Algorithms for asset replacement under limited technological forecast

Int. J. Production Economics ∎ (∎∎∎∎) ∎∎∎–∎∎∎

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Algorithms for asset replacement under limited technological forecast Yuri Yatsenko a,n, Natali Hritonenko b a b

School of Business, Houston Baptist University, 7502 Fondren, Houston, TX 77074, USA Department of Mathematics, Prairie View A&M University, Prairie View, TX 77446, USA

art ic l e i nf o

a b s t r a c t

Article history: Received 21 January 2014 Accepted 27 August 2014

The optimal asset replacement is analyzed when the future course of technological change is known on a limited future horizon. Comparison of factual and desired properties of known replacement methods leads us to the idea of how to improve their efficiency under changing technology reflected in decreasing operating and new asset costs. We introduce new modifications of the economic life and two-cycle variable-horizon methods by correcting their capital recovery factor. Next, we demonstrate that the modified methods deliver solutions equal or close to the infinite-horizon replacement under technological change. & 2014 Elsevier B.V. All rights reserved.

Keywords: Asset replacement models Economic life Technological change Finite horizon Optimization

1. Introduction The optimal replacement of productive assets under changing operating and maintenance costs caused by technological advances is a problem of enormous theoretical complexity and practical importance (Hartman and Tan, 2014). This paper focuses on a theoretic analysis of deterministic asset replacement methods when the future course of technological change is known on a limited forecast horizon only. On the basis of a comparative theoretic analysis and numeric simulation of existing replacement methods, we introduce their new modifications that deliver solutions equal (or close) to the infinite-horizon replacement policy under improving technology. The known replacement algorithms can be grouped in the following categories: (IH) The infinite-horizon minimization of the present value of the total replacement cost chooses the infinite sequence of the future lifetimes of a sequentially replaced asset (Elton and Gruber, 1976; Sethi and Chand, 1979; Bethuyne, 1998; Regnier et al., 2004; Yatsenko and Hritonenko, 2005, 2008, 2010). (EL) The economic Life method minimizes the asset's total equivalent annual cost by choosing the optimal replacement time of current asset (Thuesen and Fabrycky, 1993; Newman et al., 2004; Hartman, 2007). This method is equivalent to the infinite-horizon optimization in the case of stationary asset costs. In contrast to common opinion (Hartman, 2007) that the

n

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

EL method does not consider improving technology, we demonstrate that it can take the changing cost of new assets into account. (VH) The variable-horizon replacement method of Christer and Scarf (1994) minimizes the annual replacement cost over two future replacement cycles. (FH) The fixed-horizon replacement determines a finite sequence of the future asset replacements to minimize the present value of the replacement cost over a given finite horizon (Hritonenko and Yatsenko, 1996, Scarf and Hashem, 2002). We will skip this method from consideration because it involves significant end-of-horizon effects and is less efficient than the variable-horizon method (Scarf and Hashem, 2002, Yatsenko and Hritonenko, 2005, 2011). The goal of this paper is to construct replacement methods that use limited technological forecast data but produce the best results (at least, the time of first replacement) for the infinitehorizon technological forecast. Correspondingly, our ideal benchmark problem is the infinite-horizon replacement. It is known (Regnier et al., 2004, Yatsenko and Hritonenko, 2011) that the economic life and variable-horizon methods under technological change produce results different from the infinite-horizon replacement. At the same time, comparing factual and desired properties of these methods allows us to improve their efficiency under continuing technological improvement. Specifically, we introduce modified economic life and variable-horizon methods with a corrected capital recovery factor that compensates the influence of further unconsidered horizon. We prove that both modified methods deliver the best (infinite-horizon) solution if the observed technological change is exponential and affects equally both the operating and new asset costs (the so-called proportional technological change). Namely, the suggested methods find the

http://dx.doi.org/10.1016/j.ijpe.2014.08.020 0925-5273/& 2014 Elsevier B.V. All rights reserved.

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infinite-horizon optimal asset lifetime for proportional technological change and arbitrary age-dependent profile of deterioration. In the case of non-proportional exponential technological change, the optimal asset lifetime is non-constant and depends on the cost components: the optimal lifetime of sequentially replaced assets increases if the operating cost decreases faster than the new asset cost, and converse (Regnier et al., 2004, Yatsenko and Hritonenko, 2011). Then, we use numeric simulation to show that the modified VH method is the best option and produces the solution closest to the infinite-horizon optimization. The paper is as follows. Section 2 describes the economic life, variable-horizon, and infinite-horizon replacement methods and introduces their modifications in a continuous-time serial replacement model. Section 3 provides necessary background for theoretical analysis. Section 4 compares the methods analytically and numerically and demonstrates a superior performance of the modified replacement methods under various scenarios of improving technology. Section 5 concludes and gives some practical recommendations.

2. Formulation of asset replacement methods under study Let us consider a firm that needs to replace periodically a single asset with new assets that perform identical operations but have different replacement costs. We will describe this process in the continuous time t A [0,1). The changing economic-technological environment is represented by the following functions:

 P(t) is the cost (purchase price and installation cost) of a new asset at time t (of vintage t).

