Case Study / Scenario-Based MCQs for Sub-Topics of Topic 13: Linear Programming

Question 1. A furniture manufacturer produces tables and chairs. Producing a table yields a profit of $\textsf{₹}800$, and a chair yields a profit of $\textsf{₹}500$. The manufacturer wants to decide how many tables and chairs to produce to maximize total profit. Let $x$ be the number of tables and $y$ be the number of chairs. What is the objective function for this problem?

(A) $Z = 800 + 500$.

(B) $Z = x + y$.

(C) Maximize $Z = 800x + 500y$.

(D) Minimize $Z = 800x + 500y$.

Question 2. In the furniture manufacturing scenario from Question 1, suppose making a table requires 2 hours of assembly time and a chair requires 1 hour. The total assembly time available per week is 40 hours. What type of LPP component does the limited assembly time represent?

(A) An objective function.

(B) A decision variable.

(C) A non-negativity restriction.

(D) A constraint.

Question 3. A company is planning its production for two products, P1 and P2. The quantities of P1 and P2 to be produced are represented by $x_1$ and $x_2$ respectively. The constraints include limited machine hours and raw material. Which of the following is a fundamental concept applied to $x_1$ and $x_2$ in a typical LPP formulation of this problem?

(A) They must be integer values.

(B) They must satisfy non-negativity restrictions ($x_1 \geq 0, x_2 \geq 0$).

(C) They represent the total profit.

(D) They are coefficients in the constraints.

Question 4. A nutritionist is designing a diet using two food items, A and B, to meet minimum nutritional requirements while minimizing cost. Let the cost per kg of Food A be $\textsf{₹}100$ and Food B be $\textsf{₹}150$. Let $x_A$ and $x_B$ be the quantities (in kg) of Food A and Food B used. Which part of the LPP represents the total cost?

(A) The decision variables.

(B) The constraints.

(C) The objective function.

(D) The non-negativity restrictions.

Question 5. A small factory has 10 workers. Each worker can produce either 5 units of product X or 3 units of product Y per day. The factory wants to maximize the total number of units produced per day. Let $x$ be the number of units of X and $y$ be the number of units of Y. A crucial aspect of formulating this as an LPP is assuming the relationships are linear. Which part of this description directly leads to defining decision variables?

(A) The number of workers (10).

(B) The production rates (5 units of X, 3 units of Y per worker).

(C) The products X and Y whose production quantities need to be decided.

(D) The goal to maximize total units.

Question 6. A company seeks to minimize its transportation costs for shipping goods from a warehouse to two stores. The cost per unit to Store 1 is $\textsf{₹}10$ and to Store 2 is $\textsf{₹}15$. Let $x_1$ and $x_2$ be the number of units shipped to Store 1 and Store 2, respectively. What is the primary goal of this problem in LPP terms?

(A) Maximize profit.

(B) Minimize the objective function.

(C) Satisfy resource constraints.

(D) Find feasible solutions.

Question 1. A company produces two types of fertilizers, Grade A and Grade B. Grade A requires 2 kg of nitrogen and 1 kg of phosphate. Grade B requires 1 kg of nitrogen and 3 kg of phosphate. The company has 800 kg of nitrogen and 900 kg of phosphate. Let $x_A$ and $x_B$ be the quantities (in kg) of Grade A and Grade B produced. Write the constraint representing the limited availability of nitrogen.

(A) $2x_A + x_B \leq 800$.

(B) $x_A + 3x_B \leq 900$.

(C) $2x_A + x_B \geq 800$.

(D) $x_A + 3x_B \geq 900$.

Question 2. In the fertilizer problem from Question 1, suppose the profit per kg of Grade A is $\textsf{₹}10$ and Grade B is $\textsf{₹}12$. Formulate the objective function to maximize profit, assuming $x_A, x_B \geq 0$ are the decision variables.

(A) Maximize $Z = 800x_A + 900x_B$.

(B) Minimize $Z = 10x_A + 12x_B$.

(C) Maximize $Z = 10x_A + 12x_B$.

(D) Maximize $Z = (2x_A + x_B) + (x_A + 3x_B)$.

