CC BY
17
RELIABILITY AND SENSITIVITY ANALYSIS OF A SYSTEM WITH CONDITIONAL AND EXTENDED
WARRANTY
Kajal Sachdeva
Department of Mathematics, Maharshi Dayanand University, Rohtak, Haryana, India
kajal.rs.maths@mdurohtak.ac.in
Gulshan Taneja
Department of Mathematics, Maharshi Dayanand University, Rohtak, Haryana, India
drgtaneja@gmail.com
Amit Manocha* •
Department of Applied Sciences, TITS Bhiwani, Haryana, India amitmanocha80@yahoo.com
Corresponding Author Abstract
System reliability and maintenance cost are the most crucial and decisive factors influencing consumers' buying behaviour. The manufacturer attempts to address the consumer concern by offering a warranty in accordance with the reliability cf the system and maintenance costs. This study aims to examine the stochastic behaviour of a single unit system operating in three different time frames, namely normal, extended and expired warranty time duration. The system user can prolong the normal warranty period at an extra cost. This prolonged warranty is termed an 'Extended Warranty'. However, the manufacturer provides a warranty on a system with certain conditions. If the failures are covered under the warranty conditions, the repair/replacement is done free of cost; otherwise, all charges are borne by the system user. Markov and regenerative processes are used to derive the system's reliability and other performability measures. Time distributions used in the study are taken as arbitrary. The profit function for the manufacturer and the user is formulated and analysed. Sensitivity analysis for system availabilities in different time zones and profit functions is also done. Numerical examples for exponential, Weibull and Erlang time distributions are discussed to illustrate the derived measures.
Keywords: Reliability; Extended warranty; Regenerative process; Profit function; Sensitivity analysis.
1. Introduction
The recent advancement in various technological aspects has paved the way for endless technologies and innovations to hit the mainstream, forcing manufacturers to offer a gamut of consumer options. Despite the numerous advantages of technological development, the flip side is the system's complexity. Consequently, the consumers are apprehensive of system reliability which may adversely affect the sales of the developed product. Hence, in the pursuit of ensuring system
reliability and addressing the consumer's concerns and dilemmas, the manufacturer offers a warranty and extended warranty. System reliability and warranty may be considered as interrelated concepts. Warranty is the written agreement provided to system users for cumulative product acceptance without any strain of product manufacturing or faults. Ives and Vitale [1], Ritchken et al. [2], Singpurwalla and Wilson [3] proposed that the warranty on various products caters to risk reduction, quality bench-marking and enhanced market competitiveness. Different types of warranty policies were introduced to optimize profit. Free Replacement Warranty, Full-Service Warranty and Renewing Pro-Rata Warranty were described by Blischke, and Murthy [4], Jain and Maheshwari [5], Bai and Pham [6] respectively. Huang et al. [7] discussed the future problems and challenges in reliability and warranty. Kadyan and Ramniwas [8] proposed a probabilistic model for a single-unit system protected by warranty conditions. Rahman and Chattopadhyay [9] developed cost models on long-term/service contract policies. Pham and Bai [10] discussed warranty costs, compared different warranty policies and evaluated the warranty benefits. The warranties offered to consumers can safeguard them from exorbitant maintenance costs considering the ever-increasing maintenance cost resulting from advancements. A conditional warranty covers the cost associated with defects stipulated in the agreement at the time of purchase. Taneja [11] developed a stochastic model in which repair/replacement is done by the manufacturer on predefined warranty conditions. Lei et al. [12] discussed the characterization of warranty price policies and optimized the product price, which is profitable to the user. Further, Solkhe and Taneja [13] considered the system with conditional warranty and compared the performability measures before and after the expiry of warranty periods. Niwas [14] analysed a warranted system with waiting time for repair and cost paid by the user if failures happen due to unauthorised modifications. Hooti et al. [15] optimised the warranty duration and the repair. Another type of warranty, known as an extended warranty, is worth considering these days. There are systems used in daily life for which manufacturers, dealers or third parties provide extended warranties, such as home appliances and electronic equipment, particularly in the automotive industry. Though optional, it offers users a sense of security regarding system reliability, maintenance costs, etc. Suiter and Lorson [16] described the pros and cons of an extended warranty for the system user. Huang [17] considered a system with minor, degraded and catastrophic failures where warranty was provided for either one year at a fixed lump sum price or monthly warranty plan which may be extended for another month. Salmasnia and Hatami [18] considered a model with an extended warranty where the failures are controlled using technology and non-periodic maintenance activities.
