Final Exam Fall 2014 BSc 400 Instructor Mschmidtova 120 Minu
Final Examfall 2014 Bsc 400instructor Mschmidtova120 Minutes No
Final Exam fall 2014 – BSC 400 instructor: M. Schmidtova 120 minutes, no books/no notes, only calculators are allowed. The exam involves solving linear programming problems, formulation of problems, and network analysis related to project scheduling. Students are asked to determine optimal solutions, interpret constraints, analyze the effects of changing parameters, formulate investment and profit maximization problems, construct project networks, and analyze project completion times and critical paths.
Paper For Above instruction
The final exam for BSC 400 covers key concepts in linear programming, project management, and decision analysis. It requires a comprehensive understanding of mathematical modeling, problem formulation, and analysis of constraints and objectives.
Linear Programming Problem and Optimization Analysis
The first problem involves a furniture manufacturer producing two types of coffee tables, aiming to maximize profit under skilled and unskilled labor constraints. The decision variables are the number of circular (X) and rectangular (Y) tables. Constraints include labor availability: 3 hours of skilled labor and 6 hours of unskilled labor for each circular table, and 2 hours of each for rectangular tables. The profit functions are maximized subject to these resource constraints.
Mathematically, the problem is formulated as:
Maximize Profit Z = 50X + 30Y
Subject to:
3X + 2Y ≤ 90 (skilled labor constraint)
6X + 2Y ≤ 120 (unskilled labor constraint)
X, Y ≥ 0
Using methods such as the graphical approach or simplex method, the optimal solution involves examining corner points of the feasible region. The solution yields the optimal number of each table to produce for maximum profit, along with the total profit value. The analysis includes interpretation of slack or surplus variables, implications of changing the objective function, and dual prices.
Formulating Investment and Profit Maximization Problems
The second set of problems involve formulating linear programming models.
Investment Portfolio Optimization
The problem involves allocating a maximum of $12,000 among three investment funds: municipal bonds (7%), bank CDs (8%), and high-risk accounts (12%). Constraints include a maximum of $2,000 in high-risk investments and a requirement to invest at least three times as much in municipal bonds as in CDs. The LP formulation aims to minimize risk or optimize returns while respecting these constraints.
Variables: let M = amount invested in municipal bonds; C = amount in CDs; R = amount in high-risk. The LP is formulated as:
Maximize Return R = 0.07M + 0.08C + 0.12R
Subject to:
M + C + R ≤ 12,000
R ≤ 2,000
M ≥ 3C
M, C, R ≥ 0
Florist Flower Ordering Problem
The florist needs to order roses and carnations with constraints on cost, profit, and minimum/maximum order quantities. Variables: let R = dozens of roses; C = dozens of carnations. The profit maximization LP is formulated as:
Maximize Profit P = 20R + 8C
Subject to:
20R + 5C ≤ 450 (cost constraint)
C ≥ 20 (minimum carnations)
R ≤ 60 (maximum roses)
R, C ≥ 0
Project Network Construction and Critical Path Analysis
The scheduling involves constructing a project network diagram based on activities and their immediate predecessors. Each activity is represented as an arrow or node with associated durations. The network is used to identify the critical path, which determines the minimum project duration, and activities with slack that can be delayed without affecting overall completion time.
Expected Project Duration and Critical Path
Given activity durations, the expected project completion time is calculated by summing durations along the critical path. The critical path can be identified by calculating the earliest and latest start and finish times of activities, with activities on the longest path being critical. Activities with non-zero slack have some flexibility, and their slack value indicates the amount of delay allowed without extending the project duration.
Conclusion
This comprehensive examination tests knowledge of linear programming formulations, constraint analysis, sensitivity analysis including dual prices, and project scheduling techniques. Mastery of these topics is essential for effective decision-making in operations research and project management contexts. The ability to formulate real-world problems into LP models, interpret solutions, and analyze the effects of parameter changes is fundamental for optimizing resource allocation and project efficiency.