Skip to content
← Business Mathematics and Statistics

Business Mathematics and Statistics · Class 12 Commerce

Ch 10Operations Research (Transportation Problem, Assignment Problems, Decision Theory) — Class 12 Business Mathematics and Statistics, concept-first.

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.

38

Q&A

6

Concepts

Not available

Exam weightage

Start learning — read this chapter →

Key concepts

Hover a concept to preview it and jump to its most relevant Q&A.

Transportation Problem and Initial Basic Feasible Solution

A transportation problem moves goods from several sources (with fixed supplies) to several destinations (with fixed demands) at minimum total cost.

Most relevant Q&A

Start with this concept →

Chapter contents

The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.

1

Introduction to Operations Research and the Transportation Problem

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 compare…

2

North-West Corner Method

The North-West Corner Method is the simplest way to obtain an IBFS. It ignores cost altogether and allocates purely by position, always starting at the top-left (north-west) cell of the table.

3

Least Cost (Matrix Minima) Method

The Least Cost Method (also called the Matrix Minima Method) improves on NWC by allocating to the cheapest cells first, so the starting solution is usually closer to optimal.

4

Vogel's Approximation Method (VAM)

Vogel's Approximation Method is the most reliable of the three IBFS methods because it uses a penalty — the extra cost of not choosing the cheapest cell in a row or column — to decide where to allocat…

5

Testing the IBFS for Optimality

An Initial Basic Feasible Solution — however it was obtained — is only a starting point. It must be tested to see whether the total cost can be reduced further, and improved if it can.

6

The Assignment Problem — Meaning and Structure

An assignment problem arises whenever exactly 'workers' (people, machines, vehicles) must be matched one-to-one with exactly 'jobs' (tasks, routes, territories), so that every worker gets exactly one…

7

The Hungarian Method

The Hungarian Method finds the optimal one-to-one assignment in a square cost matrix without checking every possible permutation (which grows as and quickly becomes impractical).

8

The Assignment Problem — Maximization Case

The Hungarian Method is built to minimise a cost matrix. When the table instead gives profits (or output, or any measure to be maximised), convert it first:

9

Decision Theory — Elements and Decision-Making under Uncertainty

A decision-making situation in business mathematics has three elements: - Courses of action (alternatives) — the choices open to the decision-maker, e.g.

10

Decision-Making under Risk — EMV and EOL

When the probabilities of the states of nature are known (from past data, market research, etc.), the situation is decision-making under risk, and the standard criterion is Expected Monetary Value:

Exercises

+Show 8 questions8 questions
  1. Q4A company has three plants $O_1, O_2, O_3$ with per-unit transportation costs to three destinations $D_1, D_2, D_3$ as given below. Supplies…Free
  2. Q5State the rule used to check whether a transportation table's basic feasible solution is non-degenerate. Using the Vogel's Approximation Met…Free
  3. Q7Explain why an assignment problem is regarded as a special case of the transportation problem. Using the cost matrix of Worked Example 4 (th…Free
  4. Q8Four workers $W_1, W_2, W_3$ must be assigned to three jobs $J_1, J_2, J_3$ — one worker per job — at minimum total cost. The cost matrix (₹…Preview
  5. Q9State the condition used in the Hungarian Method to decide whether the current reduced matrix already gives an optimal assignment. Referring…Preview
  6. Q12Define the terms 'state of nature' and 'course of action' as used in decision theory, and illustrate both using the payoff table of Worked E…Preview
  7. Q13Distinguish between decision-making under risk and decision-making under uncertainty. Among the four criteria studied for decision-making un…Preview
  8. Q15Using the payoff table of Worked Example 6, suppose the probabilities are revised to $P(S_1)=0.5$, $P(S_2)=0.3$, $P(S_3)=0.2$. Compute the E…Preview

Sample & Board Papers

Sample papers and previous-year board questions for this subject.

