Inventory Modeling for Imperfect Production Process with Inspection Errors, Sales Return, and Imperfect Rework Process

Transcription

1 Vol., No. 4, 4 58, 017 ISSN: Inventor Modeling for Imperfect Production Process with Inspection Errors, Sales Return, and Imperfect Rework Process Aditi Khanna 1, Aakanksha Kishore, Chandra K. Jaggi 3* Department of Operational Research Facult of Mathematical Sciences Universit of Delhi, Delhi , India 1 dr.aditikhanna.or@gmail.com, kishore.aakanksha@gmail.com, 3 ckjaggi@ahoo.com * Corresponding author (Received November 15, 016; Accepted Januar 16, 017) Abstract In real life, due to certain machine problems, process deterioration and man other factors, production processes deliver imperfect qualit items. So, the effect of these defectives cannot be ignored in terms of ensuring good customer service. In order to sustain toda s cut-throat competition, rework process of defective items becomes a rescue to compensate for the imperfections present in the production sstem. The present model attempts to explore the traditional imperfect environment with a more practical approach b incorporating the concept of inspection errors, along with an imperfect rework process. B considering human errors as unavoidable, Tpe-I and Tpe-II errors are also incorporated in the stud. To prioritize on the customer satisfaction level, Sales returns are given full price refunds. An analtical method is emploed to maximize the expected total profit per unit time to stud the combined effect of aforementioned factors on the optimal production quantit. A numerical example along with a comprehensive sensitivit analsis has been presented to demonstrate the applicabilit of the model and also to observe the effects of ke parameters on the optimal production polic respectivel. The pertinence of the model can be found in most manufacturing industries like textile, electronics, furniture, footwear, crocker etc. Kewords- Inventor, Production, Imperfect Items, Inspection, Reworking 1. Literature Overview A common unrealistic assumption in the manufacturing sector is that all the items produced are of good qualit. This has led man researchers to stud the classical EPQ extensivel and relax the traditional assumption of perfect qualit. The first stud in this field can be dated back a centur ago and was projected b Taft (1918). (Schrad, 1967; Porteus, 1986; Rosenblatt and Lee, 1986; Lee and Rosenblatt, 1987) were some other researchers to stud the effect of imperfect qualit items on EPQ models and hence laid the foundation of vast research in this direction. Ver soon, the need for incorporating human errors along with imperfect qualit was taken into account. In view of this, Liou et al. (1994), Makis (1998) first studied the effects of inspection errors in the imperfect EMQ model. Some other researchers emphasized on reworking of defective items in order to compensate the losses incurred due to errors. Salameh and Jaber (000) carried out an important research b considering that the whole lot contains a random percentage of defective items with known p.d.f. The also assumed that whole received lot goes through 100% screening process and the sorted out defective items are sold as a single batch at a discounted price. Furthermore, Haek and Salameh (001) first examined the relevance of rework on imperfect qualit items in finite production model with allowable shortages. The assumed that all the defectives enter the rework process and emerge as perfect items after rework. So, there are no scrap items produced in their model. However, Chiu (003) considered the emergence of scrap items before the beginning of rework process and considered that a random proportion, with 4

2 Vol., No. 4, 4 58, 017 ISSN: known p.d.f., of non-reworkable items, are not sent for rework and are directl sold as scrap. Then, Chiu et al. (004) extended (Chiu, 008) for imperfect rework process without shortages. Other major additions in the area of rework and inspection plans are those of Jamal et al. (004) and Duffuaa and Khan (005). Further, Chiu et al. (006), Chiu et al. (007) and Chiu, (008) investigated the concept of random defective rate, rework, and inspection process for different situations i.e. planning optimal production lot size without derivatives, with an additional service level constraint and failure in repair explained without derivatives respectivel. Later, Liu et al. (009) provided an optimal production polic with rework process. In this paper the studied a production inventor sstem with rework, where stationar demand is either satisfied b production sstem with new raw materials or b rework sstem with defectives coming from the production process. Moreover, Cárdenas-Barrón (009) developed an EPQ-tpe inventor model with planned shortages to determine the economic production quantit for a single product manufactured in a single-stage production sstem which generates imperfect qualit products, and all defective products are reworked in the same ccle. Further, Yoo et al. (009) examined the combined effect of imperfect qualit, inspection errors, sales return and rework on optimal production policies with two different disposition methods of scrap items. Additionall, some other important works in the area of imperfect production are those of (Sana, 010a; Sana, 010b; Chung, 011). Soon after, Sana (011) assumed perfect and imperfect qualit products in an integrated productioninventor model for a three-laer suppl chain. He emphasized on the impact of business strategies such as optimal order size of raw materials, unit production cost, production rate and idle times in different sectors of collaborating market set ups. Later, Khan et al. (011) also assumed that the screening process is not error free. The took similar approach as that of Salameh and Jaber (000), to reach optimal solution in an imperfect qualit environment. Then, Konstantaras et al. (01) developed inventor models for imperfect qualit items with shortages and learning in inspection. Further, Hsu and Hsu (013a) constructed two EPQ models to elaborate the impact of the time factors of when to sell the scrap items on optimal production lot size and backordering quantit. In one of the models, he sold defectives after the end of production period and in the other defectives are sold after the end of production ccle. Adding to this, Hsu and Hsu (013) developed an EOQ model for imperfect qualit items with screening errors and full backlogged shortages along with sales returns. In the same model, the also discussed the case of no shortages under order overlapping scheme. In extension to their own model, Hsu and Hsu (014) revisited their work Hsu and Hsu (013b) to consider the case that returned items are replaced with good items instead of full price refunds. Recentl, Rezaei (016) formulated an EOQ model for imperfect items under three inspection strategies i.e. partial inspection, full inspection or no inspection. In recent times, Jaggi et al. (016) constructed a production model showcasing various relevant features related to imperfect qualit scenario viz. defectives, imperfect inspection process, rework of all sorted defectives, and imperfect inspection of reworked items.. Model Development.1 Problem Description The classic EPQ model assumes that the production unit functions perfectl and thus there is no possibilit of producing defective items. However, these defective items not onl hamper firm s profitabilit but also ruin customers trust. In order to deliver onl perfect items to customers, manufacturing firms adopt rigorous inspection/detection techniques to reduce/eliminate defects to a considerable extent. Although, it is again impractical to assume that inspection process can be 43

3 Vol., No. 4, 4 58, 017 ISSN: perfect. Unavoidable human errors like weak process control, deficient planned maintenance, inadequate work instructions etc. in the inspection process make it prone to errors, and leads to Tpe-I and Tpe-II errors. Due to Tpe-I error, there is direct financial loss to the firm as nondefectives have been classified as defectives b mistake, and then sold at a reduced price in a lesser restrictive inventor. The outcome of Tpe-II error is Sales returns due to customer s disappointment, resulting in more loss of goodwill rather than financial to the firm. To avoid shortages during the inspection period, inspection rate is assumed to be much higher than the demand rate. In order to reduce the production-inventor costs significantl and raise the count of perfect items, a fraction of the imperfect items can be reworked and repaired with a relativel smaller additional reworking and holding costs instead of simpl discarding all the defectives as scrap. To achieve supreme standards of qualit, a second inspection technique is implemented b the firm so as to remove even the slightl defective products from the reworked lot. However, the second inspection process is assumed to be error free. The non-reworkable and the scrap items are discarded at a cheaper price when the get accumulated after the end of inspection and rework process respectivel. This description indicates that presence of defectives, inspection errors, rework, imperfect rework process have considerable affect on the production behavior of manufacturing firms. The manufacturer s problem here is to find out the optimal production quantit, which is determined b maximizing the difference between total revenues and costs per unit time. Various cost components integrated in the model are Production cost, Inspection Cost, Tpe-I Error Cost, Tpe-II Error Cost, Rework Cost, Disposal Cost, and Inventor Holding Cost. Revenue includes sales from perfect, reworked and scraps with some revenue loss from sales returns.. Assumptions The proposed mathematical model is based on following assumptions. (i) Semi-finished goods are used to manufacture a product. (ii) Production rate is finite. (iii) Production and inspection processes are not perfect. (iv) Demand rate is constant, uniform and deterministic. Also demand is satisfied b perfect items onl. (v) To avoid shortages during the inspection process, the production rate of non-defective items needs to be much greater than the demand rate i.e. (1-x)P > λ. (vi) Rework process begins after the end of production process and is also assumed to be imperfect. (vii) All the defect returns are accumulated till the completion time of production and Inspection processes. (viii) Onl a fraction of the defectives are sent for rework and the remaining imperfect items are discarded at a cheaper price before the start of rework process. (ix) Time period is infinite and lead time is insignificant..3 Notations Following nomenclature is used for the paper development. Parameters P Production rate in units per unit time Rework rate in units per unit time P 1 44

