Supply chain production model with preservation technology under fuzzy environment

Article history: Received September 2 2013 Received in Revised Format November 17 2013 Accepted March 2


Introduction
Deteriorating inventory were studied in the past decades (Ghare & Schrader, 1963;Covert & Philip, 1973;Dave & Patel, 1981;Kang & Kim, 1983;Goyal & Giri, 2001;Bhunia et al., 2009;Chang et al. 2010;Singh & Singh, 2011) and the focus was usually on constant or variable deterioration rate and quantity discount.Investing on preservation technology (PT) for reducing deterioration rate has received little attention in the past years.The consideration of PT is important due to rapid social changes and the fact that PT can reduce the deterioration rate, significantly.Moreover, sales, inventory and order quantities are very sensitive to the rate of deterioration, especially for fast deteriorating products.The higher rate of deterioration would result in a higher total annual relevant cost and a lower demand rate (Yang & Wee, 2006;Johnny et al., 2007).Ouyang et al. (2006) found that if the retailer can reduce effectively the deteriorating rate of items by improving the storage facility then the total annual relevant inventory cost could be reduced.Many enterprises invest on equipment to reduce the deterioration rate and extend the product expiration date.For example, refrigeration equipment is used to reduce the deterioration rate of fruits, flowers and sea foods in the supermarket.Murr and Morris (1975) showed that a lower temperature could increase the storage life and decrease decay.Wee et al. (2010) presented a model using PT.Huang et al. (2011) discussed the comments on PT investment for deteriorating inventory.Dye and Hsieh (2012) developed an optimal replenishment policy for deteriorating items with effective investment in PT.Dye (2013) investigated on the effect of PT investment on a non-instantaneous deteriorating inventory model.Hsieh and Dye (2013) developed a production inventory model incorporating the effect of PT investment when demand is fluctuating with time.
Generally, some customers would like to wait for backlogging during the shortage period.However, the willingness is diminished with the length of the waiting time for the next replenishment.The longer the waiting time is, the smaller the backlogging rate is.The opportunity cost due to lost sales should be considered.Papachristos and Skouri (2000) developed an EOQ inventory model with time-dependent partial backlogging.They assumed the rate of backlogged demand increases exponentially while the waiting time for the next replenishment decreases.Teng et al. (2003) extended the backlogged demand to any decreasing function of the waiting time up to the next replenishment.The related analysis on inventory systems with partial backlogging have been performed by Teng and Yang (2004), San José et al. (2006), Chang and Lin (2010), etc.
Usually researchers consider different parameters of an inventory model either as constant, timedependent or probabilistic nature for the development of the economic order quantity model.However, in the real life situations, these parameters may have little deviations from the exact value, which may not follow any probability distribution.In these situations, if these parameters are treated as fuzzy parameters, then it will be more realistic.These types of problems are de-fuzzified by using a suitable fuzzy technique and then the solution procedure can be obtained in the usual manner.Several authors, namely Chang et al. (2010), Lin and Yao (2000), Singh and Singh (2011), Ruoning and Xiaoyan (2010) and Singh et al. (2011) developed inventory models in fuzzy sense by considering different parameters in fuzzy sense.Yadav et al. (2013) focused on retailer's optimal policy under inflation in fuzzy environment with trade credit.Urvashi and Singh (2013) discussed a model of inventory control with fuzzy inflation and volume flexibility under random planning horizon.Dutta and Kumar (2013) discussed about the fuzzy inventory models for deteriorating items with shortages and fully backlogged condition.
In this research, an effort is accomplished to formulate a supply chain production model for deteriorating items with fuzzy demand, fuzzy production by allowing preservation technology cost.Under real life circumstances, these parameters in the production model are uncertain and imprecise.In this situation, a suitable way to model these imprecise data is to use fuzzy sets and to formulate a production model in a fuzzy environment.We use the centroid method for defuzzifying fuzzy total average cost.For the proposed model, producer and supplier use the preservation technology.The optimal solution of the proposed model not only exists but also is unique.To illustrate the theory of the proposed model, a numerical example is provided and sensitivity analysis with respect to parameters of the system is carried out in both crisp and fuzzy sense.