 A(t,u) is the operating and maintenance cost at time u for the



asset bought at time t. Then, the variable u  t is the age of the asset, 0r u  trM, and M is the maximal physical service life of assets. S(t,u) is the salvage value of the asset of age u  t bought at time t, 0 rS(t,u) oP(t).

The technological change leads to the availability of newer assets that require less maintenance and/or are less expensive, so P(t) and A(t,u) decrease in t. The maintenance cost A(t,u) usually increases in the age u t (as the asset becomes older) because of physical deterioration, however, it can also decrease because of learning (Goetz et al., 2008). The general form of the function A(t,u) can depict various hypotheses of deterioration and learning. To calculate actual replacement costs over a finite horizon, the replacement theory commonly uses the capital recovery factor Rðr; TÞ that converts the present value of a certain cost over a specified future interval into the sequence of equivalent annual costs. Under the assumption of continuous compounding, the annual capital recovery factor over the interval [0,T] is as follows: Rðr; TÞ ¼

r 1  e  rT

ð1Þ

where r 40 is the instantaneous discount rate. One can see that Rðr; TÞ-1=T at r-0, which is consistent with the continuous-time analysis of the zero-discounting case by Scarf and Hashem (2002). To describe the process of sequential replacement of the single asset with a new asset, we introduce the endogenous lifetime Lk of the k-th asset, k¼ 1,2,…. For clarity, we assume that the first asset is purchased at time t¼0 and will be replaced at the end of its lifecycle, then the time of introducing the first asset is τ0 ¼0 and the time of its replacement with the second asset is τ1 ¼L1. Correspondingly, the time τk of the replacement of k-th asset with (k þ1)-th asset is as follows: τk ¼ τk  1 þ Lk ¼ Σ kj ¼ 1 Lj :

ð2Þ

The asset replacement cost: The present value of the total replacement cost of the k-th asset, k ¼1,2,…, over its future lifetime Lk is calculated at a given industry-wide discount rate r 40 as follows (Regnier et al., 2004, Hartman, 2007): PV k ðLk ; τk  1 Þ ¼ e  rðτk  1 þ Lk Þ ½Pðτk  1 þ Lk Þ  Sðτk  1 ; τk  1 þ Lk Þ Z τ k  1 þ Lk þ e  ru Aðτk  1 ; uÞdu τk  1

ð3Þ

The first term of (3) represents the discounted cost of the new asset minus the discounted salvage value of the current asset, and the integral is the discounted maintenance costs over the future lifetime of the current asset. Next, we provide mathematical formulations of the replacement methods under study. 2.1. Infinite-horizon replacement The infinite-horizon replacement method (Regnier et al., 2004; Hartman, 2007; Hritonenko and Yatsenko, 2007, 2008) assumes the knowledge of external technological parameters P, A, and S on the infinite horizon [0,1) and determines the infinite optimal sequence of consecutive asset lifetimes Lk, k ¼1,2,…, that minimizes the present value of the total replacement cost over [0,1): PV1 ðL1 n ; L2 n ; :::Þ ¼

min

Lk ;k ¼ 1;:::;0 o Lk r M

1

PV1 ðL1 ; L2 ; :::Þ;

PV1 ðL1 ; L2 ; :::Þ ¼ ∑ PVk ðLk ; τk  1 Þ; k¼1

ð4Þ

ð5Þ

where PVk is given by (3) and τk is determined from (2). In contrast to the infinite-horizon optimization, other replacement methods work in the case of a limited technological forecast. Namely, we assume that the technological parameters P(t), A(t,u), and S(t,u) are known for 0 rtru rT o1 on some finite future interval [0,T], where the value T is not smaller than the unknown lifetime L1 of the current asset. To ensure that, it is enough to assume T ZM. 2.2. Economic life replacement The economic life (EL) method determines the first asset lifetime that minimizes the equivalent annual cost of the first asset replacement (Thuesen and Fabrycky, 1993). C 1 ðL1 Þ ¼ Rðr; L1 ÞPV1 ðL1 ; 0Þ:

ð6Þ

By the EL method, the optimal lifetime EL1 of the first asset is determined as follows: EL1 ¼ arg minC 1 ðLÞ: 0oLrM

ð7Þ

To find the first optimal lifetime EL1, it is enough to know the cost P(t) and sequences S(0,t) and A(0,t) over the future interval [1,EL1] only. In the general case, the EL method produces different optimal lifetimes EL1, EL2,…, for sequentially replacements k ¼1,2,3,… of the asset. Finding the first optimal lifetime EL1 is the most relevant task in engineering practice. A common consensus in the operations research replacement theory is that the EL method cannot take technological change into account. Fortunately, it is true only partially. Indeed, the above version (6) of the EL method assumes replacement at the end of the current asset lifecycle and so, in fact, considers possible technological improvement as the change of the new asset cost PðL1 Þ. At the same time, the EL method (4) and (5) does not consider improvements in the maintenance cost at all. In Section 2.4 below, we will offer a modified EL method that addresses this drawback.

