## Data Summary | Location | $/Ton-Mile | Tons | X | Y | |-----------------|------------|-------|-----|------| | **Supply** | | | | | | North Haverbrook| 1.10 | 650 | 700 | 1200 | | Ogden | 1.20 | 450 | 250 | 600 | | Springfield | 1.40 | 400 | 225 | 825 | | **Market** | | | | | | Seattle | 2.25 | 300 | 50 | 1200 | | Lubbock | 2.25 | 200 | 450 | 300 | | Nashville | 2.25 | 500 | 800 | 250 | | Philadelphia | 2.25 | 500 | 950 | 1100 | --- ## a) Distance Between Ogden and Lubbock **Coordinates:** - Ogden: (250, 600) - Lubbock: (450, 300) **Distance formula:** \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] \[ d = \sqrt{(450 - 250)^2 + (300 - 600)^2} \] \[ d = \sqrt{(200)^2 + (-300)^2} \] \[ d = \sqrt{40000 + 90000} \] \[ d = \sqrt{130000} \] \[ d \approx 360.56 \] **Answer:** **The distance between Ogden and Lubbock is approximately 360.6 units.** --- ## b) Cost to Transport 400 Tons from Springfield to Seattle **Coordinates:** - Springfield: (225, 825) - Seattle: (50, 1200) **Distance:** \[ d = \sqrt{(50 - 225)^2 + (1200 - 825)^2} \] \[ d = \sqrt{(-175)^2 + (375)^2} \] \[ d = \sqrt{30625 + 140625} \] \[ d = \sqrt{171250} \] \[ d \approx 413.81 \] **Cost:** - Rate from Springfield: \$1.40/ton-mile - Tons: 400 \[ \text{Total cost} = 400 \times 413.81 \times 1.40 \] \[ = 400 \times 579.33 \] \[ = 231,732 \] **Answer:** **The cost to transport 400 tons from Springfield to Seattle is approximately \$231,732.** --- ## c) Optimal Location for the Super Assembly Center We use the **center of gravity (weighted average)** method, weighting by supply and demand: ### Step 1: Total Tonnage **Supply total:** 650 (NH) + 450 (Ogden) + 400 (Springfield) = **1,500** tons **Demand total:** 300 (Seattle) + 200 (Lubbock) + 500 (Nashville) + 500 (Philadelphia) = **1,500** tons ### Step 2: Weighted Coordinates - **Supply Side:** - \(\sum X_{\text{sup}} = 700 \times 650 + 250 \times 450 + 225 \times 400 = 455,000 + 112,500 + 90,000 = 657,500\) - \(\sum Y_{\text{sup}} = 1200 \times 650 + 600 \times 450 + 825 \times 400 = 780,000 + 270,000 + 330,000 = 1,380,000\) - **Demand Side:** - \(\sum X_{\text{dem}} = 50 \times 300 + 450 \times 200 + 800 \times 500 + 950 \times 500 = 15,000 + 90,000 + 400,000 + 475,000 = 980,000\) - \(\sum Y_{\text{dem}} = 1200 \times 300 + 300 \times 200 + 250 \times 500 + 1100 \times 500 = 360,000 + 60,000 + 125,000 + 550,000 = 1,095,000\) - **Total weighted X:** \(657,500 + 980,000 = 1,637,500\) - **Total weighted Y:** \(1,380,000 + 1,095,000 = 2,475,000\) - **Total tons:** \(1,500 + 1,500 = 3,000\) ### Step 3: Center of Gravity \[ X^* = \frac{1,637,500}{3,000} \approx 545.83 \] \[ Y^* = \frac{2,475,000}{3,000} \approx 825.00 \] **Answer:** **The optimal location for the assembly center is approximately at (545.8, 825.0).** --- ## d) Minimized Shipping Cost This requires calculating the cost from all suppliers to the assembly center and from the center to all markets. ### Step 1: Distances from Suppliers to Assembly Center Let assembly center at (545.8, 825.0): #### North Haverbrook to Center: \[ \sqrt{(700-545.8)^2 + (1200-825)^2} \approx \sqrt{23,684 + 140,625} \approx \sqrt{164,309} \approx 405.36 \] #### Ogden to Center: \[ \sqrt{(250-545.8)^2 + (600-825)^2} \approx \sqrt{87,497 + 50,625} \approx \sqrt{138,122} \approx 371.75 \] #### Springfield to Center: \[ \sqrt{(225-545.8)^2 + (825-825)^2} = \sqrt{102,922 + 0} = 320.80 \] ### Step 2: Distances from Center to Markets #### Center to Seattle: \[ \sqrt{(50-545.8)^2 + (1200-825)^2} \approx \sqrt{246,371 + 140,625} \approx \sqrt{386,996} \approx 622.10 \] #### Center to Lubbock: \[ \sqrt{(450-545.8)^2 + (300-825)^2} \approx \sqrt{9,281 + 275,625} \approx \sqrt{284,906} \approx 534.03 \] #### Center to Nashville: \[ \sqrt{(800-545.8)^2 + (250-825)^2} \approx \sqrt{64,839 + 330,625} \approx \sqrt{395,464} \approx 629.15 \] #### Center to Philadelphia: \[ \sqrt{(950-545.8)^2 + (1100-825)^2} \approx \sqrt{163,489 + 75,625} \approx \sqrt{239,114} \approx 488.98 \] ### Step 3: Costs #### From Suppliers to Assembly Center: - NH: 650 tons × 405.36 × \$1.10 = \$289,834 - Ogden: 450 × 371.75 × \$1.20 = \$201,745 - Springfield: 400 × 320.80 × \$1.40 = \$179,648 **Subtotal:** \$289,834 + \$201,745 + \$179,648 = \$671,227 #### From Assembly Center to Markets: - Seattle: 300 × 622.10 × \$2.25 = \$419,168 - Lubbock: 200 × 534.03 × \$2.25 = \$240,314 - Nashville: 500 × 629.15 × \$2.25 = \$707,794 - Philadelphia: 500 × 488.98 × \$2.25 = \$550,103 **Subtotal:** \$419,168 + \$240,314 + \$707,794 + \$550,103 = \$1,917,379 ### **Total Minimized Shipping Cost:** \[ \$671,227 + \$1,917,379 = \$2,588,606 \] --- ## **Summary Table** | Question | Answer | |----------|--------| | a) Distance Ogden–Lubbock | **360.6 units** | | b) Cost Springfield–Seattle | **\$231,732** | | c) Optimal Location | **(545.8, 825.0)** | | d) Minimized Shipping Cost | **\$2,588,606** | --- **All calculations shown. If using Excel, use the formulas above for each cell.**

### a) Distance Between Ogden and Lubbock The Euclidean distance between Ogden (250, 600) and Lubbock (450, 300) is approximately 360.6 units. This measurement is calculated using the distance formula based on their coordinates. ### b) Cost to Transport 400 Tons from Springfield to Seattle Transporting 400 tons from Springfield (225, 825) to Seattle (50, 1200) costs about \$231,732, considering the distance of approximately 413.81 miles and the rate of \$1.40 per ton-mile. The calculation multiplies the distance by the quantity and rate. ### c) Optimal Location for the Super Assembly Center The optimal site for the assembly center is at approximately (545.8, 825.0), derived by weighted averaging of supply and demand locations. This position minimizes the overall transportation distance and cost based on the combined data. ### d) Minimized Shipping Cost The total minimized shipping cost, accounting for all supplier-to-center and center-to-market routes, is about \$2,588,606. This sum reflects the most efficient routing based on calculated distances and transportation rates.