- Email: [email protected]
A note on inventory reorder point determination
Journal of Accounting Education Teaching and Educational Notes Vol. 4, No. 2 Fall, 1986
A NOTE ON INVENTORY REORDER POINT DETERMINATION
CALIFORNIA
Raman C. Pate1 STATE UNIVERSITY,
CHICO
are as important as Abstract: Managerial decisions regarding timing of inventory replenishment determining the economic order size in inventory management. Yet most of the introductory texts on cost/ managerial accounting and production/ operations management do not adequately discuss reorder point determination, especially the condition on lead time demand that is necessary for valid application of the commonly reported formula for the reorder point. A clear understanding of the reorder point determination is important to students, teachers, and practicing managers alike. This note attempts to explain the underlying condition and modification needed in the reorder point formula.
Inventory management is an important function in business. Managerial decisions concerning the amount and timing of inventory replenishment become especially crucial in periods of high interest costs and cyclical swings in the economy. A thorough understanding of the concepts and methods involved in the determination of economic order quantity (EOQ) and reorder point (ROP) is essential to students, teachers, and practicing managers in business. In view of the importance of inventory management in business, the topic of EOQ and ROP determination should be an important component of the business curriculum. Since CPA, CMA and similar professional examinations often include questions on EOQ and ROP, it is especially important to cover the topic adequately in the accounting curriculum. The topic of EOQ and ROP determination is usually included in introductory cost/ managerial accounting and production/ operations management courses. Most textbooks (see list of references) for these courses introduce the topic with the discussion of the basic EOQ model. However, most of the discussion in these texts is concerned with explaining EOQ determination. Several of the texts fail to explain clearly the ROP determination in general. The coverage of ROPin various texts is often brief and inadequate, and it is mostly confined to a stochastic demand situation. A few texts’ explicitly give the formula for ROP in the case of deterministic demand, but do not discuss the underlying condition necessary for the valid See, for example, Anderson, Sweeney, and Williams [1983]; Chase and Aquilano [1985]; Garrison 119851; Killough and Leininger [1984]; Moriarity and Allen [1984]; and Morse [1981, 19841.
132 use of such a formula.
Even in the case of stochastic demand covered in most of the texts, the limitation on the use of the commonly mentioned formula for ROPis not discussed except in Monks, Horngren [I982 and 19841, Matz and Usry, and Morse. Monks [p. 407-4101 illustrates the use of the ROPformula under different situations through examples, but does not explicitly provide the necessary modification in the formula and fails to explain the underlying condition for its validity. Horngren [p. 7651 does footnote the situation where the commonly given ROP formula is inapplicable and gives the general formula, but without adequate explanation. Only Matz and Usry provide better than average coverage of the ROPdetermination. However, they fail to explain the determination of quantities due in from previous orders (outstanding orders) required in their formula. Morse [p. 6471 briefly remarks that the ROP will be greater than EOQ if lead period is large, but provides no further explanation. The purpose of this note is to point out the limitation on the use of the commonly given ROP formula and clearly explain the general formula for determining the reorder point. After a brief introduction of the basic EOQ model, the ROP determination is first presented assuming known demand and known lead period. The case of stochastic demand is then considered assuming a known lead period. Finally, an approximate formula for ROP is suggested for the most general case where both lead period and demand are stochastic.
