THÔNG TIN TÀI LIỆU
Chapter 10.1 Engineering Economy Thomas R. Huston University of Cincinnati, Cincinnati, Ohio 1.1 INTRODUCTION Engineering is the profession that is devoted to the application of scienti®c knowledge for practical purposes. Through the application of scienti®c knowl- edge, engineers are continually developing products, processes, and services for the bene®t of society. Engineers have been instrumental in many of the advances of society. For example, the state of modern transportation can be linked to the eorts of engineers. While undertaking such pursuits, the engineer is typically faced with a variety of alternatives. These alternatives may include material selections, the degree of computer automation, the selection of an applicable safety system, and the means of manufacturing. Each alternative will have inherent technical advantages and disadvantages that the engineer must evaluate. The evaluation of any alternative will also have to consider the constraints of the particular problem or project. The engineer will typically be well informed about the technical aspects of various alternatives. However, the engineer must also have a sound understanding of the economic feasibility of the various alternatives. Indeed, money is a scarce resource that must be allo- cated in a prudent fashion. This chapter provides a foundation in the basic principles of engineering economics. Through the application of these basic principles, the engineer will be able to address economic issues. One such issue is the economic feasibility of alternatives. Engineering economics oers a means to assess any receipts and disbursements associated with an alternative. Such an assessment will consider the magnitude and timing of the receipts and disbursements. In¯ation and taxes may also be factors that enter into the economic evaluation of an alternative. The basic principles of engineering economics also provide methods for the comparison of alternatives and the subsequent selec- tion of an optimal alternative. For example, an engi- neer may be confronted with the selection of machinery from a variety of sources. As another exam- ple, the engineer may face the economic decision of manufacturing a part versus purchasing a part. It should also be recognized that there are limita- tions to engineering economics. Certain problems may not have the potential to be evaluated properly in eco- nomic terms. Some problems may be highly complex wherein economics is a minor consideration. Still other problems may not be of sucient importance to warrant engineering economic analysis. 1.2 ELEMENTARY CONCEPTS OF ENGINEERING ECONOMICS There are several fundamental concepts that form a foundation for the application of the methods of engi- neering economics. One fundamental concept is the recognition that money has a time value. The value of a given amount of money will depend upon when it is received or disbursed. Money possessed in the 829 Copyright © 2000 Marcel Dekker, Inc. present will have a greater value than the same amount of money at some point in the future. It would be preferable to receive $1000 in the pre- sent rather than receiving $1000 ®ve years hence. Due to the earning power of money, the economic value of $1000 received at the present will exceed the value of $1000 received ®ve years in the future. The $1000 received today could be deposited into an interest bear- ing savings account. During the intervening period of ®ve years, the $1000 would earn additional money from the interest payments and its accumulated amount would exceed $1000. The time value of money is also related to the purchasing power of money. The amount of goods and services that a quantity of money will purchase is usually not static. In¯ation corresponds to a loss in the purchasing power of money over time. Under the pressures of in¯ation, the cost of a good or service will increase. As an example, during the period of 1967 to 1997 the cost of a U.S. ®rst-class postage stamp rose to 32 cents from 5 cents. De¯ation is the opposite condition of in¯ation. Historically, in¯a- tionary periods have been far more common than periods of de¯ation. A fundamental concept that is related to the time value of money is interest. Money is a valuable com- modity, so businesses and individuals will pay a fee for the use of money over a period of time. Interest is de®ned as the rental fee paid for the use of such a sum of money. Interest is usually quanti®ed by the interest rate where the interest rate represents a percen- tage of the original sum of money that is periodically applied. For instance, a ®nancial institution may charge 1% per month for a borrowed sum of money. This