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International Journal of Systems Science
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Periodic review stochastic inventory models with
service level constraint
Chih-Young Hung a , Kuo-Chen Hung b , Wen-Han Tang c , Robert Lin d & Chi-Kae Wang e a
Institute of Management of Technology, National Chiao Tung University , Taiwan, R.O.C. b
Department of Logistics Management , National Defense University , Taiwan, R.O.C. c
Department of Management Science , Chinese Military Academy , Taiwan, R.O.C. d
Department of Information Management , Oriental Institute of Technology , Taiwan, R.O.C.
e
Department of Mechanical Engineering , Chung Yuan Christian University , Taiwan, R.O.C. Published online: 02 Mar 2009.
To cite this article: Chih-Young Hung , Kuo-Chen Hung , Wen-Han Tang , Robert Lin & Chi-Kae Wang (2009) Periodic review
stochastic inventory models with service level constraint, International Journal of Systems Science, 40:3, 237-243, DOI: 10.1080/00207720802298962
To link to this article: http://dx.doi.org/10.1080/00207720802298962
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Periodic review stochastic inventory models with service level constraint
Chih-Young Hunga, Kuo-Chen Hungb, Wen-Han Tangc, Robert Lind* and Chi-Kae Wangea
Institute of Management of Technology, National Chiao Tung University, Taiwan, R.O.C.;bDepartment of Logistics Management, National Defense University, Taiwan, R.O.C.;cDepartment of Management Science, Chinese Military
Academy, Taiwan, R.O.C.;dDepartment of Information Management, Oriental Institute of Technology, Taiwan, R.O.C.; eDepartment of Mechanical Engineering, Chung Yuan Christian University, Taiwan, R.O.C.
(Received 11 April 2005; final version received 12 June 2008)
This article considers the periodic review stochastic inventory models with service level constraint to provide an improved solution procedure. The previous researchers assumed that the objective function is concave down in the lead time so that the minimum must occur on the boundary points of each sub-domain. In this article, we will show that their assumption is questionable since the minimum might not occur at the boundary points of each sub-domain. In a recent paper in International Journal of Systems Science, Ouyang and Chuang studied this problem. However, their solutions contained questionable results and their algorithm might not find the optimal solution due to flaws in their solution procedure. We develop some lemmas to reveal the parameter effects and then present our improved solution procedures for finding the optimal solution for periodic review stochastic inventory models in which the lead time demand is a normal distribution. The savings are illustrated by solving the same examples from Ouyang and Chuang’s paper to demonstrate the improvement using our revised algorithm. In the direction of future research, we discuss the comparison between the reordered point being fixed and the reordered point as a new variable.
Keywords: inventory model; service level constraint; crashable lead time
Nomenclature
A ordering cost per order D expected demand per year
h holding cost per unit per year L length of lead time
T the review period R target level
X the lead time demand which has a distribution function F with finite mean L and standard derivation pffiffiffiffiLð4 0Þ
proportion of demands that are not met from stock so 1 is the service level
fraction of the demand during the stock-out period that will be backordered
C(L) lead time crashing cost
xþ maximum value of x and 0, i.e. xþ¼maxfx, 0g
Ti# the minimum solution under the constraint level, with Ti#¼maxfT
i, ðB2LiÞþg, for EACi(T, Li) for (B 2 Li) þ T
Ti for those i, that satisfies that B2Li, the
minimum solution under the constraint level, with T
i ¼MIDfB 2L
i, ðdi=bÞ1=2, ðB2Li1Þþg
for EACi(T, B2T) with (B2Li1)þ
T B2Li.
