Optimal Solutions | Mathematics Optional Notes for UPSC PDF Download

 Page 1


Edurev123 
2. Optimal Solutions 
2.1 A paint factory produces both interior and exterior paint from two raw 
materials ?? ?? and ?? ?? . The basic data is as follows : 
 Tons of raw material per ton of 
 Exterior Paint Interior Paint 
Max. Daily 
Availabiiity 
Raw Material, ?? ?? 6 4 24 
Raw Material, ?? ?? 1 2 6 
Profit per ton (Rs. 1000) 5 4  
 
A market survey indicates that the daily demand for interior paint cannot exceed 
that of exterior paint by more than 1 ton. The maximum daily demand of interior 
paint is ? tons. The factory wants to determine the optimum product mix of 
interior and exterior paint that maximizes daily profits. Formulate the LP problem 
for this situation. 
(2009 : 12 Marks) 
Solution: 
Let the required exterior paint be ?? 1
 tons and interior paint be ?? 2
 tons. 
Now as per the given problem table, 
6?? 1
+4?? 2
?=24
?? 1
+2?? 2
?=6
 
Also given that 
?? 2
-?? 1
?=1
?? 2
?=2
 
Also, we need to determine the optimum product mix of both exterior and interior paint 
that maximizes the problem. So, the LPP will be : 
Maximize :???????????????????????????????????????????????????? =5000?? 1
+4000?? 2
 
Subject to : 
Page 2


Edurev123 
2. Optimal Solutions 
2.1 A paint factory produces both interior and exterior paint from two raw 
materials ?? ?? and ?? ?? . The basic data is as follows : 
 Tons of raw material per ton of 
 Exterior Paint Interior Paint 
Max. Daily 
Availabiiity 
Raw Material, ?? ?? 6 4 24 
Raw Material, ?? ?? 1 2 6 
Profit per ton (Rs. 1000) 5 4  
 
A market survey indicates that the daily demand for interior paint cannot exceed 
that of exterior paint by more than 1 ton. The maximum daily demand of interior 
paint is ? tons. The factory wants to determine the optimum product mix of 
interior and exterior paint that maximizes daily profits. Formulate the LP problem 
for this situation. 
(2009 : 12 Marks) 
Solution: 
Let the required exterior paint be ?? 1
 tons and interior paint be ?? 2
 tons. 
Now as per the given problem table, 
6?? 1
+4?? 2
?=24
?? 1
+2?? 2
?=6
 
Also given that 
?? 2
-?? 1
?=1
?? 2
?=2
 
Also, we need to determine the optimum product mix of both exterior and interior paint 
that maximizes the problem. So, the LPP will be : 
Maximize :???????????????????????????????????????????????????? =5000?? 1
+4000?? 2
 
Subject to : 
6?? 1
+4?? 2
?=24
?? 1
+2?? 2
?=6
?? 2
-?? 1
?=1
?? 2
?=2
?? 1
,?? 2
?=0
 
2.2 Consider the following linear programming problem : 
Maximize, ?? =?? ?? +?? ?? ?? -?? ?? ?? +?? ?? ?? 
subject to 
?? ?? +?? ?? ?? +?? ?? ?? +?? ?? ?? ?=????
?? ?? +?? ?? ?? +?? ?? ?=?? ?? ?? ,?? ?? ,?? ?? ,?? ?? ?=?? 
(i) Using the definition find all its basic solutions. Which of these are degenerate 
basic feasible solutions and which are non-degenerate basic feasible solutions. 
(ii) Without solving the problem, show that it has an optimal solution. Which of the 
basic feasible solution(s) is/are optimal. 
(2015 : 20 Marks) 
Solution: 
(i) For given set of equation we make the following table : 
S.No. Basic Variables 
Non-basic 
Variables 
Solution 
Is solution 
degenerate 
….... 
1. ?? 1
,?? 2
 ?? 3
=?? 4
=0 
?? 1
=4 
?? 2
=8 
No  
2. ?? 1
,?? 3
 ?? 2
=?? 4
=0 
?? 1
=4 
?? 3
=4 
No ….. 
3. ?? 1
,?? 4
 ?? 2
=?? 3
=0 
?? 1
=-12 
?? 4
=8 
Yes …. 
4. ?? 2
,?? 3
 ?? 1
=?? 1
=0 
No 
solution 
- - 
5. ?? 2
,?? 4
 ?? 1
=?? 3
=0 
?? 2
=6 
?? 4
=2 
No …. 
Page 3


Edurev123 
2. Optimal Solutions 
2.1 A paint factory produces both interior and exterior paint from two raw 
materials ?? ?? and ?? ?? . The basic data is as follows : 
 Tons of raw material per ton of 
 Exterior Paint Interior Paint 
Max. Daily 
Availabiiity 
Raw Material, ?? ?? 6 4 24 
Raw Material, ?? ?? 1 2 6 
Profit per ton (Rs. 1000) 5 4  
 
