An integrated approach to the design and operation for spare parts logistic systems

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An integrated approach to the design and operation for spare parts logistic systems

Muh-Cherng Wu

⇑

, Yang-Kang Hsu, Liang-Chuan Huang

Department of Industrial Engineering and Management, National Chiao Tung University, Hsin-Chu, Taiwan, ROC

a r t i c l e

i n f o

Keywords:

Logistic network design Bill of material Spare parts Genetic algorithm Neural network Tabu search

a b s t r a c t

This paper attempts to solve a comprehensive design problem for a spare part logistic system. The design factors encompass logistic network design, part vendor selection, and transportation modes selection. Two approaches to solve the problem were proposed. In Approach 1, we simultaneously considered all the design factors and proposed two algorithms (SGA-1 and TGA-1). In Approach 2, the design problem was solved in two stages. Firstly, we aimed to ﬁnd a near-optimal logistic network. Secondly, with the obtained logistic network, we proposed three algorithms (SGA-2, TGA-2, and NN-GA-Tabu) to ﬁnd optimal combinations for part vendor and transportation modes selection. Numerical experiments indicate that Approach 2 outperforms Approach 1, and the NN-GA-Tabu outperforms all the other four algorithms. The proposed NN-GA-Tabu might also be a good solution architecture for solving other comprehensive space search problems.

1. Introduction

Spare part management is a very important issue for capitally-intensive industries (e.g., semiconductor manufacturing, aero-space, defense, and high-speed train). Building a leading-edge semiconductor wafer fab may cost up to 2 billion dollars; and the associated spare parts inventory may need 10–15% of the total expenditure. Other capitally-intensive industries also reveal the same characteristics. Thus, the design and operation of a spare part logistic system is very important for these industries.

A spare part logistic system (also called a logistic network) typ-ically involves a group of stations that are hierarchtyp-ically structured as shown inFig. 1. In the hierarchy, terminal stations, essentially designed to repair machines in the service ﬁeld, are equipped with machine-repairing staffs and spare parts inventory. Other higher-layer stations are designed to store and repair spare parts in order to supply spare parts to terminal stations. Parts delivery between any two stations needs a transportation time. In literature, such a logistic network is characterized as a multi-echelon system ( Sher-brooke, 1968)

As shown inFig. 2, a machine typically comprises a hierarchical assembly of parts – called bill of materials (BOM). In literature, a spare part logistic system that considers only one kind of part is called a single-indenture system. In contrast, a multi-indenture sys-tem is a spare part logistic syssys-tem that considers a BOM hierarchy involving many kinds of parts. This research is concerned with a

multi-indenture, multi-echelon (simply called MIME) spare part sup-ply chain system.

Several survey papers on spare part logistics in a MIME system have been published (Guide & Srivastava, 1997; Kennedy, Patter-son, & Fredendall, 2002). Prior studies could be essentially grouped in two categories.

One category aimed to find optimal operation policies for a given spare part logistic system; that is, how to determine optimal inventory level and repair-staff level for each station in order to re-duce the total operational cost. Some assumed that each station is equipped with an infinite staffing capacity for repairing parts; and paid attention to the decision of stocking levels. The pioneer one

is the METRIC model developed bySherbrooke (1968); many of

its extensions have been developed (e.g.,Graves, 1985; Muckstadt, 1973; Sherbrooke, 1986). Given a ﬁnite stafﬁng capacity for repair-ing parts, some others investigated the decision for optimum stock-ing levels (e.g.,Diaz & Fu, 1997; Kim, Shin, & Park, 2000; Perlman, Mehrez, & Kaspi, 2001). Extending the frontier,Sleptchenko, van

der Heijden, and van Harten (2003)aimed to solve a more complex

problem – ﬁnding an optimum combination for both repair-staff capacities and stocking levels.

The other category attempted to ﬁnd an optimal design for a spare part logistic system. Some aimed to design an optimal logis-tic network (Candas & Kutanoglu, 2007; Jeet, Kutanoglu, & Partani, 2009; Rappold & van Roo, 2009); some focused on optimal selec-tion of part vendors (Wu & Hsu, 2008); and some others examined optimal selection of transportation modes (Kutanoglu & Lohiya, 2008). Such design factors were only partially addressed in prior studies. Their obtained solutions might leave a space for further improvement if more design factors are simultaneously addressed.

⇑Corresponding author. Tel.: +886 35 731 913. E-mail address:mcwu@mail.nctu.edu.tw(M.-C. Wu).

Expert Systems with Applications 38 (2011) 2990–2997

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Expert Systems with Applications

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Yet, such a comprehensive inclusion of design factors may require formidable computational efforts.

In this paper, we attempt to solve a comprehensive design prob-lem for a spare part logistic system. The design factors encompass logistic network design, part vendor selection, and transportation modes selection. Two approaches to solve the problem were proposed.

