- Email: [email protected]
Inventory models with ramp type demand rate, partial backlogging and Weibull deterioration rate
Available online at www.sciencedirect.com
European Journal of Operational Research 192 (2009) 79–92 www.elsevier.com/locate/ejor
Production, Manufacturing and Logistics
Inventory models with ramp type demand rate, partial backlogging and Weibull deterioration rate K. Skouri
a,*
, I. Konstantaras a, S. Papachristos b, I. Ganas
c
a
Department of Mathematics, University of Ioannina, 45110 Ioannina, Greece Department of Agribusiness, University of Ioannina, 45110 Ioannina, Greece Department of Accounting, Technological Education Institute of Epirus, 48100 Preveza, Greece b
c
Received 14 September 2006; accepted 1 September 2007 Available online 11 September 2007
Abstract In this paper, an inventory model with general ramp type demand rate, time dependent (Weibull) deterioration rate and partial backlogging of unsatisfied demand is considered. The model is studied under the following different replenishment policies: (a) starting with no shortages and (b) starting with shortages. The model is fairly general as the demand rate, up to the time point of its stabilization, is a general function of time. The backlogging rate is any non-increasing function of the waiting time up to the next replenishment. The optimal replenishment policy for the model is derived for both the above mentioned policies. Ó 2007 Elsevier B.V. All rights reserved. Keywords: Inventory; Ramp type demand; Deteriorating items; Partial backlogging
1. Introduction Maintenance of inventories of deteriorating items is a problem of major concern in the supply chain of almost any business organizations. Many of the physical goods undergo decay or deterioration over time. Commodities such as fruits, vegetables, foodstuffs, are subject to direct spoilage while kept in store. Highly volatile liquids such as gasoline, alcohol, turpentine, undergo physical depletion over time through the process of evaporation. Electronic goods, radioactive substances, photographic film, grain, deteriorate through a gradual loss of potential or utility with the passage of time. A model with exponentially decaying inventory was initially proposed by Ghare and Schrader (1963). Covert and Philip (1973) and Tadikamalla (1978) developed an economic order quantity model with Weibull and Gamma distributed deterioration rates, respectively. The assumption of constant demand rate is usually valid in the mature stage of a product’s life cycle. In the growth and/ or end stage life cycle, demand rate may well be approximated by a linear function. Resh et al. (1976) and Donaldson (1977) were the first who studied a model with linearly time varying demand. Since then, several researchers have studied deteriorating inventory models with time varying demand under a variety of modeling assumptions (e.g. Dave and Patel, 1981; Sachan, 1984; Goyal, 1987; Datta and Pal, 1988; Goswami and Chaudhuri, 1991; Hariga, 1996; Yang et al., 2001).
*
Corresponding author. Tel.: +30 26510 98230. E-mail address: [email protected] (K. Skouri).
0377-2217/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2007.09.003
80
K. Skouri et al. / European Journal of Operational Research 192 (2009) 79–92
Two excellent surveys on recent trends in modeling of continuously deteriorating inventory are those by Raafat (1991) and Goyal and Giri (2001). In most of the above papers two types of time varying demand rate have been considered: (i) linear positive/negative trend in demand rate and (ii) exponentially increasing/decreasing demand rate. However, demand cannot increase continuously over time. For example, demand rate for fashionable products, increases with time up to a certain moment (only if customers are satisfied with quality and price) and then stabilizes to a constant rate. The term ‘‘ramp type’’ is used to represent such demand pattern. Hill (1995) proposed an inventory model with increasing demand (general power of time) followed by a constant demand. Mandal and Pal (1998) considered an inventory model for exponentially decaying items by allowing shortages. Wu et al. (1999) related the backlogging rate to the waiting time up to the next replenishment (partial backlogging). Wu and Ouyang (2000) studied the Mandal and Pal’s (1998) inventory model under two different replenishment policies: (a) those starting with no shortages and (b) those starting with shortages. Wu (2001) investigated an inventory model with ramp type demand rate, Weibull distributed deterioration rate and partial backlogging. Giri et al. (2003) extended the ramp type demand inventory model with a more generalized Weibull deterioration distribution. In the above cited papers, the determination of the optimal replenishment policy requires the determination of the time point, when the inventory level reaches zero. So two cases should be examined: (1) this time point occurs before the time point, at which the demand is stabilized, and (2) this time point occurs after the time point, where the demand is stabilized. To the best of our knowledge almost all of the researchers examine only the first case. So the analysis is incomplete as the investigation covers a part of the feasible solution space and the results obtained are questionable. Deng et al. (2007) revisit the inventory model considered by Mandal and Pal (1998) and Wu and Ouyang (2000). They study it by considering the two cases given above and comment on these questionable results. In the most of the above referred papers, complete backlogging of unsatisfied demand is assumed. In practice, there are customers who are willing to wait and receive their orders at the end of shortage period, while others are not. Inventory models, which consider a mixture of backorders and lost sales for non-deteriorating products, were proposed by Montgomery et al. (1973), Park (1982), and Rosenberg (1979). These authors assumed that only a fixed fraction of demand during the stockout time is backlogged and the rest is lost. In the last few years, considerable attention has been paid to inventory models with partial backlogging. The backlogging rate can be modelled taking into account the customers’ behavior. The first paper in which customers’ impatience functions are proposed seems to be that by Abad (1996). Chang and Dye (1999) developed a finite horizon inventory model using Abad’s reciprocal backlogging rate. Skouri and Papachristos (2002) studied a multi-period inventory model using the negative exponential backlogging rate proposed by Abad. Teng et al. (2002) extended the Chang and Dye’s (1999) and Skouri and Papachristos’ (2002) models, assuming as backlogging rate any decreasing function of the waiting time up to the next replenishment. Research on models with partial backlogging continues with Wang (2002) and San Jose et al. (2005, 2006). In this paper we extend the work of Deng et al. (2007) as follows: (i) we introduce a general ramp type demand rate, with its variable part being any positive function of time. Note that research appeared up to now takes this part as either linear or exponential function. (ii) We consider Weibull distributed deterioration rate, and (iii) we consider a backlogging rate, which is any non-increasing function of the waiting time up to the next replenishment. The paper is organized as follows. The notation and assumptions used in the models are given in Section 2. The model starting with no shortages is studied in Section 3 and the corresponding one starting with shortages is studied in Section 4. For each model the optimal policy is obtained. Numerical examples highlighting the results obtained are given in Section 5. The paper closes with concluding remarks in Section 6.
2. Notation and assumptions 2.1. Notation T t1 S c1 c2 c3 c4 l I(t)
the constant scheduling period (cycle) the time when the inventory level reaches zero for the model starting without shortage (Figs. 1 and 2), or the replenishment time for the model starting with shortage (Figs. 3 and 4) the maximum inventory level at the scheduling period (cycle) the inventory holding cost per unit time the shortage cost per unit time the cost incurred from the deterioration of one unit the per unit opportunity cost due to the lost sales parameter of the ramp type demand function (time point) the inventory level at time t 2 [0, T]
K. Skouri et al. / European Journal of Operational Research 192 (2009) 79–92
81
2.2. Assumptions The inventory model is developed under the following assumptions: 1. The ordering quantity brings the inventory level up to the order level S. Replenishment rate is infinite. 0 2. Shortages are backlogged at a rate b(x), which is a non-increasing function of x (b (x) 6 0) with 0 6 b(x) 6 1, b(0) = 1 and x is the waiting time up to the next replenishment. Moreover it is assumed that b(x) satisfies the relation b(x) + Tb 0 (x) P 0, where b 0 (x) is the derivate of b(x). The cases with b(x) = 1 (or 0) correspond to complete backlogging (or complete lost sales) models. 3. The time to deterioration of the item is distributed as Weibull (a, b), i.e. the deterioration rate is h(t) = abtb1 (a > 0, b > 0, t > 0). There is no replacement or repair of deteriorated units during the period T. For b = 1, h(t) becomes constant, which corresponds to exponentially decaying case. 4. The demand rate D(t) is a ramp type function of time given by DðtÞ ¼
f ðtÞ; f ðlÞ;
t < l; t P l;
where f(t) is a positive, continuous function of t 2 (0, T].
3. The mathematical formulation of the model starting with no shortages In this section the inventory model starting with no shortages is studied. The replenishment at the beginning of the cycle brings the inventory level up to S. Due to demand and deterioration, the inventory level gradually depletes during the period (0, t1) and falls to zero at t = t1. Thereafter shortages occur during the period (t1, T), which are partially backlogged. The backlogged demand is satisfied at the next replenishment. The inventory level, I(t), 0 6 t 6 T satisfies the following differential equations: dIðtÞ þ hðtÞIðtÞ ¼ DðtÞ; dt dIðtÞ ¼ DðtÞbðT tÞ; dt
0 6 t 6 t1 ; Iðt1 Þ ¼ 0;
ð1Þ
t1 6 t 6 T ; Iðt1 Þ ¼ 0:
ð2Þ
The solutions of these differential equations are affected from the relation between t1 and l through the demand rate function. To continue the two cases: (i) t1 < l and (ii) t1 > l must be considered. The realization of the inventory level for the two cases is depicted in Figs. 1 and 2, respectively.
Inventory level
S
0 t1
μ
T
Time
Fig. 1. Inventory level for the model starting with no shortage over the cycle (case t1 < l).
82
K. Skouri et al. / European Journal of Operational Research 192 (2009) 79–92
Inventory level
S
0 μ
t1
T
Time
Fig. 2. Inventory level for the model starting with no shortage over the cycle (case t1 > l).
