Optimal asset replacement: Profit maximization under varying technology

Journal Pre-proof Optimal asset replacement: Profit maximization under varying technology

Yuri Yatsenko, Natali Hritonenko PII:

S0925-5273(20)30064-5

DOI:

https://doi.org/10.1016/j.ijpe.2020.107670

Reference:

PROECO 107670

To appear in:

International Journal of Production Economics

Received Date:

19 April 2019

Accepted Date:

04 February 2020

Please cite this article as: Yuri Yatsenko, Natali Hritonenko, Optimal asset replacement: Profit maximization under varying technology, International Journal of Production Economics (2020), https://doi.org/10.1016/j.ijpe.2020.107670

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Journal Pre-proof

Optimal asset replacement: Profit maximization under varying technology

Yuri Yatsenkoa,1, Natali Hritonenkob a Dunham

College of Business, Houston Baptist University, 7502 Fondren, Houston TX 77074, USA

b Department

of Mathematics, Prairie View A&M University, Prairie View, TX 77446, USA

Abstract. The paper analyzes a profit-maximizing serial replacement problem under variable asset productivity, operating cost, and replacement cost that depend on asset age and installation time. The majority of modern asset replacement models minimize costs. The authors examine and highlight essential differences between profit-maximizing and cost-minimizing replacement strategies in the infinite-horizon framework. Next, they construct and analyze an effective multi-cycle replacement algorithm that approximates the infinite-horizon solution with non-equal asset lifetimes. Numeric simulation illustrates theoretic outcomes and demonstrates a good performance of the constructed algorithm under different patterns of changing technology. Keywords: Asset replacement, improving technology, asset productivity, infinite-horizon optimization, multi-cycle replacement, profit maximization

Corresponding author. Tel.: +1 281 649 3195, fax + 1 281 649 3436. E-mail addresses: [email protected] (Yu. Yatsenko), [email protected] (N. Hritonenko) 1

Journal Pre-proof

Optimal asset replacement: Profit maximization under varying technology

Abstract. The paper analyzes a profit-maximizing serial replacement problem under variable asset productivity, operating cost, and replacement cost that depend on asset age and installation time. The majority of modern asset replacement models minimize costs. The authors examine and highlight essential differences between profit-maximizing and cost-minimizing replacement strategies in the infinite-horizon framework. Next, they construct and analyze an effective multicycle replacement algorithm that approximates the infinite-horizon solution with non-equal asset lifetimes. Numeric simulation illustrates theoretic outcomes and demonstrates a good performance of the constructed algorithm under different patterns of changing technology. Keywords: Asset replacement, improving technology, asset productivity, infinite-horizon optimization, multi-cycle replacement, profit maximization

1. Introduction

Profit maximization and cost minimization are two fundamental alternative business strategies, which are commonly used in production economics to formally describe management objectives. Historically, the economic analysis of rational asset replacement started with the maximization goal for the present value of total revenue or profit (Malcolmson 1975, Elton and Gruber 1976, Bean et al. 1984). Compared to cost minimization, profit maximization accounts for asset productivity, which makes it more complicated for analysis and data collection in both theory and practice. Correspondingly, the focus of asset replacement has gradually shifted to the cost minimization (Sethi 1979, Christer and Scarf 1994; Bethuyne 1998, Regnier et al. 2003, Hartman and Rogers 2005, Stutzman et al 2017, Al-Chalabi 2018, and others). The cost-minimizing asset

1

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replacement is technically and conceptually simpler, so, it became a favorite subject in engineering economics textbooks (Thuesen and Fabrycky 1993, Newman et al., 2004; Hartman, 2007). As the survey (Hartman and Tan 2014) demonstrates, most recent replacement models minimize the involved costs. Only few current replacement models (Goolsbee 1998, Goetz et al 2008, Hritonenko and Yatsenko 2009, Motamedi et al 2013, Adkins and Paxson 2013) maximize the present value of profit and take asset productivity into account. The present paper analyzes a profit-maximizing serial replacement problem under evolving productivity, operating cost, and replacement cost, which depend on both the asset age and installation time. Those three exogenous dynamic parameters describe changing economictechnological environment that affects asset replacement decisions. Specifically, the productivity of assets usually decreases with their age, though can increase in time due to technological improvements. It is well known that profit maximization and cost minimization strategies have fundamental differences in real business world (Samuelson and Marks 2014) and produce different outcomes in economic analysis (Newman et al. 2004), including asset replacement. Profit maximization leads to growth and expansion, while cost minimization means contraction and is preferable during hard economic times. Therefore, a systematic comparative analysis of profit-maximizing vs cost-minimizing replacement is relevant for both theory and practice and is one of the main goals of the present paper. The real complexity of optimal asset replacement occurs under changing technology. Ongoing technological improvements affect current and future replacement costs, which leads to different durations of sequential replacements of the asset. Starting with (Terborgh, 1949), a popular approach to describe asset replacement under technological change is the infinite-horizon optimization (Grynier 1973; Elton and Gruber, 1976; Bean et al. 1984; Bethuyne, 1998; Regnier et al., 2004; Hartman, 2007; Yatsenko and Hritonenko 2005, 2017). The first theoretic analysis of consecutive replacements is provided in cost-minimizing framework by Regnier et al. (2004), 2

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who prove that the optimal lifetime of sequentially replaced assets increases when the operating cost decreases faster than the new asset cost but decreases otherwise. Such relations are at no means trivial: Cheevaprawatdomrong and Smith (2003) referred to them as a paradox in asset replacement under technological improvement, which was explained by Hritonenko and Yatsenko (2007). Profit-maximizing optimization of a chain of consecutive asset replacements is more complex and has not been even touched in the operations research. In real economy, the productivity (efficiency, effectiveness) of many asset categories depends on their age and installation time. The revenue of such assets as cruise ships, vehicle rentals, and entertainment facilities deteriorates in age with usage (Adkins and Paxson 2013). Even more important intrinsic effects are related to productivity deterioration and learning (Jovanovic and Lach 1989; Colombo and Mosconi 1995). The effectiveness of practically any sophisticated machine or equipment depends on its age, which cannot be reflected in cost minimization criterion. Examples include trucks, buses, cars, ships, aircrafts, construction equipment, manufacturing machines, oil rigs, and more (Cooley et al. 1997; Hartman 2007; Goetz et al. 2008; Scarf et al. 2007; Jovanovic and Tse 2010). An important real case for profit-maximizing approach is the replacement of airplanes (Goolsbee, 1998), whose revenue and fuel efficiency deteriorate with age and are better for newer vintages. A single airplane lasts for decades and costs tens of millions dollars. Major carriers make decisions about hundreds of planes, which are central to effective airline operation (Goolsbee, 1998). The discounted maximization of profit (or utility) has always been a favorite objective in the macroeconomic theory of economic growth. The dependence of the asset productivity on asset installation time (vintage year) is a core element of the investment-specific technological change (Greenwood et al. 1997), while its dependence on asset age is caused by deterioration and learning. Vintage capital models (Boucekkine et al. 1997, 2014, Cooley et al. 1997; Hritonenko and Yatsenko 2005, 2009, Jovanovic and Tse 2010, Jovanovic and Yatsenko 2012) explore renovation of capital assets on a large economic scale using the profit maximization objective. 3