1. Introduction
In the traditional inventory model, the lead time is considered as a predetermined constant or a variable as in Silver and Peterson (1985) such that lead time is not controllable. Liao and Shyu (1991) developed a new inventory model to decompose the lead time into several components, each having a different piecewise linear crash cost function for lead time reduction. Therefore, lead time becomes a new decision variable. Gallego (1992) created a wonderful two-point distribu-tion to serve as the most unfavourable case among the distributions with the same mean and variance to estimate the total expected cost of the lost sales such that the minmax distribution free approach of Scarf (1958) can apply to the stochastic inventory models. Moon and Gallego (1994) extended the minmax distribution free approach for stochastic inventory model with backorders and lost sales. Ouyang, Yeh, and Wu (1996) generalised Ben-Daya and Raouf’s (1994) assumption allowing shortages. Ouyang and Wu (1997) extended it to inventory model with service level constraint. Moon and Choi (1998) and Lan, Chu, Chung, and Wan (1999) pointed out the problem in Ouyang et al.’s (1996) method. Ouyang and Wu (1998)
*Corresponding author. Email: [email protected] ISSN 0020–7721 print/ISSN 1464–5319 online
ß 2009 Taylor & Francis DOI: 10.1080/00207720802298962 http://www.informaworld.com
extended the Ouyang et al. (1996) article to apply the minimax distribution free procedure. Ouyang and Chuang (1999) studied stochastic inventory models with service level constraint which are solved by the minimax distribution free procedure. Ouyang and Chuang (2000) developed periodic review stochastic inventory models involving variable lead time with a service level constraint for the lead time with normal distribution and distribution free. Wu and Tsai (2001) studied stochastic inventory models with a mixed normal distribution from different customers. Pan and Hsiao (2001) developed the model with backorder discount to ensure that customers would be willing to wait for backorders. Chu, Yang, and Chen (2005) improved the results of Ouyang and Wu (1997), first for lead time demand following a normal distribution, and then extending the minmax distribution free procedure to solve the problem.
From the stochastic inventory models with crash-able lead time, there are many generalised extensions one can apply for more realistic inventory models. However, there is a questionable result in Ouyang and Chuang (2000) that deserves more detailed discussion. Ouyang et al. (1996) proved that the expected annual cost is a concave down function in lead time so that the minimum value will occur on the boundary points of each sub-domain. It is an excellent discovery that dramatically simplifies the solution procedure. To clearly indicate this property, we denote it as follows: the minimum values for concave down functions degenerate to the boundary points on the sub-domain of the crash cost. However, Ouyang et al. (1996) considered the stochastic inventory model without service level constraint. The researchers who followed them believed that this property also holds with the service level constraint.
In this article, we will point out that the degeneracy to the boundary points for the concave down function requires more detailed examination. We will construct a correct and efficient algorithm to find the optimum order quantity and lead time, develop lemmas to reveal the parameter effects and illustrate our improvement by solving the same numerical example in Ouyang and Chuang (2000) to indicate that sometimes their algorithm does not find the optimal solution.
2. Assumptions
We use the same assumptions as Ouyang and Chuang (2000) and several new expressions to clearly indicate our solution procedure.
(A1) The inventory level is reviewed every T units of time. A sufficient ordering quantity is ordered up to the target level R and the
ordering quantity is arrived at after L units of time.
(A2) The length of the lead time L is less than the cycle length T so that there is never more than a single-order outstanding in any cycle. (A3) The target level R ¼ expected demand during protection interval þ safety stock (SS), and SS ¼ k (SD of protection interval demand), that is, R ¼ D(T þ L) þ k(T þ L)1/2 where k is the safety factor and satisfies P(X4R) ¼ q, q representing the allowable stock-out prob-ability during the protection interval and is provided.
(A4) The lead time L has n mutually independent components. The ith component has a mini-mum duration ai, and normal duration biand
a crash cost per unit time ci. Further, we
assume that c1c2 cn. The lead time
components are crashed one at a time starting with the least cicomponent and so on.
(A5) If we let L0¼Pnj¼1bjand Lithe length of lead
time with components 1, 2, . . . , i crash to their minimum durations, then Li¼Pnj¼iþ1bjþ
Pi
j¼1aj. The lead time crash cost C(L) per
cycle for a given L 2 [Li, Li1] is given by
CðLÞ ¼ ciðLi1LÞ þPj¼1i1cjðbjajÞ.
(A6) If X has a normal distribution function F(x), according to Ravindran, Phillips, and Solberg (1987), then B(r) ¼ L1/2 (k), where (k) ¼ ’(k) k[1 (k)]40, and ’, are the standard normal probability density function and distribution function, respectively. (A7) In the beginning, the domain for (T, L) is
T40 and LnL L0. However, the cost for
crashed lead time is defined for each interval L 2[Li, Li1], where i ¼ 1, . . . , n. Hence, we
need to consider EAC(T, L) for L 2 [Li, Li1],
where i ¼ 1, . . . , n. In the next section, we will only consider L 2 [Li, Li1] for the time
being.
