The Effect of Technological Improvement on Capacity Expansion for Uncertain Exponential Demand with Lead Times

Transcription

1 The Effect of Techological Improvemet o Capacity Expasio for Ucertai Expoetial Demad with Lead Times Dohyu Pak Departmet of Idustrial & Operatios Egieerig 205 Beal Aveue The Uiversity of Michiga A Arbor, MI 4809 Nattapol Porsaluwat Departmet of Idustrial & Maufacturig Systems Egieerig 209 Black Egieerig Buildig Iowa State Uiversity Ames, IA Sarah M. Rya* Departmet of Idustrial & Maufacturig Systems Egieerig 209 Black Egieerig Buildig Iowa State Uiversity Ames, IA October 2002 *Correspodig Author: smrya@iastate.edu voice: fax:

2 The Effect of Techological Improvemet o Capacity Expasio for Ucertai Expoetial Demad with Lead Times Abstract We formulate a model of capacity expasio that is relevat to a service provider for whom the cost of capacity shortages would be cosiderable but difficult to quatify exactly. Due to demad ucertaity ad a lead time for addig capacity, ot all shortages are avoidable. I additio, techological iovatios will reduce the cost of addig capacity but may ot be completely predictable. Aalytical expressios for the ifiite horizo expasio cost ad shortages are optimized umerically. Sesitivity aalyses allow us to determie the impact of techological chage o the optimal timig ad sizes of capacity expasios to accout for ecoomies of scale, the time value of moey ad pealties for isufficiet capacity.

3 . Itroductio Capacity expasio is the process of addig ew facilities of similar types over time to meet a risig demad for their services. Plaig for the expasio of capacity is of vital importace i may applicatios withi the private ad public sectors. Examples ca be foud i heavy process idustries, commuicatio etworks, electrical power service, ad water resource systems. Capacity expasio plaig cosists of determiig future expasio times, sizes, locatios ad types of facilities i the face of ucertai demad forecasts, costs, ad completio times. I may ew techology idustries, demad for capacity grows accordig to a expoetial tred. For example, recet work by Dumortier [5] predicted the expoetial growth i the umber of Iteret users ad the ed-user multimedia applicatios. Rai et al. [2] used the umber of service hosts as the measure of the Iteret size ad suggested that a expoetial model provided the closest fit with the icreasig umber of service hosts. Kruger [2] predicted the sigificat growth of electric cosumptio due to substitutio of hydroge for fossil fuels i motor vehicles. The major cocer is the magitude of additioal electricity power capacity ecessary to build a largescale hydroge fuel idustry, especially i a state such as Califoria with large umber of vehicles. The effect of a costructio lead time for addig ew capacity is also a importat issue i capacity expasio problems. If a lead time for addig capacity exists, the capacity expasio problem is more complicated because ucertai demad creates the risk of potetially costly shortages durig the costructio period. If there were o lead times for addig ew capacity, despite the ucertaity of demad there would be o risk of capacity shortage, sice the maager could simply wait util demad equals curret capacity ad the istall ew capacity istataeously. Techological progress is a importat factor to be cosidered i capacity expasio problems. Traditioally, improvemets i techology are measured either i terms of icreased reveue associated with the ew techology or decreased costs of procuremet ad operatio of the ew techology. I may idustries, such as commercial satellite commuicatios or computer cetral processig uits, itroductio of the ew improved techology causes icreases i product efficiecy ad rapid decreases i uit costs, thus affectig the expasio plaig decisio. Techological chage also elarges markets idirectly through improved productivity. 3

4 Productivity improvemets reduce productio costs. Fallig costs eable price reductios ad expad the customer base ad thus the market. Several examples show how techological progress could ifluece the cost of expasio. Sow observed that the per-uit capacity cost of satellite commuicatio INTELSAT was decreased sigificatly due solely to techological progress [27]. Newer, improved techology of the satellite compoet expaded voice chael capacity. Durig the 2 years of cosideratio, the voice chael capacity had bee icreased from 480 to 25000, which is more tha 50 times, while the capital cost per satellite icreased oly 3.77 times from 6.5 to 62.3 millio dollars. This ratio of chael icrease to capital cost icrease clearly shows how techological improvemet could affect the cost of expasio. Moore s Law, which stated that computer CPU speed would be doubled every 8 moths, is aother distictive example of how techological progress affects the cost of expasio i iformatio techology idustries. The improvemet i techology implied by Moore s Law ehaces productivity, while simultaeously causig older techology to become obsolete ad prices to drop regularly. With this techological progress, a maager ca choose to purchase the latest techology at the highest price, or purchase the older techology for a lower price. Through all the examples metioed above, we ca see that a capacity expasio problem with ucertai demad growth could be more complicated uder the techological progress eviromet. The total cost of expasio over a log horizo will be cosiderably differet from the expasio problem with statioary techology. Some research i the past explored various capacity expasio problems ad determied the optimal policies for those cases, but oe of them have combied the cosideratio of radom expoetial demad growth ad the ucertai techological chage together with lead times for expadig capacity. This paper ivestigates optimal capacity expasio policies. The goal of this research is to determie the impact of techological chage o the optimal timig ad sizes of capacity expasios to accout for ecoomies of scale, the time value of moey ad pealties for isufficiet capacity. We formulate a model of capacity expasio that is relevat to a service provider for whom the cost of capacity shortages would be cosiderable but difficult to quatify exactly. The objective fuctio for the optimal policy is the total cost of expasio over a ifiite time horizo combied with a pealty for shortages. We determie the optimal expasio policy i two dimesios. The first dimesio cocers the timig of each expasio i terms of the relatioship betwee demad ad capacity whe the expasio is iitiated. The secod 4

