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MIDDLE EAST TECHNICAL UNIVERSITY Ankara, Turkey

INSTITUTE OF APPLIED MATHEMATICS

A value-adding approach to reliabilityunder preventive maintenance costs and its

applicationsSadia Samar Ali1, Erdem Kilic2, Gerhard Wilhelm Weber3, Rameshwar Dubey4

Abstract. No equipment (system) can be perfectly reliable in spite of the utmost care

and best efforts on the part of the designer, decision maker and manufacturer. For a large

number of systems, maintenance is a must, as it is one of the effective ways of increasing

the reliability of the system. The two sides of maintenance are corrective and preventive

maintenance. It is generally assumed that a preventive maintenance action is less costly

than a repair maintenance action. Folk wisdom supports the notion that a higher quality

translates into lower maintenance costs (as well as other components of life-cycle costs)

for the users. We examine this proposition in detail on the basis of a failure-time model that

relates conformance quality to reliability. We know that maintenance plays an important

role in reliability theory and it increases the life-time of an item or system at lower cost.

Illustratively, we present reliability in the context of contracts with asymmetric informa-

tion. The model show how to overcome information rents through price distortions and

quantity rationing. In this paper, we have shown how preventive maintenance is affecting

on different life-time distributions. The paper ends with a conclusion and an outlook to

future studies.

Keywords. Reliability function under preventive maintenance, Probability distribu-

tion, Financial means, Operational Research, Manufacturing Services, Economics

Preprint No. 2013-04-19April 2013

1Fortune Institute of International Business, New Delhi 110057 India, +91-9650691133, email:

[email protected] of Business Administration and Economics, Yeditepe University, Turkey, email:

[email protected] of Applied Mathematics, Middle East Technical University, Turkey4Symbiosis Institute of Operations Management, Nashik, India



Figure 1. Representation of Reliability in a System.

1 Introduction

It is supposed to be a fact that no system or a equipment is perfect in the sense that

it will continue to function (without failure) forever, how so ever carefully it might

have been designed and manufactured. However, it is assumed that the reliability

of equipment or a system may be increased by proper maintenance. Such

maintenance is known as preventive maintenance. It is done periodically, before

the failure of the system; hence, it is different from the corrective or repair

maintenance, which is carried out only after the failure of the item or the system

[5, 19].

In this paper, we examine the effect of preventive maintenance on the reliability of

an item that functions until the first failure, which comes under support of Relia-

bility, Maintainability, Supportability, Quality (RMSQ) as shown in Figure 1. This

figure depicts a cycle of different phases of products including RMSQ. For this

purpose we consider some well-known life-time distributions in this relation.

In Section 2, we consider the effect of preventive maintenance on the reliability

of an item that follows an exponential failure time distribution. It is shown that

preventive maintenance does not improve the reliability of such an item. We prove

that the mean time to failure (MTTF) of such an item is equal to the mean life-time

without maintenance and we establish it as a characteristics property of exponentialfailure time distribution. In Section 3, we consider a power function distribution

and obtain the condition under which maintenance reliability exceeds the reliability

without maintenance. In Section 4, we discuss the effect of preventive maintenance

with respect to some other distributions with a general discussion on preventive

maintenance. Two types of relative maintenance policies had been considered by

Barlow and Hunter [4]; Crow [15]; Abdel-Hameed [1]; Tjiparuro and Thompson

[30].

Birolini [6] and Khan Malik [23] discussed reliability models which were built

for a service producing system which works intermittently, is subject to wear, and

2



can be improved through maintenance actions like leaning, lubrication, and re-

alignment, etc., short replacement. Juang and Gary [21] considered a theoreticBayesian approach to determine an optimal adaptive preventive maintenance pol-

icy with minimal repair. When the failure density is Weibull with uncertain pa-

rameters, a Bayesian approach is established to formally express and update the

uncertain parameters for determining an optimal adaptive preventive maintenance

policy. Chelbi and Ait-Kadi [12] considered a repairable production unit subject

to random failures, which supplies input to a subsequent assembly line, operating

according to a just-in-time configuration. Preventive maintenance actions are reg-

ularly performed on the production unit at instants T , 2T , 3T , .... The corrective

and preventive maintenance actions have random durations.

