A single period inventory model for incorporating two-ordering opportunities under imprecise demand information

Article history: Received 20 September 2010 Received in revised form 4 December 2010 Accepted 6 December 2010 Available online 7 December 2010 The ordering strategy for a single period inventory model is the key to achieve success in the competitive business environment. This article considers demand in a form of fuzzy number and discusses the SPIM in which the retailer has the opportunity to reorder once during the period. The entire period/season is divided into two slots and the reorder is to be made during the mid-season after the early-season demand has been observed. The objective is to find the expected optimal order quantity together with profit maximization. We illustrate the implementation of the proposed model using a numerical example and explain that the explicit consideration of this reordering opportunity could lead us to better results in terms of profitability. © 2011 Growing Science Ltd. All rights reserved


Introduction
Since the development of EOQ model, a lot of research works have been made in the inventory control system.Often uncertainties may be associated with customer demand.So the real-world inventory control problems are imprecisely defined and human interventions are often required to solve these decision-making problems.The single-period inventory control problem is one of these and it has wide applications in the real-world in assisting the decision maker to determine the optimal quantity to order.The classical single period inventory model (SPIM) is a well-known problem.In real business environment, there are various types of SPIM, namely, the stocking of spare parts, perishable items, style goods and special seasonal products, etc., which have a wide relevance in business.Most of the SPIMs describe a business strategy where an item is to be ordered only once to satisfy customer demand for a precise period (Hadley & Whiten, 1963;Khouja, 1999).Practically, in addition to the supply at the beginning of the sales period, an additional replenishment opportunity exists sometime during the sales period.Moreover, the demand is uncertain, either random or vague, so for multi-product environments like seasonal or fashionable items, there needs a huge initial investment and enormous storage space capacity to avoid loss of good will.Consequently, the optimal ordering strategy has become a major issue to overcome the organization's limitations and restrictions as well as the customer satisfaction.This paper focuses on a two ordering strategy throughout the whole season/period where the reorder is to be made during the mid-season after the early-season demand has been observed.Though, the models with such reordering opportunities are very useful in spot selling business, till now, we have not come across any work in this area except the work of Lau andLau (1997), (1998) and Dutta et al. (2007).Lau andLau (1997, 1998) developed the model with stochastic demand and divided the whole season into two slots by demand scale and allow shortages for both the slots.Dutta et al. (2007) considered the model with fuzzy demand and divided the whole season into two slots by time scale.They also allowed shortages during slot-1, but it depends on the corresponding profit function to be constructed by the decision maker with his credibility preference to the associated over stock or under stock profit.A similar concept of reordering strategy can be found in Chung and Flynn (2001).They extended the newsboy problem by introducing reactive production, i.e., production occurs in two stages, an anticipatory stage and a reactive stage.
The main purpose of this article is to recast the Dutta et al.'s model (2007) by introducing stochastic variation into the choice of slot-1's demand and reducing shortages during slot-1 as much as possible.Moreover, we aim at providing an uncomplicated reordering-model as compared with Dutta et al. (2007), especially in determining the expected resultant profit.Replenishment rate is considered as instantaneous.As the demands are linguistic in nature and the optimal order quantity in the second slot depends on the demand that arises in the first slot, so the profit function as well as the decision variable during the second slot are also fuzzy quantity.Solution procedure is presented using ordering of fuzzy numbers with respect to their possibilistic mean values (Carlsson & Fuller, 2001).The objective is to determine a personal policy that will maximize the total resultant profit under the above state of affairs.This paper is organized as follows.In Section 2, some definitions and propositions related to this study are introduced.In Section 3, we redefine the model with reordering opportunity under fuzzy demand and then obtain the optimal order quantities and expected resultant profit step by step to solve the model.Numerical examples are carried out in Section 4 to illustrate the efficiency of the model and finally, Section 5 contains some concluding remarks.