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2.3. Variable-horizon replacement method

2.5. Modified VH method

The variable-horizon (VH) replacement method was first introduced in (Christer and Scarf, 1994) to address technological development, changing discount rate, and other actual issues of replacement decision (Scarf and Hashem, 2002). It finds the optimal replacement policy ðVL1 n ; VL2 n Þ that minimizes the equivalent annual cost over two future replacement cycles

The modified VH method finds the optimal replacement policy ðVL1 n ; VL2 n Þ that minimizes the corrected equivalent annual cost over the two future replacement cycles:

ðVL1 n ; VL2 n Þ ¼ argminC 2 ðL1 ; L2 Þ;

C^ 2 ðL1 ; L2 Þ ¼ Rðr þ c; L1 þ L2 Þ ∑ PV k ðLk ; τk  1 Þ ¼

ð8Þ

L1 ;L2

ðVL1 n ; VL2 n Þ ¼ argminC^ 2 ðL1 ; L2 Þ;

ð12Þ

L1 ;L2

2

k¼1

2

r

k¼1

1  e  rðL1 þ L2 Þ

C 2 ðL1 ; L2 Þ ¼ Rðr; L1 þ L2 Þ ∑ ½PV k ðLk ; τk  1 Þ ¼  Z  e  rL1 PðL1 Þ þ e  rL1 Sð0; Lk Þ þ

L1

 Z  e  rL1 PðL1 Þ þ e  rL1 Sð0; L1 Þ þ

e  ru Að0; uÞdu

L1

e  ru Að0; uÞdu

0

þ e  rðL1 þ L2 Þ PðL1 þ L2 Þ þ e  rðL1 þ L2 Þ SðL1 ; L1 þ L2 Þ  Z L1 þ L2 e  ru AðL1 ; uÞdu þ

0

þ e  rðL1 þ L2 Þ PðL1 þ L2 Þ þ e  rðL1 þ L2 Þ SðL1 ; L1 þ L2 Þ  Z L1 þ L2 e  ru AðL1 ; uÞdu : þ

L1

r þc 1  e  ðr þ cÞðL1 þ L2 Þ

L1

ð9Þ

The more common finite-horizon replacement minimizes the discounted replacement cost over a given planning horizon of a fixed length (Hritonenko and Yatsenko, 1996, Scarf and Hashem, 1997, Hartman and Murphy, 2006, Hartman, 2007, Hartman and Tan, 2014). In this method, the length of the horizon is given by management considerations (for example, the length of a contract). The VH method (8)–(9) does not fix the horizon length and employs a finite horizon because of data availability and algorithmic logic. The fixed-horizon method coincides with the VH method only when the given horizon has the optimal length ðL1 n ; L2 n Þ. In general, it performs worse than the VH method (Scarf and Hashem, 2002, Yatsenko and Hritonenko, 2011), so, we do not consider the fixed-horizon replacement here. The two-cycle VH replacement method (8)–(9) was analyzed in (Scarf and Hashem, 1997, 2002, Yatsenko and Hritonenko, 2011) and recommended for use under improving technology, mainly, because it performs better than the EL method. Minor modifications of the two-cycle method were exploited in (Scarf et al., 2007, Mardin and Takeshi, 2012). However, a more scrupulous analysis discovers drawbacks of this method and shows simple ways to improve its efficiency in situations with continuous technological improvement. Specifically, we introduce the following new modifications of the EL and VH replacement algorithms.

ð13Þ

2.6. Asset replacement methods in discrete time Because of practical needs, the Operations Research often describes the replacement problem in a discrete time. The above replacement algorithms can be reformulated in the discrete time i, i¼1,2… (years). The discrete-time analog of the capital recovery factor (1) over the interval [0,T] is the well-known annual compounding capital recovery factor (Newman et al., 2004, p. 89; Hartman, 2007, p. 121): dð1 þdÞT ~ Rðd; TÞ ¼ ; ð1 þdÞT  1

ð14Þ

2.4. Modified EL method

where d4 0 is the given annual discount factor, related to the instantaneous discount rate r in (1) as d ¼ er  1. Discrete-time analogs of the above replacement algorithms are described by the formulas similar to (4)–(13) and obtained by using the capital recovery factor (14) instead of (1) and replacing the integrals in the above formulas with sums. Discrete-time versions of infinite-horizon, EL, and VH methods are provided in (Yatsenko and Hritonenko, 2011). The authors are planning to analyze discrete versions of the modified EL and VH methods in the next paper. We shall notice that, in the discrete time, all the above optimization problems require their unknown variables (L1, L2,…) to be integer-valued and, as a result, belong to integer programming problems. Qualitative properties of such optimization problems are very difficult to analyze (Yatsenko and Hritonenko, 2011, Regnier et al., 2004), so we restrict our analysis to the continuous time in this paper.

To address continuous technological change, we introduce the efficient capital recovery factor

3. Theoretical fundamentals

^ c; LÞ ¼ Rðr þc; LÞ; Rðr;

ð10Þ

where c is an aggregate rate of technological change. The rationale for formula (10) and estimations of the rate c for various types of technological change will be given later and based on the comparison of the factual and desired properties of the EL and VH ^ c; LÞ methods. In Section 4, we will demonstrate that using Rðr; instead of Rðr; LÞ in the EL and VH algorithms significantly improves their efficiency. The modified EL method determines the lifetime L1 that minimizes the corrected equivalent annual cost of the first asset replacement L1 ¼ arg minC^ 1 ðLÞ; 0oLrM

C^ 1 ðLÞ ¼ Rðr þ c; LÞPV 1 ðL; 0Þ;