A market survey indicates that the daily demand for interior paint cannot exceed 
that of exterior paint by more than 1 ton. The maximum daily demand of interior 
paint is ? tons. The factory wants to determine the optimum product mix of 
interior and exterior paint that maximizes daily profits. Formulate the LP problem 
for this situation. 
(2009 : 12 Marks) 
Solution: 
Let the required exterior paint be ?? 1
 tons and interior paint be ?? 2
 tons. 
Now as per the given problem table, 
6?? 1
+4?? 2
?=24
?? 1
+2?? 2
?=6
 
Also given that 
?? 2
-?? 1
?=1
?? 2
?=2
 
Also, we need to determine the optimum product mix of both exterior and interior paint 
that maximizes the problem. So, the LPP will be : 
Maximize :???????????????????????????????????????????????????? =5000?? 1
+4000?? 2
 
Subject to : 
6?? 1
+4?? 2
?=24
?? 1
+2?? 2
?=6
?? 2
-?? 1
?=1
?? 2
?=2
?? 1
,?? 2
?=0
 
2.2 Consider the following linear programming problem : 
Maximize, ?? =?? ?? +?? ?? ?? -?? ?? ?? +?? ?? ?? 
subject to 
?? ?? +?? ?? ?? +?? ?? ?? +?? ?? ?? ?=????
?? ?? +?? ?? ?? +?? ?? ?=?? ?? ?? ,?? ?? ,?? ?? ,?? ?? ?=?? 
(i) Using the definition find all its basic solutions. Which of these are degenerate 
basic feasible solutions and which are non-degenerate basic feasible solutions. 
(ii) Without solving the problem, show that it has an optimal solution. Which of the 
basic feasible solution(s) is/are optimal. 
(2015 : 20 Marks) 
Solution: 
(i) For given set of equation we make the following table : 
S.No. Basic Variables 
Non-basic 
Variables 
Solution 
Is solution 
degenerate 
….... 
1. ?? 1
,?? 2
 ?? 3
=?? 4
=0 
?? 1
=4 
?? 2
=8 
No  
2. ?? 1
,?? 3
 ?? 2
=?? 4
=0 
?? 1
=4 
?? 3
=4 
No ….. 
3. ?? 1
,?? 4
 ?? 2
=?? 3
=0 
?? 1
=-12 
?? 4
=8 
Yes …. 
4. ?? 2
,?? 3
 ?? 1
=?? 1
=0 
No 
solution 
- - 
5. ?? 2
,?? 4
 ?? 1
=?? 3
=0 
?? 2
=6 
?? 4
=2 
No …. 
6. ?? 3
,?? 4
 ?? 1
=?? 2
=0 
?? 3
=3 
?? 4
=2 
No ….. 
 
Table gives all basic solutions (?? 1
=-12,?? 4
=8) is a degenerate solution. 
Non-degenerate solutions are: 
(?? 1
,?? 2
)=(4,8)
(?? 1
,?? 3
)=(4,4)
(?? 2
,?? 4
)=(6,2)
(?? 3
,?? 4
)=(3,2)
 
(ii) ?? is optimal (from table - 1) 
 ?
(?? 1
,?? 2
)=(4,8)
(?? 2
,?? 4
)=(6,2)
 
and value of ?? is 20 . 
2.3 An agricultural firm has 180 tons of nitrogen fertilizers, 250 tons of phosphate 
and 220 tons of potash. It will be able to sell a mixture of these substances in their 
respective ratio ?? :?? :?? at a profit of Rs. 1500 per ton and a mixture in the ratio 
?? :?? :?? at a profit of Rs. 1200 per ton. Pose a linear programming problem to show 
how many tons of these two mixture should be prepared to obtain the maximum 
profit. 
(2018 : 10 Marks) 
Solution: 
Let ?? and ?? tons of mixture is brought. 
? Quantity of nitrogen is 
3?? 10
+
2?? 8
. 
Quantity of phosphate is 
3?? 10
+
4?? 8
 
Quantity of potash is 
4?? 10
+
2?? 8
 
Also, since quantity of fertilizers is limited. 
? Respective equations can be written as 
Nitrogen : 
3?? 10
+
2?? 8
=180?
3?? 10
+
?? 4
=180 
Page 4


Edurev123 
2. Optimal Solutions 
2.1 A paint factory produces both interior and exterior paint from two raw 
materials ?? ?? and ?? ?? . The basic data is as follows : 
 Tons of raw material per ton of 
 Exterior Paint Interior Paint 
Max. Daily 
Availabiiity 
Raw Material, ?? ?? 6 4 24 
Raw Material, ?? ?? 1 2 6 
Profit per ton (Rs. 1000) 5 4  
 
