Journal of Systems Engineering and ElectronicsVol. 20, No. 4, 2009, pp.768-776Optimal production lot size with process deteriorationunder an extended inspection policy*Hu Fei, Xu Genqi & Ma LiriaDept. of Mathematics, Tianjin Univ., Tianjin 300072, P. R. China(Received April 23, 2008)Abstract: A mathematical model to determine the optimal production lot 8ize for a deteriorating productionsystem under an extended product inspection policy is developed. The last- K product inspection policy is consideredso that the nonconforming items can be reduced, under which the last K products in a production lot are inspectedand the nonconforming items from those inspected are reworked. Consider that the products produced towards theend of a production lot are more likely to be nonconforming, is proposed an extended product inspection policyfor a deteriorating production system. That is, in a production lot, product inspections are performed among themiddle K1 items and after inspections, all of the last K2 products are directly reworked without inspections. Ourobjective here is the joint optimization of the production lot size and the corresponding extended inspection policysuch that the expected total cost per unit time is minimized. Since there is no closed form expression for ouroptimal policy, the existence for the optimal production inspection policy and an upper bound for the optimal lotsize are obtained. Furthermore, an eficient solution procedure is provided to search for the optimal policy. Finally,numerical examples are given to ilustrate the propoeed model and indicate that the expected total cost per unittime of our product inspection model is less than that of the last- K inspection policy.Keywords: production lot size, product inspection policy, deteriorating production system, economic manufac-turing quantity.1. Introductioninvestment in quality improvement and setup cost re-duction. Djamaludin et al3] utilized lot size to controlThe classical economic manufacturing quantitythe warranty cost per item for products under free re-(EMQ) model assumes that a production facility ispair warranty(FRW). Reference [4] reformulated thefailure free, and that all the items produced are ofmodel in Ref. [3] to consider that the production pro-perfect quality. However, in real production, a prod-cess is subject to a random deterioration from an in-uct system unavoidably deteriorates due to usage orcontrol state to an out-of-control state, taking bothage such as corrosion, fatigue, and cumulative wear,the restoration cost and the inventory holding costand as a result, substandard items are produced. Ainto account and obtaining the bounds of the optimalgreat deal of research efforts have been devoted to ex-production run length.tending the classical EMQ model under the relaxedIn order to enhance the reliability of the deterio-assumption that the production process always pro-rating production process so that the nonconformingduces the items of acceptable quality. Rosenblatt anditems can be reduced in a production lot, many reLee[1] initially studied the efects of process deteriora-searchers have focused on finding an optimnal main-tion on the traditional EMQ model, where the randomtenance/inspection policy in deteriorating produc-degeneration from the in-control state to the out-of-tion systems. Lee and Rosenblatt5l considered ancontrol state is exponentially distributed. Porteusl21inspection mechanism to monitor the imperfect pro-assumed that the production process can shift to theducti中國煤化工dermined theout-of- control state with a given probability each timeprod↓schedule. Tsengl6]an item is produced, and explored different options fordevelMYHC N M H Gnaintenance poliey* This project was supported by the National Natural Science Foundation of China (60874034).Optimal production lot size with process deterioration under an extended inspection policy769for the deteriorating production system, and com-sult, it