A multi-echelon and multi-indenture repairable item queuing model during emergencies

Math1 Comput. Modelling, Vol. 12, No. I, PP. 851-864, 1989 Printed in Great Britain. All rights resewed

0895-7177/89 $3.00 + 0.00 Copyright 0 1989 Pergamon Press plc

A MULTI-ECHELON AND MULTI-INDENTURE REPAIRABLE ITEM QUEUING MODEL DURING EMERGENCIES

M. JUNG

Pohang Institute of Science and Technology, Pohang, Korea

E. S. LEE?

Department of Industrial Engineering, Durland Hall, Kansas State University, Manhattan, Kansas, KS 66506, U.S.A.

(Received February 1988; accepted for publication November 1988)

Communicated by X. J. R. Avula

Abstract-The fluid and diffusion approximation for queuing modeling are used to investigate the emergency supply and inventory situations for repairable and expensive sub-indentured items in a large and complex multi-indentured multi-echelon system. It is shown that this approach is a very effective technique for testing emergency situations of complex systems. Various numerical examples are solved to illustrate the approach and to show the dynamics during an emergency situation.

INTRODUCTION

The supply and repair of sub-indentured expensive recoverable items in a large and multi-echelon operation is a frequently encountered practical and difficult problem. Some of the examples encountered in practice are the supply and repair of expensive airplane parts of a large airline operation, the supply and repair of sub-parts in a large trucking company, and the supply and repair of sub-parts in military logistics situations.

Various investigators have studied this problem [l-lo]. However, all these studies are concerned with regular operations where the breakdown of expensive sub-parts is very infrequent and Poisson distribution and Palm’s theorem can be applied. Furthermore, unlimited repair capacities can be assumed under these situations.

However, the most critical test of the adequacy of the system is during the worst scenario such as an emergency where the demands exceed the repair or supply capabilities. In the military situation, this emergency arises during the initial war surge period. In order to assess this emergency readiness and sustainability and in order to obtain the best or least costly policies during this worst scenario, the true dynamics during this period must be analyzed and accurately modeled based on the emergency characteristics which are completely different from regular time operations. One of the most important characteristics during emergency is the limited repair capabilities. This limitation arose from the fact that when an emergency starts the demand rate for repair increases suddenly by several folds. It would be very unrealistic and very expensive to keep the repair capabilities several folds higher than required during regular time, or to keep high inventories. Thus, in nearly all practical situations, the repair and supply capacities are certainly limited during emergency.

The problem of limited repair capacities during emergency for a complex system is very difficult to solve. This is because of the fact that there exist no effective techniques to treat even fairly simple dynamic unsteady state queues. It is completely impossible to deal with so many random queues which represent the various limited repair capacities of a complex and large system.

In order to avoid the above difficulties, a completely different approach from the traditional queuing approach, namely the fluid approximation (FA) and diffusion approximation (DA) is used in this work [l 11. These approaches avoid completely dealing with the random queues and thus avoid the abovementioned difficulties. By the use of the proposed approaches, we can establish

851

852 M. JUNG and E. S. LEE

a model for only the worst scenario and be able to study the true dynamics and estimate the desired stock level (DSL) under the most severe demand situations [12].

It should be emphasized that by avoiding dealing with the random queues, we did solve the problem completely. We only solved the problems during the other extreme, namely, during emergency which requires very high demands.

FLUID APPROXIMATION

The fluid approximation approach is based on the idea that the number of arrivals is so large that individual counts become insignificant and thus the process behaves like a fluid. For illustrative purposes, consider a single repair facility with n servers. Further assume that the repair times for the different items remain the same for all the servers. The representing dynamic equations can be obtained by simple material balance:

where

L(t) = L(t - 1) + a(t) -d(t) (1)

Q(t) = Q(t - 1) + a(t) -s(t) (2)

S(t) = s(t - 1) + s(t) - d(t) (3)

Q(t) = the number of items waiting to be repaired (queue lengths) at the end of time interval t s(t) = the number of items being repaired at the end of time interval t L(t) = the number of items in the base (including both being repaired and waiting) at the end

of time interval t = Q(r) + s(t) a(t) = number of items arrived during time interval t d(t) = number of items departed during time interval t s(t) = number of items entered into repair during time interval t.

The above simple dynamic equations can be examined by referring to Figs 1 and 2 [ll]. During the time period t&i, where the arrival rate Iz(t) is less than the repair capacity cc, equations (l)-(3) are in a dynamic build-up state. In other words, the number entering repair, s(t), is larger than the number finished repair, d(t), or the number of busy servers is increasing. At time t,, the arrival rate r(t) equals the service capacity p, and thus all the servers are busy at c,. If there are n servers, then the number of items being repaired has reached the limit n, or

S(t) = n.