Assumptions and Notation
In developing the mathematical models of the inventory system for this study, the following common assumptions were used.

Model development for producer
The inventory model is shown in Fig. 1.The planning horizon T is divided by 2 parts T 1 and T 2 , where T 1 is the production time and T 2 is the non-production time.During the time interval [0, T 1 ], the inventory level decreases due to demand and deterioration and increases due to the production.During the time interval [0 , T 2 ], there is no production, the inventory decreases due to deterioration rate and demand.This figure also shows that if we use the PT then our production is more and we can use our stock for a long time period.
Inventory level Inventory level without PT 0 T 1 T 2 Time

Fig. 1. The graphic representation of inventory model for producer
There are the following differential equations for the inventory level of producer.
We have boundary conditions 1 (0) 0 p I  and 2 2 ( ) 0 p I T  , the solution of the above differential equations are as follows, By the boundary condition I p1 (T 1 ) = I p2 (0), we have By using the boundary condition 2 (0) , we can find out the maximum inventory level with PT and without PT, which is as follows, and 2 2 1 .
The total relevant cost involves the following four factors, (a).Holding cost: The present value of the holding cost per unit time is as follows, Hence, the present value of the total relevant inventory cost per unit time is as follows,

Model development for supplier
The inventory model is shown in Fig. 2 By using the condition (0) , we have maximum inventory for the supplier is follows, The total relevant cost involves the following six factors as follows, (a).Holding cost: The present value of the holding cost per unit time is 3 0 ( ) (b).Purchasing cost: The present value of the purchasing cost per unit time is (1 ) e T e T r r (1 ) (1 ) Hence the present value of the total relevant inventory cost per unit time is (1 ) ( 1) There are two decision variables in the present value of the total inventory cost TC.The first part is associated with the production time period of producer, T 1 and the second part is associated with the time period of supplier T 3 .We can find out the value of T 2 and T 4 with the help of Eq. ( 5), Eq. ( 6) and the relation T=T/n -T 3 .

Fuzzy model for producer and supplier
In a real situation and in a competitive market situation both the production rate and the demand rate are highly uncertain in nature.To deal with such a type of uncertainties in the super market, we consider these parameters in fuzzy form.Here p and d are not known precisely and p and d are defined by triangular fuzzy numbers such that Here, by the centroid method is given as follows, The membership of the fuzzy cost function given by the extension principle is Defuzzification for the fuzzy total cost is thus given by centroid method is

dy M T T f T T T T T T T T T T y dy
, g and q f T T T T T T are given by Eq. ( 28), Eq. ( 29) and Eq. ( 30), respectively.We can minimize the total average cost per unit time for the Optimal value of T 1 (say T 1 * ) and T 3 (say T 3 * ) as defined above and the minimum total cost is given by

Solution Procedure
To derive the optimal solution, the following classical optimization technique is used.
Step 1.Take the partial derivatives of TC (T 1 , T 3 ) with respect to T 1 and T 3 and equate the results to zero to find out the optimal value of T 1 and T 3 .The necessary conditions for optimality are Step 2. The convexity of the total cost is analyzed with the help of following conditions Step 3.With T 1 * and T 3 * found in step 2, derive TC * ( T 1 * ,T 3 * ).

Numerical illustration for crisp model
A numerical example is considered to illustrate the model.The following values of parameters are used in this example.
s C = $0.30,δ = 0.061, r = 0.001,  = $200, k = 0.007, η = $400, a= 0.001.The software Mathematica5.2 is used to derive the optimal solution.The optimal value of total cost is obtained as $92.678 and production time (T 1 * ) is 8.2854 days and (T3 * ) is 3.29462 days.With the help of Eq. ( 5) and Eq. ( 6), we can find out the non-production time period (T 2 * ) and shortage period (T 4 * ) and by using the condition of optimality we can analyze that TC is a strictly convex function, as shown below.
The three dimensional producer and supplier's total cost graph is shown in Fig. 3.

