Linear Programming

If a problem requires a certain condition to be met 'exactly', the constraint will be expressed as:

In the constraint $ax + by \leq c$, the value 'c' usually represents:

The 'Divisibility' assumption in LPP means that:

In an LPP, if the objective is to maximize $Z$, $Z$ usually represents:

A merchant stocks two types of suitcases: Large ($x$) and Medium ($y$). He has ₹$18,000$ and storage for $50$ pieces. Large suitcases cost ₹$400$ each and Medium ones ₹$300$ each. He wants to maximize profit $Z = 50x + 40y$. Which is the investment constraint?

A company manufactures two products $A$ and $B$. The profit per unit of $A$ and $B$ is ₹$3$ and ₹$4$ respectively. $A$ takes $1$ minute and $B$ takes $2$ minutes on a machine. The machine is available for $7$ hours and $30$ minutes. If $x$ and $y$ are units of $A$ and $B$, the machine constraint in minutes is:

A diet for a sick person must contain at least $4000$ units of vitamins, $50$ units of minerals, and $1400$ calories. Two foods $F_1$ and $F_2$ cost ₹$4$ and ₹$3$ per unit respectively. $F_1$ contains $200$ units of vitamins, $1$ unit of mineral, and $40$ calories. $F_2$ contains $100$ units of vitamins, $2$ units of minerals, and $40$ calories. The mineral constraint is:

A winery produces two types of wine, Red ($x$) and White ($y$). Each batch of Red wine requires $500$ kg of grapes and $20$ hours of fermentation. Each batch of White requires $400$ kg of grapes and $15$ hours of fermentation. There are $5000$ kg of grapes and $200$ hours of fermentation time available. The winery decides that the fermentation time used for Red wine must be at least $60\%$ of the total fermentation time used. Which inequality represents this time constraint?

A machine shop produces two parts, $P_1$ and $P_2$. $P_1$ requires $5$ minutes of milling and $P_2$ requires $8$ minutes. The milling machine is available for $480$ minutes. The production of $P_2$ must be at least $20$ units, and the number of $P_1$ units must be at least twice the number of $P_2$ units minus $10$. Which inequality represents this part relationship?

An investment firm manages two portfolios, Aggressive ($x$) and Conservative ($y$). The Aggressive portfolio has an expected return of $15\%$ and the Conservative $8\%$. The firm wants a total return of at least $\$20,000$ from a total investment of $\$200,000$. Furthermore, the amount in the Aggressive portfolio cannot exceed $60\%$ of the amount in the Conservative portfolio. Which inequality represents this portfolio limit?

Which of the following constraints is a vertical line?

In a LPP, the constraints are $x \ge 0, y \ge 0$. This indicates that the feasible region lies in which quadrant?

If the objective function is $Z = x + y$, and the feasible region vertices are $(0,0), (2,0), (0,3)$, what is the maximum value of $Z$?

What is the shape of the feasible region for the constraints $x \ge 2, y \ge 2, x \le 5, y \le 5$?

For the objective function $Z = 3x + 2y$, if the feasible region is a triangle with vertices $(0,0), (4,0), (0,6)$, what is the maximum value?

If the constraints are $x + y \ge 1, x + y \le 4, x \ge 0, y \ge 0$, what is the area of the feasible region?

Which of the following describes an unbounded feasible region?

Maximize $Z = x + y$ subject to $2x + y \le 50$, $x + 2y \le 40$, $x \ge 0, y \ge 0$. The feasible region is the quadrilateral OABC. Find the value of $Z$ at the point of intersection of the two lines.

A chemical plant produces two types of solutions, S1 and S2. Each unit of S1 costs Rs 300 and S2 costs Rs 400. To minimize cost subject to the constraints $x + y \ge 8$ and $2x + y \ge 10$ ($x,y$ being units of S1 and S2), what is the minimum cost?

Find the maximum value of $Z = 5x + 3y$ for the region $3x + 5y \le 15$, $5x + 2y \le 10$, $x, y \ge 0$. The intersection of the lines is at point $P$. Calculate the coordinates of $P$ and then $Z$.

If the objective function $Z = x + y$ is to be maximized subject to $x \le 2$ and $y \le 3$ ($x,y \ge 0$), what is the maximum value?

The line $x + y = 5$ and $x + y = 10$ are parallel. If these are the only constraints besides non-negativity, what can be said about a region satisfying $x + y \ge 10$ and $x + y \le 5$?

What is the shape of the region defined by $x \ge 2, x \le 5, y \ge 1, y \le 3$?

In a diet problem, the goal is typically to minimize the cost while meeting nutritional requirements. This is an example of:

Given the constraints $y \le 3x$, $x + y \le 4$, $x, y \ge 0$, find the vertex of the feasible region other than the origin and the x-intercept.

Which of the following describes the objective function $Z = 2x + 3y$ for the feasible region shown if the region is bounded by $(0,0), (4,0), (0,4)$?

The vertices of a feasible region are $(0,10), (5,5), (15,15), (0,20)$. Which point minimizes $Z = x - y$?

In a linear programming problem, the objective function is $Z = 4x + y$. The feasible region is determined by $x + y \le 50$, $3x + y \le 90$, $x \ge 10$, $y \ge 0$. At which point is the objective function maximized?

A cooperative society of farmers has 50 hectares of land to grow two crops $X$ and $Y$. The profit from crops $X$ and $Y$ per hectare are estimated as Rs 10,500 and Rs 9,000 respectively. To control weeds, a liquid herbicide has to be used for crops $X$ and $Y$ at rates of 20 litres and 10 litres per hectare. Further, no more than 800 litres of herbicide should be used. How much land should be allocated to each crop to maximize total profit?

Find the maximum value of $Z = x + y$ subject to $x - y \le -1$, $-x + y \le 0$, $x, y \ge 0$.

Which point lies in the infeasible region for the constraint $3x + y \ge 9$?

Where does the corner point $(5, 0)$ lie for the region $x + y \le 5, x \ge 0, y \ge 0$?

Which of these is NOT a linear inequality?

If a feasible region is enclosed by boundary lines on all sides, it is:

What is the maximum value of $Z = 5x + 2y$ subject to $x + y \le 10, x \le 7, y \le 7, x, y \ge 0$?

A corner point $(x, y)$ of a feasible region must satisfy:

The feasible region for $x \ge 2$ and $y \le 5$ with $x, y \ge 0$ lies in which quadrant(s)?

Determine the maximum value of $Z = 3x + 2y$ subject to $x + y \le 100$ and $x, y$ are non-negative integers such that $x$ must be a multiple of 10.

A LPP has constraints $x \ge 0, y \ge 0, x + y \le 1$. If we add $x + y \le 2$, the feasible region:

Consider $Z = ax + by$. If the feasible region is bounded and $(x_1, y_1)$ and $(x_2, y_2)$ are two distinct optimal solutions, then: