Optimal trade-credit policy for perishable items deeming imperfect production and stock dependent demand

Article history: Received June 1 2013 Received in revised format August 15 2013 Accepted August 15 2013 Available online August 17 2013 Trade credit is the most succeeding economic phenomenon which is used by the supplier for encouraging the retailers to buy more quantity. In this article, a mathematical model with stock dependent demand and deterioration is developed to investigate the retailer’s optimal inventory policy under the scheme of permissible delay in payment. It is assumed that defective items are produced during the production process and delay period is progressive. The objective is to minimize the total average cost of the system. To exemplify hypothesis of the proposed model numerical examples and sensitivity analysis are provided. Finally, the convexities of the cost functions and the effects of changing parameters are represented through the graphs. © 2013 Growing Science Ltd. All rights reserved


Introduction
In long-established inventory models, it is often assumed that the purchasing cost for the items is paid by the retailer to the supplier as soon as the items have been received.In practice, a delay period known as trade credit period is offered by the supplier to the retailer, in paying for purchasing cost.Up to the end of the trade credit of a cycle, the retailer is free of charge, but he/she is charged on an interest for those items not being sold before this end.During the trade credit period, the retailer can accumulate revenues by selling items and earning interests.Goyal (1985) is the first person who developed the EOQ model under conditions of permissible delay in payments.Shah et al. (1988) studied the same model, incorporating shortages.Later on, Aggarwal and Jaggi (1995) discussed the inventory model considering deterioration and permissible delay in payment.Other motivating mechanisms in this research area are those of Teng (2002), Ouyang et al. (2006), Khanra et al. (2011), Singh et al. (2011), Teng et al. (2012), Singhal & Singh (2013) and Singh and Sharma (2013).
Deterioration of goods plays an important role in inventory system since in real life situations most of the physical goods deteriorate over time.Foods, pharmaceuticals, drugs, radioactive substances are some examples of items in which sufficient deterioration can take place during the normal cargo period and thus it plays an important role in analyzing the system.Generally, deterioration is defined as decay, damage or spoilage and obsolescence, which result in decrease of value of the original one.Ghare and Schrader (1963) presented the first model for decaying items.Covert and Philip (1973) extended their model considering Weibull distribution deterioration.Raafat (1991) presented a survey of literature on deteriorating inventory models.Hariga and Benkherouf (1994) proposed an inventory model for deteriorating items and later on Goyal and Giri (2001) provided a detailed review of deteriorating inventory literatures.Some other models dealing with the same issue are Yang and Wee (2006) and Kumar et al. (2012).
Many business practices reveal that the presence of a larger quantity of goods displayed attract customers to buy more quantity.This phenomenon implies that the demand may have a positive correlative with stock level.As Levin et al. (1972) observed that ''large piles of consumer goods displayed in a supermarket will lead the customer to buy more.Yet, too much piled up in everyone's way leaves a negative impression on buyer and employee alike".Gupta and Vrat (1986) and Baker and Urban (1988) were the first to initiate a class of inventory models in which the demand rate is inventory dependent.Mandal and Phaujdar (1989) then developed a production inventory model for deteriorating items with uniform rate of production and linearly stock-dependent demand.Other papers related to this research area are by Zhou and Yang (2005), Lee and Dye (2012).
Most of the existing production inventory models ignored the presence of the imperfect production process.However, in real life situation, it is often observed that some of the items may be imperfect in nature, which are reworked at a cost to make them perfect.The production of defective items may be due to machine breakdown, labor problem, etc. Rosenblatt and Lee (1986) presumed that the time between the beginnings of the production run until the process goes out of control is exponential and that defective items can be reworked instantly at a cost and kept in stock.Kim and Hong (1999) determined the optimal production run length in deteriorating production processes.Salameh and Jaber (2000) developed an economic production inventory model for items with imperfect quality items.Goyal and Barron (2002) extended the model presented by Salameh and Jaber's (2000).An inventory model is developed by Chung and Hou (2003) to obtain an optimal run time for a deteriorating production system with shortages.Yu et al. (2005) generalized the models of Salameh and Jaber (2000), considering deterioration and partial backordering.Later on, Kang (2010) presented an inventory model considering trade credit and items of imperfect quality.Recently, Sarkar and Moon (2011), Singh and Singh (2011), Sarkar (2012) and Singh et al. (2012) established some motivating inventory models with imperfect production processes.
An enormous work has been done in the field of trade-credit.Many previous economic order quantity inventory models are developed with trade-credit, a very few production inventory models are developed under allowable delay in payment.In addition, the inventory models for perishable items with imperfect production, stock dependent demand under trade-credit in which production rate depends on demand factor are much rare.Therefore, the present model is developed with these unique features.This model is an extension of the model Sarkar (2012) by considering deterioration and demand dependent production.The most favorable solution of the proposed model not only exists but also is unique.To obtain the optimal solution some lemmas are provided and with the help of sensitivity analysis, the effect of change in the parameters on the optimal policy is also disclosed.

