Comprehensive Study Guide: Company Location Exercises and Methods

Theoretical Foundations of Facility Location Planning

Facility Location Strategy Overview:
- Selecting the optimal location for a business is a critical strategic decision that directly impacts operational costs, supply chain efficiency, competitive advantage, and customer accessibility.
- Location models integrate qualitative and quantitative factors to determine the optimal spatial positioning of plants, warehouses, outlets, or service facilities.
Factor Weighting Method (Multi-Factor Scoring):
- A quantitative approach used to evaluate and compare candidate locations by assigning relative weights to relevant qualitative and quantitative criteria.
- Mathematical Formulation:
- Let $n$ be the total number of decision factors.
- Let $w_i$ represent the weight assigned to factor $i$ , where the sum of all factor weights equals $1$ (or $100\%$ ): $\sum_{i=1}^{n} w_i = 1.0$
- Let $s_{ij}$ represent the score assigned to location $j$ for factor i$.\n * The total composite score S_j $for location$ j is calculated as:\n S_j = \sum_{i=1}^{n} w_i \cdot s_{ij}\n * Decision Rule: Select the location j^ that yields the maximum composite score:\n j^ = \arg\max_j (S_j)\n\n* **Center of Gravity Method**:\n * A mathematical technique used to find the optimal location for a centralized facility (e.g., a distribution center or hub) that minimizes total transportation costs or distance relative to existing markets or demand points.\n * Mathematical Formulation:\n * Let (X_i, Y_i) $represent the spatial coordinates of demand point or market$ i$.
- Let $W_i$ (or $V_i$ ) represent the demand volume, workload, or estimated number of clients associated with point i$.\n * The horizontal coordinate C_x of the optimal center of gravity is computed as:\n C_x = \frac{\sum_{i=1}^{n} X_i \cdot W_i}{\sum_{i=1}^{n} W_i}\n * The vertical coordinate C_y of the optimal center of gravity is computed as:\n C_y = \frac{\sum_{i=1}^{n} Y_i \cdot W_i}{\sum_{i=1}^{n} W_i}\n\n# FRIOLERA S.L. - Industrial Ice Factory Location\n\n* **Business Context**:\n * Company: FRIOLERA S.L.\n * Sector: Ice cube production.\n * Scenario: Opening a new facility outside Madrid.\n * Candidate Locations: Parla, Fuenlabrada, Leganés, Getafe.\n\n* **Decision Criteria and Evaluation Matrix**:\n * Factors and assigned weights:\n * Labor cost: Weight = 0.35\n * Infrastructure: Weight = 0.25\n * Communication: Weight = 0.10\n * Clients: Weight = 0.30\n * Factor Evaluation Table:\n * Labor cost (0.35 $): Parla$ = 9 $, Fuenlabrada$ = 7 $, Leganés$ = 8 $, Getafe$ = 6\n * Infrastructure (0.25 $): Parla$ = 6 $, Fuenlabrada$ = 8 $, Leganés$ = 7 $, Getafe$ = 7\n * Communication (0.10 $): Parla$ = 8 $, Fuenlabrada$ = 9 $, Leganés$ = 6 $, Getafe$ = 8\n * Clients (0.30 $): Parla$ = 8 $, Fuenlabrada$ = 5 $, Leganés$ = 5 $, Getafe$ = 3\n\n* **Detailed Factor Weighting Calculations**:\n * Score for Parla:\n S_{\text{Parla}} = (0.35 \times 9) + (0.25 \times 6) + (0.10 \times 8) + (0.30 \times 8)\n