**Problem Of The
Month**

**December
2002---Reliability And Life Cycle Cost**

You must know when to **accept**
the financial risk and when to **reject** the financial risk based on reliability engineering
calculations and life cycle costs. Download
this problem of the month as a PDF file
(300KB) revised

Reliability and money are a
wonderful combination—one hand washes the other. The problem of life cycle cost is to know **when**
things fail so you can **price-out** the failure costs with an Excel spreadsheet for life
cycle cost considering the time value of money in NPV calculations. You find when things will failure by
exercising the reliability calculations.
These chicken or egg problems are the reasons why engineers need training
in Reliability Engineering
Principles and Life Cycle
Costs.

**The Problem:**

We have a situation where we can
expect one failure per year. The
consequence of failure is very high at US$20,000,000 (many situations of high
consequence of failure exist with pipelines, refineries, chemical plants,
shipping, aviation, transportation, and so forth. We can mitigate the failure costs by
installation of equipment to make the system more reliable—of course the
capital expenditures cost money. We must
balance the costs of capital with the reduction of risks—**the issue is to
find the lowest long term cost of ownership**. The lowest long term cost of ownership
involves net present value (NPV) decisions.
NPV calculations take into account the time value of money. We want to mitigate the high cost of unreliability which means
we don’t want to spend too much and we don’t want to spend too little over the
life of the equipment.

Our default consequence example has
an annual cost of US$20,000,000. This is
the default case for doing nothing to mitigate the failure. On a risk based calculation, the probability
of failure = 1.0. Probability is a
statement of unreliability. Reliability
+ Unreliability = 1 from the
complementary equation.

The amount of money at risk is **$risk
= (probability of failure)*($consequence of failure)**. For the default case of one failure per year
for taking no evasive action is 1*$20,000,000 = $20,000,000. Taking this high risk requires no expenditure
of capital.

Installing redundant instruments
would decrease the probability of failure, i.e., reliability will increase as
the unreliability decreases [reliability + unreliability = 1]. The reliability of each instrument is 95%
based on a one year mission. The
instruments can be installed in parallel for increased reliability. Installed instrument cost is US$10,000 per
instrument. **Based on a financial
justification, how many devices should be installed considering reliability and
financial consequences?**

Take these conditions for the
spreadsheet calculations: discount rate = 12%, tax provision = 38%, project
life = 20 years.

**Reliability calculations:**

The first instrument installed will function as
if it is in series. The second, third,
fourth, and so forth instruments will function as if installed in parallel
operation. The parallel reliability
calculation for the system is:

R_{system}
= 1- (1-reliablity)^{N}

where N is the number of similar items in parallel.

The probability of failure (a
statement of unreliability) is:

_{system}
= (1-reliablity)^{N}.

Table 1 shows the reliability calculations along with the probability for
failure. The financial exposure is
calculated as (column 3)*($20,000,000).

The problem can be viewed two ways:

1.
The financial calculation for each case can be converted into net
present values driven only by the $Risk for each year and the capital costs
which will be expended. The option with
the lowest calculated NPV is the winner.
In short, use the life
cycle cost spreadsheet:

A) put the capital cost from Table 1, column 5
into cell **D5**

B) put the annual cost from Table 1, column 4
into cells **E17:X17**

C) read the NPV in cell **C3** ßSee Table 2, column 6

The case with the least negative value is the winner.

2. The default case is the most
expensive and a saving can be calculated between the no instrument case and the
alternatives along with the capital costs required for each alternative. In short, use the life cycle cost spreadsheet:

A) put the capital cost from Table 1, column 5
into cell **D5**

B) put the annual cost ** saving**
($20,000,000 - $Risk/yr) from Table 1, column 4 into cells

C) read the NPV in cell

The option with the largest positive NPV is the winner in this cost difference condition which examines the savings (of course it requires a datum case).

Table 2 shows the calculations with both views on NPV. The winner case is for installing four
instruments which will operate in parallel.

You need the reliability values
which drive the probability of failure.
The probability of failure multiplied by the exposure consequence is the
amount of risk you must guard against each year. Of course for higher reliability, the greater
will be the capital equipment. Using the
NPV values you can find the conditions which represent the lowest long term
cost of ownership at four devices.

The $Risk/year will go into the
simple NPV calculations along with the capital expenditure to drive the
negative NPV values. The ($Risk/year
from the datum case of no protective instruments less the $Risk/year for the
different number of instruments) will be the annual savings for each capital
expenditure will produce the positive NPV values—of course you can see the D values by taking the differences between the
negative NPVs.

A little technology, a little money,
a simple spreadsheet, considering the time value of money along with tax
provisions, and so forth gives a short, sweet solution. This is an illustration of how reliability
and life cycle cost are a wonderful combination to achieve a decisive action
plan for reducing financial exposure. As
John Ruston said over 100 years ago: “Its foolish to spend too much money but
it’s unwise to spend too little.” It’s
foolish to take to little risk and it’s unwise to take too much risk—you get it
right for the lowest long term cost of ownership by using NPV decisions and
reliability principles.

This problem is described in “**Reliability
Issues From A Management Perspective**” prepared for the 52^{nd} API
Pipeline conference,

**Reliability systems (series and parallel) –**

When devices are placed in series, the
reliability of the systems follows the computations shown in Figure 1. Reliability for the system can plummet when
many items are placed in series as system reliability is paced by the least
reliable device in series.

With a series system, when any link
in the chain fails, the entire chain fails (remember to old fashion Christmas
tree lights where failure of one light bulb resulted in the failure of the
entire string). The simple test for a
series system is easy—take out items one-by-one and if the system fails, then
the individual component was in series.

Many very long series systems exist
and the system functions with high reliability by pushing individual
reliabilities to high levels by extensive verification testing and operating
the systems at low loads so that “freeboard” between loads and strengths is
very large. If loads and strengths do
not overlap, then high reliability is obtained for each component to provide
high system integrity.

An example of a long series system
is seen in long natural gas pipelines which lack enroute storage
capabilities. A long gas pipeline can
have ~125 welded connection per mile and the pipelines can easily be 1500 miles
long. This represents 187,500 welded
connections in series! If the system
needs an overall reliability of 90% the individual reliabilities must be R_{s}
= 0.9 = R^{187500}, or the reliability of the individual welded
connections (assuming all reliability values to be the same), R = 0.9^{(1/187500)
}= 0.9^{0.00000533} = 0.999999438 which is a spectacular value for
individual components.

Parallel systems are shown in Figure 2. The system survives if one or more items
survive. The system computation is 1
minus the product of the unreliability.

It only requires a few items in
parallel to achieve high overall system reliabilities.

Contrast the system outcome between
parallel systems in Figure 2 with the series systems in Figure 1. The numbers speak for themselves.

Figures 1 and 2 come from Reliability Engineering
Principles training course, section 4 concerning reliability models.

**Comments:**

Refer to the caveats on the **Problem
Of The Month Page** about the limitations of the following solution.
Maybe you have a better idea on how to solve the problem. Maybe you find where
I've screwed-up the solution and you can point out my errors as you check my
calculations. E-mail your comments, criticism, and corrections to: Paul
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© Barringer & Associates, Inc. 2001