Huang [Journal of Statistics and Management Systems, Vol. 6, No. 2, pp. 171-180, 2003.] studied the EOQ (Economic Order Quantity) and EPQ (Economic Production Quantity) models with backlogging and defective
items using the algebraic approach. He assumed that a 100% inspection policy and the known proportion of defective items was removed after the screening process prior to storage or use. In this paper, we will offer another simple approach to find both the optimal lot size and backorder level under the minimized total relevant cost per unit time

Hsin-Fan Chen

Ming-Hon Hwang

Yu-Cheng Tu

Yung-Fu Huang

English

HRČAK - Portal of Croatian Scientific and Professional Journals

H.-F. Chen, Y.-F. Huang, Y.-Ch. Tu, M.-H. Hwang: An easy approach to derive EOQ and EPQ models with shortage and defective itemsENGINEERING MODELLING 23 (2010) 1-4, 37-41 37SUMMARYHuang [Journal of Statistics and Management Systems, Vol. 6, No. 2, pp. 171-180, 2003.] studied the EOQ(Economic Order Quantity) and EPQ (Economic Production Quantity) models with backlogging and defectiveitems using the algebraic approach. He assumed that a 100% inspection policy and the known proportion of defectiveitems was removed after the screening process prior to storage or use. In this paper, we will offer another simpleapproach to find both the optimal lot size and backorder level under the minimized total relevant cost per unit time.Key words: arithmetic-geometric mean inequality approach, algebraic approach, EOQ, EPQ, shortage, defectiveitem.UDC 336.64:336.645:519.86:512.6Original scientific paperReceived: 30.03.2010.An easy approach to derive EOQ and EPQmodels with shortage and defective itemsHsin-Fan Chen, Yung-Fu Huang, Yu-Cheng Tu and Ming-Hon HwangDepartment of Marketing and Logistics Management, Chaoyang University of Technology,168, Jifong East Road, Wufong, Taichung 41349, TAIWANe-mail: huf@cyut.edu.tw1. INTRODUCTIONThe economic order quantity (EOQ) model with/without shortages and economic production quantity(EPQ) model with/without shortages are widely usedby practitioners as a decision-making tool for thecontrol of inventory. However, the assumptions of theEOQ / EPQ models are rarely met. This has led manyresearchers to study the EOQ / EPQ extensively underrealistic situations.A common unrealistic assumption of the EOQ /EPQ is that all units produced or purchased are ofgood quality. We know that it is difficult to produceor purchase items of a 100% good quality. Recently,Huang [1] studied the EOQ and EPQ models withbacklogging and defective items. He assumed that100% inspection policy and the known proportion ofdefective items was removed after the screeningprocess prior to storage or use. In addition, Huang[1] used the algebraic method to determine the optimalsolution by minimizing the annual relevant cost. Inprevious several papers, the EOQ and EPQ formulaefor the shortage case have been derived usingdifferential calculus and solving two simultaneousequations with the need to prove optimality conditionswith second-order derivatives. The mathematicalmethodology is difficult to many younger studentswho lack the knowledge of calculus. Grubbström andErdem [2] and Cárdenas-Barrón [3] showed that theformulae for the EOQ and EPQ with backloggingwere derived without differential calculus. Thisalgebraic approach could therefore be used easily tointroduce the basic inventory theories to youngerstudents who lack the knowledge of calculus. ButRonald et al. [4] thought that their algebraic procedurewas too sophisticated to be absorbed by ordinaryreaders. Hence, Ronald et al. [4] derived a procedureto transform a two-variable problem into two steps,and then, in each step they solve a one-variableproblem using only the algebraic method withoutcalculus. Recently, Chang et al. [5] rewrote theobjective function of Ronald et al. [4] so that the usualskill of completing the square can solve the problemwithout using their sophisticated method. AllH.