BASIC
EOQ MODEL
The basic EOQ model is based upon the assumptions of known demand, D, during a period (T, usually a year) and known lead time (L, the time elapsed between placing an order and receiving the ordered quantity). The optimum order quantity, EOQ, is determined by considering the cost of ordering relative to inventory carrying charges for a given demand during the period as follows: Q* = /2D(O/H)
(1)
where Q* = economic order quantity, EOQ D = known demand during a period 0 ordering cost per order H = inventory holding (carrying) charges per unit per period. q
The associated optimum (minimum) given by TC* = d?i%%?-
total cost of inventory
during a period is
133
where TC* = minimum total inventory cost assuming inventory replenishments of the optimum quantity Q*. The number of orders, N, to be placed during the period under consideration is given by N= D/Q* and the inventory cycle time (t, the period between or placements of orders) is determined as t= T/N or alternatively
inventory
replenishments
(4)
as
t = (TJ D)Q* by substituting
N from (3). The expression
Q* = (01
(5) can be also written
as
T)t
= dt
(6)
expressing EOQ in terms of demand per unit of time (d = D/ T), and the cycle time, t. As an example, consider a product whose demand is 4,000 units during a year. If ordering cost is $18 per order and the cost of inventory holding is 90~ per unit per year, then expressions (1) through (4) provide the following results: Economic Order Quantity: Q* = 400 units per order Minimum Total Inventory Cost: TC* = 360 dollars Number of orders during a year: N = 10 orders Inventory cycle time: t = 0.1 year or t = 5.2 weeks
REORDER
POINT
DETERMINATION
The question of when to place an order of EOQ units as determined by (1) is essentially determined by the expected demand during the lead period. Clearly, the time to place an order is when the inventory level falls to a point where sufficient inventory exists to satisfy the expected demand during the lead period. Hence the reorder point is set equal to the expected demand during the lead period. Under the assumed known demand, the demand during the lead period is basically determined by the product of lead time, L, and demand per unit of time, d. Hence the reorder point is determined as R = dL
134
where R= ROP d = the demand
rate per unit of time.
The ROP formula (7) is obviously a function of lead time when demand is assumed known; that is, when dis a fixed known quantity. Hence, the validity of the ROPformula (7) commonly reported in various texts depends upon the magnitude of lead time L relative to the inventory cycle time, t. As long as the lead time is less than cycle time, the ROP formula (7) is valid in view of (6), since L < t implies dL < dt which is equivalent dL<
to the condition
Q*
(8)
This means the time to place an order is when inventory drops to the ROP, thereby assuring sufficient stock on hand to last until the order arrives at the end of current inventory cycle (or equivalently at the beginning of a new cycle) as illustrated in Figure 1.
FIGURE
1
t Inventory Q*
--
L
Slope = 4
I
a
-IA
.---
tt= --
l 1
Lead time is less than Cycle time (Lead time Demand
is less than EOC?)
135
For the example considered earlier, suppose lead time is 0.05 year or 2.6 weeks. Since L = 0.05 ( t = 0.1, the ROP is given by (7) as R= y
(0.05)
= 200 units If L is expressed
in weeks, then ROP is obtained
R = -y
from (7) as
(2.6)
= 200 units When demand is assumed known, the ROP formula is applicable so long as the lead time is less than cycle time. If, however, lead time is greater than inventory cycle time (that is, when the condition (8) is not satisfied so that lead time demand exceeds optimum order quantity), then the ROPformula is not applicable because it is not possible to place an order at an inventory level of more than the order quantity. For instance, suppose the lead period is 0.13 year or 6.76 weeks, which is greater than the cycle time of 0. I year or 5.2 weeks. Demand during the lead period is 520 units, which is greater than the EOQ. According to the ROPformula, R=520 would not be possible to use since it is impossible for inventory to drop to 520 units when EOQ= 400 units. The ROP formula needs to be modified when lead time demand exceeds the order quantity. Since the cycle time is determined by demand and order quantity while the lead period is primarily fixed by the supply environment, there is no assurance that L will be less than t as required for the valid use of formula (7) for ROP. When the condition L>t
(9)
or equivalently, dL>
Q*
flO>
prevails, the ROP formula requires modification by considering the manner in which the lead time demand is satisfied as illustrated in Figure 2 (following page) and explained below. The higher demand during the lead period is satisfied by the on-hand inventory at the time of placing an order (that is, ROP) and by the receipts of previously placed orders (outstanding orders) which arrive during the lead period, namely, Lead time demand = ROP + Receipts of previously placed orders arriving during the lead period.
