Samacheer Kalvi Class 12 Business Maths Solutions Chapter 10 Operations Research Exercise 10.4

Tamilnadu Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Ex 10.4

Choose the correct answer

Question 1. The transportation problem is said to be unbalanced if
(a) Total supply \( \neq \) Total demand
(b) Total supply = Total demand
(c) m = n
(d) m + n - 1
Answer: (a) Total supply \( \neq \) Total demand
In simple words: A transportation problem is unbalanced if the total amount of goods available (supply) is not equal to the total amount needed (demand). If they are not equal, you can't perfectly match all supply to all demand without adjustments.

🎯 Exam Tip: Remember, for a transportation problem to be 'balanced', the total supply must always exactly equal the total demand. This is a foundational concept in operations research.

Question 2. In a non – degenerate solution number of allocation is
(a) Equal to m + n - 1
(b) Equal to m + n + 1
(c) Not equal to m + n - 1
(d) Not equal to m + n + 1
Answer: (a) Equal to m + n - 1
In simple words: In a transportation problem, if a solution is 'non-degenerate', it means the number of items you've assigned (allocations) is exactly 'm' plus 'n' minus 1, where 'm' is the number of rows and 'n' is the number of columns. This exact number of allocations is important for checking the solution's quality.

🎯 Exam Tip: The condition for a non-degenerate basic feasible solution is crucial for applying methods like the Modi method or Stepping Stone method to find the optimal solution in transportation problems.

Question 3. In a degenerate solution number of allocations is
(a) Equal to m + n - 1
(b) Not equal to m + n - 1
(c) Less than m + n - 1
(d) Greater than m + n - 1
Answer: (c) Less than m + n - 1
In simple words: If a transportation problem has a 'degenerate' solution, it means you have fewer allocations than the required 'm + n - 1' number. This situation needs a special step called assigning a very small, imaginary amount (epsilon) to an empty cell to move forward.

🎯 Exam Tip: Degeneracy can occur in the initial or subsequent solutions. To resolve it, an arbitrarily small positive value (epsilon) is assigned to one or more unoccupied cells to meet the condition for a non-degenerate solution.

Question 4. The Penalty in VAM represents difference between the first
(a) Two largest costs
(b) Largest and Smallest costs
(c) smallest two costs
(d) None of the options
Answer: (c) smallest two costs
In simple words: In the VAM method (Vogel's Approximation Method), the 'penalty' for each row or column is found by subtracting the second smallest cost from the smallest cost. This penalty helps decide where to make the first allocation to minimize total cost.

🎯 Exam Tip: VAM's penalty calculation is key to its effectiveness, as it focuses on the "cost of not making an allocation" to the cheapest cell, which often leads to a good initial basic feasible solution close to the optimal one.

Question 5. Number of basic allocation in any row or column in an assignment problem can be
(a) Exactly one
(b) At least one
(c) At most one
(d) None of the options
Answer: (a) Exactly one
In simple words: In an assignment problem, each task must be assigned to exactly one person, and each person can do exactly one task. This means there will be only one assignment (allocation) in any row or column. This ensures everyone gets one job and no job is done twice.

🎯 Exam Tip: The core principle of an assignment problem is that each resource (like a worker) is assigned to exactly one task, and each task is handled by exactly one resource. This uniqueness is what makes it distinct from transportation problems.

Question 6. North - West Corner refers to
(a) Top left corner
(b) Top right corner
(c) Bottom right corner
(d) Bottom left corner
Answer: (a) Top left corner
In simple words: In transportation problems, the North-West Corner method starts allocations by looking at the cell in the very top-left corner of the transportation table. It's a simple way to begin solving the problem, always starting from that specific cell.

🎯 Exam Tip: The North-West Corner Method (NWCM) is the simplest method for finding an initial basic feasible solution, but it doesn't consider costs, so the resulting solution is often not optimal.

Question 7. Solution for transportation problem using method is nearer to an optimal
(a) NWCM
(b) LCM
(c) VAM
(d) Row Minima
Answer: (c) VAM
In simple words: When solving transportation problems, the Vogel's Approximation Method (VAM) usually gives an initial solution that is very close to the best possible solution (the optimal one). It is often better than other starting methods because it considers the penalty costs.

