A Two-Warehouse Inventory Model with Imperfect Quality and Inspection Errors

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1 esearch Journal of Applied Sciences, Engineering and Technolog 4(0): , 0 ISSN: Maell Scientific Organization, 0 Submitted: ecember 8, 0 Accepted: April 3, 0 Published: October 5, 0 A To-Warehouse Inventor Model ith Imperfect Qualit and Inspection Errors Tie Wang School of Management, Shanghai Universit, Shanghai, 00444, China School of Mathematics, Liaoning Universit, Shenang, Liaoning 0036, China Abstract: In this stud, e establish a ne inventor model ith to arehouses, imperfect qualit and inspection errors simultaneousl. The mathematical model b maimizing the annual total profit and the solution procedure are developed. As a bproduct, e correct some technical error in developing the optimal ordering policies in the above to papers. Moreover, e find a mild condition satisfied b most common distributions to make the ETPU() concavit. The Proposition is used to determine the optimal solution of ETPU(). Keords: EOQ, imperfect qualit, misclassification errors, to-arehouse INTOUCTION The traditional EOQ (economic order quantit) model is a crucial building block of the deterministic inventor theor because of its simple and elegant structure as ell as its rich managerial insights. Hoever, some unrealistic assumptions make it not conform to actual inventories; the model must be etended or altered. Porteus (986) investigated the influence of defective items on the traditional EOQ model. He assumed that there is a fied probabilit that the process ould go outof-control. osenblatt and Lee (986) assumed that the time beteen the in-control and the out-of-control state of a process follos an eponential distribution and that the defective items are reorked instantaneousl. The suggested producing in smaller lots hen the process is not perfect. In a later stud, (Lee and osenblatt, 987) studied a joint lot sizing and inspection polic for an EOQ model ith a fied percentage of defective products. Salameh and Jaber (000) etended the traditional EOQ model b accounting for imperfect qualit items and considered the issue that poor-qualit items are sold as a single batch b the end of the 00% screening process. Along this line, (Papachristos and Konstantaras, 006) discussed the non-shortages in inventor models here the proportion of defective items is a random variable. The proposed an alternative to the Salameh and Jaber (000) b speculating on the timing of ithdraing and selling the imperfect lot, (Wee et al., 007) developed an optimal inventor model for items ith imperfect qualit and shortage backordering b assuming that all customers are illing to ait for ne suppl hen there is a shortage. (Maddah and Jaber, 008) rectified a fla (Salameh and Jaber, 000) b the reneal-reard theorem and leaded to simple epressions of the optimal order quantit and epected profit per unit time. Chang and Ho (00) revisited Wee et al. (007) and applied the ellknon reneal-reard theorem to obtain a ne epected net profit per unit time and derived the eact closed-form solutions to determine the optimal lot size, backordering quantit and maimum epected net profit per unit time ithout differential calculus. Khan et al. (0) eplored an EOQ model ith imperfective items and an imperfect inspection process That is, the inspector ma commit errors hile screening. The probabilit of misclassification errors is assumed to be knon. The inspection process ould consist of three costs: C C C Cost of inspection Cost of Tpe I errors Cost of Tpe II errors (Sarker and Kindi, 006) developed an inventor model from the Buer s perspective, to determine the optimal ordering policies in response to a discount offer settled b the vendor for five possible cases. Goal and Jaber (006) and Cardenas-Barron (009) etended and corrected (Sarker and Kindi, 006). Cardenas-Barron (009) investigated an inventor model for imperfective items under a one-time-onl discount, here the defectives can be screened out b a 00% screening process and then can be sold in a single batch b the end of the 00% screening process. The optimal order policies associated ith three kinds of effective times of the reduced price are obtained. ecentl, Yang (004) considered to-arehouse models for deteriorating items ith shortages under inflation. Yang (006) etended the above stud b incorporating partial backlogging and then compared the to to-arehouse inventor models based on the minimum cost approach. Zhou and Yang (005) presented a to-arehouse inventor model for items ith stock-level-dependent demand rate. Moreover, (Lee 3896

