Optimizing reliability and service parts logistics for a time-varying installed base

European Journal of Operational Research 218 (2012) 152–162

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European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor

Stochastics and Statistics

Optimizing reliability and service parts logistics for a time-varying installed base Tongdan Jin a,⇑, Yu Tian b a b

The Ingram School of Engineering, Texas State University, TX 78666, USA Sun Yat-Sen Business School, Sun Yat-Sen University, Guangzhou, China

a r t i c l e

i n f o

Article history: Received 18 December 2010 Accepted 5 October 2011 Available online 7 November 2011 Keywords: Reliability Repairable inventory Life cycle cost analysis Non-stationary demand Performance based logistics

a b s t r a c t Performance based contracting (PBC) emerges as a new after-sales service practice to support the operation and maintenance of capital equipment or systems. Under the PBC framework, the goal of the study is to increase the system operational availability while minimizing the logistics footprint through the design for reliability. We consider the situation where the number of installed systems randomly increases over the planning horizon, resulting in a non-stationary maintenance and repair demand. Renewal equation and Poisson process are used to estimate the aggregate fleet failures. We propose a dynamic stocking policy that adaptively replenishes the inventory to meet the time-varying parts demand. An optimization model is formulated and solved under a multi-phase adaptive inventory control policy. The study provides theoretical insights into the performance-driven service operation in the context of changing system fleet size due to new installations. Trade-offs between reliability design and inventory level are examined and compared in various shipment scenarios. Numerical examples drawn from semiconductor equipment industry are used to demonstrate the applicability and the performance of the proposed method. Ó 2011 Elsevier B.V. All rights reserved.

1. Introduction Two types of after-sales service contracting methods have been discussed in literature pertaining to capital equipment: material based contract and performance based contract (Gadiesh and Gilbert, 1998; Kim et al., 2007; Nowicki et al., 2008; Oliva and Kallenberg, 2003). Under the material contract, the equipment user or the customer pays for unit transactions of spare parts, labors, and other relevant costs. Albeit its popularity, the major concern is that the original equipment manufacturer (OEM) has less motivation to improve the system reliability. A common brief held by the OEM is that customer satisfaction can be fulfilled through extended warranty programs or effective parts replacement services. However, this brief may hurt the OEM’s market share in a long term perspective. Customers may turn to other suppliers who can provide more reliable yet less expensive systems. Throughout the paper, system and equipment are used interchangeably. Performance based contracting (PBC) emerged as a new service model aiming to improve the reliability performance while lowering the sustainment cost. This new contracting method is often referred to as ‘‘performance based logistics’’ (PBL) in defense sector, or called as ‘‘power by the hour’’ (PBH) in commercial airlines. Under the PBC scheme, the OEM and his customers reach a service ⇑ Corresponding author. Tel.: +1 512 245 4904; fax: +1 512 245 7771. E-mail address: [email protected] (T. Jin). 0377-2217/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2011.10.026

level agreement with respect to performance criteria such as system availability, repair time, and fleet readiness. Instead of paying for materials and labors, the customers pay for the system performance during its useful lifetime. As such, the OEM is incentivized to invest more resources on reliability during design and manufacturing of the system, as it will be paid off by reduced support and logistics costs. Some successful PBL contracts have been reported in the US military sectors. For instance, the US Navy has seen equipment availability improvement, from 67% under traditional material contracts to 85% for the F/A-18 aircraft under PBL, and from 62% to 94% for the Aegis cruiser (Garvey, 2005; Kratz, 2005). System availability can be improved by reducing the meantime-to-repair (MTTR). The length of MTTR primarily depends on three logistics factors: the failure diagnostics time, the spare parts stocking level, and the parts delivering time. A large body of literature has been published with the aim of optimizing the inventory operation and resources. These models can be classified into either single-echelon inventory models or multi-echelon models (Kennedy et al., 2002; Muckstadt, 2005). In particular, two continuous review policies are often used to manage the service parts logistics: (1) the lot-size/reorder-point (q, r) model, where q represents the order quantity, and r is the reorder point; and (2) the reorder-point/order-up-to-level (s, S), where s is the reorder point and S represents the order-up-to-level. To minimize the inventory cost, the demand stream of the spare parts needs to be forecasted based on historical data or the number of installed systems (Jalil et al., 2011). In

T. Jin, Y. Tian / European Journal of Operational Research 218 (2012) 152–162

153

Nomenclature Notations N(t) the number of installed systems in (0, t] Wi time instance for the ith installation with i = 1, 2, . . ., N(t) T random variable representing the failure time t the length of the inventory planning horizon k installation rate a module failure rate, decision variable F(t) cumulative distribution function of the module lifetime H(t) the expected number of renewals at time t S(t) aggregate fleet failures at time t SðtÞ the expectation of S(t) mmax maximum inventory phases or setups in (0, t] m actual inventory setups, and m 2 [1, mmax], decision variable c1 cost per setup of inventory parameters c2 ordering cost per replenishment c3 unit inventory holding cost per unit time l lead time for inventory replenishment COi estimation of the expected ordering cost for phase i, for i = 1, 2, . . ., m CHi estimation of the expected holding cost for phase i, for i = 1, 2, . . . , m L length of the ith phase for all i. Yi cumulative inventory time prior to the ith phase tij expected end time of the jth cycle in phase i expected lead time demand for the jth cycle in phase i hij qi quantity per order for phase i, and q = {q1, q2, . . . , qm}, decision variable ri reorder point for phase i, and r = {r1, r2, . . . , rm}, decision variable

