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In this paper, an environment friendly Economic Pr oduction Quantity (EPQ) model of a single item is considered in which the b usiness in each cycle starts with shortage and ends with the end of stock. The whole problem i s formulated to maximize profit of the manufacturer with random business period and the ra ndomness is removed by chance constrained method. This model involves selling pri ce dependent demand and purchased raw material dependent credit period which are desc ribed by two sets of linguistic relations under fuzzy logic. In addition, a new method of pay ment of due raw material cost (DRC) (DRC is paid as soon as it can possible) is prescri bed with supported lemma and a comparative study has been done between the new met hod of payment and the old method of payment (DRC is paid at the end of cycle). The m odel is optimized by a real coded genetic algorithm (GA) developed for this purpose w ith tournament selection, arithmetic crossover and polynomial mutation. The model is ill ustrated with different sets of numerical examples for different scenarios. A pract ic l application has also been demonstrated with real world data. Some sensitivity analysis are presented graphically. Keywords: Fuzzy logic; genetic algorithm; construction of m e bership function; delay in payment; chance constrained technique AMS Mathematics Subject Classification (2010): 90B05 1. Introduction The concept of EPQ model with production center and sale counter together is to determine the optimum produced quantity against the customers ’ d mand so that total cost involved in the system is minimum. In this process, normally the production firm pays the supplier for the raw materials as and when these are purchas ed. Now-a-days, with the advent of multi-national in the markets of developing countri es like India, Nepal, China, etc, competition between the traders / suppliers is very stiff and they take up different promotional ventures / tools to push the sale. In r eal practice, a supplier provides forward financing to the retailers i.e. offers credit perio d f r payment to attract more customers. In this systems, relaxed period for payment is given t o the firm management if the outstanding N. Chakraborty, S. Mondal and M. Maiti 74 dues are paid within a given credit period. Here cr edit period is treated as a promotional tool as it is one kind of price discount because pa ying later circuitously reduces the purchasing cost and motivates the firms to increase the ordered quantity and to go for more production. Goyal [21] firstly explored an EPQ model under the conditions of permissible delay in payment. Chung et al. [14] derived the optimal pric ing and ordering policy for an integrated inventory model when trade credit is lin ked to order quantity. Chen and Ouyang [10] developed an integrated fuzzy inventory model with permissible delay in payments. These models are developed under the assumption tha t suppliers offers a credit period to the wholesaler. This policy is named as single leve l credit system where in two level credit system, retailer gets a part of credit achieved by the wholesaler. Chen & Kang [7] developed an integrated inventory model considering permssible delay in payment and variant pricing strategy. Ho [6] presented integrat ed inventory model with price and credit linked demand under two level trade credit system. In the last two decades, the inventory models with trade credit have been widely studied b y several researchers. Recently Das et al. [1] developed an integrated production inventor y model under interactive fuzzy trade credit policy. Deterioration of units is one of the most cruc ial factor in inventory problems for deteriorating items. Over the years, there are some investigations on inventory control / supply chain of deteriorating items with permissibl e delay in payment. Aggarwal and Jaggi [20] and Chu et al. [17] presented the ordering pol icies for deteriorating item with trade credit. Jamal et al. [18] allowed shortages in the model of Aggarwal and Jaggi. Chang and Dye [16] allowed the partial backlogged shortages w ith time dependent variation in deterioration rate in Jamal et al. model. Maragatha m & Lakshmidevi [26] presents some fuzzy inventory model for deteriorating items with price dependent demand. Ouyang et al. [11] developed two inventory models for deteriorati ng items with permissible delay in payment. Some notable research papers of deteriorat ing items incorporating various types of assumptions are due to Bhunia et. al. [3], Sana [5], Chang et. al. [15] etc. Most of the above inventory models are developed with constant deterioration. Recently Sarkar et al. [2] developed an integrated inventory model with va riable lead time, defective items and delay in payment. In the existing literature, most of the invent ory models are generally developed with the assumption of infinite planning horizon. Jaggi & Khanna [19] highlights on a Supply chain model for deteriorating items with stock-depe nd nt consumption rate and shortages under inflation and