However, stochastic modelling of a system with prolonged conditional warranty and identification of key parameters influencing the most on the manufacturer/user profit is yet to be reported in the literature. So, the objective of our study is to stochastically analyze a system functioning in normal, extended and beyond warranty periods with a focus on its reliability characteristics and economic viability. The study also identifies the parameter that significantly impacts system profitability from the manufacturer and user perspectives. This article is structured as follows. System description and assumptions made are described in Section 2. Various notations used in the study are cited in Section 3. A probabilistic model for the described system is developed in Section 4. Transition probabilities and mean sojourn times are also evaluated in this Section. Expressions for reliability indices and metrics impacting the system's profitability are derived in Sections 5, 6 and 7, respectively. In Section 8, the profit functions for the manufacturer and the user are established. Section 9 focused on the sensitivity analysis of availabilities and profit functions. To illustrate the developed model, numerical examples for different density functions are discussed in Section 10. Conclusions regarding reliability, system availabilities, the profitability of the manufacturer/user and their sensitivity are also drawn in this Section. Finally, Section 11 provides several concluding insightful interpretations.
2. System Descriptions and Assumptions
Descriptions for the system under consideration and assumptions made for the analysis are as follows:
1. System has a single unit that operates in three different warranty periods named normal, extended and expiry warranty periods.
2. The failed system is inspected by the manufacturer or external source to ensure
(a) Whether faults occur in a system comes under warranty claims or not.
(b) Whether the system is repairable or needs replacement.
3. If an inspection reveals that a fault falls inside the purview of a normal or extended warranty conditions, the manufacturer is liable for paying the cost of repair/replacement. Otherwise, all the charges are borne by the user itself.
4. In the expiry (beyond warranty) period, the user manages the expenditure for repair/replacement.
5. Transition time distributions have been taken general.
Figure 1: System description.
Figure 1 shows the description of the system. Markov and regenerative processes are applied to develop a stochastic model for the system defined. Mathematical expressions for reliability, Mean time to failure (MTTF), availabilities, expected busy period of repairman, and the number of replacements are derived. Profit functions are formulated. Sensitivity analysis is also done for availabilities in three different time zones and for the profit function of the manufacturer and the user. Exponential, Weibull and Erlang distributions are used for numerical calculations. Various conclusions on reliability indices, profitability and sensitivity, are drawn.
3. Nomenclature
The notations for various probabilities/transition densities are:
WITH CONDITIONAL AND EXTENDED WARRANTY Volume 18, September 2023
So state of system at t=0
© symbol for Laplace transform
p/q probability that fault is covered/not covered by warranty terms.
pi/p2/p3 probability that system gets failed during normal/extended/
expired warranty period. r1/r2 probability that fault is repairable/ irreparable and only to be replaced.
f (t) p.d.f. of failure time.
im (t)/iu (t) p.d.f. of inspection time by the repairman engaged by manufacturer/
user itself.
gk(t)/hk(t) p.d.f. of repair/replacement time within warranty period 'k=n/et/ex'
Ik (t) P{repairman of manufacturer is engaged in inspection at instant t | S0 = i
within warranty period 'k'}.
BMk(t)(BUk(t)) P{manufacturer repairman is occupied with repair/replacement when the associated cost are to be paid by the manufacturer (user) at instant t | S0 = i within warranty period 'k'}.
RMk(t)(RUk(t)) expected number of replacement in (0,t], when expenses are met by manufacturer(user) | S0 = i during warranty period 'k'.
where k denotes normal(n), extended(et), expired(ex).
4. Stochastic Model
The probable states of described system are:
State 0 State 3 State 6 State 9
(Onw)j
(F(m) v \rrpn )l
(Fiet ); (F(m) );
(Fret v'
State 1 State 4 State 7
(Fin V' (F(u))'
rn
(F(u) )'
(Fret )'
State 2 State 5 State 8
State 12: (Fiex )' State 15: (Oexw)'
where,
Okw Fik
F(m)(F(m)) /F(m)(F(m)) Frn (Frpn )/ Fret (Frpet)
(u) /F(u)
State 10: (F^ )' State 13: (F^ )'
(F(m));
rn (F(u));
rpn (F(u) );
rpet
State 11: (Oetw )' State 14: (F^ )'
(u)
Frk / Fr
rpk
operative system in warranty period 'k'. failed system under inspection in warranty period 'k'. failed system under repair(replacement) in normal/extended warranty period, for which expenses are to be borne by manufacturer.
failed system under repair/ replacement in warranty period 'k', for which charges are to be borne by user itself.
