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Economic lot sizing with learning and continuous time discounting: Is it significant?
Int. J. Production Economics 71 (2001) 135}143
Economic lot sizing with learning and continuous time discounting: Is it signi"cant? Mohamad Y. Jaber *, Maurice Bonney School of Industrial Engineering, Department of Mechanical Engineering, Ryerson Polytechnic University, Toronto, ON, Canada M5B 2K3 School of Mechanical, Materials, Manufacturing Engineering and Management, University of Nottingham, University Park, Nottingham NG7 2RD, UK
Abstract Learning curves are a means of representing continuous improvement in "rms. Such improvements bring savings in production costs. This may also allow smaller batches to be produced more frequently and hence bring further savings in holding costs. Earlier research advocated that for more realistic modelling of inventory problems, the holding cost should be evaluated means of the internal discount rate of the "rm (C. Van Deft, J.P. Vail, International Journal of Production Economics 44 (1996) 255}265). This paper examines whether, when learning is considered, it is reasonable to ignore the e!ect of continuous time discounting of costs by investigating the e!ect of learning and time discounting both on the economic manufacturing quantity and minimum total inventory cost. Numerical examples are provided to illustrate the solution procedure for the mathematical model developed. Although the analysis yields di!erent economic order quantities, the di!erence in cost from the quantities derived using Wilson lot size formula is not signi"cant. 2001 Elsevier Science B.V. All rights reserved. Keywords: Learning; EMQ/EOQ; Continuous time discounting; Intermittent production
1. Introduction The use of the learning curve has been receiving increasing attention due to its applications in di!erent operations management areas; e.g. inventory management. In general, learning is an important consideration whenever an operator begins production of a new product, changes to a new machine or restarts production after some delay.
* Corresponding author. Tel.: 416-979-5000-Ext. 7623; fax: 416-979-5265. E-mail address: [email protected] (M.Y. Jaber).
This implies that the time/cost needed to produce a product will reduce as the individual, or group of individuals, becomes more pro"cient. The economic manufacturing quantity model with learning in production has been treated in Refs. [1}15]. These researches concluded that in the presence of learning in production, the optimal lot size policies were to produce more lots of smaller sizes, and that resulted in substantial savings in total inventory costs. Traditional inventory models [16,17] do not account for the time-value of money. The e!ects of discounting may be considered when determining the economic manufacturing quantity [18}23].
0925-5273/01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 5 - 5 2 7 3 ( 0 0 ) 0 0 1 1 3 - 4
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The basis of this research stems from the principles discussed above. First, learning has economic implications for the design of inventory systems due to the continuous improvement in production capacity. Secondly, as advocated in [18}23], discounting e!ects should be considered for the realistic modelling of the economic lot size problem. However, none of these Refs. [1}23] studied the combined e!ect of learning in production and continuous time discounting on the economic manufacturing quantity. That is the focus of this paper. The rest of the paper is organised as follows: Section 2 presents an introduction to learning curve theory, Section 3 describes the mathematical model that is used, Section 4 discusses the numerical results drawn from the mathematical model developed in Section 3, and "nally, Section 5 presents a summary and conclusions.
Fig. 1. Wright's learning curve.
2. Learning curve theory Early investigations of learning revealed that the time required to perform a task declined as experience with the task increased. The "rst attempt made to formulate relations between learning variables in quantitative form, by Wright [24], resulted in the theory of the `learning curvea. Wright's power function formulation of the `Aircraft Learning Curvea, known to some as Wright's model of progress, can be represented as ¹ "¹ j\@, H
(1)
where ¹ is the time to produce the jth unit, ¹ is H the time to produce the "rst unit (note that the initial production rate is p "1/¹ ), j is the pro duction count, and b is the slope of the learning curve when represented on a logarithmic scale. Wright's simple mathematical expression is easy to implement and understand by managers. Fig. 1 illustrates Wright's learning curve.
3. The mathematical model Consider an inventory process where an item is produced in batches, at an increasing production
Fig. 2. Variation in inventory level with learning.
rate because of learning, and consumed at a constant rate of r units per time period (e.g., units per year). In any production cycle i, de"ne t as the time to produce q units and build a G maximum inventory of Z units, and t as the G time required to deplete Z . The level of inventory G can then be expressed as a function of time, (t), G as (t)"q (t)!rt G G (t)"!rt#rt G
for 0)t)t , for t )t)t ,
(2) (3)
where t is the cycle time, i.e. the time for produ cing and consuming the q units, and equals the G sum of t and t . Fig. 2 illustrates the hypothesised variation in the inventory level given in Eqs. (2) and (3) over the cycle time t . As agreed by many researchers [2,10}12], a good approximation of the
M.Y. Jaber, M. Bonney / Int. J. Production Economics 71 (2001) 135}143
137
Fig. 3. Cash #ow for the elements of production cost in cycle i.