This equilibrium service situation remains true during the time interval f,-f4 in Figs 1 and 2. Although the service has reached equilibrium, the number of arrivals are still increasing until t2. After 2; the number of arrivals are again less than the service capacity. However, all the servers must still be kept busy until the number of items waiting is reduced to zero at t.,.

act1 JA to t1 (2 t3 t4

Time Fig. 1

t (1 tz (3 4

Time

Fig. 2

Emergency queuing model 853

Thus, equations (l)-(3) form three different states during an emergency. During time period to-tl, these equations, or the emergency situation, is in a dynamic build-up state. During f,-L,, the service rates become steady state, or d(t) = s(t). At f4, there is a dynamic reduction of busy servers.

Using the above equations, the number of units in resupply at time t can be calculated dynamically. Once this is known, the stock level required to achieve a given probability P, of no back orders can be calculated by obtaining the smallest non-negative integer, N,, such that

ito [exp ( - L(l)(L(t))‘/i!] ’ pb

where it is assumed that the arrivals are Poisson distributed. To illustrate the approach, the results of a simple scenario is summarized in the first three

columns of Table 1. It is assumed that the breakdown rate during regular operation is 0.8 unit/day and during the emergency this breakdown rate is 3 units/day and the emergency lasts 15 days. A fixed repair time of 5 days is assumed. In summary, the expected demand rate is obtained by:

dk = 0.8, k-cl, K>28

dk = 3.0, l<k<14

s

k- 14

dk = l(t) dt, k = 15, 16. . . ,28 (4) k-15

where dk is the expected demand rate at day k and is listed in the second column of Table 1. The expected demand rate returns to the regular time rate at day 28. It should be noted that these equations for daily demand are arbitrarily assumed and the exponentially decreasing demand was used by Muckstadt [l].

The results in column 3 of Table 1 were obtained by assuming unlimited repair capacity. In practice, the repair capacity is almost always limited during an emergency. The results of three different scenarios with limited repair capacity are also listed in Table 1. Notice the rapid build-up of queue length when the repair capacity is limited.

To illustrate the influence of limited repair capacity on the dynamics of emergency, the expected number of units in resupply (ENUR) is plotted in Fig. 3 for the scenario where the repair capacity is doubled after 10 days of emergency. The difference between ENUR and the number of servers in any given day gives the queue length or the number waiting for service. It is this queue length or the area inside the curves formed by the points AFBGCDE that is important from a logistics standpoint. The maximum height of the curve at point B represents the time at which the maximum queue or maximum ENUR which is not available for use and the duration from point A to point

Table I, Limited and unlimited repair capacity

Limited repair capacity Unlimited

repair Case (I)? Case (2)i Case (3B Expected capacity

Day demand DSL Queue DSL Queue DSL Queue DSL

0 0.8 6 0.0 6 0.0 6 0.0 6 I 3.0 8 1.2 8 1.2 8 0.0 8 2 3.0 II 3.4 II 3.4 II 0.4 11 5 3.0 I8 10.0 I8 10.0 18 7.0 I8 6 3.0 18 II.2 20 Il.2 20 7.0 18

IO 3.0 I8 20.0 29 20.0 29 14.0 26 I.5 3.0 I8 30.0 40 25.0 40 19.0 34 20 I.8 14 36.2 41 26.2 41 20.2 36 25 1.1 9 38.1 49 23.1 38 17.3 32 30 0.8 6 37.4 48 17.4 11.6 26 35 0.8 6 36.4 47 11.4 :: 5.6 19 40 0.8 6 35.4 46 5.4 I9 I2 41 0.8 6 34.4 45 0.0 I2

8:: 7

46 0.8 6 33.4 44 0.0 6 0.0 6 51 0.8 6 32.4 43 0.0 6 0.0 6

101 0.8 6 22.4 32 0.0 6 0.0 6 201 0.8 6 2.4 IO 0.0 6 0.0 6

tCase (1): Five servers. fCasc (2): Five servers day 1 to day 10; 10 servers after day IO. @Zase (3): Five Servers day 0; eight servers day I to day IO; ten servers after day 10.


35.0-

30.0-

: 25.0-

5

IL 0 20.0-

.8 ,’ :

,“; I’ ;

r' i ,/

: , I,,,

: ,,4

:,/ ; \

: \

I’ \

: L____ EN”*. 7

,*’ /’ \

I’ i

/

t L______

\

.--EXPECTED DEMAND

---.__ --.-__-___-.-.-_-.-_-.-

0 6 12 18 24 30 36 42 48

DAYS

Fig. 3. Fluid approximation with limited repair capacity

C represents the duration at which there are units waiting to be repaired or the duration during which the repair capacity is not adequate. Thus, both the height and the width (or duration) of this curve are important parameters. For example, if we assume the system fails any time there are 15 units or more in resupply, then, knowing the duration or width of the area ABCDE, one can estimate the mean time between system failures (MTBSF). For the scenario in Fig. 3, this MTBSF is 36 - 5 = 3 1 days. The maximum height of the area can be used to estimate the maximum shortage. Notice that because of the fixed repair time of 5 days, there is always a delay of at least 5 days during which time the unit is not available. It is this delay that makes the dynamic so severe.