-F. Chen, Y.-F. Huang, Y.-Ch. Tu, M.-H. Hwang: An easy approach to derive EOQ and EPQ models with shortage and defective items38 ENGINEERING MODELLING 23 (2010) 1-4, 37-41previously mentioned articles use algebraicoptimization.Although Huang [1] used the easily algebraicapproach to find the optimal solution, his method hadthe same problem as Grubbström and Erdem [2] andCárdenas-Barrón [3]. Recently, Tu et al. [6] usedanother simple arithmetic-geometric mean inequalityapproach to investigate the retailer’s EOQ under tradecredit period depending on the order quantity.Therefore, in this paper, we provide the same as Tu etal. [6] easy-to-understand and simple-to-applyarithmetic-geometric mean inequality approachwithout using derivatives to obtain the optimal solution.This approach could therefore be used easily tointroduce the basic inventory theories to youngerstudents who lack the knowledge of calculus.2. DERIVATION OF THE LOT SIZE ANDBACKORDER LEVELThe following notation and assumptions are thesame as Huang's [1] which will be used further on inthis paper.NotationQ − order quantity (EOQ model) / productionquantity (EPQ model) (including defectiveitems),V − maximum on-hand inventory level (includingdefective items),L − maximum shortage (backorder) level(including defective items),D − demand rate for nondefective items, units pertime,P − production rate for nondefective items, unitsper time (P>D),A − ordering cost / setup cost,I − the fixed inspection cost incurred with each lot,i − unit inspection cost,h − unit stock holding cost per unit per time,b − unit shortage cost per unit short per time,k − the known percentage of defective items in Q,TAC(V,L) − total relevant cost per unit time.Assumptions(1) Demand rate is known and constant.(2) Production rate is known and constant.(3) Time period is infinite.(4) Each lot purchased / produced contains a knownproportion of defectives removed prior to storageor use.(5) The screening time for items is so fast that we canneglect it. The inspection cost consists of a fixedper lot inspection cost and a fixed unit inspectioncost.We use here an easy and a simple arithmetic–geometric-mean-inequality (AGM) approach the sameas Tu et al. [6] to obtain the optimal lot size andbackorder level that minimizes the annual total relevantcost. The arithmetic–geometric mean inequality is asfollows. For any of two real positive numbers, say xand y, the arithmetic mean (x+y)/2 is always greaterthan or equal to the geometric mean xy . Namely,x y xy2+≥The equation holds only if x=y.Model I : EOQ model (shown in Figure 1)Fig. 1  The inventory level of Model IFrom Eq. (2) in Huang [1] we know that the total relevant cost per unit time, TAC(V,L), can be expressed as:2 2 2 2D ( 1 k ) V h ( 1 k ) L bTAC(V ,L ) A I (V L )i  (1 k )(V L ) 2D 2D⎡ ⎤− −= + + + + +⎢ ⎥− + ⎢ ⎥⎣ ⎦(1)For convenience, we let Q=V+L and substitute into Eq. (1).Then, we can obtain:H.-F. Chen, Y.-F. Huang, Y.-Ch. Tu, M.