FAQs on Linear Programming Problems (LPP)
Here’s a set of student-friendly FAQs on Linear Programming Problems (LPP):
1. What is Linear Programming (LPP)?
Linear Programming is a method used to find the best possible solution (maximum profit, minimum cost, etc.) when there are limited resources and some conditions (called constraints).
In simple words, it’s a smart way of making decisions using maths.
In other words,
An LPP is a math model where you optimize a linear objective (like maximizing profit or minimizing cost) subject to linear constraints (rules) on the decision variables. In standard form it’s written with linear equations/inequalities and nonnegativity.
2. Where is Linear Programming used in real life?
LPP is used in many areas, such as:
1. Business (to maximize profit or minimize cost)
2. Transportation (to reduce travel cost or time)
3. Agriculture (to get the best crop mix)
4. Industry (to decide production schedules)
5. Daily life (even planning your study hours can be seen like an LPP!)
3. What are the main parts of an LPP?
Every Linear Programming Problem has three important parts:
1. Decision Variables –
The things we need to decide (e.g., how many chairs and tables to make).
2. Objective Function –
What we want to achieve, what to maximize or minimize (maximize profit or minimize cost).
3. Constraints –
The rules or limits (like limited raw material, time, or money).
4. Feasible region –
The region of all points that satisfy the constraints.
For two variables the feasible region is a polygon; in general it’s a convex polyhedron.
4. Why is it called “Linear”?
It’s called “linear” because both the objective function and the constraints are written in the form of linear equations or inequalities (straight lines, not curves).
5. How do we solve LPP?
To solve LPP, we follow these steps:
1. Define the decision variables.
2. Write the objective function.
3. Write the constraints.
4. Draw the feasible region (the area that satisfies all constraints).
5. Find the corner/vertex points.
6. Check which corner point gives the best value of the objective function.
6. What is the Feasible Region?
The feasible region is the shaded area on the graph that satisfies all the given conditions (constraints).
It is the “possible zone” where the solution can lie.
7. Why do we check corner points in LPP?
Because the best solution (optimum) always lies at one of the corner points of the feasible region. That’s why we test them.
8. Where does the best answer occur?
If an optimum exists, at least one optimum occurs at a corner (vertex/extreme point) of the feasible region. If the same best value occurs at two corners, it also occurs on the whole edge between them (so there can be many optimal answers). This is the fundamental theorem of linear programming.
9. Can an LPP have more than one solution?
Yes. Sometimes two or more points give the same best value. In that case, the problem has multiple solutions.
10. What happens if there is no solution?
If the feasible region does not exist (constraints clash with each other), the problem is called infeasible.
11. Can an LPP be impossible to solve or go off to infinity?
Yes:
Infeasible: no point satisfies all constraints → no solution.
Unbounded objective: the objective can grow without limit in a feasible direction → no finite optimum.
These are standard outcomes in LP.
12. How do you usually solve an LPP?
Two classic families of methods:
Simplex method:
moves along vertices/edges of the feasible region; updates a basis each step. Excellent in practice on many problems.
Interior-point (barrier) methods:
move through the interior; strong polynomial-time theory and very good on large, sparse problems.
13. What is “standard form”? What are slack variables?
“Standard form” writes constraints as equalities with nonnegative variables (e.g., turn “≤” into “=” by adding a slack variable). This is how solution methods like simplex are set up.
14. What is the dual, and what are shadow prices?
Every LPP has a dual problem. At optimum, the primal and dual objective values are equal (strong duality). Optimal dual variables are often called shadow prices: they show the marginal value of relaxing a binding resource constraint by one unit.
15. What are common model outcomes?
Unique optimum: one best corner point.
Multiple optima: whole edge/face is optimal (objective is parallel to a binding edge).
Infeasible: no point satisfies all constraints → no solution.
Unbounded objective: the objective can grow without limit in a feasible direction → no finite optimum.
16. How do I solve small two-variable LPPs by hand?
Use the graphical method:
Plot each linear constraint.
Shade the feasible region.
List the corner points.
Evaluate the objective at those corners; the best value wins. That is, check which corner point gives the best value of the objective function. This works because of the vertex property.
17. A tiny worked example (graphical)
Maximize
z = 3x + 5y
Subject to
2x + y ≤ 8,
x +2y ≤ 8,
x ≥ 0, y ≥ 0.
