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A simple ordering policy with linear lifetime for a unit
Microelectron. Reliab., Vol. 35, No. 5. pp. 847-849, 1995 Copyright ~ 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0026 2714/95 $9.50 +.00
Pergamon
0026-2714(94)000115-4
TECHNICAL NOTE A SIMPLE O R D E R I N G POLICY WITH LINEAR LIFETIME FOR A UNIT V. SRIDHARAN
Department of Mathematics, College of Engineering, Anna University, Madras 600 025, India (Received for publication 30 July 1994)
Abstract In this paper optimum ordering policies of a one-unit system where each failed unit is scrapped and each space is only provided after a lead time by an order with a deterministic linear lifetime for the unit is considered. The optimum ordering policy minimizing the expected cost per unit time in the steady state is discussed by introducing the two types of constant lead times along with ordering costs and downtime cost. The problem concludes with a numerical example minimizing the expected cost function.
1. INTRODUCTION In reliability and maintenance engineering, failure of an operating unit (i.e. machine, equipment, etc.) takes place according to some probability law. Thus, reliability engineers must face the replacement policies such as an age replacement or a block replacement discussed by Barlow and Proschan [-1, 2] or ordering policies discussed by Wiggins [-3], Kaio and Osaki [4] have considered a one-unit model with two types of orders, namely expedited (emergency) and regular. In this paper optimum ordering policies with two types of constant lead times, three linear costs and linear lifetime pattern for the unit minimizing the expected cost are discussed.
immediately, irrespective of the state of the original unit. On the other hand, if the original unit fails before that time instant to, expedited order is made immediately at the failure instant and the spare takes over the operation as soon as it is delivered after an expedited lead time, L~. The process of cycle repeats itself. Introducing two types of constant lead times and three linear costs and linear deterministic lifetime for the unit and noting that every replacement time instant is a regeneration point, the expected cost per unit time in the steady state is derived. It is shown that there exists an optimum policy minimizing the expected cost under certain conditions for a linear deterministic lifetime for the unit.
2. THE MODEL
3. ASSUMPTIONS
This paper considers a one-unit system where each failed unit is scrapped without repair and each spare is only provided after a lead time by an order. To seek an optimum policy minimizing or maximizing some criterion, the following policy is adopted: order for a spare is made at a prespecified time instant t o during an operating period of an original unit which is called a regular order. However, if the failure of the unit takes place before that instant, emergency order is made at the failure time instant immediately. After a lead time the spare is delivered and the original unit is in operable condition or otherwise. The planning horizon is infinite and the original unit begins operating at time 0. If the original unit does not fail up to a pre-specified time instant to ~ [0, ~-~), regular order for a spare is made at time to. After a regular lead time L2, the spare is delivered and the original unit is replaced by the spare
(a) The system consists of a one-unit system whose life-time distribution is F(t) with the density function f ( t ) = a + bt (say) linear deterministic lifetime. (b) If the unit fails before the time to, an emergency (expedited) order is placed at the instant of failure which has a lead time L x. (c) If the unit is good at t o, a regular order is placed at t which has a lead time L2, different from L 1. (d) A constant cost C1 per unit time is suffered for the system downtime, a cost C 2 is suffered for the expedited order which is made at the instant t before the instant t o and a cost C3 is suffered for the regular order made at the time instant t o . Also LI <~ L 2, C2 > C3. Under these assumptions, which are reasonable, an interval which is defined as the beginning of the original unit replacement to the next replacement as one cycle, the expected cost per cycle is derived (see Fig 1).
3s:7-F
847
848
Technical Note to •-v- ~
oI
Let C be the cost incurred in a cycle. T h e n the expected cost per unit time in the steady-state is
i
m
--|
!
I
L1
O~t~to
{[
C(to) = C1 (Lt - L2)F(to) +
oI
f--~ t0
f
F(t) dt
~ to
]
l/J//f//-ff
to [91"-'-'-- L]
oI
---j t0+L2~t
r//i//ll///1
N o w let there be a deterministic linear lifetime for the unit (say) f (to) = a + bt o. Substituting f (to) = a + bto in e q u a t i o n (4) one obtains
Failure Replacement ~ ' ] State of operation 1 State of expedited order with lead time L 1 State of regularorderwith lead time L2 -- State of failure X
C(to)={CtILlat °
fLlb'~2
Fig. 1. Possible realizations of one cycle.
4. COST EQUATIONS AND LINEAR LIFETIME FOR THE UNIT In this model the following expected costs are considered. (a) W h e n the original unit fails, if no spare is available, the system is u n d e r failure until the spare is delivered. The expected cost during t h a t period is Ct
[foo
L1 dF(t) +
(to + L2 - t) dF(t)
]
+
(5)
Hence, if there is any o p t i m a for C(to) one wish to select the t o value so as to minimize C(to). A necessary condition is that
C'(to)
= 0 which leads to
o
=C,[(L1-L2,F(to)+f~y+L2F(t)dt].