(A8) fs(T ) is defined as 4((T þ Ls)/T2)3/2c/2as,
for a family of auxiliary functions.
(A9) gs(T ) is defined as as/T2þb þ c/2(T þ Ls),
for another family of auxiliary functions. (A10) MID(x, y, z) is the middle term in x, y and z. 3. Review of previous results
Ouyang and Chuang (2000) tried to solve the following problem: min EACiðT, LÞ ¼ A þ C Lð Þ T þ h 2DT þhðT þ LÞ1=2ðk þð1 ÞG kð ÞÞ ð1Þ
238 C.-Y. Hung et al.
subject to
G kð Þ DðT þ LÞ1=2 ð2Þ with GðkÞ ¼Rk1ðz kÞfzðzÞdz and CðLÞ ¼ ciðLi1LÞþ
Pi1
j¼1cjðbjajÞfor L 2 [Li, Li1].
The proper domain for the lead time should be L 2[Ln, L0], but they only considered the domain for
the lead time for L 2 [Li, Li1], with i ¼ 1, 2, . . . , n.
The reason is that the crashing costs are different in each sub-domain [Li, Li1] with i ¼ 1, 2, . . . , n, so the
minimum problem must be divided into sub-domain [Li, Li1] with i ¼ 1, 2, . . . , n. However, the crashing
costs have similar expressions, hence the results for one sub-domain can be applied to others. Therefore, Ouyang and Chuang (2000) only studied one sub-domain.
They had derived that @ @TEACiðT, LÞ ¼ A þ CðLÞ T2 þ h 2D þ h 2 ðT þ LÞ 1=2 ðk þ ð1 ÞGðkÞÞ, ð3Þ @2 @T2EACiðT, LÞ ¼ hD T þ 3T þ 4L 4T hðT þ LÞ 3=2 ðk þð1 ÞG kð ÞÞ, ð4Þ and @2 @L2EACiðT, LÞ ¼ h 4 ðT þ LÞ 3=2½k þð1 ÞG kð Þ: ð5Þ They claimed that EACi(T, L) is a convex function of T
and a concave down function of L. Hence, they assumed that, for fixed T, the minimum total expected annual cost will occur at the end points of the interval [Li, Li1]. Consequently, they only considered the
minimum problem for EACi(T, Li) and EACi(T, Li1)
for L 2 [Li, Li1].
We will accept Equations (3) and (5), but Equation (4) is questionable (that will be explained by Equation (12)) and may require revision. We will point out that their assumption that the minimum cost will occur at Lior Li1as the end points of the interval
[Li, Li1] is questionable that will be demonstrated in
Example 2. And finally, we will discuss why the minimum solution is existent and unique.
4. Our improvement
Fixing i in f1, . . . , n}, we will find the minimum value for EACi(T, L) with L 2 [Li, Li1]. We consider the
vector (T, L) in the first quadrant that satisfies the service level constraint G(k) D(T þ L)1/2, which implies that the service level constraint being satisfied
is equivalent to B2T þ L where B ¼ (G(k)/D). Hence, we may divide the first quadrant into two sub-regions, where the vectors in the sub-region f(T, L): B2T þ L} satisfy the service level constraint and the vectors in the other sub-region f(T, L): T þ L5B2} do not satisfy the service level constraint. For a fixed i in f1, 2, . . . , n}, we handle the minimum problem for L 2 [Li, Li1] as L must reduce to
boundaries and satisfies the constraint B2T L, then we divide the problem into three cases: (I) along the vertical line L ¼ Li, with T (B2Li)
þ
, (II) along the vertical line L ¼ Li1, with T (B2Li1)
þ
and (III) along the skew line T þ L ¼ B2, with (B2Li)
þ
T (B2Li1) þ
.
Moreover, owing to EACi(T, Li) ¼ EACiþ1(T, Li),
we consider Cases (I) and (II) to reveal that we may merge them into one general case as follows: Case (1), along the vertical line L ¼ Ls, with T (B
2
Ls) þ
, for s ¼ ior s ¼ i 1.