Corners of the feasible polygon:
At vertex (0,0): z = 0
At vertex (4,0): z = 12
At vertex (0,4): z = 20
Intersection of 2x+y=8 and x+2y=8: solve →
(x,y)=(8/3,8/3);
At this vertex,
z = 3(8/3)+5(8/3)
=64/3
≈21.33
→ hence optimal at (8/3,8/3), because it gives the maximum value of z.
18. Quick checklist for forming an LPP
Define clear decision variables.
Write the linear objective.
Write each constraint (≤, ≥, or =).
Add nonnegativity if required.
If needed for methods/software, convert to standard form with slacks.
19. Typical student mistakes (and how to avoid them)
Mixing units in constraints → keep units consistent.
Forgetting nonnegativity → State xi≥0.
Missing a corner point when solving graphically → list and test all vertices (including intersections of constraint lines).
20. What is the difference between feasible, infeasible, and unbounded problems?
Feasible: There is at least one solution that satisfies all constraints.
Infeasible: No solution satisfies all constraints (the feasible set is empty).
Unbounded: The feasible set exists, but the objective can grow indefinitely (no finite optimum).
21. What is the difference between equality and inequality constraints?
Inequality constraints (≤ or ≥) define half-planes or half-spaces.
Equality constraints force solutions to lie exactly on a line or plane.
In standard form, inequalities are converted to equalities by adding slack or surplus variables.
22. Why must decision variables usually be nonnegative?
Most real-world problems don’t make sense with negative amounts (e.g., you can’t produce –3 chairs). Nonnegativity keeps the solution meaningful. Also, it is part of standard form.
23. What happens if two constraint lines overlap?
Then they represent dependent constraints. The overlapping line doesn’t add new information. The feasible region is determined by the “active” constraints.
24. What is a “binding” constraint?
A binding constraint is one that holds as an equality at the optimal solution. It “binds” the feasible region at that point. If you relaxed it, the optimal value could improve. Constraints that don’t affect the optimum are non-binding.
25. What is degeneracy in LP?
A solution is degenerate if more constraints meet at a vertex than necessary (e.g., three or more lines intersecting at the same corner in 2D). Degeneracy can cause simplex to stall (cycling), but anti-cycling rules fix this.
26. What is sensitivity analysis?
It studies how changes in the input (like resource limits or profit coefficients) affect the optimal solution. Shadow prices (dual variables) are part of sensitivity analysis.
27. How is linear programming different from integer programming?
Linear programming allows variables to take fractional values.
Integer programming requires variables to be integers (e.g., number of trucks).
Integer problems are harder to solve; LP is easier and often used as a relaxation.
28. Can LPP handle more than two variables?
Yes! The theory works for any number of variables. But we can only draw graphs for 2 (or sometimes 3) variables. For larger problems, we use algorithms (like simplex or interior-point).
29. What are “basic” and “non-basic” variables in simplex?
Basic variables: currently in the solution (non-zero).
Non-basic variables: set to zero in that solution.
Each simplex step swaps one variable in/out of the basis.
30. What is the difference between primal and dual problems?
Primal: the original LPP you wrote.
Dual: another LPP constructed from it, with roles of constraints/objective reversed.
At optimum, both have the same objective value (strong duality). Dual gives economic interpretations (shadow prices).
31. Can LPP guarantee a unique solution?
Not always. There may be:
A unique optimal solution.
- Multiple optima (if the objective is parallel to a binding edge/face).
- No feasible solution (infeasible).
- Unbounded (no finite optimum).
32. Why is LPP important to learn?
Because it’s one of the simplest and most widely used optimization tools. It builds the foundation for more advanced topics (integer programming, nonlinear optimization, convex optimization). And it applies directly in business, logistics, engineering, economics, and daily decision-making.
33. Define Linear Programming.
Linear Programming is a method to optimize (maximize or minimize) a linear objective function subject to linear constraints and nonnegativity conditions.
34. What are the assumptions of LPP?
Linearity – objective and constraints are linear.
Additivity – total effect = sum of individual effects.
Divisibility – variables can take fractional values.
Certainty – coefficients are known and fixed.
Non-negativity – decision variables ≥ 0.
35. What is the difference between feasible solution and optimal solution?
Feasible solution: satisfies all constraints (may or may not be best).
Optimal solution: feasible solution that gives the best value of the objective function.