136
FIGURE 2
Inventory
Time (C--=0.1
*I
+-L=o.13+,
1
L = 0.22 -
I-
Lead time is greater (Lead time demand
than Cycle time exceeds
EOQ)
Symbolically, dL = R + K(EOQ)
(11)
where K= the number of outstanding orders or replenishments (or complete inventory cycles) during the lead period. The value of Kis the integerpart of the ratio of lead time and cycle time, that is, K
q
integer (L/t)
(12)
Substituting Kfrom (12) in (1 I), the general formula for ROP when lead time demand exceeds EOQ is given by rearranging the terms as R = dL - [LJ t]Q*
(13)
where [L/t] is an integer. For the earlier example, suppose the lead period is 0.13 year or 6.76 weeks. Since the lead period is greater than the cycle time of 0.1 year or 5.2 weeks, the demand during lead period is 520 which is greater than the EOQ. Therefore, the ROPformula (7) would be inapplicable. The general ROPformula given by (13) provides the proper ROP as
137
R=
(400)
= 520 - [ 1.3](400) = 520 - l(400) = 120 units If lead period
is expressed
in weeks, then (13) gives
= 520 - l(400) = 120 units Suppose L = 0.22 year or 11.44 weeks, the ROP according formula (13) is obtained as
to the general
(400) = 880 - [2.23(400) = 880 - 2(400) = 80 units It may be noted that the usually reported formula (7) for ROP is a special case of the general expression for ROP given by (13) above. Whenever the lead period is less than cycle time their ratio L/t, is less than I and, therefore, the integer part of the ratio in (13) is zero and the ROPformula (13) reduces to the frequently reported formula (7). For example, when lead time is 0.05 year or 2.6 weeks, the ROP according to (13) is R=
No’
---i-
(0.5) -
= 200 - [0.5](400) = 200 - O(400) = 200 units as obtained earlier using formula (7). The general formula (13) for ROP is based on the assumption that demand is known. Also, it assumes that the lead period is known and independent of demand. In other words, the demand during the lead period is considered deterministic by assuming both the demand and the lead period are known. When either the lead period or the demand is uncertain, demand during the lead period is probabilistic. Therefore, the expected demand during the lead
138
period dL in the ROPformula (13) is uncertain and needs to be replaced by the mean of the underlying probability distribution of the lead period demand. Furthermore, provision of safety stock is needed to cover higher than average demand that may occur during the lead period because of variations in demand and/or lead period. The level of safety stock would depend upon the cost of holding additional inventory of such stock relative to the desired protection against stockouts during the lead period. Hence, there is no fixed method of determining the safety stock; it would depend on the assumed risk protection against stockouts and the assumptions regarding the underlying probability distribution of demand during the lead period. Whatever the safety stock is determined, the ROP in the case of stochastic demand can be generally set at the level of inventory sufficient to meet the average demand during the lead period plus the safety stock needed to cover the possibility of higher than average demand. Symbolically, R=dL+S
(14)
where d = average demand
per unit of time.
S = safety stock. This is the formula commonly given in various texts. The validity of the formula (14) is, however, limited to the situations where the demand during the lead period is less than the order quantity. There is a tacit assumption behind the formula (14) that the lead period is smaller than the cycle time, and hence the necessary condition that the lead period demand does not exceed the economic order quantity is normally satisfied. Such an assumed condition may not, however, always hold, since the lead period is primarily determined by the supply side while the order quantity is independently determined by the demand and relevant costs. A generally applicable formula for ROPin the case of stochastic demand is given by modifying the formula (14) as R = dL + S - [L/t]Q*
(15)
where all symbols are the same as previously defined. The general formula (15) for determining the ROP is applicable in any stochastic demand situation. The formula (15) is based only on the assumption that the lead period is a known constant. Such an assumption of known lead period is not unrealistic. Most frequently, the lead period is fixed contractually. Even if the lead period varies, it can be considered nearly equal to the average lead time so long as the variation is small relative to the average, which may often be the case in practice. In any event, the general formula (15) for ROP in the case of
139
stochastic demand is valid so long as the Iead period is assumed to be a known constant regardless of its magnitude. The fast term in the general formula (15) is zero whenever the lead period is less than the cycle time; and hence the general formula reduces to the commonly reported formula (14) for computing the ROP. The discussion of a situation where both the lead period and the demand are stochastic, each having an independent probability distribution, is beyond the scope of this note. It should be noted that the safety stock determination is not considered above because it is adequately discussed in the literature and is also beyond the scope of this note.
CONCLUSION Since demand during the lead period essentially determines the reorder point, and the lead time demand in turn is a function of both the demand rate and lead time, determination of the reorder point depends upon the nature of assumptions behind these two variables. As seen above, when demand is assumed to be known, the reorder point determination quite simply depends upon the magnitude of the lead time. The usual formula (7) for the reorder point is limited to situations where the lead time is less than the inventory cycle period. A general formula for ROP given by (I 3) above is, however, applicable under any situation so long as both the demand and the lead period are deterministic. When the lead period is known but the demand is probabilistic, the general formula for ROP is given by (15). It is hoped the above presentation has clarified the limitation of the commonly used formula for RUP, and demonstrated the need for a more complete explanation of the general formula in classroom presentations.
