On the optimal production stopping and restarting times for an EOQ model with deteriorating items
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Viewpoint On the optimal production stopping and restarting times for an EOQ model with deteriorating items I have read with interest a recent paper by Yan and Cheng1 where they considered an EOQ inventory model for deteriorating items. Shortages are allowed and the production, demand and deterioration rates are assumed to be known functions of time. Unfortunately, the proposed model has some gaps which lead to inaccurate mathematical analysis. In this short note, I shall indicate these gaps and correct the theory of the Yan and Cheng model, then I will brie¯y outline the correct mathematical analysis. Here, I shall use the same notations and assumptions as in Yan and Cheng,1 ie: (i)
A single item is produced at a rate p
t, consumed at a rate q
t and deteriorates at a rate y
t so that the production rate covers both the demand and deterioration, and there is no replacement of deteriorated items. (ii) Shortages are allowed but a fraction b
0 4 b 4 1 of the demand during the stock out period is backlogged and the remaining fraction (1 ÿ b) is lost. (iii) Each cycle starts at an inventory level B which is equal to the ending inventory level of the previous cycle, B 0 for the ®rst cycle, and the set-up cost need not to be considered. (iv) The cost components are, c is the unit production cost, h is the unit inventory holding cost per unit time, b is the unit shortage cost per unit time for backlogged items, l is the unit shortage cost per unit time for lost items, and S0 is the set-up cost per cycle.
A typical cycle for the general model where the shortages are allowed is shown in Figure 1.
The inventory levels I
t at time t, as they given in Yan and Cheng,1 are
t T0 4 t 4 T01
1 I
t ÿ p
u ÿ q
udu B; T0
I
t eÿd
t
t T01
p
u ÿ q
ued
u du;
I
t eÿd
t
S t
I
t ÿb
T01 4 t 4 T1
2
q
ued
u du;
t S
q
udu
T1 4 t 4 S s 4 t 4 T2
3
4
where d
t y
tdt. The total relevant cost per unit time during the cycle T0 ; T2 as it is given in Yan and Cheng1 is underestimated. It is given by (
T1 T1 1 c p
tdt h eÿd
t R1 T2 ÿ T0 T0 T01 !
t d
u
p
u ÿ q
ue du dt T01
h
S
bb
T1
e
ÿd
t
T2
t S
S
S t
q
ue
d
u
du dt
)
T2 q
udu dt l
1 ÿ b q
tdt S
Yan and Cheng1 have dropped two terms from the above total cost in an inaccurate way. Those are (
! ) T01
t b p
u ÿ q
udu B dt T 2 ÿ T0 T 0 T0 and S0 : T2 ÿ T0
Figure 1 Typical inventory cycle for the general model.
Note that both terms are not independent of our decision variables. Dropping these two terms may happen from assumption (iii) due to Yan and Cheng1 which is inaccurate. Because, the assumption B 0 for the ®rst cycle cannot hold for the general model where shortages are allowed. Otherwise, the analysis for the ®rst cycle carnot be applied to subsequent cycles. Furthermore, the authors indicated in their conclusion that B must be negative in the gene
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