Planning and replanning the master production schedule under demand uncertainty

*Corresponding author. Paper presented at the EleventhInternational Working Seminar on Production Economics, Igls/Innsbruck, Austria, 21}25 February, 2000. Tel.: #46-13-281773; fax: #46-13-288975.E-mail address: [email protected], [email protected] (O. Tang).

Int. J. Production Economics 78 (2002) 323}334

Planning and replanning the master production schedule underdemand uncertainty

Ou Tang*, Robert W. GrubbstroK m

Department of Production Economics, Linko( ping Institute of Technology, S-581 83 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 productionplanning in a Material Requirements Planning (MRP) environment. Traditionally, an MPS is derived from a demandforecast 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 whereasa long-term frozen MPS results in a poor service level and an unfavourable inventory situation. This paper "rstinvestigates the possibility to establish a method for planning the MPS under stochastic demand. Secondly, it aims toevaluate the value of replanning actions. Finally, it provides a model for estimating appropriate MPS parameters such asthe length of the replanning interval and the length of the interval to freeze the plan. � 2002 Elsevier Science B.V. Allrights reserved.

Keywords: Master production schedule; Stochastic demand; Planning and replanning; Production-inventory system

1. Introduction

The Master Production Schedule (MPS) is essen-tial in maintaining customer service levels andstabilising production planning within a MaterialRequirements Planning (MRP) environment. Usu-ally, an MPS will face the pressure to replanbecause of the changes of operational circumstan-ces. There are basically two conditions which leadto replanning. First, there is a rolling e!ect due tothe extension of the planning period. Secondly,when demand is uncertain, there is always a fore-

cast error, and, therefore the old plan has to bemodi"ed to adapt to new information to keep theproduction cost low and maintain the service level.

When the MPS is further used for materialrequirements planning (MRP), frequent changesin the MPS lead to schedule adjustments in thesystem. Such adjustments of plans also have anaugmented e!ect in an assembly system, and this isoften referred to as system nervousness. Ner-vousness may become an obstacle in the implemen-tation of MRP and even cause a collapse of thewhole system. Therefore, to reduce the instability ofthe schedule becomes an important objective inplanning the MPS.

In one of our previous studies [1], productionrescheduling has been addressed under the assump-tion that the cost of rescheduling is determinedexternally. A decision tree method was applied

0925-5273/02/$ - see front matter � 2002 Elsevier Science B.V. All rights reserved.PII: S 0 9 2 5 - 5 2 7 3 ( 0 0 ) 0 0 1 0 0 - 6

to study the bene"t from rescheduling. The NetPresent Value approach was used and the planningtime is on a continuous scale.

In this paper, we "rst investigate the possibilityto establish a method for planning the MPS understochastic demand. Secondly, we provide a modelfor replanning MPS assuming the lower-levelschedule change cost is known. Then, using asimilar method as in GrubbstroK m and Tang [1]and taking into account the schedule change cost,we provide a model for estimating appropriateMPS parameters such as the length of intervalsto replan and freeze. Finally, we provide nume-rical examples and draw some conclusions ina summary.

2. Previous research in MPS

Basically, two approaches have been suggestedto improve the quality of an MPS, namely freezingthe MPS and adding some safety stock into theforecast. When its operation and planning condi-tions such as forecasting quality, lot sizing policy,operation cost etc. are given, a high-quality MPS isto select a proper planning horizon and a replann-ing frequency as well as a safety stock.

Lin and Krajewski [2] provided a framework forstudying the MPS. First they presented the con-cepts of a forecast window (planning horizon),a frozen interval and a replanning interval anddiscussed their relationships. Generally, the cost ofthe MPS has been divided into two parts, a forecasterror cost and an MPS change cost. The latter hasthe same properties as in the similar approach byCarlson et al. [3], in which schedule change cost isa piecewise continuous function of time. The totalcost of the MPS model depends on the forecastwindow and the frozen interval and replanninginterval. But it should be reminded that the MPSchange cost is more complicated and di$cultto estimated in reality when di!erent lot-sizingpolicies and a multi-level system are considered.However, in order to build an analytical model, weadopted this simpli"cation in our study.