Here, k denotes normal(n), extended(et), expired(ex).
By employing markov and regenerative process, the transition between various states is represented by Figure 2. The state space constitutes the set of regenerative states i.e., S={0,1, 2, ..., 15}, where 0={0,11, 15} is operative, and F={1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14} is failed state space respectively.
The transition densities from state i to state j (Qj(t)) are:
Q01W =/odF(u), Q12 (t) = pn j'^dlm (u), Q13(t) = pr2jodIm (u),
Q14(t) = ^1/0 dIm(u), Q15 (t) = dIm (u), Q20(t) = /0 dGn (u),
Q30(t) = SodHn (u), Q40 (t) = /¿dGn (u), Q50(t) = /0 dHn (u),
Q06(t) = /0f dF(u), Q67(t) = qr1 /0 dIm(u), Q68(t) = q^J^dIm(u),
Qf&(t) = p/ /g dim(u), Q6,io(0 = pr2 /J dlm(u), Qs,ii (t) = Qo,12 (t) = Ql2,13 (t)
Ql4,15 (t) = f0 dHex (u), Qi5,i2(t) = /0 dF(u).
(■t
n0 dHet(u), Qg,ii(t) = /0 dGet(u), /0 dF(u), Qii,i2(t) = fo dF(u), = /i/o diu (u), Qi2,i4 (t) = /2 /0 diu (u),
Q7,ii (t) = JodGet (u),
Qi0,ii (t) = Jo dHet(u), Qii,6 (t) = JodF(u), Qi3,i5 (t) = Jo dGex (u),
I Operative State Regenerative Point
Figure 2: State transition diagram Thus, transition probabilities from state i to j are:
dQij (t)
pij = slim til(s), where qij(t) = dt '
These probabilities follow the property of transition probability matrix for each measure of system effectiveness'
Mean sojourn time in state i are:
/•œ /*œ_
uo = tf (t)dt = F(t)dt. J 0 Jo
Similarly,
Hi = /•œ im (t)dt, Jo H2 = /•œ Gn (t)dt, J H3 = /•œ Hn (t)dt, J H4 = /•œ Gn (t)dt, J
H5 = !• œ Hn (t)dt, J H6 = !• œ Im(t)dt, J H7 = !• œ Get (t)dt, J Hs = !• œ Het (t)dt, J
H9 = f œ Get (t)dt, J Hio = f œ Het (t)dt, J Hii = f œ F(t)dt, J Hi2 = f œ Iu (t)dt, Jo
Hi3 = !• œ J Gex (t)dt, Hi4 = !• œ J Hex (t)dt, Hi5 = !• œ J F(t)dt.
Kajal Sachdeva, Gulshan Taneja, Amit Manocha
RELIABILITY AND SENSITIVITY ANALYSIS OF A SYSTEM RT&A., N°.3 (74) WITH CONDITIONAL AND EXTENDED WARRANTY_Volume 18, September 2023
Defining mij = E(qij(t))= J0°° tq^(t)dt, we have
m0i = J tq0i (t)dt = J tf (t)dt = p0. Similarly, we get the following relations
mo6 = mo,i2 = V0, m20 = V2, m30 = ^3, m40 = ^4,
mso = P5, m7,ii = P7, ms,ii = ps, m9,ii = ^9,
m10,11 = Цl0, mii,6 = mii,i2 = ViV mi2,i3 + mi2,i4 = mi3,i5 =
mi4,i5 = Vi4, mi5,i2 = Vi5, mi2 + mi3 + mi4 + mi5 = pi,
m67 + m6s + m69 + m6,i0 = V6.
The reliability and performance-indicating characteristics of the system are determined in the
following section.
5. Reliability (R(t)) and Mean Time to Failure (MTTF) Theorem 1. R(t) and MTTF are given as
R(t) = F(t), MTTF = ^Q.