This type of analysis can be used to illustrate the influences or sensitivity on the dynamics of emergency under various different conditions such as the length of repair time, transportation time, the duration and intensity of emergency, repair capacity, and the various delays due to limitations or repair capacity, transportation capacity and shortage of parts.

DIFFUSION APPROXIMATION

Although the deterministic fluid approximation discussed in the previous section is simple to calculate and appears to be ideally suited for studying the dynamics of the initial emergency surge, the stochastic property of the system is completely lost. For example, we can calculate the mean of the back orders for certain items, but we don’t know the distributions of this back order. Obviously, it is desirable to know the distribution also.

Since the dynamic queuing equations with variable arrival rates or variable demand rates are too complicated to be of practical use, various other approaches to consider the stochastic property of the system have been devised. One of these approaches is the diffusion approach which appears especially suited to model this emergency situation.

The diffusion aproach [l l-l 31 is based on the idea that the number of units in the system is so large that the integer nature of the count can be ignored. During the initial emergency surge period, this is exactly the situation where the number of units in resupply waiting to be served is large.

The behavior of the queuing system is abstractly very similar to the behavior of fluid diffusion. In queuing theory, the first moment of the probability density is the analog of the center of gravity


of the mass distribution and the variance is the analog of the moment of inertia about the center of gravity. Based on mass balance of fluid flow with diffusion and after some manipulations, the following differential equation representing the probability distribution function of the queue of an item with time-dependent arrival rate can be obtained [l I]:

(a/at)F&;t) = [(P - n(t))(al%) + (lP)(ZAt) + &Tmw?)l~q(4~~) (54

with the boundary conditions

FQ(O;t) = 0 FQ(c;t) = 1 for all t (5b) and the initial condition

Fo(q;O) = 1, (54

where

I;a(q;t) = probability distribution function of the queue at time I,

A(t) = time-dependent arrival rate at time t,

p = constant service rate,

q = the queue length,

IX = variance to mean ratio for the arrival or demand rate,

ZY = variance to mean ratio for the departure or service rate,

c = maximum queue length.

Equation (5) only represents one queue. If there are m queues, then m simultaneous equations like equation (5) must be solved.

Numerical example for limited repair capacity

Now we proceed to calculate the expected number of units in resupply [ENUR] and the desired stock level [DSL] of the simple example illustrated in case 1 of Table I by the use of diffusion approximation. Poisson demand rate in normal operating time are assumed. During the initial surge period the same demand rate as that listed in Table 1 are used. Thus the values are: I, = Zy = 1, t < I, and I, = I, = 0, t 2 1, respectively. The number of servers is 5 and the service

= 0.3

i m

2 0

E 0.2

1 \ 1 1, i \ I ‘\ I \\ i \ i ‘i, / i i( ‘\, )l

‘\, Y__ _A_ _./ ‘\_

1,

_/‘, *A ‘L__ I I

IO 20 30 40 50

OUEUE LENGTH

Fig. 4. Some typical queue length distributions.

M.C.M. 12,7--o


rate is 0.2. The software package named PDESOL which is based on Sincovec and Madsen [14, 151 is used to solve the diffusion equation (5a) with the boundary conditions (5b) and (5~). After solving the diffusion equation, the queue length distribution FQ(q;t) is obtained. Some of the typical queue length distributions are shown in Fig. 4. The important computational steps in obtaining the results are as follows.

(1) The queue probability density ps is calculated as:

Ps = F&&t) - Fo(q - l;t).

(2) The expected number of units in queue [ENUQ] is obtained as

[ENUQI = i qpq q=l

(3) The expected units in resupply [ENUR] is computed as:

[ENUR] = [ENUQ] + [number of units in service],

which is shown in column 4 of Table 2.

(4) The desired stock level is determined as follows:

(a) Obtain the smallest integer when the probability distribution is > 0.8 which represents the desired probability of no back orders. In other words, select q which satisfies F,(q;t) > 0.8.

(b) Add the number of units in repair to the integer q from (a) to get the desired stock level which is shown in column 6 of Table 2.

Table 2 shows the comparison between the fluid approximation (FA) and the diffusion approximation (DA) for the ENUR as well as the DSL to achieve 0.8 probability of no back orders.

MULTI-REPAIR STATION SYSTEM

To illustrate the multi-station nature of the repair process, Novak and Madden 1161 considered the “raw engine build-up” procedure in which a series of queues can be formed. First, a queue may be formed to wait for the test stand. A second queue for repair crew, a third queue for parts if parts shortage occurs, and a fourth queue for the test cell, etc. Thus we have a series of queues in tandem instead of a simple queuing process which was discussed previously.