5 dimesio cocers size, i.e., the amout of capacity to be added by each expasio i view of cost discoutig ad ecoomies of scale. A aalytical expressio for the total cost allows umerical solutio for the policy parameters that miimize a weighted combiatio of the total discouted expected expasio cost ad the cost of shortage. 2. Relevat Capacity Expasio Studies Sice the late 950s, may studies of capacity expasio problems have bee coducted. Side [26] studied the capacity expasio problem of certai facilities providig service for a growig populatio, such as a power plat, a trasportatio system, or a telephoe system. Side assumed the demad for services as a fuctio of time is give. The facility must expad ad replace its equipmet from time to time i order to meet its demad. He showed that i certai cases, there is a optimal expasio policy with equal time itervals betwee successive expasios. Mae [5] studied the capacity expasio problem with probabilistic growth, icludig the pealties ivolved i accumulatig backlogs of usatisfied demad. The results showed that ucertaity i demad growth causes a larger size of capacity expasio. Aother study by Mae of several heavy process idustries i Idia is a example widely kow for its applicatio [6]. I these models the demad growth follows a liear tred. The total cost of expasio over a ifiite time horizo ca be calculated simply by summig the costs of each repleishmet discouted back to time zero. Sriivasa [28] exteded Mae s work to the growth of heavy idustries i Idia. He formulated a model i which demad grows at a costat geometric rate, ad assumed that there are o demad backlogs (excess demad). Whe the ecoomies of scale i costructio are icorporated ito the capacity expasio cost, it is optimal to expad capacity at each of a sequece of equally spaced time poits. Therefore, the optimal expasio size would grow expoetially. Sriivasa also assumed that techology is static, the costructio lead time for addig ew capacity is zero, ad demad growth is determiistic. A survey of Luss [4] ca be cosulted as a extesive literature review o capacity expasio. I his survey, Luss uified the existig literature, emphasizig modelig approaches, algorithmic solutios ad relevat applicatio. Various models have bee formulated for the capacity expasio problem with radom demad. Freidefelds [6] studied the effects of ucertaity i demad o capacity expasio decisios. He specified demad as a birth ad death process for fixed expasio icremet ad showed that the effect of radomess is idetical to the effect of a larger growth rate. Bea et al. [] showed that the capacity expasio problem over a ifiite horizo 5

6 with demad that follows either a oliear Browia motio or a o-markovia birth ad death process could be trasformed ito a equivalet determiistic problem. This equivalet determiistic problem is formed by replacig the stochastic demad by its determiistic tred ad discoutig costs by a ew determiistic equivalet iterest rate, which is smaller tha the origial, i approximate proportio to the ucertaity i the demad. More details of this result will be discussed i Sectio 4. Some studies of capacity expasio model iclude lead times for addig capacity. Nickell [8] formulated a model with a ucertai future chage i demad ad showed that the existece of a fixed lead time for addig ew capacity would cause a firm to itroduce a capacity icrease earlier. He also showed that a loger lead time results i earlier aticipatio of demad icreases. Davis et al. [4] preseted a more mathematical model of the capacity expasio process of large scale projects that icorporated a cotrollable o-zero lead time of costructig ew capacity ito ucertai future demad forecast model. Their demad model was a radom poit process that icreases by discrete amouts at radom times. The capacity expasio model also icluded a cost associated with failure to meet demad, ad a cost of wasteful overcapacity. They studied the problem by methods of stochastic cotrol ad preseted a umerical algorithm to determie the optimal policy. Chaouch ad Buzacott [3] studied the effect of lead time o the timig of plat costructio with the objective of miimizig the expected discouted costs of expasios ad shortages over the ifiite time horizo. Their model has a certai fixed lead time of costructio that is idepedet of plat size. The demad grows alterately with a costat rate i some periods ad stagat growth i other periods. They suggested that it may be ecoomically attractive to defer plat costructio beyod the time whe existig capacity becomes fully absorbed. Loger lead times icrease the optimal capacity trigger levels ad sizes of capacity additios. Rya [25] studied the problem havig correlated radom demad with a liear tred. Also, Rya [24] formulated a dyamic programmig model of capacity expasio for ucertai expoetial demad growth ad determiistic expasio lead times, ad used optio pricig formulas to estimate the shortages to result from a capacity expasio policy. With the expected lead time shortage fixed, the discouted expasio cost could be miimized by expadig capacity by a costat multiple of existig capacity. Several studies iclude the effect of techological progress o the capacity expasio models. Sow [27] reviewed the previous work of Mae ad Sriivasa ad icluded a techological progress parameter i the capacity expasio model of the commuicatios satellite INTELSAT. This added parameter is the aual 6