means that at the end of a month, a fee of 1% of the amount borrowed would have to be paid to the ®nancial institution. The periodic payment of interest on a loan repre- sents a cash transaction. During such a transaction, a borrower would view the associated interest as a dis- bursement while the interest would be a receipt for the lender. In engineering economics analysis, a point of view must be selected for reference. All analysis should proceed from a sole viewpoint. Engineering economic analysis should also only consider and assess feasible alternatives. Alternatives that ordinarily would be feasible may be infeasible due to the particular constraints of a problem or project. A frequently overlooked alternative is the do-nothing alternative. Under the do-nothing alternative, the option of doing nothing is preferable to any of the other feasible alternatives. It is the inherent dierences between alternatives that must be evaluated. Indeed it is the dierences in alternatives that will lead to the selection of an optimal alternative. Such an evaluation will utilize money as a common unit of measurement to discern the dier- ences between alternatives. The evaluation of alterna- tives should also utilize a uniform time horizon to reveal the dierences in alternatives. It is essential to recognize that any decisions about alternatives will only aect the present and the future. Therefore, past decisions and any associated costs should be ignored in engineering economic analysis. The associated costs from past decisions are known as sunk costs. Sunk costs are irrelevant in engineering economic analysis. 1.3 ECONOMIC EQUIVALENCE AND CASH FLOW FORMULAS 1.3.1 Economic Equivalence In engineering, two conditions are said to be equivalent when each condition produces the same eect or impact. The concept of equivalence also pertains to engineering economics. Two separate alternatives will have economic equivalence whenever each alternative possesses the same economic value. Any prospective economic equivalence between two alternatives will be dependent upon several factors. One factor is the respective magnitudes of the cash ¯ow for each alter- native. Another factor is the timing of the receipts and disbursements for each alternative. A third factor is the interest rate that accounts for the time value of money. Through a combination of these factors, two cash ¯ows that dier in magnitude may possess the same inherent economic value. The concept of economic equivalence is revealed through the cash ¯ows asso- ciated with a routine loan. Suppose an individual borrowed $10,000 at 6% compounded annually to be repaid in annual instalments of $2374 over ®ve years. One cash ¯ow would be the sum of $10,000 at the present. The other cash ¯ow would entail ®ve annual payments of $2374 that totaled $11,870. Although each cash ¯ow occurs at distinct points in time and has a dierent magnitude, both cash ¯ows would be equiva- lent at the interest rate of 6% compounded annually. 1.3.2 Simple and Compound Interest There are dierent ways in determining the amount of interest that a sum of money will produce. One way is simple interest. Under simple interest, the amount of 830 Huston Copyright © 2000 Marcel Dekker, Inc. interest accrued, I, on a given sum of money, P,is calculated by I Pni 1 where P is the principal amount, n the number of inter- est periods, and i the interest rate. Hence with simple interest, a sum of money would increase to F P I P Pni 2 With simple interest, any interest earned during an interest period does not earn additional interest in forthcoming interest periods. In contrast, with compound interest, the interest is. determined by the principal sum of money and on any interest that has accumulated to date. So any previous interest will earn interest in the future. For example, if a sum of money, P, is deposited into an interest-bearing account at an interest rate, i, after one period the amount of money available, F, would be determined by F P1 i3 If the sum of money were deposited for two periods, the amount of money available, F , would be deter- mined by F P1 i1 iP1 i 2 4 In general, the amount of money, F, that would accu- mulate with n additional periods would be F P1 i n 5 Compound interest is more prevalent in ®nancial transactions than simple interest, although simple interest is often encountered in bonds. 