36. State the Fundamental Theorem of Linear Programming.
If an LPP has an optimal solution, at least one optimal solution occurs at a vertex (corner point) of the feasible region.
37. What is an unbounded solution?
If the objective function can increase (or decrease) without limit while still satisfying all constraints, the LPP is said to have an unbounded solution.
38. What is the difference between slack and surplus variables?
Slack variable: added to “≤” constraints to convert them to equations.
Surplus variable: subtracted from “≥” constraints to convert them to equations.
39. What are artificial variables and why are they used?
Artificial variables are temporary variables introduced to obtain an initial feasible solution for methods like the Big M method or Two-phase method when slack/surplus alone is not enough.
40. What is degeneracy in simplex method?
Degeneracy occurs when a basic variable takes value 0 in a solution. This can cause stalling or cycling in simplex.
41. What is the difference between Simplex method and Graphical method?
Graphical method: only for 2 variables, visual solution.
Simplex method: works for many variables, algebraic, systematic, widely used.
42. What is the difference between feasible region and infeasible region?
Feasible region: set of all points that satisfy all constraints.
Infeasible region: set of points that violate one or more constraints.
43. What is meant by multiple optimal solutions?
When the objective function has the same optimum value at more than one vertex (and along the line/edge joining them), the LPP has multiple optima.
44. What are the limitations of LPP?
Assumes certainty of coefficients (no randomness).
Assumes divisibility (fractions allowed).
Real-life problems may involve nonlinear relations or integer restrictions.
45. Explain the concept of duality in LPP.
Every LPP (primal) has an associated dual problem. The optimal value of the primal equals the optimal value of the dual (strong duality). Dual variables give economic meaning (shadow prices).
46. What is sensitivity analysis in LPP?
Sensitivity analysis studies the effect of small changes in coefficients or constraints on the optimal solution (important for decision-making).
47. Give two applications of LPP.
1. Production planning (maximize profit given machine hours and labor).
2. Transportation problems (minimize cost of shipping goods).
48. What is the standard form of an LPP?
Objective: Maximize
Z=cTx
Constraints:
Ax≤b, x≥0.
All constraints are inequalities of type ≤ and all variables are nonnegative.
49. What is the canonical form of an LPP?
In canonical form, all constraints are written as equalities (using slack/surplus/artificial variables), and all decision variables are nonnegative.
50. What is meant by basis and basic feasible solution (BFS)?
Basis: a set of linearly independent columns chosen from the constraint matrix.
BFS: a feasible solution obtained by setting non-basic variables = 0 and solving for basic variables.
51. What is an optimal basic feasible solution?
A basic feasible solution that also gives the best (maximum or minimum) value of the objective function.
52. State the difference between BFS and feasible solution.
Feasible solution: any solution satisfying all constraints.
Basic feasible solution: a feasible solution obtained from a basis (corner/vertex point).
53. What is the initial basic feasible solution in simplex?
The solution obtained after introducing slack (and possibly artificial) variables, which serves as the starting point for simplex iterations.
54. What are the stopping conditions of the simplex method?
When all coefficients in the objective row (for maximization) are ≤ 0, the current solution is optimal.
Or, if the solution is infeasible/unbounded, simplex terminates with that conclusion.
55. What are the advantages of the simplex method?
Works for any number of variables and constraints.
Systematic and reliable.
Provides additional info like shadow prices and sensitivity.
56. What is a convex set, and why is it important in LPP?
A set is convex if the line segment joining any two points in the set lies completely inside the set.
The feasible region of an LPP is always convex, which guarantees that an optimal solution (if it exists) occurs at a vertex.
57. What is a convex polyhedron?
The feasible region of an LPP in n-dimensional space is a convex polyhedron formed by the intersection of half-spaces defined by linear constraints.
58. What is the role of slack, surplus, and artificial variables in simplex?
Slack: convert ≤ into =.
Surplus: convert ≥ into =.
Artificial: added when necessary to start simplex (e.g., Big M or Two-phase method).
59. What is the Big M method?
A simplex method variant where artificial variables are added with a very large penalty (M) in the objective function to ensure they leave the solution quickly.
60. What is the Two-Phase method?
A simplex-based method where:
Phase I: minimize sum of artificial variables to reach a feasible solution.
Phase II: optimize the original objective function.
61. What is an unbounded feasible region?
A feasible region that extends infinitely in some direction(s). It may or may not lead to an unbounded optimum.
