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A I I E Transactions Production Lot Sizing Under A Learning Effect
Production Lot Sizing Under A Learning Effect
Fisk, John c., Ballou, Donald P.Quanto ti piace questo libro?
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Volume:
14
Anno:
1982
Lingua:
english
Pagine:
8
DOI:
10.1080/05695558208975238
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PDF, 525 KB
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This article was downloaded by: [McGill University Library] On: 27 August 2012, At: 07:37 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 Production Lot Sizing Under A Learning Effect a John c. Fisk & Donald P. Ballou a a School of Business, State University of New York at Albany, 1400 Washington Avenue, Albany, New York, 12222 Version of record first published: 09 Jul 2007 To cite this article: John c. Fisk & Donald P. Ballou (1982): Production Lot Sizing Under A Learning Effect, A I I E Transactions, 14:4, 257264 To link to this article: http://dx.doi.org/10.1080/05695558208975238 PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: http://www.tandfonline.com/page/termsandconditions 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. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae, and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand, or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material. Production Lot Sizing Under A Learning Effect JOHN C. FISK* Downloaded by [McGill University Library] at 07:37 27 August 2012 DONALD P. BALLOU School of Business State University of New York at Albany 1400 Washington Avenue Albany, New York 12222 Abstract. Models are developed for determin; ing optimal production lot sizes under a learning effect. These models consider situations in which both bounded and unbounded learning is assumed to occur. Bounded learning occurs when some nonzero lower limit is set on the time allowed to produce one unit of product. Commonly available manufacturing lotsizing procedures assume that the time to produce successive units of a product is always constant. This assumption often oversimplifies the true situation, since machine operators usually learn their skills while productively operating their machines. As they become more proficient, their perunit output increases due to the learning effect [16] . Even when experienced operators are involved, "learning" often occurs when they must machine a new or unfamiliar job [4]. Finally, operators often must "relearn" or refamiliarize themselves with jobs that are run only intermittently. Hirsch [7] indicates that operators sometimes require large amounts of time to produce each of the first several units of a lot, after which machining time per unit then drops substantially and levels off. Figure 1 shows the results of his experience with a machine center involved in the fabrication and assembly of textile machines. In this paper we present models for solving the manufacturing lotsizing problem under two different learning situations. The first situation assumes that learning conforms to the power function formulation introduced by Wright [17] . This formulation allows the time to produce a Received December 1981; revised April 1982 and August 1982. Paper was handled by the Manufacturing, Inventory, and Automated Production Department. December 1982, IIE TRANSACTIONS N U M B E R OF UNITS IN LOT '~rorn Hirsch [7,P. 1461 Fig. 1 . Total direct laborlot size relation.' unit of product to approach zero as the number of units produced approaches infinity. The power function formulation can be represented as: * On leave at the Institute pour 1'Etude des Mdthodes de Direction de I'Enterprise, Lausanne, Switzerland. 0569554/82/12000257/$2.00/0 0 1 9 8 2 IIE 257 where i = production count. y i = labor cost to produce unit i. a = labor cost to produce the first unit. b = a constant corresponding to the rate of reduc Nanda [ I , 21 analyze the effects of learning on various production lotsize models for the more general situation where the time during a production cycle when production takes place and concomitant carrying costs are significant. lhey assume that inventory levels during the time when production occurs increase linearly. This is characterized by the diagram of inventory level as a function of time shown in Fig. 2. They restrict their analysis to the case where all production lots are of the same size. tion in cost with successive units produced. Downloaded by [McGill University Library] at 07:37 27 August 2012 Similarly, the time in years