Generalized Newsboy model for MRP parameterization under uncertainties

Proceedings of the 13th IFAC Symposium on Information Control Problems in Manufacturing Moscow, Russia, June 3-5, 2009

Generalized Newsboy model for MRP parameterization under uncertainties Mohamed-Aly Louly 1 and Alexandre Dolgui 2,* 1

King Saud University College of Engineering - Industrial Engineering Department P.O. Box 800, Riyadh 11421, Kingdom of Saudi Arabia e-mail: [email protected] 2

Ecole des Mines de Saint Etienne Centre for Industrial Engineering and Computer Science 158 cours Fauriel, 42023 Saint Etienne Cedex 2, France e-mail: [email protected] Abstract: This study deals with MRP parametrization under uncertainties. The actual lead time has random deviations, so it can be considered as a random variable. MRP approach with Periodic Order Quantity (POQ) policy is considered. The aim is to find the optimal MRP time phasing. The proposed model and algorithms minimize the sum of the setup cost, backlogging cost and average holding costs. Keywords: Supply Planning, Stochastic Lead Time, Periodic Order Quantity, MRP.

Thankfully, the MRP approach can be tailored to uncertainties by searching optimal values for its parameters. An adequate choice of these parameters increases the effectiveness of MRP techniques. Thus, one of essential issues is MRP parameterization for real life companies in industrial situations. This is commonly called MRP offsetting under uncertainties. Some MRP parameters are: planned lead time, safety stock, lot-sizing rule, freezing horizon, and planning horizon. There are extensive publications concerning safety stock calculation for random demand of finished products (Porteus, 1990; Lee and Nahmias, 1993). In contrast, certain parameters seem not to be sufficiently examined as, for example, planned lead time (differences between due dates and release date) and the optimal parameterization of most used lot-sizing rules like the periodicity of the PeriodicOrder-Quantity method. If actual lead time is uncertain, the planned lead time can contain safety lead time, i.e. the planned lead time is calculated as the sum of the forecasted and safety lead time. The later should be formulated as a trade-off between overstocking and stockout while minimizing the total cost. The search for optimal value of safety lead time, and, consequently, for planned lead time, is a crucial issue in Supply chain management with MRP approach. In spite of this, as mentioned in (Porteus, 1990; Lee and Nahmias, 1993), the problem of planned lead time has been given limited attention in the literature.

1. INTRODUCTION Effective replenishment planning is a crucial problem in Supply chain management. An inadequate inventory control policy leads to overstocking or stockout situations. In the former, the generated inventories are expensive and in the later there are shortages and penalties due to unsatisfied customer demands. Material Requirements Planning (MRP) is a commonly accepted approach for replenishment planning in major companies (Axsäter, 2006). The MRP software is accepted readily, most industrial decision makers are familiar with this approach. The practical aspect of MRP lies in the fact that this provides a support clear and simple to understand, as well as a powerful information system to decision making. The state of the art on the application of this approach in industrial situation is given in (Baker, 1993; Sipper and Bulfin, 1998; Zipkin, 2000; Axsater, 2006; Tempelmeier, 2006). Nevertheless, MRP is based on the supposition that the demand and lead time are deterministic. However, most production systems are stochastic. This is because there are some random factors and unpredictable events such as machine breakdowns, transport delays, etc. which can cause random deviations from planning (Koh and Saad, 2003). Therefore, actually, the deterministic assumptions of MRP are often too restrictive.

*

author for correspondence: Prof. A. Dolgui, Ph.: +33 (0)4.77.42.01.66; Fax: +33 (0)4.77.42.66.66

978-3-902661-43-2/09/$20.00 © 2009 IFAC

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13th IFAC INCOM (INCOM'09) Moscow, Russia, June 3-5, 2009

Another newsboy generalization was examined in an earlier work (Louly and Dolgui, 2002), but it was with the simple case of Lot-for-Lot policy. In the model proposed, the backlogs are authorized and a unit backlogging cost is supposed to be known. The objective was to minimize the sum of average backlogging and holding costs. A Markovian chain was also proposed in (Louly and Dolgui, 2004) to extend the results to the Periodic Order Quantity (POQ) policy, but its transition matrix has an exponential size as a function on the length of the lead time distribution.

- periodicity (p) - planned lead time (x) In Figure 1, these parameters are equal to p=2 and x=1, respectively. In a stochastic environment, an adequate choice of these parameters is crucial, because this defines the average total cost of supply planning. In general, the calculation of the optimal values of these parameters is a very complex optimization problem.

