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52
Sensitivity and Economic Analysis of an Insured System with Extended Conditional Warranty
Kajal Sachdeva
Department of Mathematics, Maharshi Dayanand University, Rohtak, Haryana, India
Gulshan Taneja
Department of Mathematics, Maharshi Dayanand University, Rohtak, Haryana, India
Amit Manocha* •
Department of Applied Sciences, TITS Bhiwani, Haryana, India [email protected]
^Corresponding Author Abstract
Warranty and insurance are equally essential for a technological system to cover repair/replacement costs of all types of losses, i.e., natural wear/tear or unexpected external force/accidents. This paper examines the sensitivity and profitability of a stochastic model whose defects may cover under conditional warranty/insurance. The system user may extend the warranty period by paying an additional price. As a result, the system functions in normal warranty, extended warranty, and during non-warranty periods. If a system fault occurred is covered under warranty conditions, the manufacturer is responsible for all repair/replacement costs during normal/extended warranty which otherwise are paid by the insurance provider if covered under an insurance claim, or else, the user is responsible for the entire cost when coverage of fault neither falls in warranty conditions nor under the insurance policy. Using Markov and the regenerative process, various measures of system effectiveness associated with the profit of the user and the manufacturer are examined. Relative sensitivity analysis of the profit function and availability has been performed for all periods.
Keywords: Extended Conditional Warranty; Sensitivity Analysis; Profit; Insurance Cover
1. Introduction
Competitors add and offer new features to advertise their products in today's continuously expanding technological landscape. Offering a warranty on a system can be very beneficial to a company's growth. It relieves buyers' concerns, demonstrates the system's reliability, and is promotional. A warranty is a formal promise issued to the user for the free repair or system replacement if it fails. Researchers have focused their attention on warranty systems, policies, and warranty expense management in the past few decades [1-3,6]. Generally, warranties cover the cost of failures that are defined in the contract at the time of purchased. Taneja [12] described the reliability analysis of a system with predetermined warranty conditions. Further, this work has extended to warranty period and non-warranty period [9-11]. Many systems have enormous
maintenance costs and are operated for long periods. Manufacturers are offering the option of extending the warranty period with an additional charge to avoid the cost of repair/replacement for an extensive period. Jack and Murthy [4] proposed the idea of employing a game-theoretic technique to determine the length and duration of an extended warranty based on the consumer's risk attitude. Padmanabhan and Rao [7] estimated the basic warranty time to be three years, a smart option for the increasing demand for long-term service contracts. Rinsaka and Sandoh [8] discussed an extended warranty in which the manufacturer replaces the system after the first failure and only performs minor repairs on subsequent failures. They also examined the optimal pricing for such an extended warranty.
However, financial protection is provided by insurance and warranty both against unpredictable damage or loss. There is a thin strip between them. While insurance protects against unintentional damage or loss, warranty protects against defective parts. Purchasing both at the same time is likely advantageous since it provides peace of mind in knowing that including insurance covers accidental damages, the warranty will cover faulty parts, effectively including the majority of faults/accidents that emerge in a technical system. Lutz and Padmanabhan [5] investigated the impact of impartial and independent insurance providers on manufacturer price strategy. The cost analysis of an insured system with an extended conditional warranty is yet to be investigated. This paper proposes a model for a system with a conditional normal/extended warranty and long-term insurance which is structured as follows. Section 2 discusses the system's assumptions and description of the model. Section 3 describes the notations used in the analysis. Section 4 covers the system's stochastic modelling. The profitability measures and profit functions for the user, manufacturer, and insurance provider are drafted in Sections 5, 6, 7. Section 8 describes the sensitivity and relative sensitivity functions of availabilities and profit functions. Section 9 illustrates the above measurements using fixed parameter values that follow an exponential distribution. Section 10 concludes the study with interpretations.