Proof. Let JQ (t)=P[system is operative until time t | SQ = 0], then using probabilistic arguments, it can be seen from transition diagram, that
Jo (t) = pi Q01 (t) + p2 Q06 (t) + ps Qo,12 (t). (1)
The expression on R.H.S. of equation (1) shows the system transit from state 0 to failed state 1 or 6 or 12, with probability p1 Q01(t), p2Q06(t) and p3Qq,12(t) respectively in time t. Taking Laplace-Stieltjes transformation of the above equation, we get
J0* (s) = p1 Q00 (s) + p2 Q06 (s) + ps QOO12 (s). (2)
Thus,
R(t) = L-1 [ i-Jp> ]
= L-1 [1 - (p1QQ0 <s> + p2QQ0 <s> + psQQ*12(s)) ] |Subsfitufi'ng(2)]
L-1 [i] - piL— [^M] - p2L_i[^S^j - P3L-i[
i - p^ dF(n) - p2 i dF(n) - p3 / dF(M) [Using transition densities] (3) 000
i - (Pi + P2 + P3df(n) i 'dF(u) (v pi + p2 + p3 = i) i - ^V(u)du = i - F(t) = F(i).
and
MTTF = lim[i-^l] [0] s^0 s 0
-00* (0). (4)
Differentiating eqn (2) both sides w.r.t. s and then taking lim s ^ 0, we get
ro * (0) = P1 Qoi (°) + P2 Qo6 (0) + P3 Q0,*12 (0) = —
(5)
From eqn (4) and (5),
MTTF=^0.
6. System Availability
■
Theorem 2. The Laplace transformation of point-wise availability during extended warranty
period is given by
Aef* A0
(s)
Nf* (s) Df* (s) '
where
Nf * (s)=
M* (s) —q*6 (s) 0
M* (s) = M*i (s)
1 — q*7(s) —q*8(s) — q*9(s) —<10(s)
M*i(s) —q*1,6 (s)
1—f * (s)
0 0
—q*,11(s)
—q*,11(s)
—q*,11(s)
—q*0,11(s) 1
Df* (s)=
1 —q*6(s) 0 0 0 0 0 1 —q*7(s) —q*8(s) —q*9(s) —q*,10(s)
0 —q*1,6(s)
0 0
—q*,11(s)
—q*,11(s)
—q*,11(s)
—q*0,11(s) 1
The system steady-state availability is given as
A0f
Nf
Df '
where
Nf
m,
and
D1f
m + U6 + + + ^9 pr1 + ^0
0
0
0
s
Proof. Considering A0f(t) = P(system is operative at time t in extended warranty period | So = 0), then from transition diagram, we have
_ r t
A0(t) = F(t) + J qo6(u) At(t - u)du
= Mo(t) + qot(t) © Aet (t) (6)
where M0 (t) represents that the system remains operative in state 0 instead of moving to any other state. The term q06(t) denotes the system transition probability from state 0 to state 6 in time u<t and thereafter remains operative from state 6 onwards for t-u time. Similarly,
A6(t) = q67(t) © A?(t) + q68(t) © A$(t) + q69(t) © Aft (t) + q6,i0(t) © A* (t), A7 (t) = q7,ii (t) © All (t),
A8(t) = qs,ii (t) © All (t), (7)
A9(t) = q9,ii (t) © Aii (t), ()
AeUt) = qi0,ii (t) © Aii (t), Ah(t) = Mu(t)+ qnfi(t) © A6(t).
Taking Laplace Transformation of eqn(6)-(7) and solving them for A0* (s) by method of determinants, we get
\et*f„\ _ Li(s)
A0 * (s)
Mi (s)'
Li(s) = M0 (s) + Mii(s)q06(s)q67(s)q7,ii (s) + Mii (s)q*06(s)q*68 (s)q8,ii(s) +M*i (s)q*06(s)q*69 (s)q9 ,ii(s) + M*i (s)q06 (s)q6,i0 (s)q**0 ,11
(s)
-M*(s)q*67(s)qii,6(s)q7,ii(s) - M*(s)q'68(s)qh,6OOq^iiOO -M*(s)q*69(s)qii,6(s)q9,ii(s) - M* (s)qii,6(s)q6,i0(s)q*0,ii(s) = Nf * (s) (8)
Mi(s) = i - q68(s)q*i,6(s)q8,ii(s) - q69(s)q* i,6(s)q9,ii00 - q* i,6OOq6,i000q*0,ii(s) -q67 (s)q*i,6(s)q7,ii(s)
= Df * (s) (9)
Using Abel's lemma, the system's steady state availability is
lim sAet* (s) = Nit * (0) = si0 0 w Di*' (0) Df
A* = lim sA0 * (s) = = Nit, (i0)
Differentiating eqn (9) w.r.t. s,
Di *'(s) = qhfo (s)(-q67(s)q7,ii(s) - q*68(s)q*8,ii00 - q*69(s)q9,ii(s) - q6,i0(s)q*0,ii(s)) -q67'(s)q7,ii(s)q* i,6(s) - q68(s)q8,ii(s)q* i,6(s) - q69(s)q'9,ii(s)qh,6(s) -q6,i0'(s)qi*0,ii(s)q*i,6(s) - q7,ii'(s)q* i,6(s)q6,7(s) -q8,ii'(s)qii,6(s)q6,8(s) - q9,ii'(s)qii,6(s)q6,9(s) -qi0,ii' (s)q* i,6 (s)q6,ii(s)' (ii)
Setting lim s i 0 in eqn (8) and (ii), we obtain
< = pii, (i2)
Df = pn( pri + pr2 + qri + qr2) + pri pg + m69 + p^pio + m6,io + qrp +m67 + m68 + ps qr2
= Pii + P6 + P9 pri + PiQ pr2 + P7qri + ps qr2. (i3)
Similarly, availabilities during normal and expired warranty period are given as
An A0
N?
