Linear Programming

Question 1
A manufacturing unit produces two products P1 and P2. For each piece of P1 an P2, the table below provides quantities of materials M1, M2, and M3 required, and also the profit earned. The maximum quantity available per day for M1, M2 and M3 is also provided. The maximum possible profit per day is Rs __________.
\begin{array}{|c|c|c|c|c|} \hline &\text{M1 } &\text{M2}&\text{M3}&\text{Profit per piece (Rs.)} \\ \hline P1 & 2 &2 &0&150\\ \hline P2 & 3& 1&2&100\\ \hline \text{Maximum quantity available per day}& 70& 50&40\\ \hline \end{array}
A
5000
B
4000
C
3000
D
6000
GATE ME 2022 SET-2   Industrial Engineering
Question 1 Explanation: 
Let,
x_1= No. of units of product P_1
x_2= No. of units of product P_2
Max, Z=150x_1+100x_2
\begin{aligned} s.t.\;\; 2x_1+3x_2 &\leq 70 \\ 2x_1+3x_2&\leq 50 \\ 2x_2 &\leq 40 \\ x_1\geq 0, x_2&\geq 0 \end{aligned}

Point B: Intersection of

\begin{aligned} 2x_1+3x_2 &\leq 70 \\ 2x_2 &\leq 40 \\ x_2=20, x_1&=5 \end{aligned}

Point C: Intersection of
\begin{aligned} 2x_1+3x_2 &\leq 70 \\ 2x_1+x_2 &\leq 50 \\ \text{By sloving }2x_2 &=20 \\ \Rightarrow x_2=10, x_1&=20 \end{aligned}

\begin{aligned} Z_{max}&=150x_1+100x_2\\ Z|_{A(0,20)}&=150(0)+100(20)=2000\\ Z|_{B(5,20)}&=150(5)+100(20)=2750\\ Z|_{C(20,10)}&=150(20)+100(10)=4000\\ Z|_{D(25,0)}&=150(25)+100(0)=3750\\ \Rightarrow x_1&=20,x_2=10\\ Z_{max}&=4000 \end{aligned}
Question 2
In a linear programming problem, if a resource is not fully utilized, the shadow price of that resource is
A
positive
B
negative
C
zero
D
infinity
GATE ME 2022 SET-1   Industrial Engineering
Question 2 Explanation: 
In a Linear programming problem if a resource is not fully utilized. The shadow price or dual price of the particular resource is zero.
Question 3
Two business owners Shveta and Ashok run their businesses in two different states. Each of them, independent of the other, produces two products A and B, sells them at Rs. 2,000 per kg and Rs, 3,000 per kg, respectively, and uses Linear Programming to determine the optimal quantity of A and B to maximize their respective daily revenue. Their constraints are as follows: i) for each business owner, the production process is such that the daily production of A has to be at least as much as B, and the upper limit for production of B is 10 kg per day, and ii) the respective state regulations restrict Shveta?s production of A to less than 20 kg per day and Ashok's production of A to less than 15 kg per day. The demand of both A and B in both the states is very high and everything produced is sold.

The absolute value of the difference in daily (optimal) revenue of Shveta and Ashok is ________ thousand Rupees (round off to 2 decimal places)
A
6
B
7
C
12
D
10
GATE ME 2020 SET-1   Industrial Engineering
Question 3 Explanation: 
\begin{aligned} \text { Maximum } z &=2000 x_{1}+3000 x_{2} \\ A \rightarrow x_{1} \text { units } \quad x_{1} &\geq x_{2} \\ B \rightarrow x_{2} \text { units } \quad x_{2} &\geq 10 \\ x_{1} & \lt 20 \\ x_{1} & \lt 15 \\ \text { Shveta's Profit } &=\text { Rs. } 70000 \text { at }(20,10) \\ \text { Ashok's Profit } &=\text { Rs. } 60000 \text { at }(15,10) \end{aligned}
Difference Rs. 10000
Question 4
Two models, P and Q, of a product earn profits of Rs. 100 and Rs. 80 per piece, respectively. Production times for P and Q are 5 hours and 3 hours, respectively, while the total production time available is 150 hours. For a total batch size of 40, to maximize profit, the number of units of P to be produced is ____________.
A
12
B
15
C
18
D
20
GATE ME 2017 SET-1   Industrial Engineering
Question 4 Explanation: 
Max. Z=100 x P + 80 x Q

