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International Journal of Systems Science
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Improved periodic review inventory model involving
lead-time with crashing components and service level
Shing-Ko Liang a , Peter Chu b & Kuo-Lung Yang a a
Department of Industrial Engineering and Management , National Chiao Tung University , Taiwan, Republic of China
b
Department of Traffic Science , Central Police University , Taiwan, Republic of China Published online: 29 Feb 2008.
To cite this article: Shing-Ko Liang , Peter Chu & Kuo-Lung Yang (2008) Improved periodic review inventory model involving lead-time with crashing components and service level, International Journal of Systems Science, 39:4, 421-426, DOI: 10.1080/00207720701832523
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Vol. 39, No. 4, April 2008, 421–426
Improved periodic review inventory model involving
lead-time with crashing components and service level
SHING-KO LIANGy, PETER CHU*z and KUO-LUNG YANGy
yDepartment of Industrial Engineering and Management, National Chiao Tung University, Taiwan, Republic of China
zDepartment of Traffic Science, Central Police University, Taiwan, Republic of China (Received 22 April 2003; in final form 19 November 2007)
A periodic review inventory model with backorders and lost sales in which both lead-time and the periodic review length are decision variables and the production interval demand follows a normal distribution is explored in this article. Ouyang and Chuang discussed this problem in a recent paper published in the International Journal of Systems Science. However, their algorithms might not, although intended to, find the optimal solution due to questionable results in their solution procedure. The purpose of this study is 3-fold. First, the criteria for the existence and uniqueness of the critical solution for minimising the total expected annual cost are determined. Second, a correct and efficient algorithm to improve their method is constructed to find the optimal lead-time and periodic review length simultaneously. Finally, some numerical examples are provided to compare our solution procedure with that of Ouyang and Chuang’s method to demonstrate their questionable results.
Keywords:Inventory; Lead-time; Backorder; Lost sales; Periodic review
1. Introduction
The irresistible trend of dropping gross profit and shortening product life cycle has been certain in recent years while entrepreneurs have been striving to pinpoint the needs of the market under the fierce competitive environment. Most business customers bear in mind four pivotal elements: quality, price, delivery time and service on the whole. Among which delivery time has become significantly crucial nowadays since it reflects efficiency of speed intensive era and demonstrates the capability of coping with high volatile changes of demand and shorter product life cycle. That is how this particular element grabs our concentration and in the meantime is widely recognised in most real-life applications as a decision variable. For fast delivery, suppliers work overtime, add manpower, renew equip-ment, reset layout or choose better logistic means in order to cut down the lead-time in order to gain
appreciation from customers and competitive
advantages.
Lead-time has recently been used as a decision variable in some models. Liao and Shyu (1991) first presented lead-time as negotiable and decomposed it into several components, each having a different piecewise linear crash cost function for lead time reduction. Ben-Daya and Raouf (1994) extended Liao and Shyu’s (1991) work, considering both lead-time and order quantity as decision variables where shortages were neglected. Moon and Gallego (1994) assumed unfavourable lead-time demand distribution and solved both the continuous review and periodic review models with a mixture of backorders and lost sales using the minmax distribution free approach. Ouyang et al. (1996) generalised Ben-Daya and Raouf’s (1994) assumption that shortages were allowed and constructed variable
lead-time from a mixed inventory model with
backorders and lost sales. Moon and Choi (1998) and Lan et al. (1999) pointed out the problem in Ouyang
et al. (1996). They found optimal order quantities and
*Corresponding author. Email: [email protected]
International Journal of Systems Science
ISSN 0020–7721 print/ISSN 1464–5319 onlineß 2008 Taylor & Francis
http://www.tandf.co.uk/journals DOI: 10.1080/00207720701832523
optimal lead-time for a mixed inventory model, and
developed a simplified solution procedure under