Question 3. A contractor has two types of trucks, Type 1 and Type 2. Type 1 has a capacity of 20 tons and Type 2 has a capacity of 30 tons. The contractor needs to transport at least 300 tons of material. The operating cost per trip for Type 1 is $\textsf{₹}5000$ and for Type 2 is $\textsf{₹}7000$. Let $x_1$ and $x_2$ be the number of trips made by Type 1 and Type 2 trucks respectively. Formulate the constraint for the minimum total tonnage transported.

(A) $x_1 + x_2 \geq 300$.

(B) $20x_1 + 30x_2 \leq 300$.

(C) $20x_1 + 30x_2 \geq 300$.

(D) $5000x_1 + 7000x_2 \geq 300$.

Question 4. In the truck problem from Question 3, the contractor wants to minimize the total operating cost. Formulate the objective function.

(A) Maximize $Z = 20x_1 + 30x_2$.

(B) Minimize $Z = 5000x_1 + 7000x_2$.

(C) Minimize $Z = x_1 + x_2$.

(D) Maximize $Z = 5000x_1 + 7000x_2$.

Question 5. A company must produce at least 100 units of Product A and 120 units of Product B. The production of A and B are limited by a total of 300 machine hours available. Product A requires 1 hour/unit and Product B requires 1.5 hours/unit. Let $x_A$ and $x_B$ be the production quantities. Which of the following is a correct constraint based on the minimum production requirement for Product A?

(A) $x_A \leq 100$.

(B) $x_A \geq 100$.

(C) $x_A + x_B \geq 100$.

(D) $x_A + 1.5x_B \leq 300$.

Question 6. Referring to Question 5, which is the correct constraint based on the machine hour limitation?

(A) $x_A \geq 100, x_B \geq 120$.

(B) $x_A + x_B \leq 300$.

(C) $x_A + 1.5x_B \leq 300$.

(D) $x_A + 1.5x_B \geq 300$.

Question 1. A farmer has 100 acres of land. They can grow either rice or sugarcane. Growing rice requires more water but less labour per acre than sugarcane. The farmer has limitations on total water available and total labour days. The farmer wants to decide how many acres of each crop to plant to maximize total profit. This is an example of which type of LPP?

(A) Diet Problem.

(B) Transportation Problem.

(C) Manufacturing/Production Problem (or Agricultural).

(D) Blending Problem.

Question 2. A company has warehouses in Delhi and Mumbai. They need to transport goods to distribution centres in Kolkata and Chennai. There is a fixed supply at each warehouse and a fixed demand at each distribution centre. The objective is to minimize the total cost of shipping. This scenario is a classic example of which type of LPP?

(A) Production Problem.

(B) Transportation Problem.

(C) Diet Problem.

(D) Assignment Problem.

Question 3. A pet food company wants to create a new dog food mix using available ingredients like chicken meal, rice, and vegetables. Each ingredient has different costs and different levels of protein, fat, and fibre. The final mix must meet minimum percentage requirements for protein and fat, and maximum limits for fibre, while minimizing the total cost of the ingredients. This problem belongs to which category of LPP?

(A) Manufacturing Problem.

(B) Diet Problem.

(C) Blending Problem.

(D) Financial Problem.

Question 4. A restaurant wants to offer daily meal combos using available quantities of rice, lentils, and vegetables. They have limited stocks of each ingredient and want to create as many combos as possible or maximize profit from the combos, where each combo requires specific amounts of ingredients. This is most similar to which LPP type?

(A) Transportation Problem.

(B) Diet Problem.

(C) Blending Problem.

(D) Manufacturing/Production Problem.

Question 5. Consider a scenario where a company needs to allocate its monthly advertising budget among TV, radio, and online platforms to maximize customer reach. There are costs associated with each platform, and minimum amounts that must be spent on certain platforms, and a total budget limit. This is best classified as a:

(A) Transportation Problem.

(B) Diet Problem.

(C) Resource Allocation Problem (a type of Production-like problem).

(D) Blending Problem.

Question 6. A problem involves deciding the quantity of steel to produce from different types of iron ore and scrap metal, given limitations on furnace capacity and chemical composition requirements (e.g., minimum carbon, maximum impurities). The objective is to minimize the cost of raw materials. This is a type of:

(A) Manufacturing Problem.

(B) Blending Problem.

(C) Transportation Problem.

(D) Diet Problem.

Question 1. A problem has the constraints $x \geq 2$, $y \geq 3$, $x \leq 6$, $y \leq 7$, and $x, y \geq 0$. Graphically, what type of shape does the feasible region form?