62. What is the difference between primal infeasibility and dual infeasibility?
Primal infeasible: no solution satisfies primal constraints.
Dual infeasible: no solution satisfies dual constraints.
By duality theory, if one is infeasible, the other is either infeasible or unbounded.
63. What is the geometric meaning of the dual problem?
Dual variables represent the weights (shadow prices) assigned to constraints, showing how much the objective changes when resources are varied.
64. What is complementary slackness?
A condition that links the primal and dual solutions:
For each constraint, either the slack is 0 or the corresponding dual variable is 0.
This helps in verifying optimality.
65. What are the conditions for multiple optimal solutions?
If the objective function line is parallel to a constraint boundary that touches the feasible region, then all points on that boundary segment are optimal.
66. Can simplex method solve minimization problems?
Yes. Minimization can be converted to maximization by multiplying the objective function by –1, or by using the dual problem.
67. What is the difference between sensitivity analysis and post-optimality analysis?
Both study how changes affect the solution.
Sensitivity analysis: effect of small changes in coefficients or RHS values.
Post-optimality analysis: effect of larger or structural changes (new constraints, new variables).
68. What are redundant constraints?
Constraints that do not affect the feasible region (because they are automatically satisfied once other constraints are applied). They can be removed without changing the solution.
69. What is the difference between bounded and unbounded feasible regions?
Bounded region: the feasible set is enclosed (finite). Optimum is always finite (if feasible).
Unbounded region: the feasible set extends infinitely. The problem may still have a finite optimum, but it could also be unbounded.
70. What is the difference between equality constraints and inequality constraints in simplex?
≤ constraints: converted with slack variables.
≥ constraints: converted with surplus (and artificial) variables.
= constraints: directly involve artificial variables if needed.
71. What is meant by infeasible basic feasible solution (BFS)?
A BFS where one or more variables violate nonnegativity. These may appear temporarily in methods like the Big M or Two-Phase, but are not acceptable as final solutions.
72. What is the difference between simplex method and dual simplex method?
Simplex: starts with a feasible solution, maintains feasibility, improves optimality.
Dual simplex: starts with an infeasible solution, maintains optimality, moves toward feasibility.
73. What is the advantage of dual simplex method?
It is efficient when the current solution is optimal but infeasible (e.g., after adding new constraints). Saves re-starting simplex from scratch.
74. What is the main difference between primal and dual problems?
Primal: decision variables usually represent production or allocation.
Dual: variables usually represent prices or costs (shadow prices).
75. What is meant by shadow price?
The shadow price of a constraint tells us how much the objective function will change if the right-hand side (resource) increases by one unit, provided feasibility remains.
76. Why is LPP called a deterministic model?
Because all parameters (coefficients in objective and constraints) are assumed to be known with certainty (no randomness).
77. What is the difference between linear programming and nonlinear programming?
Linear programming: objective and constraints are linear.
Nonlinear programming: at least one objective or constraint is nonlinear.
78. What is the importance of convexity in LPP?
Convex feasible regions guarantee that if an optimum exists, it lies at a corner/extreme point. Without convexity, optimal solutions may lie inside, and LP theory breaks down.
79. What is the difference between isoprofit and isocost lines?
Isoprofit line: line showing constant profit (objective in maximization).
Isocost line: line showing constant cost (objective in minimization).
Used in graphical method to find the best line touching the feasible region.
80. What are the applications of Linear Programming in business?
Production planning (maximize output/profit).
Transportation and assignment problems.
Portfolio optimization in finance.
Workforce scheduling.
81. What is sensitivity range in LPP?
The range of values within which a coefficient (in objective or RHS) can change without altering the current optimal solution structure.
82. What is the role of linear independence in simplex method?
The columns chosen as basis must be linearly independent, otherwise the system cannot provide a valid basic feasible solution.
83. What is the geometric interpretation of simplex?
Simplex moves from one vertex (corner) of the feasible polyhedron to another along edges, improving the objective at each step, until optimum is reached.
84. What is the relationship between primal and dual feasibility?
If primal is feasible and bounded → dual is also feasible and bounded, with equal optimum values.
If primal is unbounded → dual is infeasible.
If primal is infeasible → dual may be infeasible or unbounded.
85. What is meant by sensitivity of resources?
It studies how the optimal value changes if resource limits (RHS values) change. The shadow price is valid only within a certain allowable range.
