- Email: [email protected]
σx value for order-point systems
Int. J. Production Economics 71 (2001) 235}245
Numerical computation of inventory policies, based on the EOQ/ value for order-point systems V Poul Alstr+m* Department of Management Science and Logistics, The Aarhus School of Business, Fuglesangs Alle 4, DK-8210 Aarhus V, Denmark
Abstract This paper examines the numerical computation of two control parameters, order size and order point in the well-known inventory control model, an (s, Q) system with a safety stock strategy. The aim of the paper is to show that the EOQ/ value is both su$cient for controlling the system and essential for the economic consequences of using V approximations in computations of optimal policies. In view of the evidence from a number of studies showing that "rms' use of statistical inventory control lags far behind academic interest in the area, this is also an important aspect. The determination of optimal values for the control variables } even in this very simple inventory control system } is a complex task, and therefore not suited to practical implementation, where one or more of the following issues are neglected: the interdependence of order size and order point, the di!erence between stock on hand and net stock, and the excess stockout included in the next period. We use the determination of optimal values for the control variables as a framework to evaluate the economic consequences of these approximations, applied in pratical inventory control. 2001 Elsevier Science B.V. All rights reserved. Keywords: Stochastic order point models; Exact and heuristic algorithms; EOQ/ : Essential for controlling the system and for using V heuristics
1. Introduction While advances in computer technology have enabled the increasing use of inventory theory in industry in the last few decades, it is still mostly very simple inventory control systems, such as EOQ models, that are used. Many of the commercially available software packages for inventory control employ this concept. Osteryoung et al. [1] "nd that 84% of the "rms surveyed use the EOQ model for inventory control, though this is di$cult to assess accurately. Wisniewski et al. [2], carried
* Tel.: #45-86102488; fax: #45-89486660.
out a survey among private "rms in Scotland, England and Denmark in 1991, 566 of which replied, a response rate of 39%. Of these, 66% were aware of stock control techniques, 75% of which employed them. A more recent study by McLaughlin et al. [3] shows that only 28% of respondents used EOQ for determining production or purchasing quantities. The aim of our paper is to show that the EOQ/ V value is both su$cient for controlling the system and essential for the economic consequences of using approximations in computations of optimal policies in an order point-order quantity system, the so-called (s, Q) inventory system. The theoretical model is a single-product inventory model with transaction reporting and stationary
0925-5273/01/$ - 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 2 2 - 5
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P. Alstr~m / Int. J. Production Economics 71 (2001) 235}245
independent yearly demand, D. When the inventory position fall to the order point, s, the quantity Q is ordered. The order will arrive after a "xed delivery lead time, so that there are no orders cross. Forecast errors have a normal distribution, and the forecast over lead time has a known average, , and stanV dard deviation, . The service strategy is the V so-called strategy [4], where is the fraction of demand satis"ed from the shelf stock. In the event that the demand cannot be met immediately, it will be back-ordered and met soon as possible from subsequent stock supplements. The ordering cost consists of an order cost, A, independent of the order size, and a proportional cost of v per unit. With a carrying charge factor of r, the holding cost per unit per year is rv. Using the results of Hadley and Whitin [5], we can calculate the mean value of the stock on hand and the expected number of back orders in this model, which makes it easy to set up the expression for the expected costs per year. This function will be minimized under the restriction that the service level is maintained. We will then use this cost function as the performance measure for the model and as a framework to evaluate the economic consequences of some approximations use in practical inventory control, namely: E an approximate estimate of physical stock, E an approximate estimate of shortages, E sequential optimization. The analysis results in a simple model for use in practical inventory control.
This sum is minimized in accordance with the service requirement when using the Lagrange multiplier method. The Lagrange function, ¸, is calculated as
where is the Lagrange multiplier and S the total expected number of units back ordered per year. Because unmet demands are back ordered, the expected number of replenishment cycles per year is D N" . Q
The only relevant costs for the chosen system is the sum of holding cost per year and order costs per year. With H as the expected on hand inventory level and N as the expected number of orders placed per year, the expected costs per year can be calculated as C"rvH#AN.
(1)
(3)
The order point, s, consists of the expected lead time demand plus the safety stock. The safety stock, b, is de"ned as the average level of the net stock just before a replenishment arrives, where the net stock is de"ned as on hand stock minus back orders. b is thus determined as
b"
(s!x) f (x) dx, (4) V where f (x) is the normal density function. Apart from the inverse function, E(k)"(z!k) I f (z) dz cf. Appendix A, the spreadsheet program Excel provides all the required values for the normal distribution. It is convenient to write b as the product k , V where k is called the e!ective safety factor. Since b"
" 2. The cost function and exact determination of the control variables
(2)
¸"C#(S!(1!)D),
V
(s!x) f (x) dx
(s!x) f (x) dx!
V "k , V we have
Q
(x!s) f (x) dx
VQ (5)
(s!x) f (x) dx"
(x!s) f (x) dx#k . (6) V V VQ In other words, the expected stock on hand at the end of a replenishment cycle, Q (s!x) f (x) dx, is V equal to the expected value of the backlog at the end of a replenishment cycle, (x!s) f (x) dx, VQ plus the safety stock, k . V
P. Alstr~m / Int. J. Production Economics 71 (2001) 235}245
To obtain the mean level of stock on hand it is therefore necessary to add the mean value of the backlog to the mean value of the net stock. In [5, Section 4}9. pp. 191}195] the calculations of the expected number of backorders at any time are based on the probability function for the number of back orders. For a continuous demand distribution, it is easier to determine the mean value of the backlogged demand graphically. As shown in Appendix B the mean value of the backlog is:
(9)
where
f (z) dz.
