Question Bank | QT Midterm

Every question from the two 2025-26 papers plus the practice problems worked in your notes and decks — organised by the skill being tested, so you can drill one pattern at a time.

Duality — convert primal to dual

Exam · Max Z = 2X₁ + 3X₂ + 4X₃; X₁+2X₂+X₃ ≤ 8; 2X₁+X₂+3X₃ ≥ 10; X₁+X₂+X₃ = 6. (Final Q1A — fully solved)
Exam · Min Z = 5X₁ + 2X₂; 3X₁+2X₂ ≤ 12; X₁+X₂ ≥ 4; X₁−X₂ = 3. (Re-Exam Q1A — fully solved)
Deck · Max Z = X₁ − X₂ + 3X₃; X₁+X₂+X₃ ≤ 10; 2X₁−X₂−X₃ ≥ 2; 2X₁−2X₂−3X₃ ≤ 6. (Deck p. 19)
Class · Max Z = 3X₁ + 5X₂ with four constraints → Min with Y₁…Y₄. (Class notes p. 19)
Notes · A larger Min primal with an equality → Max dual. (Notes p. 36)

LPP formulation

Exam · Wooden chairs and tables; carpentry 120 hrs, finishing 90 hrs; profit ₹200 / ₹300. (Final Q3B — solved)
Exam · Products P₁ and P₂ from raw materials A (150 units) and B (180 units); profit ₹40 / ₹50. (Re-Exam Q3B — solved)
Deck · Tables and chairs, Max Z = 200X₁ + 90X₂ — the standing class example. (LPP deck p. 4)
Deck · Three box types A, B, C — a minimisation formulation, Min Z = 1200X₁ + 900X₂ + 1500X₃. (p. 5)
Deck · Two products with profit per unit and assembly time. (p. 6)
Deck · Two plants × two products, Min Z = 15000X₁ + 28000X₂ + 18000X₃ + 26000X₄ — four decision variables. (p. 7)
Deck · Par Inc. golf bags — four departments, fractional coefficients (7/10, 5/6, 2/3, 1/4). (Graphical deck p. 3)
Deck · Diet problem — at least 4000 vitamins, 50 minerals, 1400 calories; a Min with ≥ constraints. (p. 5)
Deck · Crude oil blending into gasolines P and Q. (LPP deck p. 16)

Graphical solution

Max Z = 200X₁ + 90X₂; 35X₁+10X₂ ≤ 3500; 6X₁+4X₂ ≤ 1200. Corner points O, A(0,300), E(25,260), C(100,0) → Z* = ₹28,400. (worked in Unit 1)
Weekly production ≤ 25 of P1, ≤ 35 of P2, 60 worker-weeks; profit ₹60 / ₹40 → Z* = ₹2,150 at (12.5, 35). (p. 2)
Par Inc. golf bags — feasible region ABCDE, Z at C(540, 252) = 7668. (p. 4)
Diet problem — minimise cost 4X₁ + 3X₂ subject to three ≥ constraints. (p. 5)
Firm making X and Y with total capacity 9 TPD. (LPP deck p. 14)
Two more plotted on graph paper in class, with corner points marked. (Class notes pp. 8, 14)

Special cases

Exam · Identify the special case from a final simplex tableau; justify from the RHS, basic variables and Cⱼ − Zⱼ row; state one implication. (Final Q7A — solved: degeneracy)
Degenerate · Max Z = 3X₁ + 9X₂; X₁+4X₂ ≤ 8; X₁+2X₂ ≤ 4 — tie in replacement ratios. (p. 1)
Unbounded · Max Z = 5X₁ + 4X₂; X₁ ≤ 7; X₁−X₂ ≤ 8 — ratios ∞ and −1. (p. 3)
Infeasible · Max Z = 200X₁ − 300X₂; 2X₁+3X₂ ≥ 7; X₁+X₂ ≤ 400; 2X₁+1.5X₂ ≥ 900. (p. 4 · class notes p. 19)
Multiple optima · Max Z = 4X₁ + 10X₂; 2X₁+X₂ ≤ 10; 2X₁+5X₂ ≤ 20; 2X₁+3X₂ ≤ 18. (p. 5)
Conceptual: Distinguish degeneracy from multiple optimal solutions using the simplex tableau.
Conceptual: Why does an unbounded solution always indicate a badly formulated problem?