2. THE BASIC PRINCIPLES OF MRP SYSTEMS

The majority of publications are devoted to the MRP parameterization under customer demand uncertainties. As to random lead times, the number of publications is relatively small. Simulation is often used to analyze customer demand uncertainties for MRP parameterization. The majority of these works deals with the problem of nervousness, i.e. excessive change in production plans. Sridharan and Berry (1990) propose to freeze a part of the MPS to dampen instability. When frozen, the MPS is fixed, and therefore the demand is fixed. Here, the attention must be focused only on lead time uncertainties. For lead time uncertainties, some publications have already considered the safety lead time calculation. For example, Whybark and Williams (1976) show that under lead time uncertainty one is better off setting safety times than safety stocks. Molinder (1997) propose a simulated annealing approach to find appropriate safety lead time in this context. As for analytical approaches, the literature is rich with stochastic models considering again a random demand of finished product for single item production systems (Rao and Schneller, 1990; Khang and Fujiwara, 1993; Sox and Muckstadt, 1996). In this context, safety stocks can be determined by using, for example, different generalizations of the Newsboy model (Porteus, 1990; Petruzzi and Dada, 1999). Analytical models with stochastic lead time have been studied even less. The better known examples are: the paper of Kaplan (1970) which suggested a finite horizon dynamic programming approach and Liberatore (1979) who tried to extend directly the EOQ model by introducing a stochastic lead time. Yano (1987a) determined optimal planned lead time for MRP in serial production systems where one type of item is produced under random actual procurement and processing times. The finished product demand is fixed, and lot-for-lot policy is used. The sum of inventory holding and job tardiness costs is minimized. A disadvantage of this approach is the problem to express the objective function in closed form for two or more stages. Elhafsi (2002) suggested a recursive strategy for any number of stages in a serial production system without searching for a closed form. Kumar (1989) examined a one-period model (for a batch) with stochastic lead time and a fixed production due date and quantity. The timing of each order is determined so that the overall cost, composed of the holding and tardiness costs is minimized. Fujiwara and Sedarage (1997) developed a model for a

3. LITERATURE REVIEW

The goal of MRP is to determine a replenishment schedule for a given time horizon. The gross requirements for the finished product are given by the Master Production Schedule (MPS). The MRP approach deals with the calculation of these requirements for a series of sequential planning periods. One period can be a day, week, or month depending on applications. Let us introduce the following notations: x planned lead time, Q (i − x) planned orders released at the period i-x and for period i I (i ) projected on-hand inventory N (i ) net requirements G (i ) gross requirements The current inventory I (1) for the period 1 is given. The net needs of the period i are obtained as follows: N (i ) = G (i ) − I (i ) The planned released order quantity: Q(i − x) = max{0, N (i)} This is the core of the MRP approach. These rules are implemented in MRP tables. The above formulas concern the Lot-for-Lot policy, i.e. where the orders are not grouped. Often the simple lot-forlot policy is not possible (transportation constraints,…) or too costly (because of an expensive setup,…), so another lotsizing rule should be used, for example, the Periodic Order Quantity (POQ) policy. An example of MRP table for the POQ policy, which consists of grouping the needs for p consecutive periods, is given in Figure 1. Periods

1

2

3

4

5

6

7

8

Gross requirements

40

35

65

45

20

55

45

35

Projected On-hand 100

60

25

-40 45

20

55

45

35

Net Requirements

40

Planned Order receipts

85

Planned Order Releases

85

75 75

80 80

Fig. 1. An example of MRP table for POQ policy Each MRP table has several parameters: periodicity for the lot-sizing (if POQ policy is chosen), planned lead time for the time phasing, safety stock, etc. In this paper, only the two following essential parameters are considered (due to of their importance): 835