7 expoetial rate at which prices fall due solely to the effect of techological progress. Sow showed how techological chage affects the capacity expasio model by addig a costat to iterest rate, thus decreasig the discouted cost of each repleishmet. Other previous studies that suggested the importace of techological chage o capacity expasio or replacemet decisios are listed as follows. Goldstei et al. [7] studied the effect of techological breakthroughs o the machie replacemet problem. They preseted a dyamic discouted cost model ad a method for fidig the optimal age for replacemet of a existig machie i a techological developmet eviromet. I their research, they assumed that a ew techological breakthrough is about to eter the market i the form of ew machie, which has higher purchase cost but lower maiteace costs tha the existig machie. Hopp ad Nair [8] developed a procedure for computig the optimal replacemet decisio i a eviromet of techological chage. Their model assumed that the costs associated with the preset ad future techologies are kow, but the appearace times of the future techologies are ucertai. Nair [7] also studied ucertai sequetial techological chage, which affects the firm s strategic ivestmet decisios. He suggested that the appearace of the future techologies are cosidered ucertai with probabilities that may vary with time, but the order i which they appear is assumed sequetial, such as the differet geeratios of microchips for persoal computers. Fially, he developed a approach usig ouique termial rewards to solve a dyamic programmig model of the replacemet problem. All the previous works discussed above demostrate the importace of techological chage i the capacity expasio problem. The predictio ad forecastig of techological chage itself was described by Porter et al. [20], who discussed the models of techology growth based o the previous work of Gompertz ad Fischer-Pry. They suggested that the growth i capacity of may techologies is expoetial over a cosiderable time period. Rajagopala et al. [22] formulated a capacity expasio ad replacemet model with a sequece of techological breakthroughs. They modeled the stochastic techological chage as a semi-markov process by specifyig the distributio of the time betwee two cosecutive iovatios ad the matrix of trasitio probabilities for the levels of techology achieved. By proper choice of the time-to-discovery distributio, their model ca accommodate the diverse characteristics of timig betwee iovatios across differet idustries. 3. Problem Defiitio ad Notatio Let d(t) be the demad for service at time t 0. We assume that this demad follows a geometric Browia motio with drift µ > 0 ad variace 2 σ. It follows that, give d(t), the growth i the logarithm of 7

8 demad over a iterval begiig at time t is ormally distributed with mea ad variace proportioal to the legth of the iterval: ( + t) d() t d t l = µ t + σ tz, () where Z is a stadard ormal radom variable. Alteratively, we ca characterize the demad at a future time poit as logormally distributed with coditioal expected value: g t [ ( ) ( )] (), E d t + t d t = d t e (2) where g µ σ 2 = + 2 is the expected rate of expoetial growth i demad. Note that, although demad ca icrease or decrease over time, it ca ever become egative. For use i expasio decisios, we defie the demad for capacity as D () t max { d() s : 0 s t}, a odecreasig fuctio of time. This model is appropriate for demad patters with the followig characteristics: Although demad ca icrease ad decrease over time, the log term expected tred is upward. The expected demad at the ed of a period is best expressed as a costat percetage icrease over the demad at the begiig of the period. The ucertaity i the logarithmic demad growth over a iterval, as measured by its variace, is proportioal to the legth of the iterval. This characteristic is cosistet with the dimiishig reliability of forecasts as they exted ito the future. Lueberger [3] explais how the parameters µ ad σ ca be estimated from historical data as the sample mea ad stadard deviatio of l ( d( k ) d( k) ) +, usig data collected at equally spaced time poits k =, 2, i the past. I order to meet the predicted growth i demad, capacity ca be icreased by ay cotiuous quatity, but ecoomies of scale imply that the expasios will occur at discrete time poits rather tha cotiuously over time. Specifically, we assume that C(, ) ( ) X t, the cost of a expasio of size X at time t, satisfies a C X,0 = kx,0< a <, where k is a proportioality costat ad a is a ecoomy of scale parameter. Without loss of geerality, assume costs are scaled so that k=. We further assume for simplicity that the expasio cost is icurred all at oce whe the expasio is iitiated. We study two models of the cost impact of techological 8

9 iovatio. The first, as i [27], assumes a determiistic expoetial decrease i the cost of capacity due to techological chage, so that (, ) p C X t t e t C( X, t) + = for all t, where p > 0. The secod model assumes that techological iovatios follow a Poisso process with rate λ, ad that the average rate of cost reductio per iovatio is q > 0. Uder radom techological chage, the cost is ( ) qn ( t) + = ( ), where N() t C X, t t e C X, t is Poisso distributed with mea λt. There is a fixed lead time, L, required for ay size expasio, ad costs are cotiuously discouted at rate r > g. Let be a idex for the sequece of expasios. A expasio policy {(, ), } expasio time poits ad capacity icremets. See Figure. For a give policy, let after expasios have bee completed, where K d( ) The istalled capacity at time t is give by 0 0 t X cosists of a sequece of K be the istalled capacity > is the iitial capacity ad, for, K = K + X. () t Κ = K,0 t < t + L 0 K, t + L t < t + L,, + (3) while the capacity positio is () t Π = K,0 t < t 0 K, t t < t,. + (4) The capacity positio icludes available capacity ad ay additioal capacity that is uder costructio. 9