Here, we may point out that for Case (III), we only examine the minimum problem when B2Li, since if
B25Li, then (B2Li)þ¼0 and (B2Li1)þ¼0, there
is no need to consider the skew line. Therefore, we rewrite Case (III) to Case (2) as follows: Case (2), for those i that satisfies B2Li, along the skew line
T þ L ¼ B2, with B2LiT (B 2
Li1) þ
. For Case (1), the minimum problem becomes min EACiðT, LsÞ ¼ A þ CðLsÞ T þ h 2DT þ hðT þ LsÞ 1=2 ðk þ ð1 ÞGðkÞÞ ð6Þ where s ¼ i or s ¼ i 1, under the condition T (B2Ls)
þ
. To simplify the expression, we assume that as¼A þ C(Ls), b ¼(h/2)D and c ¼ hðk þ
ð1 ÞGðkÞÞ, then rewrite Equation (6) in the following: min EACiðT, LsÞ ¼ as TþbT þ cðT þ LsÞ 1=2 : ð7Þ It yields that d dTEACiðT, LsÞ ¼ as T2 þb þ c 2ðT þ LsÞ 1=2 ð8Þ and d2 dT2EACiðT, LsÞ ¼ 2as T3 c 4ðT þ LsÞ 3=2 : ð9Þ Rewriting Equation (9), d2 dT2EACiðT, LsÞ ¼ 2as 4 ðT þ LsÞ 3=2 4 T3ðT þ LsÞ 3=2 c 2as : ð10Þ
By Equation (10), we define a family of auxiliary functions, say fs(T ), as follows:
fsðT Þ ¼4 T þ Ls T2 3=2 c 2as : ð11Þ Since (d/dT )fs(T ) ¼ 6((T þ Ls)/(T2))1/2((1/T2) þ
(2Ls/T3))50, fs(T ) is a decreasing function of T, from
limT!0fsðTÞ ¼ 1to limT!1fsðTÞ ¼ ðc=2asÞsuch that
there is a unique point, say T#
s with fsðTs#Þ ¼0.
Moreover, fs(T )40 for 0 5 T 5 Ts# and fs(T )50 for
T# s 5 T so we obtain that (d 2 /dT2)EACi(T, Ls)40 for 0 5 T 5 T# s and (d 2 /dT2) EACi(T, Ls)50 for Ts#5 T.
Hence, EACi(T, Ls) is a convex function for
0 5 T 5 T#
s, and a concave down function for Ts#5 T.
For completeness, here we point out that the corrected expression for Equation (4) should be
@2 @T2EACiðT, LÞ ¼ 2 A þ C L½ ð Þ T3 h 4 ðT þ LÞ 3=2 k þð1 ÞG kð Þ ð Þ ð12Þ so that Ouyang and Chuang claimed that EACi(T, L)
being a convex function of T is questionable. This should be revised as follows: EACi(T, Ls) is a convex
function for 0 5 T 5 Ts#, and a concave down function for Ts#5 T, where s ¼ i or s ¼ i 1.
To simplify the expression, we assume another family of auxiliary functions, say gs(T ), as follows:
gsðT Þ ¼ d dTEACiðT, LsÞ ¼ as T2 þb þ c 2ðT þ LsÞ 1=2: ð13Þ Next, we will prove that gsðTs#Þ4 0.
Since ðT# sÞ 2 ¼4ða s=cÞ2=3ðTs#þLsÞ, we have that ðT# sÞ 24 4ða s=cÞ2=3Ts# and then Ts#4 4 að s=cÞ2=3: ð14Þ Using ðT# s þLsÞ1=2¼2ðas=cÞ1=3ðTs#Þ1, by Equation (14), we imply that g T s#¼b þ a1=3s c2=3Ts#2 Ts# ðas=cÞ2=3 4 0: ð15Þ Moreover, we get that limT!0gsðTÞ ¼ 1
and limT!1gsðTÞ ¼ b4 0. By Equation (13), we
know that (d/dT )gs(T ) ¼ (d2/dT2)EAC(T, Ls), then
(d/dT )gs(T )40 for 0 5 T 5 Ts# and (d/dT )gs(T )50
for T#
s 5 T. gs(T ) increases from limT!0gsðTÞ ¼ 1to
gðT#
sÞ4 0 and gs(T ) decreases from gðTs#Þ4 0 to
limT!1gsðTÞ ¼ b4 0. Therefore, there is a unique
point, say T
s with gsðTsÞ ¼0. Consequently,
(d/dT )EACi(T, Ls) is negative for T 2 ð0, TsÞ and
positive for T 2 ðT
s, 1Þ so T
s is the minimum point
for EACi(T, Ls), before we consider the constraint
(B2Ls) þ
T.