13th IFAC INCOM (INCOM'09) Moscow, Russia, June 3-5, 2009

simple Just-in-Time (JIT) system with random lead time. This is a continuous time model. The following general assumptions were used: product demand is constant and known, and the production capacity is infinite. The authors have taken into account the inventory holding, shortage and setup costs. A decomposition algorithm is proposed where each sub-problem is solved numerically. Two-level systems are studied in (Yano, 1987b). The procurement time is random continuous variable. The goal was to determine the planned lead time minimizing the sum of holding and tardiness costs. Axsäter (2005) considered a multi-level system with random operation time. The objective was to find the values of the release dates minimizing the sum of the expected holding and backlogging costs. This problem was solved by decomposition. In (Arda and Hennet, 2006), a problem with one retailer and several suppliers was considered. A queueing theoretical model was proposed. There is single type of product; the supplier lead time is exponentially distributed. The demand follows a Poisson process. The authors proposed a (S-1,S) policy and a rule for selecting the supplier of each inventory replenishment order. A system with random procurement time was studied in (Dolgui et al., 1995). The integer programming model calculates the number of products of each type to be produced during each period minimizing holding and backlogging costs. The ordered quantity calculated based on optimization via simulation. The same problem was studied in (Proth et al., 1997), a heuristic algorithm was proposed. As aforementioned, the problem studied here has already examined but for the Lot-for-Lot policy, see (Dolgui and Louly, 2002), (Louly and Dolgui, 2002), and (Louly and Dolgui, 2004). Finally, for more exhaustive reviews on the simulation and analytical studies of the parameters affecting the planned lead time calculation in MRP environment, see (Yeung et al., 1998; Mula et al., 2006; Dolgui and Prodhon, 2007, Dolgui et al., 2008).

The finished product demands are satisfied at the end of each period and unsatisfied demands are backordered and have to be satisfied during the subsequent periods. The following notations are used: h unit holding cost for row materials b unit backlogging cost for finished product; c setup cost, i.e. the cost of an supply order L probability distribution for raw materials lead time u upper value of lead time distribution for raw materials lead time of raw materials ordered at the beginning Lk of the period k D demand for finished product per period a quantity of raw materials needed to produce the end item p supply periodicity Q supply order quantity for raw materials x planned lead time for raw materials Z+

= max(Z , 0)

As the lead time is a random variable, the planned lead time can be greater than forecasted lead time to introduce safety lead time. In the considered model the quantities ordered are the same, so the planned lead time gives also initial inventory. Thus, the aim of this study can be expressed in other terms: to find the optimal values of the initial inventories aDx and parameter p, where x is the planned lead time. This approach takes into account the major factors of the supply planning with random lead time to obtain an efficient optimization algorithm for planned lead time and the periodicity calculation. 4.2 Analytical expression of the criterion Some proofs will be made here to obtain analytical expressions for the criterion as functions of decision variables. For the considered model, given that the maximal value of the lead time is equal to u, only the orders made in the previous u-1 periods may not have arrived yet. The orders made before have already arrived. Therefore, the number

4. OPTIMIZATION 4.1 Inventory model To take into account the particularities of MRP parameterization, the following model will be considered in this paper. The probability distribution of raw material procurement time (lead time) L is known. There is an upper value of the lead time distribution, i.e. L ≤ u. The finished product demand is known and constant for all periods and equal to D. The unit holding cost h of the raw material per period, and setup cost c are also known. The POQ policy is used: raw materials are ordered every p periods. The goal of this model is to search the optimal values of the parameters p and x. The orders for raw materials are made at the beginning of the periods kp+1, k=0,1,2,…, and there is no order made in the periods kp+r, r=2,3,…,p. Then, the supply orders Q of raw materials are constant Q=aDp (p is a decision variable; a is the needed row material to make one finished product).

N p, m of expected deliveries at the end of the period m=kp+r is easy to calculate.

Let Lm +1− j , j=r, r+p, r+2p,…, r+ u−1−r p, be the lead time p

of the orders made at the beginning of the periods kp+1, (k1)p+1,…, (k- u−1−r )p+1. If Lm +1− j > j , then the order p

made in the period m+1-j is delivered after the end of the period m. Let 1E be the binary function equal to 1 when the expression E is true and equal to 0 otherwise. So, if 1Lm +1− j > j is equal to 1, then the order made at the period m+1-j is delivered after the end of the period m. Thus, the random variables N p, m can be represented as follows:

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13th IFAC INCOM (INCOM'09) Moscow, Russia, June 3-5, 2009

N

p, m

=N

p, kp + r

u −1− r p

=

∑ 1L( k − j ) p +1 > jp + r ,

u −1− r p

N p, kp + r =

k ≥0,

∑ 1L( k − j ) p +1 > jp + r

j =0

j =0

p ∈ {1,2,.., u − 1} , r ∈ {1,2,.., p}

(1)

For

C ( x, p, N p, kp + r ) = c × 1r = 1 + Dha( x + p − r − pN p, kp + r ) + (2) D(h + b)( pN p, kp + r + r − p − x) +

the inverse and equal to Dha( x + p − r − pN p, kp + r ) + . Given that there is a setup cost c if r=1, the sum of the holding, backlogging and setup costs at the end of the period m is equal to:

