### <Success style="font-size:25px"> Linear Programming Problems L-2 </Success> ### <primary style="font-size:24px"> Linear programming <br> lecture-2 </primary> <hr> <main> <div style='float: left; width:99%;font-size:30px;'> </div> <div style='float: right; width:50%;'> <!----> </main> <hr> <footer style = "font-size: 18px"> $\rightarrow$ $\rightarrow$ <span style = "color:blue"> Linear programming lecture-2 </span> $\rightarrow$ Problem-1 $\rightarrow$ Solution </footer>

### <Success style="font-size:25px"> Linear Programming Problems L-2 </Success> ### <primary style="font-size:24px"> Problem-1 </primary> <hr> <main> <div style='float: left; width:99%;font-size:30px;'> - Solve the LPP graphically - Minimize \& Maximize - $z=3 x+9 y \text { (objective) }$ - Subject to constraints - $ x+3 y \leqslant 60 \text { - (i) }$ - $ x+y \geqslant 10 \text { - (ii) }$ - $ x \leqslant y \text {- (iii) }$ - $ x \geqslant 0, y \geqslant 0 \text { - (iv) }$ </div> <div style='float: right; width:1%;'> <!--  --> </main> <hr> <footer style = "font-size: 18px"> $\rightarrow$ Linear programming lecture-2 $\rightarrow$ <span style = "color:blue"> Problem-1 </span> $\rightarrow$ Solution $\rightarrow$ Solution </footer>

### <Success style="font-size:25px"> Linear Programming Problems L-2 </Success> ### <primary style="font-size:24px"> Solution </primary> <hr> <main> <div style='float: left; width:100%;font-size:30px;'> - Solve: Associated Equation - $ \begin{aligned} & x+3 y=60 \text { - (i) } \\\\ & x+y=10 \text { - (ii) } \\\\ & x=y \text { - (iii) } \end{aligned} $ - from (i) $x+3 y=60$ - put $y=0 \Rightarrow x=60$ - $x=0 \Rightarrow y=20$ - $\therefore$ points are (60, 0) & (0,20) </div> </main> <hr> <footer style = "font-size: 18px">Linear programming lecture-2 $\rightarrow$ Problem-1 $\rightarrow$ <span style = "color:blue"> Solution </span> $\rightarrow$ Solution $\rightarrow$ Solution </footer>

### <Success style="font-size:25px"> Linear Programming Problems L-2 </Success> ### <primary style="font-size:24px"> Solution </primary> <hr> <main> <div style='float: left; width:100%;font-size:30px;'> - <ins>from (ii)</ins> - $ x+y =10 $ - $ \text { put } y =0 \Rightarrow x=10$ - $ x =0 \Rightarrow y=10$ - $\therefore$ points an (10,0) \&(0,10) - <ins>from (iii)</ins> $ \quad \quad x=y$ - $ y =0 \Rightarrow x=0$ - $ x =1 \Rightarrow y=1$ - $\therefore$ points are (0,0) \&(1,1) </div> </main> <hr> <footer style = "font-size: 18px">Problem-1 $\rightarrow$ Solution $\rightarrow$ <span style = "color:blue"> Solution </span> $\rightarrow$ Solution $\rightarrow$ Solution </footer>

### <Success style="font-size:25px"> Linear Programming Problems L-2 </Success> ### <primary style="font-size:24px"> Solution </primary> <hr> <main> <div style='float: left; width:70%;font-size:30px;'> - **origin test** - $ 0+3 \times 0=0<60(T)$ - $ 0+0=0>10(F)$ $ ~ ~ $ $ 0=0(T)$ - $ \text { for } x=y$ $ ~ ~ $ $ (1,2)$ - $ 1<2$ - (10,20) $ \quad$ (T) - 10<20 (T) </div> <div style='float: right; width:1%;'> <!--   --> </div> <div style='float: right; width:30%;'>  <!----> </main> <hr> <footer style = "font-size: 18px">Solution $\rightarrow$ Solution $\rightarrow$ <span style = "color:blue"> Solution </span> $\rightarrow$ Solution $\rightarrow$ Solution </footer>