-H. Hwang: An easy approach to derive EOQ and EPQ models with shortage and defective itemsENGINEERING MODELLING 23 (2010) 1-4, 37-41 392 2 2 2D (1 k ) ( Q L ) h ( 1 k ) L bTAC( Q,L ) A I Qi  (1 k )Q 2D 2D⎡ ⎤− − −= + + + +⎢ ⎥− ⎢ ⎥⎣ ⎦(2)Then we rewrite Eq. (2) as:2( 1 k )( h b ) h ( 1 k )hbQ D( A I ) Di TAC( Q,L ) L Q2Q h b 2( h b ) (1 k )Q 1 k− + − +⎡ ⎤= − + + +⎢ ⎥+ + − −⎣ ⎦(3)It implies that when Q is given, we can set L as:hL Qh b⎛ ⎞= ⎜ ⎟+⎝ ⎠to get the minimum value of TAC(Q,L) as follows:(1 k )hbQ D( A I ) Di TAC[ Q,L(Q )]2( h b ) (1 k )Q 1 k− += + ++ − − (4)By using the AGM approach we can easily obtain that in Eq. (4):( 1 k )hbQ D( A I ) Di TAC[ Q,L( Q )]2( h b ) ( 1 k )Q 1 k( 1 k)hbQ D(A I) Di 2hbD( A I ) Di22( h b ) ( 1 - k)Q 1 k ( h b ) 1 k− += + + ≥+ − −− + +≥ × + = ++ − + −(5)When the equality:(1 k)hbQ D(A I)2( h b ) (1 - k)Q− +=+ (6)holds, TAC[Q, L(Q)] has a minimum. Then, we can find the optimal ordering quantity:1 2( A I )D h b 1 2( A I )Db 1 2( A I )DhEOQ( Q*)1 k h b 1 k h( h b ) 1 k b( h b )+ + + += = +− − + − +(7)and putting Q* into hL Qh b⎛ ⎞= ⎜ ⎟+⎝ ⎠ one can obtain the optimal allowable backorder level:1 2( A I )DhL*1 k b( h b )+=− + (8)Therefore, the minimum value of total relevant cost per unit time, TAC(Q*, L*) is:2hbD( A I ) DiTAC( Q*,L*)( h b ) 1 k+= ++ − (9)Equations (7) to (9), in this paper, are the same as Eqs. (6), (5) and (7) in Huang [1], respectively.Model II : EPQ model (shown in Figure 2)Figure 2. The inventory level of Model IIH.-F. Chen, Y.-F. Huang, Y.-Ch. Tu, M.-H. Hwang: An easy approach to derive EOQ and EPQ models with shortage and defective items40 ENGINEERING MODELLING 23 (2010) 1-4, 37-41For convenience, let D( 1 )Pρ = − , V LQρ+=  and Eq. (9) in Huang [1] we know that the total relevant cost perunit time, TAC(Q,L), can be expressed as:2 2 2 2D ( 1 k ) ( Q L ) h ( 1 k ) L b DiTAC( Q,L ) A I  +( 1 k )Q 2D 2D 1 kρρ ρ ρ⎡ ⎤− − −= + + +⎢ ⎥− −⎢ ⎥⎣ ⎦(10)Then we rewrite Eq. (10) as:22(1 k )( h b ) h (1 k )hbQ D( A I ) Di TAC( Q,L ) L Qh b 2( h b ) (1 k )Q 1 k2Qρρρ− + − +⎡ ⎤= − + + +⎢ ⎥+ + − −⎣ ⎦(11)It implies that when Q is given, we can set L as:hL Qh bρ⎛ ⎞= ⎜ ⎟+⎝ ⎠to get the minimum value of TAC(Q,L) as follows:(1 k )hbQ D( A I ) Di TAC[Q,L( Q )]2( h b ) (1 k )Q 1 kρ− += + ++ − − (12)By using the AGM approach we can easily obtain that in Eq. (12):( 1 k )hbQ D( A I ) DiTAC[ Q,L( Q )]2( h b ) ( 1 k )Q 1 k(1 k)hbQ D(A I) Di 2hbD( A I ) Di22( h b ) ( 1 - k)Q 1 k ( h b ) 1 kρρ ρ− += + + ≥+ − −− + +≥ × + = ++ − + −(13)When the equality:(1 k)hbQ D(A I)2( h b ) (1 - k)Qρ− +=+ (14)holds, TAC[Q,L(Q)] has a minimum. Then, we can find the optimal production quantity:1 2( A I )D h bEPQ(Q*)1 k h bρ+ +=− (15)and putting Q* into hL Qh bρ⎛ ⎞= ⎜ ⎟+⎝ ⎠ one can obtain the optimal allowable backorder level:1 2( A I )DhL*1 k b( h b )ρ+=− + (16)Therefore, the minimum value of total relevant cost per unit time, TAC(Q*,L*) is:2hbD( A I ) DiTAC( Q*,L*)( h b ) 1 kρ+= ++ − (17)Eqs. (15) to (17), in this paper, are the same as Eqs. (16), (15) and (17) in Huang [1], respectively.H.-F. Chen, Y.-F. Huang, Y.-Ch. Tu, M.-H. Hwang: An easy approach to derive EOQ and EPQ models with shortage and defective itemsENGINEERING MODELLING 23 (2010) 1-4, 37-41 41JEDNOSTAVAN PRISTUP ODREÐIVANJU EOQ I EPQ MODELA S NEDOSTACIMA INEISPRAVNIM PREDMETIMASA@ETAKHuang (Journal od Statistics and Management Systems, Vol. 6, No. 2, pp. 171-180, 2003.) je prou~avao EOQ(veli~ina ekonomi~nog reda) i EPQ (veli~ina ekonomi~ne proizvodnje) modele s izostalim istra`ivanjima i neispravnimpredmetima koriste}i algebarski pristup. Pretpostavio je da je stopostotnim nadziranjem i detaljnim ispitivanjembila odstranjena poznata