required to produce item i, ti, is where L = annual cost of labor. in dollars. aT = time to produce the first unit, in years. A second recognized approach to modeling learning is the bounded power function formulation proposed by De Jong [4]. This approach includes both a fixed and a variable component. The fixed component represents the minimum obtainable task time per unit produced, while the variable component is subject to learning. For the bounded power function formulation, the time to produce a unit of product approaches the fixed component time value as the number of units produced approaches infinity. The bounded power function formulation can be represented as TIME Fig. 2. Units on hand over time, linear receipt. Since, under learning, production rate increases with each additional unit produced, a more appropriate representation of inventory level over time is shown in Fig. 3. where all terms are as defined above except PC = fixed component of labor cost. a, = variable labor cost to produce the first unit. The time in years required to product item i, ti,becomes: where P,,=fixed per unit labor component, in years. a, = variable labor time to produce first unit, in years. The unbounded power function formulation has been employed by previous authors addressing the manufacturing lotsizing problem. Both Keachie and Fontana [8] and Spradlin and Pierce [4] simplify their analyses. They assume production rate to be so high relative to demand rate that usage during the time when production is taking place is insignificant in its effects upon holding costs. Adler and 258 Fig. 3. Units on hand over time, nonlinear receipt. In the forementioned woriis, t'ne authors indicate that the optimal lot size Q under conditions of learning is the same as that given by traditional square root formulas so long as regression in learning between successive lots produced is not attendant. Regression in learning refers to the loss of labor efficiency that can occur during the elapsed time between successive production runs of a product. When regression in learning does occur, Spradlin and Pierce show that successive production lots may not be of the same size and outline a dynamic programming procedure for determining the optimal number and size of production lots to produce N units given that s units have already been produced. They assume the product is to be delivered at a constant rate until the entire order is filled. We will present a procedure for solving the problem outlined by Spradlin and Pierce but account for the more general production lot sizing situation outline in Fig. 3. Under these conditions we are able to show that production lot sizes may not be equal, and traditional solution procedures are inappropriate to apply, even when regression in learning is not attendant. While Sule [15] also addresses the IIE TRANSACTIONS, Volume 14, No. 4 Downloaded by [McGill University Library] at 07:37 27 August 2012 production lotsizing problem under conditions similar to those outlined in Fig. 3, his interest is in determining a constant lotsize quantity under steadystate conditions of learning and regression in learning. Following this, we extend the production lotsizing procedure already outlined to account for the bounded power function formulation for learning. Such a procedure would address situations such as those outlined in Fig. I , where a nearly 50% rate of learning is observed over the first several units but almost no learning occurs after 2025 units have been produced. Finally, we provide computational experience using the algorithms developed. The value of f(t), to < t < to + 7,,can be expressed as the difference between the number of units produced, q(t), and the number consumed by demand during time t to. The number of units produced during time t  to depends in part upon s due to the effect of learning and can be determined by carrying out the integration displayed in expression (6). First this term should be divided by the annual cost of labor L to convert the expression from one for labor cost to one for time to produce q additional units, given that s have already been made. Solving the integrated expression for q yields an expression for q(t), the quantity produced during the time to to to + t:  Manufacturing Lot Sizing Assuming the Unbounded Power Education Learning Formulation For Fig. 3, the time during the production cycle in which production is taking place is specified as 7,,while the time during the production cycle in which no production takes place is 72.Both r1 and 72 are specified in years. In