(10)
XI
The average number of units on hand, H, is therefore
Q Q H" k # # V J(k)!J k# V 2 2Q V
,
,
(14)
Q Q #k # V J(k)!J k# V 2Q 2 V
D Q
rv
(15)
and a Lagrange function [¸(Q), k, )]
"
Q Q #k # V J(k)!J k# V 2Q 2 V
rv
D D Q V E(k)!E k# #A # Q Q V !(1!)D .
(16)
To reduce the number of separate parameters that have to be considered we normalize by multiplying both sides in Eqs. (15) and (16) by the constant term 1/vr [6] and also by inserting V
2AD EOQ eoq" " rv V V and
(11)
where J(k)!J(k#(q/ ))'0 for all arguments. V In the same way we can determine the expected number of shortages per year as I>/ NV D S" (z!k) f (z). Q V I
(13)
Q D S" E(k)!E k# V Q V
#A
This expression takes account of the fact that the number of shortages in a lead time period in excess of the order size is included in the next order period. As shown by Hadley and Whitin [5, Eqs. (4-84) and (4-85), pp. 193}194] the function, j (k), can be written as
F(k)"
(z!k) f (z) dz, I
the expected number of shortages per year S can be written as
, (7)
(8)
J(k)"(1#k)F(k)!k f (k),
E(k)"
C(Q, k)"
where f (z) is the density function for the standard normal distribution and J(k)" (z!k)f (z) dz. I
Since
which gives total relevant cost per year of
Q>/(x!s) I>/ NV f (x) dx" V (z!k)f (z) dz 2Q 2Q VQ I Q " V J(k)!J k# 2Q V
237
(12)
Q "q. V The normalized expressions for costs and the Lagrange function can now be written as q J(k)!J(k#q) eoq C " #k# # , 2 2q 2q
(17)
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P. Alstr~m / Int. J. Production Economics 71 (2001) 235}245
and J(k)!J(k#q) q ¸ " #k# , 2 2q
eoq # 1# V (E(k)!E(k#q)) 2q A !eoq
V (1!). 2A
(18)
For every value it is necessary to use a search procedure to "nd the values for q, k, and which minimize the ¸ expression. To do this we have , used the SOLVER in the spreadsheet program Excel, where the service request (E(k)!E(k#q))/ q"1!, together with q'0 are set as conditions for the minimization. To verify that SOLVER has found the correct values for q and k we also set and veri"ed the following conditions: ¸ ¸ ¸ ¸ , " , "0, , '0, , '0, k q k q
¸ ¸ ¸ , '0. ,) ,! k q kq
The results for q and k for "0.80, 0.85, 0.90, 0.95, 0.99 and eoq values in the interval 0.1 to 2.0, are shown in Figs. 1 and 2. The use of EOQ as the order size underestimates the optimal value, but it does not necessarily mean that the service level will be lower than the stipulated value, On the contrary. Fig. 2 shows that it can be optimal to operate with a negative safety stock, depending on service level and the itme's eoq values. A high eoq value means a high order size and fewer lead time periods and therefore a lower risk.
3. Heuristic models The search procedure used above to determine optimal values for the system's control variables is unsuitable in practice, where inventory control typically involves a considerable number of products. Instead of "nding the exact values for H and S in the cost and Lagrange functions we will use heuristics to "nd the value of the control variables.
Fig. 1. Optimal q value for di!erent service levels.
Fig. 2. Optimal k value for di!erent service levels.
P. Alstr~m / Int. J. Production Economics 71 (2001) 235}245
3.1. Approximate calculation of H A number of di!erent approximations can be used to get an estimate of H. Andersen [7] includes the expected amount on hand when an order arrives: From Eq. (5), the expected amount on hand when an order arrives equals the safety stock plus the expected value of the backlog, i.e.
Q
(s!x) f (x) dx"k # E(k). (19) V V Since some of the order will go to meet the backlog, Andersen only added half of the backlog to the working inventory Q/2, to get H. However, the determination of the optimal Q and k is just as di$cult as the exact determination, so this approximation has no practical importance. The most common approximation of H is simply to neglect the di!erence between the on hand stock and net stock. Putting the expected value of the on hand stock,
Q Q #k # V J(k)!J k# , V 2Q 2 V equal to the expected value of the net stock, Q #k V 2 and at the same time setting the expected number of shortages in the lead time: (E(k)!E(k#q))+ E(k). (20) V V These approximations simplify the Lagrange function to
eoq q 1# V E(k) ¸ " #k# , 2 2q A !eoq
V (1!), 2A
(21)
which, by setting the derivates of ¸ to 0 with , respect to k, and q, enables us to derive the following expression for the determination of the models optimal values for k and q:
q"eoq 1#E(k) V , A
(22)
239
2q F(k)" eoq /A V and
(23)
E(k)"(1!)q.
