Optimal pricing and inventory policies for non-instantaneous deteriorating items with permissible delay in payment: Fuzzy expected value model

Article history: Received 25 October 2011 Accepted February, 3 2012 Available online 18 February 2012 This study investigates optimal pricing and inventory policies for non-instantaneous deteriorating items with permissible delay in payment. The demand rate is as known, continuous and differentiable function of price while holding cost rate, interest paid rate and interest earned rate are characterized as independent fuzzy variables rather than fuzzy numbers as in previous studies. Under these general assumptions, we first formulated a fuzzy expected value model (EVM) and then some useful theoretical results have been derived to characterize the optimal solutions. An efficient algorithm is designed to determine the optimal pricing and inventory policy for the proposed model. The algorithmic procedure is demonstrated by means of numerical examples. © 2012 Growing Science Ltd. All rights reserved


Introduction
According to the modern view, uncertainty is considered essential to science; it is not only an unavoidable phenomenon but has, in fact, a great utility in real world applications. In essence, uncertainty occurs not only due to a lack of information but also as a result of ambiguity (impreciseness) due to the semantic statements by experts. In context of the inventory management, experts usually make interval-valued or linguistic statements about the time parameters and relevant data of inventory system. These interval-valued or linguistic statements lead to non-stochastic uncertainties. The fuzzy set theory was developed to model uncertainties in non-stochastic sense.
During last two decades, several researchers have investigated various types of inventory problems in fuzzy environments to model uncertainties in non-stochastic sense (e.g. Park;1987, Chen et al.;1996, Roy and Maiti;1997, Chang and Yao;1998, Lee and Yao;1999, Kao and Hsu;2002, Chen and Ouyang;, De & Goswami, 20062008). In aforementioned studies, the common feature is that the parameters (demand, cost coefficients etc.) were assumed to be triangular fuzzy numbers or trapezoidal fuzzy numbers. From literature survey, there are few literatures considered the parameters to be fuzzy variables. For instance, Wang et al. (2007) constructed EVM for EOQ model without backordering by characterizing the holding cost and ordering cost as fuzzy variables. Wang and Tang (2009) considered EVM for the EPQ problem with backorder in which the setup cost, the holding cost and the backorder cost are characterized as fuzzy variables, respectively. Recently, Soni and Shah (2011) developed fuzzy expected value production model by characterizing demand and production preparation time as fuzzy variables.
In recent years, researchers studied inventory problems for non-instantaneous items under different conditions. For example, Ouyang et al. (2006) studied an inventory model for non-instantaneous deteriorating items with permissible delay in payments. Geetha and Uthayakumar (2010) extended Ouyang et al.'s model incorporating time-dependent backlogging rate. However, both models consider constant demand rate and cost minimization objective. The assumption of constant demand is quite impractical in reality. It would be more realistic to consider the demand as selling price dependent. The basic idea is that price setting will influence the demand and potential profit. Therefore, we consider demand to be price sensitive.
Based on above discussion, we consider the time parameters, the holding cost rate and interest paid/earned rate in Geetha and Uthayakumar (2010) model may be varied slightly owing to some uncertainties in non-stochastic sense or uncontrolled environments. In addition, instead of constant demand rate we have assumed the demand rate as known, continuous and differentiable function of price. By incorporating above concepts we solve the new inventory model in the fuzzy sense. The main purpose of this study is to extend the paper of Geetha and Uthayakumar (2010) with a view to make the model more relevant and applicable practically.
The rest of the paper is organized as follows: In Section 2, the assumptions and notations which are used throughout the article are presented. In Section 3, fuzzy expected value model to maximize the total profit is formulated. Solution methodology comprising some useful theoretical results and algorithm to find the optimal solution is carried out in Section 4. Numerical examples are provided in Section 5 to illustrate the theory and the solution procedure. Finally, we draw a conclusion in Section 6.