Table 2. Fluid and diffusion approximations

ENUR DSL

Day Demand FA DA FA DA

0 0.8 4.0 4.0 6 6 I 3.0 6.2 6.5 8 1 2 3.0 8.4 8.3 II 9 3 3.0 10.6 10.4 13 II 4 3.0 12.8 12.4 16 13 5 3.0 15.0 14.4 18 15 6 3.0 16.2 16.4 20 17

10 3.0 25.0 24.4 29 25 15 3.0 35.0 34.4 40 36 20 1.8 41.2 41.2 47 42 25 1.1 43.1 43.4 49 45 30 0.8 42.4 42.9 48 44 35 0.8 41.4 41.9 41 43 51 0.8 31.4 38.7 43 40

101 0.8 21.4 28.7 32 30 151 0.8 17.4 18.7 23 20 201 0.8 7.4 8.6 10 10

FA-Fluid approximation. DA-Diffusion approximation.

Emergency queuing model

Table 3. Tandem aueue

857

Repair station Test cell ENUP

ENUR ENUR ENUR ENUR

Day Demand PA DA PA DA PA DA

0 0.8 4.0 4.0 4.0 4.0 8.0 8.0 I 3.0 6.2 6.5 4.0 4.0 10.2 10.5 2 3.0 8.4 8.3 4.0 4.0 12.4 12.3 3 3.0 10.6 10.4 4.0 4.0 14.6 14.4 4 3.0 12.8 12.4 4.0 4.0 16.8 16.4 5 3.0 15.0 14.4 4.0 4.0 19.0 18.4 6 3.0 16.2 16.4 4.2 4.21 20.4 20.61

IO 3.0 25.0 24.4 5.0 5.0 30.0 29.3 I5 3.0 35.0 34.4 6.0 5.9 41.0 40.3 20 1.8 41.2 41.2 7.0 6.9 48.2 48.1 25 I.1 43. I 43.4 8.0 7.9 51.1 51.4 30 0.8 42.4 42.9 9.0 8.9 51.4 51.9 51 0.8 37.4 38.7 13.2 13.1 50.6 51.9

101 0.8 27.4 28.7 23.2 23.1 50.6 51.9 151 0.8 17.4 18.7 33.2 33. I 50.6 51.9 201 0.8 7.4 8.6 43.2 43.2 50.6 51.9

The distribution of the output from this series of queues is obviously much more difficult to obtain. However, if we only consider the surge period, the dynamics of this tandem queue can be obtained in essentially the same way. The total expected number of units in pipeline [ENUP] would be the sum of all those in services and in queues.

Two workstations in series (the queues in the repair station and in the test cell) are considered in this work. The same demand rates as that shown in Table 1 are used. The following assumptions are added:

(1) There are five servers in the repair station. In other words, five items can be repaired simultaneously. Constant repair rate is assumed.

(2) The average repair time for the item is 5 days. (3) Four items can be tested simultaneously at the test cell. (4) The average test time for a repaired item is also 5 days.

The output from the repair station becomes the demand rate for the test cell. Thus, the influence of high demand rate at the start of the surge period is assumed to affect the test cell 5 days later.

For comparison, fluid approximation (FA) is first used to solve the problem. The results are shown in columns 3, 5 and 7 of Table 3 under the heading FA. Then, this problem is solved by the use of the diffusion approximation (DA) approach. Since there are two queues in tandem, the system can be represented by two differential equations. One equation is for the queue in the repair station and the other for the queue in the test cell. These equations are essentially the same equation as that shown by equation (5) except the coefficients. The computational procedure is exactly the same as before except for two queues or for two simultaneous equations, instead of one. The results are shown in columns 4, 6 and 8 of Table 3 under the heading DA. Compared to the results of Table 2, the differences are tremendous. The ENUR in Table 2 should be compared with the ENUP in Table 3. Because both represent the total expected number of units in pipeline which is not available. The maximum value of ENUR for one queue (Table 2) is approx. 43 while for maximum value of ENUP for two queues are approx. 52. Even more significant, the maximum for one queue occurs at day 25 while for maximum for two queues occurs at day 25 through 201. In other words, because of the presence of two queues in the repair station, the time needed to reduce the length of queue (or the time during which a maximum number of units is not available) is increased tremendously. As the repair steps become more complicated or require more steps, the time delay can not be predicted by using simplified repair stations.

DYNAMIC MULTI-INDENTURE LOGISTICS SYSTEM

For a multi-indenture system, the problem is more complicated. Any delay in any item in the sub-indenture influences the overall system. Let us consider a two-indenture system. The system


may be an aircraft engine and its modules. Let the expected delay time in base engine repair due to a back order on module j at base i time t be represented by G,(t).Then s,(t) can be obtained by dividing the expected back orders for module i at random points of time by the mean daily removal rate. Thus a,, after dropping t for convenience, can be expressed as follows [4]:

6, = C (xv- s,)p(x,) [

mv X,,‘Sij II

(6)

where

xii = number of module j in resupply at base i,

mu = average daily removal rate of modulej at base i,

sii = stock level of module j at base i.