S_{\text{Parla}} = 3.15 + 1.50 + 0.80 + 2.40 = 7.85\n * Score for Fuenlabrada:\n S_{\text{Fuenlabrada}} = (0.35 \times 7) + (0.25 \times 8) + (0.10 \times 9) + (0.30 \times 5)\n S_{\text{Fuenlabrada}} = 2.45 + 2.00 + 0.90 + 1.50 = 6.85\n * Score for Leganés:\n S_{\text{Legan\acute{e}s}} = (0.35 \times 8) + (0.25 \times 7) + (0.10 \times 6) + (0.30 \times 5)\n S_{\text{Legan\acute{e}s}} = 2.80 + 1.75 + 0.60 + 1.50 = 6.65\n * Score for Getafe:\n S_{\text{Getafe}} = (0.35 \times 6) + (0.25 \times 7) + (0.10 \times 8) + (0.30 \times 3)\n S_{\text{Getafe}} = 2.10 + 1.75 + 0.80 + 0.90 = 5.55\n\n* **Exercise 1 Solutions**:\n 1. **Optimal Location Selection**:\n * Parla achieves the maximum total weighted score of 7.85.\n * Ranking of candidate locations: Parla (7.85 $) > Fuenlabrada ($ 6.85 $) > Leganés ($ 6.65 $) > Getafe ($ 5.55).\n * Conclusion: Parla is the optimal location.\n 2. **Required Client Score for Getafe to become the Best Location**:\n * Let x be the unknown Client factor score for Getafe.\n * To beat Parla, Getafe's total score must exceed Parla's score (7.85):\n (0.35 \times 6) + (0.25 \times 7) + (0.10 \times 8) + (0.30 \times x) > 7.85\n 2.10 + 1.75 + 0.80 + 0.30x > 7.85\n 4.65 + 0.30x > 7.85\n 0.30x > 3.20\n x > \frac{3.20}{0.30} \approx 10.67\n * Feasibility Analysis: Since the maximum achievable factor score on this rating scale is 10 $, Getafe cannot attain a score of$ 10.67. Therefore, it is mathematically impossible for Getafe to become the top-ranked location solely by increasing its Client score.\n\n# BELLEZA NATURAL S.L. - Hotel Chain Expansion\n\n* **Business Context**:\n * Company: Hotel Belleza Natural S.L.\n * Sector: Hospitality.\n * Location region: Navarra.\n * Candidate Locations: Javier, Roncal, Izaba, Barañaín.\n\n* **Decision Criteria and Evaluation Matrix**:\n * Factors and assigned percentage weights:\n * Climate: Weight = 40\% $($ 0.40)\n * Communication: Weight = 25\% $($ 0.25)\n * Competitors: Weight = 15\% $($ 0.15)\n * Leisure: Weight = 5\% $($ 0.05)\n * Environment: Weight = 15\% $($ 0.15)\n * Factor Evaluation Table (Scores scaled from 0 to 100):\n * Climate (0.40 $): Javier$ = 60 $, Roncal$ = 40 $, Izaba$ = 50 $, Barañaín$ = 50\n * Communication (0.25 $): Javier$ = 30 $, Roncal$ = 40 $, Izaba$ = 40 $, Barañaín$ = 80\n * Competitors (0.15 $): Javier$ = 20 $, Roncal$ = 70 $, Izaba$ = 30 $, Barañaín$ = 20\n * Leisure (0.05 $): Javier$ = 5 $, Roncal$ = 20 $, Izaba$ = 15 $, Barañaín$ = 10\n * Environment (0.15 $): Javier$ = 90 $, Roncal$ = 90 $, Izaba$ = 60 $, Barañaín$ = 40\n\n* **Detailed Factor Weighting Calculations**:\n * Score for Javier:\n S_{\text{Javier}} = (0.40 \times 60) + (0.25 \times 30) + (0.15 \times 20) + (0.05 \times 5) + (0.15 \times 90)\n S_{\text{Javier}} = 24.0 + 7.5 + 3.0 + 0.25 + 13.5 = 48.25\n * Score for Roncal:\n S_{\text{Roncal}} = (0.40 \times 40) + (0.25 \times 40) + (0.15 \times 70) + (0.05 \times 20) + (0.15 \times 90)\n S_{\text{Roncal}} = 16.0 + 10.0 + 10.5 + 1.0 + 13.5 = 51.00\n * Score for Izaba:\n