Planning and replanning the master production schedule under demand uncertainty

Transcription

1 Int. J. Production Economics 78 (2002) 323}334 Planning and replanning the master production schedule under demand uncertainty Ou Tang*, Robert W. GrubbstroK m Department of Production Economics, Linko( ping Institute of Technology, S Linko( ping, Sweden Received 12 April 2000; accepted 21 August 2000 Abstract The Master Production Schedule (MPS) is essential in maintaining customer service levels and stabilising production planning in a Material Requirements Planning (MRP) environment. Traditionally, an MPS is derived from a demand forecast and aggregate production plan but the associated cost to cover demand uncertainty is usually not considered. Another di$culty in the MPS is its planning frequency. Frequent changes in the MPS reduce productivity whereas a long-term frozen MPS results in a poor service level and an unfavourable inventory situation. This paper "rst investigates the possibility to establish a method for planning the MPS under stochastic demand. Secondly, it aims to evaluate the value of replanning actions. Finally, it provides a model for estimating appropriate MPS parameters such as the length of the replanning interval and the length of the interval to freeze the plan Elsevier Science B.V. All rights reserved. Keywords: Master production schedule; Stochastic demand; Planning and replanning; Production-inventory system 1. Introduction * Corresponding author. Paper presented at the Eleventh International Working Seminar on Production Economics, Igls/ Innsbruck, Austria, 21}25 February, Tel.: # ; fax: # address: rwg@ipe.liu.se, ou.tang@ipe.liu.se (O. Tang). The Master Production Schedule (MPS) is essential in maintaining customer service levels and stabilising production planning within a Material Requirements Planning (MRP) environment. Usually, an MPS will face the pressure to replan because of the changes of operational circumstances. There are basically two conditions which lead to replanning. First, there is a rolling e!ect due to the extension of the planning period. Secondly, when demand is uncertain, there is always a forecast error, and, therefore the old plan has to be modi"ed to adapt to new information to keep the production cost low and maintain the service level. When the MPS is further used for material requirements planning (MRP), frequent changes in the MPS lead to schedule adjustments in the system. Such adjustments of plans also have an augmented e!ect in an assembly system, and this is often referred to as system nervousness. Nervousness may become an obstacle in the implementation of MRP and even cause a collapse of the whole system. Therefore, to reduce the instability of the schedule becomes an important objective in planning the MPS. In one of our previous studies [1], production rescheduling has been addressed under the assumption that the cost of rescheduling is determined externally. A decision tree method was applied /02/$ - see front matter 2002 Elsevier Science B.V. All rights reserved. PII:S ( 0 0 )

2 324 O. Tang, R.W. Grubbstro( m / Int. J. Production Economics 78(2002) 323}334 to study the bene"t from rescheduling. The Net Present Value approach was used and the planning time is on a continuous scale. In this paper, we "rst investigate the possibility to establish a method for planning the MPS under stochastic demand. Secondly, we provide a model for replanning MPS assuming the lower-level schedule change cost is known. Then, using a similar method as in GrubbstroK m and Tang [1] and taking into account the schedule change cost, we provide a model for estimating appropriate MPS parameters such as the length of intervals to replan and freeze. Finally, we provide numerical examples and draw some conclusions in a summary. 2. Previous research in MPS Basically, two approaches have been suggested to improve the quality of an MPS, namely freezing the MPS and adding some safety stock into the forecast. When its operation and planning conditions such as forecasting quality, lot sizing policy, operation cost etc. are given, a high-quality MPS is to select a proper planning horizon and a replanning frequency as well as a safety stock. Lin and Krajewski [2] provided a framework for studying the MPS. First they presented the concepts of a forecast window (planning horizon), a frozen interval and a replanning interval and discussed their relationships. Generally, the cost of the MPS has been divided into two parts, a forecast error cost and an MPS change cost. The latter has the same properties as in the similar approach by Carlson et al. [3], in which schedule change cost is