D? '
/1 ex A0
Nf
Df '
■
(14)
where
N? = po, D? = Po + P1 + P2 pri + P3 pr2 + P4 qri + ps qr2, Nf = P15, Df = P15 + P12 + P13ri + Pi4f2'
(15)
7. Expected Busy Period and Number of Replacements
Employing the definitions of BUk, BMk and ik, i £ S (defined in Section 3) and follow the same probabilistic arguments as discussed in preceding Section 6, the expected time for which repairman remain involved in repair/replacement/inspection of a failed system in different warranty zones, in steady-state given as:
BUok = D ; BMok = N ; 1o = N4 ; k=n,et,ex.
N? = q(P4r1 + P5r2); N2f = q(P7r1 + Ps r2 ); Nf = P13r1 + P14 r2;
N? = P(P2 r1 + P3r2); N3f = p(P9r1 + P1or2); N4n = P1; Nf = P6; Nf = P12;
Further, by definitions of RUk and RMk,i G S (defined in Section 3), in steady state, the expected number of replacements during different warranty time zones are:
i N , Nk
RUk = N ; RMok = Nf ; k=n,et,ex
N5n = qr21 N5et = qr2;1 N5ex = r2;
N? = pr2; N6f = pr2;
D", Df and Dix are mentioned in eq" (i3) and (i5).
Focusing on economic viability of defined system, cost-benefit analysis is performed in the succeeding section. Profitability analysis assists both the manufacturer and the user in categorizing the parameters which may cause long-term loss.
8. Cost-Benefit Analysis
To carry out a cost-benefit analysis, the profit function is defined for the manufacturer and user. Mathematically, the profit function for a system is the difference between total revenue generated and total expenditure incurred in a given period. So, in steady-state, the profit function for the manufacturer and user is formulated as follows: Manufacturer Profit
Pm = CP + EC - MC - Cf (p1 i? + P2i0f ) - Cf (p1 BM? + P2BM0f )
-Cm ( p1 RM? + p2 RM0f ). (16)
User Profit
Pu = C0(p1 A? + p2A0f + p3A0x) - CU(p310x) - C2"(p1 BU? + p2BU0f + p3BU0x)
-CU(p1 RU? + p2RU0f + p3RU0x) - CP - EC. (17)
where,
CP=Cost price of the system MC=Manufacturing cost of the system EC=Cost for extended the warranty period. C0=Revenue generated by system.
Cf (CU)= Cost of engaging the repairman by the manufacturer(user) for inspection. Cf (Cf )=Cost of engaging the repairman by the manufacturer(user) for repair/replacement. Cf (Cf )=Cost per replacement of system borne by manufacturer(user). The above-mentioned costs are considered per unit of time.
9. Sensitivity Analysis
Sensitivity analysis is a concept that determines as to which parameter (independent variable) the obtained measures (dependent variable) are highly or least affected. Relative sensitivity analysis is used to assess the impact of different parameters because the numerical values for various parameters differ significantly. A normalised form of a sensitivity function is known as relative sensitivity function. Using the eqns (i0), (i4), (i6) and (i7), the sensitivity (Dk, Zyy) and relative sensitivity functions (djy, zp for availabilities (A", AQt, AQx) and profit functions (Pm,Pu) respectively are defined as
DÎ
d(a0).
dy '
dy
Dk ;
Ak;
k = et, ex.
(18)
and
zy
d(Ps), ay ;
zyy ;
ps '
s = u, m.
(19)
z
i0. Results and Discussions
In this section, measures obtained in Sections 5-9 respectively are illustrated via numerical examples.