3.1. Case I (t1 6 l) (see Fig. 1) In this case, Eq. (1) becomes dIðtÞ þ abtb1 IðtÞ ¼ f ðtÞ; dt
0 6 t 6 t1 ; Iðt1 Þ ¼ 0:
ð3Þ
Eq. (2) leads to the following two: dIðtÞ ¼ f ðtÞbðT tÞ; t1 6 t 6 l; Iðt1 Þ ¼ 0; dt dIðtÞ ¼ f ðlÞbðT tÞ; l 6 t 6 T ; Iðl Þ ¼ Iðlþ Þ: dt
ð4Þ ð5Þ
The solutions of (3)–(5) are, respectively Z t1 b b IðtÞ ¼ eat f ðxÞeax dx; 0 6 t 6 t1 ; t Z t f ðxÞbðT xÞ dx; t1 6 t 6 l IðtÞ ¼
ð6Þ ð7Þ
t1
and IðtÞ ¼ f ðlÞ
Z
t
bðT xÞdx l
Z
l
f ðxÞbðT xÞdx;
ð8Þ
l 6 t 6 T:
t1
The total amount of deteriorated items during [0, t1] is Z t1 Z t1 atb D¼ f ðtÞe dt f ðtÞ dt: 0
0
The cumulative inventory carried in the interval [0, t1] is found from (6) and is Z t 1 Z t1 Z t1 atb axb I1 ¼ IðtÞ dt ¼ e f ðxÞe dx dt: 0
0
t
Due to (7) and (8), the time-weighted backorders due to shortages during the interval [t1, T] are I2 ¼
Z
T
½IðtÞdt ¼
t1
¼
Z
t1
Z t1
l
½IðtÞdt þ
Z
T
½IðtÞdt l
l
ðl tÞf ðtÞbðT tÞdt þ f ðlÞ
Z
T l
Z l
t
bðT xÞdx dt þ
Z
T l
Z t1
l
f ðxÞbðT xÞdx dt:
K. Skouri et al. / European Journal of Operational Research 192 (2009) 79–92
83
The amount of lost sales during [t1, T] is L¼
Z
l
ð1 bðT tÞÞf ðtÞdt þ f ðlÞ
Z
T
ð1 bðT tÞÞdt:
l
t1
The total cost in the time interval [0, T] is the sum of holding, shortage, deterioration and opportunity costs and is given by TC 1 ðt1 Þ ¼ c1 I 1 þ c2 I 2 þ c3 D þ c4 L Z t 1 Z t 1 Z b b ¼ c1 eat f ðxÞeax dx dt þ c3 0
þ c2
t
Z
l
ðl tÞf ðtÞbðT tÞdt þ f ðlÞ
t1
þ c4
Z
l
ð1 bðT tÞÞf ðtÞdt þ f ðlÞ
t1
0 T
Z
Z
l
Z
b
f ðtÞeat dt
Z
t1
f ðtÞdt 0
t
Z bðT xÞdx dt þ
l
Z
l
T
T
l
f ðxÞbðT xÞdx dt
t1
ð1 bðT tÞÞdt :
ð9Þ
l
t1
3.2. Case II (t1 > l) (see Fig. 2) In this case, Eq. (1) reduces to the following two: dIðtÞ þ abtb1 IðtÞ ¼ f ðtÞ; 0 6 t 6 l; Iðl Þ ¼ Iðlþ Þ; dt dIðtÞ þ abtb1 IðtÞ ¼ f ðlÞ; l 6 t 6 t1 ; Iðt1 Þ ¼ 0: dt Eq. (2) becomes dIðtÞ ¼ f ðlÞbðT tÞ; dt
t1 6 t 6 T ; Iðt1 Þ ¼ 0:
Their solutions are, respectively Z l Z b b IðtÞ ¼ eat f ðxÞeax dx þ f ðlÞ
b eax dx ;
ð10Þ
0 6 t 6 l;
l
t
Z
b
IðtÞ ¼ eat f ðlÞ IðtÞ ¼ f ðlÞ
t1
t1
b
eax dx;
ð11Þ
l 6 t 6 t1 ;
t
Z
t
bðT xÞdx;
ð12Þ
t1 6 t 6 T :
t1
The total amount of deteriorated items during [0, t1] is D ¼ Ið0Þ
Z
t1
DðtÞdt ¼
Z
0
l
atb
f ðtÞe dt þ f ðlÞ
0
Z
t1
atb
e dt
Z
l
f ðtÞdt f ðlÞðt1 lÞ: 0
l
The total inventory carried during the interval [0, t1] is found from (10) and (11) and is I1 ¼ ¼
Z Z
t1
IðtÞdt ¼ 0 l
eat 0
b
Z
Z
l
IðtÞdt þ 0
l
Z
t1
IðtÞdt l
b
f ðxÞeax dx þ f ðlÞ
Z
t1
Z b eax dx dt þ f ðlÞ
l
t
t1 l
eat
b
Z
t1
b eax dx dt:
t
The time-weighted backorders due to shortages during the interval [t1, T] are I2 ¼
Z
T
½IðtÞdt ¼ f ðlÞ t1
Z t1
T
Z
t
bðT xÞdx dt ¼ f ðlÞ
t1
Z t1
T
ðT xÞbðT xÞdx:
84
K. Skouri et al. / European Journal of Operational Research 192 (2009) 79–92
The lost sales in the interval [t1, T] are L ¼ f ðlÞ
Z
T
½1 bðT tÞdt:
t1
The inventory cost for this case is TC 2 ðt1 Þ ¼ c1 I 1 þ c2 I 2 þ c3 D þ c4 L Z l Z l Z atb axb e f ðxÞe dx þ f ðlÞ ¼ c1 0
t
Z T þ c2 f ðlÞ ðT xÞbðT xÞdx Z
þ c3
t1 l
atb
f ðtÞe dt þ f ðlÞ
0
Z
t1
atb
t1
Z dx dt þ f ðlÞ
l
e dt l
e
axb
t1
e
atb
Z
l
Z
t1
e
dx dt
t
l
axb
f ðtÞdt f ðlÞðt1 lÞ þ c4 f ðlÞ 0
Z
T
ð1 bðT tÞÞdt :
ð13Þ
t1
The total cost function of the system over [0, T] takes the form TC 1 ðt1 Þ; if t1 6 l; TCðt1 Þ ¼ TC 2 ðt1 Þ; if l < t1 :
ð14Þ
It is easy to check that this function is continuous at l. The problem now is the minimization of this, two branches, function TC(t1). This requires, separately, studying each of these branches and then combine the results to state the algorithm giving the optimal policy. 3.3. The optimal replenishment policy In this subsection we present the results, which ensure the existence of a unique t1, say t1 , which minimizes the total cost function for the model starting without shortages. Although the argument t1 of the functions TC1(t1), TC2(t1) is constrained, we shall search for their unconstrained minimum. The first and second order derivatives of TC1(t1) are, respectively dTC 1 ðt1 Þ ¼ f ðt1 Þgðt1 Þ; dt1 d2 TC 1 ðt1 Þ df ðt1 Þ dgðt1 Þ ¼ gðt1 Þ þ f ðt1 Þ ; dt21 dt1 dt1
ð15Þ
where b
gðt1 Þ ¼ c1 eat1
Z
t1
b
b
eat dt þ c3 ðeat1 1Þ c2 ðT t1 ÞbðT t1 Þ c4 ð1 bðT t1 ÞÞ:
ð16Þ
0
From (16) it is easily verified that, when 0 < b(x) 6 1, g(0) < 0, g(T) > 0 and further Z t1 dgðt1 Þ b b b ¼ c1 abt1b1 eat1 eat dt þ c1 þ c3 abt1b1 eat1 þ c2 ½bðT t1 Þ þ ðT t1 Þb0 ðT t1 Þ c4 b0 ðT t1 Þ > 0: dt1 0
ð17Þ
The inequality step follows from the assumption (2) made for b (x)(b(x) non-increasing so b 0 (x) 6 0 and b(x) + Tb 0 (x) P 0), and implies that g(t1) is strictly increasing. By assumption f(t1) > 0 and so the derivative dTCdt11ðt1 Þ vanishes at t1 , with 0 < t1 < T , which is the unique root of gðt1 Þ ¼ 0: For this
t1
ð18Þ
we have
2
d TC 1 ðt1 Þ df ðt1 Þ dgðt1 Þ dgðt1 Þ ¼ gðt1 Þ þ f ðt1 Þ ¼ f ðt1 Þ >0 2 dt1 dt1 dt1 dt1 so that t1 corresponds to the unconstrained global minimum of TC1(t1). If t1 is feasible, i.e. t1 6 l, the optimal value of the order level, S = I(0), is Z t 1 b S ¼ f ðtÞeat dt 0
ð19Þ
K. Skouri et al. / European Journal of Operational Research 192 (2009) 79–92
85
and subsequently the optimal order quantity Q* is Q ¼ S þ
Z
l
bðT tÞf ðtÞdt þ f ðlÞ t1
Z
T
bðT tÞdt:
ð20Þ
l
1Þ For the case b(x) = 0, dgðt > 0 and g(0) < 0, but g(T) maybe less than or greater to zero. If g(T) > 0 then t1 corresponds to dt1 the unconstrained global minimum of TC1(t1). If moreover t1 6 l, then the optimal value of the order level and the order quantity is given by (19) and (20), respectively. If g(T) < 0 then TC1(t1) is strictly decreasing attaining its minimum at l. Now we come to the branch TC2(t1). Its first and second order derivatives are
dTC 2 ðt1 Þ ¼ f ðlÞgðt1 Þ; dt1 d2 TC 2 ðt1 Þ dgðt1 Þ ¼ f ðlÞ > 0; dt21 dt1
ð21Þ ð22Þ
where the function g(t1) is given by (16). The inequality in (22) follows from (17) and ensures the strict convexity of TC2(t1). When 0 < b(x) 6 1, based on the properties of g(t1) we conclude that dTCdt21ðt1 Þ is vanished at the point t1 with 0 < t1 < T , which is the unique solution of gðt1 Þ ¼ 0:
ð23Þ
This t1 corresponds to the unconstrained global minimum of TC2(t1). Note that Eq. (23) is exactly the same as (18). If t1 is feasible i.e. t1 P l, the optimal value of the order level, S = I(0), is Z l Z t 1 b atb S ¼ f ðtÞe dt þ f ðlÞ eat dt ð24Þ 0
l
and the optimal order quantity, Q*, is Z T bðT tÞdt: Q ¼ S þ f ðlÞ
ð25Þ
t1
The previous analysis has shown that the two functions TC1(t1) and TC2(t1) have the same (unique) unconstrained minimizing point t1 2 ð0; T Þ, which is determined by (18) or (23). Based on (15) and (21) we can further establish that the function TC(t1) is differentiable at the point l and so l cannot be a corner point forTC(t1). 1Þ For the case b(x) = 0, dgðt > 0 and g(0) < 0, again g(T) maybe less or greater than zero. If g(T) > 0 then t1 corresponds dt1 to the unconstrained global minimum of TC2 (t1). If moreover t1 > l then the optimal value of the order level and the order quantity is given by (24) and (25), respectively. If g(T) < 0 then TC2(t1) is strictly decreasing attaining its minimum at T. Now we can give the procedure that leads to the optimal replenishment policy for the cases 0 < b(x) 6 1 and b(x) = 0 A. 0 < b(x) 6 1: Step 1. Compute t1 from (18) or (23). Step 2. Compare t1 to l. 2.1 If t1 6 l then the optimal order quantity and the total cost function are given from (20) and (9), respectively. 2.2 If t1 > l then the optimal order quantity and the total cost function are given from (25) and (13), respectively. B. b(x)=0 Step 1. Compute g(T). 1.1 If g(T) > 0 then go to step 1 of the previous procedure R l (case A). RT b b 1.2 If g(T) 6 0 then t1 ¼ T and consequently S ¼ Q ¼ 0 f ðtÞeat dt þ f ðlÞ l eat dt (we note that this is an expected result since g(T) 6 0 is the consequence of high lost sales cost, c4). Remark 1. The previous analysis shows that t1 is independent from the demand rate D(t). Although this seems to be a somewhat peculiar result, it agrees with the classical result that the point t1 is independent from the demand rate, known to be valid in many order level inventory systems (Naddor, 1966, p.67).
