A Logistics Specialist For Charm City Inc. Must Distr 485318
A Logistics Specialist For Charm City Inc Must Distribute Cases Of
A logistics specialist for Charm City Inc. must distribute cases of parts from 3 factories to 3 assembly plants. The monthly supplies, demands, and transportation costs per case are provided. The specialist aims to minimize total transportation cost while satisfying supply and demand constraints, with an additional requirement that at least 100 cases be distributed from factory B to assembly plant 2. You are asked to formulate a linear programming (LP) problem to model this transportation scenario.
Specifically, the task involves three main parts: (a) formulating the LP model, including decision variables, objective function, and constraints, and (b) solving the LP using software (like Excel or QM for Windows), then interpreting the optimal solution and value. Avoid solving the LP manually; only provide the formulation and insights from the software output.
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Title: A Logistics Specialist For Charm City Inc Must Distribute Cases Of
Introduction
The effective distribution of goods from multiple sources to multiple destinations is critical for minimizing costs and ensuring supply chain efficiency. In this context, Charm City Inc.'s logistics specialist faces a transportation problem involving three factories supplying three assembly plants. The goal is to minimize transportation costs while meeting supply constraints and contextual requirements. Formulating an appropriate linear programming model provides a framework for optimizing this distribution problem, which can subsequently be solved using computational tools such as Excel Solver or QM for Windows.
Decision Variables
Let the decision variables represent the number of cases transported from each factory to each assembly plant. Define:
- \(x_{ij}\): the number of cases shipped from factory \(i\) to assembly plant \(j\), where \(i = 1, 2, 3\) (factories) and \(j = A, B, C\) (plants).
Therefore, the decision variables are:
- \(x_{1A}\), \(x_{1B}\), \(x_{1C}\)
- \(x_{2A}\), \(x_{2B}\), \(x_{2C}\)
- \(x_{3A}\), \(x_{3B}\), \(x_{3C}\)
Objective Function
The objective is to minimize total transportation costs, calculated as the sum of the product of the number of cases transported and the per-case transportation cost for each factory-plant pair. Let \(c_{ij}\) denote the transportation cost per case from factory \(i\) to plant \(j\). Then, the objective function is:
\[ \text{Minimize} \quad Z = \sum_{i=1}^{3} \sum_{j \in \{A,B,C\}} c_{ij} \times x_{ij} \]
This function aims to select shipment quantities that minimize total costs while satisfying all constraints. The specific values of \(c_{ij}\) depend on provided data, which should be incorporated into the model.
Constraints
The LP formulation includes supply and demand constraints, as well as the special minimum shipment constraint from factory B to plant 2.
- Supply Constraints: For each factory, total shipments cannot exceed available supplies:
- \(x_{1A} + x_{1B} + x_{1C} \leq S_1\)
- \(x_{2A} + x_{2B} + x_{2C} \leq S_2\)
- \(x_{3A} + x_{3B} + x_{3C} \leq S_3\)
- Demand Constraints: For each assembly plant, total received cases must meet the demand:
- \(x_{1A} + x_{2A} + x_{3A} \geq D_A\)
- \(x_{1B} + x_{2B} + x_{3B} \geq D_B\)
- \(x_{1C} + x_{2C} + x_{3C} \geq D_C\)
- Additional Shipment Constraint: Enforce at least 100 cases to be shipped from factory B to plant 2:
- \(x_{2B} \geq 100\)
- Non-negativity: Shipment quantities cannot be negative:
- \(x_{ij} \geq 0\) for all \(i, j\).
These constraints collectively ensure that supply limits, demand needs, and specific shipment requirements are satisfied in the optimized solution.
Conclusion
The formulated LP model captures the essential elements of the transportation problem faced by Charm City Inc.'s logistics specialist. By defining decision variables, constructing an objective function to minimize costs, and establishing constraints to ensure supplies, demands, and special conditions are met, the model provides a comprehensive framework for optimization. With this formulation, the next step involves solving the LP using software tools to determine the optimal distribution plan that minimizes costs while satisfying all requirements.
References
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- IBM Knowledge Center. (2022). IBM ILOG CPLEX Optimization Studio Documentation.
- Google Sheets / Excel Solver Guides. (2023). Optimization Modeling Techniques.
- Murty, K. G. (1983). Linear Programming. Wiley-Interscience.
- Shapiro, J. F. (2007). Modeling the Supply Chain. Thomson.
- Jawaharlal, K. (2019). Supply Chain Management Optimization. Springer.