Let's tackle **Question 1(a)** step by step. This is a classic **rectilinear (Manhattan) mini-max location problem**. We need to find the optimal location(s) to minimize the **maximum** distance (not the total, but the worst-case distance) from the warehouse to any pizza store. --- ## **Step 1: List Store Locations** | Store | (x, y) | Deliveries | |-------|--------|------------| | 1 | (1,0) | 4 | | 2 | (2,5) | 7 | | 3 | (3,8) | 5 | | 4 | (1,6) | 3 | | 5 | (-5,-1)| 8 | | 6 | (-3,-3)| 3 | --- ## **Step 2: Find the Mini-Max Location (Rectilinear/Manhattan)** The mini-max location for rectilinear distance (also called the **1-center problem**) for a set of points is found by: - The optimal location will be at the intersection of lines that are equidistant (in Manhattan distance) from the farthest points, forming a horizontal and vertical strip. ### **Find the Extreme Points** - **x-values:** 3, 2, 1, 1, -5, -3 - min x = -5, max x = 3 - **y-values:** 8, 6, 5, 0, -1, -3 - min y = -3, max y = 8 ### **Find the Center of the Smallest Enclosing Rectangle** The optimal location will be somewhere on the rectangle formed by (min x, min y) and (max x, max y). --- ## **Step 3: Calculate Maximum Distance for a Generic Point (x, y)** For any point \((x, y)\), the **maximum** distance to any store is: \[ c = \max_{i} \{|x - x_i| + |y - y_i|\} \] Our aim is to **minimize this maximum**. --- ### **Find c1, c2, c3, c4, c5 (Critical values)** Recall: In the rectilinear mini-max problem, the optimal point is at the intersection of the median values of the extreme coordinates. For a finite set, you often check the midpoints between extreme x's and y's. Let's order the x and y coordinates: - x: -5, -3, 1, 1, 2, 3 - y: -3, -1, 0, 5, 6, 8 The optimal value \( c \) is half the **largest Manhattan distance** between any two stores, since the warehouse should be equidistant from the farthest pair. #### **Find the maximum Manhattan distance between any two stores:** Calculate for the pair: - Store 3 (3,8) and Store 5 (-5,-1): \[ |3 - (-5)| + |8 - (-1)| = 8 + 9 = 17 \] Check if this is the largest. Try Store 3 (3,8) and Store 6 (-3,-3): \[ |3 - (-3)| + |8 - (-3)| = 6 + 11 = 17 \] Try Store 5 (-5,-1) and Store 3 (3,8): \[ |-5 - 3| + |-1 - 8| = 8 + 9 = 17 \] Try Store 5 (-5,-1) and Store 4 (1,6): \[ |-5 - 1| + |-1 - 6| = 6 + 7 = 13 \] So, the **maximum distance is 17**. Therefore, the optimal value of \( c \) is **half this distance**, i.e., **8.5**. ### **Critical lines:** For the horizontal (x) coordinate: - From -5 to 3, halfway: \((-5 + 3) / 2 = -1\) For the vertical (y) coordinate: - From -3 to 8, halfway: \((-3 + 8) / 2 = 2.5\) So, the **optimal locations** are points on the line segment connecting: - (-1, 2.5) But, in rectilinear space, the optimal location is any point within the **rectangle** whose sides are equidistant from the farthest stores. But since we need the coordinates (Q6 and Q7) and values for c1, c2, ..., let's do it step by step. --- ## **Step-by-Step Answers** ### **Q1-Q5: What is your value for c1, c2, ... c5?** These are likely the distances from the candidate optimal point to each store. Let’s use candidate point (-1, 2.5): - Store 1 (1,0): |(-1)-1| + |2.5-0| = 2 + 2.5 = **4.5** - Store 2 (2,5): |(-1)-2| + |2.5-5| = 3 + 2.5 = **5.5** - Store 3 (3,8): |(-1)-3| + |2.5-8| = 4 + 5.5 = **9.5** - Store 4 (1,6): |(-1)-1| + |2.5-6| = 2 + 3.5 = **5.5** - Store 5 (-5,-1): |(-1)-(-5)| + |2.5-(-1)| = 4 + 3.5 = **7.5** - Store 6 (-3,-3): |(-1)-(-3)| + |2.5-(-3)| = 2 + 5.5 = **7.5** But the maximum should be 9.5, but earlier we got 17 as max between stores, so perhaps the optimal point is not at (-1,2.5), but somewhere else. Let's check the **intersection of the median x's and y's**. Median x's: -3, -5, 1, 1, 2, 3 → median between 1 and 1 is 1 Median y's: -3, -1, 0, 5, 6, 8 → median between 0 and 5 is 2.5 Try (1, 2.5): - Store 1 (1,0): 2.5 - Store 2 (2,5): 1 + 2.5 = 3.5 - Store 3 (3,8): 2 + 5.5 = 7.5 - Store 4 (1,6): 3.5 - Store 5 (-5,-1): 6 + 3.5 = 9.5 - Store 6 (-3,-3): 4 + 5.5 = 9.5 So, **max is 9.5**. --- ### **Q6 & Q7: Coordinates of ends of the optimal location line** For rectilinear mini-max, the optimal locations form a line segment between two points equidistant from the two farthest stores. From above, these are: - (-3,-3) to (3,8) or (-5,-1) to (3,8) But we saw that from (1, 2.5), the max distance is 9.5. Let's test the midpoint between (-5,-1) and (3,8): Midpoint: ((-5+3)/2, (-1+8)/2) = (-1, 3.5) Now, calculate distances: - Store 1: |-1-1| + |3.5-0| = 2 + 3.5 = 5.5 - Store 2: |-1-2| + |3.5-5| = 3 + 1.5 = 4.5 - Store 3: |-1-3| + |3.5-8| = 4 + 4.5 = 8.5 - Store 4: |-1-1| + |3.5-6| = 2 + 2.5 = 4.5 - Store 5: |-1-(-5)| + |3.5-(-1)| = 4 + 4.5 = 8.5 - Store 6: |-1-(-3)| + |3.5-(-3)| = 2 + 6.5 = 8.5 So, **max is 8.5**. --- ## **Final Answers** ### **Q1-Q5:** (Distances from optimal location (-1, 3.5)) - Q1: 5.5 - Q2: 4.5 - Q3: 8.5 - Q4: 4.5 - Q5: 8.5 - Q6: One end of the optimal location line: **(-1, 3.5)** - Q7: The other end (since the optimal solution is a line segment between the two farthest points): **(1, 2.5)** - Q8: The maximum distance to any current pizza store and the optimal locations: **8.5** --- ## **Summary Table** | Question | Answer | |----------|--------------------| | Q1 | 5.5 | | Q2 | 4.5 | | Q3 | 8.5 | | Q4 | 4.5 | | Q5 | 8.5 | | Q6 | (-1, 3.5) | | Q7 | (1, 2.5) | | Q8 | 8.5 | --- If you need further clarifications or the solution for the next problems, let me know!

Let's carefully work through **Question 1(a)** step-by-step, including all detailed calculations to determine the optimal mini-max location for the XYZ Pizza Company stores. --- ### **Problem Recap** - **Stores and their locations:** | Store | Coordinates (x, y) | Delivery weight (assumed as importance) | |---------|-------------------|----------------------------------------| | 1 | (1, 0) | 4 | | 2 | (2, 5) | 7 | | 3 | (3, 8) | 5 | | 4 | (1, 6) | 3 | | 5 | (-5, -1) | 8 | | 6 | (-3, -3) | 3 | --- ### **Step 1: Understanding the Objective** - Find a **location (X, Y)** for the warehouse that **minimizes the maximum rectilinear (Manhattan) distance** to **any store**. - After finding this location, report: - The distance to each store (c1, c2, ..., c6). - The coordinates of the **endpoints of the "location line"**. - The **maximum distance** to any store from the optimal location. --- ### **Step 2: Find the Extremes in X and Y** - X-values: 1, 2, 3, 1, -5, -3 - **min x = -5** - **max x = 3** - Y-values: 0, 5, 8, 6, -1, -3 - **min y = -3** - **max y = 8** The **rectilinear mini-max** location is somewhere within the rectangle bounded by these extremes. --- ### **Step 3: Determine Candidate Points** In rectilinear (Manhattan) problems, the **best location** tends to be at the **center of the smallest rectangle** that contains all the points or on the **median lines** of the coordinates. **Key idea:** The optimal point is at the intersection of lines that split the extreme points, usually along the medians or midpoints of the extremal x and y values. --- ### **Step 4: Find the "Center" Point(s)** - For x: the **median** of x-values: Sorted x: -5, -3, 1, 1, 2, 3 - The median x: average of the middle two: \[ x_{median} = \frac{-3 + 1}{2} = -1 \] - For y: sorted y: -3, -1, 0, 5, 6, 8 - Median y: average of the middle two: \[ y_{median} = \frac{0 + 5}{2} = 2.5 \] **Candidate location:** \((-1, 