Lifeng et al. [26] tries to integrate a sequential imperfect maintenance policy into

condition-based predictive maintenance (CBPM). A reliability-centered predictivemaintenance policy is proposed for a continuously monitored system subject to

degradation due to the imperfect maintenance. It is assumed that the system hazard

rate is a known function of the system condition and then can be derived directly

through CBPM.

Currit and Singpurwalla [17] explored the reliability function of a system of com-

ponents sharing a common environment. Kolowrocki [24] has a given concept for

reliability function of a homogenous series such as parallel-parallel and parallel-

series system. Barlow and Hunter [4] worked on optimum preventive maintenance

polices and Juang and Gary [21] used Bayesian method on adoptive preventive

maintenance problem.

Bloch-Mercier [10] analysed sequential checking procedure for checking proce-dure of markov deteriorating system. Kong and Frangopol [25]; Vassiliadis and

Pislikopoulos [31] used maintenance scheduling and process optimization under in

environment of uncertainty. Blanchard et al. [9], Blanchard and Fabrycky [8] and

Abdel-Hameed [1] focus many concepts on maintainability and their applications

for effective serviceability and maintenance. Cléroux et al. [14] explored the age

replacement problem with minimal repair costs. The techniques of optimal number

of minimal repairs before replacement discussed by Park [27]. Many models have

been analysed for product quality with warranty cost by Chen [13]. Kackar [22] ex-

plored Taguchi’s quality philosophy, analysed with different cases and commented

on each cases. The problem of strength-reliability of the equipment defined as the

probability that the strength of equipment exceeds the stress, instead of findingP( X > Y ) for a given set of distribution and found the required parametric values

of the assumed distributions to achieve a desired level of strength reliability. Samar

Ali and Kannan [29] and Alam and Roohi [2] assumed exponential strength and

exponential stress for this purpose. Alternatives for augmenting the exponential

strength-reliability have been suggested against the exponential distributed stress.

As an illustrative example we discuss the relevance of reliability in the context

of a manufacturer and dealer contracting model with asymmetric information re-

lated to the proposed model by Blair and Lewis [7] in the Sections 5 and 6. The

3



dealer is providing maintenance services which increase the reliability in his eco-

nomic transactions. He has full-information about the maintenance services andthe related maintenance costs. On contrary, the dealer does not have exact knowl-

edge about the final realization of the maintenance costs. The remainder of the

article is organized as follows. Section 2 analyzes the reliability for exponential

distribution under preventive maintenance. Section 3 introduces the reliability for

the power function distribution under preventive maintenance. Then, Section 4

presents reliability under preventive maintenance for other life-time distributions.

Section 5 introduces a model for optimal retail contracts under asymmetric infor-

mation. Whereas, Section 6 presents the illustration about dynamics of optimal

contracts. Finally, Section 7 concludes and gives an outlook to future studies.

2 Reliability for exponential distribution under preven-

tive maintenance

It is well-known that exponential distribution has constant failure rate and this is

the characteristics property of it. Now we consider the case of reliability under

preventive maintenance of equipment following exponential distribution [20]; [16];

[18]; [11]. If the probability density function (pdf) of failure time T is given by

f (t ) = λ e−λ t (t > 0) , (1)

then reliability of such an equipment is introduced assumed as

R (t ) = e−λ t (t > 0) . (2)

When we use the result, maintenance reliability is defined as

Rn(t ) = ( R (T ))n R(t −nT ) (nT ≤ t ≤ (n + 1)T ) .

Here, R(t ) and Rn(t ) be the reliability of a system without maintenance and with

maintenance respectively (n ∈N0).

We find reliability of that equipment with regular preventive maintenance at time

T , 2T , 3T ,..., is given by

R N (t ) =

e−λ T n

e−λ (t −nT ) for nT ≤ t ≤ (n + 1) T .

We note that this definition does not depend on the number of preventive mainte-

nance. Therefore, we conclude that preventive maintenance does not improve the

reliability of equipment having an exponential failure distribution. We present and

prove a main important result stated in the form of a theorem.

4



Theorem 2.1 Preventive maintenance does not improve the reliability of an equip-

ment or system if it has a constant failure rate.

Proof Let us suppose that the reliability does not improve after preventive main-

tenance or mean time to failure (MTTF) is constant with respect to maintenance

after any regarded time T , i.e.,

MTTF =

´ T

0 R (t ) dt

1− R (T ) = α or

ˆ T

0

R (t ) dt = α (1 − R ( T )) .