a piecewise continuous function of time. The total cost of the MPS model depends on the forecast window and the frozen interval and replanning interval. But it should be reminded that the MPS change cost is more complicated and di$cult to estimated in reality when di!erent lot-sizing policies and a multi-level system are considered. However, in order to build an analytical model, we adopted this simpli"cation in our study. An alternative and more frequently used measure for MPS quality is in terms of its stability. Sridharan et al. [4] suggested a measure of instability, which is a function of the method used to freeze the MPS, the proportion of the MPS frozen and the length of the planning horizon. This instability definition has been followed in other studies [5,2,6,7]. Still, stability is usually used together with other system performance measures, like the service level. In a similar way, Inderfurth [8] provided a de"nition of stability for this type of system. In most of the MPS studies, simulation is used due to the complexity of the problem. Analysis of Variance (ANOVA) is usually applied to determine the system performance when di!erent operation policies are tested. Capacity constraints are often excluded from MPS studies, the only exception being Kern and Wei [7]. In the following subsection, we provide a literature overview of the areas of freezing the MPS and safety stocks in MPS Freezing the MPS A rolling planning horizon is to replan MPS each period whenever information is updated. This tracks the development of demand closely, but generates instability. One solution to avoid this is to freeze a portion of the planning horizon to stabilise the schedule. In a single-level deterministic demand case, freezing this portion beyond 50% of the planning horizon could reduce the instability but increases cost tremendously [9,4]. For the demand uncertainty case, the forecast quality has a large in#uence on the optimal frozen interval and planning horizon [10]. Sridharan and Berry [11] point out that a long MPS frozen interval may increase the need for safety stocks and enlarges the lot-sizing costs at the MPS level. In a multi-level system, freezing the MPS is the most e!ective way to reduce instability and reduce total costs [6]. This result is di!erent from single-level and deterministic studies. In Kadipasaoglu's study in 1995 [12], it is indicated that freezing the MPS provides a bene"t in the stochastic case only. The length of the frozen interval is recommended to cover the cumulative lead time. The freezing portion depends on the planning horizon. A longer planning horizon actually may worsen the MRP performance in a demand uncertainty situation, whereas improve its performance in a deterministic case [10].

3 O. Tang, R.W. Grubbstro( m / Int. J. Production Economics 78(2002) 323} Chung and Krajewski [13] study the interface between aggregate planning and master production planning in a hierarchical planning environment. They concluded that the cost structure of aggregate planning has a high in#uence on the replanning frequency of the MPS Safety stocks The safety stock is used to provide a bu!er for uncertainty in the MPS. However, safety stocks should be applied with great care. Maintaining a safety stock level should not be the objective of scheduling the MPS. An MPS that allows safety stocks to be added to inventory merely in the initial scheduling can create derive more bene"ts from its operation than an MPS that carefully tracks safety stocks and reschedules accordingly. There are also studies considering safety stocks on both MPS and component levels in order to provide a #exible schedule in a multi-level system [14]. Metters [15] studies di!erent strategies for using the safety stock. An explicit approach distinguishes the safety stock from normal demand and lets this amount of stock battle the uncertainty. A new order (replanning) is trigged only when there is a real stockout. On the other hand, an implicit approach triggers a new order whenever an inventory falls down to the safety stock level. Metters also recommends a third approach, namely the stable safety policy. In this case, an explicit approach is used when there is no order in the old schedule and implicit approach when there is an order in the old schedule. Similarly, Sridharan and LaForge [5] provide an intelligent safety policy concept which does not permit a scheduling of the MPS solely for the purpose of meeting a safety stock. Even though a safety stock increases the service level, it does not necessarily provide a stable MPS. In fact, a high safety to meet demand increases instability. Studies by Sridharan and LaForge [5] and Metters [15] support such a conclusion. Hsu and El-Najdawi [16] carried out a study concerning how to determine the safety stock level. They incorporated safety stocks into the lot-sizing policies. Four techniques to determine the optimal safety stock level are examined and compared. Nevertheless, in all the studies we have mentioned above, the level of safety stocks is considered as a constant. In a situation where a portion of the planning horizon is frozen, we suggest that the level of safety stock should vary for the reason that the forecast quality is impaired along with time. Our previous studies show that the level of safety stock follows a square root of time relation when demand is a renewal process [17]. 3. The planning decision The objective of planning is to determine the production volume for a given demand forecast and an initial net inventory R. The initial net inventory is de"ned as the di!erence between cumulative production and cumulative demand so that it is either positive or negative. Instead of using techniques like exponential smoothing, we here use statistical methods for forecasting. Since our objective in this study is to investigate the planning and replanning problem under uncertainty, we use the variance of the demand forecast to express the forecasting error (forecast quality). Another assumption we have adopted is that the setup cost is relatively low. This results in a setup in each time period so that we do not need to consider the impact from the lot-sizing policy. Because of this, the system instability due to the rolling horizon e!ect is reduced. At any time, the following relation holds for a production-inventory system with backlogging: B "DM!PM #S!R. (1) However here the time period is discrete, thus i"1, 2, 2. In each interval, we must also have the following relation: E(B )"E(DM )!PM #E(S )!R. (2) Because of the uncertainty in demand, there is either an excess inventory when the cumulative production level is high, or stockouts when there is a shortage of supply. It is apparent from Eq. (1) that the system service level depends not on the demand forecast error in each period, but on the forecast error of cumulative demand. When replanning is not possible within a frozen interval, the error will

4 326 O. Tang, R.W. Grubbstro( m / Int. J. Production Economics 78(2002) 323}334 be accumulated. Our previous studies [17] have shown that the optimal safety stock may vary and be a function of time in the continuous-time case. We assume that external cumulative demand DM at the end of the ith period follows a probability density function, which depends on time explicitly: for x(0, Pr(x)DM (x#dxi)" 0 (3) f (x)dx for x*0. Its cumulative probability function, denoted F (x), is assumed to be less than unity for "nite values of x, and is shown to have the following property. Lemma 1. F (x)(f (x). Proof. Writing g as the density function for demand during the (i#1)th period, i.e. for D,we have F (x)"pr(dm #D )x) " F (y) ) g (x!y)dy )F (x) ) g (x!y)dy)f (x). Necessary for the equality to hold, we need g (x!y)dy"1, which is impossible for a "nite x. Hence the inequality is strict.. Since setup costs are disregarded, the objective of the planning is here interpreted as minimising the uncertainty cost C (a term adopted from [2]) E(C )" (h E(S )#b E(B )) " ((h #b )E(S )#b (E(DM )!PM!R )), (4) with E(S )" M (PM #R!x) f (x)dx, (5) subject to the constraints PM *PM *0 for i"1, 2, 2, n!1, (6) where h and b are the unit holding cost and stockout cost for period i, respectively, and n is the planning horizon. The Lagrangean function of this minimisation problem with the constraints Eq. (6) is " ((h #b )E(S )#b (E(DM )!PM!R )) # ξ (PM!PM ), (7) where the ξ are nonnegative Lagrangean multipliers. The necessary Kuhn}Tucker conditions for the optimal solution are "(h #b )F (PM #R )!b #ξ!ξ *0 PM for i"1, 2, 2, n, (8) "PM!PM )0 for i"1, 2, ξ 2, n!1, (9) together with the complementarily conditions PM ) "0 for i"1, 2, PM 2, n, (10) ξ ) "0 for i"1, 2, ξ 2, n!1, (11) where for convenience we de"ne ξ "ξ "0. Hence F (PM )* b #ξ!ξ for i"1, 2, h #b 2, n, (12) with equality whenever PM '0. We "rst note that due to the non-decreasing property of the PM, the optimal sequence will start with a (possibly empty) subset PM "PM "2"0 followed by a second (possibly empty) subset PM, PM,2, PM '0. A strict inequality in Eq. (12) can therefore only be valid for the "rst of these subsets (if it is nonempty). In Theorem 1 we provide a su$cient condition for determining the structure of the sequence PM, PM, 2, PM. Theorem 1. If the ratio between unit stockout cost and unit holding cost is non-decreasing in time, i.e.