(A) A triangle.

(B) A rectangle.

(C) A line segment.

(D) It is unbounded.

Question 2. Consider the constraints $x+y \leq 10$, $x \geq 0$, $y \geq 0$. Which of the following points is located within the feasible region?

(A) $(5, 6)$.

(B) $(10, 1)$.

(C) $(4, 4)$.

(D) $(-2, 5)$.

Question 3. An LPP has the constraints $x \geq 100$, $y \geq 150$, $x,y \geq 0$. When graphed, the feasible region starts at the point (100, 150) and extends upwards and to the right infinitely. What term describes this type of feasible region?

(A) Bounded.

(B) Empty.

(C) Single point.

(D) Unbounded.

Question 4. Suppose the constraints of an LPP are $x+y \leq 5$, $x \leq 2$, $y \leq 2$, $x \geq 0$, $y \geq 0$. When you graph these, the feasible region is a polygon. Which of the following points is NOT a corner point of this feasible region?

(A) $(0,0)$.

(B) $(2,0)$.

(C) $(2,2)$.

(D) $(0,2)$.

Question 5. A student is trying to graph the feasible region for the constraints $x+2y \leq 8$, $x \geq 0$, $y \geq 0$. After plotting the line $x+2y=8$, they test the point (0,0). $0 + 2(0) \leq 8$ is true ($0 \leq 8$). Which side of the line should be shaded to represent $x+2y \leq 8$?

(A) The side that does not contain the origin.

(B) The side that contains the origin.

(C) Only the line itself.

(D) The side where x and y are negative.

Question 6. If an LPP has constraints $x \leq 1$, $x \geq 5$, $y \geq 0$. What does the feasible region look like?

(A) It is bounded.

(B) It is unbounded.

(C) It is a line segment.

(D) It is empty.

Question 1. Consider an LPP with feasible region defined by $x+y \leq 6$, $x \geq 0$, $y \geq 0$. Which of the following points is a feasible solution?

(A) $(7,0)$.

(B) $(3,4)$.

(C) $(5,1)$.

(D) $(-1,5)$.

Question 2. In the problem from Question 1, which of the following points is an infeasible solution?

(A) $(0,0)$.

(B) $(2,3)$.

(C) $(6,0)$.

(D) $(4,3)$.

Question 3. Suppose the feasible region for a maximization LPP is the polygon with vertices A(0,0), B(5,0), C(4,3), D(0,2). The objective function is $Z = 3x + 5y$. What is the optimal feasible solution?

Evaluate Z at each vertex:

A(0,0): $Z = 3(0) + 5(0) = 0$

B(5,0): $Z = 3(5) + 5(0) = 15$

C(4,3): $Z = 3(4) + 5(3) = 12 + 15 = 27$

D(0,2): $Z = 3(0) + 5(2) = 0 + 10 = 10$

(A) Point A(0,0).

(B) Point B(5,0).

(C) Point C(4,3).

(D) Point D(0,2).

Question 4. In the problem from Question 3, what is the optimal value of the objective function Z?

(A) 0.

(B) 15.

(C) 27.

(D) 10.

Question 5. Suppose the feasible region of an LPP is just the single point (5, 10). What can be said about the optimal feasible solution?

(A) It is unbounded.

(B) It does not exist.

(C) The point (5, 10) is the unique feasible and optimal solution for both maximization and minimization.

(D) There are multiple optimal solutions.

Question 6. An LPP has no feasible solutions. Based on the definitions, which statement is true?

(A) It must have a unique optimal solution.

(B) It has multiple optimal solutions.

(C) It has an unbounded optimal solution.

(D) It has no optimal solution.

Question 1. A bounded feasible region for an LPP has corner points at (0,0), (10,0), (8,5), (0,6). The objective function to maximize is $Z = 2x + 3y$. To find the optimal solution using the Corner Point Method, which point(s) must you evaluate Z at?

(A) Only the point furthest from the origin.

(B) Only interior points of the feasible region.

(C) All four corner points: (0,0), (10,0), (8,5), and (0,6).

(D) Only points on the boundary but not corners.

Question 2. In the scenario from Question 1, evaluate Z at the corner points:
Z(0,0) = 0
Z(10,0) = 2(10) + 3(0) = 20
Z(8,5) = 2(8) + 3(5) = 16 + 15 = 31
Z(0,6) = 2(0) + 3(6) = 18
What is the maximum value of Z and where does it occur?

