From 7cfacd8c575601b5373810c31ecfdf08e1a27ab7 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Fabiana=20=20=E2=9A=A1=EF=B8=8F=20Campanari?= <113218619+FabianaCampanari@users.noreply.github.com> Date: Sun, 3 Aug 2025 12:40:37 -0300 Subject: [PATCH 01/10] Update README.md MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Signed-off-by: Fabiana ⚡️ Campanari <113218619+FabianaCampanari@users.noreply.github.com> --- README.md | 15 ++++++++++----- 1 file changed, 10 insertions(+), 5 deletions(-) diff --git a/README.md b/README.md index 24756f6..04f5950 100644 --- a/README.md +++ b/README.md @@ -2069,8 +2069,6 @@ Entering variable: $x_{13}$. #### [4.3](). Update Basic Variables -🏄🏄🏄🏄🏄 -
| Variable | Adjustment | New Value | @@ -2078,17 +2076,24 @@ Entering variable: $x_{13}$. | $x_{13}$ | $+100$ | $100$ | | $x_{33}$ | $-100$ | $50$ | | $x_{31}$ | $+100$ | $110$ | -| $ x_{11} $ | $-100$ | $0$ | +| $x_{11}$ | $-100$ | $0$ |
-**New Basic Variables:** -- $ x_{13} = 100 $ +[**New Basic Variables**](): + +- $x_{13} = 100$ - $ x_{21} = 10 $ - $ x_{22} = 130 $ - $ x_{31} = 110 $ - $ x_{33} = 50 $ + +
+ +🏄🏄🏄🏄🏄 + + #### [4.4](). Verify Feasibility - **Supplies**: From e5b4a0918e71ce5588576ebceb436fa5408a4d15 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Fabiana=20=20=E2=9A=A1=EF=B8=8F=20Campanari?= <113218619+FabianaCampanari@users.noreply.github.com> Date: Sun, 3 Aug 2025 12:42:50 -0300 Subject: [PATCH 02/10] Update README.md MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Signed-off-by: Fabiana ⚡️ Campanari <113218619+FabianaCampanari@users.noreply.github.com> --- README.md | 14 ++++++-------- 1 file changed, 6 insertions(+), 8 deletions(-) diff --git a/README.md b/README.md index 04f5950..ece9f3f 100644 --- a/README.md +++ b/README.md @@ -2083,19 +2083,17 @@ Entering variable: $x_{13}$. [**New Basic Variables**](): - $x_{13} = 100$ -- $ x_{21} = 10 $ -- $ x_{22} = 130 $ -- $ x_{31} = 110 $ -- $ x_{33} = 50 $ - +- $x_{21} = 10$ +- $x_{22} = 130$ +- $x_{31} = 110$ +- $x_{33} = 50$
-🏄🏄🏄🏄🏄 - - #### [4.4](). Verify Feasibility +🏄🏄🏄🏄🏄 + - **Supplies**: - Supplier 1: $ 100 $ ✔️ - Supplier 2: $ 10 + 130 = 140 $ ✔️ From 2aa3378ecc7f797cd01ae5830f110c6a39fdec67 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Fabiana=20=20=E2=9A=A1=EF=B8=8F=20Campanari?= <113218619+FabianaCampanari@users.noreply.github.com> Date: Sun, 3 Aug 2025 12:48:49 -0300 Subject: [PATCH 03/10] Update README.md MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Signed-off-by: Fabiana ⚡️ Campanari <113218619+FabianaCampanari@users.noreply.github.com> --- README.md | 14 ++++++++++---- 1 file changed, 10 insertions(+), 4 deletions(-) diff --git a/README.md b/README.md index ece9f3f..13b1f6a 100644 --- a/README.md +++ b/README.md @@ -2124,19 +2124,25 @@ Recalculating reduced costs confirms all $ \bar{c}_{ij} \geq 0 $. **Optimal solu
