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A comparison of allocation policies in a two-echelon repairable-item inventory model
International
Journal of‘ Production
291
Economics, 29 (1993) 291-302
Elsevier
A comparison repairable-item M. Demirbag
of allocation policies in a two-echelon inventory model
Btiytikkurt
Faculty of Commerce and Administration, QuPhec, Canada H3G IM8
Concordia
Universit),,
1455 de Maisonneuce
Blvd. West, Montr&al,
and
M. Parlar Faculty of Business, McMaster (Received
15 July
University. Hamilton,
Ontario, Canada L8S 4M4
1992)
Abstract The objective of this paper is to evaluate three allocation policies in a two-echelon machine repair model with state-dependent failure and repair rates. The first echelon consists of a number of bases that are served by a central depot in the second echelon. The operations at each base are modeled as a machine repair problem. Due to the complexity of the general model, a simulation study was conducted whereby the regenerative method ofdata collection was used. Two static and one dynamic allocation policy were compared with respect to five criteria. Based on the distribution of the observations, stochastic dominance tests were conducted and it was observed that static policies did not differ. The dynamic policy was found to be superior in a majority of the criteria
1. Introduction repairable-item inventory Two-echelon models are those which include a number of bases comprising the first echelon, and’ the central depot comprising the second echelon which serves all the bases. In such models, each base consists of a collection of machines which fail according to a specified probability distribution. Some of the failed machines require minor repairs which can be completed at the base, whereas the others requiring major repairs are sent to the depot. Upon completion of repair at the depot, a machine is sent to
Correspondence to: M. Parlar, McMaster University, Hamilton, 4M4.
0925-5273/93/$06.00
Faculty Ontario,
of Business, Canada L8S
f$> 1993 Elsiver Science Publishers
a base according to a prespecified allocation policy. Some of the earlier models dealing with the above problem have optimized, i.e., minimized, the expected number of backorders at all bases subject to some constraints [l]. Since the publication of Sherbrooke’s model (known as METRIC), interest in two-echelon repairable-item inventory models has increased considerably. Many researchers have extended the original results of METRIC by considering e.g., condemnation (disposal) of some failed machines [2], and multi-indentures [3]. In all these early models, one crucial assumption was the state-independence of failure and repair rates which implied that the number of machines and repair facilities were infinite. In those applications where these numbers are finite but large, state-independence serves as a convenient assumption
B.V. All rights reserved.
towards obtaining analytic solutions using queuing theory. Clearly, in many practical situations, this may not be the case when there are only a small number of machines and/or repair facilities. Examples include computer service companies that provide minor repair service on sight and major repair service at headquarters. In this paper, we aim to analyze a twoechelon repairable-item inventory model with state-dependent repair and failure rates. In particular, we compare different allocation policies and test their effectiveness based on various criteria. The policies we consider involve sending the depot-repaired machine to the base: (i) that has the least number of machines, (ii) that the machine originated from, and (iii) that has had a backorder for the longest time (first-come-first-served). The five criteria we use in evaluating the allocation policies are all measures of effectiveness. They are, (i) expected number of backorders, (ii) expected n umber of unfilled online slots, (iii) expected downtime, (iv) expected percentage of time all depot repair facilities are busy, and (v) expected number of machines per base. Based on an extensive simulation study, we find that the first policy is preferable to the others for a majority of the criteria considered. Allocation policies in multi-echelon repairable-item inventory systems have also been considered by other researchers [ 1, 41. Common to all the models studied is the assumption of state-independent repair and/or failure times which, as discussed above, implies infinite number of machines and/or repair facilities. This research is different from those above in that a more realistic model with finite machines and repair facilities is studied. Another common aspect of the above cited research is the criteria used in evaluating allocation policies. All have aimed to minimize the number of backorders, with the assumption that any policy that results in minimum backorders will also be the optimal when some other criterion is used. As discussed above, we will evaluate allocation policies using several criteria including number of backorders.
Allocation policies that have been studied in multi-echelon repairable-item inventory systems have generally been classified as static or dynamic policies. A static allocation policy (SAP) is defined to be state-independent, whereas a dynamic allocation policy (DAP) is state-dependent. An example of a DAP is Sherbrooke’s ‘marginal allocation policy’ whereby a repaired machine is sent to the base where it will cause the largest decrease in expected backorders [l]. Another such DAP is Miller’s ‘transportation time look ahead policy’ whereby a repaired machine is sent to the base with the minimum number of machines on hand plus those arriving during transportation time [4]. Such a policy is an attempt to evaluate the needs of each base and allocate the repaired machine to the ‘most needy’ base. It does not, however, involve a complete evaluation of the needs of each base since machines in stock and in repair are not considered. This is accomplished in this research. Still another DAP reported in the literature is allocation of a repaired machine to the base that has the highest percentage of its machines in the depot [S]. Two SAPS have been frequently evaluated in the literature, namely (1) first-come-first-served, and (2) sending to the base of origin. The latter has been compared to a DAP similar to the ‘transportation time look ahead policy’ by Miller [IS], and found to be inferior since it has resulted in more backorders. This research will compare both SAPS stated above to a DAP similar to the ‘transportation time look ahead policy’, but one that also considers machines in stock and in repair. In Section 2, we describe the model in detail, and discuss the solution method employed. Analysis of results are presented in Section 3, followed by the summary and conclusions in Section 4.
2. Methodology The model studied is that of a two-echelon inventory model, whereby each base that operates as the machine repair model forms the
M.D. Btiyiikkurt,
M. ParlarlA
two-echelon repairable-item
taneously placed online; otherwise it is placed in spares inventory. With probability 1 - p, the failed machine requires a major repair and is thus sent to the depot. The transportation time from the base to the depot is assumed to be negligible. There are R,, parallel repair facilities at the depot, each of which can repair a machine at an exponentially distributed time with parameter ,&. There are no spares at the depot. The depot repairable failures eventually lead to depletion of machines at the base unless orders are placed for additional machines. We use an (S - 1, S) ordering policy, whereby an order for one machine is placed when a failed machine is determined to be depot repairable. A backorder is initiated when a failed machine is determined to be depot repairable. The term backorder, in this context, has a different meaning than the one employed in inventory control literature where it is used for orders placed by a base that cannot be filled by the depot. We use the term backorder in accordance with its usage in the literature on the machine repair problem as indicated above. Once a machine is repaired at the depot it is sent to the base in accordance with the prespecified allocation policy. In this model there is no condemnation of failed machines.
first echelon, and the central depot that serves all the bases forms the second echelon. The operation in each base and its interaction with the depot is depicted in Fig. 1. There are a total of S identical machines in each base, of which a maximum of M and a minimum of L can be online (operating) at any time. When there are more than M operable machines available, the excess is placed in spares inventory. If the number of online machines decreases to L - 1, the operations at that base are shut down. The failure rate of machines is dependent on the number of online machines at the time of failure, and each machine fails according to a Poisson process with an average interfailure time of l/L. A nonoperable machine is replaced with a spare instantaneously if one is available and inspected instantaneously to determine whether it needs to be repaired at the base or the depot. With probability p, a minor repair is needed which can be performed in one of Rb parallel repair facilities at the base. The repair rate is dependent on the number of idle repair facilities at the time, and each repair time is exponentially distributed with parameter ,&. At the time of repair completion, if there are less than M online machines, the repaired machine is instan-
Base 1
r I
Base 2
----
-
Online
:
_
I L
1
M A
AP
I
fib
I 4
Repair
-.-----__I
Fig. 1. The model with N bases.