Differentiating both the sides with respect to T , we get

R (T ) = −α R (T ) ,

thus,

R (T )/ R (T ) = − 1

α ,

i.e.,

ln ( R (T )) = −T

α + c .

Therefore,

R ( T ) = e−(T / α ) +c T

≥ 0 ,

where c is constant of integration. However, R (0) = 1, therefore, c = 0. Thus,

R (T ) = e−T / α ,

which is the reliability at time T , of an item following an exponential distribu-

tion. Hence, it shows that equipment with constant MTTF follows an exponential

distribution.

Now we consider that equipment follows exponential distribution and has the reli-

ability R (T ) = e−t / α . Hence, the MTTF with respect to maintenance after time T

is given by

MTTF =

´ T 0 e−t / α

1− e−T /α = α

1− e−T /α

/

1− e−T /α

= α ,

This proves the theorem.

5



3 Reliability for the power function distribution under pre-

ventive maintenance

The reliability function for the power function distribution is given by

R (t ) = k a − t a

k a (0 ≤ t ≤ k ) ,

Reliability of such an item under preventive maintenance, is given by

R M (t ) = ( R (t )n) [ R (t −nT )] =

k a− t a

k a

k a − (t −nT )a

k a

(nT ≤ t ≤ (n + 1) T ) ,

for n ∈ {0,1,..., M −1} , where M ∈ N. In order that the reliability under pre-ventive maintenance be more than that of without maintenance, we must have

{ R M (t )/ R (t )}> 1 at the time of preventive maintenance t = nT , where n = 1,2,3..., M ,i.e.,

R M ( nT )

R(t ) =

1− T a

k a

n

1− (nT )a

k a

> 1 .

Hence,

1 − nT a

k a > 1−

nT

k

a

=

nT

k

a

− nT a

k a > 0 ;

thus,

na

− n > 0 ,

⇒ a > 1 .

This means that preventive maintenance will improve the reliability of the power

function system, if a ≥ 1. It simply means that for a < 1, preventive maintenance

may not be useful. To get a better insight into this result, Tables 2.1-2.4 present

R M (t ), the reliability with maintenance for selected values of a. Without loss of

generality, we assume k =1, doing a normalization otherwise.

Table 2.1. R (t ) and R M (t ); T =0.25, a=0.5.

t 0.1 0.25 0.3 0.4 0.5 0.6 0.75 0.8 0.9 1 R (t ) 0.6838 0.5000 0.4523 0.3675 0.2929 0.2254 0.1339 0.1056 0.0513 0

R M (t ) 0.6838 0.5000 0.3882 0.3064 0.2500 0.1709 0.1250 0.0970 0.0766 0.625

Table 2.2. R (t ) and R M (t ); T =0.25, a=1.

t 0.1 0.2 0.25 0.3 0.4 0.5 0.6 0.75 0.8 0.9 1

R (t ) 0.9 0.8 0.75 0.7 0.6 0.5 0.4 0.25 0.2 0.1 0

R M (t ) 0.9 0.8 0.75 0.7125 0.6375 0.5635 0.5063 0.45 0.4008 0.3586 0.3164

6




t 0.1 0.25 0.3 0.4 0.5 0.6 0.75 0.8 0.9 1 R (t ) 0.99 0.91 0.84 0.75 0.64 0.51 0.44 0.36 0.19 0

R M (t ) 0.99 0.9375 0.9352 0.9164 0.8789 0.8701 0.8438 0.8219 0.8054 0.7724


t 0.1 0.25 0.3 0.4 0.5 0.6 0.75 0.8 0.9 1

R (t ) 0.999 0.984 0.973 0.936 0.875 0.784 0.578 0.488 0.271 0

R M (t ) 0.999 0.9844 0.9842 0.9811 0.9689 0.9680 0.9612 0.9537 0.9506 0.9389

4 Reliability under preventive maintenance for other life-

time distributions

In this section we derive the expression for the maintenance reliability for different

life-time-distributions.

4.1 Weibull distribution

The probability density function (pdf) of Weibull distribution is given by

f (t ) =

β

θ t

θ β −1

exp−

t

θ β

(t ≥ 0),

where θ , β > 0. The reliability function for this distribution is given by

R (t ) = exp

− t

θ

β .

Reliability under preventive maintenance of such an equipment or system is defined

by

R M (t ) =

exp

−T

θ

β n

exp

−

t −nT

θ

β .