An alternative and more frequently used measurefor MPS quality is in terms of its stability. Srid-haran et al. [4] suggested a measure of instability,

which is a function of the method used to freeze theMPS, the proportion of the MPS frozen and thelength of the planning horizon. This instability def-inition has been followed in other studies [5,2,6,7].Still, stability is usually used together with othersystem performance measures, like the service level.In a similar way, Inderfurth [8] provided a de"ni-tion of stability for this type of system.

In most of the MPS studies, simulation is useddue to the complexity of the problem. Analysis ofVariance (ANOVA) is usually applied to determinethe system performance when di!erent operationpolicies are tested. Capacity constraints are oftenexcluded from MPS studies, the only exceptionbeing Kern and Wei [7]. In the following subsec-tion, we provide a literature overview of the areas offreezing the MPS and safety stocks in MPS.

2.1. Freezing the MPS

A rolling planning horizon is to replan MPSeach period whenever information is updated. Thistracks the development of demand closely, but gen-erates instability. One solution to avoid this is tofreeze a portion of the planning horizon to stabilisethe schedule. In a single-level deterministic demandcase, freezing this portion beyond 50% of theplanning horizon could reduce the instability butincreases cost tremendously [9,4].

For the demand uncertainty case, the forecastquality has a large in#uence on the optimal frozeninterval and planning horizon [10]. Sridharan andBerry [11] point out that a long MPS frozen inter-val may increase the need for safety stocks andenlarges the lot-sizing costs at the MPS level. Ina multi-level system, freezing the MPS is the moste!ective way to reduce instability and reduce totalcosts [6]. This result is di!erent from single-leveland deterministic studies. In Kadipasaoglu's studyin 1995 [12], it is indicated that freezing the MPSprovides a bene"t in the stochastic case only. Thelength of the frozen interval is recommended tocover the cumulative lead time.

The freezing portion depends on the planninghorizon. A longer planning horizon actually mayworsen the MRP performance in a demand uncer-tainty situation, whereas improve its performancein a deterministic case [10].

324 O. Tang, R.W. Grubbstro(m / Int. J. Production Economics 78 (2002) 323}334

Chung and Krajewski [13] study the interfacebetween aggregate planning and master productionplanning in a hierarchical planning environment.They concluded that the cost structure of aggregateplanning has a high in#uence on the replanningfrequency of the MPS.

2.2. Safety stocks

The safety stock is used to provide a bu!er foruncertainty in the MPS. However, safety stocksshould be applied with great care. Maintaininga safety stock level should not be the objective ofscheduling the MPS. An MPS that allows safetystocks to be added to inventory merely in the initialscheduling can create derive more bene"ts from itsoperation than an MPS that carefully tracks safetystocks and reschedules accordingly. There are alsostudies considering safety stocks on both MPS andcomponent levels in order to provide a #exibleschedule in a multi-level system [14].

Metters [15] studies di!erent strategies for usingthe safety stock. An explicit approach distinguishesthe safety stock from normal demand and lets thisamount of stock battle the uncertainty. A neworder (replanning) is trigged only when there isa real stockout. On the other hand, an implicitapproach triggers a new order whenever an inven-tory falls down to the safety stock level. Mettersalso recommends a third approach, namely thestable safety policy. In this case, an explicitapproach is used when there is no order in the oldschedule and implicit approach when there is anorder in the old schedule. Similarly, Sridharan andLaForge [5] provide an intelligent safety policyconcept which does not permit a scheduling of theMPS solely for the purpose of meeting a safetystock.

Even though a safety stock increases the servicelevel, it does not necessarily provide a stable MPS.In fact, a high safety to meet demand increasesinstability. Studies by Sridharan and LaForge [5]and Metters [15] support such a conclusion.

