A production inventory model with exponential demand rate and reverse logistics

Article history: Received January 6 2014 Received in Revised Format June 15 2014 Accepted July 17 2014 Available online July 31 2014 The objective of this paper is to develop an integrated production inventory model for reworkable items with exponential demand rate. This is a three-layer supply chain model with perspectives of supplier, producer and retailer. Supplier delivers raw material to the producer and finished goods to the retailer. We consider perfect and imperfect quality products, product reliability and reworking of imperfect items. After screening, defective items reworked at a cost just after the regular manufacturing schedule. At the beginning, the manufacturing system starts produce perfect items, after some time the manufacturing system can undergo into “out-of-control” situation from “in-control” situation, which is controlled by reverse logistic technique. This paper deliberates the effects of business strategies like optimum order size of raw material, exponential demand rate, production rate is demand dependent, idle times and reverse logistics for an integrated marketing system. Mathematica is used to develop the optimal solution of production rate and raw material order for maximum expected average profit. A numerical example and sensitivity analysis is illustrated to validate the model. © 2014 Growing Science Ltd. All rights reserved


Introduction
Mathematical modeling, analysis and calculation are important areas for research in the study of supply chain system.Supplier, producer and retailer are the members of three-layer supply chain.Supplier supplies the raw material to producer and finished goods to retailers and customers.The manufacturing process shift to out-of-control state from in-control state after some time and produced some imperfect objects.These imperfect objects reworked just after the regular manufacturing schedule.The combination among suppliers, producers and retailers is important for an integrated supply chain system.Rosenblatt and Lee (1986) considered a reverse logistic model where the probability distribution of the instant of changing from in-control situation to out-of-control situation follows an exponential distribution.They understood that the imperfect objects produced in out-of-control situation could be reworked immediately at a price and establish that the presence of imperfect products outcome in a minor set.On the beginning of reverse logistic model, Lee and Rosenblatt (1987) showed that process analysis through the manufacturing runtime could perceive the changing time and it might be restored previously.Liu and Yang (1996) investigated a particular stage manufacturing setup with defective method of supplying of reworkable and non-reworkable objects.The impact of imperfect objects on the batch sizing procedure was renowned in the works of Sana et al. (2007Sana et al. ( , 2007a)), with others.Kim and Hong (1999) considered that imperfect objects could be found after the manufacturing run and the imperfect objects were reworked at a rate.Banerjee and Kim (1995) considered the integrated inventory models in which the retailer and customer coordinate their manufacturing and ordering plan, in order to lesser the common inventory costs.Khouja and Mehrez (1994) considered three coordination systems among the members of the supply chain and showed that several coordination systems could induce to significant decrease in total price.Cardenas-Barron (2007) extended the model of Khouja and Mehrez (1994) by mathematical method, assuming n-stage multi-customer supply chain inventory coordination.Jalbar et al. (2008) developed a multi-echelon inventory coordination in which one retailer delivers an item to many customers.Chiu (2003) developed an economic production quantity model with backorders by combining the assumptions of a symmetrical of the imperfect objects were reworked to build them superior quality objects instead of reworking on all the imperfect objects and the remaining objects are sold at a low rate.Yang and Wee (2001) developed a supply chain model by integrating producer, distributer and vendor as three members of the chain.Cardenas-Barron (2008) presented a straightforward derivation to discover best production lot size with rework procedure at solitary phase manufacturing method.Cardenas-Barron (2009) developed an economic production quantity model with planned backorders for determining the manufacturing batch size and the size of backorders in an imperfect production process where all imperfect objects were reworked at the similar rotation.Sana and Chaudhuri (2010), Sana (2010Sana ( , 2010a)), Sarkar et al. (2010) and Chiu et al. (2007) showed that the imperfect objects might be reworked at a cost where overall production-inventory costs could be decreased considerably.Chiu et al. (2007) developed the job of Chiu (2003) and considered the best run-time difficulty of economic production quantity model with scrap, reworking of imperfect objects and stochastic breakdowns.Sarker et al. (2008) addressed the problem connecting to reworking of imperfect objects in a multi-stage manufacturing method by considering two operational policies: reworking of imperfect objects within the same cycle and after N cycles.Biswas and Sarker (2008) described an inventory system of a solitary manufacturing procedure with an in-cycle rework strategy of fragment with complete analysis.Sana (2011) developed an integrated production inventory model of perfect and imperfect quality products in a three-layer supply chain.Jamal et al. (2004) developed a manufacturing system with rework process consisting two cases of rework process to minimize the total production cost.At the first instant, they executed reworking in the same cycle.At the second instant, they executed reworking after N cycles.Chiu et al. (2007) considered a manufacturing system with rework, including optimal lot sizing decision, random scrap rate and service level constraint.They derived that the expected total cost of a manufacturing system is less if backlogging is allowed or equal if backlogging is not allowed.Hafshejani et al. (2012) presented a multi-product economic production quantity model with imperfect quality items and reworking, it is a nonlinear programming problem solved by genetic algorithm with limited warehouse space.Krishnamoorthi and Panayappam (2013) considered a single stage manufacturing system in which imperfect quality items produced and reworked.They developed two production models, one with shortages and second without shortages.Pasandideh et al. (2010) considered a multi-product economic production quantity model with defective products, reworking and limited warehouse space.Liao et al. (2009) examined an integrated repairable and manufacturing procedure with economic production quantity model with defective maintenance and rework upon fulfillment.Haji et al. (2009) developed a model for an inventory problem considering a single machine with imperfect quality items.They considered that all imperfect quality items are to be reworked, setup cost for rework, no shortages are allowed and waiting time for imperfect quality item.Peter Chiu et al. (2010) developed a mathematical model to establish the optimal run time for a defective fixed production rate with rework, scrap and stochastic machine wreck.Wee and Chung (2009) and Chung and Wee (2010) developed an integrated production inventory model with deterioration assuming rework and green component value design.They extended the economic production quantity model with product reliability.We developed a mathematical model of three-layer supply chain including supplier, producer and retailer.The imperfect items at retailer's level are sent back to the external supplier and the imperfect items at producer's level are reworked after regular manufacturing time.The average profit of producer at incontrol situation and out-of-control situation is considered to derive the solution of the problem.The average profits of suppliers and retailers are considered also.