In Approach 1, all the design factors are simultaneously consid-ered. That is, a new solution could be generated by varying the selection for any of the design factors. Based on such a solution representation, two meta-heuristic algorithms were proposed to solve the design problem. The two algorithms, adapted from liter-ature (Goldberg, 1989; Tsai, Liu, & Chou, 2004), are respectively called SGA-1 (simple genetic algorithm in Approach 1) and TGA-1 (Taguchi genetic algorithm in Approach TGA-1).

Approach 2 decomposes the design problems into two sub-problems. That is, we solve the design problem in two stages. In stage 1, we focus on ﬁnding a near-optimal logistic network, by the application of a technically sound heuristic rule. In stage 2, with the obtained logistic network, we proposed three meta-heuris-tic algorithms to ﬁnd optimal combinations for part vendor and transportation modes selection. The three algorithms are called SGA-2 (simple genetic algorithm in Approach 2), TGA-2 (Taguchi genetic algorithm in Approach 2), and NN-GA-Tabu (neural net-work-genetic algorithm-tabu-search).

Numerical experiments indicate that Approach 2 outperforms Approach 1. This advocates the use of a problem-decomposition approach in solving a large-scale problem, if a technically sound heuristic rule can be found. Of the three algorithms in Approach 2, the NN-GA-Tabu outperforms the other two both in solution quality and computation time. We developed the NN-GA-Tabu based on two ideas. First, we develop an efﬁcient yet rough perfor-mance evaluator to quickly justify a solution. Second, we use GA to

ﬁnd a quality solution and then use a tabu-search (a local tuning process) to obtain an improved one.

The remainder of this paper is organized as follows: Section2 describes the problem in more detail. Section 3 formulates the comprehensive design problem and analyzes possible ways to solve the problem. Section4describes the two algorithms in Ap-proach 1. Section5describes the solution architecture of Approach 2 and the proposed NN-GA-Tabu algorithm. Experiment results of all the ﬁve algorithms are compared in Section6. Concluding re-marks are in the last section.

2. Problem statement

In this research, machines are capitally-intensive and their availabilities are very important. Machine availabilities are deter-mined by the installing levels of two resources: (1) spare part inventory and (2) repair-staffs. Having a higher installing level for any of the two resources would lead to higher machine avail-abilities, yet at a price of incurring higher costs. How to make such a trade-off decision is critical to capitally-intensive industries.

As shown inFig. 2, the BOM of a machine is a hierarchy com-prising many assembly/parts. An assembly/part hereafter is called an item. The failure of each item follows a Poisson process. With long-lead times for acquisition, all items if failure need to be re-paired. Repair time is an exponential distribution and ﬁrst-come-ﬁrst-serve policy is adopted.

The failure of any item in the BOM would lead to machine-down and reduce its availability. Quick replacement of the failure item can alleviate the effect of machine unavailability. This is achievable by installing a high stocking level, yet would incur high-er inventory costs. By installing a highhigh-er level of repair-staffs, we would shorten the failure duration of items and consequently

Station (1) Station (5) Station (3) Vendor Station (6) Station (4) (15) Station (2) Layer 1 Layer 3 Layer 2 (14) (13) (12) (11) (10) (9) (8) (7) (16)

Fig. 1. The hierarchical structure of a logistic network.

Machine (A)

Assembly (B) Assembly (C) Assembly (D)

Part (E)

Level 1

Level 3 Level 2

Part (F) Part (G) Part (H) Part (I) Part (J) Part (K) Part (L) Machine (A)

Assembly (B) Assembly (C) Assembly (D)

Part (E)

Level 1

Level 3 Level 2

Part (F) Part (G) Part (H) Part (I) Part (J) Part (K) Part (L)

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require a lower stocking level, yet at a price of increasing stafﬁng costs.

A spare part logistic network is as shown inFig. 1. Each node in the network is called a logistic station (simply called station). Each station is equipped with two kinds of resources: (1) spare part inventory and (2) repair-staffs. The purpose of a logistic network is to maintain a target average availability for machines in ﬁeld, which are directly supported by terminal stations. Each station has one and only one parent station. Each station has a probability of successfully repairing an item. A failure item that cannot be suc-cessfully repaired by a station should be sent to its parent for repair.

Items-repairing may require various techniques. Therefore, dif-ferent items may need difdif-ferent types of repair-staffs. An inventory replacement policy ðs; s 1Þ is adopted in each station (Feeney & Sherbrooke, 1966). Consider a case in which an inventory level sij is installed for item i at station j. Two features of this inventory replacement policy is explained below. First, for item i at station j, its total number of stocks (including failure ones) should always be kept at sij. Second, a failure stock at station j, if sent to its parent, should get a good unit back for exchange. Likewise, receiving a fail-ure stock from its son station should give the son a good stock in exchange. A failure stock that cannot be repaired in the logistic network will ultimately be sent to its external vendor, who can al-ways successfully repair the stock but requiring much longer lead time.

