Solved Problems in Engineering Economy 2024 PDF

SOLVED PROBLEMS IN ENGINEERING ECONONOMY Charlie A. Marquez, PIE  SOLVED PROBLEMS IN ENGINEERING ECONOMY 2024 ENGINEERING ECONOMY – the analysis and evaluation of the factors that will affect the economic success of engineering projects to the end that...

SOLVED PROBLEMS IN ENGINEERING ECONONOMY Charlie A. Marquez, PIE  SOLVED PROBLEMS IN ENGINEERING ECONOMY 2024 ENGINEERING ECONOMY – the analysis and evaluation of the factors that will affect the economic success of engineering projects to the end that a recommendation can be made which will ensure the best use of capital. FORMULAS IN ENGINEERING ECONOMY SIMPLE INTEREST – the interest on a loan that is based only on the principal. Usually used for short-term loans where the period is measured in days rather than years. I = Pni (1) F = P + I = P + Pni F = P(1+ni) (2) where: I = interest P = principal or present worth n = number of interest periods i = rate of interest per interest period F = accumulated amount or future worth TYPES OF SIMPLE INTEREST ORDINARY SIMPLE INTEREST - interest is computed on the basis of 12 months of 30 days each which is equivalent to 360 days a year. In this case, the value of n that is used in the preceding formulas may be computed as: where d is the number of days the principal was invested EXACT SIMPLE INTEREST – interest is computed based on the exact number of days in a given year which is 365 days for a normal year and 366 days during a leap year (which occurs every 4 years, or if it is a century year, it must be divided by 400). Note that during leap years, February has 29 days and 28 days only during a normal year. In this case, the value of n that is used in the preceding formulas may be computed as: for a normal year for a leap year  SOLVED PROBLEMS IN ENGINEERING ECONOMY 2024 DISCOUNT – discount in simple terms is the interest deducted in advance. It is the difference between the amount a borrower receives in cash (present worth) and the amount he pays in the future (future worth). Discount = Future Worth – Present Worth D=F–P (3) Rate of discount is the discount on one unit of principal for one unit of time. d = 1 – (1 + i)-1 (4) i= (5) Where: d = rate of discount i = rate of interest for the same period COMPOUND INTEREST – interest which is based on the principal plus the previous accumulated interest. It may also be defined as ‘interest on top of interest.” This is usually used in commercial practice especially for longer periods. CASH FLOW DIAGRAMS – a graphical representation of cash flows drawn on a time scale. ↑ = receipts (positive cash flow or cash inflow) ↓ = disbursements (negative cash flow or cash outflow) F 0 F = P(1+i)n (6) 1 2 3 n P = F(1+i)-n (7) P Where: F = future amount of money P = present worth or principal i = rate of interest per interest period n = number of interest periods (1+i)n = single payment compound amount factor (1+i)-n = single payment present worth factor  SOLVED PROBLEMS IN ENGINEERING ECONOMY 2024 RATE OF INTEREST – the cost of borrowing money or the amount earned by a unit principal per unit time. TYPES OF RATES OF INTEREST NOMINAL RATE OF INTEREST – is the basic annual rate of interest. It specifies the rate of interest and the number of interest periods in one year. i= (8) Where: i = rate of interest per interest period r = nominal rate of interest m = number of compounding periods per year EFFECTIVE RATE OF INTEREST – is the actual or the exact rate of interest earned on the principal during a one-year period. ERi = (1+i)m – 1 (9) Where: ERi = effective rate of interest CONTINUOUS COMPOUNDING – based on the assumption that cash payments occur once per year but compounding is continuous throughout the year. nm F=P xnm Let x= F=P But =e Therefore, F=P (10) EQUATION OF VALUE – this is obtained by setting the sum of the values on a certain comparison or focal date of one set of obligations to the sum of the values on the same date of another set of obligations.  SOLVED PROBLEMS IN ENGINEERING ECONOMY 2024 ANNUITIES – a series of equal payments occurring at equal interval of time. TYPES OF ANNUITIES ORDINARY ANNUITY – this type of annuity is one where the payments are made at the end of each period beginning from the first period. P0 Fn 1 2 3.. 