4 Vol., No. 4, 4 58, 017 ISSN: λ Demand rate in units per unit time x Proportion of imperfect items (a random variable with known p.d.f.) θ Proportion of non-reworkable items (a random variable with known p.d.f.) θ 1 Proportion of failed reworked items (a random variable with known p.d.f.) q1 Proportion of Tpe-I imperfection error (a random variable with known p.d.f.) q Proportion of Tpe-II imperfection error (a random variable with known p.d.f.) d Production rate of imperfect items(=x*p) in units per unit time d 1 Production rate of scrap items (d 1= P 1θ 1) during rework process in units per unit time t 1 Production up time t Rework run time t 3 Inventor downtime T Ccle length E(.) Expected value operator E(α) Expected value of α K Set-up cost for each production run c Production cost per item ($/ item) i Inspection cost per item ($/ item) s Selling price ($/ item) v Salvage cost (< s) ($/ item) u Disposal cost ($/ item) cw Rework cost( inspection cost included) c r Cost of committing Tpe-I error ($/ item) c a Cost of committing Tpe-II error ($/ item) h Holding cost per unit item per unit time h 1 Holding cost for each imperfect qualit item being reworked per unit time H1 The max on-hand inventor in units, when the regular process ends H The max on-hand inventor in units, when the rework process ends Decision Variables Production size for each ccle (in units) Functions f(d) p.d.f. of defective items f(q 1) p.d.f. of Tpe-I error f(q) p.d.f. of Tpe-II error f(θ) p.d.f. of non-reworkable items f(θ 1) p.d.f. of failed reworked items T.C. Manufacturer s Total cost T.C.U. Manufacturer s Total cost per unit time T.R. Manufacturer s Total revenue T.P. Manufacturer s Total profit T.P.U. Manufacturer s total profit per unit time E.T.P.U. Manufacturer s Expected Total profit per unit time Optimal Values T* Optimal ccle length * Optimal order quantit per ccle E.T.P.U.* Manufacturer s optimal total profit per unit time 45

5 Vol., No. 4, 4 58, 017 ISSN: Model Development 3.1 Model Analsis This paper considers a finite production model in which x% of defectives are produced randoml at rate d from total produced lot. Defect percentage x is a random variable with known p.d.f. f(x). The production rate P being finite. Thus, the production rate of imperfect items can be expressed as the product of total defect proportion times the production rate i.e. d=δp. Due to the occurrence of misclassification errors in the sstem, there is generation of Tpe-I and Tpe-II inspection errors, given their respective proportions of q1=pr(items screened as defects nondefective items) and q =P r(items not screened as defects defective items)(0<q 1<q <1) following p.d.f. of f(q 1) and f(q ) respectivel. It is assumed, q 1 and q are independent of defect proportions d. So, all the items involving inspection errors are determined inter-dependentl b q 1, q, and. As an outcome of Tpe-I error, a fraction (1-x)q 1 of the total non-defective items (1-x) are wrongl classified as defectives thereb causing loss to the firm b losing an opportunit to raise the sales of perfect items. Because of Tpe-II error, there is penalt and goodwill loss to the manufacturer, since a fraction (xq ) of the total defectives (x) is wrongl classified as nondefectives and sold to customers resulting in sales return. Due to customer frustration, the wrongl classified defectives when passed out to customers reenter the sstem continuousl like demand till the end of production and inspection process and are held in the inventor for a complete ccle length T. The imperfect production and inspection processes deliver (1-q ) x and (1-x)(1-q 1) x as the actual defectives and non-defectives respectivel. A random proportion θ with p.d.f. f(θ) of the total imperfect items ((1-q ) x + xq +(1-x)q1), which includes the actual defectives, sales returns and falsel classified defectives respectivel, is discarded as scrap at a reduced price v before the start of rework process. And, the remaining (1- θ)((1-q ) x + xq +(1-x)q 1) units enter the rework process at the rate P 1. The reworked lot undergoes a second inspection process, with the inspection process being error free this time. Since the rework process is not assumed to be perfect, it produces scrap items with random proportion θ1and its p.d.f. f(θ1). The production rate d1 of scrap items can be estimated as the product of rework rate times the scrap rate during the rework process i.e. d1= P1θ1. So, the final outcome as perfect items after the end of rework and inspection processes are ((1-q ) x + xq +(1-x)q 1)(1- θ)(1- θ 1) and the items to be disposed off at a lower price v in the second stage are ((1-q ) x + xq +(1-x)q 1)(1- θ) θ 1. The sequence of events in the inventor ccle is depicted in Fig

