A centralized reverse channel structure with flexible manufacturing under the stock out situation

Article history: Received March 16 2013 Received in revised format May 9 2013 Accepted May 1


Introduction
Environmental degradation has emerged as a serious social and economic problem.In fact, several governmental policies also encourage the business organizations to re-use or re-cycle used materials with a view to prevent further environmental degradation.The impact of this consciousness on organizations is forcing them to adopt all such methods and to undertake necessary activities to prevent further degradation of the environment.Reverse manufacturing is one of the popular methods undertaken by the manufacturing organizations to recycle the goods after these have been procured from the customers and their reuse effectively for the same purpose.Re-usable and recycle-able materials/articles are procured from the customers through reverse-distribution channels and reconverted through appropriate processes to appear as new and usable.This paper has been prepared in the backdrop of a very high level of ecological consciousness on the part of the government and society.Our research work also facilitates to include implication of research topics such as flexible manufacturing system.In the present consumerist society and a cut-throat competition in the market, the manufacturers are not only employing newer methods of distribution but also newer formats of distribution.The companies are entering rural markets, semi-urban areas and reaching out to the unexpected segments of potential customers.In addition to generate a spurt in demand, the companies are using innovative marketing strategies and innovative marketing tactics with varying degrees of effectiveness.As far as distribution is concerned new departmental stores, new shopping malls are sprouting up even in the unrepresented geographical areas.Because of all this, the visibility and reach of the brand/product has increased manifolds, which causes sudden fluctuations in demand.There is a strong need for a flexible manufacturing system, which can take care of the above realities and adjusts itself to the realities of the market.
For the past few decades, reverse logistics has been receiving much attention.Schrady (1967) first studied the problem on optimal lot sizes for production/procurement and recovery.For issues in the greening process, Nahmias and Rivera (1979) studied an EPQ variant of Schrady's model (1967) with a finite recovery rate.Richter (1996aRichter ( , 1996bRichter ( , 1997) ) and Richter and Dobos (1999) investigated a waste disposal model by considering the returned rate as a decision variable.Dobos andRichter (2003, 2004) investigated a production/remanufacturing system with constant demand that is satisfied by noninstantaneous production and remanufacturing for single and multiple remanufacturing and production cycle.Dobos and Richter (2006) extended their previous model and assumed that the quality of collected returned items is not always suitable for further repairing.Konstantaras and Skouri (2010) presented a model by considering a general cycle pattern in which a variable number of reproduction lots of equal size were followed by a variable number of manufacturing lots of equal size.They also studied a special case where shortages were allowed in each manufacturing and reproduction cycle and similar sufficient conditions, as the non-shortages case, are given.
El Saadany and Jaber (2010) extended the models developed by Dobos andRichter (2003, 2004) by assuming that the collection rate of returned items is dependent on the purchasing price and the acceptance quality level of these returns.That is, the flow of used/returned items increases as the purchasing price increases, and decreases as the corresponding acceptance quality level increases.Alamri (2010) developed a general reverse logistics inventory model.Chung and Wee (2011) developed an inventory model on short life-cycle deteriorating product remanufacturing in a green supply chain model.Singh and Saxena (2012) derived an optimal returned policy for a reverse logistics inventory model with backorders.
An increase in the shelf space can influence more customers.In this connection, the observations made by Levin et al. (1972) and Silver and Peterson (1985) should be mentioned.They observed that the presence of greater quantity of the same item tends to attract more customers.The reason behind this fact is a typical psychology of the customers.They may have the feeling of obtaining a wide range for selection when a large amount is stored/displayed.Gupta and Vrat (1986) developed models for stock dependent consumption rate.Mandal and Phaujdar (1989) developed an inventory model for deteriorating items and stock dependent consumption rate.Schweitzer and Seidmann (1991) established optimizing processing rate for flexible manufacturing systems.Giri and Chaudhuri (1998) developed deterministic model of perishable inventory with stock-dependent demand rate and nonlinear holding cost and proved that the non-linear holding cost affects the total average cost.Sana et al. (2004) established a production-inventory model for a deteriorating item with trended demand and shortages.Teng and Chang (2005) proposed economic production model for deteriorating item with price and stock dependent demand.Singh and Jain (2009) worked on reserve money for an EOQ model in an inflationary environment under supplier credits.Singh and Singh (2010) worked on supply chain model with stochastic lead-time under imprecise partially backlogging for expiring items.Singh et al. (2010) contributed on an inventory model for deteriorating items with shortages and stock-dependent demand under inflation for two-shops under one management.Yadav et al. (2012) developed an inventory model of deteriorating items with stock dependent demand using genetic algorithm in fuzzy environment.Singh et al. (2013) developed a supply chain inventory model for shortages with variable demand rate.This model consists of two systems forward manufacturing and reverse manufacturing.At the beginning of each cycle, the inventory is zero.The production starts at the very beginning of the cycle.As production progresses the inventory of finished goods piles up even after meeting the market demand, deterioration/obsolescence.At the beginning of each cycle, the process of collecting returnable items in a separate store also begins.At a point where the production from the forward manufacturing system stops; the collection process of returnable items also stops at the same point (For simplicity, we assume there is no collection of used items once the remanufacturing of collected items starts).At this very point the remanufacturing of reusable items begin at a constant rate.The accumulated inventory produced from the advanced manufacturing system in the meanwhile starts getting consumed and ultimately becomes nil.The accumulated inventory of remanufacturing products, which are assumed to be as good as the newly produced products is consumed when the shortages from the forward manufacturing system begin to surface.In addition, at this stage, there is no production and inventory of remanufactured items is consumed till it becomes nil.When the inventory of remanufactured items is also nil, inventory shortages begin to accumulate for some time.Thereafter, production starts and shortages are gradually cleared after meeting demand and the cycle ends with zero inventories.Geometrical description is shown in Fig 1.Production rate is linear function of demand.