Expressions for various measures have been derived using general probability time distributions. The system user may choose specific values for the parameters involved to illustrate the model based on records of failures, repairs, costs, and available probabilities. The particular distribution can be identified by applying the appropriate test to such data. Because real data on failures, repairs, costs, etc., could not be collected in our study, we used exponential, Weibull and Erlang distributions and assumed values for parameters involved to illustrate the model numerically.
i0.i. Example i:All the Time Distributions follows Exponential Distribution
Assuming failure/inspection/replacement/repair times are exponentially distributed with their p.d.f. given as:
f (t) = Ao e-Aot, im (t) = 7me-Ymt, ¿„ (t) = 7«e-7ut, h„ (t) = ^e-^,
het(t)= pete-l}ett, gn (t) = *n.e-Knt, get(t) = aefe-aetf, gex(t) = aexe-aext, (20)
hex (t) = |8exe-^ext.
Consider the value of parameters as
p = 0.7, q = 0.3, p1 = 0.2, p2 = 0.3, p3 = 0.5, r1 = 0.7, r2 = 0.3, A0 = 0.0005,
Ym = 1.5, Yu = 1.2, an = 0.5, ad = 0.4, = 0.02, fa = 0.02, aex = 0.25,
fiex = 0.01, CP = 150, EC = 15, MC = 120, C0 = 500, Cm = 80, C2m = 100,
Cm = 15,000, Cu = 100, CU = 120, CU = 15,000. (21)
10.1.1 Effect of Time (t) and Failure Rate (A0) on Reliability Measures
Taking the values of other parameters constant as mentioned in eqn (21), the effect of parameters (t, A0) on reliability function (R(t)) is shown in Figure 3.
As both the parameters t and A0 increase, R(t) decreases. Further, Table 1 represents the values of availabilities (A^, A0t,A0x) in three different time zones for varied A0. All three availabilities decrease with an increase in A0. Also for any particular A0, the availabilities satisfy the relation An > A0 > Ae0x. In other words, during the expired warranty period, the system's availability is much influenced by A0 as compared to the other time zones.
t
Figure 3: Reliability w.r.t varied t and Aq
Table 1: Value of Availability for varied Aq
Availability
Aq An A0 AQt A ex A0
0.001 0.9832 0.9829 0.9675
0.002 0.9670 0.9663 0.9370
0.003 0.9513 0.9503 0.9083
0.004 0.9361 0.9349 0.8814
0.005 0.9214 0.9199 0.8560
0.006 0.9071 0.9054 0.8321
0.007 0.8933 0.8913 0.8094
0.008 0.8799 0.8777 0.7880
0.009 0.8669 0.8645 0.7676
0.010 0.8542 0.8517 0.7483
10.1.2 Effect of Various Rates / Costs on Profit Functions
The outcomes of profit functions, Pm for varied (EC, MC), (CP, p) and Pu for varied (CQ, C3), (A0, ri) are studied. The other parameters are kept fixed, and their values are taken as in eqn(2i). The results obtained are represented by Figure 4, 5, 6, 7 and Table 2 respectively and summarised as follows:
1. Pm goes down as MC and p increases but hike in its value is observed when EC and CP increases.
2. As the parameter CQ increases, Pu increases. Moreover, the rise in the values of A0, ri, C3 respectively, results in the decreasing Pu.
Figure 4: Variation in Pm for varied MC and EC
Figure 5: Variation in Pu for varied Cq and C^
3. The system should remain profitable for user as well as manufacturer.
Figure 6: Variation in Pu for varied A0 and r1
Figure 7: Variation in Pm for varied CP and p
Keeping that in mind, the bounds for some of the parameters are evaluated as:
(a) For EC=25, Pm > 0 iff MC < 175
(b) For Cu = 2000, Pu > 0 iff C0 > 168
(c) For r1=0.1, Pu > 0 iff A0 < 0.038
(d) For p=0.1, Pm > 0 iff CP > 105.112
Bounds for other values of EC, C3u and r1 are also mentioned in Table 2.
Table 2: Bounds for Revenwe/Cosf/Rafe
Cost/Revenue/Rate Varied Bounds For
Parameter Profitability(Pw/Pm >0)
EC=25 MC<175
MC EC=50 MC<200
EC=75 MC< 220
Cg=2000 C0 >168
C0 Cw=10000 C0 >169
Cf=20000 C0 >170
ri = 0.1 A0 < 0.038
A0 ri = 0.5 A0 < 0.039
r1 = 0.9 A0 < 0.040
p = 0.1 CP>105.112
CP p = 0.5 CP>105.784
p = 0.9 CP>106.396
10.2. Example 2: Failure Time follows Weibull Distribution
Assuming the failure time follows Weibull distribution with p.d.f.
f (t) = n.(n " ^ (22)
where 5 and n are shape and scale parameters respectively.