2. Assumptions and System Characterizations
Following are the characterizations and suppositions used in the analysis of considered system:
1. The system consists of a single insured unit dividing its whole lifetime into three periods, i.e. normal warranty, extended warranty, and non-warranty periods.
2. The manufacturer or insurance provider inspects the failed system to assure
(a) Whether system's flaws are covered by a warranty, an insurance claim, or neither.
(b) Whether the system can be repaired or needs to be replaced.
3. In the case of normal and extended warranty periods, if an inspection indicates that
(a) the defects are within the warranty domain, then the manufacturer bears the repair/replacement costs
(b) the fault is covered by insurance, the repair/replacement costs are covered by the insurance company.
(c) the defects are not covered by warranty or insurance, the user is responsible for all costs.
4. During the non-warranty period, the insurance provider or the user is solely responsible for repair/replacement costs as the case may be.
5. Transition time distributions have been taken as arbitrarily and the random variables that are involved are independent.
The system is depicted in Figure 1. The number of replacements, the availability, and the expected busy period are all calculated. The profit functions are assessed. The sensitivity analysis is also
System Manufacturer
System User
Failed Insured System in Normal Warranty Period
Failed Insured System in Extended Warranty Period
Fault Diagnosis by Manufacturer
Repair/Replacement Cost borne by Manufacturer
CED
Eligible for Warranty Claim
No
Failed Insured System in Expired Warranty Period
Fault Diagnosis by Insurance Provider
Repair/Replacement Cost borne by Insurance Provider
V«
Eligible for Insurance Claim
No
Repair/Replacement Cost borne by User
Figure 1: System description
performed for availability in three different time zones, as well as the manufacturer's and user's profit functions. Numerical estimates are based on exponential distributions. Various results are drawn about profitability and sensitivity.
3. Nomenclature The following is the nomenclature for different probabilities/transition densities:
Eo system state at time t=0
pw / pw probability that a fault is approved/ not approved under warranty conditions.
r/r probability that the fault is repairable or incurable, and that the system should be replaced.
Pn / Pet / Pex probability of a system failure within the normal/extended/non-warranty period.
Ps / ps probability that the fault is covered or not covered under the provisions of the insurance policy.
fw (t) p.d.f. of failure time.
im (t)/is (t) p.d.f. of the repairman's inspection time as contracted by the manufacturer/insurance provider.
gn (t)/get (t)/gex (t) p.d.f. of the repair time during normal/ extended/ expired warranty period
hn (t)/het (t)/hex (t) p.d.f. of the replacement time during normal/ extended/ expired warranty period
Aki (t) probability that the system is operational at time t it is given that E0 = i during warranty period 'k=N/T/X'.
IK, (t)
Bkm (t)(Bku (t)) Rkf(t)(Rkf (t))
probability that the repairman of manufacturer or insurance company is busy in inspection at time t it is given that E0 = i during warranty period 'k'.
probability that the repairman of manufacturer is busy for repair/replacement when charges are borne by manufacturer or insurance provider (user) itself at time t it is given that E0 = i during warranty period 'k'. expected number of replacement upto time t, when expenses are borne by manufacturer or insurance provider (user), given that E0 = i during warranty period 'k'.
The states of the system are specified by the following notations:
Ok
Fk! Fkis
FMrn (FMrpn ) / FMrt (FMrpt ) FSRk /FSRPk FURk /FURpk
operational unit in warranty period 'k'.
failure unit under inspection by manufacturer
in warranty period 'k'.
failure unit under inspection by insurance
provider in warranty period 'k'.
failure unit under repair(replacement) in
normal/extended warranty period, for which
charges are to be paid by manufacturer.
failure unit under repair/replacement in
warranty period 'k', for which
charges are to be paid by insurance provider.
failed system under repair/replacement in
warranty period 'k', for which
expenses are to be borne by user itself.
where k stands for normal(N), extended(T), expired(X).