\begin{aligned} 5P+3Q&\leq 150\\ P+Q&\leq 40\\ \frac{P}{30}+\frac{Q}{30}&=1 \end{aligned}
Putting value of corner point in objective function
Z(A)=3200
Z(B)=3500
Z(C)=3000
Maximum at B,
P=15, Q=25
Question 5
Maximize Z=15X_{1}+20X_{2}
subject to
12X_{1}+4X_{2}\geq 36
12X_{1}-6X_{2}\leq24
X_{1},X_{2}\geq 0
The above linear programming problem has
A
infeasible solution
B
unbounded solution
C
alternative optimum solutions
D
degenerate solution
GATE ME 2016 SET-1   Industrial Engineering
Question 5 Explanation: 
\begin{aligned} 12X_1+4X_2&\geq 36 \\ 3X_1-X_2 & \geq 9 \\ \\ 12X_1-6X_2&\leq 24 \\ 2X_1-X_2&\leq 4 \\ \text{Let},&\\ 3X_1+X_2&=9 \\ X_1=0 & X_2=9\\ X_1=3 & X_2=0\\ \\ 2X_1-X_2&=4\\ X_1=0 & X_2=-4\\ X_1=2 & X_2=0 \end{aligned}

Cylindrical plug gauge is used to measure inside diameter of straight hole.
Question 6
For the linear programming problem:
Maximize Z=3X_{1}+2X_{2}
Subject to
-2X_{1}+3X_{2} \leq 9
X_{1}-5X_{2}\geq -20
X_{1},X_{2}\geq 0
The above problem has
A
unbounded solution
B
infeasible solution
C
alternative optimum solution
D
degenerate solution
GATE ME 2015 SET-3   Industrial Engineering
Question 6 Explanation: 
\begin{aligned} z &=3 x_{1}+2 x_{2} \\-2 x_{1}+3 x_{2} & \leq 9 \\ x_{1}-5 x_{2} & \geq-20 \\ x_{1}, x_{2} & \leq 0 \end{aligned}
\therefore changing the inequalities to equalities
-2 x_{1}+3 x_{2}=9
x_{1}-5 x_{2}=-20


\therefore The soluction is unbounded.
Question 7
Consider an objective function Z(x_{1},x_{2})=3x_{1}+9x_{2} and the constraints
x_{1}+x_{2}\leq 8,
x_{1}+2x_{2}\leq 4,
x_{1}\geq 0,x_{2}\geq 0.
The maximum value of the objective function is __________
A
98
B
18
C
48
D
20
GATE ME 2014 SET-3   Industrial Engineering
Question 7 Explanation: 


\begin{aligned} (Z)_{(4,0)}&=3 \times 4+9 \times 0=12 \\ (Z)_{(0,2)}&=3 \times 0+9 \times 2=18 \\ (Z)_{(0,0)}&=0 \quad \text { (Trivial solution) } \\ Z_{\max }&=18 \end{aligned}
Question 8
A linear programming problem is shown below.
Maximize
3x+7y

Subject to
3x+7y\leq 10
4x+6y\leq 8
x,y\geq 0
It has
A
an unbounded objective function
B
exactly one optimal solution.
C
exactly two optimal solutions.
D
infinitely many optimal solutions.
GATE ME 2013   Industrial Engineering
Question 8 Explanation: 


\begin{aligned}\text { Max. } 3 x&+z y \\ \text { Sub to } 3 x&+7 y \leq 10\\ 4 x&+6 y \leq 8 \\ &x, y \geq 0 \end{aligned}
\Rightarrow It has exactly one optimal solution.
Question 9
One unit of product P_{1} requires 3 kg of resource R_{1} and 1kg of resource R_{2}. One unit of product P_{2} requires 2kg of resource R_{1} and 2kg of resource R_{2} . The profits per unit by selling product P_{1} and R_{2} are Rs.2000 and Rs.3000 respectively. The manufacturer has 90kg of resource R_{1} and 100kg of resource R_{2}

The manufacturer can make a maximum profit of Rs.
A
60000
B
135000
C
150000
D
200000
GATE ME 2011   Industrial Engineering
Question 9 Explanation: 
Since all Z_j-C_j\geq 0, an optimal basic feasible solution has been attained. Thus, the optimum solution to the given LPP is
Max \; Z=2000 \times 0+3000 \times 45=135000 \text{ with }P-1=0 \text{ and }P_2=45
Question 10
One unit of product P_{1} requires 3 kg of resource R_{1} and 1kg of resource R_{2} . One unit of product P_{2} requires 2kg of resource R_{1} and 2kg of resource R_{2} . The profits per unit by selling product P_{1} and R_{2} are Rs.2000 and Rs.3000 respectively. The manufacturer has 90kg of resource R_{1} and 100kg of resource R_{2}

The unit worth of resource R_{2} i.e., dual price of resource R_{2} in Rs. Per kg is
A
0
B
1350
C
1500
D
2000
GATE ME 2011   Industrial Engineering
Question 10 Explanation: 
Because the constraint on resource 2 has no effect on the feasible region.
There are 10 questions to complete.

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