Principles of engineering economics, concept on Micro and macro analysis, problem solving and decision making concept of simple and compound interest,interest formula for: single payment, equal payment and uniform gradient series.Nominal and effective interest rates, deferred annuities, capitalized cost.Present worth, annual equivalent , capitalized and rate of return methods , Minimum Cost analysis and break even analysis

1. Module -4
Dr. Vinay Kumar B M

3. Engineering Economics

4. What is Engineering economics?
“Engineering economics is the application of economic techniques to the
evaluation of design and engineering alternatives”.
• The role of engineering economics is to assess the appropriateness of
a given project, estimate its value, and justify it from an engineering
standpoint.

5. • It is a subset of economics concerned with the use and "...application of
economic principles" in the analysis of engineering decisions.
• As a discipline, it is focused on the branch of economics known as
microeconomics.
• It studies the behavior of individuals and firms in making decisions
regarding the allocation of limited resources.

6. Principles of Engineering Economics
• Develop the Alternatives
• Focus on differences
• Use a consistent view point
• Use a common unit to measure
• Consider all relevant criteria

7. • Make uncertainty explicit
• Revisit your decision

8. Difference between microeconomics and
macroeconomics
Microeconomics
• Is the study of particular markets,
and segments of the economy.
• It looks at issues such as consumer
behaviour, individual labour markets,
and the theory of firms.
Macro economics
• Is the study of the whole economy.
• It looks at ‘aggregate’ variables,
such as aggregate demand, national
output and inflation.

9. Micro economics is concerned with:
• Supply and demand in individual markets
• Individual consumer behaviour. e.g. Consumer choice theory
• Individual labour markets – e.g. demand for labour, wage
determination
• Externalities arising from production and consumption.

10. Macro economics is concerned with:
• Monetary / fiscal policy. e.g. what effect does interest rates have on
the whole economy?
• Reasons for inflation and unemployment.
• Economic growth
• International trade and globalization
• Reasons for differences in living standards and economic growth
between countries.
• Government borrowing

11. Decision making and problem solving
• Decision-making and problem-solving are basic elements of leadership.
• More than anything else, the ability to make sound, timely decisions
separates a leader from a non-leader.
• It is the responsibility of leaders to make high quality decisions that are
accepted and executed in a timely fashion.

12. • Leaders must be able to reason under the most critical conditions and
decide quickly what action to take.
• If they delay or avoid making a decision, this indecisiveness may create
hesitancy, loss of confidence, and confusion within the unit, and may cause
the task to fail.
• Since leaders are frequently faced with unexpected circumstances, it is
important to be flexible — leaders must be able to react promptly to each
situation.

13. • Then, when circumstances dictate a change in plans, prompt reaction
builds confidence in them.

14. Process of Decision making and problem solving
• Identify (recognize/define) the problem.
• Gather information (facts/assumptions).
• Develop courses of action (solutions).
• Analyze and compare courses of action (alternatives/solutions).
• Make a decision; select the best course of action (solution).

15. • Make a plan.
• Implement the plan (assess the results).

16. Identify the Problem
• Being able to accurately identify the nature of a problem is a crucial
undertaking.
• All leadership problems, are exploratory in nature — that is, leaders do
not always identify the right cause of a problem or develop the best
plan

17. • All the leaders should have accurate information, use their best
judgment, and make educated assumptions about the causes of a
problem.
• Then, they must consider the courses of action that will be most likely
to succeed.

18. Gather Information
• In this step, leaders must gather all available information that
pertains to or can influence the situation (identified problem) from
sources such as higher, lateral, and subordinate levels of command as
well as from applicable outside agencies.

19. Develop Courses of Action
• With the problem identified and available information gathered, you
are now ready to develop possible courses of action.
• Keep an open mind throughout this step and be prepared to anticipate
change.

20. Analyze and Compare Courses of Action
• The next step is to determine which course of action will best solve
the problem.
• Therefore, leaders should develop as many advantages and
disadvantages for each course of action as possible.
• Then, they must objectively and logically analyze the advantages and
disadvantages of each one against the advantages and disadvantages
of the others.

21. Make a Decision
• After you have carefully analyzed the possible courses of action using
all available information, consider your intuitions and emotions.
• However, never make the mistake of making decisions guided totally by
emotions or intuitions and immediately doing what “feels” right.
• Try to identify a “best” course of action that is logical and likely to
succeed.

