This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Based on the image provided, the transportation problem has the following details: Costs (C_ij): Row 1: 4, 1, 2, 6 Row 2: 8, 8, 3, 2 Row 3: 5, 9, 8, 7 Demand (D_j): D_1 = 1300 D_2 = 1000 D_3 = 1000 D_4 = 900 Total Demand = 1300 + 1000 + 1000 + 900 = 4200 Supply (S_i): S_1 = 1600 S_2 = 1200 S_3 = 1400 Total Supply = 1600 + 1200 + 1400 = 4200 Since Total Supply = Total Demand, the problem is balanced. The transportation table is: |c|c|c|c|c|c| From/To & D_1 (1300) & D_2 (1000) & D_3 (1000) & D_4 (900) & Supply (S_i) \\ S_1 & 4 & 1 & 2 & 6 & 1600 \\ S_2 & 8 & 8 & 3 & 2 & 1200 \\ S_3 & 5 & 9 & 8 & 7 & 1400 \\ Demand (D_j) & 1300 & 1000 & 1000 & 900 & 4200 \\ We will now apply the Vogel's Approximation Method (VAM). Step 1: Calculate initial penalties and make the first allocation. Calculate the difference between the two lowest costs for each row and column. Row Penalties: S_1: (2-1)=1, S_2: (3-2)=1, S_3: (7-5)=2 Column Penalties: D_1: (5-4)=1, D_2: (8-1)=7, D_3: (3-2)=1, D_4: (6-2)=4 The highest penalty is 7 (Column D_2). The minimum cost in D_2 is C_12=1. Allocate (S_1, D_2) = (1600, 1000) = 1000 units to cell (S_1, D_2). Update: S_1 = 1600 - 1000 = 600, D_2 = 1000 - 1000 = 0. Column D_2 is satisfied. Step 2: Recalculate penalties and make the second allocation. Row Penalties: S_1: (4-2)=2, S_2: (3-2)=1, S_3: (7-5)=2 Column Penalties: D_1: (5-4)=1, D_3: (3-2)=1, D_4: (6-2)=4 The highest penalty is 4 (Column D_4). The minimum cost in D_4 is C_24=2. Allocate (S_2, D_4) = (1200, 900) = 900 units to cell (S_2, D_4). Update: S_2 = 1200 - 900 = 300, D_4 = 900 - 900 = 0. Column D_4 is satisfied. Step 3: Recalculate penalties and make the third allocation. Row Penalties: S_1: (4-2)=2, S_2: (8-3)=5, S_3: (8-5)=3 Column Penalties: D_1: (5-4)=1, D_3: (3-2)=1 The highest penalty is 5 (Row S_2). The minimum cost in S_2 is C_23=3. Allocate (S_2, D_3) = (300, 1000) = 300 units to cell (S_2, D_3). Update: S_2 = 300 - 300 = 0, D_3 = 1000 - 300 = 700. Row S_2 is satisfied. Step 4: Recalculate penalties and make the fourth allocation. Row Penalties: S_1: (4-2)=2, S_3: (8-5)=3 Column Penalties: D_1: (5-4)=1, D_3: (8-2)=6 The highest penalty is 6 (Column D_3). The minimum cost in D_3 is C_13=2. Allocate (S_1, D_3) = (600, 700) = 600 units to cell (S_1, D_3). Update: S_1 = 600 - 600 = 0, D_3 = 700 - 600 = 100. Row S_1 is satisfied. Step 5: Make the final allocations. Only row S_3 and columns D_1, D_3 remain. Allocate (S_3, D_1) = (1400, 1300) = 1300 units to cell (S_3, D_1). Update: S_3 = 1400 - 1300 = 100, D_1 = 1300 - 1300 = 0. Allocate (S_3, D_3) = (100, 100) = 100 units to cell (S_3, D_3). Update: S_3 = 100 - 100 = 0, D_3 = 100 - 100 = 0. All supply and demand are met. Step 6: Calculate the total transportation cost. The allocations are: X_12 = 1000 units at cost C_12=1 X_24 = 900 units at cost C_24=2 X_23 = 300 units at cost C_23=3 X_13 = 600 units at cost C_13=2 X_31 = 1300 units at cost C_31=5 X_33 = 100 units at cost C_33=8 Total Cost = (1000 × 1) + (900 × 2) + (300 × 3) + (600 × 2) + (1300 × 5) + (100 × 8) Total Cost = 1000 + 1800 + 900 + 1200 + 6500 + 800 Total Cost = 12200 The optimum cost element of transportation using VAM is 12200. That's 2 down. 3 left today — send the next one.