Linear Programming

Linear programming is a method for the ________ of a linear objective function.

What is the shape of the region defined by $x \geq 0, y \geq 0, x \leq 2, y \leq 2$?

The maximum value of $Z = 3x + 4y$ subject to $x \geq 0, y \geq 0$ and $x + y \leq 1$ is:

Which of the following is a non-negativity constraint in a standard linear programming problem involving two products $x$ and $y$?

The optimal value of the objective function is attained at the points where the 'isoprofit' line $ax + by = k$:

If the feasible region for a LPP is unbounded, then a maximum value of the objective function $Z = ax + by$:

An objective function $Z = 3x + 2y$ is to be maximized subject to $x + y \leq 4, x \geq 0, y \geq 0$. What is the maximum value?

The feasible region for an LPP is shown in the diagram. If the objective function is $Z = px + qy$ where $p, q > 0$, and the maximum value occurs at both $A(0, 10)$ and $B(5, 5)$, what is the relationship between $p$ and $q$?

An LPP objective function is $Z = 2x + 5y$. The feasible region is bounded by the vertices $(0, 2), (0, 5), (2, 4), (5, 0)$. Which vertex represents the maximum value of the objective function?

In a linear programming problem, the constraints are $x + 2y \le 120$, $x + y \ge 60$, $x - 2y \ge 0$, and $x, y \ge 0$. The feasible region is a triangle. Find the coordinates of the vertex that provides the minimum value for $Z = 5x + 10y$.

A vendor sells sodas ($x$) and juice ($y$). He wants the number of juices to be no more than 40% of the total bottles. The constraint is:

Which of the following describes the condition where the number of units produced cannot be negative?

A constraint given as $x - y \geq 0$ means:

In a transportation problem, the total supply from a warehouse cannot exceed its capacity of 500 units. If $x$ and $y$ are units sent to two destinations, then:

In a logistics problem, a truck can carry a maximum of 1000 kg. It carries two types of boxes: Box A (20 kg) and Box B (50 kg). The number of Box A must be at least twice the number of Box B. The weight constraint is:

A company produces two types of juices, Apple and Orange. Each liter of Apple juice requires 2 kg of apples and each liter of Orange juice requires 3 kg of oranges. The company has 100 kg of apples and 150 kg of oranges. The constraint for oranges is:

A publisher sells a hardcover book for ₹600 and a paperback for ₹200. It costs ₹300 to produce a hardcover and ₹100 for a paperback. The publisher has ₹30,000 for production. Let $x$ be hardcover and $y$ be paperback. The production cost constraint is:

A mobile manufacturer makes two models: $M_1$ and $M_2$. Each $M_1$ requires 10 units of component $X$ and 5 units of component $Y$. Each $M_2$ requires 6 units of $X$ and 10 units of $Y$. There are 600 units of $X$ and 500 units of $Y$ available. If the production of $M_1$ should be at least 20% of the total production, the constraint is:

A printing company prints two types of magazines: $A$ and $B$. Magazine $A$ requires 2 hours of layout and 1 hour of printing. Magazine $B$ requires 1 hour of layout and 3 hours of printing. Total layout time is 40 hours and printing time is 60 hours. If the company must print at least 5 of each, and the total magazines cannot exceed 30, and the number of $B$ magazines must be at least 3/4 of the number of $A$ magazines, the ratio constraint is:

A company produces two types of toys: $A$ and $B$. Toy $A$ requires 4 minutes of molding and 8 minutes of painting. Toy $B$ requires 6 minutes of molding and 4 minutes of painting. The molding machine is available for 2 hours and the painting machine for 3 hours. If the number of $B$ toys must be at least twice the number of $A$ toys minus 10, the constraint is:

If a feasible region is defined by $x + y \le 2$, $x \ge 0$, $y \ge 0$, how many corner points (vertices) does it have?

At which point in the feasible region shown does the function $Z = y - x$ reach its maximum value?

The region $x \ge 0, y \ge 0$ is the first quadrant. If we add $x \le 5$, the region becomes a vertical strip. What is the $x$-coordinate of all points on the right boundary?

For the constraint $x + y \le 1$, $x \ge 0$, $y \ge 0$, identify the shaded feasible region in the plot.

In an LPP, the constraints are $x + 2y \le 10$, $3x + y \le 15$, $x, y \ge 0$. What is the corner point where the two boundary lines $x + 2y = 10$ and $3x + y = 15$ intersect?

For the objective function $Z = 200x + 500y$, the corner points are $(0, 20), (10, 10), (30, 0)$. Find the minimum value of $Z$.

The feasible region is bounded by $x+y \le 6$, $y \le 4$, $x \le 4$, $x,y \ge 0$. Identify the vertex that is NOT on the axes.

A craftsman makes wooden clocks and frames. Clock production takes 2 hours and frame production takes 1 hour. Total labor hours available are 8. Clock uses 1 unit of wood and frame uses 2 units. Total wood available is 10 units. Profit for clock is Rs 300 and frame is Rs 200. Which point in the feasible region yields the highest profit?

Two vitamins $V_1$ and $V_2$ are to be extracted from two foods $F_1$ and $F_2$. One unit of $F_1$ gives 2 units of $V_1$ and 1 unit of $V_2$. One unit of $F_2$ gives 1 unit of $V_1$ and 2 units of $V_2$. Requirements are at least 10 units of $V_1$ and 10 units of $V_2$. Costs are Rs 5 for $F_1$ and Rs 7 for $F_2$. Find the optimal mix $(x, y)$ of $F_1$ and $F_2$ to minimize cost.