6 Vol., No. 4, 4 58, 017 ISSN: Inspection Imperfect Proportion (x) Perfect Proportion (1- x) x (1-x) (1-q ) (q ) (Tpe-II error) (q 1 ) (Tpe-I error) (1- q 1 ) (1-q )x xq (1- x) q 1 (1- x) )(1-q 1) Sales returns SALES Total Imperfect Items [x+ q 1 (1- x)] = δ (θ) Non-Reworkable (1-θ) Reworkable δθ δ(1-θ) TOTAL SCRAP (θ 1) Scrap Inspection (1-θ 1) Perfect δ(1-θ) θ 1 δ(1-θ)(1-θ) Fig. 1. Sequence of events in the inventor ccle 3. Model Formulation In this segment, a mathematical model befitting the above exposed problem, assumptions and description has been formulated. The behavior of inventor ccle depicting the concerned situation is shown in Fig.. 47

7 Vol., No. 4, 4 58, 017 ISSN: Fig.. Inventor behaviors of (a) imperfect production and inspection sstem, (b) defective items sorted through inspection process, (c) total scrap items (d) sales returns 48

8 Vol., No. 4, 4 58, 017 ISSN: From the above depicted inventor ccle in Fig.., we can stud: Total items sold at selling price (s) xq 1 x1q1 x 1 xq Ccle length (T) = Total number of perfect items sold / demand rate i.e T i.e. T (1) xq 1 x 1 q () where 1 x 1 x q As x, q1, q are random variables, so, β, δ are also a random variables, with expected values given as: E[ ] E[ x] E[ q ] 1 E[ x] 1 E[ q ] (5) 1 1 E[ ] E[ x] 1 E[ x] E[ q ] As β, δ are random variables, so, α is also a random variable, with expected values given as: E[ ] E[ ] E[ ] 1 E[ ] 1 E[ ] (7) 1 As α is a random variable, so T is also a random variable, with expected value given as: E[ ] E[ T ] (8) Also, from the above described inventor sstem in Fig., t1 (9) P 1 d 1 t (10) P1 PP1 H t3 (11) H1 P d t1 (1) P d P (3) (4) (6) 49

9 Vol., No. 4, 4 58, 017 ISSN: H H P d t P P d P d 1 d 1 PP (13) 3..1 Relevant Costs Various cost components taken into account are: (i) Production Cost, which includes fixed and variable procurement costs per ccle, i.e., PC K c (14) (ii) Inspection Cost, which includes the cost of inspection, i.e., IC=i (15) (iii) Misclassification Error -1 Cost, which includes the cost of Tpe-I error. i.e., 1 EC1 c 1 x q (16) r (iv) Misclassification Error - Cost which includes the cost of Tpe-II error. i.e., EC a c xq (17) (v) Rework Cost, which includes the cost of reworking of each defective item, i.e., RC c 1 (18) w (v) Disposal Cost, which includes the cost of disposing off each scrap item, i.e. DC u 1 1 (19) (vi) Inventor Holding Cost is the cost of carring all non-defectives, defectives, reworked and items returned from the market. From Fig., it can be calculated as H t H H t Ht dt xq T Pt HC h t h t (0) Therefore, b using equations (14)-(0), total cost per ccle is given b: Total cost (T.C.) = PC + IC + EC1+ EC + RC + DC + HC (1) 50