(24)
The simultaneous determination of the two control variables can either be carried out by an iterative procedure, initiated by putting q"eoq, or by using the Excel SOLVER. The system's real normalized costs,
J(k)!J(k#q) eoq 1 # , C " q#2k# , q q 2
(25)
will be calculated and a cost index, C /C , used , , as the performance measure for the model. Here, C is the cost minimum in the exact model. , In Fig. 3, the consequences of the approximation are shown for "0.95, which gives nearly the same costs as in the exact model. While the costs are somewhat higher than the costs in the exact model for small and eoq values the service level is also higher than desired. On the face of it, it might be expected that the cost weighting of the optimization in the approximate model, which underestimates the on-hand stock and overestimates the number of shortages, would set both Q and k too high. However, the higher Q gives such a high degree of certainty that k can be set to a lower value than in the exact model. This adjustment means that the realized value of only slightly exceeds the desired value. The approximations in this model should make it more convenient for application in practice. However, the calculations, which are necessary to determine the optimal values of the control variables, are nearly as di$cult as the calculations for the exact model. The conclusion is, therefore, that the model has no advantages over the exact model. 3.2. Predetermined order size, Q The most frequently used method of determining order point and order size is to use the prescribed order size, Q"EOQ, while at the same time setting
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P. Alstr~m / Int. J. Production Economics 71 (2001) 235}245
Fig. 3. Comparison of order size and cost in a simultaneous and exact model with service level 0.95.
the stock on hand equal to the net stock and ignoring the fact that the number of shortages in a lead time period in excess of the order size is included in the next order period. This means that the optimization is based on the following Lagrange expression:
¸ "0.5 eoq#k#eoq V (E(k)!eoq(1!)) . , A (26) Setting the "rst derivate of ¸ with respect to , k and to 0, we get 2 F(k)" eoq ( /A) V and E(k)"(1!)eoq.
(27)
(28)
To "nd k we use the approximate expression in Appendix 2 for the inverse function of E(k). It is now possible to calculate the real costs by means of the expression J(k)!J(k#q) C "eoq#k# . , 2.eoq
(29)
The performance of the system is shown by the cost index, C /C , in Fig. 4 and the realized service , , (E(k)!E(k#eoq))/eoq, in Fig. 5. As mentioned above, the eoq value being smaller than the order size in the exact model, we could expect a smaller service, but the reverse is true. With small and eoq values in the particular model
released such high k values that the realized service level is much higher than speci"ed (see Fig. 5). The sequential model is easy to use in practical inventory control. As regards the cost consequences for a wide range of inventory items, using the most frequently used inventory control model leads to only slightly higher costs than the exact model. Of course, a model, where one of the two control variables is prescribed does not allow the adjustment of order size and order point, but for eoq greater than 2 and relevant value it has only a very small e!ect, so the determination of the control variables results in cost close to the minimum, which is shown in Table 1. On the other hand, the use of the model for inventory items with low eoq values and low service requests has serious cost consequences, because one pays for a service far in excess of the desired, Fig. 5. This is because the stock on hand is approximated with the net stock
Q I>/ NV Q #k # V (z!k) f (z) dz+ #k . V 2Q V 2 2 I (30) The lower the value, the worse the approximation. Added to this is the model's double counting of the number of shortages by approximating the expression to determine k, E(k)!E(k#q)"(1!)eoq with E(k)"(1!)eoq.
(31)
P. Alstr~m / Int. J. Production Economics 71 (2001) 235}245
241
Fig. 4. Cost index for sequential model.
Fig. 5. Realized service in sequential model.
Table 1 Cost index, C/C , for sequential model , , eoq
"0.8
"0.85
"0.9
"0.95
"0.99
2 3 4 5
1.086 1.058 1.045 1.038
1.064 1.041 1.031 1.025
1.045 1.028 1.021 1.016
1.027 1.017 1.012 1.009
1.012 1.007 1.005 1.004
with eoq values greater than 2, the model is suitable for practical inventory control. For inventory items with eoq value less than 2, the solution is to change the model in such a way that the simple determination of order size is kept, but where the value of the e!ective safety factor, k, is determined so as to just ensure the desired service level. 3.3. New model
The error is most pronounced partly for low eoq values and partly where the desired service level is low. McLaughlin et al. [2] points out that `many research projects have been based on simulation models which provide levels of customer service that are much higher than the parameters would suggesta. In our opinion, one explanation could be the use of formulas based on the sequential model. The analysis of the simple sequential model leads to the obvious conclusion that, for inventory items
An order size, eoq, and an e!ective safety factor, k, determined by the expression E(k)!E(k#q)"(1!)eoq, will ensure that the service requirement is just met. While the model is simple, the problem is that the expression for determining k is not explicit in k. This makes it necessary to determine k by numerical search. We have used the Excel SOLVER, where the target cell, (E(k)!E(k#q))/q, is set equal to 1!, and the changing cells contain the
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P. Alstr~m / Int. J. Production Economics 71 (2001) 235}245
Fig. 6. Cost index for di!erent service levels in `New modela.