Assumptions and Notations
The following notations and assumptions have been used in developing the mathematical model in this article. , , p t t Π : The total profit per unit time of inventory system.

Assumptions
(1) The inventory system involves single non-instantaneous deteriorating item.
Demand rate D (p) is any non-negative, continuous, decreasing function of the selling price.
During the fixed period, t 1 , the product has no deterioration. After that the on-hand inventory deteriorate with constant rate θ, where 0 < θ < 1. For simplicity, we assume that t 1 is given constant and t 1 ≤ t 2 .
There is no replacement or repair of deteriorated units during the period under consideration.
Shortages are allowed and backlogged partially. We assume the fraction of shortages backorder is ( ) , where x is the waiting time up to the next replenishment and δ is backlogging parameter 0 ≤ δ ≤ 1. This function has been utilized by many researchers (e.g. Abad (1996Abad ( , 2001, Dye (2007), Geetha and Uthayakumar (2010)).
During the trade credit period, M, the account is not settled; the revenue is deposited in an interest bearing account. At the end of the period, the retailer pays off the item ordered, and starts to pay the interest charged on the item in stock. (7) Replenishment rate is infinite and lead time is zero. (8) The system operates for an infinite planning horizon.
Holding cost rate, interest paid rate and interest earned rate are imprecise in nature and assumed to be non-interactive fuzzy variables defined on credibility space

The crisp inventory model
The inventory system evolves as follows: Q 1 units of items arrive at the inventory system at the beginning of each cycle. The inventory level is declining only due to demand rate over time interval [0, t 1 ]. The inventory level is reducing to zero owing to demand and deterioration during the time interval [t 1 , t 2 ]. After that, inventory level becomes zero and shortages begin to be accumulated during [t 2 , T]. The process is repeated as mentioned above.
Based on above description, the status of inventory at any instant of time Also, the ordering quantity over the replenishment cycle can be determined as The profit of the inventory system consists of the following components.
1. The ordering cost ( o C ) is A.
2. The inventory holding cost ( h C ) per cycle is given by The shortage cost ( s C ) per cycle due to backlog is given by The opportunity cost ( l C ) due to lost sale per cycle is given by 6. The sale revenue (R) is given by Next, based on the parameter values t 1 , t 2 and M, there are three cases to be explored.

Fuzzy Expected Value inventory model
In this article, we have considered the holding cost rate, interest paid rate and interest earned rate as fuzzy variables to tackle the reality in more effective way. When the parameters h , p i and e i (as per assumption) treated as fuzzy variables, the above inventory expressions become fuzzy and thereby the total profit per unit time becomes fuzzy variable on the credibility space ,Cr X P X . If the decision maker wants to determine optimal pricing and inventory policy such that fuzzy expected value of the total profit is maximal, a fuzzy EVM can be constructed as follows, Next section carried out the solution methodology for fuzzy EVM along with theoretical results to identify global optimal solution for ( ) 2 3 , , p t t .