86
K. Skouri et al. / European Journal of Operational Research 192 (2009) 79–92
4. The mathematical formulation of the model starting with shortages In this section the inventory model starting with shortages is studied. The cycle now starts with shortages, which occur during the period [0, t1] and are partially backlogged. At time t1 a replenishment brings the inventory level up to S. Demand and deterioration of the items depletes the inventory level during the period [t1, T] until this falls to zero at t = T. Again the two cases t1 < l and t1 P l must be examined. 4.1. Case I (t1 < l) (see Fig. 3) The inventory level, I(t), 0 6 t 6 T satisfies the following differential equations: dIðtÞ ¼ f ðtÞbðt1 tÞ; 0 6 t 6 t1 ; Ið0Þ ¼ 0; dt dIðtÞ þ abtb1 IðtÞ ¼ f ðtÞ; t1 6 t 6 l; Iðl Þ ¼ Iðlþ Þ; dt dIðtÞ þ abtb1 IðtÞ ¼ f ðlÞ; l 6 t 6 T ; IðT Þ ¼ 0: dt The solutions of (26)–(28) are respectively Z t f ðxÞbðt1 xÞdx; 0 6 t 6 t1 ; IðtÞ ¼ 0 Z l Z T atb axb axb IðtÞ ¼ e e f ðxÞdx þ f ðlÞ e dx ;
ð26Þ ð27Þ ð28Þ
ð29Þ ð30Þ
t1 6 t 6 l
l
t
and b
IðtÞ ¼ eat f ðlÞ
Z
T
b
eax dx;
ð31Þ
l 6 t 6 T:
t
The total amount of deteriorated units during [t1, T] is Z l Z l Z T atb1 atb atb D¼e f ðtÞe dt þ f ðlÞ e dt f ðtÞdt f ðlÞðT lÞ: l
t1
t1
The total inventory carried during the interval [t1, T] is found using (30) and (31) and is Z l Z T Z l Z T Z T b b b b b I1 ¼ eat f ðxÞeax dx þ f ðlÞ eax dx dt þ f ðlÞ eat eax dx dt: t1
l
t
l
t
Due (29) the time-weighted backorders during the time interval [0, t1] are Z t1 I2 ¼ ðt1 tÞf ðtÞbðt1 tÞdt: 0
Inventory level
S
0 t1
μ
T
Time
Fig. 3. Inventory level for the model starting with shortages over the cycle (case t1 < l).
K. Skouri et al. / European Journal of Operational Research 192 (2009) 79–92
87
The amount of lost sales during in [0, t1] is Z t1 L¼ ½1 bðt1 tÞf ðtÞdt: 0
The inventory cost during the time interval [0, T] is the sum of holding, shortage, deterioration and opportunity costs and is given by TC 1 ðt1 Þ ¼ c1 I 1 þ c2 I 2 þ c3 D þ c4 L Z l Z l Z atb axb ¼ c1 e f ðxÞe dx þ f ðlÞ t1
t
þ c4 ð1 bðt1 tÞÞf ðtÞdt þ c3 e
T
e
axb
Z dx dt þ f ðlÞ
l
atb1
Z
T
e
atb
Z
l l
atb
f ðtÞe dt þ f ðlÞ
T
e t
Z
T
atb
e dt l
t1
axb
Z dx dt þ
t1
½c2 ðt1 tÞbðt1 tÞ
0
Z
l
f ðtÞdt f ðlÞðT lÞ :
ð32Þ
t1
4.2. Case II (t1 > l) (see Fig. 4) The inventory level, I(t), 0 6 t 6 T satisfies the following differential equations: dIðtÞ ¼ f ðtÞbðt1 tÞ; 0 6 t 6 l; Ið0Þ ¼ 0 dt dIðtÞ ¼ f ðlÞbðt1 tÞ; l 6 t 6 t1 ; Iðl Þ ¼ Iðlþ Þ; dt dIðtÞ þ abtb1 IðtÞ ¼ f ðlÞ; t1 6 t 6 T ; IðT Þ ¼ 0: dt Their solutions are, respectively, Z t f ðxÞbðt1 xÞdx; 0 6 t 6 l; IðtÞ ¼ 0 Z l Z t IðtÞ ¼ f ðxÞbðt1 xÞdx f ðlÞ bðt1 xÞdx; 0 b
IðtÞ ¼ eat f ðlÞ
ð33Þ ð34Þ ð35Þ
ð36Þ ð37Þ
l 6 t 6 t1 ;
l
Z
T
b
eax dx;
ð38Þ
t1 6 t 6 T :
t
The number of deteriorated items and the inventory carried during [t1, T] are, respectively Z T Z T Z T atb1 atb atb axb D ¼ e f ðlÞ e dt f ðlÞðT t1 Þ and I 1 ¼ f ðlÞ e e dx dt: t1
t1
t
Inventory level
S
0
μ t1
T
Time
Fig. 4. Inventory level for the model starting with shortages over the cycle (case t1 > l).
88
K. Skouri et al. / European Journal of Operational Research 192 (2009) 79–92
The time-weighted backorders due to shortages during [0, t1] using (36) and (37) are Z l Z t Z t 1 Z l Z t f ðxÞbðt1 xÞdx dt þ f ðxÞbðt1 xÞdx þ f ðlÞbðt1 xÞdx dt: I2 ¼ 0
0
l
0
l
The amount of lost sales in the interval [0, t1] is Z l Z t1 L¼ ð1 bðt1 tÞÞf ðtÞdt þ ð1 bðt1 tÞÞf ðlÞdt: 0
l
The total cost for this case is TC 2 ðt1 Þ ¼ c1 I 1 þ c2 I 2 þ c3 D þ c4 L Z T Z T atb axb ¼ c1 f ðlÞ e e dx dt þ c2
t1
Z 0
t
Z
l
t
Z f ðxÞbðt1 xÞdx dt þ
0 b
þ c3 eat1 f ðlÞ þ c4
Z
l T
b
eat dt f ðlÞðT t1 Þ
t1
Z
t1
l
½1 bðt1 tÞf ðtÞdt þ f ðlÞ
Z
0
Z
t1
l
f ðxÞbðt1 xÞdx þ f ðlÞ 0
Z
t
bðt1 xÞdx dt
l
½1 bðt1 tÞdt :
ð39Þ
l
The total cost function of the system over [0, T] takes the form TC 1 ðt1 Þ; if t1 6 l; TCðt1 Þ ¼ TC 2 ðt1 Þ; if l < t1 :
ð40Þ
It is easy to see that TC(t1) is continuous at l. The problem now is the minimization of this, two branches, function TC(t1). 4.3. The optimal replenishment policy In this subsection we derive the optimal replenishment policy, i.e., we calculate the value, say t1 , which minimizes the total cost function. Taking the first order derivative of TC1(t1) and equating it to zero gives Z l Z T b1 atb1 axb axb e f ðxÞdx þ f ðlÞ e dx ðc1 þ c3 abt1 Þe þ
l
t1
Z
t1
½c2 bðt1 tÞ þ c2 ðt1 tÞb0 ðt1 tÞ c4 b0 ðt1 tÞf ðtÞdt ¼ 0:
ð41Þ
0
If t1 is a root of (41), for this root the second order condition for minimum is "Z # Z T l b b b1 b1 atb ax ax ½c1 c3 ðb 1Þ þ c3 abðt1 Þ e 1 abðt1 Þ e f ðxÞdx þ f ðlÞ e dx þ ½c1 þ c3 abðt1 Þb1 f ðt1 Þ þ ½c2 bð0Þ c4 b0 ð0Þf ðt1 Þ þ
t1
Z
t1
l
½2c2 b0 ðt1 tÞ þ c2 ðt1 tÞb00 ðt1 tÞ c4 b00 ðt1 tÞf ðtÞdt > 0:
ð42Þ
0
So, if (42) holds and t1 6 l then the value of order level, S, is
S ¼
Iðt1 Þ
¼e
at1
b
"Z
l
axb
e f ðxÞdx þ f ðlÞ t1
Z
#
T
e
axb
dx ;
ð43Þ
l
the ordering quantity is
Q ¼
Z
t1
f ðxÞbðt1 xÞdx þ S
0
and the total cost is TC 1 ðt1 Þ.
ð44Þ
K. Skouri et al. / European Journal of Operational Research 192 (2009) 79–92
89
Equating the first order derivative of TC2(t1) to zero gives Z T Z l Z 0 0 b atb1 axb e dx þ ½c2 bðt1 tÞ þ c2 ðt1 tÞb ðt1 tÞ c4 b ðt1 tÞf ðtÞdt þ f ðlÞ ðc1 þ c3 abt1 Þe f ðlÞ 0
t1
t1
½c2 bðt1 tÞ l
þ c2 ðt1 tÞb0 ðt1 tÞ c4 b0 ðt1 tÞdt ¼ 0:
ð45Þ
If t1 is a root of (45), for this root the second order condition for minimum is ½c1 þ þ
b1 b1 b c3 abt1 eat1 abt1 f ðlÞ
Z
l
Z
T
e
axb
dx c3 abðb
t1
ðb2Þ b 1Þt1 eat1 f ðlÞ
Z
T
t1
2c2 b0 ðt1 tÞ þ c2 ðt1 tÞb00 ðt1 tÞ c4 b00 ðt1 tÞ f ðtÞdt þ f ðlÞ
0
b1
b
eax dx þ ½c1 þ c3 abt1 f ðlÞ þ c2 f ðlÞ Z
t1
½2c2 b0 ðt1 tÞ þ c2 ðt1 tÞb00 ðt1 tÞ
l
c4 b00 ðt1 tÞdt > 0:
ð46Þ
So if (46) holds and t1 > l then the value of S is b
S ¼ Iðt1 Þ ¼ eat1 f ðlÞ
Z
T
b
eax dx;
ð47Þ
t1
the ordering quantity is
Q ¼
Z
l
f ðxÞbðt1
xÞdx þ f ðlÞ
Z
0
t1
bðt1 xÞdx þ S
ð48Þ
l
and the total cost is TC 2 ðt1 Þ. Remark 2. Due to (41) and (45) the function TC(t1) is differentiable at the point l. In the previous analysis there is no guarantee that t1 exists and corresponds to the minimum. Its uniqueness is also another issue. The proposition, which follows, provides sufficient conditions for existence, uniqueness and validity of t1 . Let us set Z T Z l b b eax dx þ ½c2 bðl tÞ þ c2 ðl tÞb0 ðl tÞ c4 b0 ðl tÞf ðtÞdt D ¼ ½c1 þ c3 ablb1 eal f ðlÞ l 0
0
0
and h(t) = c2b(t) + c2tb (t) c4b (t). The following propositions can be easily proved: 1 Proposition 1. If b 6 1, T b1 P ab , h(t) + h 0 (t) > 0t 2 [0, T], and
(1) D > 0 then the optimal value of t1 follows from (41). (2) D < 0 then the optimal value of t1 follows from (45). Proposition 1. If D > 0 then (41) has at least a root, say t1 . If for every root of (41) ðc1 þ c3 abt1b1 Þðt1b1 1Þ > c3 ðb 1Þt1b1 hðtÞþ 0 h0 ðtÞ > 0t 2 ½0; T and h(t) + h (t) > 0 then t1 is the unique optimal value of the problem. 2. If D < 0 then (45) has at least a root, say t1 . If for every root of (45) ðc1 þ c3 abt1b1 Þðt1b1 1Þ > c3 ðb 1Þt1b1 hðtÞþ h0 ðtÞ > 0t 2 ½0; T and h(t) + h 0 (t) > 0 then t1 is the unique optimal value of the problem. We note that the above propositions ensure the existence, uniqueness and validity of t1. If the conditions of this proposition do not hold then the following procedure can be used to calculate the optimal replenishment policy: Step 1. Step 1.1. Find the global minimizing point, t1 , for TC1(t1). This will be one of the following points: (a) a root of (41) (an interior point of [0, l]) which satisfies (42), (b) t1 ¼ 0, (c) t1 ¼ l. Then calculate TC 1 ðt1 Þ.