will further economize the inspection cost thatpared the difference of the system eficiency betweenall products produced at the end of a production lotadopting inspection policy and adopting maintenanceare directly reworked without inspections. In this parpolicy. A production-inventory model to considerper, we reformulate the work of Yeh and Chen!2| tovarying demand and production rates, inspection andconsider an extended product inspection policy, wheremaintenance is studied, and a heuristic solution algo-product inspections are performed for the middle K1rithm is developed to determine the production andproducts and all of the last K2 products are directlyinspection schedulesl7l. Giri and Dohi(8] determinedreworked without inspections in a production lot.jointly the optimal production run length and the2. Notation and assumptionsoptimal number of inspections during a productionrun under two different inspection policies. How-In this section, the mathematical notation and rele-ever, in many cases, it is either impossible or ex- vant assumptions are presented.pensive to interrupt the production process during a2.1 Notationproduction run, or to detect the deterioration in theprocess[l]. Kim and Hong[9] futher studied Rosenblattp: production rate;and Lee's model], and derived an optimal productiond: demand rate(< p);run length under the assumption that all defectiveh: holding cost per item per unit time;items are defected and reworked at some cost after: fixed cost to reveal the state of the productionthe production run is over. In order to reduce inspec-system;tion cost, Wang[10] reformulated the work of Kim andn: additional restoration cost;Hong9 to examine a product inspection policy, whereq: probability of the production process staying in-product inspections are performed at the end of the control state during the production of a product;production run. Djamaludin et all proposed an inCm: manufacturing cost to produce an item;spection policy under which the last certain numberC: inspection cost per item;of products in a lot are tested by burn-in method andC: rework cost per item (> C);those failed products during the test are scraped. YehW1: expected warranty cost for a conforming item;and Chen!12) employed the idea given in DjamaludinW2: expected warranty cost for a nonconforminget all and studied a production inspection scheme item (W2- W1 > Cr + C); .called last-K inspection policy for a deteriorating sys-θ1: percentage of conforming items produced whentem, where the last K products in a lot are inspectedthe process is in the in-control state;and the nonconforming items from those inspected areθ2: percentage of conforming items produced whenreworked at some cost.the process is in the out-of-control state (0 < 02 0, the fllowing results hold forx1∈[0, [].the case when production lot size is predetermined.(1)Ifρ+rnq°≥0, thenxi = 0.For a given lot size L > 0, there are two variables,(2) If0 < b1 < L, then there exists a uniqueK1 and K2, which need to be determined. To formu-ri = L-b1 that minimizes f(x1), where b1 =late our optimization problem, let us introduce threeln(-p/r)/ lnq.functions and derive some useful properties of them.(3)Ifb1≤0, thenxi = L.LetLemma 3 Let r2 = -BClnq. Given a lot sizeBδL > 0, the fllowing results hold for x2∈{0, []f(x)= (C-8)x+i_(q-*-q"), 0≤x≤L (5)(1) Ifr2q~≥C- 02Cr, thenz = 0.(2) If0< b2 < L, then there exists a uniquef(a)=p2I+ B(Wz-W)(q'“-g), 0≤q≤L x= L-b2 miming 9(x2), where b2=In [(CA-(6) 02C)rl/Inqand(3)Ifb2≤0, thenxi= L.g(22) = f(x2)- f(x2) = (O2Cr - C1)x2+According to Eqs. (5)~(7), the objective functiongiven in Eq. (4) can be rewritten a8BC(q-x2-q"), 0≤xz≤L(7)C(,K,K2)=A+血- d)hL+g號{+Let x* be the optimal value that minimizes Eq. (5)2pin the interval of [0,L], then we have the followinglemma.