A production inventory model with fuzzy random demand and with flexibility and reliability considerations Soumen Bag * , Debjani Chakraborty, A.R. Roy Department of Mathematics, Indian Institute of Technology, Kharagpur 721302, India article info Article history: Received 16 March 2007 Received in revised form 8 January 2008 Accepted 1 July 2008 Available online 15 July 2008 Keywords: Fuzzy random variable Graded mean integration value Reliability Flexibility abstract The classical inventory control models assume that items are produced by perfectly reliable production process with a fixed set-up cost. While the reliability of the production process cannot be increased with- out a price, its set-up cost can be reduced with investment in flexibility improvement. In this paper, a production inventory model with flexibility and reliability (of production process) consideration is devel- oped in an imprecise and uncertain mixed environment. The aim of this paper is to introduce demand as a fuzzy random variable in an imperfect production process. Here, the set-up cost and the reliability of the production process along with the production period are the decision variables. Due to fuzzy-randomness of the demand, expected average profit of the model is a fuzzy quantity and its graded mean integration value (GMIV) is optimized using unconstraint signomial geometric programming to determine optimal decision for the decision maker (DM). A numerical example has been considered to illustrate the model. Ó 2008 Elsevier Ltd. All rights reserved. 1. Introduction The EOQ model initially proposed by Harris in 1915 (Harris, 1915), was modified and extended later on by several researchers, changing the assumptions, with the objective to make it more real- istic (Buffa & Sarin, 1987; Hadley & Whitin, 1963; Nadoor, 1965). Inventory models with crisp, stochastic and fuzzy parameters were studied by several authors. These models have been developed by considering the parameters as crisp (Buffa & Sarin, 1987; Hadley & Whiten, 1963; Harris, 1915; Nadoor, 1965) and the remaining as either stochastic (Aggarwal, 1974; Hadley & Whitin, 1975) or fuzzy in nature (Bag, in press; Das, Roy, & Maiti, 2004; Dutta, Chakr- aborty, & Roy, 2007b, in press). Models, with inventory parameters that are fuzzy random in nature have not been studied much, as yet. Dutta, Chakraborty, and Roy (2005) were the first to incorpo- rate demand as fuzzy random variable in a simple newsboy prob- lem with fuzzy random demand. Later on, a fuzzy mixture inventory model involving fuzzy random lead time demand has been developed by Chang, Yao, and Ouyang (2004). Recently, Dut- ta, Chakraborty, and Roy (2007a) developed a continuous review inventory model in mixed fuzzy and stochastic environment; and Dey and Chakraborty (2008) developed a single period inventory problem with resalable returns in fuzzy stochastic environment. But, demand as a fuzzy random variable even in production inven- tory model is yet to be considered. It is known that, given the past data, a retailer may estimate that the demand of a commodity may follow a particular distribu- tion. However, it is very difficult to estimate the exact value of the parameters of the distribution. In this case, these parameters are vaguely defined and these can be estimated as fuzzy numbers. Consequently, the distribution is a fuzzy random distribution and we say the demand is fuzzy random. A basic assumption in the inventory management system is that set-up cost for production is fixed. In addition, the models also implicitly assume that items produced are of perfect quality. How- ever, in reality, products are not always perfect but are directly af- fected by the reliability of the production process employed to manufacture the product. In a recent paper, Cheng (1989) proposed a general equation to model the relationship between production set-up cost and process reliability and flexibility. Which was later on used by Leung (2007) and Maiti and Maiti (2005) for their respective models. The concept of fuzzy random variable and its fuzzy expectation has been presented by Kwakernaak (1978) and later by Puri and Ralescu (1986). Further, the notion of a fuzzy random variable has also been considered in (Chang et al., 2004; Das et al., 2004; Dey & Chakraborty, 2008; Dutta et al., 2005; Dutta et al., 2007a; Feng, Hu, & Shu, 2001; Kim & Ghil, 1997; Lopez-Diaz & Gil, 1998), in recent times. In the present work, an EPQ model is considered, where de- mand of the item is fuzzy random in nature with known probabil- ity distribution and the production process is assumed to be not 100% perfect, i.e. a fraction of the produced items are defective. Further, it is assumed that the defective items are sold at a reduced price and the selling price of fresh units is taken as a mark-up over the unit production cost. The model is formulated to maximize the expected average profit. Since demand is fuzzy random in nature, 0360-8352/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.cie.2008.07.001 * Corresponding author. Tel.: +91 9732653039. E-mail address: soumen@maths.iitkgp.ernet.in (S. Bag). Computers & Industrial Engineering 56 (2009) 411–416 Contents lists available at ScienceDirect Computers & Industrial Engineering journal homepage: www.elsevier.com/locate/caie