S_{\text{Izaba}} = (0.40 \times 50) + (0.25 \times 40) + (0.15 \times 30) + (0.05 \times 15) + (0.15 \times 60)\n S_{\text{Izaba}} = 20.0 + 10.0 + 4.5 + 0.75 + 9.0 = 44.25\n * Score for Barañaín:\n S_{\text{Bara\tilde{n}a\acute{i}n}} = (0.40 \times 50) + (0.25 \times 80) + (0.15 \times 20) + (0.05 \times 10) + (0.15 \times 40)\n S_{\text{Bara\tilde{n}a\acute{i}n}} = 20.0 + 20.0 + 3.0 + 0.5 + 6.0 = 49.50\n\n* **Exercise 2 Solutions**:\n 1. **Optimal Location Selection**:\n * Roncal achieves the highest weighted total score of 51.00\n * Location ranking: Roncal (51.00 $) > Barañaín ($ 49.50 $) > Javier ($ 48.25 $) > Izaba ($ 44.25).\n * Conclusion: Roncal is the optimal location.\n 2. **Sensitivity Scenario Analysis (Competitor Score Revision)**:\n * Scenario Parameters: Competitors score for Izaba increases to 55 $; Competitors score for Barañaín increases to$ 40.\n * Revised score for Izaba:\n S_{\text{Izaba, new}} = 44.25 + 0.15 \times (55 - 30) = 44.25 + 3.75 = 48.00\n * Revised score for Barañaín:\n S_{\text{Bara\tilde{n}a\acute{i}n, new}} = 49.50 + 0.15 \times (40 - 20) = 49.50 + 3.00 = 52.50\n * Comparison of revised total scores:\n * Roncal: 51.00\n * Barañaín: 52.50\n * Javier: 48.25\n * Izaba: 48.00\n * Impact: Barañaín's new score (52.50 $) surpasses Roncal ($ 51.00). Thus, Barañaín becomes the new optimal location.\n\n# COMEBIEN S.A. - Restaurant Chain Expansion in Andalusia\n\n* **Business Context**:\n * Company: Comebien S.A.\n * Sector: Restaurant / Food Service.\n * Region: Andalusia, Spain.\n * Candidate Locations: Sevilla, Huelva, Cádiz, Málaga, Jaén.\n\n* **Decision Criteria and Evaluation Matrix**:\n * Raw Factor Weights (Sum of weights = 6 + 12 + 15 + 5 + 5 + 19 + 8 + 10 + 15 + 5 = 100):\n * Clients: Weight = 6 $($ 6\%\n * Communication: Weight = 12 $($ 12\%\n * Competitors: Weight = 15 $($ 15\%\n * Suppliers: Weight = 5 $($ 5\%\n * Infrastructure: Weight = 5 $($ 5\%\n * Climate: Weight = 19 $($ 19\%\n * House rent: Weight = 8 $($ 8\%\n * Labour: Weight = 10 $($ 10\%\n * Rent per capita: Weight = 15 $($ 15\%\n * Taxation: Weight = 5 $($ 5\%\n * Factor Scores Matrix:\n * Clients: Sevilla = 4 $, Huelva$ = 5 $, Cádiz$ = 10 $, Málaga$ = 6 $, Jaén$ = 2\n * Communication: Sevilla = 6 $, Huelva$ = 4 $, Cádiz$ = 5 $, Málaga$ = 5 $, Jaén$ = 3\n * Competitors: Sevilla = 8 $, Huelva$ = 6 $, Cádiz$ = 4 $, Málaga$ = 4 $, Jaén$ = 4\n * Suppliers: Sevilla = 9 $, Huelva$ = 7 $, Cádiz$ = 3 $, Málaga$ = 9 $, Jaén$ = 4\n * Infrastructure: Sevilla = 4 $, Huelva$ = 5 $, Cádiz$ = 5 $, Málaga$ = 9 $, Jaén$ = 5\n * Climate: Sevilla = 3 $, Huelva$ = 4 $, Cádiz$ = 6 $, Málaga$ = 5 $, Jaén$ = 5\n * House rent: Sevilla = 5 $, Huelva$ = 9 $, Cádiz$ = 7 $, Málaga$ = 8 $, Jaén$ = 6\n * Labour: Sevilla = 4 $, Huelva$ = 5 $, Cádiz$ = 10 $, Málaga$ = 6 $, Jaén$ = 2\n * Rent per capita: Sevilla = 6 $, Huelva$ = 4 $, Cádiz$ = 5 $, Málaga$ = 5 $, Jaén$ = 3\n * Taxation: Sevilla = 8 $, Huelva$ = 6 $, Cádiz$ = 4 $, Málaga$ = 4 $, Jaén$ = 4\n\n* **Detailed Factor Weighting Calculations**:\n * Sevilla:\n