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Proceedings of the 2014 Industrial and Systems Engineering Research Conference

Y. Guan and H. Liao, eds.

Optimum Reliability and Maintainability Allocations for

Load-Sharing Continuous Flow System with Buffers

Khaled A. Farouk, Mohammad Younes, M. Nashat Fors,

Production Engineering Department

Alexandria University

Alexandria, Egypt

Abstract

Reliability and availability of systems are traditionally provided by using different forms of redundancies such as

multiple units in parallel or standby units. However, in oil and gas plants and other continuous flow large scale

industrial systems, applications of redundancies are neither structurally nor economically feasible. On the other hand,

multi-units sharing the load with buffers are commonly used. In these plants, buffers are essential to guarantee the

steady output flow in the cases of having failed units, or having units under maintenance and repair. In these load-sharing multi-unit systems, the total capacity of the units forming the system should be greater than the design capacity

of the plant by a specified margin, aiming at maximizing the system availability within constraints of allocated budgets.

In the proposed work, the problem of optimal allocations of reliability and maintainability to such load-sharing con-

tinuous flow multi-units systems is considered. Optimum number of units, capacity distribution among them, and

capacity of buffers will be considered as decisions variables in addition to other variables describing the performance

of the system.

KeywordsSystem availability and maintainability, imperfect maintenance, buffer allocation.

1. IntroductionProcess plants are comprehensive systems that have all combinations of system complexities (load sharing, multi-state, and in-process buffers). Designing of such systems lead to a very challenging problem which can be formulated

as: "What is the optimal components reliability and maintainability apportionment to maximize availability at the

minimum possible cost?".

During the conceptual design phase of a system, a number of choices need to be made; such as the number of units, the

required system reliability, unit maintainability, buffers, performance specifications of the chosen units, and system

attributes required to achieve the necessary performance level [1].

Process Plant is a continuous production multi-state system that consists of multiple processes. These processes are

arranged to produce a specified product based on the production sequence. The failure of any process has an immediate

effect on the final production rate. Thus, reliability and maintainability of such intermediate equipment (intermediate

process) affect the resulted production rate. To compensate for such loss of production, the redundant parallel standby

equipment approach has been introduced naturally irrespective of the involved cost. It is not economically sound

to have double sized equipment capabilities to improve the production availability, especially for high production

capacity. The approach of introducing buffers at various stages in the production line, in place of standby redundant

machines, to improve the overall production line reliability has been investigated by Sethia et al. in [ 2]. Moreover in

continuous production systems, halts due to failure or maintenance create imbalances within the production system.

Hence, in order to contain the system imbalances, due to repair and maintenance halts, buffers between processes

should be introduced [3].

The comprehensive survey on the buffer allocation problem has been conducted by Demir et al. in[4]. They concluded

that the buffering allows the equipments (processes) to operate almost independently of each other and it helps to

increase the throughput rate of the system. However, there are usually floor space and budget constraints in reality.

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Farouk, Younes, and Fors

Generally, the eventual aim is to improve system performance with minimum cost as in the case of all manufacturing

system problems. Therefore, considering the buffer allocation problem as a multi-objective problem is an important

research issue [5].

Using in-process buffer to mitigate the effect of maintenance and repair downtime, has been highlighted by many

researchers [36]. Actually, allocating an appropriate amount of buffers to meet the continuous operation constraint or

to reduce the loss caused by the preventive maintenance and occasional failure, is necessary for a prolonged operation

production system [6].

The process plant is a repairable system, which makes the maintenance function a paramount parameter that affects

the equipment availability not only by introducing the downtime but also by affecting the equipment failure rate when

maintenance is done imperfectly. Since 1979, Nakagawa highlights that the assumption that preventive maintenance

returns the equipment as good as new is often not true [7]. The equipment availability, under this assumption of perfect

maintenance, is constant with time. This relation with time indicates that the equipment will work forever as long as

it is periodically maintained, which is not true. Any repairable equipment in real life needs to be replaced by a new

equipment after a while even if it has been periodically maintained. This fact is supported by Cassady et al. [8] in

their availability simulation of one repairable equipment under imperfect repair using the Kijima model [9]. The

simulation shows that the availability function is degraded exponentially with time.