0 n A A A A Finding F when A is given: F = A{[(1+i)n – 1] / i} (11) Finding P when A is given: P = A{[1-(1+i)-n ] / i} (12) Where: F = future worth of an annuity A = a series of periodic, equal amounts of money P = present worth of an annuity i = interest rate per interest period n = number of interest periods DEFERED ANNUITY – this type of annuity is one where the first payment is made several periods after the beginning of the annuity. P0 Fn 1 2 3 4 5… 0 n A A A A A Finding F when A is given:  SOLVED PROBLEMS IN ENGINEERING ECONOMY 2024 F = A{[(1+i)n – 1] / i}(1+i)n (13) Finding P when A is given: P = A{[1-(1+i)-n ] / i}(1+i)-n (14) PERPETUITY – is an annuity wherein the payments continue indefinitely. P n →∞ P = A{[1-(1+i)-n ] / i} = A{[1-(1+i)- ∞ ] / i} P= (15) CAPITALIZED COST – this is one of the most important applications of perpetuity. The capitalized cost of any property is the sum of its first cost and the present worth of all costs for replacement, operation, and maintenance for a long period or forever. Case 1: No Replacement, maintenance and/or operation every period. CC = FC + P (16) Where: CC = capitalized cost FC = first cost P = present worth of perpetual operation and maintenance Case 2: Replacement only, no operation and maintenance CC = FC + X (17) X = S / (1+i)k-1 (18) Where: X = present worth of perpetual replacement S = amount needed to replace the property every k period  SOLVED PROBLEMS IN ENGINEERING ECONOMY 2024 k = periodic replacement GRADIENT – A series of disbursements or receipts that increases or decreases in each succeeding period by a constant amount = P PA PG P = PA + P G P = A(P/A, i%,n) + G(P/G, i%, n) (19) PG = G(P/G, i%,n) = [(1+i)n-1/i]-n}(1+i)-n Where: PA = present worth of an annuity PG = present worth of gradient CAPITAL FINANCING WITH BONDS BONDS – a financial security note issued by businesses or corporations and by the government as a means of borrowing long-term fund. It may also be defined as a long-term note issued by the lender to the borrower stipulating the terms of repayment and other conditions. BOND VALUE – the value of a bond is the present worth of all future amounts that are expected to be received through ownership of the bond. METHODS OF BOND RETIREMENT 1. The corporation may issue another set of bonds equal to the amount of bonds due for redemption. 2. The corporation may set up a sinking fund into which periodic deposits of equal amounts are made. The accumulated amount in the sinking fund is equal to the amount needed to retire the bonds at the time they are due. C F Fr Fr Fr Fr 0 1 2 n-1 0 n 1 2 3 n  SOLVED PROBLEMS IN ENGINEERING ECONOMY 2024 A A A A P A = F/(F/A, i%, n) (20) P = Fr [(1-(1+i)-n)/i] + C(1+i)-n (21) Where: A = periodic deposit into the sinking fund F = amount needed to retire the bonds, face / par value C = redemption price (often equal to F) r = bond rate i = investment rate or yield per period P = purchase price of the bond / value of the bond n periods before redemption. DEPRECIATION – the decrease in the value of a physical property with the passage of time. TYPES OF DEPRECIATION 1. Physical depreciation – this is due to the reduction of the physical ability of an equipment or asset to produce results. 2. Functional depreciation – this is due to the lessening in the demand for the function which the property was designed to render. METHODS OF DEPRECIATION 1. Straight Line Method – this method assumes that the loss in the value is directly proportional to the age of the equipment or asset. d= (22) Dn = (n){ (23) Cn = Co – Dn (24) Where: d = annual depreciation charge Co = original cost of the property CL = scrap value / salvage value  SOLVED PROBLEMS IN ENGINEERING ECONOMY 2024 L = useful life of the property Dn = accumulated depreciation up to n years. Cn = book value at the end of n years. 2. Sinking Fund Method – in this method, it is assumed that a sinking fund is established in which funds will accumulate for replacement purposes. d = (Co – CL)(i) / (1+i)L-1 (25) Dn = d(1+i)n-1 (26) Cn = Co – Dn (27) Note: all parameter definitions are the same with SLM. 3. Declining Balance Method – in this method, it is assumed that the annual cost of depreciation is a fixed percentage of the book value at the beginning of the year. This method is also called the constant percentage method or the Matheson Formula. L k=1- or k=1- (28) dn = Co(1-k)n-1(k) (29) Cn = Co(1-k)n (30) Cn = Co(CL/Co)n/L (31) Where: k = decline rate, whose value must always be < 1 and the salvage value must not be zero. 4. Double