Optimal strategies for price, warranty length, and

Scientia Iranica E (2013) 20(6), 2247{2258

Sharif University of TechnologyScientia Iranica

Transactions E: Industrial Engineeringwww.scientiairanica.com

Optimal strategies for price, warranty length, andproduction rate of a new product with learningproduction cost

S. Faridimehr and S.T.A. Niaki�

Department of Industrial Engineering, Sharif University of Technology, Tehran, P.O. Box 11155-9414, Iran.

Received 27 August 2012; received in revised form 11 March 2013; accepted 6 May 2013

KEYWORDSStatic markets;Dynamic markets;Optimal strategy;Cost functions;Maximum principle.

Abstract. This study investigates optimal strategies for price, warranty length, andproduction rate of a new product in which both static markets for non-durable and dynamicmarkets for durable products are involved. The mathematical model incorporates boththe demand and the cost functions including production, warranty length, and inventorycosts. Using the maximum principle approach, the optimal strategies and interactionsamong price, warranty length, and production rate in both markets are analyzed usingsome propositions. The analysis shows that to maximize pro�ts in all cases, the price, thewarranty length, and the production rate all must go up simultaneously, or one of themmust increase and the other two must decrease concurrently.c 2013 Sharif University of Technology. All rights reserved.

1. Introduction and literature review

Since the conditions change rapidly in the market andthe goal of many �rms is to maximize pro�ts, pricingof products has become an important decision [1]. Onone hand, high price for a product puts it out ofreach for many consumers, resulting in low revenue.On the other hand, a low price leads to a low pro�tmargin. Furthermore, in many situations, the price istreated as a sign of product quality, where standardeconomic approaches to product quality show thatthere is direct and upward sloping relationship betweenproduct quality and its price [2].

As products of several �rms are similar to eachother based on di�erent aspects, it is di�cult forconsumers to compare them just through their priceand super�cial characteristics [3]. In these situations,warranty length of a product is another important

*. Corresponding author.E-mail addresses: sina [email protected] (S.Faridimehr); [email protected] (S.T.A. Niaki)

factor for consumers purchasing the product, sinceextended warranty helps them to be more con�denttoward its quality. Buyer's willingness for a warrantystrategy that protects them against high costs ofproduct failure, makes warranty length an importantmeans in marketing and product development.

Abiding by warranty commitment and replacinga failed product with a new one need a good warrantyservice management in which product inventory isa deciding factor. Providing a new product for aconsumer in the time of request has a good e�ect on hisperspective. Adversely, lacking enough product inven-tories admittedly a�ects the consumer's perspective inan opposite way. Nonetheless, inventory managementin warranty policies, in contrast to its importance, hasbeen paid very little attention in literature [4,5].

The learning production cost refers to the e�ect ofproduction size of a good on its unit price. Arrow [6]and Rosen [7] concluded that if the cumulative pro-duction doubles, then the unit production cost couldbe decreased by a factor of 10 to 50. As a result, theproduction size a�ects the pricing strategy.

2248 S. Faridimehr and S.T.A. Niaki/Scientia Iranica, Transactions E: Industrial Engineering 20 (2013) 2247{2258

In 1960s, Fourt & Woodlock [8], Mans�eld [9]and Bass [10] proposed information di�usion modelsin attempts to make the lifecycle phenomenon forconsumer durable. These research works stimulatedlater researches, and since then, di�usion theory ofnew product has been paid attention to in marketingknowledge, as well as consumer behavioral analysis.

Economists always criticized the Bass [10] modelto be incomplete, because of not taking into accountthe e�ects of economic variables. Later, Bass etal. [11] tried to enter marketing variables in theoriginal model and proposed their generalized Bassmodel. Other extensions tried to generalize thismodel and consider important marketing variables suchas price, advertising, distribution channels, and thelike.