: Rb-
pb
...
Base N
r-lr--1
I
I
I
293
inventory model
I
I
I
I
I
I
I
I
I
I
I.*.1
I
I
I
I
I
I
I
I
I
Finally, there is no lateral supply of machines between the bases. Various versions of this model can be generated by changing some of the assumptions. For example, one can assume non-exponentially distributed interfailure and repair times. Biiyiikkurt has investigated different interfailure and repair time distributions [7], and found that the model was sensitive to a change in the distributions only when the system was highly congested. Another extension of the model could be obtained by incorporating different cost and revenue factors associated with the operation of the system, and thereby optimizing an economic criterion. This could be accomplished by making some of the parameters such as L and &! decision variables and choosing their optimal values using a search procedure. Our emphasis in this paper, however, has been comparing various allocation policies based on measures of effectiveness rather than profits. Finding the exact solution of the above model is very difficult because of the interdependence of the bases due to their joint utilization of the depot repair service. More specifically, the lead time between the placement of an order and the return of the repaired machine to the base is a function of the total number of bases and the number of backorders at each base. Thus, the interdependence between the bases results in a very complex leadtime distribution which is very difficult, if not impossible, to determine. Due to the nondeterministic repair and failure rate and the twoechelon system assumed in the model, elementary queuing theory results cannot be used to solve the stationary properties as has been done in similar research. Even a simplified approximate model for this problem as discussed by Biiyiikkurt in Ref. 8 is difficult to solve. This approximation is a continuous time Markov process which is solved for the steady state probabilities by taking advantage of the special form of the transition rate matrix. The states of the Markov process are of the form O
base (operable or in base repair), P is the number in or awaiting base repair, S is the maximum number of machines per base, and L is the minimum required number of online machines. With 8 defined as the approximate value of the rate parameter in the lead time distribution, the state transitions, the event causing the transition, and the associated rates can be summarized as follows: Transition
Event
Transition
(II. 1.) * 01. )’ + I)
Base repairable failure Base repair completion Depot repax facilities Arrival from depot
min[M,
(11.)‘) --t 01. r - I) (II. 1.) + (H - 1, r) (!I. ,‘) + (?I + I. r)
rate
(n - r)ip]
min(&. r)ic, min[M.
(II - r)]E.( 1 - p)
min [Rd. (S - n)]s
Another method that can be used when solving problems of this complexity is simulation. Since the objective of this research is to evaluate various allocation policies, it is more desirable to simulate the original complex model rather than develop exact solutions to its approximations. The model is simulated in FORTRAN using discrete event simulation. The regenerative method of collecting statistics is used since the process regenerates itself when it enters certain states. We took advantage of this property and used the regenerative method since it eliminates some of the problems associated with the traditional simulation approaches. These problems include the necessity of making decisions regarding startup conditions and detection of the steady state, and the correlation of simulated data. In regenerative method, data gathered in each cycle is independent of those gathered in other cycles. We select an approximate regeneration point that includes a group of states, as suggested by Crane and Lemoine [9]. The group of states included in the approximate regeneration point selected are those whereby a depot repairable failure at any base triggers the first backorder following a period of no backorders in the system. In all the simulation runs, this approximate regeneration point proved to be quite satisfactory in
M.D. Btiyiikkurt,
M. Parlar/A
two-echelon repairable-item
that the resulting correlation coefficients between observations in successive cycles were fairly small (approximately 0.0001-0.01) for all the statistics calculated. We employ five criteria to evaluate three allocation policies to select the base that will receive a depot-repaired machine. The first allocation policy involves sending the depotrepaired machine to the base that has the fewest number of machines, and at least one outstanding backorder. In other words, those bases that do not have machines at the depot repair facility are not considered. In calculating the fewest number of machines, we consider not only the number of machines online, but also those in stock and those in repair. For a particular base, we let 4’ be the number of machines online, u’ be the number of machines in stock and z be the number of machines in base repair. In order to determine the destination of a repaired machine, we select the base that has the lowest value of (y + u’ + z). If more than one base qualifies, then they are equally likely to receive the repaired machine. In the sequel, we will label this policy as ‘Needy’. As briefly mentioned in Section 1, compared to Miller’s ‘transportation time look ahead policy’, our policy additionally considers machines in stock and in base repair [4]. Thus, due to its generality, it provides a more complete need assessment. Also, given a fixed number of machines in the system (S), the values of J, MI and z determine the number and percentage of machines a base has in the depot repair facility. In other words, a ‘Needy’ base is one that has the most number of machines at the depot repair facility. The second allocation policy is formulated to send the repaired machine to the base it originated from. In the sequel, this policy will be labeled ‘Origin’. Finally, the third allocation policy labeled ‘FCFS’, corresponds to sending the machine to the base that has had an outstanding backorder for the longest time, i.e., first-come-first-served. In the discussion of the results below, we will index these policies as j = 1,2,3, wherej = 1 corresponds to ‘Needy’, etc.
inwntory
model
295
3. Analysis of results The first criterion we employed in evaluating allocation policies is the expected number of backorders per base. This criterion has been the focus of most of the literature on repairable-item inventory models. As stated by Sherbrooke, in evaluating a different model of this type, one with the minimum number of expected backorders also has good results with respect to other criteria [l]. The second criterion is the expected number of unfilled online slots per base, i.e., expected number of online positions per base that cannot be filled due to either base or depot repair. The next criterion is the expected downtime per base, i.e., percentage of time the operation at a base is shut down due to the availability of less than the minimum required number of machines (L). The fourth criterion is the expected percentage of time all depot repair facilities are busy. Finally, the last criterion is the expected number of operable machines per base which includes those online and in spares. In the discussion of the results below, we will index these criteria as i = 1,. . .5, where i = 1 corresponds to ‘expected number of backorders’, i = 2 corresponds to ‘expected number of unfilled online slots’, etc. We compared the three allocation policies using the five criteria mentioned above by simulating the model with the parameter settings presented in Table 1. After simulating the system for various parameter settings, we decided on the 24 problems for the following reasons. First, we selected parameters that would effect the congestion level at the depot, namely N, S, M, L, Rd, pd and p. As N, S and M increase, congestion increases, whereas as L, Rd, ,& and p increase, congestion decreases. (The system becomes more congested as more depot repair facilities become busy.) Given the regenerative point selected, as the congestion increases, the number of regeneration cycles generated within a fixed period of computer time decreases. For highly congested models, this decrease in number of cycles is so drastic that it becomes very inefficient and expensive
296
M.D.
