A deteriorating multi-item inventory model with price discount and variable demands via fuzzy logic under resource constraints q N. Chakraborty ⇑ , S. Mondal, M. Maiti Department of Applied Mathematics, Oceanology and Computer Programming, Vidyasagar University, Midnapore 721102, India article info Article history: Received 31 August 2011 Received in revised form 14 July 2013 Accepted 21 August 2013 Available online 4 September 2013 Keywords: Fuzzy logic Genetic algorithm Fuzzy inference Price discount Resource constraints Chance constrained technique abstract An inventory model of deteriorating seasonal products with Maximum Retail Price (MRP) for a whole- saler having showrooms at different places under a single management system is considered under ran- dom business periods with fuzzy resource constraints. The wholesaler replenishes the products instantaneously and earns commissions on MRP which vary with the ordered quantities following All Unit Discount (AUD), Incremental Quantity Discount (IQD) or IQD in AUD policy. Demand at showrooms are imprecise and related to selling prices by ‘verbal words’ following fuzzy logic. The wholesaler shares a part of commission with customers. The business periods follows normal distribution and converted to deterministic ones through chance constraint technique. The fuzzy space and budget constraints and fuzzy relations are defuzzified using possibility measures, surprise function and Mumdani fuzzy infer- ence technique. The model is formulated as profit maximization for the wholesaler and solved using a real coded Genetic Algorithm (GA) and illustrated through some numerical examples and some sensitiv- ity analysis. A real-life problem of a developing country is presented, solved using the above mentioned procedures and an appropriate inventory policy is suggested. Ó 2013 Elsevier Ltd. All rights reserved. 1. Introduction In the existing literature of inventory, most of the models are developed under infinite time horizon. As per Gurnani (1985), the life of a particular item is not infinite due to the change of design, technological development, variation of inventory costs, customers’ changing taste, etc. and this is very much true for the seasonal prod- ucts in developing countries where preserving facilities are not available in plenty. For these seasonal products, even though the planning horizon is assumed as finite, in every season it fluctuates depending on some extraneous factors such as climatic conditions. This time period may be assumed to be random with a probability distribution. In the literature Maiti, Maiti, and Maiti (2006) and Roy, Pal, and Maiti (2009) have solved some inventory problems with random planning horizon having exponential distribution. Also Moon and Lee (2000) have presented an EOQ model under inflation and discounting with a random product life cycle. In an inventory system, deterioration is an usual phenomenon. Mandal and Phaujdar (1989) presented an inventory model with deteriorating items. Roy, Maiti, Kar, and Maiti (2009) have done a research work of deteriorating items with stock dependent demand over random planning horizon. Also Bhunia and Maiti (1997) and Mahapatra and Maiti (2006) presented some inventory models for deteriorating items with time dependent demand and imprecise production time respectively. In the present competitive market, the demand depends on the stock directly and also inversely on the selling price. Recently Wid- yadana, Cardenas-Barron, and Wee (2011) presented a deteriorat- ing inventory problem with constant demand via a simplified approach. Also Giri, Pal, Goswami, and Chaudhuri (1996), Mandal and Maiti (2000) and others considered the demand as an indexed stock (i.e. D = dq b , d and b are constants) dependent. But there are few research works with fuzzy demand depending on stock and selling price following fuzzy inference. Recently, some inventory models with rework for the defective products (Jamal, Sarker, & Mondal, 2004; Cardenas-Barron, 2007, 2008, 2009a, 2009b; Sarker, Jamal, & Mondal, 2008; Cardenas-Barron, Trevino-Garza, & Wee, 2012) have been presented in the literature. Human knowledge is often represented imprecisely, vaguely and approximately. In our real life, some vague terms in the form of ‘words’ such as high, medium, and low, are used. The target of fuzzy inference process is to form it into natural language expres- sions of the type, IF premise ðantecedentÞ THEN conclusion ðconsequentÞ: There are two types of fuzzy inference systems: Mamdani-type (Mamdani & Assilina, 1975) and Sugeno-type (Ban, Gao, Huang, & Yin, 2007). These two types differ in the way by which output is determined. Mamdani’s effort was based on Bellman and Zadeh’s 0360-8352/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.cie.2013.08.018 q This manuscript was processed by Area Editor Alexandre Dolgui. ⇑ Corresponding author. Address: Barida, Egra, Purba Medinipur, West Bengal, India. Tel.: +91 9733932724. E-mail address: chakrabortynabakumar@yahoo.com (N. Chakraborty). Computers & Industrial Engineering 66 (2013) 976–987 Contents lists available at ScienceDirect Computers & Industrial Engineering journal homepage: www.elsevier.com/locate/caie