order to solve the type of problem outlined in Fig. 3, we begin by specifying the equation for the total cost of producing and carrying q units during the production cycle, given s units have already been produced, C!q, s): (Throughout this paper we assume that b f 1.) Let D represent annual demand rate. Then the average number of units on hand during the production phase froni to to to + r1 is the average difference between units produced and units demanded: 11(1b) s.5  D t I. dt. (9) C(q,s)= set up cost + labor cost to produce q units + carrying costs during time r1 + carrying costs during time 72. (5) The set of costs making up C(q, s) can be determined as Set up costs = S. Labor cost to produce q units given s units already produced. Using expression (1) and assuming no regression in learning between successive lots produced, the labor cost to produce q units, given s units have already been produced,can be estimated by It should be kept in mind that the symbol t i n Equation (8) measures time from the beginning of the production cycle. Multiplying expression (9) by cXT,,we obtain the carrying cost during time rl ,where c represents carrying cost per unit per year. Carrying costs during time 72. Since 7, = q 7 / ~the , average number units on hand during time 72 is where Carrying costs during time 7,. Let f(t) represent the number of units on hand at time t. Assume production starts at time to, when the s units produced during previous production cycles have been removed from stock and production is again about to begin. The average number of units on hand over 7, years of production can be expressed through a limiting argument as = maximum inventory on hand during the production cycle. Since maximum inventory on hand during the current production cycle is the number of units produced, q, less demand during time T I , average inventory on hand during 7, becomes Carrying costs during time 7, are determined by multiplying expression (1 1) by c, carrying cost per unit per year. December 1982, IIE TRANSACTIONS 259 Given the above set of equations, relation (5) outlining the total cost associated with producing and storing a production lot size q becomes Let j = production cycle, j = 1,2, . . . ,n. (12) Once the total cost equation for producing and storing q units given that s units have already been produced is known, a dynamic programming approach can be used to solve for q*, the optimal production lot size. Downloaded by [McGill University Library] at 07:37 27 August 2012 Let f*(s) = the minimum cost associated with producing and storing N  s units given s units have already been produced. qj = number of units to be produced during production cycle j. Cj(qj,S) = the cost of producing and carrying qj units, given that s units have already been produced, in j  1 lots. Using the above definitions, relation (12) is modified to represent the total cost associated with producing and storing a production lot size qj as where j1 s ~ = s f XCq i . i= 1 q*(s) = optimal production lot size given s units have already been produced, (qX(s)<N  s). The dynamic recursion, assuming s is known, is: P ( s ) = Minimum {C(q, s) + f* (s + q ) ) , q <Ns (13) Given the total cost equation above, the problem becomes one of finding the number of production lots, n, and the size of each lot, qj, j = 1 , 2 , . . . ,n such that the total cost of producing N items is minimized. Let E(Q = the cost of an optimal production schedule consisting of X units produced in n lots, evaluated for each X, s < X < N. where f*(N) 0. The recursion is applied for s = {N  1 , N  2 , . . . , 1, 0 ) , the optimal total cost for all N units q,*(X) = the lot size associated with f i ( Q . being fX(0). The optimal production lot size for the first production lot is q*(O). The number and size of the re For a given number of lots n, the dynamic programming maining production lots are easily found; e.g., the second recursion associated with the production of X units, s < production lot is of size q* [q*(O)]. X<N,is The above discussion assumes that no regression in learning occurs. For those situations where regression is expected to exist, we assume that the rate of regression between production runs is related to the cumulative number of units already produced. Spradlin and Pierce where [14] choose to represent the regression in learning occurX  q, = s, number of units previously produced in n1 ring between production cycles j  1 and j as being equal to lots, the amount learned while producing the last r units during and cycle j  1 . We represent regression as being equal to the fO(O) 0 . amount learned while producing the last fraction f of units during cycle j  1. To determine the optimal number of lots, the recursion When regression in learning is operative, the recursive equation is applied iteratively for each value of n beginning relation (13) does not apply. A different dynamic recursion with n = 1, 2 , . . . , until fl+l(N) > (N). The optimal and cost relationship are required that specifically conside~ solution is f;F(N) along with its associated n lot sizes. The the number of production cycles n required to produce N proof of this optimality criterion is outlined in Spradlin units. and Pierce [14,p. 2221 .  