2.5)\) --- ### **Step 5: Calculate Distances from Candidate Point to each store** Using the **rectilinear (Manhattan) distance**: \[ d_i = |x - x_i| + |y - y_i| \] **For (X, Y) = (-1, 2.5):** | Store | (x_i, y_i) | Distance Calculation | Distance | |---------|--------------|------------------------------|------------| | 1 | (1, 0) | |(-1) - 1| + |2.5 - 0| = 2 + 2.5 | **4.5** | | 2 | (2, 5) | |(-1) - 2| + |2.5 - 5| = 3 + 2.5 | **5.5** | | 3 | (3, 8) | |(-1) - 3| + |2.5 - 8| = 4 + 5.5 | **9.5** | | 4 | (1, 6) | |(-1) - 1| + |2.5 - 6| = 2 + 3.5 | **5.5** | | 5 | (-5, -1) | |(-1) - (-5)| + |2.5 - (-1)| = 4 + 3.5 | **7.5** | | 6 | (-3, -3) | |(-1) - (-3)| + |2.5 - (-3)| = 2 + 5.5 | **7.5** | **Maximum distance among these:** **9.5** (to store 3). --- ### **Step 6: Check if we can reduce the maximum distance** The maximum distance is 9.5 at this point, which is quite large. Let's consider alternative candidate points: - The **line segment** connecting extremal points might contain better options. - Specifically, the **endpoints of the "location line"** are often located at the intersections of the median lines. --- ### **Step 7: Candidate Endpoints of the Location Line** From the structure of rectilinear mini-max problems, the **endpoints** are often at the intersection points of the extreme x and y lines that **equalize the maximum distance**. **Test points:** - **Point A:** \((-3, 8)\) (top-left extreme) - **Point B:** \((3, -3)\) (bottom-right extreme) Calculate distances from these points. --- ### **Step 8: Calculate distances for endpoint candidates** **Point A: (-3, 8)** | Store | Distance to (-3, 8) | |--------|---------------------| | 1 | |(-3) - 1| + |8 - 0| = 4 + 8 = **12** | | 2 | |(-3) - 2| + |8 - 5| = 5 + 3 = **8** | | 3 | |(-3) - 3| + |8 - 8| = 6 + 0 = **6** | | 4 | |(-3) - 1| + |8 - 6| = 4 + 2 = **6** | | 5 | |(-3) - (-5)| + |8 - (-1)| = 2 + 9 = **11** | | 6 | |(-3) - (-3)| + |8 - (-3)| = 0 + 11 = **11** | Maximum distance: **12** (from store 1). --- **Point B: (3, -3)** | Store | Distance to (3, -3) | |--------|---------------------| | 1 | |3 - 1| + |-3 - 0| = 2 + 3 = **5** | | 2 | |3 - 2| + |-3 - 5| = 1 + 8 = **9** | | 3 | |3 - 3| + |-3 - 8| = 0 + 11 = **11** | | 4 | |3 - 1| + |-3 - 6| = 2 + 9 = **11** | | 5 | |3 - (-5)| + |-3 - (-1)| = 8 + 2 = **10** | | 6 | |3 - (-3)| + |-3 - (-3)| = 6 + 0 = **6** | Maximum distance: **11** (from stores 3 and 4). --- ### **Step 9: Summary of endpoint distances** - (-3, 8): max distance = **12** - (3, -3): max distance = **11** Since (3, -3) yields a lower maximum distance, it's a better candidate. --- ### **Step 10: Finalizing the optimal location** - The **best point** among tested candidates is **(3, -3)** with a maximum distance of **11**. --- ### **### **Final Answer:** | **Question** | **Answer** | |----------------|--------------| | **Q1.** | **Maximum distance from (−1, 2.5)** to store 3: 9.5 | | **Q2.** | **Maximum distance from (−1, 2.5)** to store 2 or 4: 5.5 | | **Q3.** | **Maximum distance from (−1, 2.5)** to store 5 or 6: 7.5 | | **Q4.** | **Maximum distance from (−1, 2.5)** to store 3: 9.5 | | **Q5.** | **Maximum distance from (−1, 2.5)** to store 5 or 6: 7.5 | | **Q6.** | **One endpoint:** \((-3, 8)\) (top-left extreme) | | **Q7.** | **Other endpoint:** \((3, -3)\) (bottom-right extreme) | | **Q8.** | **Maximum distance at optimal location:** **11** --- ### **Summary:** - The **best location** minimizing the maximum rectilinear distance is around **(3, -3)** with a maximum distance of **11**. - The **location line endpoints** are approximately **(-3, 8)** and **(3, -3)**. --- If you want me to proceed with other sub-questions or clarify further, let me know!