41.1 IEEE TRANSACTIONS ON ENGI>fEERING MANAGEMENT, VOL. 48. NO. 4, NOVEMBER 200
An ROI-Based Strategy for Implementation of Existing and Emerging Technologies in Helicopter Manufacturing Pradipta Sarkar and Christos Kassapoglou
Abstract—\n aircraft structure has several part families. These part families are currently manufactured using a baseline tech¬ nology. There are several other competing technologies, already existmg or still in the development stages, that could be applied to the structure under consideration. Each technology has a probability distribution of costs and savings (over baseline) associated with i t The problem is to determine which of the candidate technologies should be used and to what extent, to manufacture different part families. In this paper, we have obtained the optimum time-dependent technology mix over a finite planning horizon, such that the ROI is maximized. This has been done subject to the constraints that the risk resulting from investment in a combination of technologies, does not exceed, in any yppr. yom? predefined quantity, and that the quantity produced using a particular technology does not exceed its applicability in any year.
family. Unlike the process development cost, the product development cost is incurred every year (until the technology is fully implemented), irrespective of whether the technology is fully developed.
The amount ofthe structure to which each technology can be applied is termed as applicability and will vary from zero to a maximum value which is less than or equal to one (that is 100%). The applicability of each technology will be a function o f time as the amount of parts in the structure that can be manufactured using the technology will increase as the technology matures. Oiice production is leauy, each lecimuiu^^ wm ic^uii m a certain amount of (recurring) cost savings (over the baseline mean) for the structure. The recurring cost covers recurring labor inIndex Terms—t^tt cashfiow,random search and steepest ascent cluding assembly but, for this study, not including raw matemethod, time dependent technology mix. rials. This savings in recurring cost varies with die types of parts to which the technology is applied and the level of technology maturity. For this reason, the savings is assumed to have a probI . INTRODUCTION ability distribution. For a technology where limited data on poN CERTAIN manufacturing industries, such as aircraft and tential cost savings are available, (such as technologies that are automobile manufacturing, the following situation is often still underdevelopment), tiie dispersion of .savings distribuencountered: A certain structure is manufactured using a set of tion will be greater than for the ones that are already developed (baseline) technologies. A t the same time, there are several other and estabhshed. technologies, already existing or still in the development stage, The static problem, where tiie optimum combination o f techthat could be applied to the structure under consideration. nologies is selected for minimum recurring cost given a level of Each technology has a certain investment associated with risk, can be viewed under certain assumptions as a variation of it. This investment is broken into process development cost the knapsack problem or a portfolio management problem. A (to make the technology production-ready) and product develvery good survey o f the knapsack problem was given by Salkin opment cost (to set up the factory to fabricate parts with the and DeKluyver [lOJ. Lorie and Savage [9] presented a capital new technology). The process development cost is a nonrecurbudgeting problem where investment possibilities are considring cost, that includes equipment, training, and technology ered over a cert.ain period witii limited cash outlays over the pedevelopment cost. Usually, it is spent in the first few years of riod. Extensions of this problem and use of integer programming the new technology introducdon. Once the technology is fully were discussed by Weingartner [ 12], [ 13]. developed, there is no process development cost. The product A case where the returns are normally distributed (as in our deveiopment cost is also a nonrecurring cost which includes problem) but with linear constraints, was treated using dynamic drawings, tooling design and fabrication, analysis, testing, programming by Carraway et al. [2]. A similar dynamic proreport writing, and certification. The product development gramming approach was discussed by Steinberg and Parks [11]. cost has been shown by several researchers (e.g., [3], [4]) to A combination o f dynamic programming with a search procebe proportional to the weight of the structure for each part dure to solve tiie same problem with stochastic returns and linear constraints and address risk, was presented by Henig [5].