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However, such macroeconomic models oversimplify involved costs and cannot be used in production economics. The theoretic contribution of this paper is in the thorough analysis of profit-maximizing asset replacement in an infinite-horizon model, which sheds new light on real replacement strategies. The constructed model of serial asset replacement considers improvements in both the operating cost of used assets and the cost of replacement with a new asset. It allows to understand how optimal asset replacement strategies depend on the structure and dynamics of costs. The principal difference from previous papers (Yatsenko and Hritonenko 2005, 2011, 2015) is in using the profit maximization goal. Following common assumptions in asset replacement research (Hartman and Tan 2014), we restrict our theoretic analysis to exponential dependences of the productivity and costs on asset creation time, commonly known as exponential technological change. The model with detailed structure of costs allows finding important relations between the replacement times and evolving economic-technological environment. Namely, we prove that lifetimes of sequential asset replacements become shorter when productivity increases in time faster than operating and replacement costs. If the operating or replacement cost increases faster than productivity, then the replacements will stop at a certain finite time. Also, we identify a razor-thin case when the optimal asset lifetime is the same for all sequential replacements. It appears to be a proportional technological change when both costs and productivity change with the same rate. The obtained outcomes demonstrate striking differences between the profitmaximizing and cost-minimizing replacement strategies. Namely, sequential asset replacement times diverge in profit-maximizing replacement strategy as opposed to the cost-minimizing replacement in (Yatsenko and Hritonenko 2005, 2011, 2015). Our practical contribution is an effective multi-cycle replacement algorithm that calculates the profit-maximizing replacement strategy with non-equal lifetimes at limited technological forecast. The algorithm extends algorithms of (Christer and Scarf 1994; Regnier et al., 2004; Scarf et al. 2007; Yatsenko and Hritonenko 2015, 2019) for asset replacement under TC to the case when 4

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future productivity and costs are known over a longer horizon. We provide an extensive numerical simulation that includes comparison with other known replacement algorithms. Because of diverging nature of the optimal profit-maximizing strategy, the standard Economic Life method (Thuesen and Fabrycky 1993) appears to perform better than other known algorithms for certain combinations of parameters. In contrast, the multi-cycle replacement algorithm always improves the estimate of current asset lifetime under nonstationary costs. This outcome highlights the importance of calculating a multi-cycle replacement strategy. Our numeric results demonstrate that the constructed algorithm delivers solutions closer to infinite-horizon strategy than other algorithms. The practical advantage of our profit-maximizing model and algorithm is that they work for any age-dependent profiles of productivity and costs, as compared to the alternative research approach of (Adkins and Paxson 2013) restricted to geometric costs. The paper is organized as follows. Section 2 formulates and analyzes the infinite-horizon profit maximization problem in a continuous-time single asset replacement model with perfect foresight. Section 3 develops a multi-cycle algorithm for profit-maximizing replacement, which approximates the infinite-horizon solution. Numeric simulation of Section 4 demonstrates good performance of the constructed algorithm for different patterns of evolving technology. Section 5 concludes.

2. Infinite-horizon replacement under evolving technology

In this section, a profit–maximizing infinite-horizon model of serial asset replacement is constructed and analyzed. The goal of this section is to obtain a theoretic insight into profit– maximizing strategies of asset replacement under different dynamics of technological change. Analytic outcomes of this section are also used in Section 3 to construct a multi-cycle replacement algorithm, which approximates the infinite-horizon solution.

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Let us consider a firm that produces a certain output using a single asset (machine, equipment unit, or similar) that needs to be periodically replaced with new assets. For clarity, we assume that the current asset is purchased at time t = 0 and should be replaced at a certain time, and then this process will be repeated. The asset lifetime is an endogenous control variable. We assume that the replacement occurs in a changing economic-technological environment, characterized by the following functions: 

Y(t,u) is the productivity (revenue per time unit) at time u of an asset installed at time t, 0


C(t,u) is the operating and maintenance cost at time u of an asset installed at time t,



R(t) is the replacement cost (the purchase and installation costs of a new asset) at time t.

The revenue Y is generated by industry demand and measured in monetary units, for example, as the annual revenue from operating one airplane (Goolsbee, 1998). The dynamics of Y, C, and R in time t reflects the evolving technology, while their dependence on the age a=u-t is caused by asset deterioration and learning processes (Colombo and Mosconi 1995; Jovanovic and Yatsenko 2012). Assuming that the functions Y, C, and R are known over [0,), the present value of the total profit (net revenue) over [0,) is described as: 

P   Pk 1 ,

(1)

k 0

where  k 1

Pk 1  

k

e ru Y ( k , u )  C ( k , u ) du  e r k 1 R( k 1 )

(2)

is the present value of the profit over one future replacement cycle [k, k+1], r > 0 is an industrywide discount rate, k, k=1,2,…, are the replacement times when the k-asset is replaced with (k+1)-th asset, and 0 = 0. By (2), we assume that the initial incumbent asset is in place and that replacement occurs at the end of the incumbent’s lifetime.

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Alternatively, this replacement policy k, k=1,2,…, can be described by the sequence of endogenous lifetimes Lk =k k-1 of consecutively replaced assets, k=1,2,…. Then, the replacement times k are expressed via the asset lifetimes (Yatsenko and Hritonenko 2015) as k

k = k-1 + Lk   L j , k=1,2,… 0 = 0.

(3)

j 1

Following the majority of replacement research under technological change (TC), we assume an exponential TC (Malcomson 1975, Bethuyne 1998, Regnier et al. 2004, Goetz et al. 2008, Yatsenko and Hritonenko 2005, 2008, 2017, Nguyen et al. 2013). Namely, the productivity Y(t,u), maintenance cost C(t,u), and replacement cost R(t) are g t

Y (t , u )  y (u  t )e y ,

C (t , u )  c(u  t )e gct , R(t )  R0 e g Rt ,

0≤ t ≤ u < ,

(4)

where gy, gR, and gC are the TC rates and y(ut) and c(ut) are the age-dependent deterioration profiles of the asset productivity and cost. The asset productivity deteriorates because of frequent repair and longer inspection times for older assets and other reasons (Scarf et al. 2007). We assume that any change in the asset performance due to technological progress is channeled through those three attributes Y(t,u), C(t,u), and R(t). This framework simplifies economic reality, but is richer than in comparable models, for example, in (Adkins and Paxson 2013). The improving technology assumes that newer assets are more efficient and less costly: gy ≥ 0, gc ≤ 0, and gR ≤ 0. However, it is not always the case and later we investigate replacement strategies at both increasing and decreasing costs. Under conditions (4), the present value (1) is 

 k 1



P     e ru y (u   k )e  k 0  k

g y k



 c(u   k )e gc k du  e r k 1 R0 e g R k 1  . 