Biiyiikkurt,
M. Parlar/A
t\~wechelon rrpairahle-item
to simulate the system. Thus, we simulated systems with two and four bases (N). The sample problems with two bases are presented in the first half, and those with four bases are presented in the second half of Table 1. Second, as suggested in the literature on simulation methodology [ 10, 1 l] we attempted to use a 2k factorial design, where k is the number of parameters (factors). In such an experimental design, one chooses just two levels (a high and a low value) for each factor and then simulates the system for each of the 2k possible combinations of the factor levels. Since as k increases, the number of simulation runs required also increases, it is further suggested in literature that a 2k - p fractional factorial design, where p is the number of interaction effects that are of no interest [lo], or a supersaturated design be used [ll]. The impracticality of simulating very highly congested systems prevented us from using a 2k factorial design. Thus we simulated the system by selecting a low and a high value for each parameter while keeping the values of the other parameters constant. In such a design, the interest is on the linear relationships and the main effects of the factors. We selected the low and high values such that a wide range of congestion levels were generated making our findings more generalizable. These combinations of low and high values resulted in the 24 problems presented in Table 1. For a given policy j, criterion i and problem k, we collected data for 200 regeneration cycles and obtained the mean values i = 1,. . .5, j = 1, 2, 3 and k = 1,. . .24. After replicating the simulation with 100, 200, 500 and 1000 cycles with the same parameters for a few problems, we decided that 200 is an acceptable number of cycles. This decision is based on the relative precision of the confidence intervals around the estimates. We compute the relative precision by dividing the confidence interval length by the point estimate. As shown by Law, Kelton and Koenig, this measure should be at most 0.15 [12]. In the replications that we made, runs with 100 cycles resulted in relative precisions that were less than but close to 0.15. Since the computer time involved was not
Table 1 Parameter
settings
inuentor~~ model
for the simulation
S
M
L
R*
/ld
P
2 3 4 5 6 7 8 9 IO 11 12
6 10 7 7 7 7 I I 7 7 7 I
5 5 3 7 5 5 5 5 5 5 5 5
3 3 3 3 2 5 3 3 3 3 3 3
2 2 2 2 2 2 1 IO 2 2 2 2
10 10 10 10 10 10 10 10 8 18 10 10
0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.7 0.9
13 14 15 16 17 18 19 20 21 22 23 24
6 10 7 7 I 7 7 7 7 I I 7
5 5 3 7 5 5 5 5 5 5 5 5
3 3 3 3 2 4 3 3 3 3 3 3
4 4 4 4 4 4 2 5 4 4 4 4
10 10 IO 10 10 10 10 10 10 15 10 10
0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.7 0.9
Problem I
For all the problems: R, = 3, pt, = 8, i. = IO. Where, S = total machines per base, M = maximum number of online machines per base, L = minimum number of online machines per base, R,, = number of repair facilities at the depot, I(,, = mean of depot repair time distribution P = probability of minor (base) repair, Rb = number of repair facilities per base, pLb= mean of base repair time distribution, i = mean interfailure time.
Xijk,
long, we increased the number of cycles to 200 and, in return we increased our assurance that the relative precision would be within the recommended range. For cases with 500 and 1000 cycles, the computer time required was excessively long making it infeasible to generate these cycles. The mean values, are presented in Table 2. Our objective was to identify the policy j which was superior for a given criterion i. The xijk values in Table 2 were used to construct a probability density function for the sample means for given i = 1,. . .5 and j = 1, 2, 3. We decided to apply the well-known stochastic Xijk,
0.7518 1.5700 0.7168 0.9671 1.0239 0.6119
2.2839 0.6002 0.4592 2.0606 1.6620 0.3600
0.3233 0.3582 0.2051 0.3565 0.3370 0.3370
0.5779 0.3125 0.3456 0.4632 0.5492 0.1666
7 8 9 10 11 12
13 14 15 16 17 18
19 20 21 22 23 24
i=l
k
1 2 3 4 5 6
i=l
2.1810 0.7144 0.8132 1.4703 1.6170 0.3313
0.6402 1.2447 0.5839 0.8365 0.8548 0.6975
2.2839 0.7025 0.4592 2.1112 1.8170 0.3600
0.7518 1.5700 0.7168 0.9671 1.0239 0.6119
i=2
2.1543 0.7038 0.7814 1.4197 1.5988 0.3510
0.6177 1.1209 0.5828 0.8054 0.8163 0.6774
2.1758 0.6921 0.4820 2.0335 1.7474 0.3660
2.2131 2.1978 2.1622 2.2245 2.1113 2.1510
2.2118 2.1333 0.6407 4.1037 2.8725 1.4236
1.9361 1.5103 1.2019 1.8618 1.6607 1.3363
1.6820 0.9106 0.1114 3.3280 1.6246 0.4904
i=l
i=2
1.8730 1.3092 1.3524 1.5883 1.5210 1.3110
1.6308 0.8247 0.1079 3.2678 1.5330 0.9578
1.9361 1.2858 1.2019 1.8576 1.6296 1.3363
1.6820 0.9106 0.1114 3.3280 1.6246 0.4904
i=2
1.8645 1.2782 1.3062 1.5598 1.5051 1.3420
1.6256 0.801 1 0.1036 3.2457 1.4980 0.9535
1.8822 1.2944 1.2781 1.8058 1.5830 1.3422
1.7248 0.9339 0.1256 3.3924 1.6570 0.4955
i=3
(Mean values xijk; for Criterion
0.7955 1.6006 0.7550 1.0529 1.1031 0.6319
i=3
from the Simulation
Problems
Table 2 Data Obtained
0.6318 0.6514 0.6224 0.6680 0.6245 0.5845
0.6415 0.6302 0.6407 0.6362 0.5785 0.6593
0.3630 0.2856 0.1563 0.3461 0.3068 0.1907
0.2521 0.1181 0.1114 0.1999 0.0819 0.4904
i=l
i=3
0.3373 0.1720 0.1837 0.2430 0.2246 0.1835
0.2349 0.1061 0.1079 0.1830 0.0723 0.3128
0.3630 0.1688 0.1563 0.3340 0.2590 0.1907
0.2521 0.1181 0.1114 0.1999 0.0819 0.4904
i=2
k)
0.3310 0.1660 0.1716 0.2335 0.2190 0.1931
0.0996 0.1036 0.1795 0.1795 0.0675 0.3156
0.3463 0.1773 0.1631 0.3150 0.1942 0.2339
0.2656 0.1231 0.1256 0.2093 0.0849 0.4955
i=3
i, Policy j, and Problem
0.4809 0.0169 0.0597 0.1204 0.1825 0.0045
0.0502 0.0631 0.0141 0.0745 0.0667 0.0488
0.9493 0.0000 0.2329 0.8535 0.7558 0.1724
0.4284 0.6570 0.3799 0.5170 0.5354 0.3487
i=l
i=4
0.9720 0.1616 0.3810 0.7438 0.8018 0.0575
0.2503 0.5857 0.2182 0.4023 0.4123 0.2980
0.9493 0.0000 0.2329 0.8688 0.8060 0.1724
0.4284 0.6570 0.3799 0.5170 0.5354 0.3487
i=2
0.9744 0.1472 0.3490 0.7320 0.7927 0.0545
0.2202 0.5425 0.1995 0.3681 0.3750 0.2700
0.9446 0.0000 0.2384 0.8653 0.7943 0.1525
0.4336 0.6821 0.3973 0.5392 0.5433 0.3489
i=3 1
6.4221 6.6875 6.6544 6.5368 6.4508 6.8334
5.6767 9.6418 6.7949 6.6435 6.6630 6.6630
4.7161 6.3998 6.5408 4.9394 5.3380 6.6400
5.2482 8.4300 6.2832 6.0329 5.9761 6.3881
i=
i=5
4.8190 6.2856 6.1868 5.5297 5.3830 6.6687
5.3598 8.7553 6.4161 6.1365 6.1452 6.3025
4.7161 6.2975 6.5408 4.8888 5.1830 6.6400
5.2482 8.4300 6.2832 0.6329 5.9761 6.3881
i=2
4.8457 6.2962 6.2186 5.5803 5.4012 6.6490
5.3823 8.8791 6.4172 6.1946 6.1837 6.3226
4.8242 6.3079 6.5180 4.9665 5.2526 6.6340
5.2042 8.3994 6.2450 5.9471 5.8969 6.3681
i=3
dominance criterion to see which policy performed better than the others. To briefly summarize the concept of first order stochastic dominance, consider two random variables X and Y with respective density functions .f(s) and y(l)) , and distributions F(x) and G(J). Assume (as in the case of comparing, e.g., the ‘Needy’ policy (j = 1) with others for the ‘Number of Backorders’ criterion) that smaller values of the outcome are preferable to larger values. The random variable X is said to have first-degree stochastic dominance (FSD) over Y (or X is stochastically smaller than Y) if F(M’) > G(\v)for all ct~[13, 141. This means that for any \v, P[X < LV)> P( Y < ~j, i.e.. it is more likely to have a value smaller than 1~with X than with Y. If the distribution functions F and G are plotted on the same graph, then for all u’, the distribution function of X is equal to or above the distribution function of Y. In those cases where FSD does not hold, i.e., when the distributions intersect at some IV. one can introduce the concept of second-
Comparison
0.50
q
Fig. 2. Comparison
Needy
of the policies under criterion
degree stochastic dominance @SD) as follows: X is said to have SSD over Y if Z I _J
[F(w) - G(w)] dw < 0
holds for all Z, i.e., the cumulative area under F should not be more than G for each point Z over the relevant range. We note that if the larger values of the outcomes are preferred to smaller values, then the inequalities specified above reverse direction for both FSD and SSD. The Xijk values for 24 sample problems with 3 policies and 5 criteria are summarized in Table 2. With these values, we constructed the distribution functions for the first criterion ‘Number of Backorders’ (i = l), for each policy used. We divided the interval [0, 2.51 into 25 equal subintervals and plotted the resulting distributions in Fig. 2. We observed that for this criterion, the graph of the distribution for
of Three Policies
1.90 Number +
1
1.50 of Backorders Origin
200
0
FCFS
2!
