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Linear Programming Model

7.1 Linear Programming Model
 
The image is an infographic titled 'Steps to Form a Linear Programming Model.' It features three main steps, each enclosed in a box with a corresponding number and description: 1. **Identify the decision variables**: Decision variables describe the decisions that need to be made and can be represented by x & y. 2. **Identify the objective functions**: An objective function is a function that needs to be maximized or minimized. 3. **Identify the constraints**: Present the existing constraints in the form of equations or linear inequalities, which use symbols like ≤, <, =, >, and/or ≥. Constraints must be in terms of decision variables. The infographic has a clean and structured design, using blue and white colors for clarity.
 
Forming a Mathematical Model for a Situation Based on Given Constraints and Representing the Model Graphically
Description

A mathematical model can be formulated by using the variables \(x\) and \(y\) with the constraints in each situation being \(\leq\), \(\ge\), \(\lt\) or \(\gt\).

  • The region above the straight line \(ax+by=c\) satisfies the inequalities \(ax+by\ge c\) and \(ax+by\gt c\), where \(b \gt 0\).
  • The region below the straight line \(ax+by=c\) satisfies the inequalities \(ax+by\le c\) and \(ax+by \lt c\), where \(b\gt 0\).
  • The region on the right side of the line \(ax=c\) satisfies the inequalities \(ax\ge c\) and \(ax \gt c\).
  • The region on the left side of the line \(ax=c\) satisfies the inequalities \(ax\le c\) and \(ax \lt c\).
Summary

If a mathematical model involves signs like:

  • \(\ge\) or \(\le\), then a solid line \((\overline{\hspace{1cm}})\) is used.
  • \(\lt\) or \(\gt\), then a dotted line \((\text{-} \text{ -} \text{ -} \text{ -}\text{ -})\) is used.
Objective Function

An objective function is written as:

\(k=ax+by\)
 
Example \(1\)
Question
(a) Write a mathematical model for the following situation:
  The perimeter of the rectangular photo frame must not be more than \(180\) cm.
   
(b) Present the inequalities \(x-2y \geqslant -4\) graphically.
Solution (a)

Supposed \(x\) and \(y\) are the width and length of the rectangular photo frame.

Then, \(2x+2y \text{ < }180\).
Solution (b)

Given \(x-2y \geqslant -4\).

Since \(b=-2 \ (\text{< }0)\).

Hence, the region lies below the line \(x-2y=-4\).

1. A graph displaying a line with a distinct point marked on it, illustrating a specific data value or trend.
 
Example \(2\)
Question

Graph illustrating product pricing trends, showcasing a linear programming model for analysis and decision-making.

The diagram above shows the shaded region that satisfies a few constraints of a situation.

Given \(k = x+2y\), find 

(a) the maximum value of \(k\),
(b) the minimum value of \(k\).
Solution
(a) Substitute the maximum point for the shaded region, which is \((15,55)\) into \(k = x+2y\).
  \(\begin{aligned} k&=15+2(55)\\ &=125. \end{aligned}\)
  Therefore, the maximum value of \(k\) is \(125\).
   
(b) Substitute the minimum point for the shaded region, which is \((15,8)\) into \(k = x+2y\).
  \(\begin{aligned} k&=15+2(8)\\ &=31. \end{aligned}\)
  Therefore, the minimum value of \(k\) is \(31\).
 
The views expressed in the notes here are those of the contributor and don’t necessarily represent the views of pandai.

Linear Programming Model

7.1 Linear Programming Model
 
The image is an infographic titled 'Steps to Form a Linear Programming Model.' It features three main steps, each enclosed in a box with a corresponding number and description: 1. **Identify the decision variables**: Decision variables describe the decisions that need to be made and can be represented by x & y. 2. **Identify the objective functions**: An objective function is a function that needs to be maximized or minimized. 3. **Identify the constraints**: Present the existing constraints in the form of equations or linear inequalities, which use symbols like ≤, <, =, >, and/or ≥. Constraints must be in terms of decision variables. The infographic has a clean and structured design, using blue and white colors for clarity.
 
Forming a Mathematical Model for a Situation Based on Given Constraints and Representing the Model Graphically
Description

A mathematical model can be formulated by using the variables \(x\) and \(y\) with the constraints in each situation being \(\leq\), \(\ge\), \(\lt\) or \(\gt\).

  • The region above the straight line \(ax+by=c\) satisfies the inequalities \(ax+by\ge c\) and \(ax+by\gt c\), where \(b \gt 0\).
  • The region below the straight line \(ax+by=c\) satisfies the inequalities \(ax+by\le c\) and \(ax+by \lt c\), where \(b\gt 0\).
  • The region on the right side of the line \(ax=c\) satisfies the inequalities \(ax\ge c\) and \(ax \gt c\).
  • The region on the left side of the line \(ax=c\) satisfies the inequalities \(ax\le c\) and \(ax \lt c\).
Summary

If a mathematical model involves signs like:

  • \(\ge\) or \(\le\), then a solid line \((\overline{\hspace{1cm}})\) is used.
  • \(\lt\) or \(\gt\), then a dotted line \((\text{-} \text{ -} \text{ -} \text{ -}\text{ -})\) is used.
Objective Function

An objective function is written as:

\(k=ax+by\)
 
Example \(1\)
Question
(a) Write a mathematical model for the following situation:
  The perimeter of the rectangular photo frame must not be more than \(180\) cm.
   
(b) Present the inequalities \(x-2y \geqslant -4\) graphically.
Solution (a)

Supposed \(x\) and \(y\) are the width and length of the rectangular photo frame.

Then, \(2x+2y \text{ < }180\).
Solution (b)

Given \(x-2y \geqslant -4\).

Since \(b=-2 \ (\text{< }0)\).

Hence, the region lies below the line \(x-2y=-4\).

1. A graph displaying a line with a distinct point marked on it, illustrating a specific data value or trend.
 
Example \(2\)
Question

Graph illustrating product pricing trends, showcasing a linear programming model for analysis and decision-making.

The diagram above shows the shaded region that satisfies a few constraints of a situation.

Given \(k = x+2y\), find 

(a) the maximum value of \(k\),
(b) the minimum value of \(k\).
Solution
(a) Substitute the maximum point for the shaded region, which is \((15,55)\) into \(k = x+2y\).
  \(\begin{aligned} k&=15+2(55)\\ &=125. \end{aligned}\)
  Therefore, the maximum value of \(k\) is \(125\).
   
(b) Substitute the minimum point for the shaded region, which is \((15,8)\) into \(k = x+2y\).
  \(\begin{aligned} k&=15+2(8)\\ &=31. \end{aligned}\)
  Therefore, the minimum value of \(k\) is \(31\).
 
The views expressed in the notes here are those of the contributor and don’t necessarily represent the views of pandai.
Chapter : Linear Programming
Topic : Linear Programming Model
Form 5 Additional Mathematics
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