Essential assumptions and symbolizations
The mathematical model is developed on the bases of following assumptions: 1.The demand rates are exponential increasing function of time for each member of three layer supply chain.
2. The Integrated Material Run

Raw Finished
Fig. 1.The integrated material run

Mathematical Model
In this projected model, supplier delivers the raw materials at rate at be  to the producer up to manufacturing run time 1 t .The imperfect objects at supplier level are sent back after examine carefully with sale rate s w per unit item to the external dealer from where the raw materials are purchased.We assumed that the manufacturing process is "in-control" state at starting stage.The process may shift to an "out-off-control" state and may produce imperfect objects after a random time   be by retailer where the delivery charge of produced items is sustained up to time   1 kT k  .The assembled inventory at time kT decreases and accomplishes to zero level at time T .The leading differential equations at supplier, producer and retailer level are as follows:

Supplier's specific average profit
The leading differential equation is We have The inventory cost of perfect objects is

Fig. 2. Trend on inventory change
The price of screening is c S R .The earning from selling the perfect and imperfect items is  .The supplier's average profit is, using

Producer's specific average profit
If a manufacturing process does not perform well according to process designer's specification then it is said to be "out-of-control".The constraint  is a design variable and an indicator of manufactured goods consistency.Consistency of machines in a production system is usually authenticated to be an , where imperfect digits total number of working hours   . Therefore, a unit either fails or carries on and one of these two states surrogates must exist.We have , where   f t is the failure probability density function then The unit production cost is We assumed that the manufacturing process is in "in-control" state at starting stage.The process may shift to an "out-off-control" state and may produce imperfect objects after a random time   . The holding cost per unit per unit time for defective items is greater than the holding cost per unit per unit time for perfect items.Two cases are gets up in this operating system: with   From Eq. ( 8), we have From Eq. ( 9), we have e e e e e t t a a From Eq. ( 10), we have The inventory level   r Q t of defective items at time t liquidates the following differential equations: From Eq. (15a) and Eq.(15b), we have Here, Thus, the inventory cost for perfect objects is Inventory cost of defective objects is Adding Eq. ( 20) and Eq. ( 21), the total inventory cost is Thus, TIC is a function of random variable and 1 t depends on random variable .The expected number of imperfect objects in a manufacturing lot size   Total expected rework cost is The cost of arrangement is p A The income from selling the perfect and imperfect items is The cost of screening is

Case II:
When 1 t     , the system is in "in-control" state during manufacturing process The leading differential equations are as follows: From Eq. ( 25) and Eq. ( 26), we get Using Eq. ( 29), we get Here, rework cost is zero.
Adding Eq. ( 24) and Eq. ( 30), the total expected profit is . Thus, the average expected profit of the producer is 3.3.Retailer's specific average profit.
The leading differential equations are with with From Eq. ( 33) and Eq. ( 34), we have and using     Thus, For possibility of the model, 1 t kT T   must be gratified.As stated, kT T  grasps as The cost of arrangement is r A The income from selling items is at r c w be T The cost of buying objects is at p c w be T .
The cost of inventory is aT akT a The retailer's average profit is

Leader-Follower Association
Producer is the leader; supplier and retailer are the followers.
From Eq. ( 32), using The producer's expected average profit is 1) where From Eq. ( 6), using The supplier's expected average profit is From Eq. ( 39), using The retailer's expected average profit is

Integrated expected average profit
The integrated expected average profit of supply chain is Where grasps at 1 R , and then

Numerical Example
The following parameters in applicable units are considered as: .Total profit of the supply chain is $12172.9 .

Sensitivity Analysis
To analysis, how the optimal solution is influenced by the parameters, we derive the sensitivity analysis for all parameters.From the given mathematical model, we obtain the optimum result for a stable segment.
The optimum values of all parameters in the segment increases or decreases by 5% , 5%  and 10% , 10%  .The results of total profit are presented in Table 1 as follows,

Conclusion
The main conclusion draws from the sensitivity analysis is as follows: If the percentage of imperfect items  increases in the out-of-control state and the value of reliability parameter  increases then the optimal ordering size decreases and production rate increases.Hence, the expected average integrated profit decreases.The demand rate has impact on the three layers of the supply chain.The production rate has impact on the profit of producer.The production rate is demand dependent i.e. more sensitive to increases or decreases total profit of the supply chain.The values of other parameters are less sensitive to total profit of the supply chain.
In this paper, we considered three layer supply chain including supplier, producer and retailer.For mathematical model, we considered the demand rate of supplier and retailer was exponential increasing function of time.In addition, the production rate was demand dependent.The supplier supplies the material according to the demand of retailer.We have assumed to have constant product reliability; the cycle time at each stage is equal and stock-out situation at each level is insignificant.
Future research can be accomplished for including stock-out situation, shortages are allowed, multisupplier and multi-retailer may be considered.
Fig. 3.The change in inventory

Fig. 4 .
Fig. 4. The change of on-hand inventory

Fig. 5 .
Fig. 5.The change of on-hand inventory the "out-of-control" state exists during operations time.At the beginning, the production rate is 0 we have the value of 000