2 es. J. Appl. Sci. Eng. Technol., 4(0): , 0 and Hsu, 009) developed a to-arehouse inventor model for deteriorating items ith time-dependent demand. Chung et al. (009) investigated a ne inventor model ith to arehouses and imperfect qualit simultaneousl. The mathematical model b maimizing the annual total profit and the solution procedure ere developed. The above stud mentioned did not consider the inventor model ith to arehouses, imperfect qualit and inspection errors simultaneousl. Based on (Chung et al., 009; Khan et al., 0), this stud tries to develop an inventor model to incorporate concepts of to arehouses, imperfect qualit and inspection errors to establish a ne economic production quantit model. Consequentl, the inventor model in this stud is more practical than the traditional EOQ model. Of course, this stud generalizes (Chung et al., 009; Khan et al., 0). As a bproduct, e correct some technical error in developing the optimal ordering policies in the above to studies. This stud incorporates the concepts of the basic to arehouses, imperfect qualit and inspection errors. The epected total profit per unit time function ETPU() is concave. The epected total profit per unit time function ETPU() is pieceise concave. Moreover, the epected total profit per unit time function ETPU() in this stud is not pieceise concave in general. In addition, e find a mild condition satisfied b most common distributions to make the ETPU() concavit. The Proposition is used to determine the optimal solution of ETPU(). NOTATIONS AN ASSUMPTIONS It is assumed that the retailer ons a arehouse (denoted b OW) ith a fied capacit of units and an quantit eceeding this should be stored in a rented arehouse (denoted b W), hich is assumed to be available ith abundant space. The transportation cost hich transfers from W to OW is included in holding cost of the W. So it is not considered b our stud separatel. At the beginning of the period, the lot size enters the sstem ith a purchasing price of c per unit and an ordering cost of K. Out of these units, units are kept in OW and - units in W. (If #, units are onl kept in OW.) The products of the OW are sold onl after consuming the products kept in W. All of products ordered arrive at the same time. The are screened ith unit inspection cost d in the OW and W simultaneousl b the inspectors. It is assumed that each lot received contains fied percentage p of defective items. The inspection process of the lot is conducted at a rate of units/unit time. The inspectors screen out the defective items from the lot ith fied rate of misclassification. That is, a proportion m of no defective items are classified to be defective and a proportion m of defective items are classified to be no defective. It is assumed that p, m, m have independent and identical distribution function and the probabilit densit functions, f(p), f(m ) and f(m ) are knon. The selling price of good-qualit item classified b the inspectors is s per unit. The inspection process of the lot is conducted at a rate of units per unit time. It is assumed that the items that are returned from the market are stored ith those that are classified as defective b the inspectors. The are all sold as a single batch at the end each ccle at a discounted price of v/unit. After inspection time, no holding cost is considered for imperfect items classified b inspectors. This cost as scaled accordingl. There are also no holding costs for the returned products. The behavior of the inventor level ma be illustrated in Fig., here T is the ccle length, (-p) m +p (-m ) is the number of defective items classified b the inspectors ithdra from inventor, t is the total inspection time of the W, t the total inspection time of the OW and to the time to use up of the W. Figure and reveal t = (-)/ and t = /. THE MOEL From the above assumptions about the model and Fig. and, e kno the consumption process continue at the demand rate until the end of ccle time T. ue to inspection error, some of the items used to fulfill the demand are defective. These defective items are later returned to the inventor. To avoid shortages, it is assumed that the number of no defective items at least equal to the adjusted demand, that is the sum of the actual demand and items that are replaced for ones returned (pm ) from the market over T. Thus, ( p) m p( m ) T pm ( p)( m ) T So, for the limit case, the ccle length can be ritten as: ( p)( m ) T () It should note here that this behavior as suggested b Khan et al. (0). efine N (, p) and N O (, p) as the number of no defective items ith respect to arehouses W and OW, respectivel. There are to cases occur. Case : Suppose <. Then 3897

3 es. J. Appl. Sci. Eng. Technol., 4(0): , 0 Fig. : The model of products in stock of to arehouses if t 0 $ t Fig. : The model of products in stock of to arehouses if t O < t No(, p) ( p) m p( m ) () To avoid shortages, it is assumed that N O (, p) are at least equal to, the sum of the actual demand during times / and items that are replaced for ones returned from the market during time T, that is: No(, p) pm 3898