In some cases, only the price variable has beenconsidered in the Bass model. Robinson & Lakhani [12]showed that the classic marginal pricing could notbe e�ective in dynamic business environment. Sincemany �rms will be in the market for long-term period,they suggested that the use of dynamic pricing getsmore pro�ts during this period. By combining learningcurve and di�usion theory of new product, Bayus [13]tried to maximize the pro�t of a durable product, andshowed the costs and prices decrease with an increase incumulative production. However, the product demandis the one that increases �rst and then decreases inits lifecycle. Kalish [14] studied the pricing of anew product in a monopolistic market to maximizethe pro�t function. In his research, production costis a declining function of cumulative production anddemand is a function of both cumulative sales andprice.

While many researchers considered more than onevariable in the di�usion model, here we review theliterature that deals with price and warranty length. Tomaximize the pro�t of a product sold under warranty,Glickman and Berger [15] proposed a model in whichthe demand function depends exponentially on priceand warranty length. Mesak [16] o�ered some di�usionmodels that incorporate price and warranty lengthto determine the optimal price and warranty-lengthpolicies during the lifecycle of a product. Teng andThompson [17] considered a model to maximize thepro�t of a monopolist by determining the optimal priceand quality strategies. Lin and Shue [18] tried toreach the optimal price and warranty-length policiesin a dynamic market. They maximized the pro�tfunction based on some basic lifetime distributions ofthe product. Wu et al. [19] proposed a model for amanufacturer to maximize his pro�t by determiningthe optimal price and warranty-length strategies fora product with normal lifetime distribution for whichFaridimehr & Niaki [20] wrote an erratum. In addition,Wu et al. [21] developed a decision model to maximize

the pro�t of a producer in a static demand marketby determining the optimal price, warranty lengthand production rate policies based on the Weibulllifetime distribution. While the assumptions of positiveproduction rate and negative second derivatives of thedemand function with respect to price and warranty-length were taken in their research, Faridimehr andNiaki [22] extended their model to have no restrictionson both.

In this paper, we develop a model to derive theoptimal price, warranty length and production ratestrategies for a new product to maximize pro�ts inboth static and dynamic markets in which the discountrate is assumed zero and the learning productioncost is present. Although the non-zero discount rateassumption might seem unreasonable, it can be anapproximation in markets with a very low discountrate. This means that there is no di�erence betweenpresent and future pro�ts in long-term horizon. Theoptimal strategies provide guidelines for manufacturersto increase or decrease the price, the warranty lengthand the production rate that maximize their pro�tduring product lifecycle.

The remainder of the paper is organized as fol-lows: In Section 2, the mathematical formulation ofthe problem at hand is given. The detailed informationabout the model comes in Section 3. In this section,a solution approach based on maximum principle ispresented and optimal strategies are analyzed in bothstatic and dynamic markets. Finally, conclusions aremade in Section 4.

2. Problem formulation

Assuming the remaining unsold product at the end ofthe period has no value, the developed model of thissection includes the demand and the cost functions.This means that the manufacturer only pays attentionto his pro�t during the period rather than to his pro�tafter it [16]. In what comes next, the demand functionis �rst derived. Then, the cost function is developedbased on production, warranty length and inventorycosts.

2.1. Demand functionThe demand function (f) considered in this paper isanalyzed in both static and dynamic markets. Whilein both markets, this function depends on price andwarranty length, in the dynamic market it dependson cumulative sales as well. We assume the demandfunction is twice as much di�erentiable and decreaseswith an increase in price and increases with an increasein warranty length (as the case usually happens inpractice.) Moreover, the rate of increase in demandsubject to increase in warranty length decreases byincreasing price. In other words:

S. Faridimehr and S.T.A. Niaki/Scientia Iranica, Transactions E: Industrial Engineering 20 (2013) 2247{2258 2249

fp =@f@p

< 0;

fw =@f@w

> 0;

fpw =@2f@p@w

< 0;

where fp is the partial derivatives of the demandfunction with respect to the price, fw is the partialderivatives of the demand function with respect tothe warranty length, and fpw represents the secondderivatives of the demand function with respect to priceand warranty-length. The optimal strategies basedon both demand functions are discussed in subsequentsections.