22. Make a Plan
• Make a plan that includes who would do what, when, where, how, and
why.
• Be as specific as time permits, but do not leave out vital information
that could prevent mission accomplishment.

23. Implement the Plan
• Once the decision and plan are made, it is time to act.
• In this final step, you must put the plan into action, then evaluate it to ensure
that the desired results are being achieved.
• Evaluation is often a neglected step in the decision-making process.
• The key to evaluation is to seek feedback constantly on how your plan is doing.

24. Time value of money

25. • The time value of money (TVM) is the idea that money available at the
present time is worth more than the same amount in the future due to its
potential earning capacity.
• This core principle of finance holds that, money can earn interest, any
amount of money is worth more the sooner it is received.
• Time Value of Money (TVM) is an important concept in financial
management.

26. • It can be used to compare investment alternatives and to solve problems
involving loans, leases, savings.
• If a person invests his money today in bank savings, by next year he will
definitely accumulate more money than his investment. This accumulation
of money over a specified time period is called as time value of money.
• Similarly if a person borrows some money today, by tomorrow he has to
pay more money than the original loan. This is also explained by time value
of money.

27. • The time value of money is generally expressed by interest amount.
• The original investment or the borrowed amount (i.e. loan) is known as
the principal.
• The amount of interest indicates the increase between principal
amount invested or borrowed and the final amount received or owed.

28. • In case of an investment made in the past, the total amount of interest
accumulated till now is given by;
• Similarly in case of a loan taken in past, the total amount of interest is
given by;
In both the cases there is a net increase over the amount of money that
was originally invested or borrowed.

29. • When the interest amount is expressed as the percentage of the
original amount per unit time, the resulting parameter is known as the
rate of interest and is generally designated as “ i “.
• The time period over which the interest rate is expressed is known as
the interest period.
• The interest rate is generally expressed per unit year. However in
some cases the interest rate may also be expressed per unit month.

30. Example

32. Simple interest
• The interest is said to simple, when the interest is charged only on the
principal amount for the interest period.
• No interest is charged on the interest amount accrued during the
preceding interest periods.
• In case of simple interest, the total amount of interest accumulated
for a given interest period is simply a product of the principal amount,
the rate of interest and the number of interest periods.

33. • It is given by the following expression.
Simple interest reflects the effect of time value of money only on the
principal amount

34. Compound interest
• The interest is said to be compound, when the interest for any
interest period is charged on principal amount plus the interest amount
accrued in all the previous interest periods.
• Compound interest takes into account the effect of time value of
money on both principal as well as on the accrued interest also.

35. Example

39. Cash flow diagram
• The graphical representation of the cash flows i.e. both cash outflows
and cash inflows with respect to a time scale is generally referred as
cash flow diagram.

40. • The cash outflows (i.e. costs or expense) are generally represented by
vertically downward arrows whereas the cash inflows (i.e. revenue or
income) are represented by vertically upward arrows.
• In the cash flow diagram, number of interest periods is shown on the time
scale.
• The interest period may be a quarter, a month or a year.

41. • Since the cash flows generally occur at different time intervals within an
interest period, for ease of calculation, all the cash flows are assumed to
occur at the end of an interest period.
• In Fig. the cash outflows are Rs.100000, Rs.15000 and Rs.25000
occurring at end of year (EOY) “0” i.e. at the beginning, EOY “4” and EOY
“7” respectively.
• Similarly the cash inflows Rs.35000, Rs.80000 and Rs.45000 are
occurring at EOY “3”, EOY “6” and EOY “10” respectively.

42. Compound interest factors
• The compound interest factors and the corresponding formulas are
used to find out the unknown amounts at a given interest rate
continued for certain interest periods from the known values of varying
cash flows.
• The following are the notations used for deriving the compound
interest factors.

43. Unless otherwise stated, the rate of interest is compound interest
and is for the entire number of interest periods i.e. for “n” interest
periods.

44. • The present worth (P), future worth (F) and uniform annual worth (A) are
shown in Fig.

45. • In this figure the present worth, P is at the beginning and the uniform
annual series with annual value “A” is from end of year 1 till end of year 5.
• Both “P” and “A” are cash outflows.
• It may be noted that the uniform annual series with annual value “A” may
be also continued throughout the entire interest periods i.e. from
beginning till end of year 10 or for some intermediate interest periods like
commencing from end of year 3 till end of year 8.