It yields that the minimum value of EACi(T, Ls)
for (B2Ls) þ
T that occurs, say T# s, with
T#
s ¼maxfTs, ðB2LsÞþg, for s ¼ i or s ¼ i 1, and
i ¼1, 2, . . . , n.
According to above discussions, we know that it yields for 1 i n, the minimum value of EACi(T, Li) under the condition (B2Li)
þ
T that occurs at Ti#. We summarise our results in the next theorem.
Theorem 1: For Case(1), with 0 i n, the minimum value occurs at Ti#.
Next, we study the Case (2), along the skew line T þ L ¼ B2 to consider the minimum problem of EACi(T, B2T) under the condition B2Li
T (B2Li1)þ. We know that EACi T, B2T ¼di TþbT þ ei ð16Þ with di¼A þ ciðLi1B2Þ þPi1j¼1cjðbjajÞ, b ¼(hD/2)
and ei¼ciþhBðk þ ð1 ÞGðkÞÞ. We may rewrite
Equation(16) as EACiðT, B2T Þ ¼ di T 1=2 ðbT Þ1=2 !2 þ2ðdibÞ1=2þei: ð17Þ By Equation(17), it shows that without considering the condition B2LiT (B2Li1)
þ
, the minimum point is T ¼(di/b)1/2. To simplify the expression, we use a new
expression, say MIDfx, y, z}, which stands for the middle term of x, y and z. For example, MIDf7, 1, 5} ¼ 5 and MIDf6, 2, 6} ¼ 6. We combine the above discussion in the next theorem. To simplify the expression, we assume that T
i ¼MIDfB2Li, ðdi=bÞ1=2, ðB2Li1Þþg.
Theorem 2: For Case(2), the minimum solution for T ofEACi(T, B2T) occurs at Ti.
We summarise our findings in the last theorem. Theorem 3: The minimum for EAC(T, L) with L 2[Ln, L0] is expressed as (T*, L*) so that
EAC(T*, L*) is the minimum value among (a) EACiðTi#, LiÞ, for 1 i n, (b) EAC1ðT#0, L0Þ and
(c) for those i that satisfies B2Li, EACðTi , LiÞ.
5. Numerical example
Example 1: To illustrate our improvement, we consider the same numerical example as Ouyang and Chuang (2000) with the following data: D ¼ 625 units per year, A ¼ $350 per order, h ¼ $35 unit per year,
240 C.-Y. Hung et al.
¼11 units per week, ¼ 7 units per week, ( ¼ 7pffiffiffiffiffi52 units per year), ¼ 1 (backorders case), The lead time has three components with c1¼$0.4 per day, a1¼
6 days, b1¼20 days, c2¼$1.0 per day, a2¼6 days,
b2¼20 days, c3¼$5.0 per day, a3¼9 days, b3¼
16 days. It yields L0¼(8/52) ¼ 0.154 years, L1¼
(6/52) ¼ 0.115 years, L2¼(4/52) ¼ 0.077 years, and
L3¼(3/52) ¼ 0.058 years. It is assumed that q ¼ 0.2,
so the safety factor k can be found directly from the standard normal table, and it equals 0.845.
Since B ¼ (G(k)/D), it shows that B2¼0.201. Here, we must point out that if we use D ¼ 625 units per year, then our results cannot compare with those of Ouyang and Chuang (2000). We slightly modify the value of D to D ¼ 624 units per years, then our results are consistent with their results. Hence, we modify D as D ¼624 units per years.
For the first numerical example, we assume that the service level 1 ¼ 0.98, i.e. the proportion of demand that is not met from stock is at most ¼ 0.02. We compute the local minimum value for each interval L 2[Li, Li1] and list them in Table 1.