2.2. Cost functionThe cost function that is used in the mathematicalmodel involves two components: (1) the productionand warranty-length cost and (2) the inventory cost.The following two subsections are devoted to thesecomponents.

2.2.1. Production and warranty length cost functionHere, we assume that the unit cost, C, is a functionof the warranty length at time t, w(t), the cumulativeproduction volume at time t, X(t), and the productionrate at time t, q(t). Similar to Lin [23], C is assumedtwice as much di�erentiable, satisfying the followingconditions:

Cw =@C@w

> 0;

Cww =@2C@w2 > 0;

Cq =@C@q

> 0;

Cqq =@2C@q2 > 0:

In other words, the unit cost function is a strictly in-creasing convex function with respect to both warrantylength and production rate. However, it decreases withan increase in the cumulative production, where therate of decrease increases. Mathematically we have:

CX =@C@X

< 0;

CXX =@2

@X2 > 0:

Although the unit cost function increases with anincrease in the warranty length, the rate of increase

increases with an increase in the production rate anddecreases with an increase in the cumulative produc-tion. In addition, although the unit cost functionincreases with an increase in the production rate, therate of this increase decreases by an increase in thecumulative production. These characteristics can beshown as:

Cwq =@2C@w@q

> 0;

CwX =@2C@w@X

< 0;

CqX =@2C@q@X

< 0:

2.2.2. Inventory costOn one hand, having extra product in the warehousecauses extra holding cost for the manufacturer. Onthe other hand, insu�cient inventory results in delaysin o�ering warranty service to the customers. Thus,inventory management is a key component of managingservice warranty.

Let Q(t) be the cumulative sales at time t, f(t)be the sales rate at time t, and I(t) be the inventoryvolume at time t. According to Feichtinger andHartl [24], Jorgensen et al. [25] and Warburton [26],the equations of inventory balance are:

I (t) = X (t)�Q (t) ;

dIdt

= q (t)� f (t) : (1)

Moreover, similar to Jorgensen et al. [25], we proposea dynamic system with Q, X and I as state variablesin which the possibility of shortage occurrence duringthe period is considered. Besides, in contrast with Wuet al. [21], we assume the holding and the shortage costper unit of the product per period, �1, are equal. Then,the holding or the shortage cost function, h(I), can bemodeled as:

h(I) =

8><>:�1I(t); if I(t) > 00; if I(t) = 0��1I(t); if I(t) < 0

: (2)

2.3. Mathematical modelAiming to maximize the pro�t in period T , the math-ematical formulation of the problem becomes:

MaxJ =TZ

0

f[p (t)� C (w (t) ; q (t) ; X (t))]� f (t)

�h (I)g dt: (3)


Subject to:

dQdt

= f (t) ;

dXdt

= q (t) ; (4)

where p(t) is the price function of the product at timet.

3. A solution method

The mathematical model in Eqs. (3) and (4) can besolved using the maximum principle method [27]. Thepresent value of Hamiltonian, H, is formulated as:

H = (p� C) f � h (I) + �1dQdt

+ �2dXdt;

H = (p� C + �1) f � h (I) + �2q; (5)

where �1 and �2 are the current values of adjointvariables (shadow price of f and q respectively) thatmust satisfy the following di�erential equations:

d�1

dt=�HQ=� (p� C+�1) fQ+ hQ (I) ;

�1 (T )=0; (6)

d�2

dt= �HX = CXf + hX (I) ;

�2 (T ) = 0: (7)

In the above formulae HQ = @H@Q , HX = @H

@X , and fQ =@f@Q .

To have an optimal solution, the partial deriva-tives of H with respect to the main variables, p, w and

q must be zero. In other words:

Hp =@H@p

= 0;

Hw =@H@w

= 0;

Hq =@H@q

= 0: (8)

In addition, the non-singular Hessian matrix of thesecond-order partial derivatives of H with respect toprice, warranty length and production rate must benegative de�nite. This matrix is:

HM =

24 Hpp Hpw HpqHwp Hww HwqHqp Hqw Hqq

35 :Thus, the following conditions must hold:

Hpp < 0;�� Hpp HpwHwp Hww

�� > 0;�� Hpp Hpw HpqHwp Hww HwqHqp Hqw Hqq

�� < 0:

Hence, the optimal decisions can be obtained usingCramer's rule shown in Box 1 (see Appendix A for theproof).

