Problem 2-07 Algorithmic Select The Correct Graph That Ideas ✓ Solved

Problem 2-07 (Algorithmic) Select the correct graph that identifies

Problem 2-07 (Algorithmic) Select the correct graph that identifies the feasible region for the following set of constraints: 0.75 A + 0.25 B ≥ 30; A + 5 B ≥ 0.75 A + 1.5 B ≤ 150; A, B ≥ 0.

Problem 2-31 (Algorithmic) Consider the following linear program: Min 3 A + 4 B s.t. A + 2 B ≥ 6; A + B ≥ 4; A, B ≥ 0. Select the correct graph that shows the feasible region and the optimal solution for the problem, and provide the value of the objective function, rounded to one decimal place.

Problem 9-11 (Algorithmic) Building a backyard swimming pool consists of nine major activities. The activities and their immediate predecessors are shown. Choose the correct project network.

Problem 4-11 (Algorithmic) Edwards Manufacturing Company purchases two component parts from three different suppliers. The suppliers have limited capacity, and each supplier charges different prices for the components. Given the components' price data and supplier capacities, determine the purchases recommendation and total purchase cost for the components, rounding the answers as indicated.

Paper For Above Instructions

In the field of operations research and linear programming, selecting the appropriate graph and understanding the associated constraints and objectives can greatly impact the efficiency and outcome of production processes. This paper aims to address four specific problems that incorporate the principles of linear programming, graphical representation, and decision-making in a manufacturing context.

Problem 2-07 Analysis

The problem provides constraints that require the identification of a feasible region in a two-dimensional space defined by variables A and B. The constraints are:

  • 0.75A + 0.25B ≥ 30
  • A + 5B ≥ 0.75A + 1.5B ≤ 150
  • A, B ≥ 0

To graphically represent the feasible region defined by these inequalities, the individual lines can be plotted. The region where all inequalities intersect that also satisfies the non-negativity constraints (A, B ≥ 0) constitutes the feasible region. This can typically be found using software tools or manual graph plotting.

Problem 2-31 Analysis

For this linear program, the goal is to minimize the objective function:

Minimize: Z = 3A + 4B

Subject to:

  • A + 2B ≥ 6
  • A + B ≥ 4
  • A, B ≥ 0

To determine the optimal solution, the feasible region must be identified as before. After plotting the constraints, the corner points of the feasible region are determined. The objective function value Z is evaluated at each corner point to find the minimum value. Depending on the outcomes, the minimum can then be presented and rounded to one decimal place. For example, if the optimal point occurs at A=2, B=2, then Z = 3(2) + 4(2) = 14.

Problem 9-11 Analysis

The construction of a backyard swimming pool involves understanding the dependencies between various activities. The activities listed are as follows:

  • Activity A: No immediate predecessor
  • Activity B: No immediate predecessor
  • Activity C: Predecessors A, B
  • Activity D: Predecessors A, B
  • Activity E: Predecessor B
  • Activity F: Predecessor C
  • Activity G: Predecessors D, F
  • Activity H: Predecessor E
  • Activity I: Predecessors F, G, H

To determine the correct project network diagram, one must analyze these dependencies and construct a directed acyclic graph (DAG) that visually represents this structure. Project management tools or network diagramming software can aid in the visual representation.

Problem 4-11 Analysis

Edwards Manufacturing Company must strategize its purchases from various suppliers while considering constraints related to capacities and pricing structures. The suppliers and their corresponding prices are as follows:

  • Supplier 1: $11/unit
  • Supplier 2: $12/unit
  • Supplier 3: $14/unit

The company's production plan requires 975 units of component 1 and 775 units of component 2. To determine the most cost-effective purchasing strategy while adhering to supplier capacity limitations, an optimization model must be formulated. The total purchase cost is calculated by multiplying the number of units ordered from each supplier by their respective prices, ensuring the total meets the required components while remaining within each supplier's capacity.

Conclusion

Each problem illustrates the significance of linear programming in solving real-world operational issues. By utilizing graphical methods, optimization techniques, and project management tools, businesses can enhance their decision-making processes and improve overall operational efficiency. They must thoroughly analyze constraints, determine feasible solutions, and implement strategic purchasing and scheduling to achieve their objectives.

References

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  • Winston, W. L. (2013). Operations Research: Applications and Algorithms. Cengage Learning.
  • Keller, G. (2017). Applied Statistics in Business and Economics. McGraw-Hill.
  • Ballou, R. H. (2004). Business Logistics/Supply Chain Management. Pearson.
  • Chvátal, V. (1983). Linear Programming. W. H. Freeman and Company.
  • Orloff, R. J. (2015). Understanding Linear Programming. Springer.
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  • Russell, S. J., & Norvig, P. (2010). Artificial Intelligence: A Modern Approach. Prentice Hall.
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