S_{\text{Sevilla}} = \frac{(6 \times 4) + (12 \times 6) + (15 \times 8) + (5 \times 9) + (5 \times 4) + (19 \times 3) + (8 \times 5) + (10 \times 4) + (15 \times 6) + (5 \times 8)}{100}\n S_{\text{Sevilla}} = \frac{24 + 72 + 120 + 45 + 20 + 57 + 40 + 40 + 90 + 40}{100} = \frac{548}{100} = 5.48\n * Huelva:\n S_{\text{Huelva}} = \frac{(6 \times 5) + (12 \times 4) + (15 \times 6) + (5 \times 7) + (5 \times 5) + (19 \times 4) + (8 \times 9) + (10 \times 5) + (15 \times 4) + (5 \times 6)}{100}\n S_{\text{Huelva}} = \frac{30 + 48 + 90 + 35 + 25 + 76 + 72 + 50 + 60 + 30}{100} = \frac{516}{100} = 5.16\n * Cádiz:\n S_{\text{C\acute{a}diz}} = \frac{(6 \times 10) + (12 \times 5) + (15 \times 4) + (5 \times 3) + (5 \times 5) + (19 \times 6) + (8 \times 7) + (10 \times 10) + (15 \times 5) + (5 \times 4)}{100}\n S_{\text{C\acute{a}diz}} = \frac{60 + 60 + 60 + 15 + 25 + 114 + 56 + 100 + 75 + 20}{100} = \frac{585}{100} = 5.85\n * Málaga:\n S_{\text{M\acute{a}laga}} = \frac{(6 \times 6) + (12 \times 5) + (15 \times 4) + (5 \times 9) + (5 \times 9) + (19 \times 5) + (8 \times 8) + (10 \times 6) + (15 \times 5) + (5 \times 4)}{100}\n S_{\text{M\acute{a}laga}} = \frac{36 + 60 + 60 + 45 + 45 + 95 + 64 + 60 + 75 + 20}{100} = \frac{559}{100} = 5.59\n * Jaén:\n S_{\text{Ja\acute{e}n}} = \frac{(6 \times 2) + (12 \times 3) + (15 \times 4) + (5 \times 4) + (5 \times 5) + (19 \times 5) + (8 \times 6) + (10 \times 2) + (15 \times 3) + (5 \times 4)}{100}\n S_{\text{Ja\acute{e}n}} = \frac{12 + 36 + 60 + 20 + 25 + 95 + 48 + 20 + 45 + 20}{100} = \frac{381}{100} = 3.81\n\n* **Exercise 3 Solutions**:\n 1. **Optimal Location Selection**:\n * Cádiz achieves the highest score of 5.85.\n * Overall ranking: Cádiz (5.85 $) > Málaga ($ 5.59 $) > Sevilla ($ 5.48 $) > Huelva ($ 5.16 $) > Jaén ($ 3.81).\n * Conclusion: Cádiz is the optimal location.\n 2. **Required Client Score for Equal Attractiveness Across Locations**:\n * To make all locations equally attractive to Cádiz's benchmark score of 5.85 $, solve for the required client score$ c_j $for location$ j:\n S_{\text{other}, j} + 0.06 \cdot c_j = 5.85\n * Sevilla: Non-client score = 5.48 - 0.24 = 5.24 $. Solving$ 5.24 + 0.06 c = 5.85 \implies c_{\text{Sevilla}} = 10.17\n * Huelva: Non-client score = 5.16 - 0.30 = 4.86 $. Solving$ 4.86 + 0.06 c = 5.85 \implies c_{\text{Huelva}} = 16.50\n * Málaga: Non-client score = 5.59 - 0.36 = 5.23 $. Solving$ 5.23 + 0.06 c = 5.85 \implies c_{\text{M\acute{a}laga}} = 10.33\n * Jaén: Non-client score = 3.81 - 0.12 = 3.69 $. Solving$ 3.69 + 0.06 c = 5.85 \implies c_{\text{Ja\acute{e}n}} = 36.00\n 3. **Evaluation of Córdoba Location Proposal**:\n * Córdoba's evaluated score: 6.5\n * Comparison: Córdoba (6.5 $) > Cádiz ($ 5.85 $) > Málaga ($ 5.59 $) > Sevilla ($ 5.48 $) > Huelva ($ 5.16 $) > Jaén ($ 3.81).\n * Justification: Selecting Córdoba is a wise and optimal decision because its total score of 6.5 strictly exceeds the maximum score of all previously analyzed Andalusian locations.\n\n# RECICLA S.A. - Recycling Plant Location & Sensitivity Analysis\n\n* **Business Context**:\n * Company: Recicla S.A.