+Show 23 questions23 questions
  1. Q1If number of sources is not equal to number of destinations, the assignment problem is called ________. (a) unsymmetric (b) balanced (c) sym…Preview
  2. Q2In a non-degenerate solution, number of allocations is : (a) Not equal to $m + n + 1$ (b) Equal to $m + n - 1$ (c) Equal to $m + n + 1$ (d)…Preview
  3. Q3Given the following pay-off matrix (in rupees) for three strategies and two states of nature. | Strategy | $E_1$ (States-of-nature) | $E_2$…Preview
  4. Q4Give mathematical form of Assignment Problem.Preview
  5. Q5(a) Find the initial basic feasible solution of the following transportation problem. | | I | II | III | IV | Supply | | --- | --- | --- | -…Preview
  6. Q6Decision theory is concerned with : (a) decision making under certainty (b) selecting optimal decisions in sequential problem (c) analysis o…Preview
  7. Q7Select a strategy using each of the following rule. | Strategy | States-of-nature $E_1$ | States-of-nature $E_2$ | | --- | --- | --- | | $S_…Preview
  8. Q8The transportation problem is said to be unbalanced if ________. (a) $m=n$ (b) Total supply $\neq$ Total demand (c) $m+n-1$ (d) Total supply…Preview
  9. Q9North-West corner refers to ________. (a) bottom right corner (b) top left corner (c) bottom left corner (d) top right cornerPreview
  10. Q10Give mathematical form of Assignment problem.Preview
  11. Q11From the following pay-off matrix, find the optimal decision under each of the following rule (i) maxmin (ii) minimax. | Act | $S_1$ | $S_2$…Preview
  12. Q12In an assignment problem the value of decision variable $x_{ij}$ is : (a) $0$ (b) $1$ or $0$ (c) $-1$ (d) $1$Preview
  13. Q13In a non-degenerate solution number of allocations is : (a) Not equal to $m+n-1$ (b) Equal to $m+n-1$ (c) Not equal to $m+n+1$ (d) Equal to…Preview
  14. Q14What is transportation problem ?Preview
  15. Q15Given the following pay-off matrix (in rupees) for three strategies and two states of nature. | Strategy | $E_1$ | $E_2$ | | --- | --- | ---…Preview
  16. Q16Solution for transportation problem using __________ method is nearer to an optimal solution. (a) VAM (b) NWCM (c) Row Minima (d) LCMPreview
  17. Q17The transportation problem is said to be unbalanced if __________. (a) $m=n$ (b) Total supply $\ne$ Total demand (c) $m+n-1$ (d) Total suppl…Preview
  18. Q18Give mathematical form of Assignment problem.Preview
  19. Q19The research department of Hindustan Ltd., has recommended to pay marketing department to launch a shampoo of three different types. The mar…Preview
  20. Q20North-West corner refers to ________. (a) bottom right corner (b) top left corner (c) bottom left corner (d) top right cornerPreview
  21. Q21In a non-degenerate solution number of allocations is : (a) Not equal to $m + n - 1$ (b) Equal to $m + n - 1$ (c) Not equal to $m + n + 1$ (…Preview
  22. Q22Consider the following pay-off (Profit) matrix Action States. | Action | $S_1$ | $S_2$ | $S_3$ | $S_4$ | | --- | --- | --- | --- | --- | | $…Preview
  23. Q23Find the optimal solution for the assignment problem with following cost matrix. | Salesmen \ Area | 1 | 2 | 3 | 4 | | --- | --- | --- | ---…Preview

More questions

+Show 7 questions7 questions
  1. Example 1A firm has three factories $O_1, O_2, O_3$ with per-unit transportation costs to three warehouses $D_1, D_2, D_3$ as given below. Supplies a…Free
  2. Example 2For the same transportation problem in Worked Example 1, find the initial basic feasible solution using the Least Cost (Matrix Minima) Metho…Free
  3. Example 3For the same transportation problem in Worked Examples 1 and 2, find the initial basic feasible solution using Vogel's Approximation Method…Free
  4. Example 6A company must assign three workers $W_1, W_2, W_3$ to three jobs $J_1, J_2, J_3$ at minimum total cost. The cost (in ₹) of each worker doin…Preview
  5. Example 10A firm can assign any of three salesmen $W_1, W_2, W_3$ to any of three territories $J_1, J_2, J_3$. The expected profit (₹ '000) for each c…Preview
  6. Example 11A firm is considering three courses of action $A_1, A_2, A_3$ under three possible states of nature $S_1, S_2, S_3$ (demand conditions), wit…Preview
  7. Example 14Using the payoff table of Worked Example 6, with $P(S_1)=0.3$, $P(S_2)=0.5$, $P(S_3)=0.2$, compute the Expected Monetary Value (EMV) of each…Preview