current literature, most repairable inventory models (Diaz and Fu, 1997; Graves, 1985; Sherbrooke, 1968, 1992) are derived based on the assumptions that: (1) the demand for spare parts is a stationary process; and/or (2) the size of the installed systems or the installed base is static or infinite. These assumptions allow for the use of Markov model or queuing theory to estimate the steady state inventory parameters. Recently Graves and Willems (2008) investigate a manufacturing inventory placement policy considering a non-stationary demand with constant mean and time-dependent variance. When a new generation of equipment is released to the market, the spare parts inventory often operates in a non-stationary condition due to the increment of the field systems, reliability growth, and variable usage. Therefore, stationarity inventory models become ineffective in handling time-varying spare parts demand streams. System availability can also be improved by increasing the mean-time-between-failures (MTBF). In reliability literature two approaches have been extensively discussed to increase the MTBF: redundancy allocation and reliability allocation (Chen, 1992; Coit et al., 2004; Kuo and Wan, 2007; Marseguerra et al., 2005). For both techniques, the problem is formulated to maximize the system reliability subject to the resource constraint, or to minimize the total resources subject to the reliability requirement. These models usually concentrate on the costs pertaining to material acquisition, design, and manufacturing. It is estimated that the expenditure on the repair and maintenance of capital equipment may account for 60–70% of the lifecycle cost (LCC) (Berkowitz et al., 2004). Hence, the spending on the post-installation services needs to be considered and incorporated into the reliability optimization models. The PBC agreement can also be treated as a lifetime warranty service. In warranty models, the research interest is focused on the optimization of the warranty period or the minimization of

ni

expected number of replenish cycles in phase i, for i = 1, 2, . . . , m amax maximum failure rate of the module amin minimum failure rate of the module k coefficient to characterize the design difficulty in reliability growth m coefficient to characterize the production difficulty in reliability growth A baseline manufacturing cost with failure rate of amax B1 baseline design cost with the failure rate of amax B2 cost coefficient for module production c(a) unit production cost with the failure rate of a b interest rate compounded annually b1 cost for performing a regular repair cost for performing an emergency repair b2 d1 expected downtime cost incurred in a regular repair d2 expected downtime cost incurred in an emergency repair Q(a, m, q) NPV of spare parts capital cost I(a, m, q, r) NPV of inventory cost R(a, m, q, r) NPV of repair cost D(a, m, q, r) NPV of equipment downtime cost K(a) NPV of incremental design cost P(a) NPV of incremental manufacturing cost p(a, m, q, r) NPV of the total cost probability of stock-out for the jth cycle in phase i pij pout average stock-out probability during [0, t]

maintenance cost (Murthy et al., 2004; Huang et al., 2007). Reliability has been considered as a viable means to counterbalance the warranty costs. However, the spare parts logistics has long been excluded from the warranty decision models. Since most warranty models are tailored to favor the economic interest of the OEM, downtime losses and other indirect costs incurred by the customers need be re-examined and appropriately incorporated into the warranty model as well. Reliability, spare parts inventory level, and the system fleet size jointly determine the availability and the LCC of capital equipment. Along with this track, several studies have been carried out and some preliminary findings were reported (Jin and Liao, 2009; Kim et al., 2007; Öner et al., 2010). In Jin and Liao (2009), the spare parts inventory is modeled and optimized in the context of nonstationary demand scenario, taking into account the growth of the installed system population. In Kim et al. (2007), the tradeoff between product reliability and the spare parts inventory level is investigated under the game-theoretic framework. In Öner et al. (2010), the Erlang loss model (i.e. M/G/s/s queue) is used to estimate the out-of-stock probability for a single-echelon spare parts inventory, based on which the LCC is minimized by balancing the reliability with the base inventory level. The studies in Kim et al. (2007) and Öner et al. (2010) assume that the system fleet size is fixed or time-invariant over the contractual period, generating a constant spare parts demand rate. Under the PBC framework, this paper aims to minimize the net present value (NPV) of an installed base by optimizing reliability and resource allocation to design, manufacturing, and spare parts inventory. Our study extends the models in Kim et al. (2007) and Öner et al. (2010) from two aspects. First, we assume field systems randomly increase due to the market expansion. Hence the spare parts demand tends to be non-stationary. This is a more realistic

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perception as capital equipment is sequentially purchased and put to use by customers. Many times the installed base reaches a steady state, yet the spare parts demand still could be non-stationary. This occurs when OEM is implementing reliability growth program such as corrective actions and retrofits. The second aspect can be ascribed to the development of multi-phase (q, r) adaptive inventory policy. This algorithm can dynamically adjust the (q, r) parameters to meet the non-stationary demand resulted from the expanded installed base. The remainder of the paper is organized as follows. Section 2 presents the method to estimate the spare parts demand for a time-varying installed base. Section 3 concentrates on the development of a multi-phase (q, r) inventory control policy. Section 4 explores the analytical relationship between reliability and system LCC. In Section 5, we minimize the LCC through the optimal resource allocation to design, manufacturing, and after-sales sustainment. In Section 6 numerical examples and simulations are used to demonstrate the applicability and performance of the proposed method, and Section 7 concludes the paper.

failure, yet the local inventory does not have the spare part. The defective module will be directly sent to the repair center for fixing and then sent back to the original customer. The down equipment will be restored upon the reconfiguration of the fixed module. If an emergency repair is executed, the repair time, the repair cost and the customer downtime losses are larger than those in a regular repair. 2.2. Aggregate fleet repair demand As new systems are continuously acquired and deployed in the customer site, the fleet size is growing over time. Since each installed system will independently generate failures, the aggregate fleet failure stream turns out to be a non-stationary process. In the following, we introduce a quantitative approach to estimating the aggregate failures based on the work by Jin and Liao (2009). The following assumptions are made regarding the system installation and repair process: (a) After the first system is installed at time t = 0, the number of systems or the installed base, denoted as N(t) + 1, increases following the Poisson counting process with rate of k. For t > 0, N(t) has the following Poisson probability mass function:

2. Estimating non-stationary repair demand 2.1. Dynamic and Integrated Service Supply Chain Fig. 1 depicts an integrated service logistics supply chain which consists of the OEM and his customers. The OEM owns the design/ manufacturing, the repair center, and the spare parts inventory which is located close to the customers. We assume the material transportation costs are negligible compared to the material and equipment costs. We also assume the repair activity requires intensive labors or it may needs technical supports from the OEM. Hence the repair center is located in a low cost region or close to the design/manufacturing facility. As such, this integrated service model actually is a two-echelon repairable inventory system (Sherbrooke, 1968). This study concentrates on the local stocking policy and also assumes the repairable center hold zero inventories. The OEM designs and produces new capital equipment per the market demands or military contracts. After the first customer shipment, new systems are sequentially put to use at random points in time, making the fleet size increasing over time. Capital equipment is often designed with modality to facilitate the repair and maintenance activity. Upon failure, the defective module is replaced with a spare part so that the equipment can be quickly restored to the production state. Two repair options are offered by the OEM: regular repair and emergency repair. In a regular repair, the defective module will be replaced by a ready-to-use part from the local stock. The failed module is then sent back to the repair center for root-cause analysis. Emergency repair will be applied when a customer reports a

PrfNðtÞ ¼ ng ¼

ðktÞn ekt ; n!

for n ¼ 0; 1; 2; . . .

