An Inventory Model with Price and Quality Dependent Demand Where Some Items Produced Are Defective Tapan

This paper analyzes an inventory system for joint determination of product quality and selling price where a fraction of items produced are defective. It is assumed that only a fraction of defective items can be repaired/reworked. The demand rate depends upon both the quality and the selling price of the product. The production rate, unit price, and carrying cost depend upon the quality of the items produced. Quality index is used to determine the quality of the product. An algorithm is provided to solve the model with given values of model parameters. Sensitivity analysis has also been performed.


Introduction
In every business, product quality is an important factor that attracts customers.For durable goods, quality depends upon several factors.Some of such factors are type/quality of the raw materials used, type/quality of the machines used in the production process, skills of workers engaged in the production system, and so forth.It is obvious that the unit cost for a high quality product will be high.In general, unit cost increases with quality.Quality measure is an important issue in all production systems.There is no well-defined method for measuring quality.In fact, quality characteristics are not the same for all types of items.It varies from one type of items to another type.There are several research articles on quality measure.Maynes [1] described the concept of evaluating quality index as a measure of quality for durable goods.He suggested to combine characteristics of variety and the characteristics of seller to evaluate quality index.Jiang [2] defined quality index as a ratio of two different life measures based on fractile life; one represents life utilization extent and the other represents the quality improvement potential.He derived quality index formulae for several known lifetime distributions.Some authors proposed quality index method to measure quality of sea foods.Huidobro et al. [3] proposed quality determination method for raw Gilthead seabream (Sparusaurata) based on the quality parametersflesh elasticity, odor, clarity, shape of fish.Barbosa and Vaz-Pires [4] proposed the development of a sensorial scheme to measure quality of common octopus.
Though customers have the tendency to buy a high quality product, sometimes due to high price they compromise with the quality.Thus, a challenging task for a production manager is to produce units in suitable quality and setting a reasonable selling price for these units.Normally, customers' demand decreases when selling price increases.
In some production systems, all items manufactured are not good/perfect.This may be seen in failure-prone manufacturing system where the produced items are a mixture of good as well as defective items.This situation can be found in the industries where units are produced in large numbers.Some of the research articles on defective products are authored by Rosenblatt and Lee [5], Kim and Hong [6], Salameh and Jaber [7], Chung and Hou [8], Chiu [9], Sana [10], Datta [11], and Mhada et al. [12].Rosenblatt and Lee [5] studied the effect of the imperfect production process on optimal production cycle.They assumed the system deteriorates during the production process and produces some proportion of defective items.Kim and Hong [6] analyzed a production system which deteriorates randomly and shifts from in-control state to outof-control state.They determined the optimal production Advances in Operations Research run length.Salameh and Jaber [7] developed an EOQ model where all items produced are not perfect.They assumed that the imperfect items would be sold in a single batch by the end of the screening process.Chung and Hou [8] analyzed a system with deteriorating production process and allowable shortages.Chiu [9] developed a finite production rate model by assuming that a fraction of defective products can be reworked and the rest will be disposed.Sana [10] analyzed an inventory model to determine optimal product reliability, optimal production rate in a faulty production system.In his article, the demand rate is assumed to be time dependent.Datta [11] developed an inventory model with adjustable production rate and selling price sensitive demand rate.In his article, he assumed that all items produced are not perfect.This model jointly determines optimal production rate, production period, and selling price.Mhada et al. [12] proposed a model with perfectly mixed good and defective parts.But these research articles did not consider product quality as a decision parameter.Some researchers realized the importance of product quality and incorporated the quality factor in their models.Some of such articles are authored by Chen [13], Mahapatra and Maiti [14], Chen and Liu [15], and Chen [16].Chen [13] developed a model to find optimal quality level, purchase price, and selling quantity for the immediate firms, but his model is not valid in a production system where all units produced are not good.Mahapatra and Maiti [14] developed a multiobjective, multi-item inventory model with quality and stock-dependent demand rate.Their paper did not focus neither on the influence of selling price in demand nor on the defective product.Chen and Liu [15] proposed an optimal consignment policy considering a fixed fee and a per unit commission.Their model determines a higher manufacturer's profit than the traditional production system and coordinates the retailer to obtain a large supply chain profit.Chen [16] modified the model of Chen and Liu [15] by incorporating the influence of retailer's order quantity on the manufacturer's product quality.He considered the quality of the product as normally distributed.None of the above articles explained the joint determination of best selling price, product quality, and product quantity under qualitydependent production rate and costs.
In the present paper, the author has attempted to develop an inventory model to integrate the above mentioned factors.The salient features of this developed model are as follows: (i) demand rate depends upon quality and selling price (selling markup rate); (ii) a fraction of items produced are defective and only a fraction of defective items are repairable; (iii) unit cost and carrying cost are variables, dependent upon quality; (iv) production rate is quality dependent.This model maximizes the average net profit per unit time and determines the best suitable quality and the best possible markup for selling price.It also determines the optimum production quantity in each cycle.This model is suitable for a manufacturing system where the manufacturer wants to jointly determine the quality (grade) of the item that should be produced and the selling price of the items to maximize the average net profit per unit time.The model is illustrated by numerical examples.A sensitivity analysis has been performed.