10 Vol., No. 4, 4 58, 017 ISSN: h P d K c i cr 1 xq1 caxq cw 1 u 1 1 P hd h xq d P P1 P P P1 d 1 d 1 d 1 h P d hp d h P d h P d P P P P P P P d h P d P d h 3... Sales Revenue The total sales revenue consists of four parts: (i) Sales from sorted perfect items is 1 1 R1 s xq x q1 (3) () (ii) Revenue loss from Sales Return R sxq (4) (iii) Sales from reworked items 1 1 R s (5) 3 1 (iv) Sales from total scrap items R4 v 1 1 (6) Therefore, b using equations (3)-(6), total sales revenue per ccle is given b: Total Revenue (T.R.) = R 1 + R + R 3 +R 4 (7) s xq 1 x1q1 sxq s 1 11 v 1 1 (8) 3..3 Manufacturer s Total Profit For obtaining Total Profit (T.P.), we subtract equation () from (8) i.e. 51

11 Vol., No. 4, 4 58, 017 ISSN: hp d T. P. s xq 1 x 1 q sxq s 1 1 v 1 K c i cr x q1 ca xq cw u 1 P d 1 d 1 d 1 h P d hp d h P d h P d P P P P P P P d 1 d xq d 1 hp1 d1 P d h h h 1 P P P P P Manufacturer s Total Profit Per Unit Time Using equations (1) and (9), we obtain the value of total profit per unit time (T.P.U.) s xq 1 x1 q1 sxq s 1 11 v 1 1 c T. PU.. hp d i cr 1 x q1 ca xq cw 1 u 1 1 P h P d d 1 d 1 d 1 h P d h P1 d1 h P d P P P1 P P1 P P1 d 1 d xq d 1 h P1 d1 P d h h h 1 P P1 P P P1 K (9) 3..5 Manufacturer s Expected Total Profit Per Unit Time As x,, 1, q1, q,,, are random variables, therefore b using renewal-reward theorem, we get the expected value of total profit per unit time (E.T.P.U.) using (8) and (30). se[ x] E[ q ] s 1 E[ x] 1 E[ q1 ] se[ x] E[ q] s 1 E[ ] 1 E[ 1] E[ ] E. T. PU.. v E[ ] E[ ] 1 E[ ] E[ 1] E[ ] c i cr 1 E[ x] E[ q1 ] ca E[ x] E[ q ] E[ ] hp E[ x] P cwe[ ] 1 E[ ] u E[ ] E[ ] E[ 1] 1 E[ ] E[ ] P (30) 5

12 Vol., No. 4, 4 58, 017 ISSN: h P E x P E x P E E x P E h P E[ x] P h P1 E[ 1] P1 P P P1 P P1 E [ x] P 1 E[ ] E[ x] P 1 E[ ] hp E[ x] P h P1 E[ 1] P1 P E[ x] P P P1 P P1 E[ x] P E[ ] E[ x] E[ q [ ] ] E x P 1 E[ ] h h h 1 P P P1 K E [ ] [ ] [ ] 1 [ ] [ ] 1 [ ] E[ ] 4. Manufacturer s Optimal Polic The manufacturer intends to maximize his expected total profit per unit time b optimizing the production quantit. In this section, concavit of the objective function is proved and a closedform solution of the model is reached providing the required optimal production value to the manufacturer. The result is established in the form of a lemma. Lemma. The function of manufacturer s expected total profit per unit time is concave. Proof. To prove the global concavit of the expected profit function, the following second-order sufficient condition of global optimalit must hold: d.... 0; d Sufficient Condition: E T PU B taking first order derivative of E.T.P.U. with respect to, we obtain h P E[ x] P E[ x] P1 E[ ] h P E[ x] P P P P1 E [ x] P 1 E[ ] E [ x] P 1 E[ ] hp1 E[ 1] P1 h P E[ x] P d P P1 P P 1 E. T. PU.. d E[ x] P1 E[ ] E[ x] P E[ ] E[ x] E[ q] hp1 E[ 1] P1 P E[ x] P h h E[ ] P P1 P E [ x] P 1 E[ ] h1 P P1 K E (3) B taking second order derivative of E.T.P.U. with respect to, we obtain (31) 53