REFERENCES Adam, E. E., Jr. and Ebert, R. J., Production and Operations Management: Concepts, Models, and Behavior, 2nd edition, (Prentice-Hall, Inc., Englewood Cliffs, NJ, 1982). Anderson, D. R., Sweeney, D. J. and Williams, T. A., Quantitative Methodsfor Business, 2nd edition, (West Publishing, St. Paul MN, 1983): 487, 515. Bierman, H., Jr., Bonini, C. P. and Hausman, W. H., Quantitative Analysisfor Business Decisions, 6th edition, (Richard D. Irwin, Inc., Homewood, IL, 1981): 421-442. Buffa, E. S., Modern Production/Operations Management, 7th edition, (John Wiley & Sons, Inc., New York, 1983). Chase, R. B. and Aquilano, N. J., Production and Operations Management: A Life Cycle Approach, 4th edition, (Richard D. Irwin, Inc., Homewood, IL, 1985): 489,495-497. Cook, T. M. and Russell, R. A., Contemporary Operations Management: Text and Cases, 2nd edition, (Prentice-Hall, Inc., Englewood Cliffs, NJ, 1984): 358-362. Davidson, S., Maher, M. W., Stickney, C. P. and Weil, R. L., Manugeriul Accounting, 2nd edition, (The Dryden Press, Hinsdale, IL, 1985): 310-311. Deakin, E. B., and Maher, M. W., Cost Accounting, (Richard D. Irwin, Inc., Homewood, IL, 1984): 541-542.
140 Dominiak, G. F. and Louderback, J. Cl., III, ManngerialAccounri 4th edition, (Kent Publishing Company, Boston, MA, 1985): 652. Dopuch, N., Birnberg, J. G. and Demski, J. S., Cost Accounting, 3rd edition, (Harcourt Brace Jovanovich, Inc., New York, 1982): 661-662. Fischer, P. M. and Frank, W. Cl., Cost Accounting: Theory and Applications, (Southwestern Publishing Company, Cincinnati, OH, 1985): 541-545. Garrison, R., Managerial Accounting, 4th edition, (Business Publications, Inc., Plano, TX, 1985): 324-325. Horngren. C., Cost Accounting, 5th edition. (Prentice-Hall. Inc., Englewood Cliffs, NJ, 1982): 764-765. Horngren, C., Introduction to Management Accounting. 6th edition, (Prentice-Hall, Inc., Englewood Cliffs, NJ, 1984): 497-498. Kaplan, R. S., AdvancedManagement Accounting, (Prentice-Hall, Inc., Englewood Cliffs, NJ, 1982): 270-274). Killough, L. N. and Leininger, W. E., Cost Accounting, (West Publishing, St. Paul, MN, 1984): 655-658. Louderback, J. G., III, and Hirsch, M. L., Jr. (Kent Publishing Company, Boston, MA, 1982): 807-8 10. Matz. A. and Usry, M. F., Cost Accounting; Planningand Control, 8th edition. (Southwestern Publishing Company, Cincinnati, OH, 1984): 286-289. Monks, J. G., Operations Management: Theory and Problems, 2nd edition, (McGraw-Hill, Inc.. New York, 1982): 395,407-410. Moriarity, S. and Allen, C. P., Cost Accounting, (Harper & Row Publishers, New York, 1984): 264. Morse, W. J., Cost Accounting, 2nd edition, (Addison-Wesley Publishing Co., Inc., Reading, MA, 1981): 647. Morse, W. J., Davis, J. R. and Hartgraves, A. L., Management Accounring, (Addison-Wesley Publishing Company, Reading, MA, 1984): 536-537. Moscove, S. A., Crowningshield, G. R. and Gorman, K. A., Cost Accounting. 5th edition. (Houghton Mifflin Co., Boston, MA, 1985): 985. Norgaard, C. T., Management Accounting, (Prentice-Hall, Inc., Englewood Cliffs, NJ, 1985): 677-678. Rayburn, L. G., Principles of Cost Accounting: Managerial Applications, Revised edition, (Richard D. Irwin, Inc., Homewood, IL. 1982): 104-105. Shillinglaw, G., Managerial Cost Accounting, 5th edition. (Richard D. Irwin, Inc., Homewood. IL, 1982): 529-530.