260 IIE TRANSACTIONS, Volume 14, No. 4 Downloaded by [McGill University Library] at 07:37 27 August 2012 Production Lot Sizing with a Bounded Production Rate The previous section presented a procedure that accommodated a learning curve and the possibility of regression in learning but used the power function formulation asymptotic to the abscissa. If the production situation is such that learning takes place more rapidly than implied by the traditional power function, or if the learning function is similar to that shown in Fig. 1, then the learning curve should be bounded away from zero. For this, expression (4) ti = P, t a y i b ,is appropriate. The methodology for addressing this situation parallels that found in the previous section. Although we shall explicitly handle only the case s, = s, the changes required for the more general case are completely straightforward. The expression (12) for the cost of producing q units, given that s have already been made, is easily modified to include P, ,the fixed component: s+4+.5 C(q,s) = S + L has been calculated. To obtain this quantity we used numerical integration, in particular Simpson's Rule, chosen because Q(t) satisfies Q" > 0. Computational Experience The dynamic programming recursions outlined in the previous two sections were programmed in FORTRAN IV and tested on a UNIVAC 1110 computer using a series of representative problems. For each problem solved, the following set of problem parameters was assumed: N = 100 =number of units ordered. D = 300 unitslyear = annual rate of demand. s+a+.5 [Q(t)  Dtldt +  [ q  ~ / (P, t a , ib)di 20 ~+.5 Here Q(t) represents the number of units manufactured in time t , 0 < t < 71, with time measured from the start of the current production cycle. To find the expression for Q, it is necessary to solve for (T as a function of t. Carrying out the integration of (1 7) yields an expression that cannot be manipulated to produce (T as a function of t in closed algebraic form. However, the inverse function (T = Q(t) exists globally, a consequence of dr/d(T > 0 for all (T > 0. Thus it is necessary to resort to numerical techniques to find ij for a given value of t. To do this, let From Equation (17) it is necessary to find a (T such that F((T) = 0. But from a global version of the NewtonRaphson method (see [5] ,p. 79), it follows that the algorithm converges quadratically to the solution of F(4) = 0 for any choice of q, > 0. Thus the value for Q(t) can be obtained for each choice o f t . The cost expression (1 6) is known once 1' . (16) c = $700/unit/year = annual carrying cost per unit. L = $1,000,000 = annual cost of labor. A series of 13 problems was solved for the production lot sizing problem with unbounded learning rate and are included in Table 1. All problems were generated for an 85% learning curve (b = .234) and different value combinations of parameters a, S, a n d 6 The dynamic recursion (13) is referenced as algorithm I in Table 1, while the dynamic recursion (1 5) is referenced as algorithm 11. The ratio aD/L for each problem in Table 1 indicates the fraction of annual production capacity to be allocated for production of D units of product, assuming labor cost per unit produced is in dollars. As a result of the learning effect this ratio is only an upper bound on the portion of annual capacity required, though it is useful in highhghting the relative portion of annual production capacity required for a chosen value of a. Problems one through three, for example, require only a small fraction of annual production capacity so that their diagrams of inventory level as a function of time would approximate that of batch receipt. For these problems, the traditional lotsizing formula for lot size determination assuming batch receipt, (2DS/c) %, would be expected to closely approximate the optimal production lot size. For problem two in Table 1, the traditional