Sensitivity analysis

Exam · Three variables, three binding constraints; five sub-questions on binding, shadow price, RHS change, reduced cost and coefficient range. (Final Q4A — solved)
Exam · Material / Labour / Machine Hours, with X₃ non-basic and one non-binding constraint. (Re-Exam Q4B — solved)
Deck · Advertising spaces: Max Z = 50X₁ + 20X₂; the seven standard questions on budget spent, footage used, range checks and shadow price arithmetic. (Deck pp. 15–18, all worked in Unit 1)
Conceptual: Define reduced cost, objective coefficient range, shadow price, binding constraint.
Conceptual: "Optimal solution does not change" — does that mean the objective value does not change? Explain.
Conceptual: Why does a non-binding constraint always have a shadow price of zero?

Integer Linear Programming

Exam · Five projects, analyst hours in two quarters, C and E incompatible. (Final Q4B — solved)
Exam · Five projects, staff time in two terms, two projects sharing cloud infrastructure. (Re-Exam Q4A — solved)
Pure · Toy company, dolls and cars, 6 labour hours. (p. 4)
Binary · Three projects, ₹70 lakh budget. (p. 5)
Pure · Factory shift allocation — minimums per shift, 10 workers total. (p. 6 · class p. 2)
Pure · Paper mill cutting stock — five cutting patterns, minimise big rolls. (p. 7)
Binary · Four projects, ₹200 lakh + 50 manpower, at least two projects. (p. 8)
Fixed-charge · Three warehouses with opening costs and an 80-unit linking constraint. (p. 9 · class p. 5)
Binary · Four projects with four logical rules — prerequisite, either/or, at-most, at-least. (p. 10)
Mixed · Tables in multiples of 10, chairs continuous. (Note 2 p. 20)
Conceptual: Distinguish pure, mixed and binary ILP with one example each.

Transportation & Assignment

Exam · 3 workers × 3 tasks — balanced? constraints? variables? (Final Q1B — solved)
Exam · 3 warehouses × 3 plants with supply and demand — unbalanced. (Re-Exam Q1B — solved)
Exam · Hyderabad–Kolkata flight pairing, 5 outbound vs 4 return. (Final Q5B — solved)
Exam · Delhi–Chennai flight pairing with a 30-minute turnaround. (Re-Exam Q5B — solved)
Exam · Dairy company, 3 plants × 4 markets, profit matrix from selling price minus costs. (Final Q7B — solved)
Class · 4 employees × 4 tasks — full LPP conversion. (pp. 1–2, worked in Unit 3)
Class · 3 origins × 4 destinations, supply 14/16/5, demand 6/10/15/4. (pp. 3–4)
Conceptual: How do you recognise a transportation problem from an assignment problem?
Conceptual: Why are dummy costs set to zero?

⚠ Questions in the papers that are NOT in your Unit 1–3 portion

These appeared in the 2025-26 papers but belong to Units 4–7, which your notes do not cover. They are listed so you know what the full paper contains — they are not solved on this site.

Question	Topic	Syllabus unit
Final Q1C · Re-Exam Q1C	AHP pairwise comparison matrix and priority weights	Not in Units 1–7 as listed
Final Q1D · Re-Exam Q1D	Payoff matrix; Maximin / Maximax / Laplace	Unit 7 — Decision Theory
Final Q2 · Re-Exam Q2	Monte Carlo simulation of a queue / production run	Unit 6 — Simulation
Final Q3A · Re-Exam Q3A	Two-person zero-sum game payoff matrix	Unit 4 — Theory of Games
Final Q5A · Re-Exam Q5A	M/M/1 queue parameters and traffic intensity	Unit 5 — Queuing
Final Q6A · Q6B	Decision tree EMV; Hurwicz criterion	Unit 7 — Decision Theory

If your test turns out to include any of these, say so and they can be added — the papers and their synoptic keys are already rendered into the source viewer.