(ii)

N p, kp + r , k ≥ 0 is a covariance stationary process

(iii)

lim

t →∞

Cov ( N p, kp + r ,

N p, (k + t ) p + r ) = 0

Therefore, the cost can be written as follows: C ( x, p ) = 1

p

∑ E (C ( X , p, N p, kp + r ) ) p

r =1

By calculating the expectation and the probabilities, the final closed form is obtained: C ( x, p ) = cp + p −1 h + h( x − E ( N p )) + 2 p x+k −r + p (b + h) ∑ (1 − 1 ∑ F p, r ( )) p p k ≥0 r =1

(3)

Theorem 1: The average cost has the following closed form: p

random

t →∞

As shown in the previous proposition, the cost of a single period kp+r is a random variable. To study the considered multi-period problem, closed explicit forms must be obtained for the average cost and number of shortages on the infinite horizon, i.e. for the expressions:

C ( X , p) = 1

of

t ≥ u , thus, lim Cov( N k , N k + t ) = 0 .

■

)

sequence

independence between N p, kp + r and N p, ( k + t ) p + r , for

c × 1r = 1 + Dha( x + p − r − pN p, kp + r ) +

∑ C ( X , p, N

the

a stationary process. This condition is stronger than (ii), so (ii) is also valid. Finally, the condition (iii) is verified due to the

D( pN p, kp + r + r − p − x) + . The raw material inventory is

m → ∞ m k =1

r,

{ N k , k ≥ 1 } is

quantity aDp(k+1- N p, kp + r ). The number of backlogged units of the end item equals then

C ( X , p ) = lim 1

of

The condition (i) is usually verified because the criterion function is composed of measurable functions. The condition (ii) is verified because the lead times for the orders made at different periods are iid random variables. Then, the distribution of N p , kp + r does not depend on k. So,

Proof: The number of backlogged demands at the end of the period m equals the cumulative required quantity aD(kp+r) minus the initial raw material inventory aDx minus the delivered

p, k

value

, k ≥ 0 constitutes a discrete stochastic variables N process. The Law of Large Numbers cannot be applied here, because of the dependence among these variables. Which is why, this proof will be based on a covariance stationary process. In fact, it is enough to prove that: (i) the functions are measurable

The cost of the period m=kp+r, k ≥ 0 , p ∈ {1,2,.., u − 1} , r ∈ {1,2,.., p} , is equal to:

m

each

p, kp + r

The variables N p, m are independent from the decision variable x. Thus, they can be used to derive closed forms for the cost (Louly and Dolgui, 2004). Proposition:

D(h + b)( pN p, kp + r + r − p − x) +

, r ∈ {1,2,.., p}, k ≥ 0 .

▄

∑ E (C ( X , p, N p, kp + r ) ) p

4.3 Optimal solution properties The optimization problem can then be written as follows:

r =1

= c +

p p −1 h + h( x − E ( N p )) + 2 p x+k −r + p (b + h) ∑ (1 − 1 ∑ F p, r ( )) p p k ≥0 r =1

Min C ( x, p ) = cp + p −1 h + h( x − E ( N p )) + 2 p x+ k −r + p (b + h) ∑ (1 − 1 ∑ F p,r ( )) p p k ≥0 r =1

(4)

Proof: For each value of p within the set {1,2,…, u-1}, the following p stochastic process are studied:

(5)

Subject to: Np =

p

∑ N p, r

r =1

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(6)

13th IFAC INCOM (INCOM'09) Moscow, Russia, June 3-5, 2009

F p, r ( x) = Pr( N p, r ≤ x)

(7)

5. CONCLUDING REMARKS

0 ≤ x ≤ u −1 ,

(8)

1 ≤ p ≤ u −1

(9)