koli~ina neispravnih predmeta prije skladištenja ili upotrebe. U ovom radu nudimo jedandruga~iji jednostavni pristup da bi se odredila optimalna koli~ina robe kao i razina zaostalih narud`bi u uvjetimaumanjenja ukupnog relevantnog troška u jedinici vremena.Klju~ne rije~i: aritmeti~ko-geometrijski pristup srednje nejednakosti, algebarski pristup, EOQ, EPQ, nedostatak,neispravni predmeti.4. REFERENCES[1] Y.F. Huang, The deterministic inventory modelswith shortages and defective items derivedwithout derivatives, J. Stat. Managt. Sys., Vol. 6,pp. 171-180, 2003.[2] R.W. Grubbström and A. Erdem, The EOQ withbacklogging derived without derivatives, Int. J.Prod. Eco., Vol. 59, pp. 529-530, 1999.[3] L.E. Cárdenas-Barrón, The economic productionquantity (EPQ) with shortage derivedalgebraically, Int. J. Prod. Eco., Vol. 70, pp. 289-292, 2001.[4] R.J. Ronald, G.K. Yang and P. Chu, The EOQ andEPQ models with shortages derived withoutderivatives, Int. J. Prod. Eco., Vol. 92, pp. 181-184, 2004.[5] S.K.J. Chang, J.P.C. Chuang and H.J. Chen,Short comments on technical note: The EOQ andEPQ models with shortages derived withoutderivatives, Int. J. Prod. Eco., Vol. 97, pp. 241-243, 2005.[6] Y.C. Tu, K.H. Hsu and Y.F. Huang, Retailer’seconomic order quantity model under trade creditperiod depending on the order quantity withoutcalculus, J. Appl. Sci., Vol. 8, pp. 2785-2787,2008.3. CONCLUSIONSThis paper improves Huang’s [1] algebraicprocedure and offers another simple AGM approachto find the optimal ordering quantity (EOQ model) andthe optimal production quantity (EPQ model) withshortages and defective items. Using this improvedapproach presented in this paper, we can find theoptimal ordering quantity and the optimal productionquantity without using differential calculus. This shouldalso mean that this simple AGM approach is a moreaccessible approach to ease the learning of basicinventory theories to younger students who lack theknowledge of calculus.

An easy approach to derive EOQ and EPQ models with shortage and defective items

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H.-F. Chen, Y.-F. Huang, Y.-Ch. Tu, M.-H. Hwang: An easy approach to derive EOQ and EPQ models with shortage and defective itemsENGINEERING MODELLING 23 (2010) 1-4, 37-41 37SUMMARYHuang [Journal of Statistics and Management Systems, Vol. 6, No. 2, pp. 171-180, 2003.] studied the EOQ(Economic Order Quantity) and EPQ (Economic Production Quantity) models with backlogging and defectiveitems using the algebraic approach. He assumed that a 100% inspection policy and the known proportion of defectiveitems was removed after the screening process prior to storage or use. In this paper, we will offer another simpleapproach to find both the optimal lot size and backorder level under the minimized total relevant cost per unit time.Key words: arithmetic-geometric mean inequality approach, algebraic approach, EOQ, EPQ, shortage, defectiveitem.UDC 336.64:336.645:519.86:512.6Original scientific paperReceived: 30.03.2010.An easy approach to derive EOQ and EPQmodels with shortage and defective itemsHsin-Fan Chen, Yung-Fu Huang, Yu-Cheng Tu and Ming-Hon HwangDepartment of Marketing and Logistics Management, Chaoyang University of Technology,168, Jifong East Road, Wufong, Taichung 41349, TAIWANe-mail: huf@cyut.edu.tw1. INTRODUCTIONThe economic order quantity (EOQ) model with/without shortages and economic production quantity(EPQ) model with/without shortages are widely usedby practitioners as a decision-making tool for