Completing Statements MCQs for Sub-Topics of Topic 13: Linear Programming

Question 1. Linear Programming is a mathematical method used to find the optimal solution for a problem, which means finding the____

(A) average value of decision variables.

(B) minimum constraints required.

(C) best possible value of the objective function.

(D) number of feasible regions.

Question 2. The primary purpose of the objective function in an LPP is to express____

(A) the limitations on resources.

(B) the quantity of each variable produced.

(C) the goal to be optimized (maximized or minimized).

(D) the non-negativity of variables.

Question 3. Constraints in a Linear Programming Problem represent the____

(A) variables that we need to decide.

(B) profit or cost per unit.

(C) restrictions or limitations imposed by the problem conditions.

(D) optimal value of the objective function.

Question 4. The variables whose values are to be determined in an LPP are called____

(A) constraints.

(B) coefficients.

(C) objective values.

(D) decision variables.

Question 5. The "linear" aspect of Linear Programming refers to the fact that all relationships within the problem, including the objective function and constraints, are____

(A) graphed as straight lines.

(B) proportional to each other.

(C) expressed as linear equations or inequalities.

(D) non-negative.

Question 6. A common restriction in LPPs, stating that decision variables must be greater than or equal to zero, is known as the____

(A) resource constraint.

(B) non-negativity restriction.

(C) upper bound constraint.

(D) equality constraint.

Question 7. If a company wants to maximize its profit from producing two products, the objective function will be a linear function representing the total profit, and the coefficients of the decision variables in this function will be the____

(A) quantity of resources used per unit.

(B) profit per unit of each product.

(C) total available resources.

(D) minimum production requirements.

Question 8. A constraint like $5x_1 + 2x_2 \leq 100$ where $x_1$ and $x_2$ are production quantities most likely represents a limitation on the availability of a____

(A) minimum required output.

(B) resource.

(C) desired profit level.

(D) fixed cost.

Question 9. The purpose of "Programming" in Linear Programming is best described as____

(A) creating computer programs.

(B) scheduling or planning activities optimally.

(C) graphing linear functions.

(D) data analysis.

Question 10. If a constraint is stated as "at least 50 units must be produced", this implies the use of a____ Bayes' Theorem

(A) $\leq$ inequality.

(B) $\geq$ inequality.

(C) $=$ equality.

(D) $\neq$ inequality.

Question 1. The first step in mathematically formulating a real-world problem as an LPP is usually to clearly define the____

(A) optimal solution.

(B) constraints.

(C) decision variables.

(D) feasible region.

Question 2. The general structure of an LPP involves optimizing (maximizing or minimizing) a linear objective function subject to a set of linear constraints and____

(A) non-linear constraints.

(B) non-negativity restrictions on decision variables.

(C) integer value restrictions on variables.

(D) quadratic objective function.

Question 3. If a company produces two items, X and Y, and production is limited by labour hours and machine hours, the constraints for available labour hours would be formulated by considering the labour hours required per unit of X and Y and the____

(A) profit per unit of X and Y.

(B) selling price of X and Y.

(C) total labour hours available.

(D) market demand for X and Y.

Question 4. A constraint of the form $ax + by = c$ in an LPP represents a restriction where the linear combination of variables must be____

(A) at most $c$.

(B) at least $c$.

(C) exactly $c$.

(D) not equal to $c$.

Question 5. The coefficients of the decision variables in the constraints quantify the consumption or requirement of each resource or condition per unit of the corresponding____

(A) objective value.

(B) constraint limit.

(C) decision variable.

(D) total output.

Question 6. If a problem requires meeting a minimum standard, such as minimum nutrient intake in a diet, the corresponding constraint inequality will likely involve the sign____

(A) $\leq$.

(B) $\geq$.

(C) $=$.

(D) $<$.

Question 7. The non-negativity restrictions $x_i \geq 0$ for all decision variables $x_i$ are included in LPP formulations because____

(A) they make the feasible region bounded.

(B) physical quantities or levels of activity are typically non-negative in real problems.

(C) they simplify the calculation of corner points.

(D) they guarantee a unique optimal solution.

Question 8. In the mathematical formulation $Z = c_1 x_1 + c_2 x_2 + \dots + c_n x_n$, $Z$ represents the objective function, and $c_i$ represents the____

(A) $i$-th decision variable.

(B) limit of the $i$-th constraint.