M.D.
Bii~Gkkurt,
M. Purlur/A
~KW-eciwlon repairable-item
Comparison
incentory
299
model
of Three Policies
0.6
0.00
0.50
1.00
1.50 Number
0
Fig. 3. Comparison
Needy
of the policies under criterion
200
250
of Unfilled Online + Origin
3.00
3.50
4.00
4.50
Slots 0
FCFS
2.
the ‘Needy’ policy was above the other policies’ graphs, indicating that ‘Needy’ has FSD over ‘Origin’ and ‘FCFS’. For the second criterion ‘Number of Unfilled Online Slots’ (i = 2), we observe that although ‘Needy’ is dominated by ‘Origin’ and ‘FCFS’, there was no clear-cut choice between the latter two. In fact, even a SSD test did not clarify this. Therefore, either ‘Origin’ or ‘FCFS’ could be used as a preferable policy for this criterion. These results are presented in Fig. 3. For the third criterion ‘Downtime’, the ‘Needy’ policy was again found to be dominated by the others, as is seen in Fig. 4. Although ‘FCFS’ appeared to dominate ‘Origin’, it was not very significant. For the fourth criterion ‘Percentage of time all depot repair facilities are busy’ and the last ‘Number of machines per base’, the ‘Needy’ policy had FSD over the other policies, as seen in Fig. 5 and 6. We note that since larger values of the last criterion are preferred to smaller values, the distribution function of the ‘Needy’
policy is equal to or below the graphs of the distribution of the other two. The above discussion is summarized in Table 3, where we indicate the preferable policy for each criterion. In three out of five criteria, the dynamic allocation policy labeled ‘Needy’ is found to be superior to the two static policies. This finding extends the results of Miller who had concluded that a dynamic allocation policy is superior to the best static policy in his analysis of a less complex model with no base repair and state independent repair rates [6].
Table 3 Summary Criteria 1
2 3 4 5
of conclusions (i)
Preferable
policy
Needy FCFS or Origin FCFS (or Origin) Needy Needy
300
M.D. Bti~Gkkurt, M. PaularIA
two-echelon repairable-itenz
Comparison
0.9
-
0.8
-
inl?entory model
of Three Policies (Criterion W3)
0.7 ‘# z .s c” L? s ‘= 2 e h d
0.6
-
0.5
-
0.4
-
0.3
-
0.2
-
0.1
-
0.00
0.25
0
Fig. 4. Comparison
Needy
of the policies under criterion
%
Fig. 5. Comparison
0
TimeAll
Needy
of the policies under criterion
FCFS
of Three Policies
0.75
0.50
0.25
0
0.75
3.
Comparison
0.M)
0.50 Downtime + Origin
4.
Depot Repair Facilities + Origin
Busy 0
FCFS
1.00
M.D. Biiyiikkurt,
hf. Purlar:‘A two-eddon
Comparison
repnirnhlr-item
301
inventory mo&l
of Three Policies (Criterion
15)
-i 5.00
4.00
6.00 Number
I3
Fig. 6. Comparison
Needy
of the policies under criterion
am
7.00 of Machines + Origin
9.00
10.00
per Base 0
FCFS
5
4. Conclusions In this paper, we analyzed a two-echelon repairable-item inventory model with statedependent repair and failure rates. In particular, we compared three different allocation policies namely ‘Needy’, ‘Origin’ and ‘FCFS’. We tested their effectiveness based on five criteria, namely expected number of backorders, expected number of unfilled online slots, expected downtime, expected percentage of time all depot repair facilities are busy, and expected number of machines per base. Using graphical stochastic dominance tests, we analyzed the results of a simulation study. It was found that the dynamic allocation policy ‘Needy’ is superior to the two static allocation policies in three out of five criteria. It was also found that for all the criteria the static policies had no significant differences. The dynamic allocation policy considered here is only one of many possible such policies
that can be employed. A possible extension of this research would be a comparison of various dynamic allocation policies reported in the literature, such as by Miller [4]. For complex problems such as the one discussed in this paper, analytic solutions are very difficult to obtain. For this reason simulation becomes a valuable tool. Since examples of these models are frequently encountered in real applications (e.g., army bases and a central repair facility that serves all the bases), research results obtained in this paper can be useful for practitioners and researchers alike.
Acknowledgements This research by the Natural Research Council A6743 and A5872. acknowledge the SC. Albright.
was partially supported Sciences and Engineering of Canada under grants The first author wishes to guidance provided by Dr.