Preliminary concepts
The demand becomes extremely variable because of shorter product life cycles in the highly competitive market.Hence, the traditional probability theory and statistical method cannot be used properly to describe this kind of uncertainty and the fuzzy theory is employed to deal with these cases.Depending on the manager's judgments or experiences, the uncertainties and imprecision of data are described by linguistic terms such as "the demand is about , not more than ∆ and not less than ∆ ", that is, fuzzy variables.For simplification for the computation, the triangular fuzzy number is employed to describe the fuzzy demand.In this case, the equations for fuzzy total profit will be derived and the numerical analysis will provide us with the management enlightenment.
In order to consider the fuzziness of an inventory problem, we need the following definitions and property relative to this study.Let be a classical set of objects, called the universe, whose generic elements are denoted by .Membership in a classical sub-set of is often viewed as a characteristic function from to 0, 1 such that 1 for 0 otherwise .
Here 0, 1 is called a valuation set.If the valuation set is allowed to be the real interval 0,1 , is called a fuzzy set and to distinguish from classical set, it is denoted by .In this case characteristic function is called membership function of and is denoted by .The closer the value of to 1, the more belongs to .So a fuzzy set , in the universe of discourse is completely characterized by the set of pairs as , , .Clearly is a subset that has no sharp boundary and in this case it is normally written as .A fuzzy set is said to be normal if there exists at least one such that 1.A fuzzy set is said to be convex if , and 0,1 we have 1 min , .
Any convex normalized fuzzy subset on the space of real numbers with a continuous membership function : 0,1 of bounded support is called a fuzzy number (Dubois & Prade, 1978).
L-R representation of fuzzy numbers (Dubois & Prade, 1978) A fuzzy number is said to be a L-R type fuzzy number if its membership function is given by for , 0 for , 0 where L is for left and R is for right reference, is the mean value of .and are called left and right spreads, respectively.

Level set
level set (or interval of confidence at level ) of a fuzzy set in is a crisp subset of denoted by and is defined by / 0,1 .Let be the set of all fuzzy numbers.Then for any , and for any , , , where , , .,/ and for /, 0 (Bector & Chandra, 2005).

Triangular fuzzy number (TFN)
A TFN is specified by the triplet , , and is defined by its continuous membership function : 0,1 as follows: ; , where is the modal of fuzzy number , : 0,1 and : 0,1 are the left and right shape continuous functions.Hence the closure of the support of is exactly , .(Dubois & Prade, 1978;Liou & Wang, 1992) The interval-valued expectation of is defined as , where and are the left and right integral values of , respectively.The expected mean value of based on the area measurement index is defined as .

Possibilistic mean value of a fuzzy number
For a given fuzzy number , the interval valued possibilistic mean is defined as , , where and are the lower and upper possibilistic mean values of (Carlsson and Full'er , 2001) and are respectively defined by , .The possibilistic mean value of is then defined as .In other words, it can be written as .Now, if and be two fuzzy numbers, where , and , , 0,1 , then for ranking fuzzy numbers we have .

Graded mean integration value of fuzzy number
Chen and Hsieh (1999) introduced graded mean integration representation method based on the integral value of graded mean of LR-fuzzy number for defuzzifying LR-fuzzy numbers.Suppose is a LR-fuzzy number with grade , then according to Chen and Hsieh (1999), graded mean integration representation of is denoted by and is defined as , with 0 and 1.Here and are the inverse functions of and , respectively.