A NOTE ON INVENTORY REORDER POINT DETERMINATION
CALIFORNIA
Raman C. Pate1 STATE UNIVERSITY,
CHICO
are as important as Abstract: Managerial decisions regarding timing of inventory replenishment determining the economic order size in inventory management. Yet most of the introductory texts on cost/ managerial accounting and production/ operations management do not adequately discuss reorder point determination, especially the condition on lead time demand that is necessary for valid application of the commonly reported formula for the reorder point. A clear understanding of the reorder point determination is important to students, teachers, and practicing managers alike. This note attempts to explain the underlying condition and modification needed in the reorder point formula.
Inventory management is an important function in business. Managerial decisions concerning the amount and timing of inventory replenishment become especially crucial in periods of high interest costs and cyclical swings in the economy. A thorough understanding of the concepts and methods involved in the determination of economic order quantity (EOQ) and reorder point (ROP) is essential to students, teachers, and practicing managers in business. In view of the importance of inventory management in business, the topic of EOQ and ROP determination should be an important component of the business curriculum. Since CPA, CMA and similar professional examinations often include questions on EOQ and ROP, it is especially important to cover the topic adequately in the accounting curriculum. The topic of EOQ and ROP determination is usually included in introductory cost/ managerial accounting and production/ operations management courses. Most textbooks (see list of references) for these courses introduce the topic with the discussion of the basic EOQ model. However, most of the discussion in these texts is concerned with explaining EOQ determination. Several of the texts fail to explain clearly the ROP determination in general. The coverage of ROPin various texts is often brief and inadequate, and it is mostly confined to a stochastic demand situation. A few texts’ explicitly give the formula for ROP in the case of deterministic demand, but do not discuss the underlying condition necessary for the valid See, for example, Anderson, Sweeney, and Williams [1983]; Chase and Aquilano [1985]; Garrison 119851; Killough and Leininger [1984]; Moriarity and Allen [1984]; and Morse [1981, 19841.
132 use of such a formula.
Even in the case of stochastic demand covered in most of the texts, the limitation on the use of the commonly mentioned formula for ROPis not discussed except in Monks, Horngren [I982 and 19841, Matz and Usry, and Morse. Monks [p. 407-4101 illustrates the use of the ROPformula under different situations through examples, but does not explicitly provide the necessary modification in the formula and fails to explain the underlying condition for its validity. Horngren [p. 7651 does footnote the situation where the commonly given ROP formula is inapplicable and gives the general formula, but without adequate explanation. Only Matz and Usry provide better than average coverage of the ROPdetermination. However, they fail to explain the determination of quantities due in from previous orders (outstanding orders) required in their formula. Morse [p. 6471 briefly remarks that the ROP will be greater than EOQ if lead period is large, but provides no further explanation. The purpose of this note is to point out the limitation on the use of the commonly given ROP formula and clearly explain the general formula for determining the reorder point. After a brief introduction of the basic EOQ model, the ROP determination is first presented assuming known demand and known lead period. The case of stochastic demand is then considered assuming a known lead period. Finally, an approximate formula for ROP is suggested for the most general case where both lead period and demand are stochastic.
BASIC
EOQ MODEL
The basic EOQ model is based upon the assumptions of known demand, D, during a period (T, usually a year) and known lead time (L, the time elapsed between placing an order and receiving the ordered quantity). The optimum order quantity, EOQ, is determined by considering the cost of ordering relative to inventory carrying charges for a given demand during the period as follows: Q* = /2D(O/H)
(1)
where Q* = economic order quantity, EOQ D = known demand during a period 0 ordering cost per order H = inventory holding (carrying) charges per unit per period. q
The associated optimum (minimum) given by TC* = d?i%%?-
total cost of inventory
during a period is
133
where TC* = minimum total inventory cost assuming inventory replenishments of the optimum quantity Q*. The number of orders, N, to be placed during the period under consideration is given by N= D/Q* and the inventory cycle time (t, the period between or placements of orders) is determined as t= T/N or alternatively
inventory
replenishments
(4)
as
t = (TJ D)Q* by substituting
N from (3). The expression
Q* = (01
(5) can be also written
as
T)t
= dt
(6)
expressing EOQ in terms of demand per unit of time (d = D/ T), and the cycle time, t. As an example, consider a product whose demand is 4,000 units during a year. If ordering cost is $18 per order and the cost of inventory holding is 90~ per unit per year, then expressions (1) through (4) provide the following results: Economic Order Quantity: Q* = 400 units per order Minimum Total Inventory Cost: TC* = 360 dollars Number of orders during a year: N = 10 orders Inventory cycle time: t = 0.1 year or t = 5.2 weeks
REORDER
POINT
DETERMINATION
The question of when to place an order of EOQ units as determined by (1) is essentially determined by the expected demand during the lead period. Clearly, the time to place an order is when the inventory level falls to a point where sufficient inventory exists to satisfy the expected demand during the lead period. Hence the reorder point is set equal to the expected demand during the lead period. Under the assumed known demand, the demand during the lead period is basically determined by the product of lead time, L, and demand per unit of time, d. Hence the reorder point is determined as R = dL
134
where R= ROP d = the demand
rate per unit of time.