^ c; LÞ is used instead of Rðr; LÞ as in (6). in which Rðr;

ð11Þ

Five continuous-time optimization problems (4)–(5), (6)–(7), (8)–(9), (11), and (12)–(13) can be investigated using standard tools of the nonlinear optimal control theory such as gradients and derivatives. This section derives some new results and systemizes known ones, which will be used in Section 4 for comparative analysis of these problems. For clarity sake, we make two following assumptions about the external technological parameters: (A) The purchase cost P(t) and the maintenance cost A(t,u) are subjected to the gradual exponential technological improvement with different rates: PðtÞ ¼ Pe  cp t ;

Aðt; uÞ ¼ f ðu  tÞe  cq t ; 0 r t r u r T;

ð15Þ

where the function f(u  t) describes an arbitrary deterioration profile of the asset with its age u  t.

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(B) The salvage value is negligible for the analysis purposes1 : S (t,u)¼0. Under the exponential technological change (15), we will choose its aggregate rate as c¼ cq in the formula (10). Correspondingly, the efficient capital recovery factor is as follows: ^ cq ; LÞ ¼ Rðr þ cq ; LÞ Rðr;

ð16Þ

in the modified EL method (11) and the modified VH method (12)– (13). It will be shown in Section 4 that the choice c ¼cq is optimal in the sense that it produces the best possible results for exponential technological change (15).

The first step of our analysis is to derive the necessary conditions for optimality for the optimization problems (4)–(5), (6)–(7), (8)–(9), (11), and (12)–(13).

3.1.1. Optimality condition for the infinite-horizon optimization (4)–(5) If an optimal policy {Lnk , k¼ 1,2,…} exists, then it satisfies the following conditions. Z

Lk þ 1 0

e  ru f ðuÞdu ¼

r þ cp ðcq  cp Þτk Pe : r þ cq ð17Þ

for k¼ 1,2,… In particular, at k ¼1, the optimal L1 and L2 are connected by the following equation: f ðL1 Þ f ðL2 Þe  rL2  cq L1  e  cq L1 r þ cq

Z

L2 0

e  ru f ðuÞdu ¼

r þcp  cp L1 Pe : r þ cq

ð18Þ

Proof. By Theorem 1 of Yatsenko and Hritonenko (2011), the optimal Lnk , k ¼1,2,… satisfy the countable system Z τ k þ Lk ∂Aðτk ; uÞ Aðτk ; τk Þ þ Aðτk  1 ; τk Þ þ erτk e  ru du ¼ rPðτk Þ P 0 ðτk Þ: ∂τk τk ð19Þ Under assumptions (15), Eq. (19) and ∂Aðt; uÞ=∂t ¼ e cq f ðu  tÞÞ lead to Z f ðLk Þecq Lk f ð0Þ 

Lk þ 1 0

1  e  ðr þ cq ÞL f ðLÞ  e  cq L r þ cq

Z 0

L

e  ru f ðuÞdu ¼ P

  r þ cp  cp L cq  cp  ðr þ cq ÞL : e 1þ e r þ cq r þ cq

ð22Þ The proof follows from taking the derivative of the function C^ 1 ðL1 Þ given by (11) and setting it equal to zero. 3.1.4. Optimality condition for VH algorithm (12)–(13) If an optimal value policy ðL1 n ; L2 n Þ exists, then it satisfies the conditions C 2 ðL1 ; L2 Þ ¼ e  cq L1 f ðL2 Þ  ðr þ cp ÞPe  cp ðL1 þ L2 Þ ;

3.1. Optimality conditions

f ðLk Þecq Lk f ðLk þ 1 Þe  rLk þ 1  r þ cq

3.1.3. Optimality condition for modified EL algorithm (11) If an optimal value L1 n exists, then it satisfies the condition

 cq t

 Z C 2 ðL1 ; L2 Þ ¼ erL2 f ðL1 Þ ðr þ cq Þe  cq L1

0

ð23Þ

 L2 e  ru f ðuÞdu

  ðr þ cp ÞPe  cp L1 erL2 þ e  cp L2 :

ð24Þ

Excluding the unknown C2 from (20) to (21), the optimal L1 and L2 are connected by the following equation: Z L2 r þ c p  c p L1 f ðL1 Þ  f ðL2 Þe  cq L1  rL2 e  cq L1 e  ru f ðuÞdu ¼ Pe : ð25Þ r þ cq r þ cq 0 Proof. Let us rewrite (9) in the case (15) as follows:  Z L1 r  ðr þ cp ÞL1 C 2 ðL1 ; L2 Þ ¼ Pe þ e  ru f ðuÞdu 1  e  rðL1 þ L2 Þ 0  Z L2 e  ru f ðuÞdu : þ Pe  ðr þ cp ÞðL1 þ L2 Þ þ e  ðr þ cq ÞL1 0

ð26Þ

At the optimal values L1 and L2, the function (26) satisfies the standard optimality condition ∂C 2 ðL1 ; L2 Þ ¼ 0; ∂L1

