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Note on an EOQ model with a temporary sale price
international journal of
production economics ELSEVIER
lnt. J. Production Economics 37 (1994) 241-243
Note on an EOQ model with a temporary
sale price
G.E. Martin Clarkson Unirwsi@
Potsdam, NY 13699-5790, USA
Accepted 26 July I994
Abstract A flaw is revealed relatively
significant
in the formulation of a fundamental temporary when the amount of the discount is small.
Tersine [l] develops a temporary sale price model whereby a special order quantity may be purchased upon availability of a limited price discount opportunity, such that the benefit which accrues during that single, longer purchase cycle, versus rejecting the special order opportunity, is maximized. That benefit or gain is modeled as y = TC, - TC,, where TC, = (P - d)Q’ + 0.5Q’(P represents production and
- d)FQ’/R
+ C
(1)
the total relevant cost during that cycle when the special order is taken
TC,=(P-d)Q*+P(Q’-Q*)+OSQ*(P-d)FQ*/ R + OSQ*PF(Q’
- Q*)/R + CQ’/Q*
(2)
represents the total cost during the same period when the standing EOQ policy is retained, using the following notation and assumptions. P d Q’
regular unit price unit price discount special order quantity
F R C Q*
sale price model.
However,
its impact
is only
annual holding cost fraction annual demand order cost/order regular EOQ
Assumptions (supplementary to the basic EOQ model): Sale price is temporary and is available at a normal inventory replenishment time. Replenishment lead time is zero. Note that relaxation of these two assumptions to create a more general model is achieved by straightforward adjustments [l] and is not considered further here. However, as is evident in Fig. 1, Tersine’s representation of average inyentory as 0.5Q* in the fourth term of Eq. (2) is clearly flawed. While it is true that average inventory I, during the time interval [T *, T’] will equal 0.5 Q* whenever the ratio of Q’ to Q* is some integer value, in all other instances that value will understate the true I,. The smaller the ratio of Q’ to Q* (occasioned by smaller discounts) and the more different its value from integer, the greater will be the discrepancy in I,. The result will be an understated maximum g* and l
l
0925.5273!94/$07.00 .I‘ 1994 Elsevier Science B.V. All rights reserved SSDI 0925-5273(94)00069-7
G.E.
242
Martin
111 Int. J. Ptvduction
Eumomicx
37 (lW4)
241
243
+ OSQ*(P
- d)FQ*/R
+ 0.5 I CQ’/Q*i Q*Z - (Q’ - ([Q’/Q*~ + 1)Q*)2) PF/R + CQ’/Q*.
(2’)
Clearly, closed-form expressions for Q’* and g* are sacrificed with the more precise formulation of g = TC, ~ TC,. However, a simple search for maximum g can be achieved by decrementing Q’ using Tersine’s value to initialize, Q’ = dR/(P - d) F + PQ*/(P - d)
time
Fig. 1. Inventory
pattern
for a temporary
sale price
a generally overstated optimal Q*. Others, e.g., Ardalan [Z, 31 and Aull-Hyde [4], have committed this same oversight. A true representation of I, will be (see Fig. 1) 1, = (OS[Q’/Q*
- l]Q*‘/R
+ O.S(Q* + q)(T’ - t);/(T’
- T*).
(3)
where T’ is the length of the special order cycle, t is that time which elapses in the special order cycle just prior to the fractional T * component, q is the fractional portion of a Q* remaining at T ‘, and [ .] represents the greatest integer function, i.e., the largest integer equal to or less than its argument. Since q = ([Q’/Q*] + l)Q* - Q’, t = [Q’/Q*] Q*/R, T’ = Q’IR, and T* = Q*/R, substitution and simplification of Eq. (3) yields I, = 0.5 { [Q’/Q*]
Q*’
- (Q’ - (CQ’/Q*I Therefore,
Eq. (2) should
TC, = (P - d)Q*
+ f)Q*)“)/(Q’ be revised to
+ f’(Q’ - Q*)
- Q*).
(4)
(5)
and computing g using Eqs. (1) and (2’). An example provided by Tersine will illustrate the consequences of the’error: P = $lO.OO/unit, d = $l.OO/unit, R = 8000 units/yr, C = $30.00/order, F = 0.30/yr. Using Tersine’s model, Q* = 400 units, Q’* = 3407 units, and g* = $1526.26. Using the more accurate representation of I,, Q’* = 3401 units and y* = $1533.75, revealing respective errors of only 0.18% and 0.49%. Even when the holding cost fraction is increased to 50% of unit valuation per year, the discrepancies in Q’* and g* increase only to 2.76% and 0.62%. respectively. Table 1 provides results of some alternative discount values with the above parameters.