(b) The system is repairable, and a system failure corresponds to a module failure. The system is configured with different types of modules, and different module types may have different reliability levels or failure rates. Without loss of generality, the analysis below concentrates on one particular module type. Fig. 2 illustrates the system installation scenarios along with the subsequent module failures. Two mutually independent processes are involved: the Poisson counting process and the renewal process. The increment of the fleet size is modeled as a homogeneous Poisson process with rate of k. After the first installation, denoted as ‘‘0’’, new systems are randomly shipped and installed at times of W1, W2, . . . , WN(t). Meanwhile, an installed system (i.e. modules) will fail during the operation, and all failures are fixed through regular or emergency repairs. Assuming the MTTR is relatively short compared to the MTBF, the aggregate fleet failures between [0, t] can be treated as a superposition of N(t) + 1 independent renewal processes. Let F(t) be the lifetime distribution for the module under study. Based on the renewal process, the expected number of defective modules of a system, denoted by H(t), can be computed by

continuous shipments over time emergency repair repair demand

orders Product Design & Manufacturing

Repair Center

Spares Parts Inventory replenish

reliability learning

ð1Þ

System 1 System 2

System N(t) spare parts System N(t)+1 delivery

return for regular repair Fig. 1. An integrated manufacturing and service logistics model.

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first install

For the detailed derivations of Eqs. (6) and (7), readers are referred to (Jin and Liao, 2009). Obviously the mean and the variance are time-dependent functions in quadratic and cubic forms, respectively. This further proves that the aggregate fleet repairs; i.e. the demand for spare parts, is a non-stationary process with time-varying mean and variance.

failure

0 1

W1

W2

2 3. Inventory model under non-stationary demand N(t)

WN(t)

Two continuous review policies are often used in spare parts provisioning: the (q, r) policy and the (s, S) policy. Our study assumes a system failure corresponds to a module failure which requires one spare part for replacement. Therefore, the spare parts inventory depletes one unit at a time, instead of a batch of units. As such, the (q, r) policy is essentially the same as the (s, S) policy, and the (q, r) policy is adopted in our analysis. The following assumptions are made regarding the inventory control policy:

Arrival time for N(t)th new install Aggregate failures S(t)

t

0 Fig. 2. Aggregate failures for N(t) + 1 installed systems in [0, t].

HðtÞ ¼ FðtÞ þ

Z

t

Hðt  sÞdFðsÞ:

ð2Þ

0

This is the well-known renewal integral equation. H(t) has a closed form solution when F(t) is exponential or Erlang distributions. For Weibull, log-normal or other general distributions, the exact solution of H(t) is not available or is very difficult to derive. Hence numerical methods or approximations should be used to estimate H(t) (Jin and Gonigunta, 2009; Xie, 1989). To determine the spare parts inventory level, the number of renewals generated from the entire fleet is of our interest. Technically the number of aggregate fleet renewals is equal to the spare parts demand in [0, t]. Let S(t) be the aggregate renewals by time t. A generic model to estimate S(t) can be obtained as follows

SðtÞ ¼ HðtÞ þ

NðtÞ X

Hðt  W i Þ;

for t > 0;

ð3Þ

i¼1

where, H(t  Wi) represents the number of renewals for the ith system installed at Wi. Obviously S(t) is a random number due to the stochastic nature of N(t) and Wi. The properties of S(t) can be characterized by its mean and variance. If the module time-to-failure is exponential, the mean and the variance of S(t) can be derived explicitly, which will be discussed below. 2.3. Aggregate failures under exponential renewals If the lifetime is exponentially distributed with the rate of a, the cumulative distribution function is F(t) = 1  eat. By substituting F(t) into Eq. (2), the renewal equation under the exponential distribution is obtained as

HðtÞ ¼ at:

ð4Þ

Notice that H(t) is simply a linear function of t with the slope of

a. By substituting Eq. (4) into (3), the aggregate fleet repair demand

SðtÞ ¼ at þ

3.1. Multi-Phase (q, r) control policy The non-stationary demand stream poses great challenges in modeling and optimizing the spare parts inventory. Methods developed under the stationary demand condition cannot be directly used as it might lead to a sub-optimal decision. One viable approach is to monitor the demand periodically and implement an adaptive inventory replenishment policy. We propose a multiphase (q, r) control policy by dividing the planning horizon into several phases, say m, for which the stationarity assumption is appropriate in each replenish cycle. The performance of this multi-phase policy depends on the choice of m. If a shorter phase (larger m) is adopted, the Poisson approximation to the actual repair demand is more precise. The drawback is that the inventory performance may degrade due to excessive (q, r) setups. Therefore, a balance between the number of phases and the economical efficiency must be reached. Fig. 3 depicts the inventory replenish cycles over three phases with equal length of L. In each phase, a fixed (q, r) policy is adopted even if the actual demand is non-stationary. As the time evolves to the next phase, (q, r) is adjusted to meet the new demand rate. Let tij be the expected ending time for the jth cycle in phase i, then

a þ tij ¼

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   a2 þ 2ak jqi þ Sðði  1ÞLÞ

ak

;

aðt  W i Þ:

ð5Þ

i¼1

If we further apply the conditional probability theory along with some algebraic manipulations, the mean and the variance of S(t) can be obtained as

1 SðtÞ ¼ at þ akt 2 ; 2 1 1 varðSðtÞÞ ¼ at þ akt 2 þ a2 kt3 : 2 3

ð6Þ ð7Þ

for j ¼ 1; 2; . . . ; ni ;

ð8Þ

where qi is the order quantity for phase i, and ni is the expected number of replenish cycles in phase i. Detailed derivation of Eq. (8) is given in Appendix A. Let hij be the expected lead-time demand for the jth cycle in phase i, then we have

1 2 hij ¼ Sðt ij Þ  Sðt ij  lÞ ¼ al þ aklt ij  akl : 2

becomes NðtÞ X

(a) Backlogs are allowed, if occurs, an emergency repair is executed. However, it is a lost demand with respect to the spare parts inventory. (b) Replenishment lead times are known and fixed.