The goal of this paper was to study a new generalized newsboy model. The obtain model is useful for the optimization of the MRP parameterization with random procurement times and a POQ policy as lot-sizing method. The proposed model and algorithms minimize the sum of the average holding cost, backlogging cost and setup cost. This model and algorithms can be used in many industrial situations. For example, often security coefficients are introduced to calculate the planned lead time for unreliable suppliers in an MRP environment. In this case, planned lead time is equal to contractual (or forecasted) lead time multiplied by the security coefficient. This coefficient is empiric but anticipate the delay by creating safety lead time. The more unreliable a supplier is, the larger its coefficient. The model and algorithms suggested in this paper can be used to better estimate these coefficients basing on statistics on the procurement lead times for each supplier and taking into account the holding, backlogging and setup costs. This is a multi-period model with no major restriction on the type of the lead time distribution. All discrete distributions can be used. The decision variables are integer; they represent the periodicity and planned lead time for raw material. Concerning the assumption of constant demand, note that this model should be used with different possible values of the demand to examine the sensitivity of the obtained parameters to said values. If the parameter values are significantly different for the given demand levels, the approach by scenarios can be applied to choose the parameter values. In addition, the demand variations can be decoupled from planned lead time calculation by using safety stocks. This is another promising perspective for future research.

The optimal planned lead time corresponding to each periodicity p can be calculated using the following theorem. Theorem 2: For each given periodicity p, the optimal planned lead time x satisfies the following inequalities: 1 p

p

∑ F p, r (

r =1

b x + p − r −1 )≤ ≤ 1 p b+h p

p

∑ F p, r (

r =1

x + p −r ) p

(10)

Proof.

We introduce the following function G ( x, p) : G ( x, p ) = CM ( x + 1, p ) - CM ( x, p ) p

( ∑ (1 − 1p ∑ F p, r ( x + k +p p − r )) -

= h − (b + h)

r =1

k ≥0

p

∑ (1 − 1p ∑ F p, r ( r =1

k ≥0

= h − (b + h)(1 − 1

p

= h+b p

p

∑ F p, r (

r =1

)

x +1+ k + p − r )) p p

∑ F p, r (

r =1

x+ p−r )) p

x+ p−r )−b . p

(11)

G(x,p) is then an increasing function on x, because F p, r ( x) are increasing functions. Note that for each periodicity p, the optimal lead time must satisfy the following inequalities. C ( x, p ) ≤ C ( x − 1, p) ,

(12)

C ( x, p ) ≤ C ( x + 1, p) .

(13)

Acknowledgements. This work has been partially supported by Princess Fatimah Alnajras’s Research Chair of Advanced Manufacturing Technology.

The previous inequalities can be expressed using G(x,p) as follows: G ( x − 1, p) ≤ 0 and G ( x, p ) ≥ 0 . The optimal lead time x is then obtained when the increasing function G(x,p) changes its sign for negative to positive value. The later inequalities can finally be rewritten as follows: 1 p

p

∑ F p, r (

r =1

REFERENCES Arda Y, Hennet J.C., 2006. Inventory control in a multisupplier system. International Journal of Production Economics, 104 (2), pp. 249-259. Axsäter S., 2005. Planning order release for an assembly system with random operations times. OR Spectrum, 27, 459-470. Axsäter S., 2006. Inventory control, second edition, Springer. Baker K.R., 1993. Requirement planning, In: Handbooks in Operations Research and Management Science, Logistics of Production and Inventory (S.C. Graves, et al., Ed.), vol. 4, North-Holland, Amsterdam, 571-628. Dolgui A., Portmann M.C. and Proth J.M., 1995. Planification de systèmes d’assemblage avec approvisionnement aléatoire en composants. Journal of Decision Systems, 4(4), 255-279.

p b x + p − r −1 x+ p−r )≤ ≤ 1 ∑ F p, r ( ) . (14) p p p b+h r =1

▄ Not that this is a generalization of the newsboy model which is obtained when the periodicity equals one (p=1), i.e. when lot-for-lot rule is used instead of Periodic-Order-quantity method. Since, in the classical Newsboy model the demand is random and the lead time is zero, it is important to note that the Newsboy model proposed here is proposed for the symmetric case, when the demand is constant and the lead time is random.