To discuss the optimal strategies, we �rst considerthe sign of Hpwq to determine the sign of the derivativesof price, warranty length and production rate, withrespect to time. For the considered demand function,we have:

Hpwq = �Cwqfp � Cqfpw: (10)

Then, based on the assumptions on f and C givenin Subsections 2.1 and 2.2.1, Hpwq is always positive.Therefore, in order to increase the current value of

264 dpdtdwdtdqdt

375 = �HM�1

�264 f (fQ + (p� C + �1) fpQ)� CXqfp + fp (� (p� C + �1) fQ + hQ (I))f (�CwfQ + (p� C + �1) fwQ)� q (CwXf + CXfw) + fw (� (p� C + �1) fQ + hQ (I))

�CqffQ � CqXqf + CXf + hX (I)

375 : (9)

where HM�1 is the inverse of HM .

Box I.


Table 1. Optimal policies of price, warranty length and production rate in static market.

Conditions Results

(�CXqh0k)�HwwHqq �H2

wq�

+ h

k (CX � CqXq) (HpwHwq �HwwHpq)�q (CXk0 + CwXk) (HpqHwq �HpwHqq)

!> 0

and p "; w "; q "(�CXqh0k) (HpqHwq �HpwHqq) + h

k (CX � CqXq) (HpwHpq �HppHwq)�q (CXk0 + CwXk)

�HppHqq �H2

pq� !

> 0


wq�

+ h


!< 0

and p #; w "; q #(�CXqh0k) (HpqHwq �HpwHqq) + h


�HppHqq �H2

pq� !

> 0


wq�

+ h


!> 0

and p "; w #; q #(�CXqh0k) (HpqHwq �HpwHqq) + h


�HppHqq �H2

pq� !

< 0


wq�

+ h


!< 0

and p #; w #; q "(�CXqh0k) (HpqHwq �HpwHqq) + h


�HppHqq �H2

pq� !

< 0

Hamiltonian, either all of the time derivatives of price,warranty length and production rate must be positiveor two of them must be negative and one positive.

In the subsequent sections, the optimal strategyon price, warranty length and production rate arediscussed in both static and dynamic markets.

3.1. Static marketThe static market belongs to non-durable productsin which the word-of-mouth is not important in thedevelopment of the product and that almost the wholelifecycle of the product lies in the maturity phase.In this type of market, there is almost no saturatione�ect; therefore, demand is just a function of price andwarranty length. Moreover, fQ = 0, fpQ=0 and fwQ=0.

Assuming the inventory cost of an item is rel-atively low and negligible; the optimal strategies inthis type of market can be characterized through thefollowing proposition.

Proposition 1: For a given f = h(p)k(w), the optimal

policies for price, warranty length, and production rateare characterized in Table 1.

Proof: See Appendix B.If the interaction between any two of the warranty

length, the cumulative production, and the productionrate can be relatively low, then more clear results areobtained in Table 2.

Proof: See Appendix C.

3.2. Dynamic marketIn the dynamic market that is for durable products,demand exhibits di�usion and saturation e�ects, andthe word-of-mouth is an important factor in di�usion ofproduct in a market. Moreover, current demand a�ectsthe quantity of future demand; therefore, the demandis a function of price, warranty length and cumulativesales.

In addition to the assumptions considered for zerovalue of unsold products, the second assumption rising


Table 2. Optimal policies of price, warranty length and production rate in static market (low interactions between anytwo of warranty length, cumulative production and production rate).

Conditions Results

(�qh0k)�HwwHqq �H2

wq�

+ h

k (HpwHwq �HwwHpq)�qk0 (HpqHwq �HpwHqq)

!> 0

and p #; w #; q "(�qh0k) (HpqHwq �HpwHqq) + h

k (HpwHpq �HppHwq)�qk0�HppHqq �H2

pq� !