lotsizing formula yields a solution value of 18.5 versus lot sizes of 20 using either algorithm I or 11. Problems one and three compare in a similar way. Unlike problems one through three, problems four through nine in Table 1 indicate situations where significant portions of annual production capacity must be applied to the product of interest. For these problems, characterized by the diagram of inventory level as a func   December 1982, IIE TRANSACTIONS 261 Table 1 : Computational experience, production lot sizing with unbounding learning. Solution Timea Downloaded by [McGill University Library] at 07:37 27 August 2012 a aD Problem ($) L 1 2 3 4 5 6 7 8 9 10 11 12 13 100 100 100 1500 1500 1500 3000 3000 3000 1500 1500 1500 3000 .03 .03 .03 .45 .45 .45 .90 .90 .90 .45 .45 .45 .90 S f ($) (%) 200 400 600 200 400 600 200 400 600 400 400 400 400 0 0 0 0 0 0 0 0 0 20 40 60 90 Total Cost ($1 7425 8686 9661 68489 69615 70473 133862 134816 135566 71340 73054 74569 138423 Lot Sizes 13,13,13,13,12,12,12,12 20,20,20,20,20 25,25,25,25 16,14,14,14,14,14,14 22,20,20,19,19 27,25,24,24 21 ,I 7.1 6.1 6.15.1 5 30,24,23,23 38,32,30 36.25.20.20 50.25.25 70.30 100 Algorithm I .8 .8 .8 .8 .8 .8 .8 .8 .8 C   Algorithm II 15.9 9.9 7.9 13.9 9.9 7.9 12.0 7.9 5.9 8.0 6.0 4.0 2.1 "AII times in CPU sec. tion of time shown in Fig. 3 , traditional formulas no longer apply since successive lot sizes can vary in size. Since production rate tends to be low during earlier production runs, inventory levels and concomitant inventory carrying costs tend to be low also, leading to larger production lot sizes. Greater production efficiency in later lot sizes leads to smaller production runs due to the increase in the rate of inventory buildup. For problems ten through thirteen where regression in learning occurs, successive lot sizes tend to vary substantially, with the larger lots coming in initial production runs. Large initial runs are to be expected, since this is the time when most of the learning occurs and therefore most learning regression between lots could occur. Large runs tend to preserve the large increases in efficiency observed in producing the first several units of product. Comparing the efficiency of the dynamic programming algorithms I and 11, Table 1 indicates that algorithm I is significantly more efficient for each of the problems one through nine. Problems ten through thirteen consider regression in learning, which only algorithm I1 is designed to evaluate. Algorithm I would therefore be the appropriate algorithm to apply when no regression in learning is apparent, due to its increased efficiency. Algorithm I1 would necessarily be applied with regression in learning. Table 2 includes a series of thirteen problems solved for the production lot sizing problem with bounded learning rate. All problems were generated for different value com Table 2: Computational experience, production lot sizing with bounded learning. ($1 a ($) S ($1 f (%I Total Cost 20 20 20 300 300 300 600 600 600 300 300 300 600 80 80 80 1200 1200 1200 2400 2400 2400 1200 1200 1200 2400 200 400 600 200 400 600 200 400 600 400 400 400 400 0 0 0 0 0 0 0 0 0 20 40 60 90 6470 7733 8710 541 76 55332 5621 7 105265 106309 107109 56394 57614 58998 1 10780 P Problem 1 2 3 4 5 6 7 8 9 10 11 12 13 " ($1 Lot Sizes 13,13,13,13,12,12,12,12 20,20,20,20,20 25.25.25.25 16,14,14,14,14,14,14 21,20,20,20,19 26,25,25,24 20.16.1 6.1 6.16.16 28,24,24,24 28.24.24.24 30.25.24.21 35,25,20,20 50.30.20 100 Solution Timea 34.4 26.2 23.1 37.9 26.1 27.6 40.5 31.9 33.5 31.5 26.5 18.3 6.4 All times in CPU sec. 262 IIE TRANSACTIONS, Volume 14, No. 4 Downloaded by [McGill University Library] at 07:37 27 August 2012 binations of parameters a, P,, S, and f and a 70% learning curve (b = .515). While the dynamic recursion (13) could be used to solve problems of this type as long as no regression in learning is taking place, only results using dynamic recursion (15) are presented in Table 2. Problems one through three require only a small fraction of annual production capacity, while problems four through nine require larger fractions. Problems ten through thirteen again consider regression in learning. The results in Table 2 indicate that, when bounded learning is attendant, relatively equal lot sizes again occur whenever the required fraction of annual capacity necessary to produce each unit of product is small. More disparity between lot sizes occurs as the fraction of required capacity per unit increases. Lot size variation is substantial only when regression in learning is attendant. The classical