I
Manuscript received September 20, 1999; revised February 15, 2001. Review of this manuscript was arranged by Department Editor B. V. Dean. This work was supported in part by the U.S. rotorcraft industiy and govemment under RnVVNASA Cooperative Agreement no. NCCW-0076. P. Sarkar is with the United Technologies Research Center, E . Hartford, C T 06108 U S A (e-mail:
[email protected]). C. Kassapoglou is with Sikorsky Aircraft, Stratford, C T 06614 USA Publisher Item Identifier S 0018-9391(01)10259-X.
A detailed discussion of tiie risk associated witii portfolio management and optimization was provided by Bielecki and Pliska [1], where risk-sensitive control theory was used to determine optimal trading strategies. In our analysis, tiie variance ofthe returns is considered to be a measure of risk. This introduces tiie nonhnear constraint tiiat tiie variance of the overall retum should be below a target value.
415
.SARKAR AND KASSAPOGLOU: STRATEGY FOR IMPLEMENTATION TECHNOLOGIES IN HELICOPTER MANUFACTURING
The use of nonlinear constraints with continuous variables complicates the problem significantly. The present work was motivated by the solution to the static problem by Kassapoglou [6], [7], where the optimum mix of technologies was determined for a helicopter fuselage such that the cost savings is maximized, and the variance of the savings is kept below a target value. In that work, the cost savings associated with applying a certain technology to a part family was assumed to be normally distributed. Depending on the probability level selected (probability that the cost savings will be below a certain value) different optimum technology mixes resulted. An extension of the stadc problem to a dynamic situation was presented by Kassapoglou [8]. The target technology mix in that work was the optimum mix determined in [7]. Two alternative strategies for getting to the target mix as a function of time were discussed. In the first, the technology implementatfon over time takes place in such a way so as to minimize the investment required. In the second, the technology implementadon is done .so that the cost savings per yca. ^.^ liiaAnmiuu indep^..^^... ^. the investement required. A variety of intermediate scenarios switching between the two extreme cases was also examined using a stagecoach problem approach. It became clear from the work in [8] that the tradeoff between minimum investment and maximum savings is quite significant and the two should somehow be combined. Based on this situadon, th^ ; "mhiem posed is to develop a technology implementa;;.-:-; :;:,.iU'L'} liiat will determine when, and to what extent, each technology should be introduced in producdon. The decision criterion used here is the retum on investment (ROI). In this paper, we have developed a nonlinear optimization methodology to obtain the opdmum time-dependent technology mix over a finite planning horizon, so that the mean value of the ROI ; =-.;2ximize.d. This is done subject to the constraint that the risk (as measured by the variance of the cost savings population) that results from the investment in a combination of technologies does not exceed, in any year, some predefined quantity. II.
A. ROI Let be the net cash flow (assuming that the cost is incurred at the end of each year) in year k, for k = 1,... ,H. Let the interest rate or the depreciation of money be 100r% per year. Then the net present value of the cash flow is H
1
+
r
k-=l
.F^. = ^ T
[ Savings using technology j
E
-
= 0
ROI = m i n { r G ( - 1 , oo) : r is a real root of (1)} .
-
lOjfc -r {^ujjyy
B. Formulation
of the Problem
Suppose that there are T technologies in consideration, including the existing and the developing ones. Let Cjk be the process development cost for tiie j t i i technology in the fctii year, {Ch)j be the product development cost (per pound) for the j t h technology and lOOS'j be the percentage savings in recurring cost (over the mean of the baseline technology) i f technology j is used instead of the baseline. Let W and B be the weight and cost of the structure. I f Xjk is the proportion of the structure produced by the j t h technology in the year k and Uk is the number of units produced in the year k (eg., the number of aircrafts produced in the year /c), then the net cash flow in the year k is obtained as the equation at the bottom of the page, where j ' is the baseline technology and i f x j k > Xj • J otherwise if k = I and j = j' /(A^ = l a n d j = j ' ) = {J otherwise and > .Tj.A-l)
=0 for all j = 1 . 2
T.
The ROI r , for the technology mix A.. J = 1.2 T: k = 1.2, H} is obtained by equating the present value ofthe net cash flow to 0, i.e., by solving the following equation for T: See (3) at the bottom of the following page.