∂C 2 ðL1 ; L2 Þ ¼ 0: ∂L1

ð27Þ

Multiplying both parts of (26) by ð1  e  rðL1 þ L2 Þ Þ=r, taking the derivatives in L1 and L2 and using (27), we obtain that the optimal values of L1 and L2 should satisfy Z L2 e  rðL1 þ L2 Þ C 2 ðL1 ; L2 Þ ¼ e  rL1 f ðL1 Þ  ðr þ cq Þe  ðr þ cq ÞL1 e  ru f ðuÞdu 0

 ðr þ cp ÞPe  ðr þ cp ÞL1 ½1 þ e  ðr þ cp ÞL2 

0

ðf ðu  tÞ and

e  rðL1 þ L2 Þ C 2 ðL1 ; L2 Þ ¼ e  cq L1  rðL1 þ L2 Þ f ðL2 Þ ðr þ cp ÞPe  ðr þ cp ÞðL1 þ L2 Þ :

0

e  rv ðf ðvÞ  cq f ðvÞÞdv ¼ ðr þcp ÞPeðcq  cp Þτk : ð20Þ

Using the integration by parts and other transformations to simplify (19), we obtain (17). □

After routine transformations, the last two equations lead to (23)–(24). □ 3.1.5. Optimality condition for modified VH algorithm (12)–(13) If an optimal value policy ðL1 n ; L2 n Þ exists, then it satisfies the conditions

3.1.2. Optimality condition for EL algorithm (6)–(7) If an optimal value ELn exists, then it satisfies the equation

C^ 2 ðL1 ; L2 Þ ¼ ecq L2 f ðL2 Þ  ðr þ cp ÞPeðcq  cp ÞðL1 þ L2 Þ ;

  Z L 1  e  rL 1  e  rL f ðLÞ  : e  ru f ðuÞdu ¼ Pe  cp L 1  cp r r 0

C^ 2 ðL1 ; L2 Þ ¼ erL2 ecq ðL1 þ L2 Þ f ðL1 Þ  ðr þ cq Þeðr þ cq ÞL2

ð21Þ

The proof follows from differentiation of the function C 1 ðLÞ ¼ h i RL r e  rL Pe  cp L  0 e  ru f ðuÞdu obtained from (6), (1), and (3) in 1  e  rL case (15). Setting the derivative C1 (L1) equal to zero leads to the Eq. (21) for the optimal ELn. 1 A sensitivity analysis demonstrates that the salvage value does not affect the optimal life until it reaches 80–90% of the new asset cost, which is impractical (Regnier et al., 2004, Yatsenko and Hritonenko, 2011).

ð28Þ Z

L2

0 ðcq  cp ÞL2

 ðr þ cp ÞPeðcq  cp ÞL1 ½eðr þ cq ÞL2  e

e  ru f ðuÞdu

:

ð29Þ

The proof is similar to the above proof of Eqs. (23)–(24) with using (13) instead of (8). Excluding C^ 2 from (28) to (29), we obtain that the optimal L1 and L2 are connected by the same Eq. (25) as shown in the VH method. However, as the analysis of Section 4 demonstrates, the optimal L1 and L2 are different from the VH method in a general case.

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3.2. Like-with-like replacement In this section, we prove the equivalence of all five replacement methods under study in the stationary environment with no technological improvement when the asset replacement costs do not depend on the current time. The optimal replacement policy in this case is also time-independent and known as the like-with-like replacement. Theorem 1. Under stationary asset costs (that do not depend on current time) and an arbitrary age-dependent deterioration f(u), the first asset lifetime L1 is the same in the infinite-horizon, EL, modified EL, VH, and modified VH methods. The second lifetime L2 ¼L1 in the VH and modified VH methods, and Lk ¼ L1 for k¼ 2,3,… in the infinitehorizon method. Proof. This result for the EL algorithm is well known in engineering economics textbooks. The formula for the optimal L¼ L1 is as follows (Yatsenko and Hritonenko 2008): Z L 1  e  rL f ðLÞ  e  ru f ðuÞdu ¼ P: ð30Þ r 0 For the VH algorithm, the result of the theorem was earlier obtained in Scarf and Hashem (2002) in the case r ¼0. To prove that it remains true for the VH algorithm with discounting, we assume L1 ¼ L2 and demonstrate that it satisfies the optimality condition. Indeed, substituting L¼L1 ¼L2 to (25), we obtain the Eq. (30) for the optimal L. By (11)–(13), the modified EL and VH algorithms coincide with their standard counterparts in the case of like-with-like replacement. The theorem is proven. □

(4)–(5) are equal: Lk ¼Ln, k¼ 1, 2,…, where Ln is found from the nonlinear equation Z L 1  e  ðr þ cÞL f ðLÞ  e  cL e  ru f ðuÞdu ¼ Pe  cL : ð31Þ r þc 0 The modified EL and modified VH algorithms produce the same first asset lifetime L1 ¼ Ln and L2 ¼L1 ¼Ln in the modified VH method. The first asset lifetime L1 obtained by the original EL or VH algorithms is larger than Ln (and L2 4L1 in the VH algorithm). Proof. The objective function (5) of the infinite-horizon problem (3)–(5) under assumption given in (15) is as follows: 1  PW1 ðL1 ; L2 ; :::Þ ¼ ∑ e  rðτk  1 þ Lk Þ Pe  cðτk  1 þ Lk Þ k¼1 Z τ k  1 þ Lk

þ

This section contains major theoretical results of the paper that have essential practical importance. First, we consider the unlimited forecast with arbitrary age-dependent deterioration and technological improvement that affects both asset costs equally. We analytically prove that the modified EL and VH methods deliver the same optimal asset lifetime as the infinite-horizon replacement, while the original EL and VH methods do not deliver it. Next, we provide numeric simulation to demonstrate that the modified VH algorithm produces solutions very close to the infinite-horizon replacement for an arbitrary exponential technological improvement.