90
K. Skouri et al. / European Journal of Operational Research 192 (2009) 79–92
40
30
20
10
0.2
0.4
0.6
0.8
1
Fig. 5. The graphical representation of the total cost function (Example 1, model starting without shortages).
Step 1.2. Find the global minimizing, t1 , for TC2(t1). This will be one of the following points: (a) a root of (45) (an interior point of [l,T]) which satisfies (46), (b) t1 ¼ l, (c) t1 ¼ T . Then calculate TC 2 ðt1 Þ. Step 2. Find the min fTC 1 ðt1 Þ; TC 2 ðt1 Þg and accordingly select the optimum t1 .
Remark 5. The analysis shows that, in this model, t1 is dependent from the demand rate D(t). 5. Numerical examples The examples, which follow, illustrate the results obtained. Example 1. This is taken from Wu (2001) and is adapted to our models. The input parameters are: c1 = $3 per unit per year, c2 = $15 per unit per year, c3 = $5 per unit, c4 = $20 per unit, l = 0.12 year, a = 0.001, b = 2, T = 1 year, f(t) = 3e4.5t and b (x) = e0.2x. Model starting with no shortages: Using (19) (or (24)) the optimal value of t1 is t1 ¼ 0:8604 > l. The optimal ordering quantity is Q* = 4.999 (from (25)) and the minimum cost is TCðt1 Þ ¼ 6:6432 (from (13)). Fig. 5 represents the total cost function during [0, T]. Model starting with shortages: t1 ¼ 0:1631 (from (45)) ðl < t1 Þ; Q ¼ 4:999 (from (48)), and TCðt1 Þ ¼ 6:3777 (from (39)) (see Fig. 6). Example 2. This example is identical to Example 1, except that l=0.9. Model starting with no shortages: t1 ¼ 0:8604ðl > t1 Þ; Q ¼ 54:5141 (from (20)) and TCðt1 Þ ¼ 91:8508 (from (9)).
22.5 20 17.5 15 12.5 10 7.5
0.2
0.4
0.6
0.8
1
Fig. 6. The graphical representation of the cost function (Example 1, model starting with shortages).
K. Skouri et al. / European Journal of Operational Research 192 (2009) 79–92
91
Model starting with shortages: t1 ¼ 0:5673 (from (41)) ðl > t1 Þ; Q ¼ 54:56 (from (44)) and TCðt1 Þ ¼ 63:109 (from (32)). From these examples we can observe that the total cost for the model starting with shortages is less than the total cost for the model starting with no shortages. This observation agrees with known, results from literature concerning the finite horizon inventory models (Teng et al., 1997; Skouri and Papachristos, 2003). 6. Concluding remarks In this paper, an order level inventory model for deteriorating items has been studied. The model is fairly general as, the demand rate is any function of time up to the time-point of its stabilization (general ramp type demand rate), and the backlogging rate is any non-increasing function of the waiting time, up to the next replenishment. The inventory model is studied under two different replenishment policies: (a) starting with no shortages and (b) starting with shortages. Therefore, the proposed model leads to the following existing ones: 1 1. If f(t) = D0t, t 2 [0, l] and bðxÞ ¼ 1þdx , our model reduces to that of Wu (2001). 2. If additionally d = 0 (case of complete backlogging), then it further reduces to that of Wu et al. (1999).We point out that Wu et al. (1999) and Wu (2001) are restricted only to the case t1 < l. The case where t1 could be greater than l has not been examined. 3. If b = 1 (i.e. the deterioration distribution is the exponential one), f(t) = D0t and b(x) = 1, then our model reduces to those of Wu and Ouyang (2000), Mandal and Pal (1998) and Deng et al. (2007).
Acknowledgements The authors are grateful to the anonymous referees who provided valuable comments and suggestions to improve the paper. This research was co-financed by the European Union (European Social Fund, 75%) and the Hellenic Ministry of National Educational and Religions Affairs (25%) in the framework of the Operational Programme II for Educational and Initial Vocational Training, ARCHIMEDES project. References Abad, P.L., 1996. Optimal pricing and lot sizing under conditions of perishability and partial backordering. Management Science 42, 1093–1104. Chang, H.J., Dye, C.Y., 1999. An EOQ model for deteriorating items with time varying demand and partial backlogging. Journal of the Operational Research Society 50, 1176–1182. Covert, R.P., Philip, G.C., 1973. An EOQ model for items with Weibull distribution deterioration. AIIE Transaction 5, 323–326. Datta, T.K., Pal, A.K., 1988. Order level inventory system with power demand pattern for items with variable rate of deterioration. Indian Journal of Pure and Applied Mathematics 19 (11), 1043–1053. Dave, U., Patel, L.K., 1981. (T, Si) policy inventory model for deteriorating items with time proportional demand. Journal of the Operational Research Society 32, 137–142. Deng, P.S., Lin, R.H.-J., Chu, P., 2007. A note on the inventory models for deteriorating items with ramp type demand rate. European Journal of Operational Research 178, 112–120. Donaldson, W.A., 1977. Inventory replenishment policy for a linear trend in demand: An analytic solution. Operational Research Quarterly 28, 663–670. Ghare, P.M., Schrader, G.F., 1963. A model for exponentially decaying inventories. Journal of Industrial Engineering 14, 238–243. Giri, B.C., Jalan, A.K., Chaudhuri, K.S., 2003. Economic order quantity model with Weibull deterioration distribution, shortage and ramp-type demand. International Journal of Systems Science 34 (4), 237–243. Goswami, A., Chaudhuri, K.S., 1991. An EOQ model for deteriorating items with shortages and a linear trend in demand. Journal of the Operation Research Society 42, 1105–1110. Goyal, S.K., 1987. Economic ordering policy for deteriorating items over an infinite time horizon. European Journal of Operational Research 28, 298–301. Goyal, S.K., Giri, B.C., 2001. Recent trends in modeling of deteriorating inventory. European Journal of Operational Research 134, 1–16. Hariga, M., 1996. Optimal EOQ models for deteriorating items with time-varying demand. Journal of the Operational Research Society 47, 1228–1246. Hill, R.M., 1995. Inventory model for increasing demand followed by level demand. Journal of the Operational Research Society 46, 1250–1259. Mandal, B., Pal, A.K., 1998. Order level inventory system with ramp type demand rate for deteriorating items. Journal of Interdisciplinary Mathematics 1, 49–66. Montgomery, D.C., Bazaraa, M.S., Keswani, A.K., 1973. Inventory models with a mixture of backorders and lost sales. Naval Research Logistics Quarterly 20, 255–263. Naddor, E., 1966. Inventory Systems. John Wiley and Sons, Inc. Park, K.S., 1982. Inventory model with partial backorders. International Journal of Systems Sciences 13, 1313–1317. Raafat, F., 1991. Survey of literature on continuously deteriorating inventory model. Journal of the Operational Research Society 42, 27–37. Resh, M., Friedman, M., Barbosa, L.C., 1976. On a general solution of the deterministic lot size problem with time-proportional demand. Operations Research 24, 718–725.
92
K. Skouri et al. / European Journal of Operational Research 192 (2009) 79–92
Rosenberg, D., 1979. A new analysis of a lot size model with partial backlogging. Naval Research Logistics Quarterly 26, 346–353. Sachan, R.S., 1984. On (T, Si) inventory policy model for deteriorating items with time proportional demand. Journal of the Operational Research Society 35 (11), 1013–1019. San Jose, L.A., Sicilia, J., Garcia-Laguna, J., 2005. An inventory system with partial backlogging modeled according to a linear function. Asia-Pacific Journal of Operational Research 22, 189–209. San Jose, L.A., Sicilia, J., Garcia-Laguna, J., 2006. Analysis of an inventory system with exponential partial backordering. International Journal of Production Economics 100, 76–86. Skouri, K., Papachristos, S., 2002. A continuous review inventory model, with deteriorating items, time-varying demand, linear replenishment cost, partially time-varying backlogging. Applied Mathematical Modelling 26 (5), 603–617. Skouri, K., Papachristos, S., 2003. Four inventory models for deteriorating items with time varying demand and partial backlogging: A cost comparison. Optimal Control Application and Methods 24, 315–330. Tadikamalla, P.R., 1978. An EOQ inventory model for items with Gamma distribution. AIIE Transaction 5, 100–103. Teng, J.-T., Chern, M.-S., Yang, H.-L., 1997. An optimal recursive method for various inventory replenishment models with increasing demand and shortages. Naval Research Logistics 44, 791–806. Teng, J.T., Chang, H.J., Dye, C.Y., Hung, C.H., 2002. An optimal replenishment policy for deteriorating items with time-varying demand and partial backlogging. Operations Research Letters 30, 387–393. Wang, S.P., 2002. An inventory replenishment policy for deteriorating items with shortages and partial backlogging. Computers and Operations Research 29, 2043–2051. Wu, K.-S., 2001. An EOQ inventory model for items with Weibull distribution deterioration, ramp type demand rate and partial backlogging. Production Planning and Control 12 (8), 787–793. Wu, K.-S., Ouyang, L.-Y., 2000. A replenishment policy for deteriorating items with ramp type demand rate. Proceedings of the National Science Council, Republic of China, Part A: Physical Science and Engineering 24 (4), 279–286. Wu, J.-W., Lin, C., Tan, B., Lee, W.-C., 1999. An EOQ inventory model with ramp type demand rate for items with Weibull deterioration. Information and Management Science 10 (3), 41–51. Yang, H.L., Teng, J.T., Chern, M.S., 2001. Deterministic inventory lot-size models under inflation with shortages and deterioration for fluctuating demand. Naval Research Logistics 48, 144–158.