[n- B(W2- W1)](1 -g")+ f(K1+ K2)+ g(K2))} (8)Lemma 1 Letr= - Blng/(1- 02). Given anylot size L > 0, the following results hold for x∈[0, L].Note that in Eq. (8), we define the function f(K1 +(1)IfC≥8(1- rq"), thenx*=0.K2) that reflects cost reduction in a lot under the(2) If0 0, we proced to involve the decision on the opti-the first and second derivatives of f(x) with respect mal inspection policy (Ki, K2) for three cases classi-to K, we obtain f'(x) = C1-8(1- rq4-I) andfied by Lemma 1.f"(x) = 8rq-z > 0. IfCr≥8(1 - rq"), thenCase1 C≥8(1-rq')f'(x)≥0forx∈[0,]. In this case, f(x) is an in_know that f(K1 +中國煤化工creasing function of x in the interval [0, L] and henceK2):YH+ K2 and so doesx°=0. If0 0,if0< b2 < L then there exists a unique optimalKz that minimizes f(K2) and so does C(L,0, K2).inspection policy such that (Ki, K;)= (b2,L- b2); ifb2≤0 then (Ki,K&) = (0, L); otherwise if r2q"≥Therefore, we have the following theorem.Theorem 1 Under the constraintC1≥8(1-rq"),C- 02Cr then (Ki, K2)= (L,0).foragivenlotsizeL:>0,ifρ+rnq'≥0,thenProof The proof is similar to that for Theorem 2.the optimal inspection policy is (K;,K2)= (0,0). IfNote that in Theorems 1~3, δ minus Ci seems to re0 L-b, it can easily verify that Ki = 0, and therework an item without inspection, and this is theobjective function becomes C(L, 0, K2). Thus, basedcase considered by Ref. [12]. In this case, the objec-tive function in Eq. (4) is given byon Lemma 2, Kz can be obtained.Theorem 2 Given any lot sizeL> 0.Cv(L,K)=A+如2+{+(n- BC)(1-q4)+(1)if0 0, Thus we can obtain cases should be investigated.Ki = 0. In this case, the objective function becomes .(1) WhenCi≥8(1-rq4) andρ+rq'≥0, weC(L,0,K2). Furthermore, from Lemma 2, we have know that (K;, K;) = (0,0) and our objective func-K;=L-b1.tionI中國煤化工，Case3 b≤0< 0or whenb≤0In this case, by Lemma 1, we know that f(K1+K2)andTYHCN M H G=(0,L) and ouris a decreasing function of K1 + K2 and so does objective function becomes C2(L) = C(L,0, L).Optimal production lot size with process deterioration under an extended inspection policy773(3) When Cr≥8(1-rq4) and 0< br < LorG3(L) = L2(p - d)h/(2p) - d[t + nQ(I)]+whenb2 max {[4}, wherei=.2(T +n)d[n- B(W2- W)]q(ng}(p- d)hG2(L) = G3([)= G'(D) = G%(I)=Proof Let G:(L)= L2C;(L),1≤i≤7. Then,Ld(p - d)h/(d) - nq'(nq)2]G(L) has the same sign as C;(L) and is a con-tinuous function for L > 0. Further, let Q(L) =G([)= G(I)= Ld[(p - d)/(px)-1- q" + Lq' lnq for simplifying mathematical nota-tion, then we have(n - BC)g"(ng)2]G:(L) = L2(p- d)h/(2p) - d[r + nQ(D)]+Observing equation of Gl(L), ifη- B(W2-W1)≤0,Bd(W2 - W])Q(L)then G{(L)> 0for all L> 0. Hence, G(L) is anG2(L) = L2(ρ- d)h/(2p) - dl[r + nQ([)]increasing function of L and the equation G1([)= 0Table 1 Optimal policies under different values of qq(b,b1,b2)(L",K;,Ki)C(L",Ki, KE)(LigH,K*)Cy(LEH,K*0.996(59, -33, 503)(726, 444, 223)11 446(526, 467)11 4520.994(39, -22. :15)(1 246, 296, 911)11 566(755, 716)11 7190.992(29, -16, 251)(1 405, 222, 1154)11 611(1 076, 1 047)11 8260.990 .(23, -14, 200)(1 491, 177, 1291)11 635(1 253, 1 230)11 8760.985(15, -9, 133)(1 599, 118, 1466)11 9340.980(11, -7, 99)(1 650, 88, 1551)中國煤化工11 9600.970(7,-5, 66)(1 700, 59, 1634)HCNMH G11 9850.960(5, -4, 49)(1 725, 44, 1676)11 7o(1 011, 1 012)11 997774Hu Fei, Xu Genqi & Ma Liriahas a unique positive solution and so does Ci{(L)= 0.Step 3 Ifb≤ 0 < b2, then calculate C(b2). IfOtherwise, if η- B(W2- W1) < 0, then G{(L) is C%(b2) < 0 then find L4∈(0, L] such that C(L4)= 0,an increasing function of L since q decreases as Land set (L* , Ki, K;)= (L4, b2, L*-b2) and stop. Oth-increases. Thus we have G{(L) change its sign at erwise go to Step 6.most once from negative to positive, which impliesStep 4 If0 < b≤b2, then compute C(b2).that G1(L) changes its sign exactly once from negative If C(b2)≤0, then find Lζ∈[b2, [] such thatto positive. As a result, there exists a unique solution C%(L5) = 0 and set (L*,Ki, K;)= (Lζ,b2-b, L5-b2).L such that G(L) = 0 and so does C([)= 0.Otherwise calculate C%(6) and perform Steps 4.1~4.2.Simnilarly, observing equations of G2(1) and G'(L), .Step 4.1 IfC&(b) < 0, then find