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Chapter 11 Quantitative analysis

Outcome By the end of this session you should be able to: 

explain and evaluate the use of high/low analysis to separate the fixed and variable elements of total cost



explain the use of judgement and experience in forecasting



explain the learning curve effect



estimate the learning effect and apply this to a budgetary problem



calculate production times when the learning curve has reached a steady state



explain the limitations of the learning curve model

and answer questions relating to these areas.

The underpinning detail for this chapter in your Integrated Workbook can be found in Chapter 11 of your Study Text

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Chapter 11

Overview

High – low method

Cost behaviour

QUANTITATIVE ANALYSIS

Learning curves

Limitations of model

Wright’s Law

Tabular/algebraic approach

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Steady state

Applications

Quantitative analysis

High-low analysis The high-low method of analysing a semi-variable cost into its fixed and variable elements based on an analysis of historical information about costs at different activity levels. The fixed and variable costs can then be used to forecast the total cost at any level of activity.

1.1 Steps 1

Select the highest and lowest activity levels, and their costs Find the variable cost/unit

2 Variable cost/unit =

Cost at high level of activity – Cost of low level activity High level activity – Low level activity

Find the fixed cost, using either the high or low activity level 3 Fixed cost = Total cost at activity level – Total variable cost 4

Use the variable and fixed cost to forecast the total cost for a specified level of activity

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Chapter 11

Example 1

(a)

Output (units)

Total costs

200

$7,000

300

$8,000

400

$9,000

Find the variable cost per unit: Variable cost/unit =

Cost at high level of activity – Cost of low level activity High level activity – Low level activity

Variable cost per unit = ($9,000 – $7,000)/(400 – 200) Variable cost per unit = $10 per unit (b)

Find the total fixed cost Using high activity level: Total cost = $9,000 Total variable cost = 400 × $10 = $4,000 Therefore fixed costs = $5,000

(c)

Estimate the total cost if output is 350 units Variable cost = 350 × $10 = $3,500 Fixed costs = $5,000 Total cost = $8,500

(d)

Estimate the total cost if output is 600 units Variable cost = 600 × $10 = $6,000 Fixed costs = $5,000 Total cost = $11,000

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Quantitative analysis

Illustrations and further practice Now try TYU 1 from Chapter 11.

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Chapter 11

1.2 Advantages/disadvantages of the high-low method



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The high-low method has the enormous advantage of simplicity. It is easy to understand and easy to use.



It assumes that activity is the only factor affecting costs.



It assumes that historical costs reliably predict future costs.



It uses only two values, the highest and the lowest, so the results may be distorted due to random variations in these values.

Quantitative analysis

Learning curves It has been observed in some industries that there is a tendency for labour time per unit to reduce in time: as more of the units are produced, workers become more familiar with the task and get quicker per unit declines. Wright's Law Wright's Law states that as cumulative output doubles, the cumulative average time per unit falls to a fixed percentage (the 'learning rate') of the previous average time. The learning curve is 'The mathematical expression of the commonly observed effect that, as complex and labour-intensive procedures are repeated, unit labour times tend to decrease'.

Average time per unit for all units produced to date

Total number of units produced

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Chapter 11

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Quantitative analysis

Limitations of the learning curve model The learning curve model has its limitations. Learning effects are most likely to be seen if: The process is labour intensive 

Modern manufacturing environments may be very capital intensive (i.e. machine intensive) and the labour effect cannot apply if machines limit the speed of labour.

The product is new 

This may be the case in the modern environment as products have short lives and therefore new products will be introduced on a regular basis. The introduction of a new product makes it more probable that there will be a learning effect.

The product is complex 

The more complex the product the more likely that the learning curve will be significant, and the longer it will take for the learning curve to reach a steady state or 'plateau' (beyond which no more learning can take place).

Production is repetitive and there are no breaks in production 

The learning effect requires that production is repetitive with no major breaks in which the learning effect may be lost. JIT production has moved towards multiskilled and multi-tasked workers. It is possible that some of the benefits of the learning effect in a single tasking environment may be lost. The production of small batches of possible different products in response to customer demand may also lead to the loss of some of the learning effect.

