Decision-Making Techniques I: Cost-Volume-Profit and Analysis

This assignment delves into the realm of decision-making techniques, specifically focusing on Cost-Volume-Profit (CVP) analysis and limiting factor analysis. It begins with an introduction to CVP, explaining its underlying concepts and assumptions, followed by a detailed explanation of breakeven charts, including step-by-step examples of how to construct them. The assignment then extends CVP analysis to a multi-product environment, emphasizing the importance of a constant sales mix and providing formulas for calculating breakeven points and contribution to sales (C/S) ratios. Furthermore, it covers margin of safety calculations. The assignment concludes with a discussion of the limitations and advantages of CVP analysis, and an introduction to limiting factor analysis. The document contains various examples and questions to test the knowledge and understanding of the concepts.

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Decision-making techniques I
Introduction
Having familiarized ourselves with the determining costs in various strategic situations, we must
then move to make operational decisions most of them related on how a firm interacts with its
customers, the market and competitors. In this section we shall deal with two tools of decision
making: cost volume profit analysis and limiting factor analysis.
1.1Cost volume profit (CVP) analysis
Review of basic CVP analysis
Cost volume profit (CVP)/breakeven analysis is the study of the interrelationships between costs,
volume and profit at various levels of activity.
The following are underlying concepts of CVP:
Assumption of CVP
(a) Can only apply to one product or constant mix
(b) Fixed costs same in total and unit variable costs same at all levels of output
(c) Sales prices constant at all levels of activity
(d) Production = sales
1.2 Break-even charts
The concepts above can be graphed to present visual relationships between Cost Volume and Pro
Various charts are presented as follows:

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Example: Graphing CVP
A new product has the following sales and cost data.
Selling price $60 per unit ; Variable cost $40 per unit ; Fixed costs $25,000 per month
Forecast sales 1,800 units per month
Required: Prepare a breakeven chart using the above data.
Solution
Step 1 Draw the axes and label them. Your graph should fill as much of the page as possible, this
will make it clearer and easier to read. The highest value on the vertical axis will be the
monthly sales revenue. 1,800 units x $60 = $108,000
Step 2 Draw the fixed cost line and label it. This will be a straight line parallel to the horizontal ax
at the $25,000 level. The $25,000 fixed costs are incurred even with zero activity.
Step 3 Draw the total cost line and label it. The best way to do this is to calculate the total costs
for the maximum sales level (1,800 units). Mark this point on the graph and join it to the
cost incurred at zero activity, that is, $25,000.
$
Variable costs for 1,800 units (1,800 x $40) 72,000
Fixed costs 25,000
Total cost for 1,800 units 97,000
Step 4 Draw the revenue line and label it. Once again, start by plotting the revenue at the
maximum activity level. 1,800 units x $60 = $108,000. This point can be joined to the
origin, since at zero activity there will be no sales revenue.
Step 5 Mark any required information on the chart and read off solutions as required. Check that
your chart is accurate by reading off the measures: the breakeven point, the margin of
safety, the profit for sales of 1,800 units.
Step 6 Check the accuracy of your readings using arithmetic. If you have time, it is good
examination technique to check your answer and make adjustments for any errors in your
chart.
The completed graph is shown below.
1.3 Breakeven analysis in a multi-product environment
To perform breakeven analysis in a multi-product organisation, a constant product sales mix must
assumed, or all products must have the same C/S ratio.
A major assumption
Organisations typically produce and sell a variety of products and services. To perform breakeven
analysis in a multi-product organisation, however, a constant product sales mix must be assumed
other words, we have to assume that whenever x units of product A are sold, y units of product B
and z units of product C are also sold.