Cost of using technology j ]
[Xjk - Xj^k-i)i
(2)
There are two reasons to restrict our attention to the values of T that are greater tiian - 1 . First, technology mixes with ROI r < - 1 are clearly not desirable. Technology mixes with ROI - 1 < r < 0 are allowed as possibilities in intermediate years since, over tiie entire planning horizon, tiiey may yield r > 0. Second, tiie left hand side of (1) has a sigularity at r = - \ . .. ... ^ .aiiuuiji vaiiaoies ^as uic) uic in our example, as we will see later) replace J^ts by the expected values of ^A.S i n (1), and solve for tiie ROI.
r
SjBxjk
(1)
has - 1 roots, some of which may be complex. We will be concerned with only the real roots. In addition, we define the ROI to be the smallest real root r > - 1 of (1)
H^jk
FRAMEWORK
E
The ROI is a value of r for which tiie net present value of the cash flow is 0. Now, the equation
(Xjk > Xj.k-i){i nk
- J (I: = i anu j = j ))i
416
IEEE TRANSACTIONS ON ENGINEERING MANAGEMENT, VOL. 48. NO. 4. NOVEMBER 2001
Typically, the net cash üows are negative in the first few years and positive in all subsequent years. In such situations, there is at most one root r , of (3) that is > - 1 (See Result 1 in Appendix). In cases where there are two or more roots that are > - 1 , we will consider only the minimum one.
TABLE
Part Family Siiins and covers Frames and bulkheads Stringers Fittings Decks and floors Doors and fairings
In reality, the savings in recurring costs, the Sj's in (3), are random variables. Therefore, to obtain the ROI Sj's will be replaced by their respective expected values. Suppose p.{Sj) is the expectation and crj is the variance ofthe probability distribution of Sj. Then, we are interested in the following problem. Determine the optimum technology mix { x j k j - l,2,...,T;k =
1) 2) 3) 4) 5) 6) 7)
1,2,...,H < •• • < ajH
(5)
1) 2) 3) 4) 5) 6) 7)
< 1
fori=l,2,...,T J^Xjk
N/A -N/A
N/A
N/A
N/A N/A N/A
N/A N/A
N/A
skins and covers; frames and bulkheads; stringers; fittings; decks and floors; doors and fairings; miscellaneous, including seals, transparencies, etc.
Out ofthese seven part families, we will consider manufacturing of the first six families. There are seven candidate fabrication processes for tiiese part families:
(6)
T
i=i fork
N/A N/A
A helicopter fuselage can be divided into the following seven part families;
< fljfc < 1
< aj2
ALP
F U S E L A G E : D E H N m o N O F T H E PROBLEM
Mathematically, the constraints are represented as
0 HLP HSM AFP R I M PLT
i n . IMPLEMENTATION O F TECHNOLOGIES IN H E L I C O P T E R
1) all .7:^/, "s are nonnegative and do not exceed the respective applicabilities; 2) the sum of Xjk's over any year is I ; 3) the risk (as measured by the standard deviadon) associated with the savings for each unit using the technology mix in any year does not exceed a prespecified percent of the cost of the structure.
foTj =l,2,...,T;k^
S.MT
K/A means 'not applicable'.