Hsu and El-Najdawi [16] carried out a studyconcerning how to determine the safety stock level.They incorporated safety stocks into the lot-sizingpolicies. Four techniques to determine the optimalsafety stock level are examined and compared.

Nevertheless, in all the studies we have mentionedabove, the level of safety stocks is considered asa constant. In a situation where a portion of theplanning horizon is frozen, we suggest that the levelof safety stock should vary for the reason that theforecast quality is impaired along with time. Ourprevious studies show that the level of safety stockfollows a square root of time relation when demandis a renewal process [17].

3. The planning decision

The objective of planning is to determine theproduction volume for a given demand forecastand an initial net inventory R

�. The initial net

inventory is de"ned as the di!erence betweencumulative production and cumulative demand sothat it is either positive or negative. Instead of usingtechniques like exponential smoothing, we here usestatistical methods for forecasting. Since our objec-tive in this study is to investigate the planning andreplanning problem under uncertainty, we use thevariance of the demand forecast to express theforecasting error (forecast quality). Another as-sumption we have adopted is that the setup cost isrelatively low. This results in a setup in each timeperiod so that we do not need to consider theimpact from the lot-sizing policy. Because of this,the system instability due to the rolling horizone!ect is reduced.

At any time, the following relation holds fora production-inventory system with backlogging:

B�"DM

�!PM

�#S

�!R

�. (1)

However here the time period is discrete, thusi"1, 2, 2 . In each interval, we must also have thefollowing relation:

E(B�)"E(DM

�)!PM

�#E(S

�)!R

�. (2)

Because of the uncertainty in demand, there iseither an excess inventory when the cumulativeproduction level is high, or stockouts when there isa shortage of supply. It is apparent from Eq. (1) thatthe system service level depends not on the demandforecast error in each period, but on the forecasterror of cumulative demand. When replanning isnot possible within a frozen interval, the error will

O. Tang, R.W. Grubbstro(m / Int. J. Production Economics 78 (2002) 323}334 325

be accumulated. Our previous studies [17] haveshown that the optimal safety stock may vary andbe a function of time in the continuous-time case.

We assume that external cumulative demandDM

�at the end of the ith period follows a probability

density function, which depends on time explicitly:

Pr(x)DM�(x#dx�i)"�

0 forx(0,

f�(x) dx forx*0.

(3)

Its cumulative probability function, denoted F�(x),

is assumed to be less than unity for "nite values ofx, and is shown to have the following property.

Lemma 1.

F��

(x)(F�(x).

Proof. Writing g�

as the density function for de-mand 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 fora "nite x. Hence the inequality is strict. �.

Since setup costs are disregarded, the objective ofthe planning is here interpreted as minimising theuncertainty 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 andstockout cost for period i, respectively, and n isthe planning horizon. The Lagrangean functionof this minimisation problem with the constraintsEq. (6) is

¸"

��

((h�#b

�)E(S

�)#b

�(E(DM

�)!PM

�!R

�))

#

��

��(PM

�!PM

��), (7)

where the ��

are nonnegative Lagrangean multi-pliers. The necessary Kuhn}Tucker conditions forthe optimal solution are

�¸

�PM�

"(h�#b

�)F

�(PM

�#R

�)!b

�#�

�!�

��*0

for i"1, 2, 2, n,

(8)

�¸

��

"PM�!PM

��)0 for i"1, 2, 2, n!1, (9)

together with the complementarily conditions

PM�)�¸

�PM�

"0 for i"1, 2, 2, n, (10)

��)�¸

��

"0 for i"1, 2, 2, n!1, (11)

where for convenience we de"ne ��"�

�"0.

Hence

F�(PM

�)*

b�#�

��!�

�h�#b

�

for i"1, 2, 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 bevalid for the "rst of these subsets (if it is nonempty).

In Theorem 1 we provide a su$cient conditionfor determining the structure of the sequencePM�

,PM�

, 2,PM�.

Theorem 1. If the ratio between unit stockout costand unit holding cost is non-decreasing in time, i.e.