1.3.3 Cash Flow Diagrams and End-of-Period Convention In engineering, diagrams are frequently drawn to help the individual understand a particular engineering issue. A cash ¯ow diagram is often used to depict the magnitude and the timing of cash ¯ows in an engineer- ing economics issue. A cash ¯ow diagram presumes a particular point of view. A horizontal line is used to represent the time horizon, while vertical lines from the horizontal line depict cash ¯ows. An upward arrow indicates a receipt of money, while a downward arrow is a disbursement (see Fig. 1). In this chapter, there is an assumption that cash ¯ows will be discrete and will occur at the end of a period. Continuous ¯ows of cash over a period will not be considered. An extensive discussion of contin- uous cash ¯ows is oered in the references. 1.3.4 Cash Flow Patterns In ®nancial transactions, a cash ¯ow may undertake a variety of patterns. The simplest pattern is the single cash ¯ow. Under this cash ¯ow pattern, a single pre- sent amount is transformed into a single future amount (see Fig. 2). The uniform series is another cash ¯ow pattern. With this pattern, all of the cash ¯ows are of the same magnitude and the cash ¯ows occur at equally spacedtimeintervals(seeFig.3). A cash ¯ow that increases or decreases by the same amount in each succeeding period would be a uniform gradientcash¯owpattern(seeFig.4).Whereas,acash ¯ow that increases or decreases by the same percentage in each succeeding period would be a geometrical gradientcash¯owpattern.(seeFig.5). Engineering Economy 831 Figure 1 Cash ¯ow diagram. Figure 2 Present amount and future amount. Copyright © 2000 Marcel Dekker, Inc. An irregular cash ¯ow pattern would occur when- ever the cash ¯ow did not maintain one of the afore- mentioned regular patterns. Occasionally, a portion of an irregular cash ¯ow pattern may exhibit a regular pattern (see Fig. 6). In Fig. 6, the overall cash ¯ow pattern would be classi®ed as irregular but in the ®nal three years there is a uniform series pattern. Equivalent relationships between the various cash ¯ow patterns may be developed mathematically. Due to the time value of money, such relationships will be dependent upon the prevailing interest rates and the duration of the associated cash ¯ows. 1.3.5 Single-Payment Compound Amount Factor Due to the time value of money, a single cash ¯ow, P, will increase over time to an equivalent future value, F. The future value, F, will depend upon the length of time, the prevailing interest rate, and the type of inter- est. If the single cash ¯ow, P, is invested at a constant compound interest rate, i, for a given number of inter- est periods, n, then the future value, F, will be deter- mined by Eq. (5). Eq. (5) may be rewritten to introduce the following notation: F P1 i n PFjP; i; n6 The conversion factor, FjP; i; n, is referred to as the single-payment compound amount factor. It is inter- preted as ``to ®nd the equivalent future amount, F, given the present amount, P, at the interest rate, i, for n periods.'' The single-payment compound amount factor, FjP; i; n, is simply the quantity (1 i n . The evaluation of the single-payment compound amount factor is an easy calculation. Tabulated values of the single-payment compound amount factor for interest ratesof1%,8%,and10%maybefoundinTables1to 832 Huston Figure 3 Uniform series. Figure 4 Uniform gradient. Figure 5 Geometrical gradient series. Figure 6 Irregular cash ¯ow. Copyright © 2000 Marcel Dekker, Inc. 3.Note,othereconomicequivalencefactorscanalso befoundinTables1to3. Example 1. A sum of $5000 is deposited into an account that pays 10% interest compounded annually. To determine the future value of the sum of money 20 years hence, utilize Eq. (6): F 650001 0:10 20 633,637 1.3.6 Single Payment Present-Worth Amount Factor Through simple algebra, Eq. (6) can be solved for P, wherein the resulting factor, PjF; i; n, is designated as the single-payment present worth factor: P F1 i Àn FPjF; i; n7 Example 2. In the settlement of a litigation action, a boy is to receive a lump sum of $250,000 10 years in the future. What is the present worth of such a payment presuming an annual compound interest rate of 8%? P 6250,0001 0:08 À10 6115,798 Example 3. What annual rate of interest was earned if an investment of $11,000 produced a value of $21,000 after 10 years? F P1 i n 621,000 611,0001 i 10 1:909 1 i 10 i 1:909 1=10 À 1 0:066 6:67 1.3.7 Compound Amount Factor Formulas can be derived that relate a single future cash ¯ow pattern, F, to a uniform series of cash ¯ow patterns, A. The equivalent future amount, F, that a uniform cash ¯ow pattern, A, will produce is F A 1 i n À 1 i ! AF jA; i; n8 Example 4. A design engineer expects to collect $5000 per year on patent royalties. The patent remains in eect for the next 10 years. What is the future value of this series of patent royalties if it is deposited into a fund that earns 10% compounded annually? F AFjA; i; n65,000FjA; 107; 10 6500015:937 679,685 1.3.8 