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Operations Research Basics

Operations research optimises an objective Z = Σc_i·x_i subject to constraints. Linear programming (graphical/simplex) is the core tool, with the optimum at a corner of the feasible region, per industrial-engineering texts.
Exam tip: keep SI units consistent end-to-end, write the governing relation symbolically before substituting, and sanity-check magnitude and sign.

Key formulas & points

Skim these first — then read the full notes below.

  • Graphical LP: corner point theorem for 2-variable problems
  • Simplex method for multi-variable LP
  • Transportation and assignment problems are special LPs

Topic details

Introduction

Operations research applies mathematical optimisation to decision problems — product mix, transportation, assignment, and sequencing. Indian IE syllabi centre on linear programming and its special cases.

Scope in B.Tech and GATE syllabus

A linear program maximises or minimises a linear objective subject to linear constraints. The graphical method (two variables) shows the optimum at a vertex of the feasible polygon; the simplex method handles many variables algebraically.

Why this topic matters in practice

Special structures — the transportation and assignment problems — have efficient dedicated algorithms. Duality and sensitivity analysis interpret the solution economically. Formulating a problem and solving it graphically or by simplex is the recurring exam task.

Key relations & formulas

Z=c1x1+c2x2Z = c_{1}x_{1} + c_{2}x_{2}
(objective function, maximise/minimise)
a11x1+a12x2≤b1a_{11}x_{1} + a_{12}x_{2} \le b_{1}
(constraint inequality)
x1,x2≥0x_{1}, x_{2} \ge 0
(non-negativity)

Formulas (Indian textbook notation)

  • Dualprice=rateofchangeofZperunitRHSincreaseDual price = rate of change of Z per unit RHS increase

Notation and sign conventions

Relation 1 —
Z=c1x1+c2x2Z = c_{1}x_{1} + c_{2}x_{2}
Z=c1x1+c2x2Z = c_{1}x_{1} + c_{2}x_{2}
(objective function, maximise/minimise)
Write this relation with symbols exactly as in Industrial Engineering & Management — O.P. Khanna before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.
Relation 2 —
a11x1+a12x2≤b1a_{11}x_{1} + a_{12}x_{2} \le b_{1}
a11x1+a12x2≤b1a_{11}x_{1} + a_{12}x_{2} \le b_{1}
(constraint inequality)
Write this relation with symbols exactly as in Industrial Engineering & Management — O.P. Khanna before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.
Relation 3 —
x1,x2≥0x_{1}, x_{2} \ge 0
x1,x2≥0x_{1}, x_{2} \ge 0
(non-negativity)
Write this relation with symbols exactly as in Industrial Engineering & Management — O.P. Khanna before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.
Relation 4 —
Dualprice=rateofchangeofZperunitRHSincreaseDual price = rate of change of Z per unit RHS increase

Formulas (Indian textbook notation)

  • Dualprice=rateofchangeofZperunitRHSincreaseDual price = rate of change of Z per unit RHS increase
Write this relation with symbols exactly as in Industrial Engineering & Management — O.P. Khanna before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.

Fundamentals and definitions

An LP has decision variables x_i, a linear objective Z = Σc_i x_i, and linear constraints Σa_ij x_i ≤ b_j with x_i ≥ 0. The feasible region is a convex polytope; the optimum lies at a vertex (corner point).

Governing relations in practice

The graphical method plots constraints, identifies the feasible region, and evaluates Z at each corner — practical for two variables. The simplex method moves from vertex to vertex, improving Z each step, for larger problems.

Design and analysis considerations

The transportation problem minimises shipping cost from sources to destinations with supply/demand constraints, solved by NWCR/least-cost for an initial solution and MODI for optimality. The assignment problem (one-to-one) uses the Hungarian method.

Advanced theory and extensions

Duality pairs every LP with a dual whose optimum equals the primal's; shadow prices from the dual value each constraint. Sensitivity analysis shows how the solution changes with data. These formulate-and-solve skills define OR basics.

Assumptions and validity limits

State assumptions explicitly before using any relation for operations research basics — steady state, uniform properties, linear elastic material, ideal gas, incompressible flow, etc., as applicable.
Wrong assumptions invalidate the entire solution even when the formula is correct. In Industrial Engineering viva and GATE descriptive questions, listing valid assumptions often earns separate marks.

Step-by-step problem approach

1. Read the question and list given data with SI units (common in Industrial Engineering papers).
2. Draw a neat labelled diagram where applicable — examiners in Indian universities award diagram marks even when arithmetic slips.
3. Identify which relation from this topic applies to operations research basics.
4. Use equation 1:
Z=c1x1+c2x2Z = c_{1}x_{1} + c_{2}x_{2}
.
5. Use equation 2:
a11x1+a12x2≤b1a_{11}x_{1} + a_{12}x_{2} \le b_{1}
.
6. Substitute values, compute, and verify units and sign (direction).
7. State conclusion in one line — e.g. safe/unsafe, stable/unstable, feasible/infeasible.

