mathematics-class-12-unit-12-chapter-01-linear-programming-problems-l-1-6-jm1-vb10m7c

### <Success style="font-size:25px"> Linear Programming Problems L-1 </Success>
### <primary style="font-size:24px"> LPP </primary>
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- (i) Constraints are represented as linear equation/ Linear inequation (one, two or more then two variable)                
- (ii) Plan of action (programming)                
- (i) & (ii) is Linear programming.

- In all the above cases -

- (i) Constraints are represented by linear equations/inequations (in one, two or more variables)                    
- (ii) a particular plan of action from several alternatives is to be chosen (Programming)                    
- (i) \& (ii) steps together is called linear programming. So, Linear Programing is a method to determining optimum  values of a linear function subject to constraints as linear equations or inequations.
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<footer style = "font-size: 18px">Linear programming $\rightarrow$ Example $\rightarrow$ <span style = "color:blue"> LPP </span> $\rightarrow$ Some Definitions $\rightarrow$ Objective function </footer>

### <Success style="font-size:25px"> Linear Programming Problems L-1 </Success>
### <primary style="font-size:24px"> Some Definitions </primary>
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- $\rightarrow ~ $ Objective function:                         
- $\rightarrow ~ $ Decision Variables:                         
- $\rightarrow ~ $ Constraints:                         
- $\rightarrow ~ $ Optimization Problem                         
- $\rightarrow ~ $ Feasible Region                         
- $\rightarrow ~ $ Feasible Solution

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<footer style = "font-size: 18px">Example $\rightarrow$ LPP $\rightarrow$ <span style = "color:blue"> Some Definitions </span> $\rightarrow$ Objective function $\rightarrow$ Decision variable </footer>

### <Success style="font-size:25px"> Linear Programming Problems L-1 </Success>
### <primary style="font-size:24px"> Optimization problem </primary>
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- A problem which seeks to maximize / minimize under certain constraints / condition.

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<footer style = "font-size: 18px">Decision variable $\rightarrow$ Constraints Condition/hurdle $\rightarrow$ <span style = "color:blue"> Optimization problem </span> $\rightarrow$ Feasible Region $\rightarrow$ Feasible Region </footer>

### <Success style="font-size:25px"> Linear Programming Problems L-1 </Success>
### <primary style="font-size:24px"> Feasible Region </primary>
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- OABC is called feasible region satisfied by given constraints.

- Unbounded feasible region

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<footer style = "font-size: 18px">Constraints Condition/hurdle $\rightarrow$ Optimization problem $\rightarrow$ <span style = "color:blue"> Feasible Region </span> $\rightarrow$ Feasible Region $\rightarrow$ Graphical method of solving a L.P.P theorem 1 </footer>

### <Success style="font-size:25px"> Linear Programming Problems L-1 </Success>
### <primary style="font-size:24px"> Feasible Region </primary>
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- $(\alpha, \beta) \in$ feasible region. then $(\alpha, \beta)$ is called feasible solution for the given constraints simultaneously.

- **Every feasible  region must  be convex set.**

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<footer style = "font-size: 18px">Optimization problem $\rightarrow$ Feasible Region $\rightarrow$ <span style = "color:blue"> Feasible Region </span> $\rightarrow$ Graphical method of solving a L.P.P theorem 1 $\rightarrow$ Graphical method of solving a L.P.P theorem 2 </footer>

### <Success style="font-size:25px"> Linear Programming Problems L-1 </Success>
### <primary style="font-size:24px"> Graphical method of solving a L.P.P theorem 1 </primary>
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- The following theorems are fundamental in solving L.P.P.

- <span style="color:red">Theorem 1:</span>
- Let $R$ be the feasible region for an LPP and $z=a x+b y$ be the objective function, when $z$ has an optimal value (Max./Min.), subject to the constraints described by linear inequalities, this optimal value must occur at a corner point (vertex) of the feasible region.

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<footer style = "font-size: 18px">Feasible Region $\rightarrow$ Feasible Region $\rightarrow$ <span style = "color:blue"> Graphical method of solving a L.P.P theorem 1 </span> $\rightarrow$ Graphical method of solving a L.P.P theorem 2 $\rightarrow$ Optimal solution </footer>

### <Success style="font-size:25px"> Linear Programming Problems L-1 </Success>
### <primary style="font-size:24px"> Graphical method of solving a L.P.P theorem 2 </primary>
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- <span style="color:red"> Theorems 2:</span>

- Let $R$ be the feasible region for an LPP and $z=a x+b y$ be the objective function If $R$ is bounded, then the objective function $z$ has both a maximum and a minimum value on $R$ and each of these occurs at a corner point (vertex) of $R$.

* <span style="color:green">Remark</span>: If $R$ is unbounded, then a max. or a min. value of the objective function may not exist, if it exist, it must occur at a corner point of $R$(by Th-1).