The logistic network for supporting a particular group of ma-chines can be in various conﬁgurations. That is, given a generic net-work (Fig. 1), we can close some stations and reassign the parent– son relationships to create a network instance (Fig. 3). Consider a generic network that has E layers and each layer has l1, l2, . . . , lE sta-tions to open/close. Each station at layer e should be assigned to one parent in layer e 1, therefore, it has le1 possible assign-ments. As a result, all stations at layer e as a whole have lle

e1 possi-ble assignments. This implies that the possipossi-ble number of network instances isQE

e¼2l le

e1.

To reduce the total costs of a logistic network, we have two other design alternatives: (1) changing item vendors, (2) changing transportation mode for the path connecting each pair of parent and son stations. That is, consider a network instance that has s items and each item has k vendors to select. We have ks_possible choices in vendor selection. Such a vendor selection is a trade-off decision because the price of an item charged by a vendor with a lower failure rate is more expensive. Likewise, consider a network instance that has p paths connecting all pairs of

parent–son-sta-tions, and each path has m types of transportation modes. We then have mp_{possible transportation conﬁgurations.}

In summary, we have three decisions in the design of a logistic system: (1) logistic network instance selection, (2) item vendor selection and (3) transportation mode selection. Therefore, the possible number of design conﬁgurations is QEe¼2l

le

e1 k s

mp_. Noticeably, in justifying the effectiveness of each design conﬁgura-tion, we need to determine its optimal operating conditions; that is, its optimal item stocking levels and repair-stafﬁng levels. 3. Formulation and analysis

This section ﬁrstly formulates the research problem and pro-ceeds to analyze possible ways to solve it.

3.1. Formulation Sets and indices.

I = set of items, with index i 2 I;

K = set of staff-types for repairing items, with index c 2 K; Ic= set of items that could be repaired by staff-type c 2 K,

with index i 2 Ic;

P= set of possible part vendors, with index l 2P;

C= set of possible transportation modes, with index t 2C;

K= set of possible stations, with index j, m 2K;

Ks= set of terminal stations, with index j, m 2Ks;

Kup_j = set of possible parent-stations for station j 2K, with index m 2Kupj ;

Kdown

j = set of possible son-stations for station j 2K, with index m 2Kdownj ;

3.2. Decision variables

D ¼ ½x j; ½yjm; ½zjt; ½

v

il= set of decision variables for

logis-tic system design; O ¼ ½kcj; ½sij

= set of decision variables for logistic system operation;

xj¼ 1;_0; if station j 2_otherwise; Kis opened;

yjm¼

1; if station m 2Kis assigned as the parent of station j 2K; 0; otherwise;

zjt¼ 1; if the path between station j 2 K and its parent is through transportation mode t 2 C;

0; otherwise; (7) Station (1) (12) Station (5) Station (3) Vendor Station (6) Station (4) (13) (14) (15) (16) (8) (9) (10) (11) Station (2) Station (1) Station (5) Station (3)

Fig. 3. An example design of logistic networks. 2992 M.-C. Wu et al. / Expert Systems with Applications 38 (2011) 2990–2997

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vil

¼ 1; if item i 2 I is supplied by part vendor l 2P;

0; otherwise;

k ¼ ½kcj, where kcj= installed capacity of staff-type c 2 K at sta-tion j 2K;

s ¼ ½sij, where sij= the base stock level of item i 2 I at station j 2K;

3.3. Derived variables

Aavg_{D; O}_{: average machine availability for a design D operated} at O;

kij D : mean arrival rate of item i 2 I at station j 2

K

for a

deign D;

3.4. Parameters

Aobj_{= target average availability of machines;}

prep

ij = the probability that item i 2 I could be repaired at

sta-tion j 2

K

; pfailure

il = failure rate of item i 2 I while supplied by vendor

l 2

P

; nitem

i = total number of item i 2 I per machine;

nmachine

j ¼

total number of machines at station j 2

Ks

; 0; if j R

Ks

;

l

ij= the mean repair rate of item i 2 I at station j 2

K

;

cloc

j = the ﬁxed cost of opening station j 2

K

;

ctrans

j;m;t = the transportation cost from station j to station m by

transportation mode t; ttrans

j;m;t = the transportation time from station j to station m by

transportation mode t;

ccapc = cost of adding one more staff for staff-type c 2 K;

c 2 K; ð11Þ

xj; yjm; zjt;