Assumptions and Notation
The following assumptions and notations are used in the developed model.
Production Process.The production process is not completely perfect.A fraction of the items produced are defective.A fraction of the defective items can be repaired to make it perfect.The rest cannot be repaired and will be disposed.
Quality of the Product.The item can be produced in different qualities, but the manufacturer wants to market a particular quality which will be most profitable.Actually, types of raw materials used, skills of the workers working in the production line, and quality of machineries used in the production system are responsible for the quality.The quality is assumed to be under manufacturer's control.Quality is represented by a quality index , 0 <  1 ≤  ≤ 1.Here,  1 is the minimum quality that is required to market the product. = 1 indicates the top quality.
Production Rate.The production rate depends upon the quality.The production rate () is a decreasing function of .It means the rate decreases when the quality improves.This rate is taken in the following form: where  1 represents the constant part of the production rate which does not depend upon the quality and the second part  2 / decreases with quality.It can be noticed that  min =  1 +  2 and  max =  1 + ( 2 / 1 ).This type of production rate can be seen in Datta [11].Demand Rate.Demand rate (, ) depends upon both markup rate  and the quality .(, ) decreases with .Normally, (, ) is an increasing function of , but in some situations customers judge the quality by its price.If price is too low, they will doubt the quality and refuse to buy.This implies that for a given , demand will increase with  up to a certain level, and then it will decrease.

The Proposed System
Production starts just after time  = 0.Each unit will be inspected by an automated inspection system immediately after its production.Defective units will be immediately shifted to the repairing shop, repairing shop will separate the repairable and nonrepairable defective units.An amount   (disposal cost) will be charged for disposing each unit.  is the cost of repairing each unit to make it perfect.These repaired units will be brought to the selling area immediately when the stock level at the selling area is zero.A typical graph of the system is shown in Figure 1.Production and consumption will jointly continue during [0,  1 ].The level of inventory at time  =  1 is  1 .Production stops at time  =  1 .
There will be only consumption during [ 1 ,  2 ].Stock level becomes zero at time  =  2 .Immediately, repaired units will be brought which will raise the inventory level to  2 at time  =  2 .This initial stock level  2 will gradually decrease due to market demand and becomes zero at time  = .The time period [0, ] defines a complete replenishment cycle.

Revenue Calculation
The following costs are involved in the proposed inventory system: setup cost, unit cost, carrying cost, repairing cost, and disposal cost.The policy adopted here is to maximize the average net revenue (ANR) per unit time over a replenishment cycle.The decision variables are production period ( 1 ), markup rate (), and the quality level ().
The inventory level  1 at time  =  1 can be expressed as Fresh level of inventory at time  =  2 is  2 = () 1 .
The following expressions of  2 and  are obtained: Total production cost in a cycle =   ()() 1 .
Hence, the average net revenue (ANR) per unit time is (3)

Solution of the Model
The necessary conditions for the existence of a maximum value of ANR for a given value of  are Also, This implies that for given values of  and , ANR is maximum at  1 defined by ( 5).
Hence, the expression under the square root is positive.(proved) Moreover, where The sufficient conditions for maximum are The first condition (a) has already been proved.If we look at (3), we will see that (, ) ⋅ ( It is very difficult to give an analytical proof of (c).However for given values of the model parameters this can be proved.For a given value of , the most economic values  * 1 and  * of  1 and  can be obtained by jointly solving ( 5) and (8).These values of  1 and  will give the maximum value of ANR for a given quality .The optimum production quantity  * in each cycle will satisfy the following equation: An algorithm is given below for solving the problem. Algorithm.
Step 1. Select a demand pattern.Enter the values of the model parameters.
Step 2. Select the increment of , say .The value of "" can be taken as 0.1 or 0.01.Use a counter .Take  = 1.
Step 6. Repeat Steps 3, 4, and 5 until the values of  and  1 become stable.
Step 9.If  > 1, move to the next step.Else  =  + 1 and go to Step 3. Suffix " * " in ANR indicates partial derivative with respect to " * ". "" stands for the hessian at the optimal values of the decision variables.Step 10.Compare the values of ANR() and find  for which it is maximum.The corresponding values of , ,  1 will give us the solution of the model.

