Economic order quantity with imperfect quality, destructive testing acceptance sampling, and inspection errors

Transcription

1 Advnces in Mngement & Applied Economics, vol.1, no., 011, ISSN: (print version), (online) Interntionl Scientific Press, 011 Economic order quntity with imperfect qulity, destructive testing cceptnce smpling, nd inspection errors Muhmmd Al-Slmh 1 Abstrct This pper proposes nother relistic ppliction of the economic order quntity EOQ model. Reserchers hve studied EOQ models tht ssume the received lot contins items tht re perfect nd imperfect. All models in the literture ssume tht the received lot goes through 100% inspection to seprte cceptble items from the non-cceptble items. This pper rgues tht 100% inspection is not lwys chep option to the buyer; insted, the received lot is subjected to n cceptnce smpling pln before the lot cn be ccepted. In ddition, for rel life cses, destructive testing of items in the cceptnce smple must be performed to test the cceptbility of the items in the lot, s in pckged food nd electricl wires. This pper will develop n EOQ model when items re of perfect nd imperfect qulity nd single cceptnce smpling pln with destructive testing nd inspection errors is dopted. It is ssumed tht when the lot is rejected, items in the rejected lot re sold t secondry mrket t reduced price nd the buyer will plce nother order. It is lso ssumed tht there re inspection errors. 1 Mechnicl Engineering Deprtment, College of Engineering, Mjmh University, Mjmh, Sudi Arbi, emil: slmh@kfupm.edu.s Article Info: Revised: June 3, 011. Published online: October 31, 011

2 60 Economic order quntity with imperfect qulity JEL clssifiction numbers: C51 Keywords: economic order quntity, imperfect qulity, cceptnce smpling, inspection errors, destructive testing 1 Introduction Different products need different kinds of qulity ssessment tests, to insure product meets the required specifictions, such s electric current lod, sher strength, ftigue, nd het tolernce. A destructive test results in either destruction of the product or filure of product s design mrgin, for the purpose of evluting the qulity nd relibility of the product either during or fter the norml operting conditions ([1], []). A 100% inspection pln with destructive testing requires testing ll units in the lot nd destroying ll of them; hence this inspection pln does not provide vible option to both the producer nd the buyer, becuse when the lot is destroyed, the buyer will be in constnt shortge nd the producer will not gin enough on return on investment. Therefore, cceptnce smpling when destructive testing is required provides lot screening pln tht will be more cceptble ([3], [4], [5]). As it is well known, cceptnce smpling is screening strtegy used to judge the cceptbility of received lot; tht is, whether the lot contins n cceptble number of defects (red more in [6], [7], nd [8]). Acceptnce smpling plns re conveniently described by operting chrcteristic or OC curves, which estimte the probbility of cceptnce of lot for given incoming qulity level. To fit vrious needs for different OC curves, number of cceptnce smpling plns hve been developed, such s single, double, multiple, sequentil, etc (see [9] nd [10]). In this reserch, it is ssumed tht incoming lots will be subjected to single cceptnce smpling pln for ttributes, which consists of tking rndom smple of size n drwn from lot of size Q. A predetermined cceptnce number c 0 specifies the llowble number of nonconforming items within the selected

3 Muhmmd Al-Slmh 61 items in the smple. If the number of non-conforming items is less thn or equl to c, the lot will be ccepted, otherwise it will be rejected. As one of the prmeters of the OC curves, the cceptnce number c hs to be selected crefully tht will result in n cceptble smpling pln. In expensive tests nd in tests tht require destructive testing in prticulr, c becomes more criticl. Even the selection of c is rbitrry, when destructive testing is employed by the receiving gent, the cceptnce number is commonly set to zero s in [11], [1], nd [13] nd implied in [14]. In the literture, there hve not been studies tht re involved in the determintion of the optiml order quntity when items re with imperfect qulity nd destructive testing policy is being followed by the buyer. This rticle intends to investigte nd explore the reltionship between the incoming qulity level, on one side, nd the optiml lot size nd the smple size decisions, on the other side. The function tht will be minimized is the difference between the expected totl cost nd totl profit. The totl cost consists of ordering cost, holding cost, destructive testing cost, shortge cost due to defective items in the ccepted lot tht cnnot meet ny of the demnd, nd rejection cost. The profit is the return from selling the items in the rejected lot in secondry mrket. The shortge cost cn be thought of s the cost of ccepting lot, which is ssocited with the buyer s risk. The economic order quntity with imperfect incoming qulity nd destructive testing hs not been investigted by reserchers. In the literture, nondestructive cceptnce smpling plns hve been proposed for lot cceptnce; which hve been designed to fit certin OC curves. This pper will further dvnce the study of the economic order quntity for the prcticl cse when the single cceptnce smpling pln for ttributes nd destructive testing re implemented. Specificlly, this pper will develop cost function for optiml determintion of the lot size nd smple size when items re with imperfect qulity nd the buyer will impose