different conditions. Chu et al. (1999) showed some drawbacks in the Ben-Daya and Raouf’s (1994) solution procedure and used the Newton–Raphson method with an appropriate starting point to improve this problem. Wu and Ouyang (2000) assumed that an arrival order lot might contain some defective items. They derived a modified mixed inventory model in which the order quantity, order point and lead-time were decision variables. Wu and Tsai (2001) considered that the lead-time demands from different customers are not identical. They developed a mixed inventory model with backorders and lost sales for variable lead-time demand with a mixed normal distribution. Pan and Hsiao (2001) presented inventory models with backorder discount as inducement and variable lead-time to ensure that customers would be willing to wait for backorders. Pan et al. (2004) proposed integrated inventory systems with the objective to simultaneously optimising the order quantity, lead time, back ordering and reorder point. Following that Pan and Hsiao (2005) proposed two inventory models, one with normally distributed demand and another with generally distributed demand, in which lead-time crashing cost was represented as a function of reduced lead-time and the quantities in the order. Hoque and Goyal (2006) developed a heuristic solution procedure to minimise the total cost of setup or ordering, inventory holding and lead-time crashing for an integrated inventory system under controllable lead-time between a vendor and a buyer. Instead of a stock-out term in the objective function, Lee et al. (2006) added a service level constraint to the model and developed two computational algorithms to find optimal order quantity and optimal lead-time. Chang et al. (2006) developed an iterative procedure to find the optimal solution with the consideration that lead-time can be shortened at an extra crashing cost which depends on the lead-time length to be reduced and the ordering lot size. Lee et al. (2007) developed a computational algorithmic procedure to simultaneously optimise the order quantity, ordering cost, back-order discount and lead-time with the consideration that lead-time can be shortened at an extra crashing cost and allow the back-order rate as a control variable. Tempelmeier (2007) included the exact on hand inventory into the model formulation that minimises the setup and holding costs with respect to a constraint on the probability that the inventory at the end of any period does not become negative. As a result, the models are also applicable in situations with very low service levels. Wu et al. (2007) developed two algorith-mic procedures to find optimal inventory policy with the consideration that the lead-time demand follows either the mixture of normal distribution or mixture of free
distribution and the total crashing cost is related to the lead-time by a negative exponential function.
This study examines the same inventory model as Ouyang and Chuang (2000). They considered both lead-time and periodic review length as decision variables for a mixture periodic review inventory model. Ouyang and Chuang (2000) thought that it is often difficult to determine the stock-out cost in inventory systems. They added a service level constraint in the model instead of a stock-out cost term in the objective function. Their method predicted that (a) the expected annual cost is a convex function and (b) sometimes, this inventory model does not have feasible solutions. This study will show that the expected annual cost is not a convex function and this inventory model always has an optimal solution. This study also constructs correct and efficient algorithms to find the optimum order quantity and lead-time simultaneously when the protection interval demand distribution is normal. Two numerical examples are provided to explain the questionable results in Ouyang and Chuang’s algorithm.
2. Notation and assumption
We use the same notations and assumptions as Ouyang and Chuang (2000).
Notation:
A ¼Fixed ordering cost per order.
D ¼Average demand per year.
h ¼Inventory holding cost per item per year.
L ¼Length of lead-time, a decision variable.
T ¼Length of periodic review, a decision
variable.
X ¼Demand of production interval, T þ L,
which has probability density function
fX, finite mean D(T þ L) and standard
derivation pffiffiffiffiffiffiffiffiffiffiffiffiffiT þ L. xþ
¼Maximum value of x and 0,
i.e. xþ¼maxfx, 0g.
¼Proportion of demands that are not
met from stock so 1 is the service level.
¼Fraction of the demand backordered
during the stock-out period. C(L) ¼ Lead-time crashing cost. EAC(T, L) ¼ Total expected annual cost. Assumptions:
(1) The inventory level is reviewed every T units of time. A sufficient ordering quantity is ordered up
422 S.-K. Liang et al.
to the target level R, and the ordering quantity is arrived at after L units of time.
(2) The length of the lead-time L is not greater than the review period length T so that there is never more than a single order taking place in any cycle.
(3) The target level R ¼ expected demand during
protec-tion interval þ safety stock (SS), and
SS ¼ kpffiffiffiffiffiffiffiffiffiffiffiffiffiT þ L, that is R ¼ DðT þ LÞ þ kpffiffiffiffiffiffiffiffiffiffiffiffiffiT þ L
where k is the safety factor and satisfies
P(X > R) ¼ q, in which q given represents the allow-able stock-out probability during the protection interval.
(4) E½X Rþis the expected demand short at the end of
cycle. Hence, E½X Rþ are the backordered
quantities and ð1 ÞE½X Rþ are the lost sales.
Therefore, the expected net inventory level at
the beginning of the period is
R DL þ ð1 ÞE½X Rþ and the expected net
inventory level at the end of the period is
R DðT þ LÞ þ ð1 ÞE½X Rþ. Hence, the
expected holding cost per year is
h½R DððT=2Þ þ LÞ þ ð1 ÞE½X Rþ.