v

il are binary variables: ð12Þ

In the above formulation, the objective function is to minimize total logistic costs, which involve: opening costs of logistic stations, transportation costs, costs of equipping repair-staffs, and inventory holding costs of items. Constraints(1) and (2)denote that each sta-tion has only one parent. Constraint(3)ensures that a station that has been closed cannot be a parent. Constraint(4)denotes that each terminal station should be opened. Constraint(5)denotes that only one transportation mode can be selected for each path. Constraint (6)denotes that only one vendor can be selected for each item. Con-straint(7)describes a recursive formula for computing mean arrival rate for each item at each station. Constraint(8)deﬁnes the mini-mum capacity for each type of repair-staff. Constraint(9)denotes that average machine availability for a particular design/operation option can be obtained by a queuing network model. Constraint (10)deﬁnes the target availability. Constraints(11) and (12)ensure decision variables are in valid ranges.

3.6. Analysis of solution approaches

The formulation is a nonlinear program, in which constraint(7) is a recursive formula and constraint(9)is a complicated proce-dure which cannot be expressed by an explicit function. Therefore, we cannot solve the problem by analytical methods.

The problem by nature is a huge space search problem which

involves two groups of decision variables – one group

D ¼ ½x j; ½yjm; ½zjt; ½

vil

is for design optimization and the other group O ¼ ½kcj; ½sij

is for operational optimization. To effectively justify a given D, we have to know its optimal O.Sleptchenko et al.

(2003)has developed a technique to determine an optimal O for a

given D. Therefore, we consider this problem as a space search problem that involves only D; and the minimum operational costs for each D is obtainable by computing its optimal O. To solve such a space search problem, we naturally consider the use of meta-heu-ristic algorithms.

Two approaches to solve the problem were proposed. In Approach 1, all the design factors D ¼ ½xj; ½yjm; ½zjt; ½

vil

are simultaneously considered, and two meta-heuristic algorithms (SGA-1 and TGA-1) were developed. In Approach 2, we decompose the design problem into two sub-problems, which are proceeded in two stages. In stage 1, we focus on ﬁnding a near-optimal logistic network, L

¼ ½xj;½yjm

. In stage 2, with the obtained logistic network L*, we develop three meta-heuristic algorithms (SGA-2, TGA-2, NN-GA-Tabu) to ﬁnd S¼ ½zjt;½

vil

. Herein, S* denotes an optimal combination for the part vendor and transportation modes selection decisions.

Of the meta-heuristic algorithms, SGA-1 and SGA-2 are adapted from traditional GAs (Gen & Cheng, 2000; Goldberg, 1989). TGA-1 and TGA-2 are adapted from an enhanced GA, which additionally embeds the Taguchi experimental design technique in a traditional

GA. The NN-GA-Tabu, partly adapted fromWu and Hsu (2008)and

partly adapted fromChiou and Wu (2009), embeds neural network and tabu-search techniques in a traditional GA.

The reasons why we proposed the development of 1, TGA-2 and NN-GA-Tabu are described below. By a pilot study, determin-ing an optimal O for a given D requires about 6 s in computation time. This implies that applying a typical meta-heuristic algorithm such as genetic algorithm (GA) would be computationally exten-sive. For example, evaluating 20,000 solutions – a relatively small number in a typical GA application even requires about 1.5 days.

To alleviate the computational issues, two techniques are con-sidered. One technique is by embedding an experiment design par-adigm to efﬁciently generate quality solutions. The 1 and TGA-2 are applications of this technique. The other technique is by the application of neural network (NN) algorithm to develop an

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efﬁcient yet rough performance evaluator for D. With such a rough performance estimator, we can quickly justify much greater num-ber of solutions to obtain a candidate list, and then use the accurate performance evaluator to select the best one from the list. Then, the obtained solution is further reﬁned by a tabu-search process to get an improved one. The NN-GA-Tabu is an application of this technique.

4. Approach 1

This section describes the two algorithms (SGA-1 and TGA-1) in Approach 1. We first describe the chromosome (or solution) repre-sentation in the two algorithms; then present how to obtain the fitness (or quality) of a chromosome; and finally explain the basic ideas of the two algorithms.

4.1. Chromosome representation

In Approach 1, we simultaneously consider three types of de-sign factors: logistic network dede-sign, part vendor selection, and trans-portation mode selection. A design solution (called a chromosome) is a string that comprises three segments. Each segment (a smaller string) represents a design alternative for a particular type of de-sign factors. That is, a chromosome is represented by X = [Xnet

jX ven-dor_jXtrans_{] where X}net_{, X}vendor_{, and X}trans_{are its three segments, each} of which respectively models a type of design factor.

Segment Xnet_{= [g}

1, . . . , gn] is intended to represent an alternative for logistic network design, where gj2Kupj is the assigned parent of station j and n is the total number of possible stations. Each station always has a parent but not vice versa. That is, a station may not have a son. A non-terminal station (see stations 1, 3, and 5 in Fig. 3) that does not have a son should be closed. Therefore, Xnet de-notes a particular logistic network – it describes which stations are closed and how stations are supported by their possible parents.