4.1. Case of proportional technological improvement

τk  1

 e  ðr þ cÞu f ðu  τk  1 Þdu

and, after the substitution v ¼ u  τk  1 ,  Z 1 PW 1 ðL1 ; L2 ; :::Þ ¼ ∑ e  ðr þ cÞτk  1 Pe  ðr þ cÞLk þ k¼1

Lk 0

 e  ðr þ cÞv f ðvÞdv : ð32Þ

Applying the technique used in (Regnier et al., 2004; Yatsenko and Hritonenko, 2011) for discrete analogs of the IH problem (3)–(5), we rewrite the expression (32) as follows:   Z L1 PW1 ðL1 ; L2 ; :::Þ ¼ Pe  ðr þ cÞL1 þ e  ðr þ cÞv f ðvÞdv 1

0

þ e  ðr þ cÞL1 ∑ e  ðr þ cÞðτk  1  L1 Þ k¼2

 Z  Pe  ðr þ cÞLk þ

Lk

0

n

4. Replacement algorithms under exponentially improving technology

5

n

 e  ðr þ cÞv f ðvÞdv :

n

Let ðL1 ; L2 ; L3 ; :::Þ be a solution of the problem (3)–(5). Because L1 ¼L1 n is optimal, the remaining optimal sequence ðL2 n ; L3 n ; :::Þ must minimize the series   Z Lk 1 e  ðr þ cÞv f ðvÞdv : ð33Þ ∑ e  ðr þ cÞðτk  1  L1 Þ Pe  ðr þ cÞLk þ k¼2

0

Let us introduce the new (shifted by L1 to the right) sequence of the replacement times τ^ k ¼τk 1  L1 for k¼1,2,3,…, then τ^ 1 ¼ 0, τ^ 2 ¼ L2 , and τ^ k ¼ ∑kj ¼ 2 Lj by (2). Now, if we replace the summation index k in (32) with k0 ¼k 1, then (33) is identical to the original objective function (32). Therefore, the optimal ðL2 n ; L3 n ; :::Þ ¼ ðL1 n ; L2 n ; :::Þ, which is possible only if Lnk ¼Ln for k¼ 1,2,…. Substituting the last formula to formula (17), we obtain the Eq. (31) for Ln. To prove the result of the theorem for the Modified VH algorithm, we assume L1 ¼L2 and demonstrate that it satisfies the optimality condition. Indeed, substituting L¼ L1 ¼L2 to the objective function (13), we obtain  Z L r þc  ðr þ cÞL Pe þ e  ru f ðuÞdu þPe  2ðr þ cÞL C^ 2 ðL1 ; L2 Þ ¼ 1  e  2ðr þ cÞL 0  Z L e  ru f ðuÞdu þe  ðr þ cÞL 0

Definition. The exponential technological change (15) with equal rates cp ¼cq is referred to as proportional. The technological change is called non-proportional if cp a cq. The proportional technological change was earlier used by Regnier et al., (2004) in a discrete-time replacement model of type (4)–(5) and by Hritonenko and Yatsenko (2007, 2008) in continuous-time replacement problems. Theorem 2. (Proportional exponential technological change). Let cp ¼cq ¼c 40 in (15) hold over [0,1). Then, the optimal lifetimes of the consecutively replaced asset in the infinite-horizon replacement

  Z L ðr þ cÞð1 þe  ðr þ cÞL Þ  ðr þ cÞL  ru Pe þ e f ðuÞdu ¼ 1  e  2ðr þ cÞL 0   Z L r þc  ðr þ cÞL  ru Pe þ e f ðuÞdu : ¼ 1  e  ðr þ cÞL 0

ð34Þ

Substituting the formula (34) and L¼ L1 ¼L2 to the optimality conditions (24)–(25), we obtain that both Eqs. (24)–(25) lead to the same Eq. (31). The EL algorithm gives the optimal solution L found from the equation   Z L 1  e  rL r þ c  cL c  rL e e f ðLÞ  : ð35Þ e  ru f ðuÞdu ¼ P 1 r r þc r 0

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A comparative analysis of Eqs. (31) and (35) shows that the solution L is larger than the solution of (31). A similar result holds for the VH algorithm. The theorem is proven. □ Theorem 2 justifies that the choice c¼cq of the technological change rate in the corrected capital recovery factor (10) is made in an optimal way, in the sense that it leads to the same solution as the infinite-horizon replacement. It happens because the structure of optimality conditions of the modified EL and VH methods is closer to the infinite-horizon optimality condition. Namely, at c¼ cp ¼cq, the optimality conditions (22) and (28) for the modified methods coincide with the infinite-horizon optimality condition (18). A meaningful interpretation of the corrected capital recovery factor (10) is that it compensates the influence of improving technology over the further horizon [L1,1) or [L2,1), which is not considered in the original EL and VH methods. Theorem 2 has essential practical implications. If the observed technological improvement is exponential and follows the proportional rule (15), then both modified EL and VH methods can be used to find the optimal asset lifetime over the infinite horizon for arbitrary age-dependent deterioration profiles f.

