A Logistics Specialist For Charm City Inc. Must Distribute C

A Logistics Specialist For Charm City Inc Must Distribute Cases Of

Formulate a linear programming problem to minimize the total transportation cost for distributing cases of parts from 3 factories to 3 assembly plants. The problem involves defining decision variables, the objective function, and all relevant constraints, but without solving the formulation.

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A logistics specialist at Charm City Inc. is responsible for distributing cases of parts from three factories—Factory A, Factory B, and Factory C—to three assembly plants—Plant 1, Plant 2, and Plant 3. The supply capacities of each factory and the demand requirements of each plant are known, along with transportation costs per case. An essential constraint is that at least 100 cases must be shipped from Factory B to Plant 2, which must be included in the formulation. The goal is to develop a linear programming model that minimizes the total transportation cost while satisfying the supply and demand constraints and honoring the minimum shipment requirement from Factory B to Plant 2.

The model involves defining decision variables that represent the number of cases shipped from each factory to each plant. The objective function will sum the transportation costs multiplied by the decision variables, aiming to minimize total costs. The constraints include supply limitations at each factory, demand requirements for each plant, and the specific minimum shipment from Factory B to Plant 2. Additional constraints guarantee non-negativity of decision variables, ensuring that shipments are not negative, and that all shipments satisfy the supply and demand balances.

Decision Variables

Let X_ij denote the number of cases shipped from Factory i to Plant j, where i = A, B, C and j = 1, 2, 3. Therefore:

• X_A1: Cases shipped from Factory A to Plant 1
• X_A2: Cases shipped from Factory A to Plant 2
• X_A3: Cases shipped from Factory A to Plant 3

Similarly, for Factory B:

• X_B1
• X_B2
• X_B3

And for Factory C:

• X_C1
• X_C2
• X_C3

Objective Function

Minimize Total Transportation Cost = ∑_i=A,B,C ∑_j=1,2,3 C_ij * X_ij

Where C_ij is the transportation cost per case from factory i to plant j. This function aims to reduce the overall cost of distributing the cases by choosing shipment quantities optimally within the given constraints.

Constraints

• Supply Constraints:
• The total cases shipped from each factory cannot exceed its supply capacity:
• X_A1 + X_A2 + X_A3 ≤ Supply_A
• X_B1 + X_B2 + X_B3 ≤ Supply_B
• X_C1 + X_C2 + X_C3 ≤ Supply_C
• Demand Constraints:
• The total cases received by each plant must meet its demand:
• X_A1 + X_B1 + X_C1 = Demand₁
• X_A2 + X_B2 + X_C2 = Demand₂
• X_A3 + X_B3 + X_C3 = Demand₃
• Specific Shipment Minimum:
• The shipment from Factory B to Plant 2 must be at least 100 cases:
• X_B2 ≥ 100
• Non-negativity Constraints:
• All decision variables must be non-negative:
• X_ij ≥ 0, for all i, j

This comprehensive formulation captures the core of the transportation problem, ensuring cost minimization while satisfying supply, demand, and specific shipment constraints without solving the model.

References

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