(5) If X has a normal distribution function F(x), then E½X Rþ¼pffiffiffiffiffiffiffiffiffiffiffiffiffiT þ LGðkÞ where GðkÞ ¼ R1
k ðz kÞfZðzÞdz and fZ(z) is the probability density function of the standard normal random variable Z. (6) The lead-time L includes n mutually independent
components. The ith component has a minimum duration ai, and normal duration bi, and a crashing
cost per unit time ci. Further, we assume that
c1 c2 cn. The lead-time components are
crashed one at a time starting with the component of least ciand so on.
(7) If we let L0¼Pnj¼1bj and Li be the length of
lead-time with components 1, 2, . . . , i crashed to their
minimum duration, then Li¼
Pn
j¼iþ1bjþ Pi
j¼1aj. The lead-time crashing cost C(L) per cycle for
a given L 2 ½Li, Li1, is given by
CðLÞ ¼ ciðLi1LÞ þPi1j¼1cjðbjajÞ.
(8) When X has a normal distribution function, the service level constraint becomes
E½X Rþ
DðT þ LÞ ¼
GðkÞ DpffiffiffiffiffiffiffiffiffiffiffiffiffiT þ L:
3. Review of Ouyang and Chuang’s result
We review the inventory model in which the protection interval demand follows normal distribution. The total expected annual costs are the sum of the ordering cost, holding cost and lead-time crashing cost, subject to a
constraint on the service level. Hence, the problem can be formulated as Min EACðT, LÞ ¼A Tþh " DT 2 þ ffiffiffiffiffiffiffiffiffiffiffiffi T þ L p k þ1 GkÞþ # þCðLÞ T ð1Þ subject to GðkÞ DpffiffiffiffiffiffiffiffiffiffiffiffiffiT þ L, ð2Þ where 0 < T < 1 and L 2 ½Li, Li1for i ¼ 1, 2, . . . , n.
Ouyang and Chuang (2000) derived that for
L 2 ðLi, Li1Þ, i ¼ 1, 2, . . . , n, @2 @L2EACðT, LÞ ¼ h 4 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T þ L ð Þ3 q ðk þ ð1 ÞGðkÞÞ < 0: ð3Þ
They implied that EAC(T, L) is a concave function of L. They obtained @ @TEACðT, LÞ ¼ hðk þ ð1 ÞGðkÞÞ 2pffiffiffiffiffiffiffiffiffiffiffiffiffiT þ L þ hD 2 A þ CðLÞ T2 , ð4Þ and @2 @T2EACðT, LÞ ¼ 3T þ 4L 4T hðk þ ð1 ÞGðkÞÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðT þ LÞ3 q þhD T : ð5Þ
From ð@2=@T2ÞEACðT, LÞ > 0, Ouyang and Chuang
(2000) claimed that EAC(T, L) is a convex function of T. From equation (3), they knew that the minimum
would occur at L ¼ Li for i ¼ 0, 1, 2, . . . , n. For a
given Li, setting equation (4) equal to zero, they solved hðk þ ð1 ÞGðkÞÞ 2pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiTiþLi þ hD 2 ¼ A þ CðLiÞ T2 i ð6Þ
to find Ti. For each pair (Ti, Li), they computed
EAC(Ti, Li). Ouyang and Chuang (2000) first
relaxed the service level constraint and found
mini¼0,..., nEACðTi, LiÞ.
If EACðTs, LsÞ ¼mini¼0,..., nEACðTi, LiÞ and the
service level constraint ðGðkÞ=D ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiTsþLs p
Þ is
satisfied, they then accepted that (Ts, Ls) is the
optimal solution. Conversely, if the service level constraint ðGðkÞ=D ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiTsþLs
p
Þ is not satisfied,
they then assumed that EACðTt, LtÞ ¼next
mini¼0, ..., nEACðTi, LiÞ and checked whether the service level constraint ðGðkÞ=DpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiTtþLtÞ is satisfied? If it is satisfied, (Tt, Lt) is the optimal solution. Otherwise, they continued to search for a solution until the service level constraint was satisfied. They judged that this inventory model has no feasible solution if no solution for equation (6) satisfies the service level constraint.