-## Final Solution +## [Step 5]():Final Solution + +🏄🏄🏄🏄🏄 + +
+ | Variable | Value | |------------|-------| -| $ x_{13} $ | 100 | +| $x_{13}$ | 100 | | $ x_{21} $ | 10 | | $ x_{22} $ | 130 | | $ x_{31} $ | 110 | | $ x_{33} $ | 50 | +
-**Total Cost:** $\boxed{10460}$. +[**Total Cost:**](): $\boxed{10460}$. -This is the optimal solution with all reduced costs non-negative. +***This is the optimal solution with all reduced costs non-negative.***
From 09f8e624e84c3d6a7189864d6e262ed55fac385d Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Fabiana=20=20=E2=9A=A1=EF=B8=8F=20Campanari?= <113218619+FabianaCampanari@users.noreply.github.com> Date: Sun, 3 Aug 2025 12:59:22 -0300 Subject: [PATCH 04/10] Update README.md MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Signed-off-by: Fabiana ⚡️ Campanari <113218619+FabianaCampanari@users.noreply.github.com> --- README.md | 12 +++++------- 1 file changed, 5 insertions(+), 7 deletions(-) diff --git a/README.md b/README.md index 13b1f6a..cd34efe 100644 --- a/README.md +++ b/README.md @@ -2126,17 +2126,15 @@ Recalculating reduced costs confirms all $ \bar{c}_{ij} \geq 0 $. **Optimal solu ## [Step 5]():Final Solution -🏄🏄🏄🏄🏄 -
| Variable | Value | |------------|-------| -| $x_{13}$ | 100 | -| $ x_{21} $ | 10 | -| $ x_{22} $ | 130 | -| $ x_{31} $ | 110 | -| $ x_{33} $ | 50 | +| $x_{13}$ | 100 | +| $x_{21}$ | 10 | +| $x_{22}$ | 130 | +| $x_{31}$ | 110 | +| $x_{33}$ | 50 |
From 93c908ef74c781726838615d1ec7db82f442fda6 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Fabiana=20=20=E2=9A=A1=EF=B8=8F=20Campanari?= <113218619+FabianaCampanari@users.noreply.github.com> Date: Sun, 3 Aug 2025 13:02:35 -0300 Subject: [PATCH 05/10] Update README.md MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Signed-off-by: Fabiana ⚡️ Campanari <113218619+FabianaCampanari@users.noreply.github.com> --- README.md | 12 ++++++------ 1 file changed, 6 insertions(+), 6 deletions(-) diff --git a/README.md b/README.md index cd34efe..a9a8801 100644 --- a/README.md +++ b/README.md @@ -2094,13 +2094,13 @@ Entering variable: $x_{13}$. 🏄🏄🏄🏄🏄 -- **Supplies**: - - Supplier 1: $ 100 $ ✔️ - - Supplier 2: $ 10 + 130 = 140 $ ✔️ - - Supplier 3: $ 110 + 50 = 160 $ ✔️ +- [**Supplies**](): + - Supplier 1: $100$ ✔️ + - Supplier 2: $10 + 130 = 140$ ✔️ + - Supplier 3: $110 + 50 = 160$ ✔️ -- **Demands**: - - Consumer 1: $ 10 + 110 = 120 $ ✔️ +- [**Demands**](): + - Consumer 1: $10 + 110 = 120$ ✔️ - Consumer 2: $ 130 $ ✔️ - Consumer 3: $ 100 + 50 = 150 $ ✔️ From daf55fc1a64518931eb0e8c78e566b2437f24659 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Fabiana=20=20=E2=9A=A1=EF=B8=8F=20Campanari?