cumulative time to produce q units in cycle i, t , is G determined from Eq. (1) as
OG >QG > ¹ n\@ dn QG > ¹ " [(q #s #1/2)\@!(s #1/2)\@] (4) G G 1!b G
t +
where s represents the number of units produced in G the i!1 previous cycles; i.e., s " G\ q where G H H i*2. Now, de"ne K as the set-up cost per production cycle, h as the inventory carrying cost per unit per unit time, m as the material cost per unit, L as the labour cost per unit time, and as the real-rate of return. All of these are assumed to be of a constant value. In cycle i, the present value of the total cost to produce q units given that s units were producG G ed in the i!1 previous cycles, f (s , q ), is the sum of G G the present values of the total production cost, PC(s , q ), and the total holding cost, HC(s , q ); i.e. G G G G f (s , q )"PC(s , q )#HC(s , q ). Fig. 3 illustrates G G G G G G the cash #ow of production cost elements which are the set-up cost K, total material cost in cycle i, mq , G and the stream of labour cost per unit of q units G to be produced; i.e. , ,2, G where " G G O LG ¸¹ "¸¹ (s #n)\@ is the cost of producing the LG G nth unit in cycle i, 1)n)q . For example, ¹ is G G the time to producing the "rst unit in cycle i given that s units were produced in the i!1 previous G
cycles, where ¹ "¹ (s #1)\@ and the cost is G G "¸¹ . Similarly, ¹ is the time to produce G G G the second unit in cycle i at a cost of "¸¹ . G G Reasoning in the same manner, the stream of labour costs , ,2, and G occur at times ¹ , G G G O ¹ #¹ ,2, and t . Then the present value of G G the discounted stream of total labour costs is given as LC(s , q )" exp(!¹ ) G G G G # exp(!(¹ #¹ ))#2# G exp(!t ) G G G O V
\@2 L\@ OG >QG " ¸¹ x\@ eLQG> VQG > OG >QG >
¸ ¹ x\@ QG > ¹ exp ! (x\@!(s #1/2)\@) dx G 1!b
(5)
¸ ¸ " ! exp !¹
(q #s #1)(s #1/2)@!(s #1)(q #s #1/2)@ G G G G G G , (1!b)(s #1/2)@(q #s #1/2)@ G G G where '0. The present value of all production cost elements in cycle i, PC(s , q ), is the sum of the G G set-up cost K, and the present value of the
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Fig. 4. Cash #ow for the holding cost per unit of time in cycle i.
discounted stream of total labour costs LC(s , q ). G G Then PC(s , q ) is given as G G PC(q )"K#LC(s , q )#mq G G G G ¸ ¸ "K# ! exp !¹
(q #s #1)(s #1/2)@!(s #1)(q #s #1/2)@ G G G G G G (1!b)(s #1/2)@(q #s #1/2)@ G G G #mq . (6) G From Fig. 2, it is possible to construct the cash #ow of holding costs per unit of time for any cycle i given that s units were produced in the i!1 G previous cycles. This is illustrated in Fig. 4. Similarly, as in (6), the present value of the cash #ow of holding costs described in Fig. 4 at beginning of cycle i, is determined from (2)}(4) as
HC(s , q )"h G G
R
R
(q #s #1/2) G exp(!t ) #w G 2 hr h(q #r) G ! # exp(!q /r) G h(2s #1) h(2s #1) G G # exp(!t )! 2 2
R R 1!b \@ "h t#(s #1/2)\@ G ¹ R ;exp(!t) dt!h (rt#s #1/2) G ;exp(!t) dt
hq hq # G exp(!t )! G exp(!q /r), G
O P (!rt#q ) exp(!t) dt. G R G
\@ A\ 1!b #wh wj#(s #1/2)\@ G ¹ H ;exp(!wj)
(!rt#rt ) exp(!t) dt
R 1!b \@ t#(s #1/2)\@ exp(!t) dt G ¹ in (7) is of a closed form that could be approximated using the Trapezoidal Rule. Therefore, HC(s , q ) could be written as G G (s #1/2) HC(s , q )"wh G G G 2 h
(q (t)!rt) exp(!t) dt G
#h
#h
Since b is not integer valued; 0)b(1, the term
(8)
(7)
@ f (a) f (x) dx+w #f (a#w)#f (a#2w) 2 ? f (b) #2#f (a#(c!1)w) . 2
M.Y. Jaber, M. Bonney / Int. J. Production Economics 71 (2001) 135}143
where w"t /c, c is the number of trapezoids used in the approximation and it is of an arbitrary value, and t is determined from (4). Then, the present value of all costs in any cycle igiven that s units G have been produced in the i!1 previous cycles, f (s , q ), is given from (6) and (8) as G G f (s , q )"PC(s , q )#HC(s , q ) G G G G G G ¹ ¸ (q #s #1/2)\@ "K! exp ! G 1!b G
¸ ¹ (s #1/2)\@ # exp ! 1!b G
trated in Fig. 5. Eq. (9a) is valid for continuous discounting rate values di!erent from 1; i.e. '0. For the case of no discounting; i.e. "0, then the present value of all costs in any cycle i given that s units have been produced in the i!1 previous G cycles, f (s , q ), is given from [25] as G G ¹ f (s , q )"K#¸ (q #s #1/2)\@ G G G 1!b G !(s #1/2)\@#mq G G q ¹ [(q #s #1/2)\@!(s #1/2)\@] G G #h G ! G 2r (1!b)(2!b)
(s #1/2) #mq #wh G G 2
139
¹ r(s #1/2)\@ #h G . (1!b)
(9b)
\@ A\ 1!b #wh wj#(s #1/2)\@ G ¹ H (q #s #1/2) G exp(!t ) ;exp(!wj)#w G 2
For the case of continuous time discounting, '0, and no learning, b"0, the present value of all costs, f (s , q )"f (q) since s "0 and q "q G G G G ∀i3[1, n], is given as
hr h(q #r) G ! # exp(!q /r) G
f (q)"K#mq#¸
h(2s #1) h(2s #1) G G # exp(!t )! 2 2 hq hq # G exp(!t )! G exp(!q /r), G
(9a)
which is discounted to the beginning of cycle i; i.e. cycles 1, 2, 3,2, i,2, and n which are distant from time zero, respectively, by s /r"0, s /r"q /r, s /r"(q #q )/r,2, s /r" G\ q /r, G H H 2, and s /r" L\ q /r units of time, as illusL K K
O N O N e\@R dt#h (p!r)te\@R dt O P q #h !r #rt e\@R dt p O N ¸ p!r " K#mq# [1!e\@O N]#h
!h
(p!r)[q#p]e\@O N (p!r) #h
!h
q(p!r)#pr r e\O P#h e\@O N . (9c) p
Fig. 5. Cash #ow of present values of costs for n cycles occurring at beginning of cycles 1, 2, 3,2, i,2, n.