5 O. Tang, R.W. Grubbstro( m / Int. J. Production Economics 78(2002) 323} (b /h )*(b /h ), i"1, 2, 2, n!1, then we have the following conditions regarding the production sequence. If F (R )'b /(h #b ), then PM "PM "2"PM " 0; if F (R )(b /(h #b ), then PM, PM, 2, PM '0. Hence, this theorem o!ers a means for deciding where the sequence of positive cumulative production levels should start. Proof of Theorem 1. Assume (i) that F (R )' b /(h #b ), then b /(h #b ))b /(h #b )( F (R )(F (R ) for 1)i(j. Based on Eq. (12), we always need F (R #PM )*(b!ξ )/(h #b ). If PM '0, then F (R #PM )"(b!ξ )/(h #b )'F (R )'b /(h #b ) which creates a contradiction since ξ *0. Hence we obtain PM "0. Based again on Eq. (12), we always need F (R #PM )*(b!ξ #ξ )/(h #b ). If PM '0, then F (R #PM )"(b!ξ #ξ )/(h # b )'F (R )'b /(h #b ). This requires!ξ #ξ '0, therefore ξ '0, which creates a contradiction. Hence we obtain PM "0. Following the same procedure for i"3, 4,2, we end up with PM "0. Assume (ii) that F (R )(b /(h #b ). Then F (R )(F (R )(2(F (R )(b /(h #b ) (b /(h #b ) for all j)i(n. Since F (R # PM )*(b #ξ )/(h #b )*b /(h #b )'F (R ), we must have PM '0. If we assume PM "0 and take into account F (R #PM )*(b!ξ #ξ )/ (h #b ), we obtain b /(h #b )'F (R )* (b!ξ #ξ )/(h #b ), which leads to!ξ # ξ (0. Therefore ξ '0 and we have PM "PM "0. With PM "0 and b /(h #b )' F (R )*(b!ξ #ξ )/(h #b ), we obtain!ξ #ξ (0 and therefore ξ '0. Hence we have PM "PM "0. By repeating the above procedure, we obtain PM "0, which contradicts PM '0. Therefore we "nally obtain PM, PM, 2, PM '0. Theorem 2. We assume as in Theorem 1 that the ratio between unit stockout cost and unit holding cost is non-decreasing in time, i.e. b /h )b /h for i"1, 2, 2, n!1. If PM, is the earliest positive cumulative production, then the multipliers from j!1 onwards are all zero, i.e. ξ "ξ "ξ " 2"ξ "0. Proof. PM being the earliest positive production requires ξ "0. We rearrange Eq. (12) as an equality for i"j and i"j#1: F (PM #R )# ξ!ξ " b, h #b h #b F (PM #R )# ξ!ξ " b. h #b h #b Since the right-hand member is nondecreasing, so must the left-hand member also be. Assume that ξ '0, then PM "PM ; and therefore F (PM #R )(F (PM #R ), which implies (ξ!ξ )/(h #b )'(ξ!ξ )/(h #b ). Therefore, also ξ '0, which creates PM " PM. Continuing this process, we end up with!ξ /(h #b )'0, which contradicts the nonnegativity of the multipliers. Therefore we conclude that ξ "0. Analogously we get ξ "0, etc. Hence, no multiplier ξ, for i*j!1, may be positive. This concludes our proof.. By using Theorems 1 and 2, we are now in a position to construct a solution procedure to the stated problem of minimising the uncertainty cost. This procedure is stated in the following theorem. Theorem 3. As before, we assume a non-decreasing sequence of unit cost ratios b /h )b /h, i" 1, 2, 2, n!1. The solution PMH to the minimisation problem Eq. (4) subject to the constraints Eq. (6) then satisxes PMH "0 if F (R )' b, h #b F (PMH #R )" b if F (R )) b. h #b h #b If all F (x) are monotonically increasing functions, the solution must be unique. Proof of Theorem 3 follows immediately from our previous results. The theorem shows that this

6 328 O. Tang, R.W. Grubbstro( m / Int. J. Production Economics 78(2002) 323}334 optimal planning model is essentially an extension of the Newsboy problem. 4. The replanning decision The objective of replanning is to determine a rescheduled production volume in each period P, i"¹#1, ¹#2,2, for a given demand forecast, and initial net inventory R at the replanning point based on the old production plan P and the schedule change cost δ, so that the total cost can be minimised. The total cost C consists of the uncertainty cost, as discussed in the previous section, and the rescheduling cost C : C "C #C. (13) The rescheduling cost is the sum of the schedule change costs in di!erent time periods. In a practical setting, we would assume that the schedule change cost depends on the lot-sizing policy, the product structure, the rescheduling time desired, and the rescheduling quantity desired. In this study, our focus is on the MPS level. Thus we use a single schedule change cost function to take care of the aggregate information of negative e!ects of rescheduling on the lower levels. Here we assume that the schedule change cost is a function of the rescheduling time and that it is proportional to the quantity being changed. This is in accordance with the unit change cost concept in Lin and Krajewski [2]. As a result from Tang and GrubbstroK m [18] where control theory was applied, this type of cost should be a function of time as well. With this in mind, we de"ne the