Table I Optimal special order quantity discounts Tersine
and maximum
gain for various
% error
Corrected
d
Q’
Y(W
Q’
9 ($)
Q’
Y
0.10 0.25 0.50 0.75 1.00 I.25 1.50 2.00 2.50 5.00
673 1094 1825 2595 3407 4267 5176 7167 9422 27467
13.88 88.05 361.48 835.32 1526.26 2452.91 3636.08 6868. I5 11446.93 6868 1.62
635 I044 1818 2594 3401 4228 5083 7153 9419 27463
20.87 94.73 368.91 842.8 1 1533.75 2460.03 3640.90 6871.16 11454.38 68688.44
5.98 4.79 0.39 0.04 0.18 0.92 1.83 0.20 0.03 0.01
33.49 7.05 2.0
I
0.89 0.49 0.29 0.13 0.04 0.07 0.0
I
G.E. Martin
J ht.
J. Production
In conclusion, Tersine’s temporary sale price model is based on a deflated representation of average inventory during the special discount purchase cycle when the original EOQ policy is retained. However, the worst-case error does not appear to have material consequences on the objective function unless the discount is quite small, and even then, only on a relative basis. Tersine’s closed-form expressions will serve as useful approximations for the optimal values of special order quantity and discount benefit gained for most realistic circumstances.
Economics
37 (1994) 241-243
243
References Cl1 Tersine, R.J., 1994. Principles
of Inventory and Materials Management, 4th ed. Prentice-Hall, Englewood Cliffs. NJ. I21 Ardalan, A., 1988. Optimal ordering policies in response to a sale. IIE Trans., 20: 1922194. c31Ardalan, A.. 1991. Combined optimal price and optimal inventory replenishment policies when a sale results in increase in demand. Comput. Oper. Res., 18: 721-730. R.L., 1992. Evaluation of supplier-restricted II41 Am-Hyde, purchasing options under temporary price discounts. IIE Trans., 24: 184~186.
production economics ELSEVIER
lnt. J. Production Economics 37 (1994) 241-243
Note on an EOQ model with a temporary
sale price
G.E. Martin Clarkson Unirwsi@
Potsdam, NY 13699-5790, USA
Accepted 26 July I994
Abstract A flaw is revealed relatively
significant
in the formulation of a fundamental temporary when the amount of the discount is small.
Tersine [l] develops a temporary sale price model whereby a special order quantity may be purchased upon availability of a limited price discount opportunity, such that the benefit which accrues during that single, longer purchase cycle, versus rejecting the special order opportunity, is maximized. That benefit or gain is modeled as y = TC, - TC,, where TC, = (P - d)Q’ + 0.5Q’(P represents production and
- d)FQ’/R
+ C
(1)
the total relevant cost during that cycle when the special order is taken
TC,=(P-d)Q*+P(Q’-Q*)+OSQ*(P-d)FQ*/ R + OSQ*PF(Q’
- Q*)/R + CQ’/Q*
(2)
represents the total cost during the same period when the standing EOQ policy is retained, using the following notation and assumptions. P d Q’
regular unit price unit price discount special order quantity
F R C Q*
sale price model.
However,
its impact
is only
annual holding cost fraction annual demand order cost/order regular EOQ
Assumptions (supplementary to the basic EOQ model): Sale price is temporary and is available at a normal inventory replenishment time. Replenishment lead time is zero. Note that relaxation of these two assumptions to create a more general model is achieved by straightforward adjustments [l] and is not considered further here. However, as is evident in Fig. 1, Tersine’s representation of average inyentory as 0.5Q* in the fourth term of Eq. (2) is clearly flawed. While it is true that average inventory I, during the time interval [T *, T’] will equal 0.5 Q* whenever the ratio of Q’ to Q* is some integer value, in all other instances that value will understate the true I,. The smaller the ratio of Q’ to Q* (occasioned by smaller discounts) and the more different its value from integer, the greater will be the discrepancy in I,. The result will be an understated maximum g* and l
l
0925.5273!94/$07.00 .I‘ 1994 Elsevier Science B.V. All rights reserved SSDI 0925-5273(94)00069-7
G.E.