References Cl1 Sherbrooke, C.C, 1968. METRIC: A multi-echelon technique for recoverable item control, Oper. Res.. 16: 1222141. [?I Simon, R.M., 1971. Stationary properties of a twoechelon inventory model for low demand items, Oper. Res., 19: 761-773. J.A., 1973. A model for a multi-item. c31 Muckstadt, multi-echelon. multi-indenture inventory system. Manage. Sci., 20(4): 4722481. c41 Miller, B.L., 1974. Dispatching from depot repair in a recoverable item inventory system: On the optimality of a heuristic rule, Manage. Sci., 21(3): 316-325. I51 Gross. D., Kioussis L.C. and Miller D.R.. 1987. A network decomposition approach for approximating the steady-state behaviour of Markovian multi-echelon repairable-item inventory systems. Manage. Sci., 33(1 I): 145331468. Lb1 Miller, B.L.. 1968. A real time metric for the distribution of serviceable assets, RM-5687-PR, The RAND Corporation. c71 Biiyiikkurt, M.D., 1984. Analysis of a two-echelon repairable item inventory system with state
dependent failure and repair rates, Ph.D. Dissertation, Graduate School of Business, Indiana University, Bloomington. IN, USA. (1984). properties of PI Biiyiikkurt, M.D., 1988. Stationary a two-echelon repairable-item inventory system with state-dependent failure and repair rates, working paper series (# 8%009), Concordia University, Canada. c91 Crane. M.A. and Lemoine, A.J., 1977. An introduction to the regenerative method for simulation analysis, Lecture Notes in Control and Information Sciences. Vol. 4. Springer-Verlag, New York, pp. 229. Cl01 Law, A.M and Kelton, W.D., 1982. Simulation Modeling and Analysis. McGraw-Hill, New York. Techniques in Cl11 Kleijnen J.P.C., 1974. Statistical Simulation (Parts 1 and II). Marcel Dekker, Inc., New York. Cl21 Law, A.M., Kelton, W.D. and Koenig, L.W., 1981. Relative width sequential confidence intervals for the mean, Commun. Stat., B 10: 29-39. [I31 Ross, SM.. 1983. Stochastic Processes. John Wiley, New York. Analysis. [I41 Bunn, D.W., 1984. Applied Decision McGraw-Hill. New York.
Journal of‘ Production
291
Economics, 29 (1993) 291-302
Elsevier
A comparison repairable-item M. Demirbag
of allocation policies in a two-echelon inventory model
Btiytikkurt
Faculty of Commerce and Administration, QuPhec, Canada H3G IM8
Concordia
Universit),,
1455 de Maisonneuce
Blvd. West, Montr&al,
and
M. Parlar Faculty of Business, McMaster (Received
15 July
University. Hamilton,
Ontario, Canada L8S 4M4
1992)
Abstract The objective of this paper is to evaluate three allocation policies in a two-echelon machine repair model with state-dependent failure and repair rates. The first echelon consists of a number of bases that are served by a central depot in the second echelon. The operations at each base are modeled as a machine repair problem. Due to the complexity of the general model, a simulation study was conducted whereby the regenerative method ofdata collection was used. Two static and one dynamic allocation policy were compared with respect to five criteria. Based on the distribution of the observations, stochastic dominance tests were conducted and it was observed that static policies did not differ. The dynamic policy was found to be superior in a majority of the criteria
1. Introduction repairable-item inventory Two-echelon models are those which include a number of bases comprising the first echelon, and’ the central depot comprising the second echelon which serves all the bases. In such models, each base consists of a collection of machines which fail according to a specified probability distribution. Some of the failed machines require minor repairs which can be completed at the base, whereas the others requiring major repairs are sent to the depot. Upon completion of repair at the depot, a machine is sent to
Correspondence to: M. Parlar, McMaster University, Hamilton, 4M4.
0925-5273/93/$06.00
Faculty Ontario,
of Business, Canada L8S
f$> 1993 Elsiver Science Publishers
a base according to a prespecified allocation policy. Some of the earlier models dealing with the above problem have optimized, i.e., minimized, the expected number of backorders at all bases subject to some constraints [l]. Since the publication of Sherbrooke’s model (known as METRIC), interest in two-echelon repairable-item inventory models has increased considerably. Many researchers have extended the original results of METRIC by considering e.g., condemnation (disposal) of some failed machines [2], and multi-indentures [3]. In all these early models, one crucial assumption was the state-independence of failure and repair rates which implied that the number of machines and repair facilities were infinite. In those applications where these numbers are finite but large, state-independence serves as a convenient assumption
B.V. All rights reserved.
towards obtaining analytic solutions using queuing theory. Clearly, in many practical situations, this may not be the case when there are only a small number of machines and/or repair facilities. Examples include computer service companies that provide minor repair service on sight and major repair service at headquarters. In this paper, we aim to analyze a twoechelon repairable-item inventory model with state-dependent repair and failure rates. In particular, we compare different allocation policies and test their effectiveness based on various criteria. The policies we consider involve sending the depot-repaired machine to the base: (i) that has the least number of machines, (ii) that the machine originated from, and (iii) that has had a backorder for the longest time (first-come-first-served). The five criteria we use in evaluating the allocation policies are all measures of effectiveness. They are, (i) expected number of backorders, (ii) expected n umber of unfilled online slots, (iii) expected downtime, (iv) expected percentage of time all depot repair facilities are busy, and (v) expected number of machines per base. Based on an extensive simulation study, we find that the first policy is preferable to the others for a majority of the criteria considered. Allocation policies in multi-echelon repairable-item inventory systems have also been considered by other researchers [ 1, 41. Common to all the models studied is the assumption of state-independent repair and/or failure times which, as discussed above, implies infinite number of machines and/or repair facilities. This research is different from those above in that a more realistic model with finite machines and repair facilities is studied. Another common aspect of the above cited research is the criteria used in evaluating allocation policies. All have aimed to minimize the number of backorders, with the assumption that any policy that results in minimum backorders will also be the optimal when some other criterion is used. As discussed above, we will evaluate allocation policies using several criteria including number of backorders.
Allocation policies that have been studied in multi-echelon repairable-item inventory systems have generally been classified as static or dynamic policies. A static allocation policy (SAP) is defined to be state-independent, whereas a dynamic allocation policy (DAP) is state-dependent. An example of a DAP is Sherbrooke’s ‘marginal allocation policy’ whereby a repaired machine is sent to the base where it will cause the largest decrease in expected backorders [l]. Another such DAP is Miller’s ‘transportation time look ahead policy’ whereby a repaired machine is sent to the base with the minimum number of machines on hand plus those arriving during transportation time [4]. Such a policy is an attempt to evaluate the needs of each base and allocate the repaired machine to the ‘most needy’ base. It does not, however, involve a complete evaluation of the needs of each base since machines in stock and in repair are not considered. This is accomplished in this research. Still another DAP reported in the literature is allocation of a repaired machine to the base that has the highest percentage of its machines in the depot [S]. Two SAPS have been frequently evaluated in the literature, namely (1) first-come-first-served, and (2) sending to the base of origin. The latter has been compared to a DAP similar to the ‘transportation time look ahead policy’ by Miller [IS], and found to be inferior since it has resulted in more backorders. This research will compare both SAPS stated above to a DAP similar to the ‘transportation time look ahead policy’, but one that also considers machines in stock and in repair. In Section 2, we describe the model in detail, and discuss the solution method employed. Analysis of results are presented in Section 3, followed by the summary and conclusions in Section 4.