European Journal of Operational Research 192 (2009) 79–92 www.elsevier.com/locate/ejor
Production, Manufacturing and Logistics
Inventory models with ramp type demand rate, partial backlogging and Weibull deterioration rate K. Skouri
a,*
, I. Konstantaras a, S. Papachristos b, I. Ganas
c
a
Department of Mathematics, University of Ioannina, 45110 Ioannina, Greece Department of Agribusiness, University of Ioannina, 45110 Ioannina, Greece Department of Accounting, Technological Education Institute of Epirus, 48100 Preveza, Greece b
c
Received 14 September 2006; accepted 1 September 2007 Available online 11 September 2007
Abstract In this paper, an inventory model with general ramp type demand rate, time dependent (Weibull) deterioration rate and partial backlogging of unsatisfied demand is considered. The model is studied under the following different replenishment policies: (a) starting with no shortages and (b) starting with shortages. The model is fairly general as the demand rate, up to the time point of its stabilization, is a general function of time. The backlogging rate is any non-increasing function of the waiting time up to the next replenishment. The optimal replenishment policy for the model is derived for both the above mentioned policies. Ó 2007 Elsevier B.V. All rights reserved. Keywords: Inventory; Ramp type demand; Deteriorating items; Partial backlogging
1. Introduction Maintenance of inventories of deteriorating items is a problem of major concern in the supply chain of almost any business organizations. Many of the physical goods undergo decay or deterioration over time. Commodities such as fruits, vegetables, foodstuffs, are subject to direct spoilage while kept in store. Highly volatile liquids such as gasoline, alcohol, turpentine, undergo physical depletion over time through the process of evaporation. Electronic goods, radioactive substances, photographic film, grain, deteriorate through a gradual loss of potential or utility with the passage of time. A model with exponentially decaying inventory was initially proposed by Ghare and Schrader (1963). Covert and Philip (1973) and Tadikamalla (1978) developed an economic order quantity model with Weibull and Gamma distributed deterioration rates, respectively. The assumption of constant demand rate is usually valid in the mature stage of a product’s life cycle. In the growth and/ or end stage life cycle, demand rate may well be approximated by a linear function. Resh et al. (1976) and Donaldson (1977) were the first who studied a model with linearly time varying demand. Since then, several researchers have studied deteriorating inventory models with time varying demand under a variety of modeling assumptions (e.g. Dave and Patel, 1981; Sachan, 1984; Goyal, 1987; Datta and Pal, 1988; Goswami and Chaudhuri, 1991; Hariga, 1996; Yang et al., 2001).
*
Corresponding author. Tel.: +30 26510 98230. E-mail address: [email protected] (K. Skouri).
0377-2217/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2007.09.003
80
K. Skouri et al. / European Journal of Operational Research 192 (2009) 79–92
Two excellent surveys on recent trends in modeling of continuously deteriorating inventory are those by Raafat (1991) and Goyal and Giri (2001). In most of the above papers two types of time varying demand rate have been considered: (i) linear positive/negative trend in demand rate and (ii) exponentially increasing/decreasing demand rate. However, demand cannot increase continuously over time. For example, demand rate for fashionable products, increases with time up to a certain moment (only if customers are satisfied with quality and price) and then stabilizes to a constant rate. The term ‘‘ramp type’’ is used to represent such demand pattern. Hill (1995) proposed an inventory model with increasing demand (general power of time) followed by a constant demand. Mandal and Pal (1998) considered an inventory model for exponentially decaying items by allowing shortages. Wu et al. (1999) related the backlogging rate to the waiting time up to the next replenishment (partial backlogging). Wu and Ouyang (2000) studied the Mandal and Pal’s (1998) inventory model under two different replenishment policies: (a) those starting with no shortages and (b) those starting with shortages. Wu (2001) investigated an inventory model with ramp type demand rate, Weibull distributed deterioration rate and partial backlogging. Giri et al. (2003) extended the ramp type demand inventory model with a more generalized Weibull deterioration distribution. In the above cited papers, the determination of the optimal replenishment policy requires the determination of the time point, when the inventory level reaches zero. So two cases should be examined: (1) this time point occurs before the time point, at which the demand is stabilized, and (2) this time point occurs after the time point, where the demand is stabilized. To the best of our knowledge almost all of the researchers examine only the first case. So the analysis is incomplete as the investigation covers a part of the feasible solution space and the results obtained are questionable. Deng et al. (2007) revisit the inventory model considered by Mandal and Pal (1998) and Wu and Ouyang (2000). They study it by considering the two cases given above and comment on these questionable results. In the most of the above referred papers, complete backlogging of unsatisfied demand is assumed. In practice, there are customers who are willing to wait and receive their orders at the end of shortage period, while others are not. Inventory models, which consider a mixture of backorders and lost sales for non-deteriorating products, were proposed by Montgomery et al. (1973), Park (1982), and Rosenberg (1979). These authors assumed that only a fixed fraction of demand during the stockout time is backlogged and the rest is lost. In the last few years, considerable attention has been paid to inventory models with partial backlogging. The backlogging rate can be modelled taking into account the customers’ behavior. The first paper in which customers’ impatience functions are proposed seems to be that by Abad (1996). Chang and Dye (1999) developed a finite horizon inventory model using Abad’s reciprocal backlogging rate. Skouri and Papachristos (2002) studied a multi-period inventory model using the negative exponential backlogging rate proposed by Abad. Teng et al. (2002) extended the Chang and Dye’s (1999) and Skouri and Papachristos’ (2002) models, assuming as backlogging rate any decreasing function of the waiting time up to the next replenishment. Research on models with partial backlogging continues with Wang (2002) and San Jose et al. (2005, 2006). In this paper we extend the work of Deng et al. (2007) as follows: (i) we introduce a general ramp type demand rate, with its variable part being any positive function of time. Note that research appeared up to now takes this part as either linear or exponential function. (ii) We consider Weibull distributed deterioration rate, and (iii) we consider a backlogging rate, which is any non-increasing function of the waiting time up to the next replenishment. The paper is organized as follows. The notation and assumptions used in the models are given in Section 2. The model starting with no shortages is studied in Section 3 and the corresponding one starting with shortages is studied in Section 4. For each model the optimal policy is obtained. Numerical examples highlighting the results obtained are given in Section 5. The paper closes with concluding remarks in Section 6.
2. Notation and assumptions 2.1. Notation T t1 S c1 c2 c3 c4 l I(t)
the constant scheduling period (cycle) the time when the inventory level reaches zero for the model starting without shortage (Figs. 1 and 2), or the replenishment time for the model starting with shortage (Figs. 3 and 4) the maximum inventory level at the scheduling period (cycle) the inventory holding cost per unit time the shortage cost per unit time the cost incurred from the deterioration of one unit the per unit opportunity cost due to the lost sales parameter of the ramp type demand function (time point) the inventory level at time t 2 [0, T]
K. Skouri et al. / European Journal of Operational Research 192 (2009) 79–92
81
2.2. Assumptions The inventory model is developed under the following assumptions: 1. The ordering quantity brings the inventory level up to the order level S. Replenishment rate is infinite. 0 2. Shortages are backlogged at a rate b(x), which is a non-increasing function of x (b (x) 6 0) with 0 6 b(x) 6 1, b(0) = 1 and x is the waiting time up to the next replenishment. Moreover it is assumed that b(x) satisfies the relation b(x) + Tb 0 (x) P 0, where b 0 (x) is the derivate of b(x). The cases with b(x) = 1 (or 0) correspond to complete backlogging (or complete lost sales) models. 3. The time to deterioration of the item is distributed as Weibull (a, b), i.e. the deterioration rate is h(t) = abtb1 (a > 0, b > 0, t > 0). There is no replacement or repair of deteriorated units during the period T. For b = 1, h(t) becomes constant, which corresponds to exponentially decaying case. 4. The demand rate D(t) is a ramp type function of time given by DðtÞ ¼
f ðtÞ; f ðlÞ;
t < l; t P l;
where f(t) is a positive, continuous function of t 2 (0, T].
3. The mathematical formulation of the model starting with no shortages In this section the inventory model starting with no shortages is studied. The replenishment at the beginning of the cycle brings the inventory level up to S. Due to demand and deterioration, the inventory level gradually depletes during the period (0, t1) and falls to zero at t = t1. Thereafter shortages occur during the period (t1, T), which are partially backlogged. The backlogged demand is satisfied at the next replenishment. The inventory level, I(t), 0 6 t 6 T satisfies the following differential equations: dIðtÞ þ hðtÞIðtÞ ¼ DðtÞ; dt dIðtÞ ¼ DðtÞbðT tÞ; dt
0 6 t 6 t1 ; Iðt1 Þ ¼ 0;
ð1Þ
t1 6 t 6 T ; Iðt1 Þ ¼ 0:
ð2Þ
The solutions of these differential equations are affected from the relation between t1 and l through the demand rate function. To continue the two cases: (i) t1 < l and (ii) t1 > l must be considered. The realization of the inventory level for the two cases is depicted in Figs. 1 and 2, respectively.
Inventory level
S
0 t1
μ
T
Time
Fig. 1. Inventory level for the model starting with no shortage over the cycle (case t1 < l).
82
K. Skouri et al. / European Journal of Operational Research 192 (2009) 79–92
Inventory level
S
0 μ
t1
T
Time
Fig. 2. Inventory level for the model starting with no shortage over the cycle (case t1 > l).