46. • The future worth “F” is occurring at end of year 4 (cash outflow), at end
of year 6 (cash inflow) and at the end of year 10 (cash inflow).

47. Single payment compound amount factor (SPCAF)
• The single payment compound amount factor is used to compute the
future worth (F) accumulated after “n” years from the known present
worth (P) at a given interest rate ‘i’ per interest period.
• It is assumed that the interest period is in years and the interest is
compounded once per interest period.

48. The known present worth (P), unknown future worth (F) and the total
interest period “n” years are shown in Fig.

49. • The generalized formula for the future worth at the end of “n” years
is given by:
• The factor in equation is known as the single payment compound
amount factor (SPCAF).

50. Single payment present worth factor (SPPWF)
• The single payment present worth factor is used to determine the
present worth of a known future worth (F) at the end of “n” years at a
given interest rate ‘i’ per interest period.
• The present worth (P), future worth (F) and the total interest period
“n” years are shown in Fig.

51. • From the previous equation, the expression for the present worth (P)
can be written as follows;
• The factor in equation is known as single payment present
worth factor (SPPWF).

52. Uniform series present worth factor (USPWF)
• The uniform-series present worth factor is used to determine the
present worth of a known uniform series.
• Let “A” be the uniform annual amount at the end of each year, beginning
from end of year “1” till end of year “n”.
• The known “A”, unknown “P”, and the total interest period “n” years are
shown in Fig.

53. • This cash flow diagram refers to the case; if a person wants to get the
known uniform amount of return every year, how much he has to invest
now.
• The present worth (P) of the uniform series can be calculated by
considering each “A” of the uniform series as the future worth.

54. • Then the present worth (P) is calculated from the following equation:
• The factor within the bracket in equation is known as uniform series present
worth factor (USPWF).
• Thus if the value of “A” in the uniform series is known, then the present worth
P at interest rate of “i“ (per year) can be calculated by multiplying the uniform
annual amount “A” with uniform series compound amount factor.

55. Capital recovery factor (CRF)
• The capital recovery factor is generally used to find out the uniform
annual amount “A” of a uniform series from the known present worth
at a given interest rate ‘i’ per interest period.
• The cash flow diagram is shown in Fig.

56. • This cash flow diagram indicates, if a person invests a certain amount
now, how much he will get as return by an equal amount each year.
• The expression for the uniform annual amount (A) can be written as
follows;
• The factor within bracket in equation is known as the capital recovery
factor (CRF).

57. Uniform series compound amount factor
• The uniform series compound amount factor is used to determine the
future sum (F) of a known uniform annual series with uniform amount
“A”.
• The cash flow diagram is shown in Fig.

58. • This cash flow diagram states that, if a person invests a uniform amount
at the end of each year continued for “n” years at interest rate of “i” per
year, how much he will get at the end of “n” years.
• This can be calculated from the following equation:
• The factor within bracket in equation is known as uniform series
compound amount factor (USCAF).

59. Sinking fund factor
• The sinking fund factor is used to calculate the annual amount “A‟ of a
uniform series from the known future sum “F”.
• The cash flow diagram is shown in Fig.

60. • This cash flow diagram indicates that, if a person wants to get a known
future sum at the end of “n” years at interest rate of “i” per year, how
much he has to invest every year by an equal amount.
• The expression for the uniform annual amount (A) can be written as
follows;
• The factor within bracket in equation is known as sinking fund factor
(SFF).

61. Interest factors with rate of interest ‘i’ (%) and number of interest
periods ‘n’

62. Problems
1. A person is investing 7,500/- every year in a recurring deposit of 8
years. What is the amount you can expect to receive if the interest
rate is 10%.
Ans: 85770/-

63. 2. What amount a person should invest every year in order to get lumsum of
1 lakh at the end of 5 years. If the interest rate is 12%.
Ans : 15740/-
3.If a person borrows Rs.2,50,000/- now what is the uniform amount he is
expected to pay every year for next 7 years in order to repay the capital
amount borrowed? i = 10%
Ans : 51,160/-

64. 4. A person secures a loan of Rs.2,00,000 at a interest of 10% compounded
annually and starts an industry. The bank allows an free period of 3 years.
Calculate uniform end of payment to liquidate the debt for a period of 9
years. What will be the total amount paid to the bank

66. 5. A person borrows Rs 1 lakh from a bank to start a enterprise. For first
four years he doesn’t repay the loan. But at the end of 4 years he obtains
a further loan of Rs.1 lakh from the bank. After 6 years he starts
repayment of both loans and clears them in a further period of 10 years.
Calculate the yearly installment that he has to pay uniformly at 8%
interest rate.