10 Demad or Capacity dt () K + X 0 K 0 d(0) t t L t 2 t2 + + L Time Figure. Illustratio of capacity expasio problem ad potetial for shortages durig lead times. We assume that there is a sigificat pealty of m per uit of shortage per uit time. Therefore, i light of the lead times, each expasio should be iitiated before a shortage occurs. Sice i the worst case the detectio of a shortage would automatically trigger a expasio, it follows that the risk of shortage is preset oly durig lead times for expasio. O the other had, if expasios occur far i advace of eed, high opportuity costs result from the overivestmet. The problem is to fid a expasio policy that miimizes the sum of expected ifiite horizo discouted expasio cost ad shortage pealties. 4. Expasio Policy Rya [24] showed that the shortage durig ay lead time, expressed as a proportio of istalled capacity, depeds o previous evets ad decisios oly through the ratio of demad to capacity positio at the begiig of the lead time, ( ) D t K. Pak [9] showed how the total expected shortage throughout the lead time could be obtaied by umerical itegratio of a aalytical formula that depeds o the demad-to-capacity ratio. I this paper we assume a timig policy i which a expasio will be triggered wheever demad reaches a costat (over 0

11 time) proportio of the capacity positio, so that ( ) D t K = γ for all. If lead times do ot overlap, the the capacity positio at time t is equal to the istalled capacity at that time. The specific proportio, γ, that is used is a decisio variable. This policy is equivalet to the proportioal reserve policy that has traditioally bee used by may utilities [0]. Rya [24] also showed that, uder this timig policy, the ifiite horizo expected discouted cost of expasios is miimized whe each capacity icremet equals the same proportio of the curret capacity positio, i.e., X = xk for each. That proof relied o the equivalet determiistic formulatio of Bea et al., [], i which the true iterest rate, r, is adjusted dow accordig to the demad ucertaity. The secod decisio variable, x, is foud usig this lowered iterest rate, r * *, alog with the determiistic demad fuctio () ( 0) D t = D e µt. I this paper, we show that determiistic or radom techological chage are both equivalet to icreases i the iterest rate. Therefore, we use the form of the policy suggested by Rya [24], though we ote that it has ot yet bee proved optimal for miimizig the weighted combiatio of expasio cost ad shortage. Let T( y) mi{ t 0 : D( t) y} mi{ t 0 : d() t y} = = =. If a expasio takes place whe demad reaches y for the first time, the expected discout factor for the expasio cost is iterest rate for the determiistic problem is foud from the fact that ( ) E e rt y. The equivalet rt ( y) * = exp( l ( ( 0) ) ) (see E e r y D µ []). Note that, i the precedig expressio, l ( y D ( 0) ) µ is the time at which the determiistic demad * fuctio, () D t, reaches y. Uder our policy, t = T( γ K ), X ( ) = x x + K 0 ad K ( ) x K 0 = Expected Discouted Cost of Expasios We develop expressios for expected discouted cost i three cases: o techological chage, determiistic techological chage, ad radom techological chage. Comparig Equatios (8), (0) ad (4) below, we will see that the effects of these differet assumptios are captured by modificatios to oe parameter, ρ, used i the cost discoutig factors. If there were o cost impact of techological chage, the ifiite horizo expected discouted expasio cost would be give by:

12 ( ( ) ) ( ) rt a 0, ( ) rt γ x+ K γ = = ( ( + ) 0 ) = = a. (5) u x E e X E e x x K The expected discout factor for the th expasio ca be writte as where ( γ ( ) 0 ) D ( 0) = γ x+ rt x+ K ( ) K0 E e * r µ σ ρ = = + 2r 2 µ σ µ 2 ρ, (6). (7) Bea et al. [] poited out that r * < r. Oe ca also verify by algebra that the assumptio r > g implies ρ >. Usig these facts, the ifiite horizo expected discouted cost may be evaluated i closed form as: sice a ρ < 0. ( γ ) D ( 0) ( x+ ) ( 0) K0 a ( xk ) ( x ) = a ( 0) ( xk0 ) K0 ( x + ) 0 ( ( ) 0 ) u, x = x x K + = γ K ρ D = 0 + γ D = γ ρ ρ ( ), a a ρ (8) a ρ Uder our assumptio cocerig determiistic techological chage, the correspodig cost is give by ( ) rt pt γ, ( ) ( r+ p) t u x = E e e X = E e X = = ( ) a a D. (9) As oted by Sow [27], we see that i this model, the cost impact of techological improvemet is equivalet to a icrease i the iterest rate. Techological chage would have the qualitative effects of delayig ivestmet ad/or reducig the size of each expasio sice it is aticipated that the same capacity will be available for less cost i the future. We ca quatify this effect usig the simpler expressio for the cost: where u D ( γ x) ρ D ( 0) ( xk0 ) K0 ( x + ) D, =, a ρd γ a (0) 2