= <113218619+FabianaCampanari@users.noreply.github.com> Date: Sun, 3 Aug 2025 13:05:52 -0300 Subject: [PATCH 06/10] Update README.md MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Signed-off-by: Fabiana ⚡️ Campanari <113218619+FabianaCampanari@users.noreply.github.com> --- README.md | 13 +++++++------ 1 file changed, 7 insertions(+), 6 deletions(-) diff --git a/README.md b/README.md index a9a8801..e1fa437 100644 --- a/README.md +++ b/README.md @@ -2095,15 +2095,16 @@ Entering variable: $x_{13}$. 🏄🏄🏄🏄🏄 - [**Supplies**](): - - Supplier 1: $100$ ✔️ - - Supplier 2: $10 + 130 = 140$ ✔️ - - Supplier 3: $110 + 50 = 160$ ✔️ + - [Supplier 1](): $100$ ✔️ + - [Supplier 2](): $10 + 130 = 140$ ✔️ + - [Supplier 3](): $110 + 50 = 160$ ✔️ - [**Demands**](): - - Consumer 1: $10 + 110 = 120$ ✔️ - - Consumer 2: $ 130 $ ✔️ - - Consumer 3: $ 100 + 50 = 150 $ ✔️ + - [Consumer 1](): $10 + 110 = 120$ ✔️ + - [Consumer 2](): $130$ ✔️ + - [Consumer 3](): $100 + 50 = 150$ ✔️ +
#### [4.5](). Calculate Final Total Cost From 7fc45de8f4a30fcb7a0005a442c4b2e78d122f07 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Fabiana=20=20=E2=9A=A1=EF=B8=8F=20Campanari?= <113218619+FabianaCampanari@users.noreply.github.com> Date: Sun, 3 Aug 2025 13:10:16 -0300 Subject: [PATCH 07/10] Update README.md MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Signed-off-by: Fabiana ⚡️ Campanari <113218619+FabianaCampanari@users.noreply.github.com> --- README.md | 10 +++++++--- 1 file changed, 7 insertions(+), 3 deletions(-) diff --git a/README.md b/README.md index e1fa437..3e73fe7 100644 --- a/README.md +++ b/README.md @@ -2092,13 +2092,15 @@ Entering variable: $x_{13}$. #### [4.4](). Verify Feasibility -🏄🏄🏄🏄🏄 +
- [**Supplies**](): - [Supplier 1](): $100$ ✔️ - [Supplier 2](): $10 + 130 = 140$ ✔️ - [Supplier 3](): $110 + 50 = 160$ ✔️ +
+ - [**Demands**](): - [Consumer 1](): $10 + 110 = 120$ ✔️ - [Consumer 2](): $130$ ✔️ @@ -2119,9 +2121,11 @@ $$
-### [4.6](). Final Optimality Check +🏄🏄🏄🏄🏄 + +#### [4.6](). Final Optimality Check -Recalculating reduced costs confirms all $ \bar{c}_{ij} \geq 0 $. **Optimal solution reached**. +Recalculating reduced costs confirms all $\bar{c}_{ij} \geq 0$. [**Optimal solution reached**]().
From 2b5673a5535c198e903f5cde7df22e62b6278de9 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Fabiana=20=20=E2=9A=A1=EF=B8=8F=20Campanari?= <113218619+FabianaCampanari@users.noreply.github.com> Date: Sun, 3 Aug 2025 13:14:06 -0300 Subject: [PATCH 08/10] Update README.md MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Signed-off-by: Fabiana ⚡️ Campanari <113218619+FabianaCampanari@users.noreply.github.com> --- README.md | 4 +--- 1 file changed, 1 insertion(+), 3 deletions(-) diff --git a/README.md b/README.md index 3e73fe7..ba5dc1c 100644 --- a/README.md +++ b/README.md @@ -2121,15 +2121,13 @@ $$
-🏄🏄🏄🏄🏄 - #### [4.6](). Final Optimality Check Recalculating reduced costs confirms all $\bar{c}_{ij} \geq 0$. [**Optimal solution reached**]().