Lg∈[b,勾suchwe can obtain that there exists a unique solution L;that C%(L)= 0. Set (L*,K{,K)= (Lg,Li- b,0)such that G:(L) = 0 and so does C{(L)=0,2≤i≤7.and stop. Otherwise if Cg(6b)≥0, then go to Step 4.2.Finally, under the above arguments, we know that [Note that Ci(L) > C'(b)≥0 in this case.]G;(L) changes its sign exactly once from negative toStep 4.2 Search for Li∈(0, b] satisfyingpositive,1≤i≤7. Therefore, if G;(L) > 0, theni Cl(Li) =0 and set (I*,K", K&) = (L;,0,0). Stop.isanupperboundforL，1≤i≤7.More-Step 5 Ifb> 0 and C < 02C, then calculateover, it is easy to verify that Q(L[) is a continuous C%(b) and go to Step 4.1.and strictly increasing function with Q(0) = 0 andStep 6 Search for Lz∈(0, L] satisfying C{(L2) =Q(L)<1 for L> 0. Therefore, from these equations 0. Set ([",Ki, K&) = (L%, L,0) and stop.of Gl(L) ~ Gr(L), we have5. Numerical exampleG:(D)≥I2(p- d)h/(2p) - d|r + nQ([)] >In this section, we provide a numerical example toillustrate the features of the proposed model. The2(ρ- d)h/(2p)-d(τ+η)=0,1≤i≤7nominal values of the pararmeters are shown as fol-This implies that I > L;，1 ≤i≤7, that is,lows: p=1 500, d= 650, h=0.5,T= 200, η= 500,Cm=10,C1=1,Cr=5.25,W1=2,W2=10,L> max{Li}.θ1 = 0.78, 02= 0.1..3 Searching procedureTable 1 summarizes the optimal policy (L* ,Ki, K;)By employing the properties shown in Theorems 1-3,and resulting expected total cost per item C(L* ,Ki,an efficient solution procedure for searching the opti-Kz) for various of q. For comparison, Table 1 also listsmal lot size L* and the corresponding inspection pol-the results of the Yeh and Chen(2006). For example,whenq = 0.992, the optimal policy (L* ,Kt,K2) =icy (Ki, K$) is given as follows.(1 405, 222, 1154) and the corresponding expected toStep 1 Compute b, b1, b2, δ, ρ and L.Step 2 If C{≥8, then perform Steps 2.1-2.3.tal cost rate C(L*,Ki,K;)= 11611. If the Yeh andChen's model is implemented, we have (IEn,K*) =Otherwise go to Step 3.Step2.1 Ifρ≥ 0, then find L4∈(0, L] satifying(1 076,1 047) and Cy(LeH,K*)= 11 826.From Talbe 1, the following observations can beC(L;)= 0. Set (I*, K", Kz) = (Li,0,0). Stop.made.Step2.2 Ifb1 ≤0, then find功∈(0, L] satisfyingAs q decreases, the optimal number of inspectedC2(Li)= 0. Set (L*, K;,K2)= (I2,0, L2). Stop.products Ki decreases, but the optimal number of di-Step 2.3 Compute C3(b1). If C3(b1) < 0, thenrectly reworked products Kz increases and Ki+K2 in-search for Lj∈[br,L] such that C}(L3) = 0, and creases. That is, when the deterioration rate (1-q) ofset (L*,K;,Ki) = (Iz,0,Lg-b1) and stop. Oth- the sy中國煤化工'uce post-sale war-erwise search for Li∈(0,b1] such that C{(4$) = 0rantye reworked beforeand set ([*,Ki,K$) = (L;,0,0). Stop. Note thatbeingMYHC N M H Gion lot. Further-C{(b1) > C3(br)≥0 in this case]more, since an increase in the value of deteriorationOptimal production lot size with process deterioration under an ertended inspection policy775rate leads to an increase in the nonconforming itemstermnational Joumnal of Production and Economics, 1994,produced at the end of production run per cycle, more33(1-3): 97-107.products being directly reworked without any inspec- [4] Yeh R H, Ho W T, Tseng s T. Optimal production runtion will further economize inspection costs.length for products sold with warranty. European JounalThe optimal lot size L* and the total expected costof Operntional Ressearch, 2000, 120(3): 575 -582.rate C(L* ,Ki,K2) increase as q decreases. This is[5] Lee H L, Rosenblatt M J. Simultaneous determinationdue to a large lot size offsetting the setup cost of thef production cycle and inspection schedules in a pro-system as it becomes more unreliable, and a higherduction system. Management Science, 1987， 33(9):deterioration rate making a larger fraction of noncon-1125- 1136.forming items in a production lot and increasing post-6] Tseng s T. Optimal preventive maintenance policy for desale warranty cost.teriorating production systems. IIE Transactions, 1996,Comparing the