Sinking Fund Factor Similarly, an equivalent uniform series cash ¯ow pattern, A, can be obtained from a single future cash ¯ow pattern, F : A F i 1 i n À 1 ! FAjF; i; n9 1.3.9 Present-Worth Factor It is often desirable to be able to relate a present amount, P, to a uniform series cash ¯ow pattern, A. The present worth factor converts a uniform series cash ¯ow pattern, A, to a single present amount, P. The formula for the present-worth factor is P A 1 i n À 1 i1 i n ! A APjA; i; n10 1.3.10 Capital Recovery Factor The capital recovery factor is the reciprocal of the pre- sent-worth factor. This conversion factor transforms a single present amount, P, to a uniform series of cash ¯ows: A P i1 i n 1 i n À 1 ! PAjP; i; n11 Example 5. To purchase a new machine, a manu- facturer secures a $50,000 loan to be paid o in annual payments over the next ®ve years. If the interest rate of the loan is 8% compounded annually, what is the periodic payment that the manufacturer must pay? A PAjP; I; n650,000AjP; 87; 5 650,0000:2505 612,525 Engineering Economy 833 Copyright © 2000 Marcel Dekker, Inc. Table 1 One Percent: Compound Interest Factors NFjPPjFFjAAjFPjAAjPAjG 1 1.0100 0.99010 1.0000 1.00000 0.99010 1.01000 0.00000 2 1.0201 0.98030 2.0100 1.49751 1.97040 0.50751 0.49751 3 1.0303 0.97059 3.0301 0.33002 2.94099 0.34002 0.99337 4 1.0406 0.96098 4.0604 0.24628 3.90197 0.25628 1.48756 5 1.0510 0.95147 5.1010 0.19604 4.85343 0.20604 1.98010 6 1.0615 0.94205 6.1520 0.16255 5.79548 0.17255 2.47098 7 1.0721 0.93272 7.2135 0.13863 6.72819 0.14863 2.96020 8 1.0829 0.92348 8.2857 0.12069 7.65168 0.13069 3.44777 9 1.0937 0.91434 9.3685 0.10674 8.56602 0.11674 3.93367 10 1.1046 0.90529 10.4622 0.09558 9.47130 0.10558 4.41792 11 1.1157 0.89632 11.5668 0.08645 10.36763 0.09645 4.90052 12 1.1268 0.88745 12.6825 0.07885 11.25508 0.08885 5.38145 13 1.1381 0.87866 13.8093 0.07241 12.13374 0.08241 5.86073 14 1.1495 0.86996 14.9474 0.06690 13.00370 0.07690 6.33836 15 1.1610 0.86135 16.0969 0.06212 13.86505 0.07212 6.81433 16 1.1726 0.85282 17.2579 0.05794 14.71787 0.06794 7.28865 17 1.1843 0.84438 18.4304 0.05426 15.56225 0.06426 7.76131 18 1.1961 0.83602 19.6147 0.05098 16.39827 0.06098 8.23231 19 1.2081 0.82774 20.8109 0.04805 17.22601 0.05805 8.70167 20 1.2202 0.81954 22.0190 0.04542 18.04555 0.05542 9.16937 21 1.2324 0.81143 23.2392 0.04303 18.85698 0.05303 9.63542 22 1.2447 0.80340 24.4716 0.04086 19.66038 0.05086 10.09982 23 1.2572 0.79544 25.7163 0.03889 20.45582 0.04889 10.56257 24 1.2697 0.78757 26.9735 0.03707 21.24339 0.04707 11.02367 25 1.2824 0.77977 28.2432 0.03541 22.02316 0.04541 11.48312 26 1.2953 0.77205 29.5256 0.03387 22.79520 0.04387 11.94092 27 1.3082 0.76440 30.8209 0.03245 23.55961 0.04245 12.39707 28 1.3213 0.75684 32.1291 0.03112 24.31644 0.04112 12.85158 29 1.3345 0.74934 33.4504 0.02990 25.06579 0.03990 13.30444 30 1.3478 0.74192 34.7849 0.02875 25.80771 0.03875 13.75566 31 1.3613 0.73458 36.1327 0.02768 26.54229 0.03768 14.20523 32 1.3749 0.72730 37.4941 0.02667 27.26959 0.03667 14.65317 33 1.3887 0.72010 38.8690 0.02573 27.98969 0.03537 15.09946 34 1.4026 0.71297 40.2577 0.02484 28.70267 0.03484 15.54410 35 1.4166 0.70591 41.6603 0.02400 29.40858 0.03400 15.98711 36 1.4308 0.69892 43.0769 0.02321 30.10751 0.03321 16.42848 37 1.4451 0.69200 44.5076 0.02247 30.79951 0.03247 16.86822 38 1.4595 0.68515 45.9527 0.02176 31.48466 0.03176 17.30632 39 1.4741 0.67837 47.4123 0.02109 32.16303 0.03109 17.74278 40 1.4889 0.67165 48.8864 0.02046 32.83469 0.03046 18.17761 41 1.5038 0.66500 50.3752 0.01985 33.49969 0.02985 18.61080 42 1.5188 0.65842 51.8790 0.01928 34.15811 0.02928 19.04237 43 1.5340 0.65190 53.3978 0.01873 34.81001 0.02873 19.47231 44 1.5493 0.64545 54.9318 0.01820 35.45545 0.02820 19.90061 45 1.5648 0.63905 56.4811 0.01771 36.09451 0.02771 12.32730 46 1.5805 0.63273 58.0459 0.01723 36.72724 0.02723 20.75235 47 1.5963 0.62646 59.6263 0.01677 37.35370 0.02677 21.17578 48 1.6122 0.62026 61.2226 0.01633 37.97396 0.02633 21.59759 49 1.6283 0.61412 62.8348 0.01591 38.58808 0.02591 22.10778 50 1.6446 0.60804 64.4632 0.01551 39.19612 0.02551 22.43635 55 1.7285 0.57853 72.8525 0.01373 42.14719 0.02373 24.50495 60 1.8167 0.55045 81.6697 0.01224 44.95504 0.02224 26.53331 65 1.9094 0.52373 90.9366 0.01100 47.62661 0.02100 28.52167 70 2.0068 0.49831 100.6763 0.00993 50.16851 0.01993 30.47026 75 2.1091 0.47413 110.9128 0.00902 52.58705 0.01902 32.37934 80 2.2167 0.45112 121.6715 0.00822 54.88821 0.01822 34.24920 85 2.3298 0.42922 132.9790 0.00752 57.07768 0.01752 36.08013 90 2.4486 0.40839 144.8633 0.00690 59.16088 0.01690 37.87245 95 2.5735 0.38857 157.3538 0.00636 61.14298 0.01638 39.62648 834 Copyright © 2000 Marcel Dekker, Inc. Table 