Mat540 Homework Week 10 Page 1 Of 2 794079

Mat540 Homework Week 10page 1 Of 2mat540week 10 Homeworkchapter 6 1

Formulate the transportation problem as a linear programming model and solve it using the computer; solve a similar problem with given data; determine optimal shipment plans for European ports to U.S. distribution centers; and assign salespersons to regions to minimize total time based on given data.

Paper For Above instruction

Transportation problems are a subclass of linear programming models that involve the optimal allocation of resources across a network to minimize costs or maximize profits. These problems are widely encountered in logistics, supply chain management, and operational planning. The tasks presented involve formulating transportation problems as linear programming models, solving them via computational methods, and applying similar frameworks to real-world scenarios such as international shipping logistics and salesforce deployment.

Part 1: Formulating a Transportation Problem as a Linear Programming Model

The first task involves formulating a transportation problem. Suppose we have a supply node A with 130 units, and demand at two destinations B and C, with the transportation costs per unit specified. The model includes decision variables representing the amount shipped from each source to each destination. The goal is to minimize total transportation costs while satisfying supply and demand constraints.

The decision variables are defined as follows:

• x_AB: units shipped from A to B
• x_AC: units shipped from A to C

The objective function aims to minimize the total cost:

Minimize Z = 4x_AB + 3x_AC

Subject to the constraints:

• Supply constraint at A: x_AB + x_AC ≤ 130
• Demand at B: x_AB ≥ demand at B (if provided; otherwise, assume total supply equals demand)
• Demand at C: x_AC ≥ demand at C
• Non-negativity: x_AB ≥ 0, x_AC ≥ 0

Similar models can be constructed for the other problems by defining variables, objective functions, and constraints accordingly.

Part 2: Solving Transportation Problems Using Computers

These formulation models are typically solved via specialized algorithms like the Vogel's Approximation Method, stepping stone method, or using software tools such as Excel Solver, LINDO, or Lingo. Such tools efficiently handle large-scale transportation models, providing optimal shipment plans that respect supply and demand constraints while minimizing total transportation costs.

Implementation involves inputting the cost matrix, supply, and demand data into the solver, defining decision variables, and setting the objective and constraints. Optimal solutions output shipment quantities from each origin to destination that minimize total costs.

Part 3: Specific Case — European Ports to U.S. Distribution Centers

In real-world applications, such as the importation of food products by World Foods, Inc., transportation modeling ensures cost-efficient distribution. The model incorporates transportation costs from European ports (Hamburg, Marseilles, Liverpool) to U.S. cities (Norfolk, New York, Savannah), with supplies at ports and demands at distribution centers specified.

Let x_ij represent the amount shipped from port i to city j, where i = Hamburg, Marseilles, Liverpool; j = Norfolk, New York, Savannah. The objective function sums the total transportation costs:

Minimize Z = Σ (cost from i to j) * x_ij

Subject to supply constraints at each port: Σ x_ij ≤ supply_i

And demand constraints at each U.S. city: Σ x_ij ≥ demand_j.

Similar modeling applies to transportation from U.S. warehouses to distribution centers, including their respective costs, supplies, and demands. Solving this network flow problem yields the shipment quantities between each node to minimize total transportation costs while fulfilling all demands.

Part 4: Assigning Salespersons to Regions to Minimize Total Time

This optimization problem can be modeled as an assignment problem, where we assign each salesperson to a region such that the total time taken is minimized. The decision variables are binary, indicating whether a salesperson is assigned to a specific region.

Define decision variables: y_ij = 1 if salesperson i is assigned to region j; 0 otherwise. The objective function is:

Minimize Z = Σ Σ t_ij * y_ij

where t_ij is the number of days taken by salesperson i for region j.

Constraints include:

• Each region is assigned exactly one salesperson: Σ y_ij = 1 for all j
• Each salesperson is assigned to exactly one region: Σ y_ij = 1 for all i
• Binary assignment: y_ij ∈ {0, 1}

Solution using the Hungarian Algorithm or software like Excel Solver identifies the optimal assignment, minimizing total days spent.

The physical implementation of this model ensures an effective, efficient deployment of sales personnel, optimally covering regions while minimizing time commitments.

Conclusion

Transportation and assignment problems are vital tools for optimizing resource allocation in logistics and operational management. Accurate formulation, combined with computational solving techniques, provides actionable insights that improve efficiency and reduce costs. Applications like international shipping logistics and salesforce deployment demonstrate the versatility of these models. The integration of mathematical modeling with computer algorithms streamlines decision-making processes across various sectors, emphasizing the importance of operational research in business strategy.

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