(5)

An ideal management goal is to maximize the present value (5) under known Y(t,u), C(t,u), R(t), t,u[0,). The corresponding infinite-horizon optimization problem is to find the optimal sequence of replacement times k, k=1,2,…, that delivers

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max

 1 , 2 , 3 ,...



 k 1

  k 0

k



e ru y (u   k )e

g y k



 c(u   k )e gc k du  e r k 1 R0 e g R k 1  . 

(6)

The optimization problem (6) is well defined at assumptions r - gy > 0,

r - gc > 0,

r - gR > 0,

(7)

which means that the costs can also increase in time, but not faster that the discount factor r. Let us assume that a solution to (6) exists and analyze its qualitative properties. As we shall see, this solution may have the infinite or finite number of replacement cycles, depending on the given parameters gy , gc , and gR. An interior solution 1 <2 <3 <… to the infinite-horizon optimization problem (6) should satisfy the standard optimality conditions P /  k  0 at k =1,2,…. To simplify the objective function (6), we rewrite (5) as







 k 1 ~ ~ ( g  g )u ( g  g ) P     e  r u ~ y (u   k )  c~ (u   k )e c y du  e  r  k 1 R0 e R y k 1     k 0  k

~ r  r  gy ,

in terms of

g a ~ y (a)  e y y (a) ,

(8)

c~ (a )  e  gC a c(a ) .

(9)

The representation of discounted profit (5) in the equivalent form (8)-(9) also helps us to develop effective approximate algorithms for the problem (6) later in Section 4.

~

Let us denote Lk   k   k 1 , k  1,2,... . Differentiating (8) in k, we obtain the following optimality conditions for the problem under study.

~

~

~

Lemma 1. An interior solution {L1  0, L2  0, L3  0,...} to the infinite-horizon optimization problem (6) satisfies the following system of nonlinear recurrent equations for k=1,2,… ( g  g ) ( g  g ) ~ y ( Lk )  ~ y (0)  c~ ( Lk )e c y k  c~ (0)e c y k



Lk 1

0





(10)

~ ( g  g )( a  ) ( g  g ) e r a ~ y ' (a )  c~ ' (a )e c y k da   R0 (r  g y  g R )e R y k .

Proof is in Appendix.



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An exact solution to the problem (6) can be obtained by solving an associated nonlinear integral equation for the optimal replacement times. Lemma 2. Let (7) hold. If the optimal policy 1 < 2 < 3 < … exists, then

k = x-1(k-1),

Lk = k  L(k),

k=1,2,…,

(11)

where x(t) < t, t[0,), is the solution of the nonlinear integral equation



x 1 ( t ) t

 R0 e



e ru e ( g R r )t

g yt

y (u  t )  e

g y x (u )



y (u  x(u ))  e gct c(u  t )  e gc x (u ) c(u  x(u )) du

(12)

,

where the function x-1 is the inverse of x. 

Proof is in Appendix.

In Lemma 2, the function x(t) = t  L(t) is the installation time and L(t) is the lifetime of the asset being scrapped at time t. The inverse function x-1(t) = [t  L(t)]-1 describes the future replacement time of the asset installed at t, t[0,). Then, the lifetime Lk = k -k-1 for k =1,2,…, where Lk and k are related by (3). Analytic properties and numeric techniques for special cases of the equation (12) were studied in (Hritonenko and Yatsenko 1996, 2009, 2013, Motamedi et al 2013, 2014). The next statement identifies a special case gy = gc = gR when the infinite-horizon optimization (6) has a simple analytic solution: an equal-life replacement policy. Similarly to cost-minimizing replacement (Yatsenko and Hritonenko 2011, 2015; Motamedi et al. 2014), we refer to this case as the proportional TC. Theorem 1. At the proportional TC gy = gc = gR = g < r and

c~ ' (a)  ~ y ' (a)   ,

  const  0 ,

(13)

a solution to the infinite-horizon optimization problem (6) is the equal-life policy

~ ~ ~ ~ L1  L2  ...  L , where L > 0 is a unique solution of the nonlinear equation

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1  e ~y ( L)  c~( L)  ~r  ~ rL

L

0

~ e r a  ~ y (a)  c~ (a) da   R0 ~ r,

~ r  r  g.

(14) 

Proof is in Appendix.

If gy ≠ gc or gy ≠ gR, then a solution to the optimization problem (6) consists of non-equal lifetimes and can be found only numerically. The following statement reveals the structure of a variable-life replacement policy in the case of the exponential capital deterioration:  ya

y ( a )  y0 e

c(a )  c0 eca ,

,

(15)

when the productivity decreases in age a with the rate y < 0 and the operating cost increases with the rate c > 0. Theorem 2. Let (4), (7), and (15) hold,

y0  c0  rR0 ,

 y  0,

 c  0,

(16)

~ ~ ~

and a solution {L1 , L2 , L3 ,...} exist. Then: (a) At gy ≥ gc and gy ≥ gR , the optimal replacement policy includes the infinite number of

~

~

replacement cycles. If gy = gc = gR, then Li  L , i=1,2…, is defined in Theorem 1.

~

~

If gy > gc or gy > gR , then the consecutive lifetimes decrease: Li 1  Li for all i=1,2… 1,2…

~

~

~

At gy = gR and gy > gc , the consecutive lifetimes Li approach L when i. At gy > gR, Li  0 when i. (b) If gy < gc or/and gy < gR, then the optimal replacement policy includes a finite number M of

~

~

the replacement cycles {Li , i  1..., M } with the infinite lifetime LM 1   of the last asset.

~

~

The consecutive lifetimes increase: Li 1  Li for i = 1,…,M.

~

There is no replacement at all ( L1   ) when

r  gy r y

y0 

r  gc c0  (r  g R ) R0 . r  c

(17)

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

Proof is in Appendix.

Theorem 2 describes two fundamental reasons for asset replacement: (A) aging and related deterioration of assets and (B) technological change. The traditional replacement theory emphasizes aging, while our model considers both processes. In particular, the optimal policy is trivial: no replacement (L1*=) in the case when both age-dependent profiles y(a) and c(a) are constant and no technological change occurs. The asset replacement happens without improving technology because of asset aging, when y(a) decreases and/or c(a) increases indefinitely at a. The replacement also occurs under improving technology g y > g C , even when y(a) =const and c(a) = const. By Theorem 2, the balance among the increase rates g y , g C , g R of asset productivity, operating and replacement costs is a major factor that determines dynamic properties of the

~ ~ ~

variable asset lifetimes in the infinite-horizon replacement policy {L1 , L2 , L3 ,...} . Specifically, the asymptotic behavior of the optimal infinite-horizon replacement solution is qualitatively different in the case gy > max (gc, gR) and the opposite. The lifetimes of sequential asset replacements become shorter (longer) when the productivity increases faster (slower) than both operating cost and replacement cost. Moreover, if the operating cost or replacement cost increases in time faster that the productivity, then the replacements should stop at a certain finite

~

replacement instant  K and the last K-th asset has the infinite lifetime LK   .