Fig. 7. E!ective safety factor, k, as function of eoq for all integral values of service levels in the interval: 0.85 to 0.99.
numerical value of k and q, which are subject to the constraints q"eoq. The cost consequences of using the model are shown in Fig. 6, where it can be seen that, for " 0.85, 0.90, 0.95, 0.99, the model leads to only slightly higher costs than with the exact determination of the control variables. The smaller order size is paralleled with a higher value for the e!ective safety factor. The relation k"k(eoq) is almost linear, cf. Fig. 7. For instance we have for "0.90 k"!0.0392 ) eoq#1.2426, R"0.9959. Even though it is possible to manage the search procedure for determining k for a single item, it precludes the practical application of the model because inventory control typically involves a considerable number of items. The model is therefore
approximated for eoq values less than 2 so that it can be used in practice. 3.4. Final model For eoq values greater than 2, set the order size equal to EOQ and determine the e!ective safety factor, k, using the expression, E(k)"(1!) eoq. See Apppendix A for the numerical determination of the inverse E(k). For eoq values less than or equal to 2, this model results in too high costs. For these items, the order size is still determined using the EOQ formula, while the e!ective safety factor, k, is determined by substitution in the formula below. This is derived by means of a regression analysis of the correlation between the e!ective safety factor, k, determined by
P. Alstr~m / Int. J. Production Economics 71 (2001) 235}245
the expression, E(k)!E(k#q)"(1!) eoq, and eoq. The relation between k and eoq is illustrated in Fig. 7. A regression analysis for the individual values shows an approximate linear relation, a#b)eoq, with `aa and `ba dependent on . The relationship between the `aa values and and the `ba values of are determined for each integral value of the shortages percentage , de"ned as "100!100, in the interval 1}15. The results of these analyses are shown in Figs. 8 and 9. By approximating `aa with the expression 0.4722 Ln()#2.3251 and `ba with the expression 0.00015 ) ()!0.0111 ) ()!0.3151 we can "nd k as a function of eoq by the equation k"`aa#`ba ) eoq.
(32)
The cost consequences of using an (s, Q)-model, in which the order size is determined by EOQ and the e!ective safety factor by the expression,
243
k"`aa#`ba ) eoq, is shown, in Fig. 10 for selected value of . Compared with the exact model, the additional costs for relevant service levels are insigni"cant. Fig. 11 illustrates the consequences of the "nal model for the realized service level. As can be seen, the target level is almost realized.
4. Conclusion For the practical use of the inventory control system, (s, Q), the item's eoq values are essential, both for the cost consequences of the use of di!erent approximate models and also for the development of a practical model we have called the "nal model. Since statistical inventory control usually involves a large number of products, it is necessary to use an inventory control model which allows the value of the control variables to be determined
Fig. 8. Determination of `aa as function of the shortage % by regression analysis.
Fig. 9. Determination of `ba as a function of the shortage %.
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P. Alstr~m / Int. J. Production Economics 71 (2001) 235}245
Fig. 10. Cost index for "nal model.
Fig. 11. Realized service for "nal model.
directly, without the need for numerical search. We have therefore proposed a model where the order size is determined as the economic order size EOQ and the e!ective safety factor, k, depending on the eoq value. For eoq)2 k"`aa()#`ba ) eoq"0.4722 Ln()#2.3251 #[0.0015()!0.0111()!0.3151]eoq,
(33)
where is the shortages percentage (100!100)) For eoq'2 k is determined by the expression: E}INV((1!).eoq), where the function, E}INV, is the inverse function of E(k), determined by binary search, cf. Appendix A.
Appendix A. User de5ned functions in Visual Basic/Excel Option Explicit Private Upper As Double
Private Lower As Double Private Center As Double Function E(ByVal k As Double) As Double E"Application.NormDist(k, 0, 1, False) !k*(1 } Application.NormSDist(k)) End Function - - - Inverse functions - - Function f}inv(ByVal f As Double) As Double f}inv"sqr(!2* Application.Ln(Sqr(2 * Application.Pi) f)) * End Function Function E}INV(ByVal eval As Double) If eval '" 4C Then E}INV"!eval Exit Function ElseIf eval ("0.000000051 Then E}INV"5C Exit Function Else Lower "!4C Upper"5C
P. Alstr~m / Int. J. Production Economics 71 (2001) 235}245
Do Center "(Upper # Lower)/2 If (E(Center)'eval) Then Lower " Center Else Upper " Center End If Loop While (Upper ! Lower'0.00001) E}INV " (Upper # Lower)/2 End If End Function Appendix B. Estimating the mean value of backlogged demand
245
a curve whose slope depends on the random variable, x, the demand during lead time. For x greater than the order point, s, stockouts occur. The shaded area in the "gure shows for the shown value of x, the number of `stockout yearsa in a cycle period of length, t. We can calculate this area as: t (x!s). Per time unit it is: t /t(x!s). Using similar triangles, it is easy to calculate, namely: (s!x)/2Q. Thus the expected number of backorders at any time is
Q>/(x!s) I>/ NV (z!k)f (z) dz. f (x) dx" V 2Q 2Q VQ XI
References
Fig. 12.
Assuming that the demand during lead time arrives in an uniform way (see Fig. 12), we can draw
[1] J.A. Oesteryoung, D.E. McCarthy, W.J. Rheinehart, Use of the EOQ model for inventory analysis, Production and Inventory Management 27 (3) (1986) 39}45. [2] M. Wisniewski, C. Jones, K. Kristensen, H. Madsen, P. Ostergaard, Does anyone use the techniques we teach? OR Insight 7 (2) (1994) 2}7. [3] C.P. McLaughlin, G. Vasag, D.C. Whybark, Statistical inventory control in theory and practice. International Journal of Production Economics. 35 (1}3) (1994), 161}170. [4] H. Schneider, E!ects of service levels on order-points or order levels in inventory models, International Journal of Production Research 19 (6) (1981) 615}631. [5] G. Hadley, T.M. Whitin, Analysis of Inventory Systems, Prentice-Hall, Englewood Cli!s, NJ, 1963, pp. 191}195. [6] E.A. Silver, D.F. Pyke, R. Peterson, Inventory Management and Scheduling, Wiley, NewYork, 1998, pp. 348}349. [7] E.J. Andersen, The Management of Manufacturing, Addison Wesley, Reading, MA, 1994, pp. 104}105.
