A parallel redundancy has been researched in a great amount of literature on reliability theory. In the real circumstances, however, it is seldom observed except some systems that require much higher reliability. This is because most of the literature only look at the manufacturer’s point of view. The present study carries out an economic analysis of an n-unit parallel redundant system against a single unit system based on a Stackelberg game formulation considering both the consumer’s viewpoint and the manufacturer’s one. It clarifies quantitatively in what situation the manufacturer can increase his profit by dealing in the parallel redundant system.
Parallelizing of units in a system is an effective method for improving system reliability. Since this method is simple to analyze for a variety of systems, it is widely introduced in a great amount of literature on reliability theory (Barlow and Proschan, 1967 and Osaki, 2002).
Parallel systems are, however, not so widely employed in the actual environments as a great amount of literature have treated. The authors speculate that this is because most of the literature miss the consumer’s point of view and because the consumer would purchase a parallel system only when it is worth purchasing compared with a single unit system. In redundancy allocation problem (e.g. Hsieh, 2003, Kuo and Prasad, 2000 and Liang and Smith, 2004), for instance, the manufacturer seeks the optimal selection of system components in series–parallel systems considering reliability, manufacturing cost and weight, but he never takes into account the consumer’s behavior for the target system.
The present study makes a comparison between a single unit system and an n-unit parallel redundant system both from the consumer’s viewpoint and from the manufacturer’s one. In this study, we consider a situation where a manufacturer produces both single unit systems and n-unit parallel redundant systems to deal in them under a monopoly. It is postulated that the n-unit parallel redundant system consists of n identical units to that of a single unit system. It should, furthermore, be noted that since we have assumed a monopoly, the manufacturer can determine the prices of these systems (e.g. Mankiw, 2007, chap. 15). Hence we can formulate the problem within a Stackelberg game framework ( Fundenburg and Tirole, 1991, Gibbons, 1992 and Osborne and Rubinstein, 1994), letting the manufacturer and the consumers be the leader and the followers, respectively. Two cases are discussed according to the consumer’s revenue structures: Case 1 deals with a circumstance where the consumer’s revenue is proportional to the system life, while under Case 2, the consumer’s revenue significantly depends on the system reliability.
The remaining part of the paper is organized as follows. Section 2 describes the proposed model, and Sections 3 and 4 formulate and analyze the model to derive the manufacture’s optimal strategy in Cases 1 and 2, respectively. Numerical examples are also illustrated in Section 5. Section 6 concludes our paper and describes future problems. Detailed proofs are presented in the appendices.
An n-unit parallel system provides us with higher reliability than a single unit system. In this sense, it is needless to say that an n-unit parallel system is better than a single unit system. However, this is not necessarily true for manufacturer’s profit since consumers take not only system reliability but also its price into account.
This study conducted an economic analysis of an n-unit parallel system against a single unit system theoretically and quantitatively on the basis of the proposed model. The analysis was carried out considering the consumer’s viewpoint as well as the manufacturer’s one under a monopoly. Since we assumed a monopoly, we discussed the problem within a Stackelberg game framework by regarding the manufacturer and the consumer as a leader and a follower, respectively.
It was demonstrated theoretically and through numerical examples that an n-unit parallel system is not very efficient for the manufacturer in case the consumer’s revenue is proportional to the system life. Although the n-unit system could be profitable to the manufacturer on some conditions, it was clarified that when such conditions should be satisfied, other firms would enter the market and would compete with the initial firm to change the environments where the conditions would no longer be satisfied.
On the contrary, it was also demonstrated that an n-unit system is more profitable to the manufacturer than a single unit system in case the consumer’s revenue is significantly affected by the system reliability.
The proposed model can treat other revenue structures of the consumer, e.g., it is possible to introduce risk-averse utility functions for the consumer. The model can also be arranged to analyze the relationship between an n-unit standby redundant system and a single unit system in the same manner. These extension and arrangement are currently under investigation.
