1.A firm manufactures two types of products A and B. Product A requires 2 hours of milling and 1 hour of grinding. Product B requires 1 hour of milling and 2 hours of grinding. The firm has 40 hours of milling and 50 hours of grinding available. Formulate the time constraints if $x$ units of A and $y$ units of B are produced.
2.A dietitian wishes to mix two types of foods in such a way that vitamin contents of the mixture contain at least 8 units of vitamin A and 10 units of vitamin C. Food I contains 2 units/kg of vitamin A and 1 unit/kg of vitamin C. Food II contains 1 unit/kg of vitamin A and 2 units/kg of vitamin C. Formulate the constraints.
3.A merchant plans to sell two types of computers: a desktop model and a portable model. He estimates that the total monthly demand of computers will not exceed 250 units. Formulate the demand constraint.
4.In the computer business from Q3, the desktop costs ?25,000 and the portable costs ?40,000. If his maximum investment limit is ?70,00,000 (70 Lakhs), formulate the investment constraint in its simplest form.
5.A transport company has two depots A and B with 20 and 30 trucks respectively. Three factories C, D, and E need 15, 20, and 15 trucks respectively. If $x$ trucks are sent from A to C and $y$ trucks from A to D, formulate an expression for the number of trucks sent from A to E.
6.Following the transportation problem in Q5, formulate an expression for the number of trucks sent from depot B to factory C.
7.Formulate the objective function to maximize profit if a desktop (from Q3) yields a profit of ?4500 and a portable yields a profit of ?5000.
8.Write the complete set of non-negativity constraints for a 3-variable Linear Programming Problem involving decision variables $x, y,$ and $z$.
9.Maximize $Z = 3x + 4y$ subject to the constraints $x + y \le 4, x \ge 0, y \ge 0$. Find the maximum value.
10.Minimize $Z = -3x + 4y$ subject to $x + 2y \le 8$, $3x + 2y \le 12$, $x \ge 0, y \ge 0$. Find the minimum value.
11.In the LPP from Question 10, find the maximum value of $Z$.
12.Maximize $Z = 5x + 3y$ subject to $3x + 5y \le 15$, $5x + 2y \le 10$, $x \ge 0, y \ge 0$. Find the corner point formed by the intersection of the two inequalities.
13.Solve the LPP: Maximize $Z = 4x + y$ subject to $x + y \le 50$, $3x + y \le 90$, $x, y \ge 0$. Find the maximum value.
14.For the constraints $x \le 3$, $y \le 4$, $x \ge 0, y \ge 0$, list all the corner points of the feasible region.
15.Find the maximum value of $Z = 2x + 5y$ for the feasible region defined in Question 14.
16.Find the exact coordinates of the point of intersection of the constraint lines $2x + 3y = 12$ and $x + y = 5$.
17.Maximize $Z = 3x + 9y$ subject to $x + 3y \le 60$, $x + y \ge 10$, $x \ge 0, y \ge 0$. Evaluate the objective function at all corner points and find the maximum value.
18.What is the minimum value of $Z$ in the feasible region defined in Question 17?
19.Minimize $Z = 200x + 500y$ subject to $x + 2y \ge 10$, $3x + 4y \le 24$, $x \ge 0, y \ge 0$. Find the minimum value of $Z$.
20.Minimize $Z = 3x + 5y$ subject to $x + 3y \ge 3$, $x + y \ge 2$, $x, y \ge 0$. Find the minimum value.
21.In Question 20, since the feasible region is unbounded, what specific inequality must be graphed as a dotted line to confirm the minimum value?
22.Does the open half-plane from Question 21 share any common points with the feasible region?
23.Minimize $Z = x + 2y$ subject to $2x + y \ge 3$, $x + 2y \ge 6$, $x, y \ge 0$. What is the minimum value?
24.Maximize $Z = -x + 2y$ subject to $x \ge 3$, $x + y \ge 5$, $x + 2y \ge 6$, $y \ge 0$. Is there a maximum value?
25.In Question 24, justify why the maximum value does or does not exist using the unbounded region properties.
26.If the maximum value of an objective function $Z = px + qy$ occurs at two adjacent corner points $(3, 4)$ and $(4, 3)$, what is the relationship between $p$ and $q$?
27.An LPP has a feasible region determined by the constraints $x + y \le 2$ and $x + y \ge 4$, with $x, y \ge 0$. What is the maximum value of $Z = x + y$?
28.If the optimal value of the objective function is attained at the points $(0, 20)$ and $(15, 15)$, how many optimal solutions does the LPP have?
29.If the feasible region for an LPP is bounded, what does the Extreme Point Theorem guarantee regarding the objective function?
30.Given constraints $x - y \le -1$ and $-x + y \le 0$, with $x, y \ge 0$. Does this LPP have a feasible region?
