Tamkang Journal of Scence and Engneerng, Vol. 9, No 1, pp. 19 23 (2006) 19 Economc Desgn of Short-Run CSP-1 Plan Under Lnear Inspecton Cost Chung-Ho Chen 1 * and Chao-Yu Chou 2 1 Department of Industral Management, Southern Tawan Unversty of Technology, Yungkang, Tawan 710, R.O.C. 2 Department of Industral Engneerng and Management, Natonal Yunln Unversty of Scence and Technology, Toulu, Tawan 640, R.O.C. Abstract In general, the unt cost of nspecton s assumed to be constant. However, t can be argued that the unt cost of nspecton s seldom constant. In 1943, Dodge proposed the type I contnuous samplng plan (CSP-1 plan) and ndcated how to calculate ts average outgong qualty (AOQ) and average fracton nspected (AFI). In ths paper, we further propose the problem concernng the economc desgn of short-run CSP-1 plan under lnear nspecton cost. A soluton procedure s developed to fnd the unque combnaton ( *, f * ) that wll meet the average outgong qualty lmt (AOQL) requrement, whle also mnmzng the total expected cost per unt produced for the short-run CSP-1 plan when the process average p ( AOQL) and producton run length R are known. A numercal example s llustrated and the senstvty analyss of parameters s provded. Key Words: Type I Contnuous Samplng Plan (CSP-1 Plan), Average Fracton Inspected (AFI), Average Outgong Qualty (AOQ), Average Outgong Qualty Lmt (AOQL), Short-Run Producton 1. Introducton Contnuous samplng plans were frst ntroduced by Dodge [1]. The earlest type Icontnuous samplng plan s called the CSP-1 plan. Dodge [1] ndcated how to calculate the long-run average fracton nspected (AFI) and the average outgong qualty lmt (AOQL) for a CSP-1 plan. The AOQL has been wdely used n prmary ndex of the CSP-1 plan and other contnuous samplng plans as the performance of nspecton. Short-run or low-volume producton refers to processes that generate only a small number of tems at any one tme. In manufacturng, computerzaton and an ncreasng trend toward buld to order producton and quck turn around tmes have gven rse to short-run producton. Prevous researchers [2,3], McShane and Turnbull *Correspondng author. E-mal: chench@mal.stut.edu.tw [4,5], and Lu and Aldag [6]) have presented the AOQ and AFI functons for short-run CSP-1 plan. In 1998, Wang and Chen [7] have presented the problems of calculatng AOQL and mnmzng AFI for short-run CSP-1 plan. In general, the unt cost of nspecton, replacement or downstream loss s assumed to be constant. Lu [8] argued that the unt cost of nspecton, replacement or downstream loss s seldom constant. For nstance, f more nspecton s to be done, dsproportonately more equpment, staff and other facltes mght be needed, whch mght ncrease the per unt cost of nspecton or even decrease t due to economy of scale. The same argument would hold for downstream loss, as consderably more people and systems mght be needed to handle more defectve tems that are returned, or the downstream producton lne mght suffer mmensely due to an ncreased amount of defectve tems.
