Question 3 of 7: Life-Cycle Stages and Life-Cycle Cost

Question 3: Life-Cycle Stages and Life-Cycle Cost (15 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Part A — The three main stages in the life of a product

Beginning of life — creation. This stage covers design, material extraction and processing, manufacture, assembly and distribution to the customer. Almost all of the product's eventual cost and environmental burden is committed here even though very little of it is yet incurred: by the end of concept design, decisions about architecture, material and process route have fixed most of what the product will cost to make, to run and to retire. The design implication is that life-cycle thinking has to happen at the front of this stage, when it is still cheap, rather than at the end of it, when the tooling exists.

Middle of life — use and support. This stage covers installation and commissioning, operation, consumables and energy, maintenance, repair, spares, upgrade and the downtime that failures cause. For anything that consumes energy or requires servicing it is the longest stage and usually the most expensive one, and it is the stage the purchaser sees but the designer often does not. The design implications are reliability, maintainability, diagnosability, efficiency in the actual duty cycle rather than at a nameplate point, and spare-part availability over the intended service life.

End of life — retirement and recovery. This stage covers removal, collection, disassembly, reuse of components, remanufacture, recycling of materials, and final disposal of what is left, together with the regulatory obligations attached to each — refrigerant recovery, extended producer responsibility, and provincial stewardship programmes for electronics and appliances. Whether this stage is a cost or a partial recovery is decided in stage one, by choices about fastening, material variety, marking and the ease with which hazardous or valuable components can be separated.

Part B — Five factors in calculating life-cycle cost

Life-cycle cost is the discounted sum of every cash flow the product causes over its life, and the five factors below are the ones that make up essentially all of it. They are illustrated on a commercial reach-in refrigerator over a ten-year life at a six per cent discount rate, because a worked example makes the relative weight of the five visible in a way a list cannot.

Given. Purchase price CAD 4,200; energy consumption 2,150 kWh per year at CAD 0.135 per kWh; routine maintenance CAD 180 per year; expected failure rate 0.12 per year with an average consequential loss of CAD 600 per event; compressor replacement of CAD 950 in year 7; disposal with refrigerant recovery CAD 140 at year 10 against a scrap recovery of CAD 90. Discount rate 6 per cent, study period 10 years.

Find. The discounted life-cycle cost, the share carried by each factor, and whether a proposed efficiency upgrade is justified.

1. Factor 1, acquisition cost, and Factors 2 to 4 as an annual series. Acquisition is the only factor already in present-value terms: CAD 4,200. The recurring factors are energy ($2{,}150\times 0.135=\text{CAD }290.25$/yr), maintenance and spares (CAD 180/yr) and downtime and consequential loss ($0.12\times 600=\text{CAD }72.00$/yr), a combined annuity of CAD 542.25 per year. The present-value annuity factor is $$\left(\frac{P}{A},i,N\right)=\frac{1-(1+i)^{-N}}{i}=\frac{1-1.06^{-10}}{0.06}=7.3601$$ so their present value is $542.25\times 7.3601=\text{CAD }3{,}991.01$.
2. Factor 5, end of life, plus the single non-recurring overhaul. The compressor replacement discounts as $950/1.06^{7}=\text{CAD }631.80$, and the net end-of-life cost as $(140-90)/1.06^{10}=\text{CAD }27.92$. Summing all five factors, $$LCC=4200+3991.01+631.80+27.92=\boxed{\text{CAD }8{,}850.73}$$ Acquisition is 47.5 per cent of the total and energy alone is 24.1 per cent, so more than half the money the owner spends on this product is spent after the purchase order is signed.
3. Use the result to test a design decision. A specification upgrade — thicker insulation, electronically commutated fan motors and a variable-speed compressor — raises the purchase price by CAD 500 and cuts consumption to 1,450 kWh per year, saving $700\times 0.135=\text{CAD }94.50$ per year. Its present value is $94.50\times 7.3601=\text{CAD }695.53$ against CAD 500 spent, a net gain of CAD 195.53 and a simple payback of $500/94.50=5.29$ years. Solving $94.50\,(P/A,i,10)=500$ for the rate at which the upgrade breaks even gives an internal rate of return of $$\boxed{i^{*}=13.6\ \%}$$ so the upgrade is justified for any owner whose cost of capital is below 13.6 per cent, and is refused above it. That single number is the whole reason to compute life-cycle cost rather than to assert that efficiency is good.

The five factors, then, are: (1) acquisition — purchase price, freight, installation and commissioning; (2) energy and consumables over the real duty cycle, not the nameplate; (3) maintenance, spares and scheduled overhaul, including the labour and access cost of reaching the part; (4) downtime and consequential loss, which for commercial equipment is frequently larger than the repair itself; and (5) end-of-life cost net of residual value, including any statutory recovery obligation. Two conventions make the calculation defensible: discount everything to a common date at a stated rate, and state the study period explicitly, because a comparison between two products with different lives is meaningless without it.