permissible delay in payment. G urnani [22] pointed out that an infinite planning horizon is of rare occurrence because with the passage of time, the inventory cost is likely to vary disproportionately, product speci fications may be changed, etc. Also there are many real-life situations where the assumption of i finite planning horizon is not valid, i.e. the time periods of seasonal / fashionable pro ducts are normally finite and these are of single period [25] only. Moreover, the demands of c ustomers change with time, production process improves with the improvement of technology ver time, etc. In decision making problems like inventory con tr l systems being connected with the available data / possible values of the system para meters can not be always specified exactly i.e. deterministically. There are several r easons for that like lack of input information, multiple sources of data, fluctuating nature of parameter values, noise in data, bad statistical analysis, etc. For example, it is w ell known that demand of a commodity An EPQ Model for Deteriorating items Under Random P lanning Horizon with ... 75 depends on its price. Now-a-days, in the volatile m arket, the price changes very often and thus it is almost impossible to give an exact mathe matical relation between price and demand. Similar is the case with order quantity and credit period though it is a fact that offered credit period varies with order amount. But , in the society, these imprecise information / relations have been fairly communicat ed hrough human words such as high, low, large, medium, small etc. Commonly these relat ions are expressed as IF premise (antecedent) THEN conclusion (Consequent). These ty pe of fuzzy relations are handled by fuzzy inference technique. The commonly used fuzzy inference techniques areMamdani type [23] and Takagi-Sugeno type [7, 24]. These two methods are differ in the way by which the output is calculated. There are several r esearch papers using fuzzy logic in different areas of investigation. Ban et al. [9] di scussed the stability of a simplest Takagi-Sugeno fuzzy control system. Recently Chakra borty et al. [4] used Mamdani fuzzy inference technique to solved an inventory model of deteriorating seasonal products with different price discounts. Among the optimization techniques for the mode ls with fuzzy logic, the evolutionary techniques are more useful. In the literature, Ther e are several evolutionary methods such as Genetic Algorithm (GA), Particle Swarm Optimizat on (PSO), Ant Colony Optimization (ACO), etc. Maiti and Maiti [12] used simulated annealing (SA) and contractive mapping GA to solve a production invent ory model. Gupta et al. [8] used rank based selection process in a real coded GA for solv ing an inventory model with interval valued inventory cost. Recently Chakraborty et al. [4] developed a real coded GA to solve an inventory model of deteriorating multi-items wit h price discount and variable demand defined by fuzzy inference under resource constrain ts. To derive the relations between fuzzy paramet ers (such as demand and selling price) and to get the membership function of their differe nt fuzzy values it requires one’s sufficient experience about the market. Normally, t hese values are expressed verbally by human languages and due to the complexity of human language, it is difficult to derive the image idea from the above market data. Again ideas about a fact vary from man to man. Chang [13] presented a methodology of construction membership function for group opinion aggregation based on a gradation process. In spite of the above developments, there are some inventory control problems yet to be investigated such as till now, none has develope d inventory models with trade credit defined with the help of fuzzy inference which is m ore realistic. The present model defers from the others for incorp orating the following new ideas. • Normally, in the inventory models of trade credit , amount of trade credit is given deterministically through a numerical value. A rela tion is presented by a mathematical expression in crisp way. In practice, often this re lation is expressed by "words" linguistically. Here, for the first time, linguisti c relations between (price, demand) and (ordered quantity, credit period) are considered. • A new method of payment of dues of retailer to su pplier is presented and a lemma is presented which assures the validity of the new met hod. A comparative study has been done with the conventional method. • The business period of the seasonal products are finit and varies every year. Thus the time period of these products are assumed as random having a probability distribution. • The construction of membership function (MF) from the market / business data is very important for the model with fuzzy inferences. Here , a methodology is presented for the N. Chakraborty, S. Mondal and M.

journal of mathematics and informatics

An EPQ Model for Deteriorating Items under Random Planning Horizon with Some Linguistic Relations Between Demand, Selling Price and Trade Credit, Ordered Quantity