The ROP formula (7) is obviously a function of lead time when demand is assumed known; that is, when dis a fixed known quantity. Hence, the validity of the ROPformula (7) commonly reported in various texts depends upon the magnitude of lead time L relative to the inventory cycle time, t. As long as the lead time is less than cycle time, the ROP formula (7) is valid in view of (6), since L < t implies dL < dt which is equivalent dL<
to the condition
Q*
(8)
This means the time to place an order is when inventory drops to the ROP, thereby assuring sufficient stock on hand to last until the order arrives at the end of current inventory cycle (or equivalently at the beginning of a new cycle) as illustrated in Figure 1.
FIGURE
1
t Inventory Q*
--
L
Slope = 4
I
a
-IA
.---
tt= --
l 1
Lead time is less than Cycle time (Lead time Demand
is less than EOC?)
135
For the example considered earlier, suppose lead time is 0.05 year or 2.6 weeks. Since L = 0.05 ( t = 0.1, the ROP is given by (7) as R= y
(0.05)
= 200 units If L is expressed
in weeks, then ROP is obtained
R = -y
from (7) as
(2.6)
= 200 units When demand is assumed known, the ROP formula is applicable so long as the lead time is less than cycle time. If, however, lead time is greater than inventory cycle time (that is, when the condition (8) is not satisfied so that lead time demand exceeds optimum order quantity), then the ROPformula is not applicable because it is not possible to place an order at an inventory level of more than the order quantity. For instance, suppose the lead period is 0.13 year or 6.76 weeks, which is greater than the cycle time of 0. I year or 5.2 weeks. Demand during the lead period is 520 units, which is greater than the EOQ. According to the ROPformula, R=520 would not be possible to use since it is impossible for inventory to drop to 520 units when EOQ= 400 units. The ROP formula needs to be modified when lead time demand exceeds the order quantity. Since the cycle time is determined by demand and order quantity while the lead period is primarily fixed by the supply environment, there is no assurance that L will be less than t as required for the valid use of formula (7) for ROP. When the condition L>t
(9)
or equivalently, dL>
Q*
flO>
prevails, the ROP formula requires modification by considering the manner in which the lead time demand is satisfied as illustrated in Figure 2 (following page) and explained below. The higher demand during the lead period is satisfied by the on-hand inventory at the time of placing an order (that is, ROP) and by the receipts of previously placed orders (outstanding orders) which arrive during the lead period, namely, Lead time demand = ROP + Receipts of previously placed orders arriving during the lead period.