4 6 Economic order quntity with imperfect qulity destructive testing pln on the received lot. The totl cost hs the following components: Ordering cost Holding cost Destructive testing cost Shortge cost Profit from selling uncceptble lot. The cceptnce number c will be tken s zero, s it ws rgued before (see the introduction). To further limit the scope of this reserch, the following ssumptions re mde: The demnd rte is constnt nd known, The replenishment is infinite nd instntneous, There is single product, Shortges re not bck-ordered (lost sles cse), The buyer will plce new order when the received lot is rejected. For the inspection process, the following re ssumed: The number of nonconforming items follows the binomil distribution, The inspection time is negligible, The inspection pln is without replcement of destroyed items, The inspection process is not perfect. Literture Review The literture deling with the economic order quntity with imperfect incoming qulity hs exclusively been bsed on screening, 100% inspection strtegy; none of the studies on this topic hve pplied some form of cceptnce smpling. The literture proposes number of strtegies on how imperfect qulity cn be treted within the clssicl economic order quntity methodology. The first study tht includes imperfect qulity in the economic order quntity model is [15], which is

5 Muhmmd Al-Slmh 63 bsed on profit mximiztion. [16] present simple pproch for determining the economic order quntity for n item with imperfect qulity. [17] hve treted the proportion of the imperfect items s rndom vrible nd ffirmed tht profit mximiztion model will led to the sme solution drwn from cost minimiztion model. [18] consider defective products nd Tguchi s cost of poor qulity in the economic order quntity model, where the product qulity follows norml distribution function. [19] present n inventory model for imperfect items under one-time-only discount, where the defectives cn be screened out by screening process nd then cn be sold in single btch by the end of the screening process. [0] hs developed n inventory model for items with imperfect qulity nd quntity discounts in response to the request of the powerful buyer. [1] nlyze the effect of screening speed nd vribility of the supply process on the order quntity when items re imperfect nd extend the model by llowing for severl btches of imperfect qulity items to be consolidted nd shipped in one lot. [] pply the concept of entropy cost to extend the clssicl EOQ model under the ssumptions of perfect nd imperfect qulity. Stochstic nlysis of the economic order quntity with imperfect qulity hs been developed. [3] investigte the economic order quntity with imperfect qulity items, where the percentges of defective items nd poor-qulity items in ech delivered lot re ssumed to be rndom vribles, nd the inspection cost, holding cost, ordering cost re chrcterized s fuzzy vribles, respectively. [4] consider items to be of different qulities, perfect nd imperfect, nd the percentge of the yield rte from the perfect items is ssumed to be rndom vrible with known probbility distribution. Some studies hve included imperfect qulity s form of shortge tht hs to be crried to future orders. [4] nd [6] hve ssumed bd items cn be bckordered nd developed the totl profit function per cycle. [7] develop n EOQ

6 64 Economic order quntity with imperfect qulity model in which ech received lot contins some defective items nd shortges bckordered. It hs been ssumed tht 100% of the items in ech lot re inspected to seprte good from defective items, which re imperfect qulity nd scrp items. The effect of percentge defective on optiml solution is studied. [8] hve developed n economic order quntity model with defective items nd shortges in fuzzy rndom environment; where the percentge of defective items is rndom vrible nd the defective items contin imperfect qulity items nd scrp items; the rte of scrp items in defective items is fuzzy vrible. They hve written n expression for the expected profit, which hs been proven to be concve. There re studies tht exmine the effect of lerning on the economic order quntity with imperfect qulity items. The first study on the lerning effects ws presented by [9]. They hve developed models in which the percentge defective per lot is ssumed to decline ccording to lerning curve. Bsed on previous studies, [30] present the optiml lot sizes for n item with imperfect qulity when different holding costs for the good nd defective items re considered. They hve studied two situtions: without lerning effect nd with lerning effect. They hve noted tht when there is lerning effects, the lot size with different holding costs for the good nd defective items is more thn the one with sme holding costs for the good nd defective items. [31] consider lerning in inspection with three cses: prtil trnsfer of lerning, totl trnsfer of lerning, nd no trnsfer of lerning. [3] exmine cse when the screening process is imperfect nd errors cn hppen during inspection. The probbility of misclssifiction errors is ssumed to be known. In the literture review, it hs been highlighted the dvncements in the design of relistic economic order quntity models when items in the received lot re clssified s either defective or non-defective. All proposed models ssume screening, 100% inspection of the lot; uncceptble items re either bckordered or sold t discounted price. These ssumptions not necessrily hold for ll rel