\n * Sector: Glass recycling production and manufacturing.\n * Candidate Locations: Location 1, Location 2, Location 3.\n\n* **Decision Criteria and Initial Weights**:\n * Labor availability: Weight = 40\% $($ 0.40)\n * Labor cost: Weight = 15\% $($ 0.15)\n * Infrastructure: Weight = 30\% $($ 0.30)\n * Actual legislation: Weight = 100\% - (40\% + 15\% + 30\%) = 15\% $($ 0.15)\n\n* **Evaluation Scores Matrix (Scale 0 to 5)**:\n * Location 1: Labor availability = 3 $, Labor cost$ = 4 $, Infrastructure$ = 1 $, Actual legislation$ = 5\n * Location 2: Labor availability = 5 $, Labor cost$ = 1 $, Infrastructure$ = 3 $, Actual legislation$ = 2\n * Location 3: Labor availability = 1 $, Labor cost$ = 2 $, Infrastructure$ = 4 $, Actual legislation$ = 4\n\n* **Detailed Factor Weighting Calculations (Initial State)**:\n * Location 1:\n S_1 = (0.40 \times 3) + (0.15 \times 4) + (0.30 \times 1) + (0.15 \times 5)\n S_1 = 1.20 + 0.60 + 0.30 + 0.75 = 2.85\n * Location 2:\n S_2 = (0.40 \times 5) + (0.15 \times 1) + (0.30 \times 3) + (0.15 \times 2)\n S_2 = 2.00 + 0.15 + 0.90 + 0.30 = 3.35\n * Location 3:\n S_3 = (0.40 \times 1) + (0.15 \times 2) + (0.30 \times 4) + (0.15 \times 4)\n S_3 = 0.40 + 0.30 + 1.20 + 0.60 = 2.50\n\n* **Exercise 4 Solutions**:\n 1. **Optimal Initial Location**:\n * Location 2 yields the highest total score of 3.35\n * Ranking: Location 2 (3.35 $) > Location 1 ($ 2.85 $) > Location 3 ($ 2.50).\n 2. **Sensitivity Analysis (Doubling Labor Cost Weight)**:\n * Parameter Modification: Labor cost weight doubles from 15\% $to$ 30\% $($ 0.30 $) at the expense of reducing Labor availability weight from$ 40\% $to$ 25\% $($ 0.25).\n * Revised Weights: Labor availability = 0.25 $, Labor cost$ = 0.30 $, Infrastructure$ = 0.30 $, Actual legislation$ = 0.15\n * Revised Score for Location 1:\n S_{1, \text{new}} = (0.25 \times 3) + (0.30 \times 4) + (0.30 \times 1) + (0.15 \times 5)\n S_{1, \text{new}} = 0.75 + 1.20 + 0.30 + 0.75 = 3.00\n * Revised Score for Location 2:\n S_{2, \text{new}} = (0.25 \times 5) + (0.30 \times 1) + (0.30 \times 3) + (0.15 \times 2)\n S_{2, \text{new}} = 1.25 + 0.30 + 0.90 + 0.30 = 2.75\n * Revised Score for Location 3:\n S_{3, \text{new}} = (0.25 \times 1) + (0.30 \times 2) + (0.30 \times 4) + (0.15 \times 4)\n S_{3, \text{new}} = 0.25 + 0.60 + 1.20 + 0.60 = 2.65\n * Impact & Conclusion: Location 1 now achieves the highest score of 3.00 $(outranking Location 2 at$ 2.75 $and Location 3 at$ 2.65). The previous decision does **not** hold; Location 1 becomes the new optimal location.\n\n# BOLO S.L. - Bowling Center Spatial Optimization\n\n* **Business Context**:\n * Company: BOLO S.L.\n * Sector: Entertainment / Bowling Center.\n * Goal: Identify optimal geographical center of gravity.