3.1. Case I (t1 6 l) (see Fig. 1) In this case, Eq. (1) becomes dIðtÞ þ abtb1 IðtÞ ¼ f ðtÞ; dt
0 6 t 6 t1 ; Iðt1 Þ ¼ 0:
ð3Þ
Eq. (2) leads to the following two: dIðtÞ ¼ f ðtÞbðT tÞ; t1 6 t 6 l; Iðt1 Þ ¼ 0; dt dIðtÞ ¼ f ðlÞbðT tÞ; l 6 t 6 T ; Iðl Þ ¼ Iðlþ Þ: dt
ð4Þ ð5Þ
The solutions of (3)–(5) are, respectively Z t1 b b IðtÞ ¼ eat f ðxÞeax dx; 0 6 t 6 t1 ; t Z t f ðxÞbðT xÞ dx; t1 6 t 6 l IðtÞ ¼
ð6Þ ð7Þ
t1
and IðtÞ ¼ f ðlÞ
Z
t
bðT xÞdx l
Z
l
f ðxÞbðT xÞdx;
ð8Þ
l 6 t 6 T:
t1
The total amount of deteriorated items during [0, t1] is Z t1 Z t1 atb D¼ f ðtÞe dt f ðtÞ dt: 0
0
The cumulative inventory carried in the interval [0, t1] is found from (6) and is Z t 1 Z t1 Z t1 atb axb I1 ¼ IðtÞ dt ¼ e f ðxÞe dx dt: 0
0
t
Due to (7) and (8), the time-weighted backorders due to shortages during the interval [t1, T] are I2 ¼
Z
T
½IðtÞdt ¼
t1
¼
Z
t1
Z t1
l
½IðtÞdt þ
Z
T
½IðtÞdt l
l
ðl tÞf ðtÞbðT tÞdt þ f ðlÞ
Z
T l
Z l
t
bðT xÞdx dt þ
Z
T l
Z t1
l
f ðxÞbðT xÞdx dt:
K. Skouri et al. / European Journal of Operational Research 192 (2009) 79–92
83
The amount of lost sales during [t1, T] is L¼
Z
l
ð1 bðT tÞÞf ðtÞdt þ f ðlÞ
Z
T
ð1 bðT tÞÞdt:
l
t1
The total cost in the time interval [0, T] is the sum of holding, shortage, deterioration and opportunity costs and is given by TC 1 ðt1 Þ ¼ c1 I 1 þ c2 I 2 þ c3 D þ c4 L Z t 1 Z t 1 Z b b ¼ c1 eat f ðxÞeax dx dt þ c3 0
þ c2
t
Z
l
ðl tÞf ðtÞbðT tÞdt þ f ðlÞ
t1
þ c4
Z
l
ð1 bðT tÞÞf ðtÞdt þ f ðlÞ
t1
0 T
Z
Z
l
Z
b
f ðtÞeat dt
Z
t1
f ðtÞdt 0
t
Z bðT xÞdx dt þ
l
Z
l
T
T
l
f ðxÞbðT xÞdx dt
t1
ð1 bðT tÞÞdt :
ð9Þ
l
t1
3.2. Case II (t1 > l) (see Fig. 2) In this case, Eq. (1) reduces to the following two: dIðtÞ þ abtb1 IðtÞ ¼ f ðtÞ; 0 6 t 6 l; Iðl Þ ¼ Iðlþ Þ; dt dIðtÞ þ abtb1 IðtÞ ¼ f ðlÞ; l 6 t 6 t1 ; Iðt1 Þ ¼ 0: dt Eq. (2) becomes dIðtÞ ¼ f ðlÞbðT tÞ; dt
t1 6 t 6 T ; Iðt1 Þ ¼ 0:
Their solutions are, respectively Z l Z b b IðtÞ ¼ eat f ðxÞeax dx þ f ðlÞ
b eax dx ;
ð10Þ
0 6 t 6 l;
l
t
Z
b
IðtÞ ¼ eat f ðlÞ IðtÞ ¼ f ðlÞ
t1
t1
b
eax dx;
ð11Þ
l 6 t 6 t1 ;
t
Z
t
bðT xÞdx;
ð12Þ
t1 6 t 6 T :
t1
The total amount of deteriorated items during [0, t1] is D ¼ Ið0Þ
Z
t1
DðtÞdt ¼
Z
0
l
atb
f ðtÞe dt þ f ðlÞ
0
Z
t1
atb
e dt
Z
l
f ðtÞdt f ðlÞðt1 lÞ: 0
l
The total inventory carried during the interval [0, t1] is found from (10) and (11) and is I1 ¼ ¼
Z Z
t1
IðtÞdt ¼ 0 l
eat 0
b
Z
Z
l
IðtÞdt þ 0
l
Z
t1
IðtÞdt l
b
f ðxÞeax dx þ f ðlÞ
Z
t1
Z b eax dx dt þ f ðlÞ
l
t
t1 l
eat
b
Z
t1
b eax dx dt:
t
The time-weighted backorders due to shortages during the interval [t1, T] are I2 ¼
Z
T
½IðtÞdt ¼ f ðlÞ t1
Z t1
T
Z
t
bðT xÞdx dt ¼ f ðlÞ
t1
Z t1
T
ðT xÞbðT xÞdx:
84
K. Skouri et al. / European Journal of Operational Research 192 (2009) 79–92
The lost sales in the interval [t1, T] are L ¼ f ðlÞ
Z
T
½1 bðT tÞdt:
t1
The inventory cost for this case is TC 2 ðt1 Þ ¼ c1 I 1 þ c2 I 2 þ c3 D þ c4 L Z l Z l Z atb axb e f ðxÞe dx þ f ðlÞ ¼ c1 0
t
Z T þ c2 f ðlÞ ðT xÞbðT xÞdx Z
þ c3
t1 l
atb
f ðtÞe dt þ f ðlÞ
0
Z
t1
atb
t1
Z dx dt þ f ðlÞ
l
e dt l
e
axb
t1
e
atb
Z
l
Z
t1
e
dx dt
t
l
axb
f ðtÞdt f ðlÞðt1 lÞ þ c4 f ðlÞ 0
Z
T
ð1 bðT tÞÞdt :
ð13Þ
t1
The total cost function of the system over [0, T] takes the form TC 1 ðt1 Þ; if t1 6 l; TCðt1 Þ ¼ TC 2 ðt1 Þ; if l < t1 :
ð14Þ
It is easy to check that this function is continuous at l. The problem now is the minimization of this, two branches, function TC(t1). This requires, separately, studying each of these branches and then combine the results to state the algorithm giving the optimal policy. 3.3. The optimal replenishment policy In this subsection we present the results, which ensure the existence of a unique t1, say t1 , which minimizes the total cost function for the model starting without shortages. Although the argument t1 of the functions TC1(t1), TC2(t1) is constrained, we shall search for their unconstrained minimum. The first and second order derivatives of TC1(t1) are, respectively dTC 1 ðt1 Þ ¼ f ðt1 Þgðt1 Þ; dt1 d2 TC 1 ðt1 Þ df ðt1 Þ dgðt1 Þ ¼ gðt1 Þ þ f ðt1 Þ ; dt21 dt1 dt1
ð15Þ
where b
gðt1 Þ ¼ c1 eat1
Z
t1
b
b
eat dt þ c3 ðeat1 1Þ c2 ðT t1 ÞbðT t1 Þ c4 ð1 bðT t1 ÞÞ:
ð16Þ
0
From (16) it is easily verified that, when 0 < b(x) 6 1, g(0) < 0, g(T) > 0 and further Z t1 dgðt1 Þ b b b ¼ c1 abt1b1 eat1 eat dt þ c1 þ c3 abt1b1 eat1 þ c2 ½bðT t1 Þ þ ðT t1 Þb0 ðT t1 Þ c4 b0 ðT t1 Þ > 0: dt1 0
ð17Þ
The inequality step follows from the assumption (2) made for b (x)(b(x) non-increasing so b 0 (x) 6 0 and b(x) + Tb 0 (x) P 0), and implies that g(t1) is strictly increasing. By assumption f(t1) > 0 and so the derivative dTCdt11ðt1 Þ vanishes at t1 , with 0 < t1 < T , which is the unique root of gðt1 Þ ¼ 0: For this
t1
ð18Þ
we have
2
d TC 1 ðt1 Þ df ðt1 Þ dgðt1 Þ dgðt1 Þ ¼ gðt1 Þ þ f ðt1 Þ ¼ f ðt1 Þ >0 2 dt1 dt1 dt1 dt1 so that t1 corresponds to the unconstrained global minimum of TC1(t1). If t1 is feasible, i.e. t1 6 l, the optimal value of the order level, S = I(0), is Z t 1 b S ¼ f ðtÞeat dt 0
ð19Þ
K. Skouri et al. / European Journal of Operational Research 192 (2009) 79–92
85
and subsequently the optimal order quantity Q* is Q ¼ S þ
Z
l
bðT tÞf ðtÞdt þ f ðlÞ t1
Z
T
bðT tÞdt:
ð20Þ
l
1Þ For the case b(x) = 0, dgðt > 0 and g(0) < 0, but g(T) maybe less than or greater to zero. If g(T) > 0 then t1 corresponds to dt1 the unconstrained global minimum of TC1(t1). If moreover t1 6 l, then the optimal value of the order level and the order quantity is given by (19) and (20), respectively. If g(T) < 0 then TC1(t1) is strictly decreasing attaining its minimum at l. Now we come to the branch TC2(t1). Its first and second order derivatives are
dTC 2 ðt1 Þ ¼ f ðlÞgðt1 Þ; dt1 d2 TC 2 ðt1 Þ dgðt1 Þ ¼ f ðlÞ > 0; dt21 dt1
ð21Þ ð22Þ
where the function g(t1) is given by (16). The inequality in (22) follows from (17) and ensures the strict convexity of TC2(t1). When 0 < b(x) 6 1, based on the properties of g(t1) we conclude that dTCdt21ðt1 Þ is vanished at the point t1 with 0 < t1 < T , which is the unique solution of gðt1 Þ ¼ 0:
ð23Þ
This t1 corresponds to the unconstrained global minimum of TC2(t1). Note that Eq. (23) is exactly the same as (18). If t1 is feasible i.e. t1 P l, the optimal value of the order level, S = I(0), is Z l Z t 1 b atb S ¼ f ðtÞe dt þ f ðlÞ eat dt ð24Þ 0
l
and the optimal order quantity, Q*, is Z T bðT tÞdt: Q ¼ S þ f ðlÞ
ð25Þ
t1
The previous analysis has shown that the two functions TC1(t1) and TC2(t1) have the same (unique) unconstrained minimizing point t1 2 ð0; T Þ, which is determined by (18) or (23). Based on (15) and (21) we can further establish that the function TC(t1) is differentiable at the point l and so l cannot be a corner point forTC(t1). 1Þ For the case b(x) = 0, dgðt > 0 and g(0) < 0, again g(T) maybe less or greater than zero. If g(T) > 0 then t1 corresponds dt1 to the unconstrained global minimum of TC2 (t1). If moreover t1 > l then the optimal value of the order level and the order quantity is given by (24) and (25), respectively. If g(T) < 0 then TC2(t1) is strictly decreasing attaining its minimum at T. Now we can give the procedure that leads to the optimal replenishment policy for the cases 0 < b(x) 6 1 and b(x) = 0 A. 0 < b(x) 6 1: Step 1. Compute t1 from (18) or (23). Step 2. Compare t1 to l. 2.1 If t1 6 l then the optimal order quantity and the total cost function are given from (20) and (9), respectively. 2.2 If t1 > l then the optimal order quantity and the total cost function are given from (25) and (13), respectively. B. b(x)=0 Step 1. Compute g(T). 1.1 If g(T) > 0 then go to step 1 of the previous procedure R l (case A). RT b b 1.2 If g(T) 6 0 then t1 ¼ T and consequently S ¼ Q ¼ 0 f ðtÞeat dt þ f ðlÞ l eat dt (we note that this is an expected result since g(T) 6 0 is the consequence of high lost sales cost, c4). Remark 1. The previous analysis shows that t1 is independent from the demand rate D(t). Although this seems to be a somewhat peculiar result, it agrees with the classical result that the point t1 is independent from the demand rate, known to be valid in many order level inventory systems (Naddor, 1966, p.67).
