1. **Problem Statement:** We have three plants (Houston, Phoenix, Memphis) with supplies 850, 650, and 300 units respectively. Three distribution centers (Dallas, Atlanta, Denver) require 800, 600, and 200 units respectively. Shipping costs per unit are given for each plant to each center. 2. **Formulating the Transportation Tableau:** The initial tableau shows costs and supplies/demands: | From \ To | Dallas (800) | Atlanta (600) | Denver (200) | Supply | |-----------|--------------|---------------|--------------|--------| | Houston (850) | 8 | 12 | 10 | 850 | | Phoenix (650) | 10 | 14 | 9 | 650 | | Memphis (300) | 11 | 8 | 12 | 300 | | Demand | 800 | 600 | 200 | | 3. **Initial Basic Feasible Solution (using Northwest Corner Method):** - Ship 800 from Houston to Dallas (max demand 800, supply 850) - Remaining Houston supply: 850 - 800 = 50 - Ship 50 from Houston to Atlanta (remaining supply) - Atlanta demand left: 600 - 50 = 550 - Ship 550 from Phoenix to Atlanta - Phoenix supply left: 650 - 550 = 100 - Ship 100 from Phoenix to Denver - Denver demand left: 200 - 100 = 100 - Ship 100 from Memphis to Denver - Memphis supply left: 300 - 100 = 200 - Memphis still has 200 supply but no demand left, so ship 200 to Atlanta (Atlanta demand fulfilled now) 4. **Initial Transportation Tableau Shipments:** | From \ To | Dallas | Atlanta | Denver | Supply | |-----------|--------|---------|--------|--------| | Houston | 800 | 50 | 0 | 850 | | Phoenix | 0 | 550 | 100 | 650 | | Memphis | 0 | 200 | 100 | 300 | | Demand | 800 | 600 | 200 | | 5. **Calculate Total Cost for Initial Tableau:** $$\text{Cost} = 800 \times 8 + 50 \times 12 + 550 \times 14 + 100 \times 9 + 200 \times 8 + 100 \times 12$$ $$= 6400 + 600 + 7700 + 900 + 1600 + 1200 = 18400$$ 6. **Revised Tableaus and Optimization:** Using the stepping stone or MODI method, we check for improvements. - After optimization, the minimum cost solution is: | From \ To | Dallas | Atlanta | Denver | Supply | |-----------|--------|---------|--------|--------| | Houston | 800 | 0 | 50 | 850 | | Phoenix | 0 | 600 | 50 | 650 | | Memphis | 0 | 0 | 100 | 300 | 7. **Calculate Total Cost for Optimized Tableau:** $$\text{Cost} = 800 \times 8 + 0 \times 12 + 50 \times 10 + 0 \times 14 + 600 \times 14 + 50 \times 9 + 0 \times 11 + 0 \times 8 + 100 \times 12$$ $$= 6400 + 0 + 500 + 0 + 8400 + 450 + 0 + 0 + 1200 = 16950$$ 8. **Number of Tableaus Required:** The problem typically requires 2 tableaus: initial and optimized feasible solution. **Final answers:** 1. Total cost initial tableau = 18400 2. Total cost second tableau (optimized) = 16950 3. No third tableau applicable 4. No fourth tableau applicable 5. No fifth tableau applicable 6. Number of tableaus required = 2