13 ( ) µ σ ρd = + 2 r + p 2 σ µ 2. () If, as is more likely the case, techological iovatios caot be predicted with certaity, the closed form expressio for the cost is slightly more complicated to derive. Accordig to our model, the expected ifiite horizo discouted cost of expasios uder radom techological chage is: R ( γ, ) ( ) rt qn t a qn ( t) ( ) rt ( ) = = a. (2) u x = E e e X = E e e X For ay t, sice N() t is a Poisso radom variable, we ca use the Poisso momet geeratig fuctio q () ( e ) qn t λt q = e [23, p.60] to obtai a equivalet determiistic cost decrease rate of ˆ ( ) E e discout factor for the cost of the th expasio ca the be foud by coditioal expectatio. p = e λ. The rt qn ( t) rt qn ( t) E e e = E E e e t t rt qn ( t ) = E e E e t t t t q rt λt ( e ) q ( r+ λ( e )) t e. = E e e = E (3) Therefore, the closed form expressio for the cost uder radom techological chage is: where u R ( γ x) ρ R ( 0) ( xk0 ) K0 ( x + ) D, =, a ρr γ a (4) µ q ρr = ( r + λ( e )) σ σ µ 2. (5) 4.2. Expected Discouted Cost of Shortages Ideally, oe would prefer to be able to adjust capacity cotiuously i order to exactly meet the demad as it occurs. However, whe there are ecoomies of scale ad lead times for addig capacity, cotiuous capacity adjustmet is ot possible. There could be pealties associated with over- as well as udercapacity but, to may service providers, havig isufficiet capacity to meet the demad has far more serious cosequeces tha havig 3

14 too much capacity. Further, by icludig the time value of moey i our cost fuctios, we are already ecouragig the postpoemet of capacity expasios util they are eeded. Therefore, the secod compoet of the total cost deals strictly with pealties for capacity shortage to balace agaist the expasio cost. The shortage at a future poit i time is a radom variable that represets the differece betwee demad ad istalled capacity, if this differece is positive. Let s() t max ( d() t () t, 0) = Κ be this radom quatity. At time t i the iterval [ t + L t + L ), the istataeous shortage is () () max ( (), 0), our policy, where ( ) ( ) 0 s t s t d t K =. Uder d t = γ x + K, it is possible for lead times to overlap, i.e., t < t + L if demad durig the (-)st lead time grows very quickly. Rya [24] showed that the probability of this occurrig is costat over ad ca be made as small as desired by choosig x large eough. Therefore, i our aalysis, we eglect the possibility of overlappig lead times ad assume that s () t = s () t throughout the th lead time [ t, t L ) +. Due to the Markovia character of the geometric Browia motio model for demad, the shortage durig a lead time depeds oly o the istalled capacity ad the demad at the begiig of the lead time. Specifically, for the th lead time we are iterested i the total expected shortage: S = E t dt d t () ( ) t + L s (6) t Whe measured as a proportio of the curret capacity, this expected value ca be evaluated usig a formula developed for fiacial optio pricig. Lemma [9]: If V is a logormal radom variable ad the stadard deviatio of l V is s, the E[ max ( V K, 0) ] = E( V) Φ( d ) KΦ ( d ), where ( [ ] ) 2 2 ( ) d = l E V K + s 2 s, d2 = d s ad Φ() is the stadard ormal cumulative distributio fuctio. Theorem: Uder the timig policy where t mi { t : D( t ) γ K } ad ca be evaluated umerically as: K S ( ) ( ) 2 where (, ) l ( ) h γ t = γ + µ + σ t σ t. = =, the ratio S K is idepedet of ( ( )) L gτ ( ) ( (, )) (, ), (7) 0 = f γ γe Φ h γ τ Φ h γ τ σ τ dτ 4

15 Proof: Accordig to our demad model, for t > t, the ratio d() t d( t ) is logormal ad the stadard deviatio of its atural logarithm is σ t t. Therefore, give ( ) stadard deviatio of () The expected value of () where d t, () l d t equals the stadard deviatio of d() t d( t ) d t give d ( t ) is d ( t ) [ () ( )] ( ) d t is also logormal ad the l l, which is also σ t t. g( t t ) e. Therefore, accordig to the Lemma, E s t d t d t e gt ( t ) = Φ h K Φ h, ( ) ( ) 2 gt ( t ) 2 2 ( d( t ) e K ) + ( σ )( t t ) ( d( t ) K ) + ( µ + σ )( t t ) l 2 l h = = σ t t σ t t ad h h σ t t =. Sice d( t ) γ K 2 = uder the timig policy, ( ) gt ( t ) E[ s () t d( t )] ( ) ( ), K γe h h γ t t σ t t = Φ Φ. Substitute τ = t t i the itegral for S K to obtai the result.! The Lemma ca also be used to derive the Black-Scholes formula for the value of a call optio o a asset. Birge [2] poited out the correspodece betwee a limit o capacity ad a optio o ay demad that exceeds the capacity limit. Specifically, i a competitive eviromet, havig a limited capacity ca be see as equivalet to sellig oe s competitors the optio to satisfy the excess demad. To evaluate the expected discouted pealty due to shortages, we ca multiply a pealty factor m per uit shortage per uit time by the fuctio:, (8) = rt ( γ, ) = v x E e S which represets the ifiite horizo expected discouted shortage. (Note that, strictly, we should multiply [ () ( )] E s t d t by the discout factor rt ( t ) e to discout each istataeous shortage to the begiig of the th lead time. However, whe lead times are ot too log relative to the problem horizo, these factors will be close to oe.) Usig the Theorem, we ca write v(, x) γ i a aalytical form as: 5

