CAIIB ABM UNIT 8 MCQ – Linear Programming

Correct Answer: B. To allocate limited resources optimally among competing demands. Linear Programming involves mathematical methods specifically designed to find the best way to use restricted resources when faced with various needs or activities.

Correct Answer: C. They must be linear. Linear Programming requires that the relationships represented by the objective function and the constraints can be expressed as straight lines or planes, meaning they are linear equations or inequalities.

Correct Answer: C. Objective function. In Linear Programming, the goal, such as maximising profit or minimising cost, is mathematically represented by the objective function.

Correct Answer: D. Decision variables. These represent the quantities or choices that can be controlled or decided upon to achieve the objective, like the number of units of different products to produce.

Correct Answer: B. All equations must be linear. Linearity ensures that if inputs are scaled, outputs scale proportionally; for instance, if producing 1 unit requires 4 persons, producing 3 units requires 12 persons.

Correct Answer: C. The constraints are known with certainty and do not involve probabilities. Deterministic constraints mean that the values and relationships are fixed and known; the probability of their occurrence is assumed to be 1.0.

Correct Answer: C. They must be non-negative. A standard requirement in Linear Programming is that the decision variables cannot take negative values; they must be zero or positive.

Correct Answer: C. 3x + y ≤ 10. This inequality indicates that the total machine hours used for producing x units of product X (3x hours) and y units of product Y (1y hour) must not exceed the available 10 hours.

Correct Answer: B. x + y ≤ 4. This inequality represents the limitation that the total assembly time used for radios (x hours) and television sets (y hours) cannot be more than the available 4 hours.

Correct Answer: D. 1000x + 4000y ≤ 12000. The total cash needed for producing x radios (1000x) and y televisions (4000y) must be less than or equal to the available cash balance of ₹12,000.

Correct Answer: C. Graphic Approach. Plotting the constraints on a graph allows for visualisation of the feasible region and helps in identifying the optimal solution, particularly for problems with two variables.

Correct Answer: B. Slack variables are added. To transform ‘less than or equal to’ constraints into equalities, non-negative slack variables are introduced to represent the unused amount of the resource.

Correct Answer: B. 4B + 6C + m = 120. A non-negative slack variable ‘m’ is added to the left side of the ‘≤’ inequality to make it an equation, representing the difference between the limit (120) and the resources used (4B + 6C).

Correct Answer: C. The coefficients of variables in the standard LP format equations and the initial solution state. The initial tableau sets up the problem by showing the coefficients from the objective function and constraint equations, usually starting with a basic solution where only slack variables have non-zero values.

Correct Answer: C. The variable with the most negative coefficient in the Z-row. In maximization problems, the variable corresponding to the largest negative value in the objective function row (Z-row) is chosen to enter the solution because increasing it offers the greatest potential improvement in the objective function value.

Correct Answer: B. By calculating the ratios of the ‘Solution’ column entries to the corresponding positive entries in the pivot column and selecting the variable associated with the smallest non-negative ratio. This ‘minimum ratio test’ ensures that feasibility (non-negativity of variables) is maintained in the next iteration.

Correct Answer: C. The element at the intersection of the pivot row (leaving variable’s row) and the pivot column (entering variable’s column). This element is crucial for performing the row operations to update the tableau in the next iteration.

Correct Answer: B. Divide the current pivot row by the pivot entry. This operation normalises the pivot row so that the pivot element becomes 1, preparing for the elimination of other non-zero entries in the pivot column.

Correct Answer: C. New Row = Current Row – (Corresponding Pivot Column Entry) * (New Pivot Row). This formula uses row operations to make all other entries in the pivot column zero, effectively substituting the entering variable for the leaving variable in the basis.

Correct Answer: B. When there are no negative numbers in the Z-row (objective function row). For maximization, a non-negative Z-row indicates that no further improvement in the objective function is possible by bringing any non-basic variable into the solution.

Correct Answer: C. The variable associated with the largest positive value in the Z-row enters. In minimization problems, a positive value in the Z-row indicates a potential reduction in the objective function value, so the variable with the largest positive coefficient is chosen to enter.