2 Eight Percent: Compound Interest Factors NFjPPjFFjAAjFPjAAjPAjG 1 1.0800 0.92593 1.0000 1.00000 0.92593 1.08000 0.00000 2 1.1664 0.85734 2.0800 0.48077 1.78326 0.56077 0.48077 3 1.2597 0.79383 3.2464 0.30803 2.57710 0.38803 0.94874 4 1.3605 0.73503 4.5061 0.22192 3.31213 0.30192 1.40396 5 1.4693 0.68058 5.8666 0.17046 3.99271 0.25046 1.84647 6 1.5869 0.63017 7.3359 0.13632 4.62288 0.21632 2.27635 7 1.7138 0.58349 8.9228 0.11207 5.20637 0.19207 2.69366 8 1.8509 0.54027 10.6366 0.09401 5.74664 0.17401 3.09852 9 1.9990 0.50025 12.4876 0.08008 6.24689 0.16008 3.49103 10 2.1589 0.46319 14.4866 0.06903 6.71008 0.14903 3.87131 11 2.3316 0.42888 16.6455 0.06008 7.13896 0.14008 4.23950 12 2.5182 0.39711 18.9771 0.05270 7.53608 0.13270 4.59575 13 2.7196 0.36770 21.4953 0.04652 7.90378 0.12652 4.94021 14 2.9372 0.34046 24.2149 0.04130 8.24424 0.12130 5.27305 15 3.1722 0.31524 27.1521 0.03683 8.55948 0.11683 5.59446 16 3.4259 0.29189 30.3243 0.03289 8.85137 0.11298 5.90463 17 3.7000 0.27027 33.7502 0.02963 9.12164 0.10963 6.20375 18 3.9960 0.25025 37.4502 0.02670 9.37189 0.10670 6.49203 19 4.3157 0.23171 41.4463 0.02413 9.60360 0.10413 6.76969 20 4.6610 0.21455 45.7620 0.02185 9.81815 0.10185 7.03695 21 5.0338 0.19866 50.4229 0.01983 10.01680 0.09983 7.29403 22 5.4365 0.18394 55.4568 0.01803 10.20074 0.09803 7.54118 23 5.8715 0.17032 60.8933 0.01624 10.37106 0.09642 7.77863 24 6.3412 0.15770 66.7648 0.01498 10.52876 0.09498 8.00661 25 6.8485 0.14602 73.1059 0.01368 10.67478 0.09368 8.22538 26 7.3964 0.13520 79.9544 0.01251 10.80998 0.09251 8.43518 27 7.9881 0.12519 87.3508 0.01145 10.93516 0.09145 8.63627 28 8.6271 0.11591 95.3388 0.01049 11.05108 0.09049 8.82888 29 9.3173 0.10733 103.9659 0.00962 11.15841 0.08962 9.01328 30 10.0627 0.09938 113.2832 0.00883 11.25778 0.08883 9.18971 31 10.8677 0.09202 123.3459 0.00811 11.34980 0.08811 9.35843 32 11.7371 0.08520 134.2135 0.00745 11.43500 0.08745 9.51967 33 12.6760 0.07889 145.9506 0.00685 11.51389 0.08685 9.67370 34 13.6901 0.07305 158.6267 0.00630 11.58693 0.08630 9.82075 35 14.7583 0.06763 172.3168 0.00580 11.65457 0.08580 9.96107 36 15.9682 0.06262 187.1021 0.00534 11.71719 0.08534 10.09490 37 17.2456 0.05799 203.0703 0.00492 11.77518 0.08492 10.22246 38 18.6253 0.05369 220.3159 0.00454 11.82887 0.08454 10.34401 39 20.1153 0.04971 238.9412 0.00419 11.87858 0.08419 10.45975 40 21.7245 0.04603 259.0565 0.00386 11.92461 0.08386 10.56992 41 23.4625 0.04262 280.7810 0.00356 11.96723 0.08356 10.67473 42 25.3395 0.03946 304.2453 0.00329 12.00760 0.08329 10.77441 43 27.3666 0.03654 329.5830 0.00303 12.04324 0.08303 10.86915 44 29.5560 0.03383 356.9496 0.00280 12.07707 0.08280 10.95917 45 31.9204 0.03133 386.5056 0.00259 12.10840 0.08259 11.04465 46 34.4741 0.02901 418.4261 0.00239 12.13741 0.10823 11.12580 47 37.2320 0.02686 452.9002 0.l00221 12.16427 0.08221 11.20280 48 40.2106 0.02487 490.1322 0.00204 12.18914 0.08204 11.27584 49 43.4274 0.02303 530.3427 0.00189 12.21216 0.08189 11.34509 50 46.9016 0.02132 573.7702 0.00174 12.23348 0.08174 11.41071 55 68.9139 0.01451 848.9232 0.00118 12.31861 0.08118 11.69015 60 101.2571 0.00988 1,253.2133 0.00080 12.37655 0.08080 11.90154 65 148.7798 0.00672 1,847.2481 0.00054 12.41598 0.08054 12.06016 70 218.6064 0.00457 2,720.0801 0.00037 12.44282 0.08037 12.17832 75 321.2045 0.00311 4,002.5566 0.00025 12.46108 0.08025 12.26577 80 471.9548 0.00212 5,886.9354 0.00017 12.47351 0.08017 12.33013 85 693.4565 0.00144 8,655.7061 0.00012 12.48197 0.08012 12.37725 90 1,018.9151 0.00098 12,723.9386 0.00008 12.48773 0.08008 12.41158 835 Copyright © 2000 Marcel Dekker, Inc. Table 3 Ten Percent: Compound Interest Factors NFjPPjFFjAAjFPjAAjPAjG 1 1.1000 0.90909 1.0000 1.00000 0.90909 1.10000 0.00000 2 1.2100 0.82645 2.1000 0.47619 1.73554 0.57619 0.47619 3 1.1300 0.75131 3.3100 0.30211 2.48685 0.40211 0.93656 4 1.4641 0.68301 4.6410 0.21547 3.16987 0.31547 1.38117 5 1.6105 0.62092 6.1051 0.16380 3.79079 0.26380 1.81013 6 1.7716 0.56447 7.7156 0.12961 4.35528 0.22961 2.22356 7 1.9487 0.51316 9.4872 0.10541 4.86842 0.20541 2.62162 8 2.1436 0.46651 11.4359 0.08744 5.33493 0.18744 3.00448 9 2.3579 0.42410 13.5796 0.07364 5.75902 0.17364 3.37235 10 2.5937 0.38554 15.9374 0.06275 6.14457 0.16276 3.72546 11 2.8531 0.35049 18.5312 0.05396 6.49508 0.15396 4.06405 12 3.1384 0.31863 21.3843 0.04676 6.81369 0.14676 4.38840 13 3.4523 0.28966 24.5227 0.04078 7.10336 0.14078 4.69879 14 3.7975 0.26333 27.9750 0.03575 7.36669 0.13575 4.99553 15 4.1772 0.23939 31.7725 0.03147 7.60608 0.13147 5.27893 16 4.5960 0.21763 35.9497 0.02782 7.82371 0.12782 5.54934 17 5.0545 0.19784 40.5447 0.02466 8.02155 0.12466 5.80710 18 5.5599 0.17986 45.5992 0.02193 8.20141 0.12193 6.05256 19 6.1159 