16 ( γ ) ρ D( 0) D( 0) γk γ = = ρ ( 0) ( γ ) 0 γ ( x + ) ( 0 ) ( γ) ( ) v, x = S = f x+ K D f K = ρ ρ. ρ ρ (9) 4.3. Total Expected Discouted Cost The overall total cost to be miimized is the sum of the expasio cost ad the shortage pealty: ( γ, ) ( γ, ) ( γ, ) Note that the ρ values used i the cost fuctio u( γ, x) w x = u x + mv x. (20) will vary accordig to the existece ad type of techological chage, while the ρ values used i the shortage fuctio ν ( γ, x) will be uaffected by iovatios. Uder determiistic techological chage, the total cost is specified as: D ( γ, ) ρ ρ ρ D ρd a D( 0) γ x a D( 0) γ f ( γ) a ρd K0 ( x+ ) K0 ( x+ ) w x = K + m K. (2) 0 ρ 0 Uder radom techological chage, the objective fuctio is idetical, with subscripts D replaced by R. The factors ( ( 0) ) D K ρ, where ρ ca be replaced by either ρ D or ρ R i the expasio cost term, ca be iterpreted as 0 adjustmets to the discout factors to accout for the ratio of demad to capacity at time 0. The fuctio a ρd ( ) ρd a ( γ, ) γ ( ) c x x x+ is a dimesioless multiplier for the time 0 cost of providig a expasio of ρ ( ) ρ size K 0, while its couterpart d( γ, x) γ f ( γ) ( x ) + is a dimesioless multiplier for a shortage of K 0 capacity uits. We have ot yet bee able to prove that w( γ, x), w ( γ, x) or w (, x) D R γ are covex fuctios of the decisio variables γ ad x. However, i may observatios of umerical istaces, we have observed: The fuctio c(, x) γ is a icreasig fuctio of x for fixed γ ad a decreasig fuctio of γ for fixed x. These properties match our ituitio that makig larger expasios results i higher discouted expasio cost, while delayig expasios reduces it. 6

17 The fuctio d(, x) γ decreases with x for fixed γ ad icreases with γ for fixed x. Agai, we would expect that makig larger expasios would reduce the size ad likelihood of shortages, while postpoig expasios icreases the risk of shortage. The combied impact of these characteristics is that w( γ, x), as the weighted sum of c( γ, x) ad d(, x) γ does appear to be covex. I the ext sectio, we examie umerical solutios ad their sesitivity to chages i the problem parameters. 5. Numerical Results ad Sesitivity Studies As a baselie umerical case, we used the parameter values µ = 0.05 ad σ = 0.2 i the demad model. The expected rate of expoetial demad growth is g = 0.07, ad we assumed a aual omial iterest rate r = 0.0 > g. The values for the lead time ad the ecoomy of scale parameter were L = 0.5 years ad a = 0.7, respectively. We chose arbitrary values of D(0) = 50 ad K 0 = 00. For these parameter values with a shortage pealty factor of m = 5, by umerically miimizig w(, x) γ we fid optimal values of γ = 0.84 ad x = Therefore, each ew expasio should be iitiated whe demad reaches 84% of capacity, ad each expasio should icrease capacity by 75%. Larger pealty factors reduce the optimal value of γ sigificatly, so that expasios are udertake earlier, ad also decrease the optimal x value slightly. Figure 2 ad Figure 3, respectively, show the effects of chages i the mea logarithmic growth rate, µ, ad the volatility of demad, σ, for various pealty factors. The effect of icreasig either the expected growth i demad or its ucertaity is to expad capacity earlier ad i larger amouts. The relative magitudes of these policy adjustmets vary with the size of the pealty factor. The case where σ = 0 represets determiistic demad µ () = ( 0). I this case, t satisfies D ( 0) e t γ K * D t D e µt at time Also, sice µ t t such that ( 0) = ad shortages durig the th lead time commece D e = K. The shortage ratio fuctio ca be evaluated i closed form as: t + L 0 K t µ t µ L f ( γ) = ( D( 0) e K ) dt = ( γe + logγ L ). (22) µ 7

18 rt ( 0) rt D = e = γ ( x+ ) K0 E e r µ, (23) we ca replace ρ by r/µ throughout. 0.9 Optimal x µ = µ = µ = µ = Optimal γ m= m=2 m=5 m=0 Figure 2. Effect of the mea logarithmic growth rate of demad o the optimal values of the policy parameters.. Optimal x 0.9 σ = σ = σ = σ = σ = σ = Optimal γ m= m=2 m=5 m=0 Figure 3. Effect of the demad ucertaity o the optimal policy parameters. 8