> 0


wq�

+ h


!< 0

and p "; w #; q #(�qh0k) (HpqHwq �HpwHqq) + h


pq� !

> 0


wq�

+ h


!> 0

and p #; w "; q #(�qh0k) (HpqHwq �HpwHqq) + h


pq� !

< 0


wq�

+ h


!< 0

and p "; w "; q "(�qh0k) (HpqHwq �HpwHqq) + h


pq� !

< 0

here is that each person purchases the product onlyonce in the period. This assumption has a higherimpact in short planning horizons [16]. Furthermore,similar to static demand markets, to reach results thatare more tractable, we assume that the unit inventorycost of the product is relatively low and negligible. Asa result, the following proposition is made to obtainoptimal policies.

Proposition 2: For a given f = h(p)k(w)g(Q),the optimal policies for price, warranty length andproduction rate are characterized in Table 3.

Proof: See Appendix D.To reach more tractable results, one can con-

sider the case in which the production and warrantylength cost functions are independent of the cumu-lative production volume. It means that C can bede�ned as C(w(t); q(t)). Besides, if we assume thatthe interaction between the warranty length and the

production rate is relatively low, as in the staticmarket, the optimal policies can be characterized asgiven in Table 4.

Proof: See Appendix E.The above proposition shows that the optimal

price, warranty length and production rate strategiesare dependent on each other. Note that g0(Q) ispositive in the di�usion phase and negative during thesaturation phase.

4. Conclusion and recommendation for futureworks

In this paper, a mathematical decision model for a newproduct (both non-durable and durable) was proposedto determine the optimal price, warranty length andproduction rate strategies, where the discount rate wasassumed zero. Two cost functions, (1) productionand warranty length and (2) inventory were consid-


Table 3. Optimal policies of price, warranty length and production rate in dynamic market.

Conditions Di�usion e�ect Saturation e�ect

h0k�h2k � CXqh0

g0� �HwwHqq �H2

wq�

+h�h2kk0 � qh0(CXk0+CwXk)

g0

�(HpqHwq �HpwHqq)

+hh0k��Cqhk + (�CqXq+CX)

g0

�(HpwHwq �HwwHpq) > 0

and p #; w #; q " p "; w "; q "h0k

�h2k � CXqh0

g0�

(HpqHwq �HpwHqq)+h�h2kk0 � qh0(CXk0+CwXk)

g0

��HppHqq �H2

pq�


g0

�(HpwHpq �HppHwq) > 0

h0k�h2k � CXqh0


wq�


g0



g0

�(HpwHwq �HwwHpq) < 0

and p "; w #; q # p #; w "; q #h0k

�h2k � CXqh0

g0�


g0

��HppHqq �H2

pq�


g0

�(HpwHpq �HppHwq) > 0

h0k�h2k � CXqh0


wq�


g0



g0

�(HpwHwq �HwwHpq) > 0

and p #; w "; q # p "; w #; q #h0k

�h2k � CXqh0

g0�


g0

��HppHqq �H2

pq�


g0

�(HpwHpq �HppHwq) < 0

h0k�h2k � CXqh0


wq�


g0



g0

�(HpwHwq �HwwHpq) < 0

and p "; w "; q " p #; w #; q "h0k

�h2k � CXqh0

g0�


g0

��HppHqq �H2

pq�


g0

�(HpwHpq �HppHwq) < 0


Table 4. Optimal policies of price, warranty length and production rate in dynamic market (C(w(t); q(t)) and lowinteraction between warranty length and production rate).