~

Obviously, the asymptotic of solutions Lk at large k=1,2,… becomes non-relevant in the asset replacement over a finite horizon. However, as shown in the next section, qualitative properties of the optimal strategy remain the same.

3. Approximate algorithms for infinite-horizon profit-maximizing replacement

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The goal of this section is to construct an effective algorithm that calculates an optimal replacement strategy with non-equal lifetimes when future costs and productivity are known over a finite horizon. Theoretic analysis of this section demonstrates that the constructed multi-cycle algorithm approximates the infinite-horizon replacement strategy. To achieve this goal, we extend the concept of equivalent annual cost, previously used in the well-known Economic Life replacement method (Thuesen and Fabrycky 1993), its modifications (Yatsenko and Hritonenko, 2015), and two-cycle replacement algorithms of (Christer and Scarf 1994; Scarf et al. 2007; Yatsenko and Hritonenko, 2015). Theoretic background of this concept is the discounted cash flow analysis (Dayananda et al 2002), commonly used in financial practice to estimate the present value of money. Applying to asset replacement, this approach links the profit

P[ 0,T ] over a finite interval [0, T] to the corresponding equivalent annual profit AP[ 0,T ] 

~ r ~ P[ 0 ,T ] , 1  e r T

(18)

~ where ~ r is an adjusted discount rate and ~ r /(1  e  r T ) is the annual capital recovery factor1. As

r may reflect improving technology. shown below, the choice of adjusted discount rate ~ 3.1. Multi-Cycle Replacement Algorithm Let us choose T = N and apply the formula (18) to the discounted annual profit N 1

P[ 0, N ]   Pk 1 at a fixed N >=1. First, we truncate the present value (8) over [0, ) to the k 0

interval [0, N] over N replacement cycles: N 1





 k 1 ~ ~ ( g  g )u ( g  g ) P[ 0, N ]     e r u ~ y (u   k )  c~ (u   k )e c y du  e r  k 1 R0 e R y k 1  .   k  k 0

(19)

Next, applying the annual profit formula (18) to (19), we obtain the following algorithm: To find the replacement policy (L1N, L2N,…, LNN) that maximizes the annual profit The formula (18) holds for the continuous time only. The annual capital recovery factor is calculated differently in a discrete time, see (4) in (Yatsenko and Hritonenko 2011). 1

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max AP( L1 , L2 ,..., LN )  max

L1 , L2 ,..., LN

~ r  r  gy ,

~

~ r

1  e  r ( L1  L2 ... LN )

P[ 0, N ] ( L1 , L2 ,..., LN ) ,

g a ~ y (a)  e y y (a) ,

(20) (21)

over the interval [0,N], N = L1N + L2N +…+ LNN , of N > 1 future replacement cycles. The theoretical and practical advantage of the multi-cycle replacement algorithm (19)-(21) is in lifting the infinite-horizon assumption and related challenges, while still taking non-equal lifetimes into consideration. The algorithm (19)-(21) is different from the fixed horizon replacement problem (Newman et al. 2004, Hartman 2007). Here, we fix the number N of replacement cycles, while the horizon length  N 

N

L j 1

j

is unknown because the sequential

asset lifetimes Lj are unknown. The choice (21) of the adjusted discount rate ~ r is important. It is justified by the fact that the ( g  g )u term c~ (u   k )e c y in (19) becomes negligible at large k when g y  g c . When g y  g c ,

the asymptotics of the sum (19) at large k is not relevant because the profits Pk in (19) become negative after u > tcr for some critical instant tcr >=0, and, therefore, the asset replacements stop at a finite K (similarly to Theorem 2 for infinite-horizon replacement). As shown in Theorem 4 below, the multi-cycle algorithm (20) delivers the exact infinite-horizon solution at g y  g c . Multi-cycle algorithms for cost-minimizing replacement were first introduced by Christer and Scarf (1994) and extended to cost-improving TC in (Yatsenko and Hritonenko 2015, 2019). A recent addition is (Stutzman et al 2017), the only optimal stopping problem in stochastic asset replacement that simultaneously optimizes endogenous lengths of two periods.2 Now, we compare the algorithm (20) to the infinite-horizon problem (6) and demonstrate that

~ ~

~ ~

it produces a solution close to the infinite-horizon optimization. Let {L1 , L2 ,..., LN , LN 1 ,...} be a

An interested reader can find more about the one-cycle nature of stopping replacement problems in (Yatsenko and Hritonenko, 2017). 2

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N

N

N

solution of the infinite-horizon optimization problem (6) and {L1 , L2 ,..., LN } denote a ~ solution of the N-cycle optimization problem (20). Since ~ r /(1  e  r T )  ~ r at T   , the

statements of two problems (6) and (20) become similar at large T and we can expect that their solutions converge at T   . Theorem 3. The optimal values of the objective functions (6) and (20) are related by the following asymptotic formula:

~ ~ ~ ~ N N N r P ( L1 , L2 ,..., LN , LN 1 ,...) . lim AP[ 0, N ] ( L1 , L2 ,..., LN ) = ~

 N 

(22)

Proof is analogous to the proof of similar result for the cost-minimizing replacement problem in 

(Yatsenko and Hritonenko 2020). N

N

N

By Theorem 3, the N-cycle solution ( L1 , L2 ,..., LN ) at large N delivers the approximate optimal value to the infinite-horizon optimization problem (6). However, the convergence of  

N

~

N  Lk , k=1,…,N, cannot be proven analytically and we multi-cycle solution itself: Lk 

check this fact in the numeric Section 4. By Theorem 3, we can expect that qualitative properties of the optimal multi-cycle replacement policy are similar to the infinite-horizon replacement strategy. Using (2), we rewrite the N-cycle optimization problem (20) as

max AP[ 0, N ] ,

AP[ 0, N ] 

~ r ~ 1  e r  N

N 1

 k 1

  k 0

k



(23)

 1 ,..., N



~ ~ ( g  g )u ( g  g ) e r u ~ y (u   k )  c~ (u   k )e c y du  e  r  k 1 R0e R y k 1  , (24) 

in the terms of the unknown 1 < 2 <…< N . Lemma 3. The solution (L1N, L2N,…, LNN) to the optimization problem (10) satisfies the following optimality conditions for k =1,…,N-1,: ( g  g ) ( g  g ) ~ y ( Lk )  ~ y (0)  c~ ( Lk )e c y k  c~ (0)e c y k



Lk 1

0





~ ( g  g )( a  k ) ( g  g ) e r a ~ y ' (a )  c~ ' (a )e c y da   R0 (r  g R )e R y k ,

(25)

14

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( g  g ) ( g  g ) ~ y ( LN )  c~ ( LN )e c y N 1  (r  g R ) R0e R y N  AP[ 0, N ] .