A factory makes two types of bolts, A and B. Processing requires two machines, $M_1$ and $M_2$. Bolt A requires 1 hour on $M_1$ and 3 hours on $M_2$. Bolt B requires 2 hours on each. $M_1$ is available for 10 hours and $M_2$ for 15 hours. If the profit on A is Rs 5 and on B is Rs 4, find the coordinates that maximize profit.

If the constraints are $y \geq 0$ and $y \leq -2$, the feasible region is:

The feasible region for $x - y \leq 0, x \geq 0, y \geq 0$ contains the point:

In LPP, the constraints $x \geq 0, y \geq 0$ are known as:

If a problem has the constraint $x = 2$, the feasible region must lie on:

If the feasible region is bounded by $x+y=1$, $x=0$, and $y=0$, then any point $(x, y)$ inside the region satisfies:

In the LPP, the feasible region is the set of all points that satisfy:

For a system $x + y \geq 5, x \leq 3, y \leq 3, x, y \geq 0$, the corner points include:

If the objective function $Z = ax + by$ is such that $a < 0$ and $b < 0$, and the feasible region is the first quadrant, where does the maximum occur?

The region defined by $x + y \geq 4, 2x + y \leq 2, x, y \geq 0$ has how many vertices?

Consider the constraints $x + 2y \leq 8, 2x + y \leq 8, x, y \geq 0$. The region is symmetric about which line?

Given the objective function $Z = x + 2y$, and corner points $(0, 0), (4, 0), (2, 2), (0, 3)$, what is the maximum value of $Z$?

If $Z = 3x - y$, what is the value of $Z$ at the vertex $(2, 6)$?

In an LPP, the constraints are usually represented by:

For $Z = 2x + 8y$, which corner point gives the maximum value among $(0, 0), (6, 0), (4, 1), (0, 3)$?

Find the maximum of $Z = x + 2y$ given the corner points are $(0, 0), (2, 0), (1, 1),$ and $(0, 1.5)$.

Maximize $Z = 10x + 10y$ subject to $x + y \le 1, x, y \ge 0$. The maximum value of $Z$ is achieved at:

If the feasible region is bounded by $(0,0), (2,0), (0,3)$, the objective function $Z = 5x + 7y$ has its maximum value:

For the LPP: Maximize $Z = 5x + 3y$ subject to $3x + 5y \le 15, 5x + 2y \le 10, x, y \ge 0$. The maximum value of $Z$ is:

Maximize $Z = 3x + 2y$ subject to $x + 2y \le 10, 3x + y \le 15, x \ge 0, y \ge 0$. The corner points are $(0, 0), (5, 0), (4, 3), (0, 5)$. The maximum value of $Z$ is:

In a LPP, if the objective function $Z = ax + by$ is to be maximized and the feasible region is the quadrilateral with vertices $(0, 0), (5, 0), (3, 4), (0, 4)$, and $Z$ is maximum at $(3, 4)$ and $(0, 4)$, then:

In a diet problem, which constraint is likely to be used for a nutrient like 'Vitamin C'?

What is the shape of the region defined by $x \ge 2, x \le 5, y \ge 0$?

Can the objective function value be negative at the optimal solution?

The term 'Feasible' in LPP refers to solutions that are:

If $Z = x - y$, and the feasible region corner points are $(0, 0), (5, 0), (5, 5),$ and $(0, 8)$, find the minimum value of $Z$.

For the constraints $x + y \le 6, x \ge 2, y \ge 0$, which of the following is a corner point of the feasible region?

The solution set of the inequality $2x + y > 5$ is:

In a linear programming problem, the objective function to be maximized is $Z = 3x + ky$, where $k$ is a positive constant. The feasible region is determined by the constraints $x + y \le 5$, $2x + y \le 8$, $x \ge 0$, and $y \ge 0$. If the maximum value of $Z$ occurs uniquely at the corner point $(3, 2)$, which is the intersection of the boundary lines $x + y = 5$ and $2x + y = 8$, find the range of values for $k$.

For the constraints $x+y \le 4, x \ge 0, y \ge 0$, and objective function $Z = 3x + 4y$, the minimum value is:

In the context of LPP, a constraint is 'binding' or 'active' at a solution point if:

Find the minimum value of $Z = 2x + 5y$ over the region defined by $(0, 2)$, $(4, 0)$, and $(5, 5)$.

If the constraints are $x \le 5$ and $y \le 5$ with $x, y \ge 0$, the feasible region is a:

In a LPP, the objective function is always:

Calculate the value of $Z = x - y$ at the vertex $(4, 2)$ of a feasible region.

Given $Z = 5x + 10y$, and corner points $(0, 20)$, $(10, 10)$, $(15, 0)$. Where does the minimum value of $Z$ occur?

If the feasible region is bounded by $x+y=2$, $x=0$, $y=0$, what is the maximum of $Z = x + 2y$?

Which of the following describes the region $y - x \le 1$, $y \ge 0$, $x \le 2$ in the first quadrant?

Solve the following LPP graphically: Maximize $Z = 4x + y$ subject to $x + y \le 50$, $3x + y \le 90$, $x \ge 10$, $y \ge 0$. Determine the coordinates of the corner point that yields the maximum value.

A small firm manufactures necklaces and bracelets. The total number of items it can handle per day is at most 24. A bracelet takes 1 hour to make and a necklace takes 1.5 hours. The maximum time available per day is 30 hours. If the profit on a bracelet is Rs 100 and on a necklace is Rs 150, what is the maximum profit?

An advertising agency is planning a campaign. They can use Radio and TV ads. Each Radio ad costs Rs 4,000 and reaches 10,000 people. Each TV ad costs Rs 20,000 and reaches 60,000 people. The agency has a budget of Rs 100,000 and must use at least 2 TV ads. Also, the number of Radio ads must be at least the number of TV ads. To maximize reach, how many Radio ads should be used?