ð9Þ

Notice that l is the lead-time for each replenishment cycle. Upon the partitioning of the planning horizon, the lead time demand in the jth cycle of the ith phase can be approximated as a Poisson process with rate of hij/l = a + aktij  0.5akl. This assumption is reasonable in practice as long as tij  l. As the inventory progresses over time, tij becomes larger and the condition of tij  l is always satisfied. Nevertheless, differentiation must be made between Eq. (9) and the inventory policy in Jin and Liao (2009). In the latter all lead time demands within the phase are treated the same, whereas it is not in Fig. 3.

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T. Jin, Y. Tian / European Journal of Operational Research 218 (2012) 152–162

L for i=1

L for i=2

i=3

q2 q1

r2 r1 t11

0

t12

t21

t13

t22

t23

t24

t31

time

Fig. 3. A multi-phase (q, r) control policy.

3.2. Inventory cost model The inventory costs under this study include the spare parts capital cost, holding cost, ordering cost, and cost for updating (q, r) parameters. Although the shortage cost is not included in the inventory cost model, it will be considered as part of the emergency repair cost. However, the probability for spare parts shortage is still needed as it is used to estimate the emergency repair cost. Under the multiphase control policy, a set of (qi, ri) for i = 1, 2, . . . , m needs to be determined to minimize the total inventory cost.

where, ri is the re-order point for the ith phase. hij is the expected demand during the lead time as shown in Eq. (9). If j = 1, then tij1 = (i  1)L for i = 1, 2, . . . , m. 3.2.4. Total inventory cost The total inventory cost is the sum of spare parts capital cost, ordering cost, holding cost, and the cost for updating (q, r). By combining Eqs. (10)–(12) along with the setup cost, the NPV of the inventory cost, denoted as I(a, m, q, r), is obtained as

Iða; m; q; rÞ ¼ c1 3.2.1. Spare parts capital cost At time t = 0, a number of q1 spare parts is stocked in the local inventory to fulfill the failure replacement tasks. When the inventory position drops and reaches the reorder point r1, an order of q1 units is issued to the repair center for replenishing the inventory. This process may repeat multiple times before evolving to the second phase. Once entering the second phase, new decisions are made on q2 and r2, and the whole process repeats. As the installed base expands, both qi and ri need to increase in order to maintain the necessary level of quality of service. The NPV of the spare parts capital cost carried by the inventory in [0, t] can be estimated by

! m X Y i b ; Q ða; m; q; rÞ ¼ cðaÞ q1 þ r 1 þ ðqi þ r i  qi1  r i1 Þe i¼2

ð10Þ

m X

eY i b þ Qða; m; q; rÞ þ

m X ðCOi ðqi ; r i Þ

i¼1

i¼1

þ CHi ðqi ; ri ÞÞ:

ð13Þ

In the formula, c1 is the (q, r) setup cost incurred when the inventory evolves to the next phase. The decision variables are a, m, q and r for q = [q1, q2, . . . , qm] and r = [r1, r2, . . . , rm]. 4. Reliability and life cycle cost analysis The costs associated with module design, manufacturing, and equipment downtime need to be considered alongside the inventory cost in order to capture the entire lifecycle cost. The following cost analysis is developed based on one type of modules, yet the system-level cost can be estimated by aggregating the cost of individual modules.

where

Yi ¼

i X

4.1. Reliability vs. design cost

Lj  Li ;

for i ¼ 1; 2; 3 . . . ; m:

j¼1

Eq. (10) converts the capital cost into the NPV by taking into account the incremental investment on the spare parts. b and Yi represent the annual interest rate and the cumulative inventory time prior to the ith phase, respectively. Notice that c(a) is the unit manufacturing cost of the part which will be further specified in Section 4.2. 3.2.2. Ordering cost Assuming c2 is the ordering cost per replenishment cycle, the NPV of the ordering cost for the ith phase can be estimated as

SðiLÞ  Sðði  1ÞLÞ Y i b COi ¼ c2 e ; qi

for i ¼ 1; 2; . . . ; m:

ð11Þ

3.2.3. Holding cost If c3 is the unit inventory holding cost per unit time, the NPV of the expected holding cost in the ith phase can be estimated as

  ni X 1 CHi ¼ c3 ðt ij  t ij1 Þ qi þ r i  hij eY i b ; 2 j¼1

for i ¼ 1; 2; . . . ; m; ð12Þ

Mettas (2000) and Huang et al. (2007) have shown that the product design cost grows exponentially with the reliability. The exponential reliability- design cost model is also adopted by Sana (2010) and Sarkar et al. (2010) to estimate the product development cost with the goal of maximizing the profit of a production-inventory system. Based on the works in Mettas (2000) and Huang et al. (2007), the relationship between the incremental design cost, K(a), and the failure rate, a, can be modeled as

    amax  a 1 ; KðaÞ ¼ B1 exp k a  amin

for amin 6 a 6 amax ;

ð14Þ

where, amax and amin represents the maximum and the minimum failure rates, respectively. K(a) is the incremental design cost incurred if we reduce a from amax. Notice that B and k are positive parameters. In particular, B1 is the baseline design cost with amax, and k describes how difficult to further reduce the failure rate subject to design, material, and resource constraints. 4.2. Reliability vs. manufacturing cost To build a product with high reliability, it is necessary to adopt new materials, novel design and advanced manufacturing technology. Therefore, the manufacturing cost also increases with the

T. Jin, Y. Tian / European Journal of Operational Research 218 (2012) 152–162

reliability. In this paper, we adopt the reliability-manufacturing cost model proposed by Öner et al. (2010) to characterize the module production cost at different levels of a. The model is presented as follows

v ; cðaÞ ¼ A þ B2 av  amax

for amin 6 a 6 amax :