2.
The demand rate is deterministic and is a known function of the on hand inventory q.The functional relationship between the demand rate   f q and the inventory level   q t is given by the following expression: Where  denotes the shape parameter and is a measure of responsiveness of the demand to changes in the level of on hand inventory and D denotes the scale parameter.
Items are returnable and are remanufactured.Remanufactured items are as good as new ones and they are used during the shortage period of forward manufacturing.

5.
The time horizon of the inventory system is infinite.Only a typical planning schedule of length T is considered, all remaining cycles are identical.6.
Shortages are allowed and are completely backlogged.7.
The production time interval for forward production coincides with the collection time interval for reverse manufacturing.(This assumption is not applicable during the period of shortages) Notations for forward manufacturing system and reverse system: : On hand inventory level at any time t.
where l is a scale parameter,  , Ordering cost per order.  c c q t : Inventory level during the collecting process for the reverse manufacturing.

Mathematical modeling
There are five stages in the Model (in each cycle as represented in the figure).The governing differential equations are as below: Forward manufacturing process Differential equations representing reverse manufacturing collecting time & consuming time.
is the production lot size during the interval [0, p t ] in forward manufacturing system (see Appendix A) Solving Eq. (1a) and Eq.(1b), we have Solving Eq. (2a) and Eq. ( 2b)   Now to find holding cost for inventory of collected items during interval [0, s t ], we have Solving (3a),(3b), ( 3c) & (3d) and using boundary conditions The cycle consists of five stages; time for each stage and the cycle time have been calculated as below: The above expression represents time to complete one cycle.
Inventory of remanufactured items during interval   (See Appendix D to find the relation to find S 1 at time s t ).In forward manufacturing system, period of shortage starts at t = s t .It has been assumed that remanufactured items are as good as the new ones and they are used during the shortage period of forward manufacturing.Holding cost in interval [ 1 , Total holding cost and deterioration cost in interval Our problem is to find the time to stop the production when q takes optimum value Q and the time to again start the production when maximum shortages accumulate.
and using relation which is time to complete the cycle in forward manufacturing when reverse manufacturing is not included.]As 0, 1,     we have DC = 0 and ].If we assume holding cost per unit per unit of time remains same during forward and reverse manufacturing i.e.
As l increases, production occurs at a more rapid rate.Hence for large l, the model should approach the instantaneous delivery situation of the EOQ model.For large l , 1 1 1   l .Thus as l increases towards infinity, the optimal run size for the model approaches the EOQ when shortages are allowed.

Numerical example
The above theoretical results are illustrated through the numerical verification.Here we are presenting the computational results obtained using Newton Raphson Method which give insight about the behavior of optimal run size Q * , production cycle time '  T and the effects of reverse manufacturing on the total average cost TAC.To illustrate the proposed model, we have considered the following input parameters in appropriate units .5 and as we have taken in the last section 0   .
Here we derive the optimal solution for the different returned rate and holding cost.Results are presented in Tables 1 as follows,

Conclusion
When remanufacturing is undertaken, from the management standpoint there is no perceptible cost difference in terms of total average cost consisting of holding cost, shortage cost, deterioration cost & set-up cost.In view of the governments concern about ecological protection, the management can adopt the system at almost no major incremental costs.As the ratio  increases, there is very slight increase in the total average cost.Further research can be extended to consider the issue of multi objective optimization model, collection of used items during reverse manufacturing period also, inflation and discounting etc.
Appendix A B = Production lot size during forward manufacturing system = Production -deterioration Using (1as), we have, This relation can be used to find the rate of collection.

Appendix
Using (2bs), we have This relation can be used to find the rate of production of remanufactured products.

Fig. 1 .
Fig. 1.Flow of inventory in the integrated supply system unit per unit of time during the forward manufacturing.
unit per unit of time during the collecting and consuming process for the reverse manufacturing." h c : Holding cost per unit per unit of time during the remanufacturing process for the reverse manufacturing.
unit per unit of time.

Table 2
Effects of holding cost on (Q * , ' T , TAC) As  increases, Q and ' T decreases and holding cost also decreases. As  increases, shortage cost increases and total average cost slightly increase.Observations made from Table2:  Case 1: When holding cost per unit is lesser than the shortage cost per unit (0.5) (a) As holding cost increases, Q & ' T decreases.(b) As holding cost increases, there is very slight increase in the total average cost.
The above relation can be used to find the amount of inventory at time