Numerical computation of inventory policies, based on the EOQ/ value for order-point systems V Poul Alstr+m* Department of Management Science and Logistics, The Aarhus School of Business, Fuglesangs Alle 4, DK-8210 Aarhus V, Denmark
Abstract This paper examines the numerical computation of two control parameters, order size and order point in the well-known inventory control model, an (s, Q) system with a safety stock strategy. The aim of the paper is to show that the EOQ/ value is both su$cient for controlling the system and essential for the economic consequences of using V approximations in computations of optimal policies. In view of the evidence from a number of studies showing that "rms' use of statistical inventory control lags far behind academic interest in the area, this is also an important aspect. The determination of optimal values for the control variables } even in this very simple inventory control system } is a complex task, and therefore not suited to practical implementation, where one or more of the following issues are neglected: the interdependence of order size and order point, the di!erence between stock on hand and net stock, and the excess stockout included in the next period. We use the determination of optimal values for the control variables as a framework to evaluate the economic consequences of these approximations, applied in pratical inventory control. 2001 Elsevier Science B.V. All rights reserved. Keywords: Stochastic order point models; Exact and heuristic algorithms; EOQ/ : Essential for controlling the system and for using V heuristics
1. Introduction While advances in computer technology have enabled the increasing use of inventory theory in industry in the last few decades, it is still mostly very simple inventory control systems, such as EOQ models, that are used. Many of the commercially available software packages for inventory control employ this concept. Osteryoung et al. [1] "nd that 84% of the "rms surveyed use the EOQ model for inventory control, though this is di$cult to assess accurately. Wisniewski et al. [2], carried
* Tel.: #45-86102488; fax: #45-89486660.
out a survey among private "rms in Scotland, England and Denmark in 1991, 566 of which replied, a response rate of 39%. Of these, 66% were aware of stock control techniques, 75% of which employed them. A more recent study by McLaughlin et al. [3] shows that only 28% of respondents used EOQ for determining production or purchasing quantities. The aim of our paper is to show that the EOQ/ V value is both su$cient for controlling the system and essential for the economic consequences of using approximations in computations of optimal policies in an order point-order quantity system, the so-called (s, Q) inventory system. The theoretical model is a single-product inventory model with transaction reporting and stationary
0925-5273/01/$ - 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 2 2 - 5
236
P. Alstr~m / Int. J. Production Economics 71 (2001) 235}245
independent yearly demand, D. When the inventory position fall to the order point, s, the quantity Q is ordered. The order will arrive after a "xed delivery lead time, so that there are no orders cross. Forecast errors have a normal distribution, and the forecast over lead time has a known average, , and stanV dard deviation, . The service strategy is the V so-called strategy [4], where is the fraction of demand satis"ed from the shelf stock. In the event that the demand cannot be met immediately, it will be back-ordered and met soon as possible from subsequent stock supplements. The ordering cost consists of an order cost, A, independent of the order size, and a proportional cost of v per unit. With a carrying charge factor of r, the holding cost per unit per year is rv. Using the results of Hadley and Whitin [5], we can calculate the mean value of the stock on hand and the expected number of back orders in this model, which makes it easy to set up the expression for the expected costs per year. This function will be minimized under the restriction that the service level is maintained. We will then use this cost function as the performance measure for the model and as a framework to evaluate the economic consequences of some approximations use in practical inventory control, namely: E an approximate estimate of physical stock, E an approximate estimate of shortages, E sequential optimization. The analysis results in a simple model for use in practical inventory control.
This sum is minimized in accordance with the service requirement when using the Lagrange multiplier method. The Lagrange function, ¸, is calculated as
where is the Lagrange multiplier and S the total expected number of units back ordered per year. Because unmet demands are back ordered, the expected number of replenishment cycles per year is D N" . Q
The only relevant costs for the chosen system is the sum of holding cost per year and order costs per year. With H as the expected on hand inventory level and N as the expected number of orders placed per year, the expected costs per year can be calculated as C"rvH#AN.
(1)
(3)
The order point, s, consists of the expected lead time demand plus the safety stock. The safety stock, b, is de"ned as the average level of the net stock just before a replenishment arrives, where the net stock is de"ned as on hand stock minus back orders. b is thus determined as
b"
(s!x) f (x) dx, (4) V where f (x) is the normal density function. Apart from the inverse function, E(k)"(z!k) I f (z) dz cf. Appendix A, the spreadsheet program Excel provides all the required values for the normal distribution. It is convenient to write b as the product k , V where k is called the e!ective safety factor. Since b"
" 2. The cost function and exact determination of the control variables
(2)
¸"C#(S!(1!)D),
V
(s!x) f (x) dx
(s!x) f (x) dx!
