Merge pull request #550 from Quantum-Software-Development/FabianaCampanari-patch-1

Add files via upload

Author: FabianaCampanari

class__10-Northwest Corner Method/2-excel-Northwest Corner Method .md.md

@@ -0,0 +1,212 @@
+# Transportation Problem Solution in Excel (Markdown + LaTeX for README.md)
+
+This guide explains, step by step, how to solve the given transportation problem in Excel using the Solver add-in. It includes all necessary formulas and tables in Markdown and LaTeX, with ready-to-copy Excel formulas.
+
+
+**You can now copy the formulas above and use them directly in your Excel sheet. The step-by-step process will allow you to model and solve any similar transportation problem using Excel Solver.**
+
+---
+
+## **Problem Statement**
+
+Given the following transportation cost matrix, supply, and demand:
+
+
+|  | Consumer 1 | Consumer 2 | Consumer 3 | Supply |
+| :-- | :--: | :--: | :--: | :--: |
+| Supplier 1 | 12 | 22 | 30 | 100 |
+| Supplier 2 | 18 | 24 | 32 | 140 |
+| Supplier 3 | 22 | 15 | 34 | 160 |
+| **Demand** | 120 | 130 | 150 |  |
+
+**Objective:**
+Minimize the total transportation cost.
+
+---
+
+## **Mathematical Formulation (LaTeX)**
+
+Let \$ x_{ij} \$ be the number of units shipped from supplier \$ i \$ to consumer \$ j \$.
+
+**Minimize:**
+
+$$
+Z = \sum_{i=1}^{3} \sum_{j=1}^{3} c_{ij} x_{ij}
+$$
+
+**Subject to:**
+
+$$
+\sum_{j=1}^{3} x_{ij} = \text{Supply}_i \quad \forall i = 1,2,3
+$$
+
+$$
+\sum_{i=1}^{3} x_{ij} = \text{Demand}_j \quad \forall j = 1,2,3
+$$
+
+$$
+x_{ij} \geq 0 \quad \forall i, j
+$$
+
+---
+
+## **Step-by-Step Solution in Excel**
+
+### **1. Data Layout**
+
+|  | B | C | D | E | F |
+| :-- | :-- | :-- | :-- | :-- | :-- |
+|  | Cons1 | Cons2 | Cons3 | Supply |  |
+| Sup1 | 12 | 22 | 30 | 100 |  |
+| Sup2 | 18 | 24 | 32 | 140 |  |
+| Sup3 | 22 | 15 | 34 | 160 |  |
+| Demand | 120 | 130 | 150 |  |  |
+
+### **2. Decision Variables Table**
+
+Below the cost table, create a table for the shipment variables (\$ x_{ij} \$), e.g., in cells B7:D9.
+
+
+|  | Cons1 | Cons2 | Cons3 |
+| :-- | :-- | :-- | :-- |
+| Sup1 | x11 | x12 | x13 |
+| Sup2 | x21 | x22 | x23 |
+| Sup3 | x31 | x32 | x33 |
+
+**In Excel:**
+Enter `0` in each cell as the initial value.
+
+### **3. Total Cost Formula**
+
+Below the variable table, calculate the total cost using `SUMPRODUCT`:
+
+```excel
+=SUMPRODUCT(B2:D4, B7:D9)
+```
+
+- `B2:D4` is the cost matrix.
+- `B7:D9` is the variable matrix.
+
+
+### **4. Supply Constraints**
+
+For each supplier, sum the variables in the row and set equal to the supply.
+
+
+|  | Formula (Excel) |
+| :-- | :-- |
+| Sup1 | =SUM(B7:D7) |
+| Sup2 | =SUM(B8:D8) |
+| Sup3 | =SUM(B9:D9) |
+
+### **5. Demand Constraints**
+
+For each consumer, sum the variables in the column and set equal to the demand.
+
+
+|  | Formula (Excel) |
+| :-- | :-- |
+| Cons1 | =SUM(B7:B9) |
+| Cons2 | =SUM(C7:C9) |
+| Cons3 | =SUM(D7:D9) |
+
+### **6. Setting Up Solver**
+
+- **Set Objective:** The total cost cell (e.g., `B12`) - set to Min.
+- **By Changing Variable Cells:** The shipment variables (`B7:D9`).
+- **Subject to Constraints:**
+    - Each row sum equals the corresponding supply.
+    - Each column sum equals the corresponding demand.
+    - All variables \$ \geq 0 \$.
+
+
+#### **Solver Constraints (in Excel)**
+
+- `SUM(B7:D7) = F2` (Supply for Sup1)
+- `SUM(B8:D8) = F3` (Supply for Sup2)
+- `SUM(B9:D9) = F4` (Supply for Sup3)
+- `SUM(B7:B9) = B5` (Demand for Cons1)
+- `SUM(C7:C9) = C5` (Demand for Cons2)
+- `SUM(D7:D9) = D5` (Demand for Cons3)
+- `B7:D9 >= 0`
+
+
+### **7. Solving and Results**
+
+After running Solver, you will get the optimal shipment plan. For this problem, the optimal solution is:
+
+
+|  | Cons1 | Cons2 | Cons3 |
+| :-- | :-- | :-- | :-- |
+| Sup1 | 100 | 0 | 0 |
+| Sup2 | 10 | 130 | 0 |
+| Sup3 | 10 | 0 | 150 |
+
+**Total Cost:**
+
+```excel
+=SUMPRODUCT(B2:D4, B7:D9)
+```
+
+Result: **9820**
+
+---
+
+## **Excel Formulas (Copy-Paste Ready)**
+
+Assuming:
+
+- Cost matrix in `B2:D4`
+- Variables in `B7:D9`
+- Supplies in `F2:F4`
+- Demands in `B5:D5`
+
+**Total Cost:**
+
+```excel
+=SUMPRODUCT(B2:D4, B7:D9)
+```
+
+**Supply Constraints:**
+
+```excel
+=SUM(B7:D7)    // For Sup1
+=SUM(B8:D8)    // For Sup2
+=SUM(B9:D9)    // For Sup3
+```
+
+**Demand Constraints:**
+
+```excel
+=SUM(B7:B9)    // For Cons1
+=SUM(C7:C9)    // For Cons2
+=SUM(D7:D9)    // For Cons3
+```
+
+
+---
+
+## **References**
+
+For more details and visuals, see:
+
+- [Excel Easy: Transportation Problem in Excel][^2]
+- [SCM Globe: How to Solve Transportation Problems Using Excel Solver][^3]
+- [YouTube: Solving LP Transportation Problem | Excel Solver][^1]
+
+---
+
+## **Summary Table (Markdown)**
+
+|  | Cons1 | Cons2 | Cons3 | Supply |
+| :-- | :-- | :-- | :-- | :-- |
+| Sup1 | 100 | 0 | 0 | 100 |
+| Sup2 | 10 | 130 | 0 | 140 |
+| Sup3 | 10 | 0 | 150 | 160 |
+| Demand | 120 | 130 | 150 |  |
+
+**Total Cost:** 9820
+
+---
+
+