Vinber Ltd. set up a manufacturing unit at Bhiwadi in Himachal Pradesh to manufacture electric geysers and sup

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Q. Vinber Ltd. set up a manufacturing unit at Bhiwadi in Himachal Pradesh to manufacture electric geysers and supply them to dealers all over the country. Their production target was 500 geysers per week. It was decided by the management that variation in production upto 10 units would be acceptable. At the end of the first week, the production was 450 geysers. The next week, production increased to 470 geysers. A week later, production was 460 geysers. On investigation, it was found that fluctuation in production was due to irregular supply of electricity.

(a) The above para discusses some of the steps in the process of one of the functions of management. Explain these steps.

(b) Also, state the step(s) that have not been discussed in the above para.

Sikkim CbseCBSE Class XII Board 2019Subjective· 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 provided scenario illustrates the 'Controlling' function of management, specifically detailing the steps of setting performance standards, measuring actual performance, comparing it with standards, and analyzing deviations. The crucial final step of taking corrective action is not discussed.

The scenario involving Vinber Ltd. perfectly illustrates the 'Controlling' function of management. Controlling is a fundamental management process that ensures activities are performed as planned and that organisational goals are achieved. It involves setting standards, measuring actual performance, comparing it with the standards, identifying deviations, and taking corrective actions to ensure that performance aligns with objectives. Without effective control, even the best plans can fail.

Let's break down the steps discussed and those omitted in the given paragraph.

(a) Steps in the Controlling Process Discussed

The paragraph discusses several key steps of the controlling function:

Setting Performance Standards:

This is the initial step in the controlling process, where benchmarks or targets are established against which actual performance will be measured. Standards can be quantitative (like production units, sales targets, cost per unit) or qualitative (like employee morale, customer satisfaction).
- In Vinber Ltd.'s case: The management set a clear production target of 500 geysers per week. Additionally, they defined an acceptable range of deviation, stating that "variation in production upto 10 units would be acceptable." This means any production between 490 and 510 geysers would be considered satisfactory.
Measurement of Actual Performance:

Once standards are set, the next step is to monitor and measure the actual performance of employees, departments, or the organisation as a whole. This measurement should be done regularly and objectively, using appropriate metrics.
- In Vinber Ltd.'s case: The actual production was measured at the end of each week: 450 geysers in the first week, 470 geysers in the second week, and 460 geysers in the third week.
Comparison of Actual Performance with Standards:

After measuring actual performance, it is compared against the pre-established standards. This step helps in identifying whether performance is on track, exceeding expectations, or falling short.
- In Vinber Ltd.'s case:
  - Week 1: Actual production of 450 geysers is less than the standard of 500 geysers. The deviation is 50 units (500 - 450 = 50). This deviation is outside the acceptable range of 10 units.
  - Week 2: Actual production of 470 geysers is less than the standard of 500 geysers. The deviation is 30 units (500 - 470 = 30). This deviation is also outside the acceptable range.
  - Week 3: Actual production of 460 geysers is less than the standard of 500 geysers. The deviation is 40 units (500 - 460 = 40). This deviation is again outside the acceptable range.
- In all three weeks, the actual production fell short of the target and exceeded the acceptable variation limit, indicating a need for attention.
Analyzing Deviations:

When deviations occur, especially significant ones, it is crucial to analyze their causes. This step involves investigating why actual performance differed from the standards. Not all deviations require corrective action; only significant ones or those outside acceptable limits need detailed analysis. …

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(a) Steps in the Controlling Process Discussed

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