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M.Y. Jaber, M. Bonney / Int. J. Production Economics 71 (2001) 135}143
When there is no continuous time discounting, "0, (9c) reduces to
f (q)"K#mq#¸
#h
O P
O N
O N O N dt#h (p!r)t dt
q !r #rt dt p
q q "K#mq#¸ #h (1!r/p), p 2r
(9d)
where (9d) is the total cost per cycle for the traditional economic manufacturing quantity (EMQ) model. The total present value for the stream of costs shown in Fig. 5, F(q , q ,2, q ) where i3[1, n], is L the sum of the discounted values of f (s , q ) at time G G zero for every i3[1, n]; using (9a)}(9d) when k'0 and (9b) when k"0; and it is given as L F(q , q ,2, q )" f (s , q ) exp(!s /r) L G G G G L "f (s , q )# f (s , q ) G G G G\ ;exp ! q , H r H where 2)i)n.
L q "Q, G G q !q *0, G G> q *1 ∀i3[1, n]. G
(p!r)[q#p]e\@O N (p!r) !h #h
!h ;
q(p!r)#pr r e\O P#h e\@O N p
. 1!e\@O P
(12)
p"lim
1/¹ j\@PR, H (9c) reduces to
¹(q)" K#mq#h (10)
Consider the case of a "nite demand quantity of size Q units to be delivered in n lots; i.e. q #q #2#q "Q. The objective is to mini L mise (10) by "nding the set of optimal lot size quantities whose sum is Q, such that q *q *2*q due to learning e!ects [1}15, , 25]. Therefore, expressed as a non-linear programming problem, this could be written as
¸ p!r ¹(q)" K#mq# [1!e\@O N]#h
Theoretically, from (1), when the production rate approaches large values; i.e.
Minimise F(q , q ,2, q ) L L G\ " f (s , q ) exp ! q G G H r G H subject to:
The constrained non-linear programming problem, Eqs. (11a)}(11d), was solved using EXCEL spreadsheets solver from the tools menu. Numerical examples are presented with results documented and interpreted in Section 4. The total cost per unit of time of the EMQ model with continuous time discounting; i.e. (9c), is given as
(11a)
(11b) (11c) (11d)
O P (q!rt)e\@R dt ; 1!e\@O P
h " K#mq# (q!r#e\@O P) ; . 1!e\@O P (13) Van Delft and Vial [26] approximated the economic order quantity with continuous time discounting, EOQ , that minimises (13) as 2Kr EOQ + , (14) (v#h)
where v is the economic value of the item; v"m. Intuitively, the economic manufacture quantity, EOQ , that minimises (13) is given as 2Kr EMQ " , (15) (v#h)(1!r/p)
where v"m#¸/p. Jaber and Bonney [25] showed that the economic manufacturing quantity, q*, for a learning
M.Y. Jaber, M. Bonney / Int. J. Production Economics 71 (2001) 135}143
rate between 100% and 50%; that is 0)b(1, is bounded as
2Kr , when b"1 h(1!r/p)
(16a)
2Kr , when bP1. h
(16b)
q*" and q*+
Similarly, for cases of learning, 0)b(1, and continuous time discounting, '0, q*, EOQ (q*(EMQ , that minimises (11a) is bounded as
2Kr 2Kr (q*( , (m#h) (m#(¸/p)#h)(1!r/p) (17)
which, alternatively, corresponds to an optimal number of lots, n*, required to deliver a "nite demand quantity of size Q, bounded as n "Q/
EMQ (n*(n "Q/EOQ .
4. Numerical results Consider an inventory process under the e!ects of learning and continuous time discounting with
141
the following parameters: (1) "nite demand quantity, Q, of 1000 units; (2) initial production rate, p , of 1000 units per year, where ¹ "1/p "1/1000 years per unit is the time to produce the "rst unit; (3) consumption rate, r, of 500 units per year; (4) labour cost, L, of 100000 per year; (5) holding cost, h, of 50 per unit per year; (6) set-up cost, K, of 400 per production run; (7) and material cost of 150. The inventory problem described in Eqs. (11a)}(11d) was solved repeatedly for learning rates, R, of 90% (b"0.152), 80% (b"0.322), and 70% (b"0.515), where b"!log R/log 2, and continuous time discounting rates; i.e. of 0%, 5%, 10%, and 15%, with results documented in Table 1. Results in Table 1 show that the economic lot size quantities; qH∀i3(n , n ), for all values of G
learning and continuous time discounting, are bounded by the values of EOQ and EMQ which are consistent with (17). These results indicated that, in the presence of learning, the lot size quantities were more sensitive than costs to changes in values of . For example, when R"90% and increased from 0% to 15%, the optimal lot size quantity and the discounted cost, reduced respectively by 23% (from 100 lot to 1000/13 unit per lot) and 12% (from 199095 to 174579). Results in Table 2 indicated that changing batch size has little e!ect on total cost for any given values of learning and discounted rates. Thus, assuming equal lot
Table 1 Optimal lot size policies for various learning, R, and continuous time discounting, (where n "Q/EMQ , and upper,
n "Q/EOQ , lot size bounds)
R (%)
(%)
F(q , q ,2, q H ) L
n
n
90 90 90 90 80 80 80 80 70 70 70 70
0 5 10 15 0 5 10 15 0 5 10 15
199 095 190 491 182 313 174 579 174 335 166 777 159 616 152 844 164 383 157 201 150 402 143 983
8 9 10 10 8 9 10 10 8 9 10 10
11 13 14 15 11 13 14 15 11 13 14 15
Q"(q , q ,2, q,)"q ;A#q ;B#2#q ;N and n"A#B#2#N. L L
n*
Lots sizes (q , q ,2, q ) L
10 11 12 13 11 12 12 13 11 12 13 13
100 98, 92, 91, 90, 84, 83, 83, 78, 77, 91 , 90 89, 84, 83, 88, 84, 83, 79, 77, 76 91, 90 87, 83 80, 77, 76 79, 77, 76
90, 89 82 76 82 82
