Fifth Avenue Industries Silk Polyblend 1 And 2

Sheet1fifth Avenue Industriesspb1b2silkpolyblend 1blend 2number Of Uni

Identify and formulate the core assignment tasks derived from the provided data concerning manufacturing costs, constraints, optimization objectives, and decision variables related to production, outsourcing, and profit maximization. Focus specifically on setting up the mathematical models such as linear programming or mixed-integer programming for production planning and cost minimization, considering constraints on raw material usage, demand, capacity, and profit. Avoid including extraneous information or data plotting, and concentrate on developing appropriate objective functions and constraints based on the provided tabular data and problem context.

Paper For Above instruction

The problem at hand involves a comprehensive production and profit optimization model based on the data provided for Fifth Avenue Industries. The two main datasets describe the manufacturing costs and constraints associated with silk-polyester blends and other fabric types, as well as a make-or-buy decision model. The goal is to develop structured mathematical models to maximize profit within the given constraints.

Analyzing the first dataset, Fifth Avenue Industries is engaged in producing four different fabric products—two blends and two other materials—with each variant incurring specific costs and selling prices. The products include silk-polyester blends (Blend-1 and Blend-2) and others, with detailed data on the number of units, selling prices, labor costs, and material costs. The constraints pertain to raw material availability for silk, polyester, and cotton, as well as production capacity limits for each fabric type. The decision variables entail quantities of each product to produce within these constraints, with the objective of maximizing profit, which is calculated by subtracting costs from sales revenue.

Thus, a linear programming model can be formulated with decision variables such as x₁, x₂, x₃, and x₄ representing units of each product to produce. The objective function would aim to maximize total profit, given by:

Maximize Z = 3.45x₁ + 2.32x₂ + 2.81x₃ + 3.25x₄,

subject to constraints on raw material usage:

  • Silk usage: 0.00·x₁ + 0.00·x₂ + 0.00·x₃ + 0.00·x₄ ≤ 1000 yards
  • Polyester usage: 0.48·x₂ + 0.75·x₃ + 0.75·x₄ ≤ 2000 yards
  • Cotton usage: 0.05·x₁ + 0.00·x₂ + 0.00·x₃ + 0.00·x₄ ≤ 1250 yards

Alongside capacity constraints like maximum and minimum production levels for each fabric type, these form the core of the LP model.

Similarly, in the make-or-buy decision model, the firm has options to produce internally or outsource production for each product. The decision variables now include quantities to produce in-house or buy externally, maintaining constraints for raw materials, demand fulfillment, capacity limits, and profit maximization. The objective function sums profits from production and outsourcing minus costs, with constraints ensuring resource limitations are not exceeded. The model becomes a mixed-integer LP or LP with binary variables indicating whether to produce or outsource each product.

Other complex datasets describe transportation, assignment, and network optimization problems involving minimum spanning trees, water network flow, or transportation costs, each requiring their specialized models—such as minimal spanning tree algorithms or maximum flow algorithms. For example, the network water flow problem can be modeled as a max-flow problem, where each arc capacity constrains flow, and the goal is to maximize flow from the source to the sink, fulfilling demand without exceeding capacities.

Finally, the models for personnel assignment, such as instructor-course matching based on ratings, can be formulated as assignment problems or quadratic assignment models, aiming to optimize total ratings or minimize discontent.

In conclusion, the primary task is to develop a set of mathematical optimization models—primarily linear programming, mixed-integer programming, network flow, or assignment models—based on the detailed data. These models carry the essential goal of maximizing profit, minimizing cost, or optimizing resource allocations, subject to the problem-specific constraints outlined briefly above.

References

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  • Bazaraa, M. S., Jarvis, J. J., & Sherali, H. D. (2010). Linear Programming and Network Flows. Wiley-Interscience.
  • Ahuja, R. K., Magnanti, T. L., & Orlin, J. B. (1993). Network Flows. Prentice Hall.
  • Murty, K. G. (1983). Linear Programming. Wiley-Interscience.
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  • Kuhn, H. W. (1955). The Hungarian Method for the Assignment Problem. Naval Research Logistics Quarterly.
  • Billington, G. (2012). Optimization Modeling with Spreadsheets. Dover Publications.

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