Business Mathematics and Statistics · Ch 10 — Operations Research (Transportation Problem, Assignment Problems, Decision Theory)

Introduction to Operations Research and the Transportation Problem

Operations Research in Business Decision-Making

Operations Research (OR) is the branch of business mathematics that uses systematic, quantitative methods to arrive at the best possible decision when resources are limited and choices must be compared objectively. In this Tamil Nadu Class 12 Business Mathematics and Statistics chapter, three classic OR situations are studied together because they share the same spirit — finding an optimum plan of action — even though each has its own technique: the transportation problem (how to move goods from several supply points to several demand points at least cost), the assignment problem (how to match workers, machines or jobs one-to-one at least cost or greatest profit), and decision theory (how to choose the best course of action when the outcome depends on an uncertain state of nature).

This syllabus draws on the same operations-research principles taught across Indian commerce curricula, presented here with worked numericals suited to Tamil Nadu DGE Class 12 Business Mathematics and Statistics (Samacheer Kalvi) students preparing transportation problem, assignment problem and decision theory questions and answers for their board examination.

Structure of a Transportation Problem

A transportation problem has:

m sources / origins (factories, warehouses) each with a fixed supply $a_i$
n destinations (markets, depots) each with a fixed demand $b_j$
A cost matrix $C=[c_{ij}]$ , where $c_{ij}$ is the cost of transporting one unit from source $i$ to destination $j$

The objective is to decide how many units $x_{ij}$ to ship from each source to each destination so that:

Every source's full supply is shipped out,
Every destination's full demand is met, and
The total transportation cost $Z=\sum_{i}\sum_{j} c_{ij}x_{ij}$ is minimised.

Balanced vs unbalanced

Condition	Meaning	Action
Balanced	$\sum a_i = \sum b_j$ (total supply = total demand)	Solve directly
Unbalanced	$\sum a_i \neq \sum b_j$	Add a dummy row (if demand exceeds supply) or a dummy column (if supply exceeds demand) with zero cost entries, sized to absorb the difference, then solve the now-balanced table

A transportation problem can be solved only once it is balanced. The dummy row/column represents supply that is never actually shipped, or demand that is never actually met — its allocation carries zero cost and is simply a bookkeeping device that makes the table square.

Two-stage solution method

Every transportation problem is solved in two stages:

Find an Initial Basic Feasible Solution (IBFS) — a starting allocation that satisfies every supply and demand exactly. Three methods are used: the North-West Corner Method, the Least Cost Method, and Vogel's Approximation Method (studied next, each on the same worked cost matrix so the three can be compared directly).
Test the IBFS for optimality and improve it if needed, using the Modified Distribution (MODI) Method or the Stepping-Stone Method, until no further cost reduction is possible.

Definition 1Basic Feasible Solution (BFS)

An allocation of units to the cells of a transportation table that satisfies every row's supply and every column's demand exactly, using no more than $m+n-1$ occupied cells (where $m$ = number of sources, $n$ = number of destinations).

Definition 2Balanced Transportation Problem

A transportation problem in which total supply exactly equals total demand, $\sum a_i = \sum b_j$ ; only a balanced problem can be solved by the standard allocation methods.

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