Origin and development of OR, scientific method, modelling, advantages & limitations of models; Linear Programming Problem (LPP) — mathematical formulation with illustrative problems.
Topics Covered
Origin of ORNature & FeaturesScientific MethodModellingAdvantagesLimitationsApplicationsLPP Formulation
Operations Research (OR) emerged during World War II (1940s) when British and American military
leaders engaged interdisciplinary teams of scientists to optimise complex military operations — radar
deployment, anti-submarine warfare, convoy escort patterns. Hence the name: research on military operations.
After the war, the same techniques were applied to industry, business, government and economics, giving rise to modern OR.
Brief Timeline
Period
Milestone
1937
British military set up first OR team for radar deployment.
1939–45
OR teams in UK, US, Canada and Australia work on convoy systems, bombing strategies.
1947
George B. Dantzig develops the Simplex method for LPP.
1950s
Industrial OR societies founded (ORSA in USA, OR Society in UK).
Used in supply chains, scheduling, finance, healthcare, AI.
2. Definitions of OR
Churchman, Ackoff & Arnoff: "Operations Research is the application of scientific methods, techniques and tools to problems involving the operations of a system so as to provide those in control of the system with optimum solutions."
Morse & Kimball: "OR is a scientific method of providing executive departments with a quantitative basis for decisions regarding the operations under their control."
Operational Research Society (UK): "OR is the application of scientific method to complex problems arising in the direction and management of large systems of men, machines, materials and money."
3. Nature and Features of OR
3.1 Features (Characteristics)
Inter-disciplinary team approach — engineers, mathematicians, economists, psychologists collaborate.
Systems approach — examines the whole system, not isolated parts.
Quantitative basis for decisions — relies on measurable variables.
Use of scientific methods — observation, hypothesis, model, verification.
Use of computers — solves large LPPs that are intractable by hand.
Optimisation — seeks the best solution (maximum profit, minimum cost / time).
Decision-making — provides quantitative inputs to executives.
A model is an idealised representation of a real-life system. In OR, mathematical models
are central — they capture relationships among decision variables, constraints and objectives.
Classification of Models
Basis
Types
By function
Descriptive vs Prescriptive (optimisation)
By structure
Iconic, Analogue, Symbolic (Mathematical)
By nature
Deterministic vs Probabilistic (Stochastic)
By time
Static vs Dynamic
By solution
Analytical vs Simulation
5.1 Advantages of Models
Provide a logical and systematic approach to a problem.
Indicate the scope and limitations of a problem.
Reveal critical variables and their interrelationships.
Enable what-if analysis without affecting the real system.
Save time and resources compared with full-scale trials.
Permit communication of complex problems to non-specialists.
5.2 Limitations of Models
Models are abstractions — they cannot capture every real-world detail.
Construction may take time and considerable expertise.
Solutions are valid only within the model's assumptions.
Cost of model development can exceed savings if over-engineered.
Dynamic / stochastic data may render the model obsolete quickly.
6. General Solution Methods of OR Models
Analytical methods — mathematical techniques (calculus, algebra, optimisation theory).
Used in LPP, transportation, assignment, queueing.
Iterative methods — step-by-step refinement to reach the optimum (Simplex method,
Newton-Raphson).
Simulation methods — when analytical approach is impossible, mimic the system via Monte
Carlo experiments.
Heuristic methods — practical rules-of-thumb to find good (not necessarily optimal) solutions
quickly (e.g., genetic algorithms, ant colony, tabu search).
7. Applications of OR
Industry — production scheduling, inventory control, quality management.
Government / Public — disaster response, urban planning, road-traffic signal timing.
8. Linear Programming Problem (LPP)
DEFINITION
A Linear Programming Problem is the optimisation (maximum or minimum) of a linear
function of decision variables, subject to a system of linear equalities and / or inequalities, with
all variables restricted to be non-negative.
8.1 Components of an LPP
Decision Variables — quantities to be determined (e.g., \(x_1, x_2, \ldots, x_n\)).
Objective Function — the quantity to be maximised or minimised, linear in the variables:
\[ Z = c_1 x_1 + c_2 x_2 + \cdots + c_n x_n. \]
Constraints — linear inequalities / equations representing resource limits or requirements:
\[ a_{i1} x_1 + a_{i2} x_2 + \cdots + a_{in} x_n \;(\le, =, \ge) \; b_i, \quad i = 1, \ldots, m. \]
Non-negativity — \(x_j \ge 0\) for all \(j\).
8.2 General Mathematical Form
\[
\begin{aligned}
\text{Max (or Min)}\quad Z &= \sum_{j=1}^{n} c_j x_j \\[4pt]
\text{subject to}\quad \sum_{j=1}^{n} a_{ij} x_j &\;(\le, =, \ge)\; b_i, \quad i = 1, \ldots, m \\[4pt]
x_j &\ge 0, \quad j = 1, \ldots, n.
\end{aligned}
\]
8.3 Assumptions of LPP
Linearity — both objective and constraints are linear.
Certainty — coefficients \(c_j, a_{ij}, b_i\) are known with certainty.
Additivity — total contribution = sum of contributions from each variable.
Proportionality — contribution per unit is constant (no scale economies).
Divisibility — variables can take fractional values (relax for integer LP).
A furniture manufacturer makes chairs and tables. Each chair requires 5 hr of carpentry and 2 hr of painting; each table requires 8 hr of carpentry and 4 hr of painting. Profit per chair is ₹350; per table ₹600. Available time is 240 hr of carpentry and 100 hr of painting per week. Formulate the LPP to maximise weekly profit.
Formulation:
Decision variables: \(x_1\) = number of chairs, \(x_2\) = number of tables produced per week.
Objective: Max \(Z = 350 x_1 + 600 x_2\) (profit in ₹).
Constraints:
Carpentry: \(5 x_1 + 8 x_2 \le 240\)
Painting: \(2 x_1 + 4 x_2 \le 100\)
Non-negativity: \(x_1, x_2 \ge 0\).
EXAMPLE 2 (Diet Problem)
A diet must contain at least 8 units of vitamins, 12 units of minerals and 10 units of calories. Two foods F1 (₹4 per unit) and F2 (₹3 per unit) are available. F1 contains 2 V, 6 M, 4 Cal per unit; F2 contains 4 V, 2 M, 5 Cal per unit. Formulate the LPP to minimise cost.
Formulation:
Decision variables: \(x_1\) = units of F1, \(x_2\) = units of F2.
Objective: Min \(Z = 4 x_1 + 3 x_2\).
Constraints:
Vitamins: \(2 x_1 + 4 x_2 \ge 8\)
Minerals: \(6 x_1 + 2 x_2 \ge 12\)
Calories: \(4 x_1 + 5 x_2 \ge 10\)
\(x_1, x_2 \ge 0\).
9.3 More Common LPP Templates
(a) Transportation LPP
Ship goods from \(m\) origins (capacities \(a_i\)) to \(n\) destinations (demands \(b_j\)) at cost \(c_{ij}\) per unit. Decision variable \(x_{ij}\) = units shipped from origin \(i\) to destination \(j\).