🎯 Exam Tip: VAM is generally preferred for generating initial basic feasible solutions in transportation problems because it incorporates cost differences, leading to a more efficient starting point than NWCM or LCM (Least Cost Method).

Question 8. In an assignment problem the value of decision variable xij is
(a) 1
(b) 0
(c) 1 or 0
(d) none of them
Answer: (c) 1 or 0
In simple words: In an assignment problem, the decision variable \( x_{ij} \) can only be 1 or 0. It is 1 if worker 'i' is assigned to job 'j', and 0 if they are not. This binary choice shows whether an assignment happens or not.

🎯 Exam Tip: The binary nature of decision variables (0 or 1) is a defining characteristic of assignment problems, indicating a direct "yes" or "no" for each possible assignment, ensuring discrete choices.

Question 9. If number of sources is not equal to number of destinations, the assignment problem is called
(a) balanced
(b) unsymmetric
(c) symmetric
(d) unbalanced
Answer: (d) unbalanced
In simple words: If you have a different number of sources (like workers) than destinations (like jobs) in an assignment problem, it's called 'unbalanced'. To solve it, you need to add imaginary (dummy) sources or destinations to make the numbers equal.

🎯 Exam Tip: An unbalanced assignment problem requires the addition of dummy rows or columns with zero costs to make the number of rows equal to the number of columns, allowing the problem to be solved using standard methods.

Question 10. The purpose of a dummy row or column in an assignment problem is to
(a) prevent a solution from becoming degenerate
(b) balance between total activities and total resources
(c) provide a means of representing a dummy problem
(d) None of the options
Answer: (b) balance between total activities and total resources
In simple words: Dummy rows or columns are added to an assignment problem mainly to make sure the number of workers matches the number of jobs. This helps balance the problem so it can be solved correctly. They are like placeholders with no real cost.

🎯 Exam Tip: Dummy rows/columns are essential for transforming an unbalanced assignment problem into a balanced one, which is a prerequisite for applying methods like the Hungarian algorithm. The cost associated with dummy cells is always zero.

Question 11. The solution for an assignment problem is optimal if
(a) each row and each column has no assignment
(b) each row and each column has at least one assignment
(c) each row and each column has at most one assignment
(d) each row and each column has exactly one assignment
Answer: (d) each row and each column has exactly one assignment
In simple words: A perfect solution to an assignment problem means every worker has exactly one job, and every job is done by exactly one worker. This makes sure all tasks are covered without anyone being assigned multiple roles or no role at all.

🎯 Exam Tip: The optimality condition for an assignment problem emphasizes the unique one-to-one mapping between resources and tasks. This is achieved when a solution ensures exactly one assignment per row and per column in the cost matrix.

Question 12. In an assignment problem involving four workers and three jobs, total number of assignments possible are
(a) 4
(b) 3
(c) 7
(d) 12
Answer: (b) 3
In simple words: If you have four workers but only three jobs, the maximum number of actual jobs that can be assigned is three. One worker will not get a job, and a 'dummy job' would be created for that worker to balance the problem.

🎯 Exam Tip: In an unbalanced assignment problem (m workers, n jobs), the number of actual assignments will always be the minimum of m and n. The difference will be assigned to dummy tasks or workers.

Question 13. Decision theory is concerned with
(a) analysis of information that is available
(b) decision making under certainty
(c) selecting optimal decisions in sequential problem
(d) All of the options
Answer: (d) All of the options
In simple words: Decision theory helps us make the best choices by looking at all the information we have, understanding situations where we are sure or unsure, and picking the best actions even when choices happen one after another. It's a way to think smartly about decisions.

🎯 Exam Tip: Decision theory is a broad field encompassing various approaches to making choices under different conditions of certainty, risk, and uncertainty, often involving the analysis of available data to optimize outcomes.

Question 14. A type of decision – making environment is
(a) certainty
(b) uncertainty
(c) risk
(d) all of the options
Answer: (d) all of the options
In simple words: When we make decisions, we can be in different situations: sometimes we know everything for sure (certainty), sometimes we know the chances of things happening (risk), and sometimes we don't know anything at all (uncertainty). These are all common environments for making choices.

🎯 Exam Tip: Understanding the different environments (certainty, risk, uncertainty) is fundamental in decision theory, as each requires specific analytical tools and strategies for effective decision-making.