4. Stochastic Model
Figure 2: State transition diagram
Figure 2 illustrates the transition between several stages of the system. The state space is made up of the regenerative states, S={0,1, 2, ..., 23}, where 0={0, 17, 23} is operative state spaces
and F={1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 19, 20, 21, 22} is failed state space respectively. From figure 2, it may be observed that when system enters into a new state there is no continuation of inspection, repair and replacement from the previous state and hence at each time point, the process restarts probabilistically and thus the corresponding state where the system enters becomes the regenerative state. Further, it may also be observed that future state is independent of past and it depends only on present, thereby satisfying the Markov property. Therefore, the state transitions satisfy the Markov process and form the regenerative points. Thus, regenerative point technique is used to find various characteristics of the system. The transition densities qj(t) are:
qoi (t)= fw (t), q45 (t) = psriS (t),
q20 (t)= gn (t) qvo (t)= gn (t), q9,ii(t) = Pwrim (t)
qi2,15(t) = Psris (t),
q 13,17 (t) = get (t), qo,is(t) = fw (t), q 18,20 (t) = Psris (t), q20,23(t)= gex (t),
qi2 (t) = pwrim (t), q46 (t) = Psris (t), qso (t) = hn (t), q8o (t) = hn (t) q9,i2 (t)= PIT (t), qi2,16(t) = Psris (t),
q 14,17 (t) = het (t)
qi7,18(t) = fw (t), qi8,21 (t) = Psris (t),
q21,23 (t) = hex (t)
qi3(t) = p_wrim (t) q47(t) = Psris (t), q5o(t) = gn (t), qov(t) = fw (t), qi2,i3 (t) = psris (t), qio,i7 (t)= get (t) qi5,i7 (t) = get (t), qi7,9 (t)= fw (t) qi8,22 (t) = pSriS (t), q22,23 (t)= hex (t),
q i4 (t) = pwim (t),
q48(t) = PsriS (t), q6o(t) = hn (t) q9,io (t) = Pwrim (t), qi2,14 (t) = PsfiS (t), qii,i7 (t) = het (t), qi6,i7 (t) = h* (t)
qi8,19 (t) = PsriS (t), qi9,23 (t)= gex (t) q23,18 (t) = fw (t),
Mean sojourn time (^i) in state i, i G S is given as Vi=fo° t (corresponding p.d.f. of time for moving from ith state) dt Defining m.ij = /0°° tqj(t)dt, contribution to mean sourjoun time, we have
m0i = J tq0i (t)dt = J tf (t)dt = v0
f- °
mi2 + mi3 + mi4 = / tqu(t)dt W tqi^(t)dt W tqu(t)dt 000 f- °
= tpwrim(t)dt W tpwrim(t)dt + / tpWim(t)dt
000 f °
= tim (t)dt
Ц1
Similarly,
mo9 = mo,i8 = po; m3o = Ц3 m5o = Ц5'; m7o = Ц7;
m9,io + m9,ii + m9,i2 = Ц9; mii,i7 = Ц11; m 13,17 = Ц13 m 15,17 = Ц15 m17,9 = m17,18 = Pi7 m 19,23 = Pi9 m21,23 = Ц21 m 23,18 = Ц23
m2o = Ц2
m 45 + m46 + m47 + m 48 = Ц4; m6o = Ц6 m8o = Ц8 mio,17 = Pio
mi2,i3 + mi2,i4 + mi2,i5 + mi2,i6 = Ц12;
m14,17 = P14; m16,17 = P16;
m18,19 + m18,2o + m18,21 + m18,22 = Pi8; m2o,23 = P2o; m22,23 = Ц22;
In the following sections, several system profitability measures are achieved.