schedule change cost as δ " (P!P )φ if P *P, (14)!P )γ if P (P, where φ and γ both are nonincreasing functions of the time period i. This means that in order to reschedule, whether adding orders or cancelling the orders, it is always at least as expensive to change the earlier schedule as the latter one. In practical terms, these costs may relate to the time fences, which are the guidelines to note various restrictions in schedule changes related to lowerlevel items. Fig. 1. Cost structure for replanning. At a certain time, the schedule change cost is therefore a monotonically decreasing function when P (P and a monotonically increasing function when P 'P. This is illustrated in Fig. 1. Quite often in practice, to cancel an order is easier than adding an new order, so γ could then be assumed to be smaller than φ. Lemma 2. Let y (z) and y (z) be two functions of z, each having a unique minimum at z and z, respectively, and no other local minima. Then any linear combination of y (z) and y (z) with positive coezcients will have a unique minimum in the interval between z and z. The proof of this proposition is trivial. Regarding the optimal replanning decision, we state the following theorem. Theorem 4. If all F (x) are monotonically increasing, the total cost function of replanning (Eq. (13)) is convex. Then the optimal replanning decision P falls into the interval between the old production plan P and P*, the latter being the optimal solution when minimising the uncertainty cost based only on the initial net inventory R and demand forecast, see Fig. 1. Proof. The proof follows from Lemma 2 and our assumption regarding the monotonic behaviour of the schedule change cost with respect to P!P.

7 O. Tang, R.W. Grubbstro( m / Int. J. Production Economics 78(2002) 323} Combining Eqs. (13) and (14), we may write the total cost as C "E(C )#C " ((h#b )E(S )#b (E(DM )!PM!R )#δ ). (15) Its "rst-order derivative is thus C " ((h #b )F (PM #R )!b ) P if PH'P, # φ!γ if PH(P. (16) Regarding the solution to the optimal replanning decisions, we have the following theorem. Theorem 5. (i) If PH 'P and (C /P ) '0, then we have the optimal solution P "P. (ii) If PH 'P and (C /P ) (0, then the optimal P is such that (C /P ) "0. (iii) If PH 'P and (C /P ) H '0, then the optimal solution becomes P "PH. (iv) If PH (P and (C /P ) H (0 and (C / P ) (0, then the optimum will be P "P. (v) If PH (P and (C /P ) H (0, and (C /P ) '0, then the optimal value of P is such that (C /P ) "0. Proof. The proof is obtained from Theorem 4 and the de"nition of the schedule change cost in combination with Kuhn}Tucker conditions. Based on Theorems 3 and 5, we propose the following solution procedure to obtain the optimal replanning decision. (i) Start from i"¹#1. (ii) Calculate the unique solution P* according to Theorem 3. (iii) Compare P* and P and calculate P according to Theorem 5. (iv) Increase i one step and repeat the steps (ii)}(iii), if i is smaller than the planning horizon ¹#n. 5. The planning and replanning model Essentially establishing the MPS is a dynamic planning and replanning process. Fig. 2 illustrates the decision scheme following a similar interpretation as in Lin and Krajewski [2]. The replanning interval is the time period until when no replanning is under consideration. However, due to certain production constraints in practice, such as the rescheduling time fence, a portion of the old plan needs to be "xed. Sometimes, because of unfavourable economic consequences or system performance, part of the old plan needs to be kept. The total length of the "xed part of the old plan is called the frozen interval. During the time intervals beyond the frozen interval, the production plan is allowed to be changed to improve the performance of the system by using the knowledge of the state that was realised at the replanning time. In GrubbstroK m and Tang [1], a decision tree method was used to address the rescheduling problem. Here we follow the same methodology for the planning and replanning problem. However, an explicit schedule change cost has been added and the decision tree has been changed correspondingly. As in Fig. 3, the replanning may take place at time ¹. The sequences of production decisions prior to and at this point are denoted p and those beyond this point are denoted p. Let d and d represent stochastic sequences of a demand process, before and after the possible replanning time ¹ (Fig. 4). With this notation, the following planning and replanning schemes can be interpreted in the form of a decision tree. At time 0, we make a decision Fig. 2. Planning and replanning scheme (adapted from Lin and Krajewski, [2]).