136
FIGURE 2
Inventory
Time (C--=0.1
*I
+-L=o.13+,
1
L = 0.22 -
I-
Lead time is greater (Lead time demand
than Cycle time exceeds
EOQ)
Symbolically, dL = R + K(EOQ)
(11)
where K= the number of outstanding orders or replenishments (or complete inventory cycles) during the lead period. The value of Kis the integerpart of the ratio of lead time and cycle time, that is, K
q
integer (L/t)
(12)
Substituting Kfrom (12) in (1 I), the general formula for ROP when lead time demand exceeds EOQ is given by rearranging the terms as R = dL - [LJ t]Q*
(13)
where [L/t] is an integer. For the earlier example, suppose the lead period is 0.13 year or 6.76 weeks. Since the lead period is greater than the cycle time of 0.1 year or 5.2 weeks, the demand during lead period is 520 which is greater than the EOQ. Therefore, the ROPformula (7) would be inapplicable. The general ROPformula given by (13) provides the proper ROP as
137
R=
(400)
= 520 - [ 1.3](400) = 520 - l(400) = 120 units If lead period
is expressed
in weeks, then (13) gives
= 520 - l(400) = 120 units Suppose L = 0.22 year or 11.44 weeks, the ROP according formula (13) is obtained as
to the general
(400) = 880 - [2.23(400) = 880 - 2(400) = 80 units It may be noted that the usually reported formula (7) for ROP is a special case of the general expression for ROP given by (13) above. Whenever the lead period is less than cycle time their ratio L/t, is less than I and, therefore, the integer part of the ratio in (13) is zero and the ROPformula (13) reduces to the frequently reported formula (7). For example, when lead time is 0.05 year or 2.6 weeks, the ROP according to (13) is R=
No’
---i-
(0.5) -
= 200 - [0.5](400) = 200 - O(400) = 200 units as obtained earlier using formula (7). The general formula (13) for ROP is based on the assumption that demand is known. Also, it assumes that the lead period is known and independent of demand. In other words, the demand during the lead period is considered deterministic by assuming both the demand and the lead period are known. When either the lead period or the demand is uncertain, demand during the lead period is probabilistic. Therefore, the expected demand during the lead
138
period dL in the ROPformula (13) is uncertain and needs to be replaced by the mean of the underlying probability distribution of the lead period demand. Furthermore, provision of safety stock is needed to cover higher than average demand that may occur during the lead period because of variations in demand and/or lead period. The level of safety stock would depend upon the cost of holding additional inventory of such stock relative to the desired protection against stockouts during the lead period. Hence, there is no fixed method of determining the safety stock; it would depend on the assumed risk protection against stockouts and the assumptions regarding the underlying probability distribution of demand during the lead period. Whatever the safety stock is determined, the ROP in the case of stochastic demand can be generally set at the level of inventory sufficient to meet the average demand during the lead period plus the safety stock needed to cover the possibility of higher than average demand. Symbolically, R=dL+S
(14)
where d = average demand
per unit of time.
S = safety stock. This is the formula commonly given in various texts. The validity of the formula (14) is, however, limited to the situations where the demand during the lead period is less than the order quantity. There is a tacit assumption behind the formula (14) that the lead period is smaller than the cycle time, and hence the necessary condition that the lead period demand does not exceed the economic order quantity is normally satisfied. Such an assumed condition may not, however, always hold, since the lead period is primarily determined by the supply side while the order quantity is independently determined by the demand and relevant costs. A generally applicable formula for ROPin the case of stochastic demand is given by modifying the formula (14) as R = dL + S - [L/t]Q*
(15)
where all symbols are the same as previously defined. The general formula (15) for determining the ROP is applicable in any stochastic demand situation. The formula (15) is based only on the assumption that the lead period is a known constant. Such an assumption of known lead period is not unrealistic. Most frequently, the lead period is fixed contractually. Even if the lead period varies, it can be considered nearly equal to the average lead time so long as the variation is small relative to the average, which may often be the case in practice. In any event, the general formula (15) for ROP in the case of
139
stochastic demand is valid so long as the Iead period is assumed to be a known constant regardless of its magnitude. The fast term in the general formula (15) is zero whenever the lead period is less than the cycle time; and hence the general formula reduces to the commonly reported formula (14) for computing the ROP. The discussion of a situation where both the lead period and the demand are stochastic, each having an independent probability distribution, is beyond the scope of this note. It should be noted that the safety stock determination is not considered above because it is adequately discussed in the literature and is also beyond the scope of this note.
CONCLUSION Since demand during the lead period essentially determines the reorder point, and the lead time demand in turn is a function of both the demand rate and lead time, determination of the reorder point depends upon the nature of assumptions behind these two variables. As seen above, when demand is assumed to be known, the reorder point determination quite simply depends upon the magnitude of the lead time. The usual formula (7) for the reorder point is limited to situations where the lead time is less than the inventory cycle period. A general formula for ROP given by (I 3) above is, however, applicable under any situation so long as both the demand and the lead period are deterministic. When the lead period is known but the demand is probabilistic, the general formula for ROP is given by (15). It is hoped the above presentation has clarified the limitation of the commonly used formula for RUP, and demonstrated the need for a more complete explanation of the general formula in classroom presentations.