5. System Availability
1. During Extended Warranty Period
By definition of AT\(t), i=0, 9, 10, 11, 12, 13, 14, 15, 16, 17 (defined in Section 3) and the transitions that occurs during the extended warranty period, we have
ATo(t) = fw(t) + J qo9(u)AT9(t - u)du = Mo(t)+ qo9(t) © AT9(t)
(1)
The term on L.H.S. of eqn (1) denotes that the system is operational at time t given that Eo = 0. The first term on R.H.S. indicates that the system will remain in state 0 rather than transitioning to another state.The second term denotes that the system transitions from state 0 to state 9 in time u<t and and then continues operational for t-u time from state 9 onwards.
Similarly the other recurrence relations are: 'AT9(t) = 99,10(t) © AT10(t) + 99,11 (t) © ATn(t) + 99,12(0 © AT12(t)
AT10(t) = 910,17(t) © AT17(t)
ATU (t) = 911,17(0 © AT17(t)
AT12(t) = 912,13(0 © AT13(t) + 912,14(0 © AT14(t) + 912,15(0 © AT15(t) + 912,16(t) © AT16(t)
AT13(t) = 913,17(t) © AT17(t) AT14(t) = 914,17(t) © AT17(t) AT15 (t) = 915,17(0 © AT17 (t) AT16 (t) = 916,17(0 © AT17 (t) AT17(t) = M17(t) + 917,9(t) © AT9(t)
(2)
Solving e9n(1)-(2) for AT0*(s), where AT0* (s) = L[AT0(t)], we have
K1 (s)
AT0 (s)
T1(s)
K1(s) = M0 (s) + 9 09 (s)99,10 (s)910,17(s) M17 (s) + 909 (s) 99,11 (s)9n,17 (s) M17 (s)
- 917,9(s)99,10(s)910,17(s)M0(s) - 917,9(s)99,1100911,17(s)M0(s)
+ 909 0099,12 (s) 912,13 (s)913,17(s) M17 (s) + 909 (s) 99,12 (s) 912,14 (s) 914,17 (s) M1*7 (s)
- 917,9(s)99,12(s)912,13(s)913,17(s)M0(s) + 909(s)99,12 (s)912,15(s)915,17(s)Mi7(s) (3)
- 917,9 (s) 99 ,12 (s)912 ,14 (s)914 ,17 (s)M0 (s) + 9 09 (s)99,12 (s)912,16(s)916,17(s) M17(s) 917,9(s)99,12(s)912,15(s)915,17(s)M0(s) - 917,9(s)99,12(s)912,16(s)916,17(s)M0(s)
= NT1* (s)(s«y)
T1 (s) = 1 - 917,9(s)99,11(s)911,17(s) - 917,90099,12(s)912,13(s)913,17(s)
- 917,9(s)99,12(s)912,14(s)914,17(s) - 917,9(s)99,12(s)912,15(s)915,17 (s)
- 917,9(s)99,12(s)912,16(s)916,17(s) - 9i7,9(s)99,10(s)910,17(s) = DTi (s)(s«y)
The system's steady state availability is evaluated using Abel's lemma as:
AT0 = lim sATj (s) = NT?M = (5)
0 s^0 0K J DT1'(0) DT1 v '
Kajal Sachdeva, Gulshan Taneja, Amit Manocha
SENSITIVITY AND ECONOMIC ANALYSIS OF AN INSURED RT&A, No 3 (69) SYSTEM WITH EXTENDED CONDITIONAL WARRANTY_Volume I7, September 2022
Differentiating eqn (4) w.r.t. s,
DTT'00 = 911/(s)(-9670097,11 00 - q6s(s)q8,ii00 - 969^99,1100
- 96,10(s)91o,11 (s)) - 967'(s)97,11(s)911,6(s) - 96s'(s)9S,11(s)911,6(s)
- 969'(s)99,11(s)911,6(s) - 96,10'(s)91o,11 (s)911,6(s) (6)
- 97,h'(s)911,6(s)967(s) - 98,11'00911,6(s)96s(s)
- 99,11'(s)911,6(s)969(s) - 91o,n'(s)911,60096,11 (s)
Taking lim s ^ 0 in e9n (3) and (6), we get
NT1 = + p W? + pwps? + pwps? + pwps? + pw Psr) - ^o(pwr
+ pw? + pwpsr + jp^ps? + pWpsr + pwpr - 1) (7)
= V17
DT1 = m9,12 (psr + ps? + psr + p?) + m9,10 + m9,n + + pw?m1o,17
+ pw?mn,17 + pwps?m13,17 + pwps?m14,17 + pwps?m15,17 + pwps?m16,17
^17 + ^9 + ^10 pw? + ^11 pw? + ^12 pw + ^13 pwps? + ^14 p^ps? + ^15 p^ps?