V "k , V we have
Q
(x!s) f (x) dx
VQ (5)
(s!x) f (x) dx"
(x!s) f (x) dx#k . (6) V V VQ In other words, the expected stock on hand at the end of a replenishment cycle, Q (s!x) f (x) dx, is V equal to the expected value of the backlog at the end of a replenishment cycle, (x!s) f (x) dx, VQ plus the safety stock, k . V
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To obtain the mean level of stock on hand it is therefore necessary to add the mean value of the backlog to the mean value of the net stock. In [5, Section 4}9. pp. 191}195] the calculations of the expected number of backorders at any time are based on the probability function for the number of back orders. For a continuous demand distribution, it is easier to determine the mean value of the backlogged demand graphically. As shown in Appendix B the mean value of the backlog is:
(9)
where
f (z) dz.
(10)
XI
The average number of units on hand, H, is therefore
Q Q H" k # # V J(k)!J k# V 2 2Q V
,
,
(14)
Q Q #k # V J(k)!J k# V 2Q 2 V
D Q
rv
(15)
and a Lagrange function [¸(Q), k, )]
"
Q Q #k # V J(k)!J k# V 2Q 2 V
rv
D D Q V E(k)!E k# #A # Q Q V !(1!)D .
(16)
To reduce the number of separate parameters that have to be considered we normalize by multiplying both sides in Eqs. (15) and (16) by the constant term 1/vr [6] and also by inserting V
2AD EOQ eoq" " rv V V and
(11)
where J(k)!J(k#(q/ ))'0 for all arguments. V In the same way we can determine the expected number of shortages per year as I>/ NV D S" (z!k) f (z). Q V I
(13)
Q D S" E(k)!E k# V Q V
#A
This expression takes account of the fact that the number of shortages in a lead time period in excess of the order size is included in the next order period. As shown by Hadley and Whitin [5, Eqs. (4-84) and (4-85), pp. 193}194] the function, j (k), can be written as
F(k)"
(z!k) f (z) dz, I
the expected number of shortages per year S can be written as
, (7)
(8)
J(k)"(1#k)F(k)!k f (k),
E(k)"
C(Q, k)"
where f (z) is the density function for the standard normal distribution and J(k)" (z!k)f (z) dz. I
Since
which gives total relevant cost per year of
Q>/(x!s) I>/ NV f (x) dx" V (z!k)f (z) dz 2Q 2Q VQ I Q " V J(k)!J k# 2Q V
237
(12)
Q "q. V The normalized expressions for costs and the Lagrange function can now be written as q J(k)!J(k#q) eoq C " #k# # , 2 2q 2q
(17)
238
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and J(k)!J(k#q) q ¸ " #k# , 2 2q
eoq # 1# V (E(k)!E(k#q)) 2q A !eoq
V (1!). 2A
(18)
For every value it is necessary to use a search procedure to "nd the values for q, k, and which minimize the ¸ expression. To do this we have , used the SOLVER in the spreadsheet program Excel, where the service request (E(k)!E(k#q))/ q"1!, together with q'0 are set as conditions for the minimization. To verify that SOLVER has found the correct values for q and k we also set and veri"ed the following conditions: ¸ ¸ ¸ ¸ , " , "0, , '0, , '0, k q k q
¸ ¸ ¸ , '0. ,) ,! k q kq
The results for q and k for "0.80, 0.85, 0.90, 0.95, 0.99 and eoq values in the interval 0.1 to 2.0, are shown in Figs. 1 and 2. The use of EOQ as the order size underestimates the optimal value, but it does not necessarily mean that the service level will be lower than the stipulated value, On the contrary. Fig. 2 shows that it can be optimal to operate with a negative safety stock, depending on service level and the itme's eoq values. A high eoq value means a high order size and fewer lead time periods and therefore a lower risk.
3. Heuristic models The search procedure used above to determine optimal values for the system's control variables is unsuitable in practice, where inventory control typically involves a considerable number of products. Instead of "nding the exact values for H and S in the cost and Lagrange functions we will use heuristics to "nd the value of the control variables.
Fig. 1. Optimal q value for di!erent service levels.
Fig. 2. Optimal k value for di!erent service levels.
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3.1. Approximate calculation of H A number of di!erent approximations can be used to get an estimate of H. Andersen [7] includes the expected amount on hand when an order arrives: From Eq. (5), the expected amount on hand when an order arrives equals the safety stock plus the expected value of the backlog, i.e.
Q
(s!x) f (x) dx"k # E(k). (19) V V Since some of the order will go to meet the backlog, Andersen only added half of the backlog to the working inventory Q/2, to get H. However, the determination of the optimal Q and k is just as di$cult as the exact determination, so this approximation has no practical importance. The most common approximation of H is simply to neglect the di!erence between the on hand stock and net stock. Putting the expected value of the on hand stock,
Q Q #k # V J(k)!J k# , V 2Q 2 V equal to the expected value of the net stock, Q #k V 2 and at the same time setting the expected number of shortages in the lead time: (E(k)!E(k#q))+ E(k). (20) V V These approximations simplify the Lagrange function to
eoq q 1# V E(k) ¸ " #k# , 2 2q A !eoq
V (1!), 2A
(21)
which, by setting the derivates of ¸ to 0 with , respect to k, and q, enables us to derive the following expression for the determination of the models optimal values for k and q:
q"eoq 1#E(k) V , A
(22)
239
2q F(k)" eoq /A V and
(23)
E(k)"(1!)q.