13 Vol., No. 4, 4 58, 017 ISSN: d K E. T. PU.. 0 (33) 3 d E Hence, concavit of the objective function has been proved mathematicall. To determine the optimal value *, which maximizes the function of E.T.P.U., the first-order necessar condition of optimalit must be satisfied: d E. T. PU.. Necessar Condition: d 0. On setting equation (31) equal to zero, we get: * [ ] [ ] 1 [ ] [ ] 1 [ ] h P E x P E x P E E x P E hp E[ x] P hp E[ ] P P P P P P E x P E E x P E [ ] 1 [ ] [ ] 1 [ ] hp E[ x] P hp E[ ] P P E[ x] P P P P P E[ x] P [ ] 1 [ ] h h h P E[ ] E[ x] E[ q] E x P E 1 P P1 K (34) which is the optimal values of. Hence, Lemma is proved. 5. Numerical Analsis 5.1 Example This subsection validates the developed model with the help of a numerical. The optimal order quantit () and expected profit per unit time (E.T.P.U*) are determined for a given set of parameters. Let us consider a situation with the following parameters: Defect proportion (x), Proportion of Tpe-1 error (q 1), Proportion of Tpe- error (q ), Proportion of non-reworkable items (θ), Proportion of failed reworked items (θ 1) are assumed to follow Uniform distribution with their respective p.d.f. as: 5, 0.05 x ,0.01 q ,0.01 q 0.03 f (x) f (q 1) f (q ) 0, otherwise 0, otherwise 0,otherwise 5, , f ( ) f ( 1). 0, otherwise 0, otherwise Using equations (5), (6) and (7), we obtain E 0.903, E , E

14 Vol., No. 4, 4 58, 017 ISSN: Other parameters are: P = 1,00,000 units/ear, P 1 = 80,000 units/ear, λ = 50,000 units/ ear, K = $ 50/ production, c =$ 30/ unit, s = $ 60/ unit, v = $ 0/ unit, u=$ 8 /unit-, i = $10/ unit, C a= $ 1/ unit, C r= $ 10/ unit, C w= $ 15/ unit, h = $ 5/ unit, h 1 = $ 8/ unit, x= 0.08, q 1 = 0.0, q = 0.0, θ= 0.04, θ 1 = From eq. (34), (31) and (1), the optimal production lot size * 75 units/ ccle, with * * corresponding E. T. PU. $4,18,507 / ear and T 7 das respectivel. Verification of no shortage condition: (1-x)P = 9000 > λ. So, there are no shortages in the model. 5. Sensitivit Analsis Variation in the values of parameters ma happen due to uncertainties in an decision-making situations. To analze these changes, sensitivit analsis has been performed to stud the effect of changes in main parameters x, q 1, q, θ and θ 1 on the optimal production and control variables * and T.P.U.*. Results are summarized in Tables 1-5 using the data of above solved numerical. From the results shown in the Tables 1-5, we obtain the following managerial insights: x * T.C.U.* E.T.P.U.* ,094 5,56,71 4,45, ,70,694 4,31, ,84,45 4,18, ,97,95 4,05,33 Table 1. Impact of x on optimal production polic θ * T.C.U.* E.T.P.U.* ,80,574 4,0, ,84,45 4,18, ,88,89 4,16, ,9,166 4,14,593 Table. Impact of θ on optimal production polic As visible in Table 1, with the increase in the proportion of defectives (x), the production quantit (*) decreases along with the profit values (E.T.P.U.*). Therefore, it is beneficial for the manufacturer to reduce the percentage of defects b identifing the causes of imperfections as these have adverse effect on his sales and also reputation in the long run. From Table, it is evident that with the increase in the value of non-reworkable proportion (θ) of total accumulated defectives, there is decrease in the proportion (1-θ) i.e. the rework proportion. This means lesser items will be recovered after rework process reducing the maximum possible sales and hence the profit values (E.T.P.U.*). There is also increase in the production quantit (*) since more units will be required to satisf the demand with perfect items. 55