(S)
+ V16 pwpsr
2. During Normal Warranty Period
Similarly, steady-state availabilities during normal warranty period are given as
AN=m! 9
where
NN1 = vo,
DN1 = vo + V1 + P + pwrv3 + pw v4 + pwpsrv5 + pwpsrv6 + pwpsrv7 + pwpsrv8
3. During Non Warranty Period
Proceeding as above case, the steady-state availability during expired warranty period is:
AXo = N£ (10)
where
NX1 = V23, DX1 = V18 + pSrV19 + pSrV20 + pSrV21 + p^V22
6. Expected Busy Period and number of replacements
Using the definitions of Ik,, Bk™ and G S (defined in section 3) and the identical steps outlined in the preceding section, the expected time a repairman spends inspecting, repairing, or replacing a failed system in different warranty periods is given as:
Ikr, = Nk2 . Rkm = Nk3 . Rk" = Nk4. k-N T X
ik0 = Dk1' Bko = Dk1' Bko = Dk1' k-N, T X
where
NN2 = + NT2 = ^9 + pw^12; NX2 = ^1s;
NN3 = pw?^2 + pw?^3 + pwps?^5 + pwps?^6;
NT3 = pw?^10 + pw?^11 + №^13 + pWps?^14; NX3 = ps?^20 + ps?^21;
NN4 = + pwps?^s; NT4 = pwp^?^15 + pwps?^16; NX4 = pI?^19 +
Furthermore, the expected number of replacements for three warranty periods in steady-state, according to the definitions of Rkm and Rky,i G S (specified in section 3), are:
Rkm = Nk5 • Rku = Nk6 • k=N T Y
Rk0 = Dki' Rk0 = Dki' k=N, T , Y
where,
NN5 = pwr + pWpsr; NT5 = pwr + pWpsr; NX5 = psr;
NN6 = pwpr; NT6 = pwpr; NX6 = pr;
7. Cost-Benefit Analysis
The financial analysis aids both the maker and the consumer in identifying the variables that may result in long-term loss. In this section, we created profit functions to do a cost estimate. A profit function is a mathematical relationship between a system's total output and total expenditure. Thus, profit functions in steady-state are:
(ii)
(i2)
Profit to System and Insurance Provider
PM = CP + EP - MP + SP - CM1 (PnINo + PetlTo) - CS1(PnINo + PetlTo + PexIXo)
- CM2 (PnBNom + PetBTom + PexBXm) - CM3(PnRNom + PetRTm + PexRXm) Profit to System User
PU = Ro(PnANo + PetATo + PexAXo) - CU2(PnBNU + PetBTU + PexBXoU) - CU3 (PnRNU + PetRTU + PexRXoU ) - CP - EP - SP where,
CP= Expenses of purchasing the system MP= Manufacturing cost of the system EP= Expenses of extending the warranty period SP= Expenses associated with insuring the system Ro= Revenue generated by the system.
CMx/CSx= Expenses incurred by the manufacturer/insurance provider in hiring a repairman for inspection.
CM2(CU2)= Expenses incurred by the manufacturer or insurance company (user) in engaging a repairman for repair/replacement.
CM3(CU3)= Expenses incurred when a system is replaced, which are covered by the manufacturer or the insurance company (user).
All of the costs listed above are per unit time.