1-2 H}. that maximizes the minimum real root r > - 1 , of (4) (see equation at the bottom of the page), subject to the constraints that
0 2
of the six part families is less than 1. This is because fuselage parts such as seals, transparencies, etc, that form the miscellaneous part family are not included in the study. For the same reason, the sum of the cost fractions is also less than 1. Our objective is to find the optimum technology mix for each of the six part families. To do diis, we need to know the process development costs for different part families. Typically, in the helicopter industry, the company is committed to the development of a technology only i f die ROI resuhing from recurring cost savings brought about by the technology exceeds a certain hurdle (h)—known as the "hurdle test." This hurdle is set a priori widi little consideration of the overall picture, i.e., when it comes to process development decisions, an individual technology is not traded against other technologies. This is done in order to create a "bag of technologies" that can be used in die future. It may happen that, for the present applicadon, some of
the technologies have little use for a particular part family. However, the technology will still be developed for that part family (assuming it is applicable) because for other helicopter types or models which will be developed in future, these technologies have much higher use. I f the company waits to develop the technologies until the launch of those other types/models, the time to market will increase prohibitively (In this connection, we should keep in mind that often the development o f a particular type of helicopter takes a decade or more.) Therefore, the company goes forward with developing all die feasible technologies (satisfying the hurdle test) even for part families for which it is not going to be used (or to be used only marginally) in the present application. Note that all tiie seven technclogies we are discussing here have passed the hurdle test. The process development cost is the cost required to determine how to efficiently fabricate features (geometry and shape)
418 T R A N S A C T I O N S O N E N G I N E E R I N G M A N A G E M E N T , V O L . 48, N O . 4. N O V E M B E R 2001
T A B L E IV ' " M E A N S A N D V A R I A N C E S O F SAVINGS (PROPORTION) IN R E C U R R I N G C O S T S
Part Family
HLP
Skins and covers
SA'IT 0
0.0063
Frames and bulkheads
0.0001 0
0.0001
Stringers
0.0001 0
Fittings
0.0001 0
0.20 0.10 -0.05 0.0001 N/A
0.0001
N/A 0.20
N/A
ALP 0.32 0.0100 0.40 0.0800
0.05
0.40
0.0020
0.0010
0.40 0.0900
N/A
N/A
N/A
N/A
0.15 0.0100
0.15
N/A
0.0080
0.20 0.0200
0.005
Decks and floors
0
-0.01 0.0001
Doors and fairings
0.0001 0 0.0001
Manufacturing Technology HSM AFP RTM PLT 0.20 0.25 0.08 0.08 0.0200 0.0090 0.0030 0.0600 0.28 0.10 0.18 N/A 0.006O 0.0600 0.0080
0.10 0.0021
N/A N/A
0.25 0.026
-0.10 1 0.0100
N/A
0.35 0.0500
In each cell the first number is the mean and the second number is the variance N / . A means 'not applicable'.
TABLE V
cost matrix in Table V I . In addition, the process development costs are incurred in the first few years of operation. In diis example, we assume that the money for p f - : d c - , : I o p m e n t IQ spent lineariy over the first thiee years of operation. Therefore, for example, die amount of money spent for the process development of H S M for skins and covers in each of the first three years = $471 590.91/3 = $157196.97. The process development cost is 0 for every subsequent year. In this applicadon, we assume that the number of aircrafts manufactured in year k
W E I G H T AND C O S T FRACTIONS FOR T H E P A R T FAMILIES
Part Family
Weight Fractions
iikins and covers
0.249
0.109
Frames and bulkheads
0.236
0.181
Cost Fractions
Stringers
0.016
0.029
Fittings
0.043
0.091
Decks and floors
0.096
0.059
Doors and fairings
0.180
0.369
Total
0.820
0.838
Within a part family with a certain process. Since a part family has features of roughly the same complexity, the process development cost for a specific part family is proportional to the number of different features to be developed. Now, the weight of a part family is equal to the sum ofthe weights (per unit size) of different features times their size dmes die number of such repeating features. Therefore, the process development cost is approximately proportional to the part family weight. This is why the process development cost of a pai ucuiar leciiiiuiogy has been distributed among different part famdies (to which it is applicable) according to the weights o f these part families Here is one illustration. The process development cost for HSM IS $1 000000 (See Table III). HSM can be used to make skins and covers, frames and bulkheads, and fitdngs. Therefore, the S1 000 000 process development cost must be distributed among the three part families in die proportion o f the weights of the respective part families. Skins and covers account for 24.9% of die entire helicopter weight. Similarly frames and bulkheads account for 23.6% and fittings account for 4.3% o f the endre helicopter weight. Thus, the process development cost of HSM is distributed among the three part families as follows: 1) skins and covers = $1,000,000 x 0.249/(0.249-f0 236-|0.043) = $471590.91, 2) frames and bulkheads^ $1,000,000 x 0.236/(0 249 + 0.236 + Ü.Ü43) = $446 969.70, 3) fittings = $1,000.000x0.043/(0.249-f0.236-f0 043) = $81439.39. The other process development costs are distributed in a similar fashion. Wc present the part family versus process developnient