86. What are integer linear programming problems?
LPPs where some or all decision variables are restricted to integer values. These are much harder to solve than continuous LPPs.
87. What is the difference between transportation problem and general LPP?
Transportation problem: a special type of LPP with a specific structure (minimize shipping cost subject to supply and demand). Solved with special methods (Northwest Corner, Vogel’s method, MODI).
General LPP: no such special structure, solved by simplex or interior-point.
88. What is meant by “corner-point method” in LPP?
It is the graphical method of solving an LPP where the objective function is tested at all corner points of the feasible region to find the optimum.
89. What is an isoprofit (or isocost) approach in graphical LPP?
It involves drawing lines of constant objective value (profit or cost) and shifting them parallel until the last point touching the feasible region gives the optimum.
90. What are the differences between primal simplex and dual simplex?
Primal simplex: keeps feasibility, improves optimality.
Dual simplex: keeps optimality, works toward feasibility.
Used when constraints are modified or added after solving.
91. What is the dual of the dual?
The dual of the dual is the original primal problem (this property always holds in LP).
92. What is weak duality theorem?
For any feasible solutions of the primal and dual:
Primal objective ≤ Dual objective (for maximization).
This provides a check for correctness.
93. What is strong duality theorem?
If the primal (and dual) has an optimal solution, then the optimal values are equal.
94. What is complementary slackness?
At optimality, for each constraint:
Either the slack variable = 0 or the corresponding dual variable = 0.
This condition helps verify or derive optimal solutions.
95. What is the significance of shadow prices?
They indicate the marginal worth of resources — i.e., how much the objective would improve if one unit of a resource is added (within the allowable range).
96. What are sensitivity limits (allowable range)?
They define how much coefficients (objective or RHS values) can change without altering the optimal basis (structure of solution).
97. What is the effect of adding a new variable to an LPP?
Adding a variable (like a new product) requires checking its reduced cost (in simplex). If reduced cost is negative (in maximization), including it may improve the solution.
98. What is the effect of adding a new constraint?
The current solution may become infeasible → dual simplex is used to restore feasibility.
99. What are the characteristics of an LPP model?
Linear objective function.
Linear constraints.
Decision variables continuous (fractional values allowed).
Nonnegative variables.
Deterministic (no randomness).
100. What is a binding vs. non-binding constraint?
Binding constraint: active at optimum (holds as equality).
Non-binding constraint: not active at optimum (has slack > 0).
101. What is redundancy in constraints?
A constraint is redundant if removing it does not change the feasible region or solution.
102. What is the difference between feasible direction and infeasible direction?
Feasible direction: moving along it keeps the solution inside the feasible region.
Infeasible direction: moving along it takes the solution outside.
103. What is meant by optimality condition in simplex?
For maximization, when all coefficients in the objective row (Cj – Zj) ≤ 0, the solution is optimal.
104. What is meant by degeneracy in LPP?
A BFS is degenerate if at least one basic variable = 0. This may cause simplex to repeat vertices without improvement.
105. What is cycling in simplex?
Cycling occurs when simplex revisits the same solution repeatedly (due to degeneracy). Anti-cycling rules like Bland’s Rule are used to avoid it.
106. What is difference between integer programming and mixed-integer programming?
Integer programming: all variables must be integers.
Mixed-integer programming: only some variables are restricted to integers.
107. What is meant by convex combination?
A linear combination of points with nonnegative coefficients that sum to 1. Every point in the feasible region of LPP is a convex combination of its corner points.
108. Why can the simplex method stop at a degenerate solution?
Because even though improvement is possible, the pivot does not change the objective value. Further iterations eventually escape it.
109. What is the difference between bounded solution and unique solution?
Bounded solution: finite optimal value exists.
Unique solution: only one point gives that optimum.
A bounded solution may or may not be unique.
110. What are the typical exam applications of LPP?
Allocation of resources.
Product mix problems.
Transportation problems.
Diet problems (minimize cost subject to nutrition).
Workforce scheduling.
111. Is Linear Programming difficult?
Not really! At first, it may look tricky, but once you practice drawing graphs and writing equations, it becomes easy and logical. It’s like solving a puzzle step by step.
112. What’s the key to mastering LPP?
Practice with real-life examples.
Always define your variables clearly.
Carefully draw the feasible region.
Remember: The optimum value lies at the corner points.