Correct Answer: C. The corresponding resource is not fully used up (it is an abundant resource). A positive slack variable means there is unused capacity or surplus of the resource associated with that constraint.

Correct Answer: C. The resource is scarce and is being utilized to its full capacity. A zero value for a slack variable signifies that the corresponding constraint is binding, meaning the associated resource is fully consumed in achieving the optimal solution.

Correct Answer: B. Sensitivity analysis. LP results can be further analysed to see how sensitive the optimal solution and objective function value are to changes in input parameters like costs or resource limits.

Correct Answer: B. A technique where desired goals are incorporated directly into the objective function, allowing for multiple objectives. Goal Programming allows decision-makers to specify target levels for various goals and seeks a solution that minimizes deviations from these targets.

Correct Answer: B. All relationships (objective function and constraints) must be linear. LP requires that the goal (like profit or cost) and all limitations can be represented by linear equations or inequalities. For example, if cost increases directly with production quantity (e.g., Cost = 5 * Quantity), it’s linear; if it involves squared terms (e.g., Cost = 5 * Quantity^2), it’s non-linear and not suitable for standard LP.

Correct Answer: B. They represent the amount of unused resources corresponding to ‘less than or equal to’ constraints. Slack variables are added to convert ‘≤’ inequalities into equations. For instance, if a constraint is MachineHours ≤ 100, and the optimal solution uses 90 hours, the slack variable for this constraint will be 10, indicating 10 unused machine hours.

Correct Answer: C. At a corner point (vertex) of the feasible region. The feasible region is the area satisfying all constraints. Because the objective function is linear, the maximum or minimum value will occur at one of the vertices where the boundary lines of the constraints intersect. For example, if the feasible region is a triangle, the optimum will be at one of its three corners.

Correct Answer: B. When the objective function line is parallel to one of the constraint lines forming an edge of the feasible region. If the objective function slope matches the slope of a binding constraint that forms an edge of the feasible region, all points along that edge segment represent optimal solutions. For example, if Maximize Z = 2x + 2y and a constraint line x + y = 10 forms an edge of the feasible region, any combination of x and y on that line segment might yield the same maximum Z.

Correct Answer: B. To identify which corner point of the feasible region yields the best value for the objective function. By plotting the objective function (often as a line representing a certain profit/cost level) and moving it parallelly across the feasible region, one can find the corner point that the line touches last (for maximization) or first (for minimization) while still being in the feasible region, thus identifying the optimum. For example, moving the profit line outwards until it just touches the last corner of the feasible area.

Correct Answer: C. x + y >= 3000. The total volume of the mixture is the sum of the volumes of Fuel A (x) and Fuel B (y). The phrase “no less than” translates to “greater than or equal to” (>=).

Correct Answer: B. x + y <= 4000. The total amount of fuel stored, which is the sum of Fuel A (x) and Fuel B (y), cannot exceed the maximum capacity of 4000 litres. “Maximum capacity” implies “less than or equal to” (<=).

Correct Answer: A. x <= 2000 and y <= 4000. The amount of Fuel A used (x) cannot exceed the available 2,000 litres, and the amount of Fuel B used (y) cannot exceed the available 4,000 litres.

Correct Answer: B. (90x + 75y)⁄(x+y) >= 80. The weighted average octane is the total octane (90x + 75y) divided by the total volume (x+y). This average must be “no less than” (>=) 80.

Correct Answer: B. Minimise Z = 20x + 13.33y. The total cost is the sum of the cost of Fuel A (20 times the litres of A, x) and the cost of Fuel B (13.33 times the litres of B, y). The company aims to minimise this total cost.

Correct Answer: B. 600x + 900y >= 1800 AND 600x + 900y <= 3600. Total calories from Food A (600x) plus Food B (900y) must be greater than or equal to 1,800 and less than or equal to 3,600.

Correct Answer: D. 200x + 700y <= 1400. The total starch calories from Food A (200x) plus Food B (700y) must be “no more than,” which means less than or equal to (<=), 1,400.

Correct Answer: B. 400x + 100y >= 400. The total protein from Food A (400x) plus Food B (100y) must be “no less than,” meaning greater than or equal to (>=), 400 calories.