86
K. Skouri et al. / European Journal of Operational Research 192 (2009) 79–92
4. The mathematical formulation of the model starting with shortages In this section the inventory model starting with shortages is studied. The cycle now starts with shortages, which occur during the period [0, t1] and are partially backlogged. At time t1 a replenishment brings the inventory level up to S. Demand and deterioration of the items depletes the inventory level during the period [t1, T] until this falls to zero at t = T. Again the two cases t1 < l and t1 P l must be examined. 4.1. Case I (t1 < l) (see Fig. 3) The inventory level, I(t), 0 6 t 6 T satisfies the following differential equations: dIðtÞ ¼ f ðtÞbðt1 tÞ; 0 6 t 6 t1 ; Ið0Þ ¼ 0; dt dIðtÞ þ abtb1 IðtÞ ¼ f ðtÞ; t1 6 t 6 l; Iðl Þ ¼ Iðlþ Þ; dt dIðtÞ þ abtb1 IðtÞ ¼ f ðlÞ; l 6 t 6 T ; IðT Þ ¼ 0: dt The solutions of (26)–(28) are respectively Z t f ðxÞbðt1 xÞdx; 0 6 t 6 t1 ; IðtÞ ¼ 0 Z l Z T atb axb axb IðtÞ ¼ e e f ðxÞdx þ f ðlÞ e dx ;
ð26Þ ð27Þ ð28Þ
ð29Þ ð30Þ
t1 6 t 6 l
l
t
and b
IðtÞ ¼ eat f ðlÞ
Z
T
b
eax dx;
ð31Þ
l 6 t 6 T:
t
The total amount of deteriorated units during [t1, T] is Z l Z l Z T atb1 atb atb D¼e f ðtÞe dt þ f ðlÞ e dt f ðtÞdt f ðlÞðT lÞ: l
t1
t1
The total inventory carried during the interval [t1, T] is found using (30) and (31) and is Z l Z T Z l Z T Z T b b b b b I1 ¼ eat f ðxÞeax dx þ f ðlÞ eax dx dt þ f ðlÞ eat eax dx dt: t1
l
t
l
t
Due (29) the time-weighted backorders during the time interval [0, t1] are Z t1 I2 ¼ ðt1 tÞf ðtÞbðt1 tÞdt: 0
Inventory level
S
0 t1
μ
T
Time
Fig. 3. Inventory level for the model starting with shortages over the cycle (case t1 < l).
K. Skouri et al. / European Journal of Operational Research 192 (2009) 79–92
87
The amount of lost sales during in [0, t1] is Z t1 L¼ ½1 bðt1 tÞf ðtÞdt: 0
The inventory cost during the time interval [0, T] is the sum of holding, shortage, deterioration and opportunity costs and is given by TC 1 ðt1 Þ ¼ c1 I 1 þ c2 I 2 þ c3 D þ c4 L Z l Z l Z atb axb ¼ c1 e f ðxÞe dx þ f ðlÞ t1
t
þ c4 ð1 bðt1 tÞÞf ðtÞdt þ c3 e
T
e
axb
Z dx dt þ f ðlÞ
l
atb1
Z
T
e
atb
Z
l l
atb
f ðtÞe dt þ f ðlÞ
T
e t
Z
T
atb
e dt l
t1
axb
Z dx dt þ
t1
½c2 ðt1 tÞbðt1 tÞ
0
Z
l
f ðtÞdt f ðlÞðT lÞ :
ð32Þ
t1
4.2. Case II (t1 > l) (see Fig. 4) The inventory level, I(t), 0 6 t 6 T satisfies the following differential equations: dIðtÞ ¼ f ðtÞbðt1 tÞ; 0 6 t 6 l; Ið0Þ ¼ 0 dt dIðtÞ ¼ f ðlÞbðt1 tÞ; l 6 t 6 t1 ; Iðl Þ ¼ Iðlþ Þ; dt dIðtÞ þ abtb1 IðtÞ ¼ f ðlÞ; t1 6 t 6 T ; IðT Þ ¼ 0: dt Their solutions are, respectively, Z t f ðxÞbðt1 xÞdx; 0 6 t 6 l; IðtÞ ¼ 0 Z l Z t IðtÞ ¼ f ðxÞbðt1 xÞdx f ðlÞ bðt1 xÞdx; 0 b
IðtÞ ¼ eat f ðlÞ
ð33Þ ð34Þ ð35Þ
ð36Þ ð37Þ
l 6 t 6 t1 ;
l
Z
T
b
eax dx;
ð38Þ
t1 6 t 6 T :
t
The number of deteriorated items and the inventory carried during [t1, T] are, respectively Z T Z T Z T atb1 atb atb axb D ¼ e f ðlÞ e dt f ðlÞðT t1 Þ and I 1 ¼ f ðlÞ e e dx dt: t1
t1
t
Inventory level
S
0
μ t1
T
Time
Fig. 4. Inventory level for the model starting with shortages over the cycle (case t1 > l).
88
K. Skouri et al. / European Journal of Operational Research 192 (2009) 79–92
The time-weighted backorders due to shortages during [0, t1] using (36) and (37) are Z l Z t Z t 1 Z l Z t f ðxÞbðt1 xÞdx dt þ f ðxÞbðt1 xÞdx þ f ðlÞbðt1 xÞdx dt: I2 ¼ 0
0
l
0
l
The amount of lost sales in the interval [0, t1] is Z l Z t1 L¼ ð1 bðt1 tÞÞf ðtÞdt þ ð1 bðt1 tÞÞf ðlÞdt: 0
l
The total cost for this case is TC 2 ðt1 Þ ¼ c1 I 1 þ c2 I 2 þ c3 D þ c4 L Z T Z T atb axb ¼ c1 f ðlÞ e e dx dt þ c2
t1
Z 0
t
Z
l
t
Z f ðxÞbðt1 xÞdx dt þ
0 b
þ c3 eat1 f ðlÞ þ c4
Z
l T
b
eat dt f ðlÞðT t1 Þ
t1
Z
t1
l
½1 bðt1 tÞf ðtÞdt þ f ðlÞ
Z
0
Z
t1
l
f ðxÞbðt1 xÞdx þ f ðlÞ 0
Z
t
bðt1 xÞdx dt
l
½1 bðt1 tÞdt :
ð39Þ
l
The total cost function of the system over [0, T] takes the form TC 1 ðt1 Þ; if t1 6 l; TCðt1 Þ ¼ TC 2 ðt1 Þ; if l < t1 :
ð40Þ
It is easy to see that TC(t1) is continuous at l. The problem now is the minimization of this, two branches, function TC(t1). 4.3. The optimal replenishment policy In this subsection we derive the optimal replenishment policy, i.e., we calculate the value, say t1 , which minimizes the total cost function. Taking the first order derivative of TC1(t1) and equating it to zero gives Z l Z T b1 atb1 axb axb e f ðxÞdx þ f ðlÞ e dx ðc1 þ c3 abt1 Þe þ
l
t1
Z
t1
½c2 bðt1 tÞ þ c2 ðt1 tÞb0 ðt1 tÞ c4 b0 ðt1 tÞf ðtÞdt ¼ 0:
ð41Þ
0
If t1 is a root of (41), for this root the second order condition for minimum is "Z # Z T l b b b1 b1 atb ax ax ½c1 c3 ðb 1Þ þ c3 abðt1 Þ e 1 abðt1 Þ e f ðxÞdx þ f ðlÞ e dx þ ½c1 þ c3 abðt1 Þb1 f ðt1 Þ þ ½c2 bð0Þ c4 b0 ð0Þf ðt1 Þ þ
t1
Z
t1
l
½2c2 b0 ðt1 tÞ þ c2 ðt1 tÞb00 ðt1 tÞ c4 b00 ðt1 tÞf ðtÞdt > 0:
ð42Þ
0
So, if (42) holds and t1 6 l then the value of order level, S, is
S ¼
Iðt1 Þ
¼e
at1
b
"Z
l
axb
e f ðxÞdx þ f ðlÞ t1
Z
#
T
e
axb
dx ;
ð43Þ
l
the ordering quantity is
Q ¼
Z
t1
f ðxÞbðt1 xÞdx þ S
0
and the total cost is TC 1 ðt1 Þ.
ð44Þ
K. Skouri et al. / European Journal of Operational Research 192 (2009) 79–92
89
Equating the first order derivative of TC2(t1) to zero gives Z T Z l Z 0 0 b atb1 axb e dx þ ½c2 bðt1 tÞ þ c2 ðt1 tÞb ðt1 tÞ c4 b ðt1 tÞf ðtÞdt þ f ðlÞ ðc1 þ c3 abt1 Þe f ðlÞ 0
t1
t1
½c2 bðt1 tÞ l
þ c2 ðt1 tÞb0 ðt1 tÞ c4 b0 ðt1 tÞdt ¼ 0:
ð45Þ
If t1 is a root of (45), for this root the second order condition for minimum is ½c1 þ þ
b1 b1 b c3 abt1 eat1 abt1 f ðlÞ
Z
l
Z
T
e
axb
dx c3 abðb
t1
ðb2Þ b 1Þt1 eat1 f ðlÞ
Z
T
t1
2c2 b0 ðt1 tÞ þ c2 ðt1 tÞb00 ðt1 tÞ c4 b00 ðt1 tÞ f ðtÞdt þ f ðlÞ
0
b1
b
eax dx þ ½c1 þ c3 abt1 f ðlÞ þ c2 f ðlÞ Z
t1
½2c2 b0 ðt1 tÞ þ c2 ðt1 tÞb00 ðt1 tÞ
l
c4 b00 ðt1 tÞdt > 0:
ð46Þ
So if (46) holds and t1 > l then the value of S is b
S ¼ Iðt1 Þ ¼ eat1 f ðlÞ
Z
T
b
eax dx;
ð47Þ
t1
the ordering quantity is
Q ¼
Z
l
f ðxÞbðt1
xÞdx þ f ðlÞ
Z
0
t1
bðt1 xÞdx þ S
ð48Þ
l
and the total cost is TC 2 ðt1 Þ. Remark 2. Due to (41) and (45) the function TC(t1) is differentiable at the point l. In the previous analysis there is no guarantee that t1 exists and corresponds to the minimum. Its uniqueness is also another issue. The proposition, which follows, provides sufficient conditions for existence, uniqueness and validity of t1 . Let us set Z T Z l b b eax dx þ ½c2 bðl tÞ þ c2 ðl tÞb0 ðl tÞ c4 b0 ðl tÞf ðtÞdt D ¼ ½c1 þ c3 ablb1 eal f ðlÞ l 0
0
0
and h(t) = c2b(t) + c2tb (t) c4b (t). The following propositions can be easily proved: 1 Proposition 1. If b 6 1, T b1 P ab , h(t) + h 0 (t) > 0t 2 [0, T], and
(1) D > 0 then the optimal value of t1 follows from (41). (2) D < 0 then the optimal value of t1 follows from (45). Proposition 1. If D > 0 then (41) has at least a root, say t1 . If for every root of (41) ðc1 þ c3 abt1b1 Þðt1b1 1Þ > c3 ðb 1Þt1b1 hðtÞþ 0 h0 ðtÞ > 0t 2 ½0; T and h(t) + h (t) > 0 then t1 is the unique optimal value of the problem. 2. If D < 0 then (45) has at least a root, say t1 . If for every root of (45) ðc1 þ c3 abt1b1 Þðt1b1 1Þ > c3 ðb 1Þt1b1 hðtÞþ h0 ðtÞ > 0t 2 ½0; T and h(t) + h 0 (t) > 0 then t1 is the unique optimal value of the problem. We note that the above propositions ensure the existence, uniqueness and validity of t1. If the conditions of this proposition do not hold then the following procedure can be used to calculate the optimal replenishment policy: Step 1. Step 1.1. Find the global minimizing point, t1 , for TC1(t1). This will be one of the following points: (a) a root of (41) (an interior point of [0, l]) which satisfies (42), (b) t1 ¼ 0, (c) t1 ¼ l. Then calculate TC 1 ðt1 Þ.
