Matching Items MCQs for Sub-Topics of Topic 13: Linear Programming

(i) Objective Function

(ii) Constraints

(iii) Decision Variables

(iv) Non-negativity Restriction

(a) Limitations or restrictions on the problem.

(b) The quantities to be determined to achieve the goal.

(c) The function to be optimized (maximized or minimized).

(d) Requirement that variables must be greater than or equal to zero.

(i) A profit function

(ii) Resource availability

(iii) Minimum requirement

(iv) Quantity of a product

(a) A decision variable ($x_i$).

(b) A linear objective function to maximize.

(c) A constraint using a $\leq$ inequality.

(d) A constraint using a $\geq$ inequality.

(i) "Total labour hours are 500"

(ii) "Profit per unit of Product X is $\textsf{₹}100$"

(iii) "Number of units of Product Y produced"

(iv) "Minimize total cost"

(a) Decision Variable.

(b) Objective Function (Type).

(c) Constraint.

(d) Coefficient in the Objective Function.

(i) $Z = 5x + 8y$

(ii) $2x + 3y \leq 60$

(iii) $x \geq 0, y \geq 0$

(iv) $x$ and $y$

(a) Non-negativity Restrictions.

(b) A Linear Constraint.

(c) Decision Variables.

(d) An Objective Function.

(i) Coefficients in constraints

(ii) Right-hand side of constraints

(iii) Coefficients in objective function

(iv) Variables

(a) Contribution to objective per unit.

(b) Resource consumption/requirement per unit.

(c) Resource availability or minimum requirement.

(d) Quantities to be decided.

(i) $\mathbf{x}$

(ii) $\mathbf{c}^T$

(iii) $A$

(iv) $\mathbf{b}$

(a) Vector of right-hand side values.

(b) Matrix of constraint coefficients.

(c) Vector of decision variables.

(d) Row vector of objective function coefficients.

(i) "Cannot exceed 50 units"

(ii) "Must be at least 10 units"

(iii) "Exactly 25 units are required"

(iv) "Sum of $x_1$ and $x_2$"

(a) Part of a constraint expression (e.g., $x_1+x_2$).

(b) An equality constraint ($=$).

(c) A 'less than or equal to' constraint ($\leq$).

(d) A 'greater than or equal to' constraint ($\geq$).

(i) Identify variables

(ii) Define objective function

(iii) Write constraints

(iv) Add non-negativity

(a) To state the goal mathematically.

(b) To define the quantities to be controlled.

(c) To ensure variables are physically meaningful.

(d) To represent resource limits/requirements.

(i) $ax + by \leq c$

(ii) $ax + by \geq c$

(iii) $ax + by = c$

(iv) $x \geq 0, y \geq 0$

(a) Exactly equals.

(b) At most.

(c) Non-negative.

(d) At least.

(i) Coefficients of objective function (e.g., $c_i$)

(ii) Coefficients of constraints (e.g., $a_{ij}$)

(iii) Right-hand side values (e.g., $b_i$)

(iv) Variables (e.g., $x_j$)

(a) Resources available or requirements.

(b) Profit/cost per unit, etc.

(c) Unknowns to be solved for.

(d) Resource usage/contribution per unit.

(i) Manufacturing Problem

(ii) Diet Problem

(iii) Transportation Problem

(iv) Blending Problem

(a) Minimize cost of ingredients while meeting property specs.

(b) Maximize profit from production.

(c) Minimize cost of diet while meeting nutrient needs.

(d) Minimize total cost of shipping goods.

(i) Transportation Problem

(ii) Manufacturing Problem

(iii) Diet Problem

(iv) Blending Problem

(a) Minimum nutrient requirements (e.g., $\geq$).

(b) Limits on machine hours or raw material (e.g., $\leq$).

(c) Supply and demand constraints.

(d) Constraints on proportion or percentage of ingredients.

(i) Deciding how much of two food items to eat to meet daily vitamin needs at minimum cost.