Thus the expected delay time in engine repair at base i time t due to the unavailability of modules is

(7)

where

N= number of modules, n,(t) = daily engine removal rate at base i time t, and

Ri = probability that an engine will be repaired at base i time t.

Then the expected delay time in engine repair at base i due to the unavailability of modules can be incorporated with wi which is the average expected engine repair time when all modules available. The service rate p,(t) now can be represented as:

PiCr) = I/twi + 5Ar)*

Note that constant service rate, pj = l/w, was used for the single-indenture system. But in multi-indenture system, variable service rate of (8) is used because of the impact of multi-indenture structure.

Computational procedure

This two-indenture system can be solved in two steps. First, the variable service rate due to present of sub-indenture is calculated and then, the ENURs and ENUPs for the system obtained. This computation is carried out as follows:

are

(8)

the the

the are

(1) The probability distribution of the number of module j in resupply p(x,>, j = 1, 2 , . . . , N, is computed. N diffusion equations similar to equation (5) with the boundary condition for the N modules are first solved. The procedure for solving these equations is exactly the same as that for Table 2. After solving the diffusion equations, the probability distribution of the number of modules in queue for each module can be obtained. Then the probability distribution of the number of module j in resupply, p(x,), j = 1, 2, . . . , N, is calculated.

(2) For module j, the expected delay time in base engine repair due to a back order of that

(3)

(4) (5)

(6)

module, s,(t), is calculated by using equation (6). The expected delay time in engine repair at base i due to the unavailability of modules, &{t), is computed by using equation (7). Finally, the variable engine service rate at base i, pi(t), is obtained by using equation (8). Using the variable engine service rate obtained in step (4), a diffusion equation similar to equation (5) is solved. Using the queue distribution for engine repair obtained in step (5), the ENUR for engine can be obtained using the same procedure as that used for Table 2.

In solving the differential equation for the engine repair in Step (5), variable service rate must be used. This complication is caused by the occasional nonavailability of the sub-indenture modules.


Numerical results

For comparison purposes, consider again the numerical example listed under the heading “capacity (1)” in Table 1. Further assumptions are:

(1) (2) (3) (4)

(5)

Only one base in this simple system. All failed engines and failed modules are repaired at the base level, i.e. r, = 1.0. An engine is the collection of two critical modules, i.e. N = 2. 50% of the engine failures require Module 1 and 60% of the engine failures demand Module 2. The number of servers for Modules 1 and 2 are both equal to two and the average expected repair time for Modules 1 and 2 are both 4 days.

Then, the module failure rates can be calculated as:

m,,(t) = OS,(t), m,,(t) = 0.6 A,(t)

where I,(t) is the engine failure rate at base 1. To illustrate the changes in the engine repair processes caused from the stockage level of two modules, two cases are studied.

Case I. The normal operating stock levels for both modules are maintained during the whole period. In other words,

s,,=s,~=~, T>O.

Case 2. The stock levels are increased to 15 and 20 times the normal stock levels after 10 days of emergency for Modules 1 and 2, respectively. In other words,

s,, = s,> = 1, T < 10

s,, = 15; s,~ = 20, T > 10.

This problem is solved by using the above procedure and numerical values. The delay time and service rates are first obtained based on the probability distribution of the number of modules in resupply. The results are shown in Table 4 for the two cases. Based on the variable service rates listed in Table 4, the distribution of the ENUR for the engine is obtained by integrating an equation similar to equation (5).

The ENURs and DSLs in Table 4 should be compared between themselves and with columns 4 and 6 of Table 2 to demonstrate the influences of the presence of the sub-indenture level.

The effects of multi-indenture show clearly when a comparison between Case I of Table 4 and columns 4 and 6 of Table 2 is made. At day 30, the difference in ENUR due to the presence of indenture 63.42 - 42.9 = 20.5 units or about 48% increase. However 42.9 is approximately a

Table 4. Two-indenture system

Case I Case 2

Day Demand

0 0.8 I 3.0 2 3.0 3 3.0 4 3.0 5 3.0 6 3.0 7 3.0 8 3.0 9 3.0

IO 3.0 II 3.0 12 3.0 13 3.0 I4 3.0 IS 3.0 20 I.8 25 1.1 30 0.8 35 0.8 40 0.8

Delay Service time rate

0.00 0.200 I .56 0.152 2.16 0.140 2.90 0.127 3.69 0.115 4.46 0.106 5.24 0.098 5.98 0.091 6.75 0.085 1.52 0.080 8.28 0.075 9.05 0.071 9.81 0.068