(26) 

Proof is in Appendix.

One can see that the first N-1 optimality conditions (25) in the multi-cycle optimization problem (20) coincide with corresponding conditions (10) in the infinite-horizon optimization problem (6). The only difference is in the last optimality condition (26). Theorem 4. At gy = gc = gR, the solution (L1N, L2N,…, LNN) to the multi-cycle problem (20) N

N

is the equal-life policy L1  L2  ...  LN

N

~ ~  L , where L > 0 is a unique solution of the

nonlinear equation (16). 

Proof is in Appendix.

By Theorem 4, the multi-cycle optimization problem (20) produces an equal-life replacement

~

~

~

solution, which coincides with the infinite-horizon replacement policy L1  L2  ...  L of Theorem 1 in the special case when the increase rates g y , g C , and g R are the same. An extensive numeric simulation in Section 4 shows that multi-cycle replacement solutions possess similar qualitative properties to the infinite-horizon non-equal-life replacement strategy described in Theorem 2. Namely, sequential replacement cycles become shorter when the productivity increases faster than the operating cost. If the operating cost increases faster than the productivity, then sequential replacement cycles become longer and the replacements stop after some critical instant tcr >= 0 (as in Theorem 2). Moreover, the multi-cycle algorithm (20) produces a replacement solution with natural properties for any number of cycles N > 1, which is important in practice when N is small (see our numeric simulation in Section 4). Finally, when the number N >> 1 becomes larger, the multi-cycle replacement decision approaches “the optimal current decision, where optimal means the most economical if all current forecasts are accurate” (Bean et al. 1984). 3.2. One-cycle case: Economic Life replacement

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To find an optimal replacement of current asset in practice, we need to know the future dynamics of productivity Y and costs C and R, at least, over one future replacement cycle. Compared to cost minimization, profit maximization is more complicated because it considers the asset productivity and requires more input data. In the case of minimal possible forecast data, we can use only one-cycle replacement algorithms (Hartman 2007, Hartman and Tan 2014, Yatsenko and Hritonenko 2011). The one-cycle version of the algorithm (20) finds the optimal replacement time L, that maximizes the equivalent annual profit AP over the first replacement cycle:

max AP( L)  max L

~ r ~ 1  e r L

 e L

0

 ra

 y(a)  c(a)da  R0 e (  r  g ) L R



(27)

with the adjusted discount factor ~ r  r  gy . The version of (27) at ~ r = r is known in engineering economics as the Economic Life (EL) algorithm (Thuesen and Fabrycky 1993). At ~ r = r, the problem (27) does not use the TC parameters gy and gC at all. Therefore, the corresponding replacement decision will not take the TC into account. This fact is also known in economic textbooks. By (Newman et al. 2004, Hartman and Murphy 2006, Hartman 2007), the cost-minimizing EL method is theoretically justified only under so-called repeatability assumptions that include (a) stationary asset costs (gy = gR = gC = 0 in our case) and (b) the infinite planning horizon. Technological improvements lead to varying costs, so, then the first repeatability assumption fails and the Economic Life method produces non-optimal results. By analogy with (Yatsenko and Hritonenko 2011), who analyzed cost-minimizing asset replacement problem under improving technology, we refer to the algorithm (27) as the modified Economic Life algorithm for profit-maximizing asset replacement. In contrast to the standard EL method with ~ r = r , the modified EL algorithm (27) takes the TC into account. By Theorem 3, it works well for arbitrary age-dependent profiles in the special case gy = gc = gR of proportional TC and delivers exactly the same optimal solution of the infinitehorizon problem (6). We may expect that it is reasonably effective in cases when gy, gc, and gR are 16

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different but close to each other. However, the final answer will be obtained via numeric simulation in the next section. The provided theoretic analysis have a practical value in developing effective replacement algorithms. Thus, both multi-cycle algorithm (19)-(21) and modified EL algorithm (27) produce the exact infinite-horizon strategy under proportional TC and, therefore, are effective in certain domain of model parameters. An extension of (Adkins and Paxson 2013) algorithm to the proportional TC case would be desirable, so, we can analyze its effectiveness in a wider range.

4. Numeric simulation

The purpose of this section is to visualize and compare the infinite-horizon and multi-cycle policies of profit-maximizing replacement under improving technology using numeric means. The obtained outcomes demonstrate a good performance of the constructed algorithm under different dynamics of evolving technology. 4.1. Dataset description Our first simulation run is on the data from (Adkins and Paxson 2013), which is the only other attempt to optimize a profit-maximizing serial asset replacement under improving technology. Their dataset assumes the exponential TC and exponential capital deterioration:

Y (t , u )  R0 e

g yt  y ( t u )

,

C (t , u )  C0 e

g ct  ( t u )

, R(t )  R0 e g Rt ,

(28)

constants r = 0.12, Y0 = 80, C0 = 20, R0 = 100, y = -0.02, gy = gR = 0, c = 0.04, gc = 0 and gc = 0.002. After evaluation of the integrals in (19) using (28), we rewrite the N-cycle optimization problem (19)-(20) as

r  gc P[ 0, N ] ( L1 ,..., LN ) , N ( r  g c )  Li L1 ,..., LN i 1 1 e max

(29)

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k 1 k N  ( r  y ) Lk ( r  g y )  Li ( r  g R )  Li  Y C0 ( r c ) Lk ( r  gc ) ik11 Li i 1 i 1 P[ 0, N ]    0 e  e  R0 e  . (30) r   r   k 1   y c 

The optimization problem (29)-(30) with N scalar unknowns L1,…, LN is sold using Matlab3. We calculate replacement solutions for the number of cycles N = 1  5 and varying rates gc and

gy. In decision-making practice, the choice of the number N of cycles in (20) heavily depends on the horizon T of reliable forecast. We use the following heuristic rule for choosing N: After assessing T, a preliminary estimate is N0 = T/Lmax, where Lmax is the known maximal physical life of assets. After solving the problem (20) with N0 cycles, the calculated optimal asset lifetimes are shorter than Lmax. Then, one-two additional cycles can be included and calculations can be repeated. 4.2. Simulation results The time L1 of the first asset replacement is the only value that practitioners are usually interested in. Figure 1 displays the first optimal lifetime L1 obtained by the standard Economic Life method, the four-cycle algorithm (20), and the algorithm of (Adkins and Paxston 2013). Naturally, L1 = 10.99 years is the same for all three algorithms at gc = 0 (at stationary operating cost with no TC). The standard Economic Life (EL) algorithm maximizes the annual profit

AP[ 0, L1 ] over the first replacement cycle and gives the same L1 = 10.99 in all cases gc = 0, -0.002, -0.005, -0.01, -0.02. The modified EL algorithm (27) produces the same result in this case with gy = 0. The best annual profit AP[ 0, 4 ] during four replacement cycles is delivered by a four-cycle solution (20) with increasing asset lifetimes L1 < L2 < L3 < L4 shown in Fig 2. However, by Fig.1, the first lifetime L1 becomes smaller for larger TC intensity gc, which reflects a rich picture of multi-cycle replacement. Adkins and Paxson (2013) provide a solution only for one given rate gc 3