142
M.Y. Jaber, M. Bonney / Int. J. Production Economics 71 (2001) 135}143
sizes does not seem unreasonable given the di!erence in cost would be negligible. Another interesting issue is that the model of Van Delft and Vial [26], presented in (14) and (15), seems to be a good approximation for the complex model presented in (11a)}(11d). To do so, an adjustment to the production rate is necessary. Let p , p "(1!b)Q@/¹ , be the average production
F(q , q ,2, q ) L "0%
90 90 90 90 90 90 90 80 80 80 80 80 80 70 70 70 70 70 70
199 303.88 199 150.63 190 662.03 199 095.03 190 529.92 182 460.10 199 148.39 190 490.84 182 347.06 174 696.85 190 521.54 182 313.25 174 604.35 182 340.23 174 578.96 174 606.31 174 470.53 174 338.95 166 872.23 159 832.64 174 334.84 166 785.71 159 681.72 153 007.40 174 395.65 166 776.96 159 616.15 152 890.44 166 828.01 159 616.41 152 844.43 152 854.24 164 571.02 164 410.25 157 331.59 164 382.52 157 225.96 150 490.03 144 162.47 164 423.90 157 201.38 150 412.39 144 036.37 157 239.11 150 402.30 143 983.00 150 445.39 143 986.42
8 9 10 11 12 13 14 9 10 11 12 13 14 9 10 11 12 13 14
125.00 111.11 100.00 90.91 83.33 76.92 71.43 111.11 100.00 90.91 83.33 76.92 71.43 111.11 100.00 90.91 83.33 76.92 71.43
"5%
EMQl "
2Kr
¸¹ #h m# (1!b)Q@
r¹ 1! (1!b)Q@
(18)
where EMQl is the economic manufacturing quantity for learning and continuous time. Note that (18) converges to (14) when either Q or p approaches very large values, and to (15) when there is no learning; i.e., b"0. The numerical examples presented in Table 1 are reworked using Eq. (18) and the results are presented in Table 3. Comparing the results of Table 1 to those of Table 3, the di!erence in negligible. Even though these results seem disappointing because all of the preceding mathematical development ends up not making any `real-worlda di!erence, they tell managers not to worry about including both learning e!ects and discounting when choosing lot sizes.
Table 2 The behaviour of discounted costs for various values of R, , and n.
R n Q/n (%)
rate of Q units. Substituting p in (15) by p , (15) could be written as
"10% "15%
5. Summary and conclusions This paper has studied the e!ects of learning and continuous time discounting on the economic manufacturing quantity model. A new mathematical model has been developed and analysed with numerical results documented. These results indicated that, although discounting and learning a!ect the optimal batch size, suggesting that one should make in smaller batches more frequently, changing the batch size does not greatly a!ect the
Table 3 Optimal lot size policies for the approximate model
0 5 10 15
R"90%
R"80%
n*
EMQl
10 11 12 13
100.00 90.91 83.33 76.92
R"70%
F(Q)
n*
EMQl
199 095.03 190 493.64 182 315.91 174 578.96
11 12 12 13
90.91 83.33 83.33 76.92
F(Q)
n*
EMQl
F(Q)
174 334.84 166 778.86 159 617.34 152 845.34
11 12 13 13
90.91 83.33 76.92 76.92
164 382.52 157 202.35 150 403.11 143 983.34
M.Y. Jaber, M. Bonney / Int. J. Production Economics 71 (2001) 135}143
total discounted cost. As anticipated, although the lot size quantity was found to be highly sensitive to changes in learning and continuous time discounting, costs were much less sensitive. This indicates that although disregarding the continuous time discounting factor when analysing production-inventory systems might result in a non-optimal policy to be implemented, the e!ect on the total cost is small. From a decision-making point of view there is considerable #exibility in the choice of lot size and the use of discounting does not greatly a!ect the total costs. References [1] E.C. Keachie, R.J. Fontana, Production lot sizing under a learning e!ect, Management Science 13 (1966) B102}108. [2] B. Spradlin, D. Pierce, Production scheduling under learning e!ect by dynamic programming, Industrial Engineering 18 (1967) 219}222. [3] I. Steedman, Some improvement curve theory, International Journal of Production Research 8 (1970) 189}205. [4] A.W. Wortham, A.M. Mayyasi, Learning considerations with economic order quantity, AIIE Transactions 4 (1972) 69}71. [5] G.L. Adler, R. Nanda, The e!ects of learning on optimal lot determination } single product case, AIIE Transactions 6 (1974) 14}20. [6] G.L. Adler, R. Nanda, The e!ects of learning on optimal lot size determination } multiple product case, AIIE Transactions 6 (1974) 21}27. [7] D.R. Sule, The e!ect of alternate periods of learning and forgetting on economic manufactured quantity, AIIE Transactions 10 (1978) 338}343. [8] S. AxsaK ter, S. Elmaghraby, A note on EMQ under learning and forgetting, AIIE Transactions 13 (1981) 86}89. [9] D.R. Sule, A note on production time variation in determining EMQ under the in#uence of learning and forgetting, AIIE Transactions 13 (1981) 91}95. [10] J.C. Fisk, D.P. Ballou, Production lot sizing under a learning e!ect, AIIE Transactions 14 (1982) 257}264. [11] T.L. Smunt, T.E. Morton, The e!ect of learning on optimal lot sizes: further developments on the single product case, AIIE Transactions 17 (1985) 33}37.