68. 6. A Person takes a loan of 5 lakhs to start a industry at a rate of 15%. He
starts liquidating for 3 years after borrowing and opts for uniform period of
16 years. Find out amount of each payment : (a) Yearly (b) Monthly.

70. Nominal and Effective Interest
• An interest rate takes two forms: nominal interest rate and effective
interest rate.
• The nominal interest rate does not take into account the compounding
period.
• The effective interest rate does take the compounding period into
account and thus is a more accurate measure of interest charges.

71. • A statement that the "interest rate is 10%" means that interest is 10% per
year, compounded annually.
• In this case, the nominal annual interest rate is 10%, and the effective
annual interest rate is also 10%.
• However, if compounding is more frequent than once per year, then the
effective interest rate will be greater than 10%.
• The more often compounding occurs, the higher the effective interest rate.

72. • The relationship between nominal annual and effective annual interest
rates is:

73. Deferred Annuity
• A deferred annuity is a type of annuity contract that delays payments of
income, installments or a lump sum until the investor elects to receive
them.
• This type of annuity has two main phases: the savings phase in which you
invest money into the account, and the income phase in which the plan is
converted into an annuity and payments are received.
• A deferred annuity can be variable or fixed.

74. How a Deferred Annuity Works
• When funds are deposited with a life insurer, they are credited to an
accumulation account in the name of the annuity owner.
• The life insurer then credits the account balance with a fixed rate of
interest. In most cases, the fixed interest rate is guaranteed for a certain
period of time, from one year to 10 years.
• When that period expires, the interest rate is reset by the insurer,
typically for one-year periods.

75. • Most annuity contracts include a minimum rate guarantee that ensures if
interest rates fall too low, the rate credited to the account does not fall
below the minimum.
• Withdrawals are allowed in most contracts with certain limitations.
• In a typical contract, the withdrawal provisions allow for one annual
withdrawal.

76. • If any withdrawal exceeds 10% of the value of the account, the
insurer charges a surrender fee on the excess.
• This type of annuity also includes a death benefit component that
ensures the beneficiaries receive no less than the principal investment
plus any gains in the account.

77. Capitalized Cost
• Capitalized cost can be defined as an expense that is added to the cost
basis of a fixed asset on the balance sheet of a company.
• The capitalized costs are incurred while financing or building fixed
assets.
• However, these costs are not expensed in the periods of being incurred,
but identified over a time period through the way of amortization or
depreciation

78. Example of Capitalized Cost
• The capitalized cost can be exemplified as the costs related to
construction of a new factory.
• The costs related to building the asset, counting labor and other financing
costs, can be added to the asset’s carrying value on the balance sheet.
• These capitalized costs are identified in prospective time periods.

79. How to calculate Capitalized Cost
• One of the most effective ways of determining the true cost of an asset is
calculating the capitalized cost.
• Besides, it is also helpful in evaluating the long-term overall cost of a
product, service, or investment.
• The estimation of capitalized cost is helpful to consumers and businesses for
projecting future costs and liabilities.

80. • However, the only drawback to this method is that it demands a lot of
data collection for prediction of trends as well as long-term investment
costs.

81. Steps involved in calculating the capitalized costs
• Determine the time period as well as the duration of time to be used for
calculation of capitalized cost.
• Collect all the data for the specified period, and you will get the
concluding numbers readily available.
• Sum up the concluding salvage value with the capital gains thus obtaining
the final profit.

82. • Sum up the straight costs, maintenance, and any total loan interest for the
specific period thus obtaining the final cost.
• Subtract the final profit from the final cost thus obtaining the capitalized
cost for the particular transaction for the determined period.

83. Comparison of Alternatives

84. • For most of the engineering projects, equipments etc., there are more
than one feasible alternative.
• It is the duty of the project management team (comprising of
engineers, designers, project managers etc.) of the client organization
to select the best alternative that involves less cost and results more
revenue.
• For this purpose, the economic comparison of the alternatives is made.