(C) contribution of one unit of the $i$-th decision variable to the objective.

(D) required amount of the $i$-th resource.

Question 9. When writing constraints, the right-hand side values typically represent the total amount of a resource available or the minimum required level of an output, and these values must be____

(A) decision variables.

(B) coefficients.

(C) constants.

(D) optimized.

Question 10. Formulating a constraint for total labour hours used, if product A takes $l_A$ hours and product B takes $l_B$ hours, and total available hours are $L$, with $x_A$ and $x_B$ units produced, the constraint would be $l_A x_A + l_B x_B \leq L$. In this inequality, $l_A$ and $l_B$ are the____

(A) decision variables.

(B) constraint limits.

(C) coefficients of the constraint.

(D) objective function values.

Question 1. A Linear Programming Problem focused on deciding the optimal quantities of different products to manufacture to maximize profit, given limitations on resources like labour, machine time, and raw materials, is known as a____

(A) Diet Problem.

(B) Transportation Problem.

(C) Manufacturing/Production Problem.

(D) Blending Problem.

Question 2. The objective of a typical Diet Problem formulated as an LPP is to determine the quantities of various food items to include in a diet to meet minimum nutritional requirements while minimizing the____

(A) calorie intake.

(B) variety of food items.

(C) total cost of the diet.

(D) maximum nutrient intake.

Question 3. A problem dealing with the optimal movement of goods from sources with available supply to destinations with specific demand, usually minimizing total transportation costs, is classified as a____

(A) Production Problem.

(B) Diet Problem.

(C) Transportation Problem.

(D) Resource Allocation Problem.

Question 4. In a Transportation Problem, the constraints typically involve ensuring that the total amount shipped from each source does not exceed its capacity and the total amount received at each destination meets its____

(A) supply.

(B) cost.

(C) demand.

(D) profit.

Question 5. A Linear Programming Problem where the goal is to determine the best mix of different ingredients or raw materials to produce a final product with desired characteristics at minimum cost is known as a____

(A) Production Problem.

(B) Diet Problem.

(C) Blending Problem.

(D) Assignment Problem.

Question 6. Constraints in a Blending Problem often relate to achieving specific properties in the final mixture, such as minimum or maximum percentages of certain components or required levels of physical or chemical____

(A) costs.

(B) weights.

(C) volumes.

(D) characteristics.

Question 7. A Production Problem typically involves maximizing profit or minimizing cost by deciding production levels, subject to constraints imposed by the availability of production____

(A) demand.

(B) sales.

(C) resources.

(D) marketing.

Question 8. While many LPPs are related to manufacturing, diet, transportation, or blending, other applications exist such as financial planning, scheduling, and resource allocation, all of which involve optimizing a linear objective under linear____

(A) demands.

(B) prices.

(C) constraints.

(D) profits.

Question 9. In a Diet Problem, the decision variables represent the quantities of different food items to be included in the diet, and the constraints ensure that the total intake of essential nutrients meets the specified minimum____

(A) cost.

(B) requirements.

(C) calories.

(D) weight.

Question 10. A key difference between a Manufacturing Problem and a Transportation Problem is that Manufacturing focuses on production decisions within a facility, while Transportation focuses on the logistics of moving goods between different____

(A) products.

(B) costs.

(C) markets.

(D) locations.

Question 1. The Feasible Region of an LPP is the set of all points in the solution space that simultaneously satisfy____

(A) the objective function.

(B) at least one constraint.

(C) all the given constraints, including non-negativity.

(D) the corner points.

Question 2. In a 2-variable LPP, the feasible region is graphically represented by the intersection of all the half-planes defined by the____

(A) objective function.

(B) non-negativity restrictions.

(C) constraint inequalities.

(D) corner points.

Question 3. If the feasible region of an LPP can be completely enclosed within a circle, it is referred to as a____

(A) non-convex region.

(B) unbounded region.

(C) feasible solution.

(D) bounded region.

Question 4. A point $(x, y)$ is considered to be in the Infeasible Region if it____

(A) lies within the feasible region.

(B) yields a poor objective function value.

(C) satisfies all constraints.

(D) violates at least one of the constraints.

Question 5. In a 2-variable graphical representation, the feasible region is always a convex set, meaning that for any two points within the region, the line segment connecting them lies____

(A) outside the region.

(B) partially outside the region.

(C) entirely within the region.

(D) on the boundary of the region.