Solution Methodology
Using linearity of operator E the fuzzy EVM given by Eq. (5) can be reduced to following single objective crisp problem.
Case 1: 0 < M ≤ t 1 From Eq. (6), the expected value of the total profit during the replenishment cycle per unit time can be written as follows, To maximize the expected total profit per unit time, it is necessary to solve the following equations simultaneously. 1  2 3  1  2 3 and , , , , In order to identify optimal solution for ( ) 2 3 , , p t t , firstly we prove that for any given p, the optimal pair of values ( ) 2 3 , t t not only exists but also is unique. Once this is done, we shall derive the existence of p for optimal pair of values ( ) 2 3 , t t .
From Eqs. (8) and (9) we obtain respectively, Equating right hand side of Eqs. (11) and (12) we have For convenience, let ( ) Thus, t 3 is a function of t 2 and p.
Now, we substitute ( ) ( ) into Eq. (11) and making some algebraic manipulation, we obtain Motivated by Eq. (15), we assume an auxiliary function, say ( ) [ ) and t 3 is given as in Eq. (14). Differentiating ( ) 1 2 F t with respect to t 2 , and using the relation in Eqs.
(13) and (14) we get 1 , t t ∈ ∞ and it can be shown that as t 2 gets Now, the optimal value of t 2 depends on sign of ( ) 1 1 F t so we examine two sub-cases as follows: Sub-case 1.1: Let ( ) Hence, optimal value occurs at point 2 1 t t = and corresponding optimal value of 3 t can be found from Eq. (14) and is given by Summarizing the above arguments, we obtain the following result.
Next, we analyze the condition under which the optimal selling price also exists and is unique. Since, Thus, there exist unique optimal selling price * 1 p that satisfy (10). Note that the lower bound of optimal selling price (say l p ) is the solution of ( ) ( )( ) Case 2: t 1 < M ≤ t 2 From (6), the expected value of the total profit during the replenishment cycle per unit time can be written as follows.
(23) and (24) we obtain respectively, Thus, t 3 is a function of t 2 and p.
Summarizing the above arguments and as discussed earlier in case 1, we can obtain the following result.
denotes the optimal value of ( ) 2 3 , t t for case 2 then we can obtain following result. , , ⎦ is concave and attains its global maximum at point ( ) ( ) Proof: Analogous to theorem 4.1.
Next, the condition for existence and uniqueness for the optimal selling price can be derived analogously as in Case 1. Consequently, there exist unique optimal selling price, denoted by * 2 p , that satisfy ( ) Case 3: M > t 2 From Eq. (6), the expected value of the total profit during the replenishment cycle per unit time can be written as follows.
. Hence, optimal value occurs at point 2 1 t t = and corresponding optimal value of 3 t can be found from Eq. (35) and is given by denotes the optimal value of ( ) 2 3 , t t for Case 3 then we can obtain following result. , , ⎦ is concave and attains its global maximum at point ( ) ( ) Proof: Similar to theorem 4.1.
Next, the condition for existence and uniqueness for the optimal selling price can be obtained similar manner as in Case 1. Therefore, there exist unique optimal selling price, denoted by * 3 p , that satisfy ( ) Based on the concavity behavior of objective function with respect to the decision variables the following algorithmic procedure was developed to identify global optimal solutions for ( ) 2 3 , , p t t .

Algorithm 4.1:
Step 1: Input the values of all parameters. Select membership functions for holding cost rate, interest paid rate and interest earned rate with appropriate parametric values.
Step 2: Set k = 1 and initialize the value of ( ) Step 3: Compare the values of M and t 1 . If M ≤ t 1 , then go to Step 4 otherwise go to Step 5.
Step 4: Calculate by Eq. (16). Execute any one of the following cases (4.1), (4.2). Step 5: Calculate To show the efficiency of proposed computational algorithm 4.1, we run the algorithm with starting value of p = 360. The graph (Fig. 1) shows clear concave function of t 2 and t 3 for given p. Consequently, the obtained solution is a global maximum solution.

Fig. 1.
Profit function (Example 4) with respect to t 2 and t 3

Conclusion
According to the model of Geetha and Uthayakumar (2010), a new fuzzy EVM with generalized price sensitive demand is formulated. In contrast to previous studies, we characterized the holding cost rate, interest paid rate and interest earned rate as independent fuzzy variables to tackle the reality in more effective way. A solution methodology along with some useful theoretical results followed by an efficient computational algorithm is developed to determine the optimal pricing and inventory decisions. The extended model is more effective as it can help the decision maker in subjective decisions with control on selling price. In future research on this problem, it would be interesting to consider other parameters viz. variable demand rate, partial backlogging rate etc. as fuzzy or fuzzy stochastic.
Let ξ be a fuzzy variable defined on the credibility space (X, P(X), Cr). Then its membership function µ is derived from the credibility measure through µ(x) = (2Cr {ξ = x}) ∧ 1, x ∈ R.