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Chapter 11

Tabular approach

Example 2 Consider the following example of the time taken to make the first four units of a new product: Serial number of units

Time to make the units concerned (hours)

01

10 hours

02

8 hours

03

7.386 hours

04

7.014 hours

A pattern emerges when we look at the cumulative average time per unit: Serial number of units

Time to make the units concerned (hours)

Total cumulative time to make all units so far

Cumulative average time per unit

01

10 hours

10.000

10.00

02

8 hours

18.000

9.000

03

7.386 hours

25.386

8.462

04

7.014 hours

32.400

8.100

The process demonstrates a 90% learning rate.

344

90% 90%

Quantitative analysis

A new product will take 100 hours for the first unit. An 80% learning curve applies. Cumulative Units

Average time per unit

Incremental Total time

Units

Total time

Average time per unit

1

100

100

1

100

100

2

80

160

2

60

60

4

64

256

4

96

48

8

51.2

409.6

8

153.6

38.4

40.96

655.36

16

254.76

30.72

16

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Chapter 11

Algebraic approach For intermediate output levels

Y=a×xb

Where: X = cumulative number of units Y = cumulative average time per unit to produce X units a = time required to produce the first unit of output b = index of learning = log r/log 2, where r = the learning rate expressed as a decimal.

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Quantitative analysis

Example 3 The first unit of a new product is expected to take 100 hours. An 80% learning curve is known to apply. Calculate: (a)

The average time per unit for the first 16 units a = 100

b = –0.3219

x = 16

Y = 100 × 16 –0.3219 Y = 40.96 hours (b)

The average time per unit for the first 25 units x = 25 Y = 100 × 25 –0.3219 Y = 35.48 hours

(c)

The time it takes to make the 20th unit x = 20

Y = 100 × 20 –0.3219

Y = 38.12399855 hours

Total time for 20 units = 38.12399855 × 20 = 762.48 hours x = 19

Y = 100 × 19 –0.3219

Y = 38.75870124 hours

Total time for 19 units = 38.75870124 × 19. = 736.42 hours So, to make the 20th unit, 762.48 – 736.42 = 26.06 hours

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Chapter 11

Example 4 A company has started production of a new product and has found the first 10 units of production took 120 hours. The next 30 units produced took a further 150 hours. What was the learning rate? We know that every time cumulative output doubles, the cumulative average time per unit or batch falls to a fixed percentage of its previous level. Here, the cumulative average time for the first ten units (1 batch) was 120 hours. Since the incremental time for the next 30 units (3 batches) was 150 hours, the cumulative total for all 40 units (4 batches) is 270 hours (120 + 150). This means that the cumulative average time per batch is 67.5 hours (270/4). Here, the cumulative output doubles twice: once from 10 to 20 units and once more from 20 units to 40 units. So the learning effect has taken place twice. So, expressing this as an equation and then solving it to find ‘r’: 67.5 = 120 x r2 67.5/120 = r2 0.5625 = r2 √0.5625 = r And so r = 0.75 Therefore, the learning rate is 75%.

Illustrations and further practice Now try TYUs 4, 5, 6, 7 and 8 from Chapter 11.

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Quantitative analysis

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Chapter 11

Learning curve and steady state The learning curve effect will only apply for a certain range of production. For example, machine efficiency may restrict further improvements or there may be goslow arrangements in place. Once the steady state is reached the direct labour hours will not reduce any further and this will become the basis on which the target is produced.

Eventually, the learning effect will cease and the time to make each successive unit stabilises at a constant time per unit.

Example 3 (continued) The first unit of a new product is expected to take 100 hours. An 80% learning curve is known to apply. (d)

If the steady state is reached at 20 units, how long does it take to make the first 30 units? Total time to make the first 20 units = 762.48 hours (from part (c)) Time to make the remaining units = time to make the 20th unit × (30 – 20) Time to make the remaining units = 26.06 (from part c) × 10 = 260.60

Illustrations and further practice Now try Objective Test question ‘Bike Racers Co’ from Chapter 11.

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Quantitative analysis

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Chapter 11

You should now be able to answer TYU questions 1 to 8 from Chapter 11 of the Study Text as well as Objective Test question ‘Bike Racers Co’. You will be able to answer questions 160 to 168 from the Exam Kit; as well as question 298. For further reading, visit Chapter 11 from the Study Text.