(b��

/h��

)*(b�/h

�), i"1, 2, 2, n!1, then we

have the following conditions regarding the produc-tion sequence.

IfF(R

�)'b

/(h

#b

), thenPM

�"PM

�"2"PM

"

0; ifF(R

�)(b

/(h

#b

), thenPM

,PM

��, 2,PM

�'0.

Hence, this theorem o!ers a means for decidingwhere the sequence of positive cumulative produc-tion 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 needF�

(R�#PM

�)*(b

�!�

�)/(h

�#b

�). If PM

�'0, then

F�

(R�#PM

�)"(b

�!�

�)/(h

�#b

�)'F

�(R

�)'b

�/(h

�#b

�) which creates a contradiction since

��*0. Hence we obtain PM

�"0.

Based again on Eq. (12), we always needF�

(R�#PM

�)*(b

�!�

�#�

�)/(h

�#b

�). If PM

�'0, then F

�(R

�#PM

�)"(b

�!�

�#�

�)/(h

�#

b�

)'F�

(R�

)'b�

/(h�#b

�). This requires

!��#�

�'0, therefore �

�'0, which creates

a contradiction. Hence we obtain PM�"0. Follow-

ing the same procedure for i"3, 4,2, we end upwith 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

�#�

��)/(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

!�

#�

��)/

(h#b

), we obtain b

/(h

#b

)'F

(R

�)*

(b!�

#�

��)/(h

#b

), which leads to !�

#

��

(0. Therefore �'0 and we have

PM��

"PM"0.

With PM��

"0 and b��

/(h��

#b��

)'F��

(R�

)*(b��

!��

#�)/(h

��#b

��), we

obtain !��

#�(0 and therefore �

��'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 ratiobetween unit stockout cost and unit holding cost isnon-decreasing in time, i.e. b

�/h

�)b

��/h

��for

i"1, 2, 2,n!1. If PM, is the earliest positive cumu-

lative production, then the multipliers from j!1 on-wards are all zero, i.e. �

��"�

"�

��"

2"��

"0.

Proof. PM

being the earliest positive productionrequires �

��"0. We rearrange Eq. (12) as an

equality for i"j and i"j#1:

F(PM

#R

�)#

�!�

��h#b

"

b

h#b

,

F��

(PM��

#R�

)#��

!�

h��

#b��

"

b��

h��

#b��

.

Since the right-hand member is nondecreasing,so must the left-hand member also be. Assumethat �

'0, then PM

"PM

��; and therefore

F��

(PM��

#R�

)(F(PM

#R

�), which implies

(��

!�)/(h

��#b

��)'(�

!�

��)/(h

#b

).

Therefore, also ��

'0, which creates PM��

"

PM��

. Continuing this process, we end up with!�

��/(h

�#b

�)'0, which contradicts the non-

negativity of the multipliers. Therefore we concludethat �

"0. Analogously we get �

��"0, etc.

Hence, no multiplier ��, for i*j!1, may be posit-

ive. This concludes our proof. �.

By using Theorems 1 and 2, we are now ina position to construct a solution procedure to thestated problem of minimising the uncertainty cost.This procedure is stated in the following theorem.

Theorem 3. As before, we assume a non-decreasingsequence of unit cost ratios b

�/h

�)b

��/h

��, i"

1, 2, 2, n!1. The solution PM Hto the minimisation

problem Eq. (4) subject to the constraints Eq. (6) thensatisxes

PM H"0 if F

(R

�)'

b

h#b

,

F(PM H

#R

�)"

b

h#b

if F(R

�))

b

h#b

.

If all F(x) are monotonically increasing functions,

the solution must be unique.

Proof of Theorem 3 follows immediately fromour previous results. The theorem shows that this


Fig. 1. Cost structure for replanning.

optimal planning model is essentially an extensionof the Newsboy problem.