-(~)t3}{Cl[Lla+L~bt°+(;)
(b) The expected order cost is C2 dF(t) +
C 3 dF(t) =
C2F(to) +
+
C3ff(to)
o
(2)
(t + L 1) dF(t) +
f;
L2 ]
c3 ,o b,o,t fc,ILlatO f Llb~ 2
where if(to) = 1 - F(to). Moreover, the m e a n time of one cycle is
I:°
tg
(1){(L1--L2)(at°+(b2)tg)+L2+t°-(~)
+(C2- C3)(ato+(~)t2)+C3}
(t o + L2) d r ( t )
o
= (L1 - L2)F(to) + L2 +
;:°
if(t) dt.
(3)
(6)
Table 1
(a) (b) (c) (d) (e) (f) (g) (h)
LI
L2
C~
Cz
C3
a
b
5 3 5 4 3 4 3 4
8 4 8 5 5 5 5 6
0.8 0.5 0.4 0.6 0.6 0.6 0.6 0.5
50 25 30 20 25 20 25 50
30 15 20 10 15 10 15 40
3 15 3 12 6 6 7 10
55 10 10 7 3 2 2 17
Hence for t* = 32.44595 the objective value is
C(t*) =
0.3054266.
to 13.00725 14.11132 14.37633 20.01839 21.82158 30.03586 32.44595 50.05816
C(to) 1.4886780 0.7244133 0.6505462 0.4814463 0.4492928 0.3199690 0.3054266 0.3798361
Technical Note E q u a t i o n (6) can be solved by some iterative technique, for example, the bisection m e t h o d can be used.
5. NUMERICAL ILLUSTRATION F o r the various costs, lead times a n d linear deterministic lifetime for the unit Table 1 gives the relevant t o against C(to).
849 REFERENCES
1. R. E. Barlow and F. Proschan, Mathematical Theory of Reliability. Wiley, New York (1965). 2. R. E. Barlow and F. Proschan, Statistical Theory of Reliability and Life-Testing: Probability Models. Holt, Rinehart and Winston, New York (1975). 3. A. D. Wiggins, A minimum cost model of spare parts inventory control, Technometrics 13, 139-144 (1967). 4. N. Kaio and S. Osaki, Ordering policies with two types of lead times, Microelectron. Reliab. 16, 225-229 (1977).
Pergamon
0026-2714(94)000115-4
TECHNICAL NOTE A SIMPLE O R D E R I N G POLICY WITH LINEAR LIFETIME FOR A UNIT V. SRIDHARAN
Department of Mathematics, College of Engineering, Anna University, Madras 600 025, India (Received for publication 30 July 1994)
Abstract In this paper optimum ordering policies of a one-unit system where each failed unit is scrapped and each space is only provided after a lead time by an order with a deterministic linear lifetime for the unit is considered. The optimum ordering policy minimizing the expected cost per unit time in the steady state is discussed by introducing the two types of constant lead times along with ordering costs and downtime cost. The problem concludes with a numerical example minimizing the expected cost function.
1. INTRODUCTION In reliability and maintenance engineering, failure of an operating unit (i.e. machine, equipment, etc.) takes place according to some probability law. Thus, reliability engineers must face the replacement policies such as an age replacement or a block replacement discussed by Barlow and Proschan [-1, 2] or ordering policies discussed by Wiggins [-3], Kaio and Osaki [4] have considered a one-unit model with two types of orders, namely expedited (emergency) and regular. In this paper optimum ordering policies with two types of constant lead times, three linear costs and linear lifetime pattern for the unit minimizing the expected cost are discussed.
immediately, irrespective of the state of the original unit. On the other hand, if the original unit fails before that time instant to, expedited order is made immediately at the failure instant and the spare takes over the operation as soon as it is delivered after an expedited lead time, L~. The process of cycle repeats itself. Introducing two types of constant lead times and three linear costs and linear deterministic lifetime for the unit and noting that every replacement time instant is a regeneration point, the expected cost per unit time in the steady state is derived. It is shown that there exists an optimum policy minimizing the expected cost under certain conditions for a linear deterministic lifetime for the unit.