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<footer style = "font-size: 18px">Feasible Region $\rightarrow$ Graphical method of solving a L.P.P theorem 1 $\rightarrow$ <span style = "color:blue"> Graphical method of solving a L.P.P theorem 2 </span> $\rightarrow$ Optimal solution $\rightarrow$ Corner point and bounded region </footer>

### <Success style="font-size:25px"> Linear Programming Problems L-1 </Success>
### <primary style="font-size:24px"> Corner point and bounded region </primary>
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- **Corner Point:**

- Corner point of a feasible region is a point in the region which is the intersection of two boundary lines.

- **Bounded Region:**

- A feasible region of a system of linear inequalities is said to be bounded if it can be enclosed within a circle.

- Otherwise, it is called unbounded.
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<footer style = "font-size: 18px">Graphical method of solving a L.P.P theorem 2 $\rightarrow$ Optimal solution $\rightarrow$ <span style = "color:blue"> Corner point and bounded region </span> $\rightarrow$ Methods to solve LPP $\rightarrow$ Corner point method </footer>

### <Success style="font-size:25px"> Linear Programming Problems L-1 </Success>
### <primary style="font-size:24px"> Methods to solve LPP </primary>
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- (i) Simplex method

- (ii) Corner Point method

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<footer style = "font-size: 18px">Optimal solution $\rightarrow$ Corner point and bounded region $\rightarrow$ <span style = "color:blue"> Methods to solve LPP </span> $\rightarrow$ Corner point method $\rightarrow$ Corner point method </footer>

### <Success style="font-size:25px"> Linear Programming Problems L-1 </Success>
### <primary style="font-size:24px"> Corner point method </primary>
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- **Corner Point Method:**

- It is method of solving $L P P$.

- **STEPS:**

- 1. Find the feasible region of the LPP and determine its corner points either by inspection or by solving the two equations of the lines.

- 2. Evaluate $z=a x+b y$ at each coner point. Let $M$ and $m$ respectively denote the largest and smallest values.

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<footer style = "font-size: 18px">Corner point and bounded region $\rightarrow$ Methods to solve LPP $\rightarrow$ <span style = "color:blue"> Corner point method </span> $\rightarrow$ Corner point method $\rightarrow$ Corner point method </footer>

### <Success style="font-size:25px"> Linear Programming Problems L-1 </Success>
### <primary style="font-size:24px"> Corner point method </primary>
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- $Z = ax + by $

- $Z_A = p $ (smallest value)

- $Z_B = q $

- $Z_C = r $ (largest value)

- 3.(i) When the feasible region is bounded, $M$ and $m$ are the maximum and minimum values of $z$.

- (ii) In case, the feasible region is unbounded, we have

- (a) $M$ is the maximum value of $z$, if the open half plane determined by $a x+b y > M$ has no point in common with the feasible region.

- otherwise $z$ has no maximum value.

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<footer style = "font-size: 18px">Methods to solve LPP $\rightarrow$ Corner point method $\rightarrow$ <span style = "color:blue"> Corner point method </span> $\rightarrow$ Corner point method $\rightarrow$ Corner point method </footer>

### <Success style="font-size:25px"> Linear Programming Problems L-1 </Success>
### <primary style="font-size:24px"> Corner point method </primary>
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- $z_c=r \quad(\text { Maximum } )$
- if $a x+b y>r$ have no point in common with feasible region then $z_c=r$ is the maximum value.

- ie. $Z_{\text {max }}=r$ at $c$

- If $a x+b y>r$ have comman points with feasible region then

- $z=a x+b y$ have no maximum value.

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<footer style = "font-size: 18px">Corner point method $\rightarrow$ Corner point method $\rightarrow$ <span style = "color:blue"> Corner point method </span> $\rightarrow$ Corner point method $\rightarrow$ Problem </footer>

### <Success style="font-size:25px"> Linear Programming Problems L-1 </Success>
### <primary style="font-size:24px"> Solution </primary>
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- Associated eq.
- $ x+y=50 $
- &$3 x+y=90$

- from (i) $x+y=50$
- put $y=0 \Rightarrow x=50$
- $ x=0 \Rightarrow y=50$
- $\therefore$ points on $x+y=50$are

- (50,0)\&(0,50)

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<footer style = "font-size: 18px">Corner point method $\rightarrow$ Problem $\rightarrow$ <span style = "color:blue"> Solution </span> $\rightarrow$ Solution $\rightarrow$ Solution </footer>

### <Success style="font-size:25px"> Linear Programming Problems L-1 </Success>
### <primary style="font-size:24px"> Solution </primary>
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- $ Z_B=4 \times 20+30=110$

- $ Z_C=4 \times 0+50=50$

- feasible region is bounded region

- $\therefore Z_{\text {max }}=120$ at $A(30,0)$.
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<footer style = "font-size: 18px">Solution $\rightarrow$ Solution $\rightarrow$ <span style = "color:blue"> Solution </span> $\rightarrow$ Solution $\rightarrow$ Thank you </footer>