0.16351 51.1591 0.01955 8.36492 0.11955 6.28610 20 6.7275 0.14864 57.2750 0.01746 8.51356 0.11746 6.50808 21 7.4002 0.13513 64.0025 0.01562 8.64869 0.11562 6.71888 22 8.1403 0.12285 71.4027 0.01401 8.77154 0.11401 6.91889 23 8.9543 0.11168 79.5430 0.01257 8.88322 0.11257 7.10848 24 9.8497 0.10153 88.4973 0.01130 8.98474 0.11130 7.28805 25 10.8347 0.09230 98.3471 0.01017 9.07704 0.11017 7.45798 26 11.9182 0.08391 109.1818 0.00916 9.16095 0.10916 7.61865 27 13.1100 0.06728 121.0999 0.00826 9.23722 0.10826 7.77044 28 14.4210 0.06934 134.2099 0.00745 9.30657 0.10745 7.91372 29 15.8631 0.06304 148.6309 0.00673 9.36961 0.10673 8.04886 30 17.4494 0.05731 164.4940 0.00608 9.42691 0.10608 8.17623 31 19.1943 0.05210 181.9434 0.00550 9.47901 0.10550 8.29617 32 21.1138 0.04736 201.1378 0.00497 9.52638 0.10497 8.40905 33 23.2252 0.04306 222.2515 0.00450 9.56943 0.10450 8.51520 34 25.5477 0.03914 245.4767 0.00407 9.60857 0.10407 8.61494 35 28.1024 0.03558 271.0244 0.00369 9.64416 0.10369 8.70860 36 30.9127 0.03235 299.1268 0.00334 9.67651 0.10334 8.79650 37 34.0039 0.02941 330.0395 0.00303 9.70592 0.10303 8.87892 38 37.4043 0.02673 364.0434 0.00275 9.72265 0.10275 8.95617 39 41.1448 0.02430 401.4478 0.00249 9.75696 0.10249 9.02852 40 45.2593 0.02209 442.5926 0.00226 9.77905 0.10226 9.09623 41 49.7852 0.02009 487.8518 0.00205 9.79914 0.10205 9.15958 42 54.7637 0.01826 537.6370 0.00186 9.81740 0.10186 9.21880 43 60.2401 0.01660 592.4007 0.00169 9.83400 0.10169 9.27414 44 66.2641 0.01509 652.6408 0.00153 9.84909 0.10153 9.32582 45 72.8905 0.01372 718.9048 0.00139 9.86281 0.10139 9.37405 46 80.1795 0.01247 791.7953 0.00126 9.87528 0.10126 9.41904 47 88.1975 0.01134 871.8749 0.00115 9.88662 0.10115 9.46099 48 96.0172 0.01031 960.1723 0.00104 9.89693 0.10104 9.50009 49 106.7190 0.00937 1,057.1896 0.00095 9.90630 0.10095 9.53651 50 117.3909 0.00852 1,163.9085 0.00086 9.91481 0.10086 9.57041 55 189.0591 0.00529 1,880.5914 0.00053 9.94711 0.10053 9.70754 60 304.4816 0.00328 3,034.8164 0.00033 9.96716 0.10033 9.80229 65 490.3707 0.00204 4,893.7073 0.00020 9.97961 0.10020 9.86718 70 789.7470 0.00127 7,887.4696 0.00013 9.98734 0.10013 9.91125 75 1,271.8954 0.00079 12,708.9537 0.00008 9.99214 0.10008 9.94099 80 2,048.4002 0.00049 20,474.0021 0.00005 9.99512 0.10005 9.96093 85 3,298.9690 0.00030 32,979.6903 0.00003 9.99697 0.10003 9.97423 90 5,313.0226 0.00019 53,120.2261 0.00002 9.99812 0.10002 9.98306 95 8,556.6760 0.00012 85,556.7605 0.00001 9.99883 0.10001 9.98890 100 13,780.6123 0.00007 137,796.1234 0.00001 9.99927 0.10001 9.99274 Copyright © 2000 Marcel Dekker, Inc. 1.3.11 Uniform Gradient Series Factor As previously discussed, a cash ¯ow series is not always uniform. Gradient series are frequently encoun- tered in engineering economics. Formulas for conver- sion factors of gradient series have likewise been developed. Speci®cally, a uniform gradient series can be expressed as a uniform series of cash ¯ows by A G 1 i n À in À 1 i1 i n À i ! AAjG; i; n12 1.3.12 Geometrical Gradient Present-Worth Factor Cash ¯ow series that increase or decrease by a constant percentage, g, with each succeeding period can be con- verted to a present amount by the geometric gradient present worth factor. P A 1 1 g 1 g* n À 1 g*1 g) n ! A 1 1 g PjA; g*; n13 where g* 1 i 1 g À 1 ! Example 6. A manufacturer has established a new pro- duction line at an existing facility. It has been estimated that the additional energy costs for the new production line are $5,000 for the ®rst year and will increase 3% for each subsequent year. The production line is expected to have a life span of 10 years. Given an annual compound interest rate of 5% what is the present worth of the energy costs for the new production line? First calculate the value of g* given that g 3% and i 5%: g* 1 i 1 g À 1 ! 1:05 1:03 À 1 0:0194175 then P A 1 1 g 1 g* n À 1 g*1 g* n ! 65000 1:03 1:10194175 10 À 1 0:01941751:0194175 10 45 P 643,738 1.3.13 Frequency of Compounding and Its Impact on Equivalence Compounding periods may assume a variety of dura- tions. Interest can be compounded annually, semi- annually, quarterly, monthly, daily, or continuously. Perhaps, the most common compounding period is annual. Similarly, the ¯ow of funds also can occur over a variety of periods. For example, the periodic payments on a loan may be monthly. Hence, there are three conditions that can occur, concerning the frequency of the compounding periods and the frequency of the periods for the cash ¯ow. First, the frequency of the compounding periods and that of the cash ¯ow are synchronized. Secondly, the compounding periods are shorter than the periods for the cash ¯ow. Third, the compounding periods are longer than the