We have implemented an extensive computer simulation of the above replacement algorithms for different values of the cost parameters cp, cq, and cd. The simulation confirms the superior performance of the modified EL and VH algorithms. Particularly, their rounded up to one year lifetime always is the same as in the infinite-horizon optimization. In contrast, the original EL and VH algorithms produce suboptimal lifetimes under improving technology. The Table 1 and Figs. 1–3 demonstrate selected simulation results for some extreme combinations of the technological change rates cp and cq. The first group of results in Table 1 (for cp ¼ 0 and cq ¼0.01, 0.02, 0.03, 0.05, 0.1) analyzes the situation when technological improvements lead to a decrease in the maintenance cost while the new asset cost remains constant. This is the case when the modified EL and VH methods are the most advantageous. Then, the original EL method gives the same first lifetime of 20.88 years and is completely unacceptable. Both modified EL and VH methods produce good results. However, the Modified VH method gives the infinitehorizon solution for L1 with an error of less than 0.01% (and a less accurate approximation for the second lifetime L2). It is also interesting that the simulated optimal lifetime of the first asset is smaller for

4.2. Case of non-proportional technological improvement If the dynamics of the new capital cost and maintenance cost are different (i.e., cp acq), then a fundamental challenge appears that the optimal lifetimes of sequentially replaced assets are not the same in the infinite-horizon replacement (4)–(5). Properties of the replacement methods under study become more complex. The infinitehorizon replacement problem (4)–(5) under non-proportional exponential technological change is investigated in (Yatsenko and Hritonenko, 2005, 2008). The optimal dynamics of asset replacement policy appears to depend on the cost component influenced by improving technology: the varying asset lifetime decreases in time if the new asset cost decreases faster than the maintenance cost (cp 4cq), and converse. This relation for first two lifetimes L1 n and L2 n is observed below in our simulation. In this paper, we focus on the first lifetime L1 (the time of the replacement of the current asset introduced at t ¼0) for all methods under study. The practical importance of the first service life L1 n is highlighted in the replacement theory and practice (Hartman, 2007, Christer and Scarf, 1994). After the optimal first replacement L1 n is determined, one can later repeat the solution with a new refined forecast for technological data over a rolling horizon [L1 n , L1 n þT], and so on. Possibilities of qualitative analysis are limited at cp acq because of the complexity of Eqs. (22)–(29). To get an appropriate answer for practical purposes, we provide a numeric comparison of the replacement policies under study. As a simulation dataset, we choose the exponential cost functions (15) with the exponentially increasing age-dependent maintenance cost f ðuÞ ¼ A0 ecd u

Table 1 The first (L1) and second (L2) asset lifetimes for different parameters cp and cq. cp

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.01 0.02 0.03 0.05 0.1 0.00 0.01 0.02 0.03 0.05 0.1

cq

0.00 0.01 0.02 0.03 0.05 0.1 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.01 0.02 0.03 0.05 0.1

EL Modified VH method method EL method

Modified VH method

Infinite-horizon optimization

L1

L1

L1

L2

L1

L2

Ln1

Lnn 2

20.88 20.88 20.88 20.88 20.88 20.88 20.88 19.80 18.85 18.02 16.61 14.04 20.88 19.80 18.85 18.02 16.61 14.04

20.88 19.45 18.28 17.31 15.76 13.18 20.88 19.80 18.85 18.02 16.61 14.04 20.88 18.51 16.72 15.32 13.23 10.14

20.88 19.92 19.08 18.32 17.00 14.51 20.88 19.70 18.70 17.82 16.36 13.77 20.88 18.89 17.37 16.15 14.32 11.49

20.88 24.63 28.06 31.21 36.86 48.20 20.88 18.38 16.53 15.09 12.97 9.84 20.88 21.72 22.36 22.93 23.98 26.94

20.88 19.56 18.50 17.64 16.28 14.09 20.88 19.70 18.70 17.82 16.36 13.77 20.88 18.51 16.72 15.32 13.23 10.14

20.88 21.16 21.64 22.31 24.23 32.16 20.88 18.38 16.53 15.09 12.97 9.84 20.88 18.51 16.72 15.32 13.23 13.23

20.88 19.50 18.50 17.70 16.30 14.10 20.88 19.60 18.60 17.60 16.10 13.50 20.88 18.51 16.72 15.32 13.23 10.14

20.88 21.50 22.30 23.10 25.10 32.40 20.88 18.10 16.00 14.40 12.00 8.50 20.88 18.51 16.72 15.34 13.23 10.14

ð36Þ

and the parameters cp ¼ 0  0:1; cq ¼ 0  0:1; cd ¼ 0:05; P 0 ¼ 31000; A0 ¼ 2600; r ¼ 0:1: ð37Þ These parameters approximately match the replacement problem of the USPS delivery vehicles (Aitoro, 2011) with the expected 24year operational life. The optimal lifetimes for the original and modified EL and VH algorithms are calculated using MS Excel Solver. The optimal L1 and L2 for the infinite-horizon optimization (4)–(5) are obtained through numeric solution of an associated nonlinear integral equation over the planning horizon of 125 years (that includes up to five sequential asset replacements). The corresponding algorithm is described in Yatsenko and Hritonenko (2005).