A market survey indicates that the daily demand for interior paint cannot exceed 
that of exterior paint by more than 1 ton. The maximum daily demand of interior 
paint is ? tons. The factory wants to determine the optimum product mix of 
interior and exterior paint that maximizes daily profits. Formulate the LP problem 
for this situation. 
(2009 : 12 Marks) 
Solution: 
Let the required exterior paint be ?? 1
 tons and interior paint be ?? 2
 tons. 
Now as per the given problem table, 
6?? 1
+4?? 2
?=24
?? 1
+2?? 2
?=6
 
Also given that 
?? 2
-?? 1
?=1
?? 2
?=2
 
Also, we need to determine the optimum product mix of both exterior and interior paint 
that maximizes the problem. So, the LPP will be : 
Maximize :???????????????????????????????????????????????????? =5000?? 1
+4000?? 2
 
Subject to : 
6?? 1
+4?? 2
?=24
?? 1
+2?? 2
?=6
?? 2
-?? 1
?=1
?? 2
?=2
?? 1
,?? 2
?=0
 
2.2 Consider the following linear programming problem : 
Maximize, ?? =?? ?? +?? ?? ?? -?? ?? ?? +?? ?? ?? 
subject to 
?? ?? +?? ?? ?? +?? ?? ?? +?? ?? ?? ?=????
?? ?? +?? ?? ?? +?? ?? ?=?? ?? ?? ,?? ?? ,?? ?? ,?? ?? ?=?? 
(i) Using the definition find all its basic solutions. Which of these are degenerate 
basic feasible solutions and which are non-degenerate basic feasible solutions. 
(ii) Without solving the problem, show that it has an optimal solution. Which of the 
basic feasible solution(s) is/are optimal. 
(2015 : 20 Marks) 
Solution: 
(i) For given set of equation we make the following table : 
S.No. Basic Variables 
Non-basic 
Variables 
Solution 
Is solution 
degenerate 
….... 
1. ?? 1
,?? 2
 ?? 3
=?? 4
=0 
?? 1
=4 
?? 2
=8 
No  
2. ?? 1
,?? 3
 ?? 2
=?? 4
=0 
?? 1
=4 
?? 3
=4 
No ….. 
3. ?? 1
,?? 4
 ?? 2
=?? 3
=0 
?? 1
=-12 
?? 4
=8 
Yes …. 
4. ?? 2
,?? 3
 ?? 1
=?? 1
=0 
No 
solution 
- - 
5. ?? 2
,?? 4
 ?? 1
=?? 3
=0 
?? 2
=6 
?? 4
=2 
No …. 
6. ?? 3
,?? 4
 ?? 1
=?? 2
=0 
?? 3
=3 
?? 4
=2 
No ….. 
 
Table gives all basic solutions (?? 1
=-12,?? 4
=8) is a degenerate solution. 
Non-degenerate solutions are: 
(?? 1
,?? 2
)=(4,8)
(?? 1
,?? 3
)=(4,4)
(?? 2
,?? 4
)=(6,2)
(?? 3
,?? 4
)=(3,2)
 
(ii) ?? is optimal (from table - 1) 
 ?
(?? 1
,?? 2
)=(4,8)
(?? 2
,?? 4
)=(6,2)
 
and value of ?? is 20 . 
2.3 An agricultural firm has 180 tons of nitrogen fertilizers, 250 tons of phosphate 
and 220 tons of potash. It will be able to sell a mixture of these substances in their 
respective ratio ?? :?? :?? at a profit of Rs. 1500 per ton and a mixture in the ratio 
?? :?? :?? at a profit of Rs. 1200 per ton. Pose a linear programming problem to show 
how many tons of these two mixture should be prepared to obtain the maximum 
profit. 
(2018 : 10 Marks) 
Solution: 
Let ?? and ?? tons of mixture is brought. 
? Quantity of nitrogen is 
3?? 10
+
2?? 8
. 
Quantity of phosphate is 
3?? 10
+
4?? 8
 
Quantity of potash is 
4?? 10
+
2?? 8
 
Also, since quantity of fertilizers is limited. 
? Respective equations can be written as 
Nitrogen : 
3?? 10
+
2?? 8
=180?
3?? 10
+
?? 4
=180 
Phosphate: 
3?? 10
+
47
8
=250?
3?? 10
+
?? 2
=250 
Potash : 
4?? 10
+
2?? 8
=220?
2?? 5
+
?? 4
=220 
The profit; 
?? =1500?? +1200?? 
? The linear programming problem is 
 Max. ?? =1500?? +1200?? ,?? ,?? =0
3?? 10
+
?? 4
=180
3?? 10
+
?? 2
=250
2?? 5
+
?? 4
=220