4 es. J. Appl. Sci. Eng. Technol., 4(0): , 0 Substituting () into (3). We have: ( p)( m ). Case : Suppose $. Then: N (, p ) ( )( p ) m ( ) p( m ) N ( Y, P ) o ( P ) m p ( m ) (3) (4) To avoid shortages, it is assumed that N (, p) and N O (, p) are at least equal to, the sum of the actual demand during times t and t and items that are replaced for ones returned from the market during time t 0 and T-t O that is: N(, p) t ( ) pm N (, o p ) t pm 0 (5) (6) Substituting (4) and (5) in (6) and (7) and replacing t and t b (-)/, respectivel. We have: ( p)( m ) Combining the above arguments, e conclude: ( p)( m ) (7) All of products are stored in OW and W, respectivel and screen them in OW and W simultaneousl at the same time. In addition, T() denote the sum of total sales revenue of no defective qualit classified b the inspectors, imperfect qualit items classified b the inspectors and returned items per ccle; TC(), TP() and ETPU() denote the sum of total cost per ccle and the profit per ccle and average profit per ccle, respectivel. Then, TP() = TP().! TC() T( ). s( p)( m) spm) ETPU ( ). E TP T About the computations of ETPU() there are to cases to occur. Case (I): Suppose <. This case is same to the model presented in Khan et al. (0) stud. But, there is a mathematical error in () of their paper for holding cost per ccle, so the total cost per ccle, the total profit per ccle, average profit per ccle and optimal order size. The corrected total cost per ccle should be TC () = procurement cost + screening cost + holding cost: t K c d cr( p) m capm h ( t ( p) m p( m)) pm ( T t ) T. Since t = /, (8) can be ritten as follos: TC( ) K c d cr( p) m capm ( m p) p( m m) pm p( m) m h ( p)( m)( p)( m) pm (8) (9) The total profit per ccle can no be ritten as the difference beteen the total revenue and the total cost per ccle, that is: TP( ) T( ) TC( ) s( p)( m) spm v( p) m vp { K c d c pm cpm h r ( ) a ( m p) pm p( m m) ( p)( m)(( p)( m) pm) h p ( p )( m ) m (0) To obtain the eact closed-form solution to determine the optimal lot size ithout differential calculus, e can use the renealreard theorem to derive the epected net profit per unit time ETPU () as the folloing: 3899

5 es. J. Appl. Sci. Eng. Technol., 4(0): , 0 ETPU E TP ( ) ETP ( Y) ( ) T ET s E p)( E m ( s ca ) E p E m ( v cr )( E p) E m ve p ( d c) E p E m ( E m E p E p) E p E m K h { ( E p )( E m ) E p( E m E m) ( E p)( E m ) E( p) ( E m) E p( p) E m } E p G C ( H) () Equation (0) and () are corrected total profit per ccle and epected annual profit for Eq. (6) and (9) in Khan et al. (0). Note that hen m = m = 0, Eq. () gives the eact epression for Eq. (8) of Chung et al. (009) as: ETPU s ( E p ve p ( d c ) ( ) E p K h E p E p ( E p ( E p) E p () Which is same to the Eq. (6) presented in (Maddah and Jaber, 008). Case (II): Suppose $. There are to cases to be discussed. Case A: Suppose t O $t. Then TC A () = procurement cost + screening cost + holding cost: K c d c p m c pm h t r ( ) a ( ) pm ( t) t to t ( )( p) m ( ) p( m) ( t O t ) h t ( ( p) m p( m))( to t) ( ( p) m p( m )) pm (3) ( T to ) ( T t O ) Since t O = (-) (-p) (-m )/, t = (-)/, t = / and T = (-p)(-m )/, (3) can be ritten as follos TC A (): TCA( ) K c d cr( p) m capm ( ) ( m p) p( m m ) m p h ( p)( m)( p)( m) pm) h (( pm ) p( m)) (( p)( m pm)( )( p)( m) h pm ( P )( m ) (4) Case B: Suppose t O <t. Then TC B () = procurement cost + screening cost + holding cost: K c d c p m c pm h r ( ) a ( ) pm ( t) t t O t ( )( p) m ( ) p( m) ( t O t ) t ( to) h to ( ( t to))( t to) ( t to) ( p) m p( m)) ( T t ) pm ( T t O ) (5) Since t O = (-) (-p) (-m )/, t = (-)/, t = / and T = (-p)(-m )/, (5) can be ritten as follos: TCB( ) K c d cr( p) m capm ( ) ( m p) p( m m ) m p h ( p)( m(( p)( m) pm) h pm p m p ( ( ) ( ) (( p)( m)( ) pm))( p)( m) h pm ( p )( m ) So, e conclude: TC ( ) TCA( ) I t TCB( ) I ) t t t O K c d cr( p) m capm ( ) ( m p) p( m m) mp h (6) 3900