2. Methodology The model studied is that of a two-echelon inventory model, whereby each base that operates as the machine repair model forms the
M.D. Btiyiikkurt,
M. ParlarlA
two-echelon repairable-item
taneously placed online; otherwise it is placed in spares inventory. With probability 1 - p, the failed machine requires a major repair and is thus sent to the depot. The transportation time from the base to the depot is assumed to be negligible. There are R,, parallel repair facilities at the depot, each of which can repair a machine at an exponentially distributed time with parameter ,&. There are no spares at the depot. The depot repairable failures eventually lead to depletion of machines at the base unless orders are placed for additional machines. We use an (S - 1, S) ordering policy, whereby an order for one machine is placed when a failed machine is determined to be depot repairable. A backorder is initiated when a failed machine is determined to be depot repairable. The term backorder, in this context, has a different meaning than the one employed in inventory control literature where it is used for orders placed by a base that cannot be filled by the depot. We use the term backorder in accordance with its usage in the literature on the machine repair problem as indicated above. Once a machine is repaired at the depot it is sent to the base in accordance with the prespecified allocation policy. In this model there is no condemnation of failed machines.
first echelon, and the central depot that serves all the bases forms the second echelon. The operation in each base and its interaction with the depot is depicted in Fig. 1. There are a total of S identical machines in each base, of which a maximum of M and a minimum of L can be online (operating) at any time. When there are more than M operable machines available, the excess is placed in spares inventory. If the number of online machines decreases to L - 1, the operations at that base are shut down. The failure rate of machines is dependent on the number of online machines at the time of failure, and each machine fails according to a Poisson process with an average interfailure time of l/L. A nonoperable machine is replaced with a spare instantaneously if one is available and inspected instantaneously to determine whether it needs to be repaired at the base or the depot. With probability p, a minor repair is needed which can be performed in one of Rb parallel repair facilities at the base. The repair rate is dependent on the number of idle repair facilities at the time, and each repair time is exponentially distributed with parameter ,&. At the time of repair completion, if there are less than M online machines, the repaired machine is instan-
Base 1
r I
Base 2
----
-
Online
:
_
I L
1
M A
AP
I
fib
I 4
Repair
-.-----__I
Fig. 1. The model with N bases.
: Rb-
pb
...
Base N
r-lr--1
I
I
I
293
inventory model
I
I
I
I
I
I
I
I
I
I
I.*.1
I
I
I
I
I
I
I
I
I
Finally, there is no lateral supply of machines between the bases. Various versions of this model can be generated by changing some of the assumptions. For example, one can assume non-exponentially distributed interfailure and repair times. Biiyiikkurt has investigated different interfailure and repair time distributions [7], and found that the model was sensitive to a change in the distributions only when the system was highly congested. Another extension of the model could be obtained by incorporating different cost and revenue factors associated with the operation of the system, and thereby optimizing an economic criterion. This could be accomplished by making some of the parameters such as L and &! decision variables and choosing their optimal values using a search procedure. Our emphasis in this paper, however, has been comparing various allocation policies based on measures of effectiveness rather than profits. Finding the exact solution of the above model is very difficult because of the interdependence of the bases due to their joint utilization of the depot repair service. More specifically, the lead time between the placement of an order and the return of the repaired machine to the base is a function of the total number of bases and the number of backorders at each base. Thus, the interdependence between the bases results in a very complex leadtime distribution which is very difficult, if not impossible, to determine. Due to the nondeterministic repair and failure rate and the twoechelon system assumed in the model, elementary queuing theory results cannot be used to solve the stationary properties as has been done in similar research. Even a simplified approximate model for this problem as discussed by Biiyiikkurt in Ref. 8 is difficult to solve. This approximation is a continuous time Markov process which is solved for the steady state probabilities by taking advantage of the special form of the transition rate matrix. The states of the Markov process are of the form O
base (operable or in base repair), P is the number in or awaiting base repair, S is the maximum number of machines per base, and L is the minimum required number of online machines. With 8 defined as the approximate value of the rate parameter in the lead time distribution, the state transitions, the event causing the transition, and the associated rates can be summarized as follows: Transition
Event
Transition
(II. 1.) * 01. )’ + I)
Base repairable failure Base repair completion Depot repax facilities Arrival from depot
min[M,
(11.)‘) --t 01. r - I) (II. 1.) + (H - 1, r) (!I. ,‘) + (?I + I. r)
rate
(n - r)ip]
min(&. r)ic, min[M.
(II - r)]E.( 1 - p)
min [Rd. (S - n)]s
Another method that can be used when solving problems of this complexity is simulation. Since the objective of this research is to evaluate various allocation policies, it is more desirable to simulate the original complex model rather than develop exact solutions to its approximations. The model is simulated in FORTRAN using discrete event simulation. The regenerative method of collecting statistics is used since the process regenerates itself when it enters certain states. We took advantage of this property and used the regenerative method since it eliminates some of the problems associated with the traditional simulation approaches. These problems include the necessity of making decisions regarding startup conditions and detection of the steady state, and the correlation of simulated data. In regenerative method, data gathered in each cycle is independent of those gathered in other cycles. We select an approximate regeneration point that includes a group of states, as suggested by Crane and Lemoine [9]. The group of states included in the approximate regeneration point selected are those whereby a depot repairable failure at any base triggers the first backorder following a period of no backorders in the system. In all the simulation runs, this approximate regeneration point proved to be quite satisfactory in
M.D. Btiyiikkurt,
M. Parlar/A
two-echelon repairable-item
that the resulting correlation coefficients between observations in successive cycles were fairly small (approximately 0.0001-0.01) for all the statistics calculated. We employ five criteria to evaluate three allocation policies to select the base that will receive a depot-repaired machine. The first allocation policy involves sending the depotrepaired machine to the base that has the fewest number of machines, and at least one outstanding backorder. In other words, those bases that do not have machines at the depot repair facility are not considered. In calculating the fewest number of machines, we consider not only the number of machines online, but also those in stock and those in repair. For a particular base, we let 4’ be the number of machines online, u’ be the number of machines in stock and z be the number of machines in base repair. In order to determine the destination of a repaired machine, we select the base that has the lowest value of (y + u’ + z). If more than one base qualifies, then they are equally likely to receive the repaired machine. In the sequel, we will label this policy as ‘Needy’. As briefly mentioned in Section 1, compared to Miller’s ‘transportation time look ahead policy’, our policy additionally considers machines in stock and in base repair [4]. Thus, due to its generality, it provides a more complete need assessment. Also, given a fixed number of machines in the system (S), the values of J, MI and z determine the number and percentage of machines a base has in the depot repair facility. In other words, a ‘Needy’ base is one that has the most number of machines at the depot repair facility. The second allocation policy is formulated to send the repaired machine to the base it originated from. In the sequel, this policy will be labeled ‘Origin’. Finally, the third allocation policy labeled ‘FCFS’, corresponds to sending the machine to the base that has had an outstanding backorder for the longest time, i.e., first-come-first-served. In the discussion of the results below, we will index these policies as j = 1,2,3, wherej = 1 corresponds to ‘Needy’, etc.
inwntory
model
295
3. Analysis of results The first criterion we employed in evaluating allocation policies is the expected number of backorders per base. This criterion has been the focus of most of the literature on repairable-item inventory models. As stated by Sherbrooke, in evaluating a different model of this type, one with the minimum number of expected backorders also has good results with respect to other criteria [l]. The second criterion is the expected number of unfilled online slots per base, i.e., expected number of online positions per base that cannot be filled due to either base or depot repair. The next criterion is the expected downtime per base, i.e., percentage of time the operation at a base is shut down due to the availability of less than the minimum required number of machines (L). The fourth criterion is the expected percentage of time all depot repair facilities are busy. Finally, the last criterion is the expected number of operable machines per base which includes those online and in spares. In the discussion of the results below, we will index these criteria as i = 1,. . .5, where i = 1 corresponds to ‘expected number of backorders’, i = 2 corresponds to ‘expected number of unfilled online slots’, etc. We compared the three allocation policies using the five criteria mentioned above by simulating the model with the parameter settings presented in Table 1. After simulating the system for various parameter settings, we decided on the 24 problems for the following reasons. First, we selected parameters that would effect the congestion level at the depot, namely N, S, M, L, Rd, pd and p. As N, S and M increase, congestion increases, whereas as L, Rd, ,& and p increase, congestion decreases. (The system becomes more congested as more depot repair facilities become busy.) Given the regenerative point selected, as the congestion increases, the number of regeneration cycles generated within a fixed period of computer time decreases. For highly congested models, this decrease in number of cycles is so drastic that it becomes very inefficient and expensive
296
M.D.