Assumptions and notations
The following assumptions and notations are taken to discuss the model.Assumptions 1.The inventory organism deals with a single type of items.2. The replenishment rate is finite.3. The delay in payment is offered to the retailer.4. The demand is stock-dependent.5.There is no repair or replacement of the deteriorated units.6. Shortages are not permitted.7. The lead time is zero.8.The production of imperfect items is considered.Total cost of the system for i={1,2,3} ($)

Formulation of the Model
We consider an inventory model with stock-dependent demand model with different types of delay period.Depending on this policy, there may arise some cases: Case (1): If the retailer pays the purchasing cost within the time R (i.e., T ≤ R), then there is no interest charged.
Case (2): If the retailer pays the purchasing cost after R and before S (i.e., R ≤ T ≤ S), then the supplier can charge a rate of interest I c1 to the retailer.
Case (3): If the retailer pays the purchasing cost after S and before T (i.e., T ≥ S), the supplier can charge a rate of interest I c2 on the unpaid balance (see Figs. 1-3).Now, the present state of the on-hand inventory is described by the following differential equations: and Time Inventory Level with boundary conditions 1 2 (0) 0 ( ) 0 I and I T   .The solutions of the equations ( 1) and ( 2) are given as follows: (5) Now, different costs of the inventory system are as follows: Ordering cost is OC and is given by Inventory holding cost is HC and is given by Deterioration cost for deteriorating items is DC and is given by Production cost is PRC and is given by Purchasing cost is PUC and is given by Along with the trade credit, the paper considers the production of imperfect items.The lifetime of defective item follows a Weibull distribution defined as ( ) , , where α, β are two parameters and t is the time to failure.Hence, the total number of defective items is: The rework cost is RC and is given by Now, for different delay periods:

Case (1): T ≤ R
In this case, interest earned is IE 1 and is given by In this case, interest charged is IC 1 and is given by 1 0 IC  .( 14)

Case (2): R ≤ T ≤ S
In this case, interest earned is IE 2 and is given by In this case, interest charged is IC 2 and is given by