The MATLAB code of the optimization procedure for (29)-(30) is available at a reader’ request

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= -0.002. Their solution L1 = 11.36 is more distant for the EL solution L1 = 10.99 than the first lifetime L14 of four-cycle strategy at gc = -0.002. This difference can be partially explained because Adkins and Paxson (2013) maximize the profit over the endogenous interval [0, L1] rather than the annual profit. Unfortunately, they do not provide more numeric results, so we cannot compare the qualitative dynamics of their solution to ours. The family of obtained four-cycle solutions is analyzed in Figures 2 and 3 for different values

gy and gc. Figure 2 shows that the lifetimes of sequentially replaced assets become longer when the operating cost decreases in time but the productivity and replacement cost do not (gy = gR = 0). Next, we estimate about the effect of low productivity gains on the four-cycle optimal strategy, increasing the productivity rate gy from 0 to 1% under stationary operating cost gc = 0 and constant replacement cost gR = 0. Figure 3 demonstrates that, as predicted in Theorem 3, case (a), the lifetimes of sequentially replaced assets become shorter when the productivity increases in time faster than operating and replacement costs. In the second simulation run, we demonstrate the convergence of N-cycle solutions when N increases. In doing so, we keep the same parameters r, Y0, C0, R0, y, c, and vary all three TC rates gy, gc, and gR on a larger scale covering both increasing and decreasing non-equal-life replacement strategies. Different dynamics of multi-cycle replacement at non-proportional technological change are shown in Figure 4. Figure 4(a) demonstrates one-, two-, three-, four, and five-cycle solutions at output-augmenting TC at gy = gR = 0.05 and gc = 0. Then, the consecutive

~

~

optimal asset lifetimes increase: Li 1  Li for i=1,…,5, as predicted by Theorem 2. In Figure 4(b), the multi-cycle solutions at gy = gR = 0.05 and more powerful cost-saving TC with gc = 0.02 possess decreasing consecutive asset lifetimes. At the proportional TC with gy = gc = gR = 0.05, the equal-life policy L1N = L2N =…= LNN = 10.57 appears to be optimal for any N and can be calculated using the modified Economic Life algorithm (27). Figure 5 shows an example of the

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five-cycle strategy with non-monotonic consecutive asset lifetimes, which illustrates a rich and complex picture of profit-maximizing replacement compared to cost-minimizing strategy (Yatsenko and Hritonenko 2011, 2015). A comparative analysis of numeric outcomes leads to relevant observations about the applicability of algorithm. 1. The multi-cycle solution diverges from the infinite-horizon one when approaching the end of planning interval [0, N] (see examples in Figures 2, 4, and 5).

~

2. The lifetimes L1N of the first replacement converge to L1 when the number of cycles N increases (see Figure 6). This result is confirmed by all numeric experiments with different patterns of evolving technology. Therefore, if the practical goal is to optimize timing of the first replacement, the multi-cycle algorithm (20) can be used with high confidence. 3. In contrast to the cost-minimizing serial asset replacement, one-cycle replacement algorithms should be used with caution because of a diverging nature of the infinitehorizon profit-maximizing replacement. Specifically, the modified Economic Life (27) method can be recommended only when the observed technological change is close enough to the proportional TC gy = gc = gR. If no data is available for larger horizon and a expected difference in the technology rates is essential, then the standard Economic Life method may be the best choice. 4.3. Managerial recommendations The business nature of a firm’s profit maximization strategy differs from cost minimization. This difference occurs in asset replacement problems. It highlights the importance of calculating the multi-cycle replacement strategy. The profit-maximizing infinite-horizon replacement strategies appear to be quite different (both qualitatively and quantitatively) from cost-minimizing replacement (Yatsenko and Hritonenko

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2011, 2015, 2020). Both replacement policies involve non-equal asset lifetimes, however, sequential asset lifetimes in profit-maximizing solution possess more diverging nature and decrease to zero or increase indefinitely in natural cases (Theorem 2). The balance among changing asset productivity and costs appears to be a major factor that determines variable asset lifetimes in the infinite-horizon replacement policy. Specifically: 

The lifetimes of sequential asset replacements become shorter when the productivity increases faster than the operating cost and replacement cost. If the productivity increases faster that the replacement cost, then sequential asset lifetimes asymptotically decrease to zero.



If the productivity, operating cost, and replacement cost change exponentially with the same rate, then the lifetimes of all sequential asset replacements are equal.



If the operating and/or replacement cost increase in time faster than the productivity, then the lifetimes of sequential asset replacements become longer and the replacement process will stop at a certain time in the future.

The main challenge in determining an effective multi-cycle replacement strategy at changing technology/costs is caused by shortage of data about the future technology. Rational replacement of assets under changing technology critically depends on the behavior of asset–related costs in the future. The best theoretical strategy is achieved if we possess a complete knowledge of the ongoing technological change over the infinite future planning horizon. It is not possible in business practice, and managers should rely on available estimations of future costs and combine them with certain assumptions about the cost behavior on periods when estimations are not available. The managers are usually interested only in the time to replace the current asset in use, which is the most important in practice. The main insight of this paper is that, knowing even approximate trends in the future cost and productivity of new assets available on market, the managers should 21

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look out of box and predict whether they need to replace future assets more or less frequently. To do so, an analysis of multi-cycle replacement strategy is required. Such analysis is described in this paper and produces the future replacement strategy that delivers the maximal annual profit over a chosen horizon. The recommended analytic decision-making procedure is the following. First, managers should collect detailed information about the past age-dependent performance of assets. Next, they estimate the horizon of future reliable forecast data about productivity and costs and select the number N of future replacement cycles that fits this interval. Solving the optimization problem N

N

(19)-(21) produces the recommended sequence L1 , L2 ,..., L N

N

of optimal asset lifetimes. This

information can be used for replacement of the first (current) asset and for improved resource planning and production forecasting. The procedure should be repeated annually and any other time when new information about future productivity and costs becomes available.