\n\n* **Location and Client Volume Data Table**:\n * Lozoya: Estimated Clients (V_1 $)$ = 1,500 $, Coordinates$ (X_1, Y_1) = (2.3, 2.4)\n * Redueña: Estimated Clients (V_2 $)$ = 2,200 $, Coordinates$ (X_2, Y_2) = (2.2, 2.9)\n * Cabanillas: Estimated Clients (V_3 $)$ = 3,100 $, Coordinates$ (X_3, Y_3) = (2.4, 2.8)\n * Cervera: Estimated Clients (V_4 $)$ = 1,250 $, Coordinates$ (X_4, Y_4) = (2.8, 2.1)\n * Canencia: Estimated Clients (V_5 $)$ = 1,700 $, Coordinates$ (X_5, Y_5) = (2.5, 3.0)\n\n* **Detailed Calculations for Question 1 (All 5 Locations)**:\n * Total Client Demand Volume:\n \sum_{i=1}^{5} V_i = 1500 + 2200 + 3100 + 1250 + 1700 = 9,750\n * Weighted X-Coordinate Product Sum:\n \sum (X_i \cdot V_i) = (2.3 \times 1500) + (2.2 \times 2200) + (2.4 \times 3100) + (2.8 \times 1250) + (2.5 \times 1700)\n \sum (X_i \cdot V_i) = 3450 + 4840 + 7440 + 3500 + 4250 = 23,480\n * Weighted Y-Coordinate Product Sum:\n \sum (Y_i \cdot V_i) = (2.4 \times 1500) + (2.9 \times 2200) + (2.8 \times 3100) + (2.1 \times 1250) + (3.0 \times 1700)\n \sum (Y_i \cdot V_i) = 3600 + 6380 + 8680 + 2625 + 5100 = 26,385\n * Center of Gravity Coordinates:\n C_x = \frac{23480}{9750} \approx 2.4082\n C_y = \frac{26385}{9750} \approx 2.7062\n * Result 1: Optimal Center of Gravity = (2.41, 2.71).\n\n* **Detailed Calculations for Question 2 (Eliminating Two Lowest Volume Locations)**:\n * Locations with lowest client counts: Cervera (1,250 $) and Lozoya ($ 1,500).\n * Remaining locations: Redueña (2,200 $), Cabanillas ($ 3,100 $), Canencia ($ 1,700).\n * New Total Volume:\n \sum V_{\text{reduced}} = 2200 + 3100 + 1700 = 7,000\n * New Weighted X-Coordinate Sum:\n \sum (X_i \cdot V_i){\text{reduced}} = 4840 + 7440 + 4250 = 16,530\n * New Weighted Y-Coordinate Sum:\n \sum (Y_i \cdot V_i){\text{reduced}} = 6380 + 8680 + 5100 = 20,160\n * Revised Center of Gravity Coordinates:\n C_{x, \text{new}} = \frac{16530}{7000} \approx 2.3614\n C_{y, \text{new}} = \frac{20160}{7000} \approx 2.8800\n * Result 2: The new center of gravity shifts to (2.36, 2.88).\n\n# GOODPIZZA S.A. - Retail Pizzeria Location Center of Gravity\n\n* **Business Context**:\n * Company: Goodpizza S.A.\n * Sector: Fast food retail network.\n * Objective: Determine optimal store location coordinate using the Center of Gravity method.\n\n* **Location and Demand Data Table**:\n * Gravina street: Clients (V_1 $)$ = 1,800 $,$ (X_1, Y_1) = (1.7, 2.3)\n * Almirante street: Clients (V_2 $)$ = 1,100 $,$ (X_2, Y_2) = (2.1, 2.2)\n * Fuencarral street: Clients (V_3 $)$ = 2,100 $,$ (X_3, Y_3) = (1.5, 2.8)\n * Libertad street: Clients (V_4 $)$ = 1,420 $,$ (X_4, Y_4) = (1.8, 2.1)\n * Reina street: Clients (V_5 $)$ = 1,570 $,$ (X_5, Y_5) = (2.0, 1.3)\n * Pelayo street: Clients (V_6 $)$ = 1,650 $,$ (X_6, Y_6) = (1.9, 1.5)\n * Barbieri street: Clients (V_7 $)$ = 1,530 $,$ (X_7, Y_7) = (1.1, 1.9)\n * Barquillo street: Clients (V_8 $)$ = 2,100 $,$ (X_8, Y_8) = (2.2, 2.0)\n\n* **Detailed Mathematical Computations**:\n * Total Volume of Clients:\n \sum_{i=1}^{8} V_i = 1800 + 1100 + 2100 + 1420 + 1570 + 1650 + 1530 + 2100 = 13,270\n * Sum of Weighted X-Coordinates:\n \sum (X_i \cdot V_i) = (1.7 \times 1800) + (2.1 \times 1100) + (1.5 \times 2100) + (1.8 \times 1420) + (2.0 \times 1570) + (1.9 \times 1650) + (1.1 \times 1530) + (2.2 \times 2100)\n \sum (X_i \cdot V_i) = 3060 + 2310 + 3150 + 2556 + 3140 + 3135 + 1683 + 4620 = 23,654\n * Sum of Weighted Y-Coordinates:\n \sum (Y_i \cdot V_i) = (2.3 \times 1800) + (2.2 \times 1100) + (2.8 \times 2100) + (2.1 \times 1420) + (1.3 \times 1570) + (1.5 \times 1650) + (1.9 \times 1530) + (2.0 \times 2100)\n \sum (Y_i \cdot V_i) = 4140 + 2420 + 5880 + 2982 + 2041 + 2475 + 2907 + 4200 = 27,045\n * Center of Gravity Calculation:\n C_x = \frac{23654}{13270} \approx 1.7825\n C_y = \frac{27045}{13270} \approx 2.0381\n\n* **Exercise 6 Solution**:\n * Optimal Location Center of Gravity Coordinates = (1.78, 2.04).\n\n# MOVITEL S.L. - Regional Logistics Center of Gravity Analysis\n\n* **Business Context**:\n * Company: Movitel S.L.\n * Sector: Telecommunications / Commercial distribution.\n * Expansion plan: Establish new site outside Madrid across nationwide network nodes.\n\n* **Location and Estimated Client Data Table**:\n * Barcelona: Clients (V_1 $)$ = 2,800 $,$ (X_1, Y_1) = (2.7, 2.9)\n * Valencia: Clients (V_2 $)$ = 2,200 $,$ (X_2, Y_2) = (2.1, 2.6)\n * Sevilla: Clients (V_3 $)$ = 2,100 $,$ (X_3, Y_3) = (1.1, 1.8)\n * Zaragoza: Clients (V_4 $)$ = 3,500 $,$ (X_4, Y_4) = (2.8, 2.0)\n * Bilbao: Clients (V_5 $)$ = 1,500 $,$ (X_5, Y_5) = (3.1, 3.3)\n * Ciudad Real: Clients (V_6 $)$ = 1,150 $,$ (X_6, Y_6) = (1.9, 1.8)\n * Málaga: Clients (V_7 $)$ = 1,800 $,$ (X_7, Y_7) = (1.7, 1.9)\n * A Coruña: Clients (V_8 $)$ = 2,300 $,$ (X_8, Y_8) = (3.2, 2.9)\n * Badajoz: Clients (V_9 $)$ = 900 $,$ (X_9, Y_9) = (1.05, 1.4)\n * Cádiz: Clients (V_{10} $)$ = 1,000 $,$ (X_{10}, Y_{10}) = (1.8, 1.6)\n * Logroño: Clients (V_{11} $)$ = 1,200 $,$ (X_{11}, Y_{11}) = (2.0, 2.5)\n * Santander: Clients (V_{12} $)$ = 2,790 $,$ (X_{12}, Y_{12}) = (3.5, 3.2)\n * Pamplona: Clients (V_{13} $)$ = 1,120 $,$ (X_{13}, Y_{13}) = (2.9, 2.4)\n\n* **Detailed Mathematical Computations**:\n * Aggregate Volume of Clients:\n \sum_{i=1}^{13} V_i = 2800 + 2200 + 2100 + 3500 + 1500 + 1150 + 1800 + 2300 + 900 + 1000 + 1200 + 2790 + 1120 = 24,360\n * Weighted X-Coordinate Sum:\n \sum (X_i \cdot V_i) = (2.7 \times 2800) + (2.1 \times 2200) + (1.1 \times 2100) + (2.8 \times 3500) + (3.1 \times 1500) + (1.9 \times 1150) + (1.7 \times 1800) + (3.2 \times 2300) + (1.05 \times 900) + (1.8 \times 1000) + (2.0 \times 1200) + (3.5 \times 2790) + (2.9 \times 1120)\n \sum (X_i \cdot V_i) = 7560 + 4620 + 2310 + 9800 + 4650 + 2185 + 3060 + 7360 + 945 + 1800 + 2400 + 9765 + 3248 = 59,703\n * Weighted Y-Coordinate Sum:\n \sum (Y_i \cdot V_i) = (2.9 \times 2800) + (2.6 \times 2200) + (1.8 \times 2100) + (2.0 \times 3500) + (3.3 \times 1500) + (1.8 \times 1150) + (1.9 \times 1800) + (2.9 \times 2300) + (1.4 \times 900) + (1.6 \times 1000) + (2.5 \times 1200) + (3.2 \times 2790) + (2.4 \times 1120)\n \sum (Y_i \cdot V_i) = 8120 + 5720 + 3780 + 7000 + 4950 + 2070 + 3420 + 6670 + 1260 + 1600 + 3000 + 8928 + 2688 = 59,206\n * Center of Gravity Calculation:\n C_x = \frac{59703}{24360} \approx 2.4509\n C_y = \frac{59206}{24360} \approx 2.4305\n\n* **Exercise 7 Solution**:\n * Optimal Location Center of Gravity Coordinates = (2.45, 2.43).