-## [Step 5]():Final Solution +## [Step 5](): Final Solution
From c28c2fc5b1758197ad49ef385da0e8b7bb1eacba Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Fabiana=20=20=E2=9A=A1=EF=B8=8F=20Campanari?= <113218619+FabianaCampanari@users.noreply.github.com> Date: Sun, 3 Aug 2025 13:19:41 -0300 Subject: [PATCH 09/10] Update README.md MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Signed-off-by: Fabiana ⚡️ Campanari <113218619+FabianaCampanari@users.noreply.github.com> --- README.md | 17 ++++++++++------- 1 file changed, 10 insertions(+), 7 deletions(-) diff --git a/README.md b/README.md index ba5dc1c..0d02558 100644 --- a/README.md +++ b/README.md @@ -2161,7 +2161,7 @@ Recalculating reduced costs confirms all $\bar{c}_{ij} \geq 0$. [**Optimal solut ### Theoretical Explanation -The **Assignment Problem** aims to allocate *n* tasks to *n* agents (machines, workers) at minimum cost, ensuring each task and agent is assigned exactly once.** +The [**Assignment Problem**]() aims to allocate *n* tasks to *n* agents (machines, workers) at minimum cost, ensuring each task and agent is assigned exactly once. ### [Problem Statement](): @@ -2179,20 +2179,23 @@ The **Assignment Problem** aims to allocate *n* tasks to *n* agents (machines, w
+🏄🏄🏄🏄🏄 ## 1. [Hungarian Method]() (Step by Step): -### [**Step 1](): Subtract Row Minimums** +### [Step 1](): Subtract Row Minimums -#### Subtract the minimum value in each row from all elements in that row. +#### [Subtract the minimum value in each row from all elements in that row](). -- Row 1 min: 2 → [0, 2, 1] -- Row 2 min: 1 → [0, 2, 1] -- Row 3 min: 2 → [3, 0, 2] +
+ +- [Row 1 min: 2[() → [0, 2, 1] +- [Row 2 min: 1[() → [0, 2, 1] +- [Row 3 min: 2[() → [3, 0, 2]
-#### [**Matrix after row subtraction:**]() +#### [Matrix after row subtraction](): | | M1 | M2 | M3 | |---------|----|----|----| From 6db4a71d44651ade13265c6a332663d9fbff64e3 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Fabiana=20=20=E2=9A=A1=EF=B8=8F=20Campanari?= <113218619+FabianaCampanari@users.noreply.github.com> Date: Sun, 3 Aug 2025 13:23:17 -0300 Subject: [PATCH 10/10] Update README.md MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Signed-off-by: Fabiana ⚡️ Campanari <113218619+FabianaCampanari@users.noreply.github.com> --- README.md | 21 +++++++++++++++------ 1 file changed, 15 insertions(+), 6 deletions(-) diff --git a/README.md b/README.md index 0d02558..8eab8e2 100644 --- a/README.md +++ b/README.md @@ -2189,14 +2189,16 @@ The [**Assignment Problem**]() aims to allocate *n* tasks to *n* agents (machine
-- [Row 1 min: 2[() → [0, 2, 1] -- [Row 2 min: 1[() → [0, 2, 1] -- [Row 3 min: 2[() → [3, 0, 2] +- [Row 1 min: 2]() → [0, 2, 1] +- [Row 2 min: 1]() → [0, 2, 1] +- [Row 3 min: 2]() → [3, 0, 2]
#### [Matrix after row subtraction](): +
+ | | M1 | M2 | M3 | |---------|----|----|----| | Task 1 | 0 | 2 | 1 | @@ -2206,16 +2208,23 @@ The [**Assignment Problem**]() aims to allocate *n* tasks to *n* agents (machine
-### [**Step 2](): Subtract Column Minimums** +### [Step 2](): Subtract Column Minimums -### Problem Recap +### [Problem Recap](): + +
- **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 +
+ +### [Cost Table](): + +
+ | | Machine 1 | Machine 2 | Machine 3 | |---------|-----------|-----------|-----------|