10.58 0.064 II.34 0.061 12.12 0.058 24.31 0.034 42.72 0.02 I 58.90 0.016 58.18 0.016 57.41 0.016

ENUR DSL

4.00 6 6.60 8 8.68 IO

11.10 I2 13.48 14 15.95 17 18.46 I9 20.97 22 23.56 25 26.15 21 28.76 30 31.41 33 34.05 35 36.74 38 39.42 41 42.13 43 52.81 54 59.38 61 63.42 65 67.02 69 70.62 12

Delay Service time rate

0.00 0.200 I .56 0.152 2.16 0.140 2.90 0.127 3.60 0.115 4.46 0.106 5.24 0.098 5.98 0.09 I 6.75 0.085 7.52 0.080 8.29 0.075 0.04 0.198 0.11 0.196 0.26 0.190 0.54 0.181 0.99 0.167 6.29 0.089

13.25 0.055 18.15 0.043 17.34 0.045 16.48 0.047

ENUR DSL

4.00 6 6.60 I 8.68 10

II.10 I2 13.48 14 15.95 I7 IS.46 19 20.97 22 23.56 25 26.15 21 28.76 30 30.93 32 32.94 34 34.99 36 37.07 38 39.21 41 48.03 49 53.41 55 56.78 58 59.67 61 62.54 64


maximum at day 30 without indenture, while the ENUR is still increasing at day 30 for Case 1 of Table 4 and this increase is still continuing at day 40. Thus, the influence of the multi-indenture structure certainly cannot be ignored.

To reduce the impact of the emergency surge period, let us assume that additional equipment is available after 10 days. This resulted in the scenario for Case 2. Notice even though the stock levels are increased by 15 and 20 times for Modules 1 and 2, respectively, the ENURs at day 40 didn’t reduce very significantly; only 70.62 - 62.54 = 8.1 units, or 11% reduction.

DYNAMIC MULTI-ECHELON AND MULTI-INDENTURE LOGISTICS SYSTEM

Consider a multi-echelon and multi-indenture logistics system such as that illustrated in Fig. 5 during the initial surge period of an emergency. When an aircraft engine (i.e. repairable item) fails

BASE 1 BASE 2

Engine Repair II _ romovo & replace II

modal 0 I II II

I___ _q V Ir

II II II I, II

w II

Engine Repair romovo & roplaoo

modnlo s

&___A

1 Module

Transport

Fig. 5. Schematic diagram of repairable item (engine) flow in a multi-echelon and multi-indenture logistics system.


BASE1 Flight Line

BASE2 Flight Line

3.16e-'*ltdt, k=16,...,28 k-15

50% needs Module 60% needs Module

Ll Module

k-15

60% needs Module

Module cl Stock

1

RT = 5

I NS = Number of servers

’ Engine Tranrport RT = Average Repair Time

NS = 2 P 'l-T-2.5

IT = Transportation Time

Fig. 6. Input data for the multi-echelon and multi-indenture logistics system.

at the flight line in base i, it is returned to the base supply and a serviceable engine is issued. Depending on the type of failure, the failed engine may be repaired either at base i with Ri probability or at the depot with (1 = Ri) probability. If the failed engine must be repaired at the depot level, it would join in a queue at base i to be transported to the depot. The rationale behind this argument of joining in a queue is that the transportation system has only a limited capacity during the initial surge period of an emergency. After repairing the failed engine at the depot (now it is serviceable), the serviceable engine joins in a queue at the depot transportation system to be transported back to its own base.


Table 5. ENURs and DSLs for modules for two-indenture and two-echelon systems

Base 1 Base 2

ENUR DSL ENUR DSL

Day Module I Module 2 Module 1 Module 2 Module 1 Module 2 Module 1 Module 2

0 0.96 1.44 2 2 1.20 1.80 2 2 I 2.23 3.25 3 4 3.25 3.32 4 4 2 2.61 3.12 3 4 3.61 4.02 4 5 3 3.20 4.29 4 5 4.16 4.86 5 6 4 3.80 4.93 5 6 4.12 5.81 6 I 5 4.45 5.62 5 6 5.33 6.78 6 8 6 5.12 6.33 6 7 5.96 7.75 I 9 7 5.80 7.04 7 8 6.59 8.68 7 9 8 6.48 1.14 7 9 1.23 9.59 8 10 9 7.13 8.42 8 9 7.85 10.50 9 I1

10 1.18 9.09 9 IO 8.46 II.42 9 I2 11 8.42 9.15 9 II 9.06 12.36 10 13 12 9.05 10.42 10 II 9.66 13.31 11 14 20 13.22 14.74 14 16 13.36 19.35 14 20 25 14.22 15.32 15 16 13.51 20.54 I4 22 30 14.31 14.94 15 16 12.66 20.53 14 22 40 14.20 13.81 15 15 10.64 20.12 I2 21 50 14.10 12.67 15 I4 8.68 19.72 9 21

For comparison purposes, let us consider a two-echelon and two-indenture system with the same general assumptions used in obtaining the results listed in Table 1. In addition, the following assumptions are added:

(1) There are two bases in the logistics system. (2) 60% of the failed engines are repaired at the base level (i.e. R, = 0.6) and 40% of them are

sent to the depot. (3) An engine is the collection of two critical modules. (4) 50% of engine failures demand Module 1 and 60% of them require Module 2. (5) All failed modules are repaired at the base level (for simplicity). (6) There is no lateral resupply between bases.