5. Conclusion

The age-related deterioration of assets and technological change are two main economic forces that determine the necessity and intensity of industrial asset replacement. The traditional replacement theory emphases aging, while modern replacement models consider both processes. An essential increase of age-dependent costs guarantees the replacement even without improving technology. However, changing technology is becoming more relevant in the modern high-tech economy. The present paper focuses on complex dynamics of sequential asset replacements. Our main contribution is in providing an analytically trackable concise model addressing the issue. The model is simpler in other aspects such as the number of assets replaced, economies of scale, heterogeneous production and resources, multiple challengers, and similar. Such extensions are

22

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possible, but they complicate the analysis. The ideal infinite-horizon model of asset replacement provides important hints about real replacement strategies. The most important matter is that the optimal lifetime of the current asset depends on the future chain of sequential replacements of this asset. In practice, future data are known over a finite horizon, so, the optimal replacement strategy depends on our assumptions about what happens after the end of this horizon: whether production stops or continues. In most situations, the end of a known forecast interval does not mean the end of operations. Then a natural objective is to optimize the discounted annual cost or profit over one (current) replacement cycle (which leads to Economic Life method). However, one-cycle solutions have been proven to be non-optimal under general evolving technology (Regnier et al 2004; Hritonenko and Yatsenko 2007). Two-cycle replacement algorithms (Christer and Scarf 1994, Scarf et al., 2007) are an essential improvement in the sense that they improve the estimate of current asset lifetime under nonstationary costs. Our methodology is based on natural assumptions that the current dynamics of costs and productivity will continue, at least, for some time. It allows us to construct an effective multi-cycle replacement algorithm that offers a room for further improvement when future costs and productivity are known over a longer horizon. We demonstrate that a multi-cycle solution better reflects the structure of the optimal infinite-horizon policy. Naturally, the effectiveness of replacement decision depends on the accuracy of input data. Forecast data about future productivity costs are usually less accurate than their age-dependent profiles. So, in the absence of reliable forecast, the assumption about future exponential TC appears to be quite reasonable from practical point of view (Yatsenko and Hritonenko 2016). We prove analytically and illustrate numerically that sequential asset replacement times diverge in the profit-maximizing replacement problem as opposed to the cost minimizing replacement of (Yatsenko and Hritonenko 2015, 2020). At the same time, we demonstrate that the multi-cycle algorithm (20) provides a good estimate of the first replacement time, which is the

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only number that matters in practice. Therefore, this algorithm can be used with high confidence in general case of evolving technology.

5. Appendix.

Proof of Lemma 1. Let us rewrite the infinite series (8) as  k 1

P  ...  

k

k



 k 1





~ ( g  g )u e r u ~ y (u   k )  c~ (u   k )e c y du





~ ~ ( g  g )u ( g  g ) e r u ~ y (u   k 1 )  c~ (u   k 1 )e c y du  e r  k R0 e R y k  ...

(A1)

considering only the terms that contain the variable k for a specific k. Taking the derivative of (A1) in k, we obtain



~ P ( g  g ) ( g  g )  er  k ~ y ( k   k 1 )  c~ ( k   k 1 )e c y k  ~ y ( k   k )  c~ ( k   k )e c y k  k

 k 1



k

e

~ ru

~y ' (u   )  c~' (u   )e k

( gc  g y )u

k

du  ( g

R

 gy  ~ r )e

~ rk

R0 e

( g R  g y ) k



(A2)

.

Changing the variable u   k  a in the formula (A2) and setting it zero, we obtain (10). Lemma is proven.



Remark 1. Hritonenko and Yatsenko (2009) study a simpler version of the replacement problem (6) with no operating cost (at c  0). They demonstrate that the optimal replacement policy follows different patterns depending on relations between the TC rates gy and gR in the productivity and replacement cost. Adding the third major characteristic, the operating cost c, leads to much reacher qualitative picture of optimal replacement trajectories in the model (6). In this paper, we identify key relations among parameters gy, gc, and gR, which define qualitative dynamics of rational replacement policies.

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Proof of Lemma 2. Let us we introduce the functions

b(t , u )  y (u  t )e

g yt

 c(u  t )e gct and p (t )  R0 e g Rt .

(A3)

In the terms of (A3), the problem (6) becomes equivalent to the problem (3)-(4) in (Hritonenko and Yatsenko 2009). Applying Theorem 1 from the above-mentioned paper, we obtain that then the solution of problem (6) satisfies the nonlinear integral equation



x 1 ( t ) t

e ru b(t , u )  b( x(u ), u ) du  e rt p(t ),

t  [0, ),

(A4) 

which is the equation (12) in the notations (A3). The Lemma is proven.

Proof of Theorem 1. The proof follows from analysis of the recurrent nonlinear equations (12). At gy = gc = gR = g, all the equations (10) are the same for any k, which leads to the nonlinear equation for L: L ~ ~ y ( L)  c~ ( L)  c~ (0)  ~ y (0)   e r a ~ y ' (a )  c~ ' (a )da   R0 (r  g ).

(A5)

0

Denoting

the

left-hand

side

of

(A5)

as

F(L),

we

obtain

that

F(0)

=

0

and

~ F ' ( L)   ~ y ' ( L)  c~ ' ( L) (1  e  r L ) < 0. By (13), F(L)   at L  . Therefore, the nonlinear

equation (A5) has a unique solution L > 0 at the theorem conditions. It is easy to see that

~ ~ ~ L1  L2  ...  L is the only solution of the system (12). Finally, applying the integration by parts to the equation (A5), we obtain (14). Theorem is proven.



Proof of Theorem 2. The equation (14) at the conditions (18) becomes

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

x 1 ( t ) t

 

e ru y0 e

 R0 e

( g R r )t

g y t  y ( u t )

e

g y x ( u )  y ( u  x ( u ))

 c e 0

g c t  c ( u t )



 e gc x (u )c (u  x (u )) du

(A6)

.

Taking the derivative of the integral equation (A6) with respect to t and doing routine transformations, we obtain the following (not integral) equation

y0

gy y r y



 y0 1  e

1  e

( r  y ) L ( x 1 ( t ))

( g y  y ) L ( t )

 c

  c 1  e 0

0





1 gc  c ( g  g )t 1  e ( r c ) L ( x (t )) e c y r  c

( g c  c ) L ( t )

e

( gc  g y )t

 (r  g R ) R0 e

( g R  g y )t

(A7)

.

The equation (A7) connects the lifetimes L(t) = t - x(t) and L(x-1(t)) = x-1(t) - t of two sequentially

~

~

replaced assets. Namely, Li  L(t ) and Li 1  L( x 1 (t )) , where t is the time of replacement of k-th asset with (k+1)-th asset. Taking the derivative in t of the nonlinear equation (A7), using the theorem of the derivative of the inverse function, and doing routine transformations, we obtain the following equation for the derivative x' (t ) :

y ( g 0

y

  y )e

( r  y ) L ( x 1 ( t ))



 y 0 ( g y   y )e

( r  y ) L ( t )

( gc  g y )t

e

 c0 ( g c   c )e ( r c ) L (t ) e

 (r  g R )( g R  g y ) R0 e  c0 ( g c  g y )e

 c0 ( g c   c )e

( r  c ) L ( x 1 ( t )) ( g c  g y ) t

( gc  g y )t



1  x' ( x 1 (t )) x' ( x 1 (t ))

1  x' (t )

(A8)

( g R  g y )t



1  g  c 1  e ( gc c ) L (t )  c 1  e ( r c ) L ( x (t )) r  c 

. 