In order to see the effect of preventive maintenance, we have to find the values R M (t )

R (t ) at the time of preventive maintenance t = nT :

R M (nT )

R (nT ) =

exp−n

T θ

β exp

− nT

θ

β > 1 ⇒ β > 1.

This means that the preventive method is effective for the Weibull system, if for

the shape parameter β it holds β > 1.

7



4.2 Normal distribution

The probability density function of normal distribution is given by

f (t ) = 1

σ √

2Πexp

−1

2

t −µ

σ

2

(t ≥ 0) ,

where µ ,σ > 0. There is no closed-form solution for the normal reliability func-

tion. Solutions can be obtained via the use of standard normal tables. Thus,

R (t ) = 1−φ

t − µ

σ T

.

Reliability under preventive maintenance is represented by

R M (t ) =

1− φ

T − µ

σ T

n1−φ

t −nT −µ

σ T

.

4.3 Gamma distribution

The probability density function of the Gamma distribution is given by

f (t ) = t γ −1

Γ (γ ) (t ≥ 0) ,

where γ ≥ 0.Here, γ is the shape parameter, and Γ is the Gamma function given by the relation

Γ (a) =

ˆ ∞

0

t a−1 e−1dt .

There is no closed form of reliability function for the Gamma distribution also.

Thus, the reliability function is given by

R (t ) = 1− Γ t (γ )

Γ (γ ) (t ≥ 0),

with some γ > 0. Here, Γ t (a) is an incomplete Gamma function defined by rela-

tion

Γ t (a) =

ˆ t

0

t a−1e−t dt .

We can obtain the solutions using the table of incomplete Gamma Distribution

introduced by Pearson [28].

Reliability under preventive maintenance for Gamma distribution is defined as

8



R M (t ) = 1

−Γ T (γ )

Γ (γ )

n

1

−Γ t +nT (γ )

Γ (γ ) (t

≥0) .

No general comment is possible on the behaviour of the reliability under preventive

maintenance for Normal and Gamma distribution.

5 Optimal retail contract under asymmetric information

This section gives an illustrative operative example for optimal contracts with in-

corporation of maintenance costs under asymmetric information. We analyze the

effect of asymmetric information on the price and the quantity of a sale product,

resulting from the agent problem evolving between the retailer and the manufac-

turer, when incorporating the (optimal) maintenance costs or reliability in a broader

sense. The manufacturer produces a sale product at constant unit costs. The retailer

or manufacturer sells the product on the market while he provides promotion and

additional customer services, such as maintenance of a product and consumer edu-

cation. We focus on a single retailer within a monopoly framework to exclude the

agent problem from externalities.

The manufacturer determines the price of the product but he is not informed about

the exact market demand. The retailer has knowledge about the market demand

and the promotional services. The quantity of the sale product is derived by the

market demand. The retailer or dealer takes in account the repair costs prior to his

sale; he anticipates to replace the repair costs with the maintenance costs. Given

this consideration the determination of the optimal contract under asymmetric in-

formation plays a key role. The likelihood of an optimal contract is described in the

cases when maintenance costs evolve according to the certain dynamics. As main

factors we can mention the life-time utility of a product and the negotiated price in

the contract. So, the contract evolves according to dynamics of maintenance costs,

the probable life-time distribution of the product and the product’s price effects.

The optimal quantity is a consequence resulting from the mentioned dynamics.

5.1 Market demand and reliability

Market demand is determined as Q( p, X , Rt ), p being the retail price, X denoting

the total expenditure on promotion, while the reliability Rt is determined by main-

tenance costs. The promotion function is provided as X ( p, Q, Rt ).

Given our assumptions on demand, we have X p, X Q > 0 and X R < 0, which means

that promotion is increasing in price and sales, and decreasing in Rt . Here, the

dealer has perfect knowledge about Rt , and the dealer has imperfect knowledge

about Rt . Hence, for the manufacturer Rt follows a density function, because the

reliability is the realization of a random variable:

9



d dRt

(1−F ( Rt )) f ( Rt )

≤ 0 . (3)

The joint profit for the manufacturer and the dealer is defined as follows π J (Q, p) = pQ− X ( p, Q, Rt ), where the function is strictly concave in p and Q:

∂ 2π J

∂ Q∂ p> 0 . (MC)

The promotional services provided by the dealer include customer services, such

as maintenance costs, free delivery, installation and repair services. The degree of

maintenance can be presumed as reliability. The endogenous reliability is defined

by R (t ) = e−λ t

(t > 0). It states the dynamic under which the maintenance im-proves the reliability of a manufacturing system or of an equipment.