7 Muhmmd Al-Slmh 65 life cses; usully buyers do not hve the resources to exmine every item in the lot, so they re pt to dopt n cceptnce smpling pln. For certin types of products, destructive testing is required to ensure the items meet certin requirements. 3 The mthemticl Model The rticle will develop n economic order quntity model for the cse when items re clssified s defective nd nondefective nd the received lot is subjected to single cceptnce smpling pln nd destructive testing nd inspection errors. The cost function of the model is equl to the totl costs minus totl profits. Totl costs include the costs of ordering, holding, destructive testing, nd lot cceptnce. Profits include revenue from selling the remining items of the rejected lot. The vribles of the model re the lot size nd the smple size. The cceptnce number will be ssumed zero, s commonly used in destructive tests tht re bsed on cceptnce smpling. Consider buyer who plces orders of size Q from supplier with fixed ordering cost A nd unit vrible ordering cost C o nd fces constnt demnd rte D. The inventory holding cost is h nd the cycle length is T. When lot is received, the buyer uses destructive testing cceptnce smpling pln to decide if the lot should be ccepted nd plced in inventory or rejected. A smple of size n units is tken from the lot nd inspected. The probbility tht n item is defective is p. The buyer ccepts lot when the number of defective items in the smple is equl to or less thn c, the cceptnce number. The destructive testing cost is gn. ( ) The shortge cost is C per unit. The probbility of clssifying defective item s nondefective is m, with probbility distribution function f ( m ) nd cumultive distribution function Fm. ( ) When the result of destructive testing cceptnce pln is to reject the lot, the buyer cn sell the remining items in the

8 66 Economic order quntity with imperfect qulity rejected lot for K per unit nd plce new order. The behvior of the inventory level is illustrted in Figure 1. Inventory level Q Q n T Time ( Q n) p n Figure 1: Behvior of the inventory level over time The im is to find n optiml ordering quntity nd optiml smple size with zero cceptnce number tht will minimize the verge totl cost with n cceptble inspection service level: min ETCQn [ (, )] subject to P(inspection yield) p1 (1) The service level p 1 is the lest cceptble probbility tht the cceptnce smpling pln cn offset the effect of inspection errors. The service level vlue is specified by the buyer. The probbility tht lot is ccepted is equl to the probbility tht there re c or less defective prts in the lot: x c p P( xc) p (1 p) n x n x () x0

9 Muhmmd Al-Slmh 67 The received lot is rejected when the number of defects in the smple exceeds the cceptnce number c. When the lot is ccepted, the expected number of short units tht otherwise must meet prt of the demnd is equl to ( Q n) p n, nd these units will cuse the buyer to incur shortge cost. Therefore, the cost of ccepting lot is: ( ) ( ) ( ) C Qn pn P xc C Qn p n p (3) The verge expected totl cost in cycle consists of the difference between the totl costs nd the totl benefits. The costs re the fixed ordering cost, the vrible ordering cost, holding cost, cost of destructive testing, cost of lot rejection, nd cost of lot cceptnce; nd the benefits re equl to the revenue from selling the remining items in the rejected lot: AD D h( Q n) DC ( Q n) p n CD 0 gn (), p Q Q Q Q AD D KD( Q n) h( Q n) DC ( Q n) p n CD 0 gn ( ), (1 p) p (4) TC( Q, n) Q Q Q Q Q AD D KD( Q n) h( Q n) DC ( Q n) p n 3 CD 0 gn ( ), (1 p) p Q Q Q Q Q The expected totl cost cn be written s AD g( n) D KD( Q n) ( Q n) ETCQn [ (, )] p[( i1)( CD 0 ) i h i0 Q Q Q Q DC ( Q n) p n i ](1 p ) (5) Q In the expression of the expected totl cost, p is function of the smple size n nd the cceptnce number c.

10 68 Economic order quntity with imperfect qulity Theorem 3.1 The infinite series AD g( n) D KD( Q n) ( Q n) DC ( Q n) p n [( i1)( C0D ) i h ](1 p ) i0 Q Q Q Q Q converges. Proof. We cn begin by expnding the expression of the series: i0 AD g( n) D KD( Q n) ( Q n) DC ( Q n) p n [( i1)( C0D ) i h ](1 p ) Q Q Q Q Q AD g( n) D KD( Q n) ( CD 0 ) [( i1)(1 p) ] i(1 p) Q Q Q i i0 i0 ( Q n) DC ( Q n) p n [ h ] (1 p ) Q Q i0 The three resulting infinite series re convergent: i i i i i0 [( i1)(1 p ) i ] 1 p i0 i0 1 p i(1 p ) i p (1 p ) i 1 p Hence, the series converges nd it is equl to AD g n D KD Q n 1 p Q n DC ( Q n) p n ( CD ) [ ] Q Q p Q p Q Q p Substituting out the infinite series, the expected cost becomes: ( ) 1 ( ) ( ) 1 0 h AD g( n) D 1 KD( Q n) 1 p ( Q n) ETCQn [ (, )] ( CD 0 ) h Q Q p Q p Q DC ( Q n) p n Q When the cceptnce number c is set to zero, the lot cceptnce probbility becomes p (1 p) n, nd the expected totl cost reduces to (6)