The equations (A7) and (A8) at c0 = 0 were used in (Hritonenko and Yatsenko, 2009) to construct

~ ~ ~

the optimal solution {L1 , L2 , L3 ,...} . Here, we extend this technique to the case c0 ≠ 0.

~

~

Let us rewrite the recurrent equation (A7) in terms of Li and Li 1 as

y0

g y  y r  y



 y0 1  e

1  e

~ ( r  y ) Li 1

~ ( g y  y ) Li

 c

 c 1  e 0

0





~ gc c ( g  g )t 1  e ( r c ) Li 1 e c y r c

~ ( g c  c ) Li

e

( gc  g y )t

 (r  g R ) R0 e

(A9) ( g R  g y )t

.

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~ ~ ~

Next, we assume that a solution {L1 , L2 , L3 ,...} exists and consider several cases separately. Case gy > gc and gy = gR. Then, the recurrent equations (A7) approach the autonomous form

y0

g y  y r  y ~

1  e

~ ( r  y ) Li 1

 y 1  e

~ ( g y  y ) Li

0

~

  ( r  g ) R . R

0

~

at i. Therefore, Li and Li 1 approach a certain equal value L when i is large and i. Choosing

~ ~ ~ Li 1 = L at some large i >>1, we can find L from the recurrent formula (A7).

Similarly to (Hritonenko and Yatsenko 2009), this iterative process converges. To find whether

~ ~ ~

the sequence {L1 , L2 , L3 ,...} increases or decreases, we use the equation (A8) for the derivative

x' (t ) . Taking the limit of (A8) as t, we obtain that x' ( x 1 (t ))  1 and x' (t )  1 ; otherwise, ~

~

there is no solution x over [0,). Therefore, Li  Li 1 at i. At gy > gc and gy = gR, the righthand part of (A10) is positive. The only feasible situation appears when both x' ( x 1 (t ))  1 and

x' (t )  1 . It means that L' (t )  1  x' (t )  0 and, thus, the lifetime L(t) = t - x(t) monotonically ~

~

decreases. Therefore, Li 1  Li by (13). Case gy > gR and gy ≥ gc. Then, taking the limit of (A9) at i, we see that its right-hand

~

~

side approaches 0, and, therefore, both Li and Li 1 approach 0 when i. Similarly to the

~

~

previous case, we obtain that Li 1  Li . Case gc > gy. Then, the integrand of (A6) becomes negative starting with some time tcr and, therefore, the future replacements will become unprofitable and stop at a certain asset M before x(M) reaches tcr. Case gR > gy. Then, the right-hand side R0e( g R  r ) t of (A6) is asymptotically larger than the left integral of (A6), whose limit is  const  lim e t 

( g y  r )t

. Therefore, the equation (A6) cannot be

satisfied at large t, so, the replacements will become unprofitable and stop at some time tcr. 27

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

The theorem is proven.

Proof of Lemma 3. For a fixed k=1.,…,N1, each unknown variable k appears four times in the objective function (24). Differentiating (24) in k, k=1.,…,N-1, we obtain (25). Next, differentiating (25) in N, we obtain

AP[ 0, N ]  N

~ r 2e r  N  ~ (1  e r  N ) 2 ~



N 1



 k 1 ~ ~ ( g  g )u ( g  g )     e r u ~ y (u   k )  c~ (u   k )e c y du  e r  k 1 R0 e R y k 1  (A10)    k 0  k ~ ~ ~ r e r  N ~ ( g  g )  y ( N   N 1 )  c~ ( N   N 1 )e c y N 1  e r  k 1 (~ r  g ) R0 e  g k 1 . ~ rN 1 e





A solution to the optimization problem (23) should satisfy the optimality conditions

AC[ 0, N ] /  i  0 , for i =1,…,N. Using (A10) and (2), we obtain (25) and (26). 

The Lemma is proven.

Proof of Theorem 4. At gy = gc = gR, all equations (25) are the same for i =1,…,N-1. It is easy to check that N

N

L1  L2  ...  LN

N

is the only solution of the system (25)-(26) in this case. Indeed, the

N

substitution of Li  L, i  1,...N to (25) leads to the equations L ~ ~ ~ (~ y ( L)  c~ ( L))(1  e  r L )  ~ r  e  r a f (a )da   R0 ~ r, 0

i =1,…,N-1.

(A11)

Now, let us show that the equation (26) has the same form (A11). Substituting N

N

N

L1  L2  ...  LN  L and N = LN into (26) and using (24) with the new integration variable a  u   k , we obtain

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~ y ( L)  c~ ( L)  ~ r R0 

~ r ~ 1  e  r LN

N 1

e

~ r Lk

k 0

N 1

Applying the geometric series formula

 e r Lk  k 0

~ y ( L)  c~ ( L) 

~ (a))da  e  ~r L R  .  L e  ~r a ( ~ y ( a )  c 0  0  ~

(A12)

~

1  e  r LN , (A12) leads to ~ 1  e r L

~ r  L e  ~r a ( ~ y (a )  c~ (a ))da  R0  , ~ r L  0  1 e 

(A13)

which is equivalent to the equation (A11). Therefore, the equal life solution of N

Li  L, i  1,...N , satisfies all N optimality conditions (25) and (26). Theorem is proven. 

Acknowledgements. The authors are thankful to three anonymous reviewers for valuable comments that essentially improved the clarity of the paper. The paper is supported by the Ministry of Education and Science of Kazakhstan under Grant AP05131784. Natali Hritonenko acknowledges support of PVAMU FIE.

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Figure 1. The optimal asset lifetime L1 calculated by the four-cycle algorithm (20) and known one-cycle methods for different rates gc of operating cost improvement. The result is the same at stationary cost (when gc = gy = gR = 0).

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Figure 2. Increasing lifetimes L1 < L2 < L3 < L4 of sequentially replaced assets for different rates gc < 0 of cost-saving technological change and stationary productivity gy = 0.

Figure 3. Decreasing lifetimes L1 < L2 < L3 < L4 of sequentially replaced assets at outputaugmenting technological change gy > 0 and stationary costs gc = 0.

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Figure 4. Two types of the dynamics of multi-cycle replacement policy: with increasing (a) and decreasing (b) consecutive asset lifetimes. Case (a) occurs at gc = 0, gy = gR = 0.005, case (b) occurs at gc = 0.02, gy = gR = 0.005.

Figure 5. Five-cycle replacement policy at fixed rates of output-augmenting TC (gy = gR = 0.005) and varying intensity gc of cost-saving TC. .

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Figure 6. Optimal lifetimes L1N of the first replacement for N =1,2,3,4,5 at different technological change: gy = gR = 0.005 and varying gc. The third graph (gray straight line) occurs at proportional technological change gy = gc = gR = 0.005.

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Optimal asset replacement: Profit maximization under varying technology

Yuri Yatsenko: Conceptualization, Methodology, Software, Writing- Original draft preparation. Natali Hritonenko: Formal analysis, Investigation, Resources, Writing- Reviewing and Editing