(24)
The simultaneous determination of the two control variables can either be carried out by an iterative procedure, initiated by putting q"eoq, or by using the Excel SOLVER. The system's real normalized costs,
J(k)!J(k#q) eoq 1 # , C " q#2k# , q q 2
(25)
will be calculated and a cost index, C /C , used , , as the performance measure for the model. Here, C is the cost minimum in the exact model. , In Fig. 3, the consequences of the approximation are shown for "0.95, which gives nearly the same costs as in the exact model. While the costs are somewhat higher than the costs in the exact model for small and eoq values the service level is also higher than desired. On the face of it, it might be expected that the cost weighting of the optimization in the approximate model, which underestimates the on-hand stock and overestimates the number of shortages, would set both Q and k too high. However, the higher Q gives such a high degree of certainty that k can be set to a lower value than in the exact model. This adjustment means that the realized value of only slightly exceeds the desired value. The approximations in this model should make it more convenient for application in practice. However, the calculations, which are necessary to determine the optimal values of the control variables, are nearly as di$cult as the calculations for the exact model. The conclusion is, therefore, that the model has no advantages over the exact model. 3.2. Predetermined order size, Q The most frequently used method of determining order point and order size is to use the prescribed order size, Q"EOQ, while at the same time setting
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Fig. 3. Comparison of order size and cost in a simultaneous and exact model with service level 0.95.
the stock on hand equal to the net stock and ignoring the fact that the number of shortages in a lead time period in excess of the order size is included in the next order period. This means that the optimization is based on the following Lagrange expression:
¸ "0.5 eoq#k#eoq V (E(k)!eoq(1!)) . , A (26) Setting the "rst derivate of ¸ with respect to , k and to 0, we get 2 F(k)" eoq ( /A) V and E(k)"(1!)eoq.
(27)
(28)
To "nd k we use the approximate expression in Appendix 2 for the inverse function of E(k). It is now possible to calculate the real costs by means of the expression J(k)!J(k#q) C "eoq#k# . , 2.eoq
(29)
The performance of the system is shown by the cost index, C /C , in Fig. 4 and the realized service , , (E(k)!E(k#eoq))/eoq, in Fig. 5. As mentioned above, the eoq value being smaller than the order size in the exact model, we could expect a smaller service, but the reverse is true. With small and eoq values in the particular model
released such high k values that the realized service level is much higher than speci"ed (see Fig. 5). The sequential model is easy to use in practical inventory control. As regards the cost consequences for a wide range of inventory items, using the most frequently used inventory control model leads to only slightly higher costs than the exact model. Of course, a model, where one of the two control variables is prescribed does not allow the adjustment of order size and order point, but for eoq greater than 2 and relevant value it has only a very small e!ect, so the determination of the control variables results in cost close to the minimum, which is shown in Table 1. On the other hand, the use of the model for inventory items with low eoq values and low service requests has serious cost consequences, because one pays for a service far in excess of the desired, Fig. 5. This is because the stock on hand is approximated with the net stock
Q I>/ NV Q #k # V (z!k) f (z) dz+ #k . V 2Q V 2 2 I (30) The lower the value, the worse the approximation. Added to this is the model's double counting of the number of shortages by approximating the expression to determine k, E(k)!E(k#q)"(1!)eoq with E(k)"(1!)eoq.
(31)
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241
Fig. 4. Cost index for sequential model.
Fig. 5. Realized service in sequential model.
Table 1 Cost index, C/C , for sequential model , , eoq
"0.8
"0.85
"0.9
"0.95
"0.99
2 3 4 5
1.086 1.058 1.045 1.038
1.064 1.041 1.031 1.025
1.045 1.028 1.021 1.016
1.027 1.017 1.012 1.009
1.012 1.007 1.005 1.004
with eoq values greater than 2, the model is suitable for practical inventory control. For inventory items with eoq value less than 2, the solution is to change the model in such a way that the simple determination of order size is kept, but where the value of the e!ective safety factor, k, is determined so as to just ensure the desired service level. 3.3. New model
The error is most pronounced partly for low eoq values and partly where the desired service level is low. McLaughlin et al. [2] points out that `many research projects have been based on simulation models which provide levels of customer service that are much higher than the parameters would suggesta. In our opinion, one explanation could be the use of formulas based on the sequential model. The analysis of the simple sequential model leads to the obvious conclusion that, for inventory items
An order size, eoq, and an e!ective safety factor, k, determined by the expression E(k)!E(k#q)"(1!)eoq, will ensure that the service requirement is just met. While the model is simple, the problem is that the expression for determining k is not explicit in k. This makes it necessary to determine k by numerical search. We have used the Excel SOLVER, where the target cell, (E(k)!E(k#q))/q, is set equal to 1!, and the changing cells contain the
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Fig. 6. Cost index for di!erent service levels in `New modela.
Fig. 7. E!ective safety factor, k, as function of eoq for all integral values of service levels in the interval: 0.85 to 0.99.