production lot size formula, which assumes a constant production rate, is d?EqF X ~ R / ( R ) , (19) where K is deiined as the annual production rate. This formula accounts for neither learning nor regression in learning and cannot be used to optimally solve problems of the type described in this paper. Nonetheless, its simplicity suggests that it be compared against the techniques outlined in this paper, hereafter referred to as the learning models, to determine whether it might in certain circumstances be used to approximate the true optimal decision. To accomplish the comparison, we chose problems 2,5,8,10,12, and 13 from both Tables 1 and 2 for further analysis using the production lotsize formula. In the case of the chosen problems in Table 1, the value for R in the production lotsize I formula was set equal to the annual rate signified by a,, i.e., R = lla,. For the chosen problems in Table 2, R was set equal to l/(Py + a,). The value of R assigned above becomes an upper limit on unit production time whenever learning is assumed to occur. In the absence of regression in learning, it then provides a maximum expected production lot size, although some smaller value may yield lot sizes closer to the true optimal, since it would tend to reflect the effect of the learning process. Our choice of R does, however, provide a reasonable standard against which the effectiveness of the learning models can be compared under various situations. These comparisons are summarized in Table 3. Columns (I)  (3) contain labor, setup and carrying, and total costs obtained by applying the production lot size with learning models. Columns (4)  (7) of Table 3 contain the production lot sizes developed from Equation (19) for each problem tested, along with resulting labor, setup and carrying, and total costs that would be incurred. Since only 100 units were to be produced in each case, the calculated production lotsize quantities were adjusted slightly so that demand was exactly satisfied. All cost calculations were based upon these adjusted lotsize figures. Columns (8) (10) indicate the savings which result when learning and regression in learning are considered for the problems tested. Problems 2 in Table 3 (both bounded and unbounded cases) indicate that no cost savings are realized when the learning methods are applied, while, for problems 13, significant savings are possible. For problems 2, production rate is quite high relative to demand rate so that inventory carrying costs incurred during T, , which are nonlinear over I Table 3 : Comparison of lot sizing under learning with EOQ. Unbounded Learning Case Problem I Labor Setup + Carrying Total Savings (EOO  Optimal) EOO Costs Minimum Costs (see Table I ) Labor Setup + Carrying   Labor Total (19120)~ 0 69676 (25125) 0 135674 (58150) 0 67529 71425 (25125) 184 72829 76672 (25125) 4256 150964 (58150) 15636 8686 Setup + Carrying Total (99) Bounded Learning Case Minimum Costs (see Table 11) Problem Labor Setup + Carrying Total  Labor EOO Costs setup + Carrying Total Savings (EOQ  Optimal) Labor setup + Carrying Total I 3425 513 7 9 55332 102758 106309 56394 52340 541 8 8 102758. 481 0 58998 1 10780 4 ~ a l c u l a t e dproduction lot size using Equation (19)lproduction lot size used in subsequent cabculations. December 1982, IIE TRANSACTIONS I 263 Downloaded by [McGill University Library] at 07:37 27 August 2012 time, are negligible (see Fig. 2). Also, problems 2 assume no regression in learning so that labor costs are not reduced by producing in larger lots. Problems 13, on the other hand, represent a nearly opposite situation. Demand and production rates are fairly equal, so time and associated inventory carrying costs are significant. The nonlinearity of these costs over time is accounted for in the learning models, while Equation (19) assumes linearity. Regression in learning is also quite high for problems 13 (90%), and the learning models determine the need to produce in large lots so as to avoid losses in labor efficiency between production runs. It is interesting to note that for problems 13, among others, the learning models indicate the need to absorb very high setup and carrying costs in order to reduce labor cost and to thereby minimize overall costs. As the problems tested in Table 3 indicate, the most important cost savings are realized in the labor cost component when significant regression in learning is expected. Reasonable cost savings can also be realized, however, when no regression in learning is expected but demand rate is high relative to production