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A I I E Transactions The Mixed Model Learning Curve
The Mixed Model Learning Curve
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Volumen:
1
Idioma:
english
Revista:
A I I E Transactions
DOI:
10.1080/05695556908974423
Date:
June, 1969
Archivo:
PDF, 374 KB
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This article was downloaded by: [Moskow State Univ Bibliote] On: 19 October 2013, At: 07:16 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 3741 Mortimer Street, London W1T 3JH, UK A I I E Transactions Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/uiie19 The Mixed Model Learning Curve a Nick T. Thomopoulos & Melvin Lehman a b Illinois Institute of Technology , b University of Illinois , Chicago Circle Published online: 06 Jul 2007. To cite this article: Nick T. Thomopoulos & Melvin Lehman (1969) The Mixed Model Learning Curve, A I I E Transactions, 1:2, 127132, DOI: 10.1080/05695556908974423 To link to this article: http://dx.doi.org/10.1080/05695556908974423 PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content. This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sublicensing, systematic supply, or distribution in any form to anyone is expressly forbidden. Terms ; & Conditions of access and use can be found at http:// www.tandfonline.com/page/termsandconditions The Mixed Model Learning Curve Downloaded by [Moskow State Univ Bibliote] at 07:16 19 October 2013 NICK T. THOMOPOULOS Illinois Institute of Technology MELVIN LEHMAN University of Illinois at Chicago Circle Abstract: In this article an extension of the learning curve concept is introduced by considering the rate of reduction in direct labor assembly time for mixed model assembly situations. Here, more than one model is assembled on the same assembly line; hence, the repetitions of the assembly work are not always the same. Following a short description of the mathematical formulation of the learning curve (for convenience called the single model learning curve), a development of a mixed model learning curve is presented. An example is given to demonstrate the application of mixed model learning. * * * I n 1936, Wright (8) first described what is often called the "learning curve." In his paper he noted that the manufacturing costs per unit of aircraft decrease with production according to a predictable pattern. Every time the number of completed units is doubled, the manufacturing cost per unit decreases by a constant percentage. Since that time a great variety of applications of this concept in assembly have been reported in the literature; see, for example, (I), (2)) (3), (4), ( 5 ) ) and (9). I n all instances, the basic assumption is made that the units of production are exactly the same (or are from a single model). The objective of this article is to develop a learning curve for mixed model assembly lines. In these situations, two or more models are produced on one assembly line with the desire to keep each model continuously in production rather than to produce batches of each, intermittently, for inventory. Following a short review on single model learning curves is the development, with an example, for the mixed model case. is doubled. This is represented mathematically by a two parameter function. If r represents the rth unit of assembly (that is, r = 1, 2, 3,    ) and f(r) is the learning curve which yields the time required to assemble the rth unit, then f(r) = arb (r = 1, 2, 3, ..) [ll where a = assembly time for the first unit, b = a negative constant. If lOOR represents the percentage decrease in j(r) every time r is doubled, then for (0 <R I I), and R is called the learning rate. Substituting Equation 1 into Equation 2, it is seen that b = log R . log 2 For example, if the assembly time per unit decreases by 80 percent whenever the number of assemblies is doubled, then R = .80 and b =  ,322. The time required to assemble the first r units is approximated by I SINGLE MODEL LEARNING CURVES One of the most widely accepted formulations of learning continues to be based upon the power function suggested by Wright (8). The theory, as applied in assembly, states that the assembly time per unit declines by some constant percentage every time the number of assemblies June 1969 I =.arb+' [51 b + l This approximation is weakest at the smaller values of r, but improves as r increases. AIIE Transactions 127 The parameter "a" is generally some multiple of the work content time of the unit (that is, the total time required to assemble one complete unit where the time is based on standard measured time). Hence, if T represents the work content time, and k is the multiple constant (Ic>l), then [GI can be found and is here identified by PI. Similarly, E Z and pz can be obtained. I n like fashion, El2 is the collection of elements common to models MI and Mz, and f'12 is the time associated with El$.Note that PI+ F I ~ Tz Pz + f,,. and MIXED MODEL LEARNING CURVES Downloaded by [Moskow State Univ Bibliote] at 07:16 19 October 2013 TI = = The total standard time required over the N units is I n mixed model learning situations, the operators are not always performing the same operations on all models (see, for example, (6) and (7) ). Some elements of work are unique to some models, while other elements are common to two or more models. As a consequence, the rate at which operators are learning (and hence complete their work on the units) varies from element to element and from model to model. I n this section of the article, a formulation of mixed model learning curves is presented. Hence, if PI, P2, and Plz represent the fraction of N,Tj devoted to El, Ez, and El2 elemental categories, respectively, then x:_, P1 = Suppose that two models, say MI and Mz, are to be assembled on one line. Let TI and T2 represent the work content times for MI and M2, respectively. Then the time required to assemble the first unit of MI is NI~I 2 N,T, , [91 3=I PZ = ~ z f z [lo1 9 2 C NjTi 3=1 a1 = kT1 and and for M2, it is Further, if NI units of M1 and N Z units of M2 are to be assembled, then the weighted average time for the first unit is The sum of Equations 9, 10, and 11 is unity. It is noted, in observing Equation 1, that rb (r = 1, 2, 3, .  ) represents the fraction of time from "a" which is required on the rth unit. Now in mixed model learning curves, "aJ' is computed using Equation 7. It remains to find an approximation to rb. For this purpose, f(r) is restated as  where N = N1+N2. If learning curves for MI and M2 are to be obtained separately where R (and hence b) is identical for both models, then the learning curve for MI is fl(r)=alrb f(r)=ag(r) g(r) = rb (r=1,2,...,N1) (r   ). [131 = 1,2, Hence, for E l elements, the equivalent of Equation 13 and for M2, it is I n order to define a mixed model learning curve for r = 1, 2, . . . , N, it is first necessary to observe the relationships of the work elements to the models. Suppose that, in total, n elements of work are defined. Let ti (i=1, 2, , n) designate the standard time associated with the ith element. Then, if E l represents the collection of elements which are only performed on model MI, the sum of the standard times of the elements in El Also, for Ez and E12, it is 128 [121 where is  (r=1,2,...) g2(r) = rb (r = 1, 2, g12(r)= rb (r = 1, 2, . ,Nz), and . . . , N), respectively. I n order to observe the effects of gl(r) and g,(r) over N AIIE Transactions Volume I No. 2 units (where N l l N and N 2 1 N), the following transformations of r are given for gl(r): let rl N1 N = r  (r = 1,2, . . . , N) where (Nl/N) 5 n 5 Nl; hence, ql(r1) = rb($)b (r = = 1,2,  . , N). 1,2, [I4] Hence, 100(Q1) represents the percentage increase in assembly time for N units over an equivalent single model assembly process for N units (that is, where a and b are the same in f (r) as they are inl(r)). It is further possible to detect the equivalent of Q for MI and Mz separately. This is accomplished by considering" Eauation 11. the fraction of the total time devoted to elemental category Elz. First, Equation 11 is separated into Similarly, for gz(r), let rz = r N2 N (r e e e , N) For convenience, these two partitions are denoted by Downloaded by [Moskow State Univ Bibliote] at 07:16 19 October 2013 and Hence, gl(rl), gz(rz) and glz(r) are defined over the range r = l , 2 , . .  , N. Now, since PI, P2, and Plzrepresent the fraction of time devoted to El, Ez, and Elz elements, respectively, the approximation to g(r) (that is, Equation 13) for this twomodel case is = + ($>" + r b [ ~ ~ ( $  ) b~2 Hence, P12= P ( I , ~ + P ~ Now ( ~ )P(l)z . represents the fraction of the total time PIP] (r = 1, 2, . a , and N). ( 2W) [16] j=1 For convenience, let which is consumed by MI on Elz elements. Likewise, Pl(z)is the fraction for Mz. Hence, Q is Further, let q C 1 ) 2 = P(1)2 and qtt2)= q l = PI ($)b, and qlz = It is noted that (Pl+P(1)2) is the fraction of time devoted to model MI, and that (Pz+PI(z))is the corresponding fraction for model Mz. Thus, if Q1 and Qz correspond in meaning to Q for MI and Mz, respectively, P12. Then let Q = q1 + + YZ [171 q12 Q1+ Q ( 1 ) 2 > where it is readily seen that Q 1 since b <O. Hence, g(r) = &I rbQ = g(r)Q = P1 + P(l,Z [241 and (r = 1, 2, . . . ), [18I and the approximate mixed model learning curve is f(r) = > arb& = f(r)Q (r = 1, 2, . .  ). [191 The equivalent to Equation 5 for this twomodel example becomes nrwl = F(r)Q. June 1969 Do] Again it is observed that & I > 1 and Q2 1. It is also noted that lOO(Q1 1) represents the percent increase in assembly time required for units of M1 over units on a single model line (where a and b are the same in both situations). lOO(Q2 1) is the corresponding percent for Mz. The concepts developed are extended and illustrated AIIE Transactions for three modds. Suppose the models are designated by M j ( j = 1,2,3), and that the production schedule calls for N l = 700, N2 = 1000, and N3 = 300 units. Further, assume the elements and the mixed model relations are as given in Table 1. The elements and their standard times are shown in columns I and 11, respectively. Column I11 is used to identify the association between elements and models. These relations are defined by the variable eij ( i = 1, 2,  . . , 18, and j = 1, 2, 3) where Downloaded by [Moskow State Univ Bibliote] at 07:16 19 October 2013 [261 Ei Ez E3 EI3 E23 E~za Hence, element 1 is performed on all three models, element 2 is required only on MI and &I3,and so forth. The work content times per model are seen to be T1=2.89, T2=2.47, and T3 =3.49 minutes. Column IV gives the number of repetitions of the ith element over N=2000 units. This is defined by The estimated time for the first unit in this threemodel case is I n this situation, a value of k = 2 is assumed. Hence, Equation 28 yields a = 5.540 minutes. To obtain the equivalent to Q (that is, Equation 17) for Table 1 : Elemental times and model usage for threemodel example Total .00 .33 .36 .40 1.39 .64 1.10 v I11 Repetitions IV I1 XI11 5540 700 1000 300 1700 1000 1300 2000 0 330 108 680 1390 832 2200 0.000 0.060 0.020 0.123 0.251 0.151 0.395 5540 1.OOO Portion of this example, the data in Table 2 are compiled. The various elemental categories are identified in column I. Their standard times are in column 11, and the number of repetitions over N = 2000 units are in column 111. Column IV lists the total time required for each category over the given schedule. The total time for all categories is 5540 minutes. Column V shows the fraction of the 5540 minutes required by each category. In the notation used earlier, P1=.OOO, P2=.060, P3=.020, P12=.123, and SO forth. I n this example, it is assumed that the learning rate of R = .80 applies to models MI, M2, and Ma. Hence, Equation 4 yields b =  .322. Now Q, as shown in Equation 17 for the twomodel case, is extended for three models, that is, where I I1 I11 IV Standard Model Usage Repetitions Elements Times (9 (tz) (ed (e,3 (e,d (Ki)  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 I I I1 Element Standard Categories Times En 0 implies element i is not performed on model j 1 implies element i is performed on model j. i Table 2 : Seven elemental categories for threemodel example 130 .07 .20 .18 .10 .20 .14 .08 .37 .35 .30 .36 .36 .3 1 .26 .28 .19 .14 .33 1 1 1 1 0 1 0 1 1 1 0 0 1 1 1 1 1 0 1 0 1 1 1 1 1 0 0 1 1 0 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 0 0 2000 1000 2000 2000 1300 2000 1300 1000 1000 2000 1300 300 2000 1700 1000 1000 1700 1000 and I n this example, ql = .0000, q2 = .0750, q 3 = .0368, q12 = .1292, q 1 3 = .3138, q23= .I736 and q123= .3950. Hence Q=1.1234. For later convenience, Q is found from a different approach. For this purpose, note that Equation 29 is really AIIE Transactions Volume I .No. 2 3 NjTi Nb jI ' [331 Now, since the numerator is equivalent to Similar logic yields Q2= 1.0803 and Q 3 = 1.2144. These results show that the characteristics of MI cause a 12.78percent increase in assembly time, M z contributes 8.03 percent, and M3 21.44 percent. THE GENERAL CASE Downloaded by [Moskow State Univ Bibliote] at 07:16 19 October 2013 1, Suppose the assembly line has J models ( M j (j= 2,  . . , J))in production a t any one time. Again, let N , and Ti( j = 1,2,3, . . , J ) designate the number of units and work content times for the jth model, respectively. Assume, also, that lc and R are constant for all models. Then the total required number of units for all models is When many models are included in the schedule, Q in its equivalent form to Equation 34 is easier to obtain than in the form of Equation 33. For this example, the mixed model learning curve becomes Parameter a (the assembly time on unit one) is Parameter b is found as shown in Equation 4, that is, The total time required to complete the first r units is log R log 2 h =  . Q is obtained by extending Equation 34, that is, These results show that an added 12.34percent increase in assembly time is required to process these units over an equivalent single model line where a =5.540 and b =  .322. To find the percentage