corresponding periods of the cash ¯ow. If the periods of the compounding and the ¯ow of funds are synchronized, the aforementioned conver- sion factors can be utilized to determine any equivalent cash ¯ow. When the compounding periods and the periods of the cash lows are not synchronized, then intermediate steps to synchronize the periods must be undertaken prior to utilizing the aforementioned con- version factors. Example 7. What is the present value of a series of annual payments of $90,000 over 10 years at the rate of 12% compounded monthly? Convert i 1%/month to an eective annual interest rate: i e 1 0:01 12 À 1 0:126825 P APjA; 0:126825; 10 690,000 1:126825 10 À 1 0:1268251:126825 10 23 6494,622 For the condition where the compounding periods are less frequent than the cash ¯ows, it should be noted that interest is not earned on funds that are not on deposit for the entire interest period. To synchronize the timing of the ¯ows of funds with the compounding periods, any cash receipt or disbursement is moved to the end of its respective time period. With the move- ment of cash receipts and disbursements to the end of the time periods, economic equivalence can be deter- mined with the use of the aforementioned conversion factors. Engineering Economy 837 Copyright © 2000 Marcel Dekker, Inc. 1.3.14 Amortized Loans The capital needed to ®nance engineering projects will not always be available through retained earnings. Indeed, money will often have to be borrowed. There are many types of loans that exist, but this chapter will focus upon the standard amortized loan. With an amortized loan, the loan is repaid through installments over time. The most prevalent amortized loan has monthly installments with interest that is compounded monthly. Also, the monthly installments are ®xed. Each install- ment consists of a portion that pays the interest on the loan and a portion that repays the outstanding bal- ance. With each succeeding installment, the interest portion will diminish, while the portion devoted to the repayment of the outstanding balance will increase. The magnitude of an installment payment is deter- mined through the use of the capital recovery conver- sion factor. In short, the payment, A, is found by A PAjP; i; n14 Noting that each installment payment consists of an interest portion and a remaining balance portion, the following notation is introduced: I j interest payment in period j Pr j principal payment in period j B j outstanding balance at end of period j The interest portion of any installment payment is simply the product of the outstanding balance times the prevailing interest rate: I j B jÀ1 i 15 The portion of the installment that may be applied to the outstanding balance: Pr j A ÀI j 16 Example 8. A consulting ®rm obtains a $10,000 loan to purchase a computer workstation. The terms of the loan are 12 months at a nominal rate of 12% compounded monthly. What is the monthly installment payment? How does the interest portion of the installment payment vary monthly? The installment payment is calculated by merely applying the capital recovery conversion factor, (AjP; i; n: A 610,000AjP; 17; 12610,0000:08885 from Eq: 14 6888:50 The interest portion of the ®rst installment would be I 1 B 1À1 i 610,0000:01 from Eq: 15 6100:00 Hence, the portion of the ®rst installment applied to the principle would be the dierence between A and I 1 : Pr 1 A ÀI 1 6888:50 À 100:00 from 16 6788:50 The new outstanding balance would be B 1 610; 000 À 788:50 69211:50 Through an iterative process, the values for I j and B j can be found for the remaining 11 months. Obviously, the iterative nature of this problem is ideal for a computer application. Installment Payment Principal Interest Balance no. ($) ($) ($) ($) 1 888.50 788.50 100.00 9211.50 2 888.50 796.39 92.11 8415.11 3 888.50 804.35 84.15 7610.76 4 888.50 812.40 76.10 6798.36 5 888.50 820.52 67.98 5977.34 6 888.50 828.73 59.77 5148.61 7 888.50 837.02 51.48 4311.59 8 888.50 845.39 43.11 3466.20 9 888.50 853.84 34.66 2612.36 10 888.50 862.38 26.12 1749.98 11 888.50 871.01 17.49 878.97 12 888.50 870.19 8.78 0.00 There are also formulas that enable one to determine the interest portion of any installment payment with- out having to engage an iterative solution: I j B jÀ1 i APjA; i; n À j 1i 17 The corresponding remaining balance after n Àj pay- ments may also be found via the following formula: B j APjA; i; n À j18 Likewise, the principal payment for a particular install- ment would be obtained by subtracting the interest 838 Huston Copyright © 2000 Marcel Dekker, Inc. [...]