85. • The different cost elements and other parameters to be considered while
making the economic comparison of the alternatives are initial cost, annual
operating and maintenance cost, annual income or receipts, expected
salvage value, income tax benefit and the useful life.
• When only one, among the feasible alternatives is selected, the
alternatives are said to be mutually exclusive.

86. • In the economic comparison of alternatives, cost or expenses are considered
as negative cash flows, whereas the income or revenues are considered as
positive cash flows.
• From the view point of expenditure incurred and revenue generated, some
projects involve initial capital investment i.e. cash outflow at the beginning and
show increased income or revenue i.e. cash inflow in the subsequent years.
• The alternatives having this type of cash flow are known as investment
alternatives.

87. Example: Purchase of a dozer by a construction firm.
• The construction firm will have different feasible alternatives for the
dozer with each alternative having its own initial investment, annual
operating and maintenance cost, annual income depending upon the
production capacity, useful life, salvage values etc.
• Hence the differences in different parameters namely initial capital
investment, annual operation cost, annually generated revenue, expected
salvage value, useful life, magnitude of output and its quality, performance
and operational characteristics etc. may exist among the mutually
exclusive alternatives.

88. Methods of Comparison of alternatives
1. Present worth method
2. Future worth method
3. Annual worth method
In these methods all the cash flows i.e. cash outflows and cash inflows
are converted into equivalent present worth, future worth or annual
worth considering the time value of money at a given interest rate per
interest period.

89. Comparison of alternatives by present worth method
• In the present worth method for comparison of mutually exclusive
alternatives, the future amounts i.e. expenditures and incomes
occurring at future periods of time are converted into equivalent
present worth values at a certain rate of interest per interest period
and are added to present worth occurring at “0” time.
• The converted equivalent present worth values are always less than the
respective future amounts since the rate of interest is normally
greater than zero.

90. • Thus the cash flow of the mutually exclusive alternatives may consist of
future expenditures and incomes in different forms namely randomly placed
single amounts, uniform amount series commencing from end of year 1,
randomly placed uniform amount series i.e. commencing at time period other
than end of year 1.

91. • The methodology for the comparison of mutually exclusive alternatives
by the present worth method depends upon the magnitude of useful lives
of the alternatives.
• There are two cases;
a) The useful lives of alternatives are equal
b) The useful lives of alternatives are not equal.
The alternatives having equal useful lives are designated as equal life
span alternatives whereas the alternatives having unequal life spans are
referred as different life span alternatives.

92. Equal life span alternatives
• The comparison of mutually exclusive alternatives having equal life spans
by present worth method is comparatively simpler than those having
different life spans.
• In case of equal life span mutually exclusive alternatives, the future
amounts as already stated are converted into the equivalent present
worth values and are added to the present worth occurring at time zero

93. • Then the alternative that exhibits maximum positive equivalent present
worth or minimum negative equivalent present worth is selected from the
considered feasible alternatives.

94. Different life span alternatives
• In case of mutually exclusive alternatives, those have different life
spans, the comparison is generally made over the same number of
years i.e. a common study period.
• This is because; the comparison of the mutually exclusive alternatives
over same period of time is required for unbiased economic evaluation
of the alternatives.

95. • If the comparison of the alternatives is not made over the same life
span, then the cost alternative having shorter life span will result in
lower equivalent present worth i.e. lower cost than the cost alternative
having longer life span.

96. The two approaches used for economic comparison of different life span
alternatives are as follows:
1. Comparison of mutually exclusive alternatives over a time period that is
equal to least common multiple (LCM) of the individual life spans.
2. Comparison of mutually exclusive alternatives over a study period which
is not necessarily equal to the life span of any of the alternatives.