numerical value of k and q, which are subject to the constraints q"eoq. The cost consequences of using the model are shown in Fig. 6, where it can be seen that, for " 0.85, 0.90, 0.95, 0.99, the model leads to only slightly higher costs than with the exact determination of the control variables. The smaller order size is paralleled with a higher value for the e!ective safety factor. The relation k"k(eoq) is almost linear, cf. Fig. 7. For instance we have for "0.90 k"!0.0392 ) eoq#1.2426, R"0.9959. Even though it is possible to manage the search procedure for determining k for a single item, it precludes the practical application of the model because inventory control typically involves a considerable number of items. The model is therefore
approximated for eoq values less than 2 so that it can be used in practice. 3.4. Final model For eoq values greater than 2, set the order size equal to EOQ and determine the e!ective safety factor, k, using the expression, E(k)"(1!) eoq. See Apppendix A for the numerical determination of the inverse E(k). For eoq values less than or equal to 2, this model results in too high costs. For these items, the order size is still determined using the EOQ formula, while the e!ective safety factor, k, is determined by substitution in the formula below. This is derived by means of a regression analysis of the correlation between the e!ective safety factor, k, determined by
P. Alstr~m / Int. J. Production Economics 71 (2001) 235}245
the expression, E(k)!E(k#q)"(1!) eoq, and eoq. The relation between k and eoq is illustrated in Fig. 7. A regression analysis for the individual values shows an approximate linear relation, a#b)eoq, with `aa and `ba dependent on . The relationship between the `aa values and and the `ba values of are determined for each integral value of the shortages percentage , de"ned as "100!100, in the interval 1}15. The results of these analyses are shown in Figs. 8 and 9. By approximating `aa with the expression 0.4722 Ln()#2.3251 and `ba with the expression 0.00015 ) ()!0.0111 ) ()!0.3151 we can "nd k as a function of eoq by the equation k"`aa#`ba ) eoq.
(32)
The cost consequences of using an (s, Q)-model, in which the order size is determined by EOQ and the e!ective safety factor by the expression,
243
k"`aa#`ba ) eoq, is shown, in Fig. 10 for selected value of . Compared with the exact model, the additional costs for relevant service levels are insigni"cant. Fig. 11 illustrates the consequences of the "nal model for the realized service level. As can be seen, the target level is almost realized.
4. Conclusion For the practical use of the inventory control system, (s, Q), the item's eoq values are essential, both for the cost consequences of the use of di!erent approximate models and also for the development of a practical model we have called the "nal model. Since statistical inventory control usually involves a large number of products, it is necessary to use an inventory control model which allows the value of the control variables to be determined
Fig. 8. Determination of `aa as function of the shortage % by regression analysis.
Fig. 9. Determination of `ba as a function of the shortage %.
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P. Alstr~m / Int. J. Production Economics 71 (2001) 235}245
Fig. 10. Cost index for "nal model.
Fig. 11. Realized service for "nal model.
directly, without the need for numerical search. We have therefore proposed a model where the order size is determined as the economic order size EOQ and the e!ective safety factor, k, depending on the eoq value. For eoq)2 k"`aa()#`ba ) eoq"0.4722 Ln()#2.3251 #[0.0015()!0.0111()!0.3151]eoq,
(33)
where is the shortages percentage (100!100)) For eoq'2 k is determined by the expression: E}INV((1!).eoq), where the function, E}INV, is the inverse function of E(k), determined by binary search, cf. Appendix A.
Appendix A. User de5ned functions in Visual Basic/Excel Option Explicit Private Upper As Double
Private Lower As Double Private Center As Double Function E(ByVal k As Double) As Double E"Application.NormDist(k, 0, 1, False) !k*(1 } Application.NormSDist(k)) End Function - - - Inverse functions - - Function f}inv(ByVal f As Double) As Double f}inv"sqr(!2* Application.Ln(Sqr(2 * Application.Pi) f)) * End Function Function E}INV(ByVal eval As Double) If eval '" 4C Then E}INV"!eval Exit Function ElseIf eval ("0.000000051 Then E}INV"5C Exit Function Else Lower "!4C Upper"5C
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Do Center "(Upper # Lower)/2 If (E(Center)'eval) Then Lower " Center Else Upper " Center End If Loop While (Upper ! Lower'0.00001) E}INV " (Upper # Lower)/2 End If End Function Appendix B. Estimating the mean value of backlogged demand
245
a curve whose slope depends on the random variable, x, the demand during lead time. For x greater than the order point, s, stockouts occur. The shaded area in the "gure shows for the shown value of x, the number of `stockout yearsa in a cycle period of length, t. We can calculate this area as: t (x!s). Per time unit it is: t /t(x!s). Using similar triangles, it is easy to calculate, namely: (s!x)/2Q. Thus the expected number of backorders at any time is
Q>/(x!s) I>/ NV (z!k)f (z) dz. f (x) dx" V 2Q 2Q VQ XI
References
Fig. 12.
Assuming that the demand during lead time arrives in an uniform way (see Fig. 12), we can draw
[1] J.A. Oesteryoung, D.E. McCarthy, W.J. Rheinehart, Use of the EOQ model for inventory analysis, Production and Inventory Management 27 (3) (1986) 39}45. [2] M. Wisniewski, C. Jones, K. Kristensen, H. Madsen, P. Ostergaard, Does anyone use the techniques we teach? OR Insight 7 (2) (1994) 2}7. [3] C.P. McLaughlin, G. Vasag, D.C. Whybark, Statistical inventory control in theory and practice. International Journal of Production Economics. 35 (1}3) (1994), 161}170. [4] H. Schneider, E!ects of service levels on order-points or order levels in inventory models, International Journal of Production Research 19 (6) (1981) 615}631. [5] G. Hadley, T.M. Whitin, Analysis of Inventory Systems, Prentice-Hall, Englewood Cli!s, NJ, 1963, pp. 191}195. [6] E.A. Silver, D.F. Pyke, R. Peterson, Inventory Management and Scheduling, Wiley, NewYork, 1998, pp. 348}349. [7] E.J. Andersen, The Management of Manufacturing, Addison Wesley, Reading, MA, 1994, pp. 104}105.