increase in time caused from each model separately, Table 3 is used. The columns in this table correspond to those of Table 2. However, in this situation there are twelve elemental categories rather than the seven shown in Table 2. The equivalents to ql, q2, etc., again are found. These are : L Table 3: Twelve elemental categories for threemodel example I1 Standard Times I11 Repetitions IV v I1 XI11 Portion of 5540 El .OO Ez .33 .36 .40 .40 1.39 1.39 .64 700 1000 300 700 1000 700 300 1000 300 700 1000 300 0 330 108 280 400 973 417 640 192 770 1100 330 I Element Categories Es E(l)z Elcz) E(1)a El (3) E(2)3 Ez(3) .64 E ( 1 ) 2 3 1.10 1.10 1.10 Em)3 Elz(a) Hence, QI (shown in Equation 24 for the twomodel case) is now Total  5540 .OOO .060 .020 .052 .071 .I77 .074 .I15 .036 .138 .I97 .060 1.OOO 7 June 1969 AIIE Transactions 13 1 where ti and K; ( i = l , 2,  . , n ) are defined as shown earlier. Hence, the mixed model learning curve is (3) (4) (5) and the time required for the first r units is (6) (7) Downloaded by [Moskow State Univ Bibliote] at 07:16 19 October 2013 CONCLUSIONS This article shows how learning curve concepts can be extended for mixed model schedules. I n addition, the percentage increase in assembly time caused from processing the units on a mixed rather than a single model schedule is derived. The percentage increase in time contributed by each model in the schedule is also measured. The use of these formulations on mixed model lines should benefit production managers in gaining a better understanding of their assembly processes. Potential applications could include the comparison of learning costs on single and mixed model lines, and the selection of models to mixed model lines which tend to minimize the percentage increase in assembly time. The mixed model learning curve, developed in this article, predicts the time required to assemble the rth unit under the assumption that the production up to the rth unit contains the same relative proportions of models as that designated for the total production run. I n this sense, the curve can be viewed as the expected value of assembly times for mixed model lines. REFERENCES (1) AWDRESS, F. J., " T h e Learning Curve as a Production Tool," The Harvard Business Review, JanuaryFebruary, 1954. (2) BALOFF,N., "Estimating the Parameters of t h e StartUp ModelAn Empirical Approach," The Journal of Industrial Engineering, April, 1967. CONWAY,R. W., A N D SCHULTZ, A., " T h e ManufacturingProgress Function," The Journal of Industrial Engineering, JanuaryFebruary, 1959. KILBRIDGE, M . D., " A Model for Industrial Learning Costs," Management Science, Volume 8, No. 4, July, 1962. NADLER,G., A N D S M I T H ,W . D., "Manufacturing Progress Functions for T y p e s o f Processes," International Journal of Production Research, June, 1963. THOHOPOULOS, N . T., "Line BalancingSequencing for MixedModel Assembly," Management Science, Volume 14, No. 2, October, 1967. THOMOPOULO N ~. , T., "Some Analytical Approaches t o Assembly Line Problems," The Production Engineer, July, 1968. (8) W R I G H T T , . P., "Factors Affecting the Cost of Airplanes," Journal of Aeronautical Sciences, February, 1936. S. L., "Misapplications o f the Learning Curve Con(9) YOUNG, cept," The Journal of Industrial Engineering, August, 1966. Dr. Thomopoulos i s a n associate professor in the Industrial Engineering Department at the Illinois Institute of Technology where he i s currently concerned with quantitative methods in production and statistical and probability theory. Before joining the I I T industrial engineering faculty, he was a senior scientist in the Computer Science Division of the I I T Research Institute and a supervisor of operations research for the International Harvester Company. Dr. Thomopoulos holds B S and A l A degrees from the U n i versity of Illinois, and earned his P h D at I I T . H e i s a member of O R S A , T I M S , Alpha P i M u , T a u Beta P i , P i M u Epsilon, and Sigma X i . Dr. Lehman i s a n assistant professor in the Department of Systems Engineering at the University of Illinois where h i s research activities are centered around modeling of production systems, especially in the area of product assembly. Formerly, he wa,s a research engineer in the Operations Research Group at the I I T Research Institute and manager of methods engineering for Shure Brothers, Inc. Dr. Lehman received B S and M S degrees and a P h D in industrial engineering from the Illinois Institute of Technology. H e i s a member of O R S A and Sigma X i , and i s presently responsible for industrial engineering curm'culum development at the University of Illinois at Chicago Circle. AIIE Transactions