All the other time distributions follows exponential distribution with same p.d.f. and numerical value as taken in Section 10.1. Figure 8 reveals that the reliability decreases with time and for t<200, it is on the higher side for higher values of n and 5. However, the reverse trend of its values for n and 5 is noticed for t>200.
Taking 5=0.3 and n= 150, the behaviour of A£, A0f w.r.t Ym and A0x w.r.t yw is shown in Table 3. The increasing trend of availabilities with an increase in yw and ym respectively are observed.However,the system availability (A0x) during expired warranty period is lesser as compared to An, A0f for any particular value of Yw(=Ym).
t
Figure 8: ReZzabz'Zzfy w.r.f. waned f,n and 5
Table 3: Value of Availabilities for varied ym/yu
Availability
Ym/Yu A OS Aet A0 ex A0
0.1 0.9813 0.9811 0.9701
0.2 0.9848 0.9846 0.9735
0.3 0.9860 0.9857 0.9746
0.4 0.9866 0.9863 0.9752
0.5 0.9869 0.9867 0.9756
0.6 0.9872 0.9869 0.9758
0.7 0.9873 0.9871 0.9760
0.8 0.9875 0.9872 0.9761
0.9 0.9876 0.9873 0.9762
1.0 0.9876 0.9874 0.9762
10.3. Example 3: Repair Time follows Erlang Distribution
Considering the repair time in different warranty time follows Erlang distribution with p.d.f.
/ \ _ pc xkn-1 e-Qnx
gn (x) — Qn. (kn-1)!
get (X)- Qet. (23)
g (x) — Q xkcx-1.e-QexX k gex (x) — Qex. (kex-1)!
where
kn/ket/kex and Qn/Qet/Qex are shape and scale parameter during normal/extended/expired warranty period,
All of the other time distributions have the same p.d.f. ,i.e., exponential distribution and numerical values as in Section 10.1.
Table 4: Value of Availabilities for varied Xo
Availability
Xo Aon ao Aeox
0.0001 0.9907 0.9396 0.9383
0.0002 0.9817 0.8861 0.8837
0.0003 0.9727 0.8383 0.8351
0.0004 0.9640 0.7955 0.7916
0.0005 0.9554 0.7568 0.7524
0.0006 0.9469 0.7216 0.7169
0.0007 0.9386 0.6897 0.6846
0.0008 0.9304 0.6604 0.6551
0.0009 0.9224 0.6335 0.6280
0.001 0.9145 0.6087 0.6031
Taking Qn=Qet=Qex=0.5 and kn=ket=kex=7, the behaviour of An, A0t and A0x w.r.t \o is shown in Table 4. The decreasing trend of availabilities with an increase in Xo is observed.
10.4. Numerical Calculations for Sensitivity Analysis
Considering all the p.d.f. involved as exponential as taken in eqn (20) and assuming the values of parameters as mentioned in eqn (21), the sensitivity analysis is performed. Using eqn (18) and (19), the outcomes for the sensitivity and relative sensitivity functions of availability and profit
functions are summarized in Table 5, 6 and 7 respectively. The magnitude of these functions is taken into account while drawing inferences about parameters and the order in which they influence the different measures.
Table 5: Sensitivity and relative sensitivity analysis of Availabilities
Parameter Sensitivity Function Relative
Sensitivity Function
(y) Dy = dy d = Dyy
Normal Warranty Period
Ao -16.7791 -0.0085
Tm 2.1848 x 10-4 3.3052 x 10-4
an 0.0014 7.0597 x 10-4
ßn 0.3687 0.0074
P -0.0081 -0.0057
q -0.0081 -0.0025
ri -9.8315 x 10-4 -6.9408 x 10-4
-0.0246 -0.0073
Extended Warranty Period
Ao -16.1172 -0.0086
Tm 2.1840 x 10-4 3.3045 x 10-4
Kef 0.0021 8.4732 x 10-4
ßet 0.3686 0.0074
P -0.0082 -0.0058
q -0.0082 -0.0025
ri -0.0012 -8.4732 x 10-4
r2 -0.0246 -0.0074
Expired Warranty Period
Ao -32.54 -0.0166
aex 0.0053 0.0015
ßex 1.4507 0.0147
Yu 3.3582 x 10-4 4.0978 x 10-4
ri -0.0019 -0.0014
r2 -0.0484 -0.0148
It has been determined that
1. Availabilities (An, A0t, A0x) in three different periods are highly influenced by Ao. Though these are least affected by variation in yu and ym respectively.