4. Improved mathematical analysis
We begin by studying the normal distribution model. The errors in Ouyang and Chuang’s (2000) algorithm are then discussed. From equation (3), we know that for fixed T, EAC(T, L) is concave in L 2 ½Li, Li1. Hence, the problem can be simplified and formulated as
Min EACðT, LiÞ ¼A Tþh DT 2 þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi T þ Li p ðk þ ð1 ÞGðkÞÞþ þCðLiÞ T ð7Þ subject to GðkÞ D ffiffiffiffiffiffiffiffiffiffiffiffiffiffiT þ Li p , for i ¼ 0, 1, . . . , n ð8Þ We derive that @ @TEAC T, Lð iÞ ¼ h k þð ð1 ÞG kð ÞÞ 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiT þ Li p þhD 2 A þ C Lð iÞ T2 , ð9Þ and @2 @T2EAC T, Lð iÞ ¼2 A þ C Lð iÞ T3 h k þð ð1 ÞG kð ÞÞ 4 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T þ Li ð Þ3 q : ð10Þ Comparing equations (5) and (10), we know that the
result of Ouyang and Chuang (2000) for
ð@2=@T2ÞEACðT, LÞ is questionable. To examine their
result, we suppose that T#i satisfies
ð@=@TÞEACðT#i, LiÞ ¼0. Then we compute
ð@2=@T2ÞEACðT#
i, LiÞ. It shows that @2 @T2EAC T # i, Li ¼ 3T # i þ4Li 4T#i ! h k þð ð1 ÞG kð ÞÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T#i þLi 3 q þhD T#i : ð11Þ
Hence, Ouyang and Chuang (2000) showed that at the critical solution of ð@=@TÞEACðT, LiÞ ¼0, say T#i, the
second partial derivative of EAC(T, L) with respect to T is positive. Their result only implies that T#i (if it exists) is a local minimum solution but they did not prove that EAC(T, L) is a convex function of T.
From the correct expression of ð@2=@T2ÞEACðT, LiÞin
equation (11), we know that when
T ! 1,ð@2=@T2ÞEACðT, L
iÞ<0. It is clear that,
EAC(T, L) is not a convex function of T. Their prediction for EAC(T, L) being a convex function of T is then false. Now, we begin to develop our theorem for the normal distribution model. To simplify the
expression, we assume that i¼A þ CðLiÞ, !1 ¼ ðDh=2Þ,
!2¼ ðh=2Þðk þ ð1 ÞGðkÞÞ and fið Þ ¼T !1T2þ!2 T 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi T þ Li p ifor i ¼ 0, 1, . . . , n: ð12Þ
From equation (9), we realise that solving
ð@=@TÞEACðT, LiÞ ¼0 and fiðTÞ ¼0 are in fact
equivalent. Since ðd=dTÞfiðTÞ ¼2!1Tþ
ð!2T= ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðT þ LiÞ3
q
Þðð3T=2Þ þ 2LiÞ40, we derive that fi(T)
increases from fið0Þ ¼ i<0 to fið1Þ ¼ 1, then
fi(T) ¼ 0 has a unique positive solution, say Ti^.
We know that Ti^ is the minimum solution for
EAC(T, Li) without considering the constraints. There
are two constraints for the periodic review length, so we consider the following two questions:
(1) Does Ti^ satisfy the second assumption as LiTi^?
(2) Does Ti^ satisfy the service level constraint
ðGðkÞ=D ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiTi^þ Li
p
ÞÞ ? Therefore, we define
Ti ¼max Ti^, Li, G kð Þ D
2
Li
( )
: ð13Þ
It implies from T2ð@=@TÞEACðT, L
iÞ ¼fiðTÞ that ð@=@TÞEACðT, LiÞ40 on ½Ti, 1Þ. So Ti is the minimum
solution of EAC(T, Li) under these two constraints.
Finally, suppose EAC Topt, Lopt
¼ min i¼0,..., nEAC T i, Li , ð14Þ
we prove that (Topt, Lopt) is the optimal solution for this inventory model under these two constraints and the optimal solution will always exist. Hence, the prediction of Ouyang and Chuang (2000) that sometimes this inventory model has no feasible solution is false. Our improved algorithm is presented as follows: (1) Step 1: For each Li, i ¼ 0, 1, . . . , n, compute Ti^,
from the root of function fi(T).
424 S.-K. Liang et al.
(2) Step 2: Using equation (13), find T i for i ¼0, 1, . . . , n.
(3) Step 3: From equation (14), find the optimal solution (Topt, Lopt).
5. Numerical example
To demonstrate the improvement of our algorithm, we
consider an example with the following data:
D ¼500 units/year, A ¼ $400 per order, h ¼ $40 per
item per year, ¼ 7 units/week, ¼ 1, q ¼ 0.2 (in this situation, the safety factor k ¼ 0.845) and lead-time with three components. The data is shown in table 1. According to our proposed algorithm, the optimal length of a period T* ¼ 10.3142 weeks, optimal lead-time L* ¼ 6 weeks and the minimum total expected
annual cost EACðT, LÞ ¼$4984:02. The results are
listed in table 2.