4. The replanning decision

The objective of replanning is to determine a res-cheduled production volume in each period P

��,

i"¹#1, ¹#2,2, for a given demand forecast,and initial net inventoryR

�at the replanning point

based on the old production plan P��

and theschedule change cost �

�, so that the total cost can be

minimised. The total cost C�

consists of the uncer-tainty cost, as discussed in the previous section, andthe rescheduling cost C

�:

C�"C

�#C

�. (13)

The rescheduling cost is the sum of the schedulechange costs in di!erent time periods. In a practicalsetting, we would assume that the schedule changecost depends on the lot-sizing policy, the productstructure, the rescheduling time desired, and therescheduling quantity desired. In this study, ourfocus is on the MPS level. Thus we use a singleschedule change cost function to take care of theaggregate information of negative e!ects of res-cheduling on the lower levels. Here we assume thatthe schedule change cost is a function of the res-cheduling time and that it is proportional to thequantity being changed. This is in accordance withthe 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 costshould be a function of time as well. With this inmind, we de"ne the schedule change cost as

��"�

(P��

!P��

)��

ifP��

*P��

,

(P��

!P��

)��

ifP��

(P��

,(14)

where ��

and ��

both are nonincreasing functions ofthe time period i. This means that in order toreschedule, whether adding orders or cancellingthe orders, it is always at least as expensive tochange the earlier schedule as the latter one. Inpractical terms, these costs may relate to the timefences, which are the guidelines to note variousrestrictions in schedule changes related to lower-level items.

At a certain time, the schedule change cost istherefore a monotonically decreasing functionwhen P

�(P

��and a monotonically increasing

function when P�'P

��. This is illustrated in

Fig. 1. Quite often in practice, to cancel an order iseasier than adding an new order, so �

�could then be

assumed to be smaller than ��.

Lemma 2. Let y�

(z) and y�

(z) be two functions of z,each having a unique minimum at z

�and z

�, respec-

tively, and no other local minima. Then any linearcombination of y

�(z) and y

�(z) with positive coez-

cients will have a unique minimum in the intervalbetween z

�and z

�.

The proof of this proposition is trivial. Regardingthe optimal replanning decision, we state the fol-lowing theorem.

Theorem 4. If all F�(x) are monotonically increasing,

the total cost function of replanning (Eq. (13)) isconvex. Then the optimal replanning decisionP��

falls into the interval between the old productionplan P

��and P*

�, the latter being the optimal solu-

tion when minimising the uncertainty cost based onlyon the initial net inventory R

�and demand forecast,

see Fig. 1.

Proof. The proof follows from Lemma 2 andour assumption regarding the monotonic behav-iour of the schedule change cost with respect toP�!P

��. �


Fig. 2. Planning and replanning scheme (adapted from Lin andKrajewski, [2]).

Combining Eqs. (13) and (14), we may write thetotal cost as

C�"E(C

�)#C

�

"

��

((h�#b

�)E(S

�)#b

�(E(DM

�)!PM

�!R

�)#�

�).

(15)

Its "rst-order derivative is thus

�C�

�P�

"

��

((h#b

)F

(PM

#R

�)!b

)

#��

ifPH�'P

��,

!��

ifPH�(P

��.

(16)

Regarding the solution to the optimal replanningdecisions, we have the following theorem.

Theorem 5.(i) If PH

�'P

��and (�C

�/�P

�)��

'0, then wehave the optimal solutionP

��"P

��.

(ii) If PH�'P

��and (�C

�/�P

�)��

(0, then theoptimalP

��is such that (�C

�/�P

�)��

"0.(iii) If PH

�'P

��and (�C

�/�P

�)��

H�'0, then the opti-

mal solution becomesP��

"PH�

.(iv) If PH

�(P

��and (�C

�/�P

�)��

H�(0 and (�C

�/

�P�)��

(0, then the optimum will beP��

"P��

.(v) If PH

�(P

��and (�C

�/�P

�)��

H�(0, and

(�C�/�P

�)��

'0, then the optimal value ofP��

is such that (�C�/�P

�)��

"0.