( 80 nk = l5
In this context, we would like to clarify that in our helicopter example, the industry pracdce is to develop the technologies even for part families where it wdl be only marginally used, or not used at all, at the present time (provided they pa.ss the hurdle test). However, our problem formuladon is general enough and can be applied to other industrial applications as well (for example automobile), where a process will be developed only if it is of immediat&use. In addidon, the process development costs need not be proportional to weight ofthe component. To run the algorithm, one just needs to specify die process development cost for each component/part family sepaiaicl>'. In such other industrial applications, i f it turns out that the opdmum strategy recommends that a particular technology should not be used at all (for a part family), then just drop that technology from consideration and rerun the code without that particular technology. This will obtain the revised ROI. In next two sections, we will describe the algorithm and the results for the implementadon of technologies in helicopter fuselage.
IV. A L G O R I T H M
In this secdon, we will develop the algorithm using the helicopter fuselage as an example. The approach is applicable, with uiinor ciiaiiges, to ouier similar problems as w e l l
419
SARKAR AND KASSAPOGLOU: STRATEGY FOR IMPLEMENTATION TECHNOLOGIES IN HELICOPTER MANUFACTURING
TABLE VI PROCESS D E V E L O P M E N T C O S T FOR D I F F E R E N T T E C H N O L O G I E S IN D I F F E R E N T PART F A M I L I E S
Manufacturing Technology RTM AFP
P a r t Family
SMT
HLP
HSM
0
471590.91
818002.63
320463.32
446969.70
PLT
ALP
469811.32
480694.98
Skins and covers
0
Frames and bulkheads
0
0
775295.66
303732.30
N/A
455598.46
Stringers
0
0
N/A
N/A
20592.02
30188.68
30888.03
81439.39
N/A
N/A
N/A
N/A
Fittings
0
N/A
Decks and floors
0
0
N/A
315374.51
123552.12
N/A
185328.19
Doors and fairings
0
0
N/A
591327.20
231660.23
N/A
347490.35
N / A means 'not applicable'.
Obtaining the technology mix that maximizes the ROL as defined by (2) (with the net cash flows replaced by their respective expected values), for a moderate number of competing technologies (six or seven) and a moderate nuniber of years (five, six, or ten) is quite involved. The opfimization techniques, such as steepest ascent or Newton's method, that use die first and second order d e r i v > of the response surface are not useful because we have no informadon about die regularity properdes of the surface. Such algorithms are not very good for solving equations witii multiple roots. A pure grid search algorithm tiiat does die complete enumeration is computationally very intensive and requires operations that are at least in the order of where G is the number of mesh points for eaci .r, ,. ;v r en witii 20 grid points for each decision variabiv :, i c s . liogies, and a ten-year plan, 1.15 x lO'^^ calculations are needed. In our approach, we first eliminate some of tiie technology mixes by careful observations and economic arguments and, thus, reduce die sample space. Then we use a mix of steepest ascent and random search techniques on the reduced sample space to obtain the maximum ROI and the corresponding lechnology mix. A. Some
Observations
The following are a few simple observations that will be used to develop die algorithm. Note 1: The net cash flow is usually nonpositive in the first year because 1) There are no process and product development costs for tiie baseline technology, as it is already fully developed. Therefore, i f the whole structure is made of die baseline technology alone and no other technology is available, no process and product development costs are involved. Also, the mean savings in recurring costs is 0, as it is measured with respect to the baseline technology. Thus, tiie net cash flow is 0. 2) I f parts are made using other technologies, some process and product development costs are incurred the sum ofwhich is usually more than the savings in recurring costs (unless the production volume is large enough to make up for the process and product development costs). Our algorithm assumes that the net cash flow is nonpositive in the first year, which is true for most of the manufactruring technology implementation programs. In our example, we will also show what to do when it is indeed possible to obtain positive net cash flow even from year one. Note 2: For the optimum strategy, die net cash flow in tiie last year, TH can never be negative [See Result 2(a) in the A p pendix].