8 330 O. Tang, R.W. Grubbstro( m / Int. J. Production Economics 78(2002) 323} Numerical examples Fig. 3. Decision tree diagram for the planning and replanning processes. concerning the production plan p followed by p.as shown in Section 3, this requires to determine P*, for i"1, 2,2,¹, and P* for i"¹#1, ¹#2,2,¹#n. At the replanning point, a state of the net inventory R will be realised based on the outcome of the previous sequence of demand DM. We now use the replanning model given in Section 4 to decide P for i"¹#1, ¹#2,2, ¹#n. The rescheduling cost in the current model, depends on the state at which replanning occurs (essentially on R ) and this cost will be di!erent for di!erent states in general. This is one of the major di!erences from our previous study in GrubbstroK m and Tang [1]. The optimal frozen interval ¹ is therefore P "P, for all ¹(i)¹#¹. (17) The gain from replanning is the di!erence between total costs for replanning and total costs for the stay alternative. As illustrated in GrubbstroK m and Tang [1], this gain has a non-negative value C "C (R, P )!C (P R, P )*0. (18) In this equation, the state R is a result of cumulative demand DM according to Eq. (1). Since cumu- lative demand at the replanning point ¹ has a cumulative density function F (x), the expected gain from replanning is therefore E(C (¹))" C (x)df (x). (19) Similarly, we obtain the expected value of the frozen interval ¹ from E(¹ )" ¹ (x)df (x). (20) Below, we provide numerical examples to illustrate the models of previous sections. We assume that external demand follows a Stuttering Poisson process [19], in which the interval between demand events obeys an exponential distribution and the event size is geometrically distributed. The probability density function of cumulative demand at the end of the ith time period is therefore for M"0, f (M)"e e (λi) j! M!1 j!1 β(1!β) for M"1, 2,2,. (21) where λ and β are constants. The random variable M, cumulative demand, has the mean μ"λi/β and variance σ"λi(2!β)/β. This shows that for such a type of compound Poisson process, both the mean and variance of cumulative demand increase proportionally with the time period. Given the production parameters b, h, λ, β, we calculate the optimal MPS according to the model of Section 3. The safety stock SS equals the di!erence between cumulative production and expected cumulative demand immediately prior to the time of production. A "rst example is given in Table 1, which shows that the optimal safety stock increases monotonically with time, even if the demand forecast is constant in each time period. The resulting production volume is then used for the purpose of the MPS in this system. According to this production plan, the production volumes tend to be close to constant except for the "rst few periods, and they are all higher than expected demand. The increasing contribution to E(C ) shows that the system needs to pay more to cover increasing uncertainty in later time periods. Tables 2 and 3 o!er examples of the replanning model for a positive and negative initial net inventory, respectively. The frozen intervals are 1 and 3 time periods, respectively. The schedule change cost structure is of course the major component in#uencing the frozen interval. In our examples we assume a decay function Φe for adding an order