\n\n# Belle EPOQUE S.A. - Combined Multi-Method Location Evaluation\n\n* **Business Context**:\n * Company: Belle EPOQUE S.A.\n * Sector: Children's apparel and clothing retail.\n * Candidate Sites: Almirante street, Velazquez street, Ibiza street.\n\n* **Integrated Evaluation Table**:\n * Decision Factors & Raw Weights (Sum = 100):\n * Competitors: Weight = 10\n * Infrastructure: Weight = 5\n * Socioeconomic status: Weight = 15\n * Average age: Weight = 30\n * Clients: Weight = 10\n * Born rate: Weight = 20\n * Other business: Weight = 10\n * Factor Scores per Location:\n * Competitors (10 $): Almirante$ = 8 $, Velazquez$ = 4 $, Ibiza$ = 5\n * Infrastructure (5 $): Almirante$ = 8 $, Velazquez$ = 8 $, Ibiza$ = 7\n * Socioeconomic status (15 $): Almirante$ = 6 $, Velazquez$ = 8 $, Ibiza$ = 7\n * Average age (30 $): Almirante$ = 6 $, Velazquez$ = 7 $, Ibiza$ = 9\n * Clients (10 $): Almirante$ = 4 $, Velazquez$ = 5 $, Ibiza$ = 6\n * Born rate (20 $): Almirante$ = 6 $, Velazquez$ = 4 $, Ibiza$ = 8\n * Other business (10 $): Almirante$ = 3 $, Velazquez$ = 9 $, Ibiza$ = 7\n * Spatial & Volume Parameters:\n * Estimated clients (V_i $): Almirante$ = 350 $, Velazquez$ = 630 $, Ibiza$ = 510\n * Coordinates (X_i, Y_i) $: Almirante$ = (3, 2) $, Velazquez$ = (2, 9) $, Ibiza$ = (3, 5)\n\n* **Exercise 8 Solutions**:\n 1. **Factor Weighting Method Analysis**:\n * Almirante street Score:\n S_{\text{Almirante}} = \frac{(10 \times 8) + (5 \times 8) + (15 \times 6) + (30 \times 6) + (10 \times 4) + (20 \times 6) + (10 \times 3)}{100}\n S_{\text{Almirante}} = \frac{80 + 40 + 90 + 180 + 40 + 120 + 30}{100} = \frac{580}{100} = 5.80\n * Velazquez street Score:\n S_{\text{Velazquez}} = \frac{(10 \times 4) + (5 \times 8) + (15 \times 8) + (30 \times 7) + (10 \times 5) + (20 \times 4) + (10 \times 9)}{100}\n S_{\text{Velazquez}} = \frac{40 + 40 + 120 + 210 + 50 + 80 + 90}{100} = \frac{630}{100} = 6.30\n * Ibiza street Score:\n S_{\text{Ibiza}} = \frac{(10 \times 5) + (5 \times 7) + (15 \times 7) + (30 \times 9) + (10 \times 6) + (20 \times 8) + (10 \times 7)}{100}\n S_{\text{Ibiza}} = \frac{50 + 35 + 105 + 270 + 60 + 160 + 70}{100} = \frac{750}{100} = 7.50\n * Conclusion for Part 1: Ibiza street achieves the highest total factor weighting score of 7.50, making it the optimal choice under multi-factor scoring.\n 2. **Center of Gravity Method Analysis**:\n * Total Client Volume:\n \sum V_i = 350 + 630 + 510 = 1,490\n * Weighted X-Coordinate Product Sum:\n \sum (X_i \cdot V_i) = (3 \times 350) + (2 \times 630) + (3 \times 510) = 1050 + 1260 + 1530 = 3,840\n * Weighted Y-Coordinate Product Sum:\n \sum (Y_i \cdot V_i) = (2 \times 350) + (9 \times 630) + (5 \times 510) = 700 + 5670 + 2550 = 8,920\n * Center of Gravity Coordinates:\n C_x = \frac{3840}{1490} \approx 2.5772\n C_y = \frac{8920}{1490} \approx 5.9866\n * Conclusion for Part 2: The optimal geographic center of gravity coordinates are (2.58, 5.99)$$.