Optimal policy
In order to consider the SPIM with two ordering strategy under imprecise demand information the following notations are used.
Order quantity at the beginning of the season/period Order quantity at the beginning of the second slot Fuzzy demand for slot-1 Fuzzy demand for slot-2 Actual realization of demand during slot-1 Actual realization of demand during slot-2 Net purchase cost Selling price per unit Holding cost per unit for the next season Shortage/under stock cost per unit Suppose the shop-keeper has the opportunity for reorder during the middle of the season.Assuming there is no option for substitution between the products we propose the reordering strategy for an individual item and develop the model for profit maximization.Suppose the demand of the slots is characterized by the triangular fuzzy number , , 1, 2 with the membership function μ , where μ ; for ≤x ≤ ; for ≤x ≤ 0; otherwise Looking at the Dutta et al.'s (2007) fuzzy inventory model, the authors divided the whole season into two slots in time scale and considered independent fuzzy demands for both slots.It seems that the authors have solved two single-period problems with leftover items of slot-1 can be used in slot-2, which is unrealistic.Since, such a model is named as 'single-period', therefore, according to the expert's demand information, if , , be the total seasonal demand then the demand during slot-1 should be a certain percent of (θ % say).Again, most of the time a top manager of an organization depends on the experts opinions about the demand information.So the choice of slot-1's demand from the whole seasonal demand may not be unique; it may also differ from one expert to another expert.That is, the choice of may vary randomly.Thus, if stochastic variation occurs into the choice of , the notion of fuzzy random variable (FRV) must be considered.FRV is a mathematical tool in which both random behavior and fuzzy perception appear simultaneously (Puri & Ralescu, 1986;Luandjula, 2004).Thus if the expert's opinions about slot-1's demand are described by the phrases "50% of ", "60% of ", etc.Then the final choice of slot-1's demand can be derived by taking the fuzzy expectation of these fuzzy observations.If be a discrete FRV such that , 1 to , then its fuzzy expectation is given by , ∑ . Therefore, the final choice of is a summarizing fuzzy value of the central tendency of FRV.Consequently, , , becomes a fuzzy quantity.
Further, Dutta et al. (2007) allowed shortages during slot-1.But, in a competitive market, if shortages occur during the mid-season of the period and customers are going back without fulfillment of their demand, it not only affects the expected profit in the first slot but also it changes the demand rate of the next slot.To overcome the loss of goodwill, without loss of generality it can be assumed that the maximum possible demand during slot-1 is with the lowest membership grade zero.Thus if the decision-maker (DM) chooses then the possibility of shortages during slot-1 is nil rather there is obvious excess after the end of this slot.In this case, the DM is not bothering about the certain excess of products during slot-1 rather he gives the priority to customer satisfaction during the middle of the season.In addition, as replenishment is instantaneous, shortages may be backordered, if any.Therefore, if be the number of quantities left after the end of slot-1, then is defined as , where is the actual occurrence of demand in slot-1.Let be the optimal order quantity to be needed for maximizing the profit function in the second slot individually based on the fuzzy demand , if no quantities are supplied from slot-1.Then the optimal order quantity for the second slot at the beginning of slot-2 is defined as follows: Therefore, our task is to find out the only decision variables that maximizes the total profit function for the second slot.

Determination of optimal order quantity
As demand in slot-2 is imprecisely prescribed, it causes a fuzzy over stock profit and fuzzy under stock profit and hence the resultant profit function also becomes a fuzzy quantity , (say).If be the actual demand realization of fuzzy demand in slot -2, then for this slot the profit function can be formulated as , for , for . (2) Therefore the fuzzy overstock and fuzzy under stock profit functions are respectively given by .Since , is a fuzzy quantity and so it can be directly maximized.Now using several -level set and possibilistic mean value method, one can find out the required by maximizing the mean or expected value of this fuzzy profit function.If , be the -level set of , , then it can be derived as follows: Thus the possibilistic mean value of fuzzy profit function , is given by . (3) For this we need to know the -cut of the fuzzy profit function , for all 0,1 .Details analysis of deriving , can be found in Dutta et al. (2007).In order to determine the optimal , in this context, we develop the following features about .Let us first define , , then it is easy to identify the position of in , .We then have two cases, namely, , or or equivalently, 0 or 0, where and are the left and right spreads of , respectively.Case 3.2.a.When 0 then , , and then the possibilistic mean value of fuzzy profit function , is obtained as where .Therefore the optimal value of order quantity is obtained by setting the first derivative of with respect to equal to zero.That is, subject to the condition 2 0.
Again, if 2 0, then the optimal lies between and .In this case, the possibilistic mean value of the fuzzy profit function , is given by

Conclusion
This paper considered a situation in which the single-period 'newsboy type' product may be ordered twice during a period.It is supposed that the single-period problem operates under uncertainty in customer demand, which is described by imprecise terms and modeled by fuzzy sets.The reordering policy adopted here is quite relevant with real business environment.As most of the shopkeepers' store multi items in their inventory, which require more storage space as well as more initial investment, our policy reduces both the space problem and the initial investment.In this case, the DM also has the opportunity to reduce the ordering policy later in the season if the realized demand in slot-1 is low.The strategy gives both DM's achievement and customer satisfaction.The proposed model can be extended to a case in which the purchasing cost during the mid-season may differ from the initial procurement cost.

Table 3
Comparison between one order case and two order case