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[12] T.D. Klastorin, K. Moinzadeh, Production lot sizing under learning e!ects: An e$cient solution technique, AIIE Transactions 21 (1989) 2}7. [13] M.Y. Jaber, M. Bonney, Optimal lot sizing under learning considerations: The bounded learning case, Applied Mathematical Modelling 20 (1996) 750}755. [14] S.E. Elmaghraby, Economic manufacturing quantities under conditions of learning and forgetting (EMQ/LaF), Production Planning and Control 1 (1990) 196}208. [15] M.Y. Jaber, M. Bonney, The economic manufacture/order quantity (EMQ/EOQ) and the learning curve: past, present, and future, International Journal of Production Economics 59 (1999) 93}102. [16] E.A. Silver, D.F. Pyke, R. Peterson, Inventory Management and Production Planning and Scheduling, John Wiley, New York, 1998. [17] C.D.J. Waters, Inventory Control and Management, John Wiley, Chichester, 1994. [18] C.S. Park, Y.K. Son, The e!ect of discounting on inventory lot sizing models, Engineering Costs & Production Economics 16 (1989) 35}48. [19] R.W. Hall, Price changes and order quantities: Impacts of discount rate and storage costs, IIE Transactions 24 (1992) 104}110. [20] H.-P. Chao, The EOQ model and discounting, European Journal of Operational Research 59 (1992) 434}443. [21] J.L. Higle, Production planning with discounting and stochastic demands, European Journal of Operational Research 50 (1991) 257}265. [22] S. Bose, A. Goswami, K.S. Chaudhuri, An EOQ model for deteriorating items with linear time-dependent demand rate and shortages under in#ation and time discounting, Journal of the Operational Research Society 46 (1995) 771}782. [23] I. Moon, W. Yun, An economic order quantity model with a random planning horizon, Engineering Economist 39 (1993) 77}86. [24] T. Wright, Factors a!ecting the cost of airplanes, Journal of Aeronautical Science 3 (1936) 122}128. [25] M.Y. Jaber, M. Bonney, The e!ects of learning and forgetting on the optimal lot size quantity with intermitent production runs, Production Planning and Control 9 (1998) 20}27. [26] C. Van Delft, J.P. Vail, Discounted costs, obsolesence and planned stockouts with the EOQ formula, International Journal of Production Economics 44 (1996) 255}265.
Economic lot sizing with learning and continuous time discounting: Is it signi"cant? Mohamad Y. Jaber *, Maurice Bonney School of Industrial Engineering, Department of Mechanical Engineering, Ryerson Polytechnic University, Toronto, ON, Canada M5B 2K3 School of Mechanical, Materials, Manufacturing Engineering and Management, University of Nottingham, University Park, Nottingham NG7 2RD, UK
Abstract Learning curves are a means of representing continuous improvement in "rms. Such improvements bring savings in production costs. This may also allow smaller batches to be produced more frequently and hence bring further savings in holding costs. Earlier research advocated that for more realistic modelling of inventory problems, the holding cost should be evaluated means of the internal discount rate of the "rm (C. Van Deft, J.P. Vail, International Journal of Production Economics 44 (1996) 255}265). This paper examines whether, when learning is considered, it is reasonable to ignore the e!ect of continuous time discounting of costs by investigating the e!ect of learning and time discounting both on the economic manufacturing quantity and minimum total inventory cost. Numerical examples are provided to illustrate the solution procedure for the mathematical model developed. Although the analysis yields di!erent economic order quantities, the di!erence in cost from the quantities derived using Wilson lot size formula is not signi"cant. 2001 Elsevier Science B.V. All rights reserved. Keywords: Learning; EMQ/EOQ; Continuous time discounting; Intermittent production
1. Introduction The use of the learning curve has been receiving increasing attention due to its applications in di!erent operations management areas; e.g. inventory management. In general, learning is an important consideration whenever an operator begins production of a new product, changes to a new machine or restarts production after some delay.
* Corresponding author. Tel.: 416-979-5000-Ext. 7623; fax: 416-979-5265. E-mail address: [email protected] (M.Y. Jaber).
This implies that the time/cost needed to produce a product will reduce as the individual, or group of individuals, becomes more pro"cient. The economic manufacturing quantity model with learning in production has been treated in Refs. [1}15]. These researches concluded that in the presence of learning in production, the optimal lot size policies were to produce more lots of smaller sizes, and that resulted in substantial savings in total inventory costs. Traditional inventory models [16,17] do not account for the time-value of money. The e!ects of discounting may be considered when determining the economic manufacturing quantity [18}23].
0925-5273/01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 5 - 5 2 7 3 ( 0 0 ) 0 0 1 1 3 - 4
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The basis of this research stems from the principles discussed above. First, learning has economic implications for the design of inventory systems due to the continuous improvement in production capacity. Secondly, as advocated in [18}23], discounting e!ects should be considered for the realistic modelling of the economic lot size problem. However, none of these Refs. [1}23] studied the combined e!ect of learning in production and continuous time discounting on the economic manufacturing quantity. That is the focus of this paper. The rest of the paper is organised as follows: Section 2 presents an introduction to learning curve theory, Section 3 describes the mathematical model that is used, Section 4 discusses the numerical results drawn from the mathematical model developed in Section 3, and "nally, Section 5 presents a summary and conclusions.
Fig. 1. Wright's learning curve.
2. Learning curve theory Early investigations of learning revealed that the time required to perform a task declined as experience with the task increased. The "rst attempt made to formulate relations between learning variables in quantitative form, by Wright [24], resulted in the theory of the `learning curvea. Wright's power function formulation of the `Aircraft Learning Curvea, known to some as Wright's model of progress, can be represented as ¹ "¹ j\@, H
(1)
where ¹ is the time to produce the jth unit, ¹ is H the time to produce the "rst unit (note that the initial production rate is p "1/¹ ), j is the pro duction count, and b is the slope of the learning curve when represented on a logarithmic scale. Wright's simple mathematical expression is easy to implement and understand by managers. Fig. 1 illustrates Wright's learning curve.
3. The mathematical model Consider an inventory process where an item is produced in batches, at an increasing production
Fig. 2. Variation in inventory level with learning.
rate because of learning, and consumed at a constant rate of r units per time period (e.g., units per year). In any production cycle i, de"ne t as the time to produce q units and build a G maximum inventory of Z units, and t as the G time required to deplete Z . The level of inventory G can then be expressed as a function of time, (t), G as (t)"q (t)!rt G G (t)"!rt#rt G
for 0)t)t , for t )t)t ,
(2) (3)
where t is the cycle time, i.e. the time for produ cing and consuming the q units, and equals the G sum of t and t . Fig. 2 illustrates the hypothesised variation in the inventory level given in Eqs. (2) and (3) over the cycle time t . As agreed by many researchers [2,10}12], a good approximation of the
M.Y. Jaber, M. Bonney / Int. J. Production Economics 71 (2001) 135}143
137
Fig. 3. Cash #ow for the elements of production cost in cycle i.