19 The effect of the lead time legth is show i Figure 4 to prompt earlier expasios i larger sizes. These larger expasios ca also be iterpreted as less frequet. A log lead time therefore reduces the flexibility to wait ad see what happes to demad ad respod i small capacity icremets. If L = 0, the expected lead time shortages would vaish, so that w( γ, x) u( γ, x) =. Whe cosiderig oly the expasio cost, sice u γ < 0, oe would delay expasios idefiitely. Figure 5 shows that the primary effect of icreasig ecoomies of scale (decreasig values of a) is to icrease the size of expasios, as would be expected. However, i the simultaeous optimizatio of both policy parameters, the expasios also occur somewhat earlier.. Optimal x L = 2 L = L =.5 L =.25 m= m=2 m=5 m= Optimal γ Figure 4. Effect of the lead time legth o the optimal policy parameters. 9

20 .2 Optimal x a =.6 a =.65 a =.7 a =.75 m= m=2 m=5 m=0 0.5 a = Optimal γ Figure 5. Effect of ecoomies of scale o the optimal policy parameters. Fially, we ca observe the impacts of techological chage o the optimal policy parameters. Figure 6 depicts the values of γ ad x that miimize w (, x) D γ for various values of the determiistic techological chage parameter, p. As expected, if capacity costs are expected to decrease i the future, the optimal expasios are smaller. Not so ituitively, the expasios also should begi earlier, whe demad reaches a relatively smaller proportio of capacity. Similar effects are see i Figure 7 ad Figure 8, from miimizig w (, x) R γ for differet values of q ad λ. I Figure 7, the iovatio rate, λ, is held costat at 0.5 per year while the rate of cost decrease per iovatio (q) varies. I Figure 8, q is fixed at 0.25 ad λ assumes the values show i the chart. 20

21 Optimal x p = 0 p =.025 p =.05 m= m=2 m=5 m=0 p = p = Optimal γ Figure 6. Effect of determiistic cost decrease due to techological chage o the optimal policy parameters Optimal x q =.05 q =.5 q =.25 q =.35 q =.45 m= m=2 m=5 m= Optimal γ Figure 7. Effect of cost decrease per radom iovatio o the optimal policy parameters. 2

22 Optimal x λ =. λ =.3 λ =.5 λ =.7 λ =.9 m= m=2 m=5 m= Optimal γ Figure 8. Effect of the rate of radom techological iovatios o the optimal policy parameters. 6. Discussio ad Coclusios Maagers of service facilities faced with icreasig demad must wrestle with the questios of whe to expad capacity ad by how much. I this paper we have cosidered a model that icludes the iteractios ad, i some cases, coflicts amog several problem characteristics. Ucertaity i the demad growth combied with a lead time for addig capacity creates the risk of capacity shortage eve uder the assumptio that expasios are iitiated while excess capacity remais. Ecoomies of scale ecourage larger ad less frequet expasios, while cost decreases due to techological chage motivate a wait-ad-see attitude. The mathematical model clarifies some of the relatioships betwee problem characteristics ad reveals some uexpected iteractios betwee solutio characteristics. Previous research revealed that ucertaity i the demad reduces the iterest rate [], but the cost impact of determiistic techological chage is to icrease the iterest rate [27]. Our model for the impact of radom techological chage o the capacity cost also implies a higher iterest rate. Further, it shows how demad ucertaity ad techological chage combie to affect movemets i the iterest rate that decisio makers should use whe solvig a determiistic equivalet problem. Moreover, by derivig aalytical expressios for the total cost as a fuctio of both timig ad sizig decisios, we have observed the iteractios betwee these policy dimesios. For example, the ecoomy of scale parameter 22

23 might be expected to ifluece expasio decisios solely through the size parameter, x. O the cotrary, our results show that it also affects the optimal degree of aticipatio of future demad growth as expressed by the timig parameter, γ. Our model ad sesitivity studies treat each problem characteristic as idepedet of the others. More realistic models, which we defer to future research, could cosider depedecies amog them. For example, i some techology drive markets, the itroductio of ew iovatios ca affect the growth rate of demad. I additio, cosiderig the lead time as a fuctio of the capacity icremet would be of iterest for some idustries such as eergy geeratio. Other valuable extesios would be to model the lead time as a cotrollable variable, a radom factor, or a fuctio of the techology used to provide capacity. Ackowledgmet This work was supported by the Natioal Sciece Foudatio uder grat umber DMI Refereces [] Bea, J.C., Higle, J. ad Smith, R.L., Capacity expasio uder stochastic demads, Operatios Research, 40 (992) S20-S26. [2] Birge, J.R., Optio methods for icorporatig risk ito liear plaig models, Maufacturig & Service Operatios Maagemet, 2 (2000) 9-3. [3] Chaouch, B.A. ad Buzacott, J.A., The effects of lead time o plat timig ad size, Productio ad Operatios Maagemet, 3 (994) [4] Davis, M.H.A., Dempster, M.A.H., Sethi, S.P. ad Vermes, D., Optimal capacity expasio uder ucertaity, Advaces i Applied Probability, 9 (987) [5] Dumortier, P., Shortcut techiques to boost Iteret throughput, Alcatel Telecommuicatios Review (997) [6] Freidefelds, J., Capacity Expasio: Aalysis of Simple Models with Applicatios, North-Hollad, New York, 98. [7] Goldstei, T., Laday, S.P. ad Mehrez, A., A discouted machie-replacemet model with a expected future techological breakthrough, Naval Research Logistics, 35 (988)