11 Muhmmd Al-Slmh 69 AD g( nd ) 1 KDQ ( n) 1 ( Qn) ETCQn [ (, )] ( CD 0 ) ( 1) h n n Q Q (1 p) Q (1 p) Q DC ( Q n) p n Q (7) In considering the inspection yield, the number of defects in the smple is equl to pn. But becuse of the error in inspection, the reported number of defects in the smple is only pn(1 m). Therefore, we cn define the yield of the inspection to be whether pn(1 m) c 1. Therefore, the smple size n should be selected such tht: P( pn(1 m) c1) p (8) The probbility of misclssifiction of defective item m is the only rndom vrible in the inspection yield constrint; hence we cn write: c 1 P(1 m ) p1 (9) pn 1 c 1 Pm ( 1 ) p1 (10) pn c1 1 pn f ( mdm ) p 0 1 (11) The limits of the integrtion might exceed the permissible rnge of m, which cn be defined s m, m. A better description of the integrtion cn be: min mx c 1 (1) pn f ( mdm ) p m 1, min 1, mmx min c 1 F(min 1, mmx ) p1 (13) pn The economic order quntity with destructive testing cceptnce smpling nd inspection errors is found by minimizing ETCQn [ (, )], defined by Eq. (7), subject to the inspection yield constrint, defined by Eq. (13).

12 70 Economic order quntity with imperfect qulity 4 Numericl Anlysis Consider buyer who replenishes the inventory from production system tht produces perfect nd imperfect items. The relevnt costs nd prmeters re: C $0.7 C0 $ per unit A $50 h $0.5 D 1000 units per yer p 0.01 K $0.4 per unit f ( m ) norml (μ 0.00, 1) p1 0.1 c 0 In this nlysis of the model, the inspection cost function will be ssumed to be liner; gn ( ) 5n 10. A liner inspection cost function resonbly describes the ll costs ssocited with inspection, such s mngeril cost, engineering cost, lbor cost, etc. Exmples of studies tht ssume liner inspection costs include [33] nd [5]. The model is solved using GAMS-BARON solver. The optiml solution for the prmeters selected bove is Q 1,11 nd n 30. The smple size s determined by the inspection yield probbility is shown in Figure. The smple size is clculted from Eq. (13) for the cse of norml 1 1 distribution s n, where is the norml inverse cumultive 1 p(1 ( p)) distribution function, nd c 0. The plot of n vs p 1 shows tht the smple size increses when the inspection yield probbility is rised. This result is consistent with prior expecttion tht when the probbility is high, we need lrge smple in order for the number of defects in the smple which hve been correctly

13 Muhmmd Al-Slmh 71 clssified s defective to exceed the criticl number. The plot lso shows tht the smple size increses when the stndrd devition declines. For exmple, for p1 0.1, the smple size n 44 when 1, nd n 53 when 0.7. The effect of the selling price K on the lot size, smple size, nd the expected cost is illustrted in Tble 1. The lot size Q seems to increse only slightly with the increse in K, while the expected cost declines with the increse in the selling price. Tble 1. Selling price K Q n ETCQn (, ) 0.4 1, , , , , , Figure : Smple size

14 7 Economic order quntity with imperfect qulity Tble illustrtes the effect of the probbility of defective item on the lot size, smple size, nd the expected cost. With the increse in p, ll vribles decline. The decline in the lot size Q with the increse in p is consistent with the expecttion tht when the probbility of defective is high, we better order less quntity to reduce the shortge cost per cycle. Tble. Probbility of defective p Q n ETCQn (, ) , , , , Conclusion This rticle provides n economic order quntity model in which the buyer orders items from production process tht produces perfect nd imperfect products. As result, the buyer pplies n cceptnce smpling pln; becuse of the nture of the items, destructive testing is required to ssess qulity. The inspection process is not perfect; defective items cn be still clssified by the inspection s nondefective. If the received lot is rejected, the remining items re sold in secondry mrket for reduced price nd new order is plce. It hs been found tht the increse the selling price reduces the lot size, nd tht increse the probbility of defect item reduces both the lot size nd the smple size. ACKNOWLEDGEMENTS. I would like to cknowledge the support received from King Fhd University of Petroleum nd Minerls, Dhhrn, Sudi Arbi.

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