Fig. 1. Optimal asset lifetimes calculated by five replacement algorithms for decreasing rates cq ¼0  0.05 of the maintenance cost A and a constant new asset cost P. The optimal L1 of the first asset obtained by the modified VH algorithm coincides with the infinite-horizon lifetime Ln1. The lifetime L1 of the first asset decreases in time and the lifetime L2 of the second asset inclreases in all algorithms.

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5. Summary and recommendations Introducing relatively simple but theoretically justified corrections into the classic EL replacement method (Thuesen and Fabrycky, 1993) and the VH method (Scarf and Hashem, 1997, 2002), we construct their efficient modifications and demonstrate that they work well under improving technology. Specifically:

 The modified EL and VH replacement methods deliver the

 Fig. 2. Optimal asset lifetime for decreasing rates cp ¼ 0 0.05 of new asset cost and a constant maintenance cost. The solutons obtained by modified and original EL and VH algorithms coincide.



Fig. 3. Optimal asset lifetime at the proportional technological change for different rates cp ¼ cq ¼ 0 0.05. The modified EL and VH algorithms produce the infinitehorizon lifetime (indicated by a solid curve), while the original EL and VH methods do not.

a larger rate of maintenance cost (the subsequent lifetimes are larger as predicted in (Yatsenko and Hritonenko, 2008, 2011)). A theoretical justification of such asset dynamics is desirable but challenging. Fig. 1 provides a visual illustration of these simulation outcomes. The second group in Table 1 for cq ¼0, cp ¼ 0.01, 0.02, 0.03, 0.04, 0.05 and Fig. 2 demonstrate how a decrease of the only new asset cost affects the optimal lifetime. In this case, the modified EL and VH methods coincide with their standard counterparts. The modified VH method delivers a solution closer to the best IH regime than the modified EL method. At the maximum rate 5%, the modified VH method delivers the solution 16.36 years (with the relative error of approx 1%) whereas the modified EL method gives 16.61 years (with the relative error of approx 3%). Finally, the third group of Table 1 and Fig. 3 reflect the case of proportional technological change at cp ¼cq analytically explored in the Section 4.1. It is not just an illustration but also helps to clarify the parameter ranges, in which different methods produce acceptable results. In particular, the EL method delivers unacceptable results (with the relative errors of 5–20%) for all the range of 1–10% of technological change rates. Surprisingly, the original VH method also underperforms for the range of 4–10%. In particular, at cp ¼cq ¼0.05, it delivers the first lifetime of 14.32 years that differs from the infinite-horizon version of 13.23 by more than one year. In addition, it produces a completely unacceptable value of 23 years for the second asset lifetime L2 that differs from the optimal L2 by 10 years. As proven in Theorem 2, the modified EL and modified VH algorithms produce the infinite-horizon solution in this case.

acceptable first asset lifetime, close to the infinite-horizon asset lifetime, for all considered variants of exponential new asset and maintenance costs. The original EL and VH methods produce suboptimal results. Solutions obtained by the modified EL and VH methods coincide with the infinite-horizon asset lifetime when the observed technological improvement is exponential and equally affects the new asset cost and the maintenance cost (a proportional technological change). The modified VH method produces much better solution compared to the modified EL method in the case of nonproportional technological improvement.

An essential remaining task is to analyze the efficiency of those methods when the future dynamics of the new capital and maintenance costs is not exponential or is not completely known even on a limited horizon. In addition, the impact of uncertainty on the replacement policy is of primary importance (Mercier, 2008, Nguyen et al., 2013, Richardson et al., 2013).We leave these issues for a future study. Practical recommendations for choosing an efficient replacement algorithm depend on the observed dynamics of technological improvement and the data availability. Let the technological improvement be exponential and known on a future horizon [0,T]. Then:

 If the horizon length T is larger than the estimated future 

~ then the only lifetime L~ of current asset but is smaller than 2L, good possible option is the modified EL method. ~ Then: Let the horizon length T be larger than 2L. – If the observed technological improvement equally affects the new asset and the maintenance costs, then the best option is the modified EL method. – If the technological improvement affects the new asset cost and the maintenance cost differently, then the modified VH method is superior and produces much better solution compared to all other methods.

Acknowledgments The authors express their deep gratitude to two anonymous reviewers for careful reading of the paper and valuable comments. They are also thankful to participants of the 8th International Conference on Modelling in Industrial Maintenance and Reliability (University of Oxford, July 2014) for useful discussions. References Aitoro, J., 2011. USPS needs plan to replace aging vehicle fleet. Wash. Bus. J. (May 18). Bethuyne, G., 1998. Optimal replacement under variable intensity of utilization and technological progress. Eng. Econ. 43, 85–106. Christer, A.H., Scarf, P.A., 1994. A robust replacement model with applications to medical equipment. J. Oper. Res. Soc. 45, 261–275. Elton, E.J., Gruber, M.J., 1976. On the optimality of an equal life policy for equipment subject to technological improvement. Oper. Res. Q. 27, 93–99.

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