9 O. Tang, R.W. Grubbstro( m / Int. J. Production Economics 78(2002) 323} Table 1 The planning decisions with b"30, h"1, λ"5, β"0.5 i E(D) (Forecasting) PM P (MPS) SS E(S ) E(B ) E(C ) and Θe for cancelling an order. But we would be able to use any other form of schedule change cost in our replanning model. It is shown in Table 3 that the "rst two periods are frozen due to the high change cost even though stockouts exist there. In the event of a very tight production environment, within which where rescheduling is impossible, we may assign the schedule change cost an in"nitely large value so that the optimal solution always becomes identical to the old plan according to the decision rules of Theorem 5. We use di!erent values of Φ and Θ in the schedule change cost function to represent the rescheduling #exibility of the system. These vary according as costs are high (10000), (medium) 1000, (low) 100 and 50, 5, 0.5. Demand uncertainty is indicated by assigning di!erent values to λ and β. But we retain the expected demand rate as a constant. As illustrated in Fig. 5, the expected gain from replanning increases when the forecast error is raised. An optimal replanning interval is shown to exist. However, this optimal value Table 2 Replanning decision with b"30, h"1, λ"5, β"0.5, R"40 i!t E(D) φ γ P* P P (MPS) (Forecasting)

10 332 O. Tang, R.W. Grubbstro( m / Int. J. Production Economics 78(2002) 323}334 Table 3 Replanning decision with b"30, h"1, λ"5, β"0.5, R"!40 i!t E(D) φ γ P* P P (MPS) (Forecasting) Fig. 4. Expected gain from replanning vs. replanning interval on di!erent uncertainty levels. Mean demand rate"10, b"30, h"1, n"30. does not change very much with the forecast error. It remains between 11 and 12 time periods in these particular examples given by Fig. 5. Fig. 6 shows that the forecast error does not have an obvious impact on the length of the frozen interval. Intuitively, a lower schedule change cost brings more of a gain from replanning (Fig. 7). Again, from Fig. 7, we observe the existence of an optimal replanning interval and that the sequence of optimal replanning intervals is insensitive to the schedule change cost. However, the schedule change cost has a signi"cant impact on the length of the frozen interval. We have also studied examples with di!erent planning horizons. The planning horizon has a large impact on the optimal replanning interval (Figs. 8}10). With a longer planning horizon, the replanning interval is prolonged as well. Nevertheless, Figs. 8 and 10 also show that neither forecasting, nor the schedule change cost have any important in#uence on the length of the replanning interval. Fig. 5. Expected frozen interval vs. replanning interval on di!erent uncertainty levels. Mean demand rate"10, b"30, h"1, n"30. Fig. 6. Expected gain from replanning vs. replanning interval on di!erent schedule change cost levels. b"30, h"1, λ"5, β"0.5, n"30. When we study the optimal replanning interval, Fig. 9 again shows that the forecast error has little e!ect on the frozen interval. It seems as if the only

11 O. Tang, R.W. Grubbstro( m / Int. J. Production Economics 78(2002) 323} Fig. 7. Expected frozen interval vs. replanning interval on di!erent schedule change cost levels. b"30, h"1, λ"5, β"0.5, n"30. Fig. 9. Expected frozen interval vs. planning horizon on di!erent uncertainty levels when the replanning interval is optimal. Mean demand rate"10, b"30, h"1. Fig. 8. Optimal replanning interval vs. planning horizon on di!erent uncertainty levels. Mean demand rate"10, b"30, h"1. Fig. 10. Optimal replanning interval vs. planning horizon on di!erent schedule change cost levels. b"30, h"1, λ"5, β"0.5. major factor changing the frozen interval is the schedule change cost (Fig. 11). 7. Summary In this paper, we have studied planning and replanning problems at the MPS level. First, we have applied our previous research "ndings about safety stocks in an MPS system. We provide a model for MPS planning in which the safety stock is determined and is shown to be an increasing function of time. Secondly, we developed a model for replanning in which the scheduling change cost is considered and for which an e$cient solution procedure was presented. This provides Fig. 11. Expected frozen interval vs planning horizon on di!erent schedule change cost levels when the replanning interval is optimal. b"30, h"1, λ"5, β"0.5.