Proof. The proof is obtained from Theorem 4 andthe de"nition of the schedule change cost in combi-nation with Kuhn}Tucker conditions. �

Based on Theorems 3 and 5, we propose thefollowing solution procedure to obtain the optimalreplanning 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

��ac-

cording 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 dynamicplanning and replanning process. Fig. 2 illustratesthe decision scheme following a similar interpreta-tion as in Lin and Krajewski [2]. The replanninginterval is the time period until when no replanningis under consideration. However, due to certainproduction constraints in practice, such as the res-cheduling time fence, a portion of the old planneeds to be "xed. Sometimes, because of unfavour-able economic consequences or system perfor-mance, part of the old plan needs to be kept. Thetotal length of the "xed part of the old plan is calledthe frozen interval. During the time intervals be-yond the frozen interval, the production plan isallowed to be changed to improve the performanceof the system by using the knowledge of the statethat was realised at the replanning time.

In GrubbstroK m and Tang [1], a decision treemethod was used to address the rescheduling prob-lem. Here we follow the same methodology for theplanning and replanning problem. However, anexplicit schedule change cost has been added andthe decision tree has been changed correspond-ingly. As in Fig. 3, the replanning may take place attime ¹. The sequences of production decisionsprior to and at this point are denoted p and thosebeyond this point are denoted p�. Let d and d�represent stochastic sequences of a demand pro-cess, before and after the possible replanning time¹ (Fig. 4).

With this notation, the following planning andreplanning schemes can be interpreted in the formof a decision tree. At time 0, we make a decision


Fig. 3. Decision tree diagram for the planning and replanningprocesses.

concerning the production plan p followed by p�. Asshown in Section 3, this requires to determineP*�

, for i"1, 2,2,¹, and P*�

for i"¹#1,¹#2,2,¹#n. At the replanning point, a state ofthe 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 Section4 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 majordi!erences from our previous study in GrubbstroK mand 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 betweentotal costs for replanning and total costs for thestay alternative. As illustrated in GrubbstroK m andTang [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 cumu-lative demand DM

�according to Eq. (1). Since cumu-

lative demand at the replanning point ¹ has acumulative 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 thefrozen interval ¹

�from

E(¹�

)"��

��

¹�

(x) dF

(x). (20)

6. Numerical examples

Below, we provide numerical examples to illus-trate the models of previous sections. We assumethat external demand follows a Stuttering Poissonprocess [19], in which the interval between demandevents obeys an exponential distribution and theevent size is geometrically distributed. The prob-ability density function of cumulative demand atthe end of the ith time period is therefore

f�(M)"�

e��forM"0,

e��

(�i)j! �

M!1

j!1 �(1!)��

forM"1, 2,2, .

(21)

where � and are constants. The random variableM, cumulative demand, has the mean "�i/ andvariance ��"�i(2!)/�. This shows that forsuch a type of compound Poisson process, both themean and variance of cumulative demand increaseproportionally with the time period.

Given the production parameters b, h, �, , wecalculate the optimal MPS according to the modelof Section 3. The safety stock SS equals the di!er-ence between cumulative production and expectedcumulative demand immediately prior to the timeof production.

A "rst example is given in Table 1, which showsthat the optimal safety stock increases monotoni-cally with time, even if the demand forecast isconstant in each time period. The resulting produc-tion volume is then used for the purpose of theMPS in this system. According to this productionplan, the production volumes tend to be close toconstant except for the "rst few periods, and theyare all higher than expected demand. The increas-ing contribution to E(C

�) shows that the system

needs to pay more to cover increasing uncertaintyin later time periods.