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Such an assumption allows us to calculate a weighted average contribution per mix, the weighting
being on the basis of the quantities of each product in the constant mix. This means that the unit
contribution of the product that makes up the largest proportion of the mix has the greatest impa
on the average contribution per mix.
The only situation when the mix of products does not affect the analysis is when all of the product
have the same ratio of contribution to sales (C/S ratio).
Breakeven point for multiple products
The breakeven point (in number of mixes) for a standard mix of products is calculated as fixed
costs/contribution per mix.
Example: Breakeven point for multiple products
PL produces and sells two products. The M sells for $7 per unit and has a total variable cost of $2.
per unit, while the N sells for $15 per unit and has a total variable cost of $4.50 per unit. The
marketing department has estimated that for every five units of M sold, one unit of N will be sold.
The organisation's fixed costs total $36,000. Calculate the breakeven point for PL.
Solution We calculate the breakeven point as follows.
Step 1 Calculate contribution per unit
M N
$ per unit $ per unit
Selling price 7.00 15.00
Variable cost 2.94 4.50
Contribution 4.06 10.50
Step 2 Calculate contribution per mix = ($4.06 x 5) + ($10.50 x 1) = $30.80
Step 3 Calculate the breakeven point in terms of the number of mixes
= fixed costs/contribution per mix = $36,000/$30.80 = 1,169 mixes (rounded)
Step4 Calculate the breakeven point in terms of the number of units of the products
= (1,169 x 5) = 5,845 units of M and (1,169 x 1)=1,169 units of N (rounded)
Step 5 Calculate the breakeven point in terms of revenue
= (5,845 x$7) + (1,169 x $15) = $40,915 of M and $17,535 of N = $58,450 in total
It is important to note that the breakeven point is not $58,450 of revenue, whatever the
products. The breakeven point is $58,450 provided that the sales mix remains 5:1. Likewi
breakeven point is not at a production/sales level of (5,845 + 1,169) 7,014 units. Rather, it is whe
units of M and 1,169 units of N are sold, assuming a sales mix of 5:1.
1.4 Contribution to sales (C/S) ratio for multiple products
The breakeven point in terms of sales revenue can be calculated as fixed costs/average C/S ratio.
change in the proportions of products in the mix will change the contribution per mix and the ave
C/S ratio and hence the breakeven point.
Calculating the ratio
An alternative way of calculating the breakeven point is to use the average contribution to sales (
ratio for the standard mix. As you should already know, the C/S ratio is sometimes called
profit/volume ratio or P/V ratio. We can calculate the breakeven point of PL as follows.
Step 1 Calculate revenue per mix = (5 x $7) + (1x $15) = $50
Step 2 Calculate contribution per mix = $30.80 see previous example
Step 3 Calculate average C/S ratio = ($30.80/$50.00) x 100% = 61.6%
Step 4 Calculate breakeven point (total) = fixed costs ÷ C/S ratio = $36,000/0.616 = $58,
Step 5 Calculate revenue ratio of mix = (5 x $7) : (1x$15) = 35:15, or 7:3
Step 6 Calculate breakeven sales
M = $58,442 x 7/10 = $40,909 rounded ; N = $58,442 x 3/10 = $17,533 rounded

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Question 1: Test your knowledge
Alpha manufactures and sells three products, the beta, the gamma and the delta. Relevant
information is as follows.
Beta Gamma Delta
$ per unit $ per unit $ per unit
Selling price 135.00 165.00 220.00
Variable cost 73.50 58.90 146.20
Total fixed costs are $950,000.
An analysis of past trading patterns indicates that the products are sold in the ratio 3:4:5. Calcula
the breakeven sales revenue of product Beta, Gamma and Delta.
1.5 Margin of safety for multiple products
The margin of safety for a multi-product organisation is equal to the budgeted sales in the standa
mix less the breakeven sales in the standard mix. It may be expressed as a percentage of the bud
sales.
It should not surprise you to learn that the calculation of the margin of safety for multiple product
exactly the same as for single products, but we use the standard mix. The easiest way to see how
done is to look at an example which we do in this section.
Example: Margin of safety for multiple products
BA produces and sells two products. The W sells for $8 per unit and has a total variable cost of $3
per unit, while the R sells for $14 per unit and has a total variable cost of $4.20. For every five un
W sold, six units of R are sold. BA's fixed costs are $43,890 per period. Budgeted sales revenue fo
next period is $74,400, in the standard mix. Calculate the margin of safety in terms of sales
and also as a percentage of budgeted sales revenue.
Solution
To calculate the margin of safety we must first determine the breakeven point.
Step 1 Calculate contribution per unit
W R
$ per unit $ per unit
Selling price 8.00 14.00
Variable cost 3.80 4.20
Contribution 4.20 9.80
Step 2 Calculate contribution per mix = ($4.20 x 5) + ($9.80 x6) = $79.80
Step 3 Calculate the breakeven point in terms of the number of mixes
= fixed costs/contribution per mix = $43,890/$79.80 = 550 mixes
Step 4 Calculate the breakeven point in terms of the number of units of the products
= (550 x5) 2,750 units of W and (550 x 6) 3,300 units of R
Step 5 Calculate the breakeven point in terms of revenue
= (2,750 x $8) + (3,300 x $14) = $22,000 of W and $46,200 of R = $68,200 in total
Step 6 Calculate the margin of safety
= budgeted sales – breakeven sales = $74,400 – $68,200 = $6,200 sales in total, in the
standard mix
Or, as a percentage = ($74,400 – $68,200)/$74,400 x 100% = 8.3% of budgeted sales
Further reading:
Now based on the foregoing knowledge you should be able to determine how to compute target
profits for multiple products and to plot the CVP charts in a multi-product scenario.