thecontrol of inventory. However, the assumptions of theEOQ / EPQ models are rarely met. This has led manyresearchers to study the EOQ / EPQ extensively underrealistic situations.A common unrealistic assumption of the EOQ /EPQ is that all units produced or purchased are ofgood quality. We know that it is difficult to produceor purchase items of a 100% good quality. Recently,Huang [1] studied the EOQ and EPQ models withbacklogging and defective items. He assumed that100% inspection policy and the known proportion ofdefective items was removed after the screeningprocess prior to storage or use. In addition, Huang[1] used the algebraic method to determine the optimalsolution by minimizing the annual relevant cost. Inprevious several papers, the EOQ and EPQ formulaefor the shortage case have been derived usingdifferential calculus and solving two simultaneousequations with the need to prove optimality conditionswith second-order derivatives. The mathematicalmethodology is difficult to many younger studentswho lack the knowledge of calculus. Grubbström andErdem [2] and Cárdenas-Barrón [3] showed that theformulae for the EOQ and EPQ with backloggingwere derived without differential calculus. Thisalgebraic approach could therefore be used easily tointroduce the basic inventory theories to youngerstudents who lack the knowledge of calculus. ButRonald et al. [4] thought that their algebraic procedurewas too sophisticated to be absorbed by ordinaryreaders. Hence, Ronald et al. [4] derived a procedureto transform a two-variable problem into two steps,and then, in each step they solve a one-variableproblem using only the algebraic method withoutcalculus. Recently, Chang et al. [5] rewrote theobjective function of Ronald et al. [4] so that the usualskill of completing the square can solve the problemwithout using their sophisticated method. AllH.-F. Chen, Y.-F. Huang, Y.-Ch. Tu, M.-H. Hwang: An easy approach to derive EOQ and EPQ models with shortage and defective items38 ENGINEERING MODELLING 23 (2010) 1-4, 37-41previously mentioned articles use algebraicoptimization.Although Huang [1] used the easily algebraicapproach to find the optimal solution, his method hadthe same problem as Grubbström and Erdem [2] andCárdenas-Barrón [3]. Recently, Tu et al. [6] usedanother simple arithmetic-geometric mean inequalityapproach to investigate the retailer’s EOQ under tradecredit period depending on the order quantity.Therefore, in this paper, we provide the same as Tu etal. [6] easy-to-understand and simple-to-applyarithmetic-geometric mean inequality approachwithout using derivatives to obtain the optimal solution.This approach could therefore be used easily tointroduce the basic inventory theories to youngerstudents who lack the knowledge of calculus.2. DERIVATION OF THE LOT SIZE ANDBACKORDER LEVELThe following notation and assumptions are thesame as Huang's [1] which will be used further on inthis paper.NotationQ − order quantity (EOQ model) / productionquantity (EPQ model) (including defectiveitems),V − maximum on-hand inventory level (includingdefective items),L − maximum shortage (backorder) level(including defective items),D − demand rate for nondefective items, units pertime,P − production rate for nondefective items, unitsper time (P>D),A − ordering cost / setup cost,I − the fixed inspection cost incurred with each lot,i − unit inspection cost,h − unit stock holding cost per unit per time,b − unit shortage cost per unit short per time,k − the known percentage of defective items in Q,TAC(V,L) − total relevant cost per unit time.Assumptions(1) Demand rate is known and constant.