Tables 2 and 3 o!er examples of the replanningmodel for a positive and negative initial net inven-tory, respectively. The frozen intervals are 1 and3 time periods, respectively. The schedule changecost structure is of course the major componentin#uencing the frozen interval. In our examples weassume a decay function �e�� for adding an order


Table 1The planning decisions with b"30, h"1, �"5, "0.5

i E(D)(Forecasting)

PM P (MPS) SS E(S�) E(B

�) E(C

�)

1 10 22 22 12 12.10 0.10 15.062 10 36 14 16 16.14 0.14 20.193 10 49 13 19 19.17 0.17 24.114 10 62 13 22 22.18 0.18 27.415 10 74 12 24 24.20 0.20 30.336 10 86 12 26 26.22 0.22 32.967 10 98 12 28 28.24 0.24 35.388 10 110 12 30 30.25 0.25 37.619 10 122 12 32 32.26 0.25 39.72

10 10 134 12 34 34.25 0.25 41.7211 10 145 11 35 35.28 0.28 43.6112 10 157 12 37 37.27 0.27 45.4213 10 168 11 38 38.30 0.30 47.1614 10 180 12 40 40.29 0.29 48.8415 10 191 11 41 41.31 0.31 50.4416 10 202 11 42 42.32 0.32 52.0117 10 213 11 43 43.34 0.34 53.5218 10 225 12 45 45.32 0.32 54.9919 10 236 11 46 46.34 0.34 56.4120 10 247 11 47 47.35 0.35 57.80

Table 2Replanning decision with b"30, h"1, �"5, "0.5, R"40

i!T E(D)(Forecasting)

� � P* P��

P (MPS)

1 10 100.00 5.00 0 12 122 10 36.79 1.84 0 12 03 10 13.53 0.68 0 12 04 10 4.98 0.25 10 12 125 10 1.83 0.09 10 12 126 10 0.67 0.03 10 11 117 10 0.25 0.01 11 12 128 10 0.09 0.00 11 11 119 10 0.03 0.00 12 12 12

10 10 0.01 0.00 12 11 1211 10 0.00 0.00 11 11 1112 10 0.00 0.00 12 11 1213 10 0.00 0.00 11 12 1214 10 0.00 0.00 11 11 1115 10 0.00 0.00 11 11 1116 10 0.00 0.00 11 11 1117 10 0.00 0.00 11 11 1118 10 0.00 0.00 12 11 1219 10 0.00 0.00 11 11 1120 10 0.00 0.00 11 11 11

and �e�� for cancelling an order. But we would beable to use any other form of schedule change costin our replanning model. It is shown in Table 3 thatthe "rst two periods are frozen due to the highchange cost even though stockouts exist there. Inthe event of a very tight production environment,within which where rescheduling is impossible, wemay assign the schedule change cost an in"nitelylarge value so that the optimal solution alwaysbecomes identical to the old plan according to thedecision rules of Theorem 5.

We use di!erent values of � and � in the sched-ule change cost function to represent the reschedul-ing #exibility of the system. These vary accordingas costs are high (10000), (medium) 1000, (low) 100and 50, 5, 0.5. Demand uncertainty is indicated byassigning di!erent values to � and . But we retainthe expected demand rate as a constant.

As illustrated in Fig. 5, the expected gainfrom replanning increases when the forecast erroris raised. An optimal replanning interval isshown to exist. However, this optimal value


Table 3Replanning decision with b"30, h"1, �"5, "0.5,R"!40

i!T E(D)(Forecasting)

� � P* P��

P (MPS)

1 10 1000.00 5.00 62 12 122 10 367.88 1.84 63 12 123 10 135.34 0.68 36 12 124 10 49.79 0.25 27 12 295 10 18.32 0.09 19 12 276 10 6.74 0.03 16 11 237 10 2.48 0.01 12 12 188 10 0.91 0.00 12 11 149 10 0.34 0.00 12 12 15

10 10 0.12 0.00 12 11 1211 10 0.00 0.00 11 11 1112 10 0.00 0.00 12 11 1213 10 0.00 0.00 11 12 1214 10 0.00 0.00 11 11 1115 10 0.00 0.00 11 11 1116 10 0.00 0.00 11 11 1117 10 0.00 0.00 11 11 1118 10 0.00 0.00 12 11 1219 10 0.00 0.00 11 11 1120 10 0.00 0.00 11 11 11

Fig. 4. Expected gain from replanning vs. replanning interval ondi!erent uncertainty levels. Mean demand rate"10, b"30,h"1, n"30.