15 Vol., No. 4, 4 58, 017 ISSN: θ1 * T.C.U.* E.T.P.U.* ,78,768 4,, ,84,45 4,18, ,90,104 4,14, ,95,804 4,10,955 Table 3. Impact of θ 1 on optimal production polic q 1 * T.C.U.* E.T.P.U.* ,04,630 8,96, ,44,354 6,57, ,84,45 4,18, ,4,845 1,78,815 Table 4. Impact of q 1 on optimal production polic Table 3 reflects negative impact of higher value of scrap proportion (θ 1) of reworked items on total profit (E.T.P.U.*) of the inventor sstem. Consequentl, there is decrease in the proportion of perfectl repaired items (1-θ 1) which cause reduction in the number of perfect items and hence decrease in value of (E.T.P.U.*). However, the there is no change in the production quantit (*). As shown in Table 4, b committing more misclassification errors of Tpe-I (q1), the manufacturer is unable to sell all the perfect items as he is discarding some of them as scrap b mistake. Though a proportion of falsel classified defectives as non-defectives are recovered b rework process but the overall effect is the decrease in the value of (E.T.P.U.*). q * T.C.U.* T.P.U.* ,80,57 4,7, ,8,478 4,, ,84,45 4,18, ,86,370 4,14,147 Table 5. Impact of q on optimal production polic It is observed from Table 5 that as the proportion of Tpe error (q ) increases, there is increase in the units of sales returns since some defectives have been sold to customers b inspector s mistake. Due to this misclassification, the manufacturer suffers penalt cost and goodwill losses. Though all the defect returns go through the rework process but onl few are not recovered due to irreparable changes in some items and hence these are sold as scrap decreasing the value of (E.T.P.U.*). 6. Conclusion and Future Research In the present paper, a finite production model is developed to investigate the optimal production quantit b maximizing the expected total profit per unit time in an imperfect production, imperfect inspection and imperfect rework environment. In order to reduce the productioninventor costs significantl, it is beneficial for the manager to integrate a careful inspection 56

16 Vol., No. 4, 4 58, 017 ISSN: process to sort out the defectives from the produced lot. Despite the emergence of man sophisticated techniques, production of defectives is incontrovertible in most business firms. However, with little investment in the repair/rework methods, some of the defectives can be recovered and treated as perfect items after rework process. Since human errors are undeniable, the inspection team causes Tpe-I and Tpe-II errors during the inspection process. So, it is advisable for the manufacturer to identif the causes of errors to eliminate or reduce the defects. To be at par with toda s competitive market, one cannot overlook an chance to improve on qualit, so, the concept of imperfect rework along with inspection errors has been incorporated to make the imperfect qualit environment appear more relevant and applicable in practice. As per author s knowledge, such a real life set-up has not been explored in the existing literature of inventor management. A numerical example along with a comprehensive sensitivit analsis is emploed to demonstrate the practicalit of the model. For future research, it would be interesting to extend the present model for different demand functions viz., stock dependent demand, price and time dependent demand or both. Another possible direction ma be developed b incorporating different trade credit strategies in the present model. The model can also be studied under the effect shortages, deterioration, inflation, two warehousing etc. References Cárdenas-Barrón, L. E. (009). Economic production quantit with rework process at a single-stage manufacturing sstem with planned backorders. Computers & Industrial Engineering, 57(3), Chiu, S. W. (008). Production lot size problem with failure in repair and backlogging derived without derivatives. European Journal of Operational Research, 188(), Chiu, S. W., Gong, D. C., & Wee, H. M. (004). Effects of random defective rate and imperfect rework process on economic production quantit model. Japan Journal of Industrial and Applied Mathematics, 1(3), Chiu, S. W., Ting, C. K., & Chiu, Y. S. P. (007). Optimal production lot sizing with rework, scrap rate, and service level constraint. Mathematical and Computer Modelling, 46(3), Chiu, Y. P. (003). Determining the optimal lot size for the finite production model with random defective rate, the rework process, and backlogging. Engineering Optimization, 35(4), Chiu, Y. S. P., Lin, H. D., & Cheng, F. T. (006). Optimal production lot sizing with backlogging, random defective rate, and rework derived without derivatives. Proceedings of the Institution of Mechanical Engineers, Part B: Journal of Engineering Manufacture, 0(9), Chung, K. J. (011). The economic production quantit with rework process in suppl chain management. Computers & Mathematics with Applications, 6(6), Duffuaa, S. O., & Khan, M. (005). Impact of inspection errors on the performance measures of a general repeat inspection plan. International Journal of Production Research, 43(3), Haek, P. A., & Salameh, M. K. (001). Production lot sizing with the reworking of imperfect qualit items produced. Production Planning & Control, 1(6), Hsu, J. T., & Hsu, L. F. (013a). An EOQ model with imperfect qualit items, inspection errors, shortage backordering, and sales returns. International Journal of Production Economics, 143(1), Hsu, J. T., & Hsu, L. F. (013a). Two EPQ models with imperfect production processes, inspection errors, planned backorders, and sales returns. Computers & Industrial Engineering, 64(1),