Declining Balance Method – this method is very similar to DBM but the decline rate, k, is replaced by 2/L. dn = Co[1-(2/L)]n-1(2/L) (32) Cn = Co[1-(2/L)]n (33) 5. SUM-OF-THE-YEARS’ DIGIT METHOD (SYD) dn = (Co – CL)(reverse digit/ digits) (34)  digits = (35)  SOLVED PROBLEMS IN ENGINEERING ECONOMY 2024 6. SERVICE-OUTPUT METHOD – this method assumes that the total depreciation that has taken place is directly proportional to the quantity of output of the property up to that time. dn = (Co-CL / T)(Qn) (36) where: T = total units of output up to the end of life Qn = total number of units of output during the nth year DEPLETION – depletion cost is the reduction of the value of a certain natural resource such as mines, oil, quarries, etc. due to the gradual extraction of its contents. METHODS OF COMPUTING DEPLETION CHARGE 1. UNIT OR FACTOR METHOD – this method is dependent on the initial cost of the property and the number of units in the property. Depletion cost in any year = (units sold during the year) (37) 2. PERCENTAGE OR DEPLETION ALLOWANCE METHOD Depletion charge = Fixed percentage of Gross Income (38) Depletion charge = 50% of the Net Taxable Income (39) INVESTMENT OF CAPITAL METHODS OF MAKING ECONOMY STUDIES 1. RATE OF RETURN (ROR) METHOD – this method is a measure of the effectiveness of an investment of capital. When this method is used, it is necessary to decide whether the computed rate of return is sufficient to justify the investment. If the computer ROR is RORreq’d, the proposed investment is justified. Conditions: 1. A single investment of capital is made at the beginning of the first year. 2. The capital invested is the total amount of capital investment required to finance the project. 3. There is identical revenue and cost date for each year.  SOLVED PROBLEMS IN ENGINEERING ECONOMY 2024 ROR = (40) 2. ANNUAL WORTH METHOD – in this method, interest on the original investment is included as a cost. If the excess if annual cash inflows over annual cash outflows is 0, the proposed investment is justified. Same conditions apply as the ROR method. 3. PRESENT WORTH (PW) METHOD – this method is based on the concept of present worth. If the present worth of net cash flows is than 0, the proposed project is economically justified. 4. PAYBACK PERIOD METHOD – this is the length of time required to recover the first cost of an investment from the net cash flow. Payback Period (years) = (41) COMPARING ALTERNATIVES - METHODS OF COMPARING ALTERNATIVES 1. ROR ON ADDITIONAL INVESTMENT – in this method, if the ROR on additional investment is than the ROR required, then the alternative requiring an additional investment is more economical and therefore, should be chosen. ROR on additional investment = (42) 2. ANNUAL COST (AC) METHOD – to use this method, the annual cost of the alternatives including interest on capital is determined. The alternative with the least annual cost is chosen. This applies only to alternatives which has a uniform cost data for each year and a single investment of capital at the beginning of the project. 3. EQUIVALENT UNIFORM ANNUAL COST (EUAC) METHOD – in this method, all cash flows must be converted to an equivalent uniform annual cost. The alternative with the least EUAC should be selected. This method is flexible and can be used for any type of alternative selection problems. 4. PRESENT WORTH COST (PWC) METHOD – in this method, determine the present worth of the net cash outflows for each alternative for the same period of time. The alternative with the least PW should be selected.  SOLVED PROBLEMS IN ENGINEERING ECONOMY 2024 5. PAYBACK PERIOD METHOD – in this method, the payback period for each alternative is computed. The alternative with the shortest payback period is adopted. BREAK-EVEN ANALYSIS – this is used in situations where the cost of two or more alternatives may be affected by a common variable. BREAK-EVEN POINT – is the value of the variable for which the costs of the alternatives will be equal. C1 = f1(x) and C2 = f2(x) (43) To Break-Even, Income = fixed costs + variables cost (x) (44) Income = fixed costs + variables cost (x) + profit/loss (45) Income = fixed costs + variables cost (x) + dividends + profit/loss (46) Where: C1 = certain specified total cost applicable to alternative 1 C2 = certain specified total cost applicable to alternative 2 x = a common independent variable affecting alternatives 1 & 2