8. Sensitivity Analysis
Sensitivity analysis is an approach that examines whether a parameter has a high or low influence on the derived measures. Due to the wide range of numerical values for various parameters, relative sensitivity analysis is performed to compare the effects of various parameters. A relative sensitivity function is a standardized version of a sensitivity function.The sensitivity (Ark, 5rs) and relative sensitivity functions (zrk, zrs) for availabilities (ANo, ATo, AXo) and profit functions (PM,PU) are defined using the eqns (5), (9), (1o), (11) and (12) and stated as follows:
Ark = ^; Zrk = Ag k = N, T, X (13)
and
¿rs = ^; Zrs = 5-f; s = U, M (14)
dr Ps
where r is the parameter
9. Results and Discussions
The system characteristics determined in the preceding sections 5-8 are illustrated numerically in this section. Assume that all of the distributions are exponentially distributed and their probability density functions are as follows:
hwe -Xwt im (t) = Yme-Ymt, is (t) = 7se~ -Yst
fine- -ßnt, het (t) = fite-ßt\, gn (t) = an e att
ate- -att gex(t) = axe-axt hex(t) = ßxe- -ßxt
' fw (t)
^get (t)
Consider the fixed value of parameters as
pw = 0.7, pWW = 0.3, pn = 0.2, pet = 0.3, pex = 0.5, r = 0.7, r = 0.3, ps = 0.8, pS = 0.2, Xw = 0.0005, Ym = 1.5, Ys = 1.2, an = 0.5, at = 0.4, fin = 0.02, fit = 0.02, ax = 0.25, fix = 0.01, CP = 150, EP = 15, SP = 20, MP = 120, R0 = 500, CM1 = 80, CM2 = 100, CM3 = 15,000, CSi = 90, CU2 = 120, CU3 = 15,000.
(15)
(16)
n
9.1. Variation in Profit Functions
The profit function has been graphically represented as a function of various parameters in this section. Figure 3 shows the change in manufacturer's profit (PM) versus and Aw. Profit falls
Figure 3: Manufacturer's Profit (PM) for varied \w and ps rapidly as Xw and ps rises.
The variation occured in user's profit (PU) due to changes in Rq and r is depicted in Figure 4. Profit begins to rise as Rq rises, and as r rises, profit declines. Figure 5 shows the decrease in profit differential (PU-PM) versus SP and MP. As SP rises, the profit margin narrows, whereas as MP rises, the profit margin widens. Lower/upper bounds for a system's profitability can also be determined, few of them mentioned are:
1. PM > 0 if \w < 0.018 for ps = 0.5.
2. PU > 0 if R0 > 150 for r = 0.1.
3. PU > PM if SP < 132 for MP=100.
Lower/upper limits for other parameters can be interpreted in the same way.
Figure 4: User's Profit CPU) for waned Ro and r
Figure 5: Difference of Profit (PU-PM)for waned SP and MP
9.2. Numerical Calculations for Sensitivity Analysis
We compute the sensitivity analysis for profit functions and availabilities in this section by treating all transition densities as exponential and having fixed parameter values as stated in eqn (15) and eqn (16) respectively.
The results for sensitivity and relative sensitivity functions for availabilities and profit functions (specified in section 8) are shown in Tables 1, 2 and 3 respectively. The absolute value of both the functions is considered for various conclusions. It has been confirmed that
1. The availability of three alternative warranty times (ANo,ATo,AXo) is substantially influenced by Aw. Variation in ys and ym, on the other hand, appears to have the slightest effect.