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1.6 Limitations and advantages of CVP
Limitations:
(a) It is assumed that fixed costs are the same in total and variable costs are the same per unit at
levels of output. This assumption is a great simplification.
(i) Fixed costs will change if output falls or increases substantially (most fixed costs are step costs
(ii) The variable cost per unit will decrease where economies of scale are made at higher output
volumes, but the variable cost per unit will also eventually rise when diseconomies of scale
begin to appear at even higher volumes of output (for example the extra cost of labour in
overtime working).
The assumption is only correct within a normal range or relevant range of output. It is generally
assumed that both the budgeted output and the breakeven point lie within this relevant range.
(b) It is assumed that sales prices will be constant at all levels of activity. This may not be true,
especially at higher volumes of output, where the price may have to be reduced to win the extra
sales.
(c) Production and sales are assumed to be the same, so that the consequences of any increase in
inventory levels or of 'de-stocking' are ignored.
(d) Uncertainty in the estimates of fixed costs and unit variable costs is often ignored.
Advantages
(a) Graphical representation of cost and revenue data (breakeven charts) can be more easily
understood by non-financial managers.
(b) A breakeven model enables profit or loss at any level of activity within the range for which the
model is valid to be determined, and the C/S ratio can indicate the relative profitability of differen
products.
(c) Highlighting the breakeven point and the margin of safety gives managers some indication of
level of risk involved.
2.1 Limiting factor analysis
All companies are limited in their capacity, either for producing goods or providing services. There
always one resource that is most restrictive (the limiting factor).
Merging the known to the unknown
 An organisation might be faced with just one limiting factor (other than maximum sales demand)
but there might also be several scarce resources, with two or more of them putting an effective limit on
the level of activity that can be achieved.
 Examples of limiting factors include sales demand and production constraints.
– Labour. The limit may be either in terms of total quantity or of particular skills.
– Materials. There may be insufficient available materials to produce enough units to satisfy
sales demand.
– Manufacturing capacity. There may not be sufficient machine capacity for the production
required to meet sales demand.
 It is assumed in limiting factor analysis that management would make a product mix decision or
service mix decision based on the option that would maximise profit and that profit is maximised
when contribution is maximised (given no change in fixed cost expenditure incurred). In other
words, marginal costing ideas are applied.

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 Contribution will be maximised by earning the biggest possible contribution per unit of limiting
factor. For example if grade A labour is the limiting factor, contribution will be maximised by earning
the biggest contribution per hour of grade A labour worked.
– The limiting factor decision therefore involves the determination of the contribution earned
per unit of limiting factor by each different product.
– If the sales demand is limited, the profit-maximising decision will be to produce the top ranked
product(s) up to the sales demand limit.
 In limiting factor decisions, we generally assume that fixed costs are the same whatever product or
service mix is selected, so that the only relevant costs are variable costs.
 When there is just one limiting factor, the technique for establishing the contribution-maximising
product mix or service mix is to rank the products or services in order of contribution-earning ability
per unit of limiting factor.
2.2 Two potentially limiting factors
It is possible to deal with situations where two limiting factors are potentially limiting (and there are
also product/service demand limitations). The approach in these situations is to find out which factor (if
any) prevents the business from fulfilling maximum demand.
Where there is a maximum potential sales demand for an organisation's products or services, they should
still be ranked in order of contribution-earning ability per unit of the limiting factor. The contribution-
maximising decision, however, will be to produce the top-ranked products (or to provide the top-ranked
services) up to the sales demand limit.
Example: Two potentially limiting factors
Lucky manufactures and sells three products, X, Y and Z, for which budgeted sales demand, unit selling
prices and unit variable costs are as follows.
X Y Z
Budgeted sales demand (units) 550 500 400
$ $ $ $ $ $
Unit sales price 16 18 14
Variable costs: materials 8 6 2
labour 4 6 9
12 12 11
Unit contribution 4 6 3
The organisation has existing inventory of 250 units of X and 200 units of Z, which it is quite willing to
use up to meet sales demand. All three products use the same direct materials and the same type of
direct labour. In the next year, the available supply of materials will be restricted to $4,800 (at cost) and
the available supply of labour to $6,600 (at cost).
Determine what product mix and sales mix would maximise the organisation's profits in the next year.
Solution
There appear to be two scarce resources, direct materials and direct labour. This is not certain, however,
and because there is a limited sales demand as well, either of the following might apply.
 There is no limiting factor at all, except sales demand.
 􀁸There is only one scarce resource that prevents the full potential sales demand being achieved.
Step 1 Establish which of the resources, if any, is scarce.
X Y Z
Units Units Units
Budgeted sales 550 500 400
Inventory in hand 250 0 200
Minimum production to meet demand 300 500 200
Minimum production to Required materials Required labour