The dealer reports the manufacturer his maintenance services which we denote

here as reliability Rt . The dealer pays a fee A to the manufacturer which is the

franchise fee. The dealer has the right to offer a contract that the manu-facturer

either accepts or rejects. The additional profit comes from the information rent

that the dealer obtains from the knowledge about Rt which is not observable by the

manufacturer.

Then, the optimization process evolves according to the market demand and sup-

ply functions and the preferences of the dealer. The dealer has the incentive tohave a good measure of reliability, so he tries various reliability distributions as

benchmarks for the density function of R. His maximization problem is described

as

π d ( Rt | Rt ) = max

Q≤Q( Rt )

p( Rt )Q− X

p( R

t ), Q, Rt

− A( R

t ) . (4)

Here, R denotes alternative reliability distributions. The optimal retail contract

will depend on how the manufacturer’s choice of p and Q affects this marginal rate

of substitution. We assume that the following cases have to be distinguished in this

relation:

− X pR,− X QR > 0 , (P1)

− X pR,− X QR < 0 , (P2)

− X pR,− X QR = 0 . (P3)

In cases (P1) and (P2), the marginal rate of substitution depends on both price and

quantity level.

10



5.2 Manufacturer’s problem

The design of a contract with a decision consisting of a strategy { p( Rt ),Q( Rt ), A( Rt )}is given as the following optimization problem:

maximize

ˆ R H

R L

A( Rt )dR ,

subject to

π d ( Rt | Rt ) ≥ 0 , (IR)

π d ( Rt ) = π d ( Rt | Rt ) ≥ π d ( Rt | Rt ) , (IC)

Q( Rt ) ≤ argmaxQ

[ p( Rt )Q− X ( p( Rt ), Q, Rt )− A( Rt )] . (SC)

The necessary and sufficient conditions for implementing retail incentive contracts

are represented by (SC).

The individual rationality (IR) condition stipulates that the dealer must earn at leasthis reservation profit. (IC) is an incentive compatibility constraint, which implies

that the dealer maximizes her profits when he truthfully reports Rt in accord with

the revelation principle. The sales constraints (SC) indicates that the dealer cannot

be induced to sell more than the profit-maximizing level.

Lemma 1. Necessary and sufficient conditions for implementing retail incentive

contracts are SC and

(i) π d ( Rt ) = ˆ R H

R

(

− X θ ) d ˆ Rt ,

(ii) Q( Rt ), p

( Rt ) ≥ 0 for case (P1),

(iii) Q( Rt ), p

( Rt ) ≤ 0 for case (P2).

Using Lemma 1, the transfer fee can be written as:

11



A( Rt ) = p( Rt )Q( Rt )− X ( p( Rt ), Q( Rt ), Rt )−´ R H

R (− X R)d ˆ Rt .

The finalized manufacturer’s problem then becomes

maximize p( Rt ),Q( Rt )

ˆ R H

R

[ p( Rt )Q( Rt )− X ( p( Rt ), Q( Rt ), Rt ) + h( Rt ) X Rt ] dF ( Rt ) , (M’)

subject to (SC),

where h( Rt ) = (1−F ( Rt ))/ f ( Rt ).

6 Dynamics of optimal contracts

6.1 Optimality dynamics

The first best contract is described in Lemma 2. The optimal decision is given as

{ p∗( Rt ), Q∗( Rt ), A

∗( Rt )} when the manufacturer can observe the degree of relia-

bility.

Lemma 2. In the first-best contract:

(i) p∗( Rt ) = X Q,

(ii) Q∗( Rt ) = X p,

(iii) π d ∗( Rt ) = 0.

The conditions (i)-(ii) describe that price and quantity are equal to the marginal

cost. Condition (iii) states that dealer is not allowed to have excess profit at all

through the intervention of the manufacturer. A franchise fee, which is set by the

manufacturer, transfers the revenues from the dealer to the manufacturer without

modifying the price or the quantity. With full-information the manufacturer would

choose the first best solutions p( Rt )∗ and Q( Rt )

∗ to maximize the profit. This

would mean that the manufacturer could avoid potential distortions in the choice

of the price and quantity.

The optimal contract is characterized according to the cases for (P1) through (P2).

