A joint lot-sizing and marketing model with reworks, scraps and imperfect products

Article history: Received 15 July 2010 Received in revised form 16 October 2010 Accepted 30 October 2010 Available online 30 October 2010 In this paper, we establish an economic production quantity (EPQ) based inventory model by considering various types of non-perfect products . We classify products in four groups of perfect, imperfect, defective but reworkable and non-reworkable defective items. The demand is a power function of price and marketing expenditure and production unit cost is considered to be a function of lot size. The objective of this paper is to determine lot size, marketing expenditure, selling price, set up cost and inventory holding cost, simultaneously. The problem is modeled as a nonlinear posynomial geometric programming and an optimal solution is derived. The implementation of the proposed method is demonstrated using a numerical example and the sensitivity analysis is also performed to study the behavior of the model. © 2011Growing Science Ltd. All rights reserved.


Introduction
The main focus of inventory control problems is on optimizing order quantity or lot-sizing subject to some capacity limitations.In such problems, the objective is either to minimize the total costs associated with the inventory control system including ordering and holding costs or to maximize the benefits associated with the system.The EOQ model has been widely employed along these lines in inventory control systems to determine ordering or purchasing quantity .When the production takes place at a constant rate, the EOQ model is extended to economic production quantity (EPQ).One of the primary assumptions with most lot-sizing models is that demand is constant and is given in planning horizon.However, demand can be affected by different factors such as price and marketing expenditure (Lilian, 1992).The other issue on classical lot-sizing problems is the absence of quality characteristics.Product quality is normally affected by the state of the production process, which may shift from an "in-control" state to an "out-of-control" state and produce defective items (Lee & Rosenblatt, 1987).Hence defective items are produced when the quality characteristics are outside the specification limits and the products cannot be used until the necessary rework is applied.The production process may also produce imperfect quality products and items of imperfect quality could be used in another production/inventory situation that is less restrictive process and acceptance (Salameh & Jaber, 2000).Therefore, imperfect items can be sold to a particular purchaser at a discount price.However, as the production of imperfect or defective products is a natural expectation, it will be more realistic to integrate quality considerations into the classical models to cope with real life manufacturing conditions.We classify products in four groups of perfect, imperfect, defective but reworkable and non-reworkable defective items.The percentage of each type is assumed to be constant and deterministic.
The other controversial assumption with many classical EPQ model is that the unit production expenditure is assumed to be constant (Bedworth & Bailey, 1987) and all units produced are of good quality (Warets, 1994).In recent decades, researchers have tried to determine the optimal batch quantity of imperfect production systems with the consideration of different operating conditions.Gupta and Chakraborty (1984) considered the reworking option of rejected items.They considered recycling from the last stage to the first stage and obtained an economic batch quantity model.Lee et al. (1997) developed a model of batch quantity in a multi-stage production system considering various proportions of defective items produced in every stage while they ignored the rework situation.Salameh and Jaber (2000) surveyed an EOQ model where each lot contains a certain percentage of defective items with a continuous random variable.They also considered that imperfect items could be sold as a single batch at a reduced price by the end of 100% inspection but they did not address the impact of the rejected and the reworked items and the selling price.In their paper, Salameh and Jaber did not declare the appropriate cycle time for selling the imperfect products.Hayek and Salameh (2001) assumed that all produced defective items were repairable and provided an optimal point for EPQ model under the effect of reworking the imperfect quality items.Teunter and van der Laan (2002) tried to find the solution for the non -optimal condition in an inventory model with remanufacturing.Chiu (2003) considered a finite production model with random defective rate, scrap, reworking of repairable defective items and backlogging to derive an optimal operating policy including lot size and backordering levels that minimized overall inventory costs.Chan et al. (2003) provided a framework to integrate lower pricing, rework and reject situations into a single EPQ model.They also found that the time schedule for selling imperfect items is critical, as this decision would affect the inventory cost and the batch quantities.Jamal et al ( . 2004) considered a single production system with rework options incorporating two cases of rework process to minimize the total system cost .In the first case, they considered that the rework executed within the same cycle and the same stage of production.In the second case, the defective items are accumulated up to N cycles to be reworked for the following cycle where all defective products could be reworked.Ben-Daya et al. (2006) developed integrated inventory inspection models with and without replacement of nonconforming items discovered during inspection where the process includes no inspection, sampling inspection, and 100% inspection.They also proposed a solution procedure to determine the operating policies for inventory and inspection which consists of order quantity, sample size, and acceptance number.Recently, Jaber et al. (2009) applied the concept of entropy cost to extend the classical EOQ model under the assumptions of perfect and imperfect quality.
There is also another types of lot sizing where the primary objective is the maximization of profit which incorporates pricing and marketing expenditure (Fathian et al., 2009;Lee & Kim, 1993;Lee et al., 1996;Sadjadi, 2005).Sadjadi et al. (2005) presented a joint production, marketing and inventory model which determines the production lot size, marketing expenditure and product's selling price when demand and production have linear relationship.Lee and Kim (1993) formulated a model incorporating production and marketing decisions in a profit maximizing firm over a planning horizon.Their integrated model simultaneously determines price, marketing expenses, demand or production volume and lot size for a single product.
Geometric Programming (GP) is a popular mathematical programming technique for engineering design purposes (Beightler & Philip, 1976;Duffin, 1967;Sadjadi et al., 2005;Jung & Klein, 2001;Abad, 1988;Kim & Lee, 1998;Lee, 1993;Lee et al., 1996).The proposed model of this study differs from the earlier ones since they assume that all items are produced by a perfect reliability.However, we classify products in four groups of perfect, imperfect, defective but reworkable and nonreworkable defective and imperfect items can be sold at the end of processing period as a single batch at a reduced price per unit.Furthermore, unlike the earlier works, the proposed model determines the holding cost per product and the set up cost per production cycle.We also consider interest and depreciation cost as part of problem formulation.The objective of this model is to maximize the total profit with decision variables of lot size ( Q ), marketing expenditure(M), selling price ( P S ), set up cost (A) and inventory holding cost (H), where demand is a power function of selling price and marketing expenditure with constant elasticity and the cost per unit is a power function of lot size.This paper is organized as follows: Problem definition, notations and assumptions used throughout this study are presented in Section 2. In Section 3, the proposed mathematical models are derived in order to maximize the profit per unit time.The optimal solution procedure is represented in Section 4. In Section 5, a numerical example is provided to illustrate the implementation of the proposed model.Finally, in Section 6, conclusion remarks and recommendations for possible future work are presented.