The main contribution of this study is to define the generic formulation of optimal availability allocation considering

the equipment capacity margin and the buffer size under imperfect maintenance. The formulation imitates the real

design requirements of oil and gas industries.

2. Problem StatementGiven a system ofKunits (pumps) sharing the supply to satisfy the rated demand of Q cubic meters or barrels per unit

time. The units are sharing the load with proportionsrksuch thatKk=1 rk= 1. A capacity margin, where(0 < 1),

is provided for each unit of the system in order to guarantee the continuity of system supply even in case of having

some or all units down during maintenance or repair. Buffer storage should be made available to receive the excess

capacity (Q) during the up-time of the whole or part of the system. Practically, system maintenance and repair areassumed imperfect. The Kijima first model of virtual-age [9] is used to take the imperfect repair and maintenance

into consideration while building our formulation. In the Kijima model, maintenance and repair do not necessarily

restore the units back to their initial state as good as new but only remedy a part of their deteriorations. The remaining

part is conceived as virtual age of units of the system to start within the next period of its operational life. As already

proposed by Kijima, a factor , where(0

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Acronyms

BV Buffer Volume

CM Corrective Maintenance

DWNT Downtime

FC Failure Cost

MC Maintenance CostPM Preventive Maintenance

TC Total Cost

Notations

BVC Buffer volume cost

C Acquisition cost of one unit

DWNT(i) Downtime at theith period of time

DTC Cost incurred by the system downtime

ICC Cost due to the increase in capacity

L Total operation time

MBV Maximum buffer size requirednM number of maintenance actions

nF number of failure events

Q rated demand per unit time

QD Production volume per one day

q exponent for capacity increase cost

Rmin Minimum allowable reliability by the end of unit operating life

rk proportion of total capacity taken by unit k

Sk(j) state of thekth unit at the jth period of time

TBM Time between actions of PM preventive maintenance

tk j period of time taken by unitkfrom the start of period j

TTFk j time to failure of unitkin the operating period j

Tk j the length of jth uptime period

TM Time necessary to perform PM

TR Time necessary to perform CM

u uniformly distributed random number [0, 1]Ykm units cumulated virtual age

UBVC Buffer volume cost per unit time

UDTC Downtime cost per unit time

UFC CM cost per unit time

UMC PM cost per unit time

factor of Virtual Age shape factor in Weibull distribution Capacity Margin provided to each unit of the system scale factor in Weibull distribution

3. System DescriptionIn the following subsections we calculate the failure probability and the corresponding reliability and the replacement-

age. Moreover, an algorithm is given in Section3.3 to determine the maximum number of working periods dictated

by the limitation of reliability.

3.1 Unit History Timeline

As already mentioned above, the lifetime of any one of the units can be described as a series of uptime periodsTk j for

(k=1, 2, ..., Kunits)and(j=1, 2, ...,Nworking periods), during which it provides useful work. Each uptime period

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is followed by a downtime period of length either TM or TR depending on whether it is PM or CM period. The unit

is not restored back as good as new after PM or CM, but however restored with a residual life known as virtual age.

Next, a formula according to Kijima first model to evaluate the virtual age Ykm ofkth component after (m-1) periods of

operational life and at the start ofmth period is given as follows:

Ykm=

0 for m=1

m1

j=1

Tk j for m>1

(1)

Tk j is the length of jth uptime period.Ykm is the virtual age accumulated along (m-1) operating periods and attributed

to performing (m-1) imperfect maintenance and repair actions.

