Let's solve each subquestion step by step: --- ## **a) Distance between Ogden and Lubbock** **Ogden:** (250, 600) **Lubbock:** (450, 300) **Formula (Euclidean distance):** \[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] **Calculation:** \[ = \sqrt{(450 - 250)^2 + (300 - 600)^2} = \sqrt{(200)^2 + (-300)^2} = \sqrt{40000 + 90000} = \sqrt{130000} = 360.56 \text{ miles} \] --- ## **b) Cost to transport 400 tons from Springfield to Seattle** - **Springfield:** (225, 825) - **Seattle:** (50, 1200) - **Distance:** \[ = \sqrt{(50 - 225)^2 + (1200 - 825)^2} = \sqrt{(-175)^2 + (375)^2} = \sqrt{30625 + 140625} = \sqrt{171250} = 413.73 \text{ miles} \] - **Transportation cost from Springfield:** $1.4/ton mile - **Tonnage:** 400 tons **Total cost:** \[ \text{Total Cost} = \text{distance} \times \text{tons} \times \text{cost per ton-mile} \] \[ = 413.73 \times 400 \times 1.4 = 231,684 \text{ dollars} \] --- ## **c) Optimal location for the Super Assembly Center** Optimal location is the **weighted center of gravity** based on all supply and demand points. Weights: Supply/market tons × cost per ton-mile ### **Step 1: Calculate Weights for Each Location** | Location | Tons (Dn) | Cost ($/ton-mi) | Xn | Yn | Weight (Dn × Fn) | |---------------------|-----------|-----------------|------|------|------------------| | North Haverbrook | 650 | 1.1 | 700 | 1200 | 715 | | Ogden | 450 | 1.2 | 250 | 600 | 540 | | Springfield | 400 | 1.4 | 225 | 825 | 560 | | Seattle | 300 | 2.25 | 50 | 1200 | 675 | | Lubbock | 200 | 2.25 | 450 | 300 | 450 | | Nashville | 500 | 2.25 | 800 | 250 | 1125 | | Philadelphia | 500 | 2.25 | 950 | 1100 | 1125 | ### **Step 2: Calculate Weighted Average X and Y** \[ X^* = \frac{\sum (Dn \times Fn \times Xn)}{\sum (Dn \times Fn)} \] \[ Y^* = \frac{\sum (Dn \times Fn \times Yn)}{\sum (Dn \times Fn)} \] Calculate numerator and denominator: #### **Sum of Weights:** \[ \text{Total Weight} = 715 + 540 + 560 + 675 + 450 + 1125 + 1125 = 5190 \] #### **Weighted X Numerator:** \[ = (715 \times 700) + (540 \times 250) + (560 \times 225) + (675 \times 50) + (450 \times 450) + (1125 \times 800) + (1125 \times 950) \] \[ = 500500 + 135000 + 126000 + 33750 + 202500 + 900000 + 1068750 = 2924500 \] #### **Weighted Y Numerator:** \[ = (715 \times 1200) + (540 \times 600) + (560 \times 825) + (675 \times 1200) + (450 \times 300) + (1125 \times 250) + (1125 \times 1100) \] \[ = 858000 + 324000 + 462000 + 810000 + 135000 + 281250 + 1237500 = 4101750 \] #### **Compute X* and Y*:** \[ X^* = \frac{2924500}{5190} = 563.70 \] \[ Y^* = \frac{4101750}{5190} = 790.30 \] **Optimal location:** \[ (\boxed{563.7},\; \boxed{790.3}) \] --- ## **d) Minimized Shipping Cost to Satisfy All Market Demand** Assume all supply is consolidated at the optimal location, and all market demands are shipped from there. ### **Step 1: Calculate total demand** \[ \text{Total Market Demand} = 300 + 200 + 500 + 500 = 1500 \text{ tons} \] ### **Step 2: Calculate Distance from Optimal Location to Each Market** Let \((X^*, Y^*) = (563.7, 790.3)\) #### **Seattle:** (50, 1200) \[ D = \sqrt{(563.7 - 50)^2 + (790.3 - 1200)^2} = \sqrt{263,808.7 + 167,340.1} = \sqrt{431,148.8} = 656.73 \text{ mi} \] #### **Lubbock:** (450, 300) \[ D = \sqrt{(563.7 - 450)^2 + (790.3 - 300)^2} = \sqrt{12,898.7 + 240,277.3} = \sqrt{253,176} = 503.17 \text{ mi} \] #### **Nashville:** (800, 250) \[ D = \sqrt{(563.7 - 800)^2 + (790.3 - 250)^2} = \sqrt{55,796.9 + 292,449.1} = \sqrt{348,246} = 590.12 \text{ mi} \] #### **Philadelphia:** (950, 1100) \[ D = \sqrt{(563.7 - 950)^2 + (790.3 - 1100)^2} = \sqrt{149,226.7 + 95,244.1} = \sqrt{244,470.8} = 494.43 \text{ mi} \] ### **Step 3: Compute Shipping Cost to Each Market** \[ \text{Cost} = \text{Distance} \times \text{Tons} \times \text{Cost per ton-mile} \] - All markets: $2.25/ton-mile #### **Seattle:** \[ = 656.73 \times 300 \times 2.25 = 443,311.25 \] #### **Lubbock:** \[ = 503.17 \times 200 \times 2.25 = 226,426.50 \] #### **Nashville:** \[ = 590.12 \times 500 \times 2.25 = 663,885 \] #### **Philadelphia:** \[ = 494.43 \times 500 \times 2.25 = 555,733 \] #### **Total Shipping Cost:** \[ = 443,311.25 + 226,426.50 + 663,885 + 555,733 = \boxed{1,889,355.75} \] --- ## **Final Summary** - **a)** Distance between Ogden and Lubbock: **360.56 miles** - **b)** Cost to transport 400 tons from Springfield to Seattle: **$231,684** - **c)** Optimal location for the Super Assembly Center: **(563.7, 790.3)** - **d)** Minimized total shipping cost to satisfy all market demand: **$1,889,355.75** If you need a detailed Excel solution, let me know!