Fig. 5. Expected frozen interval vs. replanning interval on di!er-ent uncertainty levels. Mean demand rate"10, b"30, h"1,n"30.

Fig. 6. Expected gain from replanning vs. replanning interval ondi!erent schedule change cost levels. b"30, h"1, �"5,"0.5, n"30.

does not change very much with the forecast error.It remains between 11 and 12 time periods inthese particular examples given by Fig. 5.Fig. 6 shows that the forecast error does not havean obvious impact on the length of the frozeninterval.

Intuitively, a lower schedule change cost bringsmore of a gain from replanning (Fig. 7). Again, fromFig. 7, we observe the existence of an optimal re-planning interval and that the sequence of optimalreplanning intervals is insensitive to the schedulechange cost. However, the schedule change cost hasa signi"cant impact on the length of the frozeninterval.

We have also studied examples with di!erentplanning horizons. The planning horizon hasa large impact on the optimal replanning interval(Figs. 8}10). With a longer planning horizon, thereplanning interval is prolonged as well. Neverthe-less, Figs. 8 and 10 also show that neither forecast-ing, nor the schedule change cost have anyimportant in#uence on the length of the replanninginterval.

When we study the optimal replanning interval,Fig. 9 again shows that the forecast error has littlee!ect on the frozen interval. It seems as if the only


Fig. 7. Expected frozen interval vs. replanning interval on di!er-ent schedule change cost levels. b"30, h"1, �"5, "0.5,n"30.

Fig. 8. Optimal replanning interval vs. planning horizon ondi!erent uncertainty levels. Mean demand rate"10, b"30,h"1.

Fig. 9. Expected frozen interval vs. planning horizon on di!er-ent uncertainty levels when the replanning interval is optimal.Mean demand rate"10, b"30, h"1.

Fig. 10. Optimal replanning interval vs. planning horizon ondi!erent schedule change cost levels. b"30, h"1, �"5,"0.5.

Fig. 11. Expected frozen interval vs planning horizon on di!er-ent schedule change cost levels when the replanning interval isoptimal. b"30, h"1, �"5, "0.5.

major factor changing the frozen interval is theschedule change cost (Fig. 11).

7. Summary

In this paper, we have studied planning andreplanning problems at the MPS level. First, wehave applied our previous research "ndings aboutsafety stocks in an MPS system. We providea model for MPS planning in which the safety stockis determined and is shown to be an increasingfunction of time. Secondly, we developed a modelfor replanning in which the scheduling changecost is considered and for which an e$cientsolution procedure was presented. This provides


the possibility for further studying replanning whenseveral alternative replanning points are con-sidered.

The numerical examples demonstrate proceduresto solve the optimal replanning interval and toestimate the expected frozen interval. An interest-ing "nding is that the forecast error appears to be oflittle importance to either of them when the safetystock bu!er is optimised according to our planningmodel. But how to determine the optimal planninghorizon still remains an open question.

In this paper, we have assumed that there is onlyone replanning point inside of the "rst planninghorizon. In future studies, it is of interest to removethis assumption. Simulation studies may be neededfor dynamic replanning problems, in which re-planning decisions may be interpreted followingthe model in Section 4. Only after that, the interre-lationship between the replanning interval, thefrozen interval and the planning horizon may be-come more transparent.

Another remaining point for discussion is theschedule change costs. If in the real world multi-level production system we could develop prin-ciples for aggregating the information necessary forestimating the rescheduling cost, our replanningmodel in Section 4 would become more importantfor practical uses.

References

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[17] R.W. GrubbstroK m, O. Tang, Further developments onsafety stocks in an MRP system applying Laplace trans-forms and input}output analysis, International Journal ofProduction Economics 60}61 (1999) 381}387.

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Planning and replanning the master production schedule under demand uncertainty