The actual data used are shown in Fig. 6. The stock levels for the modules are:

(1) sl, = s,* = s,, = sz2 = 1 for k < 10

(2) s,, = sIz = sl, = s22 = 15 for k > 10,

where k is the k th day of the emergency. In other words, the stock level for the modules increased 15 times after 10 days.

Numerical results

This problem can be solved in essentially the same manner as that described for the single-echelon and multi-indenture systems except for the fact that there are six queues in this system. Based on the demand rates shown in Fig. 5, the ENURs and DSLs for the modules in Base 1 and Base 2 are first obtained. These results are shown in Table 5. The variable service rates and delay time for engine repair are then obtained (Table 6). Notice that the delay times are dropped to zero at Base 1 and to 0.024 at Base 2, respectively, due to the increase in stock level after the 10th day.

The expected number of units in repair (ENUR) for the engines at the various queue lines are finally obtained and shown in Table 7. The final column in Table 7 shows the total of all the ENURs, or the total expected number of units in pipeline (ENUP). This is the total number of engines which is not available for the effort.

It is interesting to compare the results in Table 7 with those in Table 4. At day 40, the ENUP in Table 7 is 123.5 while the ENUR in Table 4, Case 2, is 64, or the number of unavailable engines increased by 93%.

Acknowledgements-This work was supported by Air Force Logistics Command, Wright-Patterson Air Force Base, Ohio under Contract No. F33600-86-C-0398.

Tab

le

6.

Exp

ecte

d de

lay

time

and

serv

ice

rate

s in

en

gine

re

pair

fo

r th

e tw

o-in

dent

ure

and

two-

eche

lon

svst

ems

Bas

e I

Bas

e 2

Day

D

elay

tim

e Se

rvic

e ra

te

0

0.00

0 0.

2000

0 I

I.92

0 0.

1445

0 2

2.41

1 0.

I34

93

3 3.

030

0.12

453

4 3.

718

0.11

470

5 4.

461

0.10

570

6 5.

230

0.09

775

7 5.

988

0.09

101

8 6.

152

0.08

509

9 7.

500

0.08

000

IO

8.23

4 0.07556

II

0.00

0 0.

2OO

Oo

12

0.00

2 0.

1999

4 I3

0.

006

0.19

976

I4

0.01

4 0.

1994

3 I5

0.

037

0.19

853

25

1.40

8 0.

1560

5 35

I ,

281

0.15

921

45

0.83

5 0.

1713

8 50

0.

714

0.17

501

Del

ay

time

Serv

ice

rate

0.00

0 0.

2OQ

Oo

2.04

1 0.

1420

3 2.

540

0.13

262

3.14

8 0.

1227

2 3.

822

0.11

335

4.53

0 0.

1049

3 5.

256

0.09

751

5.96

2 0.

0912

2 6.

657

0.08

579

7.33

1 0.

08 I

09

8.02

2 0.

0767

9 0.

024

0.19

903

0.06

5 0.

1914

2 0.

140

0.19

456

0.26

9 0.

1898

1 0.

473

0.18

271

6.99

6 0.

0833

6 0.

865

0.07

213

7.99

8 0.

0769

3 7.

592

0.07

941

Tab

le

7. E

NU

Rs

and

DSL

s fo

r en

gine

s in

th

e tw

o-ec

helo

n an

d tw

o-in

dent

ure

logi

stic

s sy

stem

Failu

re

rate

B

ase

I B

ase

2 B

ase

I tr

ansp

ort

Bas

e 2

tran

spor

t D

epot

D

epot

tr

ansp

ort

Day

B

ase

I E

ase

2 E

NU

R

DSL

E

NU

R

DSL

E

NU

R

DSL

E

NU

R

DSL

E

NU

R

DSL

E

NU

R

DSL

E

NU

P

0 0.

48

0.60

2.

40

3 3.

00

4 0.

80

1 1.

00

2 3.

60

5 1.

80

3 12

.6

I I .

80

2.22

4.

40

5 5.

48

6 2.

27

3 2.

34

3 5.

53

6 3.

53

4 23

.6

2 1.

80

2.22

5.

61

6 6.

99

8 2.

76

4 3.

26

4 5.

87

7 3.

87

5 28

.7

3 I .

80

2.22

7.

00

8 8.

71

IO

3.69

5

4.29

5

6.08

7

4.13

5

33.9

4

I .a0

2.