### <Success style="font-size:25px"> Linear Programming Problems L-2 </Success> ### <primary style="font-size:24px"> Solution </primary> <hr> <main> <div style='float: left; width:99%;font-size:30px;'> - <ins>from (i) & (iii)</ins> - $ x+3 y=60$ - $ \Rightarrow x+3 x=60$ - $(x=y)$ - $ \Rightarrow \quad 4 x=60$ - $ \Rightarrow \quad x=15$ - $ \therefore y=15$ - $ B(15,15)$ </div> <div style='float: right; width:1%;'> <!--  --> </main> <hr> <footer style = "font-size: 18px">Solution $\rightarrow$ Solution $\rightarrow$ <span style = "color:blue"> Solution </span> $\rightarrow$ Solution $\rightarrow$ Solution </footer>

### <Success style="font-size:25px"> Linear Programming Problems L-2 </Success> ### <primary style="font-size:24px"> Solution </primary> <hr> <main> <div style='float: left; width:50%;font-size:30px;'> - from (ii) & (iii) - $ x+y=10$ - $ \Rightarrow x+x=10 $ - $\quad(x=y)$ - $ \Rightarrow 2 x=10$ - $\Rightarrow x=5$ - $\therefore y=5$ - $ \therefore A(5,5)$ </div> <div style='float: right; width:1%;'> <!--  --> </div> <div style='float: right; width:50%;'>  <!----> </main> <hr> <footer style = "font-size: 18px">Solution $\rightarrow$ Solution $\rightarrow$ <span style = "color:blue"> Solution </span> $\rightarrow$ Solution $\rightarrow$ Solution </footer>

### <Success style="font-size:25px"> Linear Programming Problems L-2 </Success> ### <primary style="font-size:24px"> Solution </primary> <hr> <main> <div style='float: left; width:99%;font-size:30px;'> - $\therefore$ Corner point of feasible region $A B C D$ are - $ A(5,5), B(15,15),$ - $C(0,20), D(0,10)$ - value of $Z=3 x+9 y$ at Corner points are - $ Z_A=3 \times 5+9 \times 5=60 \quad \text { (smallest) }$ - $ Z_B=3 \times 15+9 \times 15=45+135=180 \text { (largest) }$ - $ Z_C=3 \times 0+9 \times 20=0+180=180 \text { (largest) }$ - $ Z_D=3 \times 0+9 \times 10=0+90=90$ </div> <div style='float: right; width:1%;'> <!--  --> </main> <hr> <footer style = "font-size: 18px">Solution $\rightarrow$ Solution $\rightarrow$ <span style = "color:blue"> Solution </span> $\rightarrow$ Solution $\rightarrow$ Problem-2 </footer>

### <Success style="font-size:25px"> Linear Programming Problems L-2 </Success> ### <primary style="font-size:24px"> Solution </primary> <hr> <main> <div style='float: left; width:99%;font-size:30px;'> - Since feasible region is bounded region - $\therefore Z_{\text {min. }}=60 \text { at } A(5,5)$ - Since largest value at two points B and C. - $\therefore Z_{\text {max }}=180$ lies on the line $B C$. - <ins>i.e.</ins> Problem has multiple optimal solution. </div> <div style='float: right; width:1%;'> <!--  --> </main> <hr> <footer style = "font-size: 18px">Solution $\rightarrow$ Solution $\rightarrow$ <span style = "color:blue"> Solution </span> $\rightarrow$ Problem-2 $\rightarrow$ Solution </footer>

### <Success style="font-size:25px"> Linear Programming Problems L-2 </Success> ### <primary style="font-size:24px"> Problem-2 </primary> <hr> <main> <div style='float: left; width:99%;font-size:30px;'> - Solve LPP graphically - Minimize - $ Z=5 x+3 y$ - Subject to - $ x+y=6$ - $ x \leqslant 4$ - $ y \leqslant 5$ - $ x \geqslant 0, y \geqslant 0$ </div> </main> <hr> <footer style = "font-size: 18px">Solution $\rightarrow$ Solution $\rightarrow$ <span style = "color:blue"> Problem-2 </span> $\rightarrow$ Solution $\rightarrow$ Solution </footer>