For easy comparison with the results of Ouyang and Chuang (2000), we quote their results in Table 2. They found that the optimal review period T* ¼ 8.84 weeks, that is, T* ¼ 0.170 years and the optimal lead time L* ¼ L1¼6, with
EAC(T*, L*) ¼ 4745.68. We may say that Ouyang and Chuang (2000) and our improved method both have the same minimum solution for the first numerical example with ¼ 0.02.
Example 2: We have the same value for parameters except that ¼ 0.015. We obtain that B2¼0.358 and then based on Theorem 2, we find that
MID B2L0¼0:048, ffiffiffiffiffiffiffiffiffi d1=b p ¼0:179, B2L1¼0:086 n o ¼0:086 ¼ B2L1, MID B2L1¼0:086, ffiffiffiffiffiffiffiffiffi d2=b p ¼0:180, B2L2¼0:125 n o ¼0:125 ¼ B2L2, and MID B2L2¼0:125, ffiffiffiffiffiffiffiffiffi d3=b p ¼0:184, B2L3¼0:144 n o ¼0:144 ¼ B2L3,
such that there is no need to compute the minimum value along the skew line T þ L ¼ B2, and then list the results in Table 3.
We find that the optimal review period, T* ¼ B2L0¼0.204, the optimal lead time, L* ¼ L0¼0.154
and the minimum value EAC(B2L0, L0) ¼ 4837.378.
We compare the results with Ouyang and Chuang. From Table 2, the sixth column, according to Ouyang and Chuang’s method, without considering the service level constraint in Table 2, the fourth row, they find the absolute minimum expected average cost ETC(T1¼8.84, L1¼6) ¼ 4745.68, then
check (E(X R)þ
/D(TiþLi)) ¼ 0.01684 ¼ 0.015, so
it has not satisfied the service level constraint. Ouyang and Chuang will consider the next minimum value
Table 1. Local minimum values for each case.
Theorem Minimum point Minimum value
1 MaxfT 0¼0:169, B2L0¼0:047g 4764.731 1 MaxfT 1¼0:170, B2L1¼0:086g 4745.681 1 MaxfT 2¼0:173, B2L2¼0:125g 4771.886 1 MaxfT 3¼0:180, B2L3¼0:144g 4941.206 2 MIDfB2L 0¼0:048, ffiffiffiffiffiffiffiffiffiffiffiffiffi ðd1=bÞ p ¼0:179, B2L 1¼0:086g 5742.229 2 MIDfB2L 1¼0:086, ffiffiffiffiffiffiffiffiffiffiffiffiffi ðd2=bÞ p ¼0:180, B2L 2¼0:125g 4998.109 2 MIDfB2L 2¼0:125, ffiffiffiffiffiffiffiffiffiffiffiffiffi ðd3=bÞ p ¼0:184, B2L 3¼0:144g 5054.462
Table 2. Results from Ouyang and Chuang (2000).
i Li(weeks) C(Li) (US $) Ti(weeks) Ri(units)
EACðTi, LiÞ
ðUS $ per yearÞ EðX RÞþ=DðTiþLiÞ
0 8 0 8.80 226 4764.73 0.0158
1 6 5.6 8.84 201 4745.68 0.0168
2 4 19.6 8.97 177 4771.89 0.0180
3 3 24.6 9.37 169 4941.21 0.0184
in Table 2, as ETC(T1¼8.80, L1¼8) ¼ 4764.73 and
then check the service level constraint to find that failed again. Following this pattern, the ratios of (E(X R)þ/D(TiþLi)) are all greater than 0.015.
This means that by Ouyang and Chuang’s method all their solutions do not satisfy the service level constraint, so they concluded that the inventory system has no feasible solution. We discover that the solution procedure of Ouyang and Chuang sometimes cannot obtain the minimum solution.
6. Directions for future research
Here we want to discuss a possible direction for future research. Since Ouyang et al. (1996) devel-oped a mixed inventory model with backorders and lost sales for variable lead time, there is a trend to treat the reordered point as a decision variable as pointed out by Moon and Choi (1998), Hariga and Ben-Daya (1999) and Yang, Ronald, and Chu (2005). Ouyang and Chuang (2000), for the normal distribution model, considered the reordered point being fixed, and for the distribution free model, they used the reordered point as a new decision variable.
