Word Count : 754

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Alright — let’s break this down systematically. --- ## Step 1: Understanding the table and data The problem is a transportation problem with: Cost matrix \(C\): \[ \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix} \] Supply \(a_i\): \( [10, 5, 8] \) Demand \(b_j\): \( [6, 7, 4] \) Check total supply vs total demand: \[ \text{Total supply} = 10 + 5 + 8 = 23 \] \[ \text{Total demand} = 6 + 7 + 4 = 17 \] Supply > Demand by \( 6 \) units → unbalanced, need a dummy demand column. --- ## Step 2: Add a dummy destination We add a fourth column (Dummy) with: - Demand = \( 23 - 17 = 6 \) - Cost = \( 0 \) in dummy column. New cost matrix: \[ \begin{bmatrix} 1 & 2 & 3 & 0 \\ 4 & 5 & 6 & 0 \\ 7 & 8 & 9 & 0 \end{bmatrix} \] Supply: \( [10, 5, 8] \) Demand: \( [6, 7, 4, 6] \) Now totals match: \( 23 _________ ______ _______ ______ _________ _______ __________ ________.
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