The results found by Ouyang and Chuang are
computed in table 3. They have
EACðT1, L1Þ ¼mini¼0, ..., 3EACðTi, LiÞ. However, (T1,
L1) do not satisfy the service level constraint. They
obtained EACðT0, L0Þ ¼next mini¼0, ..., 3EACðTi, LiÞ
where (T0, L0) meets the service level constraint.
Ouyang and Chuang finally came up with the optimal periodic review length T* ¼ 9.7142 weeks, optimal lead-time L* ¼ 8 weeks and the minimum total expected
annual cost EACðT, LÞ ¼$5005:12. Corresponding to
their results, our proposed algorithm saved $21.1. When ¼ 0.01 from table 2, we know that the critical point for the first partial derivative of the total expect annual cost does not satisfy the service level constraint. Hence, Ouyang and Chuang (2000) could not find
solutions using their algorithm. They implied that when ¼0.01, there is no feasible solution for this inventory model. However, from table 4, the results from implementing proposed algorithm: the optimal length for a period T* ¼ 57.2569 weeks, the optimal lead-time L* ¼ 8 weeks and the minimum total expected annual cost EACðT, LÞ ¼$13285:53 can be effectively found.
6. Conclusion
Enterprises face the harsh challenge of short product life cycles and delivery time. The challenge forces managers to put much effort into lead-time management. For instance, building upon this conviction and under tremendous pressure from foreign brand firms (e.g. Dell, Apple, Nokia and so forth) and Integrated Device Manufacturers (IDM) (e.g. Intel, Motorola, TI, NEC and so forth), Taiwan’s leading electronic OEM and ODM factories have successfully developed 95-3, 98-3 and 10-2 delivery models. 95-3 model represents delivering 95% of total orders to those enterprise clients within three days, 98-3 model moves up the delivery volume to 98% within three days, and 10-2 model even moves forward one big step and tries to deliver 100% of total orders within two days. The accomplishment and applications of these models enable Taiwan’s OEM and ODM factories to efficiently lower the lead-time, establish competitive advantages and constantly gain orders from those international enterprise giants. In a word, flexible and solid control capability of lead-time and delivery requirement from customers is undoubtedly not only one of the most potent means of
Table 1. Lead-time data.
Lead-time component, i Normal duration, bi(days) Minimum duration, ai(days) biai (weeks) Unit crashing cost, ci ($/week) 1 20 6 2 2.8 2 20 6 2 7 3 16 9 1 35
Table 2. Summary of the optimal solution from our proposed algorithm ( ¼ 0.02). i T^ i (weeks) Li (weeks) ðGðkÞ=DÞ2Li Ti EACðT i, LiÞ 0 9.7142 8 8.3142 9.7142 5005.12 1 9.7443 6 10.3142 10.3142 4984.02 2 9.8676 4 12.3142 12.3142 5095.64 3 10.2545 3 13.3142 13.3142 5291.56
Table 3. Summary of the optimal solution from Ouyang and Chuang’s algorithm. i Li (weeks) C(Li) Ti (weeks) EAC(Ti, Li) GðkÞ=D ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiTiþLi p 0 8 0 9.7142 5005.12 0.0192 1 6 5.6 9.7443 4977.18 0.0204 2 4 19.6 9.8676 4989.89 0.0217 3 3 54.6 10.2545 5138.65 0.0222
Table 4. Summary of the optimal solution using our proposed algorithm ( ¼ 0.01). i T^ i (weeks) Li (weeks) ðGðkÞ=DÞ2Li Ti EACðTi, LiÞ 0 9.7142 8 57.2569 57.2569 13285.53 1 9.7443 6 59.2569 59.2569 13662.80 2 9.8676 4 61.2569 61.2569 14047.68 3 10.2545 3 62.2569 62.2569 14263.50
acquiring market power in the present ferocious business arena but also in the meantime the most effective way of obtaining the minimum total expected annual cost.
In the above discussions, we pointed out the questionable algorithm in the paper of Ouyang and Chuang (2000). Their approach is too complicated and lacks theoretical solidity so that the consistency and feasibility from a mathematical programming point of view and further practicability are dubitable. We provide a simplified and theoretically-rigorous algorithm to greatly improve their weaknesses and shortcomings. Our approach promises the existence and uniqueness of optimal solution.
Acknowledgements
The authors are tremendously grateful to the two anonymous referees for constructive comments that led to great improvement of this article. All their sugges-tions were incorporated directly into the text with discretion. In addition, the authors wish to express appreciation to Miss Bonnie Shuan Wang for her assistance of English improvement.
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