cumulative time to produce q units in cycle i, t , is G determined from Eq. (1) as
OG >QG > ¹ n\@ dn QG > ¹ " [(q #s #1/2)\@!(s #1/2)\@] (4) G G 1!b G
t +
where s represents the number of units produced in G the i!1 previous cycles; i.e., s " G\ q where G H H i*2. Now, de"ne K as the set-up cost per production cycle, h as the inventory carrying cost per unit per unit time, m as the material cost per unit, L as the labour cost per unit time, and as the real-rate of return. All of these are assumed to be of a constant value. In cycle i, the present value of the total cost to produce q units given that s units were producG G ed in the i!1 previous cycles, f (s , q ), is the sum of G G the present values of the total production cost, PC(s , q ), and the total holding cost, HC(s , q ); i.e. G G G G f (s , q )"PC(s , q )#HC(s , q ). Fig. 3 illustrates G G G G G G the cash #ow of production cost elements which are the set-up cost K, total material cost in cycle i, mq , G and the stream of labour cost per unit of q units G to be produced; i.e. , ,2, G where " G G O LG ¸¹ "¸¹ (s #n)\@ is the cost of producing the LG G nth unit in cycle i, 1)n)q . For example, ¹ is G G the time to producing the "rst unit in cycle i given that s units were produced in the i!1 previous G
cycles, where ¹ "¹ (s #1)\@ and the cost is G G "¸¹ . Similarly, ¹ is the time to produce G G G the second unit in cycle i at a cost of "¸¹ . G G Reasoning in the same manner, the stream of labour costs , ,2, and G occur at times ¹ , G G G O ¹ #¹ ,2, and t . Then the present value of G G the discounted stream of total labour costs is given as LC(s , q )" exp(!¹ ) G G G G # exp(!(¹ #¹ ))#2# G exp(!t ) G G G O V
\@2 L\@ OG >QG " ¸¹ x\@ eLQG> VQG > OG >QG >
¸ ¹ x\@ QG > ¹ exp ! (x\@!(s #1/2)\@) dx G 1!b
(5)
¸ ¸ " ! exp !¹
(q #s #1)(s #1/2)@!(s #1)(q #s #1/2)@ G G G G G G , (1!b)(s #1/2)@(q #s #1/2)@ G G G where '0. The present value of all production cost elements in cycle i, PC(s , q ), is the sum of the G G set-up cost K, and the present value of the
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Fig. 4. Cash #ow for the holding cost per unit of time in cycle i.
discounted stream of total labour costs LC(s , q ). G G Then PC(s , q ) is given as G G PC(q )"K#LC(s , q )#mq G G G G ¸ ¸ "K# ! exp !¹
(q #s #1)(s #1/2)@!(s #1)(q #s #1/2)@ G G G G G G (1!b)(s #1/2)@(q #s #1/2)@ G G G #mq . (6) G From Fig. 2, it is possible to construct the cash #ow of holding costs per unit of time for any cycle i given that s units were produced in the i!1 G previous cycles. This is illustrated in Fig. 4. Similarly, as in (6), the present value of the cash #ow of holding costs described in Fig. 4 at beginning of cycle i, is determined from (2)}(4) as
HC(s , q )"h G G
R
R
(q #s #1/2) G exp(!t ) #w G 2 hr h(q #r) G ! # exp(!q /r) G h(2s #1) h(2s #1) G G # exp(!t )! 2 2
R R 1!b \@ "h t#(s #1/2)\@ G ¹ R ;exp(!t) dt!h (rt#s #1/2) G ;exp(!t) dt
hq hq # G exp(!t )! G exp(!q /r), G
O P (!rt#q ) exp(!t) dt. G R G
\@ A\ 1!b #wh wj#(s #1/2)\@ G ¹ H ;exp(!wj)
(!rt#rt ) exp(!t) dt
R 1!b \@ t#(s #1/2)\@ exp(!t) dt G ¹ in (7) is of a closed form that could be approximated using the Trapezoidal Rule. Therefore, HC(s , q ) could be written as G G (s #1/2) HC(s , q )"wh G G G 2 h
(q (t)!rt) exp(!t) dt G
#h
#h
Since b is not integer valued; 0)b(1, the term
(8)
(7)
@ f (a) f (x) dx+w #f (a#w)#f (a#2w) 2 ? f (b) #2#f (a#(c!1)w) . 2
M.Y. Jaber, M. Bonney / Int. J. Production Economics 71 (2001) 135}143
where w"t /c, c is the number of trapezoids used in the approximation and it is of an arbitrary value, and t is determined from (4). Then, the present value of all costs in any cycle igiven that s units G have been produced in the i!1 previous cycles, f (s , q ), is given from (6) and (8) as G G f (s , q )"PC(s , q )#HC(s , q ) G G G G G G ¹ ¸ (q #s #1/2)\@ "K! exp ! G 1!b G
¸ ¹ (s #1/2)\@ # exp ! 1!b G
trated in Fig. 5. Eq. (9a) is valid for continuous discounting rate values di!erent from 1; i.e. '0. For the case of no discounting; i.e. "0, then the present value of all costs in any cycle i given that s units have been produced in the i!1 previous G cycles, f (s , q ), is given from [25] as G G ¹ f (s , q )"K#¸ (q #s #1/2)\@ G G G 1!b G !(s #1/2)\@#mq G G q ¹ [(q #s #1/2)\@!(s #1/2)\@] G G #h G ! G 2r (1!b)(2!b)
(s #1/2) #mq #wh G G 2
139
¹ r(s #1/2)\@ #h G . (1!b)
(9b)
\@ A\ 1!b #wh wj#(s #1/2)\@ G ¹ H (q #s #1/2) G exp(!t ) ;exp(!wj)#w G 2
For the case of continuous time discounting, '0, and no learning, b"0, the present value of all costs, f (s , q )"f (q) since s "0 and q "q G G G G ∀i3[1, n], is given as
hr h(q #r) G ! # exp(!q /r) G
f (q)"K#mq#¸
h(2s #1) h(2s #1) G G # exp(!t )! 2 2 hq hq # G exp(!t )! G exp(!q /r), G
(9a)
which is discounted to the beginning of cycle i; i.e. cycles 1, 2, 3,2, i,2, and n which are distant from time zero, respectively, by s /r"0, s /r"q /r, s /r"(q #q )/r,2, s /r" G\ q /r, G H H 2, and s /r" L\ q /r units of time, as illusL K K
O N O N e\@R dt#h (p!r)te\@R dt O P q #h !r #rt e\@R dt p O N ¸ p!r " K#mq# [1!e\@O N]#h
!h
(p!r)[q#p]e\@O N (p!r) #h
!h
q(p!r)#pr r e\O P#h e\@O N . (9c) p
Fig. 5. Cash #ow of present values of costs for n cycles occurring at beginning of cycles 1, 2, 3,2, i,2, n.