2. THE MODEL
3. ASSUMPTIONS
This paper considers a one-unit system where each failed unit is scrapped without repair and each spare is only provided after a lead time by an order. To seek an optimum policy minimizing or maximizing some criterion, the following policy is adopted: order for a spare is made at a prespecified time instant t o during an operating period of an original unit which is called a regular order. However, if the failure of the unit takes place before that instant, emergency order is made at the failure time instant immediately. After a lead time the spare is delivered and the original unit is in operable condition or otherwise. The planning horizon is infinite and the original unit begins operating at time 0. If the original unit does not fail up to a pre-specified time instant to ~ [0, ~-~), regular order for a spare is made at time to. After a regular lead time L2, the spare is delivered and the original unit is replaced by the spare
(a) The system consists of a one-unit system whose life-time distribution is F(t) with the density function f ( t ) = a + bt (say) linear deterministic lifetime. (b) If the unit fails before the time to, an emergency (expedited) order is placed at the instant of failure which has a lead time L x. (c) If the unit is good at t o, a regular order is placed at t which has a lead time L2, different from L 1. (d) A constant cost C1 per unit time is suffered for the system downtime, a cost C 2 is suffered for the expedited order which is made at the instant t before the instant t o and a cost C3 is suffered for the regular order made at the time instant t o . Also LI <~ L 2, C2 > C3. Under these assumptions, which are reasonable, an interval which is defined as the beginning of the original unit replacement to the next replacement as one cycle, the expected cost per cycle is derived (see Fig 1).
3s:7-F
847
848
Technical Note to •-v- ~
oI
Let C be the cost incurred in a cycle. T h e n the expected cost per unit time in the steady-state is
i
m
--|
!
I
L1
O~t~to
{[
C(to) = C1 (Lt - L2)F(to) +
oI
f--~ t0
f
F(t) dt
~ to
]
l/J//f//-ff
to [91"-'-'-- L]
oI
---j t0+L2~t
r//i//ll///1
N o w let there be a deterministic linear lifetime for the unit (say) f (to) = a + bt o. Substituting f (to) = a + bto in e q u a t i o n (4) one obtains
Failure Replacement ~ ' ] State of operation 1 State of expedited order with lead time L 1 State of regularorderwith lead time L2 -- State of failure X
C(to)={CtILlat °
fLlb'~2
Fig. 1. Possible realizations of one cycle.
4. COST EQUATIONS AND LINEAR LIFETIME FOR THE UNIT In this model the following expected costs are considered. (a) W h e n the original unit fails, if no spare is available, the system is u n d e r failure until the spare is delivered. The expected cost during t h a t period is Ct
[foo
L1 dF(t) +
(to + L2 - t) dF(t)
]
+
(5)
Hence, if there is any o p t i m a for C(to) one wish to select the t o value so as to minimize C(to). A necessary condition is that
C'(to)
= 0 which leads to
o
=C,[(L1-L2,F(to)+f~y+L2F(t)dt].
-(~)t3}{Cl[Lla+L~bt°+(;)
(b) The expected order cost is C2 dF(t) +
C 3 dF(t) =
C2F(to) +
+
C3ff(to)
o
(2)
(t + L 1) dF(t) +
f;
L2 ]
c3 ,o b,o,t fc,ILlatO f Llb~ 2
where if(to) = 1 - F(to). Moreover, the m e a n time of one cycle is
I:°
tg
(1){(L1--L2)(at°+(b2)tg)+L2+t°-(~)
+(C2- C3)(ato+(~)t2)+C3}
(t o + L2) d r ( t )
o
= (L1 - L2)F(to) + L2 +
;:°
if(t) dt.
(3)
(6)
Table 1
(a) (b) (c) (d) (e) (f) (g) (h)
LI
L2
C~
Cz
C3
a
b
5 3 5 4 3 4 3 4
8 4 8 5 5 5 5 6
0.8 0.5 0.4 0.6 0.6 0.6 0.6 0.5
50 25 30 20 25 20 25 50
30 15 20 10 15 10 15 40
3 15 3 12 6 6 7 10
55 10 10 7 3 2 2 17
Hence for t* = 32.44595 the objective value is
C(t*) =
0.3054266.
to 13.00725 14.11132 14.37633 20.01839 21.82158 30.03586 32.44595 50.05816
C(to) 1.4886780 0.7244133 0.6505462 0.4814463 0.4492928 0.3199690 0.3054266 0.3798361
Technical Note E q u a t i o n (6) can be solved by some iterative technique, for example, the bisection m e t h o d can be used.
5. NUMERICAL ILLUSTRATION F o r the various costs, lead times a n d linear deterministic lifetime for the unit Table 1 gives the relevant t o against C(to).
849 REFERENCES
1. R. E. Barlow and F. Proschan, Mathematical Theory of Reliability. Wiley, New York (1965). 2. R. E. Barlow and F. Proschan, Statistical Theory of Reliability and Life-Testing: Probability Models. Holt, Rinehart and Winston, New York (1975). 3. A. D. Wiggins, A minimum cost model of spare parts inventory control, Technometrics 13, 139-144 (1967). 4. N. Kaio and S. Osaki, Ordering policies with two types of lead times, Microelectron. Reliab. 16, 225-229 (1977).