22

8.46

IO

10

.47

12

4.47

5

5.39

6

6.23

7

4.35

5

39.4

5

I.80

2.

22

9.94

II

12

.26

I4

5.29

6

6.50

7

6.37

7

4.54

6

44.9

6

1.80

2.

22

II.3

2 13

14

.07

15

6.1

I 7

7.58

9

6.51

8

4.71

6

50.4

7

I .80

2.

22

12.9

4 14

IS

.91

I7

6.93

8

8.65

IO

6.

63

8 4.

87

6 55

.9

8 1.

80

2.22

14

.48

16

17.7

7 I9

7.

73

9 9.

71

II

6.67

8

4.99

6

61.3

9

I .80

2.

22

16.0

3 I7

19

.65

21

8.52

9

10.7

9 I2

6.

66

8 5.

09

7 66

.7

IO

I .80

2.

22

17.6

0 19

21

.56

23

9.30

IO

Il

.88

I3

6.66

8

5.17

7

72.2

15

I.

80

2.22

23

.69

25

28.9

0 30

13

.31

I4

17.2

7 I7

6.

66

8 5.

17

7 95

.0

20

I .09

1.

35

27.9

2 29

34

.56

36

16.0

3 I7

21

.11

22

6.66

8

5.17

1

III.

5 25

0.

66

0.82

29

.68

31

37.8

0 39

16

.91

I8

22.6

4 24

6.

66

8 5.

17

I 11

8.9

30

0.48

0.

60

30. I

I 32

39

.63

41

16.6

9 18

22

.85

24

6.66

8

5.17

7

121.

1 35

0.

48

0.60

30

.22

32

41.2

2 43

16

.29

I7

22.8

4 24

6.

66

8 5.

17

7 12

2.4

40

0.48

0.

60

30.2

1 32

42

.17

44

15.8

9 17

22

.85

24

6.66

8

5.17

7

123.

5 50

0.

48

0.60

29

.93

31

45.7

1 47

15

.09

I6

22.8

5 24

6.

66

8 5.

17

7 12

5.4

-.

__

7.

.“-

,,._

-


REFERENCES

1. J. A. Muckstadt, Comparative adequacy of steady-state versus dynamic models for calculating stackage requirements. The Rand Corp., R-2636-AF (November 1980).

2. D. Gross, H. D. Kahn and J. D. Marsh, Queuing models for spares provisioning. Nav. Res. Logisr. Q. 24, 521 (1977). 3. C. Sherbrooke, METRIC: a Multi-Echeon Technique for Recoverable Item Control. Op. Res. 16, 122 (1968); also The

Rand Corp., RM-5078-PR (November 1966). 4. J. A. Muckstadt, A model for a multi-item, multi-echelon inventory system. Mgmr Sci. 20, 472 (1973). 5. R. J. Hillestad. Dvna-METRIC: Dvnamic Multi-Echelon Technique for Recoverable Item Control. The Rand Corp.,

R-2785AF (July i982). 6. W. S. Demmy and V. J. Presutti, Multi-echelon inventory theory in the Air Force Logistics Command. TZh4S Stud.

Mgmt Sci. 16, 2789 (1981). 7. R. Pyles, The Dyna-METRIC readiness assessment model: motivation, capabilities, and uses. The Rand Corp.,

WD-892-AF (February 1982). 8. R. J. Hillestad and M. J. Carrillo, Models and techniques for recoverable item stackage when demand and the repair

process are nonstationary-Part I. Performance measurement. The Rand Corp., N-1482-AF (May 1980). 9. J. H. Bielow and K. Isaacson. Models to assess the peacetime material readiness and wartime sustainability of U.S.

Air Forces: a progress report, The Rand Corp., N-1896-MRAL (October 1982). 10. N. Sinnh and P. Vrat. A location-allocation model for a two-echelon renair-inventory system. ZZE Trans. 16,222 (1984). 11. G. F. Newell, Appkbtion of Queuing Theory, 2nd edn, Chapman & Hall, London (1982). 12. M. Jung, Queuing optimization and multi-echelon and multi-indenture logistics system with limited capacity. Ph.D.

Dissertation, Dept of Ind. Engng, Kansas State University (1984). 13. L. Kleinrock, Queuing Systems, Vol. 2, Wiley, New York (1976). 14. R. F. Sincovec and N. K. Madsen, Software for nonlinear partial differential equations. ACM Trans. Mach. Software

1, 232-260 (1975). 15. R. F. Sincovec and N. K. Madsen, Algorithm 494: PDEONE, solutions of systems of partial differential equations.

ACM Trans. Math. Software 1, 261-263 (1975). 16. R. A. Novak and J. L. Madden, TJEMS, Transport Jet Engine Management Simulator. Working Notes XRS-79-015,

AFLC, DCS/Planes and Programs (September 1982).

Documents

A multi-echelon and multi-indenture repairable item queuing model during emergencies