242
Martin
111 Int. J. Ptvduction
Eumomicx
37 (lW4)
241
243
+ OSQ*(P
- d)FQ*/R
+ 0.5 I CQ’/Q*i Q*Z - (Q’ - ([Q’/Q*~ + 1)Q*)2) PF/R + CQ’/Q*.
(2’)
Clearly, closed-form expressions for Q’* and g* are sacrificed with the more precise formulation of g = TC, ~ TC,. However, a simple search for maximum g can be achieved by decrementing Q’ using Tersine’s value to initialize, Q’ = dR/(P - d) F + PQ*/(P - d)
time
Fig. 1. Inventory
pattern
for a temporary
sale price
a generally overstated optimal Q*. Others, e.g., Ardalan [Z, 31 and Aull-Hyde [4], have committed this same oversight. A true representation of I, will be (see Fig. 1) 1, = (OS[Q’/Q*
- l]Q*‘/R
+ O.S(Q* + q)(T’ - t);/(T’
- T*).
(3)
where T’ is the length of the special order cycle, t is that time which elapses in the special order cycle just prior to the fractional T * component, q is the fractional portion of a Q* remaining at T ‘, and [ .] represents the greatest integer function, i.e., the largest integer equal to or less than its argument. Since q = ([Q’/Q*] + l)Q* - Q’, t = [Q’/Q*] Q*/R, T’ = Q’IR, and T* = Q*/R, substitution and simplification of Eq. (3) yields I, = 0.5 { [Q’/Q*]
Q*’
- (Q’ - (CQ’/Q*I Therefore,
Eq. (2) should
TC, = (P - d)Q*
+ f)Q*)“)/(Q’ be revised to
+ f’(Q’ - Q*)
- Q*).
(4)
(5)
and computing g using Eqs. (1) and (2’). An example provided by Tersine will illustrate the consequences of the’error: P = $lO.OO/unit, d = $l.OO/unit, R = 8000 units/yr, C = $30.00/order, F = 0.30/yr. Using Tersine’s model, Q* = 400 units, Q’* = 3407 units, and g* = $1526.26. Using the more accurate representation of I,, Q’* = 3401 units and y* = $1533.75, revealing respective errors of only 0.18% and 0.49%. Even when the holding cost fraction is increased to 50% of unit valuation per year, the discrepancies in Q’* and g* increase only to 2.76% and 0.62%. respectively. Table 1 provides results of some alternative discount values with the above parameters.
Table I Optimal special order quantity discounts Tersine
and maximum
gain for various
% error
Corrected
d
Q’
Y(W
Q’
9 ($)
Q’
Y
0.10 0.25 0.50 0.75 1.00 I.25 1.50 2.00 2.50 5.00
673 1094 1825 2595 3407 4267 5176 7167 9422 27467
13.88 88.05 361.48 835.32 1526.26 2452.91 3636.08 6868. I5 11446.93 6868 1.62
635 I044 1818 2594 3401 4228 5083 7153 9419 27463
20.87 94.73 368.91 842.8 1 1533.75 2460.03 3640.90 6871.16 11454.38 68688.44
5.98 4.79 0.39 0.04 0.18 0.92 1.83 0.20 0.03 0.01
33.49 7.05 2.0
I
0.89 0.49 0.29 0.13 0.04 0.07 0.0
I
G.E. Martin
J ht.
J. Production
In conclusion, Tersine’s temporary sale price model is based on a deflated representation of average inventory during the special discount purchase cycle when the original EOQ policy is retained. However, the worst-case error does not appear to have material consequences on the objective function unless the discount is quite small, and even then, only on a relative basis. Tersine’s closed-form expressions will serve as useful approximations for the optimal values of special order quantity and discount benefit gained for most realistic circumstances.
Economics
37 (1994) 241-243
243
References Cl1 Tersine, R.J., 1994. Principles
of Inventory and Materials Management, 4th ed. Prentice-Hall, Englewood Cliffs. NJ. I21 Ardalan, A., 1988. Optimal ordering policies in response to a sale. IIE Trans., 20: 1922194. c31Ardalan, A.. 1991. Combined optimal price and optimal inventory replenishment policies when a sale results in increase in demand. Comput. Oper. Res., 18: 721-730. R.L., 1992. Evaluation of supplier-restricted II41 Am-Hyde, purchasing options under temporary price discounts. IIE Trans., 24: 184~186.