In addition, segment Xvendor_{= [}

_v1

_{, . . . ,}

_vk

_{] is intended to represent} an alternative for part vendor selection, where

vi

2Pis a vendor selection for item i. Segment Xtrans_{= [h}

1, . . . , hn] is intended to rep-resent an alternative for transportation mode selection, where hj2C is a transportation mode selection for station j. Notice that each element (gj,

vi

, or hj) in a chromosome X is called a gene.

4.2. Fitness evaluation

Noticeably, a chromosome X just denotes a design solution. To evaluate the fitness of X (the solution quality), we have to obtain its optimal operating policies which include inventory stocking levels as well as repair-staffing levels at each station j. According toSleptchenko et al. (2003), such optimal operating policies can be obtained. We therefore define the total logistic cost, under opti-mal operating policies, as the fitness of X.

4.3. Algorithms: SGA-1 and TGA-1

The two meta-heuristic algorithms (SGA-1 and TGA-1) are both based on a typical GA architecture, which involves four stages: ini-tial population generation, population evolution, population updating, and evolution termination. Yet, in the stage of population evolution, the two algorithms are different in their ways of creating new chromosomes. Each of the four stages is explained below.

In stage 1 – initial population generation, we randomly create N valid chromosomes to form an initial population P0. These created chromosomes should be valid, in the sense that each gene value must be in its designated set (i.e., gj2K

up

j ,

vi

2P, hj2C). The initial population will be iteratively updated, and the updated population at iteration t is called Pt.

In stage 2 – population evolution, various kinds of genetic oper-ators are developed to create new chromosomes from Pt. In SGA-1, only two genetic operators: crossover and mutation are included. Other than the two, TGA-1 includes one more genetic operator (called the Taguchi operator). The mechanism for creating new chromosomes by each genetic operator is explained below.

In the crossover operator, we randomly choose two chromo-somes (e.g., X1and X2), by randomly sectioning each segment into two parts and swap their gene values, to create two new ones (e.g., X3and X4). Deﬁne Xi¼ Xneti Xv

endor i Xtransi h i

for i = 1, 2, 3, 4. For each segment X seg_i in Xi, deﬁne its two sectioning parts as Xsegi;a and X

seg i;b. As shown inFig. 4, new chromosomes X3and X4can be created by applying sectioning and swap operations on X3and X4.

In the mutation operator, we randomly choose one chromo-some (e.g., X1), by randomly changing one of its gene values for each segment (i.e., Xnet1 ; Xv

endor

1 ; X

trans

1 ), to create one new

chromosome.

In the Taguchi operator, we randomly choose two chromosomes (e.g., X1 and X2), by applying the Taguchi experimental design method, to create one new chromosome (e.g., X5). Suppose a chro-mosome has n genes in total. Each gene, of the two chrochro-mosomes X1and X2, has at most two levels. Then, ﬁnding a good chromo-some from all possible combinations of X1and X2can be seen as an experimental design problem, which has n factors and each fac-tor has two levels. Using the orthogonal array experimental design, the Taguchi operator can efﬁciently obtain a new and good chro-mosome X5.

In each iteration, the total number of newly created chromo-somes is N(Pcr+ Pmu+ Pta), where 0 6 Pcr, Pmu, Pta61 represent parameters of various genetic operators. This implies that, at the end of stage 2, we have N(1 + Pcr+ Pmu+ Pta) chromosomes in total. In stage 3 – population updating, we use roulette wheel selec-tion method (Michalewicz, 1996) to screen N chromosomes from the N(1 + Pcr+ Pmu+ Pta) ones to form a new population. In stage 4 – evolution termination, the population evolution/updating proce-dures will be terminated if the iteration number has achieved an upper bound (i.e., t = Tmax) or a best solution has sustained over a certain number of iterations (Tbest).

5. Approach 2

X

3

X

Fig. 4. The crossover operator and its corresponding chromosomes. 2994 M.-C. Wu et al. / Expert Systems with Applications 38 (2011) 2990–2997

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5.1. Stage 1: determining logistic network

In stage 1, the solution space of possible logistic networks can be quite large. Consider a logistic network that has E layers and l1, l2, . . . , lE stations to open/close. As stated, the possible number of network instances is Q ¼QEe¼2l

le

e1; that is. Q = 224= 1.68 107 for E = 3, l1= 2, l2= 4, and l3= 10.

