Piyush is the Production Manager of a shoe company 'Comforty'. He has 1000 workers under him in his department

Question 16 of 26

Q. Piyush is the Production Manager of a shoe company 'Comforty'. He has 1000 workers under him in his department. Piyush booked a special order for 300 pairs of gents shoes at an estimated cost price of ₹ 500 per pair plus a 20% profit margin. This order is to be completed within 10 days. After completion of the order, Piyush found out that the production cost of one pair of shoes was ₹ 700. He tried to find out the reason but he couldn't. He reported it to the Chief Executive Officer. For this the Chief Executive Officer appointed Abhay, who was not only a specialized Cost Accountant, but also a qualified Human Resource Manager. After two days, Abhay reported that the set target could have been completed by only 800 workers. This revealed that there was overmanning in the organization. The organization was not utilizing the human resources optimally which had led to high labour cost. The above case highlights an important function of management.

(i) Identify the function of management discussed above.

(ii) State any three points of importance of the function identified in

(i) above.

Uttarakhand UbseCBSE Class XII Board 2024Subjective· 4mImportance★★★★★

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Concept understanding — Controlling Definition

Controlling Definition — First Encounter

Imagine you're trying to prove that a function is continuous at a point. You've seen the epsilon-delta definition: for every $\epsilon > 0$ , there exists a $\delta > 0$ such that if $|x - c| < \delta$ , then $|f(x) - f(c)| < \epsilon$ . That's the standard definition.

Now think about what you're actually doing when you use this definition to prove continuity. You're given an arbitrary $\epsilon$ — a tolerance on the output — and you must produce a $\delta$ — a window on the input — that controls the output error. The $\delta$ you choose depends on $\epsilon$ , on the function $f$ , and on the point $c$ . The core idea: you control the input to control the output.

That's the intuition behind the term "controlling definition." It's a definition that tells you: to achieve a desired precision in the output, you must restrict the input within some range. The definition itself gives you the control — the $\delta$ — as a function of the desired precision $\epsilon$ .

The Precise Statement

A controlling definition is a definition of a limit (or continuity, or derivative) that explicitly quantifies the relationship between an output tolerance and an input restriction. For the limit $\lim_{x \to c} f(x) = L$ , the controlling definition is:

\lim_{x \to c} f(x) = L \iff \forall \epsilon > 0, \ \exists \delta > 0 \ \text{such that} \ 0 < |x - c| < \delta \implies |f(x) - L| < \epsilon

The word "controlling" highlights that $\delta$ is chosen after $\epsilon$ is given — you control the input window $\delta$ to force the output error below $\epsilon$ . The $\delta$ is not arbitrary; it must be found (or shown to exist) for every possible $\epsilon$ .

Why "Controlling" Matters

This phrasing is most common in analysis textbooks when contrasting with other equivalent definitions. For example, the "sequential definition" of a limit says: if every sequence $x_n \to c$ (with $x_n \neq c$ ) has $f(x_n) \to L$ , then the limit exists. That definition doesn't give you a direct control mechanism — it's about checking sequences.

The controlling definition is the one you actually use in proofs. It's constructive: you pick $\delta$ as a function of $\epsilon$ , often by solving inequalities. For instance, to prove $\lim_{x \to 2} (3x+1) = 7$ , you set $\delta = \epsilon/3$ . That's the control: if $|x-2| < \epsilon/3$ , then $|(3x+1)-7| = 3|x-2| < \epsilon$ . …

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The case highlights Controlling as the management function — the process of comparing actual performance (₹700 cost, 1000 workers) against the standard (₹500 cost, 800 workers needed) to detect deviations like overmanning and take corrective action.

Understanding the Concept: Why This is Controlling

Management functions are typically divided into Planning, Organising, Staffing, Directing, and Controlling. The case describes a clear sequence: a target was set (plan), work was done, actual results differed from the plan, the deviation was investigated, and a root cause (overmanning) was identified. This is the essence of Controlling — it is not about blame, but about comparing actuals with standards to improve future performance.

Notice that Abhay, the Cost Accountant, didn't just report the cost overrun. He dug deeper to find why the cost was high — the organisation had 1000 workers when only 800 were needed. This diagnostic step is what separates controlling from mere reporting. The function closes the loop between planning and execution.

Watch out

A common mistake is to confuse Controlling with Directing or Staffing. Directing involves guiding and motivating people; Staffing deals with recruitment and placement. Controlling specifically focuses on measuring performance against standards and correcting deviations — which is exactly what happened here.

Step-by-Step Solution

1. Identify the management function

The case gives us several clues:

A target was set (₹500 per pair, 10 days, 1000 workers)
Actual performance was measured (₹700 per pair)
A deviation was found (₹200 extra per pair)
The cause was investigated (overmanning — 200 excess workers)
A report was made to the CEO for corrective action

This sequence — setting standards, measuring actuals, comparing, analysing deviations, and taking corrective action — is the textbook definition of Controlling.

Controlling = Setting Standards \to Measuring Performance \to Comparing \to Analysing Deviations \to Corrective Action

2. State three points of importance of Controlling

Controlling is not a bureaucratic exercise; it serves critical purposes in any organisation. Here are three key points of importance, drawn directly from the case:

Helps in achieving organisational goals: By detecting that 200 workers were idle, the company can now redeploy them or adjust hiring. Without controlling, the company would continue paying for unproductive labour, drifting away from its profit targets. Controlling keeps everyone aligned with the planned objectives. …

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Understanding the Concept: Why This is Controlling

Step-by-Step Solution

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