(2) Production rate is known and constant.(3) Time period is infinite.(4) Each lot purchased / produced contains a knownproportion of defectives removed prior to storageor use.(5) The screening time for items is so fast that we canneglect it. The inspection cost consists of a fixedper lot inspection cost and a fixed unit inspectioncost.We use here an easy and a simple arithmetic–geometric-mean-inequality (AGM) approach the sameas Tu et al. [6] to obtain the optimal lot size andbackorder level that minimizes the annual total relevantcost. The arithmetic–geometric mean inequality is asfollows. For any of two real positive numbers, say xand y, the arithmetic mean (x+y)/2 is always greaterthan or equal to the geometric mean xy . Namely,x y xy2+ ≥The equation holds only if x=y.Model I : EOQ model (shown in Figure 1)Fig. 1  The inventory level of Model IFrom Eq. (2) in Huang [1] we know that the total relevant cost per unit time, TAC(V,L), can be expressed as:2 2 2 2D ( 1 k ) V h ( 1 k ) L bTAC(V ,L ) A I (V L )i  (1 k )(V L ) 2D 2D⎡ ⎤− −= + + + + +⎢ ⎥− + ⎢ ⎥⎣ ⎦(1)For convenience, we let Q=V+L and substitute into Eq. (1).Then, we can obtain:H.-F. Chen, Y.-F. Huang, Y.-Ch. Tu, M.-H. Hwang: An easy approach to derive EOQ and EPQ models with shortage and defective itemsENGINEERING MODELLING 23 (2010) 1-4, 37-41 392 2 2 2D (1 k ) ( Q L ) h ( 1 k ) L bTAC( Q,L ) A I Qi  (1 k )Q 2D 2D⎡ ⎤− − −= + + + +⎢ ⎥− ⎢ ⎥⎣ ⎦ (2)Then we rewrite Eq. (2) as:2( 1 k )( h b ) h ( 1 k )hbQ D( A I ) Di TAC( Q,L ) L Q2Q h b 2( h b ) (1 k )Q 1 k− + − +⎡ ⎤= − + + +⎢ ⎥+ + − −⎣ ⎦ (3)It implies that when Q is given, we can set L as:hL Qh b⎛ ⎞= ⎜ ⎟+⎝ ⎠to get the minimum value of TAC(Q,L) as follows:(1 k )hbQ D( A I ) Di TAC[ Q,L(Q )]2( h b ) (1 k )Q 1 k− += + ++ − − (4)By using the AGM approach we can easily obtain that in Eq. (4):( 1 k )hbQ D( A I ) Di TAC[ Q,L( Q )]2( h b ) ( 1 k )Q 1 k( 1 k)hbQ D(A I) Di 2hbD( A I ) Di22( h b ) ( 1 - k)Q 1 k ( h b ) 1 k− += + + ≥+ − −− + +≥ × + = ++ − + −(5)When the equality:(1 k)hbQ D(A I)2( h b ) (1 - k)Q− +=+ (6)holds, TAC[Q, L(Q)] has a minimum. Then, we can find the optimal ordering quantity:1 2( A I )D h b 1 2( A I )Db 1 2( A I )DhEOQ( Q*)1 k h b 1 k h( h b ) 1 k b( h b )+ + + += = +− − + − + (7)and putting Q* into hL Qh b⎛ ⎞= ⎜ ⎟+⎝ ⎠  one can obtain the optimal allowable backorder level:1 2( A I )DhL*1 k b( h b )+= − + (8)Therefore, the minimum value of total relevant cost per unit time, TAC(Q*, L*) is:2hbD( A I ) DiTAC( Q*,L*)( h b ) 1 k+= ++ − (9)Equations (7) to (9), in this paper, are the same as Eqs. (6), (5) and (7) in Huang [1], respectively.Model II : EPQ model (shown in Figure 2)Figure 2. The inventory level of Model IIH.-F. Chen, Y.-F. Huang, Y.-Ch. Tu, M.-H. Hwang: An easy approach to derive EOQ and EPQ models with shortage and defective items40 ENGINEERING MODELLING 23 (2010) 1-4, 37-41For convenience, let D( 1 )Pρ = − , V LQ ρ+=  and Eq. (9) in Huang [1] we know that the total relevant cost perunit time, TAC(Q,L), can be expressed as:2 2 2 2D ( 1 k ) ( Q L ) h ( 1 k ) L b DiTAC( Q,L ) A I  +( 1 k )Q 2D 2D 1 kρρ ρ ρ⎡ ⎤− − −= + + +⎢ ⎥− −⎢ ⎥⎣ ⎦ (10)Then we rewrite Eq. (10) as:22(1 k )( h b ) h (1 k )hbQ D( A I ) Di TAC( Q,L ) L Qh b 2( h b ) (1 k )Q 1 k2Qρρρ− + − +⎡ ⎤= − + + +⎢ ⎥+ + − −⎣ ⎦ (11)It implies that when Q is given, we can set L as:hL Qh bρ⎛ ⎞= ⎜ ⎟+⎝ ⎠to get the minimum value of TAC(Q,L) as follows:(1 k )hbQ D( A I ) Di TAC[Q,L( Q )]2( h b ) (1 k )Q 1 kρ− += + ++ − − (12)By using the AGM approach we can easily obtain that in Eq. (12):( 1 k )hbQ D( A I ) DiTAC[ Q,L( Q )]2( h b ) ( 1 k )Q 1 k(1 k)hbQ D(A I) Di 2hbD( A I ) Di22( h b ) ( 1 - k)Q 1 k ( h b ) 1 kρρ ρ− += + + ≥+ − −− + +≥ × + = ++ − + −(13)When the equality:(1 k)hbQ D(A I)2( h b ) (1 - k)Qρ− +=+ (14)holds, TAC[Q,L(Q)] has a minimum. Then, we can find the optimal production quantity:1 2( A I )D h bEPQ(Q*)1 k h bρ+ += − (15)and putting Q* into hL Qh bρ⎛ ⎞= ⎜ ⎟+⎝ ⎠  one can obtain the optimal allowable backorder level:1 2( A I )DhL*1 k b( h b )ρ+= − + (16)Therefore, the minimum value of total relevant cost per unit time, TAC(Q*,L*) is:2hbD( A I ) DiTAC( Q*,L*)( h b ) 1 kρ+= ++ − (17)Eqs. (15) to (17), in this paper, are the same as Eqs. (16), (15) and (17) in Huang [1], respectively.H.-F. Chen, Y.-F. Huang, Y.-Ch. Tu, M.-H. Hwang: An easy approach to derive EOQ and EPQ models with shortage and defective itemsENGINEERING MODELLING 23 (2010) 1-4, 37-41 41JEDNOSTAVAN PRISTUP ODREÐIVANJU EOQ I EPQ MODELA S NEDOSTACIMA INEISPRAVNIM PREDMETIMASA@ETAKHuang (Journal od Statistics and Management Systems, Vol. 6, No. 2, pp. 171-180, 2003.) je prou~avao EOQ(veli~ina ekonomi~nog reda) i EPQ (veli~ina ekonomi~ne proizvodnje) modele s izostalim istra`ivanjima i neispravnimpredmetima koriste}i algebarski pristup. Pretpostavio je da je stopostotnim nadziranjem i detaljnim ispitivanjembila odstranjena poznata koli~ina neispravnih predmeta prije skladištenja ili upotrebe. U ovom radu nudimo jedandruga~iji jednostavni pristup da bi se odredila optimalna koli~ina robe kao i razina zaostalih narud`bi u uvjetimaumanjenja ukupnog relevantnog troška u jedinici vremena.Klju~ne rije~i: aritmeti~ko-geometrijski pristup srednje nejednakosti, algebarski pristup, EOQ, EPQ, nedostatak,neispravni predmeti.4. REFERENCES[1] Y.F. Huang, The deterministic inventory modelswith shortages and defective items derivedwithout derivatives, J. Stat. Managt. Sys., Vol. 6,pp. 171-180, 2003.[2] R.W. Grubbström and A. Erdem, The EOQ withbacklogging derived without derivatives, Int. J.Prod. Eco., Vol. 59, pp. 529-530, 1999.[3] L.E. Cárdenas-Barrón, The economic productionquantity (EPQ) with shortage derivedalgebraically, Int. J. Prod. Eco., Vol. 70, pp. 289-292, 2001.[4] R.J. Ronald, G.K. Yang and P. Chu, The EOQ andEPQ models with shortages derived withoutderivatives, Int. J. Prod. Eco., Vol. 92, pp. 181-184, 2004.[5] S.K.J. Chang, J.P.C. Chuang and H.J. Chen,Short comments on technical note: The EOQ andEPQ models with shortages derived withoutderivatives, Int. J. Prod. Eco., Vol. 97, pp. 241-243, 2005.[6] Y.C. Tu, K.H. Hsu and Y.F. Huang, Retailer’seconomic order quantity model under trade creditperiod depending on the order quantity withoutcalculus, J. Appl. Sci., Vol. 8, pp. 2785-2787,2008.3. CONCLUSIONSThis paper improves Huang’s [1] algebraicprocedure and offers another simple AGM approachto find the optimal ordering quantity (EOQ model) andthe optimal production quantity (EPQ model) withshortages and defective items. Using this improvedapproach presented in this paper, we can find theoptimal ordering quantity and the optimal productionquantity without using differential calculus. This shouldalso mean that this simple AGM approach is a moreaccessible approach to ease the learning of basicinventory theories to younger students who lack theknowledge of calculus.

An easy approach to derive EOQ and EPQ models with shortage and defective items

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