Some Special Cases
Case 1.When  = 0, This is optimal production quantity (OPQ) when all items produced are perfect.It can be noted that this expression of OPQ is similar to the basic inventory model with finite production rate and without shortages.Further, if we take limit as () → ∞, then  * = √2  (, )/ ℎ () which is classical EOQ formula.
Case 2. When  = 0, This is OPQ when all defective items are nonrepairable.
This is OPQ when all defective items are repairable.
Case 4. If we substitute  = 0 in (5), it will give us the optimum production period in each cycle as which is similar to (2) of Datta [11] except the last term in the denominator.Datta's [11] article assumed that the defective items would be separated at the end of the production period.Due to this reason, the last term is different.

Solution.
Results are shown in Table 2. Let us analyze the model for a linear demand pattern in the form: (, ) =  + V −  where , V,  > 0. It can be observed that  satisfies the condition   < 0. One advantage of this linear form is that one can easily estimate the constants , V,  by using multiple linear regression.The demand must be positive, that is,  > 0. This condition gives  < ( + V)/ =  1 , say.Thus, the value of  lies in the open interval (0,  1 ).Parameter values are given in Table 3.

Solution.
Results are shown in Table 4.

Sensitivity Analysis
In this section, a sensitivity analysis has been performed to analyze the effects of demand parameters and decision variables ,  1 on the average net profit ANR.The previous numerical problems are considered for this sensitivity analysis.The changes of ANR for the changes of these parameters/variables are shown in Table 5.  (i) proper care should be taken while estimating the demand parameters.A small error in these parameters may mislead the production manager;

Analysis of Nonlinear
(ii) if the production manager is forced to change the optimal production time by a small amount of time, it will not make a significant difference in the optimal profit.

Concluding Remarks
This paper described how to solve a managerial problem associated with a faulty production system which produces a combination of perfect and defective units.It is assumed that only a fraction of defective products are repairable.This model jointly determines the quality of the product and selling price which will maximize the average net revenue per unit time.A general model is developed.For illustrating the model, two numerical problems are presented with two demand patterns.Sensitivity analysis is performed for both examples.An algorithm is provided for solving the model.This algorithm helps in generating computer codes.However, standard optimizing software, like MATLAB, can be used for solving this model.In MATLAB, fmincon function can be used.The model considers quality as a continuous variable described by the quality index , but a situation may arise where manufacturer/business organization may face the problem to select the appropriate quality out of a finite number of quality options.It means that the quality has a discrete set of options.The previous model can be used to tackle this situation with a minor modification.In

Figure 1 :
Figure 1: Pictorial representation of the system.

Table 1 :
Values of the model parameters.

Table 2 :
Optimum results and hessian value.

Table 3 :
Values of the model parameters.

Table 4 :
Optimum results and hessian value." in ANR indicates partial derivative with respect to " * ". "" stands for the hessian at the optimal values of the decision variables.
*Change in ANR with respect to m (%) Change in ANR with respect to k (%) Change in ANR with respect to n (%) Change in ANR with respect to t 1 (%) Demand Pattern (Example 1).It can be observed that demand parameters  and  are very sensitive parameters.ANR increases rapidly with the increase of the value of .But rate of increase is more for higher values of .The parameter  is relatively less sensitive.The parameter  is relatively more sensitive for lower values than higher values.The decision parameter  is more sensitive than the other decision parameter  1 .A change in the value of  by 12% causes more than 20% decrease in the value of ANR.The demand parameters  and  are very sensitive.The parameter  is more sensitive for higher values, but  is more sensitive for lower values.The demand parameter V is relatively less sensitive.The decision variable  1 is very less sensitive.The sensitivity of  is moderate.To get a better idea about the sensitiveness previous results are graphically presented in Figures2 and 3.The following are the managerial insights of the above analysis:Change in ANR with respect to u (%) Change in ANR with respect to w (%) Change in ANR with respect to t 1 (%) Change in ANR with respect to (%) Change in ANR with respect to k (%)