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When there is no continuous time discounting, "0, (9c) reduces to
f (q)"K#mq#¸
#h
O P
O N
O N O N dt#h (p!r)t dt
q !r #rt dt p
q q "K#mq#¸ #h (1!r/p), p 2r
(9d)
where (9d) is the total cost per cycle for the traditional economic manufacturing quantity (EMQ) model. The total present value for the stream of costs shown in Fig. 5, F(q , q ,2, q ) where i3[1, n], is L the sum of the discounted values of f (s , q ) at time G G zero for every i3[1, n]; using (9a)}(9d) when k'0 and (9b) when k"0; and it is given as L F(q , q ,2, q )" f (s , q ) exp(!s /r) L G G G G L "f (s , q )# f (s , q ) G G G G\ ;exp ! q , H r H where 2)i)n.
L q "Q, G G q !q *0, G G> q *1 ∀i3[1, n]. G
(p!r)[q#p]e\@O N (p!r) !h #h
!h ;
q(p!r)#pr r e\O P#h e\@O N p
. 1!e\@O P
(12)
p"lim
1/¹ j\@PR, H (9c) reduces to
¹(q)" K#mq#h (10)
Consider the case of a "nite demand quantity of size Q units to be delivered in n lots; i.e. q #q #2#q "Q. The objective is to mini L mise (10) by "nding the set of optimal lot size quantities whose sum is Q, such that q *q *2*q due to learning e!ects [1}15, , 25]. Therefore, expressed as a non-linear programming problem, this could be written as
¸ p!r ¹(q)" K#mq# [1!e\@O N]#h
Theoretically, from (1), when the production rate approaches large values; i.e.
Minimise F(q , q ,2, q ) L L G\ " f (s , q ) exp ! q G G H r G H subject to:
The constrained non-linear programming problem, Eqs. (11a)}(11d), was solved using EXCEL spreadsheets solver from the tools menu. Numerical examples are presented with results documented and interpreted in Section 4. The total cost per unit of time of the EMQ model with continuous time discounting; i.e. (9c), is given as
(11a)
(11b) (11c) (11d)
O P (q!rt)e\@R dt ; 1!e\@O P
h " K#mq# (q!r#e\@O P) ; . 1!e\@O P (13) Van Delft and Vial [26] approximated the economic order quantity with continuous time discounting, EOQ , that minimises (13) as 2Kr EOQ + , (14) (v#h)
where v is the economic value of the item; v"m. Intuitively, the economic manufacture quantity, EOQ , that minimises (13) is given as 2Kr EMQ " , (15) (v#h)(1!r/p)
where v"m#¸/p. Jaber and Bonney [25] showed that the economic manufacturing quantity, q*, for a learning
M.Y. Jaber, M. Bonney / Int. J. Production Economics 71 (2001) 135}143
rate between 100% and 50%; that is 0)b(1, is bounded as
2Kr , when b"1 h(1!r/p)
(16a)
2Kr , when bP1. h
(16b)
q*" and q*+
Similarly, for cases of learning, 0)b(1, and continuous time discounting, '0, q*, EOQ (q*(EMQ , that minimises (11a) is bounded as
2Kr 2Kr (q*( , (m#h) (m#(¸/p)#h)(1!r/p) (17)
which, alternatively, corresponds to an optimal number of lots, n*, required to deliver a "nite demand quantity of size Q, bounded as n "Q/
EMQ (n*(n "Q/EOQ .