2. The profit functions PM and PU are both quite sensitive to CP and Ro.
3. A relative change in PU and PM is caused by variations in CU3 and CM3.
Table 1: Relative Sensitivity Analysis of Availabilities w.r.t. different rates
Parameter (r) Sensitivity Analysis A _ a(Ao) Ar _ 3r Relative Sensitivity Analysis z _ Ar*r Zr = Ao
Normal Warranty Period
Aw -1982.8 -1
Tm 2.1842 * 10-4 3.3047 *10-4
Ys 1.0239 * 10-4 1.2393 *10-4
an 0.0014 7.0606 * 10-4
ßn 0.3686 0.0074
Extended Warranty Period
Aw -17.3586 -0.0088
Ym 2.1835 * 10-4 3.3042 * 10-4
Ys 1.0235 * 10-4 1.2390 *10-4
at 0.0021 8.4742 * 10-4
ßt 0.3685 0.0074
Expired Warranty Period
Aw -32.53 -0.0165
ax 0.0054 0.0014
ßx 1.4508 0.0148
Ys 3.3583 * 10-4 4.0977 *10-4
Table 2: Relative Sensitivity of Manufacturer's Profit w.r.t. different rates/costs
Profit for System Manufacturer
Parameter (r) Sensitivity Analysis Am = 3(Pm) °r = dr Relative Sensitivity Analysis zm = $!"* r ■^r Pm
Aw -5.5312 * 1o3 -o.o445
Ym o.o251 6.o533 * 1o—4
Ys o.o266 5.132o * 1o—4
an o.o253 2.o338 * 1o—4
ßn 6.771o o.oo22
at -o.oo14 —9.oo35 * 1o—6
ßt -o.2475 — 7.9585* 1o—5
ax o.2118 8.5131 * 1o—4
ßx 56.723o o.oo91
CMi -2.2718 1o-4 — 2.922o* 1o—4
CSi -4.32o6*1o-4 — 6.2519* 1o—4
CM2 -o.oo81 -o.o13o
CM3 — 1.289o * 1o-4 -o.o311
CP 1 2.4117
EP 1 o.2412
MP -1 -1.9293
SP 1 o.3216
Furthermore, the order in which input variables effect availabilities (An, Aof, Aox) and profit functions (Pm,P") are
Availability(ANo): Aw >ß n >an >'Ym>'Ys• Availability(AT0): Xw>ßt>at>Ym>Ys. Availability(AXo): Aw>ß x >ax > Ts •
Profit Function(Pm):CP>MP>SP>EP>Aw>CM3>CM2>ßx>ßn>ax>CSi> Ym >Ys >CMi >an >ßt >at •
Profit Function(Pu): Ro>CP>SP>EP>Aw>ßx>ßt>ßn>ax>CU3>Ys>at>Ym> an >CU2.
Table 3: Sensitivity and Relative Sensitivity of User's Profit w.r.t. different rates/costs
Profit for System User Parameter Sensitivity Analysis Relative Sensitivity Analysis
(r) ™ d(Pu) °r - dr zu = W*r ¿r — Pu
Aw -1.3446 * 105 -0.0218
ax 1.4179 0.0012
ßx 379.7890 0.0123
Ys 0.1094 4.2598 * 10-4
Ym 0.0546 2.6575 * 10-4
an 0.1396 2.2649 * 10-4
ßn 37.3800 0.0024
at 0.3270 4.2442 * 10-4
ßt 56.0506 0.0036
Ro 0.9874 1.6020
CU2 -0.0019 -7.3982 * 10-5
CU3 -1.9213 * 10-5 -9.3514 * 10-4
CP -1 -0.4867
EP -1 -0.0487
SP -1 -0.0649
10. Conclusion
The sensitivity and economic analysis of the insured system operating under normal warranty, extended warranty, and no warranty conditions were explored in this study Various profitability indicators and profit functions for the user, manufacturer, and insurance provider have been drafted using Markov and regenerative techniques. After that, the measures are assessed using numerical calculation in which the transition density follows an exponential distribution. For system profitability, lower/upper bounds of the measures involved have been identified. The failure rate has significant influence on availability and profit, whereas the inspection rate has the least. Revenue and cost pricing also significantly impacts the system's profit. This research gives optimum analysis regarding benefits for the user, the manufacturer as well as the insurance provider.
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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