ð15Þ

In this formula, A is the baseline unit manufacturing cost with amax. B2 and m capture the incremental cost if one intends to decrease the failure rate relative to amax. Both B2 and v are positive parameters. Assuming a Poisson installation process with rate of k, the NPV of the incremental manufacturing cost for the whole fleet, denoted as P(a), can be estimated as

PðaÞ ¼ ðcðaÞ  AÞ 1 þ

NðtÞ X

! ebW i ;

ð16Þ

i¼1

where, Wi is the installation time for the ith system for i = 1, 2, . . . , N(t). Manufacturing costs for later installations are converted into the NPV by multiplying exp (bWi). Notice that P(a) actually is a random variable because of the stochastic nature of N(t) and Wi. The expected incremental manufacturing cost for the entire fleet installed in [0, t] is given in the following equation, and the detailed derivation is available in Appendix B.

  k E½PðaÞ ¼ ðcðaÞ  AÞ 1 þ ð1  ebt Þ : b

ð17Þ

4.3. Repair cost

for i ¼ 1; 2; . . . ; m;

ð18Þ

ri X hxij ehij ; x! x¼0

for j ¼ 1; 2; . . . ; ni :

ð19Þ

Depending on hij, the value of pij is slightly different in different cycles within the same phase. In general, pij increases as hij becomes large. To estimate the total repair costs in [0, t], the average stock-out probability for the entire planning horizon needs to be estimated as

pout ¼

m 1 X p: m i¼1 i

ð20Þ

Let b1 and b2 be the expected cost for performing a regular and an emergency repair, respectively. Then the NPV of the fleet repair cost in [0, t] can be estimated as

E½Rða; m; q; rÞ ¼ ð1  pout ÞE½R1  þ pout E½R2 ;

ð21Þ

where

   k 1 þ ð1  ebt Þ  ktebt ; b b    ab2 k 1 þ ð1  ebt Þ  ktebt : E½R2  ¼ b b

E½R1  ¼

ab1

Let d1 and d2 be the expected customer downtime cost for a regular and an emergency repair, respectively. The total downtime cost between [0, t] is given as follows

E½Dða; m; q; rÞ ¼ ð1  pout ÞE½D1  þ pout E½D2 ;

ð24Þ

where

   k 1 þ ð1  ebt Þ  ktebt ; b b    ad2 k 1 þ ð1  ebt Þ  ktebt : E½D2  ¼ b b

E½D1  ¼

ad1

ð25Þ ð26Þ

Detailed derivations of E[D1] and E[D2] are given in Appendix C. Similar to the repair cost, the downtime cost is also jointed determined by a, m, q and r. 5. Optimization formulation 5.1. Optimization model The objective of the OEM is to allocate resources to design, manufacturing, and service logistics such that the total cost is minimized while the system performance goals are met. The following optimization model is formulated to accommodate OEM’s objective in the context of performance-driven environment.

Min

pða; m; q; rÞ ¼ KðaÞ þ PðaÞ þ Iða; m; q; rÞ þ Rða; m; q; rÞ þ Dða; m; q; rÞ

ð27Þ

Subject to

where

pij ¼ 1 

4.4. Customer downtime cost

Problem P1

Under the multi-phase (q, r) inventory control policy, the average stock-out probability in the ith phase, denoted as pi, can be estimated by ni 1 X pi ¼ p ; ni j¼1 ij

157

ð22Þ

amin 6 a 6 amax

ð28Þ

0 < r i < qi ; for all i ¼ 1; 2; . . . ; m 1 6 m 6 mmax :

ð29Þ ð30Þ

Problem P1 is formulated to minimize the NPV of the fleet cost in [0, t]. All these cost items in Eq. (27) have been derived in Sections 3 and 4. The decision variables are a, m, q, and r. Except for a which is a real number, all others are integers. Constraint (28) defines the acceptable range of the failure rate, and Eq. (30) controls the maximum numbers of inventory setups. The lifecycle of an installed base can be divided into three stages: increment, steady-state, and retirement. The lifecycle of individual systems consists of design, manufacturing, operation, and retirement. It is worth mentioning that Problem P1 is formulated to minimize the fleet LCC, yet our current focus is on the cost analysis in the increment phase. This phase is critical to new design in terms of gaining the market share, attaining the reliability target, and ensuring the customer satisfaction. This process usually lasts 5– 7 years for most commercial capital equipment or 10–15 years for defense/military systems such as aircraft. Since P(a), R(a, m, q, r) and D(a, m, q, r) are random variables, Problem P1 actually is a stochastic optimization problem. It is usually difficult to search the solution directly based on P1. To make the problem tractable, a common practice is to minimize the expected cost by transforming P1 into a deterministic model as follows.

ð23Þ

Detailed derivations of E[R1] and E[R2] are provided in Appendix C. Both E[R1] and E[R2] are the function of a. Since pout is the function of m, q and r, the repair cost is actually dependent upon the values of a, m, q and r.

Problem P2

Min E½pða; m; q; rÞ ¼ KðaÞ þ E½PðaÞ þ Iða; m; q; rÞ þ E½Rða; m; q; rÞ þ E½Dða; m; q; rÞ

ð31Þ

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T. Jin, Y. Tian / European Journal of Operational Research 218 (2012) 152–162

Subject to

amin 6 a 6 amax

ð32Þ

0 < r i < qi ; for all i ¼ 1; 2; . . . ; m

ð33Þ

1 6 m 6 mmax :

ð34Þ

 Step 1. Let k = 1, mL = 1, mH = mmax.  Step 2. For given mL and mH, determine E[p(a, mL, q, r)] and E[p(a, mH, q, r)] by solving Problem P2, respectively. Also let mkm ¼ bðmL þ mH Þ=2c, where b  c is the floor operator to take the integer part of the original value.  Step 3. Let k = k + 1. If E[p(a, mL, q, r)] P E[p(a, mH, q, r)], then let mL = mmk1 and mkm = b(mL + mH)/2c; otherwise let mH = m mmk1 and m L + mH)/2c.  k k = b(m   6 1, then stop and choose min{E[ Step 4. If mm  mk1 m p(a, mL, q, r)], E[p(a, mH, q, r)]} as the best solution at the current value of a; otherwise go to Step 2.