4. Numerical results Consider an inventory process under the e!ects of learning and continuous time discounting with
141
the following parameters: (1) "nite demand quantity, Q, of 1000 units; (2) initial production rate, p , of 1000 units per year, where ¹ "1/p "1/1000 years per unit is the time to produce the "rst unit; (3) consumption rate, r, of 500 units per year; (4) labour cost, L, of 100000 per year; (5) holding cost, h, of 50 per unit per year; (6) set-up cost, K, of 400 per production run; (7) and material cost of 150. The inventory problem described in Eqs. (11a)}(11d) was solved repeatedly for learning rates, R, of 90% (b"0.152), 80% (b"0.322), and 70% (b"0.515), where b"!log R/log 2, and continuous time discounting rates; i.e. of 0%, 5%, 10%, and 15%, with results documented in Table 1. Results in Table 1 show that the economic lot size quantities; qH∀i3(n , n ), for all values of G
learning and continuous time discounting, are bounded by the values of EOQ and EMQ which are consistent with (17). These results indicated that, in the presence of learning, the lot size quantities were more sensitive than costs to changes in values of . For example, when R"90% and increased from 0% to 15%, the optimal lot size quantity and the discounted cost, reduced respectively by 23% (from 100 lot to 1000/13 unit per lot) and 12% (from 199095 to 174579). Results in Table 2 indicated that changing batch size has little e!ect on total cost for any given values of learning and discounted rates. Thus, assuming equal lot
Table 1 Optimal lot size policies for various learning, R, and continuous time discounting, (where n "Q/EMQ , and upper,
n "Q/EOQ , lot size bounds)
R (%)
(%)
F(q , q ,2, q H ) L
n
n
90 90 90 90 80 80 80 80 70 70 70 70
0 5 10 15 0 5 10 15 0 5 10 15
199 095 190 491 182 313 174 579 174 335 166 777 159 616 152 844 164 383 157 201 150 402 143 983
8 9 10 10 8 9 10 10 8 9 10 10
11 13 14 15 11 13 14 15 11 13 14 15
Q"(q , q ,2, q,)"q ;A#q ;B#2#q ;N and n"A#B#2#N. L L
n*
Lots sizes (q , q ,2, q ) L
10 11 12 13 11 12 12 13 11 12 13 13
100 98, 92, 91, 90, 84, 83, 83, 78, 77, 91 , 90 89, 84, 83, 88, 84, 83, 79, 77, 76 91, 90 87, 83 80, 77, 76 79, 77, 76
90, 89 82 76 82 82
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sizes does not seem unreasonable given the di!erence in cost would be negligible. Another interesting issue is that the model of Van Delft and Vial [26], presented in (14) and (15), seems to be a good approximation for the complex model presented in (11a)}(11d). To do so, an adjustment to the production rate is necessary. Let p , p "(1!b)Q@/¹ , be the average production
F(q , q ,2, q ) L "0%
90 90 90 90 90 90 90 80 80 80 80 80 80 70 70 70 70 70 70
199 303.88 199 150.63 190 662.03 199 095.03 190 529.92 182 460.10 199 148.39 190 490.84 182 347.06 174 696.85 190 521.54 182 313.25 174 604.35 182 340.23 174 578.96 174 606.31 174 470.53 174 338.95 166 872.23 159 832.64 174 334.84 166 785.71 159 681.72 153 007.40 174 395.65 166 776.96 159 616.15 152 890.44 166 828.01 159 616.41 152 844.43 152 854.24 164 571.02 164 410.25 157 331.59 164 382.52 157 225.96 150 490.03 144 162.47 164 423.90 157 201.38 150 412.39 144 036.37 157 239.11 150 402.30 143 983.00 150 445.39 143 986.42
8 9 10 11 12 13 14 9 10 11 12 13 14 9 10 11 12 13 14
125.00 111.11 100.00 90.91 83.33 76.92 71.43 111.11 100.00 90.91 83.33 76.92 71.43 111.11 100.00 90.91 83.33 76.92 71.43
"5%
EMQl "
2Kr
¸¹ #h m# (1!b)Q@
r¹ 1! (1!b)Q@
(18)
where EMQl is the economic manufacturing quantity for learning and continuous time. Note that (18) converges to (14) when either Q or p approaches very large values, and to (15) when there is no learning; i.e., b"0. The numerical examples presented in Table 1 are reworked using Eq. (18) and the results are presented in Table 3. Comparing the results of Table 1 to those of Table 3, the di!erence in negligible. Even though these results seem disappointing because all of the preceding mathematical development ends up not making any `real-worlda di!erence, they tell managers not to worry about including both learning e!ects and discounting when choosing lot sizes.
Table 2 The behaviour of discounted costs for various values of R, , and n.
R n Q/n (%)
rate of Q units. Substituting p in (15) by p , (15) could be written as
"10% "15%
5. Summary and conclusions This paper has studied the e!ects of learning and continuous time discounting on the economic manufacturing quantity model. A new mathematical model has been developed and analysed with numerical results documented. These results indicated that, although discounting and learning a!ect the optimal batch size, suggesting that one should make in smaller batches more frequently, changing the batch size does not greatly a!ect the
Table 3 Optimal lot size policies for the approximate model
0 5 10 15
R"90%
R"80%
n*
EMQl
10 11 12 13
100.00 90.91 83.33 76.92
R"70%
F(Q)
n*
EMQl
199 095.03 190 493.64 182 315.91 174 578.96
11 12 12 13
90.91 83.33 83.33 76.92
F(Q)
n*
EMQl
F(Q)
174 334.84 166 778.86 159 617.34 152 845.34
11 12 13 13
90.91 83.33 76.92 76.92
164 382.52 157 202.35 150 403.11 143 983.34
M.Y. Jaber, M. Bonney / Int. J. Production Economics 71 (2001) 135}143
total discounted cost. As anticipated, although the lot size quantity was found to be highly sensitive to changes in learning and continuous time discounting, costs were much less sensitive. This indicates that although disregarding the continuous time discounting factor when analysing production-inventory systems might result in a non-optimal policy to be implemented, the e!ect on the total cost is small. From a decision-making point of view there is considerable #exibility in the choice of lot size and the use of discounting does not greatly a!ect the total costs. References [1] E.C. Keachie, R.J. Fontana, Production lot sizing under a learning e!ect, Management Science 13 (1966) B102}108. [2] B. Spradlin, D. Pierce, Production scheduling under learning e!ect by dynamic programming, Industrial Engineering 18 (1967) 219}222. [3] I. Steedman, Some improvement curve theory, International Journal of Production Research 8 (1970) 189}205. [4] A.W. Wortham, A.M. Mayyasi, Learning considerations with economic order quantity, AIIE Transactions 4 (1972) 69}71. [5] G.L. Adler, R. Nanda, The e!ects of learning on optimal lot determination } single product case, AIIE Transactions 6 (1974) 14}20. [6] G.L. Adler, R. Nanda, The e!ects of learning on optimal lot size determination } multiple product case, AIIE Transactions 6 (1974) 21}27. [7] D.R. Sule, The e!ect of alternate periods of learning and forgetting on economic manufactured quantity, AIIE Transactions 10 (1978) 338}343. [8] S. AxsaK ter, S. Elmaghraby, A note on EMQ under learning and forgetting, AIIE Transactions 13 (1981) 86}89. [9] D.R. Sule, A note on production time variation in determining EMQ under the in#uence of learning and forgetting, AIIE Transactions 13 (1981) 91}95. [10] J.C. Fisk, D.P. Ballou, Production lot sizing under a learning e!ect, AIIE Transactions 14 (1982) 257}264. [11] T.L. Smunt, T.E. Morton, The e!ect of learning on optimal lot sizes: further developments on the single product case, AIIE Transactions 17 (1985) 33}37.
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