6 es. J. Appl. Sci. Eng. Technol., 4(0): , 0 ( p)( m)(( p)( m) pm) hi t (( pm ) p( m)) O t (( p)( m) pm( )( p)( m) ( Em E p) Ep ( Em Em Em E p h ( E p( E m ) E( p) E( m ) E p( p) E m ) ( Ep )( Em E p pm p m p h I ( ( ) ( ) tot (( p)( m)( ) pm))( p)( m) h pm ( p )( m ) (7) h E( p) E( m ) ( ( E p)( E m ) E p( p) I t t E m O E p K { ( ( E p )( E m ) Ep( p) E m E p Note that hen m = m = 0, TC A () = TC B () and (7) ill be reduced to Eq. (3) presented in Chung et al. (009). Consequentl: TP( ) T( ) TC( ) s( p)( m) spm v( p) m vp { K c d h c r ( pm ) cpm a m p p m m m p h ( p)( m)( p)( m) pm) (( pm ) p( m)) h I tot (( p)( m) ( )( p)( m) ( pm ) p( m) p h I tot (( p)( m )( ) pm ))( p)( m ) pm ( p)( m ) h (8) To obtain the eact closed-form solution to determine the optimal lot size, e can use the reneal-reard theorem to derive the epected net profit per unit time ETPU () as: ETPU E TP ( ) ETP ( ) ( ) T ET s( E p( E m) ( s ca ) E p E m) ( v c E p E m ve p d c r )( ) ( )) E p) E m ) ( Em Ep ) Ep ( Em Em h Em Ep ( E p)( E m E( p) E( m ) Ep ( pem ) ( Ep )( Em ) E p ( Ep ) Em Ep Em ( h ( E p)( E m EpI t t Em E p p E m O ( ) ( E p)( E m ( Ep ) E( p) E( m h ) ( E p( E m ) ) ( Em Ep ) Ep ( Em m) Em Ep ( E p( E m E( p) E( m ) E p( p) E m ( E p)( E m ) E p J I ( L) (9) Note that hen m = m = 0, Eq. (9) gives the corrected epression for Eq. (8) of Chung et al. (009) as: ETPU s ( E p ) ve p ( d c ) ( ) ( Ep )( Em Ep E( p) h ( ( E p)( E m ( E p) ) h E p ( ) K { ( ( E p E p) h E p E( p) ( E p) ( Ep ) Ep E( p) h ( E p) ( E p) ) h E p E p) ( E p) ( Ep (0) 390

7 es. J. Appl. Sci. Eng. Technol., 4(0): , 0 Combining () and (0), e have: ETPU( ) if ETPU ( ) ETPU( ) if and, (),() E pi EpI tot m ( )( p) E pf( ( )( p) Taking derivative of ETPU () ith respect gives: ETPU () = ETPU () (3) Using the arithmetic geometric mean inequalit (AM-GM) theorem: ETPU( ) C GH E m E pf ( ( )( p) ) J detpu ( ) ( E p( ) E m E p p f ( ( )( p) ) L ( E p( E m ( ) (7) When the to terms related to in () are equal, impling: From the assumptions about the sstem parameters, e kno: * G H detpu ( ) detpu ( ) lim 0 lim L 0. 0 d 0 d K ( Ep )( Em ( Em Ep ) E p( Em Em ) E( p) ( Em ) E p E m E p( p) E m h ( E p)( E m ) Ep (4) and () reduces to equalit, that is, the maimum profit is: ETPU * ( ) C GH (5) The (3) is the corrected mathematical epression for (0) of (Khan et al., 0). Note that hen m = m = 0, (3) gives the eact epression for (4) of (Chung et al., 009) as: ** K h E p. E( p) (6) This is same to the result obtained b Maddah and Jaber (008). Note that, Ep ( pi ) t E p pi m Ot ( ) ( )( p) E p( p) F( ( )( p) ) and Hence there must eit a root for d ETPU ()/d = 0 at least, so * must satisf the folloing equation: E m E pf ( )( p) ( E p)( ) J p E m E p f ( )( p) L 0. ( E p)( E m ( ) (8) Let m = m = 0, (8) gives the eact epression for (5) of Chung et al. (009) as: ** K h E p E( p) E p ( E p E p E p E( p) h ( E p) ( E p h E p E( p) ( E p) E p (9) Taking derivative of ETPU () ith respect again gives: E m E pf d ETPU p ( ) ( )( ) 3 d ( E p)( ) p E m E p f 3 ( )( p) 3 ( ) ( E p)( E m ) E m E p J p f ( )( p) ) 3 ( E p)( E m )( ) 390