2. Pm and Pu both the profit functions are extremely sensitive to Ao.
3. Variation in CU and Cm results in a nominal change in Pm and Pu respectively.
Table 6: Sensitivity and relative sensitivity analysis of User Profit
Parameter Sensitivity Function Relative
Sensitivity Function
(y) 7u_ d(Pu ) Zy _ dy 7u _ 7Uy ¿y— pu
Ao -1.7455 x 104 -0.0268
aex 1.6730 0.0013
ßex 448.1202 0.0137
Yu 0.1003 3.6900 x 10-4
Ym 0.0545 2.5063 x 10-4
an 0.1474 2.2597 x 10-4
ßn 39.4737 0.0024
Uet 0.3453 4.2349 x 10-4
ßet 59.1900 0.0036
Co 0.9874 1.5137
u C1 -2.0489 x 10-4 -6.2816 x 10-5
u C2 -0.0093 -0.0034
u C3 -9.6069 x 10-5 -0.0044
CP -1 -0.4599
EC -1 -0.0460
P -2.0369 -0.0044
q -3.6463 -0.0034
ri -0.9001 -0.0019
T2 -26.3183 -0.0242
Pi 494.8074 0.3034
P2 494.7154 0.4551
P3 487.5415 0.7474
Moreover, the order or sequence in which different parameters influence the availabilities (Afi,
, A0x ) and profit functions (Pm,Pu ) are
• Availability(): A0 >fi n >r2> p>q>an >r1>Ym •
• Availability(A0): A0>fiet>r2>aet>p>q>r1>Ym•
• Availability(A0x): Ao>fiex>r2>Uex>ri>Yu•
• Profit Function(Pu):Co>p3>CPp2>pi>>EC>Ao>r2>fiex>CUp>>fiet>
CU >q> fin >ri >aex >aet >Yu >Ym >an >CU.
• Profit Function(Pm):CP>MC>EC>A0>p>r2>Cf>p2>p1 >C2m>fiet>Cm>
fin >ri >IXet >ttn >Ym >q.
Table 7: Sensitivity and relative sensitivity analysis of Manufacturer Profit
Parameter Sensitivity Function Relative
Sensitivity Function
(y) Zm _ 3(Pm) Zy _ dy 7my zm _ Zyy y pm
Ao —2.1458 x 103 -0.0244
Tm 0.0086 2.9373 x 10—4
an 0.0188 2.1404 x 10—4
ßn 5.0452 0.0023
Kef 0.0441 4.0166 x 10—4
ßet 7.5651 0.0034
nm C1 — 1.6524 x 10—4 —3.0100 x 10—4
nm C2 -0.0029 -0.0066
nm C3 —5.2050 x 10—5 -0.0178
CP 1 3.4155
EC 1 0.3415
MC -1 -2.7324
P -1.5181 -0.0242
q -0.0089 6.0795 x 10—5
ri -0.0387 —6.1683 x 10—4
r2 -3.4432 -0.0235
Pi -2.1573 -0.0098
P2 -2.1690 -0.0148
11. Conclusion
A stochastic model of system functioning in normal, extended and expiry warranty conditions is developed in this paper. Markov and regenerative processes are employed to derive various reliability characteristics and profit functions for the manufacturer as well as the user of the system. The derived measures are further illustrated by discussing numerical examples for exponential, Weibull and Erlang cases. System is found available for a longer period in normal as compared to extended or expiry warranty periods. Upper/ lower bounds are obtained for involved rates/ costs, which can affect the system's profitability. Availabilities and profit functions are observed to be most sensitive to the failure rate. Further, for cost consideration, manufacturer and user profit functions are influenced most by the cost price of the system (CP) and revenue generated (Co), respectively. Since the results for a described system are obtained using general probability time distribution, the finding of the study is lucrative from the standpoints of both the manufacturer and the user if they have real data on failures, repairs, costs, and so on.
Funding
The first author delightedly acknowledges the University Grants Commission (UGC), New Delhi, India for providing financial support.
Disclosure statement The authors declare that they have no conflict of interest.
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