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23-CS-1 Engineering Economics · May 2013

Question 6 of 6: Replacement Analysis — Old Packaging Machine

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

Notes on this paper

National Exams — May 2013 — 11-CS-1 Engineering Economics. Open book; non-communicating calculator permitted. Any five of the six questions constitute a complete paper; all questions are of equal value. Fully worked solutions to all six questions are given below. Standard compound-interest factors are used throughout; minor rounding differences are immaterial.

Question 6: Replacement Analysis — Old Packaging Machine (20 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.

Assumption: the current market value is the purchase price declined 15%/yr for 7 years, $45{,}000(0.85)^{7}=\$14{,}426$ (installation is not recoverable). This is the defender's opportunity cost today.

(a) Defender's Marginal Cost of Keeping It One More Year

The new machine (challenger) is justified only if its minimum EAC is below the defender's marginal cost of retaining it. That marginal cost for the coming year is the sum of the capital cost (lost interest on the retained market value plus the drop in market value) and the O&M:

$$MV_{0}=14{,}426,\quad MV_{1}=0.90(14{,}426)=12{,}983$$
$$\text{Capital cost} = i\cdot MV_{0} + (MV_{0}-MV_{1}) = 0.09(14{,}426) + (14{,}426-12{,}983) = 1{,}298 + 1{,}443 = 2{,}741$$
$$\text{Marginal cost} = 2{,}741 + \underbrace{2{,}000}_{\text{O\&M}} = \boxed{\$4{,}741/\text{yr}}$$

So a new machine is justified only if its minimum EAC over its economic life is less than about $4,741/year. If the best available new machine can deliver service for less than this, replace; otherwise keep the old one.

Check that year 1 really is the binding comparison. Repeating the same marginal-cost calculation for later years, with the market value falling 10%/yr and O&M rising $270/yr, gives $4,741 (year 1), $4,737 (year 2), $4,760 (year 3), $4,808 (year 4) and $4,878 (year 5). The series dips by about $4 in year 2 before rising steadily, so the defender's minimum equivalent annual cost over its remaining life is $4,739 — indistinguishable from the year-1 figure at the precision of the data. The threshold for the challenger is therefore about $4,740/year either way.

(b) When to Replace

Replace the old machine at the point where its marginal cost of keeping it one more year first exceeds the minimum EAC of the challenger. Because the defender's O&M rises ($270/yr) while its market value keeps falling, its marginal cost trends upward over time; each year one compares the coming year's marginal cost (recomputed as above, e.g. ≈$4,740 next year, changing modestly thereafter) with the challenger's minimum EAC and replaces in the first year the challenger becomes cheaper.

(c) Effect of Past Installation and Capital Costs

They have no effect—the original $45,000 purchase price and $3,000 installation are sunk costs. Only the machine's current market value (its opportunity cost of being kept) and its future operating costs are relevant to the replacement decision. Including the sunk purchase/installation would be the classic error of "throwing good money after bad."

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