Biiyiikkurt,
M. Parlar/A
t\~wechelon rrpairahle-item
to simulate the system. Thus, we simulated systems with two and four bases (N). The sample problems with two bases are presented in the first half, and those with four bases are presented in the second half of Table 1. Second, as suggested in the literature on simulation methodology [ 10, 1 l] we attempted to use a 2k factorial design, where k is the number of parameters (factors). In such an experimental design, one chooses just two levels (a high and a low value) for each factor and then simulates the system for each of the 2k possible combinations of the factor levels. Since as k increases, the number of simulation runs required also increases, it is further suggested in literature that a 2k - p fractional factorial design, where p is the number of interaction effects that are of no interest [lo], or a supersaturated design be used [ll]. The impracticality of simulating very highly congested systems prevented us from using a 2k factorial design. Thus we simulated the system by selecting a low and a high value for each parameter while keeping the values of the other parameters constant. In such a design, the interest is on the linear relationships and the main effects of the factors. We selected the low and high values such that a wide range of congestion levels were generated making our findings more generalizable. These combinations of low and high values resulted in the 24 problems presented in Table 1. For a given policy j, criterion i and problem k, we collected data for 200 regeneration cycles and obtained the mean values i = 1,. . .5, j = 1, 2, 3 and k = 1,. . .24. After replicating the simulation with 100, 200, 500 and 1000 cycles with the same parameters for a few problems, we decided that 200 is an acceptable number of cycles. This decision is based on the relative precision of the confidence intervals around the estimates. We compute the relative precision by dividing the confidence interval length by the point estimate. As shown by Law, Kelton and Koenig, this measure should be at most 0.15 [12]. In the replications that we made, runs with 100 cycles resulted in relative precisions that were less than but close to 0.15. Since the computer time involved was not
Table 1 Parameter
settings
inuentor~~ model
for the simulation
S
M
L
R*
/ld
P
2 3 4 5 6 7 8 9 IO 11 12
6 10 7 7 7 7 I I 7 7 7 I
5 5 3 7 5 5 5 5 5 5 5 5
3 3 3 3 2 5 3 3 3 3 3 3
2 2 2 2 2 2 1 IO 2 2 2 2
10 10 10 10 10 10 10 10 8 18 10 10
0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.7 0.9
13 14 15 16 17 18 19 20 21 22 23 24
6 10 7 7 I 7 7 7 7 I I 7
5 5 3 7 5 5 5 5 5 5 5 5
3 3 3 3 2 4 3 3 3 3 3 3
4 4 4 4 4 4 2 5 4 4 4 4
10 10 IO 10 10 10 10 10 10 15 10 10
0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.7 0.9
Problem I
For all the problems: R, = 3, pt, = 8, i. = IO. Where, S = total machines per base, M = maximum number of online machines per base, L = minimum number of online machines per base, R,, = number of repair facilities at the depot, I(,, = mean of depot repair time distribution P = probability of minor (base) repair, Rb = number of repair facilities per base, pLb= mean of base repair time distribution, i = mean interfailure time.
Xijk,
long, we increased the number of cycles to 200 and, in return we increased our assurance that the relative precision would be within the recommended range. For cases with 500 and 1000 cycles, the computer time required was excessively long making it infeasible to generate these cycles. The mean values, are presented in Table 2. Our objective was to identify the policy j which was superior for a given criterion i. The xijk values in Table 2 were used to construct a probability density function for the sample means for given i = 1,. . .5 and j = 1, 2, 3. We decided to apply the well-known stochastic Xijk,
0.7518 1.5700 0.7168 0.9671 1.0239 0.6119
2.2839 0.6002 0.4592 2.0606 1.6620 0.3600
0.3233 0.3582 0.2051 0.3565 0.3370 0.3370
0.5779 0.3125 0.3456 0.4632 0.5492 0.1666
7 8 9 10 11 12
13 14 15 16 17 18
19 20 21 22 23 24
i=l
k
1 2 3 4 5 6
i=l
2.1810 0.7144 0.8132 1.4703 1.6170 0.3313
0.6402 1.2447 0.5839 0.8365 0.8548 0.6975
2.2839 0.7025 0.4592 2.1112 1.8170 0.3600
0.7518 1.5700 0.7168 0.9671 1.0239 0.6119
i=2
2.1543 0.7038 0.7814 1.4197 1.5988 0.3510
0.6177 1.1209 0.5828 0.8054 0.8163 0.6774
2.1758 0.6921 0.4820 2.0335 1.7474 0.3660
2.2131 2.1978 2.1622 2.2245 2.1113 2.1510
2.2118 2.1333 0.6407 4.1037 2.8725 1.4236
1.9361 1.5103 1.2019 1.8618 1.6607 1.3363
1.6820 0.9106 0.1114 3.3280 1.6246 0.4904
i=l
i=2
1.8730 1.3092 1.3524 1.5883 1.5210 1.3110
1.6308 0.8247 0.1079 3.2678 1.5330 0.9578
1.9361 1.2858 1.2019 1.8576 1.6296 1.3363
1.6820 0.9106 0.1114 3.3280 1.6246 0.4904
i=2
1.8645 1.2782 1.3062 1.5598 1.5051 1.3420
1.6256 0.801 1 0.1036 3.2457 1.4980 0.9535
1.8822 1.2944 1.2781 1.8058 1.5830 1.3422
1.7248 0.9339 0.1256 3.3924 1.6570 0.4955
i=3
(Mean values xijk; for Criterion
0.7955 1.6006 0.7550 1.0529 1.1031 0.6319
i=3
from the Simulation
Problems
Table 2 Data Obtained
0.6318 0.6514 0.6224 0.6680 0.6245 0.5845
0.6415 0.6302 0.6407 0.6362 0.5785 0.6593
0.3630 0.2856 0.1563 0.3461 0.3068 0.1907
0.2521 0.1181 0.1114 0.1999 0.0819 0.4904
i=l
i=3
0.3373 0.1720 0.1837 0.2430 0.2246 0.1835
0.2349 0.1061 0.1079 0.1830 0.0723 0.3128
0.3630 0.1688 0.1563 0.3340 0.2590 0.1907
0.2521 0.1181 0.1114 0.1999 0.0819 0.4904
i=2
k)
0.3310 0.1660 0.1716 0.2335 0.2190 0.1931
0.0996 0.1036 0.1795 0.1795 0.0675 0.3156
0.3463 0.1773 0.1631 0.3150 0.1942 0.2339
0.2656 0.1231 0.1256 0.2093 0.0849 0.4955
i=3
i, Policy j, and Problem
0.4809 0.0169 0.0597 0.1204 0.1825 0.0045
0.0502 0.0631 0.0141 0.0745 0.0667 0.0488
0.9493 0.0000 0.2329 0.8535 0.7558 0.1724
0.4284 0.6570 0.3799 0.5170 0.5354 0.3487
i=l
i=4
0.9720 0.1616 0.3810 0.7438 0.8018 0.0575
0.2503 0.5857 0.2182 0.4023 0.4123 0.2980
0.9493 0.0000 0.2329 0.8688 0.8060 0.1724
0.4284 0.6570 0.3799 0.5170 0.5354 0.3487
i=2
0.9744 0.1472 0.3490 0.7320 0.7927 0.0545
0.2202 0.5425 0.1995 0.3681 0.3750 0.2700
0.9446 0.0000 0.2384 0.8653 0.7943 0.1525
0.4336 0.6821 0.3973 0.5392 0.5433 0.3489
i=3 1
6.4221 6.6875 6.6544 6.5368 6.4508 6.8334
5.6767 9.6418 6.7949 6.6435 6.6630 6.6630
4.7161 6.3998 6.5408 4.9394 5.3380 6.6400
5.2482 8.4300 6.2832 6.0329 5.9761 6.3881
i=
i=5
4.8190 6.2856 6.1868 5.5297 5.3830 6.6687
5.3598 8.7553 6.4161 6.1365 6.1452 6.3025
4.7161 6.2975 6.5408 4.8888 5.1830 6.6400
5.2482 8.4300 6.2832 0.6329 5.9761 6.3881
i=2
4.8457 6.2962 6.2186 5.5803 5.4012 6.6490
5.3823 8.8791 6.4172 6.1946 6.1837 6.3226