### <Success style="font-size:25px"> Linear Programming Problems L-2 </Success> ### <primary style="font-size:24px"> Solution </primary> <hr> <main> <div style='float: left; width:99%;font-size:30px;'> - Solution: : Associated equation for given constraints are - $ x+y=6 \text {\ldots (i) }$ - $ x=4 \text {\ldots (ii) }$ - $ y=5 \text { \ldots (iii) }$ - from (i) $x+y=6$ - $ \text { put } y =0 \Rightarrow x=6$ - $ x =0 \Rightarrow y=6$ - $\therefore$ points are (6,0) \& (0,6) </div> <div style='float: right; width:1%;'> <!--   --> </main> <hr> <footer style = "font-size: 18px">Solution $\rightarrow$ Problem-2 $\rightarrow$ <span style = "color:blue"> Solution </span> $\rightarrow$ Solution $\rightarrow$ Solution </footer>

### <Success style="font-size:25px"> Linear Programming Problems L-2 </Success> ### <primary style="font-size:24px"> Solution </primary> <hr> <main> <div style='float: left; width:70%;font-size:30px;'> - from (ii) $x=4$ is a line parallel to $y$-axis intersect- $x$-axis at- $(4,0)$ - from (iii) $y=5$ is a line parallel to $x$-axis intersect $y$ axis at $(0,5)$ - feasible region is line AB with corner points - A(4,2) \& B(1,5) - $ \therefore Z_A=5 x+3 y$ - $ = 5 \times 4 + 3 \times 2 $ - $ = 26 $ </div> <div style='float: right; width:1%;'> <!--  --> </div> <div style='float: right; width:30%;'>  <!----> </main> <hr> <footer style = "font-size: 18px">Problem-2 $\rightarrow$ Solution $\rightarrow$ <span style = "color:blue"> Solution </span> $\rightarrow$ Solution $\rightarrow$ Problem-3 </footer>

### <Success style="font-size:25px"> Linear Programming Problems L-2 </Success> ### <primary style="font-size:24px"> Solution </primary> <hr> <main> <div style='float: left; width:99%;font-size:30px;'> - $ Z_B =5 x+3 y$ - $ =5 \times 1+3 \times 5$ - $ =20 \quad \text { (smallest) }$ - $ \therefore Z_{\text {min }} =20 \text { at } B(1,5)$ </div> </main> <hr> <footer style = "font-size: 18px">Solution $\rightarrow$ Solution $\rightarrow$ <span style = "color:blue"> Solution </span> $\rightarrow$ Problem-3 $\rightarrow$ Solution </footer>

### <Success style="font-size:25px"> Linear Programming Problems L-2 </Success> ### <primary style="font-size:24px"> Problem-3 </primary> <hr> <main> <div style='float: left; width:99%;font-size:30px;'> - Solve the LPP graphically - $ \text { Minimize } Z=3 x+5 y$ - $ \text { Subject to }$ - $ x+2 y \geqslant 10$ - $ x+y \geqslant 6$ - $ 3 x+y \geqslant 8$ - $ x \geqslant 0, y \geqslant 0$ </div> <div style='float: right; width:1%;'> <!--   --> </main> <hr> <footer style = "font-size: 18px">Solution $\rightarrow$ Solution $\rightarrow$ <span style = "color:blue"> Problem-3 </span> $\rightarrow$ Solution $\rightarrow$ Solution </footer>

### <Success style="font-size:25px"> Linear Programming Problems L-2 </Success> ### <primary style="font-size:24px"> Solution </primary> <hr> <main> <div style='float: left; width:99%;font-size:30px;'> - <ins>Solve:</ins> $ ~ $ Associated equation for given constraints are - $ x+2 y=10 \text {\ldots (i) }$ - $ x+y=6 \text { \ldots (i) }$ - $ 3 x+y=8 \text {\ldots (iii) }$ - from (i) $\quad x+2 y=10$ - $ \text { put } $ - $ y =0 \Rightarrow x=10$ - $ x =0 \Rightarrow y=5$ - $\therefore$ points and (10,0) \& (0,5) </div> <div style='float: right; width:1%;'> <!--  --> </main> <hr> <footer style = "font-size: 18px">Solution $\rightarrow$ Problem-3 $\rightarrow$ <span style = "color:blue"> Solution </span> $\rightarrow$ Solution $\rightarrow$ Solution </footer>