(ii) Deciding how many units of two products to make using limited labour and material to maximize profit.

(iii) Deciding how to ship grain from storage facilities to mills to minimize shipping cost.

(iv) Deciding how to mix different grades of oil to meet viscosity specifications at minimum cost.

(a) Manufacturing Problem.

(b) Diet Problem.

(c) Transportation Problem.

(d) Blending Problem.

(i) Production quantity

(ii) Amount of food item in diet

(iii) Quantity shipped between locations

(iv) Proportion of ingredient

(a) Transportation Problem.

(b) Manufacturing Problem.

(c) Diet Problem.

(d) Blending Problem.

(i) Supply and demand are key constraints.

(ii) Minimum nutritional intake must be met.

(iii) Optimizing use of limited resources (labour, materials).

(iv) Physical or chemical properties of a mixture are important.

(a) Diet Problem.

(b) Manufacturing Problem.

(c) Transportation Problem.

(d) Blending Problem.

(i) Feasible Region

(ii) Infeasible Region

(iii) Bounded Region

(iv) Unbounded Region

(a) Can be enclosed within a finite circle.

(b) Region containing points that satisfy all constraints.

(c) Region containing points that violate at least one constraint.

(d) Extends infinitely in at least one direction.

(i) A line $ax+by=c$

(ii) Shaded half-plane

(iii) Intersection of shaded regions

(iv) A point outside the feasible region

(a) Represents the feasible solutions for a single linear inequality/equation.

(b) Represents the feasible region.

(c) Represents the boundary of a constraint.

(d) Represents an infeasible solution.

(i) $x \leq 5, y \leq 8, x+y \leq 10$

(ii) $x \geq 5, y \geq 8$

(iii) $x+y \leq 5, x+y \geq 10$

(iv) $x=2, y=3, x \geq 0, y \geq 0$

(a) Unbounded feasible region.

(b) Empty feasible region.

(c) Bounded feasible region.

(d) Feasible region is a single point.

(i) Feasible region is empty

(ii) Feasible region is bounded and non-empty

(iii) Feasible region is unbounded

(iv) Feasible region is a single point

(a) An optimal solution is guaranteed to exist.

(b) No feasible solution exists.

(c) A unique feasible and optimal solution exists.

(d) An optimal solution may or may not exist.

(i) Satisfies all constraints

(ii) Violates at least one constraint

(iii) Intersection of boundary lines in feasible region

(iv) The entire set of points satisfying all constraints

(a) Infeasible point.

(b) Feasible point.

(c) Feasible region.

(d) Corner point.

(i) Feasible solution

(ii) Infeasible solution

(iii) Optimal feasible solution

(iv) Basic feasible solution (often corner points)

(a) A feasible solution that yields the best objective function value.

(b) A solution that does not satisfy all constraints.

(c) A solution that satisfies all constraints.

(d) A feasible solution at a vertex of the feasible region.

(i) Satisfies $x \geq 0, y \geq 0$ but violates $x+y \leq 5$ where feasible region is $x+y \leq 5, x,y \geq 0$

(ii) Satisfies all constraints and maximizes Z

(iii) Any point within the feasible region

(iv) Point where boundary lines intersect within the feasible region

(a) Optimal feasible solution.

(b) Feasible solution.

(c) Corner point.

(d) Infeasible solution.

(i) Empty feasible region

(ii) Bounded feasible region

(iii) Unbounded feasible region

(iv) Feasible region is a line segment

(a) Optimal solution guaranteed to exist at a corner point(s).

(b) Multiple optimal solutions might exist along the segment.

(c) No feasible or optimal solution exists.

(d) Optimal solution may or may not exist, if it exists it's at a corner.

(i) Every feasible solution is optimal.

(ii) Every optimal solution is feasible.

(iii) If the feasible region is bounded, a unique optimal solution always exists.

(iv) If an optimal solution exists, at least one corner point is optimal.

(a) False.

(b) True.

(c) False (could be multiple optimal solutions).

(d) True (Corner Point Theorem).