Note 3: I f the net cash flow in the first year is nonpositive, then there can be only odd number of roots, r, of (4) that are greater than - 1 [See Result 2(b) in the Appendix]. Note 4: For die purpose of finding the opdmum strategy, it is sufficient to do search only on those technology mixes for which die savings in recurring costs increase from a year to next. This means that as we move from year to + 1, tiie nonrecurring cost and the net casn fiow may decrease, but the savings i n recuning costs must increase [See Result 2(d) in tiie Appendix]. Note 5: I f there is a technology with a negative mean savings over baseline (tiiat is, it actually costs more on a recurrent cost basis to switch to this strategy from the baseline), it will automatically drop out o f the optimum technology mix. B. Generating a Random Technology Mix The steps for generating a random technology mix { x j k j = 1, T, fc = 1, i / } are summarized below. 1.
Rearranging the technologies: Arrange t h e t e c h n o l o g i e s i n d e c r e a s i n g o r d e r o f mean c o s t s a v i n g s . T h e r e f o r e , a f t e r t h e r e a r r a n g e m e n t we h a v e M ^ i ) > f^iS2) > > •• > PiSr). 2 . S e t fc = 1 . 3. Maximum and minimum bounds o f t h e v a r i a n c e s of the t e c h n o l o g y mixes: Maximum p o s s i b l e v a l u e o f t h e v a r i a n c e o f the technology
mix i n y e a r fc
=MaxVfc
=
S S [
'^(T-j)i"T-j.L-)-
J=0
VI
f
i s t h e .sth o r d e r
where of
[ ( T ) , ^
=
1.2
T], al,
statistics i s the c o r r e -
s p o n d i n g v a l u e of a p p l i c a b i l i t y is
such
that
^J^(i"%.f
a n d T*
1 < 1 < I^j=o'^r-j.fc •
Minimum p o s s i b l e v a l u e o f t h e v a r i a n c e o f
420
IEEE TRANSACTIONS ON ENGINEERING MANAGEMENT, VOL. 48, NO. 4, NOVEMBER 2001
the technology
mix i n y e a r k
=MinVfc
2\
T.-l
1
-E
4 .
j=0
where i s s u c h t h a t E J = I ^ «i.A; < 1 < X ^ J l j aj^fc. I f f o r a n y y e a r k,a^ < MinV^, then t h e v a r i a n c e v a l u e i s n o t a c h i e v a b l e a n d we q u i t t h e program. I f f o r any y e a r k,a^ > MaxVjt, then s u c h a h i g h v a r i a n c e i s n o t r e a c h a b l e a n d we s e t Q = yMaxVi . 4.
