- A company has a CNC (computer numeric controlled) precision metal cutting machine and is considering a replacement for the machine. If repaired and reprogrammed with new software, the existing machine can be used for another 5 years. The existing machine will have no salvage value at the end of 5 years if kept for that period; however, the firm can sell the existing machine today to another firm for $4,500. If the existing machine is kept, it will require an immediate $2,000 overhaul to enable continued operation. (The overhaul will not extend the service life beyond 5 years or increase the value of the existing machine). Operating costs for the existing machine are estimated at $1,500 for the first year and will increase by $500 each year after the first.
A new machine with the same capabilities as the existing machine will cost $10,000 and will have operating costs of $1,500 for the first year. For each additional year (after the first year) the new machine is used, its operating costs will increase by $750 over the previous year’s operating costs. The new machine’s salvage value is $7,500 after one year and declines by 15% from the previous year’s salvage value each year.
The firm’s MARR is 12%. Should the existing CNC machine be replaced now?
- A plant expansion plan calls for the installation of additional production equipment to increase parts production. The equipment needed for the expansion costs $350,000 and will generate annual revenues of $100,000. Operation of the equipment requires (annually) $20,000 in labor, $15,000 in material and $5,000 in power and utility costs. The equipment will be classified as a 7-year MACRS property. The company will stop using the equipment at the end of 5 years, at which time it will be sold for $75,000.
- b) Determine the after-tax net present worth of the project if the firm’s MARR is 10%. Be sure to include a cash flow diagram and indicate the formulas used to “solve” the cash flow diagram to find NPW. Do you recommend that the firm make the investment in plant expansion?
- Your firm has negotiated a five-year lease on mineral rights. The annual cost stated in the lease is $250,000, to be paid at the beginning of each of the 5 years. The general inflation rate, f, is 1.7%. Find the equivalent cost of the lease in constant dollars for each of the 5 years. Arrange your computations in a table as shown below:
|End of Year||Cash flow|
|Conversion factor (multiplier) appropriate for f =1.7%||Cash flow|
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As the engineering manager for the National Capital Region Transportation Authority, you have been asked to determine how much additional road network capacity must be added over the next 20 years and when the additional capacity should be added.
To address this problem, you make the following simplifying analyses and assumptions: 1
1 Adapted from Gaver and Thompson (1973)
- Let x be the amount of road network capacity (in appropriate units, such as thousands of vehicles per hour) which will be added. A study of past records shows that the cost of adding new capacity follows the general form
C(x) = axb
C(x) is the cost of new capacity x (where C(x) is measured in millions of $ and x is measured in thousands of cars per hour),
a is a constant proportionality factor (cost is proportional to the amount of capacity added), and b is a constant economy of scale factor (for example, adding 4000 cars/hr of new capacity might only cost twice as much as adding 1000 cars/hr of new capacity).
A reasonable assumption to make is that a > 0 and 0 < b < 1.
Therefore, it is advantageous to build new capacity larger than current demand and allow for future growth of demand.
- Any new capacity must be paid for. Since money has a time value, $1 spent t years from the present is equivalent to $1(e-rt) in “present value dollars” today, where r is an appropriate discount rate and continuous compounding is assumed.
Therefore, it is advantageous to defer spending on new capacity into the future, where “future dollars” may be used to pay for the capacity expansion.
- Considering (1) and (2) above, you reason that the decision on how much new capacity to add and when depends on a tradeoff between building new capacity now to take advantage of economy of scale and deferring expansion into the future so that “future dollars” may be used to pay for the capacity expansion.
- Formulate an optimization problem to determine the optimal size and frequency of capacity expansion for the network. Do this by: 2
- Assume that demand for new capacity increases linearly. If x is demand for new capacity, then
t measures the time (in years) from some point where demand and capacity are equal.
- , an optimal time interval between expansions of capacity. (Later, afterFind , you will be able to determinefinding x, the amount of new capacity to add).
- Observe that, looking forward from t = 0, the discounted cost of adding capacity x , …, is, 4, 3 2at intervals of
C(x) axb + … axb + e-r2= axb + e-r
= axb + … ] [axb + e-raxb + e-r
= [ C(x) ]axb + e-r
Rearranging the terms in the last equation,
C(x) )= axb / ( 1- e-r
Recalling , when t = t = that x =
C(x) )b / ( 1- e-r= a(
- in the expression for C(x) above that minimizes C(x). RecallFind a value of that a > 0, b > 0, and > 0. Support your conclusion by an appropriate demonstration (or proof).
- Subsequent research and data analysis shows that appropriate values for b, r = 1.5, what may be inferred. Assuming that b = 0.5, r = 10% (0.1) and and should you recommend as the optimum interval for expanding capacity, and how much capacity should be added at each interval?
You may use Excel or MATLAB in answering (d) and (e) above.
Product code: Research-AW453
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