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Chapter Analysis
Intermediate12 pages • EnglishQuick Summary
The chapter on Linear Programming introduces the concept of optimization problems where the objective is to maximize or minimize a linear function subject to a set of linear constraints. It explains the mathematical formulation of such problems, highlighting key aspects like objective functions, constraints, and feasible regions. The chapter primarily focuses on solving linear programming problems using the graphical method, demonstrating important concepts like bounded and unbounded regions and corner-point solutions.
Key Topics
- •Linear Programming Problems
- •Objective Functions
- •Constraints and Feasible Region
- •Graphical Method
- •Bounded and Unbounded Regions
- •Corner Point Method
- •Optimization in Linear Programming
Learning Objectives
- ✓Understand the formulation of linear programming problems.
- ✓Sketch the feasible region for a set of constraints.
- ✓Apply the graphical method to solve linear programming problems.
- ✓Identify optimal solutions at corner points.
- ✓Differentiate between bounded and unbounded feasible regions.
Questions in Chapter
Maximise Z = 3x + 4y subject to the constraints: x + y ≤ 4, x ≥ 0, y ≥ 0.
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Minimise Z = -3x + 4y subject to x + 2y ≤ 8, 3x + 2y ≤ 12, x ≥ 0, y ≥ 0.
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Maximise Z = 5x + 3y subject to 3x + 5y ≤ 15, 5x + 2y ≤ 10, x ≥ 0, y ≥ 0.
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Minimise Z = 3x + 5y such that x + 3y ≥ 3, x + y ≥ 2, x, y ≥ 0.
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Maximise Z = 3x + 2y subject to x + 2y ≤ 10, 3x + y ≤ 15, x, y ≥ 0.
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Minimise Z = x + 2y subject to 2x + y ≥ 3, x + 2y ≥ 6, x, y ≥ 0. Show that the minimum of Z occurs at more than two points.
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Minimise and Maximise Z = 5x + 10y subject to x + 2y ≤ 120, x + y ≥ 60, x - 2y ≥ 0, x, y ≥ 0.
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Minimise and Maximise Z = x + 2y subject to x + 2y ≥ 100, 2x - y ≤ 0, 2x + y ≤ 200; x, y ≥ 0.
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Additional Practice Questions
Explain the concept of a feasible region in linear programming.
easyAnswer: A feasible region in linear programming is the set of all possible points that satisfy all constraints of the problem, including non-negativity constraints. It is typically represented graphically as the common area where the graphs of all constraints intersect, and solutions within this region are called feasible solutions.
What is the significance of corner points in linear programming?
mediumAnswer: Corner points, or vertices, of a feasible region are significant because, according to the Corner Point Theorem, the optimal solution (whether maximum or minimum) of a linear programming problem lies at one of the corner points if it exists. This property significantly simplifies the search for the optimal solution.
Describe how to graphically solve a linear programming problem.
mediumAnswer: To graphically solve a linear programming problem, plot the constraints on a graph to form the feasible region. Identify the corner points of this region. Evaluate the objective function at each corner point to determine which provides the optimal value.
Discuss how unbounded feasible regions affect the solution of a linear programming problem.
hardAnswer: When the feasible region is unbounded, an optimal solution might still be found, but there could be cases where no maximum or minimum exists. Additional checks are needed, such as evaluating the objective function related to boundary directions, to decide if an optimal value exists.
How can linear programming be applied in real life?
mediumAnswer: Linear programming can optimize various real-life scenarios, like resource allocation in manufacturing, profit maximization in product mix, dietary planning, and logistic problems concerning transportation and supply chain management.