20 Chung-Ho Chen and Chao-Yu Chou Chen and Chou [9] adopted the modfed Cassady et al. s [10] model and presented the economc desgn of long-run CSP-1 plan under lnear nspecton cost. In ths paper, we further propose the problem concernng the economc desgn of short-run CSP-1 plan under lnear nspecton cost. A soluton procedure s developed to fnd the unque combnaton ( *, f * ) that wll meet the AOQL requrement, whle also mnmzng the total expected cost per unt produced for the short-run CSP-1 plan when the process average p ( AOQL) and producton run length R are known. 2. Mathematcal Model and Soluton Procedure Assume that the per unt nspecton cost s lnearly proportonal to the average number of nspectons per nspecton cycle. The total expected cost functon of short-run CSP-1 plan under lnear nspecton cost ncludes the expected nspecton cost per unt produced, the expected acceptance cost per unt produced, and the expected replacement cost per unt produced. From Yang [3], Cassady et al. [10], and Chen and Chou [9], we have the mathematcal model as follows: mnmze subject to where E( C) C AFI C p(1 AFI) C pafi max AOQ( R) p 0 p 1 R, nteger 0 f 1 s a r L (1) AFI: Average fracton nspected for short-run CSP-1 AOQ( R) plan ( 1 ) P X Z Y AOQ( R) AOQ1 [ 1] 2 2R Y AOQ 1 1 X 1 f (1 f) pq f (1 f) q Y f (1 f) q fpq 2 1 1 pq (2 1) q 1 1 2 2 Z ( 1) p q fp fp E(C): The total expected cost per unt produced durng one nspecton cycle : The clearance number of the 100% nspecton stage for short-run CSP-1 plan f: The samplng frequency of the samplng nspecton stage for short-run CSP-1 plan p: The ncomng proporton defectve (ts estmate value s process average p) q: 1-p a: Constant of the unt nspecton cost composed of a fxed porton b: Constant of the unt nspecton cost composed of a varable porton U: Average number of unts produced durng 100% nspecton stage for short -run CSP-1 plan durng one nspecton cycle ( = 1 q ) pq V: Average number of unts passed durng samplng nspecton stage for short-run CSP-1 plan durng one nspecton cycle ( 1 fp ) R: Producton run length P L : The specfed AOQL value C s : The nspecton cost per unt nspected (= a + b(u + fv)) C a : The cost of acceptng a non-conformng unt C r : The cost of replacng a non-conformng unt found durng nspecton. Wang and Chen [7] has presented the detals of the calculaton of the AOQL for a short-run CSP-1 plan based on Yang s [3] renewal process approach. Hence, we can use the followng method to fnd the combnaton (, f) that satsfes the mnmum E(C) for model (1). Step 1. By adoptng the method of Appendx A of Wang and Chen [7], we can obtan all the combnatons (, f) of parameters whch reach the maxmum AOQ wth gven (R, p L ). Step 2. Compute the respectve E(C) by the combnaton (, f) of parameters from step 1.
Economc Desgn of Short-Run CSP-1 Plan Under Lnear Inspecton Cost 21 Step 3. By comparng the E(C) of step 2, the unque combnaton ( *, f * ) that has the mnmum E(C) s the optmal soluton. 3. Numercal Example and Senstvty Analyss 3.1 Numercal Example Assume that the manufacturer adopts the CSP-1 plan for controllng the product qualty of process. The producton quantty of product s fnte under the condtoned capablty. The manufacturer uses the buld to order producton. The producton run length has the specfed unts each tme. Hence, the manufacturer needs to use the short-run CSP-1 plan. Consder the example gven n Chen and Chou [9]. Let a =4,b = 0.6, C r =8,C a = 16, p L = 0.001, and p = 0.0015. Assume that R = 500. The optmal soluton of Chen and Chou s [9] work s * = 198 and f * = 0.6029717 wth E(C) = 364.2816. By solvng the above model (1), we obtan that the optmal combnaton of parameters are * = 231 and f * = 0.01636173 wth E(C) = 319.6034. Thus, the total expected cost per unt