To select an optimal one from the solution space, we need to evaluate each candidate solution. Yet, evaluating a candidate logis-tic network may be computationally extensive because it has a huge number of possible versions, due to the variations caused by the selection of item vendors and transportation modes. Con-sider a scenario with s items and p connecting paths, each item has k vendors and each path has m transportation modes to select. Then a logistic network has V = ks_mp_{possible versions; that is} V = 322= 3.14 1010 for a typical application case with m = k = 3, p = 10, s = 12. Moreover, evaluating such a version may be also time-consuming, because for each possible version we need to compute its optimal operating policies.

To reduce the computational extensive issues, this research pro-poses one heuristic rule in ﬁnding a near-optimal logistic network. The heuristic rule, which appears to be technically sound, request that the parent of each station j is the one inKup_j that is open and is the nearest one in transportation distance. With this heuris-tic rule, we only have to determine whether a station should be open or close. As a result, the solution space of possible logistic networks is greatly reduced. For a logistic network with E layers and l1, l2, . . . , lEstations, its possible number of network instances now becomes Q0¼QE1e¼1 2

le₁

; that is. Q0_{= 45 if E = 3, l} 1= 2, l2= 4, and l3= 10.

For a logistic network, the number of its possible versions is quite huge (V = ks_mp_{). We randomly sample 30 versions and compute} the total logistic cost for each version while it is under optimal oper-ating policies. That is, for a logistic network with E layers and l1, l2, . . . , lE stations, we only have to evaluate N0¼ 30Q0¼ 30QE1

e¼1 2 le

1

logistic network versions. For a case with E = 3, l1= 2, l2= 4, and l3= 1, we have N0= 1350 while the complexity of its original solution space is N = QV = 224

322= 5.2 1017_.

5.2. Stage 2: selecting vendor and transportation modes

The output of stage 1 is a near-optimal logistic network (say, L*). With the logistic network L*, in stage 2, we aim to ﬁnd an opti-mal combination for the decisions of part vendor and transporta-tion modes selectransporta-tion. For this purpose, three meta-heuristic algorithms (SGA-2, TGA-2, and NN-GA-Tabu) were developed, in which a chromosome is now represented by a smaller string, say X = [XvendorjXtrans]. The procedures of SGA-2 and TGA-2 are similar to that of SGA-1 and TGA-1 as described in Section4. Thus we only explained the procedure of NN-GA-Tabu herein.

The procedure of NN-GA-Tabu comprises four major steps. Firstly, we apply the neural network technique to develop an efﬁ-cient yet rough performance evaluator of a chromosome. Secondly, with such an efﬁcient evaluator, we use a traditional GA to obtain a list of candidate solutions. Thirdly, we use the original perfor-mance evaluator, developed bySleptchenko et al. (2003), to select

the best one from the candidate list. Finally, the obtained solution is further reﬁned by a tabu-search process, which is adapted from Glover and Laguna (1997). Each step is further explained below.

The development of the rough performance evaluator is based on the back-propagation (BP) neural network algorithm (Fausett, 1994), which has been widely used to emulate a system’s behavior by a sampled data set. That is, for a system with vectors Xias input and Zias its corresponding output, we can randomly sample n1 pairs of {Xi, Zi} to train (or establish) a BP neural network, which can compute an estimated output bZifor each sampled Xi. Then n2 pairs of {Xi, Zi} are randomly sampled to test the BP neural network. If the predicted error e ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pn2 i¼1 ZibZi 2 n2 r

is acceptable, then the BP network can be used to emulate the system’s behavior; that is, computing for bZifor other Xi. A detailed algorithm for developing such a BP neural network can be referred toFausett (1994).

Based on the BP neural network algorithm, a procedure Rough_Evaluator is developed to quickly and roughly estimate the ﬁtness of a chromosome, as stated below.

5.2.1. Procedure Rough_Evaluator

Sample Ntchromosomes, say Xi¼ XviendorjX trans i

h i

; i ¼ 1; . . . ; Nt. Use the original evaluator to compute the ﬁtness for each

sampled chromosomes, Zi= Original_Evaluate(Xi); i = 1, . . . , Nt. Use the data set {Xi, Ziji = 1, . . . , Nt} to develop a BP neural

net-work, where Xiis input and Ziis output.

Represent the developed BP neural network by bZi¼ Rough E

v

aluateðXiÞ.

Notice that, in the above procedure, the rough evaluator (or the BP neural network) is denoted by bZi¼ Rough E

v

aluateðXiÞ and the original evaluator is denoted by Zi= Original_Evaluate(Xi). The pro-cedure NN-GA-Tabu can thus be stated below.

5.2.2. Procedure NN-GA-Tabu

Step 1: Establish a rough performance evaluator, bZi¼ Rough E

v

aluateðXiÞ.