Conditions Di�usion e�ect Saturation e�ect

h0k��HwwHqq �H2

wq�� Cq (HpwHwq �HwwHpq)�

+hk0 (HpqHwq �HpwHqq) > 0and p #; w #; q " p "; w "; q "h0k ((HpqHwq �HpwHqq)� Cq (HpwHpq �HppHwq))

+hk0�HppHqq �H2

pq�> 0



+hk0 (HpqHwq �HpwHqq) < 0and p "; w #; q # p #; w "; q #h0k ((HpqHwq �HpwHqq)� Cq (HpwHpq �HppHwq))

+hk0�HppHqq �H2

pq�> 0



+hk0 (HpqHwq �HpwHqq) > 0and p #; w "; q # p "; w #; q #h0k ((HpqHwq �HpwHqq)� Cq (HpwHpq �HppHwq))

+hk0�HppHqq �H2

pq�< 0



+hk0 (HpqHwq �HpwHqq) < 0and p "; w "; q " p #; w #; q "h0k ((HpqHwq �HpwHqq)� Cq (HpwHpq �HppHwq))

+hk0�HppHqq �H2

pq�< 0

ered in the proposed model. The �rst is a functionof warranty length, production rate and cumulativeproduction volume, and the later includes both holdingand shortage costs. The maximum principle methodwas then utilized to obtain the optimal strategies.Next, the optimal strategies were discussed both forstatic market, the market of non-durable products,and dynamic market which is for durable products.Before analyzing optimal strategies in both markets, weconcluded that the product of the price derivatives withrespect to time, the warranty length derivatives withrespect to time, and the production rate derivativeswith respect to time must be positive. Moreover, weconsidered the optimal strategies, when the warrantylength, production rate and cumulative production donot interact with each other in the production andwarranty length cost function. This assumption endsin more clear and tractable results.

While the marketing variables considered in theproposed model were price and warranty length, futureworks on incorporating other marketing variables such

as advertising and distribution are recommended. Inaddition, future study can focus on various warrantystrategies, and compare them with each other. Ana-lyzing the cases of duopoly and oligopoly markets willbe an interesting �eld of research because in reality,many markets include more than one producer.

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Appendix A

In order to analyze the relationships among price,warranty length and production rate, we �rst take thetime derivatives on both sides of Eqs. (8) to reach:

ddt

�@H@p

�=@2H@p2

dpdt

+@2H@p@w

dwdt

+@2H@p@q

dqdt

+@2H@p@Q

dQdt

+@2H@p@X

dXdt

+@2H@p@�1

d�1

dt

+@2H@p@�2

d�2

dt= 0; (A.1)

ddt

�@H@w

�=@2H@w@p

dpdt

+@2H@w2

dwdt

+@2H@w@q

dqdt

+@2H@w@Q

dQdt

+@2H@w@X

dXdt

+@2H@w@�1

d�1

dt

+@2H@w@�2

d�2

dt= 0; (A.2)


ddt

�@H@q

�=@2H@q@p

dpdt

+@2H@q@w

dwdt

+@2H@q2

dqdt

+@2H@q@Q

dQdt

+@2H@q@X

dXdt

+@2H@q@�1

d�1

dt

+@2H@q@�2

d�2

dt= 0:

(A.3)

Substituting dQdt = f , dX

dt = q and d�1dt and d�2

dt fromEqs. (6) and (7) in Eqs. (A.1), (A.2), and (A.3), wehave:

ddt

�@H@p

�=@2H@p2

dpdt

+@2H@p@w

dwdt

+@2H@p@q

dqdt

+@2H@p@Q

f +@2H@p@X

q +@2H@p@�1

(�HQ)

+@2H@p@�2

(�HX) = 0;(A.4)

ddt

�@H@w

�=@2H@w@p

dpdt

+@2H@w2

dwdt

+@2H@w@q

dqdt

+@2H@w@Q

f +@2H@w@X

q +@2H@w@�1

(�HQ)

+@2H@w@�2

(�HX) = 0; (A.5)

ddt

�@H@q

�=@2H@q@p

dpdt

+@2H@q@w

dwdt

+@2H@q2

dqdt

+@2H@q@Q

f +@2H@q@X

q +@2H@q@�1

(�HQ)

+@2H@q@�2

(�HX) = 0:(A.6)

Then, the following matrix can be constructed basedon Eqs. (A.4), (A.5), and (A.6).

