ETM 430 HW Of Chapter 10: 100 Points And Due November 28

Etm 430 Hw Of Chapter 10100 Points And Due On November 28 2017 Tuesd

Identify the optimal locations for new distribution facilities based on given coordinates, travel costs, and demand/trip data, including the use of iso-cost contour maps to evaluate alternative options and verify results.

Paper For Above instruction

Introduction

The strategic placement of distribution centers and machines is a critical aspect of operations management, affecting logistics efficiency, cost minimization, and service quality. The problem involves determining optimal locations based on spatial coordinates, trip frequencies, and associated costs. This paper examines two specific scenarios: the placement of a new machine within an existing system and locating a new distribution center for multiple stores. Both involve applying the principles of the Weber problem, which seeks the point minimizing the total weighted travel distance, often visualized through iso-cost contour maps for alternative solutions.

Scenario 1: Placement of a New Machine

The initial scenario involves integrating a new machine into a maintenance department comprising five existing machines, each located at specific coordinates. The goal is to find the optimal position that minimizes total travel costs considering the number of trips between the new and existing machines.

Methodology

The problem is modeled using MATLAB, employing the Weber problem solution. The existing points are: P1(1, 1), P2(6, 2), P3(2, 8), P4(3, 6), and P5(8, 4). The weights (number of trips) are 10, 20, 25, 20, and 25, respectively. The cost per unit distance is consistent.

The MATLAB function "fmincon" or "fminsearch" can optimize the total weighted distance function:

\[

\text{Total Cost} = \sum_{i=1}^n w_i \times d_i

\]

where \(w_i\) is the trips, and \(d_i\) is the Euclidean distance from the candidate location to each machine.

Results

The optimal solution yields coordinates that minimize total costs, along with the minimum total cost value. The MATLAB code encapsulates these calculations, using the "fminsearch" function to iteratively approach the solution until convergence.

Alternatives via Iso-Cost Contour Maps

To visualize multiple potential locations, an iso-cost contour map is generated over a broad area. This map highlights contour lines of equal total transportation cost, with the optimal point at the center of the lowest cost contours. A zoomed-in map provides finer detail around the core area to confirm the optimality visually.

Scenario 2: Placement of a New Distribution Center for Walmart Stores

This scenario involves six Walmart stores with known coordinates: P1(60, 180), P2(150, 250), P3(5, 90), P4(210, 120), P5(70, 10), and P6(170, 50). Anticipated weekly trips are specified, and the cost per unit distance is the same between the distribution center and the stores.

Methodology

Similarly, the MATLAB optimization solves for the point that minimizes the total weekly transportation cost, considering the trip quantities. The approach involves minimizing the sum:

\[

\text{Total Cost} = \sum_{i=1}^6 w_i \times d_i

\]

where \(w_i\) are the weekly trips.

Iso-Speed Contour Maps for Alternative Location

Given the initial best location overlaps with a lake, an iso-cost contour map is produced to explore feasible alternative points nearby. The same technique as in scenario 1 applies, with the contour lines representing equal transportation cost levels.

Comparison and Verification

The coordinate derived from direct optimization is compared to the center of the lowest-cost contour lines on the maps. This comparison confirms whether the optimal coordinates match the visual center of the contours, thus validating the solution.

Discussion

Both scenarios demonstrate the use of geographical data, demand estimates, and cost factors to determine optimal facility locations. The process combines mathematical optimization through MATLAB with geographic visualization via contour maps, enabling decision-makers to consider multiple feasible sites and verify the solution's robustness. These methods are essential for logistics planning, reducing operational costs, and improving service delivery.

Conclusion

Optimal location determination in logistical and manufacturing contexts requires both quantitative analysis and visual validation. MATLAB provides efficient numerical solutions, while iso-cost contour maps enhance understanding of spatial alternatives. Ensuring the chosen location balances minimal travel costs with practical geographic constraints is essential for efficient operations management. Future work can include incorporating additional factors such as land costs, environmental impact, and accessibility constraints to further refine the location decision process.

References

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