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meet sales demand at cost at cost
Units $ $
X 300 2,400 1,200
Y 500 3,000 3,000
Z 200 400 1,800
Total required 5,800 6,000
Total available 4,800 6,600
(Shortfall)/Surplus (1,000) 600
Materials are a limiting factor, but labour is not.
Step 2 Rank X, Y and Z in order of contribution earned per $1 of direct materials consumed.
X Y Z
$ $ $
Unit contribution 4 6 3
Cost of materials 8 6 2
Contribution per $1 materials $0.50 $1.00 $1.50
Ranking 3rd 2nd 1st
Step 3 Determine a production plan.
Z should be manufactured up to the limit where units produced plus units held in inventory will meet
sales demand, then Y second and X third, until all the available materials are used up.
Sales demand Production Materials
Ranking Product less units held quantity cost
Units Units $
1st Z 200 200 (x $2) 400
2nd Y 500 500 (x$6) 3,000
3rd X 300 175 (x$8) *1,400
Total available 4,800
* Balancing amount using up total available.
Step 4 Draw up a budget.
The profit-maximising budget is as follows.
Product Units Material
cost /
unit $
Total
material
cost$
Contribution per
$1 of material $
Total
contribution
$
Z op.
inventory
200
Z
production
200
400 2 800 1.50 1200
Y op.
inventory
-
Y
production
500
500 6 3000 1 3000
X op.
inventory
250
X
production
175
425 8 3400 0.5 1700
Total
contribution
5900
2.3 Limiting factor analysis - make or buy decisions and scarce resources

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In a situation where a company must sub-contract work to make up a shortfall in its own in-house
capabilities, its total costs will be minimised if those units bought have the lowest extra variable cost of
buying per unit of scarce resource saved by buying.
Combining internal and external production
An organisation might want to do more things than it has the resources for, and so its alternatives would
be as follows.
(a) Make the best use of the available resources and ignore the opportunities to buy help from outside
(b) Combine internal resources with buying externally so as to do more and increase profitability
Buying help from outside is justifiable if it adds to profits. A further decision is then required on how to
split the work between internal and external effort. What parts of the work should be given to suppliers
or sub-contractors so as to maximise profitability?
In a situation where a company must sub-contract work to make up a shortfall in its own in-house
capabilities, its total costs will be minimised if those units bought have the lowest extra variable cost of
buying per unit of scarce resource saved by buying.
Example: Make or buy
TW manufactures two products, the D and the E, using the same material for each. Annual demand for
the D is 9,000 units, while demand for the E is 12,000 units. The variable production cost per unit of the D
is $10, that of the E $15. The D requires 3.5 kgs of raw material per unit, the E requires 8 kgs of raw
material per unit. Supply of raw material will be limited to 87,500 kgs during the year.
A sub contractor has quoted prices of $17 per unit for the D and $25 per unit for the E to supply the
product. How many of each product should TW manufacture in order to maximise profits?
Solution
D E
$ per unit $ per unit
Variable cost of making 10 15
Variable cost of buying 17 25
Extra variable cost of buying 7 10
Raw material saved by buying 3.5 kgs 8 kgs
Extra variable cost of buying per kg saved $2 $1.25
Priority for internal manufacture 1 2
Production plan Material used kgs
Make D (9,000 x3.5 kgs) 31,500
E (7,000 x8 kgs) 56,000
87,500
The remaining 5,000 units of E should be purchased from the contractor.
2.4 The principles of linear programming
Linear programming is a technique for solving problems of profit maximisation or cost minimisation and
resource allocation. 'Programming' has nothing to do with computers: the word is simply used to denote
a series of events. If a scenario contains two or more limiting factors, linear programming must be
applied.
A typical business problem is to decide how a company should divide up its production among the
various types of product it manufactures in order to obtain the maximum possible profit. A business
cannot simply aim to produce as much as possible because there will be limitations or constraints within
which the production must operate. Such constraints could be one or more of the following.
o Limited quantities of raw materials available
o A fixed number of labour hours per week for each type of worker
o Limited machine hours