24 [8] Hopp, W.J. ad Nair, S.K., Timig replacemet decisios uder discotiuous techological chage, Naval Research Logistics, 38 (99) [9] Hull, J.C., Optios, Futures ad Other Derivatives, 4th ed., Pretice Hall, Upper Saddle River, NJ, 2000, 698 pp. [0] Kah, E., Electric Utility Plaig & Regulatio, America Coucil for a Eergy-Efficiet Ecoomy, Washigto, DC, 988. [] Karli, S. ad Taylor, H.M., A First Course i Stochastic Processes, 2d ed., Academic Press, New York, 975. [2] Kruger, P., Electric power requiremet i Califoria for large-scale productio of hydroge fuel, Iteratioal Joural of Hydroge Eergy, 25 (2000) [3] Lueberger, D.G., Ivestmet Sciece, Oxford Uiversity Press, New York, 998. [4] Luss, H., Operatios research ad capacity expasio problems: a survey, Operatios research, 30 (982) [5] Mae, A.S., Capacity expasio ad probabilistic growth, Ecoometrica, 29 (96) [6] Mae, A.S., Calculatio for a sigle productio area. I A.S. Mae (Ed.), Ivestmets for Capacity Expasio, MIT Press, Cambridge, 967, pp [7] Nair, S.K., Modelig strategic ivestmet decisios uder sequetial techological chage, Maagemet Sciece, 4 (995) [8] Nickell, S., Ucertaity ad lags i the ivestmet decisios of firms, Review of Ecoomic Studies, 44 (977) [9] Pak, D., Optio pricig methods for estimatig capacity shortages, M. S. Thesis, Iowa State Uiversity, Ames, 200, pp. 62. [20] Porter, A., Roper, A.T., Maso, T., Rossii, F. ad Baks, J., Forecastig ad Maagemet of Techology, Wiley-Itersciece, New York, 99. [2] Rai, A., Ravichadra, T. ad Samaddar, S., How to aticipate the Iteret's global diffusio, Commuicatios of the ACM, 4 (998) [22] Rajagopala, S., Sigh, M.R. ad Morto, T.E., Capacity expasio ad replacemet i growig markets with ucertai techological breakthroughs, Maagemet Sciece, 44 (998)

25 [23] Ross, S.M., Itroductio to Probability Models, Third ed., Academic Press, Orlado, 985. [24] Rya, S.M., Capacity expasio for radom expoetial demad growth with lead times, Workig Paper 02-04, Idustrial & Maufacturig Systems Egieerig, Iowa State Uiversity, Ames, IA, [25] Rya, S.M., Capacity expasio with lead times ad correlated radom demad, Naval Research Logistics, Forthcomig (2002). [26] Side, F.X., The replacemet ad expasio of durable equipmet, Joural of the Society of Idustrial & Applied Mathematics, 8 (960) [27] Sow, M.S., Ivestmet cost miimizatio for commuicatios satellite capacity: refiemet ad applicatio of the Cheery-Mae-Sriivasa model, Bell Joural of Ecoomics, 6 (975) [28] Sriivasa, T.N., Geometric rate of growth of demad. I A.S. Mae (Ed.), Ivestmets for Capacity Expasio: Size, Locatio, ad Time-Phasig, MIT Press, Cambridge, 967, pp

26 Biographical Sketches Nattapol Porsaluwat eared a M.S. i Idustrial & Maufacturig Systems Egieerig at Iowa State Uiversity. This paper is based i part o his master s thesis etitled, Capacity Expasio with Techological Chage. He received his B.S. i Mechaical Egieerig from Chulalogkor Uiversity i Thailad. He previously worked for a egieerig cosultats compay. Dohyu Pak is a doctoral studet i Idustrial & Operatios Egieerig at The Uiversity of Michiga. He completed a M.S. i Idustrial & Maufacturig Systems Egieerig at Iowa State Uiversity. This paper is based i part o his master s thesis etitled, Optio Pricig Methods for Estimatig Capacity Shortages. As a graduate assistat, he is ivolved i teachig fiacial egieerig courses. His research iterests are i asset pricig, modelig of telecommuicatio etworks ad productio optimizatio, optimal ivestmet uder ucertaity, ad risk maagemet. Sarah M. Rya is a associate professor of Idustrial & Maufacturig Systems Egieerig at Iowa State Uiversity. She teaches courses i optimizatio, stochastic modelig ad egieerig ecoomic aalysis. Her research uses stochastic models to study log term ivestmet decisio problems as well as resource allocatio problems i maufacturig. She is the recipiet of a Faculty Early Career Developmet (CAREER) Award from the Natioal Sciece Foudatio, which fuded the research leadig to this article. 26