5.2. Solution method Problem P2 is still difficult to solve because it involves the combinatorial natures of mixed integer programs and the complexity of solving nonlinear problems (Gupta and Ravindran, 1985). Genetic Algorithm (GA) (Marseguerra et al., 2005) has been proven to be very efficient to solve large and complex non-linear mixed-integer programming problems. In this paper, a bisectional search algorithm in Jin and Liao (2009) is combined with GA to determine a near-optimal solution. Before applying the bisectional search, a is divided into multiple levels (e.g. 50) within [amin, amax]. For each level of a, a bisectional search procedure is utilized to search the optimal inventory solution. Let mL and mH be the lower and upper limits of inventory partitioning, respectively. The steps of the algorithm are given as follows:

For a given value of a, the GA is applied to search the optimal solution sets of (q, r) corresponding to mL and mH, respectively. The GA process is repeated for a considerable number of generations (e.g. 100 generations) until the cost cannot be further reduced in the current search. Then the current best solution set, denoted as (a, m⁄, q⁄, r⁄), that min{E[p(a, mL, q, r)], E[p(a, mH, q, r)]} is recorded. Now the algorithm moves to the next level of a, and the bi-sectional search process is repeated to find new (a, m⁄, q⁄, r⁄) that minimize the objective function. After all levels of a 2 [amin, amax] have been exhausted, the solution resulting in the lowest cost is treated as the best solution.

Table 1 Common parameters for HVLC and LVHC (n/a = not applicable).

6. Numerical studies

Parameters

Value

Unit

Parameters

Value

Unit

amax amin

0.06 0.03 0.05 0.1 1 1 2 or 5

Fails/month Fails/month n/a n/a n/a Week Year

mmax c1 c2 b1 b2 d1 d2

6 or 12 1500 5000 1000 2500 2000 4500

phases $/setup $/order $/repair $/repair $/down event $/down event

b k

m l t

6.1. ATE system overview Automatic Test Equipment (ATE) is a complex electronics machine widely used to test wafers in semiconductor manufacturing industry. ATE is often designed in modularity to facilitate the maintenance, repair, and upgrade. A high-end ATE system usually costs 2–3 million dollars while the price of a lower-end system is about half a million. Under the regular repair, the defective module is removed and swapped with a spare part of the same type, and the system can be recovered within 2–4 hours. If an on-hand spare module is not available, the defective module is handled via the emergency repair.

Table 2 Variable parameters for HVLC and LVHC. Parameters

Value

Unit

Comment

k c3 A B1 B2 k c3 A B1 B2

60 10,000 5000 400,000 200 30 20,000 10,000 800,000 400

install/month $/unit/year $/unit $/design $/unit Install/month $/unit/year $/unit $/design $/unit

HVLC HVLC HVLC HVLC HVLC LVHC LVHC LVHC LVHC LVHC

6.2. PBL contract optimization PBL contracts are investigated in terms of shipping quantity and unit cost for two product lines: (1) high volume low cost (HVLC); and (2) low volume high cost (LVHC). These product lines represent the typical shipping and cost profiles in the ATE industry. Parameters related to design, manufacturing, spare parts supply,

Table 3 Analytical solution for HVLC for two years planning. m

1

2

3

4⁄

5

6

a

0.0346 2,791,172 158,245 203,162 159,593 1500 187,183 338,740 583,465 1,159,284 (16, 15)

0.0353 2,715,468 127,567 190,215 175,594 2927 197,497 258,396 589,760 1,173,512 (10, 8) (18, 17)

0.0352 2,706,140 133,535 193,048 174,415 4354 214,294 222,931 590,181 1,173,382 (6, 5) (14, 12) (18, 17)

0.0348 2,699,297 150,496 200,240 164,436 5781 222,140 195,804 589,773 1,170,627 (6, 4) (10, 9) (14, 13) (17, 16)

0.0352 2,707,892 133,535 193,048 173,332 7209 233,615 190,408 595,124 1,181,621 (4, 3) (8, 7) (12, 11) (15, 14) (18, 17)

0.0351 2,705,425 136,681 194,474 173,218 8636 232,463 182,702 595,469 1,181,781 (4, 2) (7, 6) (10, 9) (13, 12) (16, 15) (18, 17)

Total cost K(a) P(a) Capital cost Set-up cost Ordering cost Holding cost Repair cost Downtime cost (q1, r1) (q2, r2) (q3, r3) (q4, r4) (q5, r5) (q6, r6)

The bold and asterisk are simply to indicate that the data set is the optimal solution.

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To verify the analytical solution, a Matlab-based simulation program is developed to mimic the inventory operation under the growing installed base up to two years. For each feasible solution in Table 3, we randomly generate its failure scenarios with corresponding a. Failure arrival times from all systems are recorded and arranged in a chronological order. Then the (q, r) policy obtained from the analytical method is applied to compute the stock-out probability, inventory, repair and downtime costs. This process is repeated for 5000 runs for each solution. Finally, the cost is averaged over 5000 iterations and the result is listed in Table 4. It shows that the simulation always yields a lower cost estimate because NPV is computed using the actual simulated time instead of the discrete time point in the analytical approach. The simulation also shows that m = 4 is the best solution among the feasible solutions.

repair and equipment failures are given in Tables 1 and 2. Notice that costs related to design, baseline manufacturing, and inventory holding are doubled for the costly module. For the HVLC product lines, the optimal reliability and the spare parts inventory level are obtained based on the optimization model in Problem P2. The optimal values of a, m, q and r are listed in Table 3 along with the total cost for the modules. Table 3 shows that E[p(a, m, q, r)] is minimized when a = 0.0348 fails/month, m = 4, and (q, r) = {(6, 4), (10, 9), (14, 13), (17, 16)} with L = 6 months in each phase. In other words, a four-phase spare parts inventory policy should be adopted to manage the service parts supply as it achieves the minimum fleet cost of $2,699,297. Table 3 also provides detailed costs for design, manufacturing, inventory, repair and downtime losses for the two-year planning horizon. Table 4 Simulations to verify HVLC analytical solution (5000 iterations). m

1

2

3

4⁄

5

6

a

0.0346 2,742,467 158,245 203,186 159,593 1500 186,479 343,838 563,547 1,126,079

0.0353 2,678,107 127,567 190,244 175,594 2927 196,784 262,982 574,286 1,147,722

0.0352 2,659,294 133,535 193,042 174,415 4354 212,417 228,589 571,361 1,141,581

0.0348 2,645,756 150,496 200,322 164,436 5781 221,136 202,712 567,389 1,133,484

0.0352 2,653,544 133,535 193,122 173,332 7209 231,826 196,549 573,108 1,144,864

0.0351 2,652,455 136,681 194,613 173,218 8636 230,715 190,296 573,274 1,145,021

Total cost K(a) P(a) Capital cost Set-up cost Ordering cost Holding cost Repair cost Downtime cost

The bold and asterisk are simply to indicate that the data set is the optimal solution.