264 @2H@p2

@2H@p@w

@2H@p@q

@2H@w@p

@2H@w2

@2H@w@q

@2H@q@p

@2H@q@w

@2H@q2

37524 dpdtdwdtdqdt

35 =

�264 @2H

@p@Qf + @2H@p@X q � @2H

@p@�1

@H@Q � @2H

@p@�2

@H@X

@2H@w@Qf + @2H

@w@X q � @2H@w@�1

@H@Q � @2H

@w@�2

@H@X

@2H@w@Qf + @2H

@w@X q � @2H@w@�1

@H@Q � @2H

@w@�2

@H@X

375 :(A.7)

So:24 dpdtdwdtdqdt

35 = �264 @2H

@p2@2H@p@w

@2H@p@q

@2H@w@p

@2H@w2

@2H@w@q

@2H@q@p

@2H@q@w

@2H@q2

375�1

264 @2H@p@Qf + @2H

@p@X q � @2H@p@�1

@H@Q � @2H

@p@�2

@H@X

@2H@w@Qf + @2H

@w@X q � @2H@w@�1

@H@Q � @2H

@w@�2

@H@X

@2H@w@Qf + @2H

@w@X q � @2H@w@�1

@H@Q � @2H

@w@�2

@H@X

375 :(A.8)

Eq. (A.8) leads to Eq. (A.9) by calculating thepartial derivatives of current Hamiltonian. Note that:

HM�1 =

264 @2H@p2

@2H@p@w

@2H@p@q

@2H@w@p

@2H@w2

@2H@w@q

@2H@q@p

@2H@q@w

@2H@q2

375�1

=1�

24 HwwHqq �H2wq HpqHwq �HpwHqq

HpqHwq �HpwHqq HppHqq �H2pq

HpwHwq �HwwHpq HpwHpq �HppHwq

HpwHwq �HwwHpqHpwHpq �HppHwqHppHww �H2

pw

35 ;(A.9)

where � is the determinant of the HM matrix.

Appendix B

Considering the fact that in the static markets, thedemand function does not depend on the cumulativesales, the partial derivatives of price, warranty length,and production rate can be found as follows:24 dp

dtdwdtdqdt

35 = � 1�

24 HwwHqq �H2wq

HpqHwq �HpwHqqHpwHwq �HwwHpq

HpqHwq �HpwHqq HpwHwq �HwwHpqHppHqq �H2

pq HpwHpq �HppHwqHpwHpq �HppHwq HppHww �H2

pw

35�24 h2k2gg0

h3kk0gg0h0�Cqh2k2gg0

35 : (B.1)

Thus,

dpdt

= � 1�

�(�CXqh0k)

�HwwHqq �H2

wq�

+h�k (CX � CqXq) (HpwHwq �HwwHpq)�q (CXk0 + CwXk) (HpqHwq �HpwHqq)

��;

(B.2)


dwdt

= � 1�

�(�CXqh0k) (HpqHwq �HpwHqq)

+h�k (CX � CqXq) (HpwHpq �HppHwq)�q (CXk0 + CwXk)

�HppHqq �H2

pq� ��

;(B.3)

dqdt

= � 1�

�(�CXqh0k) (HpwHwq �HwwHpq)

+h�k (CX � CqXq) �HppHww �H2

pw��

q (CXk0 + CwXk) (HpwHpq �HppHwq)

��:

(B.4)

Appendix C

Knowing that in the static markets the demand func-tion does not depend on the cumulative sales and alsoconsidering the interaction terms between any two ofwarranty length, cumulative production and produc-tion rate, the partial time derivatives of price, warrantylength and production are obtained as follows:24 dp

dtdwdtdqdt

35 = � 1�

24 HwwHqq �H2wq




pw

3524 �CXqh0k�CXqhk

CXhk

35 : (C.1)

In other words,

dpdt

=� CX�

�(�qh0k)

�HwwHqq �H2

wq�

+ h�k (HpwHwq �HwwHpq)�qk0 (HpqHwq �HpwHqq)