Applications & exam relevance

Operations Research Basics appears in factories, logistics, and service systems. In Indian mechanical curricula this topic is tested because it connects theory to productivity, layout, and operations.
GATE and semester exams often combine operations research basics with earlier units — revise prerequisites before attempting mixed problems.
Industry interview panels sometimes ask: "Where did you use operations research basics?" — answer with a lab, mini-project, or plant visit example if possible.

Common mistakes in exams

• Forgetting the non-negativity constraints x_i ≥ 0
• Evaluating the objective inside the region instead of at the corner points
• Using an unbalanced transportation problem without adding a dummy row/column
• Confusing maximisation and minimisation directions when reading the graph

Quick revision checklist

Before attempting operations research basics problems, confirm you can:
1. Graphical LP: corner point theorem for 2-variable problems
2. Simplex method for multi-variable LP
3. Transportation and assignment problems are special LPs
Revise the solved examples in Industrial Engineering & Management — O.P. Khanna and one previous-year GATE or university paper for this unit.

Worked examples

Try the problem first — open the solution when you are ready to check.

LP objective at a corner

Problem

Maximise Z = 3x₁ + 5x₂. A corner of the feasible region is (x₁, x₂) = (4, 6). Find Z there.

Solution

Z = 3x₁ + 5x₂ = 3(4) + 5(6) = 12 + 30 = 42 (evaluate all corners; the largest is optimal).

Conceptual check — Operations Research Basics

Problem

In a Industrial Engineering semester or GATE paper you are asked: "State the main assumption, the governing relation, and one practical consequence of operations research basics." What should a complete answer include?

Practice questions

Most-asked interview and GATE questions for this topic — expand any item for a model answer.

  1. 1
    What is Operations Research Basics, and why does it appear in B.Tech / GATE syllabi?

    Model answer

    Operations research optimises an objective Z = Σc_i·x_i subject to constraints. Linear programming (graphical/simplex) is the core tool, with the optimum at a corner of the feasible region, per industrial-engineering texts.
  2. 2
    State the relation Z = c₁x₁ + c₂x₂ and name each symbol.

    Model answer

    The governing relation is Z=c1x1+c2x2Z = c_{1}x_{1} + c_{2}x_{2}. Write every symbol with SI units before substituting numbers.
  3. 3
    State the relation a₁₁x₁ + a₁₂x₂ ≤ b₁ and name each symbol.

    Model answer

    The governing relation is a11x1+a12x2≤b1a_{11}x_{1} + a_{12}x_{2} \le b_{1}. Write every symbol with SI units before substituting numbers.
  4. 4
    State the relation x₁, x₂ ≥ 0 and name each symbol.

    Model answer

    The governing relation is x1,x2≥0x_{1}, x_{2} \ge 0. Write every symbol with SI units before substituting numbers.
  5. 5
    State the relation Dual price = rate of change of Z per unit RHS increase and name each symbol.

    Model answer

    The governing relation is Dualprice=rateofchangeofZperunitRHSincreaseDual price = rate of change of Z per unit RHS increase. Write every symbol with SI units before substituting numbers.
  6. 6
    Explain: Graphical LP: corner point theorem for 2-variable problems

    Model answer

    Graphical LP: corner point theorem for 2-variable problems — state the assumption range and one exam trap linked to this point.
  7. 7
    Explain: Simplex method for multi-variable LP

    Model answer

    Simplex method for multi-variable LP — state the assumption range and one exam trap linked to this point.
  8. 8
    Explain: Transportation and assignment problems are special LPs

    Model answer

    Transportation and assignment problems are special LPs — state the assumption range and one exam trap linked to this point.
  9. 9
    How would you correct this error in a viva: Forgetting the non-negativity constraints x_i ≥ 0?

    Model answer

    Identify the wrong assumption or unit mix-up, rewrite the correct relation, and recompute with a one-line sanity check.
  10. 10
    How would you correct this error in a viva: Evaluating the objective inside the region instead of at the corner points?

    Model answer

    Identify the wrong assumption or unit mix-up, rewrite the correct relation, and recompute with a one-line sanity check.
  11. 11
    How would you correct this error in a viva: Using an unbalanced transportation problem without adding a dummy row/column?

    Model answer

    Identify the wrong assumption or unit mix-up, rewrite the correct relation, and recompute with a one-line sanity check.
  12. 12
    How would you correct this error in a viva: Confusing maximisation and minimisation directions when reading the graph?

    Model answer

    Identify the wrong assumption or unit mix-up, rewrite the correct relation, and recompute with a one-line sanity check.

Exams & GATE

  • 1
    O.P. Khanna Ch. 12 — identify feasible region and optimal corner point.
  • 2
    Avoid: Forgetting the non-negativity constraints x_i ≥ 0
  • 3
    Avoid: Evaluating the objective inside the region instead of at the corner points
  • 4
    Avoid: Using an unbalanced transportation problem without adding a dummy row/column

📖 Standard books (India)

  • Industrial Engineering & Management — O.P. Khanna

    Read: Syllabus unit

    Work study, PPC, and OR basics

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