Table 5 Analytical solution for LVHC for two years planning. m

1

2

3⁄

4

5

6

a

0.0389 1,989,495 105,738 130,183 173,225 1500 187,350 363,489 346,056 681,954 (9, 8)

0.0392 1,917,505 98,996 125,574 188,511 2927 206,342 265,062 346,526 683,569 (5, 4) (10, 9)

0.0392 1,908,833 98,996 125,574 187,200 4354 213,972 222,430 356,356 699,952 (4, 2) (7, 6) (10, 9)

0.0382 1,916,826 122,975 140,824 186,704 5781 224,064 197,214 350,977 688,286 (3, 1) (5, 4) (8, 7) (10, 9)

0.0415 1,939,300 60,871 93,184 185,473 7209 251,203 191,747 389,006 760,606 (2, 1) (4, 3) (7, 6) (9, 8) (10, 9)

0.0391 1,935,740 104,004 129,024 185,655 8636 251,411 172,978 366,986 717,046 (2, 1) (3, 2) (5, 4) (7, 6) (9, 8) (10, 9)

Total cost K(a) P(a) Capital cost Set-up cost Ordering cost Holding cost Repair cost Downtime cost (q1, r1) (q2, r2) (q3, r3) (q4, r4) (q5, r5) (q6, r6)

The bold and asterisk are simply to indicate that the data set is the optimal solution.

Table 6 Analytical solution for LVHC for five year planning. m

5

6

7

8⁄

9

10

11

a

0.0330 7,178,666 (4, 3) (9, 8) (13, 12) (16, 15) (20, 19)

0.0329 7,170,843 (4, 2) (7, 6) (11, 10) (14, 13) (17, 16) (20, 19)

0.0323 7,173,913 (3, 2) (6, 5) (9, 8) (12, 11) (15, 14) (18, 17) (20, 19)

0.0334 7,170,725 (3, 2) (6, 5) (8, 7) (11, 10) (14, 13) (16, 15) (18, 17) (20, 19)

0.0326 7,177,034 (3, 1) (5, 4) (7, 6) (10, 9) (12, 11) (14, 13) (16, 15) (18, 17) (20, 19)

0.0336 7,198,871 (2, 1) (5, 4) (7, 6) (9, 8) (11, 10) (13, 12) (15, 14) (17, 16) (19, 18) (20, 19)

0.0325 7,197,575 (2, 1) (4, 3) (6, 5) (8, 7) (10, 9) (12, 11) (14, 13) (15, 14) (16, 15) (18, 17) (20, 19)

Total cost (q1, r1) (q2, r2) (q3, r3) (q4, r4) (q5, r5) (q6, r6) (q7, r7) (q8, r8) (q9, r9) (q10, r10) (q11, r11)

The bold and asterisk are simply to indicate that the data set is the optimal solution.

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For the LVHC, the optimal values for a, m, q and r are listed in Table 5 along with the total cost. The result in Table 5 shows a three-phase inventory control policy with failure rate a⁄ = 0.0392 is preferred as it minimizes the fleet cost. Compared with the HVLC, the analysis suggests that the reliability improvement of costly modules is not necessary if the contractual horizon is short. This finding echoes the empirical studies by Randall et al. (2010) who found that OEM is unwilling to invest on reliability unless a long-term contract is committed. Now we extend the LVHC contract to five years and re-examine the reliability decision on the product. Table 6 presents the optimal decisions on a along with m, q, r for LVHC for a 5-year contract. This result shows the OEM is motivated to reduce a from 0.0392 to 0.0334, or 15% reliability growth, as this decision will minimize the total fleet cost. It is observed from Tables 3, 5 and 6 that (q, r) approaches to (s + 1, s) inventory model as the time increases regardless of the shipping volume and the unit cost. As stated by Sherbrooke (1992) when the unit cost for spare parts is relatively higher than the cost of processing an order, the optimal decision logically leads to base stocking policy with one-for-one replenishment.

7. Conclusion Under the PBC framework, we proposed a lifecycle approach to managing reliability and service parts logistics where the demand rate is non-stationary with time-varying mean and variance. The study brought together reliability design and inventory optimization, which are often separated in literature, to improve system performance while lowering the cost through optimal design, manufacturing, and inventory policy. The research also expanded and generalized existing repairable inventory models by taking into account the stochastic increment of field systems. Renewal theory and Poisson process are used to estimate the mean and the variance of aggregate fleet failures. A multi-phase adaptive inventory policy that dynamically adjusts the stocking quality is developed to meet the non-stationary demand stream. The bi-sectional search embedded with genetic algorithm enables the logistics planner to find a sub-optimal, if not global optimal, solution in a timely manner. The solution is further tested and validated using Monte Carlo simulation. Results show that the multi-phase replenishment policy is quite effective to allocate spare parts to a changing installed base. Modeling and optimizing inventory systems under non-stationary condition usually is difficult due to the stochastic nature of the demand. This study takes an early step down the path to address non-stationary service parts logistics problems arising from the after-sales sustainment business. Our model can be applied to a variety of capital equipment, such as wind turbines, highspeed rail, computing and network servers, and new aircraft engines. They all share following features: technology-centric design, market expansion, high maintenance cost, and expensive downtime. Future research will consider the spare stocking policy in the repair center, and extend the solution to accommodate the entire fleet lifecycle from increment, steady state to retirement.