durng one nspecton cycle of short-run CSP-1 plan s less than that of Chen and Chou [9]. Table 1. Economc samplng plans under dfferent process average p (a =4,b =0.6,C r =8,C a = 16, p L = p f AFI E(C).0010 231.0163617.5287233 401.8467.0011 231.0163617.5349683 378.4256.0012 231.0163617.5413218 359.3599.0013 231.0163617.5477249 343.6136.0014 231.0163617.5542597 330.5441.0015 231.0163617.5608830 319.6034.0016 231.0163617.5676008 310.4088.0017 231.0163617.5744240 302.6707.0018 231.0163617.5813188 296.1356.0019 231.0163617.5883313 290.6557.0020 231.0163617.5954188 286.0566.0021 231.0163617.6026329 282.2565.0022 44.8772805.8973094 273.2317.0023 34.9052231.9177545 262.6023.0024 25.9301721.9372828 252.5979.0025 17.9522091.9556707 243.1755.0026 10.9713946.9726672 234.2941.0027 4.9877831.9880095 225.9189.0028 1.9960002.9960192 218.0386 3.2 Senstvty Analyss (1) The effect of the process average For a gven set of parameters, t s observed that the mnmum E(C) for a short-run CSP-1 plan has the followng propertes (as shown n Table 1): 1. The larger p, the more * lkely decreases. 2. AFI ncreases wth p. As the value of p ncreases, t s more lkely that 100% nspecton wll be encountered. 3. E(C) decreases wth p. (2) The effect of constant a For a gven set of parameters, as the constant a ncreases, both * and f * have the same value and E(C) ncreases slowly (as shown n Table 2). (3) The effect of constant b For a gven set of parameters, as the postve b n- Table 2. Economc samplng plans under dfferent constant a (b =0.6,C r =8,C a = 16, p = 0.15%, p L =0.1%, R = 500) a f AFI E(C) 1 231.0163617.5608830 317.9207 2 231.0163617.5608830 318.4816 3 231.0163617.5608830 319.0425 4 231.0163617.5608830 319.6034 5 231.0163617.5608830 320.1643 6 231.0163617.5608830 320.7251 7 231.0163617.5608830 321.2860 8 231.0163617.5608830 321.8469 Table 3. Economc samplng plans under dfferent constant b >0(a =4,C r =8,C a =16,p = 0.15%, p L = b f AFI E(C).1 231.0163617.5608830 55.1512.2 231.0163617.5608830 108.0416.3 231.0163617.5608830 160.9321.4 231.0163617.5608830 213.8225.5 231.0163617.5608830 266.7129.6 231.0163617.5608830 319.6034.7 231.0163617.5608830 372.4938.8 231.0163617.5608830 425.3842.9 231.0163617.5608830 478.2747 1.0 231.0163617.5608830 531.1651
22 Chung-Ho Chen and Chao-Yu Chou creases, both * and f * have the same value and E(C) ncreases fast (as shown n Table 3). As the negatve b decreases slowly, both * and f * have the same value and E(C) decreases slowly (as shown n Table 4). (4) The effect of replacement cost C r For a gven set of parameters, as the C r ncreases, both * and f * have the same value and E(C) ncreases lttle (as shown n Table 5). (5) The effect of acceptance cost C a For a gven set of parameters, as the C a ncreases, Table 4. Economc samplng plans under dfferent constant b <0(a =4,C r =8,C a = 16, p = 0.15%, p L = b f AFI E(C).0001 231.0163617.5608830 2.207911.0002 231.0163617.5608830 2.155020.0003 231.0163617.5608830 2.102130.0004 231.0163617.5608830 2.049239.0005 231.0163617.5608830 1.996349.0006 231.0163617.5608830 1.943459.0007 231.0163617.5608830 1.890568.0008 231.0163617.5608830 1.837678.0009 231.0163617.5608830 1.784787.0010 231.0163617.5608830 1.731897.0020 231.0163617.5608830 1.202993.0030 231.0163617.5608830.674088.0040 231.0163617.5608830.145184.0042 231.0163617.5608830.039403 Table 5. Economc samplng plans under dfferent replacement cost C r (a =4,b =.6,C a =16,p = 0.15%, p L =0.1%,R =500) C r f AFI E(C) 4 231.0163617.5608830 319.6000 5 231.0163617.5608830 319.6008 6 231.0163617.5608830 319.6017 7 231.0163617.5608830 319.6025 8 231.0163617.5608830 319.6034 9 231.0163617.5608830 319.6042 10 231.0163617.5608830 319.6051 11 231.0163617.5608830 319.6059 12 231.0163617.5608830 319.6067 13 