expected total cost rates, our opti-28(8): 687 694.mal polic (I*, Ki, K2) performs significantly better[7] Alfares H K, Khursheed S N, Noman s M. Integrating qual-than Yeh and Chen's optimal policy (LEH,K*).ity and maintenance decisions in a production-inventory6. Conclusionmodel for deteriorating items. Intermnational Jounal ofproduction Research, 2005, 43(5): 899 911.1In this paper, an extended product inspection pol-8] Giri B C, Dohi T. Inspection scheduling for imperfect pro-icy (Ki, Ki) for a deteriorating production systemduction processes under free repair warranty contract. Euis proposed, where the deteriorating production pro-ropean Journal or Operation Research, 2007, 183(1): 238-cess may go out-of-control with a given probability252.each time an item is produced. That is, in a produc-tion lot, product inspections are performed form the[9] Kim C H, Hong Y. An optimal production run length indeteriorating production system. Intermnational Jourmal of(L- Ki-K; + 1)th to (L - K)th, and all productsform the (L- K; + 1)th until the end of the pro-Production Economics, 199, 58(2): 183 -189.duction run are fully reworked without inspections. [10] Wang C H. Integrated production and product inspetionWe show that there exists a unique optimal policypolicy for a deteriorating production system. Intemational(L*,K{,K) such that the expected total cost rate :Jourmal of Production Econormics, 2005, 95(1): 123 134.is minimized. Since there is no a closed-form expres- [1] Djamaludin I, Wilon R J, Murthy D N P. Lot sizing andsion for the optimal policy, a searching procedure istesting for items with uncertain quality. Mathemnatical andprovided to fficiently find it. Moreover, numericalCormputer Modelling, 1995, 22(10-12): 35 44.examples are carried out to investigate the behavior [12] Yeh R H, Chen T H. Optimal lot size and inspection pol-of the proposed model and to compare the solutionsicy for products 8old with warranty. Puropean Jourmral ofwith the ones in Yeh and Chen's model. The numeri-Openation Research, 2006, 174(2): 766 776.cal exanmples show that our optimal policy can further [13] Chiu s w, Ting C K, Chiu Y s P. Optimal producreduce the expected total cost rate in a production lottion lot sizing with rework, scrap rate, and service levelthan Yeh and Chen's optimal policy does.constraint. Mathematical and Computer Modelling, 2007,References46(34): 535-549.[1] Rosenblatt M J, Lee H L. Economic production cycles with[14] Chen C K, Lo C C, Liao Y X. Optimal lot size with learn-ing consideration on an imperfect production system withimperfect production processe. IIE Thansactions, 1986,allowable shortages. International Journal of Production18(1): 48-55.Econormics, 2008, 113(1): 459 469.2] Porteus E L. Optimal lot sizing, process quality improvel5] Chern M s, Yang H L, Teng J T, et al. Partial backlogment and setup cost reduction. Opernational Research,中國煤化工triorating itemns with1986, 34(1); 137-144.[3] Djamaludin 1, Murthy D N P, Wilson R J. Quality con-CNMHGEuropean Journal ofOperational Research, 2008, 191(1): 127 141.trol through lot sizing for items sold with warranty. In-776Hu Fei, Xu Genqi & Ma LiriaHu Fei was born in 1977. He is the Ph. D. candidatestitute of Chinese Academy of Sciences. His researchin School of Electrical Engineering and Automation interests are spectral theory of linear operator, systemand a lecture in Department of Mathermnatics, Tian- stability and reliability analysis, etc.jin University. His research interests include systemmodeling and reliability analysis.E mail: hfzdzy@126.comMa Lixia was born in 1978. She is the Ph.D. candi-date in School of Electrical Engineering and Automa-Xu Genqi was born in 1959. Now he is a professor in tion and a lecture in Department of Mathematics,Department of Mathematics, Tianjin University. He Tianjin University. Her research interest is in systemreceived his Ph.D. degree from the System Science In- stability analysis.中國煤化工MYHCNMHG