A Last-Minute Revision Notes (Cheat Sheet) for Linear Programming Problems (LPP)
Linear Programming Problem, LPP Last minute revision notes, exam-focused, and easy to recall
🔹 Basics
LPP: Optimize (Max or Min) a linear objective subject to linear constraints and x≥0.
Objective function: Maximize
Z = c1x1 + c2x2 +… OR Minimize.
Constraints: Linear inequalities/equalities.
Feasible region: Intersection of all constraints (always convex).
Optimum solution: Lies at a corner (vertex) of feasible region (Fundamental Theorem).
🔹 Important Definitions
Feasible solution: Satisfies all constraints.
Optimal solution: Best feasible solution (max/min).
Bounded solution: Finite optimum.
Unbounded solution: Objective can grow infinitely.
Infeasible: No solution satisfies constraints.
Slack variable: Added to ≤ constraint.
Surplus variable: Subtracted from ≥ constraint.
Artificial variable: Added for simplex start (Big M / Two-phase).
🔹 Graphical Method (for 2 variables)
Plot constraints.
Find feasible region (shaded polygon).
Identify corner points.
Evaluate objective function at all corners.
Choose the optimum (max/min).
🔹 Simplex Method – Key Points
Works for many variables.
Start from an initial basic feasible solution (BFS).
Pivoting: Entering variable (improves Z), Leaving variable (maintains feasibility).
Stopping condition: In maximization, all Cj − Zj ≤ 0
Degeneracy: BFS with a basic variable = 0.
Cycling: Simplex repeats solutions → avoid with Bland’s Rule.
🔹 Duality & Sensitivity
Dual problem: Every LPP has a dual.
Strong Duality: Optimal values of primal and dual are equal.
Weak Duality: Primal value ≤ Dual value (for max).
Shadow price: Change in objective per unit change in RHS of constraint.
Complementary Slackness: For each constraint, either slack = 0 or dual variable = 0.
🔹 Common Exam Applications
Product mix: Max profit subject to resources.
Transportation problem: Min shipping cost.
Diet problem: Minimize food cost subject to nutrition.
Workforce scheduling: Allocate workers to shifts.
🔹 Common Mistakes (avoid in exam!)
Forgetting nonnegativity condition x ≥ 0.
Missing a corner point in graphical method.
Confusing slack and surplus variables.
Not checking for infeasible or unbounded cases.
Forgetting to mention assumptions of LPP (linearity, certainty, divisibility, nonnegativity, additivity).
🔹 Super Quick Formulas
Z = CjXj (Objective).
Optimality in simplex (max): If all
Cj−Zj≤0, stop.
Number of basic variables in BFS = number of constraints.
Feasible region: Convex polygon/polyhedron.
🔹 Assumptions of LPP (often directly asked)
Linearity – objective & constraints are linear.
Additivity – effects add up.
Divisibility – variables can be fractional.
Certainty – coefficients known exactly.
Nonnegativity – variables ≥ 0.
🔹 Types of solutions
Feasible solution: satisfies constraints.
Basic feasible solution (BFS): feasible solution at a vertex.
Optimal BFS: BFS with best value.
Degenerate solution: BFS with basic variable = 0.
Multiple optima: more than one best solution.
Unbounded solution: objective → infinity.
Infeasible problem: no solution exists.
🔹 Special Methods
Big M method – uses penalty (M) for artificial variables.
Two-phase method – Phase I finds feasible solution, Phase II optimizes.
Dual simplex method – maintains optimality, restores feasibility.
🔹 Theorems (asked in short notes)
Fundamental Theorem of LP: If optimum exists, it occurs at a vertex.
Weak Duality Theorem: For any feasible primal & dual, primal value ≤ dual value (for max).
Strong Duality Theorem: Optimal primal = optimal dual.
Complementary Slackness: For each constraint, either slack = 0 or dual variable = 0.
🔹 Key Exam Applications (examples they love to ask)
Diet problem → Minimize cost with nutritional constraints.
Assignment problem → Minimize cost/time of assigning jobs.
Transportation problem → Minimize shipping cost.
Blending problem → Mix raw materials at minimum cost.
Scheduling → Allocate workers/machines optimally.
🔹 Common Short Questions
Define slack/surplus/artificial variables.
What is degeneracy/cycling?
Difference between primal & dual simplex.
State assumptions of LPP.
Graphical method steps.
Applications in real life.
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