F o r i = 1 , 2 , . . . , T - 1 , do t h e f o l l o w i n g . (a) Compute Lj^k and Uj^k- S e e equat i o n a t t h e bottom o f the page. w i t h t h e u n d e r s t a n d i n g t h a t x^^ i s 0. (b) I f Lj^k>Uj^ki go t o s t e p 6. (c) I f Lj^k < Uj^k, draw Xj^k, from U n i f o n n (Lj^k,Uj,k) •
ROI [refer to the definition in (2)]. Denote these ROIs by {ROh,t = 1,2,...,2H{T-1)} 3) Fixane > 0. I f max{ROI«, i = 1 , 2 , . . . , 2 i ï " ( r - 1)} > ROIo -I- e, repeat step 2 above with the technology mix corresponding to the max{ROIt, t = 1 , 2 , . . - , 2H{T 1)} as the pivot mix. D. New Runs: Repeat Until
Convergence
Choose a different random technology mix and then repeat the procedures for searching in a neighborhood of technology mix as described in die previous two sections. Check i f the ROI has improved or not. I f the ROI has increased, keep the most recent ROI and the corresponding technology mix, odienvise discard it. Continue the whole process until the ROI value stabilizes (over a large number of runs). V. IMPLEMENTATION O F T E C H N O L O G I E S FUSELAGE:
A. Optimum Time-Dependent
IN HELICOPTER
RESULTS
Technology
Mix
We have assumed that all cost and savings distributions are normal. In this section, we will discuss in detail the results for a planning horizon of ten years. The maximum allowable risk is set to a = 0.0855. This value was selected because it corresponds to the target variance value for tiie entire fuselage that maximizes the recurring savings with a 1 % probability of the 5. XT,k = 1 Xj,kI f XT,k < aT,k, savings being lower [7]. This also assumes that all part families t h e n go t o s t e p 7 ; o t h e r w i s e go b a c k have the same variance of the cost savings population. to s t e p 4 . We have used the algoritiim developed in Section I V to obtain 6. I f A;>1, t h e n Xj^k=Xj,k-i, for j=l,...,T. the optimum technology mixes for skins and covers, frames and I f k=l,Lj^k c a n n e v e r e x c e e d C/^-^fc. bulkheads, decks and floors, and doors and fairings. We perform 7. I f k - L Proof: There is only one change of signs in tiie coefficie: sequence {Ti ,...,T}i}. Therefore, by Descartes' rule of sig there is almost one positive value ofv = r) that satisfi( (1). The result follows by noticing that r = - I. Result 2: Let {xjkJ = 1 , . . . , T; fc = 1 i ï } be a tecl nology mix and let Tk be die net (expected) cash f l o w in th year fc, for fc = 1 , . . . , i f . Assume tiiat all the process develop ment cost occurs in first H' year, where H' < H. a) If {xjkyj = l , . . . , T ; f c = 1,...,H} is tiie optimur strategy, tiie net (expected) cash flow in die last year, Ti can never be negative. b) If J^i, die net (expected) cash flow in die first year is nonpos itive and i f {xjkJ = 1 , . . . , T; fc = 1 , . . . , i f } is die op timum technology mix, tiien there can be only odd numbe of roots r, (if at all any) of (4) tiiat are greater than - 1 . ' c) Let Tl be nonpositive. Then, i f some J^j-s increase whilf tiie others remain the same, tiie ROI increases. d) Let {xjkJ = 1 , . . . , T; fc = 1 , . . . , i T } be die optimun: technology mix and let pl be the (expected) savings ic recurring cost in year k, then pl < pl.^, for ~ 1,2,...,^-L
AND KASSAPOGLOU: STRATEGY FORIMELEMENTATION TECHNOLOGIES IN HELICOPTER MANUFACTURING
Proof: b)
REFERENCES
is negative, then find the minimum value o f s, such that TH-S is non negative (The existence of s is shown later). Define J.'. = 3^
[xj^^n-,)^ \ Xjk,
if
k > H - s
Ot otherwise
Then, the technology mix {x'j^,j = l,...,T;k = 1,...,H} dominates die technology mix {xjk,\j = l,...,T;k 1,...,H} because die yearly net cashflow widi die first technology' mix is always greater or equal to that of die later one. Hence, {xjk,j - l,...,T;k = 1 , . . . , . ^ } is not die optimum technology mix. Note that the existence of s is guaranteed by the assumption tiiat H' < H. Because for all A; > l^define if j is baseline otherwise