Weibull distribution withkas a shape factor and kas a characteristic time is considered the most common distri-bution applied to model time to failure tk j because of its universality. The length of uptime periodTk j depends on

whether it is ended by PM or CM and is calculated as shown in Equation ( 2).

Tk j=

TTFk j for tk j

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N1

j=1

Tk j k

ln

1

Rmin

1k

(6)

It is clear from Equation (6) that as=0 (prefect maintenance and repair), there is a limitation on the lifetime of thecomponent.

Figure 1: Identification of Preventive and Corrective events along system operating parameters

An Algorithm, given in Figure1, is proposed to evaluate the maximum number of working periods dictated by the

limitation of reliability by the start of the last period of a lifetime. Moreover, the total number nMof PM and the total

numbernfof CM are also obtained from this algorithm.

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3.4 Buffer and Capacity Margin

Research in[36] discusses the use of in-process buffer to mitigate the effect of maintenance and repair downtime.

The buffer could not be filled without capacity margin () above the required rate of production as shown in Figure2.

0 200 400 600 800 1000 1200 1400 1600 1800 20000

2

4

6

8

10

12

14

16

18

20

Time (Days)

QD

(103

Gallon)

Unit Production rate

Buffer Volume

Buffer volume change with time

Unit production rate with time

Figure 2: Single unit production rate & buffer volume with time

It is clear from Figure2 that the availability of production does not only depend on the unit reliability and maintain-

ability, but it also depends on the available buffer volume and the rate of buffer refilling.

3.4.1 Multi-Unit System (Load-Sharing System)

When one of the units fail in load-sharing systems, the system does not fail but it does not fully operate as shown in

Figure3. The probability that the whole system falls into downtime decreases with the use of multiple units, however,

the probability that one unit fails increases.

0 200 400 600 800 1000 1200 1400 1600 1800 20000

2

4

6

8

10

12

14

16

18

20

Time (Days)

QD

(103

Gallon)

Single unit system production rate

4 units system production rate

Buffer volume

Figure 3: Multiple-unit system production rate and buffer volume with time

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3.4.2 Evaluation of Downtime and Buffer Level in Load-Sharing Systems

In this subsection we describe the algorithm that calculates the downtime and the size of buffer storage of a system

composed of multiple units. A flowchart of the algorithm is given in Figure1. Here, it should be noted that another

scale of time should be introduced. This scale starts with zero time at the start of the operating life of the system and

continues up to the systems end of life. This systems end-of-life is determined by stating the minimum allowable

reliability of the components of the system. Discrete events of PM and failure of different components are located onthis new time scale (system time scale) as discrete points idenoting the time of occurrence of the i

th event whether it

is PM or CM.

Step 1 Run the procedure as given by the algorithm described in Figure 1. The results will be a two dimensional array.

One dimension represents one of two events, either PM or CM on the units. The other corresponding dimension

represents the time of occurrence of the event (i).

Step 2 Combine the results of the K units into one array. Then sort the data in the combined array by the column of the

time of occurrence of the different events iin an ascending order. If G is the total number of events recorded inthe combined array, then 0 i G.

Step 3 Evaluate at each moment of time (i) the quantity supplied by each unit taking into account its capacity marginand its stateSk(i1)in the just previous moment of time whether it is in PM or in CM by the Equation ( 7).

Supplyk(i) = rkQ (1 + ) [(i i1) (k(i1) TM) (k(i1) TR)] ,

where N = the total number of events occurring up to the end of lifetime of the system,

1 = 0,

k(i1) =

1 i f f S k(i1) =PM

0 otherwise,

k(i1) =

1 i f f S k(i1) =CM

0 otherwise.