90
K. Skouri et al. / European Journal of Operational Research 192 (2009) 79–92
40
30
20
10
0.2
0.4
0.6
0.8
1
Fig. 5. The graphical representation of the total cost function (Example 1, model starting without shortages).
Step 1.2. Find the global minimizing, t1 , for TC2(t1). This will be one of the following points: (a) a root of (45) (an interior point of [l,T]) which satisfies (46), (b) t1 ¼ l, (c) t1 ¼ T . Then calculate TC 2 ðt1 Þ. Step 2. Find the min fTC 1 ðt1 Þ; TC 2 ðt1 Þg and accordingly select the optimum t1 .
Remark 5. The analysis shows that, in this model, t1 is dependent from the demand rate D(t). 5. Numerical examples The examples, which follow, illustrate the results obtained. Example 1. This is taken from Wu (2001) and is adapted to our models. The input parameters are: c1 = $3 per unit per year, c2 = $15 per unit per year, c3 = $5 per unit, c4 = $20 per unit, l = 0.12 year, a = 0.001, b = 2, T = 1 year, f(t) = 3e4.5t and b (x) = e0.2x. Model starting with no shortages: Using (19) (or (24)) the optimal value of t1 is t1 ¼ 0:8604 > l. The optimal ordering quantity is Q* = 4.999 (from (25)) and the minimum cost is TCðt1 Þ ¼ 6:6432 (from (13)). Fig. 5 represents the total cost function during [0, T]. Model starting with shortages: t1 ¼ 0:1631 (from (45)) ðl < t1 Þ; Q ¼ 4:999 (from (48)), and TCðt1 Þ ¼ 6:3777 (from (39)) (see Fig. 6). Example 2. This example is identical to Example 1, except that l=0.9. Model starting with no shortages: t1 ¼ 0:8604ðl > t1 Þ; Q ¼ 54:5141 (from (20)) and TCðt1 Þ ¼ 91:8508 (from (9)).
22.5 20 17.5 15 12.5 10 7.5
0.2
0.4
0.6
0.8
1
Fig. 6. The graphical representation of the cost function (Example 1, model starting with shortages).
K. Skouri et al. / European Journal of Operational Research 192 (2009) 79–92
91
Model starting with shortages: t1 ¼ 0:5673 (from (41)) ðl > t1 Þ; Q ¼ 54:56 (from (44)) and TCðt1 Þ ¼ 63:109 (from (32)). From these examples we can observe that the total cost for the model starting with shortages is less than the total cost for the model starting with no shortages. This observation agrees with known, results from literature concerning the finite horizon inventory models (Teng et al., 1997; Skouri and Papachristos, 2003). 6. Concluding remarks In this paper, an order level inventory model for deteriorating items has been studied. The model is fairly general as, the demand rate is any function of time up to the time-point of its stabilization (general ramp type demand rate), and the backlogging rate is any non-increasing function of the waiting time, up to the next replenishment. The inventory model is studied under two different replenishment policies: (a) starting with no shortages and (b) starting with shortages. Therefore, the proposed model leads to the following existing ones: 1 1. If f(t) = D0t, t 2 [0, l] and bðxÞ ¼ 1þdx , our model reduces to that of Wu (2001). 2. If additionally d = 0 (case of complete backlogging), then it further reduces to that of Wu et al. (1999).We point out that Wu et al. (1999) and Wu (2001) are restricted only to the case t1 < l. The case where t1 could be greater than l has not been examined. 3. If b = 1 (i.e. the deterioration distribution is the exponential one), f(t) = D0t and b(x) = 1, then our model reduces to those of Wu and Ouyang (2000), Mandal and Pal (1998) and Deng et al. (2007).
Acknowledgements The authors are grateful to the anonymous referees who provided valuable comments and suggestions to improve the paper. This research was co-financed by the European Union (European Social Fund, 75%) and the Hellenic Ministry of National Educational and Religions Affairs (25%) in the framework of the Operational Programme II for Educational and Initial Vocational Training, ARCHIMEDES project. References Abad, P.L., 1996. Optimal pricing and lot sizing under conditions of perishability and partial backordering. Management Science 42, 1093–1104. Chang, H.J., Dye, C.Y., 1999. An EOQ model for deteriorating items with time varying demand and partial backlogging. Journal of the Operational Research Society 50, 1176–1182. Covert, R.P., Philip, G.C., 1973. An EOQ model for items with Weibull distribution deterioration. AIIE Transaction 5, 323–326. Datta, T.K., Pal, A.K., 1988. Order level inventory system with power demand pattern for items with variable rate of deterioration. Indian Journal of Pure and Applied Mathematics 19 (11), 1043–1053. Dave, U., Patel, L.K., 1981. (T, Si) policy inventory model for deteriorating items with time proportional demand. Journal of the Operational Research Society 32, 137–142. Deng, P.S., Lin, R.H.-J., Chu, P., 2007. A note on the inventory models for deteriorating items with ramp type demand rate. European Journal of Operational Research 178, 112–120. Donaldson, W.A., 1977. Inventory replenishment policy for a linear trend in demand: An analytic solution. Operational Research Quarterly 28, 663–670. Ghare, P.M., Schrader, G.F., 1963. A model for exponentially decaying inventories. Journal of Industrial Engineering 14, 238–243. Giri, B.C., Jalan, A.K., Chaudhuri, K.S., 2003. Economic order quantity model with Weibull deterioration distribution, shortage and ramp-type demand. International Journal of Systems Science 34 (4), 237–243. Goswami, A., Chaudhuri, K.S., 1991. An EOQ model for deteriorating items with shortages and a linear trend in demand. Journal of the Operation Research Society 42, 1105–1110. Goyal, S.K., 1987. Economic ordering policy for deteriorating items over an infinite time horizon. European Journal of Operational Research 28, 298–301. Goyal, S.K., Giri, B.C., 2001. Recent trends in modeling of deteriorating inventory. European Journal of Operational Research 134, 1–16. Hariga, M., 1996. Optimal EOQ models for deteriorating items with time-varying demand. Journal of the Operational Research Society 47, 1228–1246. Hill, R.M., 1995. Inventory model for increasing demand followed by level demand. Journal of the Operational Research Society 46, 1250–1259. Mandal, B., Pal, A.K., 1998. Order level inventory system with ramp type demand rate for deteriorating items. Journal of Interdisciplinary Mathematics 1, 49–66. Montgomery, D.C., Bazaraa, M.S., Keswani, A.K., 1973. Inventory models with a mixture of backorders and lost sales. Naval Research Logistics Quarterly 20, 255–263. Naddor, E., 1966. Inventory Systems. John Wiley and Sons, Inc. Park, K.S., 1982. Inventory model with partial backorders. International Journal of Systems Sciences 13, 1313–1317. Raafat, F., 1991. Survey of literature on continuously deteriorating inventory model. Journal of the Operational Research Society 42, 27–37. Resh, M., Friedman, M., Barbosa, L.C., 1976. On a general solution of the deterministic lot size problem with time-proportional demand. Operations Research 24, 718–725.
92
K. Skouri et al. / European Journal of Operational Research 192 (2009) 79–92
Rosenberg, D., 1979. A new analysis of a lot size model with partial backlogging. Naval Research Logistics Quarterly 26, 346–353. Sachan, R.S., 1984. On (T, Si) inventory policy model for deteriorating items with time proportional demand. Journal of the Operational Research Society 35 (11), 1013–1019. San Jose, L.A., Sicilia, J., Garcia-Laguna, J., 2005. An inventory system with partial backlogging modeled according to a linear function. Asia-Pacific Journal of Operational Research 22, 189–209. San Jose, L.A., Sicilia, J., Garcia-Laguna, J., 2006. Analysis of an inventory system with exponential partial backordering. International Journal of Production Economics 100, 76–86. Skouri, K., Papachristos, S., 2002. A continuous review inventory model, with deteriorating items, time-varying demand, linear replenishment cost, partially time-varying backlogging. Applied Mathematical Modelling 26 (5), 603–617. Skouri, K., Papachristos, S., 2003. Four inventory models for deteriorating items with time varying demand and partial backlogging: A cost comparison. Optimal Control Application and Methods 24, 315–330. Tadikamalla, P.R., 1978. An EOQ inventory model for items with Gamma distribution. AIIE Transaction 5, 100–103. Teng, J.-T., Chern, M.-S., Yang, H.-L., 1997. An optimal recursive method for various inventory replenishment models with increasing demand and shortages. Naval Research Logistics 44, 791–806. Teng, J.T., Chang, H.J., Dye, C.Y., Hung, C.H., 2002. An optimal replenishment policy for deteriorating items with time-varying demand and partial backlogging. Operations Research Letters 30, 387–393. Wang, S.P., 2002. An inventory replenishment policy for deteriorating items with shortages and partial backlogging. Computers and Operations Research 29, 2043–2051. Wu, K.-S., 2001. An EOQ inventory model for items with Weibull distribution deterioration, ramp type demand rate and partial backlogging. Production Planning and Control 12 (8), 787–793. Wu, K.-S., Ouyang, L.-Y., 2000. A replenishment policy for deteriorating items with ramp type demand rate. Proceedings of the National Science Council, Republic of China, Part A: Physical Science and Engineering 24 (4), 279–286. Wu, J.-W., Lin, C., Tan, B., Lee, W.-C., 1999. An EOQ inventory model with ramp type demand rate for items with Weibull deterioration. Information and Management Science 10 (3), 41–51. Yang, H.L., Teng, J.T., Chern, M.S., 2001. Deterministic inventory lot-size models under inflation with shortages and deterioration for fluctuating demand. Naval Research Logistics 48, 144–158.