Various outcomes for the different cases are described in the next section. The

manufacturer introduces price and quantity distortions.

It is optimal to introduce price and quantity distortions into the retail contract for

cases (P1) and (P2). The following propositions describe the nature of these distor-

tions. Denote p J (Q( Rt ), Rt ) as the joint-profit-maximizing price given Q( Rt ) and

Rt , and Q J ( p( Rt ), Rt ) as the joint-profit-maximizing quantity given p( Rt ) and Rt .

12



Proposition 1. Given condition (P1) (− X pRt ,− X QRt > 0), the solution to (M)involves

(i) price ceilings with p( Rt ) ≤ p J (Q( Rt ), Rt ) ≤ p∗( Rt ) for Rt < R H ,t ,

(ii) quantity rationing with Q( Rt ) ≤ Q J ( p( Rt ), Rt ) ≤ Q∗( Rt ) for Rt < R H ,t .

6.2 Optimal pricing and output distortions

The dealer has the opportunity to derive an information rent because of his in-

sider information. The information rent that the dealer commands, depends on the

marginal rate of substitution of exogenous demand ( X ) for reliability. This interde-

pendence is increasing in both p and Q. The optimal pricing and output distortionsdetermine the substitutionality between X and Rt . This substitutionality induces

decreasing information rents of the dealer.

The manufacturer profits are higher with a price ceiling and information rents are

declining with a decrease in quantity. This means that it is optimal for the manu-

facturer to force the dealer to sell below the joint-profit-maximizing quantity.

In (P2) the substitutability between X and Rt is decreased when the price p and the

quantity Q are increased. To reduce the information rents the manufacturer aims to

set the price p and quantity Q higher than the profit maximizing level. However,

he cannot force the dealer to sell more. The mechanism is resulting in a price floor

which allows the dealer to choose the profit-maximizing quantity for the predeter-mined price.

Proposition 2. Given condition (P2) (− X pRt ,− X QRt

< 0) the solution to (M) in-

volves

(i) price floors with p( Rt ) ≥ p J (Q( Rt ), Rt ) ≤ p∗( Rt ) for Rt < R H ,t ,

(ii) quantity rationing with Q( Rt ) = Q J ( p( Rt ), Rt ) ≥ Q∗( Rt ) for Rt < R H ,t .

7 Conclusion and Outlook

We know that maintenance plays an important role in reliability theory and it in-

creases the life-time of an item or system at lower cost. In the above discussion,

the preventive maintenance effect on different time distributions has been investi-

gated. In this paper, we have shown that preventive maintenance affects different

life-time distributions. The preventive method is effective for the Weibull system,

if for the shape parameter it holds β > 1 and for Normal distribution, while for

Gamma distribution no general comments are possible.

13



We present an example where reliability in the form of maintenance costs is sub-

ject to asymmetric information. Price, quantity and market demand relationshipare incorporated in a dynamic framework which analyzes the maintenance costs as

a policy variable. The outcome is determined according to quantity rationing and

price policies.

As a further setting of the model, different markets can be chosen by analyzing the

different price setting strategies and consumer behavior in relation to maintenance

costs or to reliability in general. In addition, applications of stochastic differential

equations or stochastic hybrid systems can be considered to obtain more insight

information about the model dynamics.

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Appendix

Proof of Lemma 1

A necessary condition for local incentive compatibility is

π d ,( Rt ) = π d 2 ( Rt | Rt ) = − X Rt > 0 . (A1)

In fact, π d ( Rt ) is decreasing, individual rationality binds only at Rt = R H ,t . Thus,

(A1) implies

π d ( Rt ) = 0 +

R H ˆ

R

(− X Rt )d ˆ Rt . (A2)

16



The first-order condition for local incentive compatibility implies that

π d 1 ( Rt | Rt ) = 0 ; (A3)

the second-order conditions for local incentive compatibility require

π d 11( Rt | Rt ) ≤ 0 . (A4)

Total differentiation of (A3) with respect to Rt and employing (A4) implies that the

following is required:

0 ≥ π d 11 = −π d

12 = −π d 21 = − X Rt , p p( Rt ) + X Rt Q Q( Rt ) . (A5)

In case of (P1) or (P2), it follows that Q, p ≥

0 (Q, p ≤

0), respectively are

sufficient to satisfy (A5). The proof is completed by showing that local incentive

compatibility implies global incentive compatibility .