17 Vol., No. 4, 4 58, 017 ISSN: Hsu, J., & Hsu, L. (014). A supplement to an EOQ model with imperfect qualit items, inspection errors, shortage backordering, and sales return. International Journal of Industrial Engineering Computations, 5(), Jaggi, C. K., Khanna, A., & Kishore, A. (016, March). Production inventor policies for defective items with inspection errors, sales return, imperfect rework process and backorders. In K. Singh, M. Pande, L. Solanki, S. B. Dandin, & P. S. Bhatnagar (Eds.), AIP Conference Proceedings (Vol. 1715, No. 1, p. 0006). AIP Publishing. Jamal, A. M. M., Sarker, B. R., & Mondal, S. (004). Optimal manufacturing batch size with rework process at a single-stage production sstem. Computers & Industrial Engineering, 47(1), Khan, M., Jaber, M. Y., & Bonne, M. (011). An economic order quantit (EOQ) for items with imperfect qualit and inspection errors. International Journal of Production Economics, 133(1), Konstantaras, I., Skouri, K., & Jaber, M. Y. (01). Inventor models for imperfect qualit items with shortages and learning in inspection. Applied Mathematical Modelling, 36(11), Lee, H. L., & Rosenblatt, M. J. (1987). Simultaneous determination of production ccle and inspection schedules in a production sstem. Management Science, 33(9), Liou, M. J., Tseng, S. T., & Lin, T. M. (1994). The effects of inspection errors to the imperfect EMQ model. IIE transactions, 6(), Liu, N., Kim, Y., & Hwang, H. (009). An optimal operating polic for the production sstem with rework. Computers & Industrial Engineering, 56(3), Makis, V. (1998). Optimal lot sizing and inspection polic for an EMQ model with imperfect inspections. Naval Research Logistics (NRL), 45(), Porteus, E. L. (1986). Optimal lot sizing, process qualit improvement and setup cost reduction. Operations Research, 34(1), Rezaei, J. (016). Economic order quantit and sampling inspection plans for imperfect items. Computers & Industrial Engineering, 96(C), 1-7. Rosenblatt, M. J., & Lee, H. L. (1986). Economic production ccles with imperfect production processes. IIE Transactions, 18(1), Salameh, M. K., & Jaber, M. Y. (000). Economic production quantit model for items with imperfect qualit. International Journal of Production Economics, 64(1), Sana, S. S. (010a). A production inventor model in an imperfect production process. European Journal of Operational Research, 00(), Sana, S. S. (010b). An economic production lot size model in an imperfect production sstem. European Journal of Operational Research, 01(1), Sana, S. S. (011). A production-inventor model of imperfect qualit products in a three-laer suppl chain. Decision Support Sstems, 50(), Schrad D. A (1967). A deterministic inventor model for repairable items. Naval Research Logistics, 14(3), Taft, E. W. (1918). The most economical production lot. Iron Age, 101(18), Yoo, S. H., Kim, D., & Park, M. S. (009). Economic production quantit model with imperfect-qualit items, two-wa imperfect inspection and sales return. International Journal of Production Economics, 11(1),