Problem Definition
Consider a single economic production lot-sizing where a single product in a batch size of Q is produced at a finite production rate, P units per unit time.A 100% inspection is performed in order to identify the quality of each product at negligible cost.Demand for perfect product is a power function of selling price and marketing expenditure.Each lot produced contains p 1 percent of imperfect quality items (See Fig. 1 ) .

Fig. 2. Inventory level in stock during one cycle
The perfect and imperfect products are kept in stock when identified.The imperfect products are sold at the end of processing period, i.e., end of T p in Fig.
( 2), as a single batch at a reduced price per unit ) ( I S that is proportion of selling price of good quality products.The lot also contains a percentage of defectives, p 2 , so that these defective products can be reprocessed, or reworked, after the processing period and finally they are kept in stock.These products are assumed to be of good quality after reprocessing and they will not need any inspection.Each lot produced also contains a percentage of defectives, p 3 , so that these units are rejected when identified.In other words, a defective product which could not be reworked is rejected immediately after its work operation completes.The main objective of the present study is to maximize the total profit of the inventory system.

A
Setup cost for each lot (decision variable)

Assumptions
The model is based on the following assumptions: 1.In this model it is assumed that demand is a function of price per unit of good quality products and marketing expenditure per unit i.e., 2. The production unit cost is defined as a power function of stock level and r is the scaling constant for unit production cost.
(2) The exponent β represents lot size elasticity of unit production cost with This function is similar to the function considered by Sadjadi et al. (2005) and Panda and Maiti (2009).Van Beek and Putten (1987) the total cost of interest and depreciation per production cycle is affected by reliability, set up and holding cost.Therefore it can be assumed as: 12. Inspection cost is negligible.

Modeling
Fig. 2 presents the behavior of the inventory level in stock during one cycle.The purpose is to maximize the total profit per unit time ( π ).The total relevant cost per unit time includes setup, production (processing and reprocessing), inspection, rejection and inventory holding costs.According to Fig. 2, it can be easily shown that: To ensure that inventory level does not run into shortages, we must have: The revenue and various costs per unit time derived with respect to Eqs. 5-9 are as follows:

Revenue
The total revenue per unit time can be calculated as follows,

Setup cost
The setup cost for the production system during a cycle is designated as A. Using Eq ( .9) yields the following for the setup cost per unit time,

Production cost
Production cost in each cycle consists of two parts: Processing cost at time T p , and reprocessing or reworking cost at time T r .The quantity of products reworked during each cycle is p 2 Q.Therefore, according to the notation used and from Eq. ( 9), production cost per unit time is as follows,