Step 2: Develop a traditional GA that evaluates a chromosome by bZi¼ Rough E

v

aluateðXiÞ. By the GA, ﬁnd a candidate set of chromosomes S, where

S ¼ XiðteÞ; bZiðteÞ

n o

; i ¼ 1; . . . ; N;

teis the iteration while the GA terminates:

Step 3: Find the best chromosome Xi in S, by the original

evaluator

ZiðteÞ ¼ Original E

v

aluateðXiðteÞÞ; for i ¼ 1; . . . ; N;

i¼ Arg Min

16i6NL

ZiðteÞ:

Step 4: Use a tabu-search process to reﬁne Xi; that is Xi¼ Tabu SearchðX_iÞ; where

Xiis the solution obtained by the tabu-search process:

Table 1

Station opening cost and the number of machines at each terminal station (cost: $K).

Station ID 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Layer ID 1 1 2 2 2 2 3 3 3 3 3 3 3 3 3 3

Open cost 6800 5000 2000 1800 1900 1200 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000

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6. Numerical experiments

Numerical experiments are carried out to justify the five algo-rithms (SGA-1, TGA-1, SGA-2, TGA-2, and NN-GA-Tabu) to solve the comprehensive design problem. We first describe tested sce-nario; then present the parameters of the five algorithms and the computing hardware; and finally report and analyze the experi-ment results.

The tested scenario, involving 16 stations in a three-layer hier-archy (Fig. 1), has a target machine availability Aobj_{= 0.95.}_{Table 1} shows the opening cost for each station and the number of ma-chines at each terminal station. The transportation times/costs,

operated under transportation mode 1, are shown in Tables 2

and 3. We assume that the transportation times/costs, operated

under different modes, are proportional; that is, ttrans j;m;2 . ttrans j;m;1¼ 0:75; ttrans j;m;3 . ttrans j;m;1¼ 0:50; ctransj;m;2 . ctrans j;m;1¼ 1:20, and ctransj;m;3 . ctrans j;m;1¼ 1:50. This implies the faster is a transportation mode the more expen-sive is its transportation cost.

The BOM hierarchy for each machine involves three levels (Fig. 2). Items in level i require type i repair-staffs and their unit stafﬁng costs are shown inTable 4.Table 5shows the quantity of each item as well as its failure rates and units costs while supplied by different vendors. The higher the failure rates, the lower is the unit cost. Each station is given a repairable probability – a probabil-ity of successfully repairing an item as shown inTable 6, which also gives repairing times.

Parameters of the ﬁve algorithms are set as follows: N = 50, Tbest= 100, Tmax= 100, Pcr= 0.8, Pmu= 0.2, and Pta= 0.04. In NN-GA-Tabu, a BP neural network, with e = 0.00795, is established by using n1= 1300 and n2= 200. For each experiment, we run 10 rep-licates, by using personal computers equipped with 3.4G CPU and 504 MB RAM.

Table 7compares the experiment results of the ﬁve algorithms,

in terms of solution quality (total logistic cost) and computation time. The table indicates that Approach 2 signiﬁcantly outperforms Approach 1 both in solution quality and computation time. More-over, the qualities of solutions obtained in Approach 1 have a high-er variation. That is, Approach 2 is not only more effective but also more robust.

The reason why Approach 2 outperforms may be due to the adoption of the heuristic rule – the parent of each station is just the nearest one in its upper layer. This heuristic rule appears to be technically sound. Therefore, a chromosome violating the heu-ristic rule tends to be inferior. In Approach 1, many such violating chromosomes are likely to be generated in the population

evolu-Table 3

Transportation times/costs between layers 2 and 3, operated under transportation mode 1 (time: hour, cost: $K).

Station 3 Station 4 Station 5 Station 6 Time Cost Time Cost Time Cost Time Cost Station 7 24 2400 56 21,600 72 38,400 248 60,000 Station 8 32 5400 48 15,000 64 29,400 240 48,600 Station 9 40 9600 40 9600 56 21,600 232 38,400 Station 10 56 21,600 24 2400 40 9600 216 21,600 Station 11 64 29,400 32 5400 32 5400 224 29,400 Station 12 80 48,600 48 15,000 32 5400 240 48,600 Station 13 88 60,000 56 21,600 40 9600 248 60,000 Station 14 232 38,400 232 38,400 248 60,000 40 9600 Station 15 248 60,000 216 21,600 232 38,400 24 2400 Station 16 264 86,400 232 38,400 216 21,600 40 9600 Table 2

Transportation times/costs between layers 1 and 2, operated under transportation mode 1 (time: hour, cost: $K).

Vendor Station 3 Station 4 Station 5 Station 6

Time Cost Time Cost Time Cost Time Cost Time Cost

Station 1 240 2500 32 5400 48 15,000 64 29,400 34 15,000

Station 2 240 2500 80 48,600 48 15,000 32 5400 34 15,000

Table 4

Unit costs for various types of repair-staffs.