Question 6. If the constraints of an LPP are such that there is no point that satisfies all of them simultaneously, then the feasible region is____

(A) unbounded.

(B) a single point.

(C) empty.

(D) a line segment.

Question 7. The non-negativity restrictions $x \geq 0, y \geq 0$ limit the feasible region to the first quadrant in a 2-dimensional graph, which is a type of____

(A) bounded region.

(B) single point region.

(C) unbounded region.

(D) infeasible region.

Question 8. The vertices or corner points of the feasible region are the points where two or more boundary lines of the feasible region____

(A) are parallel.

(B) intersect.

(C) are non-linear.

(D) are outside the region.

Question 9. A feasible region could be a single point, a line segment, a polygon, or an unbounded convex set. This variation depends entirely on the nature of the____

(A) objective function.

(B) decision variables.

(C) constraints.

(D) optimal solution.

Question 10. If a feasible region is unbounded, it means that at least in one direction, the region extends infinitely, implying that some decision variables can potentially take arbitrarily large____

(A) negative values.

(B) positive values.

(C) integer values.

(D) zero values.

Question 1. A solution $(x, y)$ to an LPP is called a Feasible Solution if it satisfies____

(A) the objective function only.

(B) at least one constraint.

(C) all the constraints of the problem.

(D) the optimal value.

Question 2. An Infeasible Solution is any solution that does not belong to the feasible region, meaning it fails to satisfy at least one of the____

(A) objective function coefficients.

(B) decision variables.

(C) constraints.

(D) corner points.

Question 3. An Optimal Feasible Solution is a feasible solution that results in the best possible value (maximum for maximization, minimum for minimization) of the____

(A) decision variable.

(B) constraint.

(C) feasible region.

(D) objective function.

Question 4. Points that lie on the boundary of the feasible region are considered feasible solutions because they satisfy the constraints, often as equalities, and are part of the set of allowed____

(A) infeasible solutions.

(B) optimal solutions.

(C) potential solutions.

(D) non-negativity restrictions.

Question 5. The vertices, also known as corner points, of the feasible region are specific feasible solutions that are crucial because the optimal solution, if it exists for a bounded region, will always be located at one or more of these____

(A) interior points.

(B) extreme points.

(C) non-feasible points.

(D) unbounded points.

Question 6. If the feasible region of an LPP is empty, it means that no combination of decision variable values satisfies all constraints, and therefore the problem has no____

(A) objective function.

(B) corner points.

(C) feasible solutions.

(D) decision variables.

Question 7. For an LPP with a non-empty bounded feasible region, it is guaranteed that at least one optimal feasible solution exists, and it will be found among the____

(A) interior points.

(B) infeasible solutions.

(C) corner points.

(D) unbounded solutions.

Question 8. If an LPP has multiple optimal solutions, it means that the optimal value of the objective function is achieved at more than one feasible point, typically along a line segment connecting two adjacent____

(A) infeasible points.

(B) corner points.

(C) origins.

(D) interior points.

Question 9. Any point $(x,y)$ that lies strictly inside the feasible region is a feasible solution, but it is not necessarily the optimal solution for a linear objective function, as the optimum tends to occur at the____

(A) center of the region.

(B) boundary of the region.

(C) infeasible region.

(D) origin.

Question 10. The set of all feasible solutions to an LPP is known as the feasible region or the____

(A) optimal space.

(B) solution space.

(C) constraint space.

(D) objective space.

Question 1. The Corner Point Theorem states that if an optimal solution to a bounded, non-empty LPP exists, it must occur at least at one of the____

(A) interior points.

(B) points on the boundary.

(C) corner points of the feasible region.

(D) points outside the feasible region.

Question 2. The Corner Point Method for a bounded feasible region involves identifying all corner points, evaluating the objective function at each, and then selecting the point(s) that yield the best value (maximum for maximization, minimum for minimization) as the____

(A) feasible solution.

(B) infeasible solution.

(C) optimal solution.

(D) unbounded solution.

Question 3. If a maximization problem has an unbounded feasible region, the objective function might also be unbounded (can increase indefinitely), in which case, no finite maximum value exists, and the problem has an____

(A) unique optimal solution.

(B) multiple optimal solutions.

(C) unbounded solution.

(D) infeasible solution.

Question 4. For a minimization problem with an unbounded feasible region, if the objective function is bounded below (its value doesn't decrease indefinitely), then the minimum value will exist and occur at one of the____

(A) interior points.