(i) (2,2) (Assume it's inside)

(ii) (0,0)

(iii) (7,0) (Assume it's outside)

(iv) (3,4)

(a) A corner point and feasible solution.

(b) An interior feasible solution (not a corner).

(c) An infeasible solution.

(d) The origin, which is a corner point if feasible.

(i) Corner Point Theorem

(ii) Multiple Optimal Solutions

(iii) Unbounded Solution (Maximization)

(iv) Empty Feasible Region

(a) The objective function can be increased indefinitely.

(b) No feasible solutions exist, so no optimal solution.

(c) Optimal solution occurs at corner points (if it exists and region is convex).

(d) Optimal value is attained at more than one feasible point, including adjacent corners.

(i) Identify corner points

(ii) Evaluate objective function at corner points

(iii) Select best objective value

(iv) Find feasible region

(a) To find the set of all possible solutions.

(b) To apply the main principle of the theorem.

(c) To find the point(s) that give the optimal solution.

(d) To get the value of the function to optimize at critical points.

(i) Empty feasible region

(ii) Bounded and non-empty feasible region

(iii) Unbounded feasible region where Z (maximization) increases indefinitely

(iv) Unbounded feasible region where Z (minimization) increases as you move away from origin (assuming positive coefficients)

(a) Optimal solution does not exist (unbounded).

(b) No optimal solution exists (no feasible solution).

(c) An optimal solution exists at a corner point.

(d) An optimal solution exists at a corner point.

(i) Maximum Z value is same at two adjacent corner points.

(ii) Feasible region is empty.

(iii) Feasible region is bounded.

(iv) For maximization, Z can increase indefinitely over the feasible region.

(a) Multiple optimal solutions exist.

(b) No feasible solution exists.

(c) An optimal solution is guaranteed to exist.

(d) The problem has an unbounded solution.

(i) Point in the interior of the feasible region

(ii) Point on the boundary of the feasible region (not a corner)

(iii) Corner point of the feasible region

(iv) Point outside the feasible region

(a) Cannot be an optimal solution.

(b) Could be an optimal solution only if multiple optimal solutions exist along an edge.

(c) Where the optimal solution occurs for a bounded region.

(d) Is a feasible solution but not where the optimal solution usually occurs for linear Z.

(i) Graph constraints

(ii) Identify feasible region

(iii) Find corner points

(iv) Evaluate Z at corners

(a) Solve systems of boundary line equations.

(b) Plot lines and shade feasible side for each.

(c) Substitute corner coordinates into objective function.

(d) Find the intersection of all shaded regions.

(i) Shaded area

(ii) Vertex of the shaded area

(iii) An Iso-profit/Iso-cost line

(iv) No overlapping shaded region

(a) Feasible region.

(b) Represents points with the same objective function value.

(c) No feasible solution.

(d) Corner point.

(i) Feasible region is empty.

(ii) Feasible region is bounded.

(iii) Iso-profit line can move infinitely outwards while touching the feasible region.

(iv) Iso-cost line touches a single corner point and moves away towards origin.

(a) Optimal solution guaranteed at a corner point.

(b) Unique optimal solution for minimization found.

(c) No feasible solution exists.

(d) Unbounded solution for maximization.

(i) Corner Point Method

(ii) Iso-profit/Iso-cost Line Method

(iii) Shading technique

(iv) Test point (e.g., origin)

(a) To determine which side of a boundary line is feasible for an inequality.

(b) To visually represent the feasible region.

(c) To find the optimal value by evaluating Z at vertices.

(d) To find the optimal point(s) by moving a line parallel to itself.

(i) No feasible solution

(ii) Unbounded solution (Maximization)

(iii) Multiple optimal solutions

(iv) Feasible region is a single point

(a) Iso-profit line coincides with an edge of the feasible region.

(b) Feasible region is an empty set.

(c) Feasible region has only one point satisfying all constraints.

(d) Feasible region is unbounded and the objective function can increase indefinitely.