12 334 O. Tang, R.W. Grubbstro( m / Int. J. Production Economics 78(2002) 323}334 the possibility for further studying replanning when several alternative replanning points are considered. The numerical examples demonstrate procedures to solve the optimal replanning interval and to estimate the expected frozen interval. An interesting "nding is that the forecast error appears to be of little importance to either of them when the safety stock bu!er is optimised according to our planning model. But how to determine the optimal planning horizon still remains an open question. In this paper, we have assumed that there is only one replanning point inside of the "rst planning horizon. In future studies, it is of interest to remove this assumption. Simulation studies may be needed for dynamic replanning problems, in which replanning decisions may be interpreted following the model in Section 4. Only after that, the interrelationship between the replanning interval, the frozen interval and the planning horizon may become more transparent. Another remaining point for discussion is the schedule change costs. If in the real world multilevel production system we could develop principles for aggregating the information necessary for estimating the rescheduling cost, our replanning model in Section 4 would become more important for practical uses. References [1] R.W. GrubbstroK m, O. Tang, Modelling rescheduling activities in a multi-period production}inventory system, International Journal of Production Economics 68 (2) 123}135. [2] N.-P. Lin, L.J. Krajewski, A model for master production scheduling in uncertain environments, Decision Sciences 23 (1992) 839}861. [3] R.C. Carlson, J.V. Jucker, D.H. Kropp, Less nervous MRP systems:a dynamic economic lot-sizing approach, Management Science 25 (8) (1979) 754}761. [4] V. Sridharan, W.L. Berry, V. Udayabhanu, Measure master production schedule stability under rolling planning horizons, Decision Sciences 19 (1988) 147}166. [5] V. Sridharan, R.L. LaForge, The impact of safety stock on schedule instability, cost and service, Journal of Operations Management 8 (4) (1989) 327}347. [6] S.N. Kadipasaoglu, V. Sridharan, Alternative approach for reducing schedule instability in multistage manufacturing under demand uncertainty, Journal of Operations Management 13 (1995) 193}211. [7] G.M. Kern, J.C. Wei, Master production rescheduling policy in capacity constrained just-in-time make-to-stock environments, Decision Sciences 27 (2) (1996) 365}387. [8] K. Inderfurth, Nervousness in inventory control:analytical results, OR Spektrum 16 (1994) 113}123. [9] V. Sridharan, W.L. Berry, V. Udayabhanu, Freezing the master production schedule under rolling planning horizons, Management Science 33 (9) (1987) 1137}1148. [10] X. Zhao, T.S. Lee, Freezing the master production schedule for material requirements planning systems under demand uncertainty, Journal of Operations Management 11 (1993) 185}205. [11] V. Sridharan, W.L. Berry, Freezing the master production schedule under demand uncertainty, Decision Sciences 21 (1990) 97}120. [12] S.N. Kadipasaoglu, The e!ect of freezing the master production schedule on cost in multilevel MRP system, Production and Inventory Management 36 (1995) 30}36. [13] C.-H. Chung, L.J. Krajewski, Replanning frequencies for master production schedules, Decision Sciences 17 (1986) 263}273. [14] C.A. Yano, R. Carlson, Interaction between frequency of rescheduling and the role of safety stock in material requirements planning systems, International Journal of Production Research 25 (2) (1987) 221}232. [15] R.D. Metters, A method for achieving better customer services, lower costs, and less instability in master production schedules, Production and Inventory Management 34 (1993) 61}65. [16] J.I. Hsu, M.K. El-Najdawi, Integrating safety stock and lot-sizing policies for multi-stage inventory systems under uncertainty, Journal of Business Logistics 12 (2) (1991) 221}238. [17] R.W. GrubbstroK m, O. Tang, Further developments on safety stocks in an MRP system applying Laplace transforms and input}output analysis, International Journal of Production Economics 60}61 (1999) 381}387. [18] O. Tang, and R. W. GrubbstroK m, Rescheduling considerations for production planning using control theory, Paper for Presentation at the Fifteenth International Conference on Production Research, Limerick, Ireland, 9}13 August, [19] G. Hadley, T.M. Whitin, Analysis of Inventory System, Prentice-Hall, Englewood Cli!s, NJ, 1963.