(B) points at infinity.

(C) corner points of the feasible region.

(D) points outside the feasible region.

Question 5. If the objective function attains the optimal value at two adjacent corner points of a bounded feasible region, then the optimal solution is not unique, and any point on the line segment connecting these two corners is also an____

(A) infeasible solution.

(B) interior solution.

(C) alternative optimal solution.

(D) unbounded solution.

Question 6. The process of finding the optimal value by substituting the coordinates of each corner point into the objective function is called____

(A) graphing.

(B) identifying feasible region.

(C) evaluating the objective function.

(D) solving the constraints.

Question 7. If the feasible region of an LPP is empty, the Corner Point Method indicates that the problem has____

(A) a unique optimal solution.

(B) multiple optimal solutions.

(C) an unbounded solution.

(D) no feasible solution, and thus no optimal solution.

Question 8. The Corner Point Method principle is particularly straightforward to apply when the LPP has exactly two decision variables because the feasible region can be easily visualized and its corner points identified on a____

(A) single line.

(B) 3D space.

(C) plane (2D graph).

(D) numerical table.

Question 9. If, for a maximization problem with an unbounded feasible region, the objective function does NOT increase indefinitely, then a maximum value exists, and it will be found at a____

(A) point in the interior.

(B) point on an unbounded edge.

(C) corner point.

(D) point outside the feasible region.

Question 10. For a non-empty, bounded feasible region, the minimum value of the objective function will always occur at one or more of the corner points, similar to how the maximum value does. This is a direct application of the____

(A) feasibility criteria.

(B) non-negativity rule.

(C) duality theorem.

(D) Corner Point Theorem.

Question 1. The Graphical Method is a technique specifically designed to solve Linear Programming Problems that have only____

(A) one objective function.

(B) linear constraints.

(C) two decision variables.

(D) non-negativity restrictions.

Question 2. The first step in the graphical method is to represent the constraints as linear equations and inequalities and then graph these on a coordinate plane to determine the____

(A) objective function value.

(B) optimal solution.

(C) feasible region.

(D) corner points.

Question 3. To identify the feasible side of a linear inequality like $ax + by \leq c$ graphically, a common method is to use a test point (like the origin, if not on the line) and check if it satisfies the inequality. If it does, the feasible region for that constraint is the half-plane containing the test point; otherwise, it's the half-plane____

(A) below the line.

(B) above the line.

(C) on the boundary line.

(D) opposite to the test point.

Question 4. Once the feasible region is graphed, the corner points are found by determining the coordinates of the intersection points of the boundary lines that form the vertices of the____

(A) objective function.

(B) feasible region.

(C) test point.

(D) half-planes.

Question 5. Using the Corner Point Method with the graphical approach, after identifying all corner points, the optimal solution for a bounded feasible region is found by evaluating the objective function at each corner point and selecting the one that gives the required maximum or minimum____

(A) coordinates.

(B) variable values.

(C) feasible region.

(D) objective value.

Question 6. A special case in the graphical method occurs when the feasible region is empty, which means that there are no points satisfying all constraints, and therefore the LPP has no____

(A) objective function.

(B) decision variables.

(C) feasible solution.

(D) constraints.

Question 7. Another special case is an unbounded solution, which can occur if the feasible region is unbounded and the objective function can be increased indefinitely (for maximization) or decreased indefinitely (for minimization) over this____

(A) interior region.

(B) boundary line.

(C) feasible region.

(D) corner point.

Question 8. In the Iso-profit or Iso-cost line method, a line representing the objective function $Z=k$ is drawn for some constant $k$. The optimal solution is found by moving this line parallel to itself until it last touches the feasible region. For maximization, this movement is typically away from the origin, and for minimization, it is typically____

(A) parallel to an edge.

(B) towards the origin.

(C) perpendicular to a constraint.

(D) outside the region.

Question 9. If the Iso-profit line coincides with one of the edges of the feasible region at the optimal value, this indicates that any point on that edge segment is an optimal solution, leading to the case of____

(A) no feasible solution.

(B) a unique optimal solution.

(C) multiple optimal solutions.

(D) an unbounded solution.

Question 10. The graphical method is limited to LPPs with two variables because it relies on visualization in a 2D plane. For problems with more than two variables, analytical methods like the Simplex method are used, which do not require graphical____

(A) formulation.

(B) representation.

(C) constraints.

(D) variables.