Staff type Repairing item type Cost ($K)

Type 1 Level 1 800

Type 2 Level 2 620

Type 3 Level 3 480

Table 6

Repairable probability and repairing time for stations at each layer.

Layer 1 Layer 2 Layer 3

Repairable probability 0.8 0.7 0.6

Repairing time (h) 8 10 12

Table 5

Item failure rates and unit costs of different vendors.

Item Quantity of each item Failure rates provided by each vendor (number of failures per 106

h) Unit cost charged by each vendor ($K)

FRT1 FRT2 FRT3 C1 C2 C3 A 1 2835 2948.4 3175.2 1300 1200 1000 B 1 2205 2293.2 2469.6 220 200 190 C 1 2520 2620.8 2822.4 310 280 250 D 1 2835 2948.4 3175.2 180 170 160 E 2 1575 1638 1764 40 38 35 F 1 1890 1965.6 2116.8 50 40 30 G 1 2205 2293.2 2469.6 30 25 20 H 1 1575 1638 1764 50 42 36 I 3 1890 1965.6 2116.8 35 30 25 J 1 2205 2293.2 2469.6 55 48 45 K 2 2835 2948.4 3175.2 40 30 20 L 2 2835 2948.4 3175.2 30 28 25

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tion process. Then, the best solution in the chromosome population is less likely to be improved. As a result, the ultimately obtained solution tends to be inferior. This advocates the use of a prob-lem-decomposition approach in solving a large-scale space search problem, if a technically sound heuristic rule can be found.

Of the three algorithms in Approach 2 (Table 7), the

NN-GA-Tabu signiﬁcantly outperforms the other two in terms of com-putation time, and slightly better than the other two in terms of solution quality. This indicates that the NN-GA-tabu is an effective and efﬁcient method to solve the comprehensive logistic network design problem.

7. Concluding remarks

This paper examined a comprehensive design problem for a spare part logistic system, which involves the following decisions: logistic network design, part vendor selection, and transportation modes selection. In prior studies, these decisions were only partially addressed. A comprehensive inclusion of such design factors may require formidable computational efforts.

Two approaches were proposed to solve the problem. In Approach 1, we simultaneously considered all design factors and proposed two meta-heuristic algorithms (SGA-1 and TGA-1). In Approach 2, we decomposed the design problems into two sub-problems. The ﬁrst sub-problem is to ﬁnd a near-optimal logistic network. With the obtained logistic network, we proposed three meta-heuristic algorithms (SGA-2, TGA-2, and NN-GA-Tabu) to solve the second sub-problem – selecting vendors and transportation modes.

Numerical experiments indicate that the NN-GA-Tabu outper-forms the other four algorithms. The merits of the NN-GA-Tabu lead to three implications. First, it advocates the use of a prob-lem-decomposition paradigm in solving a comprehensive space search problem, if a technically sound heuristic rule can be found. Second, it advocates the development of an efficient (could be rough) performance evaluator to speed up the justification of a solution. Third, it advocates the use of GA-Tabu search mechanism to find a near-optimum solution. Thus, the proposed NN-GA-Tabu may also be a good solution architecture for solving a comprehen-sive space search problem. Other applications of the NN-GA-Tabu solution architecture are being considered in future research. Acknowledgements

This research was partially support by a research grant, under contract NSC94-2623-7-009-004, Taiwan.

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Table 7

Experiment results of Approach 1 and Approach 2 (time: hour, cost: $K).

Replicate Approach 1 Approach 2

SGA-1 TGA-1 SGA-2 TGA-2 NN-GA-Tabu

Cost Time Cost Time Cost Time Cost Time Cost Time

1 74,747 49.5 75,406 47.8 74,225 39.7 74,199 35.8 74,216 7.3 2 76,251 72.3 74,625 73.2 74,081 51.2 74,081 74.0 74,081 7.1 3 78,907 53.7 74,511 106.7 74,113 38.6 74,205 57.3 74,081 7.6 4 75,659 53.8 74,244 85.8 74,255 27.5 74,101 54.3 74,081 7.1 5 74,560 48.6 78,512 148.4 74,205 30.6 74,241 42.9 74,081 7.1 6 76,774 44.7 74,229 72.0 74,159 25.1 74,205 83.5 74,081 7.2 7 74,517 42.3 78,722 137.2 74,108 51.9 74,081 83.1 74,081 7.5 8 79,530 29.5 74,342 96.0 74,081 22.0 74,081 79.5 74,205 6.9 9 76,202 39.7 74,284 139.4 74,139 19.9 74,081 62.2 74,081 7.5 10 74,468 48.3 78,459 362.5 74,133 26.0 74,085 62.4 74,206 6.9 AVG 76,162 48.2 75,733 126.9 74,150 33.2 74,136 63.5 74,119 7.2 DEV 1815 11 1984 89 60 12 67 17 62 0.2