Case (3): T ≥ S
In this case, interest earned is IE 3 and is given by In this case, interest charged is IC 3 and is given by Thus, the total average cost for case (1): is Z 1 (T) and is given by, The total average cost for case (2) is Z 2 (T) and is given by The total average cost for case (3) is Z 3 (T) and is given by Our objective is to minimize the total cost of the inventory system.The necessary conditions for the existence of the optimal solutions are 1 ( ) 0, dZ T dT Using the software Mathematica-8.0, from eq. ( 22) to Eq. ( 24) we can determine the optimum values of , where i=1, 2, 3 and the optimal value Z i (T i * ), where i=1,2,3 of the total cost can be determined by ( 21) provided they satisfy the sufficiency conditions for minimizing Z i (T i * ), where i=1, 2, 3 given by For the cost minimization we may formulate the three lemmas (motivated by Sarkar ( 2012)) as follows:  where all the values of ϕ i 's are given in Appendix 1.
Proof.For minimization of the total cost Z 1 , Differentiating the above expression with respect to T, we get and again, differentiating the above expression with respect to T, we find After some simplification, we get 17 18 5 11 10 6 and 0.
Hence the proof.
Lemma 2. Z 2 (T * ) must have a minimum value at T * if it satisfies the equation and the inequality   Differentiating the above expression with respect to T, we get and again, differentiating the above expression with respect to T, we have For minimum of Z 2 , After some simplification, we get Hence the proof.
and the inequality  where all the values of ϕ i 's are given in Appendix 3.
Proof.For minimization of the total cost Z 3 , Differentiating the above expression with respect to T, we get and again, differentiating the above expression with respect to T, we have For minimum of Z 3 , After some simplification, we get

Sensitivity Analysis
Sensitivity analysis of the numerical example (1) and ( 2) are presented in Table 1 and 2 respectively as follows.The behavior of the parameters changed with respect to the total average cost is shown graphically in Fig. 10 (for example (1)) and Fig. 11 (for example (2)) and some interesting results drawn from sensitivity analysis are given as follows.
(1) The total average cost increases as the purchasing cost (C p ) increases, which is true in practical situation.As the purchasing cost per item increases, it is obvious to increase the optimal cost of the system.
(2) The total average cost of the system increases with an increase in ordering cost (C A ), which is quite natural as the per order growth of ordering cost implies an increase in total average cost of the system.
(3) When the selling price increases the total average cost of the inventory system decreases.The fact is that due to the higher selling price retailer accumulates more revenue and earns more interest during the delay period.
(4) As the demand parameters (a, m) increases the total average cost of the system increases.The motive is that more demand means more production consequently the total average cost increases.
(5) An increase in production parameter (k) shows that the retailer produces more items therefore the holding cost and deterioration cost, etc. increases as a result the optimal cost of the system increases.(6) The total average cost decreases as the deterioration rate (θ) and the deterioration cost (C d ) decreases which according to the real situation.
(7) An increase in rework cost per unit item indicates the growth of the total rework cost.To reduce the cost, the production of imperfect items will have to be reduced.(8) When the production cost (C) and the holding cost (C h ) increases the total average cost of the system increases.The reason is that per unit increase in production and holding costs increases the total production and holding costs therefore the total average cost of the proposed model increases.

Conclusion
In this research article, an inventory model for deteriorating items with stock dependent demand rate considering imperfect production and delay in payment scheme has been developed.In this model, two delay periods have been provided by the supplier to attract the retailer.During the delay period an interest was earned on accumulated revenue by the retailer selling his/her commodity.In most of the papers, the examiners have considered the production of the perfect items through different machinery systems.However, in practical situation, due to employment problems, machine breakdowns, the system produces imperfect quality items, which may rework at a cost to make it perfect.In this model, the production of the imperfect items follows Weibull distribution and the production rate depends on the demand factor.An algorithm to determine the optimal policy has also been presented.In addition, sensitivity analysis is performed to examine the effect of parameters.From sensitivity analysis it is observed that the model is enough stable with respect to the changes in system parameters.Further, the model may be generalized by considering shortages and n cycles in a finite planning horizon.

Appendix 1.
The values are given as follows: The values are given as follows:

Fig. 10 .
Fig. 10.(For example (1)) The effect of changing parameters on the optimal cost function Lemma 3. Z 3 (T * ) must have a minimum value at T * if it satisfies the equation 

Table 1
(For example (1)) The effect of changing the parameter (i) while keeping all other parameters unchanged

Table 2 (
For example (2)) The effect of changing the parameter (i) while keeping all other parameters unchanged   