Numerical illustration for fuzzy model
All the symbols in the fuzzy environment have the same numerical values as in the crisp environment with initial demand parameter and production with some uncertainty.The production rate and demand rate are considered as the triangular fuzzy numbers (300, 360, 400) and (310, 350, 410), respectively The three dimensional producer and supplier's total cost graph in the fuzzy environment is shown in Fig. 4.   show the effect of n (number of deliveries) on the total cost of the model in the crisp and fuzzy model.As we increase the n then total cost decreases up to a point and then it again increases.Therefore, we have to take an optimal value of the number of cycles to minimize the total cost.The following inferences can be observed from the sensitivity analysis based on Table 1.
1) The total cost per unit time is less sensitive on the changes in set up cost parameter.
2) The holding cost and purchasing cost has stronger effect on the optimal value of total cost than the other parameters.
3) In the fuzzy sense, optimal production time and supplier's time period is lower than the time period obtained in the crisp model.
4) Total cost obtained in the fuzzy sense is higher than the crisp total cost.

Concluding Remarks
In this paper, we have investigated a model where producer and supplier use the PT to reduce the deterioration rate.This model is analyzed under two circumstances of the crisp and the fuzzy forms.
Here, production and demand is treated as a fuzzy variable.Producer and supplier have different PT cost, which depends on the type of technology used.A solution procedure is also derived to optimize the total cost of producer and supplier.The main emphasis of this paper is on cost reduction by making effective investment in PT.Sensitivity analysis shows that while PT applies in the model, then the total cost is reduced due to the decrement in the deterioration rate.These changes can also be seen in the fuzzy sense but total cost is slightly higher than the crisp model.Effect of cost parameters on the total cost is also analyzed.Finally, the proposed model can be extended in several ways.For example, we could extent the model to allow for multi supplier.
(b).Purchasing cost: The purchasing cost includes loss due to deterioration as well as the cost of the item.The present value of the purchasing cost per unit time is as follows, .Ordering cost: The present value of the ordering cost per unit time is as follows, 20) (c).Ordering cost: The present value of the ordering cost per unit time is as follows, .Shortage cost: The present value of the shortage cost in the entire time horizon is as follows, (f).Lost sale cost: The present value of the lost sale cost occurs during the time period 0 to T 4 .During this time period, the complete shortage is 4 dT and the partial backlog is 4 d T  .Lost sales are the difference between the complete shortage and the partial backlog.Thus, the present worth lost sale cost per unit time is as follows,    are based on subjective judgments.We apply arithmetic operators based on fuzzy quantities and then defuzzify the same to convert them into crisp output.The membership functions for p  and d  is defined as follow, determined by the decision maker based on the uncertainty of the problem.Defuzzification of and d p  

Fig. 3 .
Fig. 3.The Producer and Supplier's unit time total cost in crisp model Then optimal value of the total cost is $102.973, production time (16495 days.With the help of Eq. (5) and Eq.(6) we can find out the non-production time period cost is more in the fuzzy environment.With the help of this data we can analyze that TC is a strictly convex function as shown as follows,

Fig. 4 .
Fig. 4. The Producer and Supplier's unit time total cost in fuzzy model

Fig. 6 .
Fig.6.(a)The effect of parameter ζ on the total cost in the crisp model

Fig. 6 .
Fig.6.(b)The effect of parameter ζ on total cost in the fuzzy model Fig 8.(a) The effect of parameter n on the total cost in the crisp model Fig 8.(b) The effect of parameter n on the total cost in the fuzzy model

Fig. 9 .
Fig. 9. (a) The effect of parameter r on the total cost in the crisp model Fig. 9.(b) The effect of parameter r on the total cost in the fuzzy model . The planning horizon T/n is divided in 2 parts of T 3 and T 4 , where T 4 is the shortage time.Supplier's cycle starts with the maximum inventory I ms at t = 0.During the time interval [0, T 3 ], the inventory level decreases due to demand and deterioration with PT.During the time interval [0 , T 4 ], shortage occurs, which is partial backlogged.The inventory level at time t during the time interval [0 , T 3 ], and [0 , T 4 ], is governed by the following differential equations, s I T  and 2 (0) 0, s I  the solution of the above differential equations are as follows, 3

Table 1
Sensitivty analysis of the proposed model ParametersPercentage of under-estimation and over-estimation parameters