Inventory holding cost
The inventory holding cost per cycle is obtained as the average inventory times holding cost per product per cycle.Following Jamal et al. (2004), we do not consider any inventory holding costs for the defective items while the machines are waiting for the rework.The reason for this is the low percentage of the items as well as the low level of such costs as storage, etc.The average inventory level can be evaluated as follows, As a result and using Eq. ( 5) through Eq. ( 8), the inventory holding cost will be expressed as:

Interest and depreciation cost
According to our notations and assumptions, interest and depreciation costs per unit time can be calculated as follows,

Total profit
Based on the above definitions and assumptions, we have the following formulation for the total profit per unit time (π ): ) 21 ( Substituting Eqs.11-14 and Eqs.19-20 in Eq.21 and some simplification, the total profit per unit time is as follows, ( )

The Optimal Solution Procedure
In this section, we are interested in simultaneously determining lot size, selling price, marketing expenditure, set up and holding costs.The objective function is as follows, The above objective function is a signomial geometric programming (GP) with 1 degree of difficulty.As the global optimality is not guaranteed for a signomial problem (Duffin et al., 1967), we modify the profit function into the posynomial GP problem with one additional variable and constraint .This technique was developed by Duffin et al. (1967) and it is assumed that there is a lower bound Z for the objective function where the maximization of Z (or minimizing 1 − Z ) is equivalent to maximization of the objective value.Also in order to simplify the objective function, we define the following, where i a denotes the coefficient of ith term of Eq. ( 23).Therefore, the above problem is modified as follow,  ,  ,  , , 1 Model ( 26) is a posynomial GP with 1 degree of difficulty.Hence according to Duffin et al. (1967) it can be solved globally by its dual problem which is formulated as follows,  The dual variables in Eq. ( 27) can be rewritten in terms of variable 7 w as follows, .
To have feasible solution for dual problem, we need some additional assumptions to ensure that w 1 to w 6 remain positive.) 32 ( Since the model has 1 degree of difficulty, we can eliminate 7 out of 8 dual variables by substituting Eqs. ( 28) and ( 29) into Eq.( 27) and write out the dual objective function as maximizing a function in just one dual variable (w 7 ).By taking the logarithm of substituted dual problem we obtain a function in just one variable that is proved to be a concave function by Duffin et al. (1967) and the unique solution can be detremined using a simple line search technique such as bi-section.The procedure is similar to the method used by Sadjadi et al. (2005) and Jung and Klein (2001).

A numerical example
To illustrate the implementation of the model developed in this paper, consider a production system where the parameters are as follows, P=9000 units/year, p 1 =50%, p 2 =2%, p 3 =3%,

Sensitivity Analysis
In this section we analyze the behavior of the proposed model when some of the parameters change in different intervals.The sensitivity analysis of pricing model could also help us extract some managerial implications.

Proposition 1. Let *
Q remain unchanged and suppose that changing p 1 does not have any effect on p 2 and p 3 .As p 1 increases (decreases), * P S increases (decreases) (see Fig. 3).

Proof. Given
From Eq. ( 36) and Eq. ( 37), we understand that is a concave function of p 1 when * Q remains unchanged.Hence, as p 1 increases (decreases), the selling price for good quality products must be increased (decreased).38) and Eq. ( 39), we understand that AQ K is a function of p 2 when * Q remains unchanged.Hence, as p 2 increases (decreases), the set up cost must be increased (decreased).From Eq. ( 40) and Eq. ( 41) we realize that changing p 1 or p 3 does not affect set up cost (when * Q remains unchanged).

Conclusion and future research
In this paper, a joint lot sizing and marketing problem was investigated by considering production of various types of non-perfect products.The products are classified in four groups of perfect, imperfect, defective but reworkable, and finally non-reworkable defective items.The proposed model of this paper considers demand as a power function of price and marketing expenditure and it assumes that production unit cost is a function of lot size.Furthermore, the interest and depreciation costs are also considered as part of modeling formulation.The proposed model of this paper has been solved using GP method and the implementation of the proposed method was illustrated using a numerical example.Furthermore the sensitivity analysis is presented to study the behavior of model parameters.
This research can be extended in some directions.The effect of machine breakdown on this model may be recommended for further study.Another important study is to investigate the effect of time value of money on optimal solution.Besides, it would be interesting to model the problem when various parameters are not deterministic and described in fuzzy or interval form.
selling price of imperfect products is proportion of the selling price of the good quality products idemand for the imperfect product with reduced price always exits.7.Proportions of imperfect, reworked and rejected products are constant in each cycle.8.No imperfect or defective product is produced during the rework.9.The processing and reprocessing are accomplished using the same resources at the same speed.10.No stop is allowed during the manufacturing operations of one lot.11.Production rate, setup time, etc. are constant and deterministic.
expenditureThe marketing expenditure per unit time is as follows,