4.8242 6.3079 6.5180 4.9665 5.2526 6.6340
5.2042 8.3994 6.2450 5.9471 5.8969 6.3681
i=3
dominance criterion to see which policy performed better than the others. To briefly summarize the concept of first order stochastic dominance, consider two random variables X and Y with respective density functions .f(s) and y(l)) , and distributions F(x) and G(J). Assume (as in the case of comparing, e.g., the ‘Needy’ policy (j = 1) with others for the ‘Number of Backorders’ criterion) that smaller values of the outcome are preferable to larger values. The random variable X is said to have first-degree stochastic dominance (FSD) over Y (or X is stochastically smaller than Y) if F(M’) > G(\v)for all ct~[13, 141. This means that for any \v, P[X < LV)> P( Y < ~j, i.e.. it is more likely to have a value smaller than 1~with X than with Y. If the distribution functions F and G are plotted on the same graph, then for all u’, the distribution function of X is equal to or above the distribution function of Y. In those cases where FSD does not hold, i.e., when the distributions intersect at some IV. one can introduce the concept of second-
Comparison
0.50
q
Fig. 2. Comparison
Needy
of the policies under criterion
degree stochastic dominance @SD) as follows: X is said to have SSD over Y if Z I _J
[F(w) - G(w)] dw < 0
holds for all Z, i.e., the cumulative area under F should not be more than G for each point Z over the relevant range. We note that if the larger values of the outcomes are preferred to smaller values, then the inequalities specified above reverse direction for both FSD and SSD. The Xijk values for 24 sample problems with 3 policies and 5 criteria are summarized in Table 2. With these values, we constructed the distribution functions for the first criterion ‘Number of Backorders’ (i = l), for each policy used. We divided the interval [0, 2.51 into 25 equal subintervals and plotted the resulting distributions in Fig. 2. We observed that for this criterion, the graph of the distribution for
of Three Policies
1.90 Number +
1
1.50 of Backorders Origin
200
0
FCFS
2!
M.D.
Bii~Gkkurt,
M. Purlur/A
~KW-eciwlon repairable-item
Comparison
incentory
299
model
of Three Policies
0.6
0.00
0.50
1.00
1.50 Number
0
Fig. 3. Comparison
Needy
of the policies under criterion
200
250
of Unfilled Online + Origin
3.00
3.50
4.00
4.50
Slots 0
FCFS
2.
the ‘Needy’ policy was above the other policies’ graphs, indicating that ‘Needy’ has FSD over ‘Origin’ and ‘FCFS’. For the second criterion ‘Number of Unfilled Online Slots’ (i = 2), we observe that although ‘Needy’ is dominated by ‘Origin’ and ‘FCFS’, there was no clear-cut choice between the latter two. In fact, even a SSD test did not clarify this. Therefore, either ‘Origin’ or ‘FCFS’ could be used as a preferable policy for this criterion. These results are presented in Fig. 3. For the third criterion ‘Downtime’, the ‘Needy’ policy was again found to be dominated by the others, as is seen in Fig. 4. Although ‘FCFS’ appeared to dominate ‘Origin’, it was not very significant. For the fourth criterion ‘Percentage of time all depot repair facilities are busy’ and the last ‘Number of machines per base’, the ‘Needy’ policy had FSD over the other policies, as seen in Fig. 5 and 6. We note that since larger values of the last criterion are preferred to smaller values, the distribution function of the ‘Needy’
policy is equal to or below the graphs of the distribution of the other two. The above discussion is summarized in Table 3, where we indicate the preferable policy for each criterion. In three out of five criteria, the dynamic allocation policy labeled ‘Needy’ is found to be superior to the two static policies. This finding extends the results of Miller who had concluded that a dynamic allocation policy is superior to the best static policy in his analysis of a less complex model with no base repair and state independent repair rates [6].
Table 3 Summary Criteria 1
2 3 4 5
of conclusions (i)
Preferable
policy
Needy FCFS or Origin FCFS (or Origin) Needy Needy
300
M.D. Bti~Gkkurt, M. PaularIA
two-echelon repairable-itenz
Comparison
0.9
-
0.8
-
inl?entory model
of Three Policies (Criterion W3)
0.7 ‘# z .s c” L? s ‘= 2 e h d
0.6
-
0.5
-
0.4
-
0.3
-
0.2
-
0.1
-
0.00
0.25
0
Fig. 4. Comparison
Needy
of the policies under criterion
%
Fig. 5. Comparison
0
TimeAll
Needy
of the policies under criterion
FCFS
of Three Policies
0.75
0.50
0.25
0
0.75
3.
Comparison
0.M)
0.50 Downtime + Origin
4.
Depot Repair Facilities + Origin
Busy 0
FCFS
1.00
M.D. Biiyiikkurt,
hf. Purlar:‘A two-eddon
Comparison
repnirnhlr-item
301
inventory mo&l
of Three Policies (Criterion
15)
-i 5.00
4.00
6.00 Number
I3
Fig. 6. Comparison
Needy
of the policies under criterion
am
7.00 of Machines + Origin
9.00
10.00
per Base 0
FCFS
5
4. Conclusions In this paper, we analyzed a two-echelon repairable-item inventory model with statedependent repair and failure rates. In particular, we compared three different allocation policies namely ‘Needy’, ‘Origin’ and ‘FCFS’. We tested their effectiveness based on five criteria, namely expected number of backorders, expected number of unfilled online slots, expected downtime, expected percentage of time all depot repair facilities are busy, and expected number of machines per base. Using graphical stochastic dominance tests, we analyzed the results of a simulation study. It was found that the dynamic allocation policy ‘Needy’ is superior to the two static allocation policies in three out of five criteria. It was also found that for all the criteria the static policies had no significant differences. The dynamic allocation policy considered here is only one of many possible such policies
that can be employed. A possible extension of this research would be a comparison of various dynamic allocation policies reported in the literature, such as by Miller [4]. For complex problems such as the one discussed in this paper, analytic solutions are very difficult to obtain. For this reason simulation becomes a valuable tool. Since examples of these models are frequently encountered in real applications (e.g., army bases and a central repair facility that serves all the bases), research results obtained in this paper can be useful for practitioners and researchers alike.
Acknowledgements This research by the Natural Research Council A6743 and A5872. acknowledge the SC. Albright.
was partially supported Sciences and Engineering of Canada under grants The first author wishes to guidance provided by Dr.
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