(7)

Step 4 Evaluate the quantity that should be delivered according to the demandQ (ii1)in the interval of timei1toi

Step 5 The buffer storage is assigned to receive the excess supply Q provided by capacity margin of each unit inorder to guarantee the continuous satisfaction of the demand. The quantity of fluid in the buffer storage should

be updated at each timei by the following formula:

BV(i) =

0 Q (i i1)>Kk=1 Supplyk(i)

Q (i i1) +

K

k=1

Supplyk(i) Otherwise

(8)

Step 6 When the quantity stored in the buffer storage reaches zero during the period i i1 and the bufferBV(i)isnot sufficient to satisfy the demand in this period then system is considered down. The downtime of the system

is evaluated as follows:

DWNT(i) =DWNT(i1) +

TM

BV(i1)

Q

K

k=1

k(i1) +

TR

BV(i1)

Q

K

k=1

k(i1) (9)

Step 7 The system availability or system probability of success is evaluated at i as follows:

A(i) =1DWNT(i)

i(10)

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4. Optimization ModelThe optimization has been conducted using a hybrid model Genetic Algorithm (GA) and discrete event simulation.

The hybrid model is used to search for near optimum solution. The genes of GA chromosome signify the decision

variables. The decision variables indicate the number of units in the load-sharing system, the capacity margin per each

unit, the buffer volume, and the manufacturer. The parameters of the unit; such as cost, reliability and maintainability,

are defined by the manufacturer. Hence, the manufacturer decision variable takes values from a predefined set of

manufacturers.

4.1 Assumptions

The system is a share-loading system that consists of N units. The maintenance is done imperfectly. The system has

extra capacity margin. The system has a finite volume of buffer.

4.2 Cost Analysis

The total cost is naturally affected by the following factors:

1. The extra capacity margin cost is a function in and q as a given power to be determined numerically fromstatistics of prices. The cost is evaluated as follows:

ICC= ((1 + )q1) C (11)

2. Expected value of preventive maintenance cost:

E[MC] =UMC TM K

k=1

N

j=1

P

tk j TBM

E[MC] =UMC TM K

k=1

N

j=1

e

TBM+Yk j

k

k(12)

3. Expect value of failure cost:

E[FC] =UFC TR K

k=1

N

j=1

1e

TBM+Yk j

k

k (13)4. Expected value of downtime cost:

E[DTC] =UDTC E[DWNT(G)] (14)

5. Expected cost of buffer storage:

E[BVC] =UBVC E[MBV] (15)

4.3 Objective Function

Minimize Z =ICC +E[MC] +E[FC] +E[DTC] +E[BVC]

subject to

e

Yk j

kRmin, where 1 k K

(16)

5. Computation ResultsBased on the above algorithm the simulation has been built and the interaction between the system parameters has

been investigated.

5.1 Buffer Volume versus Capacity Margin for Load-Sharing Systems

Capacity margin has been investigated at two levels; low (=0.2) and high (=0.5), versus buffer volume with fivedifferent levels; very low (VB = 4QD), low (VB = 5QD), moderate (VB = 6.7QD), high (VB = 10QD), extra high (VB

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1 2 3 4 5 6 7 8 9 100.86

0.88

0.9

0.92

0.94

0.96

0.98

1

Number of units

SystemA

vailability

BV = 20 QD

BV = 10 QD

BV = 6.7 QD

BV = 5 QD

BV = 4 QD

BV = 20 QD

BV = 10 QD

BV = 6.7 QD

BV = 5 QDBV = 4 QD

= 0.2

= 0.2

= 0.2

= 0.2

= 0.2= 0.5

= 0.5

= 0.5

= 0.5

= 0.5

Figure 4: System availability for different number of units, different capacity margin and different buffer volume

= 20QD). The resulting system availability against different load-sharing Nunits is depicted in Figure4. Note that all

other system parameters were fixed during this experiment.

It is clear how the buffer volume affect the system availability. When the buffer volume increases the system availabil-

ity increases. The Capacity margin is a critical factor in system design which shows not only the fact that increasing

the capacity margin will increase the system availability but it also shows that it is critical to the system availability

behavioral change with a number of units in load-sharing systems.