REFERENCES Adam, E. E., Jr. and Ebert, R. J., Production and Operations Management: Concepts, Models, and Behavior, 2nd edition, (Prentice-Hall, Inc., Englewood Cliffs, NJ, 1982). Anderson, D. R., Sweeney, D. J. and Williams, T. A., Quantitative Methodsfor Business, 2nd edition, (West Publishing, St. Paul MN, 1983): 487, 515. Bierman, H., Jr., Bonini, C. P. and Hausman, W. H., Quantitative Analysisfor Business Decisions, 6th edition, (Richard D. Irwin, Inc., Homewood, IL, 1981): 421-442. Buffa, E. S., Modern Production/Operations Management, 7th edition, (John Wiley & Sons, Inc., New York, 1983). Chase, R. B. and Aquilano, N. J., Production and Operations Management: A Life Cycle Approach, 4th edition, (Richard D. Irwin, Inc., Homewood, IL, 1985): 489,495-497. Cook, T. M. and Russell, R. A., Contemporary Operations Management: Text and Cases, 2nd edition, (Prentice-Hall, Inc., Englewood Cliffs, NJ, 1984): 358-362. Davidson, S., Maher, M. W., Stickney, C. P. and Weil, R. L., Manugeriul Accounting, 2nd edition, (The Dryden Press, Hinsdale, IL, 1985): 310-311. Deakin, E. B., and Maher, M. W., Cost Accounting, (Richard D. Irwin, Inc., Homewood, IL, 1984): 541-542.
140 Dominiak, G. F. and Louderback, J. Cl., III, ManngerialAccounri 4th edition, (Kent Publishing Company, Boston, MA, 1985): 652. Dopuch, N., Birnberg, J. G. and Demski, J. S., Cost Accounting, 3rd edition, (Harcourt Brace Jovanovich, Inc., New York, 1982): 661-662. Fischer, P. M. and Frank, W. Cl., Cost Accounting: Theory and Applications, (Southwestern Publishing Company, Cincinnati, OH, 1985): 541-545. Garrison, R., Managerial Accounting, 4th edition, (Business Publications, Inc., Plano, TX, 1985): 324-325. Horngren. C., Cost Accounting, 5th edition. (Prentice-Hall. Inc., Englewood Cliffs, NJ, 1982): 764-765. Horngren, C., Introduction to Management Accounting. 6th edition, (Prentice-Hall, Inc., Englewood Cliffs, NJ, 1984): 497-498. Kaplan, R. S., AdvancedManagement Accounting, (Prentice-Hall, Inc., Englewood Cliffs, NJ, 1982): 270-274). Killough, L. N. and Leininger, W. E., Cost Accounting, (West Publishing, St. Paul, MN, 1984): 655-658. Louderback, J. G., III, and Hirsch, M. L., Jr. (Kent Publishing Company, Boston, MA, 1982): 807-8 10. Matz. A. and Usry, M. F., Cost Accounting; Planningand Control, 8th edition. (Southwestern Publishing Company, Cincinnati, OH, 1984): 286-289. Monks, J. G., Operations Management: Theory and Problems, 2nd edition, (McGraw-Hill, Inc.. New York, 1982): 395,407-410. Moriarity, S. and Allen, C. P., Cost Accounting, (Harper & Row Publishers, New York, 1984): 264. Morse, W. J., Cost Accounting, 2nd edition, (Addison-Wesley Publishing Co., Inc., Reading, MA, 1981): 647. Morse, W. J., Davis, J. R. and Hartgraves, A. L., Management Accounring, (Addison-Wesley Publishing Company, Reading, MA, 1984): 536-537. Moscove, S. A., Crowningshield, G. R. and Gorman, K. A., Cost Accounting. 5th edition. (Houghton Mifflin Co., Boston, MA, 1985): 985. Norgaard, C. T., Management Accounting, (Prentice-Hall, Inc., Englewood Cliffs, NJ, 1985): 677-678. Rayburn, L. G., Principles of Cost Accounting: Managerial Applications, Revised edition, (Richard D. Irwin, Inc., Homewood, IL. 1982): 104-105. Shillinglaw, G., Managerial Cost Accounting, 5th edition. (Richard D. Irwin, Inc., Homewood. IL, 1982): 529-530.