In this article, we still take the fixed reordered point for the normal distribution model, such that one possible direction for the future research is to examine the normal distribution model where the reordered point is a new decision variable.
Taking the reordered point as a new decision variable is equivalent to saying that the safety factor, k, is a new variable such that the objective function becomes EAC(T, L, k). On the other hand, if we treat the safety factor as fixed, then the objective function is EAC(T, L). It is of little significance that the minimum value for EAC(T, L, k) is less than or equal to the minimum value of EAC(T, L). Generally speaking, if we are allowed to introduce another new parameter, say A (ordering cost per order), then EAC(T, L, k, A) will have even an smaller minimum value. Here, we face a problem how to compare the results from EAC(T, L), EAC(T, L, k) and EAC(T, L, k, A). If we
directly use the minimum value then definitely EAC(T, L, k, A) will be the best choice.
There are two methods, AIC or CAIC (Hartley and Helmbold 1995, p. 609–643), to test the difference between different methods with different numbers of parameters. The formulas for AIC and CAIC are given below, where p ¼ the number of variables, n ¼ number of constant data and Min ¼ minimum value for each method: AIC( p) ¼ n ln (Min/n) þ 2p and CAIC( p) ¼ AIC( p) þ p(ln n 1). In both statistics, the smaller the value, the better.
Up to now, we cannot solve the minimum problem for the EAC(T, L, k) for the normal distribution model so that the comparison between EAC(T, L, k) and EAC(T, L) by AIC or CAIC may be a direction for the future research.
7. Conclusion
In the above discussions, we pointed out the question-able results in the paper of Ouyang and Chuang so that the minimum may not occur at the boundary points of sub-domain for the crash lead time cost. We offered the corrected algorithm to find the optimal solution. Our refined algorithms are easy to use and mathema-tically sound and provide the optimal replenishment solution for decision makers. In the second numerical example, we illustrated that sometimes Ouyang and Chuang cannot derive the optimal solution and so our improved method still provides the optimal solution for the decision makers.
Notes on contributors
Chih-Young Hung is an Associate Professor in National Chiao Tung University. He received his PhD degree in Finance from Texas Tech University, USA, in 1990. His research interests include financial strategy and management, business valuation, value-based management and venture capital.
Kuo-Chen Hung received his BA and
MS degrees in Department of
Information Management from
National Defense Management
College, Taipei, Taiwan, in 1991 and 1997, respectively. He received the PhD degree in Department of Industrial Engineering and Enterprise Information from Tunghai University, Taichung, Taiwan, in 2006. Currently, he is an Assistant Professor at National Defense University. His research interests include fuzzy arithmetic, grey prediction, multiple criteria decision-making, fuzzy systems, vague sets, inventory management and system optimisation.
Table 3. Local minimum value for each case.
Theorem Minimum point Minimum value
1 MaxfT 0¼0:169, B2L0¼0:204g 4837.378 1 MaxfT 1¼0:170, B2L1¼0:243g 5008.922 1 MaxfT 2¼0:173, B2L2¼0:281g 5278.311 1 MaxfT 3¼0:180, B2L3¼0:300g 5520.673
242 C.-Y. Hung et al.
Wen-Han Tangis an Assistant profes-sor in R.O.C. Military Academy. He received his PhD in Institute of Management of Technology from National Chiao Tung University. He is an Army Colonel at present. His research interests include Lanchester model, inventory system, multiple cri-teria decision-making and data envel-opment analysis.
Robert Linis an Assistant Professor in Oriental Institute Technology. He
received his MS degree in
Department of Mathematics from National Central University, Chung-Li, Taiwan, in 1977. His research interests include inventory manage-ment, decision-making, fuzzy arith-metic and optimisation.
Chi-Kae Wangreceived his MS degree in Department of Information Management from National Defense Management College, Taipei, Taiwan in 1997. Currently, he is working towards the PhD degree at Chung Yuan Christian University, Taichung, Chung-Li, Taiwan, Department of Mechanical Engineering. His research interests are focussed on image processing, especially blind source separation and feature classification and inventory management.
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