... Generation of NC tapes Packaging Chip turning Sheet metal Tool fabrication and maintenance Painting Plating Heat treating Rework Assembly Testing Wire cutting Troubleshooting Rework Encapsulation Inspection of purchased parts Inspection of fabricated parts Mechanical and electronic calibration Engineering support to manufacturing M-XXXXX D-XXXXX N-XXXXX K-XXXXX C-XXXXX S-XXXXX V-XXXXX P-XXXXX L-XXXXX H-XXXXX... K-XXXXX C-XXXXX S-XXXXX V-XXXXX P-XXXXX L-XXXXX H-XXXXX R-XXXXX A-XXXXX T-XXXXX W-XXXXX G-XXXXX R-XXXXX J-XXXXX B-XXXXX F-XXXXX O-XXXXX E-XXXXX 852 Malstrom and Collins The anticipated sequence of manufacturing operations The departmental location for each operation The required machines and equipment for each routing operation A brief description of each production operation The applicable shop order... solutions density, 49 4-4 96 concentration and temperature fit, 49 6-4 98 Algebra, 12 9-1 35 groups, 12 9-1 33 rings and fields, 13 3-1 35 Allen-Bradley’s programmable controller configuration system (PCCS), 195 Alligatoring defect, 569 Allocation of functions, 76 2-7 65 Allocation analysis flowchart, 759 Alloys, 54 5-5 49 corrosion, 56 1-5 64 precipitation, 54 6-5 49 wear, 56 1-5 64 Abelian groups, 12 9-1 30 Absolute stability,... freestanding buildings, 65 1 state of mind, 644 supplier, 65 5-6 56 technology classes, 64 9-6 52 technology shift, 64 6-6 48 Automation, 338, 485 impact manufacturing-cost recovery and estimating systems, 86 1-8 62 OSHA, 731 robotics, 50 0-5 03 Autoregressive (AR) model, 244 Autoregressive moving-average (ARMA) models, 244 Axiom of Dominance I, 36 8-3 69 B-spline, 426 Back-propagation of utility, 195 Backlash, 784... property 5-year 7-year 1 0- year 15-year 20-year 27.5-year 39-year Table 5 MACRS Recovery Allowance Percentages Recovery year 3-year class 5-year class 7-year class 1 0- year class 15-year class 20-year class 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 33.33% 44.45 14.81 7.41 20.00% 32.00 19.20 11.52 11.52 5.76 14.29% 24.49 7.49 12.49 8.93 8.92 8.93 4.46 10. 00% 18.00 14.40 11.52 9.22 7.37 6.55... time horizon After securing these after-tax cash ¯ows, then one can proceed to utilize any of the previously mentioned means of evaluating alternatives For example, an after-tax present-worth analysis is simply where the present-worth technique is applied to the after-tax cash ¯ows Similarly, an after-tax rate of return utilizes the rate -of return technique on after-tax cash ¯ows The following example... product of the taxable income and the tax rate The after-tax cash ¯ow is then the sum of the before-tax, the principal paid, the interest paid, and taxes paid [add columns (2), (3), (4), and (7)] The after-tax column then is analyzed for its present worth: PW 66 210 PjF; 107 Y 1 69277
PjF; 107 ; 2 I3 630,190
PjA; 87; 2
0:08 Á Á Á 628,000
PjF; 107 ; 10 630,190
1:7833
0:08 64307 PW 66, 210 0:9091... human interfaces, 74 9-7 89 Automatic brake monitoring, 727 Automatic guided vehicle systems (AGVS), 63 3-6 35 Automatic storage and retrieval systems (ASRS), 635, 639640, 64 3-6 57 computing cycle times, 65 3-6 55 design, 64 8-6 49 design philosophies, 646 dynamic buffers, 65 2-6 55 flow vs static costs, 655 historical development, 64 3-6 44 inventory, 64 5-6 46 microltote-load systems, 652 rack-supported vs freestanding... after-tax cash ¯ow analysis Example 17 With the purchase of a $100 ,000 computer system, a consulting ®rm estimated that it could receive an additional $40,000 in before-tax income The ®rm is in the 30% income tax bracket and expects an after-tax MARR of 10% If the funds for the computer are borrowed on a 4-year direct 8% reduction loan with equal annual payments, what is the present worth of the after-tax... volume, low mix types of production environments Organizations should expect to make an investment on the order of person-months to person-years in terms of statistician and cost engineering time to develop and validate such relationships prior to their being used 2.7.4 Development and Use of Estimating Software In many cases, software is developed to assist in expediting the completion of cost estimates . known as the recovery period: 3-year, 5-year, 7- year, 1 0- year, 15-year, 20-year, 27.5-year, and 39-year. The IRS has guidelines that determine into which clas- si®cation an asset should be placed power plant 27.5-year Residential rental property 39-year Nonresidential rental property Table 5 MACRS Recovery Allowance Percentages Recovery 3-year 5-year 7-year 1 0- year 15-year 20-year year class. 7 .108 48 24 9.8497 0 .101 53 88.4973 0.01130 8.98474 0.11130 7.28805 25 10. 8347 0.09230 98.3471 0. 0101 7 9.07704 0. 1101 7 7.45798 26 11.9182 0.08391 109 .1818 0.00916 9.16095 0 .109 16 7.61865 27 13. 1100
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