Linear Programming: Formulation, Graphical Solution and Sensitivity (Basics) Questions

Question 1

Define linear programming problem (LPP).

Accepted Answer

An LPP is a problem of maximizing or minimizing a linear objective function subject to linear constraints and non-negativity restrictions.

Question 2

State any two assumptions of LPP.

Accepted Answer

Two assumptions are linearity and non-negativity (others include divisibility, certainty, additivity).

Question 3

What is feasible region?

Accepted Answer

Feasible region is the set of all points satisfying all constraints including non-negativity.

Question 4

What is a slack variable?

Accepted Answer

Slack variable represents unused resource in a ≤ type constraint.

Question 5

What is objective function?

Accepted Answer

Objective function is the function Z to be maximized or minimized (profit/cost etc.).

Question 6

State the corner point theorem.

Accepted Answer

If an optimal solution exists, it occurs at a corner point (vertex) of the feasible region.

Question 7

Formulate an LPP from a word problem: define variables, objective and constraints (outline).

Accepted Answer

To formulate: define decision variables (x,y), write objective function (Max/Min Z), write constraints from resources (≤/≥/=) and add x≥0, y≥0. This converts the word problem into a mathematical LPP.

Question 8

Explain slack and surplus variables with one example each.

Accepted Answer

Slack is added to ≤ constraint to convert into equality: 2x + y ≤ 100 ⇒ 2x + y + s = 100, s≥0 (unused resource).
Surplus is subtracted from ≥ constraint: x + y ≥ 50 ⇒ x + y − s = 50, s≥0 (excess above minimum).

Question 9

Write the steps of graphical method for solving LPP (any five steps).

Accepted Answer

1) Draw each constraint line (convert inequality to equation).
2) Decide correct half-plane for each inequality.
3) Obtain feasible region as intersection.
4) Find corner points of feasible region.
5) Evaluate Z at each corner point.
6) Select point giving max/min Z.

Question 10

In LPP, why do we evaluate only corner points?

Accepted Answer

Because objective and constraints are linear, feasible region is a polygon. By corner point theorem, optimum (if exists) occurs at a vertex. So checking corner points is sufficient.

Question 11

Solve: Max Z = 40x + 30y subject to 2x + y ≤ 100, x + 2y ≤ 80, x,y ≥ 0 (give optimal point).

Accepted Answer

Corner points are (0,0), (50,0), (0,40), (40,20). Z values: 0, 2000, 1200, 2200. So optimum is x=40, y=20 with Z=2200.

Question 12

What does a binding constraint mean? How to identify it from slack?

Accepted Answer

A binding constraint is fully used at optimum and has slack = 0 (no unused resource). If slack > 0, the constraint is non-binding.

Question 13

Explain formulation of LPP with a suitable example.

Accepted Answer

Formulation of an LPP means converting a word problem into a mathematical model.

Steps:
1) Identify decision variables (e.g., x and y for two products).
2) Write the objective function to maximize/minimize (profit, cost, etc.).
3) Convert each resource limit/requirement into a linear constraint (≤, ≥, or =).
4) Add non-negativity restrictions: x ≥ 0, y ≥ 0.

Example (profit maximization):
Let x = units of A, y = units of B.
Max Z = 40x + 30y
Subject to:
2x + y ≤ 100 (labour)
x + 2y ≤ 80 (material)
x ≥ 0, y ≥ 0

This complete set (objective + constraints + non-negativity) is the LPP model.

Question 14

Explain the graphical method of solving LPP (two variables) clearly.

Accepted Answer

Graphical method is used when there are only two decision variables.

Steps:
1) Convert each inequality constraint into an equation to draw its boundary line.
2) Plot each line using intercepts or two points.
3) Decide the correct half-plane for each inequality (test point method) and shade it.
4) The intersection of all shaded regions gives the feasible region.
5) Identify corner points (vertices) of the feasible region.
6) Evaluate the objective function Z at each corner point.
7) Choose the point that gives maximum/minimum Z as required.

Reason: By the corner point theorem, optimum occurs at a vertex (if it exists).

Question 15

Explain sensitivity analysis (basics) in LPP and why it is useful.

Accepted Answer

Sensitivity (what-if) analysis studies how the optimal solution changes when coefficients change.

Basic ideas (exam level):
- If a constraint is binding at optimum (slack = 0), increasing its RHS (more resource) can increase maximum profit.
- If a constraint has slack at optimum (unused resource), small increase in its RHS usually does not change the optimal solution.
- If profit coefficients in objective function change, the slope of objective line changes; a different corner point may become optimal.

So sensitivity tells which resources are critical and how robust the current optimal plan is.

Linear Programming: Formulation, Graphical Solution and Sensitivity (Basics)

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