(A) Max Z = 20 at (10,0).

(B) Max Z = 31 at (8,5).

(C) Max Z = 18 at (0,6).

(D) Max Z = 31, occurring at multiple points.

Question 3. A minimization problem has a bounded feasible region with corner points A, B, C. Evaluating the objective function Z at these points gives Z(A)=50, Z(B)=30, Z(C)=45. According to the Corner Point Method, what is the optimal value of Z?

(A) 50.

(B) 30.

(C) 45.

(D) Cannot be determined without knowing the objective function form.

Question 4. Suppose a maximization LPP has an unbounded feasible region. Evaluating the objective function $Z = 4x + 5y$ at the corner points (2,0) and (0,3) gives Z=8 and Z=15 respectively. However, as you move along one of the unbounded edges, Z continues to increase. What does this indicate about the optimal solution?

(A) The optimal solution is at (0,3) with Z=15.

(B) The optimal solution is at (2,0) with Z=8.

(C) The problem has multiple optimal solutions.

(D) The problem has an unbounded solution (no maximum value exists).

Question 5. An LPP has a feasible region that is a line segment between points P and Q. The objective function Z is evaluated at P and Q, and both yield the same optimal value. What does the Corner Point principle imply in this scenario?

(A) Only points P and Q are optimal solutions.

(B) No optimal solution exists because the region isn't a polygon with multiple vertices.

(C) Every point on the line segment between P and Q is an optimal solution.

(D) The optimal solution must be in the interior of the segment.

Question 6. For a minimization problem with an unbounded feasible region, if the objective function $Z = 10x + 20y$ has corner point values 100, 150, and 200 at different vertices, and the values increase as you move away from the origin along unbounded edges, where is the optimal solution located?

(A) There is no optimal solution.

(B) At the origin (if feasible).

(C) At the corner point that yielded the value 100.

(D) In the interior of the feasible region.

Question 1. To graphically solve the LPP: Maximize $Z = 4x + y$ subject to $x+y \leq 5$, $x \geq 0$, $y \geq 0$. The first step is to graph the constraints. Which line equation is part of this step?

(A) $4x + y = k$ (for some constant k).

(B) $x+y=5$.

(C) $x+y \geq 5$.

(D) $x = -y + 5y$.

Question 2. In the graphical method for a 2-variable LPP, the feasible region is identified as the area that satisfies all constraints. What is the next crucial step after finding the feasible region, according to the Corner Point Method principle used with the graphical method?

(A) Graph the objective function.

(B) Determine if the feasible region is bounded or unbounded.

(C) Identify the coordinates of all corner points of the feasible region.

(D) Check a random point inside the feasible region.

Question 3. Suppose, when using the graphical method for a maximization problem, the feasible region is unbounded. You evaluate the objective function at the corner points you found. To confirm if the solution is truly unbounded or if a maximum exists, what graphical technique can be helpful?

(A) Finding the area of the feasible region.

(B) Drawing an Iso-profit line and observing if it can move indefinitely in the maximizing direction while intersecting the feasible region.

(C) Checking if the origin is a feasible point.

(D) Calculating the slope of the constraint lines.

Question 4. When solving a minimization problem graphically, you identify the feasible region and its corner points. How do you use the objective function $Z=ax+by$ to find the minimum value at the corner points?

(A) Substitute the coordinates of each corner point into the expression $ax+by$ and find the largest resulting value.

(B) Substitute the coordinates of each corner point into the expression $ax+by$ and find the smallest resulting value.

(C) Draw the line $ax+by=0$ and find the feasible point closest to it.

(D) The minimum value is always 0 if the origin is feasible.

Question 5. A graphical solution leads to a feasible region that is a single point (a, b). What does this imply about the LPP and its solution?

(A) The problem has no feasible solution.

(B) The problem has an unbounded solution.

(C) The point (a, b) is the unique feasible and unique optimal solution for both maximization and minimization.

(D) The problem has multiple optimal solutions.

Question 6. You are using the graphical method and find that the feasible regions for two different constraints do not overlap. What is the immediate conclusion about the LPP?

(A) The problem has an unbounded solution.

(B) The problem has multiple optimal solutions.

(C) The feasible region is empty, meaning no feasible solution exists.

(D) The optimal solution is at the origin.