H4)` (should be 1)
-- I5: `=SUM(I2:I4)` (should be 1)
-
-### **B. Solver Configuration**
-- **Set Objective:** K2 (Minimize)
-- **By Changing Variables:** G2:I4
-- **Add Constraints:**
- - J2:J4 = 1
- - G5:I5 = 1
- - G2:I4 = binary
-