# Linear Programming

 Question 1
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 1 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 2
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 2 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 3
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 3 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 4
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 4 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 5
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 5 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 6
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 6 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 7
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 7 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 8
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 8 Explanation:
Because the constraint on resource 2 has no effect on the feasible region.
 Question 9
Simplex method of solving linear programming problem uses
 A all the points in the feasible region B only the corner points of the feasible region C intermediate points within the infeasible region D only the interior points in the feasible region.
GATE ME 2010   Industrial Engineering
Question 9 Explanation:
Simplex method provides an algorithm which consists in moving from one point of the region of feasible solutions to another in such a manner that the value of the objective function at the succeeding point is less (or more, as the case may be) than at the preceding point. This procedure of jumping from one point to another is then repeated. Since the number of points is finite, the method leads to an optimal point in a finite number of steps.

Therefore simplex method only uses the interior points in the feasible region.
 Question 10
Consider the following Linear Programming Problem
(LPP): Maximize $z=3x_{1}+2x_{2}$
Subject to $x_{1}\leq 4$
$x_{2}\leq 6$
$3x_{1}+2x_{2}\leq 18$
$x_{1}\geq 0$ ,$x_{2}\geq 0$
 A The LPP has a unique optimal solution B The LPP is infeasible C The LPP is unbounded D The LPP has multiple optimal solutions
GATE ME 2009   Industrial Engineering
Question 10 Explanation:
Linear Programming Problem (LPP)
Maximize, $z=3 x_{1}+2 x_{2}$
\begin{aligned} \text{Constraints }\quad x_{1} &\leq 4 &\quad \ldots(i)\\ x_{2} & \leq 6 &\quad \ldots(ii)\\ 3 x_{1}+2 x_{2} & \leq 18 &\quad \ldots(iii)\\ x_{1} & \leq 0, x_{2} \leq 0 &\quad \ldots(iv) \end{aligned}
Using graphical method

Because objective function have slope same as constraint (iii) i.e. objective function is parallel to constraint. Therefore the LPP has multiple optimal solution.
For example at Point B
\begin{aligned} \text { Maximum, } z &=3 x_{1}+2 x_{2} \\ &=3(4)+2(3)=18 \end{aligned}
and at point C,
Maximum, $z=3(2)+2(6)=18$
There are 10 questions to complete.

### 2 thoughts on “Linear Programming”

1. 6th question must give < 8. in question it has given 0.