### <Success style="font-size:25px"> Linear Programming Problems L-2 </Success> ### <primary style="font-size:24px"> Solution </primary> <hr> <main> <div style='float: left; width:100%;font-size:30px;'> - <ins>from (ii)</ins> $ \quad x+y=6$ - put $ \quad x=0 \Rightarrow y=6$ - $y=0 \Rightarrow x=6$ - $\therefore$ points are (6,0) & (0,6) - <ins>from (iii)</ins> - $ 3 x+y=8$ - $ \text { put } y=0 \Rightarrow x=8 / 3$ - Put $y=2, x=2$ - put $ x=0 \Rightarrow y=8$ - $ \therefore$ points are (8 / 3,0) & (0,8),(2,2) </div> </main> <hr> <footer style = "font-size: 18px">Problem-3 $\rightarrow$ Solution $\rightarrow$ <span style = "color:blue"> Solution </span> $\rightarrow$ Solution $\rightarrow$ Solution </footer>

### <Success style="font-size:25px"> Linear Programming Problems L-2 </Success> ### <primary style="font-size:24px"> Solution </primary> <hr> <main> <div style='float: left; width:70%;font-size:30px;'> - <ins>origin test</ins> - for (i) - $x+y \geqslant 10$ - $0+0>10(F)$ - <ins>for (ii)</ins> - $ x+y \geq 6$ - $ 0 + 0 =0>6(F)$ - <ins>for (iii)</ins> - 3x+y $\ge$ 8 - 3 $\times$ 0+0=0>8(F) </div> <div style='float: right; width:1%;'> <!--  --> </div> <div style='float: right; width:30%;'>  <!----> </main> <hr> <footer style = "font-size: 18px">Solution $\rightarrow$ Solution $\rightarrow$ <span style = "color:blue"> Solution </span> $\rightarrow$ Solution $\rightarrow$ Solution </footer>

### <Success style="font-size:25px"> Linear Programming Problems L-2 </Success> ### <primary style="font-size:24px"> Solution </primary> <hr> <main> <div style='float: left; width:70%;font-size:20px;'> - $\therefore$ Corner points of the feasible region are - A(10,0), B(2,4), - C(1,5) & D(0,8) - $ Z_A=3 \times 10+5 \times 0=30$ - $ Z_B=3 \times 2+5 \times 4=26 \text { (smallest) }$ - $ Z_C=3 \times 1+5 \times 5=28$ - $ Z_D=3 \times 0+5 \times 8=40 \text { (Largest) }$ </div> <div style='float: right; width:1%;'> <!--  --> </div> <div style='float: right; width:30%;'>  <!----> </main> <hr> <footer style = "font-size: 18px">Solution $\rightarrow$ Solution $\rightarrow$ <span style = "color:blue"> Solution </span> $\rightarrow$ Solution $\rightarrow$ Thank you </footer>

### <Success style="font-size:25px"> Linear Programming Problems L-2 </Success> ### <primary style="font-size:24px"> Solution </primary> <hr> <main> <div style='float: left; width:99%;font-size:30px;'> - $ 3 x+5 y<26$ - $ 3 x+5 y=26$ (Draw this line) - line $ \quad 3 x+5 y=26$ - have no common points with feasible region - $\therefore \quad Z_{\text {min }}=26 \text { at } B(2,4)$ </div> <div style='float: right; width:1%;'> <!--  --> </main> <hr> <footer style = "font-size: 18px">Solution $\rightarrow$ Solution $\rightarrow$ <span style = "color:blue"> Solution </span> $\rightarrow$ Thank you $\rightarrow$ </footer>

### <Success style="font-size:25px"> Linear Programming Problems L-2 </Success> ### <primary style="font-size:24px"> Thank you </primary> <hr> <main> <div style='float: right; width:1%;'> <!--  --> </main> <hr> <footer style = "font-size: 18px">Solution $\rightarrow$ Solution $\rightarrow$ <span style = "color:blue"> Thank you </span> $\rightarrow$ $\rightarrow$ </footer>