'HI:
Then, tiie technology mix {a;?j.,j = l , . . . , T ; / c = l,...,H} will produce nonnegative net (expected) cash flow in year H' + 1. c) From part (a) and die assumption tiiat Ti is nonpositive, tiiere are odd number of sign changes in the coefB c i e n t s e ^ e n c e j ^ ^ l ^ . ^ , TH J^- TTie result follows using Descane;;' ' u i . nf sign. d) We rewnie (1) as
[!] T. R. Bielecld and S. R. Pliska, "Risk-sensitive dynamic asset management," in Applied Mmliemalics Optimization, 1999, vol. 39, pp. 337-360. [2] R. L . Carraway, R. L . Schmidt, and L . R. Weatherford, "An algorithm for maximizing target achievement in the stochastic knapsack problem with normal returns," Nav. Res. Logisl., vol. 40, pp. 161-173, 1993. [3] T. G. Gutowski, E . - T . Neoh, and G. Dillon, "Design scaling laws for advanced composites fabrication cost," in Pwc. 5th NASA/DoD Advanced Composites Technology Conf., Seattle, WA, 1994, pp. 205-236. [4] T. Gutowski, D. Hoult, G. Dillon, E.-T. Neoh, S. Muter, E . Kim, and M. Tse, "Development of a theoretical cost model for advanceii composites fabrication," Composites Manuf., vol. 5, no. 4, pp. 231-239, 1994. M. I. Henig, "Risk criteria in a stochastic knapsack problem," in Oper. Res., 1990, vol. 38, pp. 820-825. [6] C. Kassapoglou, "Selection of manufacturing technologies for fuselage structures for minimum cu^l and low risk: Part A—Problem formulation," J. Composites Technol. Res., vol. 21, no. 4, pp. 183-188, Oct. 1999. [7] "Selection of manufacturing technologies for fuselage sünctures for minimum cost and low risk: Part B—Solution and results," J. Composites Technol. Res., vol. 21, no. 4, pp. 189-196, Oct. 1999. [8] "Determination of the optimum implementation plan for manufacturing technologies—The case of a helicopter fuselage." / Manuf Syst., vol. 19, nu. 2, jjp. i ^ i - u j . _uuu. [9] J. Lorie and L. J. Savage, "Three problems in capital rationing," J. Bus., 1955. [10] H. M. Salkin and C. A . DeKluyver, "The knapsack problem: A survey," Mav. Res. Logistics Quart., vol. 22, pp. 127-144, 1975. [11] E . Steinberg and M. S. Parks, "A preference order dynamic program for a knapsack problem with stochastic rewards," 7. Oper. Res. Soc, vol. 30, pp. 141-147, 1979. [12] H. M. Weinganner, "Capital budgeting and interrleated projects: Survey and_synthesisJ'JWa«age,J'c;.,j'ol.J2r4)p. 485-546,-1-968 [13] , Mathematical Programming and the Analysis of Capital Budgeting Problems. Englewood Cliffs, NJ: Prentice-Hall, 1963.
H
0
(11)
fc=2
The left hand si de of (11) is the present value of the net (expected) cash flow. As r -+ oo,tiie]efithandsideof(n)goes to Tl, since each term under the summation sign goes to 0. Therefore, as r —> oo, tiie net (expected) cash flow goes to a nonpositive value. There are oitiy odd number o f roots of (11) that are greater tiian — 1. Hence, at the minimum of these roots (which is defined as the R O I ) , die present value of net (expected) cash flow intersects the r axis from above. Therefore, if one oi the Tk s is increased while the others remain die same, the intersectionpoint shifts toward right and therefore the R O I increases, e) Let T
lil = ^ p { S j ) B x j y , k = \,...,H.
(12)
Suppose > Mfl- Consider tiie technology ™x {yjx,3 = l,-.-,T;k = 1,...,H}. where yj,k = Xj,k, for j = l,...,T;k = 1,...,H-1 and yj^H = Xj^n-i for j = l , . . . , r . Note tiiat {Vj.k-J = l,...,T;k = 1 , . . . , i f } will produce a higher R O I tiian {xjy,j = l,...,T;k = 1,...,H} and hence {xj^k ;j = l,.--,T;k= 1 , . . . , J7} is not an optimum technology mix. Therefore, M / f - i < PH- The same argument may be repe li*l
