The Cost Less Corp Supplies Its Four Retail Outlets

The Cost Less Corp Supplies Its Four Retail Outlets From Its Four Pla

The Cost Less Corp. supplies its four retail outlets from its four plants. The shipping cost per shipment from each plant to each retail outlet is given below. Plants 1, 2, 3, and 4 make 10, 20, 20, and 10 shipments per month, respectively. Retail outlets 1, 2, 3, and 4 need to receive 20, 10, 10, and 20 shipments per month, respectively. The distribution manager, Randy Smith, now wants to determine the best plan for how many shipments to send from each plant to the respective retail outlets each month.

Randy’s objective is to minimize the total shipping cost. Formulate this problem as a transportation problem on a spreadsheet and then use Solver to obtain an optimal solution.

Paper For Above instruction

Introduction

Transportation problems are a special class of linear programming problems that involve determining the most cost-effective way to transport goods from multiple sources (plants) to multiple destinations (retail outlets). The goal is to minimize the total transportation cost while satisfying supply and demand constraints. In the context of Cost Less Corp, each plant has a fixed shipment capacity, and each retail outlet has a specific demand that must be met. Formulating this problem accurately allows the use of computational tools like Excel Solver to find an optimal transportation plan that minimizes costs.

Formulation of the Transportation Problem

The problem involves defining decision variables, constraints, and an objective function:

  • Decision Variables: Let x_ij denote the number of shipments sent from plant i to retail outlet j, where i = 1, 2, 3, 4 and j = 1, 2, 3, 4.
  • Supply Constraints: These ensure that the total shipments from each plant do not exceed its supply:
  • Plant 1: x_11 + x_12 + x_13 + x_14 = 10
  • Plant 2: x_21 + x_22 + x_23 + x_24 = 20
  • Plant 3: x_31 + x_32 + x_33 + x_34 = 20
  • Plant 4: x_41 + x_42 + x_43 + x_44 = 10
  • Demand Constraints: These ensure that the shipments received by each retail outlet meet its demand:
  • Retail outlet 1: x_11 + x_21 + x_31 + x_41 = 20
  • Retail outlet 2: x_12 + x_22 + x_32 + x_42 = 10
  • Retail outlet 3: x_13 + x_23 + x_33 + x_43 = 10
  • Retail outlet 4: x_14 + x_24 + x_34 + x_44 = 20
  • Objective Function: Minimize total transportation costs:

Cost = Σ (cost_ij * x_ij) for all i, j

Using the given data, these costs are inputted into the spreadsheet to enable Solver to optimize the decision variables.

Implementation in Spreadsheet and Use of Solver

To implement this in Excel:

  1. Input the cost matrix with rows representing the plants and columns representing retail outlets.
  2. Create a grid of decision variables (x_ij) corresponding to shipments from each plant to each retail outlet.
  3. Set up supply constraints summing each row of decision variables and matching the supply capacities.
  4. Set up demand constraints summing each column of decision variables and matching the demands.
  5. Define the total cost formula as the sumproduct of the cost matrix and decision variables.
  6. Use the Solver add-in to set the total cost as the objective, set the decision variables as changing cells, and add constraints for supplies and demands, ensuring non-negativity.
  7. Run Solver to find the optimal shipment quantities that minimize total cost while satisfying all constraints.

Conclusion

Formulating and solving the transportation problem using spreadsheet tools such as Excel Solver facilitates efficient decision-making for logistics and supply chain management. Accurate modeling ensures cost reduction and optimal resource allocation, which are critical for competitiveness. The example of Cost Less Corp demonstrates the practical application of linear programming techniques in real-world logistics planning.

References

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