Sðt ij Þ  Sðði  1ÞLÞ ¼ jqi ;

Appendix A. Estimating inventory cycle time Under the assumption that the expected spare parts demand in a cycle is equal to qi, then the following equation is satisfied,

and j ¼ 1; 2; . . . ; ni ; ðA:1Þ

where Sðt ij Þ is the expected cumulative demand at tij and Sðði  1ÞLÞ is the expected cumulative demand at the end of phase i  1. If i = 1, then Sð0Þ = 0. By substituting Eq. (6) into (A.1), we have

0:5akt 2ij þ at ij  Sðði  1ÞLÞ  jqi ¼ 0:

ðA:2Þ

Solving this quadratic function in terms of tij, we obtain Eq. (8). Appendix B. Manufacturing cost When the installation is represented by a Poisson counting process with rate of k, there are N(t) + 1 units of products being installed by the time t (including the initial installation). Let Wi is the installation time for the ith unit for i = 1, . . . , N(t). The NPV of the incremental manufacturing cost for making N(t) + 1 units of modules during [0, t] can be expressed as

PðaÞ ¼ cðaÞ þ cðaÞebW i þ    þ cðaÞebW NðtÞ ¼ cðaÞ þ

NðtÞ X

cðaÞebW i ¼ cðaÞ þ

i¼1

NðtÞ X

V i;

ðB:1Þ

i¼1

where c(a) is the NPV of producing the first unit at t = 0. The coefficient of exp(bWi) transforms the production cost of the ith unit into the NPV, and V1, V2, . . . , VN (t) denote the NPV of the cost of the respective installation. Wi for all i can be treated as a nonordered sequence which is independent and uniformly distributed between [0, t] with fW i ðzÞ = 1/t. By conditioning on N(t) = n, the expectation of P(ajn) can be obtained as follow.

Z

E½V i jn ¼

t

0

1 cðaÞ ð1  ebt Þ; cðaÞebz dz ¼ t bt

for i ¼ 1; 2; . . . ; n: ðB:2Þ

We now further remove the condition of N(t) = n by substituting (B.2) into (B.1), the following expect NPV for N(t) + 1 installed units is obtained

  k E½PðaÞ ¼ cðaÞ 1 þ ð1  ebt Þ : b

ðB:3Þ

This expression is derived by realizing E[N(t)] = kt for the Poisson process with rate k. Appendix C. Fleet repair and downtime cost C.1. Repair cost for the first installation at t = 0 For the module instated at t = 0, the expected number of failures (i.e. repairs) between [0, t] would be H(t). Let T1, T2, . . . , TH(t) be the random variable representing the times of failures leading to subsequent repair actions. In addition, let U1, U2, . . . , UH(t) be the NPV of the corresponding repair costs. Let fT j ðzÞ denote the probability density function of Tj, and fT j ðzÞ = 1/t for j = 1, 2, . . . , H(t). Then the NPV for E[Uj] can be obtained by conditioning on Tj as

Acknowledgment This project is partially supported by the National Natural Science Foundation of China (Grant No. 70872117, 71172162).

for i ¼ 1; 2; . . . ; m;

E½U j  ¼

Z 0

t

E½U j jT j ¼ zfT j ðzÞdz ¼

Z 0

t

bz

be

1 b dz ¼ ð1  ebt Þ; t bt ðC:1Þ

where b is the repair cost per failure. Notice that the derivation of (C.1) is similar to (B.2). This is because a Poisson installation process mathematically is the same an exponential renewal process. The expected repair cost for the first installed base, denoted as E[R0] is the summation of all H(t) = at renewals occurred in [0, t], that is

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ith install 0

failure time

i

Wi

T1 T2 Tj t Fig. C.1. Renewals and repair scenario of the ith installation.

E½R0  ¼ HðtÞE½U j  ¼

ab b

ð1  ebt Þ:

ðC:2Þ

C.2. Repair cost for the ith installation with 1 6 i 6 N(t) Fig. C.1 depicts the failure and repair scenario for the ith installation between (0, t]. If the system is installed at Wi, no failures will occur prior to Wi. Let T1, T2, . . . , T HðtW i Þ be the random variable representing the times of failures leading to the repair action, respectively. Similarly, let U1, U2, . . . , U HðtW i Þ be the NPV of the costs for the corresponding repairs. Let fT j ðzÞ denote the probability density function of T j ; fT j ðzÞ ¼ 1=ðt  W i Þ. Then the expected value of Uj can be derived by conditioning on Tj and Wi as follows

E½U j  ¼

Z

t

Wi

¼

E½U j jT j ¼ zfT j ðzÞdz ¼

Z

t

bz

be

Wi

1 dz t  Wi

b ðebW i  ebt Þ: bðt  W i Þ

ðC:3Þ

Under the exponential renewal, we have H(t  Wi) = a(t  Wi). The expected repair cost for the ith installation is the summation of all repair costs occurred in [Wi, t], that is

E½Ri jNðtÞ ¼ n ¼ Hðt  W i ÞE½U j  ¼

ab b

ðebW i  ebt Þ;

for i ¼ 1; 2; . . . ; n:

ðC:4Þ

C.3. Total repair cost for the fleet Based on equations (C.2) and (C.4), the NPV of expected fleet repair cost during [0, t] is the sum of the NPV of N(t) + 1 renewal processes. That is

E½RjNðtÞ ¼ n ¼ E½R0  þ

NðtÞ X

E½Ri jNðtÞ ¼ n

i¼1

¼

¼

ab b

ab b

ð1  ebt Þ þ ð1  ebt Þ 

NðtÞ ab X

b

ðebW i  ebt Þ

i¼1

abNðtÞ b

ebt þ

NðtÞ ab X

b

ebW i ;

ðC:5Þ

i¼1

where b is the cost for one repair. Based on the technique used to solve equation (B.1), the NPV of the expected fleet repair cost between [0, t] is obtained as follows

E½R ¼ ¼

ab b

ð1  ebt Þ 

akb b

tebt þ

akb b2

ð1  ebt Þ

   k 1 þ ð1  ebt Þ  ktebt : b b

ab

ðC:6Þ

C.4. Downtime cost for the fleet Assuming d is the downtime cost per failure. Realizing the analogy between the repair cost and the downtime cost, the expected fleet downtime cost, denoted as E[D], can be obtained by substituting b with d in equation (C.6), which yields

E½D ¼

   k 1 þ ð1  ebt Þ  ktebt : b b

ad

ðC:7Þ

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