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Moreover, since the profits generated by different products vary, it may be better not to produce any of
a less profitable line, but to concentrate all resources on producing the more profitable ones. On the
other hand limitations in market demand could mean that some of the products produced may not be
sold.
Am sure that from your previous learning you have interacted with leaner programming at length
including some complex methodologies like simplex. I therefore urge you to practice how LP techniqes
can be used in this circumstances. The following question is a good starting point. You are required to
provide an LP solution for the same.
Q2 :LP and limiting factors
WX Co manufactures two products, A and B. Both products pass through two production departments,
mixing and shaping. The organisation's objective is to maximise contribution to fixed costs.
Product A is sold for $1.50 whereas product B is priced at $2.00. There is unlimited demand for product A
but demand for B is limited to 13,000 units per annum. The machine hours available in each department
are restricted to 2,400 per annum. Other relevant data are as follows.
Machine hours required Mixing Shaping
Hrs Hrs
Product A 0.06 0.04
Product B 0.08 0.12
Variable cost per unit $
Product A 1.30
Product B 1.70
2.5 Slack and surplus
Slack occurs when maximum availability of a resource is not used. Surplus occurs when more than a
minimum requirement is used. If, at the optimal solution, the resource used equals the resource
available there is no spare capacity of a resource and so there is no slack. If a resource which has a
maximum availability is not binding at the optimal solution, there will be slack.
Example: Slack and surplus
A machine shop makes boxes (B) and tins (T). Contribution per box is $5 and per tin is $7. A box requires
3 hours of machine processing time, 16kg of raw materials and 6 labour hours. A tin requires 10 hours of
machine processing time, 4kg of raw materials and 6 labour hours. In a given month, 330 hours of machine
processing time are available, 400kg of raw material and 240 labour hours. The manufacturing technology
used means that at least 12 tins must be made every month. The constraints are:
3B + 10T ≤ 330 ; 16B + 4T ≤ 400 6B + 6T ≤ 240 & T ≥ 12
The optimal solution is found to be to manufacture 10 boxes and 30 tins.
If we substitute these values into the inequalities representing the constraints, we can determine whether
the constraints are binding or whether there is slack.
Machine time: (3 x 10) + (10 x 30) = 330 = availability
Constraint is binding.
Raw materials: (16 x 10) + (4 x 30) = 280 ≠ 400
There is slack of 120kg of raw materials.
Labour: (6 x10) + (6 x 30) = 240 = availability
Constraint is binding.
If a minimum quantity of a resource must be used and, at the optimal solution, more than that quantity
is used, there is a surplus on the minimum requirement. This is shown here in the production of tins
where the optimal production is 30 tins but T ≥ 12. There is therefore a surplus of 18 tins over the
minimum production requirement.

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You can see from this that slack is associated with ≤ constraints and surplus with ≥ constraints. Machine
time and labour are binding constraints so they have been used to their full capacity. It can be argued
that if more machine time and labour could be obtained, more boxes and tins could be produced and
contribution increased.
Limiting factors and shadow prices
The shadow price or dual price of a limiting factor is the increase in value which would be created by
having one additional unit of the limiting factor at the original cost.
Whenever there are limiting factors, there will be opportunity costs. As you know, these are the benefits
forgone by using a limiting factor in one way instead of in the next most profitable way.
For example, suppose that an organisation provides two services X and Y, which earn a contribution of
$24 and $18 per unit respectively. Service X requires 4 labour hours, and service Y 2 hours. Only 5,000
labour hours are available, and potential demand is for 1,000 of each of X and Y. Labour hours would be
a limiting factor, and with X earning $6 per hour and Y earning $9 per hour, the profit-maximising
decision would be as follows.
Contribution
Services Hours $
Y 1,000 2,000 18,000
X (balance) 750 3,000 18,000
5,000 36,000
Priority is given to Y because the opportunity cost of providing Y instead of more of X is $6 per hour (X's
contribution per labour hour), and since Y earns $9 per hour, the incremental benefit of providing Y
instead of X would be $3 per hour.
If extra labour hours could be made available, more X (up to 1,000) would be provided, and an extra
contribution of $6 per hour could be earned. Similarly, if fewer labour hours were available, the decision
would be to provide fewer X and to keep provision of Y at 1,000, and so the loss of labour hours would
cost the organisation $6 per hour in lost contribution. This $6 per hour, the marginal contribution earning
potential of the limiting factor at the profit-maximising output level, is referred to as the shadow price (or
dual price) of the limiting factor.