rate. Problem 8 (unbounded case) in Table 3, for example, shows a savings of $858 in total carrying and setup costs when using the learning models, a 21% savings in these costs when compared to the use of Equation (19). [3] [4] [5] [6] [7] [8] [9] [lo] [Ill 1121 1131 1141 [15] [16] Conclusion Previous researchers dealing with unbounded learning curve methodology in conjunction with production lot sizing have considered only the situation where production receipt is instantaneous or where a steady state condition is assumed to exist between learning and regression in learning. This paper has extended previous work by considering more general situations. In addition, we have presented a procedure for solving the production lotsizing problem with a bounded learning rate. Significant cost savings using these models relative to classical production order quantity models occur whenever demand rate is high relative to production rate, significant regression in learning is expected, or both. The analysis considers only the production of a specified number of units N. Application of the model to an infinite horizon problem would yield satisfactory results since, as cumulative units produced becomes large, the increase in production efficiency approaches zero, and production lot sizes tend to stabilize. The user need only try successively larger values of N until lotsize stabilization occurs. References [I] [2] Adler, G. and Nanda, R., "The Effects of Learning on mal Lot Size DeterminationSingle Product Case," TRANSACTIONS 6,1,1420 (1974). Adler, G. and Nanda, R., "The Effects of Learning on mal Lot Size DeterminationMultiple Product Case," TRANSACTIONS 6,1,2127 (1974). OptiAIIE OptiAIIE 1171 Bump, E., "Effects of Learning on Cost Projections," Management Accounting 1924 (May 1974). De Jong, J. R., "The Effects of Increased Skills on Cycle Time and its Consequences for Time Standards," Economics 1 , 1 , 5161 (1957). Hemici, P., Elements of Numerical Analysis, John Wiley and Sons, N.Y. (1964). Hindmarsh, G. and Towill, D., "Theory and Application of the Time Constant Learning Curve Model," Paper WA 16.14, ORSAITIMS Joint National Meeting, Puerto Rico (1975). Hirsch, W. Z., "Manufacturing Progress Functions," The Review of Economics and Statistics 34, 14355 (1952). Keachie, E. and Fontana, R., "Effects of Learning on Optimal Lot Size," Management Science 13, 102108 (1966). Levy, F., "Adaptation in The Production Process," Management Science 1 1 , B13654 (1965). McIntyre, E., "CostVolumeProfit Analysis Adjusted for Learning," Management Science 24, 2, 149160 (1977). Morse, W., "Reporting Production Costs That Follow the Learning Curve Phenomenon," 7'he Accounting Review 76173 (1972). Morse, W., "The Use of the Learning Curve in Financial Accounting," The CPA 5157 (January 1974). Pegels, C., "Start Up or Learning CurvesSome New Approaches," Decision Science 70513 (October 1976). Spradlin, B. and Pierce, D., "Production Scheduling Under a Learning Effect by Dynamic Programming," The Journal of Industrial Engineering 18,21922 (1967). Sule, D. R., "The Effect of Alternate Periods of Learning and Forgetting on Economic Manufacturing Quantity," AIIE TRANSACTIONS 10, 3, 33843 (1978). Wortham, A. W. and Mayyasi, "Learning Consideration with Economic Order Quantity," AIIE TRANSACTIONS 4 , 1 , 6 9 70 (1972). Wright, T., "Factors Affecting the Cost of Airplanes," Journal o f Aeronautical Science 3, 11 228 (1 936). John Fisk is Associate Professor of Management Science in the School of Business of the State University of New York at Albany, where he teaches operations and distribution management. He has addressed professional associations across the United States, particularly on computer applications to business problems. He has consulted with and presented seminars for IBM, Sears Roebuck and Company, Standard Brands, Inc., and others. He received his BA, MBA, and DBA degrees from Kent State University. Dr. Fisk holds membership in the American Inventory Control Society and the Purchasing Management Association. Donald P. Ballou is Associate Professor of Management Science and Management Information Systems in the School of Business of the State University of New York at Albany. He received a BA in mathematics from Harvard University and a PhD in applied mathematics from the University of Michigan. He is a member of ACM, MAA, SMIS, TIMS, and Sigma Xi. Dr. Ballou's publications include articles in the Transactions o f the AMS, Transportation Research, Journal of Regional Science, and Management Science. IIE TRANSACTIONS, Volume 14, No. 4