8 es. J. Appl. Sci. Eng. Technol., 4(0): , 0 Em E p p f '( ( )( p) ) 4 ( E p( ) p Em E f ' ( p) ( )( p) 3 3 ( E p)( ) (30) From the (30), e can not determine the conavit of ETPU (). But, e notice that d ETPU ()/d <0 hen f ()#0. This condition is satisfied for man usuall used distributions such as normal distribution, eponential distribution and uniform distribution and so on. In the folloing discussion, e assume f ()#0. Note that I>0 and J>0 because of h is more than h. Both ETPU () and ETPU () are concave. So, * and * are the possible optimal solutions for ETPU(). Let opt * represent the optimal solution of ETPU(). Because *< and *$ must be satisfied, so e have the folloing proposition. Proposition : Under f ()#0 three cases ma occur: C if *< and *$, then opt * = * or * such that ETPU ( opt *) = ma{ ETPU ( *), ETPU ( *)} C if *< and *<, then opt * = * C if *$ and *$, then opt * = * CONCLUSION This stud incorporates the concepts of the basic to arehouses, imperfect qualit and inspection errors to generalize Chung et al. (009) and Khan et al. (0). The epected total profit per unit time function ETPU() in Salameh and Jaber (000) is concave. The epected total profit per unit time function ETPU() in Chung et al. (009) is pieceise concave. Hoever, the epected total profit per unit time function ETPU() in this stud is not pieceise concave in general. But, e find a mild condition satisfied b most common distributions to make the ETPU() concavit. The Proposition is used to determine the optimal solution of ETPU(). z c K s v NOTATIONS Number of units demanded per ear Storage capacit in OW, fied Storage capacit in W Order quantit Unit variable cost Fied ordering cost Unit selling price of items of good qualit Unit selling price of defective items, v<c Screening rate d Unit screening cost h, h Holding cost for items in the W and OW, respectivel, h $ h T The ccle length m Probabilit of Tpe I error (classifing a no defective item as defective) m Probabilit of Tpe II error (classifing a defective item as no defective) p Probabilit that an item is defective t Inspection time of the Wt inspection time of the OW t O Time to use up of the W f(p) Probabilit densit function of p f(m ) Probabilit densit function of m f(m ) Probabilit densit function of m B Number of items that are classified as defective in rented arehouse B 0 Number of items that are classified as defective in on arehouse B Number of defective items that are returnedfrom the market in rented arehouse B 0 Number of defective items that are returnedfrom the market in on arehouse c a Cost of accepting a defective item c r Cost of rejecting a non defective item T() The sum of total revenue of good qualit and imperfect qualit items per ccle TC() The sum of total costs per ccle * opt He optimal solution such that ETPU ( opt* ) ill be a maimum EFEENCES Cardenas-Barron, L.E., 009. Optimal ordering policies in response to a discount offer: Corrections. Int. J. Prod. Econ., : Cardenas-Barron, L.E., 009. Optimal ordering policies in response to a discount offer-etensions. Int. J. Prod. Econ., : Chang, H.C. and C.H. Ho, 00. Eact closed-form solutions for optimal inventor model for items ith imperfect qualit and shortage backordering. Omega, 38: Chung, K.J., C.C. Her and S.. Lin, 009. A toarehouse inventor model ith imperfect qualit production processes. Comput. Ind. Eng., 56: Goal, S.K. and M.Y. Jaber, 006. A note on Optimal ordering policies in response to a discount offer. Int. J. Prod. Econ., : Khan, M.V., M.Y. Jaber and M. Bonne, 0. An Economic Order Quantit (EOQ) for items ith imperfect qualit and inspection errors. Int. J. Prod. Econ., 33:

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