6. Conclusion and Future WorksWhen it comes to the question of determining the optimal system configuration, the choice of the system parameters

is based on the minimum total cost (initial cost + operation cost) as shown in Figure5.

1 2 3 4 5 6 7 8 9 100.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2x 10

7

Number of Units

TotalCost

= 0.2 , BV = 5 10e+4 gallon

= 0.2 , BV = 10 10e+4 gallon

= 0.2 , BV = 15 10e+4 gallon

= 0.2 , BV = 20 10e+4 gallon

= 0.2 , BV = 25 10e+4 gallon

= 0.5 , BV = 5 10e+4 gallon

= 0.5 , BV = 10 10e+4 gallon

= 0.5 , BV = 15 10e+4 gallon

= 0.5 , BV = 20 10e+4 gallon

= 0.5 , BV = 25 10e+4 gallon

Figure 5: Total Cost for different number of units, different capacity margin and different buffer volume

It is clear that when the buffer size increases, the cost decreases due to the increase in the system availability. The

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capacity margin improves the utilization of buffers, which consequently increases the availability and decreases the

cost. The number of units in a multiple-unit system affects the availability and the initial cost. The availability of the

system increases with the number of units if and only if it is combined with a proper capacity margin. On the other

hand, the initial cost increases with the number of units in a multiple-unit system. All of these relations are clearly

depicted in Figure5. It is also clear from this figure that the minimum total cost occurs when a multiple-unit system

consists of 4 units with a capacity margin=0.5 and a buffer volume = 105 gallon.

The optimal availability and maintainability allocation with both buffer volume and capacity margin have been investi-gated. The results show that the buffer and capacity margin are critical variables when deciding on the optimal system

parameters. The time to unit replacement without considering the imperfect maintenance will be infinity, which is

unrealistic . In oil and gas plants and other continuous flow large scale industrial systems, the load-sharing system

parameters must be defined carefully to assure system consistency. For future work, investigations of different im-

perfect maintenance models using the above problem is required in order to reach a better understanding of the effect

of imperfect maintenance. The proposed model can be extended to cover multiple-process system with in-process

allocation of buffers to maximize the system availability and minimize the total cost.

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pp. 05, 2012.

[2] P. C. Sethia, Enhancing reliability of a continuous manufacturing system using WIP buffers, International Jour-

nal of Simulation Modelling, vol. 7, pp. 6170, June 2008.

[3] T. Murino, E. Romano, and P. Zoppoli, Maintenance policies and buffer sizing: an optimization model,WSEAS

TRANSACTIONS on BUSINESS and ECONOMICS, vol. 6, no. 1, 2009.

[4] L. Demir, S. Tunali, and D. T. Eliiyi, The state of the art on buffer allocation problem: a comprehensive survey,

Journal of Intelligent Manufacturing, Sept. 2012.

[5] M. Amiri and A. Mohtashami, Buffer allocation in unreliable production lines based on design of experiments,

simulation, and genetic algorithm, The International Journal of Advanced Manufacturing Technology, vol. 62,

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[6] Y. Zhang, S.-Y. Gong, and J.-Y. Sheng, Optimal buffer inventory for maintenance action under random produc-

tion capacity availability, 2012 International Conference on Quality, Reliability, Risk, Maintenance, and Safety

Engineering, pp. 400404, June 2012.

[7] T. Nakagawa, Imperfect Preventive-Maintenance, IEEE Transactions on Reliability, vol. R-28, p. 402, Dec.

1979.

[8] C. R. Cassady, I. M. Iyoob, K. Schneider, and E. A. Pohl, A Generic Model of Equipment Availability Under

Imperfect Maintenance,IEEE TRANSACTIONS ON RELIABILITY, vol. 54, no. 4, pp. 564571, 2005.

[9] M. Kijima, H. Morimura, and Y. Suzuki, Periodic replacement problem without assuming minimal repair, Eu-

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