...buyer that a product is as has been represented. It may be condidered to be a contractual agreement between buyers and sellers that is entered into on the sale of the product (Blischke and Murthy, 1996). A warranty can be viewed as a signal that conveys information about product reliability and as such it serves as an important marketing tool.

Servicing a warranty involves additional costs to the manufacturer and this has an impact on the profit levels. The warranty cost is a function of the product reliability and can be reduced through the use of various techniques to improve product reliability (such as reliability improvement through development and redundancy processes). However, this improvement generally involves incurring additional costs and as a result the manufacturing cost increases, which may imply a higher selling price. The design phase must incorporate marketing variables and the design and marketing decisions must be made jointly.

In this paper, we develop a model to determine the optimal sale price, warranty period and product reliability to maximize the discounted profit for a repairable product sold with a free replace-repair warranty policy. We assume that the learning effects result in the unit manufacturing cost decreasing with total sales volume. We consider two scenarios for the pricing and warranty. In the first scenario the price and warranty period are constant whereas in the second scenario they change over the product life cycle. The outline of this paper is as follows. Section 2 presents a brief literature review on marketing and/or technical strategy decisions involving pricing and/or warranty. In Section 3, we give the mathematical details of the model formulation. Section 4 examines the optimal policies for the proposed model. Section 5 deals with illustrative numerical examples. Finally, in Section 6, we draw conclusions and highlight of some extensions worthy of future investigation.

2. Review of the literature

There are many different aspects to warranty decisions that have been studied by researchers from diverse disciplines because of their importance. Blischke and Murthy (1996) have highlighted several of these issues and a review of the literature published in the last 10 years has been performed by Murthy and Djamaludin (2002). Many researchers have investigated optimal strategies that link engineering issues (such as reliability improvement through redundancy and development, maintenance, testing policies, burn-in, etc.) with warranty to either maximize the manufacturers' profit or minimize the total cost. In this context we highlight the papers of Murthy and Nguyen (1987), Mi (1997), Monga and Zuo (1998), Pohl and Dietrich (1999), Hussain and Murthy (2003) and Shue and Chien (2005).

Many researchers have used sales models that consider the determination of optimal marketing and/or technical strategies, such as advertising, price, quality, etc. We highlight the papers of Dockner and Gaunersdorfer (1996), Mendez and Narasimhan (1996), Teng and Thompson (1996), Zhao and Zheng (2000), Chen and Chu (2001) and Kamrad et al. (2005). Price and warranty are two major commercial variables that influence sales decisions and ultimately the profit levels. Glickman and Berger (1976) presented a model to determine the optimal price and warranty period that maximizes a manufacturer's profit for a failure-free warranty policy. Nguyen and Murthy (1988) developed a model for obtaining the optimal reliability allocation taking into account the manufacturing and warranty costs. Murthy (1990) developed a model to obtain the optimal price, warranty period and product reliability to maximize a manufacturer's profit. In these models, the manufacturing cost and the price are assumed to be constant over the product life cycle. Teng and Thompson (1996) considered the optimal price and quality policies for the introduction of a new product. They assumed that the unit cost declines along the learning curve and investigated the dynamics between price and quality for the new product. Lin and Shue (2005) and Wu et al. (2006) modified the Teng-Thompson price-quality model into a price-warranty decision model in which the warranty length replaces the quality level. Wu et al. (2006) derived a normal lifetime distributed product whereas Lin and Shue (2005) investigated numerous basic lifetime distributions. DeCroix (1999) presented a game-theoretic model that represents firms in an oligopoly that choose warranty, reliability and price levels for their goods and examines the Nash equilibria for this game.

3. Model formulation

3.1. Nomenclature and Notation

[theta] = reliability parameter (design variable);

F(t, [theta]) = cumulative distribution function for the first time to failure;

f(t, [theta]) = probability density function associated with F(t, [theta]);

r(t, [theta]) = hazard function associated with F(t, [theta]);

P(t) = unit sale price at time t (marketing variable);

W(t) = duration of warranty period for products sold at time t (marketing variable);

[omega](W, [theta]) = expected warranty cost per unit sold;

c(t, [theta]) = total manufacturing cost of unit product, which includes unit development cost and production cost--to indicate that this is a function of [theta];

[c.sub.m](t, [theta]) = unit production cost--to indicate that this is a function of [theta];

[c.sub.r] = expected cost of servicing a warranty claim;

[delta] = discount rate;

L = product life cycle;

q(t) = sales (production) rate at time t, [less than or equal to] t < L;

Q(t) = accumulated sales in [0, t], [less than or equal to] t < L;

[Q.sub.0] = parameter characterizing past sales or production experience;

[Q.sub.M] = maximum sales potential;

[psi] = parameter to reflect the relative influence of innovators in the sales model;

S(t, [theta]) = expected number of failures for an item over the interval [0, t);

[[pi].sub.L] = expected discounted integral profit during the interval [0, L).

3.2. Product warranty policy

Manufacturers provide many types of warranty policies for different products. In this paper, we consider the Free Replace-repair Warrenty (FRW) policy which was defined as follows by Blischke and Murthy (1996).

"Under the policy, the seller agrees to repair or provide replacements for failed items free of charge up to a time W...

NOTE: All illustrations and photos have been removed from this article.



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