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Alright, let's work through this step-by-step. --- ## 1. Understanding the given table and extracting allocations We have the transportation problem: Cost matrix: \[ C = \begin{bmatrix} 16 & 20 & 12 \\ 14 & 8 & 18 \\ 26 & 24 & 16 \end{bmatrix} \] Supply: \( a_1 = 200, \; a_2 = 160, \; a_3 = 90 \) Demand: \( b_1 = 180, \; b_2 = 120, \; b_3 = 150 \) Allocated cells from table: - \( F_1 \to W_1 \) : \( 180 \) units - \( F_1 \to W_3 \) : \( 20 \) units - \( F_2 \to W_2 \) : \( 120 \) units - \( F_2 \to W_3 \) : \( 40 \) units - \( F_3 \to W_3 \) : \( 90 \) units Check supply: F1: 180 + 20 = 200 ✅ F2: 120 + 40 = 160 ✅ F3: 90 ✅ Check demand: W1: 180 ✅ W2: 120 ✅ W3: 20 + 40 + 90 = 150 ✅ Number of allocated cells = \( 5 \) \( m + n - 1 = 3 + 3 - 1 = 5 \) ✅ (non-degenerate) --- ## 2. Step 1 – Finding \(u_i\) and \(v_j\) (MODI / UV method) Let \( u_1 = 0 \) (reference). We use \( u_i + v_j = c_{ij} \) for allocated cells. From allocated cells: 1. \( F1W1 \): \( u_1 + v_1 = 16 \Rightarrow 0 + v_1 = 16 \Rightarrow v_1 = 16 \) 2. \( F1W3 \): \( u_1 + v_3 = 12 \Rightarrow 0 ____ ______ _______ _______ ______ ________ ________ ______.
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