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review, International Journal of Production Economics, 103, 271–285. Petruzzi N.C. and Dada M., 1999. Pricing and the newsvendor problem: a review with extensions. Operations Research, 47, 183-194. Porteus E.L., 1990. Stochastic inventory theory. In: Handbook in OR and MS, Vol 2, Heyman D.P. and Sobel M.J. (eds.), Elsevier Science Publishers, 605-652. Proth J.M., Mauroy G., Wardi Y., Chu C. and Xie X., 1997. Supply management for cost minimization in assembly systems with random component yield times. Journal of Intelligent Manufacturing, 8 (5), 385- 403. Rao S. and Schneller G.O., 1990. On the stochastic nonsequential production-planning problem. Journal of the Operational Research Society, 41, 241-247. Sipper D., Bulfin R.L. Jr, 1998. Production: planning, control and integration, McGraw Hill. Sridharan V. and Berry W.L., 1990. Master production schedule, make-to-stock products: a framework for analysis. International Journal of Production Research, 28, 541-558. Sox C.R. and Muckstadt J.A., 1996. Multi-item, multiperiod production planning with uncertain demand. IIE Transactions, 28, 891-900. Tang O. and R.W. Grubbstrom, (2003). The detailed coordination problem in a two-level assembly system with stochastic lead times. International. Journal of Production Economics 81–82 (2003) 415–429. Tempelmeier H., 2006. Inventory management in supply networks: problems, models, solutions, Books on demand GmbH. Ton G. de Kok, Jeremy W.C.H. Visschers, 1999. Analysis of assembly systems with service level constraints. International Journal of Production Economics 59. 313326. Whybark D.C. and Wiliams J.G., 1976. Material Requirement under uncertainty. Decision Science, 7, 595-606. Yano C.A., 1987a. Setting planned leadtimes in serial production systems with tardiness costs. Management Science, 33(1), 95-106. Yano C.A., 1987b. Stochastic leadtimes in two-level assembly systems. IIE Transactions, 19(4), 95-106. Yeung J.H.Y., Wong W.C.K. and Ma L., 1998. Parameters affecting the effectiveness of MRP systems: a review. International Journal of Production Research, 36, 313331. Yeung J.H.Y., W.C.K. Wong, Ma L. and Law J.S., 1999. MPS with multiple freeze fences in multi-product multilevel MRP systems. International Journal of Production Research, 37, 2977-2996. Zipkin P., 2000. Foundation of Inventory Management, McGraw-Hill.

Dolgui A., and Louly M.A., 2002. A Model for Supply Planning under Lead Time Uncertainty. International Journal of Production Economics, 78, 145-152. Dolgui A., and Prodhon C., 2007. Supply planning under uncertainties in MRP environments: A state of the art, Annual Reviews in Control, 31(2), 269-279. Dolgui A., Hnaien F., Louly A., Marian H., 2008. Parameterization of MRP for Supply Planning Under Uncertainties of Lead Times. Chapter 14, in: Supply Chain: Theory and Applications, V. Kordic (Ed.), I-Tech Education and Publishing, Vienna, Austria, (ISBN 9783-902613-22-6), p. 247-262. Elhafsi M., 2002. Optimal leadtimes planning in serial production systems with earliness and tardiness costs. IIE Transactions, 34, 233-243. Fujiwara O., and Sedarage D., 1997. An optimal (Q,r) policy for a multipart assembly system under stochastic part procurement lead times. European Journal of Operational Research, 100, 550-556. Kaplan R.S., 1970. A Dynamic Inventory Model with Stochastic Lead Times. Management Science, 16, 491 – 507. Khang D.B. and Fujiwara O., 1993. Multi-period stochastic network flow problems with service level requirements: approximation and error estimates, IIE Transactions, 25(2), 104 – 110. Koh S.C.L., and Saad S.M., 2003. MRP-controlled manufacturing environment disturbed by uncertainty, Robotics and Computer Integrated Manufacturing, 19, 157–171. Kumar A., 1989. Component Inventory Cost in an Assembly Problem with Uncertain Supplier Lead-Times. IIE Transactions, 21 (2), 112 –121. Lee, H.L. and Nahmias S., 1993. Single-product, SingleLocation Models. In: Handbooks in Operations Research and Management Science, Logistics of Production and Inventory (S.C. Graves, et al., Ed.), vol. 4, NorthHolland, Amsterdam, 3-57. Liberatore M.J., 1979. The EOQ Model under Stochastic Lead Times. Operations Research, 27, 391 – 396. Louly M.A. and Dolgui A., 2002. Newsboy model for supply planning of assembly systems. International Journal of Production Research, 40, 2002, 4401-4414. Louly M.A., and Dolgui A., 2004. The MPS parameterization under lead time uncertainty. International Journal of Production Economics, 90, 369376. Louly M.A., and Dolgui A., 2008. Optimal time phasing and periodicity for MRP with POQ policy. International Journal of Production Economics, (under reviewing). Molinder A. 1997. Joint optimisation of lot-sizes, safety stocks and safety lead times in an MRP system. International Journal of Production Research, 35, 983994. Mula J., Poler R., Garcıa-Sabater J.P., and Lario F.C., 2006. Models for production planning under uncertainty: A

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