��; (C.2)

dwdt

=� CX�

�(�qh0k) (HpqHwq �HpwHqq)

+ h�k (HpwHpq �HppHwq)�qk0�HppHqq �H2

pq� ��

; (C.3)

dqdt

=� CX�

�(�qh0k) (HpwHwq �HwwHpq)

+ h�k�HppHww �H2

pw��

qk0 (HpwHpq �HppHwq)

��: (C.4)

Appendix D

Since the demand function in dynamic markets de-pends not only on price and warranty length, but alsoon cumulative sales, the partial time derivatives ofprice, warranty length, and production rate are:24 dp

dtdwdtdqdt

35 = � 1�

24 HwwHqq �H2wq




pw

35

�

266664kgg0

�h2k � CXqh0

g0�

hgg0h0

�h2kk0 � qh0(CXk0+CwXk)

g0

�hkgg0

��Cqhk + (�CqXq+CX)g0

�377775 : (D.1)

In other words,

dpdt

= � gg0�h00BBBBBBBB@

h0k�h2k � CXqh0


wq�

+h�h2kk0� qh0(CXk0+CwXk)

g0

�(HpqHwq�HpwHqq)+hh0k

��Cqhk+ (�CqXq+CX)g0

�(HpwHwq�HwwHpq)

1CCCCCCCCA ; (D.2)

dwdt


h0k�h2k � CXqh0

g0�

(HpqHwq �HpwHqq)


g0

��HppHqq�H2

pq�

+hh0k��Cqhk+ (�CqXq+CX)

g0�

(HpwHpq�HppHwq)

1CCCCCCCCA ; (D.3)

dqdt


h0k�h2k � CXqh0

g0�

(HpwHwq �HwwHpq)


g0

�(HpwHpq�HppHwq)+hh0k

��Cqhk+ (�CqXq+CX)g0

��HppHww�H2

pw�

1CCCCCCCCA :(D.4)


Appendix E

Since the demand function in dynamic markets de-pends on price, warranty length and cumulative sales,assuming the production and warranty-length cost afunction of warranty length and production rate (nota function of the cumulative production volume) andthat the interaction between the warranty length andproduction rate in the cost function is negligible, thepartial time derivatives of price, warranty length andproduction rate are:24 dp

dtdwdtdqdt

35 =

� 1�

24 HwwHqq �H2wq




pw

35�24 h2k2gg0

h3kk0gg0h0�Cqh2k2gg0

35 : (E.1)

In other words,

dpdt

= �h2kgg0�h00BBBB@

h0k��

HwwHqq �H2wq�

�Cq (HpwHwq �HwwHpq)�

+hk0 (HpqHwq �HpwHqq)

1CCCCA ; (E.2)

dwdt


h0k�

(HpqHwq �HpwHqq)

�Cq (HpwHpq �HppHwq)�

+hk0�HppHqq �H2

pq�

1CCCCA ; (E.3)

dqdt


h0k�

(HpwHwq �HwwHpq)

�Cq �HppHww �H2pw��

+hk0 (HpwHpq �HppHwq)

1CCCCA : (E.4)

Biographies

Sina Faridimehr received his B.S. and M.S. degreesin 2010 and 2012, respectively, both in Industrial Engi-neering from Sharif University of Technology, Tehran,Iran. His research interests include economic and�nancial analysis.

Seyed Taghi Akhavan Niaki received his B.S.degree in Industrial Engineering from Sharif Universityof Technology, Tehran, Iran, in 1979, and his M.S.and Ph.D. degrees, both in Industrial Engineering,from West Virginia University, USA, in 1989 and 1992,respectively. He is currently Professor of Industrial En-gineering at Sharif University of Technology, Tehran,Iran. His research interests include: simulation model-ing and analysis, applied statistics, multivariate qualitycontrol and operations research. Before joining SharifUniversity of Technology, he worked as a SystemsEngineer and Quality Control Manager for the IranianElectric Meters Company. He is also a member of ��.

Documents

Optimal strategies for price, warranty length, and