Proof of Lemma 2

The first-best problem is written as max p,Q

π ∗ = p( Rt )Q( Rt )− X ( Rt ). Optimization

yields π ∗ p = Q∗( Rt )− X p( Rt ) = 0 and π ∗Q = p∗( Rt )− X Q( Rt ) = 0. In the first-best

case, the upstream firm only needs to provide the downstream firm with its reser-

vation profit level; i.e., π d ∗ = 0.

According to conditions (i) and (ii), price and quantity are chosen so that the

marginal revenue equals the marginal cost. Marginal cost is measured by the in-crease in promotion required when either price or quantity is increased. Condition

(iii) implies that the manufacturer extracts all excess profits from the dealer.

Proof of Propostion 1

Assume (SC) is not binding in the solution to (M). Differentiation of (M) with

respect to Q and p yields π m p = Q( Rt )− X p + h( Rt ) X Rt , p = 0 and π mQ = p( Rt )− X Q + h( Rt ) X Rt , p = 0, implying that π J

p( p( Rt ), Q( Rt ), Rt ) = −h( Rt ) X Rt , p > 0 and

π J Q( p( Rt ), Q( Rt ), Rt ) = −h( Rt ) X Rt p > 0 where π J is defined in the text.

Given (MC) and strictly concavity of π J ( p, Q, Rt ), this implies p J ( Rt )(Q( Rt ), Rt )

and Q J ( p( Rt ), Rt ) are increasing functions.

Let p−1(Q( Rt ), Rt ) be the inverse of Q J ( p( Rt ), Rt ). Strict concavity of π J impliesthat p J (Q

( Rt ), Rt ) intersects p−1(Q

( Rt ), Rt ) once at Q∗. It is now easy to verify

that

p J (Q( Rt ), Rt ) >=<

p−1(Q( Rt ), Rt ) as Q = Q∗ . (A6)

For case (P1), π J p, π J

Q > 0. Given the strict concavity of π J and (A6), this im-

plies that p( Rt ) < p J (Q( Rt ), Rt ) and ( p( Rt ),(Q( Rt )) lies at a point between the

p J ( Rt ,Q( Rt )) and p−1( Rt ,Q( Rt )) schedules. Hence, for case (P1), Q( Rt ) < Q∗ and

17



p( Rt ) < p∗. To complete our proof, notice that Q( Rt ) < Q J ( p( Rt ), Rt ) which veri-

fies that (SC) is nonbinding as we originally asserted.

Proof of Propostion 2

For case (P2), (SC) is binding by following argument. Suppose that (SC) is not

binding, then π J Q > 0 and π J

p < 0, implying that ( p( Rt ), Q( Rt )) must be located

at a point Q > Q∗, between the p−1( Rt ,Q( Rt )) and the p J ( Rt ,Q( Rt )) schedules.

However, in the proof of Proposition 1, we demonstrated that ( p( Rt ), Q( Rt )) must

be located at a point Q < Q∗, between the p J ( Rt ,Q( Rt )) and the p−1( Rt ,Q( Rt ))schedules whenever (SC) is not binding. Thus, Q( Rt ) = Q J ( p( Rt ), Rt ), since the

value of Q that maximizes the dealer profits also maximizes joint profits given

the price p( Rt ). In this instance, the manufacturer chooses the following price,

recognizing that it will influence the quantity; Q J ( p( Rt ), Rt ), which the dealer sells.The first-order condition for the choice of p( Rt ) is given by

d π m

d p= π J

p +π J Q[dQ J /d p] + h( Rt )[ X Rt , p + X Rt Q[dQ J /d p] , (A7)

π J p = −h( Rt ) X Rt , p−h( Rt ) X Rt ,Q[dQ J /d p] = 0 ,

where, the second lime follows from the fact thatπ J Q = 0 for Q J ( p( Rt ), Rt ). Solving

(A7) for π J p yields

π J p = −h( Rt ) X Rt , p−h( Rt ) X Rt ,Q[dQ J /d p] < 0 ,

because [dQ J /d p] =−π J pQ/π QQ by (MC). Consequently, we have p( Rt )> p J (Q( Rt ), Rt )

and Q( Rt ) = Q J ( p( Rt ), Rt ). This implies that ( p( Rt ), Q( Rt )) lies on the p−1( Rt ,Q( Rt ))schedule, where Q( Rt ) > Q∗ and p( Rt ) > p∗.

18

reliability under preventive maintaince

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