97. Example -1

104. Example -2

111. Example - 3

118. Comparison of alternatives by future worth method
• In the future worth method for comparison of mutually exclusive
alternatives, the equivalent future worth (i.e. value at the end of the
useful lives of alternatives) of all the expenditures and incomes
occurring at different periods of time are determined at the given
interest rate per interest period.
• The equivalent future worth of these expenditures and incomes will be
determined using different compound interest factors namely single
payment compound amount factor and uniform series compound amount
factor

119. Example -4 (Using data of Example-1)

122. Example -7

128. Example - 8

133. Comparison of alternatives by annual worth method
• In this method, the mutually exclusive alternatives are compared on the
basis of equivalent uniform annual worth.
• The equivalent uniform annual worth represents the annual equivalent
value of all the cash inflows and cash outflows of the alternatives at the
given rate of interest per interest period.
• In this method of comparison, the equivalent uniform annual worth of all
expenditures and incomes of the alternatives are determined using
different compound interest factors

134. • Since equivalent uniform annual worth of the alternatives over the useful
life are determined, same procedure is followed irrespective of the life
spans of the alternatives i.e. whether it is the comparison of equal life
span alternatives or that of different life span alternatives.
• In other words, in case of comparison of different life span alternatives
by annual worth method, the comparison is not made over the least
common multiple of the life spans as is done in case of present worth and
future worth method.

135. • The reason is that even if the comparison is made over the least common
multiple of years, the equivalent uniform annual worth of the alternative
for more than one cycle of cash flow will be exactly same as that of the
first cycle provided the cash flow i.e. the costs and incomes of the
alternative in the successive cycles is exactly same as that in the first
cycle.
• Thus the comparison is made only for one cycle of cash flow of the
alternatives

136. Example-9

141. Example -10

147. Rate of return
• The rate of return technique is one of the methods used in selecting an
alternative for a project.
• In this method, the interest rate per interest period is determined,
which equates the equivalent worth (either present worth, future worth
or annual worth) of cash outflows (i.e. costs or expenditures) to that of
cash inflows (i.e. incomes or revenues) of an alternative.

148. • The rate of return is also known by other names namely internal rate of
return (IRR), profitability index etc.
• It is basically the interest rate on the unrecovered balance of an
investment which becomes zero at the end of the useful life or the study
period.
• After determination of the rate of return for a given alternative, it is
compared with minimum attractive rate of return (MARR) to find out the
acceptability of this alternative for the project.

149. Example -11

159. Capitalized cost analysis
• Capitalized cost represents the present worth of an alternative for a
project that is going to serve for a longer period of time i.e. for an
infinite period of time.
• As the name indicates, it refers to the present worth of mainly cost or
expenditures (cash outflows) of the alternative over infinite period of
time.
• Capitalized worth refers to present worth of expenditures and revenues
of an alternative over infinite period of time.

160. • The capitalized cost method is used for comparison of mutually exclusive
alternatives which have perpetual service life (assumed to serve
forever).
• The examples of this kind of projects are bridges, dams, irrigation
projects, water supply systems for cities, pipeline projects etc.
• This method an also be used for finding out the capitalized cost of
permanent fellowship/scholarship endowment in educational institutes
and other organizations.

161. • The capitalized cost/worth of a perpetual cash flow having uniform
amount series with end of year payments “A” is obtained as follows.

163. Example 12

165. Example -12

169. Breakeven analysis
• The breakeven analysis is used to calculate the value of a factor (or
variable) at which the expenditures and revenues of a project or
alternative are equal.
• This value of the variable is known as the breakeven point.
• Corresponding to the breakeven point, profit or loss can be determined if
the expected value of the variable is higher or lower than the breakeven
value.

170. • The breakeven analysis is also used for comparing two alternatives by
determining the breakeven point i.e. the quantity of a factor (common to
both the alternatives) at which the total equivalent worth of both
alternatives are equal.
• The examples of some of the factors which are used in the breakeven
analysis are quantities produced per year, hours of operation per year,
rate of return per year and useful life etc.

171. Schematic diagram for breakeven point of a single alternative

172. Schematic diagram of breakeven point between two alternatives

173. Algebraic Relationships

175. Example-13
A company is engaged in producing fly ash bricks which are sold at
uniform price of 4/- each. The variable cost is 2.5 /unit and the fixed
cost is Rs.20,000/-.
How many units of fly ash bricks must be produced and sold so that the
company can breakeven.
Further how much sales has to be made at this level of activity if the
company desires a profit of Rs.1Lakh.

177. Example-14
A contractor is thinking of selling his present dump truck and buying new
one. The new truck costs Rs.8,50,000/- and expected to incur O and M
cost Rs.6/ton-mile. It has a life of 15 years and no salvage value.
The presently owned truck can be sold for Rs.3,50,000/- and if kept it
will cost at Rs.9/ ton-mile. It has an expected life of 5 years and no
salvage value. i = 10%.
Find BE point in terms of ton-miles per year.