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Quantitative analysis

Answers

Example 1 Output (units) 200 300 400 (a)

Total costs $7,000 $8,000 $9,000

Find the variable cost per unit: Variable cost/unit =

Cost at high level of activity – Cost of low level activity High level activity – Low level activity

Variable cost per unit = ($9,000 – $7,000)/(400 – 200) Variable cost per unit = $10 per unit (b)

Find the total fixed cost Using high activity level: Total cost = $9,000 Total variable cost = 400 × $10 = $4,000 Therefore fixed costs = $5,000

(c)

Estimate the total cost if output is 350 units Variable cost = 350 × $10 = $3,500 Fixed costs = $5,000 Total cost = $8,500

(d)

Estimate the total cost if output is 600 units Variable cost = 600 × $10 = $6,000 Fixed costs = $5,000 Total cost = $11,000

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Chapter 11

Example 2 Consider the following example of the time taken to make the first four units of a new product: Serial number of units

Time to make the units concerned (hours)

01

10 hours

02

8 hours

03

7.386 hours

04

7.014 hours

A pattern emerges when we look at the cumulative average time per unit: Serial number of units

Time to make the units concerned (hours)

Total cumulative time to make all units so far

Cumulative average time per unit

01

10 hours

10.000

10.00

02

8 hours

18.000

9.000

03

7.386 hours

25.386

8.462

04

7.014 hours

32.400

8.100

The process demonstrates a 90% learning rate.

354

90% 90%

Quantitative analysis

A new product will take 100 hours for the first unit. An 80% learning curve applies. Cumulative Units

Average time per unit

Incremental Total time

Units

Total time

Average time per unit

1

100

100

1

100

100

2

80

160

2

60

60

4

64

256

4

96

48

8

51.2

409.6

8

153.6

38.4

40.96

655.36

16

254.76

30.72

16

Example 3 The first unit of a new product is expected to take 100 hours. An 80% learning curve is known to apply. Calculate: (a)

The average time per unit for the first 16 units a = 100

b = –0.3219

x = 16

Y = 100 × 16 –0.3219 Y = 40.96 hours (b)

The average time per unit for the first 25 units x = 25 Y = 100 × 25 –0.3219 Y = 35.48 hours

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Chapter 11

(c)

The time it takes to make the 20th unit x = 20

Y = 100 × 20 –0.3219

Y = 38.12399855 hours

Total time for 20 units = 38.12399855 × 20 = 762.48 hours x = 19

Y = 100 × 19 –0.3219

Y = 38.75870124 hours

Total time for 19 units = 38.75870124 × 19. = 736.42 hours So, to make the 20th unit, 762.48 – 736.42 = 26.06 hours

Example 4 A company has started production of a new product and has found the first 10 units of production took 120 hours. The next 30 units produced took a further 150 hours. What was the learning rate? We know that every time cumulative output doubles, the cumulative average time per unit or batch falls to a fixed percentage of its previous level. Here, the cumulative average time for the first ten units (1 batch) was 120 hours. Since the incremental time for the next 30 units (3 batches) was 150 hours, the cumulative total for all 40 units (4 batches) is 270 hours (120 + 150). This means that the cumulative average time per batch is 67.5 hours (270/4). Here, the cumulative output doubles twice: once from 10 to 20 units and once more from 20 units to 40 units. So the learning effect has taken place twice. So, expressing this as an equation and then solving it to find ‘r’: 67.5 = 120 x r2 67.5/120 = r2 0.5625 = r2 √0.5625 = r And so r = 0.75 Therefore, the learning rate is 75%.

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Quantitative analysis

Example 3 (continued) The first unit of a new product is expected to take 100 hours. An 80% learning curve is known to apply. (d)

If the steady state is reached at 20 units, how long does it take to make the first 30 units? Total time to make the first 20 units = 762.48 hours (from part (c)) Time to make the remaining units = time to make the 20th unit × (30 – 20) Time to make the remaining units = 26.06 (from part c) × 10 = 260.60

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Chapter 11

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