231.0163617.5608830 319.6076 14 231.0163617.5608830 319.6084 15 231.0163617.5608830 319.6093 16 231.0163617.5608830 319.6101 both * and f * have the same value and E(C) ncreases lttle (as shown n Table 6). (6) The effect of producton run length R For a gven set of parameters, as the R ncreases, both * and f * have dfferent values. As R, t s more lkely that our soluton s the same as that of Chen and Chou [9] (as shown n Table 7). Table 6. Economc samplng plans under dfferent acceptance cost C a (a =4,b =.6,C r =8,p = 0.15%, p L =0.1%,R =500) C a f AFI E(C) 8 231.0163617.5608830 319.5981 9 231.0163617.5608830 319.5988 10 231.0163617.5608830 319.5994 11 231.0163617.5608830 319.6001 12 231.0163617.5608830 319.6008 13 231.0163617.5608830 319.6014 14 231.0163617.5608830 319.6021 15 231.0163617.5608830 319.6027 16 231.0163617.5608830 319.6034 17 231.0163617.5608830 319.6040 18 231.0163617.5608830 319.6047 19 231.0163617.5608830 319.6054 20 231.0163617.5608830 319.6060 Table 7. Economc samplng plans under dfferent producton length R (a =4,b =0.6,C r =8,C a = 16, p L =0.1%,p = 0.15%) R f AFI E(C) 100 66.3352029.7957561 354.6506 200 117.0782806.6684981 321.4296 300 160.0123270.6084124 311.8836 400 197.0281848.5865751 317.7287 500 231.0163617.5608830 319.6034 600 262.0130652.5424658 323.7298 700 291.0063717.5253710 327.3847 800 318.0033749.5119071 332.1045 900 343.0056301.5024642 338.3670 1000 367.0035412.4922667 343.5882 2000 182.5948367.6912951 366.1726 3000 187.5977062.6850925 365.6004 4000 189.6000890.6824835 365.2951 5000 190.6018443.6811115 365.1043 10000 194.6022941.6763253 364.7059 20000 196.6026063.6739271 364.4973 25000 196.6032635.6738475 364.4543
Economc Desgn of Short-Run CSP-1 Plan Under Lnear Inspecton Cost 23 4. Conclusons In ths paper, we have presented the economc desgn of short-run CSP-1 plan under lnear nspecton cost. The producton length R, the constant of the unt nspecton cost composed of a varable porton b, and the process average p have an mportant effect on the total expected cost per unt produced durng one nspecton cycle. Further drecton of study wll extend to other attrbute samplng plans or contnuous samplng plans under non-lnear nspecton cost. References [1] Dodge, H. F., A Samplng Inspecton Plan for Contnuous Producton, Annals of Mathematcal and Statstcs, Vol. 14, pp. 264 279 (1943). [2] Blackwell, M. T. R., The Effect of Short Producton Runs on CSP-1, Technometrcs, Vol. 19, pp. 259 263 (1977). [3] Yang, G. L., A Renewal-process Approach to Contnuous Samplng Plans, Technometrcs, Vol. 25, pp. 59 67 (1983). [4] McShane, L. M. and Turnbull, B. W., Probablty Lmts on Outgong Qualty for Contnuous Samplng Plans, Technometrcs, Vol. 33, pp. 393 404 (1991). [5] McShane, L. M. and Turnbull, B. W., New Performance Measures for Contnuous Samplng Plans Appled to Fnte Producton Runs, Journal of Qualty Technology, Vol. 24, pp. 153 161 (1992). [6] Lu, M. C. and Aldag L., Computerzed Contnuous Samplng Plans wth Fnte Producton, Computers and Industral Engneerng, Vol. 25, pp. 445 448 (1993). [7] Wang, R. C. and Chen, C. H., Mnmum Average Fracton Inspected for Short-run CSP-1 Plan, Journal of Appled Statstcs, Vol. 25, pp. 733 738 (1998). [8] Lu, M.-C., Multple Crtera Optmzaton of an Economcally-based Contnuous Samplng Plan, Unpublshed Ph.D. Dssertaton, Arzona State Unversty, U.S.A., p. 45 (1987). [9] Chen, C.-H. and Chou, C.-Y. Economc Desgn of Contnuous Samplng Plan Under Lnear Inspecton Cost, Journal of Appled Statstcs, Vol. 29, pp. 1003 1009 (2002). [10] Cassady, C. R., Mallart, L. M., Rehmert, I. J., and Nachlas, J. A., Demonstratng Demng s kp Rule Usng an Economc Model of the CSP-1, Qualty Engneerng, Vol. 12, pp. 327 334 (2000). Manuscrpt Receved: Sep. 23, 2004 Accepted: Dec. 2, 2004